Beyond digital interference cancellation

Beyond digital interferencecancellation

Joint RF-baseband interference cancellationto reduce receiver complexity and power consumption.

Vijay Venkateswaran

Beyond digital interference cancellation

PROEFSCHRIFT

ter verkrijging van de graad van doctoraan de Technische Universiteit Delft,

op gezag van de Rector Magnificus Prof. ir. K.C.A.M. Luyben,voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 20 september om 12.30 uurdoor

Vijay Venkateswaran

MSc in de Electrotechniek,The University of Arizona,geboren te Chennai, India.

Dit proefschrift is goedgekeurd door de promotor:

Prof. dr. ir. A-J van der Veen

Samenstelling promotiecommissie:

Rector Magnificus voorzitterProf. dr. ir. A-J van der Veen Technische Universiteit Delft, promoterProf. dr. ir. R.L. Lagendijk Technische Universiteit DelftProf. ir. P. Hoogeboom Technische Universiteit DelftProf. dr. K. A. A. Makinwa Technische Universiteit DelftProf. dr. ir. J. P. M. G. Linnartz Technische Universiteit Eindhoven /

Philips Research, EindhovenProf. dr. B. Ottersten Royal Institute of technology, Stockholm,

SwedenProf. dr. D. T. M. Slock Eurecom, Sophia-Antipolis, France

Copyright c© 2010 by Vijay Venkateswaran

All rights reserved. No part of the material protected by this copyright noticemay be reproduced or utilized in any form or by any means, electronic ormechanical, including photocopying, recording or by any information storageand retrieval system, without the prior permission of the author.

Author email: [email protected]

Summary

One of the major drawbacks towards the realization of MIMO and multi-sensorwireless communication systems is that multiple antennas at the receiver eachhave their own separate radio frequency (RF) front ends and analog to digitalconverter (ADC) units, leading to increased circuit size and power consumption.Improvements in RF and ADC technology happen at a much slower pace whencompared to digital circuits, so that this problem is likely to be more critical infuture.

In a dense multi-user wireless communication setup, these multiple RF frontends and ADCs spend most of their power in processing signals from interferingusers. The purpose of this research is to look at alternative mobile receiver archi-tectures, from the joint perspective of a digital signal processing engineer as wellas that of an RF designer.

We start by specifying the need for a communion of RF and DSP techniques.We propose that advanced signal processing algorithms can be used in combin-ation with existing circuit configurations, such as integrated phased arrays andmulti-channel feedback ADCs, to perform analog interference cancellation. In-terference cancellation allows for a reduced number of receiver chains and lowresolution ADCs, hence reduced circuit size and power consumption.

In summary, the research addresses the following questions:

1. Can we potentially reduce the cost and power dissipation of MIMO trans-ceivers, by optimization across the RF-baseband borderline?

2. Can we design a flexible baseband platform that is tailored to low powercircuits, demonstrating a potential for low cost in a dense multi-user setup?

One approach to cancel interference in RF and to reduce the number of re-ceiver chains in antenna array systems is to design RF phase shift combiners. Analternative is to integrate existing ADCs with a feedback beamformer (this setup

i

SUMMARY ii

is especially compatible with Sigma-Delta ADCs) to identify and cancel the in-terferer. Interference cancellation in the RF and in the mixed signal componentsof the receiver allows ADC units to represent the desired user more effectively fora fixed precision.

For both the above mentioned architectures, we consider the hardware lim-itations and propose closed form solutions minimizing the overall mean squareddistortion between the transmitted signals and its received estimate, and illus-trate significant power savings in the receiver. In both the cases we also specifyapproximate solutions, when the closed form solutions are not feasible.

Given such architectures, we propose techniques to estimate the changes instate of the wireless channel. Finally, we also specify that these approaches havethe capacity to cancel the intermodulation products arising from the non-linearityof the RF components.

On a higher level, it is imperative for the DSP engineer to abandon looking atADCs and RF components as "black boxes" within a sensing/ communicationssystem. For example, viewing a digitally assisted Sigma-Delta ADC as an equal-izer or viewing multi-antenna RF circuits as integrated phased arrays to cancelinterference may result in highly efficient joint solutions for mapping radio wavesinto the digital domain.

Clearly, such hybrid architectures will result in DSP techniques driving thewireless revolution rather than being an afterthought for coping with the imper-fections.

Contents

Summary i

List of Figures vii

List of Tables xi

1 Introduction 11.1 Shannon’s communication system abstraction . . . . . . . . . . . . 11.2 Capacity achieving MIMO and its realization . . . . . . . . . . . . 2

1.2.1 The promise of MIMO . . . . . . . . . . . . . . . . . . . . . 31.2.2 Listening in current systems . . . . . . . . . . . . . . . . . . 3

1.3 Trends in analog and mixed signal components . . . . . . . . . . . 51.3.1 Trends in RF operation . . . . . . . . . . . . . . . . . . . . 61.3.2 Trends in ADC operation - digitally assisted ADCs . . . . . 71.3.3 Shift in MIMO paradigm . . . . . . . . . . . . . . . . . . . 7

1.4 Thesis theme: The need for DSP assistance in integrated architec-tures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . 81.4.2 Scope of the thesis . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 RF beamformers to reduce number of receive chains in antennaarray communication systems . . . . . . . . . . . . . . . . . . . . . 9

1.6 Feedback beamformer (FBB) with multi-channel oversampling ADCs 131.7 Wideband interference cancellation with APN . . . . . . . . . . . . 161.8 Digital beamformer - Asynchronous interferers in OFDM . . . . . 18

2 Concept of phase shifters: Narrow-band RF interference cancellation 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

iii

CONTENTS iv

2.1.1 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.1.2 Contributions and Outline . . . . . . . . . . . . . . . . . . . 24

2.2 System setup and data model . . . . . . . . . . . . . . . . . . . . . 252.2.1 RF data processing . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 Received data model . . . . . . . . . . . . . . . . . . . . . . 252.2.3 High-resolution digital beamforming . . . . . . . . . . . . . 262.2.4 Analog preprocessing network (APN) . . . . . . . . . . . . 262.2.5 Problem formulation . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Preprocessor design–APN not quantized . . . . . . . . . . . . . . . 282.3.1 Conditions on W to minimize the MSE . . . . . . . . . . . 282.3.2 Maximizing the SQNR . . . . . . . . . . . . . . . . . . . . . 30

2.4 Preprocessor design–APN with discrete phase shifts . . . . . . . . 342.4.1 Matching the cross-correlation vector . . . . . . . . . . . . . 342.4.2 Quantized Matching pursuit (QMP) . . . . . . . . . . . . . 35

2.5 Online Correlation Estimation . . . . . . . . . . . . . . . . . . . . . 372.6 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.6.1 Finite sized APN dictionaries . . . . . . . . . . . . . . . . . 412.6.2 Effect of the ADC resolution . . . . . . . . . . . . . . . . . 412.6.3 Effect of the number of APN outputs . . . . . . . . . . . . 422.6.4 Effect of source spacing . . . . . . . . . . . . . . . . . . . . 432.6.5 Effect of the channel estimation . . . . . . . . . . . . . . . . 442.6.6 Communication setup and channel estimation with LRB’s . 442.6.7 Communication theoretic view of APN and coarse ADCs . 46

2.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 472.8 Appendix: APN design using cross spectral projections . . . . . . . 49

3 RF impairments - IP3 model 513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.1 Problem statement and summary . . . . . . . . . . . . . . . 523.2 Intermodulation product model . . . . . . . . . . . . . . . . . . . . 52

3.2.1 Non-linear LNA output . . . . . . . . . . . . . . . . . . . . 523.2.2 Third order IP and the effective radiation pattern . . . . . 533.2.3 Extension to multi-user and wide-band models . . . . . . . 553.2.4 Beamforming to cancel IP3 componenrs . . . . . . . . . . . 55

3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Multi-channel ADCs with feedback: Mixed signal interference cancellation 574.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.1.1 Interference cancellation at the ADC . . . . . . . . . . . . . 584.1.2 Setup and Objectives . . . . . . . . . . . . . . . . . . . . . 594.1.3 Outline and Contributions . . . . . . . . . . . . . . . . . . . 60

4.2 Predictive quantization with Σ∆ ADCs . . . . . . . . . . . . . . . 61

4.2.1 Signal sampling and reconstruction . . . . . . . . . . . . . . 614.2.2 Oversampled Σ∆ ADCs . . . . . . . . . . . . . . . . . . . . 634.2.3 Generalized higher order Σ∆ ADC . . . . . . . . . . . . . . 644.2.4 Problem formulation . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Single channel feedback beamformer design . . . . . . . . . . . . . 664.3.1 Data model . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3.2 Estimation of w . . . . . . . . . . . . . . . . . . . . . . . . 684.3.3 Iterative refinement . . . . . . . . . . . . . . . . . . . . . . 69

4.4 Multi-channel feedback beamformer design . . . . . . . . . . . . . 694.4.1 Data model . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.4.2 Estimation of W . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5 Construction of the training sequence . . . . . . . . . . . . . . . . 734.5.1 Single-channel ADC . . . . . . . . . . . . . . . . . . . . . . 734.5.2 Multi-channel ADC . . . . . . . . . . . . . . . . . . . . . . 744.5.3 Digital postprocessing . . . . . . . . . . . . . . . . . . . . . 74

4.6 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.6.1 Effect of fixed precision DAC feedback . . . . . . . . . . . . 764.6.2 Effect of source spacing . . . . . . . . . . . . . . . . . . . . 784.6.3 Effect of the ADC oversampling factor . . . . . . . . . . . . 784.6.4 Extent of ADC power savings . . . . . . . . . . . . . . . . . 78

4.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 Wideband RF interference cancellation 835.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.1.1 Delay lines in RF . . . . . . . . . . . . . . . . . . . . . . . . 845.1.2 Intermodulation distortion . . . . . . . . . . . . . . . . . . . 855.1.3 Problem statement . . . . . . . . . . . . . . . . . . . . . . . 855.1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2 System Setup and data model . . . . . . . . . . . . . . . . . . . . . 865.2.1 RF system model . . . . . . . . . . . . . . . . . . . . . . . . 865.2.2 Wide-band discrete time data model . . . . . . . . . . . . . 865.2.3 High resolution digital beamforming . . . . . . . . . . . . . 875.2.4 Analog preprocessing setup . . . . . . . . . . . . . . . . . . 88

5.3 Wideband Preprocessor Design . . . . . . . . . . . . . . . . . . . . 895.3.1 Minimizing the overall MSE . . . . . . . . . . . . . . . . . . 905.3.2 Marginal estimate of W . . . . . . . . . . . . . . . . . . . . 915.3.3 Improved estimate of W . . . . . . . . . . . . . . . . . . . . 92

5.4 Alternative approaches to update W . . . . . . . . . . . . . . . . . 945.4.1 Approach 1: Iterative least squares . . . . . . . . . . . . . . 945.4.2 Approach 2: Closed form solution for W when ND = m . . 95

5.5 Simulation and experimental results . . . . . . . . . . . . . . . . . 975.5.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . 97

CONTENTS vi

5.5.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . 985.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.7 Appendix 1: Partitioned matrix cost function . . . . . . . . . . . . 102

6 Digital interference cancellation: asynchronous OFDM 1056.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.1.1 Background on OFDM receivers . . . . . . . . . . . . . . . 1066.1.2 Setup and problem description . . . . . . . . . . . . . . . . 1066.1.3 Contributions and outline . . . . . . . . . . . . . . . . . . . 108

6.2 Data model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2.1 Single user OFDM model . . . . . . . . . . . . . . . . . . . 1096.2.2 Asynchronous OFDM - Frequency domain model . . . . . . 1116.2.3 Time-domain data model and superimposed training setup 112

6.3 Beamformer design . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.3.1 Beamformer design for asynchronous OFDM systems . . . . 1146.3.2 Joint offset and beamformer estimation . . . . . . . . . . . 1156.3.3 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.4 Beamformer updates . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.4.1 Prefiltering . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.4.2 Post processing - Alternating LS . . . . . . . . . . . . . . . 118

6.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7 Conclusions 1237.1 Fundamental questions and summary of main results . . . . . . . . 123

7.1.1 Context: IOP-Gencom project . . . . . . . . . . . . . . . . 1257.2 Suggestions for future research . . . . . . . . . . . . . . . . . . . . 126

7.2.1 Compressive sampling and analog front-ends . . . . . . . . 128

Bibliography 131

Samenvatting 139

Acknowledgements 141

List of Figures

1.1 Trends and power consumption of band-pass communication sys-tems: A simple experiment performed by NXP semiconductorsshows the ADCs consume nearly 40 % of the receiver power. . . . 4

1.2 Moore’s law and ADC power trends: ADC power is obtained bylinear curve fitting of power ratings from different manufacturers.For details, please refer to [1]. . . . . . . . . . . . . . . . . . . . . . 5

1.3 Proposed receiver setup with APN transforming Nr = 4 antennaarray signals to ND = 2 ADC inputs, where Nr > ND. . . . . . . . 10

1.4 Different interference cancellation architectures: (a) digital base-band processor operating on the antenna array signals; (b-d) pos-sible APN techniques (b) phase shifter operating in RF, (c) phaseshifter operating in non-zero IF followed by down conversion, (d)Integrated phase shifter with the mixer arrangement. . . . . . . . . 11

1.5 (a) Oversampled ADC operating at a sampling frequence fs � f0

and (b) ∆ modulator used with an oversampling ADC setup. AΣ∆ modulator has an integrator embedded before the quantizer. . 14

1.6 (a) Single order Σ∆ ADC with a 1-bit quantizer operating atsampling frequency fs (b) its Kth order discrete time equivalentwith a K × 1 feedback vector w = [w1, · · · , wK ]. . . . . . . . . . . 15

1.7 (a) APN operating on RF antenna array signals partially cancelsinterference and reduces the number of receiver chains (b) discretetime equivalent followed by a digital space-time beamformer ϑ. . . 17

1.8 Multiple un-coordinated users operating in a wireless channel. Thedesired user (user 1) offset from the receiver by time τ , and user 2is a neighboring WLAN system with a random offset from the re-ceiver. User 3 can be a bluetooth device operating in the frequencyband of WLAN transmission . . . . . . . . . . . . . . . . . . . . . 19

vii

LIST OF FIGURES viii

1.9 Dependence among various chapters in the thesis . . . . . . . . . . 20

2.1 Proposed receiver architecture: the analog preprocessing network(RF beamformer) cancels interference and reduces the number ofantenna signals to a smaller number of ADC chains. . . . . . . . . 22

2.2 (a) Proposed receiver architecture with RF beamformer (b) dis-crete time equivalent. . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 J(W)asafunctionofp = ‖u2‖ and θ . . . . . . . . . . . . . . . . . 342.4 (a) Architecture 1, where each antenna has its own ADC. (b) Ar-

chitecture 2, containing time varying beamformers Ti, i ∈ {1, · · · , p}for different training periods. (c) Architecture 3 with p low resol-ution beamformers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5 Performance comparison of the APN setup for different RW andR = 10 as function of transmit SNR (a) SINR at the input of firstADC, (b) MSE at the output of baseband receiver. . . . . . . . . . 42

2.6 MSE performance comparison at the output of the baseband re-ceiver as a function of transmit SNR (a) for varied R (b) for variousnumbers and resolutions of ADCs . . . . . . . . . . . . . . . . . . . 43

2.7 MSE performance comparison at the output of the baseband re-ceiver for different ND. . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.8 Performance comparison for ND = 2 setup as a function of spacingbetween desired user and 2 interferers: (a) SINR at the input ofthe first ADC (b) MSE at the output of the baseband receiver. . . 45

2.9 Average estimation error between the true and observed channelstatistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.10 Performance comparison, when Rx and rxs are estimated for 2-bitLRBs and varied channel lengths: (a) MSE (b) BER . . . . . . . . 47

2.11 Achievable throughputs for interference limited MIMO system andAPN setup (a) SIR = -10 dB (b) SIR = 0 dB . . . . . . . . . . . . 48

2.12 (a) Full rank Wiener beamformer (b) reduced rank beamformer . . 50

3.1 Spectrum of interfering signals with frequencies f1 = 2.42 GHzand f2 = 2.44 GHZ and their third order IP terms coinciding withfc = f0 = 2.40GHz . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1 (a) Antenna array configuration with desired and interfering usersignals quantized by oversampled ADCs followed by baseband com-bining to estimate the desired user; (b) Proposed multichannel(MC) ADC architecture with a feedback beamformer (FBB) toidentify and cancel interfering user signals. . . . . . . . . . . . . . 59

4.2 (a) Nyquist and (b) oversampled ADC set-up. . . . . . . . . . . . . 624.3 (a) First order continuous time Σ∆ modulator with 1-bit output,

(b) Discrete time equivalent model. . . . . . . . . . . . . . . . . . 63

4.4 Discrete time equivalent model of a Kth order Σ∆ ADC withweighted feedback, represented by w = [w0, · · · , wK−1]T . . . . . . 64

4.5 Interference cancellation with a FBB W operating on a first ordermulti-channel Σ∆ ADC. . . . . . . . . . . . . . . . . . . . . . . . . 70

4.6 Performance comparison as a function of the resolution of x[n] =DAC(WHbK [n]): (a) average SINR at the ADC input, (b) MSEat the receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.7 Performance comparison as a function of the angular spacing betweenthe desired user and 2 interferers: (a) average SINR at the ADCinput, (b) MSE at the receiver output. . . . . . . . . . . . . . . . . 79

4.8 Performance comparison as a function of varying oversampling ra-tios with 1-bit DAC: (a) average SINR at the ADC input, (b) MSEat the receiver output. . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.9 Performance comparison illustrating oversampling ratios and ADCresolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.1 (a) APN operating on RF antenna array signals partially cancelsinterference and reduces the number of receiver chains (b) discretetime equivalent followed by a digital space-time beamformer ϑ. . . 88

5.2 (a) SINR performance comparison (b) MSE performance compar-ison with Nt = 3 users received by an 4×2 APN for 500 randomlygenerated Rayleigh fading channels with L = 3 . . . . . . . . . . . 99

5.3 MSE performance comparison with Nt = 4 users received by a 4×2APN and oversampled twice for 500 randomly generated Rayleighfading channels with L = 3 . . . . . . . . . . . . . . . . . . . . . . 100

5.4 (a) Response of the wireless channel to a raised cosine pulse (b)relative energies of the channel response for a line of sight and nonline of sight scenario . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.5 (a) SINR performance comparison (b) BER performance compar-ison with 4× 2 APN for multi-path channel width 40 MHz at 2.4GHz and 0.1 µs delay spread . . . . . . . . . . . . . . . . . . . . . 102

6.1 OFDM transmission with inverse discrete Fourier transform (IDFT)followed by cyclic prefix (CP) insertion to combat multipath echoesin a frequency selective fading environment. . . . . . . . . . . . . . 106

6.2 OFDM transmission (a) the IDFT-CP combination leads to a cir-cular channel in the time domain (b) A DFT of circular code leadsto a point wise representation in frequency domain. . . . . . . . . . 107

6.3 Multiple un-coordinated users operating in a wireless channel. Thedesired user (user 1) offset from the receiver by time τ , and user 2is a neighboring WLAN system with a random offset from the re-ceiver. User 3 can be a bluetooth device operating in the frequencyband of WLAN transmission . . . . . . . . . . . . . . . . . . . . . 107

LIST OF FIGURES x

6.4 OFDM transmission in (a) time-domain: the IDFT-CP cyclic struc-ture destroyed by asynchronous interferer (b) frequency domain:A DFT does not have any structure in frequency domain. . . . . . 108

6.5 Superimposed training algorithm (STA) used with an asynchron-ous OFDM system to estimate time offset and to cancel interferingusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.6 Pictorial representation of the superimposed training algorithm toestimate the beamfomer taps θ when desired user is synchronizedto the receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.7 Itrerative postprocessing to update the space-time beamformerweights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.8 Incorrectly estimated delays for subspace fitting . . . . . . . . . . 1206.9 Comparison of SNR with output SINR for joint delay and source

separation with SIR = 0dB . . . . . . . . . . . . . . . . . . . . . . 1206.10 BER performance of joint offset estimation and source separation

L = 3 and β = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

List of Tables

2.1 Quantized Matching Pursuit (QMP) algorithm . . . . . . . . . . . 37

4.1 Operation of multi-channel interference canceling Σ∆ ADCs . . . . 76

5.1 A greedy approach to design wide-band RF phase shifter weightswhen m = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2 A one-step subspace approach to design wide-band RF phase shifterweights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.1 Pre-filtering and alternating least squares approach to update thebeamformer taps θ . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

xi

LIST OF TABLES xii

List of Abbreviations

ADC Analog-to-Digital ConverterAPN Analog Preprocessing NetworkAGWN Additive Gaussian White NoiseBER Bit Error RateBPSK Binary Phase Shift KeyingCDMA Code Division Multiple AccessCMOS Complementary Metal-Oxide-SemiconductorCP Cyclic PrefixCS Compressive SamplingDAC Digital-to-Analog ConverterDFT discrete Fourier TransformdB DecibelDPCM Differential Pulse Code ModulationFBB Feedback BeamformerFIR Finite Impulse ResponseFM Frequency ModulatedFoM Figure of MeritFT Fourier TransformGHz Giga HertzHPF High Pass Filter

1

IF Intermediate FrequencyIDFT Inverse Discrete Fourier TransformISI Inter-Symbol InterferenceIP Intermodulation productIP3 Third Order intermodulation productLMS Least Mean SquaresLNA Low Noise AmplifierLOS Line-of-SightLPF Low-Pass FilterLRB Low Resolution BeamformersMAC Medium Access controlMCSD Multi-channel Σ∆ ADCMIMO Multiple-Input Multiple-OutputMRE Mutually Referenced EqualizersMSE MeanSquaredErrorNB Narrow-bandNBI Narrowband Interference or Narrowband InterfererOFDM Orthogonal Frequency-Division MultiplexingOSR OverSampling RatioPL Path LossPCM Pulse Code ModulationPSD Power Spectral DensityQPSK Quadrature phase shift keyingQAM Quadrature amplitude modulationRF Radio FrequencyΣ∆ ADCs Sigma Delta ADCsSIMO Single-Input Multiple-OutputSISO Single-Input Single-OutputSNR Signal-to-Noise RatioSINR Signal to Interference-plus-Noise RatioST space-timeULA Uniform linear arrayUWB Ultra-widebandWB wide-bandWLAN Wireless Local Area NetworkWSN Wireless Sensor Network

Notations

x Single elementx Vector of elementsx Multi-channel vectorX Matrixx ConjugatexT TransposexH Hermitianvec(X) vector operation of a matrixvec−1(x) convert a N2 × 1 matrix into N ×N matrixX−1 Inverse of matrix XX† Pseudo-inverse of matrix X

x(t) continuous time signalx[k] discrete time signal (t = kT ) for sampling period T‖x‖2f Frobenius norm of xxf frequency domain representation of xx1(t) ∗ x2(t) convolution of x1(t) and x2(t)x1(t)� x2(t) point-wise productx1(t)⊗ x2(t) Kronecker product

3

Chapter 1Introduction

One important factor in the design of mobile communication and sensing sys-tems is the power consumed in the batteries. Consider a scenario where a userdownloads movies into his handheld device or a set of remote sensors that com-municate with each other. It may be impossible to receive/transmit all messageswhile keeping the power consumption within the limits of the battery potential. Insuch circumstances, the aim is to trade off the cost of the communication againstthe quality of signal reproduction. More precisely, the cost of the communicationis specified by the power consumed in the transceivers, the bandwidth used for thetransmission, the circuit size and the latency required to reproduce the message.The quality of signal reproduction is specified by optimization criteria such asminimizing the mean squared error or the bit error rate between the transmittedsignal and its received estimate. The fundamentals of communication theory ex-plained below takes these factors into account to propose a mathematical modelrepresenting the cost and quality terms.

1.1 Shannon’s communication system abstraction

Shannon formulated the root of communication theory as [2]:The fundamental problem of communication is that of reproducing at one point,either exactly or approximately a message selected at another point.Shannon’s abstraction of a communication system consists of five fundamentalentities as follows:

• The information source and the transmitter, respectively producing the mes-sage and the corresponding signals intended for communicating with thereceiver.

• The channel as the physical medium containing additive perturbations from

1

CAPACITY ACHIEVING MIMO AND ITS REALIZATION 2

other transmitters and noise terms.

• The receiver and decoder intended to receive and decode the transmittedmessage.

This model is very general, but the fundamental problems pertaining to mo-bile transceivers are already apparent. The task of the communication engineer isusually to design the entities labelled transmitter and receiver, given the character-istics of the source and the channel. The criteria are to perfectly or approximatelyreproduce the signals at the receiver while minimizing the power consumption orone of the other costs mentioned in the beginning of this chapter.

The power constraints in the communication system as proposed by Shannonrevolve around the transmit power. Information theory literature follows thisproblem statement to propose fundamental results that optimize data rate fora given transmit power [3]. Specific results in the direction of channel-codingand rate-distortion theory leading to communication systems operating close tocapacity are well explained in [3] . All these techniques focus mostly on encodingstrategies at the transmitter, that either maximize the data rate or minimize thepower consumption in the transceivers.

Currently, mobile users spend more time downloading or streaming a largeamount of data and we expect this trend to continue in the foreseeable future.Given that mobile systems predominantly download data, the information sourceis usually a fixed station unconstrained in power and transmission rates. Thereforethe power consumption in the mobile receiver becomes more critical to the batterylife of the device. The bottlenecks are then the physical medium and the mobilereceiver. For these reasons, we address problems that are mostly concerned withthe power consumption at the mobile receiver.

Consider a household scenario, where a user buys a wireless local area network(WLAN) transceiver equipment and expects it to work with his laptop/handhelddevice. However his neighbors might cause interference with their uncoordinatedWLAN systems. He might also own other equipment (e.g. bluetooth) that oper-ates in the same frequency band. In this event, the mobile receiver attemptingto reconstruct a message from the physical medium may fail due the presence ofinterferers and deep fades in the physical medium.

1.2 Capacity achieving MIMO and its realization

One approach to improve the throughput and quality of the received signal bycompensating for fading and interferers, is to have antenna arrays at both thetransmitter and the receiver.


1.2.1 The promise of MIMO

Digital communication using multiple input multiple output (MIMO) systems hasemerged as one of the most signicant technical breakthroughs in the past decade.Research in antenna arrays to generate spatial diversity dates back well beforethe MIMO boom seen in the previous decade [4]. Consider a link for which thetransmitting end as well as the receiving end are equipped with multiple antennaelements. The idea behind MIMO is that the signals on the transmit antennasat one end and the receive antennas at the other end are "combined" in such away that the quality (bit error rate - BER) or the throughput (bits/sec) for eachMIMO user will be improved.

A MIMO communication setup architecture can be described in more detail asfollows; A digital source in the form of a binary data stream is fed to a transmittingblock encompassing the functions of error control coding and (possibly joinedwith) mapping to complex modulation symbols (QPSK - quadrature phase shiftkeying, QAM - quadrature amplitude modulation, etc.), and is then mappedonto one of the multiple transmit antennas. Mapping may include linear spatialweighting of the antenna elements or linear antenna space-time pre-coding. Afterupward frequency conversion, filtering and amplification, the signals are launchedinto the wireless channel. At the receiver, the signals are captured by possiblymultiple antennas and the demodulation and decoding operations are performedto recover the message.

The underlying mathematical nature of MIMO, where data is transmitted overa matrix rather than a vector channel, creates new and enormous opportunitiesbeyond just the added diversity or array gain benefits. This has been shownin [5]. For an overview of MIMO systems and its potential to transform conven-tional communication systems, refer to [6]. As a consequence of the potential ofmulti-antenna wireless channels, an extraordinary number of publications haveflourished in the open literature proposing a variety of efficient communicationschemes.

Moore’s law [7] specifies that the size of the transistors reduce by half forevery 18 months. A direct result of this is the improvement in performance of thedigital circuits and computers over the past two decades. The transition towardsmobile MIMO communications is also enabled by Moore’s law, with receivers thatimplement advanced digital signal processing algorithms and capable of operatingat high data rates. This has meant that MIMO communication techniques havebeen introduced in WLAN standards [8], and that we can now buy a WLANreceiver equipment. However, the rates as promised by MIMO theory are rarelyachieved in practice.

1.2.2 Listening in current systems

Shannon’s model involved only the transmit power and implied that talking isthe most power consuming act in communications. While that is true for most


Multipath channel

Interferer

Baseband processin

g Desired user

estimate

Q

Q

RF-IF

RF-IFBaseband proc

essing

DAC

DAC

IF-RF

Desired user

300

in 5 years 400

State of the art

Trend

120

500300

30

Baseband RFDAC

100

in 5 years 30

State of the art

Trend

120

300300

50

RF BasebandADC

Typical power consumption in mW*Tx Rx

IF-RF

* Courtesy Johan van den Heuvel, TU Eindhoven

Figure 1.1: Trends and power consumption of band-pass communication systems: Asimple experiment performed by NXP semiconductors shows the ADCs consume nearly40 % of the receiver power.

of the communication systems, in a household WLAN setup the base stationtransmitting signals (usually fixed) does so without much concern or restrictionson the power consumption.

On the other hand, the receiver is mobile and limited by a finite battery life.The radio frequency (RF) signals at the antenna array of the receiver containscontributions from the desired user, interfering users and noise. These signalsare amplified, downconverted to baseband (BB), and followed by analog to di-gital conversion. In current multi-user communication systems such as WLANor code division multiple access (CDMA), the RF front end and the analog todigital converter (ADC) units spend most of their dynamic range and circuitryin quantizing the interfering user signals. In a multiple antenna setup, multipleRF front-ends and ADCs process the interfering users, which leads to increasedpower consumption and circuit size.

Although advanced interference cancellation algorithms exist in the digitalbaseband to perfectly cancel interfering user as explained in [4, 6], this straight-forward approach of digitizing a signal with multiple antennas (with or withoutoversampling), and subsequently applying spatial and/or temporal filtering in thedigital baseband would lead to excessive power consumption in the RF front-end

TRENDS IN ANALOG AND MIXED SIGNAL COMPONENTS 5

1990 1992 1994 1996 1998 2000 2002 200410

0

101

102

Year of publication

Pow

er c

onsu

mpt

ion

(slo

pe)

Comparison of Moore’s law with ADC power consumption

ADC power

Moore’s law

Figure 1.2: Moore’s law and ADC power trends: ADC power is obtained by linear curvefitting of power ratings from different manufacturers. For details, please refer to [1].

and the ADCs.Consider Fig. 1.1 showing the power consumption information for a 2 × 2

MIMO WLAN system as determined in trials performed by NXP semiconductors.It was found that for this specific case the ADCs consume nearly 40% of the powerin the receiver. Improvements in the RF and the ADC technology happen at amuch slower pace [1,9] when compared to digital circuits, so that this problem islikely to get larger rather than smaller as shown by Fig. 1.2.

In the following sections, we briefly specify some recent trends in the RF andADC component designs and motivate the need to use DSP techniques that followMoore’s law in combination with analog and mixed signal components. Suchintegrated architectures would facilitate advanced signal processing techniquesused in combination with the constraints of analog electronics.

1.3 Trends in analog and mixed signal components

As noted earlier, over the past decade trends in very large scale implementations(VLSI) implementations of digital design and signal processing algorithms areshaped according to Moore’s law. However, powerful DSP algorithms need ef-fective analog front-ends and high resolution ADCs. The analog front-end circuitdoes not follow Moore’s law. Moreover, recent advances and future directions ofanalog and mixed signal components cannot be listed under one single umbrella.In this section, we specify some of the interesting advancements in RF and ADC


architectures.

1.3.1 Trends in RF operation

Limitations on the availability of spectrum

Consider an ideal wireless receiver unconstrained by power. In this case, oneimportant limitation is the availability of spectrum. Off the shelf RF devicescannot adapt to the intense use of the spectrum caused by the proliferation ofmobile wireless systems.

One approach is to look for gaps or holes in the current spectrum, and this is anarea of active publication known as cognitive radio or spectrum sensing [10]. Thisis facilitated by typically digitizing ultra wide-band signals and looking for unusedspaces in the spectrum. More precisely, the transceiver checks for the frequencybands that are inactive and starts transmitting on these specific frequencies.

Such cognitive radio systems might require an adaptive front-end combiningmultiple antennas in RF and should also be capable of transmitting on differentfrequencies, both significant challenges to realize. Some relevant work on recon-figuring the RF front-end based vacancies in the spectrum has been done recentlyin [11].

Migration to higher frequencies

An alternative to the dense RF spectrum below 10 GHz is to look at the 7GHzunlicensed band around 60 GHz [12]. However, the migration towards higherfrequencies leads to higher path loss, and requires more expensive hardware andsignificant changes in technology. Intuitively, we should also require multipleantennas to supplement for higher path loss and canceling interferers over a widebandwidth.

As we go towards higher frequencies, because the wavelengths are so short theantennas are small enough to fit on the transceiver chip. In this case, it is possibleto offset the path loss by integrating many antennas onto a single chip so thattogether they can form a beam to steer transmissions in a particular direction.Such phased-array antennas can also be used to boost reception and to cancelinterferers [13].

Adaptive front-ends

The unifying theme of the above two significant directions is the need to have anadaptive front-end integrating the multiple antennas in RF with digital assistance.This adaptive front-end can be used to either detect signals in different bands asin [11], or to cancel interferers, as [13]. Preprocessing the signals from multipleantennas in RF also allows us to use a reduced number of receiver chains.


1.3.2 Trends in ADC operation - digitally assisted ADCs

In 0.13-micron complimentary metal oxide semiconductor (CMOS) technology,a digital logic gate requires roughly 5 femto Joules per operation [14]. On theother hand, a state of the art 12-bit ADC consumes roughly 5 nano Joules perconversion, which corresponds to an energy equivalent of 1 million logic gates [14].

The power consumed in the ADCs follow Padc ∝ fs22res, where fs is thesampling frequency and res is the number of bits used in the ADC [1]. AdvancedDSP algorithms require high resolution ADCs, especially in the presence of lowsignal to noise and signal to interference ratios. While digital gates only need todistinguish two logic levels, high-resolution ADCs must resolve minute differencesin voltage or current at high speed.These are some factors that contribute toincreased power consumption in the ADCs.

An obvious direction for improving the performance of ADCs is to exploitwhat has transformed them into a bottleneck in the first place: powerful digitalcomputing and signal processing. Some significant improvements in ADC tech-nology, and more specifically on digital assistance in ADCs can be found at [15].The digitally assisted ADCs either use a feedback from the digital side as in Σ∆ADCs [16] or utilize the inherent non-linearities of the mixed signal devices incombination with digital signal processing algorithms [9].

At a higher level, it is imperative to abandon looking at ADCs as "blackboxes" within a system. For example, viewing a digitally assisted ADC as partof an equalizable, nonideal communication channel may result in highly efficientjoint solutions for mapping radio waves into the digital domain.

1.3.3 Shift in MIMO paradigm

In summary we require MIMO communication systems and multiple antennas to

• exploit spatial diversity and perform close to capacity,

• effectively use spectrum and cancel interferers in a multi-user scenario,

• migrate to higher frequencies.

Advances in digital processing and existence of adaptive analog and mixed signalfront-ends adds a new dimension: the design of architectures that

• enable digital feedback to compensate for the analog imperfections and in-terferers.

Clearly, the key to such innovations will be broad, multidisciplinary approachesthat use digital processing capabilities as a driver, rather than an afterthought intrying to cope with imperfections in evolutionary analog circuits.

THESIS THEME: THE NEED FOR DSP ASSISTANCE IN INTEGRATEDARCHITECTURES 8

1.4 Thesis theme: The need for DSP assistance in integrated ar-chitectures

One well known approach to cancel interference as well as to minimize analogimpairments, is to digitize wide band signals followed by digital cancellation.In this case, the wide-band signals containing desired user signals and interfer-ers are corrupted by wireless channel and RF imperfections. These signals aredown-converted to baseband followed by digital compensation and calibrationtechniques to cancel the interfering users and imperfections [17,18]. However, op-erating on a wide-band signal would require large sampling frequencies and highresolution ADCs. Thus this approach would be at odds with our motive to reducereceiver power and circuit size.

1.4.1 Problem statement

An alternative approach to handle the RF and ADC challenges and bottlenecks asmentioned in the previous section is to integrate signal processing techniques withnovel analog and mixed signal architectures. By integration, we refer to coarselyprogrammable RF and ADC components combined with digital assistance. Asimple example of such an architecture would be the well known Sigma-Delta (orΣ∆) ADC [16].

Our approach is to consider the integrated architecture of a crude spatialbeamformer operating in the analog domain (before the ADCs). This is especiallysuited for a multi-user scenario, where such beamformers can be designed to relievethe ADCs and the RF components from spending most of their dynamic range inprocessing the interferers.

An integrated architecture offers us the possibility of using advanced digitalsignal processing algorithms to bridge the ever increasing gap between the powerconsumption of the Moore’s law enabled digital circuits and analog circuits. How-ever, realizing these architectures in silicon gives rise to a new set of challenges.A majority of these challenges would be in the area of analog circuit design, andis an area of active research in the solid state community.

Given that such architectures exist, this thesis explores the possibility of usingDSP to assist the analog circuit designer. In a way, this work can be seen as aDSP perspective to systematically design flexible analog front-ends, and to relaxsome of the challenges observed by the RF designer.

1.4.2 Scope of the thesis

The focus throughout this thesis is to look at the possibilities and usefulness ofinterference cancellation before the ADCs. Indeed, if we can cancel most or evenonly a part of the signals from the interfering users before the RF/ADC units, wecan achieve a significant improvement in the dynamic range of the ADCs.

RF BEAMFORMERS TO REDUCE NUMBER OF RECEIVE CHAINS IN ANTENNAARRAY COMMUNICATION SYSTEMS 9

Instead of one particular solution, this thesis considers multiple ways to sys-tematically perform analog interference cancellation. In the following sections,we illustrate some of the architectures that facilitate a reduced number of RFchains and low resolution ADCs. Subsequently, we give a small preview of eachapproach, its advantages and its limitations. To perform interference cancellationbefore the ADCs and in RF, the questions that we need to answer on a higherlevel are as follows:

• What are the possibilities of cancelling interference before the ADCs usingintegrated architectures?

– Here we look for the compatible architectures that allow interferencecancellation

• What are the design constraints and limitations of these architectures?

– What are possible communication scenarios that can be used in com-bination with these architectures?

– How do we satisfy the necessary and sufficient conditions required tocancel interferers?

– How do we estimate the state of the wireless channel?

• Additionally, is it also possible to account for circuit impairments?

– Can we design the architectures to account for non-linearities in RFamplifiers?

The following sections outline the important chapters of this thesis. In eachcase, we consider a specific multi-user communication mode and a multi-antennareceiver setup, illustrate the challenges in designing the receiver, and proposesolutions to jointly cancel interference and reduce the overall power consumption.

1.5 RF beamformers to reduce number of receive chains in an-tenna array communication systems

As noted earlier, one approach to reduce ADC power is to reduce the numberof receiver chains. One sub-optimal technique to achieve this in a multi-antennasystem is to use antenna/diversity selection. This is a well known approach, wherebasically we select the antenna with the highest signal energy [19]. This techniqueis a good starting point, but does not enable interference cancellation before theADCs.

An advancement over the antenna selection is the use of analog preprocessingnetworks (APNs) for linearly combining the antennas. The APN acts as a beam-former in the analog domain, and operates on the RF or intermediate frequency(IF) signals from the antenna array as shown in Fig. 1.3 to cancel the signals


Desired user signal

Multipath channel

User 2

Baseband processing

Desired user

estimate

Q

Q

Analog preprocessing

network

z1[k]

x1(t)s(1)[k]

s(1)[k]

s(2)[k]ADC units

W

RF-BB

RF-BBϑ

xNr=4(t)zND=2[k]

Figure 1.3: Proposed receiver setup with APN transforming Nr = 4 antenna arraysignals to ND = 2 ADC inputs, where Nr > ND.

from the interfering users. For example, we see here that the APN configurationmaps Nr = 4 antenna array signals to ND = 2 ADC inputs. Typically, the APNhas fewer outputs than the inputs (Nr > ND), as shown in Fig. 1.3 and is usedin combination with low resolution ADCs.

The common underlying theme in the design of the APNs is that, for a narrow-band (NB) system, the propagation delay of the signals received at the antennacan be represented in baseband as a phase shift in the range of [−π, π]. In amulti-user NB scenario, the received antenna array signals can then be seen asa combination of distinct phase shifted signals from the desired and interferingusers. Under certain conditions, an estimate of the desired signals can be obtainedusing a phase shift beamforming vector operating on the antenna array signals.

An overview of programmable phase shift vectors is specified in [13]. Onemust note that such RF phase shifts are usually limited by hardware to 3-4 bitresolution. Our primary objective is to check if such an architecture can cancel theinterferers either in the RF or in the intermediate frequency (IF), and to designthe beamformer coefficients.

The specific location of the phase shift networks in the receiver chain is usuallydefined by the end objective and by the hardware constraints. Fig. 1.4 shows pos-sible APN architectures. For a setup without any limitations on power and withoptimal performance with respect to the bit error rate, it is always preferred to usedigital beamforming and signal processing techniques with a topology specified inFig. 1.4(a).

However, in a dense multi-user scenario with many interferers, the digitalbeamforming arrays would require high resolution ADCs which are currently thebane of MIMO systems. As an alternative, in direct conversion receivers [20],phase shift networks can be used in RF as demonstrated by [21,22] (Fig. 1.4(b)).Another option is to operate phase shifters at a non-zero IF (Fig. 1.4(c) [22]),or to combine phase shift networks with the down-conversion process using localoscillators as proposed in [13] (Fig. 1.4(d)).

Given any of these architectures, our objective is to design the APN phase


+ Q

φ1

φ2

X DSP

+ Q

φ1

φ2

X

DSP

X

IF-BBQ

X

DSP

X

IF-BB

φ1

φ2

+

Q

DSP

digital beam-former

X

X Q

(a) (b)

(c) (d)

Figure 1.4: Different interference cancellation architectures: (a) digital baseband pro-cessor operating on the antenna array signals; (b-d) possible APN techniques (b) phaseshifter operating in RF, (c) phase shifter operating in non-zero IF followed by downconversion, (d) Integrated phase shifter with the mixer arrangement.

shifts that estimate the desired user signals. Note that in Fig. 1.4(b-d), we showthe APN arrangement with only 1 ADC (ND = 1). However, our approach wouldcontain an arbitrary number of ADCs. These ND low resolution ADC outputsare later combined in digital baseband to obtain a high resolution estimate of thedesired user signal.

Choosing the phase shift vectors that lead to a reasonable estimate of thedesired user signal requires knowledge of the wireless channel. Note that intro-duction of an APN means that this information is not directly available at thereceiver and that the APN design problem is not trivial. We approach the APNdesign in the following order:

P2-0 Assume for the time being that the APN and the ADCs are not quantizedand the channel state is available at the receiver. Is it possible to design theoptimal APN that minimizes the overall MSE at the receiver? How manyADCs (ND) are needed here to cancel the interfering users?

P2-1 Incorporating the ADC resolution (which relates to power consumption),can we design the APN to minimize the overall mean squared error andmaximize the desired signal to quantization noise ratio?


P2-2 Given that the APN weights are poorly quantized, can we approximate thepreviously designed optimal APN matrix as a set of limited resolution phaseshifters and the hardware setup?

P2-3 Is it possible to track for changes in the multi-user setup and to modify theweights of the APN accordingly?

The problems, [P2-0 - P2-3] and their solutions form chapter 2 of the thesis.For a given ADC resolution, receiver chain and hardware limitations, we willspecify possible ways to design the optimal analog phase shifter that minimizesthe mean squared error between the desired user and its received estimate. Wetake into consideration that the APN is coarsely quantized (typically the elementsof the APN have 3-4 bit resolution), and may also contain phase errors [23] whenoperating in RF.

Taking the above mentioned constraints into account we specify a two-stepbeamforming technique, and combine the APN outputs digitally to obtain a highresolution estimate of the desired user. The publications that are associated withchapter 2 are:

• V. Venkateswaran and A. J. van der Veen, Analog beamforming for MIMOcommunications – with phase shift networks and online channel estimation,accepted for publication in IEEE Transactions on Signal Processing, Aug2010.

• V. Venkateswaran and A. J. van der Veen, Optimal RF phase shifter designto reduce ADC power consumption in multi-antenna systems, In Proc. ICASSP,2010.

• V. Venkateswaran and A.J. van der Veen, Partial beamforming to reduceADC power consumption in antenna array systems, In Proc. SPAWC, 2008.

RF beamformer to cancel intermodulation products

In conventional receivers, the RF circuits are coarsely designed to obtain a WBsignal, of which only a small part is made up of desired user. This is due to thefact that designing NB RF components with fine channel selectivity is expensive.

In such a setup, the low noise amplifier (LNA) operates at a bandwidth that isconsiderably larger than that of the digital baseband. Strong interferers, operatingin the adjacent bands may drive the LNA into a non-linear mode of operation [20],resulting in intermodulation products (IP) of carrier frequencies.

RF component design to cancel the IP terms is an area of active research inthe solid state circuit community [24]. Our approach in chapter 3 is to checkwhether the APN architecture of chapter 2 can be designed to also account fornon-linear IP terms. To realize this we consider an APN arrangement operatingon a WB communication setup and

FEEDBACK BEAMFORMER (FBB) WITH MULTI-CHANNEL OVERSAMPLING ADCS13

1. our first challenge is to model the effect of non-linearity?

2. does the model provide us with any intuition to cancel the IP terms?

These questions are reviewed in chapter 3. However, chapter 3 is not an originalcontribution of the thesis, and for details refer to [25, 26]. This chapter is onlyintended to illustrate the scope of using APN like circuits to account for non-linearities in RF amplifiers.

1.6 Feedback beamformer (FBB) with multi-channel oversamplingADCs

The RF preprocessing architecture using the APN arrangement described inchapter 2 is one approach to reduce the ADC power by starting from an antennaarray setup with more antennas and reducing it to fewer receiver chains.

An alternative approach to reduce the ADC power consumption would be tocheck for existing ADC architectures that facilitate interference cancellation. Thiswould require us to have more insight into the analog to digital conversion. TheADCs are usually specified by two separate operations:

• sampling - usually uniform in time and an invertible operation, providedthe Nyquist sampling theorem is satisfied.

• quantization - non-invertible operation transforming the sampled signalsonto finite representations.

Based on the above two operations, the resolution of the ADC can be increasedeither through (a) amplitude quantization i.e. reducing the quantization step sizeor (b) over-sampling with rates much higher than the Nyquist rate. The ADCscan thus be classified on the basis of operation as (a) Nyquist rate ADCs and (b)oversampling ADCs.

Consider an oversampling ADC operating at a sampling frequency fs = Rd2f0e,where f0 is the highest frequency of the input signal and R is the oversamplingratio (OSR) [27]. In a straightforward oversampling ADC for every increase in R,the quantization noise variance reduces by a factor 1/R. From the ADC powerrelation Padc ∝ fs22res we see that for a given power consumption, the over-sampling ADC results in same quantization noise variance as that of a Nyquistrate ADC [27].

However, the circuit size of an oversampling ADC is considerably less thanthat of a Nyquist rate ADC. For this reason the oversampling ADCs (with lowresolution say res = 1 bit) are preferred in the case of slowly varying inputsignals. The advantage of using oversampling or ADCs is that the sophisticatedpost-processing (such as converting low resolution 1-bit signals to a high resolutionsignal) can be performed digitally, which is more robust and consumes less power.Further, more efficient representations of band-limited (BL) signals are possible


Anti-aliasingx(t) + LPF

fc = f0fsquantizer

High resolution signal

+

DAC

-

+x(t) +

quantizer

LPF

fc = f0High resolution

signal

e[n]e[n]

sampling frequency fs

fc = f0 max frequency of BL signal

Figure 1.5: (a) Oversampled ADC operating at a sampling frequence fs � f0 and (b)∆ modulator used with an oversampling ADC setup. A Σ∆ modulator has an integratorembedded before the quantizer.

by using a feedback arrangement from the quantizer output, somewhat similar toa ∆ modulator as illustrated by Fig. 1.5.

One subclass of popular oversampling ADCs is a Σ∆ modulator [16] employinga loop filter (integrator) followed by a non-linear mapping (1-bit quantizer). Thequantizer output is fed back and subtracted from the input signal using a digitalto analog converter (DAC), and this forces the average value of the oversampledand quantized signal to follow the average value of the input signal. The lowresolution ADC output is then averaged over the oversampling instants to obtaina high resolution estimate.

Traditionally, the feedback loop in the Σ∆ ADCs and the DAC arrangementare used to suppress the low frequency terms in the quantization noise. For thisreason, the Σ∆ ADCs are also referred to as noise shaping ADCs. The feedbackDAC included in the ADC architecture offers us the interesting possibility to alsosuppress the interferers. As an example, Philips e. a. [28] perform interferencecancellation with a digital high pass filter (HPF) in the feedback loop of Σ∆ADCs.

Extending such an architecture, we replace the HPF with an adaptive filteri.e. a beamformer in the feedback loop. Consider a first order Σ∆ ADC operatingon a band-limited input signal as in Fig. 1.6(a) with an integrator in the forwardloop and a DAC in the feedback loop. Fig. 1.6(b) shows a discrete-time equivalentof a higher order (Kth order) ADC.

The objective now is to design the K×1 feedback beamforming (FBB) vectorw operating with the DAC to cancel the interfering users.

The underlying theme behind this design is that, the oversampling ADCsprovide a redundant signal representation of the BL input signal. This framework,when used with some statistical information of the desired user signal, allows usto cancel the interfering users. The FBB and the DAC setup associated with theADC subsequently cancels the interferer signals in the analog domain.

As noted before, cancelling the interfering users before the ADCs leads to amore efficient representation of the desired user signal. On a higher level, this


LPF +

+ Q

DAC

-

+ ∫

fs

(a)

(b)

S/H

fs

Continuous time

oversampled discrete time instants Nyquist rate

Continuous amplitude Quantized

x[n]

x(t)

x(t)

LPF

fc = f0

fc = f0

p(t) d[n] b[n]

+ +

-+

b[n]

z−1

z−1+

z−1

+

+

w1

wK

Figure 1.6: (a) Single order Σ∆ ADC with a 1-bit quantizer operating at samplingfrequency fs (b) its Kth order discrete time equivalent with a K × 1 feedback vectorw = [w1, · · · , wK ].

architecture allows us to combine quantization and interference cancellation intoone operation.

We extend this FBB architecture with Σ∆ ADCs to multiple antennas (Nr).Thus one can utilize the spatial diversity offered by the multiple channels (orMIMO systems) in cancelling the interferers. This ADC setup, referred to as amulti-channel Σ∆ ADC is detailed in chapter 4. In this chapter, our objective isto design the FBB such that the ADC output leads to an interference and noisefree estimate of the desired user. We accomplish to answer the following questionsand perform interference cancellation via the following steps:

• Given the ADC architecture and a NB communication setup, what are theconditions that allow us to cancel interfering users?

• Subsequently, is it possible to design a unique FBB that minimizes theoverall MSE between the target signal and the ADC output?

– This is not a trivial problem, since the FBB is present in the feedbackpart of the ADC. Such an architecture will pose questions on the ex-

WIDEBAND INTERFERENCE CANCELLATION WITH APN 16

istence of a unique w minimizing the overall MSE. We will show thatusing some elegant techniques in statistical signal processing, the ADCarchitecture can be reduced to a simpler model, leading to a uniqueFBB design.

Additionally, one must note that the design has to incorporate the additionalconstraint that the ADC and DAC outputs are 1-bit coarsely quantized signals.

In the signal processing literature, several types of equalizers have been de-signed where the equalizer output is fed back to cancel the incoming signals [29].One related context is the classical least mean squares (LMS) technique [30]. Ourapproach can also be seen as integrating a feedback equalizer architecture witha quantizer. However, the design of our approach as detailed in chapter 4 isfundamentally different from the LMS like approaches.

Designing ADCs with the ability to suppress the interfering user signals posesa whole new set of questions on the possibility of digitally assisted ADCs incommunication systems. The main contributions associated with chapter 4 are

• V. Venkateswaran and A. J. van der Veen, Multi-channel Interference can-celing Sigma-Delta ADCs, Submitted to IEEE Trans. Signal Processing,April 2010.

• V. Venkateswaran and A. J. van der Veen, Feedback beamformer design tocancel interference in oversampling ADCs, In proc. ICASSP 2010.

• V. Venkateswaran, A. J. van der Veen and D. Slock, Sigma Delta interferencecanceling ADCs for antenna array systems, In proc. SPAWC 2009.

1.7 Wideband interference cancellation with APN

The previous two sections considered a NB multi-user wireless channel and pro-posed integrated architectures to reduce receiver complexity. The trend in wirelesscommunication systems is more in the direction of wide-band (WB) and ultra-wideband (UWB) signals, and this is expected to continue [31]. Designing a wide-band receiver that combines multiple time-delayed replicas of the received signaldigitally allows us to extract the desired user immersed in noise. In addition, onecan compensate for imperfections in the analog front-end.

Consider an APN setup operating on a wideband communication system. TheRF APN designed in Chapter 2 to estimate the desired user signal would fail inthe wide and ultra-wide band case due to the following reason. In a narrow-bandsignal, the propagation delays are by definition much smaller than the inverse ofthe channel bandwidth.

The fundamental difference between the NB and the WB setup is that in thelatter case, the propagation delays are comparable to inverse bandwidth, andcannot be represented using phase shifts in the range [−π, π]. An alternative is

WIDEBAND INTERFERENCE CANCELLATION WITH APN 17

APNW

x(t)

NDNr

ej2πfct

e[k]

z[k]QX

z(t) z(t)

APNW

Nr+

x[k]

ND

z[k]y[k]

ϑ

(a)

(b)

LNA

D

D

y[k]ϑD

D

Figure 1.7: (a) APN operating on RF antenna array signals partially cancels interfer-ence and reduces the number of receiver chains (b) discrete time equivalent followed bya digital space-time beamformer ϑ.

to use delay lines and cancel interference, however when implemented in RF theirlength would run into meters and is impractical to incorporate in mobile systems.

In addition, the LNA and the mixer of a band-pass communications systemoperate at a bandwidth considerably more than that of the desired user signals.For example, typically in a WLAN setup with operating bandwidth of 20 MHz,the RF LNA operates at 100 MHz. In this case, the antenna array signals at theLNA input might also contain interferers from the adjacent bands and with it theIP terms as explained in chapter 3.

In this chapter, we consider a WB communication setup where an RF phase shiftbeamformer operates in combination with a digital space-time (ST) beamformerto reconstruct the desired user signal. Our objective is to jointly estimate the RFAPN as well as the digital ST beamformer to cancel the interfering users.

• Assuming that we can use a space-time beamformer in digital basebandas shown in Fig. 1.7, how can we design the APN W operating only onthe spatial antenna array signals and minimizing the overall mean squarederror?

– We will conclude that only partial interference cancellation is possible,and propose a sub-optimal approximation to minimize the overall MSE.

• We will also apply our algorithms to an experimental WB setup and observethe amount of interference cancellation.

The main contributions associated with chapter 5 is

DIGITAL BEAMFORMER - ASYNCHRONOUS INTERFERERS IN OFDM 18

• V. Venkateswaran and A. J. van der Veen, Wideband RF preprocessing toreduce MIMO receiver complexity - in preparation for IEEE Trans. SignalProcessing June 2010.

1.8 Digital beamformer - Asynchronous interferers in OFDM

The previous sections proposed integrated architectures with beamformers to can-cel interference and to reduce receiver complexity. Chapter 6 considers an altern-ative and a potentially simpler approach, where we consider small additions to anexisting receiver setup and WLAN standard [8], and design the receiver to cancelinterferers.

As specified earlier, WB channels are characterized by multiple echoes of thetransmitted signals, giving rise to intersymbol interference (ISI) at the receiver.One well known approach in current systems is to design a digital filter (space-timebeamformer) to cancel the multi-user as well as inter-symbol interference.

From a practical viewpoint, to keep the mobile receiver power and complex-ity to a minimum, it is desired that the receiver design be simplified and thatthe transmitter covers for the imperfections in the wireless channel. Orthogonalfrequency division multiplexing (OFDM) is tailored for this purpose, where a cyc-lic prefix (CP) is introduced at the transmitter to transform the multipath fre-quency selective fading channel into a set of narrow-band channels. Subsequently,the transmitted signals can be reconstructed directly at the receiver. This prop-erty allows for equalization of very wide-band signals and has made digital videobroadcasting (DVB) and WLAN very popular.

In the presence of multiple users, one indispensable condition is that the cyclicprefix of all users must be synchronized at the receiver. This condition is difficultto satisfy, especially in the presence of un-coordinated interferers like a neighborusing WLAN. Even if this condition is met, channel estimation and equalizerdesign is not straightforward since the training signals from different users areindistinguishable.

For this reason, it is interesting to consider a small modification of the OFDMsignal where we code the transmitted data signals by a small additive sequence.We denote this sequence as a superimposed training sequence. For simplicity, weassume that such a training sequence is unique for each user (such as a CDMAsystem) and that the receiver knows the training sequence of the desired user.

We consider a communication setup where multiple users occupy a commonwireless channel. The receiver is not synchronized to the user of interest anddoes not have any coordination with other users as shown in Fig. 1.8. Thesuperimposed training sequence is then used to jointly estimate the signal fromthe desired user as well as its time offset. Subsequently we propose a series ofalgorithms to jointly design the beamformer taps as well as to estimate the timeoffset of the desired user. The research contributions associated with this werepresented in


symbol iCP symbol i+1CP

CP CP

Desired user

User 2

User 3

i-1

CP

τ

symbol offset

Figure 1.8: Multiple un-coordinated users operating in a wireless channel. The desireduser (user 1) offset from the receiver by time τ , and user 2 is a neighboring WLAN systemwith a random offset from the receiver. User 3 can be a bluetooth device operating in thefrequency band of WLAN transmission

• V. Venkateswaran, A. J. van der Veen and M. Ghogho, Joint source separ-ation and offset estimation of asynchronous OFDM systems using subspacefitting, In Proc. SPAWC, 2007.

• V. Venkateswaran and A. J. van der Veen, Source separation of asynchronousOFDM signals using superimposed training, In Proc. ICASSP, 2007.

Outline of this dissertationThe dependence among various chapters in the thesis is visible from Fig. 1.9.For example, before reading chapter 5 one must read chapters 2 and 3. However,chapters 2, 4 and 6 are independent of each other and can be read directly afterchapter 1.

In addition to the above mentioned research contributions, the other peerreviewed contributions that came out of this research are:

• V. Venkateswaran, J. H. C. Heuvel, P. Baltus, J. P. Linnartz and A. J.van der Veen, Beyond digital interference cancellation Joint RF-basebandinterference cancellation in preparation for white paper in IEEE Signal Pro-cessing magazine.

• V. Venkateswaran, Analog preprocessor mapping for antenna arrays to re-duce ADC power, Asilomar conference on Signals, Systems and Computers,2008.


Chapter 1Introduction

Chapter 2:Narrow-band RF

interference cancellation

Chapter 4Multi-channel ADCs +


Chapter 6Asynchronous OFDM


Chapter 7Conclusions

Chapter 5: Wide-band RF interference cancellation

Chapter 3: IP3 model

Figure 1.9: Dependence among various chapters in the thesis

Chapter 2Concept of phase shifters:Narrow-band RF interferencecancellation

If a man will begin withcertainties, he shall end indoubts; but if he will be contentto begin with doubts, he shall endin certainties.

Francis Bacon

In MIMO systems, the use of many RF and ADC chains at the receiver iscostly. Analog beamformers operating in the RF domain can reduce the numberof antenna signals to a feasible number of baseband channels. Subsequently,digital beamforming is used to capture the desired user signal. In this chapter,we consider the design of the analog and digital beamforming coefficients, for thecase of narrowband signals. We aim to cancel interfering signals in the analogdomain, thus minimizing the required ADC resolution. For a given resolution, wewill propose the optimal analog beamformer to minimize the mean squared errorbetween the desired user and its receiver estimate. Practical analog beamformersemploy only a quantized number of phase shifts. For this case, we propose adesign technique to successively approximate the desired overall beamformer bya linear combination of implementable analog beamformers. Finally, an onlinechannel estimation technique is introduced to estimate the required statistics ofthe wireless channel on which the optimal beamformers are based.

21

INTRODUCTION 22

Desired user signal

Multipath channel

User 2

Baseband processing

Desired user

estimate

Q

Q

Analog preprocessing

network

z1[k]

zND[k]

xNr (t)

x1(t)s(1)[k]

s(1)[k]

s(2)[k]ADC units

W

RF-BB

RF-BBϑ

Figure 2.1: Proposed receiver architecture: the analog preprocessing network (RF beam-former) cancels interference and reduces the number of antenna signals to a smallernumber of ADC chains.

2.1 Introduction

Multiple input multiple output (MIMO) and multi-sensor communication sys-tems employ multiple receive antennas to exploit selection diversity and improvemultiplexing gains. The aim is to achieve reliable communication close to theor-etical limits [6]. However, the introduction of multiple antennas at the receiverleads to separate radio frequency (RF) front ends and analog to digital converter(ADC) units, i.e., increased circuit size and power consumption as explained inthe chapter 1.

If the number of receiver antennas is given, one well known sub-optimal tech-nique to reduce the number of RF and ADC chains is to use antenna/diversityselection. Basically, we select the antenna with the highest signal energy [19].This technique does not enable interference cancellation before the ADC.

An advancement over antenna selection is the use of analog preprocessing net-works (APNs) for linearly combining the antennas, i.e., beamforming. Currenthardware developments offer many possibilities. In [21, 32, 33], a phase shift pre-processor is implemented, which uses active and passive weighting elements tocombine signals from the antenna array in the RF domain. Hajimiri e.a. [13]propose a design where the required phase delays are implemented in the RF tobaseband demodulation step, by using a bank of several phase-shifted local os-cilators. Typically these designs provide about 16 possible phases (4-bit phaseresolution) and no variable amplitude, thus implement a poorly quantized setof possible beamformers. These papers focus on the hardware design and onlybriefly touch upon the question how these beamformer coefficients should be se-lected. E.g., in [13], a set of beamforming vectors is precomputed to steer beamsin predefined directions, with a resolution of about 22 degrees. This only allowsto select the direction with highest energy, which does not necessarily result inthe desired signal in the presence of multipath and interference.

Improvements are possible by considering multiple output streams. Shownin Fig.2.1 is an architecture where an analog preprocessing network (APN) dir-

INTRODUCTION 23

ectly operates on the RF signals, mapping Nr antenna array signals to ND < Nrreceiver chains (i.e. ADCs). A digital beamformer ϑ subsequently combinesthe ADC outputs to generate the desired user estimate. Zhang e.a. [34] con-sidered such an architecture, and proposed several MIMO transmitter/receiverbeam steering techniques.

Our aim in this chapter is to design an optimal APN beamforming matrix. Ourfocus is to minimize the interference at the input of the ADCs, so that reducedresolution is possible, leading to reduced power consumption. Some design issuesare (1) to choose ND, (2) to select the beamforming coefficients, and (3) to de-termine how many bits are needed in each of the ADCs. A design constraint is thepoor resolution of the APN coefficients. Design criteria are the mean square error(MSE) at the output of the digital beamformer and the ADC power consumption.

As an example, consider a wireless channel with one interfering user trans-mitting signals with the same energy as that of the desired user, with Nr = 4antennas and ND = 2 receiver chains. It will be seen from the simulation resultsin Sec. 2.6, that RF interference cancellation with an APN can reduce the ADCpower consumption by half for the same MSE at the receiver output.

2.1.1 Connections

In the array signal processing literature, several types of preprocessing matriceshave been designed to reduce the number of receiver chains. One related contextis “beamspace array processing”, where a preprocessing is done on the receiveantennas to reduce dimensionality, see e.g., [35]. In earlier work, this is called“partially adaptive beamforming”, where the preprocessor is fixed and the digitalbeamformer is adaptive, see e.g., [36]. In this literature, the design of the beam-space transformation matrix is based on prior knowledge of the location of thesignal of interest and/or on the interference scenario. Reduced dimension trans-formations, to cancel interferers using statistics from the desired user, have alsobeen proposed [37]. In the present chapter, we aim to design the APN usingfeedback from the baseband processor, so that it can be optimized for the actualsituation on a block-by-block basis.

Transformation preprocessors have also been pursued in direction of arrival(DOA) estimation problems, e.g., [38]. In that paper, a set of preprocessors isapplied over time, and the results are combined to estimate the DOA. In contrast,our aim is to minimize interference and reconstruct the signal of interest; however,we will apply techniques from [38] to estimate the required channel parameters.

In practice, the APN coefficients are quantized. There exists a significantamount of literature on (adaptive) beamforming using variable phase only (cf.equal gain combining). Most literature considers a single weight vector (withvariable phases only) that should be designed to match certain performance cri-teria, e.g., [32, 39, 40]. The paper [34] considers an APN with multiple outputs,and it is shown that any desired weight vector can always be obtained by linearlycombining two phase-only beamformers, thus ND = 2 is sufficient. The work

INTRODUCTION 24

of [34] has generated some followup work, focussing on phase shift beamformingat the transmitter and receiver to linearly combine the signals. However, theseapproaches do not emphasize on interference cancellation nor on implementableAPN weights. cancellation and assume perfect channel knowledge at transmitter.We will consider a more restricted case where the APN weights and ADC tapsare severely quantized.

2.1.2 Contributions and Outline

In the chapter, we progressively study various aspects of the APN beamformerdesign. In Sec. 2.2, the system set-up and the data model is specified. In Sec. 2.3we consider the case where the APN and ADC have sufficiently high resolution.We will aim to design an Nr×ND preprocessing matrix that minimizes the MSE.This leads to a non-unique design. To make it unique, we also take the ADCquantization error into account and we design the APN to maximize the signalto quantization noise ratio (SQNR). For an APN with infinite precision, we willderive that it is sufficient to consider only ND = 1 output chain.

However, as mentioned earlier, in practice the APN is a programmable discretephase shifter with a coarse quantization. The RF operation is further corruptedby phase errors (these can be about 5 − 7% [23] when operating in a frequencyrange of 2− 3 GHz). In this case, using ND > 1 allows us to extract different lowresolution streams, some representing the user of interest and some the interferers,followed by digital combining. Thus, in Sec. 2.4, we study the case where the APNquantization is the limiting factor in the design. To minimize the MSE, we tryto match an optimal beamformer to a linear combination of ND vectors from theset of available quantized beamformers; for this we propose a quantized versionof the matching pursuit (MP) algorithm [41,42]. All the above mentioned designsdepend on knowledge of the channel statistics: the antenna covariance matrix andthe antenna cross-correlation vector with the desired user signal. Note that thedigital receiver does not have direct access to the antenna signals. In Sec. 2.5,we provide an algorithm to deduce this information from various observationsof beamformed outputs (the algorithm is related to that of Sheinvald e.a. [38]).The results are combined in the digital beamformer to obtain a high resolutionestimate and illustrated using simulation results.

Notation

Vectors and matrices are represented in lower and upper case bold letters. (.)T ,(.)H and (.)† represent transpose, complex conjugate transpose and pseudo in-verse, respectively. ⊗ and ‖.‖2 represent Kronecker product and Frobenius norm.The operation f = vec(F) transforms a matrix F to a vector f by stacking itscolumns, while F = vec−1(f) does the opposite. Continuous time signals arerepresented with round braces as in (·) and sampled signals as [·].

SYSTEM SETUP AND DATA MODEL 25

2.2 System setup and data model

2.2.1 RF data processing

Consider an RF signal x(t) received at an antenna. Assuming suitable bandpassprefiltering, only a narrow frequency band around a carrier frequency fc is ofinterest, and we can write

x(t) = real{x(t)ej2πfct}where x(t) is the complex envelope or baseband signal. In the receiver, the “RF tobaseband” processing block recovers x(t) using quadrature demodulation. Thissignal subsequently enters the ADC unit. Here it is sampled at time instantst = kT (where T is the sampling period) leading to x[k] and quantized using Rbits, leading to Q(x[k]). We will always assume that the Nyquist condition holds.The ADC unit includes an automatic gain control (AGC) that scales the inputsignal such that its amplitude matches the range of the ADC without overload.

If the RF signal x(t) is delayed by τ , we obtain

x(t− τ) = real{x(t− τ)ej2πfc(t−τ)} ≈ real{x(t)e−j2πfcτej2πfct}.The approximation x(t − τ) ≈ x(t) is valid if fτ � 1 for all frequencies f inthe bandwidth of x(t), i.e, the “narrowband condition”. After RF to basebandconversion, the delayed baseband signal is x(t)e−j2πfcτ and the sampled signal isx[k]e−j2πfcτ .

If we have an array with Nr receive antennas, it will be convenient to stackall signals into Nr × 1 vectors x(t), x(t) and x[k], respectively.

2.2.2 Received data model

Consider now a communication set-up, whereNt user signals s(j)(t), j ∈ {1, · · · , Nt}are transmitted over the same carrier fc, propagate over a multipath channel, andare received by the array with Nr antennas. Without loss of generality, let s(1)(t)is the desired user signal, and the other signals are considered interferers. Assumethat the narrowband condition holds for all propagation delays (except for a bulkdelay that we will ignore here) so that they can be represented by phase shifts.We can write the equivalent discrete time data model as

x[k] = Hs[k] + n[k]

where s[k] = [s(1)[k], · · · , s(Nt)[k]]T is an Nt × 1 vector of user signals, and n[k]is an Nr × 1 vector of noise signals. H is a Nr ×Nt matrix denoting the MIMOchannel response with complex entries hij , representing the channel coefficients forthe propagation of the jth user signal to the ith receive antenna, which includesthe transmit/receive filters, array response, amplitude scalings, and phase delays.

Throughout the chapter, we will make several standard assumptions on thismodel:


• The user signals s(j)[k] are modeled as random processes that are zero mean,independent, wide-sense stationary, with equal powers normalized to 1.

• The noise signal vector n[k] is sampled from an i.i.d. Gaussian process, zeromean, with unknown covariance matrix Rn.

2.2.3 High-resolution digital beamforming

Given observations x[k], our goal is to obtain an estimate s(1)[k] of the desireduser signal s(1)[k]. We will consider linear beamforming and a minimum meansquare error (MMSE) criterion. Thus let θ be an Nr × 1 weight vector, then thedigital beamforming output is

y[k] = θHx[k]

and the MMSE beamformer is obtained as the solution of

θ0 = arg minθ

E‖s(1)[k]− θHx[k]‖2. (2.1)

As is well known, the solution is given by the Wiener beamformer [43]

θ0 = R−1x rxs (2.2)

where Rx = E{x[k]xH [k]} and rxs = E{x[k]s(1)[k]}. Estimates of Rx and rxs areobtained from the sample covariance matrix and sample cross-correlation vector;this requires access to all antenna signals and the availability of a reference signal(training sequence) for the desired user.

The above solution (2.2) will serve as our reference design. In its derivation,the effect of the quantizer operator Q(·) was ignored.

2.2.4 Analog preprocessing network (APN)

As mentioned in the Introduction, it is expensive to insert a complete RF receiverchain and high-rate high-resolution ADC unit for each antenna. We thus consideran analog preprocessing unit (APN) inserted in the RF domain immediately afterthe low-noise amplifiers and bandpass filter (see Fig. 2.2(a) and its discrete timeequivalent Fig. 2.2(b)). Although there are various implementations, we willmodel the APN as an analog beamformer that constructs linear combinationsof slightly delayed antenna signals xi(t). This results in output signals zj(t),j = 1, · · · , ND, where the number of outputs ND < Nr. As before, the outputsignals are stacked into ND × 1 vectors z(t), downconverted (leading to basebandsignals z(t)), sampled and quantized (leading to z[k]). The jth ADC has resolutionRj bits.

The effect of the APN on the baseband signal is modeled using a discrete timeequivalent matrix operation

z[k] = Q(WHx[k])


APNW

x(t)

NDNr

ej2πfcte[k]

z[k] y[k]QX

z(t) z(t) ϑAPNW

Nr+

x[k]

ND

z[k] y[k]ϑ

(a) (b)

LNA

Figure 2.2: (a) Proposed receiver architecture with RF beamformer (b) discrete timeequivalent.

where W = [w1, · · · ,wND ] is a matrix of sizeNr×ND and wj = [w1j , · · · , wNrj ]Tis a Nr × 1 vector. Each entry wij corresponds to the phase delay introducedby the APN for the ith receive antenna and the jth output signal. Practicalimplementations limit wij to a small set of possible phase shifts, perhaps 8 to 16choices (3 to 4 bits). Amplitude changes are usually not possible. The weightswij can be controlled by the baseband processor but note that x[k] is not directlyavailable at the processor, making the design of wij a challenge.

The digital baseband signals z[k] are subsequently combined using a digitalbeamformer ϑ = [ϑ1, · · · , ϑND ]T , resulting in the output signal

y[k] = ϑHz[k].

To obtain an estimate of s(1)[k], ϑ can be designed as the MMSE beamformersolving

ϑ0 = arg minϑ

E‖s(1)[k]− ϑHz[k]‖2 (2.3)

leading to a Wiener beamformer ϑ0 = R−1z rzs specified in terms of correlations

of z[k] where Rz = E{z[k]zH [k]} and rzs = E{z[k]s(1)[k]}. For given W, z[k] isknown and this problem can be solved, hence ϑ0 is a function of W.

2.2.5 Problem formulation

Our aim in this chapter is to design the APN W. After that, z[k] is fixed andthe design of ϑ is relatively straightforward. For the design, there are a numberof side conditions or assumptions:

[A1 ] The APN circuits consist of a limited number of phase shift combina-tions, hence the elements of W are selected from a finite set, denoted as adictionary D.

[A2 ] Each ADC performs uniform quantization with a resolution of Rj bits,hence Q{zj(t)} ∈ [−2−(Rj−1), 2−(Rj−1)].

The ADC power consumption can be approximated as Padc ∝ fs22R, wherefs is the sampling frequency (in our case the Nyquist rate) and R is the ADC

PREPROCESSOR DESIGN–APN NOT QUANTIZED 28

resolution in bits. We wish to minimize the number of bits in the ADC, since thisis directly related to the power consumption in the analog section of the receiver.The number of bits is determined by the required dynamic range, which is partiallycontrolled by W. Indeed, if more interference cancellation is performed, fewer bitsare needed for the same MSE performance.

Design objectives are (1) minimizing the MSE at the output of the digitalbeamformer, including the quantization noise, and (2) minimize the energy con-sumption in the ADCs, represented by

∑NDj=1 22Rj . Several problems can be for-

mulated using these objectives, but they do not all have feasible solutions.We therefore approach the APN design in the following order:

[P0 ] We initially relax [A1] and [A2] and assume a perfect and continuous APN,and high resolution ADCs. What are then the constraints on the design ofW? (It will follow that W is not unique.)

[P1 ] Assuming low-resolution ADCs, each with Rj = R bits, how does thedesign change? Can we compute a unique W?

[P2 ] Now considering the discrete nature of the APN, select W from a fixed setof discrete phase shifts present in D, such that the overall MSE is minimized.Here it is assumed that the ADC resolution is not limiting.

The above design problems [P1] and [P2] form the core of this chapter and arecovered respectively in Sections 2.3 and 2.4.

The design techniques will assume knowledge of the Nr × Nr antenna arraycovariance matrix Rx = E{x(t)xH(t)} and a Nr × 1 cross covariance vectorrxs = E{x(t)s(1)(t)}. Note that the introduction of the APN implies that thiscovariance matrix is not available in the digital part of the receiver. In Sec. 2.5, weexplain a technique to estimate Rx and rxs from a set of low-rate beamformers indigital baseband, assuming that a training sequence of the desired user is available.

2.3 Preprocessor design–APN not quantized

In this section, we consider problem [P1]: design the APN considering only thequantization by the ADCs, and design the number of bits which each ADC shoulduse. We do not consider the limited choice of phase shifts for the APN—this caseis deferred to Sec. 2.4.

2.3.1 Conditions on W to minimize the MSE

Let us also ignore the quantization operation by the ADCs for the moment, i.e.,consider problem [P0]. The problem is to design

θ = Wϑ , W : Nr ×ND.


We have z[k] = WHx[k], and the output y[k] = ϑHz[k] should be close to thedesired signal s(1)[k] in MMSE sense, i.e., we require

θ = θ0 = R−1x rxs .

Thus, W implements a rank reduction on the space spanned by x[k]. Somecommon but sub-optimal designs based on rank reduction are listed in Appendix2.8 (also refer to [34,44]).

Instead, we will now consider which conditions W has to satisfy such that wecan optimize the MSE. Note that, once W is specified, we know z[k] = WHx[k],and if we want to minimize the MSE of the output y[k] = ϑHz[k], we know wewill select

ϑ0 = Rz−1rzs = (WHRxW)−1WHrxs

whereRz = WHRxW , rzs = WHrxs . (2.4)

Thus, the MSE is a function of W only. We can see immediately that W willnot be unique: e.g., we could choose W = θ0, ϑ = 1 (for ND = 1), or W =[θ0, θ0], ϑ = [1/2, 1/2]T (for ND = 2), among many other possibilities.

Define the “whitened” correlation matrices

rxs = R−1/2x rxs , W = R1/2

x W . (2.5)

The following lemma characterizes all solutions W that lead to MMSE-optimalbeamformers θ = Wϑ.

Lemma 1. Consider the scenario [P0]: the APN is not quantized, and the quant-ization error of the ADCs is ignored. Then all beamformers W that lead to theMMSE-optimal solution θ0 = R−1

x rxs are characterized by the condition

rxs ∈ colspan{W} ⇔ rxs ∈ colspan{RxW}

Proof. θ = R−1x rxs = Wϑ implies that R−1

x rxs is in the column span of W.Hence a necessary and also sufficient condition on W is rxs ∈ colspan{W}.

An alternative proof is as follows. Define the orthogonal projection matrix

PW = W(WHW)−1WH

For any W, and corresponding optimal ϑ0 = (WHRxW)−1WHrxs, the MSE isgiven by

E‖s(1)[k]− (R−1z rzs)Hz[k]‖2 = E‖s(1)[k]− rHzsR

−1z WHx(t)‖2

= 1− rHxsW(WHRxW)−1WHrxs

= 1− rHxsW(WHRxW)−1WHrxs

= 1− rHxsPWrxs (2.6)


Thus, the MMSE solution W0 satisfies

W0 = arg maxW

rHxsPWrxs (2.7)

Clearly, this only specifies that rxs ∈ colspan{W}, and there can be many solu-tions.

2.3.2 Maximizing the SQNR

The next question is whether we can narrow down the set of available solutionsfor W, by satisfying additional design objectives. Our approach is to incorporatethe power consumption or ADC resolution in (2.7), and minimize the MSE whilemaximizing the signal to quantization noise ratio (SQNR) of the output estimate.This leads to a design for W, and also to criteria on the number of bits whicheach ADC should use.

We thus consider problem [P1]. Incorporating the effect of the ADC on thesignal at the output of the analog beamformer, we have

z[k] = Q(WHx[k])

The effect of the quantizer will be modeled by an ND × 1 additive noise vectore[k] = [e1[k], · · · , eND ]T , i.e.,

z[k] = WHx[k] + e[k].

As usual, e[k] is modeled as uniformly distributed noise, entrywise independent,and uncorrelated to x[k]. The corresponding covariance matrix of z[k] is (replacingEq. (2.4))

Rz = WH(Rx + Re)W , rzs = WHrxs , (2.8)

where Re is a diagonal matrix whose diagonal entries represent the quantizationnoise variance. Suppose that the ith ADC has a resolution of Ri number ofbits. We assume that an automatic gain control (AGC) is used such that thedynamic range of the ADC is optimally used. The quantization noise varianceσ2ei = E{ei[k]ei[k]} will then depend on the signal variance at the input of the

ADC, with some abuse of notation1 denoted as σ2zi . Using the well known Lloyd-

Max equation [45], we have

σ2ei = σ2

ziγ2−2Ri

12

where γ is an AGC scaling factor that models the difference between the averageinput power and the peak input power.

1since zi[k] denotes the signal at the output of the ADC in this subsection.


To reach a feasible optimization problem, we will limit ourselves to the casewhere all ADCs use an equal number of bits, Ri = R. The noise covariance matrixcan then be expressed as

Re = Dzγ2−2R

12where Dz = diag{WHRxW} .

Given W, the optimal digital beamformer, acting on z[k], is still ϑ = Rz−1rzs.

At the output of the beamformer, the average energies of the desired user signaland quantization noise are respectively

E‖s(1)[k]‖2 = E‖ϑHz[k]‖2 = rHzsR−1z rzs

E‖ϑHe[k]‖2 = rHzsR−1z ReRz

−1rzs

and the corresponding SQNR of the output signal is

SQNR =E‖ϑHz[k]‖2E‖ϑHe[k]‖2

=rHzsR

−1z rzs

rHzsR−1z DzR−1

z rzsγ2−2R

12

(2.9)

where Rz and rzs are functions of W as specified in (2.8). The objective is todesign W, first of all, to minimize the MSE as discussed in the previous sub-section, and this leads to design freedom, which is used to maximize the SQNR.

Regarding the minimization of the MSE, we can follow the derivation that ledto (2.6), however, Rz is now slightly different since it also involves the quant-ization noise Re. For a reasonable number of bits, we will have that ‖Re‖ =‖Dz‖γ2−2R

12 � ‖WHRxW‖. In that case, we can ignore the influence of Re onRz, Eq. (2.6) still holds, and the MMSE is obtained for any W satisfying (2.7),

rxs ∈ colspan{W}. (2.10)

Using the whitened quantities (2.5), the numerator of the SQNR expression canbe written as

rHzsR−1z rzs = rHxsW

H(WRxW)−1WHrxs = rxsW(WHW)−1WHrxs

and the denominator (up to scaling by γ2−2R/12) as

rHzsR−1z DzR−1

z rzs = rHxsW(WHW)−1Dz(WHW)−1WHrxs

where Dz = diag{WHW}. Subject to (2.10), the numerator is equal to ‖rxs‖2,which is a constant independent of W. It suffices to minimize the denominator.It further follows from the expression of the SQNR that the scaling of W is notimportant.

The SQNR optimization problem (subject to optimal output MSE) becomes:

W0 = arg minW

rHxs‖rxs‖

W(WHW)−1Dz(WHW)−1WH rxs

‖rxs‖(2.11)

subject to rxs ∈ colspan{W}We will solve this problem in closed form for the case ND = 2.


Theorem 1. Consider the scenario [P1]: the APN is not quantized, the ADCsare quantized at R bits. Assume ND = 2. Then the optimal APN that minimizesthe MSE and maximizes the SQNR (subject to optimal MSE) is obtained if allcolumns of W are equal to the MMSE beamformer, R−1

x rxs, up to scaling andcertain linear transformations.

Proof. To solve (2.11), we first parametrize W such that the constraint is satisfied.Thus let

W = UVH = u1vH1 + u2vH2

where V is a 2× 2 unitary matrix since ND = 2, and U = [u1 u2] is an Nr × 2matrix such that

u1 = rxs/‖rxs‖ .

Further definep = ‖u2‖ , α = uH1 u2/p

(Note that |α| ≤ 1.) Then

WHW = V[

1 αpαp p2

]VH

(WHW)−1 = V1

p2(1− |α|2)

[p2 −αp−αp 1

]VH

rHxs‖rxs‖

W = uH1 UVH = [1, αp]VH

rHxs‖rxs‖

W(WHW)−1 = [1, αp]1

p2(1− |α|2)

[p2 −αp−αp 1

]VH = vH1

Dz = diag{WHW} = diag{v1vH1 + p2v2vH2 + αpv2vH1 + αpv1vH2 }

Introduce a sufficiently general parametrization (θ,Φ1,Φ2) for V as

V =[ejΦ1

ejΦ2

] [cos(θ) − sin(θ)sin(θ) cos(θ)

]=:[φ1

φ2

] [c −ss c

](2.12)

where Φ1,Φ2 and θ are in the range (−π, π]. (A completely general parametriza-tion would also have two complex phase factors at the right, but one phase can beabsorbed in v2, and the other can be extracted to multiply the complete matrixV; the form of the cost function shows that that phase will cancel.) Then

Dz =[c2 + p2s2 − αpsc− αpsc 0

0 s2 + p2c2 + αpsc+ αpsc

]

=[c2 + p2s2 − 2βpsc 0

0 s2 + p2c2 + 2βpsc

]


where β = Re(α); note that −1 ≤ β ≤ 1. The cost function (2.11) becomes

J(W)=vH1 Dzv1

=[c s][c2 + p2s2 − 2βpsc

s2 + p2c2 + 2βpsc

] [cs

]

=c2(c2 + p2s2 − 2βpsc) + s2(s2 + p2c2 + 2βpsc)=(c4 + 2p2s2c2 + s4) + 2βp(s3c− sc3)

Since −1 ≤ β ≤ 1, minimizing the cost function will require choosing β at ex-tremes,

β = −sign(s3c− sc3)

Subsequently, optimal values for θ will follow as well, as a function of p. Thelocation of the minima of J(W) = J(θ, p) is shown in Fig. 2.3. The value of theminimum is 0.5. Although there are multiple minima, in any case, we will have|α| = 1: the correlation coefficient between u1 and u2 has absolute value 1, whichimplies that u1 and u2 are equal, up to a scaling and phase rotation.

In summary we derived that, given a specific resolution of the ADCs and thenumber of ADCs, the optimal approach to minimize the MSE and maximize theSQNR is to choose the columns of W all parallel to rxs. Translated to W, itmeans that each beamformer in the APN is parallel to the Wiener beamformerR−1

x rxs. In actuality, they should differ by at least a phase shift such that thequantization noise on the ND beamformer outputs becomes uncorrelated. Thedigital beamformer will simply average the results, i.e., average out the additivenoise and quantization noise.

The result was obtained for ND = 2, however it seems reasonable that itgeneralizes to larger ND. Also, the result was obtained for all ADCs havingthe same resolution Ri = R, but the same result will follow also for unequalresolutions. In this case the diagonal terms in Dz in J(W) will have unequalscaling, but it can be seen that the optimization still leads to |β| = 1, so that thesame conclusion follows.

Finally, let us consider the effect of the APN on the power consumption bythe ADCs. It is clear that by inserting an APN, interference cancellation be-comes possible, leading to reduced requirements on ADC resolution and henceenabling power reduction. As function of the interference power, the benefits canbe arbitrarily large compared to a setup without APN.

Next, we compare ND = 1 to ND = 2, while keeping a constant output MSEafter digital beamforming. For the optimal APN, the digital beamformer willsimply be averaging the outputs of the ADCs, so that the quantization noisepower at the output is halved. Quantization noise of one channel is proportionalto 2−2Ri . Thus, for ND = 2 and the same SQNR, each ADC needs half a bit lessthan for ND = 1. However, power consumption is also proportional to 22Ri . For2 ADCs, each with half a bit less, the total power consumption is constant. Thus,there is no particular advantage to choose ND > 1 from this perspective.

PREPROCESSOR DESIGN–APN WITH DISCRETE PHASE SHIFTS 34

0 1 2 3 4 5 6−3

−2

−1

0

1

2

3

Location of the minima J(W) = J(!. p) for ND=2

p

! "

[−#

: #]

Figure 2.3: J(W) as a function of p = ‖u2‖ and θ

More generally, ND > 1 allows us to use multiple ADCs with lower resol-ution in situations where high rate high resolution ADCs cease to exist due tofundamental limitations [1].

2.4 Preprocessor design–APN with discrete phase shiftsIn the previous section, we did not take the quantization of the APN coefficientsinto account. In practice, the elements of W can only be selected from a discretealphabet, usually only from a set of possible phase shifts. We will now studythis case, meanwhile ignoring the quantization of the ADCs, i.e., assuming theirresolution is high enough such that it does not dominate the design.

2.4.1 Matching the cross-correlation vector

Since the elements of W are quantized, let D represent the set of all possibleNr × 1 phase shift vectors that a column in W can take. The entries of D aredenoted as D = {φm}Mm=1, whereM is the size of the dictionary. If the APN tapsare quantized by RW bits then M = 2NrRW . Typically, RW would be 2 to 4 bits.Similarly, call DND the set of all possible APN matrices W.

The APN design is now transformed into a problem of selecting W ∈ DNDsuch that the MSE distortion is kept at a minimum,

W = argminϑ,W∈DND

E‖s(1)[k]− ϑHWHx[k]‖2 . (2.13)

Lemma 2. The MMSE beamformer solving (2.13) is obtained as W = Rx−1/2W,

whereW = argmin

ϑ,W∈DND‖rxs −Wϑ‖2 (2.14)


and where D = Rx1/2D = {Rx

1/2φm}Mm=1.

Proof. Following (2.7), solving (2.13) is equivalent to solving

W = arg maxW∈DND

rHxsPWrxs = arg maxW∈DND

‖PWrxs‖2

This is equivalent to

W = arg minW∈DND

‖(I−PW)rxs‖2 = arg minW∈DND

‖rxs −W(WHW)−1WHrxs‖2

which is equivalent to (2.14), since for given W the optimal choice for ϑ is ϑ =W†rxs = (WHW)−1WHrxs.

Thus, the problem becomes to match rxs in least squares sense to linear com-binations of columns of W, each of which can assume only values in a discreteset. Equivalently, the columns of W should span a subspace to which rxs is close.The selection complexity is exponential in Nr, ND, and RW.

2.4.2 Quantized Matching pursuit (QMP)

To reduce the complexity, the columns of W are selected one-by-one. The match-ing pursuit (MP) technique [41] is a greedy technique that recursively chooses thedictionary elements to obtain the best approximation of an input vector, in thiscase rxs. Indeed, write

‖rxs −Wϑ‖2 = ‖rxs −w1ϑ1 −w2ϑ2 − · · · −wNDϑND‖2.

In the greedy approach, we first solve

{ϑ1,w1} = arg minϑ1,w1∈D

‖rxs −w1ϑ1‖2. (2.15)

Given w1, the optimal solution for ϑ1 is ϑ1 = w†1rxs = (wH1 w1)−1wH

1 rxs, so thatthe problem reduces to

w1 = arg maxw1∈D

|wH1 rxs|‖w1‖

.

The solution requires a search in the dictionary, at a complexity exponential inNr and RW. To facilitate the search, we can first normalize the vectors in D tounit norm, and then search for the vector with maximal correlation to rxs.

After selecting w1 and ϑ1, we compute the residual vector r(1)xs = rxs −w1ϑ1

and proceed similarly as (2.15),

{ϑ2,w2} = arg minϑ2,w2∈D

‖(rxs −w1ϑ1)−w2ϑ2‖2 = arg minϑ2,w2∈D

‖r(1)xs −w2ϑ2‖2

where

r(1)xs = rxs −w1ϑ1 = rxs −

(wH

1 rxs

‖w1‖2)

w1.


After selecting w2 and ϑ2, we could continue the process with the residualvector

r(2)xs = r(1)

xs −w2ϑ2.

However, w1 and w2 are not orthogonal, and better coefficients ϑ1 and ϑ2 can becomputed at this point, leading to a smaller residual. This requires to solve

{ϑ1, ϑ2} = arg minϑ1,ϑ2

∥∥∥∥rxs − [w1,w2][ϑ1

ϑ2

]∥∥∥∥ . (2.16)

Define W(2) = [w1,w2] and introduce a QR factorization

W(2) = Q(2)R(2) (2.17)

where Q(2) is a Nr × 2 orthonormal matrix and R(2) a 2 × 2 upper triangularmatrix. The solution to (2.16) is

[ϑ1

ϑ2

]= (R(2))−1Q(2)Hrxs

and the corresponding (smaller) residual is

r(2)xs = r(1)

xs −Q(2)Q(2)Hr(1)xs = P⊥

W(2)r(1)xs

which is the projection onto the orthogonal complement of the column span ofW(2). The recursion follows in an obvious way. Note that the QR factorization(2.17) is easily updated once new vectors wi are added, and that in fact it isnot necessary to explicitly compute ϑ at intermediate steps. The algorithm issummarized in Table 2.1.

Further refinements of this algorithm are possible. The update of the QRfactorization and the computation of the residual r(i)

xs can be integrated into asingle update step. In principle, a better selection of wi could be obtained bycomputing the residuals for all possible wi from the dictionary and selecting theone that gives smallest residual, however the complexity of this is probably toohigh.

The processing structure is reminiscient of Goldstein e.a. [37], where the focusis to design orthogonal projections of w1, · · · ,wND (not quantized) such that itslinear combination using ϑ to estimate s(1)[k] does not require computations ofR−1

x .In practical implementations, the beamforming weights are usually quantized

phase shifts with unit amplitude. In this case, it may be more accurate to first splita desired weight vector w into two weight vectors w1 and w2, each with entries onthe unit circle, such that a linear combination of them gives the desired w. Zhange.a. [34] showed that such a partitioning is always possible. Subsequently, thephase vectors w1 and w2 are each quantized into discrete phase shifts, resulting

ONLINE CORRELATION ESTIMATION 37

Table 2.1: Quantized Matching Pursuit (QMP) algorithm

Objective: Select the discrete phase vectors W of the APNGiven: Input covariance matrix Rx and cross-correlation matrix rxs; dictionaryD = {φm}Mm=1.

• Transform the dictionary vectors into the “whitened” domain:

D = Rx1/2D = {Rx

1/2φm}Mm=1

• W(0) = [·], r(0)xs = Rx

−1/2rxs

• Recursion: for i = 1, · · · , ND,

– wi = arg maxwi∈D

|wHi r(i−1)

xs |‖wi‖

– W(i) = [W(i−1) wi]

– Update QR factorization: Q(i)R(i) := W(i)

– r(i)xs = r(i−1)

xs −Q(i)Q(i)Hr(i−1)xs

• W = Rx−1/2W(ND)

in the “greedy” selection, now consisting of a pair of vectors. The process continuesas before to ND recursions.

In this section, APN has been designed from (2.13) assuming that the ADCquantization is negligible. It is later shown using simulations in Sec. 2.6 that fora small RW (as is the case in practice) and R ≥ 6 bits, the MSE is dominated bythe quantization of the W.

2.5 Online Correlation EstimationThe APN phase shift design requires the knowledge of Rx and rxs. However,since the digital baseband processor has only access to the beamformed outputsz[k] and not to the individual antenna signals x[k], it is not possible to directlycompute these correlations from the available observations (and, in the case ofrxs, a training signal s1[k]). In the context of direction of arrival estimation, asimilar problem was studied by Sheinvald e.a. [38] and Tabrikian e.a. [46].

Regarding system architectures, there are a number of options as enumeratedin Fig. 2.4.

1. Each antenna has its own ADC, operating at a low rate and low resolution.In this way, full (but noisy) information is available and estimates of Rx, rxs

can be computed straightforwardly.


e[k]

APNW

Nr+

x[k]

ND

z[k] y[k]ϑ

(a)

IQ APN Estimation

e[k]

APNW

Nr+

x[k]

ND

z[k] y[k]ϑ

Q APN EstimationTi

(b)

e[k]

APNW

Nr+

x[k]

ND

z[k] y[k]ϑ

(c)

QAPN

Estimation

T1

Tq

Figure 2.4: (a) Architecture 1, where each antenna has its own ADC. (b) Architecture2, containing time varying beamformers Ti, i ∈ {1, · · · , p} for different training periods.(c) Architecture 3 with p low resolution beamformers.

2. During a training phase, a range of beamformers W = T1, · · · ,Tq areapplied instead of the optimal W, resulting in output sequences li[k] =Q(TH

i x[k]), for i = 1, · · · , q. Here Ti is a Nr × p matrix with p > 1 andli[k] is a p × 1 vector denoting the beamformer output during the trainingphase. The corresponding correlation statistics of l1[k], · · · , lq[k] are com-puted and the statistics of x[k] are inferred, as discussed below. Note thatthe automatic gain controls may have to re-adjust because the interferingsignals will not be suppressed during the estimation phase.

3. A separate APN and bank of low-rate/low-resolution ADCs is used tomonitor the inputs, resulting in output sequences li[k] = Q(TH

i x[k]), fori = 1, · · · , q. We refer to this setup as low resolution beamformers (LRB’s)and the quantization is typically with 1-2 bit ADCs. This is quite similar tocase 2, except that there is more flexibility in the number of APN outputs(can be different than ND) and ADC resolutions. The APN could simply bea set of switches, making a selection of the antennas towards a low numberof ADCs.

The choice of system architecture depends on various criteria, such as avaiable


space for additional hardware, the stationarity of the received signals, and theduration and density of the training periods in the desired signal. Regarding thetraining signal, there are also issues related to acquisition and synchronization tothe desired signal; we will not discuss this further.

Without loss of generality, we will consider case 3 and discuss how Rx andrxs are inferred. Let p > 1 be the number of APN outputs (dimension of eachli[k]). From the LRB output sequences li[k] = Q(TH

i x[k]), we will be able toform estimates of R(i)

l , of size p× p, with model

R(i)l = TH

i RxTi, i = 1, · · · , q

As in [38], we subsequently stack the columns of each of these matrices into vectorsvec(R(i)

l ), with model

vec(R(i)l ) = (Ti ⊗Ti)Hvec(Rx)

where the identity vec(ABC) = (CT⊗A)vec(B) was used. Stacking these vectorsresults in the model

vec(R(1)l )

...vec(R(q)

l )

=

(T1 ⊗T1)H...

(Tq ⊗Tq)H

vec(Rx)

⇔ refl = Tefrx (2.18)

Assuming Tef has a left inverse, we can estimate the data covariance matrix usingLeast Squares2 as Rx = vec−1(T†efr

efl ). The complexity of this step is in the order

of q2p4N2r .

The Hermitian property of Rx can be exploited by introducing a vectorizationoperator “vech(Rx)” that separately stacks the real and imaginary components ofthe upper triangular (resp. strictly upper triangular) part of its argument. Similarto [38,47], this refinement reduces the computational complexity and ensures thatthe resulting estimate is Hermitian.

A necessary condition for the left invertibility of Tef is that this is a “tall”matrix, or qp2 ≥ N2

r . Once this condition is satisfied, simple designs for the Ti

are already sufficient to obtain invertibility. E.g., for Nr = 4, p = 2, q = 4,matrices of the form

Ti =

1 0wi,1 0

0 10 wi,2

, wi,1 6= 1, wi,2 6= 1 ,

2 The paper [38] proposes to use a weighted Least Squares, but it can be shown that, for thisunparametrized estimate of Rx, the weight does not change anything.

SIMULATION RESULTS 40

with distinct wi,1 and wi,2, lead to a full rank Tef. Such low complexity selectionmatrices have also been used for DOA applications in [46].

If only switches are used, then (for p = 2) each Ti gives access to one cross-correlation entry in Rx. For Nr = 4 there are 6 such entries, and a minimaldesign is (q = 6)

T1 =

1 00 10 00 0

, T2 =

1 00 00 10 0

, T3 =

1 00 00 00 1

,

T4 =

0 01 00 10 0

, T5 =

0 01 00 00 1

, T6 =

0 00 01 00 1

.

The vector rxs can be estimated in a similar way from estimates of r(i)ls :=

E(l(i)[k]s(1)[k]) via the model equations

r(1)ls...

r(q)ls

=

TH1...

THq

rxs

This requires the matrix in the RHS to be tall (qp ≥ Nr) and full column rank,which is a milder condition than what we had for the estimate of Rx. As men-tioned, we need the desired user to be synchronized to the receiver and the re-ceiver must have knowledge of the training sequence transmitted at the start ofthe packet.

2.6 Simulation resultsTo assess the performance of the proposed algorithms, we have applied it to amulti-user/antenna setup and computer generated data. We present simulationresults that incorporate the impairments, discrete design and channel parameterestimation as covered in Sections 2.3, 2.4 and 2.5.

The input SNR is the signal to noise power ratio for the desired signal andthe noise as received at antenna 1; it is the same for all antennas. The input SIRis the signal to interference power ratio for the desired signal and the sum of allinterference signals as received at antenna 1; it is the same for all antennas. Allusers transmit QPSK signals, with zero mean and unit variance as assumed inSec. 2.2.2 and the interferers have equal powers. The performance indicators areusually

1 SINR at the first ADC input – a high SINR indicates that less power isspent in quantizing the interferers for a given ADC resolution.


2 MSE – observed at the output of the digital receiver.

All results are obtained by averaging 100 Monte Carlo runs, each with inde-pendent Rayleigh fading channel realizations and independently generated datasignals. Each run transmits data packages of size 8192 symbols, as in a WLANtransmission packet. The QMP algorithm we proposed in table 2.1 and Sec. 2.4is used to design the APN weights from the training sequence. Unless specifiedotherwise, we used Nt = 4 transmitters, Nr = 4 receive antennas, and ND = 2ADC receiver chains. The ADC resolution is R = 10 bits and transmit SIR is-5 dB. The APN is represented by a dictionary with RW = 4 bits. In the caseswhere the receiver is based on estimated channel coefficients, these are estimatedfrom a training sequence of length 256 symbols incorporated in the data packet.

2.6.1 Finite sized APN dictionaries

Figs. 2.5(a) and (b) show the SINR at the input of the first ADC and the MSEat the receiver output respectively. A training sequence of length 256 is usedfor the design of the APN. We show curves for varying dictionary size RW, andfixed ADC resolution of R = 10 bits. Consider the SINR plot (Fig. 2.5(a)), curve1 corresponds to a case with no APN and Nr = 4 ADCs operating with floatprecision and curves 2-6 show the performance of the APN setup for increasingRW. Comparing curves 1-4, the results show that the introduction of the APNwith dictionary size RW = 4 bits improves the SINR at the first ADC input upto a factor of 20 dB. For increasing SNR, the performance saturates: it is limitedby the residual interference power. The performance can be further improved byincreasing RW. Consider Fig. 2.5(b) comparing the MSE at the receiver. For aMSE 0.05, the setup with RW = 4 bits and ND = 2 ADCs each with R = 10bits (curve 4) performs 2 dB worse than the optimal Wiener beamformer with NrADCs and float precision (curve 1).

2.6.2 Effect of the ADC resolution

Figs. 2.6(a) and (b) show the MSE performance at the receiver for a similar setupas in Sec. 4.6.1, where the APN resolution is RW = 4 bits. In Fig. 2.6(a),curve 1 corresponds to a case with Nr = 4 ADCs, with float precision and curves2-5 correspond to ND = 2 ADCs and varying ADC resolution Ri = R bits. Thetransmit SIR is -5 dB. We observe that for R ≥ 6 bits, MSE curves ovelap and thefinite precision APN leads to an error floor. In other words, for higher resolutionADCs, the APN resolution is the limiting factor.

Fig. 2.6(b) gives an idea of ADC power savings with the introduction ofan APN, for an APN designed with RW = 5 bits and varying ADC resolutionR. Comparing curves 2-3 and 4-5 respectively, we see that the introduction ofan APN with ND = 2 ADCs operating with R = 4 and R = 6 bits results ina similar MSE values as that of a receiver without APN, Nr = 4 ADCs with


0 5 10 15 20 25 30 35−10

−5

0

5

10

15

20

25

Input SNR

SIN

R a

t the

firs

t AD

C in

put

SINR performance comparison at the ADC’s with and without APNas a function of finite sized APN dictionaries for Nt = Nr = 4 and ND=

1: Avg SINR at ADC’s without APN2: SINR at first ADC RW = 2 APN3: SINR at first ADC RW = 3 APN4: SINR at first ADC RW = 4 APN5: SINR at first ADC RW = 5 APN6: SINR at first ADC RW = 6 APN

1

2

3

4

56

0 5 10 15 20 25 30 3510−3

10−2

10−1

100

Input SNR ( in dB)

MSE

at t

he re

ceiv

er

MSE performance comparison at the receiver as a function of finite sized APN dictionary for Nt=Nr=4 and ND=2

1: Nr = 4 ADC’s and Wiener−Hopf2: MSE with RW = 23: MSE with RW =34: MSE with RW = 45: MSE with RW = 56: MSE with RW = 6

61

4

5

3

2

Figure 2.5: Performance comparison of the APN setup for different RW and R = 10as function of transmit SNR (a) SINR at the input of first ADC, (b) MSE at the outputof baseband receiver.

precision R = 4 and R = 6 bits followed by optimal Wiener beamformer. SinceADC power consumption is related to

∑NDi=1 22Ri , this suggests that the use of an

APN can reduce the ADC power consumption by half.

2.6.3 Effect of the number of APN outputs

Fig.2.7 shows the MSE performance at the output of the baseband receiver for asimilar setup as in Sec. 4.6.1, where the numberND of ADCs is varied, RW = 4−5bits and ADC resolution R = 10 bits. Typically, for APN with perfect interferencecancellation, MSE ∝ 1

ND. However, fixed precision APN and interfering users

limit the performance gains. From curve 3, we see that even with ND = 4, theMSE curves lead to an error floor and this suggests that the APN with RW = 4


0 5 10 15 20 25 3010−2

10−1

100

Input SNR (in dB)

MSE

at t

he re

ceiv

er

MSE performance comparison as a function of ADC resolution fortransmit SIR = −5 dB, Nt=Nr=4 and ND=2 with R bits each

1: Float precision with Nr = 4 ADCs

2: ADC resolution R = 2 bits3: ADC resolution R = 4 bits4: ADC resolution R = 65: ADC resolution R = 86: ADC resolution R = 10

1

2

4−63

0 5 10 15 20 25 3010−2

10−1

100

Input SNR (in dB)

MSE

at t

he re

ceiv

er

MMSE performance using ADC resolution for transmit SIR = −5 dBAPN designed with RM=5 bits, Nt=Nr=4 and ND=2 with R bits

1: Float precision with Nr = 4 ADCs2: Nr=4 ADCs with resolution R = 4 bits3: ND=2 ADCs with resolution R = 4 bits4: Nr=4 ADCs with resolution R=6 bits 5: ND = 2 ADCs with resolution R = 6 bits 1

2354

Figure 2.6: MSE performance comparison at the output of the baseband receiver as afunction of transmit SNR (a) for varied R (b) for various numbers and resolutions ofADCs

might be ill-conditioned. Increasing the APN Resolution to RW = 5 bits leads toimproved MSE performance as is obvious from curves 4 and 5. However, to limitthe APN circuit size it is suggested to choose RW = 4 and ND = Nr/2.

2.6.4 Effect of source spacing

Figs. 2.8(a) and (b) show the SINR and MSE performance as a function of thespacing between two adjacent sources. The simulations consider line of sightscenarios without multi-path, and results are observed for Nt = Nr = 3 with thedesired user transmitting from an angle say θ1 = 0◦. The MSE is a function oftransmit SNR= 20 dB, transmit SIR = -5 dB, and angular spacing θ betweenthe desired user and interferers. We consider 2 interferers equidistant from the


0 5 10 15 20 25 3010−2

10−1

100

Transmit SNR (in dB)

MSE

at t

he re

ceive

r

MSE performance comparison for various ND, RW transmit SIR = −5 dB, Nt=Nr=4 and ND=2 with R bits

1: Optimal Wiener beamformer Nr = 42: RW = 4 bits APN outputs ND=23: RW = 4 bits APN outputs ND = 44: RW = 5 bits APN outputs ND = 25: RW = 5 bits APN outputs ND =4 1

2345

Figure 2.7: MSE performance comparison at the output of the baseband receiver fordifferent ND.

desired user and in opposite directions transmiting from angles θ2 = θ − θ1 andθ3 = θ1 − θ. From Fig. 2.8(a) we see that for θ > 20◦, the APN improvesthe SINR at the first ADC by a factor of 15 dB. In both the cases, we see thatthe APN performs poorly for angular spacing < 5◦, and the performance can beimproved by increasing the number Nr of the antennas.

2.6.5 Effect of the channel estimation

Fig. 2.9 shows the mean squared estimation error between the true channel para-meters Rx and the estimated channel parameters at the receiver for a setupwith RW = 4 bits as a function of training lengths. To reduce the estimationtime/complexity, we limit the length of the training sequence to 256 as in curve4.

2.6.6 Communication setup and channel estimation with LRB’s

The previous sections have given indications on the improvements in SINR, powerconsumption and MSE for phase shifter based APN as functions of RW and Ri.Here we focus on channel estimation using LRBs as specified in Sec. 2.5. We selectthe architecture type 3 in Sec. 2.5 and choose q = 6 LRB’s with p = 2 outputs.As specified in that section, switches T1, · · · ,T6 are used to estimate Rx.

Figs. 2.10(a) and (b) compare the MSE and BER performance of the fixedprecision RW = 4, ND = 2 APN estimated using LRBs as a function of traininglengths. The results are compared with the reference Nr = 4 ADC case andbeamformer designed using true channel parameters. The ADC resolution is keptat R = 6 bits.


0 20 40 60 80−10

0

10

20

30

Spacing between each antenna element in degrees

SIN

R a

t the

firs

t AD

C in

put (

in d

B)

SINR performance at the first ADC for transmit SIR = −5 dB, transmit SNR = 20 dB, Nt=3, Nr=4, Nr=6 and ND=2

1: Average SINR at the input without APN2: Average SINR at the ADC input with Nr=43: Average SINR at the ADC input with Nr=6

0 20 40 60 8010−3

10−2

10−1

100

101


Mea

n sq

uare

d er

ror

MSE performance comparison at the first ADC for transmit SIR = −5 dB, transmit SNR = 20 dB, Nt=4, Nr=4 and ND=2

1: Optimal Wiener beamformer with Nr=4 ADCs2: APN with Nr=4 and ND=23: APN with Nr=6 and ND=2

Figure 2.8: Performance comparison for ND = 2 setup as a function of spacing betweendesired user and 2 interferers: (a) SINR at the input of the first ADC (b) MSE at theoutput of the baseband receiver.

In Fig. 2.10(a), the curves 1 and 3 show that that going from Nr = 4 toND = 2 ADCs leads to performance degradation of 2 dB at MSE of 0.05. For thesake of completeness, we also show the BER performance between the transmittedand received QPSK signals and see that curves 1 and 3 in Fig. 2.10(b) show thatgoing from Nr = 4 to ND = 2 ADCs APN leads to performance degradation of 2dB at BER 10−3.

Considering that the above setup reduces the interferer contributions at theADCs, these results suggest that architectures with reduced RF chains, limitedby RF imperfections can perform close to theoretical MIMO while consuming afraction of the power.


−5 0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Transmit SNR (in dB)

Nor

mal

ized

MSE

MSE between true and est E ||R_x − \hat{R}_x ||for N_t=4, N_r=4 N_D=2 and SIR = −5dB

1: Training length = 322: Training length = 643: Training length = 1284: Training length = 2565: Training length = 512

12345

Figure 2.9: Average estimation error between the true and observed channel statistics.

2.6.7 Communication theoretic view of APN and coarse ADCs

The need for preprocessing with respect to ADC dynamic range, RF interferencecancelation and power consumption is well documented in the earlier part ofthe chapter. Here we observe the effect of preprocessor on the achievable datathroughputs for fixed precision ADCs. In other words, given a setup with lowresolution ADCs and multi-user transmission scenario, does the APN lead toperformance close to Shannon’s channel capacity.

Achievable throughputs with APN

In [48,49], the authors observed the achievable throughputs when a MIMO setuplimited by interferers, is used with MMSE detection at the receiver. We extendthose results for fixed precision ADCs and observe that introducing APN leadsto the improved data throughputs of low resolution ADCs. In Fig. 2.11(a) thecurve 1 corresponds to the Shannon channel capacity (ideally coded signals) foran interference free SIMO (Nr = 4, Nt = 1) case C = log2(1 + SINR) and curve2 to the achievable throughputs for uncoded transmission float precision ADCswithout APN in a dense multi-user scenario. The curves 1 and 2 are reproducedfrom [48] for our setup and transmit SIR = 20 dB. For the same setup, usinglow resolution Nr = 4 3-bit ADCs, the maximum transmission rates saturate fortransmit SNR > 15dB as seen in curve 3. Introduction of APN with ND = 2and limited by 5◦ phase errors and 3-bit ADCs improves the transmission ratesfor by 25 % as seen by comparing curves 4 and 3, in addition to reduced powerconsumption in ADCs by half. As is obvious, for high resolution ADCs, the

CONCLUDING REMARKS 47

0 5 10 15 20 25 3010−2

10−1

100

Input SNR (in dB)

MSE

at t

he re

ceiv

er

MSE performance when APN is estimated from 2−bit LRB and different training lengths for transmit SIR = −5 dB Nt=Nr=4 ND=

1: No APN; optimal Wiener beamf 2: LRB’s Training length 1283: LRB Training length 2564: LRB Training length 3845: LRB Training length 512 1

2−34−5

0 5 10 15 20 25 3010−4

10−3

10−2

10−1

100

Input SNR (in dB)

Bit e

rror r

ate

at th

e re

ceiv

er

BER performance comparison as a function of 2−bit LRB and training lenghts for transmit SIR = −5 dB and Nt=Nr=4 and ND=2

1: Nr=4 ADC’s with Wiener beamformer

2: LRB Training length 1283: LRB Training length 2564: LRB Training length 3845: LRB Training length 512 1

23−45

Figure 2.10: Performance comparison, when Rx and rxs are estimated for 2-bit LRBsand varied channel lengths: (a) MSE (b) BER

achievable rates for APN and no-APN cases as seen by curves 2 and 5 coincidewith each other. However, note that in curve 5, the power consumption is halfof curve 2. Similar experiments are repeated for different interferer setup andcompared in Fig. 2.11(b) for transmit SIR 0 dB. To summarize, the interferencelimited MIMO capacity [49] can be improved with the introduction of APN, inaddition to power consumption for low precision ADCs. In a way, proper designof APN preserves diversity as well as reducing ADC power.

2.7 Concluding remarks

In this chapter, we have proposed a MIMO receiver employing an analog prepro-cessing network (APN) or multichannel beamformer in the RF domain, followed


−20 −15 −10 −5 0 5 10 15 20 25 300

2

4

6

8

10

12

Throughput and Channel capacity for a Nr=4, Nt=4, ND=2 APN setup

Transmit SNR

bits

/cha

nnel

/Hz

1: Shannon channel capacity2: No APN SIR −10 dB 4 10−bit ADC’s3: No APN SIR = −10 dB 4 3−bit ADC’s4: APN SIR =−10dB 2 3−bit ADC’s5: No APN SIR = 20 dB 4 10 bit ADC’s6: No APN SIR = 20 dB 4 3−bit ADC’s7: APN SIR −10 dB 10−bit ADC’s

−20 −15 −10 −5 0 5 10 15 20 25 300

2

4

6

8

10

12

SNR ( in dB)

bits

/cha

nnel

/Hz

Throughput and channel capacity for aNr=4, Nt=4, ND=2 APN setup

1: Shannon channel capacity − SIR = !2: No APN SIR = 0 dB 10 bit ADC3: No APN SIR = 0 dB 3−bit ADC4: APN SIR = 0 dB 10 bit ADC5: APN SIR = 0 dB 3−bit ADC

Figure 2.11: Achievable throughputs for interference limited MIMO system and APNsetup (a) SIR = -10 dB (b) SIR = 0 dB

by a digital beamformer in baseband. The prime advantage of this architecture isthat it reduces the number of antenna elements to a smaller number of mixers andADC chains. Further, it can reduce the interference at the inputs of the ADC,so that less dynamic range and fewer bits are required. Overall, significant powersavings are possible.

An optimal preprocessor to minimize the MSE and maximize the desired userSQNR at the digital baseband was derived. It was shown that, if the APN quant-ization is very fine, it is sufficient to consider only ND = 1 analog beamformingoutput. In practice the quantization is poor, and a larger number of outputs isrequired such that the cross-correlation vector rxs is well approximated. Further

APPENDIX: APN DESIGN USING CROSS SPECTRAL PROJECTIONS 49

research is needed in the following directions:

• In practice, the APN coefficients have poor accuracy; implementation errorsof up to 7% of phase have been reported [23]. This affects both the chan-nel estimation and the APN design. How can this effect be modeled andincorporated into the design?

• Initially, we are not synchronized to the source of interest. It may then becomplicated to estimate rxs and design the APN; subsequently, the inter-ference may overwhelm the ADCs and make acquisition impossible. Whatis a good initialization strategy?

An alternative to using an APN to reduce ADC power, is to exploit spatialand temporal oversampling with a predictive Sigma-Delta ADC as in chapter 4(also refer to [50]).

2.8 Appendix: APN design using cross spectral projectionsTo obtain some intuition on the APN design problem [P0] as specified in section2.3.1, we first consider a few sub-optimal techniques before deriving a closed formAPN in section 2.3.2. Introduce an eigenvalue decomposition of Rx:

Rx = UΛUH

where U = [u1, · · · ,uNr ] is anNr×Nr unitary matrix containing the eigenvectors,and Λ is a diagonal matrix containing the eigenvalues of Rx, sorted from largeto small. W can be chosen as the ND dominant eigenvectors, corresponding tothe ND largest eigenvalues, W = [u1, · · · ,uND ]. This is somewhat similar to oneof the approaches proposed in [34]. In this way, computation of z[k] = WHx[k]retains the components with the largest energy and drops the components withless power. However, this does not distinguish between desired and interferingusers. If the interferers are strong, the desired user could be projected out.

This is avoided in the following design, based on “cross spectral projections”.Instead of selecting ND dominant eigenvectors, selecting the eigenvectors thatcontain a large correlation with the desired user array response given by rxs resultsin a better approximation. This approach is similar to one technique in [44] wherethe authors sort the basis vectors based on the cross spectral norm, defined as

cxs = Λ−1UHrxs

More precisely, let s(1)[k] = θHx[k] with θ = R−1x rxs. Then we can write

θ = UΛ−1UHrxs = Ucxs

and obtain the energy of the output signal

E{|s(1)[k]|2} = θHRxθ = cHxsΛcxs =Nr∑

j=1

|cj |2λj (2.19)

APPENDIX: APN DESIGN USING CROSS SPECTRAL PROJECTIONS 50

x1(t)

xNr (t)

U

Nr ×Nr

cxs

Nr × 1

Q

Q

Q

Q

x1(t)

xNr (t)

W

z1(t)

zNd(t)

Nr ×Nd Nd × 1

(a) (b)θ

ϑ s(1)r [k]s

(1)f [k]

Figure 2.12: (a) Full rank Wiener beamformer (b) reduced rank beamformer

where cxs = [c1, · · · , cNr ]T and λj is the j-th eigenvalue in Λ. Selecting the NDcolumns of U corresponding to the largest (weighted) cross spectral norm terms,i.e., the ND largest terms among

|cj |2λj =|uHj rxs|2

λj, j = 1, · · · , Nr

would lead to the “best” ND × 1 representation of the desired user signal, in thesense of maximizing the output energy of the desired user.

Collect the selected ND columns of U into a matrix U′, and likewise for c′xs.Although it seems natural to choose W = U′ and ϑ = c′xs, the above techniquedoes in fact not prescribe a partitioning of θ into W and ϑ. Moreover, by selectingcolumns of U we have also limited our choice: x[k] is projected on a subspace ofeigenvectors. This is not necessarily optimal.

It will be shown in the main text in Sec. 2.3.1(Lemma 1) that the APN leadingto MMSE has to satisfy the condition rxs ∈ col span{RxW}. Choosing W equalto the dominant ND eigenvectors of Rx is optimal if rxs is in this subspace, whichoccurs only if there are at most ND sources in white noise. This also implies that,for the same scenario, the solution from cross spectral projections, where W = U′

consists of eigenvectors maximizing the cross spectral norm would be optimal onlyif the dominant ND eigenvectors are selected.

Chapter 3RF impairments - IP3 model

Strong interferers may drive the RF components into a non-linear mode of oper-ation. In this chapter, we consider a non-linear low noise amplifier (LNA) andmodel the non-linear output of the LNA as signals from interfering users. Fromthis model and the APN architecture as proposed in chapter 2, we conclude thatthe proposed setup of chapter 2 can cancel the intermodulation products arisingdue to the non-linearity of the RF components. This is not an original contribu-tion of the thesis, and for details refer to [20,25,26]. This chapter is only intendedto illustrate the scope of using APN like circuits.

3.1 Introduction

Designing RF components to account for imperfections such as device non-linearitiesin the front-end of communication systems is an important challenge. One popu-lar technique is to sample and quantize the WB signals, followed by interferencecancellation and digital compensation to account for the RF impairments. Theseapproaches are popularly known as dirty RF [18].

One important application of such dirty RF approaches is to compensate forthe non-linearities in RF devices such as a low noise amplifier (LNA). In conven-tional receivers, the RF circuits are coarsely designed to obtain a WB signal, ofwhich only a small part is made up of desired user. This is due to the fact thatdesigning NB RF components with fine channel selectivity is expensive.

In such a setup, the LNA operates at a bandwidth that is considerably largerthan that of the digital baseband. Typically, for a WLAN environment, theoperating bandwidth of the desired user is 20 MHz, while the LNA containssignals in the bandwidth of 100 MHz. Strong interferers, operating in the adjacentbands may drive the LNA into a non-linear mode of operation [20], resultingin intermodulation products (IP) of carrier frequencies. As a consequence, the

51

INTERMODULATION PRODUCT MODEL 52

digital baseband system might also contain harmonic products of these stronginterferers [20].

3.1.1 Problem statement and summary

Recent developments in the solid state community [51,52] offers many possibilitiesto combat intermodulation products in RF circuits. However, these papers focuson the circuit level.

This chapter systematically models the IP3 terms as RF interferers [25, 26]and to design the APN [13] to cancel them. It will turn out that signals frominterfering users in adjacent frequency bands of a weakly non-linear system canlead to interferers superimposed on the desired user signal. It will also turn out,that the spatial signature of the harmonic product will be different from that of theadjacent band interferers. We will conclude that the harmonic intermodulationproducts can be viewed as additional interfering user signals coming from differentdirections.

3.2 Intermodulation product model

LNAs are usually designed to operate in a linear mode, however strong interferersfrom multiple interferers may force them into a non-linear mode of operation [20,53]. Our first objective in this section is to understand the effect of LNA amplitudenon-linearities in the presence of strong interfering signals. This provides us witha starting point on possible approaches to cancel IP components.

3.2.1 Non-linear LNA output

We assume a weakly non-linear LNA and will show that the intermodulationproducts of beams from interfering users will radiate in directions different fromthe principal beam patterns of the interferers.

Let v1(t) = a1 cos(2πf1t+φ1) and v2(t) = a2 cos(2πf2t+φ2) be two interferinguser signals at the antenna array, operating respectively at the RF frequencies f1

and f2 (f1, f2 6= fc), where ai and φi are respectively the scaling factor and thephase shift for i ∈ {1, 2}. A linear LNA would result in

xl(t) = κ(a1 cos(ω1t+ φ1) + a2 cos(ω2t+ φ2))

where κ is a scaling factor denoting the gain of the LNA and ωi = 2πfi. In thelinear mode of operation, the LNA output containing terms with carrier frequen-cies with ω1, ω2 6= ωc are canceled after down-convesion, followed by band passfiltering.

Consider a scenario where the LNA is operating in a non-linear mode. Assumefor simplicity, that the LNA output follows a weakly non-linear model and can be


expressed as a power series

xnl(t) =∞∑

i=0

κi[v1(t) + v2(t)]i

where κi is a constant denoting the power series coefficients. For simplicity, weconsider only up to i = 3 in the above power series expression. Using the identitiescos2 x = 1

2 (1 + cos 2x) and cos3 x = 14 (3 cosx + cos 3x), we proceed to represent

the non-linear LNA output as

xnl(t) = κ0 + κ1[a1 cos(ω1t+ φ1) + a2 cos(ω2t+ φ2)] +κ2

2(a2

1 + a22) (3.1)

+34κ3[a3

1 cos(ω1t+ φ1) + a32 cos(ω2t+ φ2)]

+12κ2[a2

1 cos 2(ω1t+ φ1) + a22 cos 2(ω2t+ φ2)]

+14κ3[a1 cos 3(ω1t+ φ1) + a2 cos 3(ω2t+ φ2)]

+32κ3a1a2[a1 cos(ω1t+ φ1) + a2 cos(ω2t+ φ2)]

+κ2a1a2[cos(ω1t− ω2t+ φ1 − φ2)]

+κ33a1a2

2[a1 cos(2ω1t− ω2t+ 2φ1 − φ2) + a2 cos(2ω2t− ω1t+ 2φ2 − φ1)].

In the above expression (3.1), the terms in the first three lines contain signalsoperating at the DC or at multiples of ω1, ω2. Typically, the LNA output isdown-converted, low pass filtered and quantized using the ADC. For multiples ofω1, ω2 6= ωc, the terms in the first 3 lines of (3.1) are always filtered out. The lasttwo expressions of (3.1) contain the intermodulation harmonic products of v1(t)and v2(t) [51, 53] operating at frequencies n1ω1 ± n2ω2 where n1, n2 ∈ {±1,±2}are positive/negative integers. These terms are of significance, especially whenn1ω1 ± n2ω2 ≈ ωc.

3.2.2 Third order IP and the effective radiation pattern

In band-pass communication systems, the third order distortion (also referred toas IP3) of components (e.g. n1 = ±2, n2 = ±1) might end up near the frequencyband of interest [21]. Higher order distortion terms tend to be weaker, while theeven order terms will end up far from the frequency band of interest [20].

Let n1 = 2, n2 = 1, f1 = 2.42 GHz and f2 = 2.44 GHz be the respective ordersand frequencies of strong interferers. As seen from Fig. 3.1, 2f1 − f2 ≈ fc = 2.40GHz, and in this example the dominant IP3 component falls into the operatingbandwidth centered around fc. The non-linear LNA output near fc is

xnl(t)|ωc = κ33a2

1a2

2cos(2ω1t− ω2t+ 2φ1 − φ2).


f1 − f2

2f1 − f2

f0 = 2.40 f1 = 2.422f2 − f1

f2 = 2.44

Figure 3.1: Spectrum of interfering signals with frequencies f1 = 2.42 GHz and f2 =2.44 GHZ and their third order IP terms coinciding with fc = f0 = 2.40GHz

The other terms of the LNA output in (3.1) are annihilated when the down-converted signals are used with a low pass filter. For 2ω1 − ω2 ≈ ωc, the down-converted complex baseband signal can then be represented as

xnl(t) = e−jωctxnl(t)|fc = Av(t)

whereA = κ33a2

1a24 e−j(2φ1−φ2) is a complex baseband term independent of ω1, ω2 and t,

and v(t) denotes a complex baseband signal and independent of ω1 and ω2.Assume for the time being, that the RF interferers v1(t) and v2(t) are NB

signals. The IP3 term v(t) is also a NB signal centered around 2ω1 − ω2 ≈ ωc.The LNA output xnl(t) can also be viewed as v(t) modulated by a phase shiftof 2φ1 − φ2. Going back to our NB frame-work, the non-linearity in the LNAmodulates the point source v(t) with a radiation pattern different from that ofv1(t) and v2(t). In addition, note that xnl(t) operates at a frequency fc.

Alternatively, the phase shift 2φ1 − φ2 can also be seen as a NB propagationdelay as long as (2φ1 − φ2)f � 1, where f is in the range of frequencies in thebandwidth of v(t). Extending the setup to an array of Nr antenna and assumingfor simplicity a uniform linear array (ULA) the IP3 components at the antennaarray can be simplified as:

xnl(t) = A

1e−j(2φ1−φ2)

...e−j(Nr−1)(2φ1−φ2)

v(t) = av(t) (3.2)

where a is a Nr × 1 vector denoting the array response and is a function of2φ1 − φ2. For an arbitrary antenna array configuration, the elements of a arecomplex entries corresponding to propagation delays. The consequence of theexpression is that Rx, nl = E{xnl(t)xHnl(t)} is of rank one.

CONCLUSION 55

3.2.3 Extension to multi-user and wide-band models

Consider a case, where multiple non-linear combinations of interferers leads toNip IP3 terms represented by a Nip × 1 vector v(t) = [v(1)(t), · · · , v(Nip)(t)].The NB wireless channel from user j to the antenna array can be denoted as aNr × 1 vector a(j) similar to a in (3.2) with j ∈ {1, · · · , Nip}, and the basebandequivalent IP3 terms can be seen as a MIMO channel output

xnl(t) =[a(1), · · · , a(Nt)

]v(t).

The conclusion is that the covariance matrix for a multi-user case Rx, nl =E{xnl(t)xHnl(t)} is of rank Nt.

3.2.4 Beamforming to cancel IP3 componenrs

From the NB array processing literature [4], a Nr × 1 phase shift vector presentin the null space of Rx, nl can be used as a spatial filter to cancel xnl(t). Giventhe APN setup W containing phase shift vectors and the statistics of the wirelesschannel, one approach to account for non-linearity in the RF components is todesign the phase shift vectors canceling xnl(t).

Spatial distribution of intermodulation products has already been specifiedin [26]. The systematic modeling of the IP3 terms and the necessary conditionsfor using spatial filter made of spatial filter in digital baseband is specified in [25].

Our objective in this mini chapter is to suggest that instead of digital beam-forming, an RF phase shift beamformer can be designed as in chapters 2 and 5to partially cancel the IP3 terms xnl(t). This would mean that the IP3 signalssuperimposed with the desired user are not translated into digital baseband, andthat the ADCs do not spend any energy in processing the IP3 signals, leading toreduced power consumption.

3.3 ConclusionIn this chapter, we have briefly illustrated some prior work [25,26] on antenna ar-ray systems that design a beamformer to cancel the IP terms due to non-linearitiesin the LNA. The conclusion is that the APN arrangement proposed in chapters 2and 5 can also be extended to cancel the IP3 terms.

CONCLUSION 56

Chapter 4Multi-channel ADCs with feedback:Mixed signal interference cancellation

Beauty of style and harmony andgrace and good rhythm dependon simplicity.

Plato

In multi-user MIMO communication receivers, the use of high resolution ADCsis costly. In the presence of (strong) interference, more bits are used than would benecessary for quantizing only the signal of interest. Thus, if it is possible to cancelthe interference in the analog domain, considerable savings can be realized. In thischapter, we exploit the fact that a MIMO receiver consists of a bank of ADCs, andpropose a new architecture, wherein a feedback beamformer (FBB) takes a linearcombination of the ADC outputs and feeds back the result to be subtracted at theinput. This ADC architecture is especially compatible with existing Σ∆ ADCs,that already consist of a digital to analog converter (DAC) in the feedback loop,and enables sophisticated source separation algorithms designed in the digitalbaseband to cancel the interfering users in the analog domain. Subsequently, theΣ∆ ADCs digitize only the desired user signals, and achieve considerable savingsin power consumption. Using a Mean Squared Error criterion and assuming atraining sequence is available, we present an algorithm to design the weights of theFBB. The interference suppression and power savings of the proposed approachare demonstrated via simulation results.

57

INTRODUCTION 58

4.1 Introduction

4.1.1 Interference cancellation at the ADC

Antenna arrays in MIMO receivers exploit spatial diversity to achieve reliable com-munications at low signal energies and in the presence of interferers [6]. However,multiple receive antennas lead to multiple RF and ADC chains, which increasesthe circuit area and the power consumption as explained in chapter 1.

In this chapter, we exploit the fact that MIMO receivers have multiple anten-nas and ADC chains. One well known sub-optimal approach to reduce the numberof ADC chains, thereby reducing the ADC power, is to select the antennas withthe largest signal energies and only quantize these [19]. However, such techniquesdo not fully exploit the advantages of MIMO systems, cannot track variations inthe wireless channel, and fail in the presence of interferers. Instead of antennaselection, if we can cancel the interference a priori in the analog domain, thenconsiderable power savings can be achieved. It is possible to integrate analogphase shifters within the RF architecture [13], so that we obtain a set of analogbeamformers to cancel interferers. In [54] and chapter 2, we detailed one suchapproach. We discussed algorithms for estimating the channel and designing thebeamforming coefficients to minimize the mean squared error (MSE) between thedesired user and its estimate at receiver.

The present chapter introduces an alternative approach. Here we consider abank of ADCs connected to the antenna array. A digital beamformer operateson the ADC outputs to identify the interferer signals. These interferer estimatesare fed back via a digital to analog converter (DAC) to cancel the interferinguser signals before the quantization operation. For a given ADC resolution, in-terference cancellation would allow a more faithful digital representation of thedesired user signals; alternatively, we can quantize with fewer bits to achieve thesame performance. The space-time beamforming coefficients are designed using atraining sequence from the desired signal.

One subclass of ADCs, especially compatible with the above DAC feedbackarchitecture, is the family of Σ∆ ADCs. These have been extensively studiedin the literature [16, 55]. Typically, the Σ∆ ADCs sample at a frequency muchhigher than the Nyquist frequency, and obtain coarsely quantized signals (say1-bit), which are subsequently integrated to obtain a high resolution signal in thedigital baseband. Further improvements in signal prediction and reconstruction,exploiting the bandlimited (BL) nature of the input, have been studied previouslyin [56–58]. The DAC feedback included in the Σ∆ architecture offers possibilitiesto suppress the BL interferers. As an example, Philips e.a. [28] perform interfer-ence cancellation with a high pass filter in the DAC feedback loop to cancel someclass of interferers. This is a single-channel and non-adaptive solution. Interfer-ence cancellation utilizing the spatial diversity offered by multi-antenna receivershas not been addressed yet. However [28] offers a starting point for the realizationof such architectures in silicon.

INTRODUCTION 59

Multipath channel

Interferer

DSP Post-

processing Desired user

estimate

Q

Q

RF-IF

RF-IF

Desired user

sampling frequency

1-bit Quantizer

ADC

fs

x1(t)

xNr (t)

c(1)[k]

c(2)[k]

b1[k]

bNr [k]

c(1)[k]

digital baseband

post-processin

g

Σ−∆

Σ−∆

MMSE detector

+

+

x1(t)

xNr (t)

b1

bNr

fs

fs

s(1)

s(2)

DAC

-

-

Q

Q

α1 αNr

αi Antenna response

Figure 4.1: (a) Antenna array configuration with desired and interfering user signalsquantized by oversampled ADCs followed by baseband combining to estimate the desireduser; (b) Proposed multichannel (MC) ADC architecture with a feedback beamformer(FBB) to identify and cancel interfering user signals.

4.1.2 Setup and Objectives

We consider a narrow-band (NB) multi-user setup, where the desired and interfer-ing users transmit over a shared wireless channel as in Fig. 4.1(a). These signalsare received by an array of Nr antennas, downconverted to baseband, sampled andcoarsely quantized using an oversampled ADC. In Fig. 4.1(a), the digital post-processor does beamforming to suppress the interfering signal. Alternatively, inFig. 4.1(b) the multiple ADC outputs are fed-back via a digital beamformer tocancel the interferers. We will refer to this architecture as a multi-channel (MC)Σ∆ ADC with a feedback beamformer (FBB). Note that the Σ∆ ADC is just onerelevant architecture to cancel the interferers; this could be generalized to othertypes of oversampled ADCs.

With the objective to cancel the interferer before the ADC operation, our aimin this chapter is to design the FBB coefficients. If done successfully, this willlead to reduced power consumption. Even a partial cancellation of the interferinguser energy would lower the requirements on the dynamic range of the ADC andlead to an overall power reduction in the ADC units. As will be shown usingsimulation results in Sec. 4.6, introducing the FBB may lead to a reduction inpower consumption by a factor of four.

INTRODUCTION 60

In the signal processing literature, several types of equalizers have been de-signed where the equalizer output is fed back to cancel the incoming signals [29].One related context is the classical least mean squares (LMS) technique [30]. Feed-back equalization techniques in the context of blind channel estimation have beenproposed in [59, 60]. In these methods, a feedback beamformer operating on theequalizer outputs estimates and cancels the redundancy in the incoming signals.Our aim is to integrate such techniques within the ADC architecture. Somewhatsimilar work on designing a quantizer with a single channel FBB in the contextof sub-band coding is given by [56]. In this work the authors specify the ADCoutput as an interpolation of bandlimited signals, and design the quantizer (asan oversampled frame) to cancel the redundancy in the incoming signals. Morerecently [61] design FBBs in the context of a compressive sensing framework.

The above mentioned approaches aim to perform noise shaping or to cancelredundancy in the incoming signals. In contrast, our aim is to design the quantizeroutput to cancel the interferers and to reconstruct the signal of interest. Thissignal is identified via a training sequence.

4.1.3 Outline and Contributions

We progressively address the different design issues of an FBB arrangement usedwith a Σ∆ ADC. In Sec. 4.2, we specify the oversampled Σ∆ ADC set-up andprovide general background on higher-order Σ∆ ADCs that provide a filter in thefeedback loop. In Sec. 4.3, we then consider a single-channel Σ∆ ADC operatingon the incoming band limited signals. We formulate a mean squared error (MSE)cost function on the difference between the desired user training signal and itsestimate at the output of the ADC. Solving this cost function is not directlyfeasible, but it can be modified into a prediction error cost function that can besolved in closed form. The original cost function can then be solved iterativelyusing the previous solution as a starting point. In Sec. 4.4, we consider a bankof ADCs receiving contributions from the desired and the interfering users, andderive an extension of the single channel FBB design to a multi-channel FBBdesign. Again, the FBB can be designed in a closed form for a prediction errorcost function, and the minimum MSE solution can be obtained iteratively. Alldesigns depend on the availability of a training signal, i.e., the output of the ADCsin the absence of interferers and noise. In Sec. 4.5, we consider some aspects ofthis. Finally, in Sec. 4.6, we show simulations to indicate the performance of theproposed algorithms. It is seen that in many cases an interference suppression ofover 20-25 dB can be achieved, leading to comparable power savings.

Notation: (.), (.)T , (.)H and (.)† denote conjugation, transpose, Hermitiantranspose, and pseudo-inverse operations. ⊗ denotes the Kronecker product,vec(·) a vectorization operator (stacking all columns of its argument into a vec-tor), and ‖.‖ denotes the Frobenius norm. Vectors and matrices are representedin lower and upper case bold letters, and we use an underscore to denote somemulti-channel vectors and matrices (for distinction with the single-channel case).

PREDICTIVE QUANTIZATION WITH Σ∆ ADCS 61

IK denotes the identity matrix of size K ×K, 1K and 0K are K × 1 vectors ofones and zeros respectively.

4.2 Predictive quantization with Σ∆ ADCsIn this section, we review the Σ∆ ADC architecture, and in particular extensionswith a higher-order feedback loop. We derive a transfer function model, andformulate the design problem to be solved in subsequent sections.

4.2.1 Signal sampling and reconstruction

Consider an ADC operating on a continuous time input signal x(t). We willassume that x(t) is band limited by the highest frequency of operation f0, andtime-limited to a given observation interval t ∈ [0, T ), where T would correspondto the duration of a transmission packet. Outside this interval, the signal issupposed to be zero.

If x(t) is uniformly sampled N times in the interval, then its samples aredenoted as x[n] = x( nN T ), for n = 0, · · · , N−1, and they are stacked in an N×1vector x = [x[0], · · · , x[N − 1] ]T .

Let M correspond to the number of samples of x(t), obtained when sampledat the Nyquist rate of 2f0 in [0, T ), i.e. M = b2f0T c. If sampled at this rate, theinput signal is uniquely represented by its samples and can be reconstructed as

x(t) =M−1∑

m=0

x[m]sinc(t−m(T/M)

T/M

). (4.1)

This reconstruction can be implemented via a lowpass filter. as shown in Fig.4.2(a). If the input signal has variance σ2

x and the ADC quantizes the signal atres bits, the quantization noise e[n] is modeled to be uniformly distributed andindependent of x[n] with a variance σ2

e = c2−2resσ2x, where c is a constant [62].1

In oversampling ADCs, the sampling frequency is much greater than theNyquist rate. Let N be an integral multiple of M and define the oversamplingratio (OSR) R = N

M . If sampled at this rate, the input signal is redundantlyspecified by its samples. Indeed, given the M × 1 Nyquist sample vector xM , theoversampled sample vector xN of size N × 1 is given by inserting t = nT/N in(4.1), leading to the interpolation formula (ignoring quantization noise)

xN = IxMwhere I is a tall N ×M matrix with entries

Inm = sinc(n(T/N)−m(T/M)

T/M

). (4.2)

1We assume that the ADC unit includes an automatic gain control (AGC) which scales theinput signal to match the range of the ADC without overload.


Q LPF

Q LPF

Q LPF

x(t)

z−1x(t)

z−1

fcutoff = f0

fcutoff = f0

fcutoff = f0

OSI 1

OSI R

fN ⇒ Nyquist frequency(a)

+

(b)

Critically sampled

R levels

{−1, 1}

{−1, 1}

fN = 2f0

xM (t)

xR(t)

Figure 4.2: (a) Nyquist and (b) oversampled ADC set-up.

Conversely, given xN , the Nyquist sample vector xM can be obtained by prem-ultiplying xN by any left inverse of I, leading to an infinite number of possiblereconstructions that differ in the way that the quantization noise is processed.One straightforward reconstruction of the Nyquist samples can be implementedvia a lowpass filter followed by a downsampler. as shown in Fig. 4.2(b). In thiscase, the quantization noise on the samples in xN has variance σ2

e,N = c2−2resσ2x,

whereas after reconstruction the samples of xM have quantization noise varianceσ2e,M = 1

Rc2−2resσ2

x. This is a factor R lower.

However, it is possible to do much better. Since the input signal is bandlimited,it is possible to predict the current input sample x[n] from previous samplesx[n − 1], x[n − 2], · · · . By subtracting this prediction from the input signal andquantizing only the prediction error p[n], the required dynamic range at the inputof the quantizer is much smaller, and so will be the quantization error. This isthe principle of differential quantization, or predictive ADCs [56, 62]. A specialcase is the Σ∆ ADC, reviewed in the next subsection.


+ +

z−1

z−1-+

+ Q

DAC

-

+ ∫

fs

(a)

(b)

+x[n]

x(t)

e[n]

LPF

fc = f0

LPF

fc = f0

p(t) d[n] b[n]

p[n] b[n]d[n]

x[n]

Down-sample

Down-sample

Figure 4.3: (a) First order continuous time Σ∆ modulator with 1-bit output, (b) Dis-crete time equivalent model.

4.2.2 Oversampled Σ∆ ADCs

Consider a first order Σ∆ ADC operating at a sampling frequency fs = 2Rf0 as inFig. 4.3(a). Its discrete-time equivalent is shown in Fig. 4.3(b). In this setup, theprediction error p[n] is integrated, and the quantizer Q{·} digitizes the integratoroutput d[n] to obtain b[n] = Q{d[n]}. The quantizer output b[n] is fed back witha one-sample delay and subtracted from the input through a DAC to obtain aprediction of x[n]. As before, Nyquist-rate samples are obtained after low-passfiltering and decimation by a factor R. For modeling purposes, we replace thequantizer operator Q{·} by an i.i.d. additive noise source e[n] as in Fig. 4.3(b),such that b[n] = d[n] + e[n]. For details, see [16].

As indicated in the figure, x[n] = b[n− 1] is the prediction of x[n], and p[n] =x[n]− x[n] is the prediction error. The discrete-time integrator operating on p[n]has output d[n], satisfying the difference equation

d[n] = p[n] + d[n− 1] = (x[n]− b[n− 1]) + d[n− 1] . (4.3)

We assume, for simplicity throughout the chapter, that the initial states ared[−1] = 0 and b[−1] = 0. The ADC output b[n] can be rewritten from (4.3),utilizing b[n] = d[n] + e[n], as

n∑

j=0

b[j] =n∑

j=0

x[j] + e[n] . (4.4)


+ +

-

+ +x[n] b[n]d[n]

e[n]

p[n]

z−1

z−1+

z−1

+

+

w0

wK−1

x[n]

Figure 4.4: Discrete time equivalent model of a Kth order Σ∆ ADC with weightedfeedback, represented by w = [w0, · · · , wK−1]T .

Stack the Σ∆ modulator output b[n] for n = 0, · · · , N − 1 as a N × 1 vector b =[b[0], · · · , b[N − 1]]T , and likewise for the input vector x = [x[0], · · · , x[N − 1]]T

and the quantization noise vector e = [e[0], · · · , e[N − 1]]T . From (4.4), therelation between b and x can be written as

Lb = Lx + e (4.5)

where L is an N ×N lower triangular matrix whose non-zero elements are equalto 1; it corresponds to the integration operation.

It is thus seen thatb = x + L−1e ,

where L−1 is a lower bi-diagonal matrix with +1 and −1 on the main diagonaland off diagonal positions, respectively. One can interpret L−1 as a high passfilter, suppressing the low frequency quantization noise terms. For this reason,Σ∆ ADCs are commonly referred to as noise shaping ADCs. The remaininghigh frequency quantization noise terms will be canceled by a LPF with cut-offfrequency f0 prior to the decimation of the output b. It is well known that thismakes the quantization noise power drop off with a factor 1/R3, rather than1/R as we had with a straightforward oversampling ADC [27]. Since the powerconsumption of an ADC can be approximated as Padc ∝ fs22res, we see that itis advantageous to maximize R and use a 1-bit converter.

4.2.3 Generalized higher order Σ∆ ADC

The discrete time equivalent model specified by Fig. 4.3 can be generalized to ahigher order Σ∆ ADC [16, Ch. 6]. Consider Fig. 4.4, explaining a Kth order Σ∆ADC. In this case, the predictor of x[n] is formed by the output of a Kth order


FIR filter with coefficients wk, k = 0, · · · ,K − 1, which we will stack in a vectorw = [w0, · · · , wK−1]T . The prediction of x[n] is x[n] = wHbK [n − 1], wherebK [n− 1] = [b[n− 1], · · · , b[n−K]]T .

Similar to the first order case, the prediction error p[n] = x[n]−wHbK [n−1] isintegrated and subsequently quantized, represented by the additive quantizationnoise e[n]. The output thus becomes

b[n] =n∑

j=0

p[j] + e[n] .

Similar to (4.4), the ADC output can be written as

b[n] +n∑

j=0

wHbK [j − 1] =n∑

j=0

x[j] + e[n] . (4.6)

Stack the LHS and RHS terms of (4.6) for n = 0, · · · , N − 1, and assume thatb[j] = 0 for j < 0. It follows that the ADC output vector b satisfies

Mb = Lx + e (4.7)

where M is an N×N matrix whose elements are made of w (as specified in detailin the next section), and L is an N×N matrix as defined in (4.5). As an example,a first order (K = 1) Σ∆ ADC with w = w0 leads to M described by

M =

1w0 1...

. . . . . .w0 · · · w0 1

.

For the special case where w0 = 1, M is equal to L, and (4.7) reduces to (4.5).Following (4.7), we see that b is given by

b = (M−1L)x + M−1e .

The main question with this generalized architecture is how to design w. E.g.,we can select the objective to minimize the low-frequency components of thequantization noise terms M−1e, i.e., noise shaping. Alternatively, we can designw such that b[n] is an estimate of a target signal b0[n].

4.2.4 Problem formulation

The recurring theme in this chapter is to design the feedback weights specifiedby w, focusing on the cancellation of unwanted interfering signals. This reducesthe required dynamic range at the quantizer, leading to lower requirements on

SINGLE CHANNEL FEEDBACK BEAMFORMER DESIGN 66

the oversampling ratio or ADC resolution, and consequently minimizes the powerconsumption.

For the design of w, it is clear that the receiver needs a way to distinguishinterfering signals from the desired signal. To simplify the design procedure, wemake the assumption that there is a training phase, namely one entire transmis-sion packet during which the receiver has knowledge of a training signal specifiedby a N × 1 vector b0 = [b0[0], · · · , b0[N − 1]]T , along with the correspondingmeasured input signal x. This enables us to design w that minimizes the meansquared error (MSE) between the desired and observed signal at the output ofthe quantizer, i.e., such that the ADC quantizes mostly the desired signal. Afterthis, the FBB w is kept fixed and is used for subsequent transmission packets.

We approach the design problem in the following order:

• We initially consider a single channel ADC. Can we design w to minimizethe MSE?

• Subsequently, we extend the single channel set-up to a multi-channel scen-ario. Each channel has an ADC, and can use the feedback from the otherADCs as well. How is the feedback beamformer W designed?

• Finally, we consider the application of this multichannel ADC in a multi-user communication scenario, and discuss how a suitable training sequenceb0 is obtained.

4.3 Single channel feedback beamformer designWe consider the single channel feedback set-up in Fig. 4.4. Our aim will be toderive an estimation algorithm for the feedback weights w. To start, we will firstrederive the data model (4.7) in more details.

4.3.1 Data model

Let x(z) and b(z) denote the z-transformations of the sequence x and b (de-noted by N × 1 vectors) respectively, i.e. x(z) =

∑N−1n=0 x[n]z−n and b(z) =∑N−1

n=0 b[n]z−n. Similarly, let w(z) represent the z-transform of w: w(z) =∑K−1n=0 wnz

−n. The transfer function in Fig. 4.4 of the set-up is representedin the z-domain as

(1 +

z−1w(z)1− z−1

)b(z) =

11− z−1

x(z) + e(z)

⇔ B(z)b(z) = L(z)x(z) + e(z) (4.8)

whereL(z) =

11− z−1


is the transfer function of the integrator and

B(z) = 1 +z−1w(z)1− z−1

is the transfer function of the feedback arrangement. Define (with some abuse ofnotation, since as written here the matrix Z is not invertible)

Z−1 =

01 0

. . . . . .1 0

, Z =

0 1. . . . . .

0 10

.

We can then make the correspondence

L(z)x(z) ↔ Lx =

11 1...

.... . .

1 1 · · · 1

x[0]x[1]· · ·

x[N − 1]

where L = (I− Z−1)−1 = I + Z−1 + · · ·+ Z−(N−1). Similarly,

w(z)b(z) ↔ Wb =

w0 0... w0

wK−1. . . . . .. . . . . . . . .

0 wK−1 · · · w0

b[0]b[1]· · ·

b[N − 1]

where W = w0I + w1Z−1 + · · ·+ wK−1Z−(K−1). The transfer function (4.8) canthus be written in matrix form as

(I + Z−1LW)b = Lx + e (4.9)

which corresponds to our earlier model (4.7), setting M = I + Z−1LW.

Since Mb is a linear function of the entries of w, we can write this alternatively


as a matrix multiplied by w. Specifically,

Wb =

b[0] 0b[1] b[0]...

. . . . . .b[N − 1] · · · b[1] b[0]

w0

...wK−1

0...

=

b[0] 0

b[1]. . .

.... . . b[0]

......

...b[N − 1] · · · b[N −K]

w0

...wK−1

=: Bw.

It follows thatMb = b + Z−1LWb = b + Z−1LBw.

Starting from (4.7), an alternative model formulation is thus

(Z−1LB)w = (Lx− b) + e. (4.10)

4.3.2 Estimation of w

To estimate w, we assume that a desired output (training) sequence b0 is given,along with measured input data x. Ideally, we would aim to minimize the expectedmean squared error at the output, i.e., minimize

J(w) = E‖b− b0‖2 (4.11)

where b is given by the model (4.9), written as

b = M−1Lx + M−1e ,

and the expectation is with respect to the quantization noise e. A complicationwith this formulation is that w enters the model via the denominator, M−1, i.e., ina nonlinear way. The same complication occurs in auto-regressive moving-average(ARMA) model identification [63], where the direct Least Squares problem isconsidered not attractive. More practical is to formulate, as in the Prony method[63], a prediction error problem, i.e., to premultiply (4.10) by M and move thedenominator to the other side and minimize

J ′(w) = E‖Mb−Mb0‖2 = E‖Lx + e−Mb0‖2 . (4.12)

Write Mb0 as a function of w, i.e., Mb0 = b0+(Z−1LB0)w, where B0 is obtainedusing the delayed samples of b0. We obtain

J ′(w) = E‖(Lx− b0)− (Z−1LB0)w + e‖2 .

MULTI-CHANNEL FEEDBACK BEAMFORMER DESIGN 69

This is a Least Squares cost function. Minimization of J ′(w) to w leads to theclosed-form solution

w = (Z−1LB0)†(Lx− b0) . (4.13)

A unique solution exists if Z−1LB0 has a left inverse, i.e., if this matrix has fullcolumn rank. Since Z−1 and L commute, we can write Z−1LB0 = LZ−1B0. Thematrix L is square and invertible and does not change the column rank of Z−1B0.The matrix Z−1 introduces a zero row on top of B0 and drops the last row. Wethus require that the (N − 1)×K Toeplitz matrix

b0[0] 0

b0[1]. . .

.... . . b0[0]

......

...b0[N − 2] · · · b0[N −K − 1]

is full column rank. This requires at least that N > K. Under this condition,it is sufficient that, e.g., b0[0] 6= 0. More generally, it is sufficient that one ofb0[0], b0[1], · · · , b0[N −K − 1] is not zero.

4.3.3 Iterative refinement

As in the Prony method, we can go back to solve the original cost function (4.11) inan iterative fashion. This relates to the iterative prefiltering technique by Steiglitzand McBride [63], and is also known as Iterative Weighted Least Squares. Theconvergence of such techniques is shown in [64].

Thus, given the solution w in (4.13), form W and M = I + Z−1LW. Due toits structure (lower triangular with main diagonal equal to I), the matrix M isalways invertible. Define Λ = M−1. Then

J(w) = E‖b− b0‖2 = E‖M−1(Mb−Mb0)‖2 = E‖Λ(Lx + e−Mb0)‖2.

We now freeze Λ and solve the resulting Weighted Least Squares problem for wpresent in M:

w = (ΛZ−1LB0)†Λ(Lx− b0) .

If necessary, this process is repeated until convergence. In this manner, we obtaina good approximation to the solution of the original cost function (4.11).

4.4 Multi-channel feedback beamformer design

4.4.1 Data model

We now consider an extension of the FBB design to a multi-channel (MC) setup,where a bank of Nr Σ∆ ADCs is used to quantize an Nr × 1 vector x[n] =


+ +

z−1-+

b[n]

e[n]

p[n]+S/H

fs

x[n]x(t)

z−1

z−1

LPF

fc = f0

++

+

ϑ

s(1)[n]Nr

Nr

W0

WK−1

y[m]

m ∈ {1, · · · , M}

R

x[n]

Figure 4.5: Interference cancellation with a FBB W operating on a first order multi-channel Σ∆ ADC.

[x1[n], · · · , xNr [n] ]T , as shown in Fig. 4.5. The Kth order feedback filter w isreplaced by a feedback beamformer (FBB) or space-time filter W of sizeKNr×Nr.

As before, Nr low-resolution quantizers digitize the integrator output d[n] asb[n] = Q{d[n]} = d[n] + e[n], where e[n] is the quantization noise, and theseare now Nr × 1 vectors. The ADC outputs are fed back via the FBB matrixW. The prediction of x[n] is a Nr × 1 vector x[n] = WHbK [n − 1], wherebK [n − 1] = [bT [n − 1], · · · , bT [n − K] ]T is a KNr × 1 vector, and W is aKNr × Nr filtering matrix W = [WT

0 , · · · , WTK−1 ]T with Wj the Nr × Nr

beamforming matrix for the jth lag, j = 0, · · · , K − 1. Similar to Sec. 4.3, theNr × 1 prediction error vector is

p[n] = x[n]−WHbK [n− 1] (4.14)

and the ADC output satisfies

b[n] +n∑

j=0

WHbK [j − 1] =n∑

j=0

x[j] + e[n]. (4.15)

Let b be a NNr × 1 vector stacking the b[n] as b = [bT [0], · · · , bT [N − 1] ]T ,and likewise for x and e. Stacking the LHS and RHS terms of the ADC outputin (4.15) for n = 0, · · · , N − 1, and assuming that b[j] = 0 for j < 0, leads to aMC relation

M b = Lx + e (4.16)

where L = L⊗INr is aNNr×NNr lower triangular matrix and M is aNNr×NNrlower triangular matrix constructed from the elements of W (see below). L andM denote the MC equivalents of L and M respectively. For example, whenK = 1,

M =

INrWH INr

WH . . . INrWH · · · WH INr

.


More specifically, we can generalize (4.9) to the MC context. Indeed, using mul-tichannel z-transforms, we can write

d(z) = (I− z−1I)−1p(z)b(z) = (I− z−1I)−1x(z) − (I− z−1I)−1WH(z)z−1b(z) + e(z)

⇔ [I + z−1(I− z−1I)−1WH(z)]b(z) = (I− z−1I)−1x(z) + e(z)

Let Z−1 = Z−1 ⊗ I, then we can rewrite this as

M b = L x + e (4.17)

whereM = I + Z−1LWH ,

and

WH =

WH0 0... WH

0

WHK−1

. . . . . .

. . . . . . . . .0 WH

K−1 · · · WH0

.

To generalize (4.10) to the MC context, we first write WHb in terms of entries ofW, as follows:

WHb =

WH0 b[0]

WH0 b[1] + WH

1 b[0]...

=

(b[0]T ⊗ I)vec(WH0 )

(b[1]T ⊗ I)vec(WH0 ) + (b[0]T ⊗ I)vec(WH

1 )...

= (B ⊗ I)w

where

B =

b[0]T 0

b[1]T. . .

.... . . b[0]T

......

...b[N − 1]T · · · b[N −K]T

, w = vec(WT ) =

vec(WT0 )

vec(WT1 )

...vec(WT

K−1)

.

(B has size N ×KNr, whereas w has size KN2r × 1.) Furthermore,

Z−1LWHb = (Z−1 ⊗ I)(L⊗ I)(B ⊗ I)w = (Z−1LB ⊗ I)w


so thatM b = b + (Z−1LB ⊗ I)w . (4.18)

We can thus write the MC data model (4.17) as

M b = L x + e ⇔ (Z−1LB ⊗ I)w = (L x− b) + e . (4.19)

Due to the Kronecker structure, we can also rearrange this in a more compactform as

(Z−1LB)W = (LX−B) + E (4.20)

where

X =

x[0]T

x[1]T......

x[N − 1]T

, B =

b[0]T

b[1]T......

b[N − 1]T

,

and with a similar definition for E.

4.4.2 Estimation of W

Similar to the single-channel case, we aim to design the FBB W such that b[n]approximates a known training sequence b0[n], i.e., we aim to minimize

J(W) = E‖b− b0‖2 (4.21)

where b is given by the model (4.17), i.e.,

b = M−1Lx + M−1e .

Similar to before, this minimization is complicated, and we replace it by theminimization of a modified cost function

J ′(W) = E‖M b−M b0‖2 = E‖L x+e−M b0‖2 = E‖(LX−B0)−(Z−1LB0)W+E‖2F .(4.22)

The closed form solution is given by

W = (Z−1LB0)†(LX−B0) . (4.23)

Using this as a starting point, we can solve (4.21) iteratively by setting M =I + Z−1LW, defining Λ = M−1, and writing

J(W) = E‖b− b0‖2 = E‖Λ(Lx + e−Mb0)‖2 .

For a fixed Λ, we can rewrite Mb0 using (4.18), and solve for W present in w:

w = (Λ[Z−1LB0 ⊗ I])†Λ(Lx− b0) . (4.24)

CONSTRUCTION OF THE TRAINING SEQUENCE 73

If necessary, this can be repeated iteratively a number of times.A few remarks are in order. First of all, the solution (4.23) is seen to treat

each channel “independently”: each column of W depends only on a correspondingcolumn of the input data X and training data B0. This is different for thesolution of the weighted problem (4.24), because the weighting Λ does not havea Kronecker structure (i.e., it cannot be factored into a Kronecker product of twomatrices). The weighting relates the channels to each other.

Secondly, we can verify the conditions for the existence of a unique left inverse.For the first solution (4.23), we require B0 at least to be tall, i.e., N > KNr.However, this is a necessary but not sufficient condition. If each channel has thesame training sequence, then b0[n] = 1b0[n], and it is seen that each column B0 isNr times repeated: we can write B0 = B0⊗1T . Thus, for such training sequences,the first solution is not unique. However, the Kronecker structure may also be anadvantage, as it will lead to simpler calculations: we can take as the (non-unique)left inverse

(Z−1LB0)† = [(Z−1LB0)⊗ 1T ]† = (Z−1LB0)† ⊗ 1/N .

For the second solution (4.24), the matrix to be left-inverted has size NNr×KN2r ,

and it is tall if N > KNr. If B0 has repeated columns, then the same will holdfor Λ[Z−1LB0 ⊗ I]. Thus, in general also this solution will not be unique, andagain, the structure can be exploited to facilitate the computations.

4.5 Construction of the training sequence

The FBB designs as discussed in the previous sections all depend on the criticalassumption that a training sequence b0 (or b0) is available. In this section, wediscuss a few cases under which such a training sequence may be obtained.

4.5.1 Single-channel ADC

We consider first the single-channel case. Ideally, we should have b0 = x0: thetraining sequence is equal to an interference- and noise-free version of the incomingsignal. For the case where the channel is instantaneous and not convolutive, thismeasured signal is (up to a complex scaling α) equal to the transmitted signal,presumably an oversampled version of the Nyquist-rate symbol sequence s(1) ofM entries (the superscript (1) denotes the desired user index). In this case, wecan set

b0 = αIs(1) , (4.25)

where I is the interpolation function as defined in (4.2). If the channel is convo-lutive, then this generalizes to

b0 = G(1)s(1) , (4.26)

CONSTRUCTION OF THE TRAINING SEQUENCE 74

where the N ×M -matrix G(1) is the oversampled channel response of the de-sired user. This matrix can be estimated using standard channel estimationtechniques [65], although the presence of the interference may complicate this.Alternatively, the estimation of α or G(1) can be integrated in the MSE estima-tion of w, changing the problem into a more general ARMA prediction estimationproblem. The details of this estimation are omitted here and the reader is referredto [63].

Note that the FBB ADC structure consists of a feedback filter and is capableof doing some equalization. If the channel is convolutive but we still use (4.25)and set b0 = Is(1), then the FBB will attempt to do interference cancellationand equalization. The quality of the result depends on how well the equivalentKth order AR filter can equalize the convolutive channel. Presumably, this wouldwork best for a channel that is FIR of order less than K after oversampling.

4.5.2 Multi-channel ADC

In principle, the multi-channel case is a straightforward generalization of thesingle-channel case, although we have somewhat more design freedom. We will as-sume that all channel outputs are to reconstruct the same desired user sequences(1), up to complex scalings denoted by αi for the ith channel. In that case,we obtain for the instantaneous channel case as suitable training sequence thegeneralization of (4.25) as

b0 = Is(1) ⊗ a = (I ⊗ a)s(1) , (4.27)

where a = [α1, · · · , αNr ]T . We can recognize that a is the array response vector(or direction vector) of the desired user.

For a convolutive channel, we obtain as generalization

b0 = G(1)s(1) , (4.28)

where G(1) is an NNr ×M matrix, with little structure, that contains the over-sampled channel responses of the desired user to each of the Nr antennas.

As in the single-channel case, if G(1) (or a) is unknown, it will have to beestimated using a prior channel estimation phase, or the estimation has to beintegrated in the estimation of W, making this an ARMA estimation problem.

4.5.3 Digital postprocessing

After the ADC outputs b have been obtained, typically two postprocessing stepsare applied:

1. The outputs are low-pass filtered and downsampled. This operation can berepresented by a wide rectangular matrix F, e.g. F = I†. The downsampledADC output as shown in Fig. 4.5 is aMNr×1 vector y

M: y

M= (F⊗I)b.


2. These down-sampled signals yM

= [yT [0], · · · , yT [M − 1] ]T are used witha Nr × 1 digital beamforming vector ϑ to obtain an estimate of the desireduser symbol sequence s(1) = [s(1)[0], · · · s(1)[M −1] ]T as s(1)[m] = ϑHy[m].A reasonable approach to estimate ϑ is to minimize the output MSE

ϑ1 = arg minϑ‖(s(1))T − ϑHY‖2

⇒ ϑ1 = (Y YH)−1Y s(1) (4.29)

where Y = [y[0], · · · , y[M − 1] ] and ϑ is the well known Wiener beam-former. In this case s(1) is known from training.

Ideally, we should have designed the FBB using a MSE criterion based onthe resulting single output s(1), and jointly design W and ϑ. It can be seenthat this leads to a design problem with (too) many degrees of freedom and nounique solution, unless additional constraints are taken into account, such as thequantization errors.

For a related design problem, we were able to derive that the output MSEis minimized if all Nr ADC outputs are nominally equal to the same signal, upto complex scaling, but with independent (quantization) noise [54]. This thenmotivates the design of the training as in the previous subsection. In the simplestcase, all ADCs reconstruct the same output signal, and the optimal (Wiener)beamformer ϑ in that case becomes a simple average of the ADC outputs, aftercorrecting for any phase differences. For convolutive channels, we can model theoutput as (4.28) and base the design of the beamformer on this model.

The effect of the lowpass filtering should be integrated in the MSE cost func-tions (4.11) and (4.21), i.e., we do not minimize ‖b − b0‖ but rather filteredversions ‖F(b − b0)‖2. The generalization is straightforward and will lead to aprojection matrix based on F. It will enter in the equations in a similar way asΛ.

4.6 Simulation results

To assess the performance of the proposed algorithms, we have applied them to amulti-user/antenna set-up and computer generated data. We present results thatincorporate a first order FBB in a bank of first order Σ∆ ADCs as presented inSection 4.4.

In the simulations, the input signal to noise ratio (SNR) is the signal to noisepower ratio between the desired user signal and the thermal noise as receivedat antenna 1; it is the same for all antennas. The input signal to interferenceratio (SIR) is defined as the ratio of the power of the desired signal to the sumof powers of all interference signals as received at antenna 1; it is the same forall antennas. All users transmit QPSK signals with zero mean and unit variance,and the interferers have equal powers. The performance indicators are


Table 4.1: Operation of multi-channel interference canceling Σ∆ ADCs

Objective: Cancel interference with multi-channel Σ∆ ADCsStep 1: Given: Input signal x

• Initialize: d[0] = 0; b[0] = 0

• For OSI n = 1 to N and antennas i = 1 to Nr

– p[n] = x[n]−WHbK [n− 1]

– Σ∆ modulator output b[n] =∑ni=1 p[i]

Step 2: MMSE detector ϑ in digital baseband operating on b[n]: s(1)[n] = ϑHb[n],to reconstruct s(1)[n].

1. The average signal to interference and noise ratio (SINR) of the predictionerror signal p[n] in the MC Σ∆ ADC. A high SINR indicates that less poweris spent in quantizing the interferers for a given ADC resolution.

2. The MSE of s(1)[n], observed after digital post-processing as in Sec. 4.5.3.

All results are obtained by averaging 1000 Monte Carlo runs, each with in-stantaneous independent Rayleigh fading channel realizations and independentlygenerated data signals. Each run transmits data packages of length 8192 symbolsas in a WLAN transmission, where the first 256 symbols are used for training.The beamformer design techniques proposed in Sec. 4.4 are used to design theFBB weights from the training sequence, followed by the digital beamformer ϑ.Unless specified otherwise, we used Nt = 4 transmitters and Nr = 4 receive an-tennas. The SIR at the antennas is −5 dB and the oversampling Σ∆ ADCs eachhave 1-bit resolution. After downsampling the effective resolution of each ADCat the Nyquist rate would be log2R, which corresponds to 6 bits for R = 64 (nottaking into account the increase in resolution due to noise shaping).

4.6.1 Effect of fixed precision DAC feedback

Fig. 4.6 shows the basic performance of the MC FBB ADC receiver, where weset R = 256. We show the SINR at the input of the ADC, i.e. prediction errorp[n], and the output MSE as a function of the input SNR. We further considerthat in practice the feedback loop uses poorly quantized signals: the DAC in thefeedback loop often has only 1 bit. Therefore, we plot several curves, for varyingresolution of x[n] =DAC(WHbK [n − 1]). A 1 bit DAC output has two possiblelevels, corresponding to the sign of of the elements of x[n]. For a precision greaterthan 1 bit, most DAC implementations are not linear, which is why they are lesspopular.


0 5 10 15 20 25 30 35−10

−5

0

5

10

15

20

25

Transmit signal to noise ratio (in dB)

Aver

age

SIN

R a

t the

AD

C in

put (

in d

B)

SINR performance at the ADC for transmit SIR = −5 dB, Nt=4, Nr=4 R = 256 and varying DAC resolutions

1: Average SINR − ADC input no FBB2: Average SINR − ADC input with 1−bit DAC FBB3: Average SINR − ADC input with 4−bit DAC FBB

1

23

20 25 3010−3

10−2

10−1

100


Mea

n sq

uare

d er

ror a

t the

rece

iver

MSE performance at the receiver for transmit SIR = −5 dB, Nt=4, Nr=4, R = 256 and varying DAC resolutions

1: MSE at the receiver float ADCs Wiener beamf.2: MSE at the receiver with 1−bit DAC and R=256 FBB3: MSE at the receiver with 4−bit DAC and R=256 FBB

1

23

Figure 4.6: Performance comparison as a function of the resolution of x[n] =DAC(WHbK [n]): (a) average SINR at the ADC input, (b) MSE at the receiver.

For reference, curve 1 in Fig. 4.6(a) plots the SINR performance for the casewithout FBB i. e. W = I when K = 1. Similarly, curve 1 in Fig. 4.6(b) plotsthe MSE performance for the optimal (Wiener) beamformer acting in the digitaldomain with full precision ADCs.

Fig. 4.6(a) compares the average SINR at the input of the ADCs as a functionof transmit SNR for different resolutions of the DAC. We see that the introductionof the FBB (curves 2, 3) improves the SINR by a factor of 20 to 25 dB, whencompared to a set-up without the FBB (curve 1). However, the introduction ofthe fixed precision DAC in the feedback loop introduces a small performance lossat the receiver as seen in Fig. 4.6(b). In this case, curve 1 acts as the reference andcurves 2–3 show the MSE performance of the MC Σ∆ ADC set-up for differentDAC precisions.


4.6.2 Effect of source spacing

Figs. 4.7(a) and (b) show the SINR and MSE performance as a function of theangular spacing between two adjacent sources. The simulations consider a lineof sight scenario without multi-path, and the results are observed for Nt = 3sources, with the desired user transmitting from an angle of θ0 = 0◦ and receivedby an Nr = 3 uniform linear array whose elements are spaced half-wavelengthsλ/2 apart. The DAC output x[n] =DAC(WH bK [n− 1]) is of 4-bit precision andR = 256. Two interferers are located at varying equidistant angles θ1 = θ andθ2 = −θ. The plots show the SINR and MSE as function of θ, where the transmitSNR is 24 dB, and the transmit SIR is −3 dB. For reference, curve 1 plots theSINR performance for the case without FBB, and the MSE performance for theoptimal (Wiener) beamformer, for Nr = 3 antennas.

Looking at Fig. 4.7(a) and comparing curves 1 and 2, we see that for θ > 20◦

the introduction of the FBB improves the SINR at the first ADC by a factor 25dB. The FBB setup cannot suppress the interferers for angular spacing θ < 10◦.Fig. 4.7(b) shows the MSE performance after digital post-processing. It is senthat the performance of the FBB follows that of the optimal Wiener beamformerclosely, both cannot suppress interferers for small angular deviations.

4.6.3 Effect of the ADC oversampling factor

We now keep the DAC precision at 1 bit and vary the oversampling ratio R fortransmit SNR = 24 dB and Nt = Nr = 4. Figs. 4.8(a) and (b) show the SINR atthe ADC input and the MSE performance at the receiver for varying R (curve 2).For reference, curve 1 plots the SINR performance for the case without feedback,and the MSE performance for the optimal (Wiener) float precision beamformer.

In Fig. 4.8(a), we observe that the introduction of the FBB improves the SINRby a factor of 20 dB (when R ≥ 8). In Fig. 4.8(b), we see that MSE saturatesfor a FBB with R ≥ 64 due to the limited resolution of the DAC. Note thatalthough the MSE performance with the 1-bit DAC is considerably worse whencompared with the optimal antenna array set-up and float precision, the energyconsumption in the proposed set-up is considerably less than that of the referenceset-up.

4.6.4 Extent of ADC power savings

We now keep the DAC precision fixed at 1-bit and compare the MSE performanceof our Σ∆ ADC setup with a fixed precision first order Σ∆ ADC without FBB.The ADC resolution is kept fixed at 1-bit. In Fig. 4.9, curves 1, 2 and 4respectively show the MSE performance at the receiver for float precision ADCs,R = 32 and R = 128 with no FBB i.e. W = I. Curve 3 corresponds to a MC Σ∆ADC setup with R = 32, and a FBB W with 1-bit DAC arrangement as in Sec.4.4.


0 20 40 60 80−5

0

5

10

15

20


Aver

age

SIN

R a

t the

AD

C in

put (

in d

B)

SINR performance at ADC input for transmit SIR = −3 dB, trasmit SNR = 24 dB, Nt=3 Nr=3 4−bit DAC and R=256

1: Average SINR ADC input Nt=3 Nr=3 no FBB2: Average SINR ADC input Nt=3 Nr=3 with FBB

0 20 40 60 8010−3

10−2

10−1

100

101


Mea

n sq

uare

d er

ror

MSE performance at receiver transmit SIR = −3 dB, transmit SNR = 25 dB, Nt=3,Nr=3 and R=256

1: MSE at receiver Float ADCs Wiener beamf.2: MSE at receiver with 4−bit DAC and FBB

Figure 4.7: Performance comparison as a function of the angular spacing between thedesired user and 2 interferers: (a) average SINR at the ADC input, (b) MSE at thereceiver output.

From Fig. 4.9, we observe that the introduction of the FBB leads to a 5-bit(R = 32) ADC with MSE performance close to that of an equivalent 7-bit (R =128) ADC without FBB. From the power consumption relation Padc ∝ R(f022res),we can conclude that interference cancellation with fixed precision ADCs in adense multi-user setup leads to quadruple improvement in power consumption ata similar performance.

4.7 Concluding remarks

In this chapter, we have proposed a multi-channel ADC set-up, employing a space-time feedback beamformer that complements the usual DAC feedback. The prime


0 50 100 150 200 250 300−5

0

5

10

15

20

Oversampling ratio R

Aver

age

SINR

at t

he A

DC in

put (

in d

B)

SINR performance comparison at the ADC input for transmit SIR = −5 dB, transmit SNR = 24 dB, Nt=4, Nr=4 1−bit DAC and varying R

1: Average SINR − ADC input Nt=4 Nr=4 no FBB2: Avg. SINR − ADC input Nt=4 Nr=4 1−bit DAC with FBB

1

2

0 50 100 150 200 250 30010−2

10−1

100

Oversampling ratio R

Aver

age

SINR

at t

he A

DC in

put (

in d

B)

MSE performance comparison at the receiver for transmit SIR =−5dtransmit SNR =25 dB, Nt=Nr=4, 1−bit DAC and varying R

1: MSE at receiver Nt=4 Nr=4 float ADCs & Wiener beamf2: MSE at receiver with Nt=4, Nr = 4 1−bit DAC with FBB

21

Figure 4.8: Performance comparison as a function of varying oversampling ratios with1-bit DAC: (a) average SINR at the ADC input, (b) MSE at the receiver output.

advantage of this architecture is that it reduces the interference at the input ofthe ADCs, so that less dynamic range and fewer bits are required to reconstructthe desired user signals. These requirements can lead to significant power savingsin the ADCs.

For this architecture, we designed the optimal feedback beamformer coeffi-cients that minimize the output MSE compared to a training sequence. Simu-lations showed that the SINR at the input of the quantizer can be improved bymore than 10 − 25 dB (depending on the SNR), thus indicating the potentialpower savings.

Further research is required along the following directions to implement theproposed class of ADCs in practice:

• We did not analyze the effect of the coarse quantization of the ADC. In


16 18 20 22 24 26 28 3010−2

10−1

100


Aver

age

MSE

at t

he re

ceiv

er

MSE performance at the receiver for transmit SIR = −5 dB, Nt=4, Nr=4 DAC resolution − 1 bit and varying OSR − R

1: MSE receiver − float ADCs & Wiener beamf.2: MSE at the receiver with R=32 !−" no FBB3: MSE at the receiver with R=32 !−" with FBB4: MSE at the receiver with R=128 !−" no FBB

14

2

3

Figure 4.9: Performance comparison illustrating oversampling ratios and ADC resol-ution.

practice, this set-up operates on 1 bit signals of the Σ∆ ADC output, andthe simulations indicated that this is adequate.

• We also did not analyze the effect of the quantization of the feedback DAC.When the DAC resolution is greater than 1 bit, the conversion may not belinear in practice, thus showing a preferred resolution of 1 bit. Simulationsshowed that this is sufficient for improving the SINR at the input of thequantizer, with a small loss in MSE.

• The beamformer estimation employs training signals corresponding to thedesired user. This is not practical in all cases. Furthermore, we need to doa forward channel estimation (or equivalently, replace the design problemof the AR feedback channel by an ARMA prediction error problem). Thisextension is a topic for future research.

• Initially, we are not synchronized to the desired user, and in a dense set-up,interference may overwhelm the ADCs. This requires a good initializationstrategy.

Chapter 5Wideband RF interferencecancellation

In multi-antenna systems, designing multiple wide-band (WB) receiver chains toaccount for RF impairments and interferers is a well known approach. However,this approach would require use the of high resolution ADCs and increased numberof receiver chains leading to an increase in power consumption. To reduce thenumber of receiver chains, we introduce an analog phase shifter matrix (alsoreferred to earlier as APN) operating on the antenna array signal as specifiedin chapter 2 for the narrowband case. The objective of this chapter is to usethe same setup of chapter 2 and design the APN but this time, to suppress thewide-band interferers and to minimize the overall mean squared error between thetransmitted signals and its received estimate.

One fundamental limitation of the APN arrangement is that in a WB setup,the delays of the multi-band components of the signals are larger than the in-verse bandwidth of the transmitted signals. Thus the received signals cannotbe represented by phase shifts of the transmitted signals in the range [−π, π].Moreover implementing delays in the RF for the required space-time beamformeris impractical. For this reason, we propose to jointly design a 2-step beamformerminimizing the overall mean squared error between the transmitted signals andits received estimate. Here multiple phase shifted combinations of the RF antennaarray signals are combined with a space-time beamformer in digital baseband tocancel the interfering users.

5.1 IntroductionWide-band (WB) and ultra wide-band receivers have become increasingly popularin recent years, because of increase in achievable data rates [31]. At the receiver,

83

INTRODUCTION 84

equalization is required. If this is done in digital baseband, then we can combinedelayed replicas of signals transmitted at low SNR’s to reconstruct the transmittedsignal at the receiver. In addition, processing in digital baseband also allows usto combine code-division and temporal diversity techniques to reconstruct thedesired user signals effectively [6]. However, designing RF components to accountfor imperfections in such systems is an important challenge.

One popular technique is to sample and quantize the WB signals, followed byinterference cancellation and digital compensation of the RF impairments. Theseapproaches are known as dirty RF [18]. However, the increased dynamic rangemeans that the receiver spends a considerable amount of energy in processing theinterferers. This is especially critical in a multi-user mobile WB scenario, whereusually the transmitted user signal energy is very low.

Integrated phased array systems [13, 21] offers us some techniques to addressRF interference cancellation and reduced power consumption. For simplicity,we have assumed a narrow-band (NB) multi-antenna system in chapter 2 (alsorefer to [54]), documented the need for alternative architectures, and proposedtechniques that perform joint RF-baseband beamforming.

5.1.1 Delay lines in RF

Consider for the time being that the RF signals are downconverted to digitalbaseband, and sampled at the Nyquist frequency. For a narrow-band setup, thepropagation delay spread is sufficiently smaller than the sample interval and thedelays can be represented as phase shifts in [−π, π]. This model allows us tocombine the antenna array signal using a beamformer consisting of a phase shiftvector to estimate the desired user signal.

One fundamental difference between the NB and the WB set-up is that inthe latter case, the multi-path delays are comparable and in many cases greaterthan the sampling interval. The phase shift approximation of the propagationdelays does not hold and the interference cancellation using a beamformer madeof phase shifts is not quite effective. Consider a WLAN setup with a band-widthBW = 20 MHz at 2.4 GHz. The average delay spread in indoor environmentsdue to multipath reflections at 1 − 2 GHz operating frequencies and 20-40 MHzbandwidth is of the order of ∆ = 0.1µs [66], while the WLAN symbol duration is50ns.

One common approach to cancel the WB RF interferers is through the use ofdelay lines and phase shift combiners, the RF equivalent of a digital space-time(ST) beamformer. Implementing a delay in RF using a transmission line wouldrequire a length of L = ∆c = 0.1 · 10−6 × 3 · 108 ≈ 30 meters. This limitation canbe somewhat compensated by approximating delays using RC networks, howevercanceling the interference in mobile RF systems is not practical using RF delays.

INTRODUCTION 85

5.1.2 Intermodulation distortion

In conventional receivers, the RF circuits are coarsely designed to obtain a WBsignal, of which only a small part is made up of desired user. This is due to the factthat designing NB RF components with fine channel selectivity is expensive. Insuch a setup, the LNA and the APN operate at a bandwidth that is considerablylarger than that of the digital baseband. Typically, for a WLAN environment,the operating bandwidth of the desired user is 20 MHz, while the LNA containssignals over a bandwidth of 100 MHz.

Strong interferers operating in the adjacent bands may drive the LNA into anon-linear mode of operation [20], resulting in intermodulation products (IPs) ofthe various carrier frequencies. As a consequence, the digital baseband systemmight also contain harmonic products of these interferers [20].

To get a better understanding of the non-linearity, we refer to the non-linearmodel of the LNA in chapter 3. As seen there, it turns out that the interferingsignals from the adjacent frequency bands in a weakly non-linear system can leadto interferers superimposed on the desired user signal. It also turns out that thespatial signature of the IP term will be different from that of the adjacent bandinterferers. The conclusion is that the harmonic IPs can be viewed as additionalinterfering user signals coming from different directions.

5.1.3 Problem statement

Our aim in this chapter is to design a phase only APN beamforming matrix oper-ating on the WB antenna array signals. The focus is to minimize the interference(and the IP terms), so that the ADCs and the receiver chains spend less energy inprocessing the interferers. Some design issues are to choose the number of ADCs(ND) and the beamformer weights (W). The design criterion is to minimize theoverall mean squared error (MSE) between the desired user and its received es-timate. In this chapter, we do not consider the fixed precision of the ADCs northe fixed precision of the APN, and assume that the quantization noise and phaseerrors are negligible.

5.1.4 Contributions

In this chapter we consider the above mentioned challenges and design phaseshift beamformers that cancel the interferers and minimize the overall MSE at thereceiver. In Sec. 5.2, the system setup and the data model is specified. In Sec. 5.3we specify the conditions for interference cancellation, and propose an APN designtechnique used in combination with a linear ST beamformer operating in thedigital baseband to reconstruct the desired user signal. The major contribution isto jointly design the APN and a linear space-time beamformer operating on theAPN outputs in the baseband to reconstruct the desired user signal.


We will specify that a direct solution of the problem is not feasible, and modifythe design to a reasonable approximation that successively updates the columnsof the APN to minimize the overall MSE. Subsequently, we will also specify al-ternative approaches to design the APN minimizing the overall MSE and a closedform solution in Sec. 5.4. Finally, we illustrate the performance of the abovesetup using simulations and experimental results at 2.4 GHz.

5.2 System Setup and data model

5.2.1 RF system model

Consider an RF signal x(t) received at an antenna. If x(t) is centered at a carrierfrequency fc we can write

x(t) = real{x(t)ej2πfct}

where x(t) denotes the complex envelope or baseband signal. As mentioned inchapter 2 ( refer to Sec. 2.2), in a NB environment the propagation delay spreadτ of the multiple paths at the antenna is considerably smaller than the inverse ofthe bandwidth i.e. fτ � 1, where f corresponds to frequencies in the bandwidthof x(t). This allows us to approximate the replicas of the received signal delayedby τ using phase shifts i.e. x(t− τ) ≈ real{x(t)e−j2πfcτej2πfct}. Thus we can themodel the received NB signal as a phase shift of transmitted signals.

In a WB scenario, the propagation delays does not satisfy the relation fτ � 1.In this case, the above phase shift approximation does not hold.

5.2.2 Wide-band discrete time data model

Consider now a band-pass communication setup withNt transmitting users centeredaround fc, and represented by their equivalent baseband representation s(j)(t), j ∈{1, · · · , Nt}. Let s(1)(t) be the desired user signal, whereas other signals are con-sidered interferers. These signals are modified by the wireless channel and receivedat the antenna as x(t).

A thorough treatment of the WB wireless channel between the users and thereceiver would involve the use of Maxwell equations, and is not very useful forthe design of signal processing algorithms. For this reason, we simplify the WBchannel from user j to the receiver as a finite impulse response (FIR) filter h(j)(t)which includes the transmit/receive filters, array response, propagation delaysand multi-path echoes. The equivalent baseband signal received at the antennaarray can be specified as

x(t) =Nt∑

j=1

h(j)(t) ∗ s(j)(t) + n(t)


where n(t) denotes thermal noise. For a more detailed wireless channel model,refer to [67].

If we have an array of Nr antennas at the receiver, it is convenient to stackall signals into a Nr × 1 vector x(t) such that

x(t) =Nt∑

j=1

h(j)(t) ∗ s(j)(t) + n(t).

Here h(j)(t) is a Nr × 1 channel response vector from user j and n(t) is a Nr × 1vector denoting the thermal noise terms at the antennas. The down-convertedx(t), is sampled at time t = kT (T refers to the sampling period) and modeled asa Nr × 1 vector x[k] denoting the discrete time ADC output:

x[k] =[

H[0], · · · , H[L]]

s[k]...

s[k − L]

+ n[k] (5.1)

where H[l] = [ h(1)[l], · · · , h(Nt)[l] ] is a Nr × Nt matrix denoting the discretetime-channel impulse response at lags l. s[k] = [ s(1)[k], · · · , s(Nt)[k] ]T and n[k]are Nt× 1 and Nr× 1 vectors denoting respectively the discrete-time transmittedsignals and thermal noise. We assume:

A1 The source signals and thermal noise terms are independent of each other.

A2 The desired user training sequence is known at the receiver.

In this model, s[k] contains the desired and interfering user signals. From chapter3 it can be deduced that s[k] can be generalized to also contain IP terms due tothe non-linearities in the RF amplifiers.

5.2.3 High resolution digital beamforming

Given the observations x[k], the objective is to estimate the desired user signals(1)[k]. One well known approach to cancel the multi-path echoes is to exploitthe shift invariant nature of the wireless channel impulse response. Stack the msuccessive channel outputs of x[k] as seen in (5.1) as a mNr × 1 vector xm[k] =[xT [k] · · · xT [k+m−1] ]T . Our focus now is to design a mNr×1 ST beamformingvector θ = [θT1 , · · · , θTm]T operating on xm[k] to obtain an estimate of the desireduser signal free of multipath echoes and interferers as y[k] = θHxm[k].

A reasonable estimate of s(1)[k] can be obtained by designing θ to minimizethe MSE between y[k] = θHxm[k] and s(1)[k].

θ0 = arg minθE‖s(1)[k]− θHxm[k]‖2 (5.2)


APNW

x(t)

NDNr

ej2πfct

e[k]

z[k]QX

z(t) z(t)

APNW

Nr+

x[k]

ND

z[k]y[k]

ϑ

(a)

(b)

LNA

D

D

y[k]ϑD

D

Figure 5.1: (a) APN operating on RF antenna array signals partially cancels interfer-ence and reduces the number of receiver chains (b) discrete time equivalent followed bya digital space-time beamformer ϑ.

The solution of (5.2) is given by the Wiener Hopf equation θ = R−1X rXs [43],

where RX = E{xm[k]xHm[k]} is a mNr ×mNr matrix and rXs = E{xm[k]s(1)[k]}is a mNr × 1 vector. Estimates of R−1

X and rXs are obtained from the samplecovariance matrix, and requires access to all antenna signals and time delay m.The above solution (5.2) will act as our reference design and m is usually chosensuch that mNr ≥ Nt(L + m), i.e. the well known relation in linear estimationtheory that enables finding an inverse of the wireless channel [4].

5.2.4 Analog preprocessing setup

We consider the receiver setup as shown in Fig. 5.1(a), where an analog prepro-cessing network (APN) operates on the RF antenna array signals immediatelyafter the LNAs as in chapter 2 and [54]. For simplicity, we model the APN as ananalog beamforming matrix with ND outputs computing linear combinations ofthe antenna array signal where ND < Nr.

As in chapter 2, the APN outputs are stacked as a ND × 1 vector z(t), down-converted into baseband and sampled/quantized using ND ADCs. The effect ofthe APN is modeled as (refer to Fig. 5.1(b))

z[k] = WHx[k] + e[k].

Here W = [w1, · · · ,wND ] is an Nr × ND matrix, wj = [w1j , · · · , wNrj ]T is anNr × 1 vector and e[k] is a ND × 1 vector denoting the quantization noise terms.As in chapter 2, the digital baseband does not have direct access to the antenna

WIDEBAND PREPROCESSOR DESIGN 89

array signals. In this chapter, we assume that the variances of the quantizationnoise terms are negligible when compared to that of z[k]. We do not consider thequantization of the APN and the ADCs.

Since delay lines are not practical, we cannot stack x[k] to xm[k] before theAPN and no spatio-temporal processing is possible in RF. For this reason, W isa phase only beamformer which operates only on the Nr × 1 vector x[k].

Given the APN setup, we follow a two-step approach to estimate the desireduser. The first step is to design W and estimate z[k]. Once W is designed, z[k]is kept fixed and the second step is to design a mND × 1 beamforming vectorϑ = [ϑT1 , · · · , ϑTm] operating on zm[k] = [zT [k], · · · , zT [k+m− 1] ]T to estimatethe desired user signal. The design criterion is to minimize the overall MSE

ϑ0 = argminϑ

E‖s(1)[k]− ϑHzm[k]‖2 (5.3)

As before, the baseband space-time beamforming vector ϑ in (5.3) is designedusing the Wiener-Hopf criterion as ϑ0 = R−1

Z rZs where RZ = E{zm[k]zHm[k]} isan mND ×mND matrix and rZs = E{zm[k]s(1)[k]} is a mND × 1 vector.

The design of ϑ is thus straightforward, and the rest of the chapter will focuson the APN design W to minimize the overall error.

5.3 Wideband Preprocessor DesignWe design the phase only APN to suppress the interferers and to minimize theoverall MSE. Throughout this section, we will be using the following propertiesfrequently:Let Q be a positive definite matrix, partitioned with blocks of compatible sizesas

Q =[Q11 Q12

QH12 Q22

]

(P-1) - The matrix inversion Lemma:[Q11 Q12

QH12 Q22

]−1

=[0

Q−122

]+[

I−Q−1

22 QH12

][Q11 −Q12Q−1

22 QH12

]−1[I −Q12Q−1

22

]

=[Q−1

11

0

]+[−Q−1

11 Q12

I

][Q22 −QH

12Q−122 Q12

]−1[−QH12Q

−111 I

]

(P-2) The Schur complement:

det[

Q11 Q12

QH12 Q22

]= det(Q11) det(Q22 −QH

12Q−111 Q12)

The matrix Q22 −QH12Q

−111 Q12 is the Schur complement of Q11, where Q11 and

its Schur complement are positive definite [68].


5.3.1 Minimizing the overall MSE

Given z[k] = WHx[k] at the digital baseband construct

zm[k] =[zT [k], · · · , zT [k +m− 1]

]T= (Im ⊗W)Hxm[k].

Define RZ = E{zm[k]zHm[k]} and rZs = E{zm[k]s(1)[k]}. Also define RX =E{xm[k]xHm[k]} and rXs = E{xm[k]s(1)[k]}.

Note that z[k], RZ = (Im ⊗W)HRX (Im ⊗W) and rZs = (Im ⊗W)HrXsare functions of W. Let ϑ0 be designed using the Wiener-Hopf equation oper-ating on zm[k] as in (5.3): ϑ0 = R−1

Z rZs. For a special case, when m = 1,ϑ0 = R−1

z [0]rzs[0] where Rz[0] = E{z[k]zH [k]} = WHRxW and rzs[0] =E{z[k]s(1)[k]}. In this case, we can factorize Rx and absorb it in the whitenedAPN matrix W as in chapter 2.3 and the solution of W minimizing the overallMSE follows the result of the NB APN case of chapter 2.3.

For a general case when m > 1, we start from the space-time beamformingvector ϑ as designed using the Wiener-Hopf criterion ϑ0 = R−1

Z rZs. The resultingMSE is

D = E‖s(1)[k]− ϑH0 zm[k]‖2 = σ2s − rHZsR

−1Z rZs (5.4)

= σ2s − rHXs(Im ⊗W)

[(Im ⊗W)HRX (Im ⊗W)

]−1(Im ⊗W)HrZs

where σ2s = E{s(1)[k]s(1)[k]}.

The MSE D is a function of W, but is not easily minimized over W. For ageneral case, whenm > 1, absorbing RX in (Im⊗W) would destroy the Kroneckerstructure.

To obtain a feasible solution of W, we decompose the original MSE expressionof (5.4) into resolvable MSE sub-expressions and proceed to optimize each sub-expression successively. For simplicity let us consider m = 2, so that zm[k] =[zT [k], zT [k + 1] ]T and

rZs =[

rzs[0]rzs[1]

]RZ =

[Rz[0] Rz[−1]Rz[1] Rz[0]

],

with Rz[l] = E{z[k + l]zH [k]}, rzs[l] = E{z[k + l]s(1)[k]} and Rz[1] = RHz [−1].

Using the matrix inversion lemma (P-1), the inverse of the covariance matrixRZ can be rewritten as

[Rz[0] RH

z [1]Rz[1] Rz[0]

]−1

=[R−1

z [0]0

]+[−R−1

z [0]RHz [1]

I

].

[Rz[0]−Rz[1]R−1

z [0]RHz [1]

]−1 [−Rz[1]R−1z [0] I

].(5.5)


The original cost function (5.4) can be rewritten using (5.5) as

D = σ2s − rHZsR

−1Z rZs

=[σ2s − rHzs[0]R−1

z [0]rzs[0]]

︸︷︷︸e2

− (rHzs[1]− rHzs[0]R−1z [0]RH

z [1])︸︷︷︸fH

[Rz[0]−Rz[1]R−1

z [0]RHz [1]

]−1

︸︷︷︸G−1

·

(rzs[1]−Rz[1]R−1z [0]rzs[0])︸︷︷︸

f

= e2 − fHG−1f = D1 −D2 (5.6)

where D1 = e2 and D2 = fHG−1f are positive terms and G is positive definite.As specified before, rZs and RZ are functions of W. Starting from cost (5.6), thenext step is to design W that minimizes D1 −D2 for a case where ND = m = 2.

To keep the optimization fairly simple and to find a feasible solution, in thenext sub-section we design W by minimizing D1 followed by using the remainingdesign freedom to maximize D2. Although this two step approach in generalwould lead to an approximation of the true minimum, it is expected that thetwo-step approach will work quite well in this case. This is due to the fact thatoptimizing D1 will specify only one column of W, so that we can use the optionof D2 to specify the remaining column of W. Moreover the nesting relation of(5.6) can be extended to larger m by entering into recursion.

An alternative and intuitively elegant approach to reach (5.6) starting fromthe original cost function (5.4) is explained in Appendix 5.7. This approachrecursively utilizes the Schur complement relation to partition the matrix RZinto sub-matrices and optimizes each-partition independently.

5.3.2 Marginal estimate of W

The overall cost reduced from (5.3) to (5.6) does not lead to a closed form expres-sion of W. However, it gives the following starting point to design W. In thissub-section, we design W considering only the minimization of D1. Let Rz[0] andrzs[0] be represented using whitened representations as follows:

rzs[0] = WHrxs[0] = WHrxs[0]Rz[0] = WHRx[0]W = WHW

where W = Rx1/2W and rxs[0] = Rx

−1/2rxs[0]. Using the above expressions,D1 can be rewritten as

D1 = σ2s − rHxs[0]W(WHRxW)−1WHrxs[0]

= σ2s − rHxs[0]W(WHW)−1WHrxs[0]

(5.7)


Designing W utilizing the whitened expression to minimize D1 leads to

W0 = arg minW

σ2s − rHxs[0]W(WHW)−1WHrxs[0]

= argmaxW

rHxs[0]W(WHW)−1WHrxs[0]

= argmaxW

rHxs[0]PWrxs[0] (5.8)

where PW = W(WHW)−1WH is a projection matrix. The optimum is attainedfor any W such that

rxs[0] ∈ col span{W}.The conclusion of the result (5.8) is that one marginal condition to minimize theoverall MSE must satisfy

rxs[0] ∈ col span{RxW}. (5.9)

For simplicity, let ND = 2 and W = [w1, w2] where w1 is an Nr × 1 vector.Thus, without loss of generality we can choose the first column of W as

w1 = R−1x rxs[0].

The conclusion is that is that the APN W minimizing the MSE must have onecolumn vector parallel to R−1

x rxs to optimize D1. We can subsequently use thesecond column of W (i.e. w2) to optimize D2.

5.3.3 Improved estimate of W

This sub-section improves on the marginal estimate of W in (5.9) by including theadditional constraint of maximizing D2. Let Rz[1] and rzs[1] be denoted utilizingthe whitened representations

rzs[1] = WHrxs[1] = WHrxs[1]Rz[1] = WHRx[1]W = WHRx[1]W

where W = Rx1/2W, rxs[1] = Rx

−1/2rxs[1] and Rx[1] = R−1/2x Rx[1]R−1/2

x .Using the above expressions, D2 can be rewritten as

D2 = fHG−1f

where

f = (rzs[1]−Rz[1]R−1z [0]rzs[0])

= WH[rxs[1]−Rx[1]W(WHRxW)−1WHrxs[0]

]

= WH[rxs[1]−Rx[1]W(WHW)−1WHrxs[0]

]

= WH[rxs[1]−Rx[1]PWrxs[0]

]= WHv (5.10)


where v is a Nr × 1 vector. Since rxs[0] ∈ col span{W0} by our earlier design ofW we can specify that

v := rxs[1]−Rx[1]rxs[0] or v = R−1/2x

[rxs[1]−Rx[1]R−1

x rxs[0]]

In particular, v is not a function of the second column of W.Similarly, the ND ×ND matrix G can be expressed using W as

G = Rz[0]−Rz[1]R−1z [0]RH

z [1]= WH

[Rx −Rx[1]W(WHRxW)−1WHRH

x [1]]W

= WH[I−Rx[1]PWRH

x [1]]

W (5.11)

Let M = Rx[1]PWRHx [1], which is a function of W. The residual squared dis-

tortion D2 can be rewritten using (5.10) and (5.11) as a function of W:

D2 = fHG−1f

= vHW[WH(I−M)W

]−1

WHv. (5.12)

The next step is to design W such that such that the mean squared term D2

is maximized

W0 = arg maxW

vHW[WH(I−M)WH

]−1

WHv. (5.13)

To simplify the optimization problem (5.13) we temporarily ignore the fact thatM is a function of W. Let us replace M by an approximation M e.g. some ofthe possible values are

1. M = 0

2. M = Rx[1]RHx [1]

3. M = Rx[1]PWRHx [1].

In the option 3 of the above approximations, PW is based on the column vectorw1 i.e. PW = w1(wH

1 w1)−1wH1 . Let

W = (I− M)1/2W and v = (I− M)−1/2v.

Proceeding in the same way as in (5.8) leads to the conclusion that W maximizingD2 must satisfy

v ∈ col span{W} or (I− M)−1v ∈ col span{W}

We can thus choose the second column of W as

w2 = (I− M)−1v or w2 = (I− M)−1[

rxs[1]−Rx[1]rxs[0]].

ALTERNATIVE APPROACHES TO UPDATE W 94

For this choice of W = [w1, w2], we can now recompute M = Rx[1]PWRHx [1]

and recompute w2 until convergence. In the original domain, we have

w2 = Rx−1/2(I− M)−1R−1/2

x

[rxs[1]−Rx[1]R−1

x rxs[0]]

(5.14)

Combining the two separate conditions for the optima of D1 and D2 i.e. theconclusions of (5.9) and (5.14) leads to a W that approximately minimizes theoverall distortion D as

W =

[R−1

x rxs[0]︸︷︷︸w1

, Rx−1/2(I− M)−1R−1/2

x

[rxs[1]−Rx[1]R−1

x rxs[0]]

︸︷︷︸w2

]

If we choose to keep the approximation (1) for M = 0 then we can obtain

W = R−1x (rxs[0], rxs[1]−Rx[1]R−1

x rxs[0]) (5.15)

These results can be extended to a case of arbitrary m, where W can bedesigned to minimize residual costs D2, · · · , Dm and an approximate W minim-izing the overall distortion D. The stepwise description of the greedy approachfor m = 2 is briefly mentioned in table 5.1.

Table 5.1: A greedy approach to design wide-band RF phase shifter weights when m = 2

Objective: Select the weights of WB APNGiven: Input covariance matrix RX , cross-correlation matrix rXs and initial valueof M = 0.

• Design w1 = rxs[0] in whitened domain minimizing D1

• For a fixed w1

– w2 = Rx−1/2(I− M)−1R−1/2

x

[rxs[1]−Rx[1]R−1

x rxs[0]]

– Iterate until convergence

∗ W = [w1, w2].∗ update M = Rx[1]PWRH

x [1]∗ recompute w2

• W = Rx−1/2W

5.4 Alternative approaches to update W

5.4.1 Approach 1: Iterative least squares

Given an initial or marginal estimate of the APN W as in (5.15), the columnsof W = [w1, w2] can be iteratively updated using the cost (5.3) in combinationwith the shift invariant nature of the marginal estimate of W operating on x[k].


For example from the solution of (5.9), we see that one coarse estimate ofs(1)[k] can be obtained as wH

1 x[k], corrupted by interferers and noise.Alternatively, starting from (5.14), and approximating M = 0 and neglecting

Rx[1] another coarse estimate of the desired user signal s(1)[k] can be obtainedfrom wH

2 x[k + 1]. It is reasonable to assume that wH1 x[k] and wH

2 x[k + 1] areapproximately equivalent up to a constant phase multiplying factor and both leadto coarse estimates of s(1)[k]. Stacking the two marginal estimates for the entirepacket of length N � Nr leads to the following approximation (equivalent up toa scaling factor):

wH1 [x[1], · · · , x[N − 1]] ≈ wH

2 [x[2], · · · , x[N ]]⇒

xH [1]...

xH [N − 1]

w1 ≈

xH [2]...

xH [N ]

w2

X1w1 ≈ X2w2 (5.16)

From the above expression for a given w1 = R−1x rxs[0], w2 can be designed as a

least squares fitw2 = arg min

w2‖X1w1 −X2w2‖2 (5.17)

In this case, w2 can be updated as w2 = X †2X1w1. Once w2 is estimated, we cango to the approximation specified by (5.16), fix w2 = w2 and iteratively updatew1 to minimize the cost (5.17) as

w2 = X †2X1w1

w1 = X †1X2w2

This approach can also be seen as a modification of the mutually referencedequalizers (MRE) approach as proposed Gesbert e.a. [69]. In [69], the authorsuse a two-step equalization approach. In this paper the first step consists of usingMREs to obtain multiple coarse estimates of the desired user signals. These signalsare subsequently combined with a super-equalizer to obtain a high resolutionestimate of the desired user signal.

In our case, the APN is a variation of the MREs and the digital space-timebeamformer ϑ is a variation of the super-equalizer. For a better understanding ofsuch two-step equalization approaches, refer to [70,71].

5.4.2 Approach 2: Closed form solution for W when ND = m

The underlying thread in the joint optimization of W and ϑ is to partition theweights W = [w1, · · · , wND ] such that a linear combination of W and ϑ obtainsa reasonable match of the mNr × 1 space-time Wiener-beamforming vector θ i.e.

{W0, ϑ0} = arg minW, ϑ

‖θ − (I⊗W)ϑ‖2.


The above expression leads to the least squares solution

θ1

...θm

=

Wϑ1

...Wϑm

⇒

[θ1, · · · , θm

]︸︷︷︸

Θ

= W[ϑ1, · · · , ϑm

]︸︷︷︸

Υ

(5.18)

where Θ and Υ are Nr ×m and ND ×m matrices. Note that for ND = m, Θis a tall matrix since Nr > ND and Υ is a square matrix. From (5.18), we canobtain W for the square matrix Υ as follows:

• For Nr = ND ⇒ choose W = Θ as the closed form solution.

• For Nr ≥ m ⇒ choose W as the ND dominant left singular vectors of Θ.

Note that at no stage in the previous section does the APN operate on thespatio-temporal antenna array signals. In some ways these marginal estimatesz[k] of the multiple echoes of the desired user signal are connected to estima-tion techniques in distributed signal processing literature [72]. Typically, in theseapproaches the marginal beamformers are designed using only the partial inform-ation of the received signals as in [73], followed by a fusion center to obtain a highresolution signal.

Alternatively, this proposed approach can also be interpreted as starting froma NB APN design as explained in chapter 2 followed by the matching pursuitapproximation of multi-path echoes.

In all the above techniques a closed form solution is not possible for arbitrarym and ND, due to the nature of the frequency selective channel and the RFconstraints involved. At best, we can only show that a good starting as in (5.9)when used in combination with iterative least square fitting techniques such as(5.16) can converge to the optimal W minimizing D.

Though we specify that alternative approaches to design W are possible, weuse only the greedy approach as explained in the previous section (Sec. 5.3) andthe closed form approach (Sec. 5.4.2) due to its simplicity and application forarbitrary Nr, m and ND.

Table 5.2: A one-step subspace approach to design wide-band RF phase shifter weights

Objective: Select the weights of WB APNGiven: Input covariance matrix RX and cross-correlation matrix rXs: θ = R−1

X rXs.

• Obtain Θ =[θ1, · · · , θm

]as specified in (5.18).

• Choose the columns of W as the ND dominant left singular vectors of Θ.

SIMULATION AND EXPERIMENTAL RESULTS 97

5.5 Simulation and experimental results

To assess the performance of our setup with respect to simulations and experi-mental data, we have applied it to a multi-user/antenna setup.

In the simulations, the input SNR is the signal to noise power ratio for thedesired signal and the noise as received at the antenna 1; it is the same for allantennas. The input SIR is the signal to interference ratio for the desired signaland the sum of all interference signals as received at the antenna 1; it is thesame for all antennas. All users transmit QPSK signals, with zero mean and unitvariance. The performance indicators are

1. SINR at the first ADC input - a high SINR indicates that less power is spentin quantizing the interferers.

2. MSE of the desired user - observed at the output of the digital receiver.

5.5.1 Simulation results

The computer generated results are obtained by averaging 500 Monte Carlo runs,each with independent Rayleigh fading channel realizations and independentlygenerated data signals. Each run transmits data packages of size 8192 symbolsas in WLAN transmission, where the first 256 symbols are used for training andestimation of the correlation vector rXs. The APN design algorithms used forevaluation are the greedy approach (table 5.1) and the subspace approach (table5.2).

Unless specified otherwise, the receiver contains Nr = 4 antennas, ND = 2receiver chains with ADC resolution R = 8 bits. It is assumed that length ofthe wireless channel L = 3 is same for all transmitting users. The ADC outputsare used in combination with a space-time beamformer ϑ operating on m = 2temporal samples to obtain an estimate of the desired user. These results arecompared with the reference setup with θ with Nr = 4 ADCs and designed using(5.2).

Fig. 5.2(a) and (b) compare the performance of the average SINR observedat the input of the first ADC and the MSE at the receiver output respectivelyfor Nt = 3 users. Consider the SINR plot in Fig. 5.2(a), curve 1 correspondsto a case with no APN and Nr = 4 ADCs. Curve 2 shows the performance duethe introduction of the APN setup with ND = 2 ADCs and using the greedyapproach. Curve 3 corresponds to ND = 2 ADCs and the closed form approach.Comparing curve 1 with that of curves 2-3, the results show an improvement inthe SINR by a factor of 2 dB.

The relatively low value of SINR improvement at the ADC input when com-pared with a NB APN setup specified in chapter 2 is due to the residual ISI termsof the wide-band desired user. Note that the SINR performance of marginalestimates using the greedy design APN is better than the closed form design.


The marginal estimates at the ADC output can be further improved by adigital baseband space-time beamformer ϑ with m = 2. This is illustrated by theMSE performance at the receiver as in Fig. 5.2(b). For a MSE of 0.05, the APNdesigned with closed form approach performs 2 dB worse than the optimal Wienerbeamformer θ with Nr = 4 ADCs and m = 4. The greedy approach performs afurther 2 dB worse than the closed form design.

Fig. 5.3 shows the computer generated simulations for Nt = 4 users andchannel length L = 3. To improve the performance, the ADC inputs are usedin combination with a raised cosine filter [27] (roll-off factor β = 0.1) and over-sampled twice the symbol rate (i.e. OSR R = 2). The ADC outputs (RND × 1vector) are subsequently used with a space-time beamformer ϑ. For details of theoversampled data-model, refer to [70]. Consider the curves 2 and 3 in Fig. 5.3. Inthis case, choosing W as the dominant left-singular vectors of Θ with Nr < RNDleads to a poor performance of the closed form approach.

The conclusion from the above observations is that as long as Nr ≥ RND,the closed form approach performs better than the greedy approach. In all othercases, the greedy approach followed by a space-time beamformer leads to a morereasonable estimate of the desired user.

5.5.2 Experimental results

The proposed algorithms are also compared using wireless channel response ob-tained from off-line experiments. In these experiments, we simulate the receptionthrough an indoor wireless channel at 2.4 GHz. The channel impulse responsesare derived from experimental data measured in an office at the Delft Universityof Technology 1 [74].

The office has dimensions 3.5m × 5m and a height 3.5m. The actual meas-urement setup had a transmit antenna at the center of the room and a receivingantenna 2m away. We have used data from two such locations, one with a directline of sight (LOS) and one without a direct LOS as shown in Fig. 5.4.

Fig. 5.4(a) shows the amplitude of the channel impulse response to a raisedcosine pulse with symbol rate T = 20ns and β = 0.1, when demodulated tobaseband from a carrier frequency of 2.4 GHz. These values are normalized to unitpower. Fig. 5.4(b) shows the frequency domain representation of the RF channelimpulse response centered at 2.4 GHz. We did not have any specific applicationin mind with these numbers, they are chosen to provide an approximate scenariosuch as a WLAN transmission.

In this experiment we chose Nt = 2 sources, transmitted over the above chan-nels and received using Nr = 4 antennas. The received power of the desired andinterfering user signals are scaled to be equal and we added complex white Gaus-sian noise with variance σ2 such that the received SNR := E‖x[k]‖2/(Nrσ2). Thenoisy antenna array signals are transformed to ND = 2 ADC outputs using a

1We are grateful to Dr. Z. Irahhauten for sharing his measurement data


! " #! #" $! $" %! %"!&

!'

!"

!(

!%

!$

!#

!

!"#$%&'()%'*$#+)(,)$,'%-)"#(',).'$)/01

23-"#*-)4567)#()(8-)29:)'$;<().'$)/01

4567)#()(8-)29:)=,")("#$%&'()457)>)!?)/0@)6(>?@)6

">A)6

9>B)=,")C8#$$-+)+-$*(8)D>?

)

)

#*)+,-./0-)1234)/5)56-)+78)9:;<5)=956><5)+?3)/../:0-@-:5

$*)+,-./0-)1234)/5)56-)+78)9:;<5)0.--AB)/;;.>/C6)+?3

%*)+,-./0-)1234)/5)56-)+78)9:;<5)CD>E-A)F>.@)+?3

#

%

$

0 5 10 15 20 25 30 3510−4

10−3

10−2

10−1

100


MSE

at t

he re

ceiv

er

MSE at the receiver SIR = −3 dB, Nt=3, Nr=4 ND=2 for channel L=3

1: MSE at receiver with ND=4 ADCs without APN2: MSE at receiver ND=2 ADCs greedy APN3: MSE at receiver ND=2 ADCs closed form APN

1

32

Figure 5.2: (a) SINR performance comparison (b) MSE performance comparison withNt = 3 users received by an 4 × 2 APN for 500 randomly generated Rayleigh fadingchannels with L = 3

Nr ×ND (4× 2) APN matrix.Fig. 5.5(a) and (b) compare the performance of the average SINR observed

at the input of the first ADC and the MSE at the receiver output. Considerthe SINR plot in Fig. 5.5(a), curve 1 corresponds to a case with no APN andNr = 4 ADCs. Curve 2 shows the performance due the introduction the APNsetup using ND = 2 ADCs using the greedy approach and curve 3 with the closedform approach. Comparing curve 1 with that of curves 2-3, the results show an

CONCLUSION 100

0 5 10 15 20 25 30 3510−4

10−3

10−2

10−1

100


MSE

at t

he re

ceiv

er

MSE at the ADC input for Nt=4, Nr=4, ND=2 OSR=2 channel with L=3

1: MSE ND=4 ADCs space−time Wiener equalizer2: MSE at receiver ND=2 ADCs greedy APN3: MSE at receiver ND=2 ADCs closed form APN

3

2

1

Figure 5.3: MSE performance comparison with Nt = 4 users received by a 4× 2 APNand oversampled twice for 500 randomly generated Rayleigh fading channels with L = 3

improvement in the SINR by a factor of 3 dB.Fig. 5.5(b) compares the MSE performance for the same scenario, used in

combination with a space-time beamformer (m = 2). Curve 1 corresponds to acase Nr = 4 ADCs followed by Wiener beamformer θ. Curves 2 and 3 respectivelycorrespond to the APN designed using greedy and closed form approaches forND = 2 ADCs. For an MSE 0.01, the APN setup with ND = 2 ADCs performsless than 2 dB worse when compared with the reference (curve 1) with Nr = 4ADCs.

5.6 Conclusion

In this chapter, we have proposed a MIMO wide-band receiver employing an ana-log preprocessing network (APN) comprising of RF phase shifts, followed by aspace-time digital beamformer in baseband. The prime advantage of this archi-tecture is that it reduces the number of antenna elements to a smaller number ofmixers and ADC chains. Further, it can reduce the interference at the input ofthe ADCs, so that less dynamic range and fewer bits are required. As specifiedalso in chapter 2, overall power savings is possible with such an APN architecture.

The WB phase shifter and the digital space-time beamformer are jointly de-signed to suppress the interferers in the RF. We specify the conditions for inter-ference cancellation, and propose different approaches to design the phase shifterweights. We compare the performance of these approaches using simulations and

CONCLUSION 101

0 20 40 60 80 100 1200

0.15

0.3

0.45

0.6

0.75

0.9

1.0

Time (nano Seconds)

Puls

e re

spon

se(A

mpl

itude

) with

! =

0.1

Response to a raised cosine pulse with ! =0.1 and T=20 nS

1: LOS measurements2: NLOS measurements

2.28 2.32 2.36 2.40 2.44 2.48 2.52 2.56−20

−18

−16

−14

−12

−10

−8

−6

−4

−2

0

Frequency in GHz

Relative power (in dB)

Relative power of indoor channels with pulse duration T=20 ns

non line of sightLine of sight

Figure 5.4: (a) Response of the wireless channel to a raised cosine pulse (b) relativeenergies of the channel response for a line of sight and non line of sight scenario

experimental WB multi-user wireless setup. In addition to the challenges spe-cified in the conclusion of chapter 2, further research is required to study the RFimpairments and non-linearities when realized in RF hardware.

APPENDIX 1: PARTITIONED MATRIX COST FUNCTION 102

0 5 10 15 20 25 30 35−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0


Aver

age

SIN

R a

t AD

C in

put (

in d

B)SINR performance at ADC input for transmit SIR = −3 dB,

Nt=2, Nr=4 ND=2 BW=20 MHz delay spread 0.1 µ s at fc=2.4 GHz

1: Average SINR at ADC without APN 2: Average SINR at ADC greedy APN3: Average SINR at ADC closed form

2, 3

1

0 5 10 15 20 25 30 3510−4

10−3

10−2

10−1

100


MSE

at t

he re

ceiv

er

MSE at receiver for transmit SIR = −3 dB, Nt=2, Nr=4 ND=2 BW=20 MHz delay 0.1 µ s at fc=2.4 GHz

1: MSE at receiver with ND=4 ADCs and no APN2: MSE at receiver ND=2 ADCs greedy APN3: MSE at receiver ND=2 ADCs closed form APN

1

3

2

Figure 5.5: (a) SINR performance comparison (b) BER performance comparison with4× 2 APN for multi-path channel width 40 MHz at 2.4 GHz and 0.1 µs delay spread

5.7 Appendix 1: Partitioned matrix cost functionConsider the MSE relation specified by (5.4), where ϑ is designed using theWiener-Hopf criterion ϑ0 = R−1

Z rZs. Using the Schur complement (P-2), thisexpression can be rewritten as a product of determinants:

D = E‖s(1)[k]− ϑH0 zm[k]‖2 = σ2s − rHZsR

−1Z rZs

=1

det(RZ)det[σ2s rHZs

rZs RZ

](5.19)


where RZ is a non-singular matrix, σ2s = E{s(1)[k]s(1)[k]} and det corresponds

to the determinant operation. For simplicity let m = 2 as before, leading to

D =1

det(RZ)det

σ2s rHzs[0] rHzs[1]

rzs[0] Rz[0] RHz [1]

rzs[1] Rz[1] Rz[0]

=1

det(RZ)det[

A BC D

]. (5.20)

Again using the property (P-2):

D =1

det(Rz[0]) det(Rz[0]−Rz[1]R−1z [0]Rz

H [1]︸︷︷︸G

)det(A) det(D−CA−1B).

(5.21)Note that G = Rz[0]−Rz[1]R−1

z [0]RzH [1] as defined before in section 5.3.1.

Consider the sub-matrix A, applying (P-2) again on the determinant of Agives

det(A) = det[

σ2s rHzs[0]

rzs[0] Rz[0]

]

= det(Rz[0])[σ2s − rHzs[0]R−1

z [0]rzs[0]]

︸︷︷︸e2

(5.22)

where e2 corresponds to the mean squared scaling error associated with the MSEdistortion D1. Now let us consider the matrix D−CA−1B:

D−CA−1B = Rz[0]−[rzs[1]Rz[1]

] [ σ2s rHzs[0]

rzs[0] Rz[0]

]−1 [ rHzs[1]RH

z [1]

]. (5.23)

Using the matrix inversion lemma (P-1), the above expression (5.23) can berewritten as

D−CA−1B = Rz[0]− [rzs[1] Rz[1] ] .[[0

R−1z [0]

]+[

I−R−1

z [0]rzs[0]

]e−2[I −rHzs[0]R−1

z [0]]].

[rHzs[1]RH

z [1]

]

= Rz[0]−Rz[1]R−1z [0]Rz

H [1]︸︷︷︸G

− (rzs[1]−Rz[1]R−1z [0]rzs[0])︸︷︷︸

f

e−2 (rHzs[1]− rHzs[0]Rz−1[0]Rz[0])︸︷︷︸

fH

. (5.24)


The determinant of the above expression (5.24) used in combination with (P-2)leads to

det(D−CA−1B) = det(G− fe−2fH)

= det(G)(e2 − fHG−1f)1e2

(5.25)

Inserting the relations (5.22) and (5.25) in the original cost function (5.21)cancels the determinant terms and the overall cost can be rewritten as:

D =1

det(Rz[0]) det(G)det(Rz[0])e2 det(G)(e2 − fHG−1f)

1e2

= e2︸︷︷︸D1

− fHG−1f︸︷︷︸D2

(5.26)

The expression (5.26) reduces to the final result of Sec. 5.3.1 as specified by(5.6). Approaching the MSE problem using the property (P-2) intuitively showsthat the overall distortion D specified by (5.19) can be rewritten as a nesting ofsmaller determinants. Optimizing each determinant then leads to a feasible Was specified in section 5.3.Such a structure allows us to extend the relation (5.6) to an arbitrary value ofm.

Chapter 6Digital interference cancellation:asynchronous OFDM

In the presence of un-coordinated interfering users as in a wireless LAN setup, theorthogonal frequency division multiplexing (OFDM) transmission fails. For thisreason, it is interesting to consider a minor changes in the OFDM transmissionwhere we code the transmitted data signals by a small additive sequence. Wedenote this sequence as a super-imposed training sequence.

We consider an asynchronous multi-user wireless LAN system, where the re-ceiver is offset in time by an unknown value with respect to the desired user. Theuser signals are transmitted with superimposed training and an antenna array isused in the receiver to jointly estimate the time-offset and perform space-time(ST) source separation using subspace fitting. Smoothing is used to improve onthe offset estimate and alternating projections are used to improve the beam-former weights. Simulation results indicate that the propose semi-blind sourceseparation scheme converges to the optimal beamformer, when the receiver issynchronized to the desired user and has complete knowledge of its signal.

6.1 IntroductionA user buys a WLAN transceiver equipment and expects it to work, howeverhis neighbors might cause interference with their uncoordinated WLAN systems.He might also own other equipment (e.g. bluetooth) that operates in the samefrequency band. The low-receiver complexity OFDM driving the WLAN standard[8] would fail in the presence of multiple users.

Due to the wide-spread use of WLAN systems in household environments,future WLAN systems based on OFDM might be limited by interferers and maynot be able to deliver increased data rates as promised by MIMO. For this reason,

105

INTRODUCTION 106

CP insertion

Multipath channelIDFT

Input OFDM signal

OFDM demodulator beamform

er per freq. bin

OFDM demodulator

sf [1]

sf [N ]sf [N ]

sf [1]

yNr [k]

y1[k]

P/S

Figure 6.1: OFDM transmission with inverse discrete Fourier transform (IDFT) fol-lowed by cyclic prefix (CP) insertion to combat multipath echoes in a frequency selectivefading environment.

it is interesting to see whether small additions to current setup would help inreducing interference, while remaining compatible with the legacy equipment.

We consider an OFDM communication setup that requires neither slot syn-chronization nor coordination among different users.

6.1.1 Background on OFDM receivers

The idea of OFDM modulation technique is to define a symbol sequence in thefrequency domain, transmit it in the time-domain and map the received samplesback into the frequency domain.

Consider a cyclic prefix (CP) based OFDM [75] communication system forwireless LAN as shown in Fig. 6.1. Here the input sequence is modulated by aninverse discrete fourier transform (IDFT) followed by appending the last part ofthe input sequence as a prefix [8]. This overall sequence has a cyclic structurewith a CP, and is then transmitted over the wireless channel.

At the receiver, the signal from the transmitted user corrupted by the multi-path echoes is received using an array of antennas. When the CP is synchron-ized to the receiver, the DFT demodulation in combination with the circular CPstructure transforms the received multi-path channel with echoes into a set ofNB channels as shown in Fig. 6.2. The prime advantage of using the OFDMtransmission approach is that the discrete Fourier transform (DFT) output at thereceiver antenna array can be used with a linear combination of spatial weightsto reconstruct the transmitted signal. This setup can be extended to a MIMO-OFDM or to a multi-access OFDM [75], as long as the CP from multiple inputsignals are synchronized and co-ordinated to the receiver.

6.1.2 Setup and problem description

Consider an asynchronous multiuser-OFDM system as shown in Fig. 6.3 wheremultiple users operate on a common wireless channel. In this scenario, the in-terfering users can enter and exit the wireless channel on an ad-hoc basis as in ahousehold WLAN setup. The receiver is not synchronized to the user of interest,

INTRODUCTION 107

Desired user channel response diagonal

Desired user OFDM signal

OFDM demodulated signalDesired user channel response

cyclic structureDesired user

signalReceive antenna

signal

(a) (b)

Figure 6.2: OFDM transmission (a) the IDFT-CP combination leads to a circularchannel in the time domain (b) A DFT of circular code leads to a point wise represent-ation in frequency domain.

symbol iCP symbol i+1CP

CP CP

Desired user

User 2

User 3

i-1

CP

τ

symbol offset

Figure 6.3: Multiple un-coordinated users operating in a wireless channel. The desireduser (user 1) offset from the receiver by time τ , and user 2 is a neighboring WLAN systemwith a random offset from the receiver. User 3 can be a bluetooth device operating in thefrequency band of WLAN transmission

or to have any coordination with the interfering users. The signal from the desireduser is received at the antenna array corrupted by interference from other usersand noise.

When un-coordinated interferers exist in the common wireless channel, theOFDM transmission fails [76]. In this case, the low complexity spatial frequencydomain beamforming in the receiver cannot retrieve the desired user signal. Thisis due to the property that when the CPs of different un-coordinated users are notaligned, the circular convolution property of the IDFT-CP operations is destroyedas illustrated by the Fig. 6.4.

Our objective is to consider the above scenario and propose novel receiverdesign algorithms that jointly estimate the time-offset as well as the space-timebeamformer taps to cancel interference. To facilitate the receiver to separate the

INTRODUCTION 108

Desired user channel response cyclic structure

Desired user signal

Interferer channel response No cyclic structure

Interferer signalReceive antenna signal

combined channel response not cyclic

Desired user channel response block diagonal

Desired user OFDM signal

OFDM demodulated signal Interfering user channel response

No structureDesired user OFDM signal

(a)

(b)

Figure 6.4: OFDM transmission in (a) time-domain: the IDFT-CP cyclic structuredestroyed by asynchronous interferer (b) frequency domain: A DFT does not have anystructure in frequency domain.

desired and interfering users, it is preferred to superimpose the information signalsfrom the desired user with a distinct color code. Superimposed training has beenrecently considered by several authors [77,78], where the training (a periodic nonrandom) sequence is arithmetically added to the transmitted sequence. In thisway no bandwidth is lost when sending the training data.

6.1.3 Contributions and outline

Joint estimation techniques using superimposed training of un-coordinated OFDMsystems can achieve both interference cancellation and maximize spectral effi-ciency(since no bits are lost due to training and multiple users can operate at thesame time). In this case, the receiver does not require any slot synchronizationand co-ordination with the medium access control (MAC) layers.

Superimposed training algorithms (STA) [77, 79] consider a non random (notnecessarily periodic) code sequence being arithmetically added to the transmittedsequence. It is assumed that each user transmits with a unique superimposedsequence. These sequences impose distinct statistics of the transmitted signals

DATA MODEL 109

at the receiver antenna array. This statistical information (time varying mean)property can be exploited to separate the required user and to estimate the chan-nel.

In such approaches [80, 81], the desired user is assumed to be synchronizedto the receiver and a beamformer is designed utilizing the correlation betweenthe receive antenna array and the superimposed sequence. In this chapter, theproblem is extended for OFDM systems where the desired user is not synchronizedto the receiver.

Subspace or eigen-structure methods [82] are known to have high resolu-tion capabilities and yield accurate estimates. It has been shown that suchschemes have the same asymptotic properties as that of maximum likelihood(ML) schemes. In this chapter we propose a joint estimation algorithm basedon subspace fitting techniques. We show that

(a) a nonlinear least squares(NLLS) approach leads to a reasonably accurateestimate of the time offset and the beamformer coefficients to cancel inter-ferers.

(b) subsequently smoothing is used to eliminate spurious local maxima followedby improved estimation in frequency domain using alternating projections.

The rest of this chapter is organized as follows: In Sec. 6.2 we describe thedata model of an OFDM system and show how the CP misalignment destroys theorthogonality in the receiver. Subsequently, we specify the conditions to cancelthe interferers using a spatio-temporal beamformer. In Sec. 6.3, we propose thesuperimposed training algorithm to jointly estimate the asynchronous OFDMsignal offset and the space-time beamformers, followed by simulation results andthe conclusions.

6.2 Data model

6.2.1 Single user OFDM model

Let sf [b] be a N×1 vector denoting the information sequence of the OFDM blockb, where the subscript f denotes the frequency domain component. The IDFToutput sequence s[b] is an N ×1 vector (s[b] = [s[(b−1)N + 1], · · · , s[bN ] ]T ) andobtained from the frequency domain information sequence as

s[b] = FHsf [b]

where F is a N ×N DFT matrix.For the time being, let us assume that the transmitted signal be synchronized

to the receiver. As specified in Fig. 6.1, a redundant CP of length LCP is insertedbetween successive OFDM symbol blocks. After the CP insertion, the transmitteddata block is of length N + Lcp: s′ = [s[bN − LCP + 1], · · · , s[bN ], s[(b− 1)N +

DATA MODEL 110

1], · · · , s[bN ] ] is transmitted over the wireless channel. For simplicity, we willdrop the block index b and denote the transmitted signal with a time index k :=s′[k] .

Consider a Nr×1 frequency selective fading channel modeled by an equivalentdiscrete time FIR filter h(1)[l] operating on the transmitted signals, where thesuperscript (1) corresponds to desired user (user 1).

The channel response from an arbitrary user i includes the transmit/receivefilters, array response, amplitude scalings, and phase delays and is denoted ash(i)[l] = [hi1[l], · · · , hiNr [l] ]T . The channel response h(i)[l] is assumed to be offinite duration and zero outside the interval [0,L), where L is the order of thechannel and the CP length is chosen to satisfy L ≤ LCP .

The received antenna array signals are sampled at t = kT as an Nr × 1 vectory[k] given by

y[k] =[

h(1)[0] h(1)[1] · · · h(1)[L]]

s′[k]s′[k − 1]

...s′[k − L]

+ w[k] (6.1)

where w[k] is an Nr × 1 vector corresponding to thermal noise at the antennasand time instant k.

Stack the received antenna array signals of length N for a given block anddiscarding the CP terms leads to

y = Hc,N s

where y = [yT [1], · · · , yT [N ] ]T is a NNr × 1 vector and the underline denotesthe multi-antenna (multi-channel) response. Hc,N is a NNr ×N circular matrixcorresponding to resultant channel response obtained due to the CP insertion.For example, when N = 4, L = 1 and LCP = 2:

Hc,N =

h(1)[0] h(1)[1]h(1)[1] h(1)[0]

h(1)[1] h(1)[0]h(1)[1] h(1)[0]

. (6.2)

The received block of signals are mapped back to the frequency domain usinga N DFT operation in the receiver

yf

= [(INr ⊗FN )Hc,N ]s = Λsf (6.3)

where Λ is a NNr×N block diagonal matrix (each column has Nr non-zero entriesand each row has one non-zero entry) and corresponds to the DFT of the circularconvolution between channel response. Note that a circular convolution in timedomain is equivalent to a pointwise multiplication in frequency domain. From

DATA MODEL 111

(6.3), an estimate of the transmitted signal at each sub-carrier can be obtainedusing a Nr×1 spatial beamformer operating on the Nr×1 antenna array signals.

Thus the step involving the complicated space-time equalization is avoideddue to the data structure in the OFDM transmission.

To synchronize the receiver and to estimate the taps of the spatial beamformer,the WLAN standard [8] transmits a set of dedicated training signals at the startof the transmission packet. In this well known pilot tone assisted modulation(PTAM) approach for OFDM systems, the dedicated training pilots are placed inspecific frequency bins to acquire the channel state information (CSI). The PTAMapproach requires synchronized pilots to estimate the channel, and results in aloss of bandwidth.

6.2.2 Asynchronous OFDM - Frequency domain model

Let us consider the scenario, where Nt users occupy the common wireless channel.Let s(i) and s(i)

f correspond to the time and frequency domain representations ofthe information sequence from user i. The signal from user i, i ∈ {1, · · · , Nt} istransmitted in blocks of length N + LCP after the CP insertion.

For the sake of simplicity let us consider that there is one interfering WLANsymbol (i.e. Nt = 2) occupying the common wireless channel. The interferer doesnot have any coordination with the desired user or with the receiver. The userdata and asynchronous interferer are received by a set of Nr antennas, corruptedby the frequency selective environment and noise.

If the desired and interfering users are synchronized to each other, then thetime-domain transmitted signals of the desired and the interfering users are ex-pressed using N × 1 vectors as

[s(1)

s(2)

]= (INt ⊗FH)

[s(1)f

s(2)f

].

After the CP insertion, the time-domain signals s′[k] = [s′(1)[k], s′(2)[k]]T is re-ceived at the antenna array as y[k] modulated by the multipath echoes as in(6.5).

In this case, the time-domain cyclic structure as in Fig. 6.2 is satisfied andthe transmitted signals at each sub-carrier can be estimated at the receiver in amanner somewhat similar to (6.3), provided Nr ≥ Nt. Subsequently, after theFFT operation

yf

= Λ

[s(1)f

s(2)f

].

In this case, Λ is a NNr × NNt block diagonal matrix corresponding to themulti-channel wireless channels.

Consider an asynchronous user scenario, where the interfering user begins itstransmission with an unknown offset τ2 with respect to the receiver. If τ2 + L ≤

DATA MODEL 112

LCP , then the cyclic property still holds. The asynchronous interfering signal canbe represented in the frequency domain as a phase shift version of the synchronousinterferer and utilizing the shift-invariance property, leads to

s(2)f = F(ej2πλτ2 � s(2))

where λτ2 = [1, τ2, · · · τN−12 ]T is a N × 1 vector.

However for an arbitrary value of the delay τ2, the cyclic time-domain structureof (6.3) is not satisfied as shown in Fig. 6.4 and as explained in [76]. In this case,per-carrier frequency domain beamforming as in [75] is not possible.

6.2.3 Time-domain data model and superimposed training setup

To simplify the data model in the presence of asynchronous interfering users, werepresent the received signals with a time-domain representation. For simplicity,we assume that all user employ OFDM transmission and the transmitted signalis represented after CP insertion in time domain as s(1)[k] 1. We also assume forthe time being that the desired user is synchronized to the receiver as shown inFig. 1. This assumption will be lifted in the next section.

In an OFDM transmission system using superimposed training, a known pilotsequence c(1)[k] specific to the desired user and with period P is added to thetime domain OFDM signal represented, and represented with a time index k as

x(1)[k] = c(1)[k] + s(1)[k]. (6.4)

As specified before, these superimposed training schemes are essentially used toacquire the CSI for any block transmission systems. The PTAM scheme can beseen as a special case of superimposed training where a pilot sequence is insertedinto selected frequency bins which is equivalent to a training sequence being addedonto the time domain OFDM signal.

We make the following assumptions about the transmitted signals from de-sired and interfering users:

A1 The information sequence of the desired user s(1)[k] is zero mean and whitewith E{‖s(1)[k]‖2} = σ2

s .

A2 The superimposed sequence of the desired user c(1)[k] is a non-random peri-odic sequence with period P such that c(1)[k] = c(1)[k +mP ]∀m, k.

A3 The information sequence of the asynchronous interfering users is zero mean.The interfering users may or may not use superimposed training. For in-terfering users using superimposed training it is assume that their periodP ′ 6= P .

1With a slight abuse of notation, henceforth we denote the time domain transmitted signalafter CP insertion as s(1)[k] and not s′(1)[k]

BEAMFORMER DESIGN 113

CP insertion

Multipath channel

User 2

User 3

IDFT&

P/SInput

OFDM signal

S/P Receiver block

+

c(1)[k] mod P

x(1)[k]

s(1)f [1]

s(1)f [N ]

Figure 6.5: Superimposed training algorithm (STA) used with an asynchronous OFDMsystem to estimate time offset and to cancel interfering users

A3 The block length N is a multiple of the superimposed training period P i.e.M = N+LCP

P , where M is an integer.

Each user is associated with one antenna and the receiver is supplied with Nrreceive antennas. The channel response from Nt users can be extended as aNr × Nt matrix H[l] = [h(1), · · · , h(Nt)] of order L. The received Nr × 1 signalis given by

y[k] =[

H[0] H[1] · · · H[L]]

x[k]x[k − 1]

...x[k − L]

+ w[k] (6.5)

where x[k] = [x(1)[k], · · · , x(Nt)[k]]T is a Nt × 1 vector of the transmitted signals(superimposed with training) from the desired and interfering users at time instantk. For simplicity of expression, we will assume that the wireless channel lengthsfrom multiple users is the same.

6.3 Beamformer design

Fig. 6.5 shows a multi-user discrete-time baseband schematic of an OFDM trans-mission system using the proposed superimposed training algorithm (STA). Theinformation sequence from the user of interest is added with a non-periodic se-quence (known at the receiver) and transmitted over a wireless channel corruptedby interference from other users. The setup in Fig. 6.5 will cancel the asyn-chronous interference using a time domain space time (ST) beamformer. Thissuperimposed sequence is utilize to design the space time beamformer at the re-ceiver to detect and filter out the desired user among other interfering users.


6.3.1 Beamformer design for asynchronous OFDM systems

The objective is to estimate the transmitted information sequence from the desireduser using an mNr × 1 space-time beamforming vector θ = [θT1 , · · ·θTm], wherem is the length of the beamformer which combines Nr × 1 receive antenna arraysignals at successive time instants. For time instants k, k ∈ {1, · · · , N + LCP },stacking the successive antenna array signals from time k to k −m+ 1 gives

ym[k] = [yT [k], · · · , yT [k −m+ 1] ]T

= HmxL+m[k] + wm[k] (6.6)

Hm =

H[0] H[1] · · · H[L]. . . . . .

H[0] H[1] · · · H[L]

,

and xL+m[k] and wm[k] are obtained by stacking x[k] and w[k] over successivetime instants similar to ym[k]. For this case Hm is a mNr × (m+L+ 1)Nt blocktoeplitz matrix. In the above expression (6.6), the negative time indices K < 0correspond to samples from the previous block.

The ST beamformer tries to design aMNr×1 filtering matrix θ = [θT1 , · · · , θTm]T

such that θHym[k] is an estimate of the transmitted superimposed signal x(1)[k]as in (6.4). Note that x(1)[k] = s(1)[k] + c(1)[k] and from an estimate of x(1)[k],the desired user signal can be estimated as s(1)[k].

A necessary condition for space-time beamforming is Hm should be a tallmatrix [83]. The instantaneous error of the ST beamformer θ is

ε[k] = yHm[k]θ − x(1)[k]

=[

yT [k], · · · , yT [k −m+ 1]]θ1

...θm

− (s(1)[k] + c(1)[k])(6.7)

Stacking the error εk for P successive time instants, to a P × 1 vector, we obtainfor k, k ∈ {1, 2, · · · , N} the corresponding least squares (LS) problem:

ε[k]...

ε[k + P − 1]

=

yT [k] yT [k − 1] · · · yT [k −m+ 1]...

. . ....

yT [k + P − 1] yT [k + P − 2] · · · yT [k + P −m]

θ

−

s(1)[k] + c(1)[k]...

s(1)[k + P − 1] + c(1)[k + P − 1]

(6.8)

To remove the effect of unknown symbols s(1)[k], we propose to use the assump-tions A1, A2 and A3 and average out (6.8) overM superimposed sequence periods


M = N+LCPP , where M is an integer and N is the number of subcarriers. The

superimposed training periods M can be made sufficiently large by stacking sig-nals from multiple OFDM blocks, and the cost function is can be averaged forε[j] = 1

M

∑M−1i=0 ε[iP + j] for all j = 0, · · · , P − 1. Exploiting the time varying

mean property A1 and A2, the resulting block average cost function is given by

ε[0]...

ε[P − 1]

=

zT [0] · · · zT [−m+ 1]...

. . ....

zT [P − 1] · · · zT [P −m]

θ1

...θm

−

c(1)[0]...

c(1)[P − 1]

⇔ ε = Zθ − c(1) (6.9)

where zj = 1M

∑M−1i=0 y[iP + j]. Given the superimposed training sequence and

the time-averaged observations of the received signal, the LS problem can bespecified as

θ0 = arg minθ‖ε‖2.

Minimizing the error vector ε with respect to θ leads to a LS estimate for θ givenby

θ = Z†c(1) (6.10)

6.3.2 Joint offset and beamformer estimation

In the previous section we assumed that the OFDM receiver was synchronizedto the user of interest. Now we consider a situation where the receiver is notsynchronized to the user of interest. Assume that the desired user sequence isoffset from the interferer by an unknown time offset τ . Our aim is to jointlyestimate the offset τ as well as θ that satisfies

{θ, τ} = arg minθ, τ

∥∥∥∥∥∥∥

zT [0] · · ·zT [−m+ 1]...

. . ....

zT [P − 1]· · · zT [P −m]

θ −

c(1)[τ ]...

c(1)[(τ + P − 1)(N+LCP )]

∥∥∥∥∥∥∥

2

= arg minθ, τ‖Zθ − c(1)

τ ‖2. (6.11)

Here c(1)τ is a circular shift of the superimposed sequence c(1) by integer τ . We

exploit the shift invariance property that for τ < P , a delay in time domaincorresponds to a phase progression in frequency domain. Similar to [84]

FP c(1)τ = FP c(1) � φτ

where φτ = [1, φ, · · · , φP−1]T , φ = exp −j2πτP and FP is a P × P DFT matrix.Let Z = [diag(FP c(1))]−1FP .


c(1)0 c

(1)P−1

Z

Estimate of user signals +

superimposed trainingbeamformer taps stack m successive antenna array signals

block average of signal estimate

user data cancels out leaving only superimposed training

block average of antenna array

x(1)[k] x(1)[k + P ] x(1)[k + MP ]

x(1)[k]

x(1)[k + P ]

x(1)[k + MP ]

ym[k + MP ]ym[k + P ]ym[k]

ym[k]

ym[k + P ]

θH

θH

θH

z[0] z[P − 1]

z[P −m]

Figure 6.6: Pictorial representation of the superimposed training algorithm to estimatethe beamfomer taps θ when desired user is synchronized to the receiver.

Applying the DFT matrix FP to (6.11) leads to the LS problem

{θ, τ} = argminθ,τ

‖FPZθ −FP c(1) � φτ‖2

= argminθ,τ

‖Zfθ − C(1)f φτ‖2 (6.12)

where Zf = FPZ and C(1)f = diag(FP c(1)). For a fixed C

(1)f , the minimum of

(6.12) with respect to τ and θ is a measure of how well we can estimate τ suchthat C(1)

f φτ is in the range of Zf .

Thus (6.12) can be viewed as fitting φτ and θ to minimize (6.11), a 1-dimensional case of the weighted subspace fitting problem or a non-linear leastsquares (NLLS) [82,85].


By substituting the pseudoinverse solution θ = Z†fC(1)f φτ back in (6.12)

τ0 = argminτ‖ZfZ†fC

(1)f φτ − C(1)

f φτ‖2

= argminτ‖(Πz − I)C(1)

f φτ‖2

= argmaxτ

‖ΠzC(1)f φτ‖2

= argmaxτ{φHτ C(1)H

f ΠzC(1)f φτ} (6.13)

where Πz = ZfZ†f is the projection on the column span of Zf . Optimization overa single parameter τ is easily carried out using signal classification techniquessuch as [86].

6.3.3 Smoothing

Due to the time dispersive nature of the channel, the received signals containseveral shifts of the superimposed sequence resulting in multiple solutions (L +m−1 local maxima) that satisfy (6.13) at τ0, · · · , τ0+L+m−2, where τ0 is the trueoffset. This can be explained from the data model. Stacking the antenna arrayoutput ym[k] for P successive time instants to compute the periodic mean similarto (6.6) gives the model for τ = τ0:

z0 · · · zP−1

.... . .

...z−m+1· · ·zP−m

≈

h(1)[0]. . .h(1)[L]. . . . . .

h(1)[0] . . .h(1)[L]

c(1)τ · · · c

(1)τ+P−1

.... . .

...c(1)τ−m−L+2· · ·c

(1)τ+P−m−L+1

⇔ ZT ≈ H(1)m C(1)T

τ

where C(1)τ is a P×L+m−1 matrix and H(1)

m is a mNr×L+m−1 channel matrixcontaining taps only from user 1 (the channel taps from other users cancel outsince they do not have periodic mean property). Assuming that H(1)

m is tall andhas full rank, rowspan(Z) = rowspan(C(1)

τ ). Thus each row of (C(1)τ ) of dimension

1× P is in the row span(Z). Similar to [84],[c(1)τ · · ·c(1)

τ+P−1

]∈ row span(Z)

[c(1)τ−1 · · · c

(1)τ+P−2

]∈ row span(Z)

[c(1)τ−m−L+2 · · · c

(1)τ+P−m−L+1

]∈ row span(Z)

This results in L + m − 1 consecutive values of τ satisfying rowspan(Z). Wecan use a moving average filter of order L + m − 1 to smooth out the local

BEAMFORMER UPDATES 118

maxima essentially, if f(τ) = argmaxτ{φHτ C(1)Hf ΠzC

(1)f φτ} is the cost function

in (6.12) then we form J (τ) = f(τ) + · · · + f(τ − L + m + 2) and search forargmaxτ J (τ). Simulation results show that moving average operation results inglobal convergence.

Given an estimate of τ , the space-time beamformer taps θ can be estimatedfrom (6.12): θ = Z†fC

(1)f φτ . This value is used as initial point θ = θSTA for the

alternating LS as explained in Sec. 6.4.

6.4 Beamformer updates

6.4.1 Prefiltering

If Y = [ym[l], · · · , ym[N + l − 1] ] the received antenna matrix, is rank defi-cient additional nullspace solutions exist, where l = τ + LCP and denotes thestart of OFDM symbol. In this case there exist beamformers θ0 such thatYTθ0 = 0. These nullspace solutions lead to two independent ST beamformersθ and θ + θ0 reconstructing the same signal, and can be avoided by using a pre-filter bF (obtained from SVD(Y)), which reduces the number of rows from mNrto Nt(L + m − 1). The prefilter whitens the antenna array signal Y = FHY.This prewhitening improves the conditioning of the received matrix, since the STbeamformers in the whitened domain are approximately orthogonal and some-times result in the convergence of the iterative algorithms to independent solu-tions.

6.4.2 Post processing - Alternating LS

The OFDM signal in frequency domain is constant modulus (CM) and this canbe exploited in a block iterative algorithm. In frequency domain the source sep-aration can be modeled as a LS problem with cost function (CP removal followsdefinition of Y)

argmins∈CM; θ

‖FN (YTθ − c(1)

mod P)− s‖

where CM = {s|sk| = 1; for k = 0 to , N − 1}, s = [s(0), · · · , s(N−1)]T is

the estimate of transmitted OFDM symbol in frequency domain, c(1)

mod Pis a

repeating sequence of length N of c(1). With an initial value of θ = θSTA from thesuperimposed training algorithm specified by (6.13), an alternating LS algorithmoperates as follows:(a) restricting s to CM signals estimate s given τ and θ:

s = FN (YTθ − c(1)

modP )

s = s� |s| (6.14)


Space-time Equalizer

OFDM demod+

Alternate projection

Y

Nt(L + m + 1)

estimated signalsused back as training

S/P

Desired user

estimate

θ

s

y[k] x(1)[k]

c(1)[k]

-

Figure 6.7: Itrerative postprocessing to update the space-time beamformer weights.

(b) estimate θ given s and τ :

θ = (Y†)T (FHN s + c(1))

Table 6.1: Pre-filtering and alternating least squares approach to update the beamformertaps θ

Objective: Given received signals at Nr antennas, update ST equalizer coefficientsand OFDM output.

• SVD: Y = UΣV

• Prewhitening filter: FH = Σ−1UH/√N

• Prefilter output: Y = FHY =√NV

• Initialize θ = θSTA

• Begin iterations

– z = YTθ − c(1)

modP– z = z� |z|– θ = Y†T (QHz− c(1)

modP )

• End iterations

6.5 Simulation resultsThe performance of the subspace fitting and the alternating LS scheme is veri-fied for a 2-user multi-user OFDM system. All users transmit constant modulus


0 2 4 6 8 10 12 14 16 18 2010!3

10!2

10!1

100

Reco

very

failu

re ra

te

Incorrect estimation of desired user offset for 2!user OFDM with channel length 3

subspace fitting Beta = 0.3MUSIC type search Beta = 0.3Subspace fitting + smoothing Beta = 0.2

Figure 6.8: Incorrectly estimated delays for subspace fitting

0 5 10 15 20 25 30!2

0

2

4

6

8

10

12

14

16

SNR

SINR

SNR versus SINR for joint offset and beamformer estimationfor 2!user system averaged over 128 OFDM symbols for SIR = 0 dB

Subspace fitting + alternating LS (Beta = 0.5)subspace fitting + alternating LS (beta = 0.25)Wiener beamformer using known desired user signal

Figure 6.9: Comparison of SNR with output SINR for joint delay and source separationwith SIR = 0dB

(QPSK) information sequences, with superimposed training. The training poweroverhead introduced by superimposed sequences is β given by β = σ2

c

σ2c+σ2

s, where

σ2c is the variance of superimposed sequence c(1). The desired and interfering

users are neither synchronized, nor any form of coordination exist between themand the receiver. The information sequences, corrupted by a multi-user channeland noise is received using Nr = 2 antennas. We consider a L=3 Rayleigh fadingchannel with signal to interference ratio 0 dB. To satisfy Hm to be a tall matrix,an oversampling ratio of 2 is added [83]. We choose N = 64 subcarriers and


0 5 10 15 20 25 30 3510!5

10!4

10!3

10!2

10!1

100

101

SNR

BER

BER performance comparison of 2!user OFDM with desired user not synchronized to receiver

1: STA for beta = 0.52: Subspace fitting + alt.LS (start of transmission)3: Subspace fitting + alt. LS (continuous data)4: Wiener equalizer using known desired user signal

Figure 6.10: BER performance of joint offset estimation and source separation L = 3and β = 0.5

P = 16. This period of superimposed training is used to ensure that Z is a tallmatrix. The time varying mean is averaged over 128 OFDM symbols, which isapproximately equal to the packet length of a wireless LAN system. All userstransmit information sequences superimposed with periodic sequences known tothe receiver. We consider a Rayleigh fading channel for Nt = 2 users, Nr = 2 andsignal to interference ratio (SIR) = 0 dB. An oversampling ratio of 2 is used toincrease the dimensions of Hm. Fig. 3 shows the recovery failure rate perform-ance for L = 3 channel, where the training power overhead is β = 0.3 for subspacefitting schemes. A recovery failure occurs when the estimated τ rounded to aninteger is different from the correct (integer) τ . This performance is comparedwith a similar algorithm but based on MUSIC type search [81] for the same chan-nel and power overhead. The recovery failures can be improved by performingsmoothing as shown in Fig. 3.

Fig. 4 compares the SINR values of the received OFDM symbol, for differentvalues of power overhead when the subspace fitting scheme gives correct offset es-timates followed by alternating LS. The Wiener beamformer is designed assumingthat signal and offset of the desired user is known. We observe that the subspacefitting scheme with β = 0.5 converges to the Wiener beamformer.

Fig. 5 shows the BER performance for joint offset and beamformer estimationusing subspace fitting followed by alternating LS (curve 2), where the transmis-sion starts at beginning of observation window. Curve 3 considers a continuoustransmission of data. Curve 2 performs around 6 dB better than the superim-posed training scheme (curve 1)as explained in [81].The performance of curve 3 isbetter than curve 2 for high SNR’s, since the edge effect is removed and converges

CONCLUSION 122

with Wiener beamformer. In the simulations, the BER computations are basedon trials where the offset is correctly estimated.

6.6 ConclusionIn this chapter, an algorithm to perform joint estimation of offset and sourceseparation for asynchronous WLAN is proposed. From simulation results, it isverified that the subspace fitting followed by alternating LS converges to theperformance obtained via Wiener beamformer designed using a known desireduser signal.

Chapter 7Conclusions

7.1 Fundamental questions and summary of main results

Signals are analog in nature and present in infinite forms. However, the possib-ilities to efficiently process and store them in their original domain is limited.Digital signal processing algorithms with the help of Moore’s law and algebraictechniques offer us limitless possibilities to play with the signals as well as theirreconstructions. One very important application for DSP techniques is in mobilewireless communications, where the receivers and transmitters have to be de-signed to reconstruct signals efficiently in the presence of impairments in wirelesschannels, interferers and non-linearities.

This thesis has touched on a range of issues associated with interference can-cellation in wireless communication systems. In this context, we have tried toexplore the possibilities, the limitations, the scope of future wireless receivers andthe role of DSP.

Our objective is to view the analog front-end and mixed signal hardware fromthe eyes of a DSP engineer. This would allow us to realize adaptive RF and ADCcomponents that offer the flexibility of utilizing advanced DSP algorithms, andultimately reducing the complexity of analog and mixed signal circuits. This isnot a trivial problem and requires us to sequentially answer the following broadquestions:

• Are there possible RF and ADC architectures that can be designed to beused in combination with wireless communication systems, accounting forinterference and impairments?

• Given the existence of such analog front-ends, how do we integrate interfer-ence cancellation techniques with these front-ends?

123

FUNDAMENTAL QUESTIONS AND SUMMARY OF MAIN RESULTS 124

• What are the orders of magnitude improvement in power consumption andcircuit size that can be achieved with such integrated approaches?

Throughout this thesis, we consider only MIMO communication systems. Mul-tiple antennas at the receiver provides us with the design freedom to cancel theinterferers and to exploit the spatial diversity. Undoubtedly, digital interferencecancellation is appealing when the receiver power and complexity are not factored.This scenario is not practical, considering the need to make MIMO communica-tions mobile, more compact and less power consuming.

In chapter 2 of the thesis, we consider an architecture with an RF phaseshift combiner operating on the antenna array signals. The antenna array signalscontain contributions of desired and interfering users, transmitted over a NBwireless channel. The phase shift combiners reduce the number of receiver chains,and are designed to minimize the overall MSE between the transmitted signal andits received estimate, while maximizing the desired SQNR at the receiver.

The closed form solutions proposed in chapter 2 to estimate the beamformertaps and to estimate the channel state lead to a robust approach to reconstruct thedesired signals at a minimum cost. Further, multi-channel APN can be utilized incombination with existing MIMO architectures that use spatio-temporal diversityand coding techniques to further enhance the performance. We illustrate usingsimulation results in Sec. 2.6 that the MSE of the setup with coarse RF phaseshifters transforming Nr = 4 antennas to ND = 2 receiver chains, coincides withthe MSE performance of a setup with Nr = 4 ADCs. This leads to at-least afactor of two improvement in power.

An alternative to designing phase shifters in RF is to use a digital beamformerwith a DAC feedback and to perform interference cancellation before the ADCs.In chapter 4, we consider such an architecture where a beamformer is integratedwith multi-channel Σ∆ ADCs. This is a novel concept, combining quantizationand interference cancellation in one step. Using a multi-channel setup connectedto multiple antennas allows us to exploit the spatial diversity to perform the abovetwo operations more efficiently.

Performing interference cancellation before the quantizer allows us to use lowresolution ADCs, thereby reducing the power consumption. Integrating a di-gital FBB with the ADCs provides the flexibility of advanced signal processingalgorithms. The compatibility of existing Σ∆ ADCs with the proposed FBB ar-chitecture leads to a very elegant and robust approach to cancel the interference.The closed form solution for the interference canceling Σ∆ ADCs is related to wellknown classical feedback equalization approaches such as the LMS algorithm [29].

The FBB design can be modified to account for either a NB or a WB in-terferers. On a higher level, these integrated architectures allow us to view theADC components as equalizers or beamformers that achieve substantial powerreduction through canceling the interfering users.

The above two approaches discussed in chapters 2 and 4 focus mostly on NBinterference cancellation. In chapter 2, the NB property allows us to approxim-

FUNDAMENTAL QUESTIONS AND SUMMARY OF MAIN RESULTS 125

ate propagation delays as phase shifts. This specifies that a linear combinationof phase shifts from the antenna array can cancel the interfering user. Theseproperties do not hold for a wide-band scenario. The NB approach of chapter 2serves as an excellent starting point to get ourselves familiar with front-end ar-chitectures and the beamforming problem. However, the current trend in wirelesscommunications is to move towards wide and ultra-wide band signals.

Given such wide-band transmission scenarios, how do we design the RF phaseshifter to cancel the interference and to reduce the number of receiver chains. Inaddition to the multi-path echoes of the interferers, the impairments in the RFcircuitry may lead to non-linearities. The objective must be to use the same setupas described in chapter 2 and incorporate additional challenges usually seen in aWB case.

We perform WB interference cancellation with the RF phase shifter arrange-ment in chapter 5. We have shown that the RF phase shifters cannot cancel theinterference in RF and that a closed form solution to design the APN is not al-ways possible. However, it is possible to design an RF phase shifter matrix tapsto partially cancel the interfering users, and its multipath echoes. This is thenfollowed by digital space-time combining to obtain a high resolution estimate ofthe desired user signal. To test the effectiveness of the algorithm, we use measureddata from offline experiments and estimate the phase shifter weights to cancel RFinterference.

7.1.1 Context: IOP-Gencom project

This research was supported by the Senter-Novem under the IOP-Gencom pro-gram (IGC-0502B) and was intended to be a joint collaboration between Delft Uni-versity of Technology, Eindhoven University of Technology and Industrial partnerPhilips Research.

The objective of this project was to come up with novel architectures andalgorithms leading to cheap and power efficient implementations of wireless com-munication systems. During the course of our research, we realized the need tojointly optimize the RF and baseband signal processing components to achievethe stated goals.

In this context, my contribution had been to propose novel architectures andthe corresponding signal processing algorithms. Chapter 2 provides a systematicapproach to design the RF phase shifter taps with a specific optimization criteriato cancel the interference. As a part of the joint work with Eindhoven Universityof Technology, a Nr × ND phase shifter (with Nr = 4 and ND = 2) based onchapter 2 is currently being designed in 2.4GHz 65 nm CMOS technology by Johanvan den Heuvel e.a. [87].

Given that similar phase shift architectures exist such as [13, 87], my uniquecontribution had been to design the APN and incorporating the optimizationcriteria (interference cancellation), power requirements (receiver chains and ADCresolution) and hardware requirements (phase shift dictionaries). This theme was

SUGGESTIONS FOR FUTURE RESEARCH 126

continued in chapters 4 and 5 to perform mixed signal interference cancellationand wide-band interference cancellation.

7.2 Suggestions for future research

So far, most of the architectures and the algorithms proposed in this thesis havebeen verified with simulations and in some cases with offline experiments. Al-though we incorporate the hardware constraints such as quantization, phase errorsand phase shift dictionaries in our calculations, this research is far from completeuntil one can implement such integrated architectures in CMOS or other RFtechnologies. The implementation might provide answers to some of the followingopen problems:

• RF implementation of the discrete phase shifters may give us an idea of RFparasitics/coupling. It could also provide us with a more precise value onthe amount of power savings.

• Phase errors vary with respect to the implementation of the analog front-ends. They have to be estimated i.e. the APN must also be calibrated.

• The LRB architectures in chapter 2 require additional hardware. Alternat-ive approaches to obtain a simple snapshot must also be investigated.

• In addition, we reiterate some of the remaining digital challenges mentionedin the chapter 2 as follows:

– How can we model the effect of phase errors and incorporate them inthe APN design?

– Initially, we are not synchronized to the source of interest. It maythen be complicated to estimate the state of the wireless channel; sub-sequently, the interference may overwhelm the ADCs and make acquis-ition impossible. What is a good initialization strategy?

Nevertheless, we hope that the ongoing RF implementation by my colleague Jo-han v. d. Heuvel would answer some of the questions and leave the remainingquestions for future researchers.

Integrated phased array architectures is one important approach that can real-ize adaptive analog front-ends with a specific optimization criteria (However, thedesign of adaptive front-ends involves challenges and constraints.). In additionto these approaches, integrating Σ∆ quantizer with LMS like beamforming al-gorithms as proposed in chapter 4 can lead to more efficient sampling of thedesired user signals.

The auto-regressive Σ∆ ADCs with a feedback loop provides us with manypossibilities to effectively design the ADC output. This has led to some elegant


and popular mixed signal implementations [16]. However, a more systematic ap-proach to the design of such ADC circuits with a specific optimization criteriais lacking and the academia has always followed the industry [88] in the under-standing of Σ∆ ADC architectures.

Computing the closed form optimization solutions for Σ∆ ADCs as in chapter4 to cancel interference is one step towards systematically understanding suchADC architectures. However designing a multi-channel auto-regressive structurein hardware to estimate and cancel the interferers would require further investig-ation into some of the following open problems:

• If the wireless channel is convolutive, the architecture is supposed to doboth equalization and interference cancellation. This is possible only upto a certain channel length. The question is, what is this maximal lengthand what happens if the channel is longer. Presumably the channel can beestimated and convolved with the training sequence so that the equalizationis not taking place at this level.

• We did not analyze the effect of the coarse quantization of the ADC. Inpractice, the set-up of chapter 4 operates on 1 bit signals of the Σ∆ ADCoutput, and the simulations indicated that this is adequate.

• Stability of Σ∆ ADCs is an important factor, and has been thoroughly in-vestigated for Σ∆ ADCs with fixed feedback loop [89]. In chapter 4, wepartially address for an adaptive feedback loop using a closed form FBB ex-pression designed from measurements over an entire training packet. How-ever, realizing such ADCs would require a closer look in this area.

• We did not analyze the effect of the quantization of the feedback DACs.

• We did not consider the latency introduced by the Σ∆ ADC. This is not aserious issue for a NB environment, however it would be critical for a WBenvironment and might require a higher order Σ∆ ADC.

• The beamformer estimation employs training signals corresponding to thedesired user. This is not practical in all cases.

We briefly mentioned in chapter 3 that intermodulation products can beviewed as interferers and cancelled using the APN. In chapter 5, we extend theRF phase shifter design to also cancel the WB interference. To realize the APNarrangements in RF and to account for circuit non-linearities, more research isneeded along the following directions:

• Although we have modeled the non-linearity in the LNAs approximately asa Taylor series expansion in chapter 3, a more thorough examination of IP3cancelation, in combination with RF implementation is required.


• The greedy WB approach as explained in chapter 5 (Table 5.1) is sub-optimal in nature and converges to the optimal approach only for a largenumber of APN outputs. More investigation is required along the directionof replacing this approximation with more effective APN design techniques.

• Practical issues such as the availability of a training signal, choice of thetraining length and performance analysis for very long and short channellengths must be thoroughly examined. It might also be interesting to con-sider these architectures in combination with the superimposed trainingsequence approaches as in chapter 6.

• The RF implementations in this thesis have been limited to phase shift ar-chitectures and can be easily integrated to existing phased array systems.There is a strong motivation to explore other possible RF and IF architec-tures integrated with mixers [13] or other analog front-end components.

7.2.1 Compressive sampling and analog front-ends

On a higher level, some very interesting questions originate.Recently, interest in sampling and quantization has spiked due to advance-

ments in compressive/ sparse sampling (CS) theory [90, 91]. This emerging fieldis based on the revelation that a small collection of non-adaptive (even random)linear projections of a compressible signal contains enough information for digitalsignal reconstruction and processing. The implications of CS are promising formany applications and enable the design of new kinds of ADCs, imaging systems,cameras, and distributed source coding algorithms for sensor networks.

CS theory specifies that sub-Nyquist sampling can perfectly reconstruct thedesired signals. Their backbone is to choose a set of random projections andsample/ quantize the analog signals. One fundamental question is how do weimplement such samplers and ADCs in practice?

Maybe one needs to design ADCs for a specific scenario. More specifically,ADCs for ultra-wide band systems might be different from that of a NB scenario.However having a separate front-end to account for different wireless channelrealizations might not be a practical solution.

Consider the communication scenarios in chapters 2 and 5. We can intuitivelysee that for the same RF hardware, two distinct APN design approaches andthe respective digital baseband algorithms in chapters 2 and 5 can effectivelyreconstruct the desired user signal. Similarly, consider the multi-channel Σ∆ADC design in chapter 4. The target sequence can be modified to account fordifferent NB and WB wireless channels as explained in Sec. 4.5.

The author’s intuition is that with the help of integrated architectures, it ispossible to target the ADCs and analog front-end for a specific application andscenario. Some relevant work in designing the ADCs for a CS framework has beenreported recently in [61].


However, whether multi-user communications satisfy the necessary conditionsfor CS theory is an altogether different question.

I have finished with more questions than I started with, however I hope the an-swers (or partial answers) in this thesis lead to realizing low power communicationcircuits and digitally assisted electronic components.

Bibliography

[1] R. Walden, “Analog to Digital converter survey and analysis,” IEEE Journalon Selected areas in Communications, vol. 17, no. 4, pp. 539–550, Apr 1999.

[2] C. E. Shannon, “A mathematical theory of communication,” Bell system tech-nical journal, vol. 27, 1948.

[3] T. Cover and J. Thomas, Elements of Information Theory. Wiley Inter-science, 2006.

[4] H. Krim and M. Viberg, “Two decades of array signal processing research,”Signal Processing Magazine, vol. 13, no. 4, pp. 67–94, 1996.

[5] E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans-actions on Telecommunications, vol. 10, no. 6, pp. 585–595, 1999.

[6] D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “From theory topractice: an overview of MIMO space-time coded wireless systems,” IEEEJournal on Selected areas in Communications, vol. 21, no. 3, pp. 281–302,Mar 2003.

[7] G. E. Moore, “Cramming more components onto integrated circuits,” Elec-tronics, vol. 38, no. 8, April 19.

[8] IEEE Task P802.11n, Wireless LAN medium access and physical layer spe-cifications: Amendment 4 enhancements for higher throughputs. pub-IEEE-STD, Jun 2008.

[9] B. Murmann, “Digitally assisted analog circuits,” IEEE Micro, vol. 26, no. 2,pp. 38–47, Mar-Apr 2006.

[10] I. Mitola, J. and J. Maguire, G.Q., “Cognitive radio: making software radiosmore personal,” Personal Communications, IEEE, vol. 6, no. 4, pp. 13 –19,Aug 1999.

131

BIBLIOGRAPHY 132

[11] D. Cabric, M. S. Chen, D. A. Sobel, S. Wang, J. Yang, , and R. Broder-sen, “Novel Radio Architectures for UWB, 60 GHz, and CognitiveWirelessSystems,” EURASIP Journal on Wireless Communications and Networking,Special Issue on CMOS RF Circuits for Wireless Applications, vol. 2006, Apr2006.

[12] B. Razavi, “Gadgets gab at 60 GHz,” Spectrum, IEEE, vol. 45, no. 2, pp. 46–58, feb. 2008.

[13] A. Hajimiri, H. Hashemi, A. Natarajan, X. Guang, and A. Babakhani, “In-tegrated phased arrays systems in silicon,” Proceedings of the IEEE, vol. 93,no. 9, pp. 1637–1655, Sep 2005.

[14] B. Murmann, “A/d converter trends: Power dissipation, scaling and digit-ally assisted architectures,” in Custom Integrated Circuits Conference, 2008.CICC 2008. IEEE, 21-24 2008, pp. 105 –112.

[15] B. Razavi, Principles of Data Conversion System Design. Wiley - IEEE,1994.

[16] S. R. Norsworthy, R. Schreier, and G. C. Temes, Delta-Sigma Data convert-ers: Theory, design and simulation. Wiley Interscience, 1997.

[17] T. C. Schenk and J. P. Linnartz, RF Imperfections in High-rate WirelessSystems: Impact and Digital Compensation. Springer, 2008.

[18] T. Hentschel, M. Henker, and G. Fettweis, “The digital front-end of softwareradio terminals,” Personal Communications, IEEE, vol. 6, no. 4, pp. 40 –46,Aug 1999.

[19] S. Sanayei and A. Nosratinia, “Antenna selection in MIMO systems,” IEEECommunications Magazine, vol. 42, no. 10, pp. 68–73, Oct 2004.

[20] B. Razavi, RF Microelectronics. Prentice Hall, 1998.

[21] P. Baltus and R. Dekker, “Optimizing RF front ends for low power,” Pro-ceedings of the IEEE, vol. 88, no. 10, pp. 1546–1559, Oct 2000.

[22] H. Zarei, C. Charles, and D. Allstot, “Reflective-type phase shifters formultiple-antenna transceivers,” Circuits and Systems I: Regular Papers,IEEE Transactions on, vol. 54, no. 8, pp. 1647–1656, Aug. 2007.

[23] N. Fourikis, Phased Array-Based Systems and Applications. Wiley, 1997.

[24] B. Razavi, “Challenges in Portable RF Transceiver Design,” Circuits andDevices Magazine, IEEE, vol. 12, no. 5, pp. 12–25, Sep 1996.

BIBLIOGRAPHY 133

[25] A. J. Boonstra and S. v. d. Tol, “Spatial filtering of interfering signals atthe initial Low Frequency Array (LOFAR) phased array test station,” RadioScience, p. R5S09(16 pages), 2005.

[26] W. Sandrin, “Spatial distribution of intermodulation products in activephased array antennas,” Antennas and Propagation, IEEE Transactions on,vol. 21, no. 6, pp. 864 – 868, Nov 1973.

[27] J. G. Proakis and D. G. Manolakis, Digital Signal Processing : Principles,Algorithms, and Applications. Upper Saddle River, EUA : Prentice-Hall,1996.

[28] K. Philips, P. Nuijten, R. Roovers, A. van Roermund, F. Chavero, M. Pal-lares, and A. Torralba, “A continuous time Sigma Delta ADC with increasedimmunity to interferers,” IEEE Journal on Solid State Circuits, vol. 39,no. 12, pp. 2170–2178, Dec 2004.

[29] J. G. Proakis, Digital Communications. McGraw Hill, 2000.

[30] B. Widrow, J. M. McCool, M. G. Larimore, and C. R. Johnson, “Station-ary and nonstationary learning characteristics of the LMS adaptive filter,”Proceedings of the IEEE, vol. 64, no. 8, pp. 1151–1162, Aug 1976.

[31] M. Ghavami, L. B. Michael, and R. Kohno, Ultra Wideband signals andsystems in communication engineering. Wiley, 2004.

[32] S. Denno and T. Ohira, “Modified constant modulus algorithm for DSP ad-aptive antennas with microwave analog beamforming,” IEEE TransactionsAntennas Propagation, vol. 50, no. 6, pp. 850–857, Jun 2002.

[33] R. Eickhoff, R. Kraemer, I. Santamaria, and L. Gonzalez, “Developing energy-efficient MIMO radios,” IEEE Vehicular Technology Magazine, vol. 4, no. 1,pp. 34–41, Mar 2009.

[34] X. Zhang, A. F. Molisch, and S. Y. Kung, “Variable phase shift based RFbaseband codesign for MIMO antenna selection,” IEEE Transactions SignalProcessing, vol. 53, no. 11, pp. 4091–4103, Nov 2005.

[35] M. D. Zoltowski, S. D. Silverstein, and C. P. Mathews, “Beamspace rootMUSIC for minimum redundancy linear arrays,” IEEE Transactions SignalProcessing, vol. 41, no. 7, pp. 2502–2507, Jul 1993.

[36] B. D. van Veen and R. Roberts, “Partially adaptive beamformer design viaoutput power minimization,” IEEE Transactions Signal Processing, vol. 11,pp. 1524–1532, Nov 1987.

[37] J. S. Goldstein, I. S. Reed, and L. L. Scharf, “A multistage representationof the Wiener filter based on orthogonal projections,” IEEE TransactionsInformation Theory, vol. 44, no. 7, pp. 2943–2959, Nov 1998.

BIBLIOGRAPHY 134

[38] J. Sheinvald and M. Wax, “Direction finding with fewer receivers via timevarying preprocessing,” IEEE Transactions Signal Processing, vol. 47, no. 1,pp. 2–9, Jan 1999.

[39] S. T. Smith, “Optimum phase-only adaptive nulling,” IEEE TransactionsSignal Processing, vol. 47, no. 7, pp. 1835–1843, Jul 1999.

[40] G. M. Kautz, “Phase-only shaped beam synthesis via technique of approx-imated beamaddition,” IEEE Transactions Antennas Propagation, vol. 47,no. 5, pp. 887–894, May 1999.

[41] S. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionar-ies,” IEEE Transactions Signal Processing, vol. 41, no. 12, pp. 3397–3415,Dec 1993.

[42] V. K. Goyal, M. Vetterli, and N. T. Thao, “Quantized overcomplete ex-pansions in RN : Analysis, Synthesis and Algorithms,” IEEE TransactionsInformation Theory, vol. 44, no. 1, pp. 16–31, Jan 1998.

[43] L. L. Scharf, Statistical Signal Processing. Reading, MA: Addison-Wesley,1991.

[44] J. S. Goldstein and I. S. Reed, “Reduced rank adaptive filtering,” IEEE Trans-actions Signal Processing, vol. 45, no. 2, pp. 492–496, Feb 1997.

[45] A. Gersho and R. Gray, Vector quantization and signal compression. Kluweracademic publishers, 1994.

[46] D. Tabrikian and A. Faizakov, “Optimal preprocessing for source localizationby fewer receivers than sensors,” in Proc. IEEE workshop on SSP 2001, pp.213–216.

[47] A. J. van der Veen and A. Paulraj, “An analytical constant modulus al-gorithm,” IEEE Transactions Signal Processing, vol. 44, no. 5, pp. 1136–1155,May 1996.

[48] S. Catreux, P. F. Driessen, and L. J. Greenstein, “Data throughputs usingMIMO techniques in a noise-limited cellular environment,” IEEE Transac-tions Wireless. Communications, vol. 1, no. 2, pp. 226–235, Apr 2002.

[49] R. S. Blum, J. H. Winters, and N. R. Sollenberger, “On the capacity ofcellular systems with MIMO,” IEEE Communications Letters, vol. 6, no. 6,pp. 242–244, Jun 2002.

[50] V. Venkateswaran and A. J. van der Veen, “Source separation in Multi-channel sigma-delta ADC’s,” Submitted to IEEE Transactions on Signal Pro-cessing.

BIBLIOGRAPHY 135

[51] E. Mensink, A. M. Klumperink, and B. Nauta, “Distortion cancelation bypolyphase multipath circuits,” IEEE Transactions on Circuits and Systems- vol 1, vol. 52, no. 9, pp. 1785–1794, Sep 2005.

[52] E. Keehr and A. Hajimiri, “Equalization of third-order intermodulationproducts in wideband direct conversion receivers,” Solid-State Circuits, IEEEJournal of, vol. 43, no. 12, pp. 2853 –2867, dec. 2008.

[53] W. Sansen, “Distortion in elementary transistor circuits,” IEEE TransactionsCircuits and Systems, vol. 46, no. 3, pp. 315–325, Mar 1999.

[54] V. Venkateswaran and A. J. van der Veen, “Analog beamforming in MIMOcommunications with phase shift networks and online channel estimation,”To appear in IEEE Transactions on Signal Processing, Aug 2010.

[55] P. A. Aziz, H. V. Sorensen, and J. van der Spiegel, “An overview of Sigma-Delta convertors,” IEEE Signal Processing magazine, vol. 1, no. 1, pp. 61–84,Jan 1996.

[56] H. Boelcskei and F. Hlawatsch, “Noise reduction in oversampled filterbank using predictive quantization,” IEEE Transactions Information The-ory, vol. 47, no. 1, pp. 155–172, Jan 2001.

[57] N. Thao and M. Vetterli, “Deterministic analysis of oversampled A/D con-version and decoding improvement based on consistent estimates,” IEEETransactions Signal Processing, vol. 42, no. 3, pp. 519–531, Mar 1994.

[58] S. Hein and A. Zakhor, “Reconstruction of oversampled bandlimited signalsfrom Sigma-Delta encoded single bit binary sequences,” IEEE TransactionsSignal Processing, vol. 42, no. 4, pp. 799–811, Apr 1994.

[59] K. Abed-Meraim, E. Moulines, and P. Loubaton, “Prediction error methodfor second order blind identification,” IEEE Transactions Signal Processing,vol. 45, no. 3, pp. 694–705, Mar 1997.

[60] D. T. M. Slock, “Blind fractionally spaced equalization, perfect reconstructionfilter-banks and multi-channel linear prediction,” in IEEE ICASSP, no. 4,1994, pp. 585–588.

[61] P. Boufounos and R. G. Baraniuk, “Sigma delta quantization for compressivesensing,” in SPIE Wavelets XII, vol 6701, Aug 2007.

[62] N. Jayanth and P. Noll, “Digital coding of waveforms,” Prentice Hall, 1964.

[63] M. H. Hayes, Statistical Digital Signal Processing and Modeling. John Wileyand Sons, 2002.

BIBLIOGRAPHY 136

[64] J. McClellan and D. Lee, “Exact equivalence of the Steiglitz-Mcbride iterationand IQML,” Signal Processing, IEEE Transactions on, vol. 39, no. 2, pp.509–512, Feb 1991.

[65] G. Giannakis, Y. Hua, P. Stoica, and L. Tong, Eds. Signal Processing Ad-vances in Wireless and Mobile Communications, Volume 1: Trends in Chan-nel Estimation and Equalization. Upper Saddle River, NJ, USA: PrenticeHall PTR, 2000.

[66] A. Paulraj and C. Papadias, “Space-time processing for wireless communic-ations,” Signal Processing Magazine, IEEE, vol. 14, no. 6, pp. 49 –83, nov1997.

[67] W. C. Jakes, Microwave mobile communications- II Ed. Wiley - IEEE, 1994.

[68] R. Horn and C. R. Johnson, Matrix analysis. Cambridge University Press,1990.

[69] D. Gesbert, P. Duhamel, and S. Mayrague, “Online blind multi-channel equal-ization based on mutually referenced filters,” IEEE Transactions Signal Pro-cessing, vol. 45, no. 9, pp. 2307–2317, Sep 1997.

[70] A. J. van der Veen, S. Talwar, and A. Paulraj, “A subspace approachto blind space-time signal processing for wireless communication systems,”IEEE Transactions Signal Processing, vol. 45, no. 1, pp. 173–190, Jan 1997.

[71] D. Gesbert, A. J. van der Veen, and A. Paulraj, “On the equivalence of blindequalizers based on MRE and subspace intersections,” IEEE TransactionsSignal Processing, vol. 47, no. 3, pp. 856–859, March 1999.

[72] Z.-Q. Luo, M. Gastpar, J. Liu, and A. Swami, “Distributed signal processingin sensor networks [from the guest editors],” Signal Processing Magazine,IEEE, vol. 23, no. 4, pp. 14 – 15, July 2006.

[73] M. Gastpar, P. L. Dragotti, and M. Vetterli, “The Distributed KarhunenLoeve transform,” IEEE Transactions Information Theory, vol. 52, no. 12,pp. 5177–5196, Dec 2006.

[74] Z. Irahhauten, H. Nikookar, and G. Janssen, “An oveview of ultra-wide bandindoor channel measurements and modeling,” IEEE Microwave and WirelessCommunications Letters, pp. 386–388, Aug 2004.

[75] G. Stuber, J. Barry, S. McLaughlin, Y. Li, M. Ingram, and T. Pratt, “Broad-band MIMO-OFDM wireless communications,” Proceedings of the IEEE,vol. 92, no. 2, pp. 271 – 294, Feb 2004.

[76] T. Thomas and F. Vook, “Asynchronous interference in broadband cyclic pre-fix communications,” in Proc. IEEE wireless communications and networkingconference 2003, no. 1, pp. 568–572.

BIBLIOGRAPHY 137

[77] A. Orozco-Lugo, M. Lara, and D. McLernon, “Channel estimation using im-plicit training,” IEEE Transactions on Signal Processing, vol. 52, no. 1, pp.240–254, Jan.

[78] J. Tugnait and X. Meng, “On superimposed training for channel estima-tion:performance analysis, training power allocation and frame synchroniza-tion,” IEEE Transactions on Signal Processing, vol. 54, no. 2, pp. 752–765,Feb 2006.

[79] V. Venkateswaran, A. J. van der Veen, and M. Ghogho, “Joint source sep-aration and offset estimation of asynchronous OFDM signals using subspacefitting,” in Proc. SPAWC 2007, pp. 1–5.

[80] A. Orozco-Lugo, G. M. Galvan-Tejada, M. Lara, and D. McLernon, “A newapproach to achieve multiple packet reception for ad hoc networks,” in Proc.IEEE ICASSP 2004, no. 4, pp. 429–432.

[81] V. Venkateswaran and A. J. van der Veen, “Source separation of asynchronousofdm signals using superimposed training,” in Proc. IEEE ICASSP 2007,no. 3, pp. 385–388.

[82] M. Viberg and B. Ottersten, “Sensor array processing based on subspacefitting,” IEEE Transactions Signal Processing, vol. 39, no. 5, pp. 1110–1121,May 1991.

[83] E. Moulines, P. Duhamel, J. Cardoso, and S. Mayrague, “Subspace meth-ods for blind identification of multichannel FIR filters,” IEEE TransactionsSignal Processing, vol. 43, no. 2, pp. 516–525, Feb 1995.

[84] A. van der Veen, M. Vanderveen, and A. Paulraj, “Joint angle and delay es-timation using shift invariance techniques,” IEEE Transactions Signal Pro-cessing, vol. 46, no. 2, pp. 405–418, Feb 1998.

[85] T. K. Moon and W. Stirling, Mathematical Methods and Algorithms for Sig-nal Processing. Prentice-Hall, 2000.

[86] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,”IEEE Transactions on Antennas and Propagation, vol. 34, no. 3, pp. 276–280,Mar 1986.

[87] J. H. C. Heuvel, J. P. M. G. Linnartz, and P. G. M. Baltus, “Optimizing RFfrontends for low power,” in Proc. of ISCAS 2010.

[88] R. M. Gray, “Oversampled Sigma-Delta modulation,” IEEE Transactions onCommunications, vol. 35, no. 5, pp. 481–489, May 1987.

[89] S. Hein and A. Zakhor, “On the stability of Sigma-Delta modulators,” IEEETransactions Signal Processing, vol. 41, no. 7, pp. 2322–2348, July 1993.

BIBLIOGRAPHY 138

[90] R. Baraniuk, E. Candes, R. Nowak, and M. Vetterli, “Compressive Sampling,”IEEE Signal Processing magazine, vol. 13, no. 3, pp. 12–13, March 2008.

[91] Y. Eldar and T. Michaeli, “Beyond band limited sampling,” IEEE SignalProcessing magazine, vol. 26, no. 5, pp. 48–68, May 2009.

Samenvatting

Een van de grote nadelen van de realisatie van MIMO en multi-sensor draad-loze communicatie-systemen is dat meerdere antennes op de ontvanger elk huneigen aparte radiofrequentie (RF) front-ends en analoog naar digitaal convert-ers (ADCs) hebben, wat leidt tot een hogere circuit-omvang en stroomverbruik.Verbeteringen in RF-technologie en ADCs gebeuren in een veel trager tempoin vergelijking met digitale schakelingen, zodat dit probleem in de toekomstwaarschijnlijk nog nijpender wordt.

In een dichte multi-user setup draadloze communicatie gebruiken deze meer-dere RF-front-ends en ADCs het grootste deel van hun vermogen voor de ver-werking van storende signalen van andere gebruikers. Het doel van dit onderzoekis te kijken naar alternatieve mobiele ontvanger-architecturen, vanuit het geza-menlijke perspectief van de digitale signaalverwerkings-ingenieur en die van deRF-ontwerper.

We beginnen met een vermelding van de noodzaak van een gemeenschappelijkgebruik van RF- en DSP-technieken. De stelling is dat geavanceerde signaalverwerkings-algoritmen kunnen worden gebruikt in combinatie met kleine aanpassingen aanbestaande circuit-configuraties, zoals de introductie van in het analoge domeingeintegreerde phased arrays en van feedback bij multi-channel ADCs, om interfer-entie annulering in het analoge domein uit te voeren. Hierdoor hebben we minderdigitale ontvangers nodig, en per ontvanger volstaat een lagere resolutie (aantalbits). Dit zorgt voor een navenant kleinere circuitomvang en stroomverbruik.

In het kort richt het onderzoek zich op de volgende vragen:

1. Kunnen we de kosten en het energieverlies van MIMO transceivers mogelijkverminderen, door optimalisatie in de interface tussen RF en basisband?

2. Kunnen we een flexibel MIMO basisband platform ontwerpen, toegesnedenop low power circuits, met een potentieel voor lage kosten?

139

SAMENVATTING 140

Zoals hierboven genoemd is een aanpak om de invloed van RF storingen teannuleren en het aantal digitale ontvangers te beperken, het ontwerpen van di-gitaal bestuurde analoge RF beamformers. Een alternatief is om de bestaandeADCs te integreren met een feedback beamformer (deze setup is vooral compati-bel met bestaande Sigma-Delta ADCs), voor het identificeren en annuleren van destoorbronnen. Door deze aanpak kan de benodigde precisie van de ADCs beperktblijven, wat leidt tot een kostenreductie.

Voor de bovengenoemde aanpakken ontwikkelen we geschikte kleinste-kwadratenoptimaliseringsproblemen en gesloten-vorm oplossingen hiervoor, en illustrerendat dit leidt tot een aanzienlijke energiebesparing in de ontvangers. Voor beidegevallen hebben we tevens benaderende oplossingen, wanneer de gesloten vormoplossingen niet haalbaar zijn. Tevens ontwerpen we algoritmes voor het schattenvan de draadloze communicatiekanalen. We geven ook aan hoe dezelfde aanpakgebruikt kan worden om de effecten van niet-lineaire intermodulatie-producten teannuleren.

Op een hoger niveau stellen we dat het noodzakelijk is voor DSP-ingenieursom niet meer te kijken naar ADCs en RF-componenten als "zwarte dozen" ineen systeem. Een geintegreerde aanpak van bijvoorbeeld Sigma-Delta ADCs meteen digitale equalizer en feedback loop, of het bekijken van RF-schakelingen metgeintegreerde phased arrays om storingen in het analoge domein te annuleren, kanin zeer efficiente oplossingen resulteren. In deze optiek zal een dergelijke hybridearchitectuur er toe leiden dat DSP-technieken de draadloze revolutie sturen, inplaats van beschouwd worden als een bijzaak voor het omgaan met de onvolko-menheden van RF.

Acknowledgements

I would like to thank Prof.Alle-Jan van der Veen for giving me a chance to doPhD in his group. Alle-Jan’s door has always been always open, both literallyand metaphorically for endless discussions. Though thesis writing is a solitaryact, the innovations of this thesis would not be possible without his assistance.As a person, he is probably the nicest one can come across. As a supervisor,he has been my conscience and a constructive critic. As an editor-in-chief, hehopes to promote high quality path-breaking technical journals. All of these areexceptional qualities, difficult to emulate in reality and rarely seen in academia.Its an honor to have been associated with such a thorough scientist and hope thatthe originality of this work is a testimony of his efforts.

When Alle-Jan is busy or away, one needs to look no further than the chair ofsignal processing for communications, Prof.Geert Leus. Geert provides instant-aneous feedback your questions, is up-to date with latest research and is verynatural in teaching and guiding students. As a group, they (Alle-Jan and Geert)make you feel very comfortable with other professors and visitors and are one ofthe best set of gurus of Signal Processing.

Association with Alle-Jan and Geert opened the door for working closely withsome very good researchers. Prof.Dirk Slock had excellent intuition gave a goodinitial point for one problem. Profs. Jean-Paul Linnartz and Peter Baltus intro-duced me to the practical world of communication circuits. Stimulating discus-sions with Profs.Amir Leshem and Urbashi Mitra improved my ability to system-atically approach open research problems.

I would also like to thank for the enthusiasm and advice of Prof. Bane Vasicduring my stay in Tucson, and for initiating interest in communication systemsand related algorithms.

Johan van den Heuvel has been an enthusiastic colleague, and proposed somevery original questions and challenges in the world of RF and analog circuits.My research took a very positive turn, once I was exposed to the challenges in

141

ACKNOWLEDGEMENTS 142

solid state community. I would also like to than the Dutch Ministry of EconomicAffairs for their financial support.

Vaibhav Maheshwari and Claude Simon made my stay in Delft more enjoyable.Last but probably the most relevant, the love and affection of my parents andSuha is so outstanding that I often fail the need to acknowledge it. Suha has beennothing short of amazing and in many ways, is my inspiration.

Vijay VenkateswaranDelft, 2010

Beyond digital interference cancellation

Documents

Transcript of Beyond digital interference cancellation