Journal of Sensors

Volume 2015 (2015), Article ID 964098, 7 pages

http://dx.doi.org/10.1155/2015/964098

## Variable Step-Size Method Based on a Reference Separation System for Source Separation

College of Communications, PLA University of Science and Technology, Nanjing 210007, China

Received 9 December 2014; Revised 7 April 2015; Accepted 9 April 2015

Academic Editor: Jian-Nong Cao

Copyright © 2015 Pengcheng Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Traditional variable step-size methods are effective to solve the problem of choosing step-size in adaptive blind source separation process. But the initial setting of learning rate is vital, and the convergence speed is still low. This paper proposes a novel variable step-size method based on reference separation system for online blind source separation. The correlation between the estimated source signals and original source signals increases along with iteration. Therefore, we introduce a reference separation system to approximately estimate the correlation in terms of mean square error (MSE), which is utilized to update the step-size. The use of “minibatches” for the computation of MSE can reduce the complexity of the algorithm to some extent. Moreover, simulations demonstrate that the proposed method exhibits superior convergence and better steady-state performance over the fixed step-size method in the noise-free case, while converging faster than classical variable step-size methods in both stationary and nonstationary environments.

#### 1. Introduction

Blind source separation (BSS) aims at extracting the latent unknown source signals from their observed mixtures by an array of sensors without a priori knowledge of the original source signals and the mixing coefficients. In the separating process, nothing can be used except for the observation sequences and the statistical characteristic assumptions of the sources. This makes BSS become a versatile tool used in many multisensor systems such as antenna arrays in acoustics or electromagnetism, chemical sensor arrays, and electrode arrays in electroencephalography [1].

Several optimization algorithms have been proposed for BSS [2] and can be generally categorized into batch-based algorithms and adaptive (sequential) algorithms. Batch-based algorithms are block-wise and will not work until a block of data samples is received, such as the fast fixed-point algorithm [3]. In this paper, we consider the latter, which have particular practical advantages due to their computational simplicity and latent ability in tracing a nonstationary environment [4].

However, the traditional adaptive BSS algorithms such as equivariant adaptive separation via independence algorithm (EASI) [5] and natural gradient algorithm (NGA) [6] usually assume that the step-size is a small positive constant, leading to an inevitable conflict between the learning rate and stability performance, that is, slow convergence speed or large steady state error. A simple way to solve the conflict is reducing the learning rate as the iteration goes on [7, 8], but it brings about another new problem: if the learning rate decreases to be too small before source components are extracted, the separation system will fail to separate sources properly. To improve the learning rate and stability performance, variable step-size algorithms have been proposed. The variable step-size algorithms can exploit the online measurements of the state of the separation system from the outputs and the parameter updates. In [4, 9, 10], variable step-size algorithms have been derived according to the gradient of different contrasts, that is, NGA, EASI, and S-NGA algorithms. Zhang et al. put forward a grading learning algorithm based on the measurements of correlation of the separating signals, whose learning rate is updated by the state of separating [11]. Hsieh et al. proposed an effective learning rate adjustment method based on an improved particle swarm optimizer [12]. But the separating performance of these variable step-size algorithms is usually sensitive to the initial parameter settings. As a result, the convergence is still slow and improper initial value of learning rate results in large steady state error or even divergence. Ou et al. proposed a variable step-size algorithm based on an auxiliary separation system [13]. The step-size is updated by estimating a pseudo-performance index in the light of the index descending in an exponential form. Compared to classical variable step-size methods, the separation performance of Ou’s method is less sensitive to the initial settings.

In order to improve the initial convergence and stability performance, we consider using a reference separation system based on MSE of the instantaneous outputs to update the step-size. This technique is shown to improve the convergence speed and the steady-state performance. Moreover, the use of “minibatches” can reduce the whole computational load of the algorithm. The remainder of this paper is organized as follows. In Section 2, the principle of adaptive source separation methods is briefly summarized. Our algorithm is proposed in Section 3. Numerical stimulation results and discussion are provided in Section 4. At the end of the paper, a concise conclusion is given. What is more, this paper can be regarded as an important complement for Ou’s method in [13].

#### 2. Adaptive Algorithms for BSS

In the noise-free instantaneous case, we assume that unknown statistically independent zero mean source signals, with at most one having a Gaussian distribution, contained within pass through an unknown mixing system ; therefore mixed signals can be modeled aswhere is the time index and is the vector transpose operator. To simplify the problem, we further assume that the number of sources matches the number of mixtures, that is, , an exactly determined problem.

The blind separation problem is then to recover original source signals from observations , which is equivalent to estimate an separating matrix that performs the inverse operation of the mixing process, as subsequently used in separation model. Figure 1 shows a block diagram of adaptive BSS model. Then the output signal vector is obtained:where is an estimate of to within the well-known permutation and scaling ambiguities.