Optimal combination of fourth-order cumulant based contrasts for blind separation of noncircular signals

Abstract

In this paper, we have considered the problem of blind source separation of noncircular signals. We have proposed a new Jacobi-like algorithm that achieves optimization of the optimal combination of complex fourth-order cumulant based contrasts. We have investigated the application to separation of non-Gaussian sources using fourth order cumulants. And computer simulations applied to noncircular signals illustrate the proposed approach.

Introduction

The problem of the blind source separation (BSS) appears in many signal processing application such as telecommunications, RADAR and the bio-medical field. The goal is to restore a set of source signals through observation values of a mixture of these source signals without using explicit knowledge. The BSS problem has given rise to numerous theoretical and applied works, see [1] for a review of recent results and applications. The instantaneous linear mixture model, where no time delays occur in the propagation from sources to sensors, may be very useful and can be accurate in many of BSS applications.

In this paper, we have considered an instantaneous mixing system for which some solutions were proposed in [2], [3], [4]. And, we have focused our attention on contrast functions based on high-order statistics. In this field, it was shown that maximizing contrast solves the source separation problem e.g. [5], [6], [7], [8]. A previous whitening process helps to reduce the number of unknown parameters and it results in a set of normalized uncorrelated components in relation with the sources through a unitary transformation. The resulting problem is then to identify the unitary matrix.

One way to optimize contrast to estimate the unknown unitary mixing matrix is to use Jacobi optimization techniques because of their favorable stability, satisfactory practical convergence and computational parallelism, see [9], [10]. Moreover, the analytic solution of the optimal Jacobi angles can be done in the case of two source signals. The Jacobi technique was spread out to the joint approximate diagonalization of matrix sets. It has yielded the two popular algorithms JADE [5] and SOBI [11] while it has led later to the algorithm STOTD [12] and the ones proposed in [6] for tensor sets. A generalization of these algorithms has been done in [7]. It was shown that optimal combination of contrasts, proposed in several works see [8], [7], [13], [28], allows to improve blind source separation. In [13], in the real valued case, a generalized ICA algorithm, based on Jacobi technique, shows very efficient results optimizing an optimal combination of contrasts based on fourth-order cumulants. In the two-source case, we have derived the large-sample approximations of the mean square error (MSE) in order to reach an optimal weight.

Noncircular signals have attracted a high interest these last 20 years, e.g. in wireless telecommunication applications. In this context, more statistics may be used than we can use for circular random variables. See the initial works of Picinbono on circularity [14] and on the second-order statistics of complex signals described by the covariance and the relation functions (the last one is also named pseudo-covariance matrix) [15]. In the case of complex signals, and for a given order, cumulants depend on the considered relative number of complex conjugates. Hence, in [16], [17] two fourth-order cumulants based on different statistics are exploited in order to identify the mixing matrix. This is done in the context of blind separation of noncircular sources whatever the number of sources and sensors (within a limited range). In [18], [19], it was pointed out the interest of second order optimal widely linear filters for cyclo-stationarity signals under non-circularity conditions. In [20], the covariance and the relation matrices are used in order to perform the MUSIC-like estimation of direction of arrival for noncircular sources. Algorithms were proposed in the case of parametric entropy rate estimator using a widely linear autoregressive model [21] and in the case of second order statistics for blind separation of noncircular sources [22]. In [23], the NC-FastICA algorithm was proposed in order to improve the separation of noncircular sources. This algorithm, built under unitary constraint, is based on fixed-point update for the constrained optimization problem. The idea is to take into account the pseudo-covariance matrix, that is the second-order noncircular information, which is nonzero if the sources are second-order noncircular. Several functionals were proposed in the cost function. One of them is even motivated by kurtosis. Nevertheless it does not offer a cost function depending on all the fourth-order cumulants. A comparison to the NC-FastICA algorithm is done in the simulation experiment section.

In this paper, we have focused on fourth-order cumulants with complex valued sources. In this context, there are three different fourth-order cumulants without taking into account the conjugate cumulants. The idea is, that with noncircular signal, at least one of the two nonclassical cumulants is nonzero. So its use should lead to the improvement of the performance of the BSS. We have generalized the contrast proposed in [12] by using noncircular statistics.

The purpose of this paper is to provide

• a generalized contrast which exploits all the statistics available through the R-order cumulants in the context of blind noncircular signal separation,

• an original and optimal combination of the different statistics obtained through the minimization of the asymptotical error variance on the Jacobi parameters in the context of two sources,

• an adapted Jacobi-like algorithm in the context of noncircular sources (more than two sources) under unitary constraint.

The paper is organized as follows. First, in Section 2, recalling the BSS model, we have proposed to associate a contrast for each statistic available in the context of complex source signals. Thus, a new contrast is defined based on a combination of the different contrasts previously proposed. In Section 3, we have proposed, for each contrast, a Jacobi-like algorithm in order to optimize the corresponding criterion. And we have presented a generalized ICA algorithm based on the Givens rotation in the case of more than two complex valued sources. In Section 4, a statistical local study of the Jacobi angles is presented in the case of two sources in order to estimate an optimal combination of the available contrasts. It allows us to compare, in Section 5, different algorithms thanks to computer simulations. The latter illustrate the usefulness of considering an optimal combination of the different statistics in fourth-order cumulant based contrasts in the context of noncircular signals even in the case of more than two sources.