Blind Identification Of Underdetermined Mixtures Research Articles

A Promising Technique for Blind Identification: The Generic Statistics

The generic statistics, related to lower-order derivatives of the log characteristic function (CAF) evaluating at the processing points located away from the origin, are frequently used in multivariate statistical signal processing. In comparison with the conventional statistics, e.g., cumulants, the generic statistics have the following advantages: (a) they can offer the structural simplicity and controllable statistical stability of lower-order statistics, and retain higher-order statistical information; (b) if the derivatives of the log CAF were evaluated at all (infinitely many) possible processing-points, a complete description of the joint CAF would be obtained. Furthermore, we show in this paper that even if a random process is symmetrically distributed, the odd-order generic statistics are not equal to zero, while in such a case the odd-order cumulants are equal to zero. For these reasons, a family of blind identification (BI) methods, in which the mixing matrix is obtained by decomposing the tensor constructed by the higher order derivatives of the log CAF of the observations, is proposed to achieve BI of underdetermined mixtures. Simulation results show that the BI methods based on generic statistics have superior performance to the existing cumulant-based method, such as FOOBI method, especially, when the SNR of the observations is high and/or the data block is short.

Blind Identification of Underdetermined Mixtures with Complex Sources Using the Generalized Generating Function

The generalized generating function (GGF), which can exploit the statistical information carried on complex-valued signals efficiently by treating its real and imaginary parts as a whole and offer an elegant algebraic structure, has been proposed by authors for blind identification of mixtures. In this paper, we extend the GGF-based method to be able to deal with the challenging underdetermined mixtures with complex-valued sources. A new algorithm named ALSGGF, in which the mixing matrix is estimated by decomposing the tensor constructed from the higher conjugate derivative of the second GGF of the observations with alternating least squares algorithm, is proposed. Furthermore, we show that the conjugate derivatives of different orders of the second GGF can be used jointly in a simple way to improve the performance. Simulation experiments validate the superiority of the proposed ALSGGF algorithms.

Blind Identification of Underdetermined Mixtures with Complex Sources Using the Generalized Generating Function

Blind Identification of Underdetermined Mixtures with Complex Sources Using the Generalized Generating Function

PARAFAC-Based Blind Identification of Underdetermined Mixtures Using Gaussian Mixture Model

This paper presents a novel algorithm, named GMM-PARAFAC, for blind identification of underdetermined instantaneous linear mixtures. The GMM-PARAFAC algorithm uses Gaussian mixture model (GMM) to model non-Gaussianity of the independent sources. We show that the distribution of the observations can also be modeled by a GMM, and derive a maximum-likelihood function with regard to the mixing matrix by estimating the GMM parameters of the observations via the expectation-maximization algorithm. In order to reduce the computation complexity, the mixing matrix is estimated by maximizing a tight upper bound of the likelihood instead of the log-likelihood itself. The maximum of the tight upper bound is obtained by decomposition of a three-way tensor which is obtained by stacking the covariance matrices of the GMM of the observations. Simulation results validate the superiority of the GMM-PARAFAC algorithm.

Sequential blind identification of underdetermined mixtures using a novel deflation scheme.

In this brief, we consider the problem of blind identification in underdetermined instantaneous mixture cases, where there are more sources than sensors. A new blind identification algorithm, which estimates the mixing matrix in a sequential fashion, is proposed. By using the rank-1 detecting device, blind identification is reformulated as a constrained optimization problem. The identification of one column of the mixing matrix hence reduces to an optimization task for which an efficient iterative algorithm is proposed. The identification of the other columns of the mixing matrix is then carried out by a generalized eigenvalue decomposition-based deflation method. The key merit of the proposed deflation method is that it does not suffer from error accumulation. The proposed sequential blind identification algorithm provides more flexibility and better robustness than its simultaneous counterpart. Comparative simulation results demonstrate the superior performance of the proposed algorithm over the simultaneous blind identification algorithm.

CONFAC Decomposition Approach to Blind Identification of Underdetermined Mixtures Based on Generating Function Derivatives

This work proposes a new tensor-based approach to solve the problem of blind identification of underdetermined mixtures of complex-valued sources exploiting the cumulant generating function (CGF) of the observations. We show that a collection of second-order derivatives of the CGF of the observations can be stored in a third-order tensor following a constrained factor (CONFAC) decomposition with known constrained structure. In order to increase the diversity, we combine three derivative types into an extended third-order CONFAC decomposition. A detailed uniqueness study of this decomposition is provided, from which easy-to-check sufficient conditions ensuring the essential uniqueness of the mixing matrix are obtained. From an algorithmic viewpoint, we develop a CONFAC-based enhanced line search (CONFAC-ELS) method to be used with an alternating least squares estimation procedure for accelerated convergence, and also analyze the numerical complexities of two CONFAC-based algorithms (namely, CONFAC-ALS and CONFAC-ELS) in comparison with the Levenberg-Marquardt (LM)-based algorithm recently derived to solve the same problem. Simulation results compare the proposed approach with some higher-order methods. Our results also corroborate the advantages of the CONFAC-based approach over the competing LM-based approach in terms of performance and computational complexity.

Blind Identification of Underdetermined Mixtures Based on the Characteristic Function: The Complex Case

Blind identification of underdetermined mixtures can be addressed efficiently by using the second ChAracteristic Function (CAF) of the observations. Our contribution is twofold. First, we propose the use of a Levenberg-Marquardt algorithm, herein called LEMACAF, as an alternative to an Alternating Least Squares algorithm known as ALESCAF, which has been used recently in the case of real mixtures of real sources. Second, we extend the CAF approach to the case of complex sources for which the previous algorithms are not suitable. We show that the complex case involves an appropriate tensor stowage, which is linked to a particular tensor decomposition. An extension of the LEMACAF algorithm, called then proposed to blindly estimate the mixing matrix by exploiting this tensor decomposition. In our simulation results, we first provide performance comparisons between third- and fourth-order versions of ALESCAF and LEMACAF in various situations involving BPSK sources. Then, a performance study of is carried out considering 4-QAM sources. These results show that the proposed algorithm provides satisfying estimations especially in the case of a large underdeterminacy level.

Blind Identification of Underdetermined Mixtures by Simultaneous Matrix Diagonalization

In this paper, we study simultaneous matrix diagonalization-based techniques for the estimation of the mixing matrix in underdetermined independent component analysis (ICA). This includes a generalization to underdetermined mixtures of the well-known SOBI algorithm. The problem is reformulated in terms of the parallel factor decomposition (PARAFAC) of a higher-order tensor. We present conditions under which the mixing matrix is unique and discuss several algorithms for its computation.

Fourth-Order Cumulant-Based Blind Identification of Underdetermined Mixtures

In this paper we study two fourth-order cumulant-based techniques for the estimation of the mixing matrix in underdetermined independent component analysis. The first method is based on a simultaneous matrix diagonalization. The second is based on a simultaneous off-diagonalization. The number of sources that can be allowed is roughly quadratic in the number of observations. For both methods, explicit expressions for the maximum number of sources are given. Simulations illustrate the performance of the techniques

Blind identification of under-determined mixtures based on the characteristic function

Linear mixtures of independent random variables (the so-called sources) are sometimes referred to as under-determined mixtures (UDM) when the number of sources exceeds the dimension of the observation space. The algorithms proposed are able to identify algebraically a UDM using the second characteristic function (c.f.) of the observations, without any need of sparsity assumption on sources. In fact, by taking higher order derivatives of the multivariate c.f. core equation, the blind identification problem is shown to reduce to a tensor decomposition. With only two sensors, the first algorithm only needs a SVD. With a larger number of sensors, the second algorithm executes an alternating least squares (ALS) algorithm. The joint use of statistics of different orders is possible, and a LS solution can be computed. Identifiability conditions are stated in each of the two cases. Computer simulations eventually demonstrate performances in the absence of sparsity, and emphasize the interest in using jointly derivatives of different orders.

Fourth-order blind identification of underdetermined mixtures of sources (FOBIUM)

For about two decades, numerous methods have been developed to blindly identify overdetermined (P/spl les/N) mixtures of P statistically independent narrowband (NB) sources received by an array of N sensors. These methods exploit the information contained in the second-order (SO), the fourth-order (FO) or both the SO and FO statistics of the data. However, in practical situations, the probability of receiving more sources than sensors increases with the reception bandwidth and the use of blind identification (BI) methods able to process underdetermined mixtures of sources, for which P>N may be required. Although such methods have been developed over the past few years, they all present serious limitations in practical situations related to the radiocommunications context. For this reason, the purpose of this paper is to propose a new attractive BI method, exploiting the information contained in the FO data statistics only, that is able to process underdetermined mixtures of sources without the main limitations of the existing methods, provided that the sources have different trispectrum and nonzero kurtosis with the same sign. A new performance criterion that is able to quantify the identification quality of a given source and allowing the quantitative comparison of two BI methods for each source, is also proposed in the paper. Finally, an application of the proposed method is presented through the introduction of a powerful direction-finding method built from the blindly identified mixture matrix.