Complex-Valued Matrix Derivatives: With Applications in Signal Processing and Communications

by Are Hjørungnes

• ISBN13: 9780521192644
• ISBN10: 0521192641
• Format: Hardcover
• Copyright: 2011-03-31
• Publisher: Cambridge University Press

Summary

In this complete introduction to the theory of finding derivatives of scalar-, vector- and matrix-valued functions with respect to complex matrix variables, Hjirungnes describes an essential set of mathematical tools for solving research problems where unknown parameters are contained in complex-valued matrices. The first book examining complex-valued matrix derivatives from an engineering perspective, it uses numerous practical examples from signal processing and communications to demonstrate how these tools can be used to analyze and optimize the performance of engineering systems. Covering un-patterned and certain patterned matrices, this self-contained and easy-to-follow reference deals with applications in a range of areas including wireless communications, control theory, adaptive filtering, resource management and digital signal processing. Over 80 end-of-chapter exercises are provided, with a complete solutions manual available online.

Author Biography

Are Hjorungnes is a Professor in the Faculty of Mathematics and Natural Sciences at the University of Oslo, Norway. He is an Editor of the IEEE Transactions on Wireless Communications and has served as a Guest Editor of the IEEE journal of Selected Topics in Signal Processing and the IEEE journal on Selected Areas in Communication.

Table of Contents

Preface	p. xi
Acknowledgments	p. xiii
Abbreviations	p. xv
Nomenclature	p. xvii
Introduction	p. 1
Introduction to the Book	p. 1
Motivation for the Book	p. 2
Brief Literature Summary	p. 3
Brief Outline	p. 5
Background Material	p. 6
Introduction	p. 6
Notation and Classification of Complex Variables and Functions	p. 6
Complex-Valued Variables	p. 7
Complex-Valued Functions	p. 7
Analytic versus Non-Analytic Functions	p. 8
Matrix-Related Definitions	p. 12
Useful Manipulation Formulas	p. 20
Moore-Penrose Inverse	p. 23
Trace Operator	p. 24
Kronecker and Hadamard Products	p. 25
Complex Quadratic Forms	p. 29
Results for Finding Generalized Matrix Derivatives	p. 31
Exercises	p. 38
Theory of Complex-Valued Matrix Derivatives	p. 43
Introduction	p. 43
Complex Differentials	p. 44
Procedure for Finding Complex Differentials	p. 46
Basic Complex Differential Properties	p. 46
Results Used to Identify First- and Second-Order Derivatives	p. 53
Derivative with Respect to Complex Matrices	p. 55
Procedure for Finding Complex-Valued Matrix Derivatives	p. 59
Fundamental Results on Complex-Valued Matrix Derivatives	p. 60
Chain Rule	p. 60
Scalar Real-Valued Functions	p. 61
One Independent Input Matrix Variable	p. 64
Exercises	p. 65
Development of Complex-Valued Derivative Formulas	p. 70
Introduction	p. 70
Complex-Valued Derivatives of Scalar Functions	p. 70
Complex-Valued Derivatives of f(z, z*)	p. 70
Complex-Valued Derivatives of f(z, z*)	p. 74
Complex-Valued Derivatives of f(Z, Z*)	p. 76
Complex-Valued Derivatives of Vector Functions	p. 82
Complex-Valued Derivatives of f(z, z*)	p. 82
Complex-Valued Derivatives of f(z, z*)	p. 82
Complex-Valued Derivatives of f(Z, Z*)	p. 82
Complex-Valued Derivatives of Matrix Functions	p. 84
Complex-Valued Derivatives of F(z, z*)	p. 84
Complex-Valued Derivatives of F(z, z*)	p. 85
Complex-Valued Derivatives of F(Z, Z*)	p. 86
Exercises	p. 91
Complex Hessian Matrices for Scalar, Vector, and Matrix Functions	p. 95
Introduction	p. 95
Alternative Representations of Complex-Valued Matrix Variables	p. 96
Complex-Valued Matrix Variables Z and Z*	p. 96
Augmented Complex-Valued Matrix Variables Z	p. 97
Complex Hessian Matrices of Scalar Functions	p. 99
Complex Hessian Matrices of Scalar Functions Using Z and Z*	p. 99
Complex Hessian Matrices of Scalar Functions Using Z	p. 105
Connections between Hessians When Using Two-Matrix Variable Representations	p. 107
Complex Hessian Matrices of Vector Functions	p. 109
Complex Hessian Matrices ofMatrixFunctions	p. 112
Alternative Expression of Hessian Matrix of Matrix Function	p. 117
Chain Rule for Complex Hessian Matrices	p. 117
Examples of Finding Complex Hessian Matrices	p. 118
Examples of Finding Complex Hessian Matrices of Scalar Functions	p. 118
Examples of Finding Complex Hessian Matrices of Vector Functions	p. 123
Examples of Finding Complex Hessian Matrices of Matrix Functions	p. 126
Exercises	p. 129
Generalized Complex-Valued Matrix Derivatives	p. 133
Introduction	p. 133
Derivatives of Mixture of Real- and Complex-Valued Matrix Variables	p. 137
Chain Rule for Mixture of Real- and Complex-Valued Matrix Variables	p. 139
Steepest Ascent and Descent Methods for Mixture of Real- and Complex-Valued Matrix Variables	p. 142
Definitions from the Theory of Manifolds	p. 144
Finding Generalized Complex-Valued Matrix Derivatives	p. 147
Manifolds and Parameterization Function	p. 147
Finding the Derivative of H(X, Z, Z*)	p. 152
Finding the Derivative of G(W, W*)	p. 153
Specialization to Unpatterned Derivatives	p. 153
Specialization to Real-Valued Derivatives	p. 154
Specialization to Scalar Function of Square Complex-Valued Matrices	p. 154
Examples of Generalized Complex Matrix Derivatives	p. 157
Generalized Derivative with Respect to Scalar Variables	p. 157
Generalized Derivative with Respect to Vector Variables	p. 160
Generalized Matrix Derivatives with Respect to Diagonal Matrices	p. 163
Generalized Matrix Derivative with Respect to Synunetric Matrices	p. 166
Generalized Matrix Derivative with Respect to Hermitian Matrices	p. 171
Generalized Matrix Derivative with Respect to Skew-Symmetric Matrices	p. 179
Generalized Matrix Derivative with Respect to Skew-Hermitian Matrices	p. 180
orthogonal Matrices	p. 184
Unitary Matrices	p. 185
Positive Semidefinite Matrices	p. 187
Exercises	p. 188
Applications in Signal Processing and Communications	p. 201
Introduction	p. 201
Absolute Value of Fourier Transform Example	p. 201
Special Function and Matrix Definitions	p. 202
Objective Function Formulation	p. 204
First-Order Derivatives of the Objective Function	p. 204
Hessians of the Objective Function	p. 206
Minimization of Off-Diagonal Covariance Matrix Elements	p. 209
MIMO Precoder Design for Coherent Detection	p. 211
Precoded OSTBC System Model	p. 212
Correlated Ricean MIMO Channel Model	p. 213
Equivalent Single-Input Single-Output Model	p. 213
Exact SER Expressions for Precoded OSTBC	p. 214
Precoder Optimization Problem Statement and Optimization Algorithm	p. 216
Optimal Precoder Problem Formulation	p. 216
Precoder Optimization Algorithm	p. 217
Minimum MSE FIR MIMO Transmit and Receive Filters	p. 219
FIR MIMO System Model	p. 220
FIR MIMO Filter Expansions	p. 220
FIR MIMO Transmit and Receive Filter Problems	p. 223
FIR MIMO Receive Filter Optimization	p. 225
FIR MIMO Transmit Filter Optimization	p. 226
Exercises	p. 228
References	p. 231
Index	p. 237
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