Complex-valued matrix derivatives : with applications in signal processing and communications

Select an Action

Place Hold(s)
Add to My Lists
Email
Print

Title:

Personal Author:

Hjorungnes, Are

Publication Information:

Physical Description:

xxi, 247 pages : illustrations ; 25 cm.

ISBN:

9780521192644

Abstract:

"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"-- Provided by publisher.

Subject Term:

Matrix derivatives

Systems engineering

Signal processing -- Mathematical models

Telecommunication -- Mathematical models

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010297754	TA347.D4 H56 2011	Open Access Book	Book	Searching... Unknown

On Order

Summary

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 Notes

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.

Preface	p. xi
Acknowledgments	p. xiii
Abbreviations	p. xv
Nomenclature	p. xvii
1 Introduction	p. 1
1.1 Introduction to the Book	p. 1
1.2 Motivation for the Book	p. 2
1.3 Brief Literature Summary	p. 3
1.4 Brief Outline	p. 5
2 Background Material	p. 6
2.1 Introduction	p. 6
2.2 Notation and Classification of Complex Variables and Functions	p. 6
2.2.1 Complex-Valued Variables	p. 7
2.2.2 Complex-Valued Functions	p. 7
2.3 Analytic versus Non-Analytic Functions	p. 8
2.4 Matrix-Related Definitions	p. 12
2.5 Useful Manipulation Formulas	p. 20
2.5.1 Moore-Penrose Inverse	p. 23
2.5.2 Trace Operator	p. 24
2.5.3 Kronecker and Hadamard Products	p. 25
2.5.4 Complex Quadratic Forms	p. 29
2.5.5 Results for Finding Generalized Matrix Derivatives	p. 31
2.6 Exercises	p. 38
3 Theory of Complex-Valued Matrix Derivatives	p. 43
3.1 Introduction	p. 43
3.2 Complex Differentials	p. 44
3.2.1 Procedure for Finding Complex Differentials	p. 46
3.2.2 Basic Complex Differential Properties	p. 46
3.2.3 Results Used to Identify First- and Second-Order Derivatives	p. 53
3.3 Derivative with Respect to Complex Matrices	p. 55
3.3.1 Procedure for Finding Complex-Valued Matrix Derivatives	p. 59
3.4 Fundamental Results on Complex-Valued Matrix Derivatives	p. 60
3.4.1 Chain Rule	p. 60
3.4.2 Scalar Real-Valued Functions	p. 61
3.4.3 One Independent Input Matrix Variable	p. 64
3.5 Exercises	p. 65
4 Development of Complex-Valued Derivative Formulas	p. 70
4.1 Introduction	p. 70
4.2 Complex-Valued Derivatives of Scalar Functions	p. 70
4.2.1 Complex-Valued Derivatives of f(z, z*)	p. 70
4.2.2 Complex-Valued Derivatives of f(z, z*)	p. 74
4.2.3 Complex-Valued Derivatives of f(Z, Z*)	p. 76
4.3 Complex-Valued Derivatives of Vector Functions	p. 82
4.3.1 Complex-Valued Derivatives of f(z, z*)	p. 82
4.3.2 Complex-Valued Derivatives of f(z, z*)	p. 82
4.3.3 Complex-Valued Derivatives of f(Z, Z*)	p. 82
4.4 Complex-Valued Derivatives of Matrix Functions	p. 84
4.4.1 Complex-Valued Derivatives of F(z, z*)	p. 84
4.4.2 Complex-Valued Derivatives of F(z, z*)	p. 85
4.4.3 Complex-Valued Derivatives of F(Z, Z*)	p. 86
4.5 Exercises	p. 91
5 Complex Hessian Matrices for Scalar, Vector, and Matrix Functions	p. 95
5.1 Introduction	p. 95
5.2 Alternative Representations of Complex-Valued Matrix Variables	p. 96
5.2.1 Complex-Valued Matrix Variables Z and Z*	p. 96
5.2.2 Augmented Complex-Valued Matrix Variables Z	p. 97
5.3 Complex Hessian Matrices of Scalar Functions	p. 99
5.3.1 Complex Hessian Matrices of Scalar Functions Using Z and Z*	p. 99
5.3.2 Complex Hessian Matrices of Scalar Functions Using Z	p. 105
5.3.3 Connections between Hessians When Using Two-Matrix Variable Representations	p. 107
5.4 Complex Hessian Matrices of Vector Functions	p. 109
5.5 Complex Hessian Matrices ofMatrixFunctions	p. 112
5.5.1 Alternative Expression of Hessian Matrix of Matrix Function	p. 117
5.5.2 Chain Rule for Complex Hessian Matrices	p. 117
5.6 Examples of Finding Complex Hessian Matrices	p. 118
5.6.1 Examples of Finding Complex Hessian Matrices of Scalar Functions	p. 118
5.6.2 Examples of Finding Complex Hessian Matrices of Vector Functions	p. 123
5.6.3 Examples of Finding Complex Hessian Matrices of Matrix Functions	p. 126
5.7 Exercises	p. 129
6 Generalized Complex-Valued Matrix Derivatives	p. 133
6.1 Introduction	p. 133
6.2 Derivatives of Mixture of Real- and Complex-Valued Matrix Variables	p. 137
6.2.1 Chain Rule for Mixture of Real- and Complex-Valued Matrix Variables	p. 139
6.2.2 Steepest Ascent and Descent Methods for Mixture of Real- and Complex-Valued Matrix Variables	p. 142
6.3 Definitions from the Theory of Manifolds	p. 144
6.4 Finding Generalized Complex-Valued Matrix Derivatives	p. 147
6.4.1 Manifolds and Parameterization Function	p. 147
6.4.2 Finding the Derivative of H(X, Z, Z*)	p. 152
6.4.3 Finding the Derivative of G(W, W*)	p. 153
6.4.4 Specialization to Unpatterned Derivatives	p. 153
6.4.5 Specialization to Real-Valued Derivatives	p. 154
6.4.6 Specialization to Scalar Function of Square Complex-Valued Matrices	p. 154
6.5 Examples of Generalized Complex Matrix Derivatives	p. 157
6.5.1 Generalized Derivative with Respect to Scalar Variables	p. 157
6.5.2 Generalized Derivative with Respect to Vector Variables	p. 160
6.5.3 Generalized Matrix Derivatives with Respect to Diagonal Matrices	p. 163
6.5.4 Generalized Matrix Derivative with Respect to Synunetric Matrices	p. 166
6.5.5 Generalized Matrix Derivative with Respect to Hermitian Matrices	p. 171
6.5.6 Generalized Matrix Derivative with Respect to Skew-Symmetric Matrices	p. 179
6.5.7 Generalized Matrix Derivative with Respect to Skew-Hermitian Matrices	p. 180
6.5.8 orthogonal Matrices	p. 184
6.5.9 Unitary Matrices	p. 185
6.5.10 Positive Semidefinite Matrices	p. 187
6.6 Exercises	p. 188
7 Applications in Signal Processing and Communications	p. 201
7.1 Introduction	p. 201
7.2 Absolute Value of Fourier Transform Example	p. 201
7.2.1 Special Function and Matrix Definitions	p. 202
7.2.2 Objective Function Formulation	p. 204
7.2.3 First-Order Derivatives of the Objective Function	p. 204
7.2.4 Hessians of the Objective Function	p. 206
7.3 Minimization of Off-Diagonal Covariance Matrix Elements	p. 209
7.4 MIMO Precoder Design for Coherent Detection	p. 211
7.4.1 Precoded OSTBC System Model	p. 212
7.4.2 Correlated Ricean MIMO Channel Model	p. 213
7.4.3 Equivalent Single-Input Single-Output Model	p. 213
7.4.4 Exact SER Expressions for Precoded OSTBC	p. 214
7.4.5 Precoder Optimization Problem Statement and Optimization Algorithm	p. 216
7.4.5.1 Optimal Precoder Problem Formulation	p. 216
7.4.5.2 Precoder Optimization Algorithm	p. 217
7.5 Minimum MSE FIR MIMO Transmit and Receive Filters	p. 219
7.5.1 FIR MIMO System Model	p. 220
7.5.2 FIR MIMO Filter Expansions	p. 220
7.5.3 FIR MIMO Transmit and Receive Filter Problems	p. 223
7.5.4 FIR MIMO Receive Filter Optimization	p. 225
7.5.5 FIR MIMO Transmit Filter Optimization	p. 226
7.6 Exercises	p. 228
References	p. 231
Index	p. 237

Available:*

On Order

Summary

Summary

Author Notes

Table of Contents