MIMO Instantaneous Blind Identiﬁcation and Separation based on Arbitrary Order Temporal Structure in the Data

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MIMO Instantaneous Blind Identification and

Separation based on Arbitrary Order Temporal

Structure in the Data

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de

Technische Universiteit Eindhoven, op gezag van de

Rector Magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor

Promoties in het openbaar te verdedigen

op woensdag 21 november 2007 om 16.00 uur

door

Jakob van de Laar

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prof.dr.ir. J.W.M. Bergmans

en

prof.dr.ir. M. Moonen

Copromotor:

dr.ir. P.C.W. Sommen

c

°

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission from the copyright owner.

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Laar, Jakob van de

MIMO Instantaneous Blind Identification and Separation based on Arbitrary Order Temporal Structure in the Data / by Jakob van de Laar. - Eindhoven : Technische Universiteit Eindhoven, 2007.

Proefschrift.

ISBN 978-90-386-1159-4 NUR 959

Trefwoorden: digitale signaalverwerking / tijdreeksen / algebra¨ısche vergelijkingen / homotopie / datacommunicatie / identificeerbaarheid.

Subject headings: array signal processing / MIMO systems / blind source separation / higher order statistics / homotopy.

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MIMO Instantaneous Blind Identification and

Separation based on Arbitrary Order Temporal

Structure in the Data

“There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened.” Douglas Adams (19522001)

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-Prof. dr. ir. A.C.P.M. Backx (voorzitter)

Department of Electrical Engineering, Technische Universiteit Eindhoven (TU/e)

Prof. dr. ir. J.W.M. Bergmans (eerste promotor)

Signal Processing Systems Group, Department of Electrical Engineering, Technische Universiteit Eindhoven (TU/e)

Prof. dr. ir. M. Moonen (tweede promotor)

Digital Signal Processing, Department of Electrical Engineering, Katholieke Universiteit Leuven (KUL)

Dr. ir. P.C.W. Sommen (co-promotor)

Signal Processing Systems Group, Department of Electrical Engineering, Technische Universiteit Eindhoven (TU/e)

Prof. dr. ir. A.-J. van der Veen

Circuits and Systems Group, Faculty of Electrical Engineering, Mathematics & Computer Science, Technische Universiteit Delft (TUD)

Prof. dr. ir. H.P. Urbach

Optics Research Group, Department Imaging Science & Technology, Technis-che Universiteit Delft (TUD)

Prof. dr. A.G. Tijhuis

Electromagnetics, Department of Electrical Engineering, Technische Univer-siteit Eindhoven (TU/e)

Ing. C.P. Janse

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Summary

MIMO Instantaneous Blind Identification and Separation

based on Arbitrary Order Temporal Structure in the Data

This thesis is concerned with three closely related problems. The first one is called Multiple-Input Multiple-Output (MIMO) Instantaneous Blind Identification, which we de-note by MIBI. In this problem a number of mutually statistically independent source signals are mixed by a MIMO instantaneous mixing system and only the mixed signals are observed, i.e. both the mixing system and the original sources are unknown or ‘blind’. The goal of MIBI is to identify the MIMO system from the observed mixtures of the source signals only. The second problem is called Instantaneous Blind Signal Separation (IBSS) and deals with re-covering mutually statistically independent source signals from their observed instantaneous mixtures only. The observation model and assumptions on the signals and mixing system are the same as those of MIBI. However, the main purpose of IBSS is the estimation of the source signals, whereas the main purpose of MIBI is the estimation of the mixing system. If the number of source signals is not larger than the number of sensors, the source signals can be recovered by applying the inverse of the estimated mixing system to the observed mix-tures. Hence, from this point of view IBSS is merely a direct application of MIBI. The third problem is called Instantaneous Semi-Blind Source Localization (ISBSL) and concerns the localization of a set of narrowband sources from their observed instantaneous mixtures only. In this case, the instantaneous mixing system is parameterized by the source position para-meters. Hence, narrowband ISBSL can be considered as a parameterized version of MIBI in which more a priori knowledge is available than in the general ‘fully blind’ MIBI case. For this reason we call this problem ‘semi-blind’. Because MIBI is a kind of abstraction or generalization of both IBSS and ISBSL, the main focus in this work is on MIBI, while IBSS and ISBSL are considered as applications or examples.

Many methods and algorithms for performing MIBI, IBSS and ISBSL have been devel-oped during the last decade. Until now, mainly three different approaches have been used. These are based on the following properties of the source signals, several of which are related to the guiding principle of statistical independence: non-Gaussianity, second order spatial un-correlatedness in combination with temporal non-whiteness/un-correlatedness, and second order spatial uncorrelatedness in combination with second order non-stationarity. What is lacking from those approaches is the exploitation of higher order temporal structure in the data, such as higher order non-whiteness/correlatedness and non-stationarity. Some methods for ex-ploiting Higher Order Temporal Structure (HOTS) exist, but usually these are quite specific. In addition, most blind methods described in the literature cannot deal with a MIBI scenario of great interest, viz. one with more sources than sensors. In this work we present a unifying framework for exploiting arbitrary order temporal structure in the signals by means of cumu-lant functions, which possess convenient mathematical properties such as multilinearity. The MIBI problem is formulated in such a way that any kind of temporal structure in the data, such as arbitrary order non-stationarity and non-whiteness, is exploited in a unified manner. Based

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on physically plausible assumptions on the temporal structure of the source and noise signals, and applying subspace techniques to a subspace matrix containing cumulant values arranged in a specific manner, it is shown that the MIBI problem can be ‘projected onto’ two dual mathematical problems in the sense that solving these problems solves the MIBI problem. In the first problem, MIBI is projected onto the problem of solving a system of multivariate homogeneous polynomial or so-called polyconjugal (polynomial-like) equations. The num-ber of variables and the degree (of homogeneity) of the functions in the system equal the number of sensors and the order of the considered statistics, respectively. In the second prob-lem, MIBI is projected onto the problem of solving a Multi-Matrix Generalized Eigenvalue Decomposition (MMGEVD) problem that is dual to the first problem. Possible solution ap-proaches for those problems are described. In particular, a so-called homotopy method is used for solving the system of equations. Because of the connection between the system of homogeneous equations on the one hand, and the MMGEVD problem on the other hand, this in fact solves both mathematical problems.

The theory developed in this thesis is unifying in several senses. Firstly, it is general with respect to the order of the exploited temporal structure in the sense that the mathematical problem formulation has the same structure for any considered order. Secondly, all types of statistical variability in the data, such as arbitrary order non-stationarity and non-whiteness, are exploited in a unified manner. Finally, for complex-valued signals, the conjugation pattern of the arguments of the involved cumulant functions can be chosen arbitrarily. In practice, this should be done in accordance with the characteristics of the involved signals. Our approach allows the identification of a mixing system with more sources than sensors, even with second order statistics. Depending on the number of sensors, the order and type of the exploited temporal structure, the chosen conjugation pattern, and the arrangement of the statistics in the subspace matrix, a certain maximum number of mixing matrix columns can be determined that usually exceeds the number of sensors for orders larger than one. We provide insight into the computation of the maximum number of identifiable columns as a function of the number of sensors, the employed statistical order, and the conjugation pattern. The rationale behind the work presented in this thesis is based on providing insight and highlighting the geometric and algebraic structure of the different problem formulations that are developed. Therefore, the practical problems associated with the use of estimated sensor cumulants are not discussed in great detail. Nevertheless, the theory is directly applicable to many practical scenarios. This is demonstrated by various examples, including the identification of an instantaneous mixing system for different types of signals, the separation of instantaneously mixed artificial random signals, speech signals and images, Direction Of Arrival (DOA) estimation, etcetera. Finally, the theory allows us to make trade-offs between various related quantities such as the arrangement of the statistics in the subspace matrix, maximum number of identifiable mixing matrix columns, number of sensors, exploited type(s) of temporal structure, exploited order(s) of temporal structure, employed conjugation pattern(s), number of samples required for reliable estimation of the involved statistics, and so on.

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List of Figures

1.1 MIBI problem setup. . . 3

1.2 IBSS problem setup. . . 5

1.3 Linear instantaneous mixing with two sources and two sensors. . . 6

1.4 Stationary Gaussian AR(1) source signals (left) and their mixtures (right). . . 7

1.5 Speech sources (left) and their mixtures (right). . . 8

1.6 Image sources (left) and their mixtures (right). . . 9

1.7 Source Localization problem setup. . . 11

1.8 Determination of array response vector for ULA. . . 13

1.9 Radio Telescope Array. . . 15

1.10 Deployments of hydrophone arrays. . . 15

1.11 Bijective mappings between functions, row vectors, function-valued vectors and matrices. . . 26

2.1 Two-stage approach to ICA. . . 47

2.2 Feedback system for H´erault-Jutten algorithm. . . 59

3.1 Coplanar DOA estimation problem setup. . . 84

3.2 Ideal and estimated MUSIC spectra PSM(θ) for D = 4 and S = 3. . . . 100

3.3 Ideal and estimated MUSIC spectra PSM(θ) for D = 5 and S = 3. . . . 100

3.4 Ideal (¦) and estimated (×) roots of the ROOT-MUSIC polynomial pRM(z) for D = 4 and S = 3. . . . 109

3.5 Ideal (¦) and estimated (×) roots of the ROOT-MUSIC polynomial pRM(z) for D = 5 and S = 3; left: all roots, right: area around unit circle enlarged. . 109

3.6 Ideal and estimated Min-Norm spectra PSMN(θ) for D = 4 and S = 3. . . . . 110

3.7 Ideal (¦) and estimated (×) roots of the Root-Min-Norm polynomial pRMN(z) for D = 4 and S = 3. . . . 110

3.8 Ideal and estimated Min-Norm spectra PSMN(θ) for D = 5 and S = 3. . . . . 111

3.9 Ideal (¦) and estimated (×) roots of the Root-Min-Norm polynomial pRMN(z) for D = 5 and S = 3. . . . 111

3.10 Ideal and estimated HOS-SPECTRAL-MUSIC spectra PHSM(θ) for D = 3 and S = 4. . . . 126

4.1 2 × 2 MIBI problem setup. . . . 132

4.2 Regions of support in the domain of lags k. . . . 135

4.3 Source and noise auto-correlation functions on Ωs_k. . . 136

4.4 Source and noise auto-correlation functions on Ωs\ν_k . . . 137

4.5 Graph of f (z1, z2) for example with full rank mixing matrix in Section 4.4.2.1. . . 177

4.6 Contour plot of f (z1, z2) for example with full rank mixing matrix in Section 4.4.2.1. . . 177

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4.7 Stationary Gaussian AR(1) source signals (left) and their noisy mixtures

(right) for example with full rank mixing matrix in Section 4.4.2.2. . . 179

4.8 Contour plot of ˆf (z1, z2) for example with full rank mixing matrix in Section 4.4.2.2. . . 179

4.9 Contour plots of f1(z1, z2) (left) and f2(z1, z2) (right) for example with rank-deficient mixing matrix in Section 4.4.3.1. . . 182

4.10 Graph of fA(z1, z2) for example in Section 4.4.3.1 with rank-deficient mixing matrix. . . 183

4.11 Contour plot of fA(z1, z2) for example in Section 4.4.3.1 with rank-deficient mixing matrix. . . 183

4.12 Stationary Gaussian AR(1) source signals (left) and their noisy mixtures (right) for example in Section 4.4.3.2 with rank-deficient mixing matrix. . . . 185

4.13 Contour plots of ˆf1(z1, z2) (left) and ˆf2(z1, z2) (right) for example in Section 4.4.3 with rank-deficient mixing matrix. . . 185

4.14 Contour plots of f (z1, z2) (left) and ˆf (z1, z2) (right) for example with speech signals in Section 4.4.4. . . 186

4.15 Contour plots of f (z1, z2) (left) and ˆf (z1, z2) (right) for example with images in Section 4.4.5. . . 188

4.16 Zero contour plot of function f (z1, z2) occurring in full rank example of Section 4.4.2 together with unit-norm constraint. . . 189

4.17 Gaussian AR(1) source signals (left) and noise-free estimated source signals (right) for ‘full rank example’ in Section 4.4.2.2. . . 195

4.18 Original speech source signals (left) and estimated speech signals (right) for example with speech signals in Section 4.4.4. . . 197

4.19 Original source images (left) and separated images (right) for example with image signals in Section 4.4.5. . . 198

4.20 Real and imaginary parts of z1(|λ|)-components (solid lines) and z2 (|λ|)-components (dotted lines) of homotopy solution paths of h(z, λ) = 0 for Example 1 in Section 4.4.2 with ideal subspace matrix. . . 206

4.21 Real and imaginary parts of z1(|λ|)-components (solid lines) and z2 (|λ|)-components (dotted lines) of homotopy solution paths of h(z, λ) = 0 for Example 1 in Section 4.4.2 with estimated subspace matrix. . . 206

4.22 Pseudo-spectrum P (θ) for Example 1 in Section 4.4.2. . . . 209

4.23 Pseudo-spectrum P (θ) for Example 2 in Section 4.4.3. . . . 209

5.1 MIBI problem setup. . . 216

5.2 Regions of support in the domain of time index pairs (n1, n2). . . 220

5.3 Correlation functions and regions of support for scenario with four AR(1) source signals and additive white noise. . . 224

5.4 Correlation functions on Noise-Free ROS for scenario with four AR(1) source signals and additive white noise. . . 226

5.5 Zero contour level cones of f_3,12 (z1, z2, z3), f3,22 (z1, z2, z3) and f3,32 (z1, z2, z3) for 3 × 3 AR(1) example with ideal subspace matrix. . . . 273

5.6 Zero contour spherical ellipses of f_3,12 (z1, z2, z3), f3,22 (z1, z2, z3) and f2 3,3(z1, z2, z3) for 3 × 3 AR(1) example with ideal subspace matrix. . . . 273

5.7 Stationary Gaussian AR(1) source signals (left) and their noise-contaminated mixtures (right) for 3 × 3 example. . . . 274

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List of Figures xi

5.8 Zero contour spherical ellipses of ˆf2

3,1(z1, z2, z3), ˆf3,22 (z1, z2, z3) and

ˆ f2

3,3(z1, z2, z3) for 3 × 3 AR(1) example with estimated subspace matrix. . . 275 5.9 Zero contour level cones of f_3,12 (z1, z2, z3) and f3,22 (z1, z2, z3) for 3 × 4

AR(1) example with ideal subspace matrix. . . 277

5.10 Zero contour spherical ellipses of f_3,12 (z1, z2, z3) and f3,22 (z1, z2, z3) for 3×4 AR(1) example with ideal subspace matrix. . . 277

5.11 Stationary Gaussian AR(1) source signals (left) and their noise-contaminated mixtures (right) for 3 × 4 example. . . . 278

5.12 Zero contour spherical ellipses of ˆf_3,12 (z1, z2, z3) and ˆf3,22 (z1, z2, z3) for 3×4 AR(1) example with estimated subspace matrix. . . 279

5.13 Two-dimensional auto-correlation function of first source signal used in speech examples. . . 281

5.14 Two-dimensional auto-correlation function of second source signal used in speech examples. . . 281

5.15 Speech sources (left) and their noisy mixtures (right) for 3 × 3 example. . . . 283

5.16 Zero contour spherical ellipses of ˆf_3,12 (z1, z2, z3), ˆf3,22 (z1, z2, z3) and

ˆ f2

3,3(z1, z2, z3) for 3 × 3 speech example. . . . 283 5.17 Speech sources (left) and their noise-contaminated mixtures (right) for 3 × 4

example. . . 284

5.18 Zero contour spherical ellipses of ˆf_3,12 (z1, z2, z3) and ˆf3,22 (z1, z2, z3) for 3×4 speech example. . . 285

5.19 Spherical ellipses and estimated solutions of system©fˆ2

3,1(z1, z2, z3),

ˆ f2

3,2(z1, z2, z3), ˆf3,32 (z1, z2, z3)

ª

for 3 × 3 example with AR(1) signals. . . . 287

5.20 Spherical ellipses and estimated solutions of system©fˆ_3,12 (z1, z2, z3),

ˆ f2

3,2(z1, z2, z3)

ª

for 3 × 4 example with AR(1) signals. . . . 287

5.21 Spherical ellipses and estimated solutions of system©fˆ2

3,1(z1, z2, z3),

ˆ f2

3,2(z1, z2, z3), ˆf3,32 (z1, z2, z3)

ª

for 3 × 3 example with speech signals. . . . 288

5.22 Spherical ellipses and estimated solutions of system©fˆ_3,12 (z1, z2, z3),

ˆ f2

3,2(z1, z2, z3)

ª

for 3 × 4 example with speech signals. . . . 288

5.23 Speech sources (left) and noise-free estimated source signals (right) for 3 × 3 example. . . 289

6.1 Regions Of Support in the domain of time index tuples nl. . . 298

6.2 Gamma probability density function for α = 2 and λ = 1. . . . 344

6.3 Stationary Gamma AR(1) source signals (left) and their noise-contaminated mixtures (right) for 2 × 2 example. . . . 345

6.4 Contour plots of ˆf_2,13 (z1, z2) (left) and ˆf2,23 (z1, z2) (right) for D = 2, S = 2, and l = 3. . . . 346

6.5 Stationary Gamma AR(1) source signals (left) and their noise-contaminated mixtures (right) for 2 × 3 example. . . . 347

6.6 Contour plot of ˆf3

2,1(z1, z2) for D = 2, S = 3, and l = 3. . . . 348 6.7 Contour plots of ˆf4

2,1(z1, z2) (top left), ˆf2,24 (z1, z2) (top right) and ˆf2,34 (z1, z2) (bottom) for D = 2, S = 2, and l = 4. . . . 350

6.8 Contour plots of ˆf4

2,1(z1, z2) (left) and ˆf2,24 (z1, z2) (right) for D = 2, S = 3, and l = 4. . . . 351

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6.9 Contour plot of ˆf4

2,1(z1, z2) for D = 2, S = 4, and l = 4. . . . 352

6.10 Minimum number of sensors D(◦)2_min (S). . . . 367

7.1 Example 1: Source pole positions for S = 6. . . . 377

7.2 Example 1: Normalized sensor positions for D = 3. . . . 378

7.3 Example 1: Ideal TIME-SPECTRAL-MUSIC spectrum P₃◦∗(θ) and its esti-mate ˆP◦∗ 3 (θ) for D = 3, S = 6 and c2= (◦, ∗). . . . 378

7.4 Minimum number of sensorsθDmin(◦)2(S) ≡θD2min(S). . . . 381

7.5 Minimum number of sensorsθDmin(◦)3(S) ≡θD 3 min(S). . . . 381

7.6 Minimum number of sensorsθDmin(◦)4(S) ≡θD 4 min(S). . . . 381

7.7 Example 2: Normalized array configuration with D = 10, 4 and 3 sensors for l = 1, 2 and 3 respectively. . . . 382

7.8 Example 2: Ideal TIME-MUSIC spectra P₁₀(◦)1(θ), P4(◦)2(θ) and P3(◦)3(θ). . . 383

7.9 Example 2: Ideal TIME-MUSIC spectra P₁₀(◦)1(θ), P₄(◦)2(θ) and P₃(◦)3(θ). . . 383

7.10 Example 3: Pole-zero pairs of ‘ALLPASS(1)’ source signals for S = 3. . . . 385

7.11 Example 3: Probability density functions of wrand wr. . . 386

7.12 Example 3: Ideal TIME-SPECTRAL-MUSIC spectra P₂◦◦∗(θ) and its esti-mate ˆP◦◦∗ 2 (θ) for D = 2, S = 3 and c3= (∗, ◦, ◦). . . . 386

7.13 ULA: Minimum number of sensorsθDc1min(S) ,θD 1 min(S). . . . 391

7.17 ULA Example 1: Ideal ULA-TIME-SPECTRAL-MUSIC spectrum P₄◦∗(θ) and its estimate ˆP◦∗ 4 (θ) D = 4, S = 6 and c2= (◦, ∗). . . . 392

7.18 ULA Example 2: Ideal ULA-TIME-MUSIC spectra P₁₃(◦)1(θ), P₇(◦)2(θ), P₅(◦)3(θ) and P₄(◦)4(θ) for S = 11. . . . 394

7.19 ULA Example 2: Ideal ULA-TIME-MUSIC spectra P(◦)1 13 (θ), P (◦)2 7 (θ), P (◦)3 5 (θ) and P₄(◦)4(θ) for S = 12. . . . 394

7.20 Ideal (¦) and estimated (×) roots of the ULA-TIME-ROOT-MUSIC polyno-mial pUTRM(z) for Example 3 of Section 7.1.4.3. . . . 398

A.1 Index set notation. . . 408

B.1 Platykurtic (uniform), mesokurtic (Gaussian) and leptokurtic (Super-Gaussian) distributions. . . 424

E.1 Possible continuation parameter paths in the complex plane. . . 464

E.2 Solution paths of homotopy system h(z, λ) = 0 with h(z, λ) defined in (E.2.4) and p(z) in (E.2.12). . . . 468

E.3 Intersecting zero contour curves of bivariate homogeneous polynomial (solid blue line) and unit-norm constraint function (dashed green line). . . 470

E.4 Real and imaginary parts of z1(λ)-components (solid lines) and z2 (λ)-components (dotted lines) of solution paths of h(z, λ) = 0 in (E.3.11). . . . . 471

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List of Tables

5.1 Number M_u,D2 of unique products and sensor correlation functions. . . 237

5.2 Number M_u,Dc1c2 of unique products and sensor correlation functions. . . 241

6.1 Number M_u,Dl of unique products and sensor cumulant functions. . . 310

6.2 Number Mcl u,Dof unique products and sensor cumulant functions. . . 314

6.3 Values of Smax(◦)l(D) for Q (◦)l min,D= D − 1, 1 ≤ l ≤ 4, and 1 ≤ D ≤ 5. . . . 366

7.1 θSmax,D(◦)l values for 1 ≤ l, D ≤ 4. . . . 380

7.2 Uniform Linear Array:θSmax,Dl = l(D − 1) for 1 ≤ l ≤ 4, 1 ≤ D ≤ 14. . . . 390

7.3 Uniform Linear Array:θDcminl (S) = 1 + § S/l¨for 1 ≤ l ≤ 4, 1 ≤ S ≤ 14. . . 390

A.1 Multiset coefficients Mp,G= D G p E =¡p+G−1 p ¢ =¡p+G−1 G−1 ¢ . . . 411 B.1 Example of partitioning I = {1, 2, 3}. . . . 426 B.2 Example of partitioning I = {1, 2, 3, 4}. . . . 428

B.3 Example of joint moment definition for G = 3 and l = 2. . . . 436

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List of Algorithms

2.1 IBSS based on minimizing mutual information. . . 54

2.2 H´erault-Jutten algorithm for IBSS based on zeroing nonlinear de-correlation. 60 2.3 Cichocki-Unbehauen algorithm for IBSS based on zeroing nonlinear de-correlation. . . 61

2.4 Gradient ascent algorithm for IBSS based on kurtosis. . . 62

2.5 Blind identification of Q and s in yw= Q s and y = ˆs = QTywbased on EVD of cumulant matrix of whitened data. . . 67

2.6 Blind identification of Q and s in yw= Q s and y = ˆs = QTywbased on JADE. . . 68

2.7 Blind identification of Q and s in yw= Q s and y = ˆs = QTywbased on optimizing objective functions incorporating cumulant criteria. . . 69

2.8 Blind identification of A and s based on the GEVD of two cumulant matrices. 70 2.9 Molgedey and Schuster algorithm for blind identification of A and s based on GEVD of two correlation matrices. . . 74

2.10 AMUSE algorithm for blind identification of Q and s in yw= Q s based on temporal correlation structure. . . 75

2.11 SOBI algorithm for blind identification of Q and s in yw = Q s based on JAD of correlation matrices. . . 75

2.12 Blind identification of A and s based on JAD of correlation matrices. . . 76

3.1 SPECTRAL-MUSIC. . . 98

3.2 ROOT-MUSIC. . . 104

3.3 Spectral-Min-Norm. . . 108

3.4 Root-Min-Norm. . . 108

3.5 HOS-SPECTRAL-MUSIC. . . 124

3.6 Key ingredients of a general subspace algorithm. . . 129

4.1 Pre-overview of the 2 × 2 MIBI method developed in this chapter. . . . 132

4.2 Overview of results obtained so far. . . 161

4.3 High-level algorithm for subspace based real-valued 2 × 2 MIBI. . . . 171

4.4 Homotopy continuation method employed in thesis. . . 202

4.5 Overview of our 2 × 2 MIBI method developed in this chapter. . . . 214

5.1 Pre-overview of the D × S MIBI method developed in this chapter. . . . 216

5.2 High-level algorithm for D × S MIBI exploiting Second Order Temporal Structure with arbitrary conjugation pairs (c1, c2). . . 268

6.1 Pre-overview of the D × S MIBI method based on l-th order statistics and conjugation tuple cl. . . 296

6.2 High-level algorithm for D × S MIBI exploiting l-th order temporal structure with arbitrary conjugation tuple cl. . . 341

7.1 Arbitrary order TIME-SPECTRAL-MUSIC algorithm for D × S DOA esti-mation scenario with arbitrary conjugation tuple cl. . . 374

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E.1 Overview of homotopy continuation method for solving a nonlinear system of equations by an increment-and-fix method with constant step size. . . 463

E.2 Overview of homotopy continuation method with ‘generalized gamma trick’ and ’generalized continuation parameter’. . . 465

E.3 Homotopy continuation method for finding all roots of the univariate poly-nomial p(z) =PP_p=0ap(z)pby an increment-and-fix method with constant step size. . . 467

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Glossary

List of Acronyms

AMUSE Algorithm for Multiple Unknown Signals Extraction

ASP Array Signal Processing

BI Blind Identification

BSE Blind Signal Extraction

BSP Blind Signal Processing

BSS Blind Signal Separation

CBSS Convolutive Blind Signal Separation

cdf cumulative distribution function

CDMA Code Division Multiple Access

CGF Cumulant Generating Function

CLT Central Limit Theorem

cpdf conditional probability density function

DOA Direction Of Arrival

EFOBI Extended Fourth Order Blind Identification

EVD Eigenvalue Decomposition

FCF First Characteristic Function

FJCF First Joint Characteristic Function

FOBI Fourth Order Blind Identification

FOSS Fourth Order Signal Subspace

GEVD Generalized Eigenvalue Decomposition

HOS Higher Order Statistics

HOTS Higher Order Temporal Structure

IBI Instantaneous Blind Identification

IBSS Instantaneous Blind Signal Separation

ICA Independent Component Analysis

ISBSL Instantaneous Semi-Blind Source Localization

JAD Joint Approximate Diagonalization

JADE Joint Approximate Diagonalization of Eigenmatrices

JAND Joint Approximate Non-Orthogonal Diagonalization

JAOD Joint Approximate Orthogonal Diagonalization

jcdf joint cumulative distribution function

jpdf joint probability density function

MCBI Multiple-Input Multiple-Output Convolutive Blind Identification

MGF Moment Generating Function

MIBI Multiple-Input Multiple-Output Instantaneous Blind Identification

MIMO Multiple-Input Multiple-Output

MIN-NORM Minimum-Norm

mjpdf marginal joint probability density function

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mpdf marginal probability density function

MUSIC Multiple Signal Classification

PCA Principal Component Analysis

pdf probability density function

PHD Pisarenko Harmonic Decomposition

ROOT-MUSIC Root Multiple Signal Classification

ROS Region Of Support

SCF Second Characteristic Function

SJCF Second Joint Characteristic Function

SKAI Square Kilometer Array Interferometer

SOBI Second Order Blind Identification

SOS Second Order Statistics

SOTS Second Order Temporal Structure

TRM TIME-ROOT-MUSIC

TSM TIME-SPECTRAL-MUSIC

UTRM ULA-TIME-ROOT-MUSIC

UTSM ULA-TIME-SPECTRAL-MUSIC

SVD Singular Value Decomposition

ULA Uniform Linear Array

VLA Very Large Array

List of Abbreviations used in Lists

AS ASsumptions on MIBI scenarios (pages 90, 114, 138, 222, 299)

ASM ASsumptions on Mixing system (page 31)

ASN ASsumptions on Noise Signals (page 33)

ASS ASsumptions on Source Signals (page 32)

CFP Contrast Function Property/Requirement (page 42)

DPS Desired Properties of Pseudo-Spectrum (page 99)

JP Desired Properties of objective/cost function J (y) (page 41)

MAS Main ASsumption of MIBI problem formulations in thesis (page 23)

MNC Min-Norm algorithm Constraints (page 107)

MSA Main Subspace Assumption (page 127)

PIF Possible InFluences on pseudo-spectrum (page 101)

PR Instantaneous Blind Identification PRinciples (page 16)

ST Two-STage approach to instantaneous blind problems (page 46)

TPS Thesis Problem Statement (page 22)

List of Textual Abbreviations

e.g. exempli gratia: for example

i.e. id est: that means, in other words

i.i.d. independently and identically distributed

s.t. subject to

viz. videlicet: namely

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Glossary xix

List of Mathematical Symbols and Variables

aj_i Transfer coefficient from j-th source to i-th sensor, i.e. element in the i-th row and j-th column of the mixing matrix

A Mixing or array response matrix

A¡˜θ¢ Array response matrix parameterized by DOA θ

A¡˜θ, ˜ζ¢ Array response matrix parameterized by azimuth θ and elevation ζ

A¡˜θ, ˜ζ, ˜ρ¢ Array response matrix parameterized by azimuth θ, elevation ζ and range ρ ˜

ai i-th row of A

aj _{j-th column or array response vector of A} a(θ) Array response vector parameterized by DOA θ

a¡θ, ζ¢ Array response vector parameterized by azimuth θ and elevation ζ

a¡θ, ζ, ρ¢ Array response vector parameterized by azimuth θ, elevation ζ and range ρ

c Signal propagation velocity

cp Conjugation tuple cp, (c1, . . . , cp) of length p, where each of the symbols

c1, . . . , cpcan either be ‘∗’ or ‘◦’, meaning ‘conjugation’ and ‘no conjuga-tion’ respectively

¯cp Complement of conjugation tuple cp, (¯c1, . . . , ¯cp), where ¯∗ = ◦ and ¯◦ = ∗ C Set of complex-valued numbers

C[Ω] Space of complex-valued functions defined on Ω CN _{Vector space of complex row vectors of length N} CM Vector space of complex column vectors of length M CN

M Vector space of complex matrices of size M by N

CM[Ω] Space of length-M complex function-valued column vectors defined on Ω

d Sensor spacing of ULA

dn Sensor spacing of ULA that is normalized w.r.t. the carrier wavelength

d(V) Rank of V

D Number of sensors

DV Ordered set of diagonal elements of V

ei _{i-th standard basis vector of R}

M ˜ej j-th standard basis vector of RN

fc Frequency of a carrier

¯ Hadamard product

ip Length-p tuple (i1, . . . , ip) with indices i1, . . . , ip

im,q m-th index of a numbered length-p index tuple ip,q(1 ≤ m ≤ p)

Kx,cl

t,D Set of all sensor cumulant functions with conjugation tuple cl

Mx,cl

t,D Cardinality of K

x,cl

t,D

Kx,cl

u,D Set of all unique sensor cumulant functions with conjugation tuple cl

Mx,cl

u,D Cardinality of K

x,cl

u,D

M(◦)l

u,DhZi Set of all l-multisets of Z , {z1, . . . , zD}

M(◦)l

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Mcl

u,DhZi Set of all l-multisets of Z

◦∗ _{, {Z, Z}∗_{} with l − n}

cl elements from

Z = {z1, . . . , zD} and nclelements from Z ∗_{= {(z}

1)∗, . . . , (zD)∗}

Mcl

u,D(Z) Set of unique l-multisets of Z

◦∗ _{, {Z, Z}∗_{} with l − n}

cl elements from

Z = {z1, . . . , zD} and nclelements from Z ∗_{= {(z}

1)∗, . . . , (zD)∗}

N Cardinality of a Region Of Support containing time, lag, or time-lag tuples #{·} Operator that yields the number of elements in the argument set

¦ Khatri-Rao product

⊗ Kronecker product

L (·) Linear span of a set of vectors in a vector space

Lr(·) Linear span of the conjugated functions in the argument of a function-valued

column vector

λc Wavelength of a carrier

ωc Angular frequency of a carrier

Ωs

k ROS in the domain of lag indices on which source correlation functions exist Ων

k ROS in the domain of lag indices on which noise correlation functions exist Ωs\ν_k Noise-Fee ROS in the domain of lag indices

Ωs,c1c2

n;k ROS in the domain of time-lag index pairs (n; k) on which source correlation functions with conjugation pair (c1, c2) exist

Ων,c1c2

n;k ROS in the domain of time-lag index pairs (n; k) on which noise correlation functions with conjugation pair (c1, c2) exist

Ωs\ν,c1c2

n;k Noise-Fee ROS in the domain of time-lag index pairs (n; k) Ωs,cl

nl ROS in the domain of time index tuples nl on which source

cumulant functions with conjugation tuple clexist Ων,cl

nl ROS in the domain of time index tuples nl on which noise

cumulant functions with conjugation pair clexist Ωs\ν,cl

nl Noise-Fee ROS in the domain of time index tuples nl

˜

pi Position of the i-th sensor of a sensor array ˜

pi,n Position of the i-th sensor of a sensor array that is normalized w.r.t. the carrier wavelength pRM(z) ROOT-MUSIC polynomial pRMN(z) ROOT-Min-Norm polynomial PHSM(θ) HOS-SPECTRAL-MUSIC pseudo-spectrum PSM(θ) SPECTRAL-MUSIC pseudo-spectrum PSMN(θ) Spectral-Min-Norm pseudo-spectrum rj_i Ratio rj_i , zj/ziof zjto zi R Set of real-valued numbers

R[Ω] Space of real-valued functions defined on Ω RN _{Vector space of real-valued row vectors of length N} RM Vector space of real-valued column vectors of length M RN

M Vector space of real-valued matrices of size M by N

RM[Ω] Space of length-M real function-valued column vectors defined on Ω

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Glossary xxi

ρ Range of a source in a polar, cylindrical or spherical coordinate system ˜

ρ Row vector of the ranges of all sources incident on an array

S Number of sources

θ Direction Of Arrival (DOA) or azimuthal angle ˜

θ Row vector of the DOA’s of all sources incident on an array Θ Parameter space of interest for DOA θ

u(θ) Unit vector pointing in DOA θ v Notation for a (general) column vector

vi i-th element of the column vector v

vM Notation for a column vector v with explicitly denoted length M ˜

v Notation for a (general) row vector

vj _{j-th element of the row vector ˜}_v ˜

vN _{Notation for a row vecto ˜}_{v with explicitly denoted length N} V Notation for a (general) matrix

v_ij Element in the i-th row and j-th column of the matrix V VN

M Notation for V with explicitly denoted size M × N VT _{Transpose of V}

VH _{Hermitian of V}

V−1 _{Inverse of the square matrix V}

V† _{Pseudo-inverse (Moore-Penrose inverse) of V}

kVk_F Frobenius norm of V ˜

vi i-th row of V

vj _{j-th column of V}

V Notation for a (general) set

v := w w is assigned to v

v + w w is isomorphic to v

Vp_¯ p-th order Hadamard product of V

Vcp

¯ p-th order Hadamard product of V with conjugation tuple cp Vp¦ p-th order Khatri-Rao product of V

Vcp

¦ p-th order Khatri-Rao product of V with conjugation tuple cp Vp⊗ p-th order Kronecker product of V

Vcp

⊗ p-th order Kronecker product of V with conjugation tuple cp V m Nl

¡

W¢ Rows of V span the left null space of W 0N

M Zero matrix of size M × N 0M Zero column vector of length M ˜

0N _{Zero row vector of length N}

ζ Elevation angle

˜

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List of Mathematical Operator Names

cum(·, · · · , ·) Cumulant function

diag(V) Ordered set constructed from the diagonal elements of V diag(D) Diagonal matrix constructed from the ordered set D

ddiag(V) Diagonal matrix constructed from the diagonal elements of V mom(·, · · · , ·) Moment function

off(V) Sum of squared absolute values of the off-diagonal elements of V offd(V) Matrix V with zeroed diagonal elements

rank(V) Rank of V tr(V) Trace of V

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C

HAPTER

1

Introduction

In this chapter, we introduce, contextualize and motivate the main topics considered in this thesis. These topics concern the so-called Multiple-Input Multiple-Output Instantaneous Blind Identification (MIBI), the Instantaneous Blind Signal Separation (IBSS), and the In-stantaneous Semi-Blind Source Localization (ISBSL) problems; see Figures 1.1, 1.2 and 1.7 respectively. MIBI, IBSS and ISBSL all concern systems where multiple source signals prop-agate simultaneously to multiple sensors via a certain physical transfer mechanism or mixing system. Such systems are commonly called Multiple-Input Multiple-Output (MIMO) sys-tems. The kinds of topics mentioned above typically are investigated in a research area called Array Signal Processing (ASP) [62, 84, 147, 179, 184]. In general, ASP can be defined as the part of signal processing that deals with the extraction of information from signals col-lected by an array of sensors, where each sensor observes a different mixture of the source signals. In this work, we will only consider passive sensor arrays, i.e. arrays that only re-ceive signals as opposed to active arrays that also transmit signals. The type of sensor that is employed in a specific application depends on the physical characteristics of the source(s) and propagation medium [84, 119, 179]. For example, microphone arrays are used for mea-suring acoustic signals in air, e.g. speech, whereas hydrophone arrays are used for meamea-suring acoustic signals in water, e.g. submarine engine/noise signals. Similarly, phased-array radar systems or radio antenna arrays measure electromagnetic signals that can be used for com-munication purposes or localization of sources or targets. Likewise, X-ray, radio and optical telescope arrays measure electromagnetic radiation from our universe for astronomical inves-tigation [3, 66, 67, 136, 142, 160]. The significance of the ASP research area is evident from the wide variety of practical applications.

The information of interest can be either the transfer system from the sources to the sen-sors, the (waveforms of the) source signals, or the number and location of the sources. For example, in radio and speech communication applications the main interest usually is in the particular waveforms of the signals, whereas in submarine and defense applications the main interest usually is in the location of the sources or emitters. Also, for several applications, both the signals themselves, as well as the locations of the sources are of interest, e.g. point-ing a video camera to a speakpoint-ing person, determinpoint-ing the types and locations of submarines, etc. For MIBI the main interest is in the transfer system from the sources to the sensors, for IBSS in the source signals, and for ISBSL in the positions of the sources. Depending on the amount of information that is known a priori, a certain problem can be characterized as blind, semi-blind, or non-blind, if little, medium, or a lot of information is available respectively. The less information is available, i.e. the more blind a problem is, the more strict the assump-tions on the ‘remaining information’ need to be, and vice versa. For example, we call MIBI and IBSS ‘blind’ because both the source signals and the transfer system are unknown; see also [47, 75, 80] and the references therein. This considerable lack of information has to be compensated for by strong assumptions on the statistics‡ of the signals and the form of the

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transfer system. Specifically, the results in this thesis are mainly based on the common Blind Signal Processing (BSP) assumptions that the source signals are mutually statistically inde-pendent and that the transfer or mixing system is instantaneous and linear. Many researchers have contributed to the development of, and current interest in, BSP. Some of them are: Jutten and H´erault [55,91], Cardoso [26,27,31,33–35], Comon [36,52], Bell and Sejnowski [11,12], Hyv¨arinen [79, 80, 83], and Cichocki and Amari [6, 7, 47].

The fact that many blind or semi-blind problems with multiple sources can be tackled by processing multiple sensor signals is due to the different kinds of diversity assumed to be present in the system. On the one hand, the mixing system has to possess a certain kind of spatial diversity, i.e. each of the sensors should observe a mixture of the source signals that is sufficiently different from the others. In practice, this is usually achieved by prop-erly positioning the sensors. On the other hand, the source signals are usually required to possess a certain kind of statistical diversity regarding either the probability distribution(s) of, or the temporal relation(s) between, the signals samples, e.g. blind identification meth-ods may require that the source signals have different kurtoses. In this work, we are par-ticularly interested in exploiting another type of statistical diversity, namely any kind of temporal structure in the source signals of any chosen order, such as arbitrary order non-whiteness/correlatedness and non-stationarity. For instance, when Second Order Statistics (SOS) are employed [14, 41, 43, 45, 46, 48], the required temporal diversity means that the source signals have auto-correlation functions that are sufficiently different from each other. Our main focus in this thesis will be on the development of a unified method for exploiting the temporal structure in the data of any kind and any order for solving the MIBI, IBSS, and ISBSL problems. We do not consider the conventional (fixed and/or adaptive) beamforming approach to array signal processing that is mainly concerned with the design of a beampat-tern, i.e. the (power) response to a monochromatic plane wave with a certain frequency and arriving from a certain direction. This topic, which in some sense is complementary to our work, is treated in several excellent works, e.g. see [57, 84, 90, 97, 180].

It is widely recognized that many possible applications exist for blind identification, blind separation and source localization problems. Since these three topics are strongly related, their application areas overlap. In fact, we will show in the coming sections that both IBSS and ISBSL are applications or examples of the more general MIBI problem. Therefore, only examples of IBSS and ISBSL will be given in Sections 1.1.2 and 1.1.3 respectively. Several examples of IBSS can be found in the field of biomedical engineering, where the purpose of various applications is to reveal independent sources in different kinds of biological signals like EEG’s and ECG’s. Other examples can be found in the separation of speech signals from competing speakers in the so-called ‘cocktail party’ problem, images, data communication signals, etcetera. Examples of ISBSL, which essentially is a semi-blind MIBI problem with a known parameterization of the mixing system, can be found in many sensor array systems. Typical examples include high-resolution Direction Of Arrival (DOA) and range estimation in sonar and phased-array radar systems.

The outline of this chapter is as follows. In Section 1.1, the main problems considered in this thesis, i.e. MIBI, IBSS and ISBSL, are described and some of their application areas are mentioned. Next, in Section 1.2 the motivations, objectives, and main contributions of the thesis are formulated. In Section 1.3 we give an overview of the contents of our publications. Then, in Section 1.4 we discuss some notational issues. Subsequently, in Section 1.5 an outline of the thesis is given. Finally, conclusions are drawn in Section 1.6.

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1.1 Main thesis topics and applications 3

1.1 Main thesis topics and applications

The next sections describe each of the three main thesis topics and their possible applications in turn. The structure and main assumptions of the mathematical models associated with these problems are presented. Without loss of generality, everywhere in the thesis we will assume that all signals have zero mean. In addition, we assume that all involved signals are sampled at discrete time instants. We will emphasize the relationships and differences between the various topics. In order to clearly distinguish the objectives associated with the different problems, and for future reference, a problem statement is formulated for each of them. A general thesis problem statement will be formulated in Section 1.2.8.

1.1.1 Multiple-Input Multiple-Output Instantaneous Blind

Identification (MIBI)

One of the main problems considered in this thesis is the so-called Input Multiple-Output (MIMO) Instantaneous Blind Identification (IBI) problem. We will concisely de-note this problem by the acronym MIBI. The MIBI problem setup is shown in Fig. 1.1. A set {s1[n], . . . , sS[n]} of S mutually statistically independent source signals is mixed by a Multiple-Input Multiple-Output (MIMO) linear instantaneous (i.e. memory-less) mixing sys-tem A, and only a set {x1[n], . . . , xD[n]} of D mixed sensor/observation signals corrupted by a set of D additive noise signals {ν1[n], . . . , νD[n]} is available. Hence, both the mixing system and the original sources are unknown. For this reason, the MIBI problem is called ‘blind’ [47,80]. The main conditions and purpose of MIBI can now be formulated as follows.

MIBI problem statement:

The purpose of MIBI is to identify the MIMO instantaneous mixing system from the observed mixtures of the source signals only. The main assumptions are that the source signals are mutually statistically independent and that they possess sufficient statistical diversity.

As we will see in the next chapter, the type of statistical diversity that is assumed is algorithm-specific. For example, one algorithm may exploit the non-Gaussianity of the source signals, whereas another may exploit their temporal auto-correlation structure (see Chapter 2). De-pending on the considered application, the signals and mixing system can be real- or complex-valued. In this thesis, we will always denote the number of sources by S and the number of sensors by D. Mathematically, the MIBI observation model can be stated as follows:

x[n] = S X j=1 aj_s j[n] + ν[n] = A s[n] + ν[n] ∀ n ∈ Z , (1.1.1) s₁[n] sS[n] Mixing system A Unknown ν₁[n] ν_D[n] x₁[n] x_D[n]

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where: x[n] ,    x1[n] .. . xD[n]    , s[n] ,    s1[n] .. . sS[n]    , ν[n] ,    ν1[n] .. . νD[n]    , and aj,    aj1 .. . aj_D   

are column vectors of sensor signals, source signals, additive noise signals and mixing ele-ments, respectively. Subscript indices are used to index the components of a column vector, whereas superscript indices are used to index the components of a row vector‡. Furthermore, the symbol n denotes discrete time. The vectors x[n], ν[n], and a1, . . . , aS _{are elements of}

RD

¡ CD

¢

, i.e. the vector space of real (complex) column vectors of length D‡. The length-S source signal vector s[n] is an element of RS

¡ CS

¢

. The coefficient aj_i denotes the instan-taneous transfer from the j-th source to the i-th sensor. As model (1.1.1) shows, the whole unknown transfer system is modelled by a mixing matrix A that is an element of RS_D¡CS_D¢, i.e. the vector space of matrices containing real- or complex-valued elements with D rows and S columns‡. For convenience, A is often written in terms of its columns as follows

A = £a1_{· · · a}S¤_{; see also (2.2.2) on page 31 for other expressions. As is shown in} Sec-tion 2.4, two ambiguities are inherent to the MIBI model. Namely, the norms and the order of the columns of the mixing matrix cannot be resolved. In general, these two indeterminacies do not cause serious problems because for many applications the most relevant information is contained in the ‘directions’ of the columns rather than in their magnitudes or order.

In order to be able to identify the mixing system and/or recover the source signals, the apparently large lack of information needs to be compensated for by means of a strong hy-pothesis on the source signals. This hyhy-pothesis is the so-called statistical independence§ assumption on the source signals [34, 80]. Although it is strong, it is physically plausible in many practical situations because the source signals are often generated independently of each other by different sources. Usually, also some other assumptions that characterize the statistical properties of the individual source signals are required. Typical examples can be found in Section 2.2.2. In addition to assumptions on the source signals, assumptions on the mixing system and noise signals are required. Each sensor should receive a combination of the source signals that is sufficiently different from the others. In other words, there should be sufficient spatial diversity in the mixing system. In practice, this is often achieved by po-sitioning the sensors properly. Assumptions on the diversity of the mixing system often are formulated in terms of the size and rank of the mixing matrix. Typical examples can be found in Section 2.2.1. Finally, in order to be able to properly cope with additive observation or sensor noise, certain assumptions about the statistical structure of the noise signals have to be made. See Section 2.2.3 for characteristic examples.

As we have stated earlier and will explain in the next two sections, MIBI is a kind of abstraction or generalization of both IBSS and ISBSL. This implies that applications of IBSS and ISBSL are applications of MIBI as well. Therefore, the IBSS and ISBSL examples presented in the next sections can be considered as examples of MIBI. Finally, we note that many practical problems can be described more adequately by more complex mixing models such as convolutive or nonlinear models. However, apart from being applicable to many practical problems that do (approximately) satisfy the model described above, MIBI can also be seen and used as a stepping stone for more complicated identification problems.

‡_{See Appendix A for our notational conventions.}

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1.1 Main thesis topics and applications 5

1.1.2 Instantaneous Blind Signal Separation (IBSS)

A problem closely related to MIBI is Instantaneous Blind Signal Separation (IBSS), which deals with the separation/recovery of mutually statistically independent source signals from their observed instantaneous mixtures only. The problem setup is depicted in Fig. 1.2 for S sources and D sensors. The observation model and assumptions on the signals and mixing system are the same as those of MIBI. However, while the main purpose of MIBI is the estimation of the mixing system (only), the main purpose of IBSS is the estimation of the (waveforms of the) source signals [47,75,80]. Summarizing, the main conditions and purpose of IBSS can be formulated as follows.

IBSS problem statement:

The purpose of IBSS is to recover the source signals from their observed mixtures only. The main assumptions are that the source signals are mutually statistically independent and that they possess sufficient statistical diversity.

As with MIBI, the type of statistical diversity that is assumed is algorithm-specific. Mathe-matically, the IBSS problem can be formulated as the estimation of a separation/de-mixing matrix W in such a way that the output vector y[n] defined by:

y[n] , Wx[n] (1.1.2)

contains estimates of the waveforms of the source signals s1[n], . . . , sS[n]. The IBSS prob-lem is typically tackled by designing a de-mixing system W in such a way that the (statisti-cal) properties of the source signals are restored at the output. For instance, the output signals

y1[n], . . . , yS[n] should be statistically independent. Since the data observation model is the same as that of the MIBI model (1.1.1), two IBSS indeterminacies are present that correspond to the MIBI indeterminacies, viz. the scaling and order of the original source signals cannot be recovered. Taking these into account, the goal of IBSS is to recover the source signals in some arbitrary order and with some arbitrary nonsingular scaling. Hence, the de-mixing matrix W should satisfy WA ≈ PD, where P is some permutation matrix and D is some nonsingular diagonal matrix. See Section 2.4 for more information about this issue.

As stated above, contrary to MIBI the main interest in IBSS is in the source signals instead of the mixing system. In fact, once MIBI has been performed, the source signals can be recovered (approximately) by applying the (pseudo-)inverse of the estimated mixing system (provided it is nonsingular) to the observed mixtures. Hence, from this point of view IBSS is merely a direct application of MIBI. Therefore, the main focus in this work will be on MIBI, while IBSS will be considered as an application or example.

s₁[n] sS[n] Mixing system system A Unknown ν₁[n] νD[n] x₁[n] xD[n] De-mixing W y₁[n] yS[n]

MIMO Instantaneous Blind Identiﬁcation and Separation based on Arbitrary Order Temporal Structure in the Data

MIMO Instantaneous Blind Identification and

Separation based on Arbitrary Order Temporal

Structure in the Data

ter verkrijging van de graad van doctor aan de

Technische Universiteit Eindhoven, op gezag van de

Rector Magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor

Promoties in het openbaar te verdedigen

op woensdag 21 november 2007 om 16.00 uur

door

Jakob van de Laar

prof.dr.ir. J.W.M. Bergmans

en

prof.dr.ir. M. Moonen

Copromotor:

dr.ir. P.C.W. Sommen

c

°

MIMO Instantaneous Blind Identification and

Separation based on Arbitrary Order Temporal

Structure in the Data

Summary

MIMO Instantaneous Blind Identification and Separation

based on Arbitrary Order Temporal Structure in the Data

Contents

List of Figures

List of Tables

List of Algorithms

Glossary

List of Acronyms

List of Abbreviations used in Lists

List of Textual Abbreviations

List of Mathematical Symbols and Variables

List of Mathematical Operator Names

C

HAPTER

1

Introduction

1.1

Main thesis topics and applications

1.1.1

Multiple-Input Multiple-Output Instantaneous Blind

Identification (MIBI)

1.1.2

Instantaneous Blind Signal Separation (IBSS)