A Unified Framework for Communications through MIMO Channels, Thesis of Wireless and Satellite Communication

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©c Copyright by Daniel P´erez Palomar 2003 All Rights Reserved

Ph.D. committee:

Gregori V´azquez Ana P´erez-Neira Sergio Barbarossa Mats Bengtsson Emilio Sanvicente

Technical University of Catalonia - UPC Technical University of Catalonia - UPC University of Rome “La Sapienza” Royal Institute of Technology - KTH Technical University of Catalonia - UPC

Qualification: Cum Laude

Abstract

M

ULTIPLE-INPUT MULTIPLE-OUTPUT (MIMO) CHANNELS constitute a unified way of modeling a wide range of different physical communication channels, which can then be handled with a compact and elegant vector-matrix notation. The two paradigmatic examples are wireless multi-antenna channels and wireline Digital Subscriber Line (DSL) channels.

Research in antenna arrays (also known as smart antennas) dates back to the 1960’s. However, the use of multiples antennas at both the transmitter and the receiver, which can be naturally modeled as a MIMO channel, has been recently shown to offer a significant potential increase in capacity. DSL has gained popularity as a broadband access technology capable of reliably delivering high data rates over telephone subscriber lines. A DSL system can be modeled as a communication through a MIMO channel by considering all the copper twisted pairs within a binder as a whole rather than treating each twisted pair independently.

This dissertation considers arbitrary MIMO channels regardless of the physical nature of the channels themselves; as a consequence, the obtained results apply to any communication system that can be modeled as such. After an extensive overview of MIMO channels, both fundamental limits and practical communication aspects of such channels are considered.

First, the fundamental limits of MIMO channels are studied. An information-theoretic ap- proach is taken to obtain different notions of capacity as a function of the degree of channel knowledge for both single-user and multiuser scenarios. Specifically, a game-theoretic framework is adopted to obtain robust solutions under channel uncertainty.

Then, practical communication schemes for MIMO channels are derived for the single-user case or, more exactly, for point-to-point communications (either single-user or multiuser when coordination is possible at both sides of the link). In particular, a joint design of the transmit- receive linear processing (or beamforming) is obtained (assuming a perfect channel knowledge) for systems with either a power constraint or Quality of Service (QoS) constraints.

For power-constrained systems, a variety of measures of quality can be defined to optimize the performance. For this purpose, a novel unified framework that generalizes the existing results in the literature is developed based on majorization theory. In particular, the optimal struc- ture of the transmitter and receiver is obtained for a wide family of objective functions that

i

ii

can be used to measure the quality of a communication system. Using this unified framework, the original complicated nonconvex problem with matrix-valued variables simplifies into a much simpler convex problem with scalar variables. With such a simplification, the design problem can be then reformulated within the powerful framework of convex optimization theory, in which a great number of interesting design criteria can be easily accommodated and efficiently solved even though closed-form expressions may not exist. Among other results, a closed-form expression for optimum beamforming in terms of minimum average bit error rate (BER) is obtained. For other design criteria, either closed-form solutions are given or practical algorithms are derived within the framework of convex optimization theory.

For QoS-constrained systems, although the original problem is a complicated nonconvex prob- lem with matrix-valued variables, with the aid of majorization theory, the problem is reformulated as a simple convex optimization problem with scalar variables. An efficient multi-level water-filling algorithm is given to optimally solve the problem in practice.

Finally, the more realistic situation of imperfect channel knowledge due to channel estimation errors is considered. The previous results on the joint transmit-receive design for MIMO channels are then extended in this sense to obtain robust designs.

Acknowledgements

I

FEEL EXTREMELY LUCKY to have met all the people I have met during the intriguing and appealing journey of my Ph.D. Foremost, I would like to thank my advisor Prof. Miguel Angel Lagunas for his guidance and support over the past four years. His office door has always been open, both literally and metaphorically, and I could always talk to him no matter how busy he was. I can only feel flattered for the confidence he has always shown in me. Fortunately, I have benefited from his extraordinary motivation, great intuition, and technical insight. I just hope my thinking and working attitudes have been shaped according to such outstanding qualities.

I wish to thank my mentor at Stanford University, Prof. John M. Cioffi, for he welcomed me into his research group and treated me like another member of the group with no distinction, making my stay at Stanford a pleasant and rewarding experience. He has always shown confidence in me and has given me a continuous support which I appreciate. Very special and deep thanks go to Joice DeBolt for being such a great human being always ready to listen and give advice.

My thanks also extend to Prof. Javier R. Fonollosa, for he guided my first steps as a Ph.D. student and with whom I have had the pleasure of working in several European projects. I want to thank him for the support he has always given to me.

I will always be indebted to the people who have boldly reviewed this dissertation: Toni Pascual, Gonzalo Seco, Xavi Mestre, and Carlos Aldana. I appreciate very much the enormous help of the people who, at some time or another, have reviewed some of my papers (in chronological order): Dana Brooks, Laurent Schumacher, Carlos Aldana, J´erˆome Louveaux, Wonjong Rhee, Wei Yu, Young-Han Kim, Jonathan Levin, Mats Bengtsson, George Ginis, and Jeannie Lee Fang. I would also like to thank the people with whom I have had the pleasure to collaborate: John M. Cioffi, Javier R. Fonollosa, Ana P´erez Neira, Montse N´ajar, Toni Pascual, and Diego Bartolom´e.

Special thanks go to some people who I have encountered during my Ph.D. journey: Gonzalo Seco for introducing me to the wonderful world of matrix theory at my early stage as a Ph.D. student, Wei Yu and Wonjong Rhee for the encouraging and enthusiastic “wireless and convex” discussions, and Mats Bengtsson for sharing with me the joy for convex optimization theory and for his always interesting comments on my work.

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Contents

Notation xi

• 1 Introduction Acronyms xv
  • 1.1 Motivation
    • 1.1.1 Wireless Multi-Antenna Channels
    • 1.1.2 Wireline DSL Channels
    • 1.1.3 A Generic Approach: MIMO Channels
  • 1.2 Outline of Dissertation
  • 1.3 Research Contributions
• 2 MIMO Channels: An Overview
  • 2.1 Basic MIMO Channel Model
  • 2.2 Examples of MIMO Channels
    • 2.2.1 Frequency-Selective Channel
    • 2.2.2 Multicarrier Channel
    • 2.2.3 Multi-Antenna Wireless Channel
    • 2.2.4 Wireline DSL Channel
    • 2.2.5 Fractional Sampling Equivalent Channel
    • 2.2.6 CDMA Channel
  • 2.3 Gains and Properties of MIMO Channels
    • 2.3.1 Beamforming Gain
    • 2.3.2 Diversity Gain
    • 2.3.3 Multiplexing Gain
    • 2.3.4 Tradeoffs Between Gains
  • 2.4 State-of-the-Art of Transmission Techniques for MIMO Channels
    • 2.4.1 Transmission Techniques with no CSIT
    • 2.4.2 Transmission Techniques with Perfect CSIT
    • 2.4.3 Transmission Techniques with Partial CSIT
    • 2.4.4 The Multiuser Case
  • 2.5 Linear Signal Processing for MIMO Channels viii Contents
    • 2.5.1 Linear Signal Processing
    • 2.5.2 Transmit Power Constraint
    • 2.5.3 Canonical Channel Model
    • 2.5.4 Figures of Merit: MSE, SINR, BER, and Capacity
    • 2.5.5 Optimum Linear Receiver: The Wiener Filter
  • 2.6 Chapter Summary and Conclusions
  • 2.A Analysis of the Error Probability Function
• 3 Mathematical Preliminaries
  • 3.1 Convex Optimization Theory
    • 3.1.1 Convex Problems
    • 3.1.2 Solving Convex Problems
    • 3.1.3 Duality Theory and KKT Conditions
    • 3.1.4 Sensitivity Analysis
  • 3.2 Majorization Theory
    • 3.2.1 Basic Definitions
    • 3.2.2 Basic Results
  • 3.3 Miscellaneous Algebra Results
• 4 Capacity of MIMO Channels
  • 4.1 Introduction
  • 4.2 Instantaneous Capacity
    • 4.2.1 Capacity of the Single-User Channel
    • 4.2.2 Capacity Region of the Multiple-Access Channel (MAC)
  • 4.3 Ergodic and Outage Capacities
    • 4.3.1 Capacity of the Single-User Channel
    • 4.3.2 Capacity Region of the Multiple-Access Channel (MAC)
  • 4.4 Worst-Case Capacity: A Game-Theoretic Approach
    • 4.4.1 Capacity of the Single-User Channel
    • 4.4.2 Capacity Region of the Multiple-Access Channel (MAC)
  • 4.5 Chapter Summary and Conclusions
  • 4.A Proof of Proposition 4.1
  • 4.B Proof of Theorem 4.1
  • 4.C Mixed Strategy Nash Equilibria
  • 4.D Proof of Theorem 4.2
  • 4.E Proof of Lemma 4.1
• with a Power Constraint: A Unified Framework 5 Joint Design of Tx-Rx Linear Processing for MIMO Channels
• 5.1 Introduction
• 5.2 Design Criterion
• 5.3 Single Beamforming
  • 5.3.1 Single MIMO Channel
  • 5.3.2 Multiple MIMO Channels
• 5.4 Multiple Beamforming
  • 5.4.1 Single MIMO Channel
  • 5.4.2 Multiple MIMO Channels
• 5.5 Analysis of Different Design Criteria: A Convex Optimization Approach
  • 5.5.1 Minimization of the ARITH-MSE
  • 5.5.2 Minimization of the GEOM-MSE
  • 5.5.3 Minimization of the Determinant of the MSE Matrix
  • 5.5.4 Maximization of Mutual Information
  • 5.5.5 Minimization of the MAX-MSE
  • 5.5.6 Maximization of the ARITH-SINR
  • 5.5.7 Maximization of the GEOM-SINR
  • 5.5.8 Maximization of the HARM-SINR
  • 5.5.9 Maximization of the PROD-(1+SINR)
  • 5.5.10 Maximization of the MIN-SINR
  • 5.5.11 Minimization of the ARITH-BER
  • 5.5.12 Minimization of the GEOM-BER
  • 5.5.13 Minimization of the MAX-BER
  • 5.5.14 Including a ZF Constraint
• 5.6 Introducing Additional Constraints
• 5.7 Simulation Results
• 5.8 Chapter Summary and Conclusions
• 5.A Proof of Theorem 5.1
• 5.B Proof of Lemma 5.1
• 5.C Proof of Corollary 5.2
• 5.D Proof of Schur-Convexity/Concavity Lemmas
• 5.E Gradients and Hessians for the ARITH-BER
• 5.F Proof of Water-Filling Results
• with QoS Constraints 6 Joint Design of Tx-Rx Linear Processing for MIMO Channels
• 6.1 Introduction
• 6.2 QoS Requirements
  • 6.3 Single Beamforming x Contents
    • 6.3.1 Single MIMO Channel
    • 6.3.2 Multiple MIMO Channels
  • 6.4 Multiple Beamforming
    • 6.4.1 Single MIMO Channel
    • 6.4.2 Multiple MIMO Channels
  • 6.5 Relaxation of the QoS Requirements
  • 6.6 Simulation Results
    • 6.6.1 Wireless Multi-Antenna Communication System
    • 6.6.2 Wireline (DSL) Communication System
  • 6.7 Chapter Summary and Conclusions
  • 6.A Proof of Theorem 6.1
  • 6.B Proof of Proposition 6.1
  • 6.C Proof of Theorem 6.2
  • 6.D Proof of Proposition 6.2
  • 6.E Proof of Lemma 6.1
  • 6.F Proof of Lemma 6.2
• 7 Robust Design against Channel Estimation Errors
  • 7.1 Introduction
  • 7.2 Worst-Case vs. Stochastic Robust Designs
  • 7.3 Worst-Case Robust Design
    • 7.3.1 Error Modeling and Problem Formulation
    • 7.3.2 Power-Constrained Systems
    • 7.3.3 QoS-Constrained Systems
  • 7.4 Stochastic Robust Design
    • 7.4.1 Error Modeling and Problem Formulation
    • 7.4.2 Power-Constrained Systems
    • 7.4.3 QoS-Constrained Systems
  • 7.5 Simulation Results
  • 7.6 Chapter Summary and Conclusions
  • 7.A Proof of Lemma 7.1
  • 7.B Proof of Lemma 7.2
• 8 Conclusions and Future Work
  • 8.1 Conclusions
  • 8.2 Future Work

Notation

Boldface upper-case letters denote matrices, boldface lower-case letters denote column vectors, and italics denote scalars.

XT^ , X∗, XH^ Transpose, complex conjugate, and conjugate transpose (Hermitian) of matrix X, respectively.

(·)
^ Optimal value.

X^1 /^2 Hermitian square root of the Hermitian matrix X, i.e., X^1 /^2 X^1 /^2 = X.

Px, P⊥ x Projection matrix onto the subspace spanned by the columns of X and the orthogonal subspace, respectively. Tr (X) Trace of X.

|X| or det (X) Determinant of matrix X.

|x| Absolute value (modulus) of the scalar x.

‖·‖ A norm.

‖x‖ 2 Euclidean norm of vector x: ‖x‖ 2 =

xH^ x.

‖X‖F Frobenius norm of matrix X: ‖X‖F =

Tr (XH^ X).

d (X) Vector of diagonal elements of matrix X.

λ (X) Vector of eigenvalues of matrix X.

λmax (X), λi (X) Maximum eigenvalue and ith eigenvalue (in increasing or decreasing order), respectively, of matrix X.

umax (X), ui (X) Eigenvector associated to the maximum and to the ith eigenvalue (in increasing or decreasing order), respectively, of matrix X. [X]i,j or [X]ij The (ith,jth) element of matrix X.

[X]:,j The jth column of matrix X.

diag ({Dk}) Block-diagonal matrix with diagonal blocks given by the set {Dk}. In partic- ular, if the Dk’s are scalars, it reduces to a diagonal matrix. xi

Notation xiii

sup, inf Supremum (lowest upper bound) and infimum (highest lower bound). ⋂, ⋃ (^) Intersection and union.

(x)+^ Positive part of x, i.e., max (0, x). For matrices it is defined elementwise.

g′^ (a) Derivative of function g (x) evaluated at x = a.

∇xf (x) Gradient of function f (x) with respect to x.^2

dom f Domain of function f.

(^2) If x is a complex-valued vector and function f (x) is not analytic, the well-known definition of the complex gradient operator is used, since it is very convenient, among other things, to determine the stationary points of a real-valued scalar function of a complex vector [Bra83].

xiv Notation