Complex-Valued Matrix Derivatives

Transcription

1 Complex-Valued Matrix Derivatives With Applications in Signal Processing and Communications ARE HJ0RUNGNES University of Oslo, Norway Я CAMBRIDGE UNIVERSITY PRESS

2 Preface Acknowledgments Abbreviations Nomenclature page xi xiii xv xvii Introduction Introduction to the Book Motivation for the Book Brief Literature Summary Brief Outline 5 Background Material Introduction Notation and Classification of Complex Variables and Functions Complex-Valued Variables Complex-Valued Functions Analytic versus Non-Analytic Functions Matrix-Related Definitions Useful Manipulation Formulas Moore-Penrose Inverse Trace Operator Kronecker and Hadamard Products Complex Quadratic Forms Results for Finding Generalized Matrix Derivatives Exercises 38 Theory of Complex-Valued Matrix Derivatives Introduction Complex Differentials Procedure for Finding Complex Differentials Basic Complex Differential Properties Results Used to Identify First-and Second-Order Derivatives 53

3 viii 3.3 Derivative with Respect to Complex Matrices Procedure for Finding Complex-Valued Matrix Derivatives Fundamental Results on Complex-Valued Matrix Derivatives Chain Rule Scalar Real-Valued Functions One Independent Input Matrix Variable Exercises 65 Development of Complex-Valued Derivative Formulas Introduction Complex-Valued Derivatives of Scalar Functions Complex-Valued Derivatives of f(z,z*) Complex-Valued Derivatives of f(z, z*) Complex-Valued Derivatives of f(z,z*) Complex-Valued Derivatives of Vector Functions Complex-Valued Derivatives of f(z, z*) Complex-Valued Derivatives of f(z, z*) Complex-Valued Derivatives of f{z,z*) Complex-Valued Derivatives of Matrix Functions Complex-Valued Derivatives of F(z,z*) Complex-Valued Derivatives of F(z,z*) Complex-Valued Derivatives of F(Z, Z*) Exercises 91 Complex Hessian Matrices for Scalar, Vector, and Matrix Functions Introduction Alternative Representations of Complex-Valued Matrix Variables Complex-Valued Matrix Variables Z and Z* Augmented Complex-Valued Matrix Variables Z Complex Hessian Matrices of Scalar Functions Complex Hessian Matrices of Scalar Functions Using Z and Z* Complex Hessian Matrices of Scalar Functions Using Z Connections between Hessians When Using Two-Matrix Variable Representations Complex Hessian Matrices of Vector Functions Complex Hessian Matrices of Matrix Functions Alternative Expression of Hessian Matrix of Matrix Function Chain Rule for Complex Hessian Matrices Examples of Finding Complex Hessian Matrices Examples of Finding Complex Hessian Matrices of Scalar Functions Examples of Finding Complex Hessian Matrices of Vector Functions 123

4 ix Examples of Finding Complex Hessian Matrices of Matrix Functions Exercises 129 Generalized Complex-Valued Matrix Derivatives Introduction Derivatives of Mixture of Real- and Complex-Valued Matrix Variables Chain Rule for Mixture of Real- and Complex-Valued Matrix Variables Steepest Ascent and Descent Methods for Mixture of Real- and Complex-Valued Matrix Variables Definitions from the Theory of Manifolds Finding Generalized Complex-Valued Matrix Derivatives Manifolds and Parameterization Function Finding the Derivative of H(X, Z,Z*) Finding the Derivative of G(W, W*) Specialization to Unpatterned Derivatives Specialization to Real-Valued Derivatives Specialization to Scalar Function of Square Complex-Valued Matrices Examples of Generalized Complex Matrix Derivatives Generalized Derivative with Respect to Scalar Variables Generalized Derivative with Respect to Vector Variables Generalized Matrix Derivatives with Respect to Diagonal Matrices Generalized Matrix Derivative with Respect to Symmetric Matrices Generalized Matrix Derivative with Respect to Hermitian Matrices Generalized Matrix Derivative with Respect to Skew-Symmetric Matrices Generalized Matrix Derivative with Respect to Skew-Hermitian Matrices Orthogonal Matrices Unitary Matrices Positive Semidefmite Matrices Exercises 188 Applications in Signal Processing and Communications Introduction Absolute Value of Fourier Transform Example Special Function and Matrix Definitions Objective Function Formulation 204

5 x First-Order Derivatives of the Objective Function Hessians of the Objective Function Minimization of Off-Diagonal Covariance Matrix Elements MIMO Precoder Design for Coherent Detection Precoded OSTBC System Model Correlated Ricean MIMO Channel Model Equivalent Single-Input Single-Output Model Exact SER Expressions for Precoded OSTBC Precoder Optimization Problem Statement and Optimization Algorithm Optimal Precoder Problem Formulation Precoder Optimization Algorithm Minimum MSE FIR MIMO Transmit and Receive Filters FIR MIMO System Model FIR MIMO Filter Expansions FIR MIMO Transmit and Receive Filter Problems FIR MIMO Receive Filter Optimization FIR MIMO Transmit Filter Optimization Exercises 228 References 231 Index Л 237