Synopses & Reviews
Blind identification consists of estimating a multidimensional system through the use of only its output. Source separation is concerned with the blind estimation of the inverse of the system. The estimation is generally performed by using different statistics of the outputs.
The authors consider the blind estimation of a multiple input/multiple output (MIMO) system that mixes a number of underlying signals of interest called sources. They also consider the case of direct estimation of the inverse system for the purpose of source separation. They then describe the estimation theory associated with the identifiability conditions and dedicated algebraic algorithms. The algorithms depend critically on (statistical and/or time frequency) properties of complex sources that will be precisely described.
Synopsis
Blind Identification and Separation of Complex-valued Signals introduces the reader to the blind estimation of a multiple input/multiple output (MIMO) system, showing how it mixes a number of underlying signals of interest called sources. In addition, it deals with the case of direct estimation of the inverse system for the purpose of source separation. The book describes the estimation theory associated with the identifiability conditions and dedicated algebraic algorithms. The algorithms depend critically on statistical and/or time-frequency properties of complex sources and these are also described precisely.
Synopsis
Blind identification consists of estimating a multidimensional system through the use of only its output. Source separation is concerned with the blind estimation of the inverse of the system. The estimation is generally performed by using different statistics of the outputs.
The authors consider the blind estimation of a multiple input/multiple output (MIMO) system that mixes a number of underlying signals of interest called sources. They also consider the case of direct estimation of the inverse system for the purpose of source separation. They then describe the estimation theory associated with the identifiability conditions and dedicated algebraic algorithms. The algorithms depend critically on (statistical and/or time frequency) properties of complex sources that will be precisely described.
Table of Contents
PREFACE ix
ACKNOWLEDGMENTS xi
CHAPTER 1. MATHEMATICAL PRELIMINARIES 1
1.1. Introduction 1
1.2. Linear mixing model 1
1.3. Problem definition 3
1.4. Statistics 4
1.4.1. Statistics of random variables and random vectors 4
1.4.2. Differential entropy of complex random vectors 7
1.4.3. Statistics of random processes 7
1.4.4. Complex matrix decompositions 11
1.5. Optimization: Wirtinger calculus 13
1.5.1. Scalar case 14
1.5.2. Vector case 18
1.5.3. Matrix case 23
1.5.4. Summary 25
CHAPTER 2. ESTIMATION BY JOINT DIAGONALIZATION 27
2.1. Introduction 27
2.2. Normalization, dimension reduction and whitening 27
2.2.1. Dimension reduction 28
2.2.2. Whitening 30
2.3. Exact joint diagonalization of two matrices 31
2.3.1. After the whitening stage 31
2.3.2. Without explicit whitening 33
2.4. Unitary approximate joint diagonalization 35
2.4.1. Considered problem 35
2.4.2. The 2 × 2 Hermitian case 38
2.4.3. The 2 × 2 complex symmetric case 40
2.5. General approximate joint diagonalization 42
2.5.1. Considered problem 42
2.5.2. A relative gradient algorithm 44
2.6. Summary 45
CHAPTER 3. MAXIMUM LIKELIHOOD ICA 47
3.1. Introduction 47
3.2. Cost function choice 48
3.2.1. Mutual information and mutual information rate minimization 49
3.2.2. Maximum likelihood 52
3.2.3. Identifiability of the complex ICA model 53
3.3. Algorithms 57
3.3.1. ML ICA: unconstrained W 57
3.3.2. Complex maximization of non-Gaussianity: ML ICA with unitary W 63
3.3.3. Density matching 67
3.3.4. A flexible complex ICA algorithm: Entropy bound minimization 75
3.4. Summary 81
BIBLIOGRAPHY 83
INDEX 93