In this complete introduction to the theory of finding derivatives of scalar-, vector- and matrix-valued functions with respect to complex matrix variables, Hjørungnes 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(s): Are Hjorungnes
Publisher: Cambridge University Press
Year: 2011
Language: English
Pages: 271
Tags: Приборостроение;Обработка сигналов;
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
Book Overview......Page 14
Acknowledgments......Page 15
Abbreviations......Page 17
Nomenclature......Page 19
1.1 Introduction to the Book......Page 25
1.2 Motivation for the Book......Page 26
1.3 Brief Literature Summary......Page 27
1.4 Brief Outline......Page 29
2.2 Notation and Classification of Complex Variables and Functions......Page 30
2.2.2 Complex-Valued Functions......Page 31
2.3 Analytic versus Non-Analytic Functions......Page 32
2.4 Matrix-Related Definitions......Page 36
2.5 Useful Manipulation Formulas......Page 44
2.5.1 Moore-Penrose Inverse......Page 47
2.5.2 Trace Operator......Page 48
2.5.3 Kronecker and Hadamard Products......Page 49
2.5.4 Complex Quadratic Forms......Page 53
2.5.5 Results for Finding Generalized Matrix Derivatives......Page 55
2.6 Exercises......Page 62
3.1 Introduction......Page 67
3.2 Complex Differentials......Page 68
3.2.2 Basic Complex Differential Properties......Page 70
3.2.3 Results Used to Identify First- and Second-Order Derivatives......Page 77
3.3 Derivative with Respect to Complex Matrices......Page 79
3.3.1 Procedure for Finding Complex-Valued Matrix Derivatives......Page 83
3.4.1 Chain Rule......Page 84
3.4.2 Scalar Real-Valued Functions......Page 85
3.4.3 One Independent Input Matrix Variable......Page 88
3.5 Exercises......Page 89
4.2.1 Complex-Valued Derivatives of f(z, z*)......Page 94
4.2.2 Complex-Valued Derivatives of f(z, z*)......Page 98
4.2.3 Complex-Valued Derivatives of f(Z, Z*)......Page 100
4.3.3 Complex-Valued Derivatives of f(Z, Z*)......Page 106
4.4.1 Complex-Valued Derivatives of F(z, z*)......Page 108
4.4.2 Complex-Valued Derivatives of F(z, z*)......Page 109
4.4.3 Complex-Valued Derivatives of F(Z, Z*)......Page 110
4.5 Exercises......Page 115
5.1 Introduction......Page 119
5.2.1 Complex-Valued Matrix Variables Z and Z*......Page 120
5.2.2 Augmented Complex-Valued Matrix Variables Z......Page 121
5.3.1 Complex Hessian Matrices of Scalar Functions Using Z and Z*......Page 123
5.3.2 Complex Hessian Matrices of Scalar Functions Using Z......Page 129
5.3.3 Connections between Hessians When Using Two-Matrix Variable Representations......Page 131
5.4 Complex Hessian Matrices of Vector Functions......Page 133
5.5 Complex Hessian Matrices of Matrix Functions......Page 136
5.5.2 Chain Rule for Complex Hessian Matrices......Page 141
5.6.1 Examples of Finding Complex Hessian Matrices of Scalar Functions......Page 142
5.6.2 Examples of Finding Complex Hessian Matrices of Vector Functions......Page 147
5.6.3 Examples of Finding Complex Hessian Matrices of Matrix Functions......Page 150
5.7 Exercises......Page 153
6.1 Introduction......Page 157
6.2 Derivatives of Mixture of Real- and Complex-Valued Matrix Variables......Page 161
6.2.1 Chain Rule for Mixture of Real- and Complex-Valued Matrix Variables......Page 163
6.2.2 Steepest Ascent and Descent Methods for Mixture of Real- and Complex-Valued Matrix Variables......Page 166
6.3 Definitions from the Theory of Manifolds......Page 168
6.4.1 Manifolds and Parameterization Function......Page 171
6.4.2 Finding the Derivative of H(X, Z, Z*)......Page 176
6.4.4 Specialization to Unpatterned Derivatives......Page 177
6.4.6 Specialization to Scalar Function of Square Complex-Valued Matrices......Page 178
6.5.1 Generalized Derivative with Respect to Scalar Variables......Page 181
6.5.2 Generalized Derivative with Respect to Vector Variables......Page 184
6.5.3 Generalized Matrix Derivatives with Respect to Diagonal Matrices......Page 187
6.5.4 Generalized Matrix Derivative with Respect to Symmetric Matrices......Page 190
6.5.5 Generalized Matrix Derivative with Respect to Hermitian Matrices......Page 195
6.5.6 Generalized Matrix Derivative with Respect to Skew-Symmetric Matrices......Page 203
6.5.7 Generalized Matrix Derivative with Respect to Skew-Hermitian Matrices......Page 204
6.5.8 Orthogonal Matrices......Page 208
6.5.9 Unitary Matrices......Page 209
6.5.10 Positive Semidefinite Matrices......Page 211
6.6 Exercises......Page 212
7.2 Absolute Value of Fourier Transform Example......Page 225
7.2.1 Special Function and Matrix Definitions......Page 226
7.2.3 First-Order Derivatives of the Objective Function......Page 228
7.2.4 Hessians of the Objective Function......Page 230
7.3 Minimization of Off-Diagonal Covariance Matrix Elements......Page 233
7.4 MIMO Precoder Design for Coherent Detection......Page 235
7.4.1 Precoded OSTBC System Model......Page 236
7.4.3 Equivalent Single-Input Single-Output Model......Page 237
7.4.4 Exact SER Expressions for Precoded OSTBC......Page 238
7.4.5.1 Optimal Precoder Problem Formulation......Page 240
7.4.5.2 Precoder Optimization Algorithm......Page 241
7.5 Minimum MSE FIR MIMO Transmit and Receive Filters......Page 243
7.5.2 FIR MIMO Filter Expansions......Page 244
7.5.3 FIR MIMO Transmit and Receive Filter Problems......Page 247
7.5.4 FIR MIMO Receive Filter Optimization......Page 249
7.5.5 FIR MIMO Transmit Filter Optimization......Page 250
7.6 Exercises......Page 252
References......Page 255
Index......Page 261
