Alain Geens - vubirelec.be

Vrije Universiteit BrusselFaculteit Toegepaste Wetenschappen

Departement ELECPleinlaan 2, B-1050 Brussels, Belgium

MEASUREMENT AND MODELLING OF THE NOISE BEHAVIOUR OF HIGH-FREQUENCY NONLINEAR ACTIVE

SYSTEMS

Alain Geens

Mei 2002

Promotor: Prof. Dr. Ir. Yves Rolain Proefschrift ingediend tot het behalen vande academische graad van doctor in detoegepaste wetenschappen

Vrije Universiteit BrusselFaculteit Toegepaste Wetenschappen

Departement ELECPleinlaan 2, B-1050 Brussels, Belgium

MEASUREMENT AND MODELLING OF THE NOISE BEHAVIOUR OF HIGH-FREQUENCY NONLINEAR ACTIVE

SYSTEMS

Alain Geens

Voorzitter:Prof. G. Maggetto (Vrije Universiteit Brussel)

Vice-voorzitter:Prof. J. Vereecken (Vrije Universiteit Brussel)

Promotor:Prof. Y. Rolain (Vrije Universiteit Brussel)

Secretaris:Prof. R. Pintelon (Vrije Universiteit Brussel)

Jury:Prof. R. Pollard ( University of Leeds, United Kingdom)Prof. D. Van Hoenacker (Université Catholique de Louvain)Prof. J.C. Pedro (Universidade de Aveiro, Portugal)Prof. A. Barel (Vrije Universiteit Brussel)

Voor Wendy, papa en mama

Table of Contents i

Preface v

List of Symbols ix

CHAPTER 1Noise, Linear Systems and Nonlinear Systems 11.1 Introduction 21.2 Sources of noise 5

1.2.1 Thermal noise or Johnson noise 51.2.2 Shot Noise 91.2.3 Other noise sources 10

1.3 Linear time invariant systems 111.3.1 Definition of a linear time invariant system 111.3.2 Spectral properties of a LTI system 111.3.3 Description of LTI systems at high frequencies 12

1.4 Noise and linear time invariant systems 151.4.1 The presence of noise in a LTI system 151.4.2 Noise figure 171.4.3 Input noise temperature dependence 181.4.4 Noise Figure measurements concepts: the Y-factor technique 181.4.5 Noise temperature 21

1.5 Nonlinear systems 241.5.1 Definition of a nonlinear time invariant system 241.5.2 Spectral properties of a NICE system 261.5.3 Importance of the absolute phase spectra for NICE systems 27

1.6 Noise and nonlinear systems 311.6.1 The presence of noise in a NICE system 311.6.2 Applying signal and noise together to a NICE system 33

1.7 Conclusion 421.8 Appendices 43

Appendix 1.A : Cross-correlation of deterministic signals and ergodic noise43

Appendix 1.B : Transfer function of a LTI system 45Appendix 1.C : Z and Y matrix of a n-port 47Appendix 1.D : Output PSD of the noisy LTI system 47Appendix 1.E : Determining the noise figure with the Y-factor method 48Appendix 1.F : Signal-to-noise ratio deterioration for other input noise levels

49Appendix 1.G : The noise figure of a cascade of noisy LTI systems:

Friis’formula 50Appendix 1.H : Combinatory analysis to determine the discrete output

spectrum of a -th order Volterra operator 52α

i

CHAPTER 2Noise figure measurements on NICE systems 552.1 Introduction 562.2 A very simple model for the NICE system 572.3 Determining the output spectrum of the modelled system 602.4 Determination of the noise figure 652.5 Discussion on the yielded noise figures 732.6 Experimental results 802.7 Conclusion 842.8 Appendices 85

Appendix 2.A : Autocorrelation of band-limited, white noise 85Appendix 2.B : Fifteen ways of partitioning six random variables in products

of averages of pairs 86Appendix 2.C : The combined contributions of the auto-correlation

87Appendix 2.D : The convolution of the noise spectrum with itself 89

CHAPTER 3Extension of the “Noise Figure” towards NICE systems 933.1 Introduction 94

3.1.1 Goal 943.1.2 The model for the noisy NICE system up to the 1 dB compression point 97

3.2 Variation of the NICE noise figure, as a function of the input amplitude 993.2.1 Determining the output power spectral density 993.2.2 Signal-to-noise ratio variation: the NICE noise figure 1023.2.3 Special case: a noiseless NICE system 1103.2.4 Experimental results 1123.2.5 Conclusion 114

3.3 Variation of the NICE noise figure, as a function of the input amplitude and the input noise power 1163.3.1 Introduction 1163.3.2 Determining the analytical expression 116

3.3.3 First case: 118

3.3.4 Second case: 121

3.3.5 Third case: 122

3.3.6 Variation of the NICE noise figure in hard compression 1253.3.7 Conclusion 1373.3.8 Experimental results 138

3.4 Variation of the noise power gain, as a function of a general periodic or ergodic input signal 143

Rηη τ( )

PSDnA

1( ) f( ) Nin⁄ G ′nu»

PSDnA

1( ) f( ) Nin⁄ G ′nu≈

PSDnA

1( ) f( ) Nin⁄ G ′nu«

ii

3.4.1 Introduction 1433.4.2 Determining the output power spectral density 1443.4.3 Experimental results 149


Appendix 3.A : Calculation of the 1 dB compression point for the third degree polynomial model 155

Appendix 3.B : Calculation of the output power spectral density for an input consisting of a single tone and thermal noise 156

Appendix 3.C : Variation of over a small bandwidth 164

Appendix 3.D : Taylor series expansion of an atan function 164Appendix 3.E : Probability of creating a zero crossing 165Appendix 3.F : Boundaries of the linear region 167Appendix 3.G : Autocorrelation of the noisy NICE system’s output for a

general input waveform 167Appendix 3.H : Auto-correlation of a non zero-mean signal 169

CHAPTER 4Noise-like signals and NICE systems 1714.1 Introduction 1724.2 Considerations about the output spectrum 1744.3 Discussion on fundamental issues of NPR measurements 179

4.3.1 Existing measurement techniques 1794.3.2 General framework 1814.3.3 Properties of the Frequency Response Function 1844.3.4 Reconciling the NPR and the CCPR method 1874.3.5 Proposed measurement method 1894.3.6 Experimental results 1914.3.7 Conclusion 197

4.4 Extension towards multi-port systems: mixers 1984.4.1 Introduction 1984.4.2 A simple mixer model 1984.4.3 Defining an FRF or transmission parameter for the mixer 2014.4.4 Getting extra information about the nonlinear mechanism of the mixer 2064.4.5 Experimental results 211

4.5 The mixer as a real three-port device: phase noise example 2184.5.1 Introduction 2184.5.2 Considerations about the LO multisine 2194.5.3 Properties of the FRF 2214.5.4 Experimental results 228


PSDny

1( ) f( )

iii

Appendix 4.A : Absence of an underlying linear system for amplifiers exhibiting cross-over distortion 234

Appendix 4.B : Systematic and stochastic contributions of the FRF 236Appendix 4.C : IF spectrum of the ideal mixer 237Appendix 4.D : Comparing two models for a nonlinear multiplier 239Appendix 4.E : Output spectral components of a two input NICE system

where a single tone signal is applied at both input ports 241Appendix 4.F : Systematic and stochastic contributions of the mixer’s FRF

239Appendix 4.G : The three noise sources are uncorrelated 245

Conclusions and Ideas for Further Research 247

References 253

Publications 259

iv

PREFACE

During the last years, nonlinear system theory has gained importance due the ever increasing

demand for high performance circuits that operate using ever decreasing DC power. The

consequence of these demands is that real-world devices often operate at (or close to) the limits

of their linear region. Hence, the knowledge of the behavior of those devices operating in their

nonlinear region is very important, and has been (and is still being) studied. To validate these

theories, new types of measurement instruments were developed, especially designed to

measure the nonlinear behavior of the studied Devices Under Test (DUT). However, minimal

attention was paid to the noise generated or processed by these nonlinear devices. Since the

noise power is usually much smaller than the signal power, noise would at first sight never

drive a system into its nonlinear operation mode. Similarly, since the noise power is that small,

one could intuitively think that this noise will be linearly processed by the DUT, even when the

system operates in its nonlinear mode due to a large input signal power. Furthermore, noise is

often considered as the “trash” of the system: when a circuit does not behave as expected,

disturbances and noise are often blamed to cause the failure.

However, sometimes a trash can “overflow”: If the input signal-to-noise ratio of the nonlinear

system is quite small, it is important to know the degradation of this signal-to-noise ratio

through the system. For Linear Time Invariant (LTI) systems, this degradation is quantified by

using the noise figure [6]. The open question is of course if this noise figure is also able to

v

quantify the signal-to-noise ratio degradation of a nonlinear system. A bad luck scenario could

be that the signal-to-noise ratio degradation for a nonlinear system is several orders of

magnitude larger than for the underlying LTI system. In telecommunication equipment, some

effects of the noise behavior of nonlinear systems are well known, and described using figures

such as the Noise Power Ratio [20], Co-Channel Power Ratio [23], Adjacent Channel Power

Ratio [34], etc.

The goal of this work is to get an insight in the way a nonlinear system will deal with noise.

This insight will be built up throughout the chapters:

In chapter 1, an overview will be given of the possible sources of noise, and the way the noise

behavior of LTI systems is characterized. Since the class of nonlinear systems contains a too

large variety of systems (i.e. all systems that do not always obey the superposition property),

main focus will be set on the class of NICE systems, which converge in mean square sense to a

Volterra model [7] of eventually infinite order.

Chapter 2 will investigate the possibility to use classical noise figure measurement methods in

order to determine the noise behavior of NICE systems. This chapter will also give an answer

to the question: “What errors will I make when I measure the noise figure of a nonlinear

system using classical techniques designed for LTI systems?”

One intuitively sees that simple figures of merit used for LTI systems, such as the noise figure,

are not rich enough to fully describe the noise behavior of a nonlinear system. In chapter 3, the

concept of noise figure is extended towards a nonlinear figure of merit, i.e. the NICE noise

figure, that describes the signal-to-noise ratio deterioration of NICE systems, as function of the

input signal power and the input noise power spectral density.

In chapter 4, the constraint of deterministic input signals (as assumed in all previous chapters)

will be removed, and the noise behavior of a NICE system, excited with the superposition of

noise and noise-like signals will be investigated. These noise-like signals are typically

encountered in telecommunication links, where the stochastic content of information itself is

vi

responsible for the noise-like properties of these input signals. The obtained results for two-

port systems such as amplifiers will be extended towards multi-port systems such as mixers.

Contributions of this work:

This work fills a gap that existed in the nonlinear world: the interaction of noise and nonlinear

RF systems.

Signal-to-noise ratio evolution, for a Continuous Wave (CW) input signal:

• A simple model for a noisy NICE system is proposed (chapter 1).

• The effect of classical noise figure measurements on NICE systems is

described (chapter 2).

• An extension of the concept “noise figure” towards NICE systems is

proposed (chapter 3).

• The evolution of this “NICE noise figure” is described as function of the

input signal and noise power (chapter 3).

• A measurement method for the NICE noise figure is proposed (chapter 3).

In-band distortions for noise-like (modulated) input signals:

• A model for the NICE system is given, based on systematic and stochastic

nonlinear contributions (chapter 4).

• A unified view on the NPR and the CCPR method is proposed (chapter 4).

• A measurement method is developed to measure systematic and

stochastic contributions (chapter 4).

• The theory and measurement techniques for systematic and stochastic

nonlinear contributions are extended towards mixers (chapter 4).

vii

Words of thank

I wish to thank everybody who has contributed to the realisation of this Ph. D. thesis.

First of all, Yves Rolain, Rik Pintelon and Johan Schoukens, for their helpful suggestions and

advice.

All my colleagues, for exchanging interesting ideas.

Our “technical staff”: Wilfried, Jean-Pierre and Wim, for realizing the Printed Circuit Boards,

and for their help concerning several mechanical and technical problems.

And last but not least, my dear wife Wendy, who has always supported and encouraged me,

with an infinite patience.

And, of course, everybody that I may have forgotten to mention.

Alain Geens

viii

LIST OF SYMBOLS

OPERATORS AND NOTATIONAL CONVENTIONS

number of repetitive combinations of

elements, taken -by- .

Discrete Fourier Transform of the samples

vector ,

time mean of the waveform

mathematical expectation

continuous Fourier transform:

continuous inverse Fourier transform

system operator

Cnp n p 1–+( )!

p! n 1–( )!⋅----------------------------= n

p p

DFT x mTs( )( ) 1M----- x mTs( )e

j2πkM

---------m–

m 0=

M 1–

∑=

x mTs( ) m 0 1 … M 1–, , ,=

E x t( ) 1T---

T ∞→lim x t( ) td

T 2⁄–

T 2⁄∫= x t( )

E

ℑ

ℑ x t( ) x t( )e j– ωt td∞–

∞∫=

ℑ 1–

ℑ 1– X f( ) X f( )ej2πft fd∞–

∞∫=

H [ ] y t( ) H u t( )[ ]=

ix

-th order Volterra operator

Volterra operator (of a two input system) of the

-th order in the first input ( ), and of the

-th order in the second input ( )

Briggs logarithm of the quantity

ordo, an expression is when

with .

probability density function of the waveform

auto-correlation of

cross-correlation of and

real part of the quantity

root mean square value of the waveform

magnitude of the complex value

number of repetitive variations of elements,

taken -by- .

subscript B bias or systematic nonlinear contribution

Hα [ ] α

Hαβ u1 t( ) u2 t( ),[ ]

α u1 t( )

β u2 t( )

x( )log x( )log10= x

O x( ) O x( )O x( )

x------------

x 0→lim c= 0 c ∞< <

pdf x( )

x t( )

Rxx τ( ) E x t( )x t τ+( ) =

1T---

T ∞→lim x t( )x t τ+( ) td

T 2⁄–

T 2⁄∫=

x t( )

Rxy τ( ) E x t( )y t τ+( ) =

1T---

T ∞→lim x t( )y t τ+( ) td

T 2⁄–

T 2⁄∫=

x t( ) y t( )

Re X( ) X

xrms E x2 t( ) = x t( )

X X X*⋅= X

Vnp np= n

p p

x

subscript S stochastic nonlinear contribution

superscript complex conjugate

time variance of the waveform

variance of the quantity

convolution,

for continuous signals:

for discrete signals:

number of

*

σx t( )2 E x t( ) E x t( ) –

2

= x t( )

σX2 E X E X – 2 = X

*

x t( )*y t( ) x τ( )y t τ–( ) τd∞–

∞∫=

X k( )*Y k( ) X κ( )Y k κ–( )κ∑=

#

xi

SYMBOLS

incident wave at port

reflected wave at port

bandwidth of the system

set of the complex numbers

e natural base:

frequency

intermediate frequency

local oscillator frequency

RF frequency

sample frequency

Discrete frequency response function

set of all the functions dependent on the time,

whose range is

noise power gain

noise power gain of the underlying noiseless

system.

ai i

bi i

B

C

e 1 1x---+

x

x ∞→lim=

f

fIF

fLO

fRF

fs

FRF k( )

FR

Gnu

G ′nu

xii

signal power gain

h Planck’s constant Js

impulse response of a LTI system

-th order symmetrized Volterra kernel

frequency response function of a LTI system

-dimensional Laplace transform of

j imaginary unit

k Boltzmann’s constant J/K

single sided standard noise power spectral

density dBm/Hz.

single sided input noise power spectral density

noise generated by the system itself

discrete Fourier transform of the samples

,

, single sided power spectral density of a cold

and hot noise source respectively

Gu0

h 6.546 34–×10=

h t( )

hα τ1 … τα, ,( ) α

H ω( )

Hα f1 … fα, ,( ) α

hα τ1 … τα, ,( )

j2 1–=

k 1.38 23–×10=

N0

N0 kT0 174–= =

Nin

nA t( )

NA k( )

nA mTs( ) m 0 1 … M 1–, , ,=

Nc Nh

xiii

disturbing time domain noise of the input

and output signals, respectively. Note that

is part of


and , ,

respectively

Noise Figure

NICE Noise Figure

disturbing time domain noise on the

instantaneous phase function


,

set of the natural numbers

single sided power spectral density of the

waveform

double sided (mathematical) power spectral

density of the waveform

total power of the waveform

set of the real numbers

continuous time variable

nu t( ) ny t( ), u t( )

y t( )

nA t( ) ny t( )

NU k( ) NY k( ),

nu mTs( ) ny mTs( ) m 0 1 … M 1–, , ,=

NF ω( )

NNF ω( )

nθ t( )

θ t( )

Nθ k( )

nθ mTs( ) m 0 1 … M 1–, , ,=

N

PSDx1( ) ω( )

2PSDx2( ) ω( ) ω 0≠⇔

PSDx2( ) ω( ) ω⇔ 0=

=

x t( )

PSDx2( ) ω( )

x t( )

Px x t( )

R

t

xiv

absolute temperature

absolute standard temperature

effective noise temperature

operational noise temperature

sampling period

input and output time signals respectively

input and output signals, respectively,

free from the disturbing time domain noise.

,

Fourier transform of and respectively


and ,

part of the output signal due to the -th order

Volterra operator: .

set of the integer numbers

characteristic impedance

set of the integer numbers, except zero

T

T0 T0 290K=

Te

Top

Ts

u t( ) y t( ),

u0 t( ) y0 t( ), u t( ) y t( )

u t( ) u0 t( ) nu t( )+= y t( ) y0 t( ) ny t( )+=

U ω( ) Y ω( ), u t( ) y t( )

U k( ) Y k( ),

u mTs( ) y mTs( ) m 0 1 … M 1–, , ,=

y α( ) t( ) α

y α( ) t( ) Hα u t( )[ ]=

Z

Z0

Z0

Z0 Z \ 0 =

xv

δ t( ) 0 for t 0≠=
Dirac impulse
discrete Dirac impulse ,

frequency grid spacing

output signal of the noiseless system

. Note that still

contains a portion of .

instantaneous phase (time dependent)

phase function of ,

angular frequency

δ t( )δ t( ) td

∞–

∞∫ 1=

δ k( ) δ k( ) 1= for k 0=

δ k( ) 0= for k 0≠

∆f

η t( )

η t( ) y t( ) nA t( )–= η t( )

ny t( )

θ t( )

φX ω( ) X ω( )

X ω( ) X ω( ) ejφX ω( )⋅=

ω ω 2πf=

xvi

ABBREVIATIONS

ADSL Asymmetric Digital Subscriber Line

AWG Arbitrary Waveform Generator

CCPR Co-Channel Power Ratio

CW Continuous Wave

DC Direct Current

DUT Device Under Test

e.g. for example

ENR Excess Noise Ratio

FRF Frequency Response Function

gcd greatest common divisor

GSM Global System for Mobile

i.e. id est (in other words)

IF Intermediate Frequency

LO Local Oscillator

LTI Linear Time Invariant

xvii

NF Noise Figure

NNF NICE Noise Figure

NPR Noise Power Ratio

NVNA Nonlinear Vectorial Network Analyzer

pdf probability density function

PSD Power Spectral Density

RF Radio Frequency

rms root mean square

RLDS Related Linear Dynamic System

SNR Signal-to-Noise Ratio

xviii

CHAPTER 1

NOISE, LINEAR SYSTEMS ANDNONLINEAR SYSTEMS

Abstract: If a circuit does not behave as expected, disturbances

and noise are often blamed as the cause of the failure. The sources

of these phenomena are often not quite well understood, which

makes the noisy results even more “mystical”. The interaction

between noise and linear systems has been described both

qualitatively and quantitatively in earlier work [6]. But the

interaction between noise, signals, and nonlinear systems remains

mainly an open problem.

This chapter serves as a foundation to construct a possible answer

to that question. Hitter to, several questions about noise and

nonlinearities require an answer. In this chapter, the fundamental

questions that lead to a possible description of the noise-signal

system interaction for nonlinear systems are stated. First, the

linear noise theory is rehearsed. The main focus here will be on the

similarities and differences between this linear and a nonlinear

framework. Next, an extension is proposed towards soft

nonlinearities with simple excitation signals, such as sine waves.

1

Noise, Linear Systems and Nonlinear Systems

1.1 IntroductionEven if in a theoretical framework, the concept of noise is well defined as a stochastic process,

in practical applications, the concept “noise” is a very vague concept, because people always

tend to tag all disturbing signals as “noise”. Noise is the common denominator of every

spontaneous fluctuation in electronic circuits. This is merely due to the fact that in the days that

electronic circuitry was mainly used in audio applications, noise showed itself in cracks, pops

and hisses in speakers. Because all physical processes deal in one way or another with

spontaneous fluctuations, noise is omnipresent, and it is impossible to completely eliminate

noise phenomena.

In this work, the kind of disturbances that will be studied, is the ergodic noise.

Definition 1.1

Ergodic noise is a random variable, whose -th order moments are identical, when taken over

time or realization, or

(1-1)

where represents the mathematical expectation over the realizations of , and

represents the time mean of .

Examples of disturbances that are not ergodic noise are e.g. induction in an electrical circuit of

a 50 Hz spectral component coming from the power grid, fluorescent lights or computer

equipment. The presence of dust between the sliding contacts of a sliding resistor also can

cause bad contact and sparkling between the contacts.

An advantage of working with ergodic noise, is that the cross-correlation between the -th

power of a deterministic periodic signal , and the -th power of the ergodic noise ,

can be split into the product of both time averages:

α

n t( ) is ergodic noise ⇔

α N∈∀ :E nα t( ) E nα t( ) =

E nα t( ) nα t( )

E nα t( ) 1T---

T ∞→lim nα t( ) td

T 2⁄–

T 2⁄∫= nα t( )

α

u0α t( ) β nu

β t( )

2

Introduction

(1-2)

(see Appendix 1.A). This property will be very useful when determining the power spectral

density of the output of a nonlinear system.

Since noise is always present, the design rules will be oriented towards the minimization of the

signal degradation that is caused by this noise. Noise elimination will therefore be replaced by

signal-to-noise ratio maximisation in the design rules. The reason therefore is quite obvious: if

the signal is amplified with a factor 100, this is very good, and everybody will acclaim the

amplification properties of the system. But if that same nonlinear system amplifies the noise

with a factor , the signal will be drown in the noise. Hence, while it is important to know

the signal behavior of a system, the importance of the knowledge of its noise behavior should

not be underestimated. Understanding noise behavior, can help people to adequately minimize

its influence or propagation through electrical systems.

To reach this goal, it is no longer sufficient to know what factors cause the noise. One also has

to know the type of system and the class of excitation signals that are used. For a very long

time, systems were assumed to be linear, and hence the properties of these systems together

with their noise behavior was studied in detail. The main reason for this assumption is the

simplicity of the mathematics associated to the response calculation of such systems. One

single convolution equation is sufficient to describe linear systems under all operating

conditions. This easy equation has led to a wide variety of design frameworks that allow

straightforward translation of a series of system specifications to be obtained into a circuit that

reaches the design goals. Thereto, safety margins were included in the designs to ensure that

the basic linearity assumption was met. During the last decade, applications became more and

more demanding. The portable telecommunication market pushed designers towards ever

increasing levels of power efficiency and complex modulations requiring high linearity. As a

consequence, the safety margins shrunk and nonlinear distortions played an ever increasing

role in the system performance.

E u0α t( )nu

β t τ+( )

E u0α t( )

E nuβ t τ+( )

⋅=

104

3


Several approaches and theories were developed to describe the signal properties of such

mildly nonlinear systems [1]. However, to our knowledge only very few attempts were made

to study the noise properties of such nonlinear systems. As explained earlier, the signal-to-

noise ratio of a device output is the main parameter governing the noise performance.

In this chapter, the answer to the noise description of a linear time independent system will be

given. What is noise? What is a linear system? How to quantify and qualify the influence of

noise in a linear time invariant (LTI) circuit? Starting from this framework, an extension will

be selected. A class of nonlinear systems will be defined to replace the LTI systems, and the

behavior of this system class will then be studied.

4

Sources of noise

1.2 Sources of noiseIn this section, some sources of noise will be highlighted, together with their properties.

Noise is generated by some stochastic process and, therefore, is a stochastic quantity itself.

These stochastic processes generating the noise can be caused by several mechanisms, leading

to various sources of noise. Thermal noise (also known as Johnson noise) and shot noise are

the two most important ones in electronic circuits.

1.2.1 Thermal noise or Johnson noiseThermal noise is caused by the thermal agitation of electrons in conductors. This agitation is a

pure stochastic process. The statistically fluctuating charge displacements create a varying

potential difference between the terminals of the conductor. Because the electron agitation is

proportional to the temperature, a noise source is present in every conductor whose

temperature is different from zero Kelvin (0 K).

The probability density function (pdf) of the voltage across the terminals of an

impedance having a resistive part, due to the thermal noise source, has a Gaussian distribution:

(1-3)

where , the variance of the noise voltage.

Furthermore, thermal noise has zero mean, hence . This implies that for noise,

the root mean square (rms) value and the standard deviation yield the same value.

Nyquist has theoretically deduced that the rms value (and hence the standard deviation) of the

noise voltage of a resistor with resistance R and at an absolute temperature is given by [2]:

(1-4)

nu t( )

pdf nu t( )( ) 12πσnu t( )

-------------------------e

nu t( ) E nu t( ) –( )2

–

2σnu t( )2

-------------------------------------------------------

=

σnu t( )2 E nu t( ) E nu t( ) –( )

2

=

E nu t( ) 0=

σnu t( )

T

σnu t( ) nurms E nu

2 t( ) 4kTRB= = =

5


where J/K (Boltzmann’s constant) and [Hz] is the bandwidth of the

system that carries the noise.

From equation (1-4) it follows that when the bandwidth tends to infinity, the rms value of

the noise source also becomes infinitely large. Clearly, this means that the proposed model (1-

4) is not valid for extremely large bandwidths. This is explained by the fact that in reality the

rms value of the noise source is given by Planck’s black body radiation law [3]:

(1-5)

where Js (Planck’s constant) and is the center frequency at which the

noise rms value is measured. Planck’s black body radiation law is an extension of Nyquist’s

formula (1-4), that is valid for all bandwidths and frequencies, and does not yield an infinite

rms value when the system bandwidth tends to infinity.

Figure 1-1 shows that because negative frequencies are not considered, the maximal system

bandwidth is twice the center frequency , or in any case, . Hence, considering

the maximum bandwidth, when tends to infinity, will also tend to infinity. The

denominator in (1-5) contains an exponential function of the frequency, which will tend faster

towards infinity than the numerator (containing a polynomial function of the frequency).

Hence, the result will be finite.

FIGURE 1-1. Illustration of the relation between maximal bandwidth and frequency.

k 1.38 23–×10= B

B

nurms 4hf0BR

ehf0( ) kT( )⁄

1–-----------------------------------=

h 6.546 34–×10= f0

freq

B

f0f0 B 2⁄– f0 B 2⁄+

B f0 f0 B 2⁄≥

B f0 B 2⁄=

6

Sources of noise

Nyquist’s formula (1-4) is an approximation of the general black body radiation law (1-5), for

low frequencies and high temperatures. Assuming that , the exponential function in

(1-5) can be approximated by its Taylor series up to the first degree:

(1-6)

Substituting (1-6) into (1-5) yields Nyquist’s formula (1-4).

The power spectral density (PSD) corresponding to a noise source with rms value can

easily be calculated using the formulae (1-4) and (1-5). The noisy resistor can be modelled

through its Thévenin equivalent circuit, consisting of the noise source and a noiseless resistor.

The noisy resistor will transfer a maximal power, if it is connected to a resistor with the same

value R (see Figure 1-2). Note that all results can be extended to impedances instead of

resistances, but one has to keep in mind that only the resistive part of the impedance will

generate the noise. Ideal inductors and capacitors do not generate thermal noise [2].

In the figure above, the noisy resistor will deliver to the load resistor a total power that is

given by general network analysis rules:

(1-7)

or,

FIGURE 1-2. The noisy resistor delivering power to an identical load resistor

hf0 kT«

e hf0( ) kT( )⁄ 1 hf0( ) kT( )⁄+≈

nurms

R

Rnurmsthe noisy

resistor

Pnu

Pnu

nurms( )

2

4R------------------

hf0B

e hf0( ) kT( )⁄ 1–-----------------------------------= =

7


(1-8)

when assuming that .

To get the power spectral density, one has to calculate , yielding:

(1-9)

or, again for :

(1-10)

This means that where the approximation is valid, the power spectral density of thermal noise

is constant throughout the whole frequency spectrum, and depends linearly on the absolute

temperature.

In order to have an idea of the frequency and temperature range for which the approximated

formula (1-10) is valid both the latter and the correct expression based on Planck’s black body

radiation law (1-9) are plotted in Figure 1-3 for several temperatures (300 K: room

temperature, 273 K: melting temperature of ice and 77 K: boiling point of liquid nitrogen) and

in the frequency range from 10 GHz to 10 THz. The plain horizontal lines represent the

FIGURE 1-3. Thermal noise power spectral density versus frequency

PnukTB=

hf0 kT«

PnuB⁄( )

B 0→lim

PSDnu

1( ) hf0e hf0( ) kT( )⁄ 1–-----------------------------------=

hf0 kT«

PSDnu

1( ) kT=

101

102

103

104

0

1

2

3

4

5x 10

-21

frequency [GHz]

T = 300KT = 273KT = 77K

PSD

n u

(1)

[W/H

z]

8

Sources of noise

approximation (1-10), while the dotted curves represent (1-9). The figure illustrates again the

fact that equation (1-10) is valid for low frequencies and high temperatures. Table 1-1 shows

the relative error that is made, for several temperatures and frequencies, when assuming that

the power spectral density is given by (1-10) instead of (1-9).

Hence, assuming that thermal noise has a flat (or white) frequency spectrum, leads to errors

smaller than 1% up to 100 GHz and for temperatures above the melting point of ice.

One can also conclude that the total thermal noise power varies linearly with the measurement

bandwidth and the absolute temperature , thus, cooler systems with smaller bandwidths

will collect less thermal noise power.

1.2.2 Shot NoiseShot noise finds its origin in the discrete character of the charge carriers themselves. When

crossing a potential barrier in semiconductor junctions, these discrete charge carriers create

currents that are independent of each other. Hence, this current as a function of time is a

stochastic quantity.

The probability density function (pdf) of the generated current has a Gaussian distribution.

Shot noise appears as noise current superimposed on the current through the

semiconductor junction. It has an rms value of

(1-11)

with [C] the charge of a single charge carrier, [A] the average current through the

semiconductor junction, and [Hz] is the bandwidth of the system.

1 GHz 10 GHz 100 GHz 1 THz300 K 0.008% 0.079% 0.795% 8.340%273 K 0.009% 0.087% 0.874% 9.214%77 K 0.031% 0.309% 3.144% 38.234%

TABLE 1-1. Relative error made by assuming that the thermal noise spectrum is flat

B T

in t( )

inrms 2q I0 B=

q I0

B

9


1.2.3 Other noise sourcesIt is clear that there are also several other stochastic processes that can cause the generation of

noise. A number of them will be briefly mentioned in what follows.

A. Flicker NoiseIn many components, the noise seems to contain an extra contribution whose power spectral

density is inverse proportional to the frequency. Hence, Flicker noise is also often called

noise. At high frequencies, this noise contribution is insignificant compared to the thermal

noise. The frequency at which thermal noise and flicker noise have the same power spectral

density varies from several Hz to several MHz, depending on the type of electrical component.

B. Plasma noiseThis type of noise is caused by the random motion of charges in ionized gasses, such as the

ionosphere or sparkling electrical contacts.

1 f⁄

10

Linear time invariant systems

1.3 Linear time invariant systemsIn the following section, the definition and properties of a linear time invariant (LTI) system

will be given.

1.3.1 Definition of a linear time invariant system

Definition 1.2

A linear time invariant system is a system whose properties do not change with time, and that

obeys the superposition principle. In other words, a linear combination of input signals must

result in the same linear combination of output signals, and this has to be independent of the

moment at which the experiment is performed.

Or,

(1-12)

where represents the set of the real numbers, and represents the set of all the functions

dependent of the time , whose range is .

1.3.2 Spectral properties of a LTI systemSince the Fourier Transform is a linear operator, the response spectrum of a LTI system

to an input spectrum can be written as (see Appendix 1.B):

(1-13)

where represents the transfer function of the LTI system. The response spectrum

is only the multiplication of the input spectrum with the transfer function . It only

contains energy at those frequency intervals where the input spectrum contains energy.

H is the operator of a linear system ⇔

ci R ui t( ) F :H ui t( )[ ] yi t( )= H ciui t( )

i 1=

N

∑⇒ ciH ui t( )[ ]

i 1=

N

∑ ciyi t( )

i 1=

N

∑= =∈∀,∈∀

R Ft R

Y ω( )

U ω( )

Y ω( ) U ω( ) H ω( )⋅=

H ω( ) Y ω( )

U ω( ) H ω( )

U ω( )

11


Hence, the output spectrum at angular frequency (i.e. ) is solely determined by the

input spectral component at angular frequency (i.e. ). No extra spectral components

will be created. For a discrete input spectrum, such as a sine wave, one obtains the following

relation (see Figure 1-4).

Furthermore, as opposed to nonlinear systems (see further), the transfer function does

not depend on the input spectrum . Note that, by definition of the LTI system, the

response to a general periodic waveform consists of the sum of the responses of its discrete

frequency components. Knowledge of the transfer function of a LTI system is hence sufficient

to predict the response of the device to a general periodic input. (This property is used in -

parameters.)

1.3.3 Description of LTI systems at high frequenciesAt high frequencies (i.e. frequencies for which the wavelength of the signals of in the order of

magnitude or smaller than the dimensions of the considered system), the system cannot be

considered as lumped any more. The system becomes a distributed system, which means that

voltage and current are both time and position dependent. It is therefore no longer possible to

characterize the system by means of position-independent voltages and currents. The linear

relation as expressed in the or matrix (see Appendix 1.C) can still be used if a reference

position is selected at each port of the device. The description of the system hence becomes

(with and the position where the voltages and currents are considered):

FIGURE 1-4. Input and output spectrum of a LTI system

ωi Y ωi( )

ωi U ωi( )

LTISystem

Y ω( )

ω

U ω( )

ωω1 ω1

H ω( )

U ω( )

S

Z Y

x1 x2

12

Linear time invariant systems

Phenomena occurring in distributed systems (such as reflections) are easier described using the

concept of (travelling) waves. These waves describe the energy flowing in and out the port of

the device under test (DUT).

Definition 1.3

The incident voltage wave and the reflected voltage wave at port of a multiple port LTI

system are defined as:

(1-14)

where and represent respectively the voltage and the current at the reference plane of

port , and the characteristic impedance to which the waves are referred.

The following figure illustrates the definition of the waves for a two-port LTI system:

FIGURE 1-5. Definition of reference planes at the input and output of a LTI system.

FIGURE 1-6. Definition of incident and reflected waves for a two-port system.

LTISystem

V1 x1 t,( ) V2 x2 t,( )

I1 x1 t,( ) I2 x2 t,( )

ai bi i

aiVi Z0Ii+

2---------------------= bi

Vi Z0Ii–2

---------------------=

Vi Ii

i Z0

LTISystem

a1 ξ x1,( ) a2 ξ x2,( )

b1 ξ x1,( ) b2 ξ x2,( )

13


The waves are a function of the independent variable , which represents either time or

frequency, and of the position.

Since the waves and are obtained as a linear combination of the voltages and

currents at the reference ports (see (1-14)), the relation between the incident and

reflected waves is also a linear combination of the form:

(1-15)

The matrix still fully describes the DUT, since it is only a linear transformation of the -

matrix. The matrix is the well-known -parameter representation, which is defined as

follows:

Definition 1.4

The -matrix of a LTI system with ports is a complex n-by-n matrix whose elements are

defined as follows [36]:

(1-16)

Thus the -matrix fully describes a LTI system at high frequencies.

ξ

ai bi Vi xi ξ,( )

Ii xi ξ,( )

b1…bn

Sa1…an

⋅=

S Z

S S

S n

Si j,biaj----

m j:am≠∀ 0=

=

S

14

Noise and linear time invariant systems

1.4 Noise and linear time invariant systemsFirst, a model for a noisy LTI system is introduced. Based on this model, the classical method

used to quantify the noise behavior of a noisy LTI system is built up. Care is taken to clearly

state and explain all the required assumptions. This can then be used as a sound basis for

extension towards nonlinear systems.

1.4.1 The presence of noise in a LTI systemSince any practical electronic system contains electrical components, such as resistors or

semiconductor devices, the output signal of the system will contain noise generated in these

components. As seen in section 1.2, the resistors will mainly create Johnson noise, while the

semiconductor devices will mainly be responsible for the shot noise. Other sources can also

contribute to the noise present at the output of the system. Note that, according to the definition

of a LTI system (Definition 1.2), a LTI system must be noiseless. Since a LTI system has no

contribution to the noise of its own, it can only process the input noise as an additional signal

source. From (1-12), it follows that if the input signal equals zero, the output signal

must also be zero. In practice, it is clear that even when no input signal is present, the noise

sources in the system still exist, and noise will be present at the output of the system. Hence,

the LTI model has to be enhanced for noise.

Assumption 1: The noise produced by the noisy LTI system is purely additive.

The output of the system consists of the superposition of the output of the LTI system and a

noise source (see Figure 1-7).

FIGURE 1-7. Model for the noisy LTI system.

u t( ) y t( )

nA t( )

LTISystem +

noise sourcenoisy “LTI” system

y t( )u t( )

nA t( )

15


Assumption 2: There is a perfect impedance match at the input and the output of the system, i.e.

The input and output impedances of the system are equal to .

By assumption 1, the output of the system can be written as:

(1-17)

where represents the noise superimposed on the output of the LTI system.

Some additional hypotheses are made about the noise source :

1. Stationarity of the noise: The properties of the noise source remain constant in time.

Furthermore, disturbances (such as e.g. a noise spark due to lightning) are not taken into

account as stated previously.

2. The noise source is statistically independent of the input signal . Of course,

can depend on the bias current of the system (e.g. in the case of a transistor). It is

clear that a larger DC bias current can yield more thermal noise (due to the warming up

of the resistors) and shot noise (whose rms current is directly proportional to the square

root of the DC current through the device). However, the bias current of the system will

be considered as a constant property of the system, and not as an extra input signal.

Since the noise source is not correlated with the input signal, the cross-correlation between

signal and noise is zero, i.e. . This implies that (1-17) can be rewritten

in terms of power spectral densities as (see Appendix 1.D):

(1-18)

where , and represent respectively the (double sided)

power spectral densities of the output signal , the input signal and the noise added by

the noisy linear system .

Z0

y t( ) h t( )*u t( ) nA t( )+=

nA t( )

nA t( )

nA t( ) u t( )

nA t( )

E u t( )nA t τ+( ) 0=

PSDy2( ) ω( ) H ω( ) 2PSDu

2( ) ω( ) PSDnA

2( ) ω( )+=

PSDy2( ) ω( ) PSDu

2( ) ω( ) PSDnA

2( ) ω( )

y t( ) u t( )

nA t( )

16


1.4.2 Noise figureIn order to quantify the power spectrum of the noise source present in the noisy linear system,

one has to define a figure of merit. As stated in the introduction of this chapter, a figure of

merit that is given in terms of the deterioration of the signal-to-noise ratio (SNR) will be the

best choice.

Definition 1.5

The Noise Figure (NF) of a noisy linear system quantifies the system-induced degradation of

the signal-to-noise ratio between the input and the output of the device. It is the ratio of the

signal-to-noise ratio at the input of the system to the signal-to-noise ratio at the output of the

system, when the noise component of the input signal consists of thermal noise generated at

290 K, and the system is ideally matched at the input and output [6].

(1-19)

The noise figure of a noisy LTI system (as shown in Figure 1-8) will be determined next.

No prerequisites are made about the properties of the noiseless input signal itself, since

for LTI systems, it will be shown that the noise figure is independent of the properties of .

The thermal noise power spectral density (which is constant over the frequency, see section

1.2.1) at 290 K can easily be calculated using (1-10):

FIGURE 1-8. Signal and noise applied to a noisy LTI system

NFSNRinSNRout------------------

T0 290K=

=

u0 t( )

nu t( )u t( )

Z0

Z0

LTISystem +

nA t( )

noisy LTI system

u0 t( )

u0 t( )

17


1 (1-20)

The noise power spectral density at the output of the system at angular frequency can be

written as: . Hence, the noise figure can be calculated using its

definition (1-19):

(1-21)

Note that the noise figure cannot be smaller than one. In the limit case where it is equal to one,

the system does not produce any added noise, i.e. . Hence, the signal-to-noise

ratio can only deteriorate, and in the best case (when the system is noiseless) it remains

constant. Note also that the noise figure is independent of the input signal , it depends only

on the transfer function of the noisy LTI system.

1.4.3 Input noise temperature dependenceIf the input noise source is at absolute temperature , instead of , but the temperature of all

the other noise sources in the system remains unchanged, the signal-to-noise ratio degradation

can easily be calculated (see Appendix 1.F) as:

(1-22)

1.4.4 Noise Figure measurements concepts: the Y-factor techniqueThe most straightforward approach to measure the noise figure would be to measure the signal

and noise power spectral densities at the input and output of the system and to compare them

with each other. This is an impossible approach, because it would require measurement

1. dBm means dB as referred to a standard power of 1 mW. A value of W corresponds to.

PSDnu

1( )

T0 290K=kT0 4 21–×10 W Hz⁄ 174dBm Hz⁄–= = =

ξ10 ξW 1mW⁄( )log⋅

ω

H ω( ) 2 kT0⋅ PSDnA

1( ) ω( )+

NF ω( )

2 U ω( ) 2 Z0⁄kT0

--------------------------------

2 H ω( ) 2 U ω( ) 2 Z0⁄

H ω( ) 2kT0 PSDnA

1( ) ω( )+--------------------------------------------------------------- --------------------------------------------------------------------- 1

PSDnA

1( ) ω( )

H ω( ) 2 kT0⋅---------------------------------+= =

PSDnA

1( ) ω( ) 0=

u t( )

T T0

SNRinSNRout------------------

T

1 NF 1–( )T0T------+=

18


equipment with an extremely large dynamic range to measure both the signal and the noise.

Moreover, the measurement itself would add so much noise that the measured noise power

would be drown in the noise of the measurement system.

Since it is impossible to measure directly the signal-to-noise ratio in order to determine the

noise figure, an indirect measurement method is used. Suppose that only noise is fed at the

input of the system (hence ). This noise is Johnson noise generated by

a resistor whose absolute temperature is variable. The power spectral density of the signal at

the input of the system will then be a flat spectrum, whose magnitude is temperature dependent

and given by (see (1-10)). The output power spectral density at angular frequency is

then given by:

(1-23)

and is a linear function of the absolute temperature (see Figure 1-9).

If one knows the slope and the intersection point of the straight line with the axis, these

values can be filled in into (1-21), and the noise figure can be determined. Furthermore, having

the coordinates of two points lying on this straight line is enough to determine the line and its

parameters. By bringing the resistor, which generates thermal noise at the input of the system,

at two different temperatures (while the system is kept at a constant temperature, in order not

to modify ) and by measuring the noise power spectral densities at the output of

FIGURE 1-9. Output PSD as a linear function of the absolute temperature

u t( ) nu t( )= u0 t( ), 0=

T

kT ωi

PSDy1( ) ωi( ) H ωi( ) 2kT PSDnA

1( ) ωi( )+=

T

T

PSDy1( )

PSDnA

1( )H ωi( ) 2k

slope

T

T 0=

PSDnA

1( ) ωi( )

19


the system, two points on the straight line are obtained. The two temperatures are often

referred to as the “cold” or lower temperature , and the “hot” or higher temperature .

Both output power spectral densities are called in the literature [5] and ,

corresponding respectively to the temperatures and . Introducing the Y-factor

as the ratio of both output power spectral densities, one can

calculate that the noise figure can be determined out of the measurements as (see Appendix

1.E):

(1-24)

Notes:

1. Although the concept considers that the noise source is a resistor, which produces

thermal noise, commercial noise sources rather use diodes than resistors to produce the

hot noise spectrum. (The cold noise spectrum on the other hand is created by Johnson

noise in a resistor at room temperature ). This implies that the hot noise spectrum

won’t be perfectly flat as a function of the frequency. However, the described

measurement method does not require the spectrum to be flat, but only to be known for

each frequency. A description of this frequency dependency is provided by the

manufacturer as the quantity “Excess Noise Ratio” (ENR) and is defined as the ratio

. It hence gives the equivalent hot noise temperature at which a resistor

has to be brought to produce the same noise power spectral density as the diode

containing noise source does in “hot” operating mode.

2. The measurement equipment used to determine the output hot and cold noise power

spectral densities also adds noise to the measurement. However, it is possible to

eliminate mathematically the contribution of this measurement noise, by first

determining the noise figure of the measurement equipment, and using Friis’formula [6]:

(1-25)

Tc Th

N1 ωi( ) N2 ωi( )

Tc Th

Y ωi( ) N2 ωi( ) N1 ωi( )⁄=

NF ωi( )

ThT0------ 1–

Y ωi( )TcT0------ 1–

–

Y ωi( ) 1–--------------------------------------------------------------=

Tc=

Th ω( ) T0⁄ 1–

NFtot ω( ) NF ω( )NFms ω( ) 1–

H ω( ) 2--------------------------------+=

20


, and represent respectively the noise figures of the DUT

plus the measurement system, the DUT alone and the measurement system alone.

represents the transfer function of the DUT (see Appendix 1.G).

1.4.5 Noise temperatureAn alternative quantity that describes the noise power spectral density, generated by a device,

is the noise temperature.

Definition 1.6

A device has a noise temperature , when it generates noise with a power spectral density

.

Hence, the noise figure can indeed be defined as the ratio of the signal-to-noise ratio at the

input of a device to the signal-to-noise ratio at the output of that device, when the device is

excited with a generator that has a frequency independent noise temperature of 290 K. If the

noise temperature of the source impedance differs too much from 290 K (e.g. for satellite

communications where at unclouded sky), the noise figure is not such a practical

quantity to describe the signal-to-noise evolution through the system. In such cases, it is more

convenient to use the concepts “operational noise temperature” and “effective noise

temperature” .

A. Operational noise temperatureThe operational noise temperature is defined as the absolute temperature that has to be

assigned to the source impedance, in order to get a noise power spectral density at the output of

the noiseless LTI system equal to the noise power spectral density at the output of the noisy

NFtot ω( ) NF ω( ) NFms ω( )

H ω( )

Tn ω( )

PSDn1( ) ω( ) kTn ω( )=

Tn 10K≈

Top

Te

Top ω( )

21


LTI system, when the source impedance has a noise temperature . Hence, will vary for

different .

B. Effective noise temperatureThe effective noise temperature is defined as the absolute temperature that has to be

assigned to the source impedance in order to get a noise power spectral density at the output of

the noiseless LTI system equal to the noise power spectral density at the output of the noisy

LTI system, when the source impedance has a noise temperature of 0 K ( ). Hence,

(1-26)

FIGURE 1-10. Illustration of operational noise temperature.

Ts Top

Ts

Z0 LTISystem +

nA t( )

noisy LTI system

Ts

P P1=

Z0 LTISystem

noiseless LTI system

Top Ts>

P P1=

Te ω( )

nu t( ) 0=

Top ω( ) Ts Te ω( )+=

22


can be interpreted as the apparent increase of the temperature of the input impedance,

due to the fact that the LTI system is noisy instead of noiseless.

can also be written as:

(1-27)

Or, in terms of the noise figure:

(1-28)

FIGURE 1-11. Illustration of effective noise temperature.

Te ω( )

Z0 LTISystem +

nA t( )

noisy LTI system

0K

P P1=

Z0 LTISystem

noiseless LTI system

Te

P P1=

Te ω( )

Te ω( )PSDnA

1( ) ω( )

k H ω( ) 2⋅---------------------------=

Te ω( ) NF ω( ) 1–( )T0=

23


1.5 Nonlinear systemsJust as for LTI systems, one has to know the definition and the properties of the subclass of

systems that will be used when dealing with nonlinear systems. The subclass of systems that is

considered in this text is introduced here, and is selected such as to allow a gentle departure

from the linear behavior. This allows consideration of systems that are close to be linear. It is

then a good engineering practice to extend the methods explained for linear system noise

characterization to cope with this extended class of systems that enclose the LTI systems. The

importance of knowing the phase relation between spectral components at different

frequencies is also pinpointed.

1.5.1 Definition of a nonlinear time invariant systemStrictly speaking, a nonlinear system is a system that does not obey the definition of a linear

system, i.e. Definition 1.2. This means that a nonlinear time invariant system is a system

whose properties do not change in time, and that does not obey the superposition principle. In

other words, a linear combination of input signals does not always result in the same linear

combination of output signals.

Stated mathematically,

(1-29)

It is clear that the class of nonlinear systems, as defined above, is too complex to be studied in

one single framework because it contains every conceivable nonlinearity such as chaotic

systems, systems with bifurcations, hysteresis, etc....

Clearly, a small subclass of nonlinear systems has to be chosen and studied. This class has to

be selected such as:

• to allow a gentle departure from linearity

H is the operator of a nonlinear system ⇔

c∃ i R ui t( ) F :H ui t( )[ ] yi t( )= H ciui t( )

i 1=

N

∑⇒ ciH ui t( )[ ]

i 1=

N

∑≠ ciyi t( )

i 1=

N

∑=∈∃,∈

24

Nonlinear systems

• to include linear systems as a special case

• to describe the behavior of many practical nonlinear circuits

• to have a suitable, simple mathematical model to allow design and

analysis

The choice falls on the subclass that can be tagged as “NICE” systems, which is an extension

of the Volterra systems.

Definition 1.7

A NICE system is a system whose output converges in a least squares sense to a Volterra

series, as the order of the Volterra series tends towards infinity. Or,

is a NICE system the output of can be approximated in a least squares sense

as:

(1-30)

the approximation will be better as increases. represents the -th order Volterra

operator, and can be written as:

(1-31)

with the -th order symmetrized Volterra kernel of the system [7]. (This

corresponds to a multi-dimensional impulse response.) Note that systems with bifurcations,

hysteresis, subharmonics or chaotic systems are no part of the NICE systems, they fall beyond

the scope of this work. To check if this NICE system class fulfills the requirements as stated

before, the spectral properties of the NICE system will be studied.

H ⇔ y t( ) H

u∀ t( ) F :y t( )∈ H u t( )[ ] Hα u t( )[ ]

α 1=

αmax

∑= =

αmax Hα [ ] α

Hα u t( )[ ]

Hα u t( )[ ] …∞–

∞∫ hα τ1 … τα, ,( )u t τ1–( )…u t τα–( ) τ1d … ταd

∞–

∞∫=

hα τ1 … τα, ,( ) α

25


Note that the definition does not state that the output of the NICE system equals a Volterra

series. It only tells that the NICE system output can be approximated in a least squares sense as

a Volterra series. Hence, Volterra systems are a subset of the NICE systems. A similar

reasoning is the fact that a function can be approximated in least squares sense as a

polynomial. The Taylor series expansion is a special polynomial approximation of a function,

that uniformly converges to the given function within its convergence circle.

Note also that, NICE systems can also be defined as systems that convert a periodic input

signal to a periodic output signal, with the same period.

1.5.2 Spectral properties of a NICE systemTo ease the explanation, and without loss of generality, consider the following static NICE

system, that can be described by a simple three term Volterra series:

(1-32)

For the special case of a sinusoidal excitation signal, , the output signal of

this static NICE system is given by:

(1-33)

For NICE systems, the output spectral components at angular frequency will no longer

solely be determined by the input spectral components at angular frequency . Furthermore,

the NICE system will create additional spectral components that were not present in the input

signal. From equation (1-33), it is clear that when a single tone is applied to a NICE system,

the output of the system will not only contain that single tone, but also higher order harmonics

of the tone will appear (This is illustrated in Figure 1-12). Furthermore, the linear relation

between the input and output spectrum at angular frequency does not longer exist. Indeed,

in the example system (1-32), the ratio between output and input spectrum at angular

frequency becomes , which clearly depends on the power of the input signal

.

y t( ) H u t( )[ ] k1 u t( )⋅ k2 u2 t( )⋅ k3 u3 t( )⋅+ += =

u t( ) A ω0t( )cos⋅=

y t( ) k1Ak33A3

4---------------+

ω0t( )cos k2A2

2------ k2

A2

2------ 2ω0t( )cos k3

A3

4------ 3ω0t( )cos+ + +=

ωi

ωi

ω0

ω0 k1 k33A2 4⁄+

u t( )

26

Nonlinear systems

Since the ratio depends on the input power, the -parameters will also become

power dependent. Hence, one can make important mistakes by measuring the -parameters of

a NICE system while supposing one is dealing with a LTI system. By definition, -parameters

and transfer functions are independent of the input signal’s properties, and thus applicable for

LTI systems only, and not for NICE systems.

If the excitation signal happens to contain more than one spectral tone, e.g.

where and are not harmonically related (i.e. they

are not integer multiples of each other), the situation is even more complex. Besides the

harmonics of and (located at angular frequencies and , ),

intermodulation products also appear at angular frequencies (with ).

1.5.3 Importance of the absolute phase spectra for NICE systemsFor LTI systems, the knowledge of the phase relation between the spectral components is not

necessary. The phase of an output spectral line at angular frequency can only depend on the

phase of the input spectral line at angular frequency (and of the phase of the transfer

function of course):

(1-34)

where , and represent respectively the phase of , and

. Hence the phase of the transfer function at angular frequency equals the difference

of the phases of the output and the input spectrum at angular frequency .

FIGURE 1-12. Spectral response of a NICE system to a single tone input.

NICESystem

Y ω( )

ω

U ω( )

ωω1 ω1 2ω1 3ω1

Y ω0( ) U ω0( )⁄ S

S

S

u t( ) A1 ω1t( )cos A2 ω2t( )cos+= ω1 ω2

ω1 ω2 nω1 mω2 n m Z∈,

nω1 mω2+ n m Z0∈,

ωi

ωi

φY ωi( ) φU ωi( ) φH ωi( )+=

φY ωi( ) φU ωi( ) φH ωi( ) Y ωi( ) U ωi( )

H ωi( ) ωi

ωi

27


For NICE systems however, if the input signal is a sine wave the output

signal can be e.g.

(1-35)

where , and are factors that can be function of . , and are the phases of

the output spectrum at angular frequencies , and compared to the phase of the

input spectrum at angular frequency . In this case, it is important to know the phases ,

and , otherwise the output time waveform cannot be determined. Since a

nonlinearity essentially operates on the instantaneous value of the time signal, not much can be

said about a DUT’s nonlinear behavior if this time waveform is unknown.

In order to illustrate the importance of knowing the phases between the spectral components,

two input signals and , whose amplitude spectra are identical, but with different

phase spectra, are applied to a NICE system, whose behavior can be described with Volterra

series as .

FIGURE 1-13. Amplitude and phase spectrum of both input signals.

u t( ) A ω0t( )cos⋅=

y t( ) c1A ω0t ϕ1+( )cos c2A2 2ω0t ϕ2+( )cos c3A3 3ω0t ϕ3+( )cos+ +=

c1 c2 c3 A ϕ1 ϕ2 ϕ3

ω0 2ω0 3ω0

ω0 ϕ1

ϕ2 ϕ3 y t( )

u1 t( ) u2 t( )

y t( ) H u t( )[ ] u3 t( )= =

0 10 20 30 40

-150

-100

-50

0

50

100

150

Frequency bins

Phase [deg]

u1 t( )

u2 t( )phase of

phase ofAmplitude spectra

are identical[V]

28

Nonlinear systems

These spectra correspond to the following time signals:

The blue dashed line corresponds to , while the red solid line corresponds to .

When applying both signals to a LTI system, the output amplitude spectra will be identical,

since it follows from (1-13) that and the input amplitude spectra

are identical. As shown in (1-34), a modification of into will only

result in a modification of the phase of the output spectrum into .

Applying the signals and to the NICE system, results in the following output

signals:

FIGURE 1-14. Time waveforms of both signals.

FIGURE 1-15. Both output signals of the NICE system in time and frequency domain.

0 5000 10000 15000

-5

0

5

10

Time samples

Am

plit

ude

[V]

u1 t( ) u2 t( )

Y ωi( ) H ωi( ) U ωi( )⋅=

φU ωi( ) φU ωi( ) ∆φU ωi( )+

φY ωi( ) φY ωi( ) ∆φU ωi( )+

u1 t( ) u2 t( )

0 5000 10000 15000

-200

0

200

400

600

800

1000

1200

Time samples

Am

plit

ude

[V]

[V]

29


Figure 1-15 shows the output signals, in time and frequency domain. The blue dashed lines

correspond to the output response to signal , while the red solid lines correspond to the

output response to signal . The plot of the frequency spectra clearly shows that the

amplitude of the output spectral lines differs for both output signals, even if the input

amplitude spectra are identical. This underlines the importance of the knowledge of the phase

relation between spectral lines when dealing with NICE systems.

Two solutions exist to measure this phase relation:

1. Time domain solution: The time waveform is measured and the Fourier transform is

taken to obtain the spectrum [25].

2. Frequency domain solution: One output harmonic is measured at a time, together with

the input signal. Comparing both phases yields the phase of the output harmonic

compared to the input signal [8].

u1 t( )

u2 t( )

30

Noise and nonlinear systems

1.6 Noise and nonlinear systemsKnowing the noise power spectral density at the output of a NICE system is as important as

knowing the signal power spectral density. The reason therefore is quite obvious: in many

applications the signal-to-noise ratio is a very critical parameter that has to be maximized.

Using an identical approach as earlier described for the noisy LTI system, a noisy NICE system

is introduced. Next, it is shown that the output spectrum of a noisy NICE system, excited by

the sum of signal and noise can be divided into four disjunct sets of terms, according to the

behavior of these terms. Based on this classification, different setups are introduced.

1.6.1 The presence of noise in a NICE systemLike LTI systems, real-world NICE systems also consist of electrical components, including

resistors or semiconductor devices. Again, noise is generated in the electrical components of

the NICE system, and appears at the output of the system. Like a LTI system, a NICE system

(Definition 1.7), must be noiseless by definition. (1-30) and (1-31) show that for a zero input

signal , the output signal must also be zero. An enhanced model is defined for the

noisy NICE system in similarity to the noisy LTI system. Practically, the structure of the model

will be the same as the one for the noisy LTI system, with the LTI block replaced by a NICE

system.

The output of the noisy NICE system is:

(1-36)

but this time is the operator of a NICE system instead of a LTI system.

FIGURE 1-16. Model for the noisy NICE system.

u t( ) y t( )

NICESystem +

noise source

noisy NICE system

y t( )u t( )

nA t( )

y t( ) H u t( )[ ] nA t( )+=

H [ ]

31


Note that for real-world systems, noise sources are present inside the NICE system, that cannot

be represented as a superimposed noise source on the output signal (see Figure 1-17).

The output signal of this system is given by:

(1-37)

However, in this case, terms are created that consist of the multiplication of input signal and

noise. Similar terms will also be encountered when applying signal and noise together at the

input of the system shown in Figure 1-16. Hence, the presence of noise sources inside the

NICE system (as shown in Figure 1-17) does not fundamentally modify the noise behavior of

the NICE system. They yield similar contributions as the contributions due to the presence of

input noise. Therefore, the model of the noisy NICE system as proposed in Figure 1-16 will be

used in all the following.

The same hypotheses about the noise source are made as for the noisy LTI system, i.e.

the noise source is stationary and independent of the input signal. Hence, this noise source

behaves similarly to the one that was present in the LTI case. Up till now, the noise behavior of

a LTI and of a NICE system do not differ much. Another important question is to know how

the signal-to-noise ratio will change from the input to the output of the system. For noisy LTI

systems, the change is quantified by the Noise Figure (see sections 1.4.2 and 1.4.4). For noisy

NICE systems however, some deeper investigation is required to verify if this is still the case.

FIGURE 1-17. Example of a noisy NICE system that cannot be modelled as shown in Figure 1-16

+

noise sourcenA t( )

u t( ) y t( )X

y t( ) u t( ) u t( ) nA t( )+( )⋅ u2 t( ) u t( )nA t( )+= =

nA t( )

32


1.6.2 Applying signal and noise together to a NICE systemTo analyze the evolution of the signal-to-noise ratio through the system, consider the following

setup:

To ease the calculation, and without loss of generality, assume that the noisy NICE system is

perfectly matched at both the input and the output port for all frequencies.

A. The need for discrete spectraAs shown in the previous figure, signal and noise are applied to a noisy NICE

system. The most logical next step would be to say that the input spectrum of the system

consists of the sum of the noiseless signal spectrum , and the noise spectrum .

However, the Fourier transform of a noise signal does not converge and hence

does not exist. This problem will be solved by using discrete windowed signals and

discrete spectra instead of continuous ones.

In practical measurements, continuous spectra will yield several problems. The main problem

is that since the spectrum is continuous, every frequency interval contains an infinite, non-

countable, number of frequencies. To correctly measure the spectrum at all these frequencies, a

measurement system with an infinitesimal small measurement bandwidth is required. This

means that the measurement time has to be infinity. A few “minor” problems are the need for

an analytical signal expression to evaluate the integral in the Fourier transform, or the fact that

computers have only a finite amount of memory to store the infinite number of samples.

Hence, one has to work with discrete spectra and signals instead of continuous ones.

FIGURE 1-18. Signal and noise are applied to a noisy NICE system.

System

u0 t( )

NoisyNICE

nu t( )

u t( ) nu t( ) u0 t( )+= Z0

Z0

y t( )

u0 t( ) nu t( )

U ω( )

U0 ω( ) Nu ω( )

ℑ nu t( )

Nu ω( )

33


However, some important considerations have to be taken into account:

First, the time signal has to be sampled, i.e. only the instantaneous values of with

are retained. represents the sampling period and is the inverse of the sampling

frequency . To avoid alias, the sample frequency has to obey Shannon’s theorem, i.e.

, where represents the highest frequency component present in . Since

contains noise whose bandwidth was supposed to be infinity by approximation (see

section 1.2.1), the problem arises that at first sight . However, since bandwidth of

the NICE system is finite, using a lowpass filter with a bandwidth equal to that of the NICE

system, and setting will do the job. At this point, the time waveform is discrete, but

the spectrum of is still a continuous one, it is the lowpass filtered spectrum of ,

that repeats itself each integer multiple of the sample frequency .

Next, the spectrum has to be discretised. This can be achieved by selecting only a finite

number ( ) of consecutive time samples and applying the Discrete Fourier

Transform [37] to those samples.

(1-38)

Since the DFT yields a discrete spectrum, this implies that the time signal was a periodic

signal, i.e. that the obtained spectrum corresponds to a time signal consisting of the chosen

time samples that continuously repeat themselves.

If is a periodic signal, the frequency grid has to be chosen such that all the discrete

spectral components of lie on grid lines. If the frequencies of all the spectral components

of are related to each other as rational numbers, this requirement can be met by choosing

the frequency grid spacing equal to the greatest common divisor (gcd) of all these

frequencies. If on the other hand, there exist two frequencies of all the spectral components of

whose ratio is an irrational number, it is impossible to lay all the spectral components of

on grid lines (since their gcd is zero, which implies that ). (Note that the relation

between , and is given by )

u t( ) u mTs( )

m Z∈ Ts

fsfs 2fmax> fmax u t( )

u t( ) nu t( )

fmax ∞= B

fs 2B=

u mTs( ) u t( )

fs

M u mTs( )

DFT x mTs( )( ) 1M----- x mTs( )e

j2πkM

---------m–

m 0=

M 1–

∑=

M

u0 t( )

u0 t( )

u0 t( )

∆f

u0 t( )

u0 t( ) M ∞→

M ∆f fs ∆f fs M⁄=

34


Since the input noise is an aperiodic signal, one theoretically needs an infinite number of

time samples to adequately describe spectrum using the DFT, yielding a frequency grid

spacing of 0 Hz. Practically, one has to choose a frequency grid such that the noise spectrum

does not vary too much from one grid line to another. Since by assumption was thermal

noise that has a flat power spectral density, the grid spacing obtained by guaranteeing that the

spectral components of lie on grid lines, will often be sufficient to describe the

variations in the noise spectrum. Note however, that the resulting noise spectrum obtained with

the DFT is a spectrum of a periodical signal, and thus one is dealing with periodic noise.

If on the other hand, is aperiodic, or contains irrationally related frequency components,

a frequency grid has to be chosen that is dense enough to adequately describe the variations of

the signal spectrum.

B. The discrete output spectrum of NICE systemsAs stated before, the input spectrum of the system ( ) consists of the

sum of the noiseless signal spectrum , and the noise spectrum .

For a LTI system, the signal-to-noise ratio evolution through the system at a certain frequency

is only determined by the signal and noise spectra, the noise figure and the matching

conditions at that particular angular frequency. The knowledge of or with

does not contribute to the behavior of the circuit at frequency .

For NICE systems, this is no longer the case. According to the Volterra theory, the output

spectrum at frequency is given by:

FIGURE 1-19. Signal and noise spectra at the input of the system.

nu t( )

M

nu t( )

u0 t( )

u0 t( )

U k( ) DFT u mTs( )( )=

U0 k( ) Nu k( )

Nu k( )

U0 k( )

Amplitude

k∆f

l ∆f⋅

U0 k( ) Nu k( ) k l≠

l ∆f⋅

l ∆f⋅

35


(1-39)

and can be written as:

(1-40)

where are chosen so that . Hence, the portion of the

output spectrum at grid line (lying at frequency ), due to the Volterra kernel of the -th

degree, is the sum of all possible combinations of input spectral components, lying at grid

lines , such that their sum equals . Thus, the signal-to-noise ratio at the output

of the NICE system at frequency is no longer only a function of the input signal and noise

at that particular frequency , but it will also depend on the properties of and

where . (1-39) and (1-40) will be illustrated through the following example.

Example 1.8

Consider the following static NICE system that can be described through its Volterra series:

(1-41)

According to the Volterra theory, the output spectrum at angular frequency is given by

(1-39):

(1-42)

where

Y l( ) Y α( ) l( )

α 1=

∞

∑= where Y α( ) l( ), DFT Hα u mTs( )[ ]( )k l=

=

Y α( ) l( )

Y α( ) l( ) … Hα k1 … kα 1– L, , ,( )U k1( )…U kα 1–( )U L( )

kα 1–M2-----–=

M2----- 1–

∑k1

M2-----–=

M2----- 1–

∑=

k1 … kα 1– L, , , k1 … kα 1– L+ + + l=

l l ∆f⋅ α

α

k1 … kα 1– L, , , l

lf0lf0 U0 k( ) Nu k( )

k l≠

y t( ) H u t( )[ ] H1 u t( )[ ] H3 u t( )[ ]+ u t( ) u3 t( )+= = =

l ∆f⋅

Y l( ) Y 1( ) l( ) Y 3( ) l( )+=

36


(1-43)

can also be written as:

(1-44)

According to (1-40), and knowing that , can be expressed as:

(1-45)

where the three grid lines are chosen so that . This means that there

are two grid lines that can be independently chosen ( and ) and the third grid line ( ) has

to be such that . Since time samples are chosen, the DFT will also yield

frequency lines. Hence, the sums in (1-45) have to be taken over these grid lines, instead as

over an infinite number. Consider also , with equal to the

frequency grid spacing . In that case, the DFT spectrum is given by:

(1-46)

Equation (1-45) with becomes:

(1-47)

Y 1( ) l( ) DFT H1 u mTs( )[ ]( )k l=

DFT u mTs( )( )k l=

= =

Y 3( ) l( ) DFT H3 u mTs( )[ ]( )k l=

DFT u3 mTs( )( )k l=

= =

Y 1( ) l( )

Y 1( ) l( ) H1 l( )U l( ) U l( )= =

H3 k1 k2 k3, ,( ) 1= Y 3( ) l( )

Y 3( ) l( ) U k1( )U k2( )U L( )

k2M2-----–=

M2----- 1–

∑k1

M2-----–=

M2----- 1–

∑=

k1 k2 L, , k1 k2 L+ + l=

k1 k2 L

k1 k2 L+ + l= M M

M

u t( ) A 2πf0t ϕ+( )cos⋅= f0∆f

U 1( ) A2---ejϕ=

U 1–( ) A2---e jϕ–=

U k( ) 0= k 1±≠⇔

l 1=

Y 3( ) 1( ) U 1( )U 1( )U 1–( ) U 1( )U 1–( )U 1( ) U 1–( )U 1( )U 1( )+ +=

3U 1( )U 1( )U 1–( )= 3A3

8---------ejϕ=

37


For , (1-45) yields that , i.e. there are no combinations of only non-zero

spectral lines that yield .

For , (1-45) becomes:

(1-48)

Two more non-zero output spectral lines will be obtained through similar reasoning:

(1-49)

Combinatory analysis can predict the coefficients and the number of distinct combinations in

the output spectrum (see Appendix 1.H).

C. Hypotheses about the signal and the noise spectrum1. The signal spectrum is a deterministic spectrum and information-carrying

(random) signals such as used in telecommunication systems, are not taken into account.

If different realisations of the input signal are considered, will always be the

same, since it is deterministic. Mathematically expressed, this implies that

(1-50)

at those lines where signal power is present.

2. The input noise spectrum however, has also become a non-random spectrum due

to the usage of the DFT, which represents the noise spectrum as the spectrum of periodic

noise instead of random noise. If different realisations of the noisy input signal are

considered, other noise samples will be used to calculate the DFT, and hence the noise

spectrum will vary randomly over the different realisations. The noise spectrum is

a stochastic spectrum. At each spectral line the noise is circular complex normally

distributed and independent over the frequencies [9]. This is especially true for fine

frequency grids (large ). Hence,

l 2= Y 3( ) 2( ) 0=

Y 3( ) 2( )

l 3=

Y 3( ) 3( ) U 1( )U 1( )U 1( ) A3

8------ej3ϕ= =

Y 3( ) 1–( ) 3A3

8---------e j– ϕ= Y 3( ) 3–( ) A3

8------e j– 3ϕ=

U0 k( )

U0 k( )

E U0 k( ) U k( ) 0≠=

Nu k( )

Nu k( )

M

38


(1-51)

D. Grouping the output spectrum terms as function of their propertiesThe output spectrum of the NICE system calculated with (1-39) and (1-40) contains a lot of

terms, of which some contribute to the signal output spectrum and others to the output

noise spectrum . Again, some terms exist only due to the interaction of the signal and the

noise. In order to better understand how a NICE system treats the noisy input signal, these

terms will be grouped into several sets, depending on their properties.

Before starting immediately with the formulae and criteria for noisy NICE systems, a simple

example using a noisy LTI system will be given first in order to enhance the readability of the

text.

Starting from

, (1-52)

collect all the terms that depend solely on the noise free input signal . These terms are

grouped in . Hence, for the LTI case,

(1-53)

The remainder contains terms that are solely a function of the input noise or the noise

added by the noisy LTI system itself . These terms will be labelled as :

(1-54)

E Nu k( ) 0=

E Nu2 k( )

0=

E Nu k( )Nu* l( )

0= for k l≠

E Nu k( ) 2

σNu

2 k( )=

Y0 k( )

Ny k( )

Y k( ) H1 k( )U0 k( ) H1 k( )Nu k( ) NA k( )+ +=

U0 k( )

A U0( )

A U0( ) H1 k( )U0 k( )=

Nu k( )

NA k( ) B N( )

B N( ) H1 k( )Nu k( ) NA k( )+=

39


Note that (1-54) contains two noise sources: the input noise and the noise added by the

system itself , therefore these terms are tagged as instead of as .

There are no terms in the output spectrum that are due to an interaction between the noiseless

input signal and the input noise .

is the noiseless (deterministic) output spectrum , while is the spectrum of

the noise on the output signal .

For noisy NICE systems, we will start from (1-39) and (1-40) with .

First, all the terms that depend solely on the noise free input signal are grouped in

. Some terms of are: , coming from the linear Volterra operator,

and , coming from the higher order Volterra operators.

is a part of the noiseless output spectrum . It is the noiseless output spectrum that

would be obtained if only was applied at the input of the NICE system.

Second, the terms that are solely a function of the input noise or the noise added by the

noisy NICE system itself , are grouped into . Some terms of are: ,

. is a part of the output noise spectrum . It is the

output noise spectrum that would be obtained if only was applied at the input of the

NICE system.

The remaining terms in the output spectrum are terms that depend jointly on and

. They are grouped depending on their realization average.

The terms that depend jointly on and , and whose realization average is non-

zero, are grouped into . Note that out of (1-50) follows that

(1-55)

Nu k( )

NA k( ) B N( ) B Nu( )

U0 k( ) Nu k( )

A U0( ) Y0 k( ) B N( )

Ny k( )

U k( ) U0 k( ) Nu k( )+=

U0 k( )

A U0( ) A U0( ) H1 k( )U0 k( )

Hα k1 … kα, ,( )U0 k1( )…U0 kα( )

A U0( ) Y0 k( )

U0 k( )

Nu k( )

NA k( ) B N( ) B N( ) NA k( )

Hα k1 … kα, ,( )Nu k1( )…Nu kα( ) B N( ) Ny k( )

Nu k( )

U0 k( )

Nu k( )

U0 k( ) Nu k( )

A ′ U0 N,( )

E Hα k1 … kα, ,( )U0 k1( )…U0 kα1( )Nu kα1 1+( )…Nu kα( )

Hα k1 … kα, ,( )U0 k1( )…U0 kα1( )E Nu kα1 1+( )…Nu kα( ) =

40


and this latter realization average can only be different from zero if it consists of pairs of

complex conjugate noise spectral lines (see (1-51)). Hence, a term of is e. g.

. Since these terms have a non-zero realization

average, they contribute to the deterministic output spectrum of the system . They hence

create a bias on the deterministic part of the system’s output spectrum. This bias depends on

, and hence on the input noise power spectral density.

The remaining terms that depend jointly on and , and that have a zero realization

average, are grouped into . Out of (1-51) and (1-55) follows that the realization

average is different from zero if it does not consist of pairs of complex conjugate noise spectral

lines. A term of is e. g. . Since these terms

have a zero realization average, they only contribute to the variability of the output spectrum of

the system . They hence contribute to the noise at the output of the NICE system.

Hence, the output spectrum of the noisy NICE system can be written as:

(1-56)

Where

(1-57)

A′ U0 N,( )

H3 k1 k2 k2–, ,( )U0 k1( )Nu k2( )Nu k2–( )

Y0 k( )

σNu

2

U0 k( ) Nu k( )

B ′ U0 N,( )

B ′ U0 N,( ) H3 k1 k2 k3, ,( )U0 k1( )Nu k2( )Nu k3( )

Ny k( )

Y k( ) Y0 k( ) Ny k( )+=

Y0 k( ) A U0( ) A′ U0 N,( )+=

Ny k( ) B N( ) B′ U0 N,( )+=

41


1.7 ConclusionIn this chapter, an introduction is given about noise and its influence on a linear as well as on a

nonlinear system. First of all, several sources of noise are described and the properties of the

two most important ones (thermal noise and shot noise) are highlighted. Assuming that thermal

noise has a flat power spectral density, leads to errors smaller than 1% for frequencies up to

100 GHz and temperatures above 273 K.

Next, the definition and spectral properties of a noisy LTI system were given. The noise figure

that describes the deterioration of the signal-to-noise ratio through the LTI system proved to be

a very good figure to describe the noise behavior of a LTI system. A common measurement

technique (the Y-factor method) to determine the noise figure was also explained.

Finally, NICE systems, which are a subclass of nonlinear systems are introduced. NICE

systems, whose behavior can be described through Volterra series were chosen to be the target

class to extend the noise behavior analysis of LTI systems to.

Studying the output spectrum of a NICE system, when applying a deterministic signal

superimposed on stochastic (ergodic) noise at its input, allows to prove that the output can be

split into four disjunct sets. This also illustrates the interactions between the deterministic input

signal and the input noise.

42

Appendices

1.8 Appendices

Appendix 1.A: Cross-correlation of deterministic signals and ergodic noiseConsider to be a periodic, deterministic signal, and ergodic noise. Let

. will still be a periodic, deterministic signal. The time cross-correlation

between and is in that case (per definition) given by:

(1-58)

Using the definition of the Riemann Integral [32], this yields:

(1-59)

Since tends to infinity, one can assume that is an integer multiple of the period of .

If this is not the case, the contribution to the Riemann sum of that part, that is only a fraction of

the period of can be neglected anyway, since . Let be the period of ,

in that case, . Normally, the Riemann sum is made sequential over the intervals

. However, one can also first add all the contributions of every interval, located at the same

position, relative to one period of , and then add all these contributions over all the

relative positions of the intervals in one period. This is illustrated in Figure 1-20, for a simple

sine wave. Instead of sequential adding the contributions of the intervals, first all the surfaces

of the same color and pattern are added, and eventually, the sum is made over all colors and

patterns.

u0 t( ) nu t( )

α β, N0∈ u0α t( )

u0α t( ) nu

β t( )

E u0α t( )nu

β t τ+( ) 1

T--- u0

α t( )nuβ t τ+( ) td

T 2⁄–

T 2⁄∫T ∞→

lim=

1T--- u0

α ti( )nuβ ti τ+( )∆ti

i∑∆tj 0→

limT ∞→lim=

T T u0α t( )

u0α ti( ) T ∞→ T

u0α u0

α t( )

T kT Tu0

α⋅=

∆tiu0

α t( )

43


Hence, (1-59) becomes:

(1-60)

This time, the sum is taken over the variable , i.e. over one period of . is a

periodic signal, with period , hence , and (1-60) yields:

(1-61)

Hence, in part of the expression, has to be added times, and divided by .

Because is ergodic, the summation can be made over realizations, instead of time. Due

to the limit , this yields the same result as making the sum of realizations, dividing

by , and scaling the result with a factor , i.e.

FIGURE 1-20. Illustration of taking the Riemann sum over the periods.

1T--- u0

α

kT

∑ tj kTTu0

α+ nu

β tj kTTu0

α τ+ + ∆tj

j∑∆tj 0→

limT ∞→lim=

j u0α t( ) u0

α t( )

Tu0

α u0α tj kTT

u0α+

u0α tj( )=

1T--- u0

α

kT

∑ tj( )nuβ tj kTT

u0α τ+ +

∆tj

j∑∆tj 0→

limT ∞→lim=

1T--- u0

α tj( ) nuβ tj kTT

u0α τ+ +

∆tj

kT

∑j∑∆tj 0→

limT ∞→lim=

u0α tj( ) 1

T--- nu

β tj kTTu0

α τ+ +

kT

∑T ∞→lim

∆tjj∑∆tj 0→

lim=

nuβ t( ) kT T T

u0α⁄= T

nuβ t( )

T ∞→ T

T 1 Tu0

α⁄

44

Appendices

(1-62)

And due to the ergodicity of , the time and realization average are the same, thus (1-

61) becomes:

(1-63)

The latter integral in (1-63), divided by represents the mean value of , over one

period, and hence the time mean of .

Conclusion:

Appendix 1.B: Transfer function of a LTI systemConsider to be the impulse response of a LTI system, i.e. is the output signal of the

system, when the input signal is a Dirac impulse : .

Using the pincet property of the Dirac distribution, one can write a general input signal as

follows:

(1-64)

The output signal is:

1T--- nu

β tj kTTu0

α τ+ +

kT

∑T ∞→lim E nu

β t τ+( ) 1

Tu0

α--------=

nuβ t τ+( )

u0α tj( ) E nu

β t τ+( ) 1

Tu0

α--------

∆tjj∑∆tj 0→

lim=

E nuβ t τ+( )

1

Tu0

α-------- u0

α tj( )∆tjj∑∆tj 0→

lim=

E nuβ t τ+( )

1

Tu0

α-------- u0

α t( ) t d0

Tu0

α

∫=

Tu0

α u0α t( )

u0α t( )

E u0α t( )nu

β t τ+( )

E nuβ t τ+( )

E u0α t( )

⋅=

h t( ) h t( )

δ t( ) h t( ) H δ t( )[ ]=

u t( )

u t( ) u τ( )δ t τ–( ) τd∞–

∞∫=

y t( )

45


(1-65)

Since the integration is a linear operation, and is a linear time-invariant operator,

(1-66)

This last integral is the definition of a convolution integral, hence:

(1-67)

To calculate the output spectrum, one has to take the Fourier transform of the above

expression:

(1-68)

Switching the integration order, one gets:

(1-69)

Using the shift property of the Fourier transform, and defining the Fourier transform of the

impulse response as the transfer function of the LTI system yields:

(1-70)

Hence, the convolution in time domain corresponds to a multiplication in frequency domain.

Note that if one wants to treat the sine wave as the superposition of two complex

signals and , the definition of an LTI system (Definition 1.2) has to be modified.

In that case, the constants have to be complex numbers, and the range of the functions of

has to be instead of . But then it has to be specified that the LTI system’s impulse

response is a real function and not a complex one.

y t( ) H u t( )[ ] H u τ( )δ t τ–( ) τd∞–

∞∫= =

H [ ]

y t( ) u τ( )H δ t τ–( )[ ] τd∞–

∞∫ u τ( )h t τ–( ) τd

∞–

∞∫= =

y t( ) u t( )*h t( )=

Y ω( ) ℑ y t( ) y t( )e jωt– td∞–

∞∫ u τ( )h t τ–( ) τd

∞–

∞∫ e jωt– td

∞–

∞∫= = =

Y ω( ) u τ( ) h t τ–( )e jωt– td∞–

∞∫ τd

∞–

∞∫ u τ( )ℑ h t τ–( ) τd

∞–

∞∫= =

Y ω( ) u τ( )H ω( )e jωτ– τd∞–

∞∫ H ω( ) u τ( )e jωτ– τd

∞–

∞∫ H ω( ) U ω( )⋅= = =

t( )cos

ejt 2⁄ e jt– 2⁄

ci FC R

h t( )

46

Appendices

Appendix 1.C: Z and Y matrix of a n-portLet and be respectively the voltages and currents at the n ports of the

system.

Definition 1.9

The -matrix and the -matrix of the system are the complex n-by-n matrices that satisfy

(1-71)

where and are the Laplace or Fourier transforms of and

respectively.

Out of the definition follows that the and matrices are each other inverse, or

(1-72)

Appendix 1.D: Output PSD of the noisy LTI systemThe theorem of Wiener-Kinchin [4] states that the power spectral density of a signal can

be calculated as the Fourier transform of the auto-correlation, hence:

(1-73)

The reference impedance is often selected to be in microwave environments. The

reason for this rescaling is that the auto-correlation is expressed in volts square, and hence the

unit of its Fourier transform is . In order to obtain a power spectral density (in

), the latter result must be divided by the characteristic impedance.

U1 … Un, , I1 … In, ,

Z Y

U1…Un

Z11 … Z1n… … …Zn1 … Znn

I1…In

⋅=I1…In

Y11 … Y1n… … …Yn1 … Ynn

U1…Un

⋅=

I1 … In, , U1 … Un, , i1 t( ) … in t( ), ,

u1 t( ) … un t( ), ,

Z Y

Y11 … Y1n… … …Yn1 … Ynn

Z11 … Z1n… … …Zn1 … Znn

1–

=

y t( )

PSDy2( ) 1

Z0------ ℑ Ryy τ( ) =

Z0 50Ω=

V2 Hz⁄

W Hz⁄

47


The auto-correlation is defined as:

(1-74)

Defining as the output of the noiseless LTI system, one can write that:

(1-75)

Because the input signal and the noise are uncorrelated (as assumed in section

1.4.1), and will also be uncorrelated. Rewriting (1-75) in terms of

power spectral density yields [4]:

(1-76)

Since the PSD of a signal can also be written as , considering

that the Fourier transform exists, (1-76) can also be expressed as (assuming

the Fourier transform of exists):

(1-77)

Appendix 1.E: Determining the noise figure with the Y-factor methodWith the coordinates of the two points ( , ) and ( , ), one can write down the equation

of the straight line in the , plane going through these two points:

(1-78)

The slope of this function is given by: , while the intersection with the axis is

given by: . Filling in these values in (1-21) yields:

Ryy τ( ) E y t( )y t τ+( ) =

η t( ) h t( )*u t( )=

Ryy τ( ) E η t( ) nA t( )+( ) η t τ+( ) nA t τ+( )+( ) =

E η t( )η t τ+( ) E nA t( )nA t τ+( ) + Rηη τ( ) RnAnAτ( )+= =

u t( ) nA t( )

η t( ) h t( )*u t( )= nA t( )

PSDy2( ) ω( ) PSDη

2( ) ω( ) PSDnA

2( ) ω( )+ H ω( ) 2PSDu2( ) ω( ) PSDnA

2( ) ω( )+= =

x t( ) PSDx2( ) ω( ) X ω( ) 2 Z0⁄=

X ω( ) ℑ x t( ) =

u t( )

PSDy2( ) ω( ) H ω( ) 2 U ω( ) 2

Z0------------------- PSDnA

2( ) ω( )+=

Tc N1 Th N2

T PSDy1( )

PSDy1( ) N1–

N2 N1–Th Tc–------------------- T Tc–( )⋅=

N2 N1–Th Tc–------------------- T 0=

N1N2 N1–Th Tc–-------------------Tc+

48

Appendices

(1-79)

with . Further simplification and regrouping terms gives:

(1-80)

Appendix 1.F: Signal-to-noise ratio deterioration for other input noise levelsUsing (1-21) follows that the power spectral density of the noise added by the LTI system can

be written as:

(1-81)

Since, by hypothesis, the temperature of the system is kept constant, will not

change for other input noise temperatures. The signal-to-noise ratio deterioration for other

input noise power levels is then given by:

(1-82)

Substituting (1-81) into (1-82) yields:

(1-83)

NF 1N1

N2 N1–Th Tc–-------------------Tc+

N2 N1–Th Tc–-------------------T0

--------------------------------------+ 11 1 Y–

Th Tc–-----------------Tc+

Y 1–Th Tc–-----------------T0

---------------------------------+= =

Y N2 N1⁄=

NF 1Th YTc–( )T0 Y 1–( )---------------------------+

ThT0------ 1–

YTcT0------ 1–

–

Y 1–---------------------------------------------------= =

PSDnA

1( ) ω( ) NF ω( ) 1–( ) H ω( ) 2kT0=

PSDnA

1( ) ω( )

SNRinSNRout------------------

T

H ω( ) 2kT PSDnA

1( ) ω( )+

H ω( ) 2kT------------------------------------------------------------=

SNRinSNRout------------------

T

H ω( ) 2kT NF ω( ) 1–( ) H ω( ) 2kT0+

H ω( ) 2kT--------------------------------------------------------------------------------------------- 1 NF ω( ) 1–( )

T0T------+= =

49


Appendix 1.G: The noise figure of a cascade of noisy LTI systems: Friis’formulaConsider the cascade of two noisy LTI systems, called “DUT” and “ms” as shown in Figure 1-

21:

Suppose that the power spectral density of the input signal is given by:

(1-84)

In this case, the power spectral density of the output signal is given by:

(1-85)

And hence, the noise figure of the cascade of the two noisy LTI systems is (the

dependency is omitted to enhance the readability of the equations):

(1-86)

FIGURE 1-21. Cascade of two noisy LTI systems.

+

noisy LTI system

y t( )u t( )

nA t( )

+

nAms t( )

H ω( ) Hms ω( )

noisy LTI system “ms”

u t( )

PSDu1( ) ω( ) PSDu0

1( ) ω( ) kT0+=

PSDy1( ) ω( )

PSDu0

1( ) ω( ) kT0+ H ω( ) 2 PSDnA

1( ) ω( )+ Hms ω( ) 2 PSD

nAms1( ) ω( )+

NFtot ω

NFtotSNRinSNRout------------------

PSDu0

1( )

kT0------------------

kT0 H 2 Hms2 PSDnA

1( ) Hms2 PSD

nAms1( )+ +

PSDu0

1( ) H 2 Hms2

--------------------------------------------------------------------------------------------------------= =

1PSDnA

1( )

kT0 H 2-------------------

PSDnA

ms1( )

kT0 H 2 Hms2

------------------------------------+ +=

50

Appendices

The first two terms of (1-86) equal the noise figure of the first noisy LTI system , while the

last term in (1-86) can be rewritten as:

(1-87)

Hence, out of (1-86) and (1-87) follows that:

(1-88)

The last step is to determine . This can be done without performing more measurements

than those needed to determine and . To determine , using the Y-factor

method, the power spectral densities and are applied to the measurement system (noisy

LTI system “ms”) only, and the corresponding and are measured. These measured

values can be written as:

(1-89)

Similarly, to determine , using the Y-factor method, the power spectral densities and

are applied to the cascade of the DUT and the measurement system, and the corresponding

and are measured. These measured values can be written as:

(1-90)

can the be determined out of these four measured power spectral densities, as:

(1-91)

NF

PSDnA

ms1( )

kT0 Hms2

--------------------------

H 2-------------------------------

NFms 1–

H 2-----------------------=

NFtot NFNFms 1–

H 2-----------------------+=

H 2

NFtot NFms NFms

Nc Nh

N1ms N2

ms

N1ms Nc Hms

2 PSDnA

ms1( )+= N2

ms Nh Hms2 PSD

nAms1( )+=

NFtot Nc

Nh

N1tot N2

tot

N1 2,tot Nc h, H 2 PSDnA

1( )+ Hms

2 PSDnA

ms1( )+=

H 2

H 2 N2tot N1

tot–

N2ms N1

ms–--------------------------=

51


Appendix 1.H: Combinatory analysis to determine the discrete output spectrum of a -th order Volterra operatorThe discrete output spectrum of a -th order Volterra operator consists of all the combinations

that can be made by multiplying input spectral lines , rescaled by

(the -th order Laplace transform of the Volterra kernel). Hence,

(1-92)

considering that results at different grid lines cannot be added, since they represent coefficients

of different spectral lines.

Consider that the input spectrum contains non-zero spectral lines. The number of

terms in (1-92) is then given by the number of repetitive variations of elements, taken -

by- , i.e.

(1-93)

In the case of Example 1.8, where only and were present in the input spectrum,

the number of terms created by the third order Volterra operator is , i.e.:

Not all these terms are different, e.g. . All the

permutations of the factors in a term yield the same term. Hence, the number of different terms

in (1-92) is given by the number of repetitive combinations of elements, taken -by- ,

i.e.:

(1-94)

α

α

α U k1( ) … U kα( ), ,

Hα k1 … kα, ,( ) α

… Hα k1 … kα, ,( )U k1( )…U kα( )

kαM2-----–=

M2----- 1–

∑

k1M2-----–=

M2----- 1–

∑

U k( ) M ′

M ′ α

α

#terms VM ′α M ′α= =

U 1( ) U 1–( )

V23 8=

U 1( )U 1( )U 1( ) U 1( )U 1( )U 1–( ) U 1( )U 1–( )U 1( ) U 1( )U 1–( )U 1–( )U 1–( )U 1( )U 1( ) U 1–( )U 1( )U 1–( ) U 1–( )U 1–( )U 1( ) U 1–( )U 1–( )U 1–( )

U 1( )U 1( )U 1–( ) U 1( )U 1–( )U 1( )=

M ′ α α

# different terms CM ′α M ′ α 1–+( )!

α! M ′ 1–( )!⋅--------------------------------= =

52

Appendices

In the case of Example 1.8, the number of different terms is . These terms

are:

The last question is to find out for each different term, how many identical siblings it had. This

will be its coefficient in the resulting output spectrum. Hence, the number of distinct

permutations of the factors of each term is required. Each term consists of factors

, but not all the are different. All the factors of the term can be

grouped in sets of identical factors. Suppose that there exist such sets of identical factors,

and that the number of identical factors in each set is given by (with

). In that case, the number of distinct permutations that can be obtained

with the factors is given by:

(1-95)

In the case of Example 1.8, the number of distinct permutations that can be obtained with

terms and is given by , and the number of

distinct permutations that can be done with terms and is

given by .

C23 4! 3!⁄ 4= =

U 1( )U 1( )U 1( ) U 1( )U 1( )U 1–( )U 1( )U 1–( )U 1–( ) U 1–( )U 1–( )U 1–( )

α

U k1( ) … U kα( ), , k1 … kα, ,

m ′

α1 … αm ′, ,

α1 … αm ′+ + α=

α

α!α1! … αm ′!⋅ ⋅----------------------------------

U 1( )U 1( )U 1( ) U 1–( )U 1–( )U 1–( ) 3! 3!⁄ 1=

U 1( )U 1( )U 1–( ) U 1( )U 1–( )U 1–( )

3! 2! 1!⋅( )⁄ 3=

53


54

CHAPTER 2

NOISE FIGURE MEASUREMENTSON NICE SYSTEMS

Abstract: The noise figure, which is defined as the deterioration of

the signal-to-noise ratio from the input to the output of a LTI

system can easily be measured using the Y-factor technique when

dealing with LTI systems. For NICE systems, however, the method

for determining the noise figure remains a big question mark. This

chapter investigates if the Y-factor method is also valid for

determining the “noise figure” of NICE systems.

55

Noise figure measurements on NICE systems

2.1 IntroductionSince the noise figure is a very useful and widely used figure to describe the noise behavior of

a LTI system [10], [11], the question arises if it can be used for NICE systems. Except for the

fact that the noise contribution at angular frequency is not only a function of the input noise

spectrum at the same angular frequency , but also depends on the noise at other angular

frequencies, there is almost no difference between the noise behavior of a LTI and a NICE

system. Hence, one can assume that the noise figure is also applicable for NICE systems. At

first glance, the Y-factor method appears to be still valid for NICE systems. If it really is the

case, that would be great! If the Y-factor method can really be applied to NICE systems, it is

important to know if all the assumptions and calculations used to derive the method still hold.

One can then determine the eventual errors made if the Y-factor method is used on a NICE

system, as if it were a LTI system.

The Y-factor technique is a method that implicitly assumes that there is no interaction between

the signal and the noise that are applied to the system and neither between the noise added by

the system itself and the input signal. Or in terms of the sets defined in section 1.6.2,

and are zero, or at least negligible compared to and .

Hence the question about the validity of the Y-factor method can be rephrased as follows: Is it

possible to give an accurate description of the noise produced by a NICE system under the

hypothesis or constraint that the interaction between signal and noise can be neglected?

This chapter will give an answer to those questions, using the -factor technique on a very

simple NICE system, and studying the results.

ωi

ωi

A ′ U0 N,( ) B ′ U0 N,( ) A U0( ) B N( )

Y

56

A very simple model for the NICE system

2.2 A very simple model for the NICE systemIn order to determine the influence of a NICE system on the Y-factor method, a model for the

NICE system is required. The correct approach would be to use the property that the output of

a NICE system can be approximated as a Volterra series (see also (1-30) and (1-31)):

(2-1)

The problem is that this model contains an large sum of Volterra operators, and is therefore

quite difficult to deal with. Therefore, we will select a special (simple) NICE system to

determine if the nonlinearity of a NICE system influences the Y-factor method. In order to be

of practical use, the NICE system has to be selected such as to come close to a practical

system. To ensure that the system is of practical meaning, both the signal and noise need to be

large enough to allow for compression1 of the signal. For the noise properties, band limited

noise signals will be selected. To ease the calculations, it will be assumed that no noise shaping

will be additionally introduced: the noise will be left unchanged in-band and perfectly filtered

out, out of the selected frequency band.

A very simple model for a NICE system is shown in Figure 2-1

1. Compression means that the output spectrum of the NICE system is smaller than predicted by the linear model,describing the NICE system in its linear region. Hence, gain compression means that the gain of the NICEsystem becomes smaller than the linear gain.

FIGURE 2-1. A noisy Wiener-Hammerstein system as model for the NICE system.

y t( ) … hα τ1 … τα, ,( )u t τ1–( )…u t τα–( ) τ1…d ταd∞–

∞∫∞–

∞∫

α 1=

αmax

∑=

+fB fB

u t( ) y t( )

nA t( )

k1 u t( )⋅

k3 u3 t( )⋅

u t( ) +η t( )

57


This model is a Wiener-Hammerstein system [12] with noise superimposed on its output. A

Wiener-Hammerstein system consists of a static nonlinearity with at its input and output a LTI

system. The input and output LTI systems are both ideal lowpass filters, that perfectly block

every frequency above Hz, and have a unity transfer function below Hz. We assume that

the noise power spectral density at the input of the noisy NICE system is a flat spectrum that is

frequency independent (and given by (1-10)). For a practical system, the

noise spectrum will only propagate through the system up to the bandwidth of the system.

Above this frequency, the noise power spectral density will decrease. In the proposed model,

this decrease is replaced by an immediate perfect attenuation. Furthermore, a flat noise power

spectral density, that is frequency independent implies an infinite total noise power .

Indeed, . This remark has already been made in section 1.2.1,

and the solution was to use the correct expression (1-9) for the power spectral density instead

of the frequency independent approximation. Hence, with this model, it is allowed to assume

that contains a noise term, whose power spectral density is given by , instead of using

the complex formula. The output filter is required because every system has a limited output

bandwidth. Above this frequency, the system stops to produce spectral output after a certain

frequency. The properties of the noise source are the same as discussed in section 1.6.1.

An additional constraint is placed on the noise source by assuming that its power spectral

density is zero above Hz.

If is the input signal of the system (assuming that the complete spectrum of fits in

the frequency interval ), will be the output signal of the

system (again, assuming that the complete spectrum of fits in the frequency interval

). Note furthermore that only the first and third order Volterra operator are used in the

model, while the second order operator is omitted. When the system is excited with a pure sine

wave, this results in compression of the fundamental (at frequency ) and existence of energy

at three times the frequency of the fundamental, and no distortion at two times the fundamental

frequency. In practical systems, it is often observed that the contribution at frequency is

much smaller than the contribution at . The reason is that the presence of an even order

Volterra operator compresses the maxima and the minima of a sine wave in a different way.

This asymmetric treatment of the maxima and minima is typically due to a bad biasing of the

B B

PSDnu

1( ) ω( ) kT=

Pnu

PnukT fd0

∞∫ kT fd0

∞∫ ∞= = =

u t( ) kT

nA t( )

B

u t( ) u t( )

0 B,[ ] y t( ) k1u t( ) k3u3 t( )+=

u3 t( )

0 B,[ ]

f0

2f03f0

58

A very simple model for the NICE system

NICE system. It will not be taken into account in this very simple proposed model, and will

remain small for a well designed model. The way an even distortion operates is more easily

understood with an example.

Figure 2-2 shows the effect of even and odd distortions on a sine wave. The thin blue line

represents the sine wave . The red dashed line represents the response of the system

to a sine wave: , while the green solid line represents the

response of the system to the same signal: . This figure

clearly shows that when using even order Volterra operators, the maxima and minima of the

sine wave are treated differently, while this is not the case for the odd orders. This could be

expected: as an odd function treats positive and negative parts the same way due to its

definition:

(2-2)

FIGURE 2-2. The effect of even and odd Volterra operators on a sine wave.

0 200 400 600 800 1000-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Timesamples

Am

plit

ude

0.5 t( )cos

x x2– 0.5 t( )cos( )1 0.5 t( )cos( )2–

x x3– 0.5 t( )cos( )1 0.5 t( )cos( )3–

f is an odd function f x–( )⇔ f x( )–=

59


2.3 Determining the output spectrum of the modelled systemUsing the simplified model for the NICE system, the Y-factor technique will be applied to this

model in order to determine if it will successfully yield its noise figure. Note that the algorithm

used is identical as for the determination of the noise figure of an LTI system. As explained in

section 1.4.4, the noise figure (determined using the Y-factor technique) can be computed as:

(2-3)

where the Y-factor is obtained from a hot and a cold noise power measurement ( )

with (see also Figure 2-3). and are respectively the hot ( )

and the cold ( ) noise power spectral densities of the noise source used. These power

spectral densities are considered to be generated by thermal noise and are thus frequency

independent. Although this frequency independency is not a strict requirement (see the notes in

section 1.4.4), it will be assumed here to ease the calculations. and are the

output noise power spectral densities when the input power spectral densities are respectively

and . represents the thermal noise power spectral density at a temperature of

290 K and is about dBm/Hz in magnitude.Furthermore, a perfect impedance match at

the input and output of the device for all frequencies is assumed.

Next, analytical expressions for and will be derived. Assuming that the device

noise is additive, one can say that

FIGURE 2-3. Schematic representation of the -factor technique setup.

NF ω( )

ThT0------ 1–

Y ω( )TcT0------ 1–

–

Y ω( ) 1–-------------------------------------------------------------

NhN0------ 1–

Y ω( )NcN0------ 1–

–

Y ω( ) 1–--------------------------------------------------------------= =

u0 t( ) 0=

Y ω( ) N2 ω( ) N1 ω( )⁄= Nh Nc kTh

kTc

N2 ω( ) N1 ω( )

Nh Nc N0 kT0=

174–

“cold”

“hot”

PowerMeter

Nc

Nh

Noise Sources

NICESystem +

noise sourcenA t( )

Noisy NICE System

u t( ) y t( )nu t( )=

Y

N1 ω( ) N2 ω( )

60

Determining the output spectrum of the modelled system

(2-4)

with the power spectral density of the noise source that is independent of

the input power and uncorrelated with the input noise (see section 1.6.1). The function

has to be determined next.

The power spectral density of the output of the modelled NICE system is obtained by taking

the Fourier transform of the auto-correlation of the system output .

(2-5)

The Fourier transform is divided by , because the power is calculated in a

reference impedance as commonly used in microwave environments. The auto-correlation

of the device output signal consists of the contributions of the noise added by the

system itself and the auto-correlation of the system if were equal to zero (because

and the input noise are uncorrelated). Hence:

(2-6)

Where represents the auto-correlation of the output signal of the noiseless NICE

system (see Figure 2-3). In the case of the example system described in section 2.2, if the

input signal consists only of noise , can be written as:

(2-7)

(2-7) consists of four terms:

(2-8)

N1 2, ω( ) Φ Nc h,( ) PSDnA

1( ) ω( )+=

PSDnA

1( ) ω( ) nA t( )

Φ Nc h,( )

y t( )

PSDy2( ) f( ) 1

Z0------ ℑ Ryy τ( ) =

Z0 50Ω= 50Ω

Ryy τ( )

nA t( )

nA t( )

Ryy τ( ) Rηη τ( ) RnAnAτ( )+=

Rηη τ( )

η t( )

u t( ) nu t( ) Rηη τ( )

Rηη τ( ) E k1nu t( ) k3nu3 t( )+( ) k1nu t τ+( ) k3nu

3 t τ+( )+( )

=

Rηη τ( ) E k12nu t( )nu t τ+( )

E k32nu

3 t( )nu3 t τ+( )

+=

E k1k3nu t( )nu3 t τ+( )

E k1k3nu3 t( )nu t τ+( )

+ +

61


Hence, the following terms have to be evaluated: the linear, the cubic and the combined

contribution. The linear contribution can be rewritten as:

(2-9)

with the auto-correlation function of the input noise . Because the system has a

bandwidth , and is white, thermal noise, one can consider to be band-limited,

flat, zero-mean Gaussian noise with auto-correlation (see Appendix 2.A.)

(2-10)

The cubic contribution can be rewritten as:

(2-11)

The problem now is to determine the expected value of the product of six (ergodic) random

variables. The literature [7], [38] tells that the expected value of the product of an even number

( ) of zero-mean jointly Gaussian random variables can be written as:

(2-12)

The symbol stands for the summation over all distinct ways of partitioning the

random variables into products of averages of pairs, where within a pair the permutation of the

variables does not yield another contribution. The number of distinct ways to obtain this is

given by:

(2-13)

E k12nu t( )nu t τ+( )

k12 Rnunu

τ( )⋅=

Rnunuτ( ) nu t( )

B nu t( ) nu t( )

Rnunu

c h, τ( ) Z0Nc h, B 2πBτ( )sin2πBτ

--------------------------=

E k32nu

3 t( )nu3 t τ+( )

k32E nu t( )nu t( )nu t( )nu t τ+( )nu t τ+( )nu t τ+( ) =

1 2 3 4 5 6

2M n τ1( ) … n τ2M( ), ,

E n τ1( ) … n τ2M( )⋅ ⋅ ΣΠ E n τ i( ) n τ j( )⋅ =

ΣΠ 2M

2M( )!

M!2M---------------

62

Determining the output spectrum of the modelled system

For six variables, , and (2-13) yields that the number of ways to partition the variables

equals fifteen. Appendix 2.B shows these fifteen ways and rewrites the terms as products of

auto-correlations. One will find that:

(2-14)

The auto-correlation is known and given by (2-10), hence the cubic contribution can

eventually be written as:

(2-15)

Similar calculations (see Appendix 2.C) show that the combined contributions can be written

as:

(2-16)

Hence, the auto-correlation of the output signal of the system is:

(2-17)

In a last step, (2-5) will be applied, in order to determine the output power spectral density of

the system. Using (2-39) and (2-35) of Appendix 2.A,

(2-18)

M 3=

E k32nu

3 t( )nu3 t τ+( )

k32 9Rnunu

2 0( )Rnunuτ( ) 6Rnunu

3 τ( )+ =

Rnunuτ( )

E k32nu

3 t( )nu3 t τ+( )

k32Z0

3Nc h,3 B3 9 2πBτ( )sin

2πBτ-------------------------- 6 2πBτ( )sin

2πBτ--------------------------

3+

=


3k1k3Z02Nc h,

2 B2 2πBτ( )sin2πBτ

--------------------------=


3k1k3Z02Nc h,

2 B2 2πBτ( )sin2πBτ

--------------------------=

Ryy τ( )

Ryy τ( ) k12Z0Nc h, B 2πBτ( )sin

2πBτ-------------------------- 6k1k3Z0

2Nc h,2 B2 2πBτ( )sin

2πBτ-------------------------- + +=

k32Z0

3Nc h,3 B3 9 2πBτ( )sin

2πBτ-------------------------- 6 2πBτ( )sin

2πBτ--------------------------

3+

RnAnAτ( )+

PSDy2( ) f( ) ℑ Nc h, B 2πBτ( )sin

2πBτ--------------------------

=

63


The output power spectral density of the system thus becomes:

(2-19)

Where * denotes the convolution. (Note that the Fourier transform of a product equals the

convolution of the Fourier transforms.) After a few calculations (see Appendix 2.D), one

obtains that the single sided power spectral density (from DC to ) of the output of the

nonlinear system is:

(2-20)

Note that during the calculation of the convolution, spectral components with a frequency

larger than were taken into account. This represents no problem since the model of the

internal nonlinear circuit is not band-limited and hence allows frequency components beyond

frequency . Real-world devices often behave similarly. Inside the system, high frequency

components exist that are filtered out at the output, due to the lowpass filter behavior of

connectors, transmission lines or the package of the device.

PSDy2( ) f( ) k1

2PSDnu

2( ) f( ) 6k1k3Z0Nc h, BPSDnu

2( ) f( ) + +=

9k32Z0

2Nc h,2 B2PSDnu

2( ) f( ) 6k32Z0

2PSDnu

2( ) f( )*PSDnu

2( ) f( )*PSDnu

2( ) f( ) PSDnA

2( ) f( )+ +

B

PSDy1( ) f( ) k1

2Nc h, 6k1k3Z0Nc h,2 B

k32Nc h,

3 Z02

2----------------------- 27B2 3f2–( ) PSDnA

1( ) f( )+ ++=

B

B

64

Determination of the noise figure

2.4 Determination of the noise figureAt this point, the power spectral density at the output of the NICE system excited by input

noise only is known (2-20), and it is possible to determine the results of a classical noise figure

measurement on the proposed example. The terms in (2-20) can be grouped in terms

corresponding to the linear and the nonlinear behavior:

(2-21)

Where represents the contribution of the linear part of the system to the output power

spectral density, i.e.

(2-22)

contains the contributions to the output power spectral density due to the nonlinear

part of the NICE system.

(2-23)

Hence the measured noise figure at frequency will be:

(2-24)

It is possible to split this expression for the noise figure in the sum of a linear noise figure

(i.e. the noise figure for the linear system, i.e. ) and a correction term:

(2-25)

PSDy1( ) f( ) N1 2,

L f( ) N1 2,Nice f( )+=

N1 2,L f( )

N1 2,L f( ) k1

2Nc h, PSDnA

1( ) f( )+=

N1 2,Nice f( )

N1 2,Nice f( ) 6k1k3Z0Nc h,

2 Bk3

2Nc h,3 Z0

2

2----------------------- 27B2 3f2–( )+=

f

NF f( )

NhN0------ 1– N2

L f( ) N2Nice f( )+

N1L f( ) N1

Nice f( )+------------------------------------------

NcN0------ 1–

–

N2L f( ) N2

Nice f( )+

N1L f( ) N1

Nice f( )+------------------------------------------ 1–

-------------------------------------------------------------------------------------------=

NFlin k3 0=

NF f( ) NFlin f( )b f( ) NFlin f( ) d f( )⋅–

c f( ) d f( )+---------------------------------------------------+=

65


where , and

. Hence, this correction term is a function of the hot and cold

noise power spectral densities and , the noise power spectral density generated by the

NICE system itself (through the linear noise figure), the bandwidth of the system

and the system parameters and . Note that (2-25) can also be written as:

(2-26)

In this case, one can tell that the measured noise figure will be a rescaled and biased version of

the linear noise figure, where the rescaling and the biasing are independent of the linear noise

figure, i.e. of the noise power spectral density generated by the system itself.

In the next example, it will be shown that when the noise figure is measured with a typical

solid state noise source which has an excess noise ratio ( ) of about

15.2 dB [13], and if , the correction term of the noise figure will be extremely small.

Example 2.1

Consider a solid state noise source with = 15.2 dB, and . Consider

furthermore the bandwidth of the system GHz, the system parameters and

, and suppose the system itself generates noise with a flat power spectrum

dBm/Hz. Figure 2-4 shows the behavior of this system, in terms

of 1 dB compression point and third harmonic.

FIGURE 2-4. Illustration of the 1 dB compression point of the amplifier in the example.

b f( ) N1Nice f( ) Nh N0⁄ 1–( ) N2

Nice f( ) Nc N0⁄ 1–( )–= c f( ) N2L f( ) N1

L f( )–=

d f( ) N2Nice f( ) N1

Nice f( )–=

Nh Nc

PSDnA

1( ) f( )

B k1 k3

NF f( ) NFlin f( ) c f( )c f( ) d f( )+-------------------------⋅ b f( )

c f( ) d f( )+-------------------------+=

ENR Nh N0⁄ 1–=

Nc N0=

Nh N0⁄ 1– Tc T0=

B 4= k1 10=

k3 60V 2––=

PSDnA

1( ) f( ) 550N0= 147–≈

-25 -20 -15 -10 -5 0-10

-5

0

5

10

15

20

Input power [dBm]

Ou

tpu

t p

ow

er

[dB

m]

Linear system Fundamental Third harmonic

1dB

66


The 1 dB compression point (i.e. the point where the output power of the system is 1 dB

smaller than the output power of the underlying linear system) is reached for an input power of

about dBm. The polynomial function reaches its maximum for

, corresponding to 0.24 V or dBm for the example system. This

illustrates also that the third order model is valid for input powers up to only a few dB above

the 1 dB compression point.

With these numbers, the correction term of the noise figure in (2-25) is about as

compared to . Calculations using (2-22) and (2-23) show that for the system of

Example 2.1:

(2-27)

It is clear that the effect of the nonlinearity in the system on the output noise power spectral

density can be neglected. The classical noise figure measurement technique simply doesn’t

detect the nonlinearity in the system. This result could be expected, because the cold and hot

noise powers are too small for the cubic contribution to be noticed. Note also that are

negative, this is because they represent the mathematical correction terms for , and since

, the considered system will reach gain compression, i.e. the gain will become smaller

than in the linear case.

Hence, only when the noise power spectral densities are large compared to , the nonlinear

part of the system will have an effect on the measured noise figure through the correction term

(2-25). Figure 2-5 shows the evolution of the noise figure as it would be measured with the

classical Y-factor method, as a function of . was chosen to be 12 times larger than

. The same data was used as in Example 2.1 (except for the noise source of course).

6– y k1x k3x3+=

x k1–( ) 3k3( )⁄= 2.6–

6.5 6–×10

NFlin 6.5=

N1 N1L N1

Nice+ 650N0 2.88 6–×10 N0–= =

N2 N2L N2

Nice+ 3961N0 3.35 3–×10 N0–= =

N1 2,Nice

N1 2,L

k3 0<

N0

Nc N0⁄ Nh

Nc

67


Figure 2-5 shows that when the noise figure measurement is done with a noise source for

which the hot and cold noise power spectral densities and are much larger than , the

noise figure will vary. This is in opposition to the linear case, where the noise figure is

independent of the hot and cold noise temperatures. In a first zone, where the ratio is

smaller than , the classical technique yields a noise figure that is constant and equal to the

noise figure of the linear system. In a second zone, where , the noise figure

suddenly increases as , reaching extremely high values up to . This is not an behaviour

of the polynomial expression that was chosen to model the nonlinear behavior. Indeed, one

could object that the cause of the polynomial increase of the noise figure is due to the fact that

for , the input noise power may be larger than the 1 dB compression point. In

that case, the polynomial function decreases towards , as the input power

further increases. However, this objection can be rejected using simple calculations to show

that for , the input power is still smaller than the 1 dB compression point. Since

was chosen to be 12 times larger than , and increases up to in Figure 2-5,

the largest value for is dBm/Hz. The bandwidth of the system was

4 GHz, yielding a total input hot noise power of dBm, i.e. smaller than the

1 dB compression point. The real behaviour happening when the total input noise power

becomes larger than the 1 dB compression point (i.e. a polynomial decrease of the noise figure

towards ) are shown in Figure 2-6.

FIGURE 2-5. Noise figure as a function of

100 102 104 106100

102

104

106

Nc/N

0

Noi

se F

igur

e

Nc N0⁄

Nh Nc N0

Nc N0⁄

103

Nc N0⁄ 103>

xn 106

Nc N0⁄ 103>

y k1x k3x3+= ∞–

Nc N0⁄ 103>

Nh Nc Nc N0⁄ 106

Nh 12 6×10 N0 103–= B

12 6×10 N0B 7–=

∞–

68


Figure 2-6 shows that below the 1 dB compression point (where the third degree polynomial is

a good model for the NICE system), the measured noise figure increases. Hence, the results

shown in Figure 2-5, are no behaviour of the polynomial behavior for very large arguments.

If the system exhibits gain expansion1 (i.e. ) instead of gain compression ( ), an

erroneous behavior of the measured noise figure is noted.(see Figure 2-7)

FIGURE 2-6. Behaviour due to the invalidity of the polynomial model above the 1 dB compression point.

1. Expansion is the opposite of compression. Expansion means that the output spectrum of the NICE system islarger than predicted by the linear model, describing the NICE system in its linear region. Hence, gainexpansion means that the gain of the NICE system becomes larger than the linear gain.

FIGURE 2-7. Noise figure as a function of for gain expansion.The right figure zooms in on the region where the measured noise figure is larger than .

0 1 2 3 4

x 107

-4

-3

-2

-1

0

1x 10

7

Nc/N

0

Nois

e F

igure

1dB compression point

k3 0> k3 0<

0 2 4 6 8 10

x 105

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

5

Nc/N

0

Nois

e F

igure

100

101

102

103

104

-10

-5

0

5

10

Nc/N

0

Nois

e F

igure

Nc N0⁄10–

69


When the ratio is smaller than , the classical technique will yield a noise figure

that is constant and equal to the noise figure of the linear system. On the other hand, when

, the noise figure suddenly exhibits a polynomial decrease, reaching extremely

negative values such as . Clearly, these are faulty results.

Note that gain expansion can indeed occur in real-world systems. A cubic system , e.g.

will eventually reach a maximal power output, as plotted in Figure 2-8. .

In this case, the system first exhibits gain expansion, followed by gain compression. If

and are both in the gain expansion region, the Y-factor method will yield an erroneous

negative noise figure, while when they are in the gain compression region, the yielded noise

figure will be a very large positive number.

The question to know what happens to the measured noise figure if a real-world system goes

into really deep compression such as clipping remains open. Since the proposed simple model

and the behavior of the system do not agree for high compression levels, a new model has to be

FIGURE 2-8. Possible gain behavior of an system with gain expansion.

Nc N0⁄ 103

Nc N0⁄ 103>

10– 5

y u3=

Pin

Poutcompressionexpansion

NhB

NcB

70


chosen to simulate this extreme case. The proposed piece wise linear model is shown in Figure

2-9.

In this setting, the system is linear up to a certain input power. Above this input power, the

output power remains constant. The effect of this model on the noise figure obtained by the Y-

factor method is now analysed. If and are large enough, the output power spectral

densities and will both be equal to . This implies for the Y-factor method that

(2-28)

Hence, the noise figure will be:

(2-29)

This means that for a system that goes into deep saturation, the noise figure as measured with

the Y-factor method tends to infinity, as suggested by the simple model with the polynomial

expression.

Finally, let’s highlight the main conclusions of this section. The goal of this section was to

determine the results yielded by the Y-factor method, when applied to a NICE system.

FIGURE 2-9. Input-output behavior of a system in deep saturation.

Pin

Pout

Pmax

Nh Nc

N1 N2 Pmax

YN2N1------

PmaxPmax------------- 1= = =

NF

NhN0------ 1–

YNcN0------ 1–

–

Y 1–-----------------------------------------------------

Y 1→lim

Nh Nc–N0 0⋅

------------------- ∞= = =

71


• Using a model with a third degree polynomial, the NICE system can be

modelled for input noise powers up to the 1 dB compression point. During

a classical noise figure measurement, the NICE system is excited with a

noise source for which the cold ( ) and hot ( ) power spectral

densities are of the order of magnitude of the standard thermal noise

power spectral density ( ). The nonlinear part of the system will be

ignored by this measurement technique, and the noise figure of the

underlying linear system will be obtained. This is because the cold and

hot noise powers are too small to excite the nonlinear part of the system.

When and are made several orders of magnitude larger than ,

in order to excite the nonlinearity, the yielded noise figure increases up to

.

• When using noise powers below the 1 dB compression point, and when

the system exhibits gain expansion, the yielded noise figure decreases

towards , having no more physical meaning. Hence, the Y-factor

method will fail, when applied to systems exhibiting gain expansion.

• To get an idea about the measured noise figure for input noise powers

above the 1 dB compression point, a piecewise linear model is proposed.

This model predicts that the measured noise figure will become infinity in

deep compression.

Nc Nh

N0

Nc Nh N0

106

∞–

72

Discussion on the yielded noise figures

2.5 Discussion on the yielded noise figuresIn this section the results obtained with the Y-factor method applied to a NICE system instead

of a LTI system (see section 2.4) will be discussed. Will the Y-factor method applied to a NICE

system really yield the noise figure as it was originally defined in (1-19) or not?

(2-30)

First of all, one can object that the yielded noise figure for a NICE system is varying with the

power spectral densities of and . However, the Y-factor method (for a LTI system) is

designed to yield the same noise figure, even if . Furthermore, the measured noise

figure for a NICE system can become very large, when measured with high input noise powers

as shown in Figure 2-10. (For details about the measurement, see section 2.6)

Secondly, if the system exhibits gain expansion instead of gain compression, the noise figure

becomes negative (see Figure 2-7). Yielding a negative noise figure is clearly a severe fault of

the Y-factor method. Since the noise figure represents the ratio of two signal-to-noise ratios, it

cannot in any way be negative. Hence, this negative noise figure obtained through the Y-factor

method has to be rejected.

FIGURE 2-10. Illustration of high noise figure values obtained with the Y-factor method.


T0 290K=

=

Nc Nh

Nc N0≠

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8

9

10x 10

6

Noi

se F

igur

e

frequency [GHz]

73


Finally, the predicted noise figure in the case of deep saturation with clipping is an

overestimation. Suppose that the NICE system is in deep compression such that the output

power has reached a maximum. At this point, the noise figure depends on the actual value of

the output noise power. Two cases can be considered:

1. The noise output power will be equal to the maximal signal power . This means

that the signal-to-noise ratio at the output of the system equals:

. Hence, the noise figure in this scenario is given by:

(2-31)

2. The output noise power reaches a certain level that is different from

( ). Hence, the noise figure in this scenario is given by:

(2-32)

The only way for (2-31) and (2-32) to become infinity, like the Y-factor technique predicts, is

that the input signal-to-noise ratio becomes infinitely large.

Hence, one can conclude that the results obtained with the classical Y-factor method are no

longer valid for NICE systems.

Pmax

SNRout Pmax Pmax⁄ 1= =


T0 290K=

SNRin1

--------------- SNRin T0 290K== = =

NoutB Pmax

NoutB Pmax<


T0 290K=

SNRinPmax NoutB( )⁄-------------------------------------

T0 290K=

= =

74


The reason why the Y-factor method yields the results given in section 2.4, can be intuitively

understood through the following reasoning: The Y-factor method is based on the assumption

that the measured system is a LTI system. As a matter of fact, the algorithm uses only two

points, that are assumed to be on a straight line, and determines some parameters of this

straight line, out of these two points (see section 1.4.4).

The determined parameters are the slope of the line and the intersection with the

vertical axis . Next, the algorithm puts these parameters in (1-21), i.e.

(2-33)

and yields the noise figure.

In the case of a NICE system, the relation between the input noise temperature and the output

noise power spectral density is no longer a simple straight line, but rather a curve as shown in

Figure 2-12.

FIGURE 2-11. Output noise power spectral density versus input noise temperature for a LTI system.

T

PSDy1( )

PSDnA

1( )

H ωi( ) 2kslope

Tc Th

N1 ωi( )

N2 ωi( )

H ωi( ) 2k

PSDnA

1( ) ωi( )

NF ωi( ) 1PSDnA

1( ) ωi( )

H ωi( ) 2 kT0⋅----------------------------------+=

75


For , the output noise power spectral density will be equal to the power spectral density

of the noise generated by the system itself . In the vicinity of , the curve can be

approximated by a straight line. This assumption is confirmed by the fact that for small input

noise temperatures, the Y-factor method yields the linear noise figure. For very large values of

the input noise temperature, one can assume that the amplifier will go into deep saturation, and

that the output noise power spectral density will reach a maximum value.

All the results of section 2.4 can be explained with the simple curve shown in Figure 2-12. If

and are in the order of magnitude of , and are in the region where the curve

is mainly linear, and the straight line passing through the points and

will be a very good approximation of the first part of the input versus output

noise curve (see Figure 2-13). Hence, the intersection of the straight line with the vertical axis

and the slope of this straight line will be very good approximations of respectively

and . Thus the measured noise figure will indeed be equal to the noise

figure of the underlying linear system.

FIGURE 2-12. Output noise power spectral density versus input noise temperature for a NICE system.

T

PSDy1( )

PSDnA

1( )

T 0=

PSDnA

1( ) T 0=

Nc Nh N0 Tc Th

Tc N1 ωi( ),( )

Th N2 ωi( ),( )

PSDnA

1( ) ωi( ) H1 ωi( ) 2k

76


If and are several orders of magnitude larger than , and are in the region

where the nonlinearity becomes important, and where the curve can no longer be linearly

approximated. However, the Y-factor method does not take this nonlinear behavior into

account, and simply determines a straight line through the measured points and

(see Figure 2-14). This time, the intersection of the straight line with the vertical

axis will be larger than , and the slope of this straight line is less steep than

. Hence, the Y-factor method makes an overestimation about the power spectral

density of the noise generated by the system itself, and an underestimation of the gain of the

underlying linear system. (2-33) shows that the numerator will be too large, while the

denominator is too small, leading towards a larger measured noise figure than was the case for

the low noise temperatures. This explains the increase in measured noise figure described in

the previous section.

FIGURE 2-13. Situation for and in the order of magnitude of .

FIGURE 2-14. Situation for and several orders of magnitude larger than .

T

PSDy1( )

PSDnA

1( )

Tc Th

N1 ωi( )N2 ωi( )

Tc Th T0

Nc Nh N0 Tc Th

Tc N1 ωi( ),( )

Th N2 ωi( ),( )

PSDnA

1( ) ωi( )

H1 ωi( ) 2k

T

PSDy1( )

PSDnA

1( )

Tc Th

N1 ωi( )N2 ωi( )

Tc Th T0

77


Eventually, if and become that large, that they push the system into deep compression,

the situation is even worse. Again, the Y-factor method simply determines the straight line

through the measured points and (see Figure 2-15), not taking into

account the nonlinear behavior of the system. The intersection of the straight line with the

vertical axis will be approximately the maximum output noise power spectral density, and the

slope of the straight line tends towards zero. Hence, the Y-factor method determines that the

power spectral density of the noise generated by the system itself is about the maximal noise

output power spectral density, and that the gain of the system is very small. (2-33) shows that

these wrong suppositions yield an extremely large noise figure, tending towards infinity, as

was predicted using the piece wise linear model in the previous section.

The negative noise figure that was obtained for a system exhibiting gain expansion, can also be

understood through a similar reasoning (see Figure 2-16).

FIGURE 2-15. Situation for and in the deep compression region.

FIGURE 2-16. Situation for gain expansion, yielding a negative noise figure.

Nc Nh

Tc N1 ωi( ),( ) Th N2 ωi( ),( )

T

PSDy1( )

PSDnA

1( )

Tc Th

N1 ωi( )N2 ωi( )

Tc Th

T

PSDy1( )

PSDnA

1( )

Tc Th

N1 ωi( )

N2 ωi( )

78


As can be seen in Figure 2-16, the intersection with the vertical axis lies in the negative part of

the vertical axis. Hence, the Y-factor method determines that the power spectral density of

is negative. This yields then a negative noise figure, that has no longer a physical

meaning.

nA t( )

79


2.6 Experimental resultsIn order to validate the theory, the noise figure of a Sonoma 330 amplifier, with a frequency

range from 20 kHz to 2 GHz was measured using the Y-factor technique.

First the noise figure of the amplifier is measured using a solid state noise source (HP346B

[13]), for which . The measurement setup is shown in Figure 2-17.

The Sonoma 330 amplifier is excited by a HP346B calibrated noise source [13], that can be put

in two modes, the hot and the cold mode. In the cold mode, the noise source generates thermal

noise, with a power spectral density , with the room temperature, which is

measured to be or 296 K ( dBm/Hz). The Excess Noise Ratio (see section

2.4) of the noise source is about 15.2 dB, hence dBm/Hz. The HP8565E

Spectrum Analyser is used as a power meter, and is put in “noise marker” mode [14]. This

mode corrects the measured power for the imperfections of the power detector and the

resolution bandwidth filter of the spectrum analyser [15], [16]. Strictly speaking, the usage of

this noise marker mode is not really required, because the Y-factor method only needs ratios of

powers, and hence the influence of the imperfections will disappear by taking the ratio of the

powers.

Another more severe problem is that the Spectrum Analyser itself also produces noise

(see Figure 2-17), and hence, it has a noise figure of its own. The assumption is made that the

Spectrum Analyser itself is a noisy LTI system. Because the noise power levels produced by

FIGURE 2-17. Measurement setup for the classical noise figure measurement using the Y-factor technique.

Nc N0≈

“cold”

“hot”

Nc

Nh

Noise Source

DUT+

Sonoma330

HP8565ESpectrum AnalyzerHP346B

HSA ω( )

nASA t( )

HP83006ASystemAmplifier

Nc kTc= Tc

23°C Nc 173.9–≈

Nh 158.7–≈

nASA t( )

80

Experimental results

the noise source are quite small, and because the Spectrum Analyser produces quite a lot of

noise itself, it is very difficult to detect a change in output noise of the DUT. Therefore, a

HP3006A System Amplifier is put between the output of the DUT and the input of the

Spectrum Analyser. This amplifier will amplify the noise power coming out of the DUT to a

level where the Spectrum Analyser will be able to measure those noise power levels. This

amplifier has of course also a certain power gain, and produces some noise. It is however

possible to consider this system amplifier as a part of the Spectrum Analyser, and to embed its

transfer function into the transfer function of the Spectrum Analyser, and to add the effect of

the noise produced by this amplifier to the noise source . This presents no problem

when assuming that both the Spectrum Analyser and the system amplifier behave in their

linear region. In order to remove the effect of the noise source (i.e. the effect of the

noise from the system amplifier and the spectrum analyser), Friis’ formula (1-25), section 1.4.4

is used:

(2-34)

is calculated as explained in Appendix 1.G.

Figure 2-18 shows the classically measured noise figure of the amplifier, which is about 6.4

from DC to 1.2 GHz. Note that this measurement technique ignores the nonlinearity of the

amplifier.

FIGURE 2-18. Result of a classical noise figure measurement.

nASA t( )

nASA t( )

NFtot ω( ) NF ω( )NFms ω( ) 1–

H ω( ) 2--------------------------------+=

H ω( ) 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 25.6

5.8

6

6.2

6.4

6.6

6.8

7

7.2

7.4

frequency [GHz]

Noi

se F

igur

e

81


In a second experiment, the noise figure is again measured with the Y-factor technique, but this

time with noise sources for which >> . In order to create these high input noise powers,

an extra amplifier (Mini-Circuits ZHL-42), with a power gain of 30 dB was put between the

noise source and the DUT (see Figure 2-19)

With this measurement setup, the cold and hot noise power spectral densities at the input of the

Device Under Test are respectively dBm/Hz and dBm/Hz, which is

respectively 65 dB and 75 dB larger than dBm/Hz. The same measurement

procedure was followed as described above, and the measured noise figure as a function of the

frequency is shown in Figure 2-20.

Figure 2-20 illustrates that a Y-factor measurement, where the hot and cold noise power

spectral densities are much larger than the thermal noise power spectral density at

FIGURE 2-19. Measurement setup for the Y-factor technique with and >> .

FIGURE 2-20. Noise figure measurement with and >>

Nc h, N0

HP83006A

Noise

DUT+

Sonoma330

HP8565ESpectrum AnalyzerHP346B

HSA ω( )

nASA t( )Source

SystemAmplifier

ZHL-42

Nc Nh N0

Nc 109–= Nh 99–=

N0 174–=

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8

9

10x 10

6

Noi

se F

igur

e

frequency [GHz]

Nc Nh N0

82

Experimental results

K, yields a much too large noise figure (in this case in the order of magnitude of

) as was predicted in Figure 2-5.

T0 290=

106

83


2.7 ConclusionThis chapter gives an answer to the following important question: Is it possible to use the

classical Y-factor measurement technique, that proved its usefulness in determining the noise

figure of LTI systems, to determine the noise figure for a NICE system?

Based on a very simple NICE system, consisting of a linear and a cubic subcircuit, an analysis

of the Y-factor method is performed. This model choice is motivated by its ability to add gentle

(soft) nonlinear contributions to a linear band-limited system. This is the case of most

amplifiers that go into soft compression.

Based on this model, an analytical form for the system’s output power spectral density is

calculated. Combining this expression and the Y-factor technique, leads to the following

observation:

When exciting the NICE system with a noise source for which the cold ( ) and hot ( )

power spectral densities are in the order of magnitude of the standard thermal noise power

spectral density ( ), the nonlinear part of the system will be ignored by the measurement

technique, hence yielding a false noise figure. When and become several orders of

magnitude larger than , the yielded noise figure exponentially increases towards infinity. If

the modelled system exhibits gain expansion instead of gain compression, the situation is even

worse. In this case the Y-factor technique (when using large and ), will become

negative, having no longer any physical meaning.

For an amplifier in deep compression (i.e. an amplifier that has reached its maximal output

power), the noise figure yielded by the Y-factor technique, actually equals infinity. A simple

example, however, contradicted this result.

Hence, the Y-factor technique will yield false noise figures, when it is applied to a NICE

system. Or, the interaction between the signal and the noise cannot be neglected for NICE

systems.

Nc Nh

N0

Nc Nh

N0

Nc Nh

84

Appendices

2.8 Appendices

Appendix 2.A : Autocorrelation of band-limited, white noiseThe auto-correlation of the white, band-limited thermal noise will be determined:

(2-35)

is shown in Figure 2-21, in its single and double sided representation.

Indeed, when one tells that the noise power spectral density at a certain frequency equals ,

one is using the single sided representation. However, mathematically, negative frequencies

also exist, and the restriction that the total noise power (which is the integral of the power

spectral density over all frequencies) has to be the same for both representations, the power

spectral density of the double sided representation is half as large as the one for the single sided

power spectral density. Hence, (2-35) can be rewritten as:

(2-36)

This integral yields:

(2-37)

FIGURE 2-21. Single and double sided representation of the noise.

PSDnu

2( ) f( ) 1Z0------ ℑ Rnunu

τ( ) = Rnunuτ( )⇒ ℑ 1– Z0PSDnu

2( ) f( )

=

PSDnuf( )

f

Nc h,2

-----------

f

Nc h,

B BB–single sided representation double sided representation

PSDnu

1( ) f( ) PSDnu

2( ) f( )

kT

Rnunuτ( ) Z0

Nc h,2

-----------ej2πfτ fdB–

B∫=

Rnunuτ( )

Z0Nc h,4jπτ

----------------- ej2πBτ e j– 2πBτ–( )=

85


using the fact that

(2-38)

(2-37) becomes:

(2-39)

Appendix 2.B : Fifteen ways of partitioning six random variables in products of averages of pairsThe cubic contribution is written as (see (2-11))

(2-40)

The six random variables are tagged “1” to “6”. The fifteen ways of partitioning the six tags

are:

(2-41)

ejα e jα––2j

------------------------ α( )sin=

Rnunuτ( ) Z0Nc h, B 2πBτ( )sin

2πBτ--------------------------=

E k32nu

3 t( )nu3 t τ+( )

k32E nu t( )nu t( )nu t( )nu t τ+( )nu t τ+( )nu t τ+( ) =

1 2 3 4 5 6

121212131313141414151515161616

343536242526232526232426232425

564645564645563635463634453534

86

Appendices

Since terms , and , pairs of tags pointing to equal terms will yield

factors:

, (2-42)

and pairs of tags pointing to different terms will yield factors:

(2-43)

Hence, (2-41) eventually yields:

(2-44)

Appendix 2.C : The combined contributions of the auto-correlation

The first combined contribution can be rewritten as:

(2-45)

The time average of the product of four random variables equals the sum over all distinct ways

of partitioning the four random variables into products of averages of pairs. The number of

distinct ways to do this is given by (2-13) with , yielding three ways:

(2-46)

Since terms , pairs of tags pointing to equal terms will yield factors:

, (2-47)

1 2 3= = 4 5 6= =

E nu t( )nu t( ) E nu t τ+( )nu t τ+( ) Rnunu0( )= =

E nu t( )nu t τ+( ) Rnunuτ( )=

E nu3 t( )nu

3 t τ+( )

6Rnunu

3 τ( ) 9Rnunu

2 0( )Rnunuτ( )+=

Rηη τ( )


k1k3E nu t( )nu t τ+( )nu t τ+( )nu t τ+( ) =1 2 3 4

M 2=

12 3413 2414 23

2 3 4= =

E nu t( )nu t( ) E nu t τ+( )nu t τ+( ) Rnunu0( )= =

87


and pairs of tags pointing to different terms will yield factors:

(2-48)

Hence, (2-46) eventually yields:

(2-49)

The second term can be calculated, using the anti-symmetry property of

the cross-correlation function:

(2-50)

Hence,

(2-51)

But since the auto-correlation is an even function: , both terms of the

combined contribution are equal, and hence:

(2-52)

E nu t( )nu t τ+( ) Rnunuτ( )=

E nu t( )nu3 t τ+( )

3Rnunu0( )Rnunu

τ( )=

E nu3 t( )nu t τ+( )

Rαβ τ( ) Rβα τ–( )=

E nu3 t( )nu t τ+( )

E nu t( )nu3 t τ–( )

3Rnunu0( )Rnunu

τ–( )= =

Rαα τ( ) Rαα τ–( )=



+ 6Rnunu0( )Rnunu

τ( )=

88

Appendices

Appendix 2.D : The convolution of the noise spectrum with itselfThe goal is to calculate , where is shown in the

double sided representation in Figure 2-21. First the convolution will

be calculated. Figure 2-22 shows a geometric interpretation of this convolution.

The spectrum of is drawn two times. Both spectra are at a “distance” from each

other, and for each “distance” , the area of the overlapping spectra is calculated. One

immediately sees that if , the overlapping area will be zero, and for the

overlapping area will increase linearly, resulting in the convolution having a triangle shaped

spectrum. Mathematically, one obtains that

(2-53)

But since the power spectral density is zero outside the frequency interval , and

inside this interval,

(2-54)

Defining the integration variable yields:

FIGURE 2-22. Geometric interpretation of the convolution .

PSDnu

2( ) f( )*PSDnu

2( ) f( )*PSDnu

2( ) f( ) PSDnu

2( ) f( )

PSDnu

2( ) f( )*PSDnu

2( ) f( )

f

Nc h,2

-----------

BB– f

Nc h,2

-----------

BB–

*f

2B– 2B

PSDnu

2( ) f( )*PSDnu

2( ) f( )

PSDnu

2( ) f( ) f

f

f 2B> f :2B 0→

PSDnu

2( ) f( )*PSDnu

2( ) f( ) PSDnu

2( ) φ( )PSDnu

2( ) f φ–( ) φd∞–

∞∫=

B B,–[ ]

Nc h, 2⁄

PSDnu

2( ) φ( )PSDnu

2( ) f φ–( ) φd∞–

∞∫

Nc h,2

----------- PSDnu

2( ) f φ–( ) φdB–

B∫=

ψ f φ–=

89


(2-55)

If , the integration interval lies outside , and the integral is zero. If lies in the

interval , the lower integration limit will always be smaller than , hence the

lower integration limit can be set equal to . (The upper integration limit won’t exceed , so

there is no problem.) Thus for , the integral becomes:

(2-56)

If lies in the interval , the upper integration limit will exceed smaller , hence

the lower integration limit has to be set equal to . Thus for , the integral becomes:

(2-57)

If , the integration interval lies outside , and the integral is zero.

Define as , Figure 2-23 shows a plot of the calculated

spectrum.

FIGURE 2-23. Power spectral density of

Nc h,

2----------- PSDnu

2( ) ψ( ) ψdf B–

f B+∫=

f 2B–< B B,–[ ] f

2B 0,–[ ] f B– B–

B– B

f 2B 0,–[ ]∈

Nc h,2

----------- PSDnu

2( ) ψ( ) ψdf B–

f B+∫

Nc h,2

-----------

2ψd

B–

f B+∫

Nc h,2

-----------

2f 2B+( )= =

f 0 2B,[ ] f B+ B

B f 0 2B,[ ]∈

Nc h,2

----------- PSDnu

2( ) ψ( ) ψdf B–

f B+∫

Nc h,2

-----------

2ψd

f B–

B∫

Nc h,2

-----------

22B f–( )= =

f 2B> B B,–[ ]

PSD*2( ) f( ) PSDnu

2( ) f( )*PSDnu

2( ) f( )

f

Nc h,2

-----------

22B f–( )

2B2B–

Nc h,2

-----------

22B f+( )

PSD*2( ) f( )

90

Appendices

The spectrum we are looking for is the convolution of this spectrum with the

original noise spectrum . Again, a geometric interpretation of this convolution can

be given as shown in Figure 2-24:

This time the area of overlap will increase in a square law for , and will be zero for

. Mathematically, one obtains that

(2-58)

Since is zero outside the frequency interval , and inside this

interval,

(2-59)

This last expression can again be rewritten with as:

(2-60)

If , the integration interval lies outside , and the integral is zero. If lies in

the interval , the lower integration limit will always be smaller than ,

FIGURE 2-24. Geometric interpretation of the convolution

PSD*2( ) f( )

PSDnu

2( ) f( )

f

Nc h,2

-----------

BB–

*f

f

Nc h,2

-----------

22B f–( )

2B2B–

Nc h,2

-----------

22B f+( )

PSDnu

2( ) f( )*PSD*2( ) f( )

f :3B 0→

f 3B>

PSDnu

2( ) f( )*PSD*2( ) f( ) PSDnu

2( ) φ( )PSD*2( ) f φ–( ) φd

∞–

∞∫=

PSDnu

2( ) f( ) B B,–[ ] Nc h, 2⁄

PSDnu

2( ) φ( )PSD*2( ) f φ–( ) φd

∞–

∞∫

Nc h,2

----------- PSD*2( ) f φ–( ) φd

B–

B∫=

ψ f φ–=

Nc h,2

----------- PSD*2( ) ψ( ) ψd

f B–

f B+∫

f 3B–< 2B 2B,–[ ] f

3B B–,–[ ] f B– 2B–

91


hence the lower integration limit has to be set equal to . Thus for , the

integral becomes:

(2-61)

A similar reasoning can be followed for the interval , yielding:

(2-62)

If lies in the interval , the lower and upper integration limit fall in the interval

, where . Hence, there is no need to modify the integration

boundaries.

(2-63)

If , the integration interval lies outside , and the integral is zero.

The power spectral density of is shown in Figure 2-25

FIGURE 2-25. Power spectral density of

2B– f 3B B–,–[ ]∈

Nc h,2

----------- PSD*2( ) ψ( ) ψd

f B–

f B+∫

Nc h,2

-----------

3ψ 2B+( ) ψd

2B–

f B+∫

Nc h,2

-----------

312--- f 3B+( )2= =

B 3B,[ ]

Nc h,2

----------- PSD*2( ) ψ( ) ψd

f B–

f B+∫

Nc h,2

-----------

312--- f 3B–( )2=

f B B,–[ ]

2B 2B,–[ ] PSD*2( ) f( ) 0≠

Nc h,2

----------- PSD*2( ) ψ( ) ψd

f B–

f B+∫

Nc h,2

-----------

32B ψ+( ) ψd

f B–

0∫ 2B ψ–( ) ψd

0

f B+∫+

=

Nc h,

2-----------

33B2 f2–( )=

f 3B> 2B 2B,–[ ]

PSDnu

2( ) f( )*PSDnu

2( ) f( )*PSDnu

2( ) f( )

f

Nc h,2

-----------

33B2 f2–( )

3B3B–

Nc h,2

-----------

3 f 3B–( )2

2----------------------

BB–

Nc h,2

-----------

3 f 3B+( )2

2----------------------

PSDnu

2( ) f( )*PSDnu

2( ) f( )*PSDnu

2( ) f( )

92

CHAPTER 3

EXTENSION OF THE “NOISEFIGURE” TOWARDS NICE

SYSTEMS

Abstract: Since the behavior of a NICE system is dependent on the

power of its input signal, it is clear that the “noise figure” (defined

as the signal-to-noise ratio deterioration) will also strongly depend

on the power of the input signal, including the input noise. In this

chapter, the dependency of the power of the excitation signal (i.e.

the superposition of the input signal and the noise) on the “noise

figure” of the noisy NICE system is analysed in detail.

93

Extension of the “Noise Figure” towards NICE systems

3.1 Introduction

3.1.1 GoalSince the Y-factor technique is not able to yield correct noise figures for NICE systems, other

methods have to be developed to replace it. The design goal for such an alternative method is

that it must give an accurate description of the deterioration of the signal-to-noise ratio through

the NICE system. On top of that, a more fundamental question arises: Is the noise figure still a

convenient figure to describe the signal-to-noise ratio deterioration through a NICE system?

For Linear Time Invariant systems, the knowledge of the noise figure allows prediction of the

signal-to-noise ratio deterioration for any type of input signal, (since the noise figure is

independent of the input signal ), and for noise sources at any temperature

K. However, since the power of the input signal has an effect on the ratio

for NICE systems (see section 1.5.2), one can already suspect that deterioration

of the signal-to-noise ratio for these systems will be a function of the power contained in the

excitation signal (and thus of both the power of the input noise and the

power of the input signal). Note also that just as in the previous chapters, the noise added by

the NICE system itself, that combines with the input terms will be omitted.

Hence, the noise figure, as defined for linear systems , has to be extended to a NICE

noise figure , that describes the signal-to-noise ratio deterioration for NICE systems. As

opposed to the linear noise figure, that was defined for an input noise temperature of 290 K,

and that was independent of the input signal , the NICE noise figure is defined as the

ratio of the signal-to-noise ratio at the input of the noisy NICE system, to the signal-to-noise

ratio at the output of the noisy NICE system.

(3-1)

This NICE noise figure will be a function of (among others) the following variables: the total

input signal power , the total input noise power , and of course the frequency . With

this definition, the linear noise figure is a special case of the NICE noise figure, applied on a

u0 t( )

T T0 290=≠

Y ωi( ) U ωi( )⁄

u t( ) u0 t( ) nu t( )+=

NFlin

NNF

u0 t( )

NNFSNRinSNRout------------------=

Pu0Pnu

f

94

Introduction

LTI system instead of a NICE system, and only defined for an input noise spectral density,

corresponding to a noise temperature of 290 K.

The following questions have to be answered:

1. What is the variation of the NICE noise figure with the total input signal power and

the total input noise power ?

2. Is it possible to predict the NICE noise figure for an arbitrary input noise and signal

power, based on its knowledge of that NICE noise figure for a limited number of power

levels?

3. How can one measure this NICE noise figure?

Note that for a general case, where the input signal is a general signal, and where the

only constraint on the input noise is that it is ergodic noise, the noise behavior of the

NICE system will depend on the probability density function of , and the type of input

signal . In this chapter, the description of the NICE Noise figure will be given for an

input signal consisting of a pure sinewave, and the input noise being Gaussian noise.

The first question asks to describe the variation of the NICE noise figure as a function of

and . To answer that question, the - plane where the NICE noise figure will be

evaluated, can be divided into three zones, depending on how far the NICE system goes into

compression (see Figure 3-1)

Pu0

Pnu

u0 t( )

nu t( )

nu t( )

u0 t( )

Pu0

PnuPu0

Pnu

95


Since the level of compression of a NICE system is dependent on the total input power

, lines of constant compression levels are lines given by equations of the type

= constant or = constant, as shown in Figure 3-1. Three zones can hence be

distinguished:

1. Zone 1 is the zone where the total input power is so small that the nonlinear behavior of

the NICE system is negligible as compared to the linear behavior. Hence, in this zone,

the NICE system can be modelled as a LTI system, and its noise behavior can be

described using the linear noise theory as explained in section 1.4.

2. Zone 2 is the zone where the NICE system exhibits weak compression (i.e. up to the

1 dB compression point). In this zone, the NICE system will be modelled using a static

Wiener-Hammerstein model as described in section 3.1.2. To describe the noise behavior

in this zone, the variation of the NICE noise figure will first be determined as a function

of the input signal power, under standard noise conditions, i.e. for a input noise

temperature of 290 K (see section 3.2). Next, the variation of the NICE noise figure will

be studied as a function of the input signal power, for different input noise power levels,

and the variation of the NICE noise figure over these noise power levels will be

discussed (see section 3.3).

FIGURE 3-1. Illustration of the different compression zones in the - plane.

Pu0

Pnu

NNF

zone 1:zone 2:

zone 3:hard compression

linearweak compression

Pu0Pnu

Pu Pu0Pnu

+=

Pu Pu0Pnu

+

96

Introduction

3. Zone 3, is the zone where the NICE system exhibits hard compression (above the 1 dB

compression point), and where the simple third degree polynomial model as described in

section 3.1.2 is no longer valid. To describe the noise behavior of the NICE system in

this zone, simulations using an arctan model will be performed (see section 3.3.6).

The study of the variation of the NICE noise figure will be done using a sine wave as input

signal . In section 3.4, the variation of the NICE noise figure as function of a general

input signal instead of a sine wave will be studied.

3.1.2 The model for the noisy NICE system up to the 1 dB compression pointThe model for the NICE system is still the very simple one as depicted in Figure 2-1, and

repeated in Figure 3-2. For a full description of this model, refer to section 2.2.

This system will also be frequently used in the following sections, with system parameters

, , = 4 GHz and a flat power spectral density of the noise added by

the system = in the frequency interval . Figure 3-3 shows the behavior

of this example system.

FIGURE 3-2. A noisy Wiener-Hammerstein system as model for the NICE system.

u0 t( )

+fB fB

u t( ) y t( )

nA t( )

k1 u t( )⋅

k3 u3 t( )⋅

u t( ) +η t( )

k1 10= k3 60V 2––= B

PSDnA

1( ) 550N0 0 B,[ ]

97


The 1 dB compression point (i.e. the point where the output power of the system is 1 dB

smaller than the output power of the underlying linear system) is reached for an input power of

about dBm (see Appendix 3.A). The polynomial function reaches its

maximum for , corresponding to 0.24 V or dBm for the example.

This illustrates also that the third order model is valid for input powers up to the 1 dB

compression point.

To determine the NICE noise figure, one will consider that the excitation signal of this system

consists of a deterministic part and an additive noise source . This excitation

signal can be written as:

(3-2)

where can be considered as band-limited, flat, zero-mean Gaussian noise with power

spectral density . ( represents the input noise temperature.)

FIGURE 3-3. Illustration of the 1 dB compression point of the system in the example.

-25 -20 -15 -10 -5 0-10

-5

0

5

10

15

20

Input power [dBm]

Ou

tpu

t p

ow

er

[dB

m]


1dB

6– y k1x k3x3+=

x k1–( ) 3k3( )⁄= 2.6–

u t( ) u0 t( ) nu t( )

u t( ) u0 t( ) nu t( )+ A 2πf0t( )cos⋅ nu t( )+= =

nu t( )

Nin kTin= Tin

98

Variation of the NICE noise figure, as a function of the input amplitude

3.2 Variation of the NICE noise figure, as a function of the input amplitudeFirst, the NICE noise figure as a function of the power of the input signal will be

studied, when a pure sinewave, superimposed on white thermal noise (generated by a resistor

at standard temperature K) is applied to a noisy NICE system.

3.2.1 Determining the output power spectral density

The output signal of the noisy NICE system is:

(3-3)

or,

(3-4)

It is possible to predict for each term of (3-4), if it will contribute to the output noise or

to the output signal , without prior knowledge of the spectra or , by using

section 1.6.2. According to this section, all the terms in (3-4) will generate spectral

contributions, that can be split into four disjunct sets, based on their dependency on and

. By checking in which set the spectral contributions generated by each term of (3-4)

fall, one is able to predict the effect of each term of (3-4).

• The term will only generate spectral contributions that contain

one single component of . Hence, it produces elements of the set

that are not affected by the properties of . This is quite

obvious, because comes from the linear part of the system.

u0 t( )

T0 290=

y t( )

y t( ) k1 A 2πf0t( )cos nu t( )+( ) k3 A 2πf0t( )cos nu t( )+( )3 nA t( )+ +=

y t( ) k1nu t( ) k1A 2πf0t( )cos k3nu3 t( ) 3k3Anu

2 t( ) 2πf0t( )cos + + + +=

3k3A2nu t( ) 2πf0t( )cos( )2 k3A3 2πf0t( )cos( )3 nA t( )+ +

ny t( )

y0 t( ) Nu k( ) U0 k( )

Nu k( )

U0 k( )

k1nu t( )

Nu k( )

B N( ) u0 t( )

k1nu t( )

99


• The term will only generate spectral contributions that

contain one single component of . Hence, it produces elements of

the set that are not affected by the properties of . Again, this

is quite obvious, because comes from the linear part of

the system.

• The term generates output spectral components created by the

combination of three spectral components of and thus also creates

terms of set .

• The term on the other hand generates output

spectral components created by combining two noise spectral components

and one signal spectral component. It produces not only elements of set

(such as ) that are noise contributions,

affected by the properties of , but it will also produce elements of

set (e.g. ), that are corrections on the

output signal spectrum, due to the noise.

• Term generates output spectral components

created by combining two signal spectral lines, and one noise spectral

line. It produces only elements of set (such as

) that are noise contributions, affected by the

properties of .

• Term at last, generates output spectral components

created by combining three spectral components of and thus also

creates terms of set .

Next, the output power spectral density of the noisy NICE system will be analytically

determined. Equation (3-4) can also be rewritten as:

(3-5)

k1A 2πf0t( )cos

U0 k( )

A U0( ) nu t( )

k1A 2πf0t( )cos

k3nu3 t( )

Nu k( )

B N( )

3k3Anu2 t( ) 2πf0t( )cos

B′ U0 N,( ) 3k3Nu l( )Nu m( )U0 k( )

U0 k( )

A′ U0 N,( ) 3k3N l( )N l–( )U k( )

3k3A2nu t( ) 2πf0t( )cos( )2

B′ U0 N,( )

3k3Nu l( )U0 k–( )U0 k( )

U0 k( )

k3A3 2πf0t( )cos( )3

U0 k( )

A U0( )

y t( ) k134---k3A2+

A 2πf0t( )cos 14---k3A3 2π3f0t( )cos + +=

k132---k3A2+

nu t( ) 32---k3A2nu t( ) 2π2f0t( )cos 3k3Anu

2 t( ) 2πf0t( )cos k3nu3 t( ) nA t( )+ + + +

100


Following section 2.3, the output power spectral density will be calculated by taking the

Fourier transform of the auto-correlation of the system output (see (2-5)). Since the noise

added by the system itself is uncorrelated with the input signal and input noise, (2-6) (i.e.

) remains valid. If the terms in (3-5) are called respectively

to and , (i.e. the auto-correlation of the output of the noiseless NICE system)

can be written as (using the anti-symmetry property of the cross-correlation:

):

(3-6)

Where

(3-7)

In Appendix 3.B, all the terms in (3-6) are calculated, and the corresponding power spectral

densities are determined. The output power at frequency is given by:

(3-8)

With the deterministic part of the output signal (note that , as

shown in (3-8)). When there is no input noise ( ), the latter two terms in (3-8) are

absent. This means that they are not taken into account when performing a classical modeling

y t( )

Ryy τ( ) Rηη τ( ) RnAnAτ( )+= ς1

ς6 nA t( ) Rηη τ( )


Rηη τ( ) Rςiςiτ( )

i 1=

6

∑ Rςiςjτ( ) Rςiςj

τ–( )+( )

j i 1+=

6

∑i 1=

5

∑+=

ς1 k134---k3A2+

A 2πf0t( )cos=

ς214---k3A3 2π3f0t( )cos=

ς3 k132---k3A2+

nu t( )=

ς432---k3A2nu t( ) 2π2f0t( )cos=

ς5 3k3Anu2 t( ) 2πf0t( ) cos=

ς6 k3nu3 t( )=

f0

PSDy0

1( ) f0( )

δ f f0–( )---------------------------- k1

34---k3A2+

2 A2

2Z0---------

3k3A( )2

2--------------------Z0N0

2B2 3k3 k134---k3A2+

A2N0B+ +=

y0 t( ) y t( ) y0 t( ) H u0 t( )[ ]≠

N0 0=

101


of the device (that assumes ). However, those terms yield a (systematic) contribution

in the output signal power at frequency , and are therefore added to the signal output power.

A third harmonic of the input signal will appear at frequency , and with a

power given by:

(3-9)

The output noise power spectral density will be a frequency dependent quantity (see Appendix

3.B), given by:

(3-10)

where if , and if . on the other hand will be given by the following

criterion (see (3-85) and (3-86)):

(3-11)

Note that represents the stochastic part of the output signal .

3.2.2 Signal-to-noise ratio variation: the NICE noise figureAt this point, the signal and noise power spectral densities are known at the input and at the

output of the considered system. It is thus possible to apply the definition of the NICE noise

figure (3-1) i.e.

N0 0=

f0

A 2πf0t( )cos⋅ 3f0

PSDy0

1( ) 3f0( )

δ f 3f0–( )------------------------------- 1

4---k3A2

2 A2

2Z0---------=

PSDny

1( ) f( ) k132---k3A2+

2N0

k32N0

3Z02

2------------------ 27B2 3f2–( ) 6 k1

32---k3A2+

k3Z0N02B+ +=

32---k3A2

2

2-----------------------N0β 3k3A( )2Z0

N02

------

22 2B φ–( ) PSDnA

1( ) f( )+ + +

φ f0= f f0< φ f= f f0> β

f0B2---<

β 1= f B 2f0–≤⇔

β 1 2⁄= f B 2f0–>⇔⇒

f0B2--->

β 0= f 2f0 B–<⇔

β 1 2⁄= f 2f0 B–≥⇔⇒

ny t( ) y t( )

102


(3-12)

to the obtained results. However, because the signal power spectral density is a Dirac impulse,

(i.e. all the power is concentrated at one single frequency) while the noise spectrum is a

spectral density (i.e. the power is not concentrated at one frequency, but is smeared out in a

frequency band), the signal and noise power will be compared in a frequency interval of 1 Hz

around frequency . Note that choosing this bandwidth equal to 1 Hz can be done without

loss of generality. One could object that a bandwidth of a few kHz around frequency is

more realistic. But in that case, the input noise power is , while the output noise power in

the frequency interval can be approximated as (see

Appendix 3.C), if the noise power spectral densities are flat inside this frequency interval.

Hence, the value of will be eliminated in the division of the signal-to-noise ratios, or in

other words, stating = 1 Hz can be done without loss of generality.

The signal-to-noise ratio at frequency is given by:

(3-13)

and the signal-to-noise ratio at frequency is given by the ratio of (3-8) divided by

(3-10). In (3-10), , while the criterion (3-11) for , becomes (with ):

(3-14)

By defining the signal power gain

, (3-15)

the noise power gain

NNFSNRinSNRout------------------=

f0B0 f0

B0N0

f0 B0 2⁄– f0 B0 2⁄+,[ ] B0PSDnyf0( )

B0

B0

SNRin f0

SNRin f0( ) A2

2Z0 N0⋅--------------------=

SNRout f0φ f0= β f f0=

f0 B 3⁄≤ β⇒ 1=

f0 B 3⁄> β⇒ 1 2⁄=

Gu0

PSDy0

1( ) f0( )

PSDu0

1( ) f0( )----------------------------=

103


, (3-16)

and the noise power gain of the underlying noiseless system

, (3-17)

the NICE noise figure can be rewritten as:

(3-18)

With

(3-19)

and

(3-20)

Note that some terms in (3-19) and (3-20) can be neglected compared to others, because is

a very small quantity (recall that W/Hz). The terms in (3-19) and (3-20) can be

expressed in orders of magnitude of the dimensionless1 total noise power

. For (3-19) this yields:

Gnu

PSDny

1( ) f0( )

PSDnu

1( ) f0( )----------------------------=

G ′nu

PSDny

1( ) f0( ) PSDnA

1( ) f0( )–

PSDnu

1( ) f0( )--------------------------------------------------------------=

NNFA2 2Z0 N0⋅( )⁄

Gu0

A2

2Z0---------⋅

GnuN0⋅( )⁄

----------------------------------------------------------Gnu

Gu0

---------G ′nu

PSDnA

1( ) f0( ) N0⁄+

Gu0

--------------------------------------------------------= = =

Gu0k1

34---k3A2+

23k3Z0N0B( )2 6k3 k1

34---k3A2+

N0BZ0+ +=

k134---k3A2 3k3Z0N0B+ +

2

=

G ′nuk1

32---k3A2+

2 k3

2N02Z0

2

2------------------ 27B2 3f0

2–( ) 6 k132---k3A2+

k3Z0N0B+ +=

32---k3A2

2

2-----------------------β 3k3A( )2Z0

N02

------ 2B f0–( )+ +

N0

N0 4 21–×10=

ΠnuPnu

1W( )⁄ B N0⋅( ) 1W( )⁄= =

104


(3-21)

Since is very small (e.g. W for = 4 GHz), the second and third term

of (3-21) can be neglected with respect to the first one. Calculating the terms of (3-21) for the

example system given in section 3.1.2, and with = 0.16 V (this corresponds to a total power

of the sinewave of dBm/Hz, i.e. the 1 dB compression point) yields:

. This illustrates the fact that the second and third term in (3-

19) can be neglected with respect to the first one. Hence, can be approximated as:

(3-22)

Looking at the orders of magnitude of the terms of (3-20) yields:

(3-23)

Again, since is very small, the second, third and fifth term can be neglected with respect

to the others. Hence, the noise power gain can be approximated as:

(3-24)

1. because it is impossible to compare Watts with .Watts2

Gu0k1

34---k3A2+

23k3Z0N0B( )2 6k3 k1

34---k3A2+

N0BZ0+ +=

O Πnu

0 O Πnu

2 O Πnu

1

PnuPnu

1.6 11–×10= B

A

6–

Gu078 2.5 6–×10– 2 14–×10+=

Gu0

Gu0k1

34---k3A2+

2

=

G ′nuk1

32---k3A2+

2 k32N0

2Z02

2------------------ 27B2 3f0

2–( ) 6 k132---k3A2+

k3Z0N0B+ +=

32---k3A2

2

2-----------------------β 3k3A( )2Z0

N02

------ 2B f0–( )+ +

O Πnu

2 O Πnu

0 O Πnu

1

O Πnu

0 O Πnu

1

Πnu

Gnuk1

32---k3A2+

232---k3A2

2

2-----------------------β

PSDnA

1( ) f0( )

N0----------------------------+ +=

105


Hence, the NICE noise figure is then given by:

(3-25)

It is possible to write the NICE noise figure as an expression containing the linear noise figure

(i.e. the noise figure for the simple system with ). Knowing that the linear noise

figure is given by:

(3-26)

(3-18) can be written in terms of the linear noise figure and a correction term as:

(3-27)

or, in terms of a scaling and biasing of the linear noise figure as:

(3-28)

The correction term in (3-27) is function of the linear noise figure , the bandwidth of the

system , the system parameters and , and as expected, the amplitude of the input

NNF f0( )k1

32---k3A2+

2 1

2--- 3

2---k3A2

2β+

N0 PSDnA

1( ) f0( )+

k134---k3A2+

2N0

------------------------------------------------------------------------------------------------------------------------=

NFlin k3 0=

NFlin

NFlin

k12N0 PSDnA

1( ) f0( )+

k12N0

----------------------------------------------=

NNF NFlin

G ′nuk1

2– NFlin Gu0k1

2– ⋅–

Gu0

-------------------------------------------------------------------------+ =

NFlin

18 9β+8

------------------- k3A2( )2

3k1k3A2+ NFlin

916------ k3A2( )

2 32---k1k3A2+

–

k134---k3A2+

2--------------------------------------------------------------------------------------------------------------------------------------------------------+=

NNF NFlink1

2

Gu0

---------⋅G ′nu

k12–

Gu0

---------------------- +=

NNF NFlink1

2

k134---k3A2+

2

-----------------------------------⋅

18 9β+8

------------------- k3A2( )2

3k1k3A2+

k134---k3A2+

2

------------------------------------------------------------------------+=

NFlin

B k1 k3 A

106


signal (and thus the input power). The correction term in (3-27) is, however, not negligible as

compared to the linear noise figure (see Figure 3-4), and is responsible for the difference

between the linear noise figure and the NICE noise figure, as will be shown in the next

example.

Example 3.1

Consider a system that can be modelled as described in section 3.1.2. Suppose that the

excitation signal of the system consists of a deterministic Continuous Wave (CW) signal

(i.e. a pure sinewave) superimposed on a noise source as described in (3-2). First

the linear noise figure of the system is calculated, and next the NICE noise figure is

plotted as a function of the amplitude and as a function of the signal input power of the

input CW signal. The frequency can be considered to be smaller than one third of (hence

).

The linear noise figure is given by (3-26), and equals dB1. To plot the

figure, (3-27) or (3-25) can be used. Note that the 1 dB compression point of this system is

reached for an input signal amplitude of 0.16 V, i.e. an input signal power of dBm.

1. expressed in dB is since the noise figure is a power ratio.

FIGURE 3-4. NICE Noise figure as function of the input CW amplitude (left figure). NICE Noise Figure (in dB) as function of the input signal power (in dBm, right figure).

u t( )

u0 t( ) nu t( )

NFlin

A Pu0

f0 B

β 1=

NFlin 6.5= 8.13≈

NFlin 10 NFlin( )log⋅

6–

-50 -40 -30 -20 -10 08

8.5

9

9.5

Input signal power Pu0

[dBm]

NIC

E N

ois

e F

igure

[dB

]

0 0.05 0.1 0.15 0.26.5

7

7.5

8

8.5

9

Input CW signal amplitude A [V]

NIC

E N

ois

e F

igure

107


Figure 3-4 clearly shows that the correction term cannot be neglected compared to the linear

noise figure, in the 1 dB compression region. For input signal powers far below the 1 dB

compression point, the NICE noise figure is given by the linear noise figure (since the NICE

system can be modelled as a LTI system in that region of input powers), but once the power of

the CW signal reaches the compression region, the noise figure also increases.

In section 2.4, the noise figure yielded by the Y-factor technique increased or decreased

depending on the presence of gain compression or gain expansion (i.e. depending on the sign

of the system parameter ). A similar question arises in the present case: Can the NICE noise

figure decrease instead of increase? Hence, can the NICE noise figure be smaller than the

linear noise figure? In order to answer that question, the sign of the function that describes the

correction term (3-27) is analysed.

The denominator of (3-27) is always positive. Hence, one has to analyse the sign of the

numerator:

(3-29)

(3-29) clearly shows that when the sign of changes, the correction term will also change

sign if . This means that for a system

exhibiting gain expansion instead of gain compression, the NICE noise figure will be smaller

than the linear noise figure.

A more important question is: Can the NICE noise figure be smaller than the linear noise

figure, when the system exhibits gain compression ( )? To answer this question, the zero-

crossings of (3-29) have to be calculated. Calculations show that the correction term is zero for

input signal amplitudes given by:

k3

18 9β+8

------------------- k3A2( )2

3k1k3A2+ NFlin

916------ k3A2( )

2 32---k1k3A2+

–

3k3A2 k1 1 NFlin 2⁄–( ) 98---k3A2 2 β NFlin 2⁄–+( )+

=

k3

k1 1 NFlin 2⁄–( ) 98---k3A2 2 β NFlin 2⁄–+( )>

k3 0<

A

108


(3-30)

For the system given in Example 3.1, (3-30) tells that the correction term becomes zero for

= 1.15 V, (an input power of 11.2 dBm), and hence, far above the 1 dB compression point,

in the hard compression region where the model is invalid. (3-30) also shows that the linear

noise figure plays an important role in the value of where the correction term changes sign.

The argument of the square root in (3-30) changes sign for and

. Using (3-30) and (3-29), the following conclusions can be made (for

and ):

• If , the , until a certain input amplitude (lying

below the hard compression region) is reached. Above this amplitude,

.

• If , no real value of will satisfy (3-30). Hence, the

NICE noise figure will always be larger than the linear noise figure.

• If , the input amplitude at which the correction term

changes sign always lies in the hard compression region. Hence, the NICE

noise figure will always be larger than the linear noise figure.

In Figure 3-5, the amplitude (and corresponding input signal power) for which the

correction term changes sign is plotted as function of , to illustrate the above statements.

FIGURE 3-5. Zones where the correction term is positive or negative.

A8k1 NFlin 2⁄ 1–( )

9k3 2 β NFlin 2⁄–+( )-----------------------------------------------------=

A

A

NFlin 2=

NFlin 2 2 β+( )=

k1 0> k3 0<

NFlin 2< NNF NFlin< A

NNF NFlin>

2 NFlin 2 2 β+( )< < A

NFlin 2 2 β+( )> A

A

NFlin

2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

linear Noise Figure NFlin

Inp

ut

am

plit

ud

e A

[V

]

0 2 4 6 8 10-20

-15

-10

-5

0

5

10

15

20

linear Noise Figure NFlin

[dB]

Input sig

nal pow

er

[dB

m]

1 dB compression point

1 dB compression point

NNF NFlin>NNF NFlin>

NNF NFlin<

NNF NFl in<

NNF NFlin<

NNF NFl in<

109


As an illustration, Figure 3-6 shows the noise figure as function of the input amplitude and

input signal power for the example system with dB.

Figure 3-6 shows that for the given system parameters and linear noise figure, the NICE noise

figure first becomes smaller than the linear noise figure (up to V). When becomes

larger than 0.15 V, the NICE noise figure becomes larger than the linear noise figure, as

predicted in Figure 3-5. Hence, an amplifier has not necessarily to exhibit gain expansion for

the NICE noise figure to become smaller than the linear noise figure.

3.2.3 Special case: a noiseless NICE systemSince the region of input signal amplitudes where is larger for smaller linear

noise figures, the special case with the smallest possible linear noise figure ( , see

section 1.4.2) is considered. occurs when the considered NICE system is noiseless,

i.e. it does not produce any noise by itself, or . The previous

section showed that for , the NICE noise figure is smaller than the linear noise

figure: . Hence, a noiseless NICE system has a very special property, that is

impossible to obtain for an LTI system: The NICE noise figure of a noiseless NICE system can

be smaller than one. This means that the signal-to-noise ratio at the output of the system is

larger than the signal-to-noise ratio at the input of the system. Or in other words, a noiseless

NICE system improves the signal-to-noise ratio instead of deteriorating it.

For , the expression for the NICE noise figure (3-25) becomes:

FIGURE 3-6. Noise figure vs. input CW amplitude and power.

A

Pu0NFlin 1.8= 2.55≈

0 0.05 0.1 0.15 0.21.78

1.8

1.82

1.84

1.86

1.88

1.9


NIC

E N

ois

e F

igu

re

-50 -40 -30 -20 -10 02.5

2.55

2.6

2.65

2.7

2.75

2.8

Input signal power Pu0

[dBm]

NIC

E N

ois

e F

igu

re [

dB

]

A 0.15≈ A

NNF NFlin<

NFlin 1=

NFlin 1=

nA t( ) 0= PSDnA

1( ) f( ) 0=

NFlin 1=

NNF 1<

PSDnA

1( ) f( ) 0=

110


(3-31)

The NICE noise figure of a noiseless NICE system is the ratio of the noise power gain to the

signal power gain. (3-31) can also be written as:

(3-32)

Note that although the numerator and the denominator of (3-32) look very alike, they are

however different. This means that the signal and the noise are no longer equally amplified by

a NICE system.

Figure 3-7 shows the variation of the signal and noise gain as a function of the sinewave

amplitude and as a function of the total signal power , for the example system. Note that

the noise power gain (red dashed line) is smaller than the signal power gain (blue solid line).

This means that will be larger than , because the signal is more amplified than

the noise. Hence, a noiseless NICE system is sometimes able to enhance the signal-to-noise

ratio, while a LTI system could only deteriorate it.

FIGURE 3-7. Signal and noise power gain vs. input signal amplitude and power.

NNFk1

32---k3A2+

2 12--- 3

2---k3A2

2β+

k134---k3A2+

2-------------------------------------------------------------------------

G ′nu

Gu0

-----------Gnu

Gu0

---------= = =

NNFk1

2 3k1k3A2 β 2+( )98---k3

2A4+ +

k12 3

2---k1k3A2 9

16------k3

2A4+ +-------------------------------------------------------------------------=

0 0.05 0.1 0.15 0.240

50

60

70

80

90

100

Input CW signal peak amplitude A [V]

Sig

na

l &

no

ise

po

we

r g

ain

Gu

0

& G

nu

Gu

0G

nu

-50 -40 -30 -20 -10 016

17

18

19

20

21

Input CW signal power Pu

0

[dBm]

Sig

na

l &

no

ise

po

we

r g

ain

Gu

0G

nu

[dB]

A Pu0

SNRout SNRin

111


3.2.4 Experimental resultsAs already stated in section 3.2.2, the NICE noise figure of the modelled NICE system is given

by (3-27), i.e.

(3-33)

and this expression is a function of the linear noise figure , the system parameters

and , and as expected, the power of the input signal. With the knowledge of , ,

and , the noise figure can thus be calculated. is known, because it is the amplitude of the

sine wave that is applied at the input of the system. can be measured using a classic Y-

factor method, as discussed in chapter 2. The only unknowns are the system parameters and

. Classical noise figure analysers usually do not only measure the noise figure of a device,

but they also determine the power gain of the device (see Appendix 1.G). To do so, the

measurement instrument uses the following formula:

(3-34)

and are the output noise power spectral densities when the input power spectral

densities are respectively and as defined in section 2.3. For a LTI system, (3-34) indeed

yields the power gain of the system, since for an LTI system, ,

and hence (3-34) becomes:

(3-35)

In section 2.3, an analytical expression was derived for the output power spectral density when

the input consists of band-limited white Gaussian noise (2-20):

(3-36)

NF f0( ) NFlin

18 9β+8

------------------- k3A2( )2

3k1k3A2+ NFlin

916------ k3A2( )

2 32---k1k3A2+

–

k134---k3A2+

2

--------------------------------------------------------------------------------------------------------------------------------------------------------+=

NFlin k1

k3 NFlin k1 k3

A A

NFlin

k1

k3

H f( ) 2

H f( ) 2 N2 f( ) N1 f( )–Nh Nc–

---------------------------------=

N2 f( ) N1 f( )

Nh Nc

N2 1, f( ) k12Nh c, PSDnA

1( ) f( )+=

H f( ) 2k1

2Nh PSDnA

1( ) f( ) k12Nc PSDnA

1( ) f( )+ –+

Nh Nc–--------------------------------------------------------------------------------------------------- k1

2= =

N1 2, f( ) k12Nc h, 6k1k3Z0Nc h,

2 Bk3

2Nc h,3 Z0

2

2----------------------- 27B2 3f2–( ) PSDnA

1( ) f( )+ ++=

112


Substituting this expression in (3-34) yields:

(3-37)

Where denotes the measured “gain” of the NICE system when using (3-34), and

. ( represents the total hot noise power.) The second and third

term of (3-37) are respectively and , with respect to the first term that is

. This means that the second and third term in (3-37) can be neglected as compared to

the first term. Calculations show indeed that for , the excess noise ratio of the noise

source being 15.2 dB, the bandwidth of the system GHz, and the

system parameters and , the predicted power gain of the system using

(3-34) is about . Hence, the power gain as calculated with (3-34) is a good

measure for , just for the same reason that the classic Y-factor method, when applied to a

NICE system does not detect the nonlinearities in the system. The input noise powers are too

small for the nonlinearity to be detected.

The last unknown system parameter can be determined through system identification

techniques [9], but for the simple static model that is considered, a more simple technique

exists. It is sufficient to measure the power of the third harmonic of the CW signal (at

frequency ) and to divide this power by the input power of the CW signal. The result of this

operation is (see (3-9)) from which can easily be calculated. The parameter

can also be calculated out of the output power of the fundamental at frequency . Dividing

this output power by the input power yields (see (3-21)), from which can

easily be calculated since and are known.

In order to validate the theory, the noise figure of a Sonoma 330 Amplifier, with a frequency

range for 20 kHz to 2 GHz was measured for a CW signal at 600 MHz, whose power was

swept from dBm to dBm, in steps of 1 dB. The classical noise figure measurement

using the Y-factor method yields the following results at 600 MHz: = 6.4 and = 112.

H f( ) m2 k1

2 6k1k3Z0 Nh Nc+( )Bk3

2 Nh2 NcNh Nc

2+ +( )Z02

2-------------------------------------------------------- 27B2 3f2–( )++=

O Πh( ) O Πh2( )O Πh

0( )

H f( ) m2

Πh Ph 1W( )⁄= Ph BNh=

O Πh( ) O Πh2( )

O Πh0( )

Nc N0=

ENR Nh N0⁄ 1–= B 4=

k1 10= k3 60–= V 2–

100 1 4–×10–

k12

k3

3f0k3A2( ) 4⁄( )

2k3

k3 f0k1 3k3A2 4⁄+( )

2k3

k1 A

20– 1–

NFlin k12

113


Comparing the power of the third harmonic to the input CW power for = 0.165 V results in

. Using (3-33), the corrected noise figure for the nonlinear amplifier is

calculated and extrapolated (also with (3-33)) for other input CW signal amplitudes. This

result is shown by the solid line in Figure 3-8. Calculating for other CW input powers will

yield different results, because the amplifier does not perfectly behave as the proposed Wiener-

Hammerstein model as described in section 3.1.2. With the knowledge of , was

calculated for a set of input amplitudes by comparing the output power of the fundamental

tone to the input CW power using (3-21). The noise figure was then recalculated using (3-33),

and the result is shown by the dashed line in Figure 3-8. There is a very good agreement

between both traces, indicating that the proposed Wiener-Hammerstein model is a valid

approximation for the measured amplifier.

3.2.5 ConclusionIn this section (3.2), the variation of the NICE noise figure as a function of the input signal

amplitude was studied, when the input power spectral density equals , as is the case in

most problems. The NICE noise figure (i.e. the signal-to-noise ratio deterioration) can be

written as the ratio of the noise power gain to the signal power gain.

The effect of the input noise on the signal power gain can be neglected, and hence, the signal is

treated as if there was no noise present at the input (or ). The noise power gain

on the other hand, is essentially a function of the input signal power and the noise generated by

the NICE system itself.

FIGURE 3-8. Measured noise figure vs. at 600 MHz

A

k3 65V 2––=

k3

k3 k1

A

0 0.05 0.1 0.15 0.2 0.25 0.36

7

8

9

10

11

12

Input CW signal amplitude [V]

Noi

se F

igur

e

α and G vary with A α and G are constantsk1k1

k3k3

NIC

E

A

A N0

A′ U0 N,( ) 0≈

114


The NICE noise figure can be written as the sum of the linear noise figure, and a correction

term. If the linear noise figure is larger than 2, this correction term is always positive, yielding

a worse signal-to-noise ratio deterioration for the NICE system than for its underlying linear

system. However, if the linear noise figure is smaller than 2, the correction term will be

negative, indicating that the NICE system’s signal-to-noise ratio deterioration is less than that

of the underlying linear system. In the special case of a noiseless NICE system, there will be a

signal-to-noise ratio improvement instead of a deterioration, which is impossible for LTI

systems.

Hence, the minimum noise figure for nonlinear amplifiers can be smaller than for linear

amplifiers, and can theoretically even be smaller than one. Driving the amplifier into low

saturation can possibly enhance the signal-to-noise ratio.

115


3.3 Variation of the NICE noise figure, as a function of the input amplitude and the input noise power

3.3.1 IntroductionIn this section, the variation of the NICE noise figure as a function of the input amplitude and

the input noise power will be studied. This study will be done in two steps.

First, the variation of the NNF will be studied in the linear and the weak compression zone,

using the noisy NICE system model, proposed in section 3.1.2. This NNF variation will be

studied as a function of the input signal power, for different input noise powers, and an

variation over the noise powers will be sought (sections 3.3.2 to 3.3.5).

Next, the variation of the NNF will be studied in the hard compression zone, using simulations

performed on a (nonlinear) arc-tangent (atan) model (section 3.3.6).

3.3.2 Determining the analytical expressionFor LTI systems, the noise figure is defined at an input thermal noise power spectral

density, generated by a resistor at an absolute temperature of K. By applying a

transformation (1-22) on that noise figure, the signal-to-noise ratio deterioration can be

obtained for other levels of input noise power spectral densities. This transformation is based

on (1-21), that considers that the gain of the signal power and the noise power spectral density

are equal. Since section 3.2 showed that this hypothesis does not hold for NICE systems, the

transformation (1-22) will not be applicable to NICE systems. However, a similar reasoning as

in Appendix 1.F can be followed to determine the dependency of the NICE noise figure on the

input noise power levels.

The power spectral density of the noise added by the noisy NICE system itself can be written

in a similar form as (1-81):

(3-38)

NFlin

T0 290=

PSDnA

1( ) f( ) NFlin f( ) 1–( )k12N0=

116

Variation of the NICE noise figure, as a function of the input amplitude and the input noise

which depends on the linear noise figure , the power gain of the underlying linear

system and the standard thermal noise power spectral density . Note that by assumption,

the properties of the noise added by the system itself is independent of the input signal of the

system (see section 1.6.1). Hence (3-38) is not only valid for an LTI system, but also for the

considered noisy NICE system. The NICE noise figure for a certain input noise power spectral

density can be expressed as (see also (3-18)):

(3-39)

and are respectively the signal power gain, and noise power gain of the underlying

noiseless system, that are functions of the input signal amplitude (and thus ), and the

input noise power spectral density (and thus ). The expression for the signal power gain

is:

(3-40)

and for the noise power gain of the underlying noiseless system :

(3-41)

Or, in a worst case scenario, i.e. (thus ), the noise power gain will be maximized

as function of the frequency,

NFlin

k12 N0

Nin

NNF Nin A,( )G ′nu

PSDnA

1( ) f( ) Nin⁄+

Gu0

-------------------------------------------------------G ′nu

Gu0

-----------NFlin 1–( )k1

2 N0Nin--------

Gu0

------------------------------------------+= =

Gu0G ′nu

A Pu0

Pnu

Gu0

Gu0k1

34---k3A2+

2

3k3( )2Z02Nin

2 B2 6k3 k134---k3A2+

NinBZ0+ +=

k134---k3A2 3k3NinBZ0+ +

2

k132---k3Z0 Pu0

2Pnu+( )+

2

= =

G ′nu

G ′nuk1

32---k3A2+

2 k3

2Nin2 Z0

2

2-------------------- 27B2 3f0

2–( ) 6 k132---k3A2+

k3Z0NinB + + +=

32---k3A2

2

2-----------------------β 3k3A( )2Z0

Nin2

-------- 2B f0–( )+

f0 0→ β 1=

117


(3-42)

Hence, (3-39) is the general formula that describes the variation of the NICE noise figure as a

function of and , up to the 1 dB compression point.

In (3-39), three cases can be considered, mainly depending on the input noise power level:

1.

2.

3.

3.3.3 First case:

The noisy NICE system produces much more noise (that does not mix with the input) than the

amplified input noise. This situation can be due to two factors. First, the noisy NICE system

produces a tremendous amount of noise. This situation is highly improbable and usually occurs

when the system is malfunctioning. (Typical values for are a few hundred times ).

A second possibility is that the input noise power spectral density is very low, as is the case for

satellite communications, where the input noise temperature is about 10 K (i.e.

dBm/Hz). In this case, the approximated formula (3-22) for the signal power gain

can be used, and can be neglected with respect to . The NICE noise

figure (3-39) for very low input noise power spectral densities hence becomes:

(3-43)

Since the signal and noise power gain and both have the same order of magnitude

(see Figure 3-7), the denominator of (3-43) will also be much smaller than the numerator,

G ′nuk1

32---k3A2 3k3Z0NinB+ +

2 9k3

2

2-------- A2

2------ Z0NinB+

2 k32

2-----9A2Z0NinB+ +=

k1 3k3Z0 Pu0Pnu

+( )+( )2 9k32Z0

2

2--------------- Pu0

Pnu+( )2 9k3

2Z02Pnu

Pu0+ + =

Pu0Pnu

PSDnA

1( ) f( ) Nin⁄ G ′nu»

PSDnA

1( ) f( ) Nin⁄ G ′nu≈

PSDnA

1( ) f( ) Nin⁄ G ′nu«

PSDnA

1( ) f( ) Nin⁄ G ′nu»

PSDnA

1( ) N0

Nin 189–≈

Gu0G ′nu

PSDnA

1( ) f( ) Nin⁄

NNFNFlin 1–( )k1

2 N0Nin--------

Gu0

------------------------------------------NFlin 1–( )k1

2 T0Tin-------

k134---k3A2+

2------------------------------------------= =

Gu0G ′nu

118


leading to a very large NICE noise figure. If the input signal power increases, will

decrease (if , gain compression), leading to an even more important increase of the

NICE noise figure. Hence, for an amplifier exhibiting gain compression, and being excited

with a CW signal superimposed on white noise that has a very low noise temperature

( K), the NICE noise figure will be very large, and will increase even more with

increasing power of the CW signal, as illustrated in Figure 3-9.

As a result, the NNF for low input noise temperatures is essentially determined by the noise

added by the NICE system itself. The variation of the NICE noise figure over the low input

noise temperatures is also given by (3-43), which can be rewritten as

(3-44)

where is an increasing function of the power of the input signal . Figure 3-10

shows a schematic representation of the variation of NNF, as a function of the input signal

amplitude and the input noise power spectral density , for .

FIGURE 3-9. NICE Noise Figure versus input signal amplitude and input signal power, for an input noise temperature of 10 K.

Gu0

k3 0<

Tin 290<

0 0.05 0.1 0.15 0.2160

170

180

190

200

210

220

230

240


NIC

E N

ois

e F

igu

re

-50 -40 -30 -20 -10 022

22.5

23

23.5

24

Input signal power Pu

0

[dBm]

NIC

E N

ois

e F

igu

re [

dB

]

NNF

NFlin 1–( )k12N0

k134---k3A2+

2

----------------------------------------

Nin---------------------------------------------- Φ A2( )

Nin----------------= =

Φ A2( ) u0 t( )

A Nin Nin N0«

119


Note that for LTI systems, when the noise generated by the system itself is much larger than

the amplified input noise, the noise behavior of the system is characterized in terms of

operational noise temperature, rather than by using the noise figure (see section 1.4.5). The

reason for this is that the noise figure for LTI systems also starts to increase as , when

goes towards zero. Defining an operational noise temperature for noisy NICE systems

would be a bad idea, because it requires the knowledge of the noise power gain. Indeed,

consider that the output noise power spectral density is given by:

(3-45)

since . Insert the quantity “operational noise temperature”:

(3-46)

Where k is Boltzmann’s constant. Hence, if one knows the operational noise temperature, the

output noise power spectral density still remains unknown because it is a function

of the operational noise temperature itself, and of the input signal amplitude. Thus, the

analytical expression (or a table with values) of the noise power gain has to be known.

FIGURE 3-10. Schematic representation of the signal-to-noise ratio deterioration vs. and for input noise temperatures much smaller than 290 K

A

NNF

Nin

k200K

0.2V

200

ANin

1 Nin⁄

Nin

PSDny

1( ) f( ) G ′nuNin PSDnA

1( ) f( )+ PSDnA

1( ) f( )≈=

PSDnA

1( ) f( ) G ′nuNin»

PSDny

1( ) f( ) PSDnA

1( ) f( ) k Top f( ) G⋅ ′nukTop A,( )⋅= =

PSDny

1( ) f( )

120


3.3.4 Second case:

The noise produced by the noisy NICE system is of the same order of magnitude as the

amplified input noise. Since is typically a few hundred times , one can consider

that is in the order of magnitude of . In this case, the approximated formulae for

(3-22) and (3-24) can be used. The expression for the NICE noise figure is given by (3-

39) itself, i.e.

(3-47)

This situation is similar to the case for as described in section 3.2, except that

is divided by instead of (hence is multiplied with a factor

). This implies that the considered situation, and behavior of the NNF as a function of

the input signal amplitude will be identical to the situation described in section 3.2. However,

the power spectral density of the noise added by the system itself will seem to be

instead of . And the linear noise figure will seem to be

(3-48)

rather than , just as described in the transformation formula (1-22). Hence, the

contribution of the noise added by the system to the NNF is decreasing with increasing input

noise power.

In section 3.2, it was also shown that when the linear noise figure is smaller than two, there

exists an interval of input signal amplitudes , where the NNF is smaller than the

linear noise figure. Since the linear noise figure seems to decrease for increasing input noise

power levels as given by (3-48), the special behavior with will eventually occur.

PSDnA

1( ) fi( ) G ′nuNin≈

PSDnA

1( ) f( ) N0

Nin N0 Gu0

G ′nu

NNF Nin A,( )G ′nu

PSDnA

1( ) f( ) Nin⁄+

Gu0

-------------------------------------------------------G ′nu

Gu0

-----------NFlin 1–( )k1

2 N0Nin--------

Gu0

------------------------------------------+= =

Nin N0=

PSDnA

1( ) f( ) Nin N0 PSDnA

1( ) f( )

N0 Nin⁄

N0 Nin⁄( ) PSDnA

1( ) f( )⋅ PSDnA

1( ) f( )

1 NFlin 1–( ) N0 Nin⁄( )+

NFlin

A 0 ξ,[ ]=

NNF NFlin<

121


Example 3.2

If = 6.5, according to (3-48), the apparent linear noise figure will decrease below 2 if

, or, in terms of , if .

Figure 3-11 shows a schematic representation of the variation of the NICE noise figure, as a

function of the input signal amplitude and the input noise power spectral density , for

.

Hence, the more increases, the less the noise added by the system itself becomes

important in the NNF, and the more the noise behavior of the system resembles that of a

noiseless NICE system.

3.3.5 Third case:

The noise produced by the noisy NICE system is much smaller than the amplified input noise.

This means that is several orders of magnitude larger than . The expression for the

NICE noise figure is then given by:

(3-49)

This expression is independent of the noise produced by the noisy NICE system itself. Note

that due to the polynomial model of third degree that was chosen to represent the NICE

FIGURE 3-11. Schematic representation of the signal-to-noise ratio deterioration vs. and for input noise temperatures around 290 K

NFlin

Nin NFlin 1–( )N0≥ 5.5N0= PSDnA

1( ) f( ) Nin PSDnA

1( ) f( ) k12⁄≥

A Nin

Nin N0≈

A

NNF

Nin

N0

0.2V2

ANin

Nin

PSDnA

1( ) fi( ) G ′nuNin«

Nin N0

NNFG ′nu

Gu0

-----------=

122


system, the constraint W = dBm (see Appendix 3.A) still

holds. This constraint tells that if the input signal power is zero, the maximal noise input power

spectral density at the input of the system is about dBm/Hz, or 6. W/Hz. Since

, the validity of the approximated formulae for the signal and noise power gain can be

questioned. In section 3.2, it was shown that the terms of and that are , are

in the order of magnitude of , as compared to the terms in , that are in the order

of magnitude of . This means that for the terms in to become non-negligible, the

input noise power has to increase with at least a factor or 60 dB. As a result, the

approximated formulae for and (and hence the noise behavior of the system, as

described in section 3.3.4) are valid up to dBm/Hz.

For dBm/Hz, the input signal amplitude can vary from 0 V up to 0.153 V in

order to stay below the 1 dB compression point. For dBm/Hz and = 0.153 V,

is given by:

(3-50)

and for = 0.153 V, equals:

(3-51)

These results show that is indeed a negligible term in the noise power gain,

and that the terms in and can no longer be neglected. In order to get an idea

of the variation of the NICE noise figure, for input power spectral densities higher than

dBm/Hz (= W/Hz), the ratio (i.e. the NNF) is plotted in Figure 3-12,

as a function of the total input signal power and the total input noise power, using the

constraint that the total excitation power has to be smaller than the 1 dB compression point.

Pu Pu0Pnu

+= 2.5 4–×10< 6–

102– 3 14–×10

Nin N0»

Gu0G ′nu

O Πnu( )

10 6– O Πnu0( )

101 O Πnu( )

106

Gu0G ′nu

Nin 114–≈

Nin 114–≈ A

Nin 114–≈ A

Gnu

GnuO Πnu

0 O Πnu

1 O Πnu

2 PSDnA

1( ) f( ) Nin⁄+ + +=

64.5 1.67– 31 3–×10 55 5–×10+ + =

A Gu0

Gu0O Πnu

0 O Πnu

1 O Πnu

2 + +=

80 2.58– 21 3–×10 +=

PSDnA

1( ) f( ) Nin⁄

O Πnu1( ) O Πnu

2( )

114– 4 15–×10 GnuGu0

⁄

123


Equation (3-48) shows that for dBm/Hz = W/Hz, the NICE noise

figure behaves the same as at , with an apparent linear noise figure of

. The NICE noise figure decreases below one for increasing signal input power.

When the input noise power spectral density increases, one can see from Figure 3-12 that the

NICE noise figure also increases (hence the noise power gain becomes larger than the signal

power gain), but remains in the vicinity of 1. For a given input noise PSD, the NICE noise

figure still decreases as a function of the input signal power.

The increase of the NICE noise figure, for high input noise power spectral densities, suggests

that the NNF becomes worse again in the hard compression region, where the power of the

input noise increases further.

FIGURE 3-12. Variation of the NICE noise figure for dBm/Hz.

Input signal power [W] Input noise PSD [W/Hz]

Nin 114–≥

Nin 114–= 4 15–×10

Nin N0=

1 5.5 6–×10+

124


To illustrate the conclusions of sections 3.3.3 to 3.3.5, Figure 3-13 shows the variation of the

NICE noise figure as a function of the input signal amplitude and the input power spectral

density . The plot was made using the correct formulae (3-40) and (3-41), together with the

complete expression for the NICE noise figure (3-39).

3.3.6 Variation of the NICE noise figure in hard compression

A. Motivation and modelThe previous sections described the noise behavior of a NICE system in terms of the NICE

noise figure, up to the 1 dB compression point. This 1 dB compression point limit follows

from the third degree polynomial model used in the calculations. To describe the variation of

the NICE noise figure in the hard compression zone, another model for the NICE system has to

be proposed. The most straightforward approach would be to choose a polynomial model that

has a higher (odd) degree than three. However, all the polynomial models suffer the same

problem: when their input argument goes towards infinity, their function value also tends

towards infinity. Hence, even when choosing a very high degree polynomial model, this model

will only be valid for a certain range of input power levels. Hence, instead of a polynomial

FIGURE 3-13. NICE noise figure versus input signal amplitude and input noise power spectral density, up to the 1 dB compression point.

Input noise PSD [dBm/Hz] Input signal amplitude [A]

A

A

Nin

125


model, an atan function will be used to model the static nonlinearity in the Wiener-

Hammerstein model (see Figure 3-14).

This is a good choice since an atan function is a function that is able to describe deep

compression and clipping due to its horizontal asymptotes, i.e.:

(3-52)

Furthermore, for , the function can be approximated by , which is a

linear behavior. The atan function therefore smoothly extends linear behavior to nonlinear

behavior.

B. Choosing parameters and Since the arctan model has to extend the input power range of the third degree polynomial

model given in section 3.1.2 (or vice versa, the polynomial model has to be an approximation

of the arctan model around zero), the following question arises: What is the connection

between the polynomial parameters and , and the arctan parameters and ? To answer

this question, the arctan function is expanded in its Taylor series (definition: see Appendix

3.D) up to the third degree:

(3-53)

Comparing this expression to the polynomial expression , yields:

FIGURE 3-14. Wiener-Hammerstein model with arctan function to describe the noisy NICE system.

+fB fBu t( ) y t( )

nA t( )

u t( )

α β u t( )( )atan⋅

α βu( )atanu ∞→lim α π

2--- ∞≠=

u 0≈ α βu( )atan αβu≈

α β

k1 k3 α β

α βu t( )( )atan αβ u t( ) 13---αβ3 u3 t( )⋅–⋅≈

k1 u t( )⋅ k3 u3 t( )⋅+

126


(3-54)

In Figure 3-15, both static nonlinearities are plotted, and it is clear that the polynomial model

approximates the atan model around zero.

Note that since the atan function can be approximated by the third polynomial for small values

of , the noise behavior of the system modelled using the atan function will be identical to

the system modelled with the third degree polynomial for input power levels below the 1 dB

compression point. Figure 3-16 illustrates the behavior of this system modelled using the atan

function.

The 1 dB compression point is reached for an input power of about dBm.

FIGURE 3-15. Third degree polynomial and atan approximate each other around zero.

FIGURE 3-16. Illustration of the 1 dB compression point of the atan modelled system.

β 3k3–( ) k1⁄= α k1 k1 3k3–( )⁄=

-1 -0.5 0 0.5 1-10

-5

0

5

10

u t( )

-25 -20 -15 -10 -5 0 5 10-10

-5

0

5

10

15

20

25

30

Input power [dBm]

Outp

ut pow

er

[dB

m]


1dB

5.2–

127


C. Simulations with the atan modelBecause there exists no closed form for the expression [32], only simulated

results will be shown and discussed in this section. All the simulations used the arctan model

with the parameters and given by (3-54), where and . The power

spectral density of the noise added by the system itself was given by:

dBm/MHz. The frequency bins have a

width of 1 MHz, the bandwidth of the system is 4096 bins or 4.096 GHz, and the total

number of bins in the simulation is = 131072 bins. The simulations were done at five

different input noise power spectral densities: dBm/Hz (i.e. ), dBm/Hz (in the

weak compression zone), dBm/Hz (the boundary where the NNF started to increase),

dBm/Hz and dBm/Hz (two power levels in the hard compression zone). 32

realisations of the input noise were used to calculate the spectra. The input power of the CW

signal was increased up to 30 dBm, which corresponds to V (hence, also in the hard

compression zone). The frequency of the input CW signal is 500 MHz. The following figures

show, for each input noise power spectral density, the NICE noise figure, the input signal

power gain , the noise power gain , the output noise power and the signal output

power. The output signal power was determined at the fundamental frequency. All these

quantities are plotted in dB, dBm or dBm/MHz, versus the input signal CW power (in dBm). In

the plot where both and are drawn, the red curve represents and the blue curve

.

u t( ) n t( )+( )atan

α β k1 10= k3 60V 2––=

PSDnA

1( ) 550N0 146.6dBm Hz⁄– 86.6–= = =

B

217

174– N0 144–

114–

94– 74–

A 10=

Gu0Gnu

GnuGu0

Gnu

Gu0

128


FIGURE 3-17. Simulation results for dBm/Hz = dBm/MHz.


-30 -20 -10 0 10 20 30 40-86.6

-86.4

-86.2

-86

-85.8

Input CW power [dBm]N

ois

e O

ut

[dB

m/M

Hz]

-30 -20 -10 0 10 20 30 40-20

0

20

40

Input CW power [dBm]

Sig

na

l O

ut

[dB

m]

-30 -20 -10 0 10 20 30 40-20

-10

0

10

20


Gu

0

[d

B]

-30 -20 -10 0 10 20 30 4027

27.5

28

28.5


Gn

u

[d

B]

-30 -20 -10 0 10 20 30 40-20

0

20

40


Gu

0

& G

nu

[d

B]

-30 -20 -10 0 10 20 30 400

20

40

60


NN

F [

dB

]

Nin 174–= 114–

-30 -20 -10 0 10 20 30 40-90

-80

-70

-60


No

ise

Ou

t [d

Bm

/MH

z]

-30 -20 -10 0 10 20 30 40-20

-10

0

10

20

30


Sig

na

l O

ut

[dB

m]

-30 -20 -10 0 10 20 30 40-5

0

5

10

15


NN

F [

dB

]

-30 -20 -10 0 10 20 30 40-20

-10

0

10

20


Gu

0

[d

B]

-30 -20 -10 0 10 20 30 40-10

0

10

20


Gn

u

[d

B]

-30 -20 -10 0 10 20 30 40-20

-10

0

10

20


Gu

0

& G

nu

[d

B]

Nin 144–= 84–

129




-30 -20 -10 0 10 20 30 40 50-70

-60

-50

-40

-30


No

ise

Ou

t [d

Bm

/MH

z]

-30 -20 -10 0 10 20 30 40 50-20

0

20

40


Sig

na

l O

ut

[dB

m]

-30 -20 -10 0 10 20 30 40 50-40

-20

0

20


Gu

0

[d

B]

-30 -20 -10 0 10 20 30 40 50-20

-10

0

10

20


Gn

u

[d

B]

-30 -20 -10 0 10 20 30 40 50-40

-20

0

20


Gu

0

& G

nu

[d

B]

-30 -20 -10 0 10 20 30 40 50-5

0

5

10

15


NN

F [

dB

]

Nin 114–= 54–

-30 -20 -10 0 10 20 30-40

-30

-20

-10


No

ise

Ou

t [d

Bm

/MH

z]

-30 -20 -10 0 10 20 30-20

0

20

40


Sig

na

l O

ut

[dB

m]

-30 -20 -10 0 10 20 30-10

0

10

20


Gu

0

[d

B]

-30 -20 -10 0 10 20 30-10

0

10

20


Gn

u

[d

B]

-30 -20 -10 0 10 20 30-10

0

10

20


Gu

0

& G

nu

[d

B]

-30 -20 -10 0 10 20 30-1

0

1

2

3


NN

F [

dB

]

Nin 94–= 34–

130


For dBm/Hz = dBm/MHz (see Figure 3-17), the noise power gain for

small input signal power levels is about 8 dB larger than the signal power gain . The

reason therefore is that the noise power gain shown in the simulations is indeed , which

contains per definition (3-16) the term , i.e. the noise PSD added by the system

itself. The presence of becomes also visible when the input signal power increases.

Indeed, for high input signal powers, the output noise power does not decrease further, but

reaches a constant value of about dBm/MHz, which corresponds to (see the

beginning of this section). Furthermore, the NICE noise figure is a monotone increasing

function, as predicted by the polynomial model.

For dBm/Hz = dBm/MHz (see Figure 3-18) and for dBm/

Hz = dBm/MHz (see Figure 3-19), a range of input signal power levels exists for which

the output signal-to-noise ratio is better than the input signal-to-noise ratio. The NNF is

smaller than 0 dB, because . The presence of this interval was also predicted in

section 3.3.4, using the third degree polynomial model. When the input signal power increases


-10 0 10 20 30 40-25

-20

-15

-10

Input CW power [dBm]N

ois

e O

ut

[dB

m/M

Hz]

-10 0 10 20 30 40-10

0

10

20

30


Sig

na

l O

ut

[dB

m]

-10 0 10 20 30 40-15

-10

-5

0


Gu

0

[d

B]

-10 0 10 20 30 40-10

-5

0

5


Gn

u

[d

B]

-10 0 10 20 30 40-15

-10

-5

0

5


Gu

0

& G

nu

[d

B]

-10 0 10 20 30 400

1

2

3


NN

F [

dB

]

Nin 74–= 14–

Nin 174–= 114– Gnu

Gu0

Gnu

PSDnA

1( ) f( ) Nin⁄

PSDnA

1( ) f( )

86.6– PSDnA

1( ) f( )

Nin 144–= 84– Nin 114–=

54–

GnuGu0

<

131


above the 1 dB compression point, the signal power gain further decreases, while the noise

power gain reaches a limit value: . Hence, the NNF becomes again larger than

0 dB and increases further.

In the case of dBm/Hz = dBm/MHz, the system is in hard compression, since

the input noise power itself is already higher than the 1 dB compression point. Hence, the third

degree polynomial model is no longer valid. For small input signal powers, the noise power

gain is larger than the signal power gain. An indication of this behavior was also found with

the polynomial model, where the NNF started to increase for dBm/Hz.

Furthermore, both the noise and signal power gain are smaller than for the previous input noise

PSD dBm/Hz, because the input noise power itself is large enough to push the

system in compression. There still exists an input power range for which the NNF is smaller

than 0 dB, but this range is located at higher input signal power levels (between 0 dBm and

15 dBm input signal power) than was the case for lower input noise power levels.

Finally, for dBm/Hz = dBm/MHz, the NICE noise figure curve has a more

noisy character. The reason is that with 32 realisations, the relative error made on the noise

power spectrum is about .5 dB. The simulations predict that for input noise powers as high

as dBm/Hz, the NICE noise figure increases again, because the whole NNF curve

is shifting upwards. However, an input power range at which the NNF curve suddenly makes a

dip still exists. The location of this dip increases with increasing input noise power.

Note that in all cases, for very large input signal power levels, the NICE noise figure (in dB)

eventually increases linearly as a function of the increasing input signal power (expressed in

dBm). This behavior has already been intuitively predicted in section 2.5. The NICE noise

figure is by definition the ratio of the SNR at the input of the system to the SNR at the output

of the system. The signal output power will reach a maximal power since the amplifier can

only produce a limited output power, while the output noise power reaches . Hence,

the SNR at the output of the system will tend towards a constant number. The input noise

power is a fixed quantity for each simulation, but the input signal power can still increase.

PSDnA

1( ) f( ) Nin⁄

Nin 94–= 34–

Nin 114–>

Nin 114–=

Nin 74–= 14–

1±

Nin 74–=

PSDnA

1( ) f( )

132


Thus the NICE noise figure (in dB) will be equal to the input noise power (in dBm) plus a

constant value, i.e. the NICE noise figure increases linearly with the input signal power.

D. Notes on the output signal and noise power behavior.Figure 3-22 shows the signal and noise output powers for all simulated input noise power

spectral densities.

The output noise power spectral density has the following properties:

1. Seemingly, the output noise power spectral density decreases with increasing input

signal power, to reach a constant value: .

(3-55)

Hence, when the input signal power tends to infinity, the output noise power tends to the

noise produced by the system itself. Due to the compression and clipping behavior of the

system, the input noise is “cancelled” by the system. At high compression levels, the

extrema of the sine wave are compressed. The output noise at these extrema will hence be

smaller than at the zero crossings. This results in a decrease of the overall noise variance.

When the amplitude of the input sine wave rises further above the clipping boundary (this

clipping boundary is ), the output noise will essentially only

remain at the zero-crossings of the output waveform. The output waveform tends to a

square wave, and at the horizontal parts, the noise is almost inexistent. In the extreme

FIGURE 3-22. Signal and noise output powers for all simulated input noise power spectral densities.

-40 -20 0 20 40 60-100

-50

0


No

ise

Ou

t [d

Bm

/MH

z]

-40 -20 0 20 40 60-20

0

20

40


Sig

na

l O

ut

[dB

m]

Nin

Nin

PSDnA

(1)

PSDnA

1( )

PSDny

1( )

Pu0∞→

lim PSDnA

1( )=

α βx( )atan( )x ∞→lim απ 2⁄=

133


limit that the amplitude of the input sine wave tends towards infinity, the output wave

will be a perfect square wave, containing no noise coming from the input noise, since the

slope at the zero-crossings is infinitely large. The noise , produced by the system

itself will be superimposed on this square wave of course. Figure 3-23 illustrates the

presence of the noise at the zero crossings of the output signal.

In Figure 3-23, the input waveform is shown for an input noise power spectral

density dBm/Hz, and an input signal power of 8 dBm (left plot) and

20 dBm (right plot). The output signal and the noise on the output signal are also

drawn (red line). As described, the noise on the output signal is smaller at the extrema of

the amplified signal than at its zero crossings. If the input amplitude of the sine wave

increases, the slope of the curve at the zero crossings becomes steeper, and the curve

reaches faster the zone where the noise amplitude is smaller. The total output noise power

for the left plot is dBm, where for the right plot, it is dBm.

2. The total noise output power reaches an asymptotic value for increasing input noise

power.

(3-56)

This follows from the fact that the system cannot produce an infinite output power. In the

extreme limit that the total input noise power tends to infinity, the output wave will be a

random sequence of (i.e. plus or minus the clipping boundary). But since the

noise has a maximal frequency , this random sequence will resemble a random bit

FIGURE 3-23. Input (blue curve) and output (red curve) waveform, and noise on the output waveform (green curve), for dBm/Hz and respectively 8 dBm and 20 dBm.

nA t( )

0 500 1000 1500 2000-4

-3

-2

-1

0

1

2

3

4

0 500 1000 1500 2000-4

-3

-2

-1

0

1

2

3

4

Ampl

itude

[V]

Am

plitu

de [V

]

sample number sample number

Nin 114–= Pu0

u t( )

Nin 114–=

y t( )

5.7– 11.8–

PnyPnu∞→

lim c ∞<=

απ 2⁄±

B

134


stream of bits per second. The total power in such a bit stream equals the total power

present in a square wave that has an amplitude equal to , i.e. . This

corresponds to about 24.4 dBm, or when equally distributed over 4096 MHz:

dBm/MHz.

3. When the input noise power spectral density increases, the output noise power spectral

density remains flat over a longer interval of input signal power before it starts to

decrease (see Figure 3-22). The reason therefore is that, due to the noise on the input

signal, extra zero crossings will be created, with a probability that decreases with

increasing input signal power and with increasing instantaneous value of the input signal

(see Appendix 3.E). In other words, for a constant input signal power, the extra zero

crossings will essentially be concentrated around the zero crossings of the sine wave,

when the input noise power is smaller than the signal power. Extra zero crossings will

occur anywhere in the sine wave, if the noise power is larger than the signal power. In the

linear region, these extra zero crossings cause no problem, because each point of the

input waveform is equally amplified. In the compression region, however, the part of the

signal that changes sign will be amplified more than the noise near the extremum of the

sine wave (because of the compression). Hence, if the input noise power is large, and if

the system goes into compression, the output noise is no longer concentrated in a range

around the zero crossings of the sine wave (as was the case in Figure 3-23), but the noise

will remain equally present in the sine wave. Hence, the output signal looks noisy over a

larger interval of input signal powers, than was the case for lower input noise powers.

Eventually, the total signal power will become larger than the total noise power, yielding

a decrease in probability of the extra added zero crossings at the extrema of the sine

wave, and concentrating them around the zero crossings of . The output noise

power will then decrease. This is illustrated in Figure 3-24.

B

απ 2⁄ απ 2⁄( )2 Z0⁄

11.7–

u0 t( )

135


Three very similar properties can be found concerning the output signal power:

1. The output signal power (for a constant input signal power) decreases with increasing

input noise power. The reason therefore is analogue to the decrease of the noise power

for increasing signal power. In the extreme limit that the input noise power tends to

infinity, the output wave will be a perfect digital random bit stream, containing no more

information about the signal . Hence, for increasing noise input power, the effect of

the signal on the output waveform will decrease due to the clipping.

2. The signal output power at the fundamental frequency reaches an asymptotic value for

increasing input signal power. In the extreme limit that the input signal power tends to

infinity, the output waveform will be a square wave, and the power at the fundamental

frequency will be the power of the fundamental spectral component of that square wave.

3. When the input signal power increases, the output signal power remains flat over a

longer interval of input noise power spectral densities before it starts to decrease (see

Figure 3-22). Again, a similar explanation as for the noise can be given. If the input

signal power is high, at least an equally high input noise power is required to create extra

zero crossings, and to eliminate the signal influence on the output waveform, using the

compression and clipping of the system.

FIGURE 3-24. Input (blue curve) and output (red curve) waveform, for and respectively 20 dBm and 30 dBm.

0 500 1000 1500 2000-10

-5

0

5

10

0 500 1000 1500 2000-20

-15

-10

-5

0

5

10

15

20

Nin 74dBm Hz⁄–=Pu

u0 t( )

136


Note that the elimination of the effect of an input signal on the output waveform can never

occur in the linear region, since it is a property of the nonlinear mechanism itself.

3.3.7 ConclusionIn order to describe the variation of the NICE noise figure of a noisy NICE system that is

excited by the superposition of a pure sine wave and band-limited Gaussian noise, the -

plane where the NICE noise figure will be evaluated, can be divided into three zones,

depending on how deep the NICE system goes into compression (see Figure 3-1).

In the linear zone, the total input power is that small that the NICE system can be modelled as

a LTI system and its noise behavior can be described using the linear noise theory as explained

in section 1.4. The boundary of this zone is reached when the terms in the signal power gain

(3-40) and noise power gain (3-42), that are due to the nonlinear behavior of the system, can no

longer be neglected as compared to the linear terms (i.e. about dBm total input power, see

Appendix 3.F).

In the weak compression region, the NICE noise figure is given by the ratio of the noise power

gain to the signal power gain: . Hence, the

NICE noise figure consists of two contributions: the effect of the noiseless NICE system

( ) and the effect of the noise produced by the NICE system itself:

. The latter effect decreases as , for increasing noise input PSD.

Up to about 10 dB under the 1 dB compression point, the signal power gain and the noise

power gain of the underlying noiseless system are constant as function of the input noise

PSD . This implies that the variation of the NNF for increasing input noise PSD is the

same as the behavior of that system for input noise PSD equal to , but having an apparent

linear noise figure (i.e. a ) that decreases with increasing . Since it was shown in

section 3.2, that a linear noise figure which is smaller than two, implies the existence of a range

of input signal amplitudes where , an identical behavior will be encountered

when the input noise PSD increases. Hence the apparent linear noise figure decreases below

two. For input noise power spectral densities at about 10 dB under the 1 dB compression point,

the system will actually improve the signal-to-noise ratio. Around the 1 dB compression point,

Pu0

Pnu

33–

NNF GnuGu0

⁄ G ′nuPSDnA

1( ) Nin⁄+ Gu0

⁄= =

G ′nuGu0

⁄

PSDnA

1( ) Nin⁄ Gu0

⁄ 1 Nin⁄

Gu0

G ′nu

Nin

N0

PSDnA

1( ) Nin

NNF NFlin<

137


the noise power gain will become larger than the signal power gain, yielding again an increase

of the NNF.

Hence, with the knowledge of the signal power gain , the noise power gain of the

underlying noiseless system , the linear noise figure and the power gain of the

underlying linear system (needed to obtain ), it is possible to predict the NICE

noise figure for a certain range of input power spectral densities. This range of input power

spectral densities is the range where the noise and signal power gain do not vary much, i.e.

where the total input noise power stays about 10 dB below the 1 dB compression point.

In the zone where the NICE system exhibits hard compression (above the 1 dB compression

point), simulations show that a local minimum in the NNF curve as a function of the input

signal amplitude still exists. However, this local minimum is shifting towards higher input

amplitude values, as the input noise PSD increases. Furthermore, the whole NNF curve shifts

upwards with increasing . For a given , the NNF (in dB) will eventually linearly

increase as function of the (high) input signal power (in dBm).

3.3.8 Experimental resultsAs for the linear case (see chapter 1), the most straightforward approach to measure the NICE

noise figure as a function of the input signal amplitude and the input noise power spectral

density , would be to measure simply the signal-to-noise ratio at the input and at the output

of the system, for a range of values for and . The frequency dependent behavior imposes

this measurement to be repeated for a set of frequencies, covering the bandwidth of the

considered system. However, it is very difficult to measure signal-to-noise ratios when the

input noise power spectral density is very small ( ). Therefore, another method has to

be used. For total input excitation power levels below the 1 dB compression point, it is

possible to use the formula that describes the variation of the NICE noise figure with the input

noise power spectral density (3-39). Hence, with the knowledge of the signal power gain ,

the noise power gain of the underlying noiseless system , the linear noise figure

and the power gain of the underlying linear system (needed to obtain ), it is

possible to predict the NICE noise figure in the linear and the weak compression region.

Gu0

G ′nuNFlin

k12 PSDnA

1( )

Pnu

Nin Nin

A

Nin

A Nin

Nin N0≈

Gu0

G ′nuNFlin

k12 PSDnA

1( )

138


The values for and can be determined using a classical noise figure measurement

with the Y-factor method (see section 2.4). Since and are constant functions of the

input noise power spectral density, they can be measured at any . However, using a large

input noise PSD has the advantage that the ratio of the output noise PSD to the input noise

PSD, i.e. , since . Furthermore, using a large input

noise PSD makes the measurement of and easier, than is the case for .

A Sonoma 330 Amplifier, with a frequency range from 20 kHz to 2 GHz was excited with a

CW signal at 1400 MHz. The classical noise figure measurement using the Y-factor method

yields the following results at 1400 MHz: = 6.3 and = 104. Hence, according to (3-

38), . The excitation signal of the amplifier consists of a CW signal at

1400 MHz with superimposed additive white thermal noise, having a power spectral density

dBm/Hz. The apparent linear noise figure for this input noise power spectral

density is 1.0008 (according to (3-48)), and equals 0.088.

Figure 3-25 shows the measurement setup. The Spectrum Analyzer (SA) is configured in zero-

span mode with a resolution bandwidth of 2 MHz. It operates in its frequency selective power

meter mode with a measurement bandwidth of 2 MHz. This Spectrum Analyzer mode avoids

any transient effects in the measured power which occur in the frequency sweep mode. A

load can be put at the input of the Spectrum Analyzer in order to measure the extra noise power

that the SA itself adds to the noise power that has to be measured. This measurement needs to

FIGURE 3-25. Measurement setup.

NFlin k12

Gu0G ′nu

Nin

GnuG ′nu

≈ GnuG ′nu

PSDnA

1( ) Nin⁄+=

Nin PSDny

1( ) Nin N0≈

NFlin k12

PSDnA

1( ) 551N0= u t( )

Nin 136–=

PSDnA

1( ) Nin⁄

DUT

+

50 Ω

Sonoma330

HP8565ESpectrum Analyzer

HP8648BSignal Generator

HP346BNoise Source

HP83006ASystem Amplifier

50Ω

139


be performed for every setting of the SA and is to be subtracted from the measured device

input and output power. The mismatches of the Spectrum Analyzer, generator and the input

and output of the DUT also have to be compensated for. Therefore, these mismatches are

measured, using a calibrated HP8753 Network Analyzer, and the obtained reflection factors

are used to calculate the available power gain [33] for both signal and

noise, with

(3-57)

the available power at the output of the DUT, and

(3-58)

the available power from the source. , , and are the reflection factors of

respectively the input and output of the DUT, the Spectrum Analyzer and the signal source

[33]. and are respectively the input and output reflection parameters of the DUT.

and are respectively the measured output and input powers, already corrected

for the bias of the Spectrum Analyzer.

FIGURE 3-26. Noise and signal power gain and , as a function of the input signal peak amplitude , at 1400 MHz.

Gav Pavn Pavs⁄=

Pavn1 S22ΓSA– 2 1 ΓsΓ in– 2

1 ΓSA2–( ) 1 S11Γ s– 2 1 Γo

2–( )-------------------------------------------------------------------------------------Pout

meas=

Pavs1 ΓsΓSA– 2

1 Γs2–( ) 1 ΓSA

2–( )-------------------------------------------------------Pin

meas=

Γ in Γo ΓSA Γs

S11 S22

Poutmeas Pin

meas

0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

50

60

70

80


Ava

ilabl

e G

ain

[Lin

ear]

Gsignal

Gnoise

100

110

90

80

70

60

50

40

30

Sig

nal &

Noi

se P

ower

Gai

n

Gu0G ′nu

G ′nuGu0A

140


The compensated measured curves for the noise and signal power gain are shown in Figure 3-

26. These curves are in qualitative agreement with the signal and noise gain curves in Figure 3-

18 and Figure 3-19. When the CW signal is very small, the signal and noise power gain are

equal. When the amplifier goes into saturation, the noise gain decreases faster than the signal

gain.

Figure 3-27 shows the measured NICE noise figure. Clearly a range of powers exists where

> . Hence, the signal-to-noise ratio at the output of the amplifier will be better

than the signal-to-noise ratio at the input of the amplifier. Using the measured curves for the

signal and noise power gain, the NICE noise figure can also be calculated for other input noise

power spectral densities using (3-39). Figure 3-28 shows the calculated NICE noise figure for

the measured Sonoma 330 amplifier.

FIGURE 3-27. Variation of the NICE noise figure as a function of the input CW peak amplitude , for dBm/Hz.

FIGURE 3-28. NICE noise figure as function of the input signal peak amplitude and the input noise power spectral density at 1400 MHz.

0 0.1 0.2 0.3 0.4−2

−1.5

−1

−0.5

0

0.5

1


SN

Ri/S

NR

o [dB

]N

ICE

Noi

se F

igur

e [d

B]

A Nin 136–=

SNRout SNRin

00.1

0.20.3−160

−140

−4

0

4

8

12

16

Input CW signal A [V]Nin

[dBm/Hz]

SN

Ri/S

NR

o [dB

]N

ICE

Noi

se F

igur

e [d

B]

Input CW signal amplitude A [V]Input noise PSD [dBm/Hz]

ANin

141


At the experimentally obtained level of dBm/Hz, the noise added by the DUT is

small compared to the amplified input noise. For small values of , and large (but still

below the 1 dB compression point), = , due to the linearity of the device. For

larger values of , the nonlinearity of the DUT becomes significant, and the NICE noise

figure will decrease for moderate compression levels, because the noise gain is smaller than

the signal gain. For , there is only a deterioration of the signal-to-noise ratio with

increasing amplitude of the sine wave. The zone where the signal-to-noise ratio was enhanced,

has completely disappeared, and the behavior is equivalent to the one described in section

3.3.3.

Note that in order to have a complete noise description of the amplifier, these NICE noise

figure surfaces have to be measured for every frequency of interest.

Nin 136–=

A Nin

SNRin SNRout

A

Nin N0≈

142

Variation of the noise power gain, as a function of a general periodic or ergodic input signal

3.4 Variation of the noise power gain, as a function of a general periodic or ergodic input signal

3.4.1 IntroductionIn the previous sections, the variation of the NICE noise figure was studied, when the

excitation signal of a noisy NICE system consists of the superposition of band-limited,

Gaussian noise and a CW signal . The effects of the signal

and noise on the signal and noise power gain were studied. However, practical

amplifiers are mostly used to amplify modulated signals, or even signals whose spectrum is

continuous.

This section investigates the effects on the noise power gain of a noisy NICE system, when its

excitation signal consists of white, Gaussian noise , superimposed on a general

waveform that is zero-mean, and that can be

1. a periodic signal

2. an ergodic signal

3. the product or the sum of a periodic and an ergodic signal.

These restrictions are imposed on the waveform , to be able to use (1-2) in the

calculations. Equation (1-2) states that if is a periodic signal, and is ergodic noise,

(3-59)

If on the other hand, is an ergodic signal, that is uncorrelated with , the proof of (3-

59) is trivial.

It is intuitively clear that both signal and noise power gain will depend on the properties of

. The model used for the noisy NICE system will be again the modified Wiener-

Hammerstein model, where the static nonlinearity is described using a third degree polynomial

model, as discussed in section 3.1.2.

u t( )

nu t( ) u0 t( ) A 2πf0t( )cos= u0 t( )

nu t( )

u t( ) nu t( )

u0 t( )

u0 t( )

u0 t( ) nu t( )

α β, N∈∀ :E u0α t( )nu

β t τ+( )

E u0α t( )

E nuβ t τ+( )

⋅=

u0 t( ) nu t( )

u0 t( )

143


3.4.2 Determining the output power spectral densityThe output signal of the noisy NICE system is:

(3-60)

or,

(3-61)

As in sections 2.3 and 3.2.1, the output power spectral density will be calculated by taking the

Fourier transform of the auto-correlation of the system output (see (2-5)). The noise

added by the system itself is assumed to be uncorrelated with the input signal and input noise.

After a few calculations (see Appendix 3.G), the autocorrelation can be written as:

(3-62)

Since and , (3-62) can be rewritten as:

(3-63)

y t( )

y t( ) k1 u0 t( ) nu t( )+( ) k3 u0 t( ) nu t( )+( )3 nA t( )+ +=

y t( ) k1nu t( ) k1u0 t( ) k3nu3 t( ) k3u0

3 t( ) 3k3u02 t( )nu t( ) 3k3nu

2 t( )u0 t( ) nA t( )+ + + + + +=

y t( )

Ryy τ( )

Ryy τ( ) Ru0u0τ( ) k1 3k3Rnunu

0( )+( )2⋅=

Rnunuτ( ) k1 3k3Rnunu

0( )+( ) k1 3k3 Rnunu0( ) 2Ru0u0

0( )+( )+( )⋅+

9k32R

u02u0

2 τ( )Rnunuτ( ) 6k3

2Rnunu

3τ( ) 18k3

2Ru0u0τ( )Rnunu

2 τ( )+ + +

k3 k1 3k3Rnunu0( )+( ) R

u0u03 τ( ) R

u0u03 τ–( )+

k32R

u03u0

3 τ( )+ +

RnAnAτ( )+

Rnunu0( ) Z0Pnu

= Ru0u00( ) Z0Pu0

=

Ryy τ( ) Ru0u0τ( ) k1 3k3Z0Pnu

+( )2 ⋅=

Rnunuτ( ) k1 3k3Z0Pnu

+( ) k1 3k3Z0 Pnu2Pu0

+( )+( )⋅+

Rnunuτ( ) 9k3

2Ru0

2u02 τ( ) 6k3

2Rnunu

2τ( ) 18k3

2Ru0u0τ( )Rnunu

τ( )+ + ⋅+

k3 k1 3k3Z0Pnu+( ) R

u0u03 τ( ) R

u0u03 τ–( )+

k32R

u03u0

3 τ( ) + +

RnAnAτ( ) +

144


The first term in (3-63) shows that the auto-correlation of the input signal , and hence the

input signal’s power spectral density, is amplified with a constant term that depends on the

total noise power. The second term shows a scaling of the autocorrelation of the input noise

, and hence of the input noise power spectral density. This scaling of the input noise is

dependent on both the total noise power and the total signal power. The third line in (3-63)

contains the autocorrelation of the noise , multiplied by a factor that is dependent on

the properties of the signal and the noise. Hence, in the spectral domain, this means that the

third line of (3-63) will yield extra noise contributions, since it contains respectively,

• : the noise power spectral density convolved with

the spectrum of . This yields extra noise contributions, in the order

of magnitude of . Since , its time average

, and due to the properties of the auto-correlation function

which states that , with

, (see Appendix 3.H)

(3-64)

where . Hence, the first term in the third line

of (3-63) will add an extra correction on the scaling of , i.e.

, on top of the extra noise that will be generated.

• : the noise power spectral density, convolved three times

with itself, yielding extra noise, in the order of magnitude of .

• : the noise power spectral density, convolved with

itself and the spectrum of , yielding extra noise, in the order of

magnitude of .

The fourth line in (3-63) contains only signal contributions. These signal contributions are

quite annoying, because they are cross-correlations of the square and the third power of the

input signal , and thus dependent on the properties of this input signal. Hence, not much

u0 t( )

nu t( )

Rnunuτ( )

9k32R

u02u0

2 τ( )Rnunuτ( )

u02 t( )

PnuPu0

2 u02 t( ) 0≥

E u02 t( )

0>

Rαα τ( ) E α t( ) ( )2

Rββ τ( )+=

β t( ) α t( ) E α t( ) –=

Ru0

2u02 τ( ) E u0

2 t( )

2

Ru u

τ( )+ Z0Pu0( )2 R

u uτ( )+= =) ) ) )

u t( ) u02 t( ) E u0

2 t( )

–=)

Rnunuτ( )

9k32Z0

2Pu0

2

6k32Rnunu

3τ( )

Pnu

3

18k32Ru0u0

τ( )Rnunu

2 τ( )

u0 t( )

Pu0Pnu

2

u0 t( )

145


can be said about their general behavior without further knowledge of those properties. In

general, they could possibly yield:

• in-band distortion terms [20], [34], (see chapter 4)

• spectra around (with the fundamental frequency, or center

frequency of )

• extra signal power gain corrections, dependent on the total signal power

(see chapter 4)

Note that these terms only contain signal contributions and that they create signal distortions,

but do not amplify or interact with the input noise . The last term in (3-63) is due to the

noise added by the system itself. To illustrate all the terms in (3-63), the above theory will be

applied on a given input signal.

Example 3.3

Consider . This is the special case for that was studied in the

previous sections. Calculations show that in this case,

(3-65)

(3-66)

(Hence, for this input signal is )

(3-67)

Substituting (3-65) to (3-67) into (3-63), yields for the signal:

3f0 f0u0 t( )

Pu0

nu t( )

u0 t( ) A 2πf0t( )cos⋅= u0 t( )

E u02 t( )u0

2 t τ+( ) A4

4------ A4

8------ 4πf0τ( )cos+ Z0Pu0

( )2 A4

8------ 4πf0τ( )cos+= =

E u0 t( )u03 t τ+( )

3A4

8--------- 2πf0τ( )cos 3A2

4---------Ru0u0

τ( )= =

Ru0u0

3 τ( ) Ru0u0

3 τ–( )=

E u03 t( )u0

3 t τ+( ) A6

32------ 6πf0τ( )cos 9 2πf0τ( )cos+( )=

146


(3-68)

which is in agreement with (3-40). Similar calculations can be done for the noise contributions,

yielding:

(3-69)

Further mathematical computations (essentially the same as those done in Appendix 3.B), will

show that (3-69) and (3-41) are in agreement.

Since (3-63) was obtained using the third degree polynomial model, the obtained results are

only valid under the 1 dB compression point. Hence, this means that the total excitation power

must be smaller than dBm, i.e. the 1 dB compression point. In section 3.3.5, it

was also shown that when the input noise power spectral density is smaller than

dBm/Hz, the terms in the noise power gain that are and , can be

neglected as compared to the terms that are . Hence, for dBm/Hz, the

auto-correlation of the output signal (3-63) can be simplified to:

(3-70)

As a result, the output power spectral density is given by:

Ru0u0τ( ) k1 3k3Z0Pnu

+( )2⋅ Ru0u0τ( ) 2k3 k1 3k3Z0Pnu

+( )3A2

4---------⋅ Ru0u0

τ( ) 9k32A4

16------⋅+ +

Ru0u0τ( ) k1 3k3Z0Pnu

34---k3A2+ +

2⋅=

Rnynyτ( ) Rnunu

τ( ) k1 3k3Z0Pnu+( ) k1 3k3Z0 Pnu

A2

Z0------+

+

9k32A4

4------+

⋅=

9k3

2Rnunuτ( )

8------------------------------A4 4πf0τ( )cos 6k3

2Rnunu

3τ( ) 9k3

2Rnunu

2τ( )A2 2πf0τ( )cos+ + +

Pu0Pnu

+ 6–

Nin

114– O Πnu

1 O Πnu

2

O Πnu

0 Nin 114–<

Ryy τ( ) Ru0u0τ( ) k1

2⋅ k1k3 Ru0u0

3 τ( ) Ru0u0

3 τ–( )+ k3

2Ru0

3u03 τ( )+ +=

Rnunuτ( ) k1

2 6k1k3Z0Pu0+

Rnunuτ( ) 9k3

2Ru0

2u02 τ( )⋅+⋅+

RnAnAτ( )+

147


(3-71)

Not much can be told about the signal portion in the output spectrum, because it is too

dependent on the properties of the input signal . For the noise output, however, when

terms in can be neglected as compared to terms in , the expression is very

simple. In this case, it seems that the noise power gain of the underlying noiseless NICE

system, is only a function of the total input signal power. Thus, if , then

(3-72)

At the 1 dB compression point, the relative error made by approximating the noise power gain

of the underlying noiseless system by using (3-72), is about 3.02 %, while 3 dB below the 1 dB

compression point, this relative error is about 0.55 %. Note that the relative errors are

calculated for being a sine wave. Hence, up to the 1 dB compression point, (3-72) will

be a good approximation of the noise power gain of the underlying noiseless system.

According to (3-72), the noise power gain is an expression that is in first approximation only

dependent of the total power of the input signal . The expression for the noise power gain

consists of the power gain of the underlying linear system , corrected with an extra term,

which varies linearly with the total power of the input signal . Hence, the noise power

gain of the underlying noiseless system is a linear function of the total power of the input

signal. Due to the generality of the formulae, the noise power gain will be the same for any

type of input signal . Hence, by measuring the noise power gain as a function of the total

power in a chosen waveform , the noise behavior can be predicted for every other input

signal, having the same total power as .

PSDy2( ) f( ) PSDu0

2( ) f( ) k12⋅

k1k3Z0

----------- ℑ Ru0u0

3 τ( ) Ru0u0

3 τ–( )+ k3

2

Z0------ℑ R

u03u0

3 τ( )

+ +=

Nin2

-------- k12 6k1k3Z0Pu0

+ ⋅

9k32

Z0-------- ℑ Rnunu

τ( )Ru0

2u02 τ( )

P+ SDnA

2( ) f( )+ +

u0 t( )

O Πu0

2 O Πu0

1

G ′nu

O Πu0

2 O Πu0

1 «

G ′nuk1

2 6k1k3Z0Pu0+=

u0 t( )

u0 t( )

k12

u0 t( )

Pu0

u0 t( )

u0 t( )

u0 t( )

148


3.4.3 Experimental resultsIn order to validate the theory, the noise power gain of a Sonoma 330 Amplifier is measured

for four different input signals ( , , and ) superimposed on white,

thermal noise, which has a power spectral density of . The measurement setup

used for this experiment is the same as described in section 3.3.8. The four different input

signals are phase modulated signals with a carrier frequency of 1400 MHz and a maximal

phase deviation of 40 radians. The details for each signal are the following:

• for , the carrier is modulated with a saw-tooth shaped signal, that

has a fundamental frequency of 15 kHz.

• for , the carrier is modulated with a square wave, that has a

fundamental frequency of 15 kHz.

• for , the carrier is modulated with noise.

• for , the carrier is not modulated.

Figure 3-29 shows the measured power spectra of the four input signals in order to show the

difference between the input signals.

The noise gain that is experimentally obtained through the measurements is again instead

of . But just as in section 3.3.8, the contribution of can be neglected, since

is quite large. Hence, the measured noise power gain is a good estimate of . The same

FIGURE 3-29. Power spectra of the four input signals.

u1 t( ) u2 t( ) u3 t( ) u4 t( )

136dBm Hz⁄–

u1 t( )

u2 t( )

u3 t( )

u4 t( )

1398 1400 1402−100

−50

0

frequency [MHz]1398 1400 1402

−100

−50

0

frequency [MHz]

1398 1400 1402−100

−50

0

frequency [MHz]1398 1400 1402

−100

−50

0

frequency [MHz]

u1

u3 u4

u2

Pow

er s

pect

rum

[dBm

]Po

wer

spe

ctru

m [d

Bm]

Gnu

G ′nuPSDnA

1( ) Nin⁄

Nin G ′nu

149


measurement setup as shown in Figure 3-25 was used, and the available noise power gain was

calculated and shown in Figure 3-30.

In agreement with the previous theory, for small input powers (hence in weak compression),

the behavior of the noise gain is identical for the different input signals, and obeys a linear law.

For extremely small input powers however (see the beginning of the curves near W),

this linear law is no longer obeyed, and the noise power gain seems constant (see Figure 3-31).

When dealing with a logarithmic input power scale (in dBm), this would be normal. However,

note that the scale for the input signal power is a linear one (in Watts). A possible explanation

is that for µW, the behavior of the considered amplifier is extremely linear (at least

more than as can be described by a third degree polynomial model). Note that a similar

behavior of this amplifier was observed in Figure 3-27, when determining the NICE noise

figure as a function of the input CW amplitude . The curve in Figure 3-27 seemed to be more

or less flat from 0 V to 0.05 V, corresponding to (for a single tone) a total input power going

from 0 W to 25 µW. On the other hand, when the input power becomes larger than 0.15 mW,

the curves start diverging, and seem to exhibit a more quadratic behavior, as could be expected

since the term in becomes important.

Hence it is possible, using an arbitrary signal, to predict the noise power gain of a DUT in

weak compression. Even a very simple signal such as a CW signal can be used to predict .

Suppose for example that the noise power gain was only measured using the input signal

, i.e. the CW signal with a frequency of 1400 MHz. In that case, according to (3-72), a

FIGURE 3-30. Measured available noise power gain for the four input signals.

0 0.2 0.4 0.6 0.8 1 1.2−3

20

40

60

80

100

Total input power [W]

Ava

ilabl

e N

oise

Gai

n [li

near

]

s1(t)

s2(t)

s3(t)

s4(t)

3–×10

u3 t( )u4 t( )

u2 t( )u1 t( )

Pu00=

Pu025<

A

O Πu0

2

G ′nu

u4 t( )

150


first estimate of the system parameters and can be made using linear regression

techniques [35] on the first (linear) part of the noise power gain curve. Applying these

techniques yields that the linear part of the measured noise power gain curve can be modelled

as:

(3-73)

Comparing (3-73) to (3-72), one obtains that and .

Figure 3-31 is a detail of Figure 3-30 and shows again that the four curves behave identically

and linearly when the DUT goes into weak compression.

Figure 3-32 shows the difference between the measured noise power gain curves and the linear

model. As mentioned before, the linear regression curve approximation is invalid for input

signal powers near 0 W. In the region where the curves can be approximated, using the linear

regression curve, the difference between the measured and the modelled noise power gain is

FIGURE 3-31. Detail of the linear part of the noise power gain, and linear regression curve.

k1 k3

G ′nu116.2 4.3225 5×10 Pu0

–=

k1 10.8≈ k3 134– V 2–≈

0 0.5 1 1.5−4

60

70

80

90

100

110


Ava

ilabl

e N

oise

Gai

n [li

near

]

s1(t)

s2(t)

s3(t)

s4(t)

u3 t( )u4 t( )

u2 t( )u1 t( )

4–×10

Linear regression curve

151


very small: maximum 3, when the gain varies between 60 and 100, yielding a relative error

between 3% and 5%).

FIGURE 3-32. Difference between the measured noise power gain curves and the linear model.

0 0.5 1 1.5−4

−5

−4

−3

−2

−1

0

1

2

3

4

5


Diff

eren

ce [l

inea

r]

s1(t) − model

s2(t) − model

s3(t) − model

s4(t) − model

4–×10

u3 t( )u4 t( )

u2 t( )u1 t( )

152

Conclusion

3.5 ConclusionThe linear noise figure is unable to describe adequately the noise behavior of a NICE system,

since it is by assumption a quantity that is the same for each input signal power. As shown in

previous chapters, the output of a NICE system is dependent on the properties of its input

signal, and essentially on the power of this input signal. Hence, the signal-to-noise ratio

deterioration of a NICE system is dependent on the input signal power. Therefore, the

definition of the noise figure is extended towards the NICE noise figure, that is in fact an input

signal power and input noise power dependent noise figure.

In this chapter, the variation of the NICE noise figure is studied as a function of the input noise

power spectral density and the power of the input signal. Using a Wiener-Hammerstein model

with a third degree static nonlinearity, first the effect of the power of a CW input signal on the

NICE noise figure was studied for a standard input noise power spectral density, i.e.

dBm/Hz. Under these circumstances, the NICE noise figure, as a function of the

input signal power, becomes larger or smaller than the noise figure of the underlying linear

system, depending upon the value of this linear noise figure. If , the NICE noise

figure will be larger than the linear noise figure, for increasing input signal power. For

, a decrease in NICE noise figure occurs. In the special case of a noiseless NICE

system, where , the nonlinearity is thus actually able to improve the signal-to-noise

ratio.

Next, the NICE noise figure was determined as a function of both the input signal power and

the input noise power spectral density. Up to 10 dB under the 1 dB compression point, an

increase in input noise power spectral density can be translated into a similar behavior as for

the standard input noise power spectral density, but with a decreasing linear noise figure.

Hence, for large input noise power spectral densities, the apparent linear noise figure tends

towards 1, implicating that the NICE noise figure will be better for the NICE system going into

weak compression than for the underlying linear system. As a result, noisy NICE systems are

able to improve the signal-to-noise ratio under certain conditions. Because the third degree

polynomial model was unable to describe the system’s behavior for input signal and noise

powers, above the 1 dB compression point, simulations were done with a Wiener-

Nin 174–=

NFlin 2>

NFlin 2<

NFlin 1=

153


Hammerstein model where the static nonlinearity is described using an arctan function.

Simulations showed that when the input signal power becomes very large, the NICE noise

figure tends towards infinity. For an increase in noise power spectral density, the NNF curve as

a function of the input signal power shifts upwards.

Finally, the output spectrum of the noisy NICE system with the third degree static nonlinearity

was studied when the input signal is an arbitrary periodic or ergodic waveform. Not much

could be told about the signal behavior, because this behavior was too much dependent on the

signal’s properties. The noise power gain is, however, mainly a linear function of the total

power of the input signal waveform, and is independent of its other properties. Hence, the

noise power gain will, in a first order approximation, be the same for all input waveforms that

have the same total power. This property can be used to predict the noise power gain for any

input signal by measuring the response for only one signal.

154

Appendices

3.6 Appendices

Appendix 3.A : Calculation of the 1 dB compression point for the third degree polynomial modelLet be the input signal of the third degree polynomial model described

by . The output signal is then given by:

(3-74)

The 1 dB compression point is defined as that input power where the linear output power at the

fundamental frequency is exactly 1 dB higher than the actual output of the NICE system.

Hence, it is the solution of the equation:

(3-75)

i.e.,

(3-76)

or, in terms of power:

(3-77)

For parameters , , the 1 dB compression point is about 0.24 mW, or

dBm.

u t( ) A 2πf0t( )cos⋅=

y t( ) k1u t( )= k3u3 t( )+

y t( ) k1A 3k3A3

4------+

2πf0t( )cosk3A3

4------------ 6πf0t( )cos+=

10k1A( )2

2-----------------

log 10k1A 3k3

A3

4------+

2

2---------------------------------------

log– 1=

A2

2------

2k1 101–

20------

1–

3k3-----------------------------------=

P1dB

2k1 101–

20------

1–

3k3Z0-----------------------------------=

k1 10= k3 60V 2––=

6–

155


Appendix 3.B : Calculation of the output power spectral density for an input consisting of a single tone and thermal noiseGiven:

(3-78)

one has to calculate

(3-79)

and .

Since the Fourier transform is a linear operator, one can take the Fourier transform of each

term of (3-79), in order to obtain . Hence, the auto-correlation and corresponding

spectral contributions are:

First and second term:

(3-80)

ς1 k134---k3A2+

A 2πf0t( )cos=

ς214---k3A3 2π3f0t( )cos=

ς3 k132---k3A2+

nu t( )=

ς432---k3A2nu t( ) 2π2f0t( )cos=

ς5 3k3Anu2 t( ) 2πf0t( ) cos=

ς6 k3nu3 t( )=

Rηη τ( ) Rςiςiτ( )

i 1=

6

∑ Rςiςjτ( ) Rςiςj

τ–( )+( )

j i 1+=

6

∑i 1=

5

∑+=

PSDηη2( ) f( ) 1

Z0------ ℑ Rηη τ( ) =

PSDηη2( ) f( )

Rς1ς1τ( ) 1

2--- k1

34---k3A2+

2A2 2πf0τ( )cos=

Rς2ς2τ( ) 1

2--- 1

4---k3A3

22π3f0τ( ) cos=

156

Appendices

(3-81)

Third term:

(3-82)

Fourth term:

(3-83)

(3-84)

Since the Fourier transform of a product equals the convolution of the Fourier transforms.

Figure 3-33 shows the convolution of the noise spectrum with a cosine wave with frequency

. In the upper right corner of Figure 3-33, the two shifted noise spectra are shown. In

the frequency interval , the two shifted noise spectra overlap, indicating that

they have to be added. Eventually, all spectral components above frequency will be cut off

due to the output filter, yielding a spectrum as shown in the lower right corner of Figure 3-33.

PSDς1ς1

2( ) f( )k1

34---k3A2+

2A2

2Z0------------------------------------------1

2--- δ f f0–( ) δ f f0+( )+( )=

PSDς2ς2

2( ) f( )

14---k3A3

2

2Z0-----------------------1

2--- δ f 3f0–( ) δ f 3f0+( )+( ) =

Rς3ς3τ( ) k1

32---k3A2+

2Rnunu

τ( )=

PSDς3ς3

2( ) f( ) k132---k3A2+

2N0

2------ =

Rς4ς4τ( ) 3

2---k3A2

2Rnunu

τ( )12--- 2π2f0τ( )cos=

PSDς4ς4

2( ) f( )

32---k3A2

2

2-----------------------

N02

------*12--- δ f 2f0–( ) δ f 2f0+( )+( )=

2f0 B<

2f0 B– B 2f0–,[ ]

B

157


Hence, (3-84) can be rewritten as:

(3-85)

This means that the nonlinearity will modulate the input noise (which has a bandwidth ) with

a continuous wave signal (at frequency ), resulting in a frequency translation of the noise.

Because , the translated part of the noise band that falls in the negative frequency

region will fold back into the frequency band from DC to .

If on the other hand , hence , the situation will be as shown in Figure 3-34. In

this case, the frequency band will still undergo a frequency translation, but this time, no part of

the shifted noise band will remain in the negative frequency region.

FIGURE 3-33. Convolution of the noise spectrum with a sine wave with .

f

N02

------

BB–

f

12---

2f0

*12---

2f0–

f

N04

------

B 2f0–

f

N04

------

N02

------

B 2f0+2f0 B–B– 2f0–

BB– 2f0 B– B 2f0–

add +output filter

2f0 B<

PSDς4ς4

2( ) f( )

32---k3A2

2

2-----------------------

N02

------ f [0 B 2f0–, [∈⇔

32---k3A2

2

2-----------------------

N04

------ f ]B 2f0– B, ]∈⇔

=

B

2f02f0 B<

B 2f0–

f0 B 2⁄> 2f0 B>

158

Appendices

In this case, (3-84) can be rewritten as:

(3-86)

Fifth term:

(3-87)

In order to determine , (2-12) has to be used. (2-12) states that

, for zero-mean jointly

Gaussian random variables. After similar calculation as described in Appendix 2.C, one will

obtain that:

(3-88)

FIGURE 3-34. Convolution of the noise spectrum with a sine wave with

f

N02

------

BB–

f

12---

2f0

*12---

2f0–

f

N04

------

B– 2f0+ B 2f0+2f0 B–B– 2f0–

f

N04

------

B– 2f0+ B2f0 B–B–

output filter

2f0 B>

PSDς4ς4

2( ) f( ) 0 f 0 2f0 B–,[ ] ∈⇔=

PSDς4ς4

2( ) f( )

32---k3A2

2

2-----------------------

N04

------ f 2f0 B– B,[ ]∈⇔=

Rς5ς5τ( ) 3k3A( )2E nu

2 t( )nu2 t τ+( )

1

2--- 2πf0τ( )cos=

E n2 t( )n2 t τ+( )

E n τ1( ) … n τ2M( )⋅ ⋅ ΣΠ E n τ i( ) n τ j( )⋅ = n τ1( ) … n τ2M( ), ,

Rς5ς5τ( )

3k3A( )2

2-------------------- 2Rnunu

2 τ( ) Rnunu

2 0( )+ 2πf0τ( )cos=

159


(3-89)

Note that has already been calculated in Appendix 2.D.

Figure 3-35 shows the convolution of with a cosine wave with frequency . In the

frequency interval , the two shifted noise spectra will add, yielding a

spectrum as shown in the lower right corner of Figure 3-35.

The output filter however, will cut off frequency components higher than frequency , and

since , frequency will fall in the interval . Hence only the flat part of the

spectrum and the steep sloped part are of importance. Calculations show that:

(3-90)

FIGURE 3-35. Convolution of with a sine wave.

PSDς5ς5

2( ) f( )3k3A( )2

2--------------------Z0N0

2B212--- δ f f0–( ) δ f f0+( )+( )=

3k3A( )2Z0PSDnu

2( ) f( )*PSDnu

2( ) f( )*12--- δ f f0–( ) δ f f0+( )+( )+

PSDnu

2( ) f( )*PSDnu

2( ) f( ) PSD*2( ) f( )=

PSD*2( ) f( ) f0

f0 2B– 2B f0–,[ ]

N02

------

22B f–( )

2B2B–

N02

------

22B f+( )

ff

12---

f0

12---

f0–

*

ff0f0–

2B f0–f0 2B–

f0 2B+f0 2B–– ff0f0–

2B f0–f0 2B–

f0 2B+f0 2B––

add

PSD*2( ) f( )

B

f0 B< B f0 2B f0–,[ ]

PSDς5ς5

1( ) f( )3k3A( )2

2--------------------Z0N0

2B2δ f f0–( )3k3A( )2Z0

N02

------

22 2B f0–( ) f 0 f0,[ ]∈⇔

3k3A( )2Z0N02

------

22 2B f–( ) f f0 B,[ ]∈⇔

+=

160

Appendices

Sixth term: the autocorrelation and power spectral density of this term have already been

calculated in section 2.3.

(3-91)

(3-92)

Next, the crosscorrelations have to be calculated:

(3-93)

because a sine wave and its harmonically related tone (except the fundamental, of course) are

orthogonal over 1 period, i.e. the integral over one period of their product is zero. (3-93) also

implies that .

(3-94)

since input signal and noise are uncorrelated. (Hence, )

(3-95)

and this expression is also zero, again due to the fact that harmonics of a sine wave are

orthogonal over 1 period, and that the noise is zero-mean . (Thus

)

(3-96)

Rς6ς6τ( ) k3

2 9Rnn2 0( )Rnn τ( ) 6Rnn

3 τ( )+( )=

PSDς6ς6

1( ) f( )k3

2N03Z0

2

2------------------ 27B2 3f2–( )=

Rς1ς2τ( )

k34----- k1

34---k3A2+

A4E 2πf0t( ) 3π2f0 t τ+( )( )coscos 0= =

Rς1ς2τ–( ) 0=

Rς1ς3τ( ) k1

34---k3A2+

A k1

32---k3A2+

E 2πf0t( )nu t τ+( )cos 0= =

Rς1ς3τ–( ) 0=

Rς1ς4τ( ) 3

2---k3 k1

34---k3A2+

A3E 2πf0t( )nu t τ+( ) 2π2f0 t τ+( )( )coscos =

32---k3 k1

34---k3A2+

A3E 2πf0t( ) 2π2f0 t τ+( )( )coscos E nu t τ+( ) =

Et nu t τ+( ) 0=

Rς1ς4τ–( ) 0=

Rς1ς5τ( ) 3k3 k1

34---k3A2+

A2E nu2 t( )

E 2πf0t( )cos 2πf0 t τ+( )( )cos =

3k3 k134---k3A2+

A2Rnunu0( )1

2--- 2πf0τ( ) cos=

161


and since , .

(3-97)

(3-98)

because a sine wave has zero-mean over 1 period, and the odd moment of a Gaussian random

variable is also zero [7]. (Thus )

(3-99)

because input signal and noise are zero-mean. (Thus )

(3-100)

because the input noise is zero-mean, and harmonics of a sine wave are orthogonal over 1

period. (Also )

(3-101)

because harmonics of a sine wave are orthogonal over 1 period. (Also )

(3-102)

because and . (Thus )

(3-103)

because . (Thus )

α( )cos α–( )cos= Rς1ς5τ–( ) Rς1ς5

τ( )=

PSDς1ς5

2( ) f( ) 32---k3 k1

34---k3A2+

A2N0B1

2--- δ f f0–( ) δ f f0+( )+( )=

Rς1ς6τ( ) k1

34---k3A2+

Ak3E 2πf0t( )cos E nu3 t τ+( )

0= =

Rς1ς6τ–( ) 0=

Rς2ς3τ( )

k34----- k1

32---k3A2+

A3E 2π3f0t( )cos E nu t τ+( ) 0= =

Rς2ς3τ–( ) 0=

Rς2ς4τ( )

3k32A6

8---------------E 2π3f0t( )cos 2π2f0 t τ+( )( )cos E nu t τ+( ) 0= =

Rς2ς4τ–( ) 0=

Rς2ς5τ( )

3k32A4

4---------------E 2π3f0t( )cos 2πf0 t τ+( )( )cos E nu

2 t τ+( )

0= =

Rς2ς5τ–( ) 0=

Rς2ς6τ( )

k32A3

4------------E 2π3f0t( )cos E nu

3 t τ+( )

0= =

E 2π3f0t( )cos 0= E nu3 t τ+( )

0= Rς2ς6τ–( ) 0=

Rς3ς4τ( ) k1

32---k3A2+

32---k3A2E nu t( )nu t τ+( ) E 2π2f0 t τ+( )( )cos 0= =

E 2π2f0 t τ+( )( )cos 0= Rς3ς4τ–( ) 0=

162

Appendices

(3-104)

because and [7]. (Thus

)

(3-105)

But was determined in Appendix 2.C, hence:

(3-106)

Thus

(3-107)

(3-108)

because a sine wave is not correlated with its harmonic, and [7].

(Thus )

(3-109)

because .

(3-110)

because and [7].

Rς3ς5τ( ) k1

32---k3A2+

3k3AE nu t( )nu

2 t τ+( )

E 2πf0 t τ+( )( )cos 0= =

E 2πf0 t τ+( )( )cos 0= E nu t( )nu2 t τ+( )

0=

Rς3ς5τ–( ) 0=

Rς3ς6τ( ) k1

32---k3A2+

k3E nu t( )nu

3 t τ+( )

=


Rς3ς6τ( ) k1

32---k3A2+

k33Rnn 0( )Rnn τ( )=

Rς3ς6τ–( ) Rς3ς6

τ( )=

PSDς3ς6

2( ) f( ) 32--- k1

32---k3A2+

k3Z0N02B=

Rς4ς5τ( ) 9

2---k3

2A3E nu t( )nu2 t τ+( )

E 2π2f0t( )cos 2πf0 t τ+( )( )cos 0= =


0=

Rς4ς5τ–( ) 0=

Rς4ς6τ( ) 3

2---k3

2A2E nu t( )nu3 t τ+( )

E 2π2f0t( )cos 0= =

E 2π2f0t( )cos 0=

Rς5ς6τ( ) 3k3

2AE nu2 t( )nu

3 t τ+( )

E 2π2f0t( )cos 0= =

E 2π2f0t( )cos 0= E nu2 t( )nu

3 t τ+( )

0=

163


Appendix 3.C : Variation of over a small bandwidth

In order for

(3-111)

to be true, the variation of over the frequency interval

must be small. From (3-10), it follows that can be

written as:

(3-112)

where is , is and

is . (With ) Hence, essentially the variation of the

power spectral density of the noise produced by the NICE system will determine if the

approximation (3-111) is valid. For high frequency NICE systems, one can indeed assume that

does not vary significantly over an interval of a few kHz.

Appendix 3.D : Taylor series expansion of an atan function

Definition 3.4

Let , and has a ( )-th derivative ( ) that is

continuous over and differentiable over ] [ then, such that

(3-113)

Expression (3-113) is called the Taylor series expansion of in interval .

PSDny

1( ) f( )

PSDny

1( ) f( ) fdf0 B0 2⁄–

f0 B0 2⁄+

∫ B0 PSDny

1( ) f0( )⋅≈

PSDny

1( ) f( ) f0 B0 2⁄– f0 B0 2⁄+,[ ]

PSDny

1( ) f0 B0 2⁄+( ) PSDny

1( ) f0 B0 2⁄–( )–

PSDnA

1( ) f0 B0 2⁄+( ) PSDnA

1( ) f0 B0 2⁄–( )– 3k32N0

3Z02f0B0

94---k3

2A2N02Z0B0––

PSDnA

1( ) f0 B0 2⁄+( ) PSDnA

1( ) f0 B0 2⁄–( )– O ν0( ) 94---k3

2A2N02Z0B0– O ν0

2( )

3k32N0

3Z02f0B0– O ν0

3( ) ν0 N0 1W( )⁄=

PSDnA

1( )

PSDnA

1( ) f( )

a b, R∈ a b≠ f: ab[ ] R→ n 1+ n N0∈

ab[ ] ab c ]ab[∈∃

f b( ) f a( ) f ′ a( )1!

----------- b a–( ) f″ a( )2!

------------ b a–( )2 …+ + +=

f n 1–( ) a( )n 1–( )!

------------------------ b a–( )n 1– f n( ) c( )n!

---------------- b a–( )n+ +

f x( ) ab[ ]

164

Appendices

Expanding the function in its Taylor series up to degree three, yields:

(3-114)

Appendix 3.E : Probability of creating a zero crossingConsider a sine wave on which zero mean Gaussian noise is superimposed.

For the noise to create an additional zero crossing, the instantaneous noise value at time ,

has to be larger than the absolute value of the sine wave at the same moment , i.e.

, and opposite in sign. If , the probability of creating an

additional zero crossing is given by:

(3-115)

while for , this probability is given by:

(3-116)

FIGURE 3-36. Schematic representation of the probability density function of Gaussian noise and a sine wave.

f x( ) α βx( )atan=

α βx( )atan αβ x 13---αβ3 x3⋅–⋅ …+=

t0nu t0( ) t0A 2πf0t0( )cos A 2πf0t0( )cos 0>

1

2πσnu t( )2

------------------------- e

ξ2–

2σnu t( )2

-----------------

ξd∞–

A 2πf0t0( )cos–

∫

A 2πf0t0( )cos 0<

1

2πσnu t( )2

------------------------- e

ξ2–

2σnu t( )2

-----------------

ξdA– 2πf0t0( )cos

∞∫

165


Where represents the variance of the noise, which can be calculated out of the total

noise power as follows: . Because of the property that the probability

density function of Gaussian noise is an even function, (3-115) and (3-116) can be combined to

one expression. The probability that an extra zero crossing is created, is given by:

(3-117)

Example 3.5

Calculate for and , and for sine waves

with total power equal to 8 dBm, 10 dBm and 20 dBm, the probability that the noise

superimposed on the sine wave generates an extra zero crossing. The bandwidth of the

considered system is 4096 MHz.

dBm/Hz and dBm/Hz correspond to total noise powers of

dBm and dBm respectively. This corresponds then again to noise variances of

and respectively. The total sine wave powers are due to sine waves

with peak amplitudes of respectively 0.79 V, 3.16 V and 10 V. Using (3-117), one obtains the

following results:

FIGURE 3-37. Probability of creating an extra zero crossing for dBm/Hz (left figure) and dBm/Hz (right figure)

σnu t( )2

Pnuσnu t( )

2 Z0Pnu=

1

2πσnu t( )2

------------------------- e

ξ2–

2σnu t( )2

-----------------

ξd∞–

A 2πf0t0( )cos–

∫1

2πσnu t( )2

------------------------- e

ξ2–

2σnu t( )2

-----------------

ξdA 2πf0t0( )cos

∞∫=

Nin 74dBm Hz⁄–= Nin 144dBm Hz⁄–= A ωt( )sin

Pu0

Nin 74–= Nin 144–=

47.9– 22.1

8.15 7–×10 V2 8.15V2

0 45 90 135 1800

0.1

0.2

0.3

0.4

0.5

Argument [deg]

Pro

ba

bili

ty

8 dBm 10 dBm20 dBm

0 45 90 135 180

10-300

10-200

10-100

100

Pro

ba

bili

ty

8 dBm 10 dBm20 dBm

Argument [deg]

Nin 74–=Nin 144–=

166

Appendices

At the zero crossings of the sine wave, the probability of creating an extra zero crossing is

50 %, i.e. total uncertainty. For dBm/Hz, and dBm, the probability at

the top of the sine wave is smaller than the numerical precision of the simulator, i.e. .

Appendix 3.F : Boundaries of the linear regionUsing (3-40) and (3-42), and omitting terms in and with (since the

total signal and noise power are small because considering the linear region), one obtains:

(3-118)

(3-119)

The boundary of the linear region will be there where the second term in (3-118) and (3-119)

will not be negligible any more as compared to . Suppose that this is the case when the ratio

of the second term to the first becomes larger than , hence:

(3-120)

(3-121)

yielding for the example system: dBm and dBm. Hence, If

the total input power of the NICE system becomes larger than about dBm, the system

cannot be modelled as a LTI system any more.

Appendix 3.G : Autocorrelation of the noisy NICE system’s output for a general input waveformBecause the noise added by the system itself is not correlated with the input signal or the

input noise , (2-6) is valid (i.e. ). Because the noiseless

output has six terms (see (3-61)), the autocorrelation has thirty-

Nin 144–= Pu020=

10 324–

O Πu0

n O Πnu

n n 2≥

Gu0k1

2 3k1k3Z0 Pu02Pnu

+( )+=

G ′nuk1

2 6k1k3Z0 Pu0Pnu

+( )+=

k12

10 3–

3k3Z0 Pu02Pnu

+( )

k1----------------------------------------------- 10 3–≥ Pu0

2Pnu+

k110 3–

3k3Z0-----------------≥⇒

6k3Z0 Pu0Pnu

+( )

k1------------------------------------------- 10 3–≥ Pu0

Pnu+

k110 3–

6k3Z0-----------------≥⇒

Pu02Pnu

+ 30–≥ Pu0Pnu

+ 33–≥

33–

u0 t( )

nu t( ) Ryy τ( ) Rηη τ( ) RnAnAτ( )+=

η t( ) y t( ) nA t( )–= Rηη τ( )

167


six terms (that are not all different). Because the input signal and the input noise

are uncorrelated and zero-mean, because of the anti-symmetry property of the autocorrelation

function ( ), and because for Gaussian noise, it was shown that

only the following fourteen terms persist:

(3-122)

Using (2-49), (2-14) and Appendix 2.C, (3-122) can be rewritten as:

u0 t( ) nu t( )


Rαβ τ( ) Rβα τ( )=

k12E nu t( )nu t τ+( ) 2k1k3E nu t( )nu

3 t τ+( )

+

6k1k3Et nu t( )nu t τ+( ) E u02 t τ+( )

k12E u0 t( )u0 t τ+( ) + +

6k1k3E u0 t( )u0 t τ+( ) E nu2 t( )

k1k3 E u0 t( )u03 t τ+( )

E u03 t( )u0 t τ+( )

+

+ +

k32E nu

3 t( )nu3 t τ+( )

6k32E nu

3 t( )nu t τ+( )

E u02 t τ+( )

+ +

9k32E nu

2 t( )nu2 t τ+( )

E u0 t( )u0 t τ+( ) +

3+ k32E nu

2 t( )

E u0 t( )u03 t τ+( )

3+ k32E nu

2 t( )

E u03 t( )u0 t τ+( )

9k32E u0

2 t( )u2 t τ+( )

E nu t( )nu t τ+( ) k32E u0

3 t( )u03 t τ+( )

+ +

168

Appendices

(3-123)

Appendix 3.H : Auto-correlation of a non zero-mean signal

Theorem 3.6

If is a periodic or ergodic signal, and , then

(3-124)

proof

If is an ergodic signal, the proof is trivial.

If is a periodic signal,

(3-125)

Using the definition of , it follows that , or:

k12Rnunu

τ( ) 6k1k3Rnunu0( )Rnunu

τ( )+

6k1k3Rnunuτ( )Ru0u0

0( ) k12Ru0u0

τ( )+ +

6k1k3Ru0u0τ( )Rnunu

0( ) k1k3 Ru0u0

3 τ( ) Ru0u0

3 τ–( )+ + +

k32 9Rnunu

τ( )Rnunu

2 0( ) 6Rnunu

3 τ( )+ 6k3

23Rnunu0( )Rnunu

τ( )Ru0u00( )+ +

9k32 Rnunu

2 0( ) 2Rnunu

2 τ( )+ Ru0u0

τ( ) 3k32Rnunu

0( ) Ru0u0

3 τ( ) Ru0u0

3 τ–( )+ + +

9k32R

u02u0

2 τ( )Rnunuτ( ) k3

2Ru0

3u03 τ( )+ +

α t( ) β t( ) α t( ) E α t( ) –=

Rαα τ( ) E α t( ) ( )2

Rββ τ( )+=

α t( )

α t( )

Rαα τ( ) 1T--- α t( )α t τ+( )

T 2⁄–

T 2⁄∫ dt

T ∞→lim=

β t( ) α t( ) β t( ) E α t( ) +=

169


(3-126)

Due to the linearity property of the integral, (3-126) can be rewritten as:

(3-127)

Again, using the definition of , it follows that ,

hence,

(3-128)

And since , the theorem is proven. G

Rαα τ( ) 1T--- β t( ) E α t( ) +( ) β t τ+( ) E α t τ+( ) +( )

T 2⁄–

T 2⁄∫ dt

T ∞→lim=

1T--- (β t( )β t τ+( ) E α t( ) β t τ+( ) β t( )E α t τ+( ) + +

T 2⁄–

T 2⁄∫T ∞→

lim=

E α t( ) E α t τ+( ) + )dt

Rαα τ( ) E β t( )β t τ+( ) E α t( ) E β t τ+( ) +=

E α t τ+( ) E β t( ) E α t( ) E α t τ+( ) + +

β t( ) E β t( ) E α t( ) E α t( ) – 0= =

Rαα τ( ) E β t( )β t τ+( ) E α t( ) ( )2

+=

E β t( )β t τ+( ) Rββ τ( )=

170

CHAPTER 4

NOISE-LIKE SIGNALS AND NICESYSTEMS

Abstract: In telecommunication systems, most processed signals

are narrow band noise-like signals, due to the stochastic nature of

information itself. Intuitively, it can be assumed that the nonlinear

mechanism creates extra noise using these noise-like input signals.

In the literature, the existence of such terms is quantified by the

Noise Power Ratio. This figure quantifies in-band nonlinear

distortions as function of the power of the noise-like input signal.

In this chapter, the response of NICE systems to these noise-like

input signals will be studied and determined experimentally. The

measurement techniques and theory for two-port devices (such as

amplifiers) will then be extended towards multi-port devices (such

as mixers).

171

Noise-like signals and NICE systems

4.1 IntroductionIn the previous chapters, the input signals were all deterministic signals. This hypothesis

is made in section 1.6.2, and allows to discriminate between the deterministic signal output and

the stochastic noise output. However, Shannon’s information theory [17] states that a

deterministic signal (or at least always the same instance of a modulated signal) does not

contain any information, and hence, there is no point in transmitting or amplifying an

informationless signal in a telecommunication channel. Most signals in telecommunication,

such as GSM, NICAM-728 [18], ADSL [42] are noise-like signals, due to scrambling of the

bits or due to the stochastic content of the information itself [17]. Hence, these signals have

properties that are similar to the properties of noise. Note that, in this case, a different instance

is obtained as a function of the information content, and not as a function of time. It is

intuitively clear that the presence of these noise-like signals at the input of a NICE system will

be responsible for an increased noise presence at the output of the noisy NICE system.

Hence, two noise contributions can be found at the output of the NICE system. The first is due

to the noise that is superimposed on the input signal, and that is processed by the system. This

is the kind of noise that was studied in previous chapters. The second contribution is noise that

is created by the nonlinear mechanism itself, out of the randomness of the input signal. The

spectrum of this noise is uncorrelated with the spectrum of the input signal , i.e.

. The power of both noise contributions has to be compared, in order to

compare their relative importance, and to focus on the most important one. Intuitively, one can

already assume that for narrowband devices (as usually used in telecommunications) the total

input noise power will be quite small, and so will be the effect of this noise on the output

signal. This will be mathematically verified later in this chapter.

Another confirmation of the assumption that NICE systems create output noise out of the

noise-like input signal, is the existence of the “Noise Power Ratio” [44], [45]. This is a well-

known quantity in telecommunications, that gives a measure for the power of the in-band

distortions versus the input signal power applied to the NICE system. These figures show that

if the power of the noise-like input signal tends towards zero, the in-band distortions also tend

u0 t( )

NNL k( ) U k( )

E U k( )NNL k( ) 0=

Pnu

172

Introduction

towards zero. This demonstrates that these in-band distortions are created by the input signal,

due to the nonlinear mechanism itself.

In order to adequately describe the noise-like signals that are fed to the NICE systems, these

input signals need to be modelled first. An analysis of the processing of these modelled signals

by the NICE system will give a better insight in the output noise creation mechanisms.

Another important component used in telecommunications is the mixer. This is a “two-input-

port, one-output-port” device where the output signal ideally consists of the multiplication of

the two input signals. This device is mainly used to shift spectral components from one

frequency to another, and is thus essentially a nonlinear device. In this chapter, the possibility

to extend the obtained results for two-port NICE systems towards multi-port systems such as

mixers will also be investigated. Hence, answers to the following questions are required:

• How does a frequency translating device affect signals that are shifted

towards another location in the frequency spectrum?

• How much in-band distortion is created during this frequency translation?

• What is the effect on the shifted signals if a sinewave affected by phase

noise, instead of a pure sinewave, is used to shift the signals in the

frequency spectrum?

Note that in this chapter, is a noise-like signal, that is the signal of interest, and is

the disturbing time domain noise. can be fully controlled and is stochastic over different

instances, due to the stochastic nature of information contained in the signal. Hence, a different

transmitted symbol will yield a different instance of the signal. , on the other hand, is true

noise that cannot be controlled.

u0 t( ) nu t( )

u0 t( )

nu t( )

173


4.2 Considerations about the output spectrum

Consider that is a noise like signal. Suppose also that the power spectral density of

is flat and equal to , centered around frequency , and has a bandwidth (see

Figure 4-1).

The auto-correlation of this signal is then given by (similar calculations as performed in

Appendix 2.A):

(4-1)

The input noise waveform has the same properties as in the previous chapters, i.e. flat,

Gaussian, band-limited with power spectral density and bandwidth . The

system that will be considered is still the same Wiener-Hammerstein system with the third

degree polynomial, describing the static nonlinearity (see also sections 2.2 and 3.1.2). In order

to find the system’s output spectrum, the auto-correlation of the output signal has

to be determined first. According to (3-63), the auto-correlation of the output signal of the

system is given by:

FIGURE 4-1. Power spectral density spectrum of the considered signal .

u0 t( )

u0 t( ) PSDu0

1( ) f0 B0

f0

PSD [W/Hz] B0

f

PSDu0

1( )

(1)

u0 t( )

Ru0u0τ( ) Z0PSDu0

B0πB0τ( )sin

πB0τ------------------------- 2πf0τ( )cos=

nu t( )

PSDnu

1( ) Nin= B

Ryy τ( ) y t( )

174

Considerations about the output spectrum

(4-2)

In chapter 3, it was shown that if dBm/Hz (for GHz), (4-2) simplifies to

(see also (3-70)):

(4-3)

If has a Gaussian probability density function, similar calculations as those in Appendix

2.B and Appendix 2.C, yield the following results for the auto-correlations:

(4-4)

(4-5)

(4-6)

Substituting (4-4) to (4-6) into (4-3), and knowing that , yields:

Ryy τ( ) Ru0u0τ( ) k1 3k3Z0Pnu

+( )2 ⋅=

k3 k1 3k3Z0Pnu+( ) R

u0u03 τ( ) R

u0u03 τ–( )+

k32R

u03u0

3 τ( ) + +

Rnunuτ( ) k1 3k3Z0Pnu

+( ) k1 3k3Z0 Pnu2Pu0

+( )+( )⋅+

Rnunuτ( ) 9k3

2Ru0

2u02 τ( ) 6k3

2Rnunu

2τ( ) 18k3

2Ru0u0τ( )Rnunu

τ( )+ + ⋅+

RnAnAτ( ) +

Nin 114–< B 4=


2⋅ k1k3 Ru0u0

3 τ( ) Ru0u0

3 τ–( )+ k3

2Ru0

3u03 τ( )+ +=

Rnunuτ( ) k1

2 6k1k3Z0Pu0+

Rnunuτ( ) 9k3

2Ru0

2u02 τ( )⋅+⋅+

RnAnAτ( ) +

u0 t( )

Ru0u0

3 τ( ) 3Ru0u0τ( )Ru0u0

0( ) Ru0u0

3 τ–( )= =

Ru0

2u02 τ( ) Ru0u0

2 0( ) 2Ru0u0

2 τ( ) +=

Ru0

3u03 τ( ) 9Ru0u0

τ( )Ru0u0

2 0( ) 6Ru0u0

3 τ( )+=

Ru0u00( ) Z0Pu0

=

175


(4-7)

Note that the hypothesis is not that unrealistic in the case considered. Until now, all

systems considered had quite large system bandwidths . Most commonly GHz was

used in examples and simulations. However, when designing an amplifier for e.g. mobile

telephony signals, whose frequency band goes from 890 MHz to 960 MHz in Europe,

designing an amplifier with a bandwidth of several GHz would be of no use. Hence, a

narrowband amplifier, optimized in the frequency band of interest, will be the best solution.

The total input noise power is given by , and hence, this total noise power will be

much smaller for narrow band amplifiers than for broadband ones, considering the same input

noise power spectral density .

The second and third line of (4-7) are output noise contributions. The first line of (4-7) shows

the amplification of the input signal’s spectrum and the addition of an extra term ,

which corresponds to the triple convolution of the input signal spectrum. When , this

last term cannot be neglected as compared to the largest noise term, which is .

Indeed, the ratio of both terms (at ) is about , or, with

, , W and W, this yields about . Note

furthermore that W corresponds to dBm/Hz over a bandwidth of

4 GHz, or dBm/Hz over a bandwidth of 25 MHz. Hence, W is

quite a lot of noise power for small bandwidths. This means that when considering standard

input noise power spectral density and narrowband amplifiers, the total input noise power will

be even smaller, and thus the term will be even more important as compared to

.

One question remains: What does the term represent? Since it results from

given by (4-6), it is the spectrum of that part of , that is not correlated with


2 6k1k3Z0Pu09k3

2Z02Pu0

2+ + ⋅ 6k3

2Ru0u0

3 τ( ) +=

R+ nunuτ( ) k1

2 6k1k3Z0Pu09k3

2Z02Pu0

2+ + 18k3

2Rnunuτ( )Ru0u0

2 τ( )+⋅

RnAnAτ( ) +

PnuPu0

«

B B 4=

PnuB Nin⋅

Nin

6k32Ru0u0

3 τ( )

PnuPu0

«

Rnunuτ( ) k1

2⋅

τ 0= 6k32Z0

3Pu03( ) k1

2Z0Pnu( )⁄

k1 10= k3 60– V 2–= Pnu10 10–= Pu0

10 5–= 6 1÷

Pnu10 10–= Nin 166–=

Nin 144–= Pnu10 10–=

6k32Ru0u0

3 τ( )

Rnunuτ( ) k1

2⋅

6k32Ru0u0

3 τ( )

Ru0

3u03 τ( ) u0

3 t( )

176

Considerations about the output spectrum

. This power spectrum, corresponding to the expression is generated by

combining three input DFT spectral lines , and , with , for all

, and taking the square of the norm of the resulting amplitude spectrum, divided

by . But since the input lines are stochastic quantities over the instances, the resulting DFT

spectrum is also a stochastic quantity over the instances. Therefore, it differs from a scaled

version of the stochastic input DFT spectrum [24]. In other words, the term

generates extra stochastic contributions, uncorrelated with the input DFT. Or, in other words:

disturbing noise. This noise is generated by the nonlinearity itself. It is always present if

is noise-like, even if there is no input noise ( ), or if the system itself is noiseless

( ). The next figure (Figure 4-2) shows the spectrum of , for

MHz, mW/MHz and MHz.

For narrowband amplifiers, with input signals consisting of noise-like signals, the noise

present in the output signal of the system is essentially generated due to the nonlinearity itself

(i.e. coming from ). In what follows, the studied system will hence be considered

noiseless ( ), and the input noise will be considered as not present .

Note that this noise contribution only occurs when the input signal is a stochastic signal

over the instances. It is not sufficient that is one instance of an arbitrary, randomly

chosen signal, that does not change over different measurements. In the latter case,

creates extra spectral contributions on top of the amplified signal, that are functions of the

randomly chosen properties of , just like before. But the system is excited by the same

FIGURE 4-2. Spectrum of

u0 t( ) 6k32Ru0u0

3 τ( )

U0 k1( ) U0 k2( ) U0 k3( ) ki k– j≠

i j, 1 2 3, , ∈

Z0

6k32Ru0u0

3τ( )

u0 t( )

nu t( ) 0=

nA t( ) 0= Ru0u0

3 τ( )

f0 100= PSDu0

1( ) 0.1= B0 40=

0 100 200 300 400 5000

0.5

1

1.5

2

2.5x 10

-6

frequency [MHz]

PS

D(1

) [W

/MH

z]

Ru0u0

3 τ( )

Ru0u0

3 τ( )

nA t( ) 0= nu t( ) 0=

u0 t( )

u0 t( )

Ru0

3u03 τ( )

u0 t( )

177


signal at every measurement. As a consequence, the contributions of will be the

same in each case. Hence, gives a deterministic contribution instead of a stochastic

one.

Furthermore, it is clear that since depends on the properties of the input signal, the

properties of the noise generated by this term, will also depend on the properties of the input

signal [19].

Note that every NICE system containing an odd nonlinear contribution, suffers from these

extra noise contributions that are located in the frequency band of interest, called in-band

distortions. This can easily be shown. Consider e.g. that a ( )-th order contribution is

present ( ). In that case, spectral lines from the DFT spectrum are

combined in the output. If one chooses spectral lines in the

vicinity of frequency , and spectral lines in the vicinity of , such that

(i.e. not all the indices are opposite in

sign to the indices ), the resulting amplitude spectral contribution will lie in the

neighbourhood of frequency (i.e. in the band of interest), but it will be a stochastic

contribution, uncorrelated with where .

Ru0u0

3 τ( )

Ru0u0

3 τ( )

Ru0

3u03 τ( )

u0 t( )

2n 1+

n N∈ 2n 1+ U0 k( )

n 1+ U0 k1( ) … U0 kn 1+( ), ,

f0 n U l1( ) … U ln( ), , f0–

p 1 … n, , q 1 … n, , :lp= k– q∈∃,∈∀( )¬ l1 … ln, ,

k1 … kn 1+, ,

f0U ν( ) ν k1 … kn 1+ l1 … ln+ + + + +=

178

Discussion on fundamental issues of NPR measurements

4.3 Discussion on fundamental issues of NPR measurements

4.3.1 Existing measurement techniquesIn order to quantify the in-band distortions, several techniques have been developed during the

last decades. The noise power ratio (NPR) method, and the co-channel distortion power ratio

(CCPR) method will be highlighted here.

A. The noise power ratio (NPR) methodThe most commonly figure used to quantify in-band distortions is the Noise Power Ratio. The

method was developed several decades ago, and automated commercial equipment is available

to measure the NPR [20].

Definition 4.1

The Noise Power Ratio (NPR) is the ratio between the power spectral density of the in-band

distortions and the power spectral density of the output signal, when a slice is removed from

the in-band noise spectrum.

Figure 4-3 illustrates this definition. A system or device under test (DUT) is excited with an

input signal consisting of bandpass filtered white Gaussian noise with a narrow notch in the

center of the frequency band. The nonlinearities in the DUT will transport spectral components

within the notch. The power spectral density of these components, compared to the power

spectral density of the signal outside the notch is a measure for the in-band distortions of the

FIGURE 4-3. Illustration of the definition of Noise Power Ratio.

NPR

DUT

freqfreq

PSD

PSD

179


DUT. Note that the noise power spectral density in the notch is assumed to be a measure for the

distortion within the whole frequency band of interest.

Instead of using a noise source and a set of filters to create the desired input signal, i.e.

bandpass filtered noise with a notch in the center of the frequency band, it is also possible to

use an arbitrary waveform generator (AWG). With this AWG, one can synthesize a large

number of CW-tones with equal magnitudes and random phases. Such a signal has

approximately the same properties as the filtered noise [9], but this technique has the

advantage that the notch width and the signal type (e.g. OFDM or CDMA [43]) can easily be

controlled [21].

B. Co-channel distortion power ratio (CCPR) method.Recently published papers [22], [23] presented an alternative method to quantify in-band

distortions of two-port devices. The main idea in these papers is to tag everything that differs

from the linear output of the system as “noise”. The measurement setup of this method is

shown in Figure 4-4.

The required input signal is again bandpass filtered, spectral flat, Gaussian noise. However,

there is no need for a notch in the middle of the frequency band. This input signal is fed to the

DUT, and to the underlying linear system of the DUT. Then the output of the underlying linear

system is subtracted from the actual output of the DUT, and finally the power of this resulting

signal is measured.

FIGURE 4-4. Measurement setup for the CCPR method.

+freq

PSD(1)

DUTPowerMeter

UnderlyingLinear System

of DUT

+

-

180


At first glance, two methods exist to determine in-band distortions of two-port systems.

However, when applied to an identical system under (almost) identical experimental

conditions, both methods yield different results. The CCPR method objects against the validity

of the NPR method to be a good measure for the in-band distortions. An explanation for these

different results was sought in the influence of the notch on the measurement. It was found that

the presence of the notch in the input signal produces a non negligible underestimation (up to

7 dB) of the in-band distortion. This underestimation is due to the fact that by creating a notch,

important spectral components which create output noise in the notch’s frequency band are

omitted.

To put both methods in perspective, the fundamental issues of these in-band distortion

measurements are discussed in the next sections.

4.3.2 General framework

A. A model for the considered systemsA general framework is defined, to base the theory and conclusions of this chapter on. First of

all, a model for the (narrowband) system has to be defined. The theory and techniques

described here are valid for the class of NICE systems, that can be described as shown in

Figure 4-5.

Figure 4-5 shows that the output signal of the system consists of a linear contribution

and a contribution solely due to the nonlinear part of the system . Hence:

(4-8)

FIGURE 4-5. General system model.

+Linear System

NONLINEARSYSTEM

y t( )u t( )

Underlying

y t( )

yL t( ) yNL t( )

y t( ) yL t( ) yNL t( )+=

181


Most of the systems that were designed to be linear, but have non-idealities, can be described

by this model. For most well-designed amplifiers, the underlying linear system does indeed

exist. The linear output contribution dominates the nonlinear contribution for sufficiently small

input powers, i.e.

(4-9)

With , the root-mean-square value of signal . Even if (4-9) is not

valid, as is the case for crossover distortion, the theory will still be applicable [9], but in that

case the underlying linear system will not exist (see Appendix 4.A).

Note that many NICE systems can indeed be modelled by Figure 4-5, since the terms of the

Volterra series that describes their input/output relation can be split into two groups: the

operator of first order which characterizes the underlying linear system, and all the other

Volterra operators which characterize the purely nonlinear part of the system.

B. A model for the input signalsThe considered input signals are multi-carrier signals with noise-like properties. When

periodic signals are considered, this class becomes the class of random multisines [9]. These

random multisines will be used as excitation signals for the DUT.

Definition 4.2

A random multisine is a signal consisting of the superposition of sine waves, whose

frequencies are commensurate (i.e. the ratio of two frequencies is a rational number), and

whose phases are instances of an independent distributed random variable over the interval

[ [, such that .

The multisines used in this chapter will be given by:

(4-10)

yNL( )rms

yL( )rms-----------------------

urms 0→lim 0=

xrms E x2 t( ) = x t( )

u t( )

N

ϕ

0 2π, E ejϕ 0=

u0 t( ) 1N

-------- Uκ 2π fmin κ 1–( )fmax fmin–

N 1–---------------------------+

t ϕκ+ cos

κ 1=

N

∑=

182


and are respectively the maximum and the minimum frequency of the excitation

signal, is the number of frequency components, is the amplitude of the -th frequency

component of the multisine, and the phases are a instance of an independent distributed

random process on the interval [ [, such that . This uniform distribution

can be a continuous or a discrete one. The scaling is performed in order to keep the

total power of the multisine finite, for .

Note that according to (4-10), the -th frequency component of the multisine does not

necessarily lie at frequency grid line . The -th frequency component of the multisine lies at

frequency . If is the frequency grid spacing (see also

section 1.6.2), chosen such that the spectral components of the multisine lie exactly on

frequency grid lines, the grid line number where the -th frequency component lies on is

given by

(4-11)

Assuming that . The frequency difference between two adjacent components of the

multisine is not necessary equal to , but can be an integer multiple of . When working in

narrow bands at high frequencies, the indices of the grid lines on which the components of the

multisine lie can become quite large. (see Example 4.3)

Example 4.3

Consider a multisine consisting of 51 components in the frequency band between

= 900 MHz and = 901 MHz. The frequency grid spacing is kHz. In this

case, the input spectral components lying at the positive side of the frequency axis will be

denoted as .

The advantage of using random multisines instead of a noise source, is their flexibility. The

amplitude and phase of each spectral component can be arbitrary chosen and/or modified. The

only problem that arises is that one instance of a random multisine is a deterministic periodic

fmax fmin

N Uκ κ

ϕκ

0 2π, E ejϕκ 0=

1 N⁄

N ∞→

κ

κ κ

fmin κ 1–( ) fmax fmin–( ) N 1–( )⁄+ ∆f

k κ

kfmin κ 1–( )

fmax fmin–N 1–

---------------------------+

∆f-------------------------------------------------------------=

k N∈

∆f ∆f

fmin fmax ∆f 10=

U0 90000( ) U0 90002( ) … U0 90100( ), , ,

183


signal instead of a stochastic one, and hence, with this signal it is difficult to give an accurate

estimate of the in-band distortions of a system, as stated in section 4.2.

However, if several instances of the multisine are considered, then the spectrum of the

multisine is a random quantity when looking over the different instances of the multisine.

Hence, instead of having a signal that varies randomly in time, the signal varies randomly over

the instances. The average of different measurements of a spectral component of one single

multisine instance is no longer zero, since it is a deterministic quantity. The ensemble

(instance) average of a spectral component taken over different multisine instances drawn from

a same amplitude distribution is zero, since the phase of the spectral components is random

distributed over the instances, with , just as was the case for the discrete noise

spectrum discussed in section 1.6.2.

4.3.3 Properties of the Frequency Response FunctionThe considered input signals are noise-like signals, and the study of the simple

polynomial model in section 4.2, revealed that apparently the output signal consists of a

part that is correlated with the input signal, and a part that is not correlated with the input

signal. This means that the frequency response function, will contain terms with similar

properties. But before taking a closer look to the frequency response function, the concepts

“underlying linear system” and “related linear dynamic system” will be highlighted.

The underlying linear system is introduced in Figure 4-5. It is the system that describes the

output signal, when the input signal tends towards zero. In the polynomial model

, the underlying linear system is described by: , i.e. a system that

scales the input signal with a factor . In the model , the underlying linear

system is described by the first term in the Taylor series expansion: , i.e. a system

that scales the input signal with a factor . The underlying linear system is not dependent on

the input signal . Note that the underlying linear system does not always exist, as is the

case for amplifiers exhibiting cross-over distortion (see Appendix 4.A).

E ejϕ 0=

u t( )

y t( )

y k1u k3u3+= y k1u=

k1 y α βu( )atan=

y αβu=

αβ

u t( )

184


The related linear dynamic system, is the best linear approximation of the system, in mean

squares sense, for a given input signal. In the model , if is a sinewave

with amplitude , the output spectrum at the fundamental frequency will also be a sinewave,

with amplitude (as shown in the previous chapters). Hence, the system has

amplified the original sinewave with a factor . The linear system that best

approximates, in mean squares sense, the input-output relation for the given example, is a

system that has a voltage gain of . Hence, the related linear dynamic system is

dependent on the input signal . In this example, this is illustrated that for input signals

with different amplitudes, the gain of the related linear dynamic system will be different.

Definition 4.4

The Frequency Response Function (FRF) is the ratio of the complex output DFT spectrum

to the complex input DFT spectrum , for the frequency ranges where , or

(4-12)

For NICE systems, this Frequency Response Function will be dependent on the properties of

the input spectrum . For LTI systems, the FRF is independent of , and is simply the

transfer function of the LTI system. For microwave systems, the FRF is usually the ratio

, or an -parameter that depends on the properties of the incident wave for NICE

systems.

When a system is excited with a random multisine, the non parametric frequency response

function (or , see section 1.3.3) can be split into systematic contributions, also called

bias contributions, and stochastic contributions [24] (see also Appendix 4.B).

(4-13)

or, in terms of -parameters:

(4-14)

y k1u k3u3+= u t( )

A

k1 3k3A2 4⁄+( )A

k1 3k3A2 4⁄+

k1 3k3A2 4⁄+

u t( )

Y f( ) U f( ) U f( ) 0≠

FRF f( ) Y f( )U f( )-----------

U f( ) 0≠=

U f( ) U f( )

H f( )

b2 a1⁄ S21 a1

b2 a1⁄

FRF k( ) FRFlin k( ) FRFB k( ) FRFS k( ) NFRF k( )+ + +=

S

b2 k( )a1 k( )------------- S21 k( ) S21 B, k( ) S21 S, k( ) NS21

k( )+ + +=

185


Where:

• (or ) is the FRF (or -parameter) of the underlying

linear system (if it exists), i.e. the transfer function .

• (or ) represents the bias or systematic nonlinear

contributions to the FRF, coming from the odd nonlinearities. This value

is independent of the random phases of the multisine.

• (or ) is the stochastic nonlinear contribution to the

FRF, coming from the even and the odd nonlinearities. This value is a

function of the random phases of the multisine and is zero mean, circular

complex normally distributed.

• (or ) is a stochastic component on the FRF. It consists

of input noise , noise added by the NICE system itself , or

the noise added by the measurement system. These contributions are

assumed to be extremely small, as discussed in section 4.2.

Based on (4-13), the output spectrum of a NICE system can be written as:

(4-15)

If the input spectrum of the NICE system can be written as , the

following can be told in terms of the functions , , and

defined in section 1.6.2.D.:

• contains those terms of and which

contain the factor combined only with other terms that have no

phase contribution. The other factors in can hence be grouped into

complex conjugated pairs.

• contains those terms of and which

contain the factor and that are not part of .

FRFlin k( ) S21 k( ) S21

H1 k( )

FRFB k( ) S21 B, k( )

FRFS k( ) S21 S, k( )

NFRF k( ) NS21k( )

Nu k( ) NA k( )

Y k( )

Y k( ) FRFlin k( )U0 k( ) FRFB k( )U0 k( ) FRFS k( )U0 k( ) NS k( )+ + +=

U k( ) U0 k( ) Nu k( )+=

A U0( ) A ′ U0 N,( ) B N( ) B′ U0 N,( )

FRFB k( )U0 k( ) A U0( ) A ′ U0 N,( )

U0 k( )

U0 l( )

FRFS k( )U0 k( ) A U0( ) A ′ U0 N,( )

U0 k( ) FRFB k( )U0 k( )

186


• contains the effects of the disturbing stochastic noise, and can be

written as . This term will always be

stochastic, even if is a deterministic spectrum over the instances.

4.3.4 Reconciling the NPR and the CCPR methodDepending on the fact that the goal is to characterize the best linear approximation or the

underlying linear system, two different interpretations of the Frequency Response Function

can be distinguished:

A. Goal: determine the best linear approximation is independent of the random phases of the multisine. As a result, will be

the same for every instance of the multisine. In this case, the systematic nonlinear contribution

to the FRF cannot be separated from the frequency response function of the underlying linear

system ( ). The sum of both, called , is defined as the FRF of the related

linear dynamic system (RLDS) of the device under test:

(4-16)

This is the best linear approximation of the nonlinear system in mean squares sense for the

considered class of excitation signals [24]. The output spectrum of the system is then given by:

(4-17)

where represents the stochastic nonlinear contribution to the output

spectrum. Note that since has a stochastic character with respect to the different

instances of the multisine, also behaves as a noise source on the output of the system.

Because the input power is independent of the phase instance of the multisine, is a

constant quantity over the different instances. (4-17) can also be written as

(4-18)

NS k( )

NS k( ) B N( ) B′ U0 N,( )+=

U0 k( )

FRFB k( ) FRFB k( )

FRFlin k( ) FRFR k( )

FRFR k( ) FRFlin k( ) FRFB k( )+=

Y k( ) FRFlin k( ) FRFB k( )+( )U0 k( ) YS k( ) NS k( )+ +=

FRFR k( )U0 k( ) YS k( ) NS k( ) + +=

YS k( ) FRFS k( )U0 k( )=

FRFS k( )

YS k( )

FRFR k( )

Y k( ) Y0 k( ) Ny k( )+=

187


where represents the output spectrum of interest, i.e. a filtered

version of the input spectrum , and represents a disturbing

noise term of the output spectrum. Hence, the NICE system can be replaced by a linear system

plus a noise source (see Figure 4-6). The variance (or power) of this noise source depends on

the input signal power spectral density.

Basically, the NPR measurement determines the output spectrum in a notch in the spectrum,

this is the output spectrum at those frequencies where the input was zero. For

(in the notch), (4-17) tells that the output spectrum in the notch is given by:

(4-19)

which is the sum of the stochastic nonlinear contributions and the measurement noise (i.e. all

the disturbing noise).

Conclusion: When the goal is to determine the best linear approximation, the classical NPR

measurement will determine the power of the noise source (due to the stochastic nonlinear

contributions) in the equivalent model of the NICE system shown in Figure 4-6. The NPR

method hence gives a correct figure for the nonlinear distortions, which appear as an additional

noise source in the measurements.

B. Goal: determine the underlying linear systemAlthough is constant over the different phase instances of the multisine , it

does depend on the input power of . Hence, when the operating point of the amplifier

FIGURE 4-6. Equivalent “best linear approximation” model of the NICE system.

Y0 k( ) FRFR k( )U0 k( )=

U0 k( ) Ny k( ) YS k( ) NS k( )+=

+NOISESOURCE

“Linear”H ω( ) FRFR ω( )=

u t( ) y t( )NoiseSource

σYS

2

Y k( ) U0 k( )

U0 k( ) 0=

Y k( ) YS k( ) NS k( )+ Ny k( )= =

FRFB k( ) u0 t( )

u0 t( )

188


changes, i.e. the amplifier deals with different input signals that have different input power

spectral densities, will also change. In this case, one might prefer to determine the

underlying linear system instead of the best linear approximation. Hence, the stochastic and the

systematic nonlinear contributions are both quantified as distortions. Every output that

diverges from the response of the underlying linear system is considered to be a distortion (see

Figure 4-7).

In this case, a figure such as yielded by the CCPR measurement method [23], that is able to

separate the output of the NICE system and the output of the underlying linear system gives a

correct figure for the nonlinear distortions.

4.3.5 Proposed measurement methodIn order to measure the different contributions to the FRF, one can excite the device under test

with multi carrier signals having noise like properties, i.e. random multisines, generated by an

Arbitrary Waveform Generator (AWG). The incident and reflected wave spectra at both ports

of the device under test are measured with the Nonlinear Vectorial Network Analyzer (NVNA)

(NNMS-HP85120A-K60) [25].

An absolute calibration is needed to correct for systematic errors, since nonlinear system

characterisation requires the knowledge of the absolute waves at the ports of the DUT. Hence,

the relative calibration, as used for -parameter measurements, has to be extended with a

power meter calibration, which sets the absolute power level of the waves, and a phase

FIGURE 4-7. Equivalent “underlying linear system” model of the NICE system.

FRFB k( )

NOISESOURCE

LinearH ω( ) H1 ω( )=

u t( ) y t( )Distortions

FRFB FRFS+

+

S

189


reference calibration, which gives the phase relations between the wave components on a

harmonic frequency grid relative to a single time origin [25].

As mentioned in section 4.3.4, if the power of the DFT spectrum1 of the input signal is

constant, and have the same behavior, and therefore cannot be measured

separately. Only their sum , which represents the “gain” of the

amplifier in compression, can be determined. To split both contributions, the following

approach is proposed:

In a first step, can be determined with a classical FRF (or ) measurement, but

the amplitude of the multisine (and hence its total power) has to be kept low enough to neglect

nonlinear contributions. In this case, a good approximation for the underlying linear system

will be measured, due to the assumption that the linear output contribution dominates the

nonlinear contribution for sufficiently small input powers (4-9).

has to be determined as function of the input power and can be

measured by averaging the FRF (of measured ) over a large number of phase instances

at each power level of the input multisine. This will indeed eliminate the stochastic nonlinear

contributions, since they behave as noise (circular complex normally distributed). The

resulting is the power gain of the amplifier for that specific class of

input signals, as a function of the input power.

is a stochastic quantity that has noise-like properties. The stochastic nonlinear

contributions reveal themselves as an additional noise source superimposed on the output of

the amplifier in compression. The variance (or root-mean-square, since it is zero-mean) of the

noise source is determined by taking the sample variance over different phase instances of the

multisine, i.e.

1. I.e. the square of the magnitude of the DFT spectrum, divided by . Note the difference with the powerspectrum, that is the instance average of the power of the DFT spectrum.

Z0

FRFlin k( ) FRFB k( )

FRFlin k( ) FRFB k( )+

FRFlin k( ) S21

FRFlin k( ) FRFB k( )+

b2 a1⁄

FRFlin k( ) FRFB k( )+ 2

FRFS k( )

190


(4-20)

Another way to calculate this distortion power is, knowing that and are

uncorrelated:

(4-21)

The measurement noise power can also be determined by calculating the variance of

the output spectrum over a number of successive periods, i.e. over a number of identical

instances of the multisine. For identical instances of the multisine, is a constant

quantity. Note that it has been assumed that is negligible as compared to . This is

the place to check whether this assumption holds or not. To minimize the influence of ,

(if required) the average of the output spectrum can be taken over a number of successive

periods, i.e. over a number of identical instances of the multisine.

4.3.6 Experimental resultsThe measurements are performed on a power amplifier of type MAR6 (Mini-Circuits) [27].

This power amplifier has a supply voltage of 12 V and is terminated in a 50 load

impedance. The measurement setup is shown in Figure 4-8.


σYS

2 k( ) Z0⁄ Distortion Power=

YS k( ) b2 k( ) E b2 k( ) a1 k( )⁄ a1 k( )⋅–=

YS k( ) Y k( ) E Y k( ) U0 k( )⁄ U0 k( )⋅–=

YS k( ) U0 k( )

E YS k( ) 2

Z0-------------------

E Y k( ) 2

Z0----------------

FRFR k( ) 2E U0 k( ) 2

Z0--------------------

–=

NS k( )

Y k( )

FRFS k( )

NS k( ) YS k( )

NS k( )

Ω

DUT

50Ω

NVNA

IQ-modulator

AWG AWG

RF source

I Q

reference clock

191


The amplifier is excited by a random multisine consisting of 64 tones, spaced 5 kHz apart, and

with a center frequency of 900 MHz. Spectral components 1 up to 4 and 22, 24, 26, 28,

symmetrical to the carrier at 900 MHz are omitted, creating notches where the in-band

distortions will be measured according to the NPR method (see Figure 4-9).

The random multisines are generated with a Rohde & Schwarz SMIQ06B Vector Signal

Generator [28], driven by two VXI Arbitrary Waveform Generators (HPE1445) at its I and Q

ports. The power of the tones is swept from dBm to dBm in steps of 1 dB. Twenty

different instances of the random multisine are used. Figure 4-10 shows the power spectrum of

the measured incident wave at the input of the amplifier, for all input power levels.

Figure 4-11 shows the power spectrum of the output waveform. The data in this figure can be

used to determine the in-band distortions, by looking at the power in the notches.

FIGURE 4-9. Spectral content of the multisine.

FIGURE 4-10. Power spectrum of the multisine centered around 900 MHz.

899.8 899.9 900 900.1 900.20

0.5

1

frequency [MHz]

Spe

ctra

l con

tent

47– 27–

-200 -100 0 100 200-80

-70

-60

-50

-40

-30

-20

Relative frequency [kHz]

Pow

er [d

Bm

]

192


FIGURE 4-11. Power spectrum of the output waveform: (a) shows the spectral regrowth, while (b) zooms in on the band of interest

FIGURE 4-12. Magnitude and phase of , obtained by averaging the FRF.

-200 -100 0 100 200-80

-70

-60

-50

-40

-30

-20

-10


Pow

er [d

Bm

]

(a) (b)

-45 -40 -35 -3012

12.5

13

13.5

14

14.5

15

15.5

RF Source Power [dBm]

|FR

FR

|2 [dB

]

-50

-40

-30

-20 -200-100

0100

200

12

13

14

15

16

Relative frequency [kHz]RF Source Power [dBm]

|FR

FR

|2 [dB

]

-45 -40 -35 -30105

110

115

120

125

130

135

140


phas

e of

FR

F' R

[deg

]

-50

-45

-40

-35

-30

-25 -200-100

0100

200

100

110

120

130

140


phas

e of

FR

F' R

[deg

]

FRFR k( )

193


In Figure 4-12, (obtained by averaging the measured FRF) is shown in magnitude

and phase, as function of the input power of the tones and the relative frequency. This

illustrates that the amplifier goes into compression when the input power increases, and that

the power range used extends well above the 1 dB compression point.

Figure 4-13 shows the distortion power, calculated using (4-20) (red lines) and (4-21) (green

and black dots), as a function of the input power of the tones. The black dots are the data points

obtained where (4-21) yields negative power levels, i.e. faulty results. As expected, the

distortion increases with the power of the input signal. For low input powers, however, (4-20)

shows the presence of a constant noise power of about dBm, and (4-21) yields sometimes

negative distortion power levels, represented by the black dots to stress that they have no

physical meaning. This can be caused by errors on . These negative power levels are

constant over the input power and also correspond to about dBm. Hence, one can

conclude that the error made on the distortion power is about dBm, and in the region

where these negative power levels occur, the true distortion power will be smaller than

dBm.

When performing a classical NPR measurement, one looks at the power of the output wave

in the notch, and claims that this power contains a measure for the distortion power. Figure 4-

14 shows the output power in the notches (red curves) together with the data of Figure 4-13.

FIGURE 4-13. Distortion power as a function of the input power of the tones, according to (4-21) (green and black dots) and (4-20) (red lines).

FRFR k( )

60–

FRFR k( )

60–

60–

60–

b2

194


The blue dots represent the result of (4-21), and the green dots represent the noise power

calculated with (4-20). Two conclusions can be made:

1. Above dBm, the proposed measurement method is in good agreement with the

classical NPR measurement.

2. For classical NPR measurements, the location or coherence of the notch seems to be of

minor importance.

To calculate the in-band distortions, as proposed by the CCPR measurement method, no new

measurements are needed. First, the gain of the underlying linear system is determined as

described section 4.3.5. This power gain turned out to be 15 dB here (see Figure 4-

15).

FIGURE 4-14. Comparison between the output power in the notches (red curves), and the calculated distortion power with (4-21)(blue dots) and (4-20)(green dots).

FIGURE 4-15. Magnitude (a) and phase (b) of the FRF of the underlying linear system.

60–

H1 ω( ) 2

-200 -100 0 100 20014.6

14.8

15

15.2

|FR

Flin

|2 [dB

]

-200 -100 0 100 200100

120

140


phas

e of

FR

Flin

[deg

]

(a)

(b)

195


After subtraction of the linear part of the output spectrum from the actual measured amplifier

output spectrum, one obtains the following figure, representing or

.

For low input power levels, a constant noise power of about dBm is obtained. This again

indicates the precision of the obtained results. Hence, the obtained curve has again an error of

about dBm.

Figure 4-17 shows the output power in the notches (red data), the power of the stochastic

nonlinear contributions (green data), the distortions according to the CCPR method (blue

data), and the spectral regrowth (black data). The proposed measurement method for constant

FIGURE 4-16. Distortion power according to the CCPR method.

FIGURE 4-17. Distortion power according to the NPR method (red), CCPR method (blue), and the power of the stochastic nonlinear contributions (green). The black stars represent

the spectral regrowth (out-band).

E b2 S21a1– 2 Z0⁄

E Y k( ) H1 k( )U0 k( )– 2 Z0⁄

-50 -45 -40 -35 -30 -25-70

-60

-50

-40

-30

-20N

oise

Pow

er [d

Bm

]


60–

60–

-50 -45 -40 -35 -30 -25-80

-70

-60

-50

-40

-30

-20

Noi

se P

ower

[dB

m]

RF Source Power [dBm] Relative frequency [kHz]

YS

YS

196


input power, agrees very good with the NPR method, but does not need the presence of notches

in the spectrum. For high input powers, the CCPR method predicts a distortion power that is

much (about 8 dB) larger than the classical NPR method or the constant input power

hypothesis predicts. This is due to the systematic nonlinear contribution which is

tagged as a distortion by the CCPR method.

4.3.7 ConclusionThe NPR method will yield a good measure for the in-band distortions, according to the “best

linear approximation” model for the NICE system. On the other hand, the CCPR measurement

method yields a good measure for the in-band distortions, according to the “underlying linear

system” model for the NICE system.

A third possibility is to measure the systematic and stochastic contributions of the FRF, as

proposed in section 4.3.5. Using the latter method, it is possible to quantify the in-band

distortions, according to both the NPR and the CCPR method.

FRFB k( )

197


4.4 Extension towards multi-port systems: mixers

4.4.1 IntroductionAll the techniques mentioned in section 4.3 were originally developed for two-port devices

such as amplifiers. The methods to measure the in-band distortions of mixers are stagnating

somewhat since the 70s [29], due to the complexity of the problem. However, when

considering the mixer as a two-port device instead of a three-port device, it is possible to use

the NPR method and the CCPR methods [30], to measure in-band distortions of three-port

devices such as mixers. This requires, however, some assumptions.

This section investigates the possibility to transfer the techniques developed to measure in-

band distortions in two-port devices to three-port devices such as mixers.

4.4.2 A simple mixer modelThe aim of frequency translating devices is to shift a band of spectral components (centered

around the RF frequency ) towards another location in the frequency spectrum, i.e. a band

centered around an intermediate frequency , with the frequency of the

local oscillator. Ideally, this frequency shift should not modify in any way the amplitudes or the

phases of the spectral components, nor should it create additional spectral components

anywhere else in the frequency spectrum.

The required frequency translation can be achieved using a mixer. The ideal mixer is a

nonlinear three-port device, as represented in Figure 4-18.

FIGURE 4-18. Representation of the ideal mixer.

fRF

fIF fRF fLO–= fLO

port 1

port 2

port 3RF

LO

IFideal multiplier

a3

b3a1

b1

b2a2

198

Extension towards multi-port systems: mixers

Its behavior in terms of waves can be mathematically written as:

(4-22)

or, in terms of voltages as:

(4-23)

where and represent the input signals at the RF and LO port respectively, and

represents the output signal at the IF port. If the LO input signal is a pure sine wave

, this ideal mixer performs an almost ideal frequency

translation. The problem is that the IF spectrum will be a scaled version of the RF spectrum,

with complex scaling factor , and that a spectrum scaled with will

also be created around the “image frequency” (see Appendix 4.C).

In reality, however, a mixer is never ideal, but rather a device that can be modelled as shown in

Figure 4-19.

The three linear systems at the ports of the mixer are at least partly due to the packaging of the

device. The nonlinear multiplier is a three-port device, having two inputs and one output,

whose relation is assumed to be described as:

FIGURE 4-19. A model for the non-ideal frequency translating device.

b3 t( ) a1 t( ) a2 t( )⋅=

yIF t( ) uRF t( ) uLO t( )⋅=

uRF t( ) uLO t( )

yIF t( )

uLO t( ) ALO 2πfLOt ϕLO+( )cos⋅=

ALOe j– ϕLO 2⁄ ALOejϕLO 2⁄

fRF fLO+

Xnon-ideal multiplier

RF

LO

IFLinearSystem

LinearSystem

LinearSystem

HRF

HLO

HIF

uRF t( )

uLO t( )

yIF t( )uRF ′ t( ) yIF ′ t( )

uLO ′ t( )a3

b3a1

b1

b2a2

199


(4-24)

Where represents the ( )-th order Volterra operator, with orders

and referring respectively to the inputs and . When dealing with

periodic signals, the output spectrum due to the -th order Volterra operator

can be written as:

(4-25)

Note that this two-input one-output nonlinear multiplier model is more general than the model

shown in Figure 4-20, that states that the nonlinear multiplier is in fact the multiplied output of

two nonlinear systems (see Appendix 4.D).

Hence, the time waveform , coming out of the IF port, can be written as:

(4-26)

(4-26) and (4-25) show that the spectral components at the IF port are created by combining

spectral lines centered around with spectral lines at frequency . The spectral

components at the IF port are hence located at frequencies , with .

FIGURE 4-20. Alternative model for the non-ideal multiplier

yIF ′ t( ) Hαβ uRF ′ t( ) uLO ′ t( ),[ ]

β 0=

∞

∑α 0=

∞

∑=

Hαβ uRF′ t( ) uLO ′ t( ),[ ] α β+

α β uRF ′ t( ) uLO ′ t( )

YIF ′ αβ( ) k( ) α β,( )

Hαβ uRF′ t( ) uLO ′ t( ),[ ]

… Hαβ k1 … kα β+, ,( )URF ′ k1( )…URF ′ kα( )ULO ′ kα 1+( )…ULO ′ kα β+( )

kα β+M2-----–=

M2----- 1–

∑

k1M2-----–=

M2----- 1–

∑

X yIF ′ t( )

ideal multiplieruRF ′ t( )

uLO ′ t( )

non-linear

non-linear

yIF t( )

yIF t( ) HIF Hαβ HRF uRF t( )[ ] HLO uLO t( )[ ],[ ]

β 0=

∞

∑α 0=

∞

∑=

α

fRF± β fLO±

α fRF βfLO+ α β, Z∈

200


Hence, spectral components will be created throughout the whole frequency spectrum.

Furthermore, the spectrum around is a copy of the RF spectrum that is modified in

amplitude and phase, and contains in-band distortions and spectral regrowth (see Figure 4-21).

Since in general, one is able to eliminate the spurious frequency components by filtering (i.e.

those spectral components of the output spectrum that do not fall inside the frequency band of

interest around ), only the frequency band around will be considered in the next

sections.

4.4.3 Defining an FRF or transmission parameter for the mixerIn analogy to the amplifier case, the goal of this section is to determine the in-band distortions

in the frequency band around . The signals typically applied to mixers in

telecommunication are multiple carrier signals with noise-like properties. Hence the multisines

(see Definition 4.2 and (4-10)) used in the previous sections are still suited to model the RF

input signals applied to the mixer. The signal applied at the LO port of the mixer will still be

assumed to be a single tone carrier at frequency .

If one is dealing with a two-port device instead of a three-port device, the solution to the

problem is to determine the stochastic nonlinear contributions to the output of the system, as

described in section 4.3. It is still possible to use this technique, because the mixer problem is

very similar to the amplifier problem. Indeed, one can examine the FRF or transmission

parameter between the output of the ideal frequency shifter around the IF frequency , and

FIGURE 4-21. Effect of a real-world mixer and an ideal frequency shifter.

fIF

fIF fIF

f

f

f

RF

IF

IF

idealfrequency

realmixer

shifter

fIF

fLO

fIF

201


the actual output of the real-world device around . Examining the contributions of the LO

and the RF input waveforms to the frequency band around motivates this decision.

Despite the fact that modeling the nonlinear multiplier as a multiplication of two NICE

systems, is not general enough (see Appendix 4.D), it gives however insight in the operation of

the real-world mixer. A similar insight can be obtained by considering the real-world mixer as

a static device. To get a better insight in the energy conversions occurring in the mixer, the

general equation will be simplified. Consider that the mixer is a purely static device, then for

all angular frequencies , and the kernel impulse

responses are multidimensional Dirac impulses .

In that case, (4-26) can be rewritten as:

(4-27)

or in terms of spectra,

(4-28)

Since is a pure sinewave, the convolution of the LO signal only contains energy at

harmonics of . This means that in order to obtain the actual spectrum at the IF port, one

has to take the spectrum of the RF port, apply it to a NICE system, and shift it over every

integer multiple of (i.e. the frequency of the sine wave at the LO port). Finally, all these

shifted spectra have to be added. Hence, the frequency band around at the IF port, contains

two distinct types of contributions (see also Appendix 4.F):

fIF

fIF

ω HRF ω( ) HLO ω( ) HIF ω( ) 1= = =

hαβ τ1 … τα β+, ,( ) hαβδ τ1( )…δ τα β+( )=

yIF t( ) hαβ uRFα t( )uLO

β t( )⋅

β 0=

∞

∑α 0=

∞

∑ uLOβ t( ) hαβuRF

α t( )

α 0=

∞

∑⋅

β 0=

∞

∑= =

YIF k( ) DFT uLOβ mTs( )( )*DFT hαβuRF

α mTs( )

α 0=

∞

∑

β 0=

∞

∑=

ULO k( )*…*ULO k( )* hαβURF k( )*…*URF k( )

α 0=

∞

∑

β 0=

∞

∑=

α timesβ times

uLO t( )

uLO t( )

fLO

fIF

202


1. The spectral components, downconverted with the fundamental tone of the LO signal

and coming from the frequency band around at the output of the nonlinear system at

which was applied. This frequency band contains systematic and stochastic

components, as shown in section 4.3.3.

2. Spectral components, downconverted by harmonics of the local oscillator signal.

Originally, these spectral components lie around integer multiples of , and hence,

they only contain stochastic nonlinear contributions. Systematic contributions are only

present on excitation lines [24].

Hence, the spectrum at the IF port, around can be written as:

(4-29)

or in terms of waves and transmission parameters:

(4-30)

with ( ), the transfer function of the related linear dynamic system,

( ) the stochastic nonlinear contributions to the IF spectrum, and

( ) the ideally shifted spectrum coming from the RF port (see Figure 4-22).

FIGURE 4-22. Schematic representation of the RF spectrum and the ideally shifted RF spectrum.

fRF

uRF t( )

fRF

fIF

YIF k( ) FRFR k( ) URF ideal, k( )⋅ YS k( )+=

b3 k( ) S31 R, k( )a1 ideal, k( ) b3 S, k( )+=

FRFR k( ) S31 R, k( ) YS k( )

b3 S, k( ) URF ideal, k( )

a1 ideal, k( )

k

k

URF

URF ideal, l– l+

203


If represents the frequency grid line corresponding to the positive , Figure

4-22 shows that ideally, both the positive and the negative part of the RF spectrum is shifted

grid lines towards 0 Hz. Hence, the relation between and is given by:

(4-31)

Just as proposed in section 4.3.5, can be determined by averaging the measured

FRF, or over the different phase instances of the RF multisine. Hence,

can also be determined for all the instances of the multisine, and the variance of the

stochastic nonlinear contributions , i.e. the in-band distortion can be calculated. In this

case, the real-world mixer can be modelled as an ideal frequency translating device, followed

by a noisy LTI system as shown in Figure 4-23

However, before one can simply apply the above method, a very important remark has to be

made. Due to the fact that the phase of the local oscillator is different from zero, the mixer will

not only shift the spectral components in frequency, but it will also give them a phase shift (see

Appendix 4.C and Appendix 4.F). This has no influence on the stochastic nonlinear

contributions: their phases will be shifted, but they will remain circular complex normally

distributed. But for the systematic nonlinear contributions, this means that the phase shift will

be incorporated into . Hence,

(4-32)

FIGURE 4-23. Equivalent model of the real world mixer.

l fLO ∆f⁄= fLO

l

URF ideal, k( ) URF k( )

URF ideal, k( )URF k l+( ) k 0>⇔

URF k l–( ) k 0<⇔=

FRFR k( )

YIF k( ) URF ideal, k( )⁄

YS k( )

σYS

2 k( )

+NOISESOURCE

“LTI system”H f( ) FRFR f( )=uRF t( )

yIF t( )

NoiseSource

σYS

2 f( )

fRF fIF→f-shifter

FRFR k( )

FRFR k( ) FRFR ′ k( ) e jϕLO–⋅=

S31 R, k( ) S31 R, ′ k( ) e jϕLO–⋅=

204


where ( ) represents that part of that is independent of the

phase of the local oscillator signal.

In order to determine , one has to average the measured FRF (or -parameter)

over different instances of the RF multisine. At this point, four situations can occur:

1. One is able to measure the absolute phase of the local oscillator signal , and of the

RF and IF signals, for every instance of the multisine. In this case, can be de-

embedded out of , and can be determined as:

(4-33)

2. There is a constant phase difference between the phase of the local

oscillator signal, and the RF multisine, by e.g. synchronizing the internal clocks of the

LO source and the RF source. Furthermore, the measured waves are not calibrated in

phase, hence, , and are known except for a constant value. In other words,

(4-34)

where , and represent respectively the measured phases of the LO, RF and

IF signal, and , and are respectively the measurement errors made on the

phases of the RF, LO and IF signal. In this case, by using (4-33), can be

determined except for a constant phase shift. The systematic contributions of the FRF

will always have the following phase (see Appendix 4.F):

(4-35)

Hence, when applying (4-33) on measurements, the result will have the following phase:

(4-36)

FRFR ′ k( ) S31 R, ′ k( ) FRFR k( )

FRFR k( ) S31

ϕLO

e jϕLO–

FRFR k( ) FRFR ′ k( )

FRFR ′ k( ) E FRF k( )ejϕLO

=

∆φ ϕRF ϕLO–=

ϕRF ϕLO ϕIF

ϕRFm ϕRF ϕRF

e+=

ϕLOm ϕLO ϕLO

e+=

ϕIFm ϕIF ϕ IF

e +=

ϕRFm ϕLO

m ϕIFm

ϕRFe ϕLO

e ϕIFe

FRFR ′ k( )

φHαβk1 … kα l1 … lβ, , , , ,( ) ϕRF ϕLO–+( ) ϕRF–

φHαβk1 … kα l1 … lβ, , , , ,( ) ϕRF ϕLO– ϕIF

e ϕRF ϕRFe+( ) ϕLO ϕLO

e+( )+–+ +

φHαβk1 … kα l1 … lβ, , , , ,( ) ϕIF

e ϕRFe ϕLO

e+–+=

205


which is the phase of , except for a constant . If the phase

measurement errors do not vary significantly over the frequency band of interest,

is a constant, and the phase relations between the different points of

the transfer function are known.

3. One can guarantee that the phase of the local oscillator will be the same for each

measurement, but this constant phase remains unknown. In this case, will

be a constant, and averaging the FRF will yield the transfer function of the related linear

dynamic system . (4-32) shows that the LO phase independent transfer

function is then known, except for a constant factor ( ) that is frequency

independent. But the phase relations between the different points of the transfer function

are known.

4. If one cannot guarantee that the phase of the local oscillator will be the same for every

instance of the random multisine, and that this phase cannot be determined nor

synchronized with the RF multisine, then or will also be a

stochastic quantity, since it contains the factor . In general will be a instance

of an uniformly distributed random process on [ [. Hence, averaging the FRF (or

) will lead to an erroneous result (i.e. zero). In this case, another measurement

procedure has to be followed in order to determine the transfer function of the related

linear dynamic system. This technique is explained in the following section.

4.4.4 Getting extra information about the nonlinear mechanism of the mixerWhen the phase of the local oscillator is unknown, cannot be determined by

averaging the FRF. In this case, one has to create notches in the RF spectrum. The power of the

spectral components that will fill the notches at the IF port will be a measure for the in-band

distortions, as predicted by the NPR method, and explained in section 4.3.4. This technique is

based on the philosophy of the NPR method, but instead of using one notch in the center of the

frequency band (as the NPR method requires), it is possible to distribute the notches over the

whole band of interest. The question to be answered at this point is: Does an optimal way to

distribute the notches over the frequency band exist in order to get some extra information

about the mixer?

FRFR ′ k( ) ϕIFe ϕRF

e ϕLOe+–

ϕ IFe ϕRF

e ϕLOe+–

ϕLO e jϕLO–

FRFR k( )

e jϕLO–

FRFR k( ) S31 R, k( )

e jϕLO– ϕLO

0 2π,

S31

FRFR ′ k( )

206


For two-port devices, this problem has already been studied and solved in the literature [31].

This paper states that on one hand, special multisines can be designed to quantify the even and

odd nonlinear contributions of the device, and on the other hand, guidelines exist to give an

optimal estimate of the in-band distortions.

A. The even and odd nonlinearities.The key idea behind the discrimination of even and odd nonlinearities relies on a proper choice

of the excitation frequencies. If the excitation signal is composed only of tones with

frequencies that are an odd multiple of the grid frequency spacing , it is clear that any

product of an even number of frequency lines will be translated to frequency grid lines that are

an even multiple of the frequency grid spacing. This kind of spectral product appears in the

spectral products associated with even order Volterra kernels. On the other hand, if the product

of an odd number of spectral lines is made, the resulting frequency will appear at an odd

multiple of the frequency grid spacing. This is the case for the spectral products associated

with odd order Volterra kernels. Hence, the output spectrum of the even order Volterra

operators will lie at even frequency grid lines, while the output spectrum of the odd order

Volterra operators will lie at the odd frequency grid lines.

For three-port devices such as mixers, applying this simple rule to the RF signal is not correct

because the parity (i.e. the fact that a spectral component lies on an even or on an odd

frequency grid line) of the spectral combinations will no longer only depend on the parity of

the RF signal. The parity of the local oscillator also has to be taken into account. Furthermore,

the Volterra operators as presented for three-port devices (4-25) are no longer simply

even or odd. They can be even or odd both with respect to the RF spectral components and the

LO spectral components.

Depending on the parity of the RF and the LO spectral components, four different spectral

location of , given by

(4-37)

∆f

Hαβ [ ]

Hαβ uRF t( ) uLO t( ),[ ]

URF ki( )

i 1=

α

∏ ULO lj( )

j 1=

β

∏

207


can be encountered:

1. The RF multisine lies on even grid lines and the LO spectral components lie on odd grid

lines. Since a sum of even numbers is always even, the value of will determine if the

result lies on even or on odd frequency grid lines. Hence, it is possible to discriminate

between the even and odd order nonlinearities of the LO path in the nonlinear

mechanism:

(4-38)

2. The RF multisine lies on odd grid lines and the LO spectral components lie on even grid

lines. Since a sum of even numbers is always even, only the value of will determine if

the result lies on even or on odd frequency grid lines. Hence, it is possible to discriminate

between the even and odd order nonlinearities of the RF path in the nonlinear

mechanism:

(4-39)

3. Both the RF multisine and the LO spectral components lie on odd grid lines. This time,

will determine if the result lies on even or on odd frequency grid lines. Hence, it is

possible to discriminate between the even and odd order nonlinearities of the ( )-th

order Volterra operator:

(4-40)

4. Both the RF multisine and the LO spectral components lie on even grid lines. In this

case, all the spectral components of the IF spectrum will also fall on even grid lines, and

nothing will be detected at the odd frequency grid lines.

Hence, depending if the goal is to quantify the even and odd nonlinearities of the LO

contributions, or the RF contributions, or the combined ( ) contributions, one of the first

three above mentioned cases can be used.

β

β 2Z∈ Hαβ uRF t( ) uLO t( ),[ ] lies on even grid lines⇒

β 2Z 1+∈ Hαβ uRF t( ) uLO t( ),[ ] lies on odd grid lines⇒

α

α 2Z∈ Hαβ uRF t( ) uLO t( ),[ ] lies on even grid lines⇒

α 2Z 1+∈ Hαβ uRF t( ) uLO t( ),[ ] lies on odd grid lines⇒

α β+

α β+

α β+ 2Z∈ Hαβ uRF t( ) uLO t( ),[ ] lies on even grid lines⇒

α β+ 2Z 1+∈ Hαβ uRF t( ) uLO t( ),[ ] lies on odd grid lines⇒

α β+

208


B. Estimating the in-band distortions.As already stated above, notches have to be created in the RF spectrum in order to get a

measure for the in-band distortions. In section 4.3.4, it was shown that the power of the

spectral components present in the output spectrum at the notches, is in fact the power of the

stochastic nonlinear contributions, or the in-band distortions. It was also experimentally shown

that these notches could be placed anywhere in the frequency band of interest, without

significantly affecting the measured in-band distortions. Note, however, that in the special case

that the notches are chosen to be on odd frequency grid lines, and that the LO frequency lies on

an even grid line, nothing will be detected in these notches at the IF port. This is the situation

of case 4, in the previous section.

Similarly, in the absurd extreme case that the notch is chosen to be the entire band of interest,

except for the lowest and the highest frequency component of the multisine, it is clear that the

IF output power in the notches will be a poor estimate of the real in-band distortion. Hence, a

restriction must be imposed on the RF multisine to make sure that a minimum percentage of

the spectral lines is excited. On the other hand, the more notches (or non-excited lines) are

present in the signal, the more information can be obtained about the frequency dependency of

the nonlinearity. Dividing the RF multisine in blocks of components (with ), with one

non-excited line (also called “detection line”) in each block, implies that the power at the

detection lines in the IF spectrum is a good estimate of the in-band distortions on the excitation

lines [31]. Randomizing the position of the notch in these blocks is also advised to obtain a

better estimation of the in-band distortion on the excited lines. Figure 4-24 illustrates the

described multisines.

n n 6>

209


The knowledge of the power of the in-band distortions also permits determination of

. From (4-29) and (4-32), it follows that:

(4-41)

And hence, because and are uncorrelated:

(4-42)

Since , can easily be found to be:

(4-43)

Hence, the magnitude of the frequency response function of the related linear dynamic system

can be determined experimentally.

FIGURE 4-24. Illustration of a multisine divided in blocks of 8 components (7 excited +1 absent), and with one detection line in each block. In (a), this detection line is always the same

component of the block, while in (b) the position of the detection line is randomized.

(a)

(b)

f

f

FRFR ′ k( )

YIF k( ) FRFR ′ k( ) e jϕLO– URF ideal, k( )⋅ ⋅ YS k( )+=

YS k( ) URF ideal, k( )

E YIF k( ) 2 FRFR ′ k( ) 2E e jϕLO– 2 E URF ideal, k( ) 2 E YS k( ) 2 +=

E e jϕLO– 2 1= FRFR ′ k( )

FRFR ′ k( )E YIF k( ) 2 E YS k( ) 2 –

E URF ideal, k( ) 2 ------------------------------------------------------------------------=

210


C. Even and odd nonlinearities + in-band distortionsNote that it is obvious that the two techniques mentioned above can be combined. If one wants

to determine the even and odd nonlinear effects of the RF path, together with the in-band

distortions, the RF multisine has to lie on odd frequency grid lines and the LO components on

even frequency grid lines (see section 4.4.4.A.). Furthermore, in order to obtain an optimal

knowledge of the in-band distortions on the excitation lines, the multisine has to be divided in

blocks of ( ) spectral components lying on odd grid lines, and in each block, one odd

spectral component has to be omitted.

4.4.5 Experimental resultsThe measurements are performed on a mixer of type ZEM-4300 (Mini-Circuits) [27]. The

measurement setup is shown in Figure 4-25.

The local oscillator signal is a sine wave with a frequency of 900 MHz, and a power of

7 dBm. The RF signal is a random multisine generated with a Rohde & Schwarz

SMIQ06B Vector Signal Generator [28], driven by two VXI Arbitrary Waveform Generators

(HPE1445) at its I and Q ports. The multisine has a center frequency of 1300 MHz, lying on a

frequency grid with a grid spacing of 5 kHz, and consisting of 64 components. This multisine

is an odd multisine, relative to the grid spacing of 5 kHz. Since the components lie on odd

frequency grid lines, the frequency difference between two consecutive lines is at least 10 kHz.

Since the frequency grid spacing kHz, it can be shown that the components of the


n n 6>

50Ω

NVNA

IQ-modulator

AWG AWG

RF source

I Q

XRF

LO

IF

reference clock

uLO t( )

uRF t( )

∆f 5=

211


multisine lie on odd grid lines 25999, 260001, 260003, etc..., while the LO sine wave lies on an

even grid line ( ). According to section 4.4.4.A., using this multisine, it is

possible to discriminate between the contributions of the even and odd nonlinearities of the RF

path. The in-band distortions will be determined by making notches in the multisine, as

described in section 4.4.4.B. The multisine is divided in 8 blocks of 8 components, and with

one detection line in each block, the position of this detection line is randomized. Figure 4-26

shows the power spectrum of the measured incident wave at the RF port, for all input power

levels.

The power of the SMIQ06B is swept from dBm to 10 dBm in 26 equidistant steps in the

logarithmic scale, resulting in the power of the individual tones being swept from dBm to

dBm in 26 steps. Twenty different instances of the random multisine were generated.

FIGURE 4-26. Power spectrum of the RF multisine centered around 1300 MHz.

kLO± 180000±=

5–

51–

36–

212


Figure 4-27 shows the power spectrum of the IF waveform, centered around MHz.

The data in this figure can be used to determine the in-band distortions, by looking at the

power in the notches, as shown in Figure 4-28.

Figure 4-28 shows the power in the notches of the IF spectrum. In the left plot, this noise

power is shown for all frequencies on one plot, while in the right plot, the variation of the

power in the notches is shown as a function of the RF source power and the frequency relative

to . The bold lines in Figure 4-28 are the noise powers at the odd detection lines,

representing the stochastic contributions due to odd nonlinearities in the RF path, while the

FIGURE 4-27. Power spectrum of the IF waveform, centered around 400 MHz: (a) shows the spectral regrowth, while (b) zooms in on the band of interest.

FIGURE 4-28. Noise power in the notches. The bold lines represent the noise power in the odd notches, and the thin lines represent the noise power in the even notches.

(a) (b)

fIF 400=

-5 0 5 10-95

-90

-85

-80

-75

-70

-65

Noi

se P

ower

[dB

m]

RF Source Power [dBm]-5

0

5

10

-400

-200

0

200

400-95

-90

-85

-80

-75

-70

-65

RF Source Power [dBm]Relative frequency [kHz]

Noi

se P

ower

[dB

m]

fIF

213


thin lines represent the noise power at the even detection lines, and hence the stochastic

contributions due to even nonlinearities in the RF path. As expected, the noise contributions

due to the even nonlinearities are much smaller than the ones due to the odd nonlinearities. The

reason therefore is that in theory, no even nonlinearities of the RF path can be found at the IF

frequency. Since according to Theorem 4.6 in Appendix 4.E, the couples , needed to

generate spectral contributions around , are: ,

with . Furthermore, according to (4-39), an odd can only contribute to spectral

components lying on odd grid lines. Hence, one can assume that the noise power found at the

even detection lines is the power of the disturbing stochastic noise and the power of the

even input lines (see Figure 4-26). Figure 4-26 also shows that the input power at these even

lines increases about 10 dB, with increasing RF power. This increase of about 10 dB is also

detected in the thin lines in Figure 4-28.

The noise power at the odd detection lines, on the other hand, is a measure for the in-band

distortions. These in-band distortions can be interpolated to get the in-band distortions at the

excitation lines. Using (4-43) one obtains the power gain of the related linear dynamic system

.

Figure 4-29 shows the resulting power gain of the related linear dynamic system.

The left figure shows this power gain versus the input RF power of the IQ-modulator, for all

frequencies, on one figure. The right figure shows as function of both frequency

FIGURE 4-29. Calculated power gain of the related linear dynamic system.

α β,( )

fIF fRF fLO–= α β,( ) 1 1,( ) 2 z1 z2,( )+=

z1 z2, Z∈ α

NS k( )

FRFR ′ k( ) 2

-5 0 5 10-18

-17

-16

-15

-14

-13

-12


|FR

FR

|2 [dB

]

-5

0

5

10 -400-200

0200

400

-18

-17

-16

-15

-14

-13

-12


|FR

FR

|2 [dB

]

FRFR ′ k( ) 2

FRFR ′ k( ) 2

FRFR ′ k( ) 2

214


and input power. Clearly, the mixer goes into compression, and the magnitude of

is a constant over the (small) band of interest.

The method explained in point 2 at the end of section 4.4.3, can also be applied to obtain

information about the noise behavior of the mixer. Therefore, the errors made when measuring

the phases of the signals may not vary over the measurements. Using (4-33), i.e.

(4-44)

can be determined in magnitude, and - except for a constant value - also in phase,

as shown in Figure 4-30.

With the knowledge of , the in-band distortions on the excitation lines can be

determined out of (4-42) as:

FIGURE 4-30. Magnitude and phase of , obtained by averaging the FRF.

FRFR ′ k( ) 2

FRFR ′ k( ) E FRF k( )ejϕLO =

FRFR ′ k( )

-5 0 5 10-18

-17

-16

-15

-14

-13

-12


|FR

FR

|2 [dB

]

-5

0

5

10 -400-200

0200

400

-18

-17

-16

-15

-14

-13

-12


|FR

FR

|2 [dB

]

-5 0 5 10-95

-90

-85

-80

-75

-70

-65

-60

-55


phas

e of

FR

F' R

[deg

]

-5

0

5

10 -400-200

0200

400

-100

-90

-80

-70

-60

-50


phas

e of

FR

F' R

[deg

]

FRFR ′ k( )

FRFR ′ k( ) 2

215


(4-45)

These calculated in-band distortions are shown in Figure 4-31.

In order to compare the results obtained on the one hand by using the output power in the

notches, and on the other hand by averaging the FRF, the results obtained using both methods

are drawn on one figure.

FIGURE 4-31. Noise power at the excitation lines.

FIGURE 4-32. Comparision of both methods for (a) and the noise power (b).

E YS k( ) 2 E YIF k( ) 2 FRFR ′ k( ) 2 E URF ideal, k( ) 2 ⋅–=

-5 0 5 10-100

-95

-90

-85

-80

-75

-70

-65

-60

Noi

se P

ower

[dB

m]

RF Source Power [dBm] -5

0

5

10

-400

-200

0

200

400-100

-90

-80

-70

-60

RF Source Power [dBm]Relative frequency [kHz]

Noi

se P

ower

[dB

m]

-5 0 5 10-100

-95

-90

-85

-80

-75

-70

-65

-60

Noi

se P

ower

[dB

m]


(a) (b)

FRFR′ k( ) 2

216


The red lines in Figure 4-32, are the results obtained by using the output power in the notches,

while the blue dots and lines represent the results obtained using the averaging of the FRF.

Clearly, both methods are in good agreement. Hence, when using the designed multisine, the

output power in the notches is a good estimate for the in-band distortions on the excitation

lines.

217


4.5 The mixer as a real three-port device: phase noise example

4.5.1 IntroductionIn the previous sections, the mixer was considered as a two-port device, whose FRF clearly

depends on the amplitude and the phase of the local oscillator. The local oscillator signal was

described as

. (4-46)

In reality however, the local oscillator signal is not a pure sine wave, because of the phase

noise. In other words, the spectrum of is not an spectral line of infinitesimal small

width, but is rather a spectrum as shown in Figure 4-33:

Hence, the spectral component has a finite width, e.g. dBc1 at 100 kHz away from the

carrier. The instantaneous phase of the local oscillator signal contains a

small stochastic component , that is assumed to be Gaussian and zero-mean:

(4-47)

hence,

(4-48)

FIGURE 4-33. Real-world spectrum of an RF source.

1. dBc is “dB compared to the carrier power”. Hence, dBc means that the power of the consideredcomponent is 60 dB lower than the power of the carrier.


uLO t( )

zoom

fLOfLO

60–

60–

θ t( ) 2πfLOt ϕLO+=

nθ t( )

θ t( ) 2πfLOt ϕLO nθ t( )+ +=

uLO t( ) ALO 2πfLOt ϕLO nθ t( )+ +( )cos⋅=

218

The mixer as a real three-port device: phase noise example

4.5.2 Considerations about the LO multisine

A. Hardware requirementsThe goal of the measurements is to investigate the effect of the phase noise of the local

oscillator signal on the IF output signal. The main idea is to construct a multisine that

approximates the real shape of the spectrum of the local oscillator, and to apply this LO

multisine to the mixer, together with the RF multisine. It is clear that such a measurement

requires the usage of a signal source whose phase noise is much smaller than the phase noise of

the local oscillator to be simulated. Otherwise, it will be impossible to create the multisine. A

second remark is that the frequency grid spacing of the LO multisine will be much smaller than

the frequency grid spacing of the RF multisine. Hence, the measurement equipment needs a

very high spectral resolution.

B. Construction and properties of the multisineThe LO multisine has to be an approximation of the phase noise spectrum. Its amplitude can be

constructed using the data sheets of the local oscillator, to match the actual phase noise

spectrum. What about the phases of the multisine? First, one can assume, without loss of

generality, that the phase of the component at the local oscillator fundamental frequency is

zero ( ). Indeed, by guaranteeing , the phase shift of the local oscillator

doesn’t need to be de-embedded out of the measured FRF, when performing mixer

measurements (see section 4.4.3). Next, one will determine the phase relations in the phase

noise spectrum. (4-48) can be rewritten as:

(4-49)

This represents a Quadrature Amplitude Modulated signal [43], with the in-phase component

being , and the quadrature component being . Consider that the variance

of is much smaller than , so that both and can be

approximated by the first term in their Taylor series expansion, i.e. and

. In that case, (4-49) can be written as:

(4-50)

ϕLO 0= ϕLO 0=

uLO t( ) ALO nθ t( )( ) 2πfLOt ϕLO+( ) ALO nθ t( )( )sin 2πfLOt ϕLO+( )sin–coscos=

nθ t( )( )cos nθ t( )( )sin

nθ t( ) π 2⁄ nθ t( )( )cos nθ t( )( )sin

nθ t( )( )cos 1≈

nθ t( )( )sin nθ t( )≈

uLO t( ) ALO 2πfLOt ϕLO+( ) ALOnθ t( ) 2πfLOt ϕLO+( )sin–cos=

219


Taking the DFT of (4-50), over an integer number of periods, yields

(4-51)

where is the frequency grid line corresponding to , represents the discrete Dirac

impulse (or Kronecker delta) ; . represents the

DFT spectrum of , and this DFT spectrum is a narrowband spectrum (typically a few

MHz [39]).

The phase noise spectrum consists of two parts: the spectrum of the pure sinewave

, superimposed on the spectrum of , centered around , and

given a phase shift of .


(4-52)

When , one can state that the phase of simply equals , (or zero in

the considered case).


(4-53)

Hence, the phases of the spectral components symmetrical to are not simply opposite in

sign. They are opposite in sign and each component has a phase shift of , see

Figure 4-34.

ULO k( )ALO

2----------ejϕLO δ k kLO–( ) δ k kLO+( )+( )=

Nθ k( )*ALO

2---------- δ k kLO–( )e

j π2--- ϕLO+

δ k kLO+( )ej π

2--- ϕLO+ –

+

+

kLO fLO δ k( )

δ k( ) 1= for k 0= δ k( ) 0= for k 0≠ Nθ k( )

nθ mTs( )

ALO 2πfLOt ϕLO+( )cos nθ t( ) fLO

90°+

k kLO=

ULO kLO( )ALO

2----------ejϕLO

ALO2

----------ej π

2--- ϕLO+

Nθ 0( )+=

Nθ 0( ) 1« ULO kLO( ) ϕLO

k kLO≠

ULO k( )ALO

2----------e

j π2--- ϕLO+

Nθ k kLO–( )ALO

2----------e

j– π2--- ϕLO+

Nθ k kLO+( )+=

fLO

π 2⁄ ϕLO+

220


Otherwise told, for , components symmetrical to the center frequency have

supplementary angles, i.e. the sum of the angles is always .

Since is Gaussian noise, the spectral components of are circular complex

normally distributed, and hence their phases are uniformly distributed over [ [. Hence the

phases of the spectral components of the phase noise spectrum will also be uniformly

distributed over [ [. Note that the phases of the lower part of the spectrum are

supplementary angles of a uniform distribution over [ [, yielding also a uniform

distribution over [ [.

Conclusion: the constructed LO multisine will be a random multisine, except for the

component at , which is a deterministic one.

4.5.3 Properties of the FRFThe goal of this section is to determine the in-band distortions in the frequency band around

. Hence, the FRF for a mixer as defined in section 4.4.3, will be studied, when a random

multisine is applied at both the RF and the LO port. The output spectrum at the IF port is given

by:

(4-54)

FIGURE 4-34. Schematic representation of the construction of the LO multisine, for .

ϕ π2---+ϕ– π

2---+ 0Phases

Amplitudes

ϕLO 0=

ϕLO 0=

π

nθ t( ) Nθ k( )

0 2π,

0 2π,

0 2π,

0 2π,

fLO

fIF

YIF k( ) YIF αβ( ) k( )β∑

α∑=

221


with . The smallest values for and to obtain spectral components around

is . All the other values for and that will yield spectral contributions

around are then given by with (see Theorem 4.6 in

Appendix 4.E). is a (positive) frequency grid line on which a component of the RF multisine

lies, and is the (positive) frequency grid line corresponding to .

The frequency response function of the mixer is given by (for ):

(4-55)

(4-55) consists of terms of the following form:

(4-56)

with the constraint . When not considering very high degrees

of nonlinearity, the constraint has to be split into two constraints and

(see Theorem 4.6 in Appendix 4.E).

Six distinct terms will contribute to the FRF:

(4-57)

1. is the FRF of the “underlying linear system”, i.e. the result of the Volterra

operator , or the term

(4-58)

Its phase is .

2. represents the bias or systematic nonlinear contributions to the FRF, both

with respect to the RF and the LO port. This value is independent of the random phases

of both the RF and the LO multisines. It consists of the terms:

α β, 2N 1+∈ α β

fIF α β 1= = α β

fIF α β,( ) 1 1,( ) 2 γ ε,( )+= γ ε, N∈

k

l fLO

k 0>

FRF k( ) Y k( )URF ideal, k( )--------------------------------

YIF αβ( ) k( )β∑

α∑

URF ideal, k( )----------------------------------------= =

Hαβ k1 … kα l1 … lβ, , , , ,( )URF k1( )…URF kα( )ULO l1( )…ULO lβ( )

URF ideal, k( )----------------------------------------------------------------------------------------------------

k1 … kα l1 … lβ+ + + + + k=

k1 … kα+ + k= l+

l1 … lβ+ + l–=

FRF k( ) FRFlin k( ) FRFBB k( ) FRFBS k( ) FRFSB k( ) FRFSS k( ) NFRF k( )+ + + + +=

FRFlin k( )

H11 uRF t( ) uLO t( ),[ ]

H11 k l+ l–,( )URF k l+( )ULO l–( )

URF ideal, k( )------------------------------------------------ H11 k l+ l–,( )ULO l–( )=

φH11k l+ l–,( ) ϕLO–

FRFBB k( )

222


where one of the indices ( ), and the other s are grouped in

pairs of opposite indices (this is possible since is odd). One of the indices

( ) and the other ‘s are grouped in pairs of opposite indices (this is

possible since is odd). The phase of these terms is .

3. represents the nonlinear contributions to the FRF, that are systematic with

respect to the RF port, and stochastic with respect to the LO port. This value is

independent of the random phases of the RF multisine, but depends on the random

phases of the LO multisines. It consists of the terms:

where one of the indices ( ), and the other s are grouped in

pairs of opposite indices (this is possible since is odd). It is not true that one of the

indices ( ) and the other s are grouped in pairs of opposite

indices. The phase of these terms is

4. represents the nonlinear contributions to the FRF, that are stochastic with

respect to the RF port, and systematic with respect to the LO port. This value is

independent of the random phases of the LO multisine, but depends on the random

phases of the RF multisines. It consists of the terms:

where it is not so that one of the indices ( ), and the other s

are grouped in pairs of opposite indices. But, it is true that one of the indices

( ) and the other s are grouped in pairs of opposite indices. The phase of

these terms is


URF ideal, k( )----------------------------------------------------------------------------------------------------

ki k l+= i 1 … α, , ∈ ki

α lj l–=

j 1 … β, , ∈ ljβ φHαβ

k1 … kα l1 … lβ, , , , ,( ) ϕLO–

FRFBS k( )


URF ideal, k( )----------------------------------------------------------------------------------------------------

ki k l+= i 1 … α, , ∈ ki

α

lj l–= j 1 … β, , ∈ lj

φHαβk1 … kα l1 … lβ, , , , ,( ) φULO

lj( )

j 1=

β

∑+

FRFSB k( )


URF ideal, k( )----------------------------------------------------------------------------------------------------

ki k l+= i 1 … α, , ∈ ki

lj l–=

j 1 … β, , ∈ lj

223


5. represents the nonlinear contributions to the FRF, that are stochastic with

respect to both the RF port and the LO port. This value depends on both the random

phases of the RF and the LO multisines. It consists of the terms:

where it is not so that one of the indices ( ), and the other ‘s

are grouped in pairs of opposite indices. And it is not so that one of the indices

( ) and the other ‘s are grouped in pairs of opposite indices. The phase of

these terms is

6. is a stochastic component on the FRF, due to the effect of the input noise, the

noise added by the mixer itself , or due to the noise added by the measurement

system. These contributions are assumed to be extremely small, as discussed in section

4.2

If the phases of the RF and the LO random multisines vary randomly, three circular complex

normally distributed noise sources created by the stochastic character of the input signals will

deteriorate the IF output signal. These noise sources are , and coming

respectively from , and . They hence characterize

respectively the stochastic contributions coming from the LO port, the stochastic contributions

coming from the RF port, and the stochastic contributions coming from both ports. These noise

sources are uncorrelated, and hence their powers can be added (see Appendix 4.G).

As proposed in section 4.3.5, the FRF of the related linear dynamic system

can be determined by averaging the measured FRF over

φHαβk1 … kα l1 … lβ, , , , ,( ) φURF

ki( )

i 1=

α

∑ ϕLO– φURFk( )–+

FRFSS k( )


URF ideal, k( )---------------------------------------------------------------------------------------------------

ki k l+= i 1 … α, , ∈ ki

lj l–=

j 1 … β, , ∈ lj

φHαβk1 … kα l1 … lβ, , , , ,( ) φURF

ki( )

i 1=

α

∑ φULOlj( )

j 1=

β

∑ φURFk( )–+ +

NFRF k( )

NA k( )

YBS k( ) YSB k( ) YSS k( )

FRFBS k( ) FRFSB k( ) FRFSS k( )

FRFR k( ) FRFlin k( )= FRFBB k( )+

224


the different phase instances of the RF and the LO multisines. Hence,

can also be determined for all instances of the multisine, and thus

, the variance of all the stochastic nonlinear contributions, i.e. the in-band

distortion can be calculated. Hence, the mixer (including the effect of the phase noise of the

local oscillator) can be modelled as an ideal frequency translating device, followed by a noisy

LTI system as shown in Figure 4-23. This time, the noise source also contains the effect of the

phase noise of the LO.

It is also possible to determine the variances , and (or powers) of

each of the three noise sources separately. Two methods can be used to determine these noise

powers. The first method requires the possibility to measure all the powers and phases at all the

ports, while the second method only requires a power measurement at the IF port.

A. By measuring the FRF• is obtained by applying a random multisine at the RF and the

LO port. From these measurements, one also obtains ,

using

(4-59)

and hence .

• If the same instance of the RF multisine is fed at the RF port, while the

LO multisine varies from instance to instance, the phases of the RF

multisine will be deterministic over the measurements, and hence the

average of the FRF will be:

(4-60)

From these measurements, one also obtains , using

(4-61)

and hence .

YSB k( ) YBS k( ) YSS k( )+ +

σYBS YSB YSS+ +2 k( )

σYBS

2 k( ) σYSB

2 k( ) σYSS

2 k( )

FRFR k( )

YBS YSB YSS+ +

YBS YSB YSS+ + YIF FRFR URF⋅–=

σYBS YSB YSS+ +2

E FRF k( ) FRFlin k( ) FRFBB k( ) FRFSB k( )+ +=

YBS YSS+

YBS YSS+ YIF E FRF URF⋅–=

σYBS YSS+2

225


• Next, if the same instance of the LO multisine is fed at the LO port, while

the RF multisine varies from instance to instance, the phases of the LO

multisine will be deterministic over the measurements, and hence the

average of the FRF will be:

(4-62)

From these measurements, one also obtains , using

(4-63)

and hence .

From the three measured variances , and , the variances

of each individual noise source can easily be calculated:

(4-64)

B. By creating notches in the multisinesSuppose that spectral component in the RF multisine. This means that

there will be no systematic contributions coming from the RF multisine in the spectral

component at the IF spectrum (see also section 4.3). Therefore,

and, : the

output spectrum in the notch will contain the powers of only two of the three noise sources, i.e.

the effect of the stochastic properties of the RF input and the effect of the combined RF and LO

stochastic input. Hence, will be measured. However, the effect of the stochastic

properties of the LO input signal, i.e. the disturbing effect of the phase noise on the systematic

nonlinear contributions, will not be measured in the notch.

Alternatively, when omitting a spectral component in the LO multisine, these notches will be

filled in the IF spectrum with only stochastic contributions of the LO multisine, i.e.

E FRF k( ) FRFlin k( ) FRFBB k( ) FRFBS k( )+ +=

YSB YSS+

YSB YSS+ YIF E FRF URF⋅–=

σYSB YSS+2

σYBS YSB YSS+ +2 σYBS YSS+

2 σYSB YSS+2

σYBS

2 σYBS YSB YSS+ +2 σYSB YSS+

2–=

σYSB

2 σYBS YSB YSS+ +2 σYBS YSS+

2–=

σYSS

2 σYBS YSS+2 σYSB YSS+

2 σYBS YSB YSS+ +2–+=

URF ideal, k0( ) 0=

YIF k0( )

FRFlin k0( ) FRFBB k0( ) FRFBS k0( ) 0= = = YIF k0( ) YSB k0( ) YSS k0( )+=

σYSB YSS+2

226


. Again, only the variance of two of the three noise

sources can be measured.

Finally, when a spectral component is omitted in both the RF and the LO multisine, there will

be a spectral line in the IF spectrum, that contains only stochastic contributions from the RF

and IF multisine, and no systematic contributions, i.e. . This means that the

variance in this notch corresponds to the variance of only one of the three noise sources.

With the knowledge of , and , the variance of each of the three

noise sources separately can be determined, was obtained immediately in the last

measurement, and the other two can be determined as follows:

(4-65)

Note that measuring the output power in the notches in order to get an estimate of the in-band

distortions is only valid if the phase noise of the local oscillator can be assumed to be

negligible.

YIF k( ) YBS k( ) YSS k( )+= σYBS YSS+2

YIF k( ) YSS k( )=

σYSS

2

σYSB YSS+2 σYBS YSS+

2 σYSS

2

σYSS

2

σYBS

2 σYBS YSS+2 σYSS

2–=

σYSB

2 σYSB YSS+2 σYSS

2–=

227


4.5.4 Experimental resultsThe measurements are performed on a mixer of type ZEM-4300 (Mini-Circuits) [27]. The

measurement setup is shown in Figure 4-35.

The local oscillator signal is a random multisine with a center frequency of 900 MHz,

lying on a frequency grid with a grid spacing of 1 kHz, and consisting of 12 components

(having a power of dBc) plus the center frequency component (this center frequency

component has a power of 7 dBm). Spectral components at kHz are omitted. Fifteen

FIGURE 4-35. Measurement setup

50Ω

NVNA

IQ-modulator

AWG AWG

RF source

I Q

XRF

LO

IF

reference clock

AM-modulator

AWG

RF source

uLO t( )

19–

900 3±( )

228


different instances of the random multisine were generated. Figure 4-36 shows the power

spectrum of the measured incident wave at the LO port, for all RF input power levels.

The RF signal is random multisine generated with a Rohde & Schwarz SMIQ06B

Vector Signal Generator [28], driven by two VXI Arbitrary Waveform Generators (HPE1445)

at its I and Q ports. The multisine has a center frequency of 1300 MHz, lying on a frequency

grid with a grid spacing of 5 kHz, and consisting of 64 components. This multisine is an odd

multisine, relative to the grid spacing of 5 kHz. Since the components lie on odd frequency

grid lines, the frequency difference between two consecutive lines is at least 10 kHz. The

multisine is divided in 8 blocks of 8 components, and with one detection line in each block, the

position of this detection line is randomized. Figure 4-37 shows the power spectrum of the

measured incident wave at the RF port, for all input power levels.

FIGURE 4-36. Power spectrum of the LO multisine centered around 900 MHz

FIGURE 4-37. Power spectrum of the RF multisine centered around 1300 MHz.

-40 -20 0 20 40-70

-60

-50

-40

-30

-20

-10

0

10


Pow

er [d

Bm

]

uRF t( )

229


The power of the SMIQ06B is swept from dBm to 10 dBm in 13 equidistant steps in the

logarithmic scale. Fifteen different instances of the random multisine were generated.

Figure 4-38 shows the power spectrum of the IF waveform, centered around MHz.

The data in this figure can be used to determine the in-band distortions, by looking at the

power in the notches. Using (4-65), the power of each of the three noise sources can be

determined. This result is shown in Figure 4-39.

The magenta lines show the power of the noise source , i.e. the effect of the stochastic

nonlinearities coming from the RF port only. The thin black and green lines show the power of

the noise source , i.e. the effect of the stochastic nonliearities coming from the LO port

FIGURE 4-38. Power spectrum of the IF waveform, centered around 400 MHz: (a) shows the spectral regrowth, while (b) zooms in on the band of interest.

FIGURE 4-39. Power of the three noise sources.

5–

(b)(a)

fIF 400=

-5 0 5 10-75

-70

-65

-60

-55

-50

-45

-40

Noi

se P

ower

[dB

m]


σYSB

2

σYBS

2

230


only, and the bold red and green lines show the power of the noise source , i.e. the

combined stochastic contributions coming from both the RF and the LO ports.

For the designed LO phase noise signal, the RF stochastic nonlinearities are about 10 dB larger

than the other noise sources. Hence, one can say that in this case, the in-band distortions are

essentially due to the stochastic nonliearities coming from the RF port.

σYSS

2

231


4.6 Conclusion When noise-like signals (such as the ones that are used in telecommunication environments)

are fed to NICE systems, the output noise (i.e. that part of the output spectrum that is not

correlated with the noise-like input signal) consists of two contributions. The first contribution

is due to the noise that is superimposed on the input signal , and that is processed by the

system, or the noise generated by the noisy NICE system itself . This is the noise that

was studied in previous chapters. The second contribution is noise that is created by the

nonlinear mechanism itself, based on the randomness of the input signal. The DFT spectrum of

this noise is uncorrelated with the DFT spectrum of the input signal :

. For narrowband NICE systems, the total power of the noise

superimposed on the input signal is rather small, and hence, the contribution of this noise to the

output noise can be neglected compared to the noise that the NICE system creates out of the

noise-like input signal.

A general framework using random multisines to model the noise-like input signals was

chosen. Multisines have the advantage that the magnitude and phase of each of their spectral

components can be modified at will, and the randomness of the phases assures a noise-like

behavior. Under this framework, it was shown that the output spectrum of the NICE system

contains two types of contributions. Systematic contributions yield a filtered version of the

input random multisine, while stochastic contributions can be modelled as a frequency

dependent noise source, that will be responsible for the in-band distortions. With this

description of the NICE system, using systematic and stochastic contributions, it can be shown

that the NPR and the CCPR method, two apparently contradictory methods in the literature,

used to quantify in-band distortions, both quantify the in-band distortions, but under different

assumptions. Hence, a NICE system excited by noise-like signals, can be modelled as a LTI

system and a noise source, i.e. as a noisy LTI system!

To extend the theory of the systematic and stochastic contributions towards multi-port devices,

the behavior of a mixer, excited with random multisines at its RF port was studied. The local

oscillator was in a first step assumed to be part of the mixer. Hence, the mixer which is a three-

port device could be treated as a two-port device. A frequency response function can then be

nu t( )

nA t( )

NNL k( ) U k( )

E U k( )NNL k( ) 0=

232

Conclusion

defined between the output of the ideal frequency translator, and the actual output of the mixer.

Hence, a mixer that has to shift noise-like signals whose power is constant in the frequency

band, can be modelled as an ideal frequency shifter, followed by a LTI system and a noise

source, i.e. as the cascade of an ideal frequency translator and a noisy LTI system!

In a second step, the local oscillator signal is no longer considered as a pure sinewave, but as a

real sinewave affected by phase noise. In order to know the effect of this phase noise on the

output spectrum of the mixer, the phase noise spectrum is modelled as a random multisine. A

random multisine is now applied at both input ports of the mixer. This time, the contributions

at the output spectrum cannot be simply tagged as stochastic or systematic: they will be

systematic or stochastic with respect to each input port. This leads to four contributions, with

each of the contributions being systematic or stochastic with respect to the RF port and the LO

port. The contribution that is systematic to both ports represents a filtered version of the

convolution of the RF and the LO multisine, and the three other contributions are responsible

for in-band distortions. They represent three independent noise sources: the effect of the

random character of the RF signal alone, the effect of the random character of the LO port

alone (i.e. the phase noise), and the combined random characters. Hence, a mixer that has to

shift noise-like signals whose power is constant in the frequency band, and whose local

oscillator is affected by phase noise, can be modelled as an ideal frequency shifter, followed by

a LTI system and three independent noise sources, i.e. as the cascade of an ideal frequency

translator and a noisy LTI system!

For all these models, two measurement methods are proposed. The first method requires the

measurement of both power and phase at each port of the device, and yields a frequency

response function in magnitude and phase, plus a noise source to quantify the in-band

distortions. The second method only requires the measurement of the power at each of the

ports, and the usage of notches in the random multisine. The latter measurement method only

yields the magnitude of the frequency response function, and the noise source.

233


4.7 Appendices

Appendix 4.A : Absence of an underlying linear system for amplifiers exhibiting cross-over distortionThe effect of cross-over distortion can be idealised as follows: when the input signal is smaller

than a certain threshold, the output signal will be zero. For a sine wave, this means that in a

certain interval around its zero crossings, the output signal will be zero. This is illustrated in

the following figures. An amplifier exhibiting cross-over distortion with a voltage gain of one

(i.e. 0 dB) and a threshold of 1 V (and a bandwidth of 2 kHz) was simulated. The red curve

represents the output signal, and the blue curve the input signal.

The criterion for existence of the underlying linear system is given by (4-9):

(4-66)

For cross-over distortion, when the root-mean-square value of the input signal tends towards

zero, the output signal will be zero, whenever the input signal amplitude is smaller than the

threshold value. Hence, one has to take a look at the evolution of the ratio

for decreasing .

FIGURE 4-40. Time waveform and amplitude spectrum for an input amplitude of 3 V.

yNL( )rms

yL( )rms----------------------

urms 0→lim 0=

yNL( )rms yL( )rms⁄

urms

0 0.01 0.02 0.03 0.04 0.05 0.06-3

-2

-1

0

1

2

3

time [s]

Am

plitu

de [V

]

-2000 -1000 0 1000 20000

0.5

1

1.5

Frequency [Hz]

Am

plitu

de [V

]

234

Appendices

When the input amplitude is larger than the threshold, the most power of the output signal is

located at the fundamental frequency of the sine wave. Hence the ratio is

very small.This is illustrated in Figure 4-40. The simulated ratio is there

0.11, i.e. dB power ratio.

When the input decreases towards the threshold value, the output signal resembles a pulse with

a small non-zero width that is repeated every time the sine wave reaches an extremum (see

Figure 4-41). The frequency spectrum resembles thus a discrete sinc function, and clearly, the

sum of the power at the higher order ( ) harmonics of the fundamental becomes larger than

the power at the fundamental frequency itself. (The ratio is here 2.33 or

7 dB power ratio).

FIGURE 4-41. Time waveform and amplitude spectrum for an input amplitude of 1.01 V

FIGURE 4-42. Time waveform and amplitude spectrum for an input amplitude equal to the threshold voltage.



19–

0 0.01 0.02 0.03 0.04 0.05 0.06-1

-0.5

0

0.5

1

time [s]

Am

plit

ud

e [

V]

-2000 -1000 0 1000 20000

0.1

0.2

0.3

0.4

0.5

Frequency [Hz]A

mp

litu

de

[V

]

2>


0 0.01 0.02 0.03 0.04 0.05 0.06-1

-0.5

0

0.5

1

time [s]

Am

plit

ud

e [

V]

-2000 -1000 0 1000 20000

0.1

0.2

0.3

0.4

0.5

Frequency [Hz]

Am

plit

ud

e [

V]

235


In limit that the input amplitude equals the threshold value, coming from large values (i.e. the

right limit), (this is the smallest value that can be given to the input signal amplitude in order to

obtain a non zero output signal), the output signal is a repeated impulse (with infinitesimal

small width, if the bandwidth of the system is infinity). The spectrum of this output signal is a

discrete “flat” spectrum, with the same amplitude at each integer multiple of the fundamental

frequency. Hence the ratio tends to infinity (5.56 or 15 dB power ratio for

the simulated signal), and one can conclude that the underlying linear system does not exist.

Appendix 4.B : Systematic and stochastic contributions of the FRFThe frequency response function at the -th frequency grid line is given by [9]:

(4-67)

Where represents the contribution of the -th order Volterra kernel to the output

spectrum. is given by (1-40):

(4-68)

With

(4-69)

and

(4-70)

Hence, the -th order contribution to the FRF is given by or,


k

FRF k( ) Y k( )U k( )------------

Y α( ) k( )

α 1=

∞

∑

U k( )-------------------------------= =

Y α( ) k( ) α

Y α( ) k( )

Y α( ) k( ) … Hα k1 … kα 1– L, , ,( )U k1( )…U kα 1–( )U L( )

kα 1– M 2⁄–=

M 2⁄ 1–

∑k1 M 2⁄–=

M 2⁄ 1–

∑=

L k k1 … kα 1–+ +( )–=

U k( ) DFT x mTs( )( ) 1M----- x mTs( )e

j2πkM

---------m–

m 0=

M 1–

∑= =

α Y α( ) k( ) U k( )⁄

236

Appendices

(4-71)

The term will be a stochastic one if its phase is a

function of the random phases of the input spectral lines. On the other hand, if the phase of this

term is not a function of the random phases of the input spectral lines, the term will represent a

deterministic contribution, and hence be a part of the systematic contributions to the FRF.

Since the denominator has a random phase, a first requirement for (4-71) to be a

deterministic term is that one of the numerator’s factors is equal to the denominator. Take e.g.

, hence (4-69) becomes: . A second constraint has to be put on the

remaining factors: since has random phase, the only way to cancel this random

phase is that one of the remaining indices equals . In other words, the

remaining factors have to be grouped in pairs , such that their phases cancel.

Hence, if is even (i.e. is odd), there will be terms that are systematic contributions of

the FRF.

If the above constraints are not fulfilled, the term will be a stochastic contribution of the FRF.

On the other hand, if is odd (i.e. is even), only stochastic contributions to the FRF will

be created.

Appendix 4.C : IF spectrum of the ideal mixerConsider the RF input signal of the mixer to be a random multisine

(4-72)

FRF α( ) k( ) … Hα k1 … kα 1– L, , ,( )U k1( )…U kα 1–( )U L( )

U k( )---------------------------------------------------------

kα 1– M 2⁄–=

M 2⁄ 1–

∑k1 M 2⁄–=

M 2⁄ 1–

∑=

Hα k1 … kα 1– L, , ,( )U k1( )…U kα 1–( )U L( )

U k( )---------------------------------------------------------

U k( )

L k= k1 … kα 1–+ + 0=

α 1– U k1( )

k2 … kα 1–, , k1– α 1–

U ki( )U ki–( )

α 1– α

α 1– α

uRF t( ) Uκ 2πfκ t ϕκ+( )cosκ 1=

N

∑=

fκ fmin κ 1–( )fmax fmin–

N 1–---------------------------+=

237


(see also Definition 4.2 and (4-10)). If , the output signal

at the IF port of the ideal mixer will then be given by:

(4-73)

Using Simpson’s formula , (4-73) becomes:

(4-74)

showing that the original multisine is duplicated, and these duplicates are shifted in

frequency and scaled. Around , the original multisine, scaled with

can be found, while around , the original multisine, scaled with

is present.


uIF t( ) ALO 2πfLOt ϕLO+( )cos Uκ 2πfκ t ϕκ+( )cosκ 1=

N

∑=

ALO Uκ 2πfκ t ϕκ+( ) 2πfLOt ϕLO+( )coscosκ 1=

N

∑=

α( )cos β( )cos α β+( )cos α β–( )cos+( ) 2⁄=

uIF t( )ALO

2---------- Uκ 2π fκ fLO–( )t ϕκ ϕLO–+( )cos

κ 1=

N

∑=

ALO

2---------- Uκ 2π fκ fLO+( )t ϕκ ϕLO+ +( )cos

κ 1=

N

∑+

uRF t( )

fIF fRF fLO–=

ALO ejϕLO–

⋅ 2⁄ fRF fLO+

ALO ejϕLO⋅ 2⁄

238

Appendices

Appendix 4.D : Comparing two models for a nonlinear multiplierThe two models for the nonlinear multiplier that are compared, are shown in Figure 4-43.

The first model is a nonlinear (NICE) system with two inputs and one output, while the second

model is the multiplication of two nonlinear (NICE) systems.

The output signal of the first model can be written as (4-24):

(4-75)

where

(4-76)

While for model 2, the output signal can be written as:

FIGURE 4-43. Two models for the nonlinear multiplier.

Xnonlinear multiplier

uRF ′ t( )

yIF ′ t( )

uLO ′ t( )

uRF ′ t( )

uLO′ t( )

yIF ′ t( )non-linear

MODEL 1

X yIF ′ t( )

ideal multiplieruRF ′ t( )

uLO ′ t( )

non-linear

non-linear

MODEL 2

yIF ′ t( )

yIF ′ t( ) Hαβ uRF′ t( ) uLO ′ t( ),[ ]

β 0=

∞

∑α 0=

∞

∑=

Hαβ uRF ′ t( ) uLO ′ t( ),[ ] =

… hαβ τ1 … τα β+, ,( )uRF ′ t τ1–( )…uRF ′ t τα–( ) …⋅

∞–

∞

∫∞–

∞

∫

uLO ′ t τα 1+–( )…uLO ′ t τα β+–( )dτ1…dτα β+

yIF ′ t( )

239


(4-77)

where

(4-78)

The two models will be equivalent, if (compare (4-78) to (4-76)):

(4-79)

In other words, the two-input single output NICE system can be represented by a

multiplication of two NICE systems, only if all its Volterra kernels can be written as the

product of a kernel of the RF NICE system and a kernel of the LO NICE system. Since this is

not always possible, model 1 is a more general model than model 2.

The fact that the separability is not always valid can be shown through a simple counter-

example: Suppose that a two-input ( ), one-output ( ) NICE system multiplies

both input signals, and then applies this product to a lowpass filter, with impulse response .

In this case, the output signal can be written as:

(4-80)

Or, in order to obtain the kernel , (4-80) equals:

yIF ′ t( ) HαRF uRF′ t( )[ ]

α 0=

∞

∑

HβLO uLO′ t( )[ ]

β 0=

∞

∑

=

HαRF uRF ′ t( )[ ] Hβ

LO uLO ′ t( )[ ]

β 0=

∞

∑α 0=

∞

∑=

HαRF uRF ′ t( )[ ] Hβ

LO uLO ′ t( )[ ] =

… hαRF τ1 … τα, ,( )hβ

LO τ1 ′ … τ β′, ,( )uRF ′ t τ1–( )…uRF ′ t τα–( ) …⋅

∞–

∞

∫∞–

∞

∫

uLO ′ t τ1 ′–( )…uLO ′ t τβ′–( )dτ1…dταdτ1′…dτβ′

hαβ τ1 … τα β+, ,( ) hαRF τ1 … τα, ,( ) hβ

LO τ1 ′ … τ β′, ,( )⋅=

u1 t( ) u2 t( ), y t( )

e t–

y t( )

y t( ) e t– * u1 t( )u2 t( )( ) e τ1– u1 t τ1–( )u2 t τ1–( ) τ1d∞–

∞∫= =

h11 τ1 τ2,( )

240

Appendices

(4-81)

Yielding the kernel , or in its symmetrized form:

(4-82)

Hence, it is clear that (4-82) cannot be written as product of two first order kernels

Appendix 4.E : Output spectral components of a two input NICE system where a single tone signal is applied at both input portsConsider a two input NICE system, where a spectral component at frequency is applied at

port 1, and a spectral components at frequency is applied at port 2. Suppose that , and

that the frequency difference between the components is small as compared to their

frequencies, i.e. . This is typically the situation that occurs for mixers, where

port 1 is the RF port, and port 2 is the LO port. The frequencies of the signals applied at these

ports are situated in the orders of magnitude of the GHz, while 11.1 MHz is a very commonly

used IF frequency. In that case, one can rename frequency as frequency , and frequency

as frequency , with . The output spectrum of the system will then consist

of clusters of frequency components, that are centered around frequency

( ), or, in the vicinity of , since . Note that for very high degrees of the

nonlinearity, all these clusters will cross each other, since mathematically speaking, the

( )-th degree nonlinearity of and the ( )-th degree nonlinearity of

will fall at the same frequency grid line. This will first occur at the smallest common

multiple of and . However, since both frequencies are high, and lie close to each

other, this smallest common multiple will be an extremely high frequency. (e.g. the smallest

common multiple of 1.5 GHz and 1.45 GHz is 43.5 GHz, needing a nonlinearity of at least

29th degree to produce a frequency component that high.) Hence, one can conclude that the

output spectrum are distinct clusters of spectral components.

e τ1– δ τ1 τ2–( )u1 t τ1–( )u2 t τ2–( ) τ1d τ2d∞–

∞∫∞–

∞∫

h11 τ1 τ2,( ) e τ1– δ τ1 τ2–( )=

h11 τ1 τ2,( )e τ1– δ τ1 τ2–( ) e τ2– δ τ2 τ1–( )+

2-----------------------------------------------------------------------------=

h1A τ1( ) h1

B τ2( )⋅

f1f2 f2 f1<

f1 f2– f1 f2,«

f2 f0f1 f0 ∆+ ∆ f1 f2–=

z f0 ∆ 2⁄+( )⋅

z Z∈ zf0 ∆ 2⁄ f0«

f0 1Hz⁄ f0 ∆+ f0 ∆+( ) 1Hz⁄

f0f0 f0 ∆+

241


For each output spectral component, it is possible to determine what degrees of each input port,

or which order Volterra operator contributes to that component.

Theorem 4.5

1. The smallest values of and that are needed to obtain a frequency component at

frequency with is , , .


frequency (with ) is , , .


frequency (with ) is , , .

proof.

1. . Hence,

2. . Hence,

3. . Hence,

Theorem 4.6

If the Volterra operator contributes to a certain frequency component in the output

spectrum of a two input NICE system, then all the Volterra operators with

, will also contribute to that frequency component.

Hαβ [ ]

α β

mf0 µ∆+ µ 0 1 … m, , , ∈ α β+ m= α µ= β m µ–=

α β

mf0 m ν+( )∆+ m ν, N∈ α β+ m 2ν+= α m ν+= β ν=

α β

mf0 ν∆– m ν, N∈ α β+ m 2ν+= α ν= β m ν+=

mf0 µ∆+ mf2 µ f1 f2–( )+ µf1 m µ–( )f2+= =

α µ =β m µ–=α β+ m=

mf0 m ν+( )∆+ mf2 m ν+( ) f1 f2–( )+ m ν+( )f1 νf2–= =

α m ν+=β ν =

α β+ m 2ν+=

mf0 ν∆– mf2 ν f1 f2–( )– ν– f1 m ν+( )f2+= =

α ν =β m ν+=

α β+ m 2ν+=

Hαβ [ ]

Hγε [ ]

γ ε,( ) α β,( ) 2 z1 z2,( )+= z1 z2, N∈

242

Appendices

proof

The frequency component is created by combining spectral components from port

1, lying at frequency grid lines and spectral components from port 2, lying at

frequency grid lines , such that

(4-83)

It is clear that (4-83) also can be written as:

(4-84)

where and are arbitrarily chosen natural numbers. Or, by combining spectral

components from port 1, and spectral components from port 2, it is still possible to

obtain frequency grid line . G

Note that for narrow band signals, centered around and , one can assume that the

conclusions of this appendix are still valid, if the bandwidth of those signals is much smaller

than . The motivation for this claim is similar to the first part of this appendix: In

the previous part of this appendix, the smallest frequency spacing between two consecutive

spectral components of the output spectrum was . Hence, for narrow band signals, one can

consider that the smallest frequency spacing between two consecutive center frequencies in the

output spectrum is also . Thus, assuming that the largest input bandwidth of both narrow

band signals is , at least a ( )-th degree nonlinearity is needed before the spectra

around two consecutive center frequencies overlap. But since was assumed, this degree

will be very high.

Appendix 4.F : Systematic and stochastic contributions of the mixer’s FRFConsider the RF input signal of the mixer to be a random multisine

Y αβ( ) k( ) α

k1 … kα, , β

l1 … lβ, ,

k kii 1=

α

∑= ljj 1=

β

∑+

k kii 1=

α

∑= lj ki ′ ki ′–( )+( ) lj ′ lj ′–( )+( )

j ′ 1=

z2

∑+i ′ 1=

z1

∑+j 1=

β

∑+

z1 z2 α 2z1+

β 2z2+

k

f1 f2

∆ f1 f2–=

∆

∆

B0 B0 ∆⁄

B0 ∆«

243


(4-85)

(see also Definition 4.2 and (4-10)), and consider the LO input signal to be a pure sine wave:

. The output spectrum at the IF port at frequency

will then be given by:

(4-86)

with (see Appendix 4.E for motivation of this choice for and ). Assume

that is the (positive) frequency grid line of the local oscillator sine wave ( , with

the frequency grid spacing).

In this case, the frequency response function of the mixer is given by:

(4-87)

Just as shown in Appendix 4.B, (4-87) will consist of terms of the following form:

(4-88)

with the constraint . When not considering very high degrees

of the nonlinearity, the constraint can be split into two constraints and

(see the first part of Appendix 4.E). Knowing that

(this follows out of the definition of ), and assuming that

is a constant, all these terms (4-88) can be split into two disjunct sets:

1. The term (4-88) will be a stochastic one if its phase is a function of the random phases of

the RF input spectral lines, and hence be a part of the stochastic contributions of the FRF.

uRF t( ) Uκ 2πfκ t ϕκ+( )cosκ 1=

N

∑=

fκ fmin κ 1–( )fmax fmin–

N 1–---------------------------+=


fIF fRF fLO–=

YIF k( ) YIF αβ( ) k( )β∑

α∑=

α β, 2N 1+∈ α β

l fLO l∆f=

∆f

FRF k( ) Y k( )URF ideal, k( )--------------------------------

YIF αβ( ) k( )β∑

α∑

URF ideal, k( )----------------------------------------= =


URF ideal, k( )----------------------------------------------------------------------------------------------------

k1 … kα l1 … lβ+ + + + + k=

k1 … kα+ + k l+=

l1 … lβ+ + l–=

ULO l–( ) ALOe jϕLO– 2⁄= uLO t( )

ϕLO

244

Appendices

2. The term (4-88) will be a deterministic one if its phase is not a function of the random

phases of the RF input spectral lines, and hence be a part of the systematic contributions

to the FRF.

Note that no extra constraint is required for the factor . The reason

therefore is that since is a pure sinewave, the indices can only be , and due

to the constraint and the constraint that is an odd natural number, the

expression must be:

(4-89)

and hence the phase of will always be .

Appendix 4.G : The three noise sources are uncorrelated

In this appendix, the independence of the noise sources , and will be

shown. In other words, the power of the sum of the noise sources has to be equal to the sum of

the power of each noise source. The power of the combined noise sources is given by (the

dependence of the frequency grid line is omitted to enhance readability):

(4-90)

Next, it will be shown that each of the terms of the second line of (4-90) is zero:

1. The phase of depends both on the random phase of the RF multisine (through

)and on the random phase of the LO multisine (through ). Hence, its

expectation over the instances will be zero.

2. For , it might be possible that the random phases of the RF multisine cancel

each other, e.g.:

ULO l1( )…ULO lβ( )

uLO t( ) l1 … lβ, , l±

l1 … lβ+ + l–= β

ULO l1( )…ULO lβ( )

ULO l1( )…ULO lβ( ) ULO l–( )( )β 1+

2------------

ULO l( )( )β 1–

2------------

⋅=

ULO l1( )…ULO lβ( ) ϕLO–

YBS k( ) YSB k( ) YSS k( )

k

E YBS YSB YSS+ + 2 E YBS2 E YSB

2 E YSS2 + +=

2Re E YSBYBS* ( ) 2Re E YSBYSS

* ( ) 2Re E YBSYSS* ( )+ + +

YSBYBS*

YSB YBS*

YSBYSS*

245


(4-91)

But even in that case, remains a function of the random phases of the LO

multisine, and thus its mean value will be zero.

3. For , a similar reasoning as for can be done. This time, the phases of

the LO multisine might cancel for a special case, but will remain a function of

the random phases of the RF multisine, yielding a zero average value.

Conclusion:

YSB H31 k1 … k3 l–, , ,( )URF k1( )URF k2( )URF k3( )ULO l–( )=

YSS* H33

* k1 … k3 l1 … l3, , , , ,( )URF* k1( )URF

* k2( )URF* k3( )ULO

* l1( )ULO* l2( )ULO

* l3( )=

YSBYSS*

YBSYSS* YSBYSS

*

YBSYSS*

E YBS YSB YSS+ + 2 E YBS2 E YSB

2 E YSS2 + +=

246

CONCLUSIONS AND IDEAS FORFURTHER RESEARCH

This work tried to describe the way noise is treated by a particular class of nonlinear systems,

i.e. the NICE systems. This class of NICE systems is general enough to describe most of the

systems and circuits used in practical telecommunications equipment. In opposition to the

linear systems, a separated study of the signal and the noise behavior cannot be done when

dealing with NICE systems. The interaction between the signal and the noise due to the

nonlinearity of the NICE system itself makes this separated approach impossible. Since the

noise power is usually much smaller than the signal power, the effect of the noise on the signal

output will be quite small, and hence the signal behavior can be studied as if there was no noise

present, as is done in the literature [41]. However, the noise behavior strongly depends on the

signal properties, and therefore can only be studied together with the signal.

The linear system theory describes the noise behavior of a linear system, using the noise figure.

This quantity describes the signal-to-noise ratio deterioration of a signal, when it is applied to a

linear system, and can be measured using the well-known Y-factor technique. This technique

applies two noise sources at the input of a system, and determines the noise figure out of the

ratio of the respective output powers. However, simply applying measurement techniques

designed for LTI systems to NICE systems can have serious consequences! In its commercial

form, the Y-factor technique is blind for the nonlinearities in a NICE system, because the

power of the noise excitation signals used is far too small. Hence, the measurement technique

247

only sees the underlying linear system of the NICE system and returns the linear noise figure.

Boosting the power levels of the excitation signals in order to detect the nonlinearities is also a

bad idea as it can yield completely meaningless results such as negative power ratios.

Two cases have to be considered when describing the noise behavior of noisy NICE systems,

depending on the input signal.

The first case consideres an input signal consisting of a pure sinewave. Since the noise figure is

independent of the input signal, this quantity is not suited to describe adequately the noise

behavior of NICE systems. Therefore, the definition of the noise figure is extended towards the

NICE noise figure, that is defined as an input signal and noise power dependent signal-to-noise

ratio degradation. For a given input noise power spectral density, the NICE noise figure is an

increasing (when ) or decreasing (when ) function of the input signal

power. Hence, under some special circumstances ( ), it is even possible that the

signal-to-noise ratio at the output of the NICE system is larger than the signal-to-noise ratio at

the input of the NICE system. This is a phenomenon that is impossible when dealing with

linear systems. For input noise power spectral densities up to 10 dB back-off, the NICE noise

figure variation as function of the input signal power is identical to the behavior at ,

but with a modified linear noise figure that decreases with increasing input noise PSD.

Calculations and experimental results also showed that below the 1 dB compression point, the

noise power gain (i.e. the ratio of noise power spectral density at the output to the noise power

spectral density at the input of the system) is a quantity that in first approximation only

depends on the total power of the input signal.

In the second case, random, noise-like signals are applied to the noisy NICE system, as is often

the case in telecommunication equipment. The randomness of these input signals lies in the

stochastic nature of information itself. The nonlinear mechanism in the NICE systems is able

to create output noise from the random property of these noise-like signals. This type of noise

is much larger than the noise that is stochastic with respect to time, and will hence be the main

term for distortions at the output of the system. The nonlinear mechanism produces two types

of output contributions: systematic contributions and stochastic contributions. The systematic

NFlin 2> NFlin 2<

NFlin 1≈

Nin N0=

248

ones are responsible for a filtering effect of the input signals, while the stochastic contributions

present themselves as the noise source that is responsible for in-band distortions. This theory

can be extended towards multi-port devices such as mixers. If the input power of the signals is

constant, a mixer can be modeled as the cascade of an ideal frequency translator and a noisy

LTI system. The noise source inside this LTI system is generated by the randomness of the

signals at the RF and the LO port of the mixer (i.e. due to the stochastic content of the

information, and due to the phase noise of the local oscillator).

Further research

A. Extension of the systematic and stochastic contributions towards MIMO systems.In the last chapter, the extension of the technique involving the systematic and the stochastic

nonlinear contributions was already extended towards a “two-input one-output” NICE system,

i.e. a mixer. This theory can easily be extended to Multiple Input Multiple Output (MIMO)

systems. Since a MIMO system is a parallel circuit of a number of Multiple Input Single

Output (MISO) systems, the theory only needs to be extended towards MISO systems.

Consider e.g. a MISO system with input ports, and at each input port, a random multisine is

applied. In this case, the Frequency Response Function will consist of significant terms,

each having a label that tells whether it contains stochastic or systematic contributions for each

input port. The obtained results can then be used to characterize e.g. an IQ modulator such as

the SMIQ06B [28], used in measurements in chapter 4. This IQ modulator is a “three-input

one-output” NICE system.

B. A full multi-port description.All the techniques in this work assumed that there was a perfect match at all the ports of the

considered device, and only the effect of the noise on the forward gain parameter ( for a

two-port system) was studied. In reality however, mismatch can occur e.g. at the output of the

system, resulting in the noise power being reflected into the output port of the system, and

hence re-appearing (modified) in the reflected input wave. Multiple reflections can hence lead

to a complex noise behavior of the studied system. The use of the noise models for nonlinear

systems have to be generalized to full multi-port operation. Using these true multi-port models,

M

2M

S21

249

accurate predictions of high-level parameters such as the bit-error-rate (BER) and their

dependency on nonlinearities and impedance matching conditions can be obtained.

Hence, in order to obtain a full multi-port model that is valid for a wide range of compression

levels, signal-to-noise ratios and matching impedances, two essential noise sources need to be

extended towards true multi-port behavior:

1. The noise, stochastic with respect to time, i.e. the noise generated by random physical

processes. A closed theoretical framework for the interaction between the nonlinearity

and this type of noise is required. To obtain this, the influence of the wide-band matching

conditions on the noise behavior needs to be investigated.

Based on the obtained theoretical framework, a modelling approach can be built up. For

fairly low noise powers (such as the noise at standard temperature), it is expected that the

noise part of the model can be measured based on linear techniques. Experimental

verification requires the integration of the standard noise-gain analysis and the vectorial

network analyser for nonlinear systems (NVNA) [25] to extract a full signal and noise

model in the first place. Using the considered system in different, but model compatible,

settings then allows to evaluate the prediction power of the model, based on an additional

set of measurements.

2. The noise, created by the nonlinear mechanism itself, out of the stochastic properties of

the input signals. For these noise contributions, depending on the random character of

the input signal, over the different realisations, the modelling is based on the measured

response of the system to a random multisine excitation. The in-band distortions can be

measured by e.g. creating notches in the considered multisine, as described in chapter 4.

The results obtained can then be extended towards multi-port systems such as mixers.

C. The non-harmonic phase calibration problem.In this work, the NVNA was often used in a special mode, where spectral rich, non-

harmonically related multisine signals are used. One of the open problems in this mode of

operation is the absolute phase calibration of the measured signals. For harmonically related

250

signals, the phase calibration was performed using a Step Recovery Diode, that produces a

fundamental tone and all its harmonics [40]. The phase relations of the harmonics of this

fundamental tone are well-known quantities. When measuring the phases of the multisines, the

center frequency was calibrated in phase, and since the multisines were narrowband signals, it

was reasonable to assume that the same correction can be applied to all the components of the

multisine. It is however clear that for a broadband multisine, this hypothesis will no longer be

true.

To improve the accuracy of the phase calibration, a model based approach has to be used.

Using a parametric model for the calibration signal as a function of the fundamental sinewave,

used to generate the reference impulses, will allow removal of the assumptions and provide

calibration at arbitrary interpolated frequencies, where the spectrum of the phase standard is

not specified in the current non parametric approach.

251

APPENDIX A

References

253

[1] G. Vendelin, A. Pavio and U. Rohde, “Microwave Circuit Design using Linear and

Nonlinear Techniques”, John Wiley & Sons, New York, 1990.

[2] W. Davenport Jr. and W. Root, “An Introduction to the Theory of Random Signals and

Noise”, IEEE press, New York, 1987.

[3] H. Young, “University Physics”, 8th edition, Addison Wesley, New York, 1992

[4] A. Papoulis, “Probability, Random Variables and Stochastic Processes”, McGraw-Hill,

London, 1965“

[5] B.M. Oliver, “Noise Figure and Its Measurement”, Hewlett-Packard Journal, Vol. 9, No.

5, Jan 1958, pp. 3-5

[6] H. Friis, “Noise Figures of Radio Receivers”, Proceedings of the IRE, Jul 1944, pp. 419-

422.

[7] Martin Schetzen, “The Volterra and Wiener theories of nonlinear systems”, John Wiley

& Sons, New York, 1980

[8] B. Thibaud, C. Arnaud, M. Campovecchio et al. “Measurements of time domain voltage/

current waveforms at R.F. and microwave frequencies, based on the use of a Vector

Network Analyser, for the characterization of nonlinear devices. Application to high

efficiency power amplifiers and frequency mulipliers”, IEEE Transactions on

Instrumentation and Measurement, Vol. 47, pp.1259-1264.

[9] R. Pintelon and J. Schoukens, “System Identification: A Frequency Domain Approach”,

IEEE Press, New Jersey, 2001

[10] A. Di Paola, M. Sanninno, “Noise figure test-set for simultaneous measurements of gain

and noise figure of microwave devices”, 8th Mediterranean Electrotechnical Conference

MELECON ‘96, 1996, Volume 3, pp. 1368-1370.

[11] Wait Randa, “Amplifier noise measurements at NIST”, IEEE Transactions on

Instrumentation and Measurement, Vol. 46, pp. 482-485.

[12] K. Narendra and P. Gallman, “An iterative method for the identification of nonlinear

systems using a Hammerstein model”, IEEE Transactions on Automatic Control, 1966,

Vol. 11, pp. 546-550.

254

[13] “HP 346A/B/C Noise Source Operating and Service Manual”, Hewlett Packard, USA,

1992

[14] “HP8565E-series and EC series Spectrum Analyzers User’s Guide”, Hewlett Packard,

USA, 1998

[15] “Spectrum Analysis ... Noise Figure Measurements”, Hewlett Packard Application Note

150-9, April 1976

[16] “Spectrum Analyzer Measurements and Noise”, Hewlett Packard Application Note

1303, 1998

[17] Claude E. Shannon, “A Mathematical Theory of Communication”, The Bell System

Technical Journal, Vol. 27, pp.379-423, 623-656, July, October 1948

[18] A. Bower, “NICAM 728 - Digital Two-Channel Sound for Terrestrial Television”, BBC

Research Department Report, 1990

[19] R. Thoma, H. Groppe, U. Trautwein and J. Sachs, “Statistics of Input Signals for

Frequency Domain Identification of Weakly Nonlinear Systems in Communications”,

Instrumentation and Measurement Technology Conference, Brussels, Belgium, 1996

[20] “Noise Power Ratio (NPR) Measurements Using the HP E2507B/E2508A Multi-format

Communications Signal Simulator”, HP Product Note E2508-1, 1997

[21] T. Reveyrand, D. Barataud, J. Lajoinie et al., “A novel experimental noise power ratio

characterization method for multicarrier microwave power amplifiers”, 55th ARFTG

Conference, Boston, USA, June 2000

[22] J.C. Pedro and N.B. Carvalho, “On the Use of Multitone Techniques for Assessing RF

Components’ Intermodulation Distortion”, IEEE Transactions on Microwave Theory

and Techniques, Vol. 47, pp. 2393-2402, Dec. 1999

[23] J.C. Pedro and N.B. Carvalho, “A novel set-up for co-channel distortion ratio

evaluation”, IEEE MTT-S Symposium, Boston, USA, June 2000

[24] J. Schoukens, T. Dobrowiecki and R. Pintelon, “Parametric and Nonparametric

Identification of Linear Systems in the Presence of Nonlinear Distortions - A Frequency

Domain Approach”, IEEE Transactions on Automatic Control, Vol. 43, pp. 176-191,

Feb. 1998

255

[25] T. Van Den Broeck and J. Verspecht, “Calibrated Vectorial Nonlinear Network

Analyzer”, IEEE-MTT-S Symposium, San Diego, USA, 1994, pp. 1069-1072.

[26] P. Guillaume, J. Schoukens, R. Pintelon and I. Kollar, “Crest-Factor Minimization Using

Nonlinear Chebychev Approximation Methods”, IEEE Transactions on Instrumentation

and Measurement, Vol. 40, pp. 982-989, December 1991.

[27] “RF/IF Designer’s Guide”, Mini-Circuits, USA, 2000

[28] “SMIQ02B/03B/04B/06B Vector Signal Generator Operating Manual”, Rohde &

Schwarz, Germany, 1999

[29] R. Swerdlow, “Analysis of Intermodulation Noise in Frequency Convertors by Volterra

Series”, IEEE Transactions on Microwave Theory and Techniques, Vol. 26, April 1978

pp. 305-313

[30] J. C. Pedro and N. B. Carvalho, “Analysis and Measurement of Multi-tone

Intermodulation Distortion of Microwave Frequency Converters”, IEEE MTT-S

Symposium, Phoenix, USA, May 2001

[31] K. Vanhoenacker, T. Dobrowiecki and J. Schoukens, “Design of Multisine Excitations to

Characterize the Nonlinear Distortions During FRF-measurements”, Instrumentation

and Measurement Technology Conference, Baltimore, USA, May 2000

[32] L. Råde and B. Westergen, “BETA Mathematics Handbook”, Chartwell-Bratt Ltd, 1990,

England

[33] D. M. Pozar, “Microwave Engineering”, 2nd. edition, John Wiley & Sons, New York,

1998

[34] N.B. Carvalho and J.C. Pedro, “Compact Formulas to relate ACPR and NPR to Two-

tone IMR and IP3”, Microwave Journal, December 1999

[35] M. Kendall and A. Stuart, “The Advanced Theory of Statistics, Volume 2: Inference and

Relationship”, 4th edition, Charles Griffin & Company Limited, London, 1979

[36] K. Kurokawa, “Power Waves and the Scattering Matrix”, IEEE Transactions on

Microwave Theory and Techniques, March 1965, pp. 194-202

[37] E. Oram Birgham, “The Fast Fourier Transform”, Prentice-Hall, New Jersey, 1974.

256

[38] B. Picinbono, “Random Signals and Systems”, Prentice Hall Inc., New Jersey, 1993.

[39] “The ARRL handbook for Radio Amateurs”, The American Radio Relay League,

Newington, USA, 1996

[40] J. Tatum, “Comb Generators and Amplifiers form Active Multipliers”, Microwaves &

RF Magazine, August 1996

[41] W. Van Moer, “Development of New Measuring and Modelling Techniques for RFICs

and their Nonlinear Behaviour”, Ph. D. Disseration, Vrije Universiteit Brussel, June

2001.

[42] W. Chan, “DSL Simulation Techniques and Standards Development for Digital

Subscriber Line Systems”, Macmillan Technical Publishing, Indianapolis, 1998.

[43] J. Proakis, “Digital Communications”, 3rd edition, McGraw-Hill, New York, 1995.

[44] G.L. Heiter, “Characterization of Nonlinearities in Microwave Devices and Systems”,

Transactions on Microwave Theory and Techniques, December 1973, pp. 797-805.

[45] J.C. Gillespie, “Noise Loading of FM Systems. The Noise-Power-Ratio (NPR) and

Customer Requirements”, 1973 G-MTT International Microwave Symposium Digest of

Technical Papers, p. 113

257

APPENDIX B

Publications

259

A. Periodical Papers[1] A. Geens and Y. Rolain, “Noise Figure Measurements on Nonlinear Devices”, IEEE

Transactions on Instumentation and Measurement, Vol. 50, No 4, August 2001, pp. 971-

975.

[2] W. Van Moer, Y. Rolain and A. Geens, “Measurement Based Modelling of Spectral

Regrowth”, IEEE Transactions on Instumentation and Measurement, Vol. 50, No 6,

December 2001.

[3] A. Geens, Y. Rolain, K. Vanhoenacker and J. Schoukens, “Discussion on Fundamental

Issues of NPR Measurements”, IEEE Transactions on Instumentation and Measurement,

under review.

B. Conference Papers[4] A. Geens and Y. Rolain, “Noise Figure Measurements on Nonlinear Devices”,

Proceedings of the 17th Instrumentation and Measurement Technology Conference,

Baltimore, USA, May 2000, pp. 796-801.

[5] A. Geens and Y. Rolain, “Noise Behavior of an Amplifier in Compression”, 55th ARFTG

Conference Digest, Boston, USA, June 2000, pp. 127-134.

[6] W. Van Moer, Y. Rolain and A. Geens, “Measurement Based Nonlinear Modelling of

Spectral Regrowth”, IEEE-MTT-S Digest, Boston, USA, June 2000, pp. 1467-1470.

[7] A. Geens and Y. Rolain, “On the Global Noise Behavior of a Third Order Nonlinearity”,

30th European Microwave Conference Proceedings, Paris, France, October 2000, pp.

159-162.

[8] A. Geens, Y. Rolain, K. Vanhoenacker and J. Schoukens, “Discussion on Fundamental

Issues of NPR Measurements”, Proceedings of the 18th Instrumentation and

Measurement Technology Conference, Budapest, Hungary, May 2001, pp. 160-165.

[9] A. Geens, Y. Rolain and W. Van Moer, “NPR & Co-Channel Distortion Ratio: A Happy

Marriage?”, IEEE-MTT-S Digest, Phoenix, USA, May 2001, pp. 1675-1678.

[10] Y. Rolain, W. Van Moer and A. Geens, “PAVO: A Simple Wideband Nonlinear

Component Model”, 57th ARFTG Conference Digest, Phoenix, USA, May 2001, pp.

100-103.

260

[11] A. Geens, Y. Rolain, W. Van Moer, “Discussion on in-band Distortions of Mixers”, 58th

ARFTG Conference Digest, San Diego, USA, November 2001.

[12] A. Geens, Y. Rolain, K. Vanhoenacker and J. Schoukens, “Measuring in-band

Distortions of Mixers”, Proceedings of the 19th Instrumentation and Measurement

Technology Conference, Anchorage, USA, May 2002.

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