Complex-valued_matrix_derivatives.pdf

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Cover

Half-title

Title

Dedication

Contents

Preface

Book Overview

Acknowledgments

Abbreviations

Nomenclature

1 Introduction

1.1 Introduction to the Book

1.2 Motivation for the Book

1.3 Brief Literature Summary

1.4 Brief Outline

2 Background Material

2.1 Introduction

2.2 Notation and Classification of Complex Variables and Functions

2.2.1 Complex-Valued Variables

2.2.2 Complex-Valued Functions

2.3 Analytic versus Non-Analytic Functions

2.4 Matrix-Related Definitions

2.5 Useful Manipulation Formulas

2.5.1 Moore-Penrose Inverse

2.5.2 Trace Operator

2.5.3 Kronecker and Hadamard Products

2.5.4 Complex Quadratic Forms

2.5.5 Results for Finding Generalized Matrix Derivatives

2.6 Exercises

3 Theory of Complex-Valued Matrix Derivatives

3.1 Introduction

3.2 Complex Differentials

3.2.1 Procedure for Finding Complex Differentials

3.2.2 Basic Complex Differential Properties

3.2.3 Results Used to Identify First- and Second-Order Derivatives

3.3 Derivative with Respect to Complex Matrices

3.3.1 Procedure for Finding Complex-Valued Matrix Derivatives

3.4 Fundamental Results on Complex-Valued Matrix Derivatives

3.4.1 Chain Rule

3.4.2 Scalar Real-Valued Functions

3.4.3 One Independent Input Matrix Variable

3.5 Exercises

4 Development of Complex-Valued Derivative Formulas

4.1 Introduction

4.2 Complex-Valued Derivatives of Scalar Functions

4.2.1 Complex-Valued Derivatives of f(z, z*)

4.2.2 Complex-Valued Derivatives of f(z, z*)

4.2.3 Complex-Valued Derivatives of f(Z, Z*)

4.3 Complex-Valued Derivatives of Vector Functions

4.3.1 Complex-Valued Derivatives of f(z, z*)

4.3.2 Complex-Valued Derivatives of f(z, z*)

4.3.3 Complex-Valued Derivatives of f(Z, Z*)

4.4 Complex-Valued Derivatives of Matrix Functions

4.4.1 Complex-Valued Derivatives of F(z, z*)

4.4.2 Complex-Valued Derivatives of F(z, z*)

4.4.3 Complex-Valued Derivatives of F(Z, Z*)

4.5 Exercises

5 Complex Hessian Matrices for Scalar, Vector, and Matrix Functions

5.1 Introduction

5.2 Alternative Representations of Complex-Valued Matrix Variables

5.2.1 Complex-Valued Matrix Variables Z and Z*

5.2.2 Augmented Complex-Valued Matrix Variables Z

5.3 Complex Hessian Matrices of Scalar Functions

5.3.1 Complex Hessian Matrices of Scalar Functions Using Z and Z*

5.3.2 Complex Hessian Matrices of Scalar Functions Using Z

5.3.3 Connections between Hessians When Using Two-Matrix Variable Representations

5.4 Complex Hessian Matrices of Vector Functions

5.5 Complex Hessian Matrices of Matrix Functions

5.5.1 Alternative Expression of Hessian Matrix of Matrix Function

5.5.2 Chain Rule for Complex Hessian Matrices

5.6 Examples of Finding Complex Hessian Matrices

5.6.1 Examples of Finding Complex Hessian Matrices of Scalar Functions

5.6.2 Examples of Finding Complex Hessian Matrices of Vector Functions

5.6.3 Examples of Finding Complex Hessian Matrices of Matrix Functions

5.7 Exercises

6 Generalized Complex-Valued Matrix Derivatives

6.1 Introduction

6.2 Derivatives of Mixture of Real- and Complex-Valued Matrix Variables

6.2.1 Chain Rule for Mixture of Real- and Complex-Valued Matrix Variables

6.2.2 Steepest Ascent and Descent Methods for Mixture of Real- and Complex-Valued Matrix Variables

6.3 Definitions from the Theory of Manifolds

6.4 Finding Generalized Complex-Valued Matrix Derivatives

6.4.1 Manifolds and Parameterization Function

6.4.2 Finding the Derivative of H(X, Z, Z*)

6.4.3 Finding the Derivative of G(W, W*)

6.4.4 Specialization to Unpatterned Derivatives

6.4.5 Specialization to Real-Valued Derivatives

6.4.6 Specialization to Scalar Function of Square Complex-Valued Matrices

6.5 Examples of Generalized Complex Matrix Derivatives

6.5.1 Generalized Derivative with Respect to Scalar Variables

6.5.2 Generalized Derivative with Respect to Vector Variables

6.5.3 Generalized Matrix Derivatives with Respect to Diagonal Matrices

6.5.4 Generalized Matrix Derivative with Respect to Symmetric Matrices

6.5.5 Generalized Matrix Derivative with Respect to Hermitian Matrices

6.5.6 Generalized Matrix Derivative with Respect to Skew-Symmetric Matrices

6.5.7 Generalized Matrix Derivative with Respect to Skew-Hermitian Matrices

6.5.8 Orthogonal Matrices

6.5.9 Unitary Matrices

6.5.10 Positive Semidefinite Matrices

6.6 Exercises

7 Applications in Signal Processing and Communications

7.1 Introduction

7.2 Absolute Value of Fourier Transform Example

7.2.1 Special Function and Matrix Definitions

7.2.2 Objective Function Formulation

7.2.3 First-Order Derivatives of the Objective Function

7.2.4 Hessians of the Objective Function

7.3 Minimization of Off-Diagonal Covariance Matrix Elements

7.4 MIMO Precoder Design for Coherent Detection

7.4.1 Precoded OSTBC System Model

7.4.2 Correlated Ricean MIMO Channel Model

7.4.3 Equivalent Single-Input Single-Output Model

7.4.4 Exact SER Expressions for Precoded OSTBC

7.4.5 Precoder Optimization Problem Statement and Optimization Algorithm

7.4.5.1 Optimal Precoder Problem Formulation

7.4.5.2 Precoder Optimization Algorithm

7.5 Minimum MSE FIR MIMO Transmit and Receive Filters

7.5.1 FIR MIMO System Model

7.5.2 FIR MIMO Filter Expansions

7.5.3 FIR MIMO Transmit and Receive Filter Problems

7.5.4 FIR MIMO Receive Filter Optimization

7.5.5 FIR MIMO Transmit Filter Optimization

7.6 Exercises

References

Index

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Complex-Valued Matrix Derivatives In this complete introduction to the theory of ﬁnding derivatives of scalar-, vector-, and matrix-valued functions in relation 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. Self-contained and easy to follow, this singular reference uses numerous practical examples from signal process- ing and communications to demonstrate how these tools can be used to analyze and optimize the performance of engineering systems. This is the ﬁrst book on complex- valued matrix derivatives from an engineering perspective. It covers both unpatterned and patterned matrices, uses the latest research examples to illustrate concepts, and includes applications in a range of areas, such as wireless communications, control the- ory, adaptive ﬁltering, resource management, and digital signal processing. The book includes eighty-one end-of-chapter exercises and a complete solutions manual (available on the Web). Are Hjørungnes 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 Communications.

This book addresses the problem of complex-valued derivatives in a wide range of contexts. The mathematical presentation is rigorous but its structured and comprehensive presentation makes the information easily accessible. Clearly, it is an invaluable reference to researchers, professionals and students dealing with functions of complex-valued matrices that arise frequently in many different areas. Throughout the book the examples and exercises help the reader learn how to apply the results presented in the propositions, lemmas and theorems. In conclusion, this book provides a well organized, easy to read, authoritative and unique presentation that everyone looking to exploit complex functions should have available in their own shelves and libraries. Professor Paulo S. R. Diniz, Federal University of Rio de Janeiro Complex vector and matrix optimization problems are often encountered by researchers in the electrical engineering ﬁelds and much beyond. Their solution, which can some- times be reached from using existing standard algebra literature, may however be a time consuming and sometimes difﬁcult process. This is particularly so when compli- cated cost function and constraint expressions arise. This book brings together several mathematical theories in a novel manner to offer a beautifully uniﬁed and systematic methodology for approaching such problems. It will no doubt be a great companion to many researchers and engineers alike. Professor David Gesbert, EURECOM, Sophia-Antipolis, France

Complex-Valued Matrix Derivatives With Applications in Signal Processing and Communications ARE HJØR UNGNES University of Oslo, Norway

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521192644 C Cambridge University Press 2011 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2011 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library. Library of Congress Cataloguing in Publication data Hjørungnes, Are. Complex-Valued Matrix Derivatives : With Applications in Signal Processing and Communications / Are Hjørungnes. p. cm. Includes bibliographical references and index. ISBN 978-0-521-19264-4 (hardback) 1. Matrix derivatives. 2. Systems engineering. 3. Signal processing – Mathematical models. 4. Telecommunication – Mathematical models. TA347.D4H56 621.382 2010046598 I. Title. 2011 2 – dc22 ISBN 978-0-521-19264-4 Hardback Additional resources for this publication at www.cambridge.org/hjorungnes Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To my parents, Tove and Odd

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