A Curtosis Based Criterion for Solving the Permutation Ambiguity in Convolutive Blind Source Separation

In this work, we present a modification of an algorithm for solving the permutation ambiguity in convolutive blind source separation. A well known approach for separation of convolutive mixtures is the transformation to the time-frequency domain, where the convolution becomes a multiplication. With this approach it is possible to use well known instantaneous ICA algorithms independently in each frequency bin. This simplification leads to reduced computational costs and better separation in each frequency bin. However, this simplification has the major drawback of arbitrary permutation in each frequency bin. Without a correction of this permutation the restored time domain signals still remain mixed. An often used approach for solving this permutation problem is the dyadic sorting, where groups of bins are consecutively depermuted. By recursively joining growing groups all bins gets sorted. In recent works we presented a criterion for the depermutation, which was based on sparsity in the time domain of the restored subband signals. In this work we modify this approach to use a curtosis based criterion which is an alternative measurement for the non-gaussianity of speech signals. Introduction Blind Source Separation (BSS) of linear and instantaneous mixtures can be performed using the Independent Component Analysis (ICA). For this case, numerous algorithms have been proposed [1, 2]. When dealing with real-world recordings of speech, this simple approach is not effective anymore. As the signals arrive multiple times with different delays, the mixing procedure becomes convolutive. These characteristics can be modeled using FIR filters. In this case, the separation is only possible when the unmixing system is again a set of FIR filters. As the direct calculation of the unmixing fitlers in time domain is very demanding, time-frequency approaches are often used. Here, the convolution becomes a multiplication and each frequency bin can be separatated using an instantaneous method. However, this simplification has a major disadvantage. The separated signals usually have arbitrary scaling and are randomly permuted across the frequency bins. Without the correction of the scaling, only filtered versions of the signals are restored. This ambiguity is often solved using the minimal distortion principle [4]. This method accepts the filtering done by the mixing system without adding new distortions. The random permutation of the single frequency bins has an even bigger impact. Without a correct alignment, different signals appear in the single outputs and the whole process fails. Many different approaches for solving this problem have been proposed. Often, the time structure of the separated bins is used and the assumption of high correlation between neighboring bins is utilized. This has been used for example in [3] and [8]. Other approaches include a statistical modeling of the single bins using the generalized Gaussian distribution. Small differences of the parameters lead to a depermutation criterion in [5] and [6]. The method from [8] introduced the so called dyadic sorting, where in every stage growing sets of bins are depermuted using the correlation method. Using multiple bins for comparison resulted in a more robust criterion. This approach has been extended in [7], where time domain signals for the sets of bins have been used. This modification allows for a evenmore robust criterion, as only one coefficient has to be considered for the depermutation. Additionally, in [7] a sparsity based criterion has been introduced. In this work we modify this approach. The calculation of the sparsity of the time domain signals can be interpreded as an measurement of non-gaussianity. Here, we propose to use a kurtosis based criterion which is an alternative measurement for the non-gaussianity. BSS for instantaneous mixtures The instantaneous mixing process of N sources into N observations is modeled by an N × N matrix A. With the source vector s(n) = [s1(n), . . . , sN (n)] and negligible measurement noise, the observation signals x(n) = [x1(n), . . . , xN (n)] T are given by x(n) = A · s(n). (1) The separation is again a multiplication with a matrixB: