Blind Identification of Underdetermined Mixtures Based on the Hexacovariance and Higher-Order Cyclostationarity

International audienceIn this work, we consider the problem of blind identification of underdetermined mixtures in a cyclostationary context relying on sixth-order statistics. We propose to exploit the cyclostationarity at higher orders by taking into account the knowledge of source cyclic frequencies in the sample estimator of the observation hexacovariance. Two blind identification algorithms based on the proposed estimator are considered and their performances are tested by means of computer simulations. Our simulation results show that significant improvements can be obtained when both second and fourth-order cyclo-stationarities are exploited

Blind identification of underdetermined mixtures of complex sources based on the characteristic function

International audienceIn this work we consider the problem of blind identification of underdetermined mixtures using the generating function of the observations. This approach had been successfully applied on real sources but had not been extended to the more attractive case of complex mixtures of complex sources. This is the main goal of the present study. By developing the core equation in the complex case, we arrive at a particular tensor stowage which involves an original tensor decomposition. Exploiting this decomposition, an algorithm is proposed to blindly estimate the mixing matrix. Three versions of this algorithm based on 2nd, 3rd and 4th-order derivatives of the generating function are evaluated on complex mixtures of 4-QAM and 8-PSK sources and compared to the 6-BIOME algorithm by means of simulation results

Blind identification of possibly under-determined convolutive MIMO systems

Blind identi¯cation of a Linear Time Invariant (LTI) Multiple-Input Multiple-Output (MIMO) system is of great importance in many applications, such as speech processing, multi-access communication, multi-sensor sonar/radar systems, and biomedical applications. The objective of blind identi¯cation for a MIMO system is to identify an unknown system, driven by Ni unobservable inputs, based on the No system outputs. We ¯rst present a novel blind approach for the identi¯cation of a over-determined (No ¸ Ni) MIMO system driven by white, mutually independent unobservable inputs. Samples of the system frequency response are obtained based on Parallel Factorization (PARAFAC) of three- or four-way tensors constructed respectively based on third- or fourth-order cross-spectra of the system outputs. We show that the information available in the higher-order spectra allows for the system response to be identi¯ed up to a constant scaling and permutation ambiguities and a linear phase ambiguity. Important features of the proposed approaches are that they do not require channel length information, need no phase unwrapping, and unlike the majority of existing methods, need no pre-whitening of the system outputs.While several methods have been proposed to blindly identify over-determined convolutive MIMO systems, very scarce results exist for under-determined (No < Ni) case, all of which refer to systems that either have some special structure, or special No, Ni values. We propose a novel approach for blind identi¯cation of under-determined convolutive MIMO systems of general dimensions. As long as min(No;Ni) ¸ 2, we can always ¯nd the appropriate order of statistics that guarantees identi¯ability of the system response within trivial ambiguities. We provide the description of the class of identi¯able MIMO systems for a certain order of statistics K, and an algorithm to reach the solution.Finally we propose a novel approach for blind identi¯cation and symbol recovery of a distributed antenna system with multiple carrier-frequency o®sets (CFO), arising due to mismatch between the oscillators of transmitters and receivers. The received base-band signal is over-sampled, and its polyphase components are used to formulate a virtual MIMO problem. By applying blind MIMO system estimation techniques, the system response is estimated and used to subsequently decouple the users and transform the multiple CFOs estimation problem into a set of independent single CFO estimation problems.Ph.D., Electrical Engineering -- Drexel University, 200

Analyse en Composantes Indépendantes Multidimensionnelles via des cumulants d’ordres variés

The author deals with the problem of multidimensional independent component analysis (MICA) which is the natural generalization of the ordinary problem of independent component analysis (ICA). First, in order to facilitate the use of higher-order cumulants, we present new formulas for the cumulant matrices of a real random vector from its moment matrices. In addition to the usual matrix operations, these formulas use only the Kronecker product, the vec operator and some commutation matrices. These formulas lend themselves to examine more closely the specific structures of cumulant matrices and provide results on the ranks of these matrices that characterize the dependence between random variables composing the random vector. The main practical interest of our matrix formulas lies in much easier cumulant evaluation and faster computation than the conventional method based on repeated use of the Leonov and Shiryaev formulas. In the second part of this thesis, we show that under the usual assumptions of the independent multidimensional component analysis, contracted cumulant matrices at any statistical order are all block diagonalizable in the same basis. We derive an algorithm for solving MICA by block diagonalization and compare the results obtained to the orders 3-6, between them and with other methods, on several synthetic signals. Simple examples are developed to justify the need to combine different levels to ensure the best separation. We also prove that the easiest case to deal with is the case of mixtures of sources that have different dimensions. In the last part of this thesis we propose a set of methods that operate only the higher- order statistics. Under certain additional assumptions, these methods are shown to completely solve the noisy MICA problem without second-order whitening by joint block diagonalization of a cumulant matrices set coming from statistics of orders strictly higher than four. A comparison with the second-order based whitening MICA methods for the separation of fetal and maternal electrical activities (measured using three electrodes placed on the mother’s abdomen) shows that this new family is better suited to this application : it allow an almost perfect separation of these two contributions.L’auteur s’intéresse au problème de l’analyse en composantes indépendantes multidimensionnelles (ACIM) qui est la généralisation naturelle du problème ordinaire de l’analyse en composantes indépendantes (ACI). Tout d’abord, afin de faciliter l’utilisation des cumulants des ordres supérieurs, nous présentons de nou- velles formules pour le calcul matriciel des matrices de cumulants d’un vecteur aléatoire réel à partir de ses matrices de moments. Outre les opérations matricielles usuelles, ces formules utilisent uniquement le produit de Kronecker, l’opérateur vec et des matrices de commutation. Nous pouvons immédiatement à partir de ces formules examiner de plus près les structures particulières des matrices de cumulants et ainsi donner des résultats sur les rangs de ces matrices qui caractérisent la dépendance entre les variables aléatoires constituant le vecteur aléatoire. L’intérêt pratique principal de nos formules matricielles réside certainement dans une évaluation des cumulants beaucoup plus aisée et rapide qu’avec la méthode usuelle basée sur une utilisation répétée des formules de Leonov et Shiryaev. Dans la deuxième partie de cette thèse, nous montrons que sous les hypothèses usuelles de l’analyse en composantes indépendantes mul- tidimensionnelles, les matrices de cumulants contractées à un ordre statistique quelconque sont toutes bloc-diagonalisables dans la même base. Nous en déduisons des algorithmes de résolution d’ACIM par bloc-diagonalisation conjointe et comparons les résultats obtenus aux ordres 3 à 6, entre eux et avec d’autres méthodes, sur quelques signaux synthétiques. Des exemples simples ont élaborés afin de justifier la nécessité de combiner des ordres différents pour garantir la meilleure séparation. Nous prouvons aussi que le cas le plus simple à traiter est celui de mélanges de sources qui ont différentes dimensions. Dans la dernière partie de cette thèse nous proposons une famille de méthodes qui exploitent uniquement les sta- tistiques d’ordres supérieurs à deux. Sous certaines hypothèses supplémentaires, ces méthodes permettent après un blanchiment d’ordre quatre des observations de résoudre complètement le problème ACIM bruité en bloc diagonalisant conjointement un ensemble de matrices de cumulants issues des statistiques d’ordres supérieurs strictement à quatre. Une comparaison avec les méthodes ACIM à blanchiment d’ordre deux pour la séparation des activités électriques foetale et maternelle (mesurées via trois électrodes placées sur l’abdomen de la mère) montre que cette nouvelle famille est mieux adaptée à cette application : elles permettent une séparation quasi parfaite de ces deux contributions