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enemer@ieee.org

New algorithmic approach for estimating the frequency and phase offset of a QAM carrier in AWGN conditionsUsing HOC

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E. Nemer - 2

• Context and Motivation• Problem statement• Carrier Estimation with HOC

• Frequency offset • Phase offset

•Performance analysis• Simulation Data• Analysis

•Conclusion

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E. Nemer - 3

LPF

tffc )(2cos

tffc )(2sin

In

Qn

)()( ttr

RxNyq

RxNyq

LPF

Carrier Estimation

K samples / symbol

FFE

PLL

FBE

Slicer

Equalizer

Often done in 2 stages : First :

blind (no reliance on apriori symbol knowledge)does not require a proper timing estimationuses the output of the RxNyq (oversampled)

Second :decision-directedcan only correct for small offsetuses typically 1 sample / symbol

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E. Nemer - 4

ML-based schemes

Feed-forward , feedback

• Criteria in designing algorithms

• Performance degradation in low SNR conditions

•Complexity issues : simplified ML approaches are often used

• Issues related to acquisition time, hang up are key in burst receivers

• Non-decision directed recovery

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E. Nemer - 5

- Symmetric process --> third cumulant = zero

- Gaussian process --> All cumulants > order 2 are = zero

y(n) = x(n) + g(n) --> HOS (y) = HOS (x)

• Inherent suppression of Gaussian and symmetric processes

• Detection of Non-linearity and phase coupling

• Peculiar boundariesC4

C2 2-------------------

- 2 < < inf

rand binary 2 sinusoids

Unif. phase

GaussianLaplacian

Impulseprocess

inf

Normalized kurt

H(f)GaussianLinear ? HOS = 0

HOS != 0 ? Non-linear !

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E. Nemer - 6

Symbol rate (but not timing) is knownChannel is equalizedStatistics of the sent symbols are known (but not exact values)

Estimate the frequency offset

Assuming:

LPF

tffc )(2cos

tffc )(2sin

In

Qn

)()( ttr

RxNyq

RxNyq

LPF

Carrier Estimation

K samples / symbol

nnTfj

nnsesz )2(

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E. Nemer - 7

)()()( * LrLreEzzELR QILjsLnnz

)()()cos(

)sin(arctan)(arg

LrLrLE

LELR

QIs

sz

)cos(

)sin(arctan)(arg

LE

LELR

s

sz

A demodulated M-PSK signal can be represented as :

sTf 2Freq offset:

1,...,1,0 Nn

The freq offset may be estimated from the autocorrelation of Zn at lag L, assumed to be a multiple of K, the symbol time

In-phase and quadrature components of the noise are small (zero) at the epok

nnj

nn esz )(

sLnn EssE *

[4]

Freq offset

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E. Nemer - 8

2*22*224 2)( LnnnLnn zzEzEzzELC

• Generalized M-QAM signal at the receiver output :

sTf 2Freq offset:

1,...,1,0 Nnn

njnn esz )(

22222 LnjLnLn eSZ

LjLnnLnn eSSEZZE

**

Ljnnnn eSSEZZE 2*22*22

• Consider the 1D slice of the 4th order cumulant

• Where the sub-terms are :

Freq offset

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E. Nemer - 9

jaaS 1 jaaS 2

jaaS 3 jaaS 4

snn ESSE *

sLnn ESSE *

2*22 5.1 snn ESSE 2*22 5.1 sLnn ESSE

LjeC

LCK 2

4

4 5.015.1

1

)0(

)(

)cos()sin()(cos5.05.1

1 2 LLjLK

5.1/5.0)(

)(arctan1ˆ

Kreal

Kimag

L

• Assume the diagonal symbols are used (S1, S3)

• Then higher order moments can be expressed in terms of the signal energy

• The (normalized) 4th-order cumulant becomes :

• From which , the frequency offset may be deduced

Freq offset

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E. Nemer - 10

2*3 )( LnnZZELC

nLjLnn eSSELC 22*

3 )(

)cos()sin()sin()cos(2)( 33 xxjxxaLC nLx 2

 

)()( 33 LCimagLCrealT

)2cos(4 3 nLaT

• If the frequency offset is zero (or known), then the phase may be estimated as:

)2cos(222/3

nLE

T

s

2/32

2arccosˆ

sE

T

• Consider the 1D slice of the 3rd order cumulant, defined as : Phase offset

• and can be shown to be (for diagonal symbols) :

• Consider the sum of the real and imaginary parts :

• and the normalized sum (by the energy) :

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E. Nemer - 11

),(

)(

)(),( / qpK

h

hqpK Wqp

k

qk

k

pk

Y

Relations between Normalized Cumulants

h(k)

W Yk

pkWY hpCpC )()()(

The Cumulants at the output of a channel / filter may be written in terms of that at the input and the coefficients :

Effect of Nyquist Filter and Channel

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E. Nemer - 12

  

Method SNR Df=200 Hz

2 kHz 10 kHz

2nd order

40dB

199 Hz 2000 Hz 10000 Hz

4th order

200 2007 10026

2nd order

20 190 2002 9999

4th order

198 2009 10024

2nd order

15 205 1982 9992

4th order

202 1990 10016

2nd order

8 221 1954 9992

4th order

198 1972 10016

 

4th order yields better accuracy at small freq offsets

… but not at high freq offsets

Freq offset

Fc = 5 MHzFsym = 160 ksym/s Block size : 100 symbols

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E. Nemer - 13

• Bias (sine wave)

• Bias (Gaussian noise)

• Variance Gaussian noise

- Occurs when frame length not an integer number of periods

- Time estimator is only asymptotically unbiased. - Need to define a new unbiased estimator of 4th stat

- Function of underlying process (noise) energy - May be reduced by increasing segment length

nnj

nn esz )(

Complex sinusoid Gaussian noise

Nn

LnnLnn zzN

zzE:1

*22*22 1

When computing the HOC of Zn, using time averages, added terms occur due to :

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E. Nemer - 14

- Occur when the frame length not an integer number of periods

- Can be reduced by computing HOC over several segments and averaging them.

1

sin

]1[cossin2

2

2

2 TwN

NTwTwNaM

6

sin

2]1[cossin8

2sin

4]1[2cos2sin2

16

4

4 TwN

NTwTwN

TwN

NTwTwNaM

- More bias terms are present in the 4th order moment than the 2nd order

2nd and 4th order moments of a sine wave

Bias : Case of a Sinewave

T: sample timew : frequency: Phase

Nn

LnL z

NM

:14,3,2

1

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E. Nemer - 15

- Time-average based estimator is only asymptotically unbiased.

- Need to define a new unbiased estimator of 4th stat :

Biased !

Define a (new) unbiased estimator :

Valid only for white Gaussian noise

2244 32

1)0( MMN

UC

22*224 3)0( nnn zEzzEC 2244 3)0( MMC

)0()0( 44 CCE

*2

:1

24

1n

Nnn zz

NM

Nn

nnzzNM

:1

*2

1

Bias : Case of Gaussian Noise

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E. Nemer - 16

- Function of underlying process (noise) energy

- May be reduced by increasing segment length

N

MVar2

2

2

NMVar

3

3

15

NMVar

4

4

96

Nn

LnL z

NM

:14,3,2

1

The segment size N has to be significantly increased (by 48 ) in order to bring the variance of the 4th order moment at par with the 2nd order.

Variance : Case of Gaussian Noise

2

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E. Nemer - 17

New estimators for the carrier phase and frequency offset were developed based on newly established expressions of the 3rd and 4th-order cumulants of the demodulated QAM signal.

The higher order estimator is more robust to noise for small values of frequency offsets, though it is not the case for larger ones.

Clearly the improvement depends on the ability to find better ways to compute the HOS in a way to reduce the large bias and variance, when using a finite data set.

more noise-robust methods for (blind) carrier estimation may be developed based on higher order statistics.