Blind source extraction based on generalized autocorrelations and complexity pursuit

Abstract

Blind source extraction (BSE) is a special class of blind source separation (BSS) method. Due to its low computation load and fast processing speed, BSE has become one of the promising methods in signal processing and analysis. This paper addresses BSE problem when a desired source signal has temporal structures. Based on the generalized autocorrelations of the desired signal and the non-Gaussianity of its innovations, we develop an objective function. Maximizing this objective function, we present a BSE algorithm and further give its stability analysis in this paper. Simulations on image data and electrocardiogram (ECG) data indicate its better performance and the better property of tolerance to the estimation error of the time delay.

Introduction

The problem of blind source separation (BSS) [1], [2], [3], [4] forms part of the independent component analysis (ICA), which is an active research field that has attracted great interest in the field of biomedical signal analysis and processing, geophysical data processing, data mining, speech analysis and image recognition, etc. [5], [6], [7], [8], [9], [10], [11], [12]. BSS is a powerful statistical technique in blind signal processing, which considers the simultaneous recovery of all the independent components from the observations. However, in practice, extracting all the source signals from a large number of observed sensor signals, for example, a magnetoencephalographic (MEG) measurement which may output hundreds of recordings, can take a long time and in these signals only a very few are desired with given characteristics. Therefore it is necessary to develop an reliable, robust and effective method to obtain only the desired signals that contain useful information. For this purpose, another technique—blind source extraction (BSE)—was proposed, which has become one of the promising methods due to its low computation load and fast processing speed. Many source extraction algorithms [1] can extract a specific signal as the first output, in which some special characteristics of the signals have to be exploited, such as linear predictability or smoothness [1], [13], non-Gaussianity [14], [15], sparseness [16] and generalized autocorrelations [17], etc.

When the desired source signals are periodic or quasi-periodic, one convenient way to exploit this property is to employ a linear predictor within the BSE structure. Many source extraction algorithms have considered this case. For example, Barros and Cichocki [13] provided a simple algorithm (simplified with “BCBSE” algorithm) which can quickly extract a desired source signal with a specific period. This algorithm in all the cases could extract the desired sources, whether they are colored as long as they are decorrelated and have temporal structures. However, this method only carries out the constrained minimization of the mean squared error, which cannot well describe the probability distribution of the innovations of the signals. In addition, it needs a prior information about the optimal time delay and is very sensitive to the estimation error of the time delay. To overcome these drawbacks, Shi et al. [15] developed a BSE algorithm (simplified with “SemiBSE” algorithm), which is based on the non-Gaussianity and the autocorrelations of the source signals and contains the mean squared error objective function presented by Barros and Cichocki [13]. This method improves the performance of BCBSE algorithm, and its tolerance to large estimated errors of the period makes the desired signal extracted robustly. However, it must be noted that its better tolerance lies the fact that the choice of initial weight is not random but unit vector. When we initialize the weight randomly, SemiBSE algorithm becomes more sensitive to the estimation error of time delay.

The alternative approach, which addresses the BSE problem when a desired source signal has linear or nonlinear autocorrelations, was first introduced in literature [17]. Based on the generalized (i.e., linear or nonlinear) autocorrelations of the primary sources, authors proposed a BSE algorithm (called “GABSE” algorithm). It has been shown that this method has good stability and linear convergence speed. Furthermore, the convergence is global except for a zero measure region. This method only assumes the sources are decorrelated with each other and every source has different temporal structures, but dose not necessarily have to be statistically independent. GABSE algorithm has been applied to many cases directly and the performance is satisfying to a certain extent. Whereas, it also suffers some disadvantages, for example, its tolerance to estimated errors of time delay is not very robust.

In this article, we aim to develop an efficient algorithm, which incorporates the generalized autocorrelations of the desired signal and the non-Gaussianity of its innovations, for extraction of the desired signal. Based on these priori special characteristics, we first develop an objective function. Then a gradient BSE algorithm for finding a maximum of this objective function is proposed. Numerical computation and theoretical analysis show that such a combination both improves the performance of existing algorithms and has better robustness to the estimated error of time delay.

This manuscript is organized as follows. The next section describes the objective function and proposes the BSE algorithm, then gives its stability analysis. Section 3 demonstrates the present technique with experiments using image data and electrocardiogram (ECG) data. The final section provides some discussions and conclusions.