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

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 24-Feb-2011
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781139037044
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Complex-Valued Matrix Derivatives: With Applications in Signal Processing and CommunicationsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 24-Feb-2011
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781139037044
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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.

Arvustused

'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 (UFRJ) 'Complex vector and matrix optimization problems are often encountered by researchers in the electrical engineering fields and much beyond. Their solution, which can sometimes be reached from using existing standard algebra literature, may however be a time consuming and sometimes difficult process. This is particularly so when complicated 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 ' an excellent monograph The book is suitable as a textbook in graduate-level studies and research.' Zentralblatt MATH

Muu info

Introduces a powerful set of mathematical tools that can be used to analyze and optimize the performance of engineering systems.
Preface xi
Acknowledgments xiii
Abbreviations xv
Nomenclature xvii
1 Introduction
1(5)
1.1 Introduction to the Book
1(1)
1.2 Motivation for the Book
2(1)
1.3 Brief Literature Summary
3(2)
1.4 Brief Outline
5(1)
2 Background Material
6(37)
2.1 Introduction
6(1)
2.2 Notation and Classification of Complex Variables and Functions
6(2)
2.2.1 Complex-Valued Variables
7(1)
2.2.2 Complex-Valued Functions
7(1)
2.3 Analytic versus Non-Analytic Functions
8(4)
2.4 Matrix-Related Definitions
12(8)
2.5 Useful Manipulation Formulas
20(18)
2.5.1 Moore-Penrose Inverse
23(1)
2.5.2 Trace Operator
24(1)
2.5.3 Kronecker and Hadamard Products
25(4)
2.5.4 Complex Quadratic Forms
29(2)
2.5.5 Results for Finding Generalized Matrix Derivatives
31(7)
2.6 Exercises
38(5)
3 Theory of Complex-Valued Matrix Derivatives
43(27)
3.1 Introduction
43(1)
3.2 Complex Differentials
44(11)
3.2.1 Procedure for Finding Complex Differentials
46(1)
3.2.2 Basic Complex Differential Properties
46(7)
3.2.3 Results Used to Identify First-and Second-Order Derivatives
53(2)
3.3 Derivative with Respect to Complex Matrices
55(5)
3.3.1 Procedure for Finding Complex-Valued Matrix Derivatives
59(1)
3.4 Fundamental Results on Complex-Valued Matrix Derivatives
60(5)
3.4.1 Chain Rule
60(1)
3.4.2 Scalar Real-Valued Functions
61(3)
3.4.3 One Independent Input Matrix Variable
64(1)
3.5 Exercises
65(5)
4 Development of Complex-Valued Derivative Formulas
70(25)
4.1 Introduction
70(1)
4.2 Complex-Valued Derivatives of Scalar Functions
70(12)
4.2.1 Complex-Valued Derivatives of f(z, z)
70(4)
4.2.2 Complex-Valued Derivatives of f(z, z)
74(2)
4.2.3 Complex-Valued Derivatives of f(Z, Z)
76(6)
4.3 Complex-Valued Derivatives of Vector Functions
82(2)
4.3.1 Complex-Valued Derivatives of f(z, z)
82(1)
4.3.2 Complex-Valued Derivatives of f(z, z)
82(1)
4.3.3 Complex-Valued Derivatives of f(Z, Z)
82(2)
4.4 Complex-Valued Derivatives of Matrix Functions
84(7)
4.4.1 Complex-Valued Derivatives of F(z, z)
84(1)
4.4.2 Complex-Valued Derivatives of F(z, z)
85(1)
4.4.3 Complex-Valued Derivatives of F(Z, Z)
86(5)
4.5 Exercises
91(4)
5 Complex Hessian Matrices for Scalar, Vector, and Matrix Functions
95(38)
5.1 Introduction
95(1)
5.2 Alternative Representations of Complex-Valued Matrix Variables
96(3)
5.2.1 Complex-Valued Matrix Variables Z and Z
96(1)
5.2.2 Augmented Complex-Valued Matrix Variables Z
97(2)
5.3 Complex Hessian Matrices of Scalar Functions
99(10)
5.3.1 Complex Hessian Matrices of Scalar Functions Using Z and Z
99(6)
5.3.2 Complex Hessian Matrices of Scalar Functions Using Z
105(2)
5.3.3 Connections between Hessians When Using Two-Matrix Variable Representations
107(2)
5.4 Complex Hessian Matrices of Vector Functions
109(3)
5.5 Complex Hessian Matrices of Matrix Functions
112(6)
5.5.1 Alternative Expression of Hessian Matrix of Matrix Function
117(1)
5.5.2 Chain Rule for Complex Hessian Matrices
117(1)
5.6 Examples of Finding Complex Hessian Matrices
118(11)
5.6.1 Examples of Finding Complex Hessian Matrices of Scalar Functions
118(5)
5.6.2 Examples of Finding Complex Hessian Matrices of Vector Functions
123(3)
5.6.3 Examples of Finding Complex Hessian Matrices of Matrix Functions
126(3)
5.7 Exercises
129(4)
6 Generalized Complex-Valued Matrix Derivatives
133(68)
6.1 Introduction
133(4)
6.2 Derivatives of Mixture of Real- and Complex-Valued Matrix Variables
137(7)
6.2.1 Chain Rule for Mixture of Real- and Complex-Valued Matrix Variables
139(3)
6.2.2 Steepest Ascent and Descent Methods for Mixture of Real- and Compl ex-Valued Matrix Vari ables
142(2)
6.3 Definitions from the Theory of Manifolds
144(3)
6.4 Finding Generalized Complex-Valued Matrix Derivatives
147(10)
6.4.1 Manifolds and Parameterization Function
147(5)
6.4.2 Finding the Derivative of H(X, Z, Z)
152(1)
6.4.3 Finding the Derivative of G(W, W)
153(1)
6.4.4 Specialization to Unpatterned Derivatives
153(1)
6.4.5 Specialization to Real-Valued Derivatives
154(1)
6.4.6 Specialization to Scalar Function of Square Complex-Valued Matrices
154(3)
6.5 Examples of Generalized Complex Matrix Derivatives
157(31)
6.5.1 Generalized Derivative with Respect to Scalar Variables
157(3)
6.5.2 Generalized Derivative with Respect to Vector Variables
160(3)
6.5.3 Generalized Matrix Derivatives with Respect to Diagonal Matrices
163(3)
6.5.4 Generalized Matrix Derivative with Respect to Symmetric Matrices
166(5)
6.5.5 Generalized Matrix Derivative with Respect to Hermitian Matrices
171(8)
6.5.6 Generalized Matrix Derivative with Respect to Skew-Symmetric Matrices
179(1)
6.5.7 Generalized Matrix Derivative with Respect to Skew-Hermitian Matrices
180(4)
6.5.8 Orthogonal Matrices
184(1)
6.5.9 Unitary Matrices
185(2)
6.5.10 Positive Semidennite Matrices
187(1)
6.6 Exercises
188(13)
7 Applications in Signal Processing and Communications
201(30)
7.1 Introduction
201(1)
7.2 Absolute Value of Fourier Transform Example
201(8)
7.2.1 Special Function and Matrix Definitions
202(2)
7.2.2 Objective Function Formulation
204(1)
7.2.3 First-Order Derivatives of the Objective Function
204(2)
7.2.4 Hessians of the Objective Function
206(3)
7.3 Minimization of Off-Diagonal Covariance Matrix Elements
209(2)
7.4 MIMO Precoder Design for Coherent Detection
211(8)
7.4.1 Precoded OSTBC System Model
212(1)
7.4.2 Correlated Ricean MIMO Channel Model
213(1)
7.4.3 Equivalent Single-Input Single-Output Model
213(1)
7.4.4 Exact SER Expressions for Precoded OSTBC
214(2)
7.4.5 Precoder Optimization Problem Statement and Optimization Algorithm
216(1)
7.4.5.1 Optimal Precoder Problem Formulation
216(1)
7.4.5.2 Precoder Optimization Algorithm
217(2)
7.5 Minimum MSE FIR MIMO Transmit and Receive Filters
219(9)
7.5.1 FIR MIMO System Model
220(1)
7.5.2 FIR MIMO Filter Expansions
220(3)
7.5.3 FIR MIMO Transmit and Receive Filter Problems
223(2)
7.5.4 FIR MIMO Receive Filter Optimization
225(1)
7.5.5 FIR MIMO Transmit Filter Optimization
226(2)
7.6 Exercises
228(3)
References 231(6)
Index 237
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.
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