Introduction

Blind source separation (BSS) is an important problem in the signal processing area, with a number of applications in communications, speech processing and biomedical signal processing. The goal of BSS is to recover the $n$ source signals from their $m$ observed linear or nonlinear mixtures [1,2].

The first BSS algorithms focused only on linear mixtures. Different approaches were taken depending on the number of mixtures, $m$, versus the number of sources, $n$. For the case where as many mixtures as unknown sources are available ($m=n$), a number of techniques have been developed. Most of them stem from the theory of independent component analysis (ICA) [3,4], a statistical technique whose goal is to represent a set of random variables as linear functions of statistically independent components. If there are more mixtures than sources ($m>n$), the redundancy in information can be used to achieve additional noise reduction [5].

On the other hand, if there are fewer mixtures than sources ($m < n$), we have an underdetermined BSS problem, which can only be solved if we rely on a priori information about the sources. Specifically, a number of algorithms that assume sparse sources have been proposed for underdetermined BSS [6,7,8].

A considerable amount of research has also been done on the so-called post-nonlinear BSS problem (PNL BSS), in which the sources are first mixed linearly and then transformed nonlinearly. For an equal number of mixtures and sources ($m=n$), some algorithms have been proposed [9,10,11,12]. However, these algorithms cannot deal with the more restricted problem of underdetermined PNL BSS. An underdetermined PNL BSS algorithm was recently proposed in [13]; it nevertheless requires the number of active sources at each instant to be lower than the number of mixtures $m$ and assumes noiseless mixtures. The approach presented in this letter relaxes these restrictions on the sources and it is able to work with noisy mixtures.

The rest of the letter is organized as follows: a description of the mixing process is given in Section II. The proposed algorithm consists of two major stages: the first one is a spectral clustering method, which is described in Section III. The second part, which deals with the estimation of the inverse nonlinearity through a set of multilayer perceptrons, is explained in Section IV, and in Section V simulation results are presented. Finally, Section VI summarizes the main conclusions of this work.