On multipath spatial diversity in wireless multiuser communications

On Multipath Spatial Diversity in

Wireless Multiuser

Communications

Haley M. Jones

B.E. (Hons.), B.Sc. (Adelaide)

December 2001

A thesis submitted for the degree of Doctor of Philosophy of The Australian National University

Department of Telecommunications Engineering

Research School of Information Sciences and Engineering

Declaration

The contents of this thesis are the results of original research and have not been submitted for a higher degree to any other university or institution.

Much of the work in this thesis has been published or has been submitted for publication.

Conference Papers

• H. M. Jones, P. B. Rapajic, and R. A. Kennedy. Bounds on Capacity Im-provements Using Spatial Filtering. InProceedings of the 5th Int. Symposium on Sig. Proc. and its Applications, pages 987-990, Brisbane, Australia, Au-gust 1999.

• H. M. Jones, R. A. Kennedy, B. D. Hart, and P. B. Rapajic. Limits on Linear Detector Performance for Close Users in a Wireless Multipath Environment. In Proceedings of Int. Conf. on Acoustics, Speech and Sig. Proc., Salt Lake City, Utah, May 2001.

• H. M. Jones, R. A. Kennedy, and T. D. Abhayapala. On Dimensionality of Multipath Fields: Spatial Extent and Richness. Proceedings of Int. Conf. on Acoustics, Speech and Sig. Proc., pages 2953-2956, Orlando, Florida, May 2002.

• H. M. Jones, R. A. Kennedy, and T. D. Abhayapala. Bounds on the Spatial Richness of Multipath. Proceedings of the 3rd Australian Communications theory Workshop, pages 76-80, Canberra, Australia, Feb 2002.

Journal Papers

• H. M. Jones, R. A. Kennedy, B. D. Hart, and P. B. Rapajic. Limits on Linear Detector Performance for Close Users in a Wireless Multipath Environment.

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• H. M. Jones, R. A. Kennedy, and T. D. Abhayapala. Bounds on the Spa-tial Richness and Dimension of Multipath. IEEE Transactions on Signal Processing (submitted).

The research in this thesis has been performed jointly with Prof. R. A. Kennedy, Dr P. B. Rapajic, Dr B. D. Hart and Dr T. D. Abhayapala. The majority, approx-imately 75%, of this work was my own.

Haley M. Jones

Department of Telecommunications

Research School of Information Sciences and Engineering The Australian National University

Acknowledgements

There are many people without whom this thesis would not have been possible and to whom I am exceedingly grateful.

• I would like to thank Prof. Rod Kennedy for his sense of humour and technical insight, and Drs Predrag Rapajic, Brian Hart and Thushara Abhayapala for their contributions, advice and technical know-how.

• I would like to thank Maria Davern, our departmental administrator, and a really good friend, who has kept me sane over the past three years, let me cry on her shoulder and listened to my problems, on top of being a great administrator.

• I would also like to thank the amazing James Ashton, without whom, not a computer in the building would be working. His patience, good humour and expertise are to be held in awe.

• I would like to acknowledge the support of the Australian Research Coun-cil and NEC Australia for provision of an Australian Postgraduate Award (Industry).

• Last, but definitely not least, I would like to thank my family: mum and dad for their moral support, love and for always being there when I needed them, Kelli and Brendan for their love and support and, especially, Raj who has looked after me so well over the past few months while I’ve been too stressed to think of anything but this thesis.

Abstract

The study of the spatial aspects of multipath in wireless communications environ-ments is an increasingly important addition to the study of the temporal aspects in the search for ways to increase the utilization of the available wireless channel capacity. Traditionally, multipath has been viewed as an encumbrance in wireless communications, two of the major impairments being signal fading and intersymbol interference. However, recently the potential advantages of the diversity offered by multipath rich environments in multiuser communications have been recognised. Space time coding, for example, is a recent technique which relies on a rich scat-tering environment to create many practically uncorrelated signal transmission channels. Most often, statistical models have been used to describe the multipath environments in such applications. This approach has met with reasonable success but is limited when the statistical nature of a field is not easily determined or is not readily described by a known distribution.

Our primary aim in this thesis is to probe further into the nature of multipath environments in order to gain a greater understanding of their characteristics and diversity potential. We highlight the shortcomings of beamforming in a multipath multiuser access environment. We show that the ability of a beamformer to resolve two or more signals in angle directly limits its achievable capacity.

We test the probity of multipath as a source of spatial diversity, the limiting case of which is co-located users. We introduce the concept of separability to define the fundamental limits of a receiver to extract the signal of a desired user from interfering users’ signals and noise. We consider the separability performances of the minimum mean square error (MMSE), decorrelating (DEC) and matched filter (MF) detectors as we bring the positions of a desired and an interfering user closer together. We show that both the MMSE and DEC detectors are able to achieve acceptable levels of separability with the users as close as λ/10.

In seeking a better understanding of the nature of multipath fields themselves, we take two approaches. In the first we take a path oriented approach. The

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effects on the variation of the field power of the relative values of parameters such as amplitude and propagation direction are considered for a two path field. The results are applied to a theoretical analysis of the behaviour of linear detectors in multipath fields. This approach is insightful for fields with small numbers of multipaths, but quickly becomes mathematically complex.

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Glossary

2D Two Dimensional

3D Three Dimensional

AOA Angle of Arrival

AWGN Additive White Gaussian Noise

BER Bit Error Rate

BLAST Bell Laboratories Layered Space Time

BPSK Binary Phase Shift Keying

CCI Co-Channel Interference

cdf Cumulative Density Function

CDMA Code Division Multiple Access

DEC Decorrelating Detector or Decorrelator

DFE Decision Feedback Equalizer

DFT Discrete Fourier Transform

DS-CDMA Direct Sequence Code Division Multiple Access

FDMA Frequency Division Multiple Access

FFT Fast Fourier Transform

GHz Gigahertz, 109 _{Hertz frequency}

ISI Intersymbol Interference

MF Matched Filter

MMSE Minimum Mean Square Error

MSE Mean Square Error

pdf Probability Distribution Function

RHS Right Hand Side

RMS Root Mean Square

SDMA Space Division Multiple Access

SINR Signal to Interference plus Noise Ratio

SIR Signal to Interference Ratio

SNR Signal to Noise Ratio

STC Space-Time Coding

STBC Space-Time Block Coding

STTC Space-Time Trellis Coding

STS Space-Time Spreading

Contents

List of Figures xvi

1 Introduction 1

1.1 Thesis Motivation . . . 1

1.2 Multiuser Access Techniques . . . 2

1.3 Multipath Environments . . . 4

1.4 Antenna Arrays . . . 7

1.5 Questions to be Answered in this Thesis . . . 13

1.6 Thesis Overview . . . 13

2 Limits to Performance Improvements via Spatial Filtering: Ca-pacity Results 15 2.1 Introduction . . . 15

2.2 Beamforming . . . 17

2.2.1 Antenna Arrays . . . 17

2.2.2 Choosing the Sensor Weights . . . 19

2.2.3 Uniform Sensor Weights . . . 19

2.3 Information Capacity for One User . . . 20

2.3.1 Analysis of Capacity with One Interfering User . . . 22

2.3.2 Analysis of Capacity with Multiple Interfering Users . . . . 25

2.4 Weights for Minimum Sidelobe Level . . . 27

2.4.1 Chebyshev Weights . . . 27

2.4.2 Comparisons of Gain and Capacity Functions with Cheby-shev and Uniform Weights . . . 28

2.5 Conclusions . . . 28

3 Probity of Multipath as a Source of Signal Diversity 31 3.1 Introduction . . . 31

x Contents

3.2.1 Separability Concept Development . . . 34

3.2.2 Metrics . . . 36

3.2.3 Separability Definitions for Specific Metrics . . . 38

3.3 Application of Separability . . . 39

3.3.1 Separability Applied to Multipath Diversity Environments . 39 3.3.2 Model Formulation . . . 40

3.4 Receiver Signal Processing . . . 41

3.4.1 Beam Pattern Generation . . . 42

3.4.2 Linear Detectors . . . 42

3.4.3 Theoretical Considerations of Beamforming . . . 47

3.5 Performance Measures . . . 52

3.5.1 Bit Error Rate . . . 53

3.5.2 Mean Square Error . . . 54

3.6 Simulation Results . . . 55

3.6.1 Simulation Model Description . . . 55

3.6.2 Assessment of Multipath Parameter Influence . . . 56

3.7 Conclusions . . . 61

4 Linear Detector Behaviour in the Spatial Domain 65 4.1 Introduction . . . 65

4.2 Background . . . 66

4.3 Review of Plane Waves . . . 68

4.3.1 Average Power of a Single Plane Wave Field . . . 69

4.3.2 Conditions for Zero Average Power in a Two Plane Wave Field 70 4.3.3 Conditions for Minimizing the Peak to Peak Average Power Distance for a Two Plane Wave Field . . . 70

4.4 Signal and Linear Detector Processing Preliminaries . . . 74

4.4.1 Representation of Users’ Multipath Signals . . . 74

4.4.2 Receiver Signal Processing . . . 75

4.5 Linear Detector Behaviour in a One Path Field . . . 75

4.5.1 Input Power Ratios . . . 76

4.5.2 Output Power Ratios . . . 76

4.5.3 MMSE Detector Behaviour in a One Path Field . . . 77

4.6 Linear Detector Behaviour in a Two Multipath Field . . . 79

4.6.1 Zeroing the Interfering User’s Output Power . . . 79

Contents xi

4.7 MMSE Detector Behaviour in a Two Multipath Field . . . 82

4.7.1 Zeroing the Interfering User’s Average Output Power . . . . 82

4.7.2 Analysis of MMSE Behaviour with Varying SNR . . . 83

4.8 Some Diversity Results in a Multipath Environment . . . 86

4.8.1 Average Minimum Distance Between Adjacent Peaks or Troughs 86 4.8.2 Minimum Number of Paths for Maximum Angular Diversity 88 4.9 Conclusions . . . 90

5 Limits on the Dimensionality of a Multipath Field 93 5.1 Introduction . . . 93

5.2 General 2D Multipath Field . . . 95

5.3 Dimensionality of General Multipath Environment . . . 97

5.3.1 Approximating a General Multipath Field . . . 97

5.3.2 Bounding the Relative Error . . . 99

5.3.3 Error Improvement with Increasing N . . . 101

5.3.4 Simulation of Error Bound and Dimensionality . . . 102

5.4 Plane Wave Synthesis . . . 104

5.5 Conclusions . . . 106

6 Conclusions and Further Work 107 6.1 Conclusions . . . 107

6.2 Further Work . . . 109

Appendix A Upper Bound on JN(z) 111 Appendix B Bounds Relating to exp(·) 113 B.1 Bounds on Remainder . . . 113

List of Figures

1.1 Illustrations of the concepts of FDMA, TDMA and CDMA. . . 3 1.2 Example of a rectangular array with 4 sensors. Each sensor output

is weighted then they are linearly combined and processed as required. 7 1.3 Beam pattern magnitude using polar coordinates for a 10 element

linear antenna array with uniform weights. In this case the beam pattern is symmetric with respect to both the vertical and horizontal axes. There are two main beams, one at 0◦ _{and the other at 180}◦_{. . 8}

1.4 Illustration of pure SDMA using pencil beams, ideally one for each user. . . 9 2.1 Linear, uniform, antenna array with an even number of sensors

placed at intervals of length,d. A plane wave front is arriving at the array at an angle of θ. The sensors are labelled according to their distance and direction from the centre of the array. . . 18 2.2 Calculation of phase of wavefront at sensor n due to delay (positive

in this case) in arrival compared with the centre of the antenna array. The phase at the sensor shown is ahead of the phase at the centre of the array, for the angle of arrival,θ, of the given wave. . . 19 2.3 Beam pattern in both linear and log scales from a linear uniformly

spaced antenna array with 2N = 10 sensors with uniform weights,

wn= 1,∀n ∈ {−N . . . N, n6= 0}. . . 20

2.4 Capacity, calculated using normalised signal powerP(θmax) = 1 and

bandwidth B = 1, of desired user’s signal in the presence of one interfering signal of the same received power with varying AOA of the interfering user’s signal. SNR = 20dB. . . 21 2.5 Actual and approximation to capacity of desired signal in the

xiv List of Figures 2.6 Comparison of power patterns for a linear uniform array of 10 sensors

using uniform and Chebyshev weights, in decibels. . . 27 2.7 Comparison of capacity for a linear uniform array of 10 sensors using

uniform and Chebyshev weights. . . 28 3.1 Generic system model for separability . . . 34 3.2 Illustration of both an advantage and a shortfall of beamforming

in a multiuser environment. Interfering signal #1 is cancelled by forming a null at its AOA. The same technique cannot be used for interfering signal #2 which has an AOA which is too close to that of the desired signal. . . 40 3.3 Block diagram of the decorrelator showing a matched filter bank

followed by decorrelating block. . . 45 3.4 Probability distribution function of binary estimate, ˆb1, of desired

user’s signal, b₁, given that b₁ = 1. . . 54 3.5 Illustration of simulation model used, with users a distance,d, apart,

a distance, DÀ d, from the antenna array receiver with scatterers in an arc of radius, r, centred on the users. The distance, d, has been exaggerated for clarity. . . 56 3.6 MSE linearε-separability performance of the MF, DEC and MMSE

detectors with increasing distance between the user positions for various numbers of multipaths, P. ε = 0.01. . . 57 3.7 MSE linearε-separability performance of the MF, DEC and MMSE

detector with increasing distance between the user positions for var-ious numbers of array sensors, L. ε = 0.01. . . 58 3.8 MSE linearε-separability performance of the MF, DEC and MMSE

detector with increasing distance between the user positions for var-ious values of scatterer distribution radius, RD. ε = 0.01. . . 59

3.9 MSE linearε-separability performance of the MF, DEC and MMSE detector with increasing distance between the user positions for var-ious multipath angular spreads, ∆θu, with respect to the users. ε =

0.01. . . 60 3.10 MSE linearε-separability performance of the MF, DEC and MMSE

detector with increasing distance between the user positions for var-ious multipath angular spreads, ∆θa, with respect to the array. ε =

List of Figures xv 3.11 Comparison of BER and MSE linear ε-separability performance of

the MMSE detector with increasing distance between the user posi-tions for various numbers of multipaths, P. εBER = 0.0025 . . . 62

4.1 Typical contour plots of the average power of the superposition of a number of plane waves (multipaths) propagating in different, ran-domly selected directions in a region of space. . . 67 4.2 Polar coordinates in 2-D . . . 69 4.3 The average power for a superposition of a pair of sinusoidal plane

waves is sinusoidal. . . 71 4.4 Average power over a region of 4λ×4λ for the superposition of a

pair of plane waves with varying relative propagation directions, ∆θ. 74 4.5 Phase difference between thepth _{multipath of the desired and}

inter-fering users. . . 81 4.6 Changes in amplitude ratio (w1/w2)/(a2/a1), with increasing SNR,

for d= 0.25λ. . . 84 4.7 Distance in degrees of the argument of the cosine term of the average

power of the interfering user in (4.62) from an odd integer multiple of π, with increasing SNR, for d= 0.25λ. . . 85 4.8 Contour plots showing the average input power for the desired(*)

and interfering (o) users and the average output power for three val-ues of SNR: 100dB, 30dB and 0dB. The MMSE detector effectively zeros the interfering user’s output power for SNR = 100dB and 30dB but is not quite so successful for 0dB when the noise power is more significant. . . 86 4.9 The average distance between a maximum and a minimum of the

average power of the sum of sinusoidal plane waves along an arbitrary direction in space with increasing numbers of waves. . . 88 4.10 Probability distribution function (pdf) and cumulative distribution

function (cdf) of propagation directions of multipaths. . . 88 4.11 Probability that all multipaths are within the given angular range

of the first multipath. . . 90 5.1 Low pass character of the Bessel function J0(z) and highpass

char-acter of the Bessel functions J8(z) and J80(z) versus argument z

xvi List of Figures 5.2 Example of accuracy of truncation in (5.10). The actual field (5.4)

has 30 component plane waves in random directions. The approx-imate field (5.10) has 2N + 1 = 15 plane waves. The approximate field is within 0.001 of the actual field for R/λ≤ 0.3890 (indicated by the dashed circle). . . 98 5.3 Minimum N required for different error thresholds εN, for the

er-ror bound with exact remainder (5.26), and the erer-ror bound with bounded remainder (5.27), taking into account the bound on N, (5.28), for increasing values of R/λ. . . 102 5.4 Average error between actual and approximate fields, as in (5.4) and