Blind Identification and Separation of Complex-valued Signals
Published on 2. October 2013
112 pages
978-1-118-57973-2 (ISBN)
Description
Blind identification consists of estimating a multi-dimensionalsystem only through the use of its output, and source separation,the blind estimation of the inverse of the system. Estimation isgenerally carried out using different statistics of theoutput.
The authors of this book consider the blind identification andsource separation problem in the complex-domain, where theavailable statistical properties are richer and includenon-circularity of the sources - underlying components. Theydefine identifiability conditions and present state-of-the-artalgorithms that are based on algebraic methods as well as iterativealgorithms based on maximum likelihood theory.
Contents
1. Mathematical Preliminaries.
2. Estimation by Joint Diagonalization.
3. Maximum Likelihood ICA.
About the Authors
Eric Moreau is Professor of Electrical Engineering at theUniversity of Toulon, France. His research interests concernstatistical signal processing, high order statistics andmatrix/tensor decompositions with applications to data analysis,telecommunications and radar.
Tülay Adali is Professor of Electrical Engineering andDirector of the Machine Learning for Signal Processing Laboratoryat the University of Maryland, Baltimore County, USA. Her researchinterests concern statistical and adaptive signal processing, withan emphasis on nonlinear and complex-valued signal processing, andapplications in biomedical data analysis and communications.Blind identification consists of estimating a multidimensionalsystem through the use of only its output. Source separation isconcerned with the blind estimation of the inverse of the system.The estimation is generally performed by using different statisticsof the outputs.
The authors consider the blind estimation of a multipleinput/multiple output (MIMO) system that mixes a number ofunderlying signals of interest called sources. They alsoconsider the case of direct estimation of the inverse system forthe purpose of source separation. They then describe the estimationtheory associated with the identifiability conditions and dedicatedalgebraic algorithms. The algorithms depend critically on(statistical and/or time frequency) properties of complex sourcesthat will be precisely described.
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10/2013
Wiley
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Eric Moreau | Tulay Adali
Blind Identification and Separation of Complex-valued Signals
Book
09/2013
1st Edition
Wiley-ISTE
€206.42
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Persons
Eric Moreau is Professor, University of Sud Toulon Var, France.
Ms. Tulay Adali is Professor at University of Maryland, Baltimore County, USA.
Content
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 rateminimization 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 withunitary W 63
3.3.3. Density matching 67
3.3.4. A flexible complex ICA algorithm: Entropy boundminimization 75
3.4. Summary 81
BIBLIOGRAPHY 83
INDEX 9
