1 Laplacian Mixture Modeling for Overcomplete Mixing Matrix in Wavelet Packet Domain by Adaptive EM-type Algorithm and Comparisons Behzad Mozaffary Mohammad A. Tinati Faculty of Electrical and Computer Engineering Univercity of Tabriz 29 Bahman Blvd., Tabriz, East Azerbaijan IRAN Abstract-- Speech process has benefited a great deal from the wavelet transforms. Wavelet packets decompose signals in to broader components using linear spectral bisecting. In this paper, mixtures of speech signals are decomposed using wavelet packets, the phase difference between the two mixtures are investigated in wavelet domain. In our method Laplacian Mixture Model (LMM) is defined. An Expectation Maximization (EM) algorithm is used for training of the model and calculation of model parameters which is the mixture matrix. And then we compare estimation of mixing matrix by LMM-EM with different wavelet. Therefore individual speech components of speech mixtures are separated. Keywords: ICA, Laplacian Mixture Model, Expectation Maximization, wavelet packets, Blind Source Separation, Speech Processing 1. Introduction Blind source separation techniques using independent component analysis (ICA) have many potential applications including speech recognition systems, telecommunications, and biomedical signal processing. The goal of ICA is to recover independent sources given only sensor observation datum that are unknown linear mixtures of the unobserved independent source signals [1]–[6]. The standard formulation of ICA requires at least as many sensors as sources. Lewicki and Sejnowski [7], [8] have proposed a generalized ICA method for learning overcomplete representations of data that allows more basis vectors than dimensions in the input. Several approaches have been investigated to address the overcomplete source separation problems in the past. Lewicki [9] provided a complete Baysian approach assuming Laplacian source prior to estimating both the mixing matrix and the source in the time domain. Clustering solutions were introduced by Hyvarinen [10] and Bofill-Zibulesky [11]. Davies and Miltianoudis [12] employed modified discrete cosine transform (MDCT) to obtain a sparse representation. They proposed a two-state Gassian mixture model (GMM) to represent the source densities and the possible additive noise and used an expectation-maximization, (EM)-type algorithm, to perform separation with reasonable performance. In this paper, we explore the case of two-sensor setup with no additive noise, where the source separation problem becomes a one-dimensional optimal detection problem. The phase difference between the two-sensor data is employed. A Laplacian mixture model (LMM) is fitted to the phase difference between the two sensors, using an EM-type algorithm in each wavelet packet. The LMM model can be used for source separation and source localization. Since in the overcomplete model of source separation estimation of mixture matrix is very important in this paper, therefore we use LMM model for each wavelet packet with phase differences. Note that wavelet packets are obtained from decomposition of two mixtures. 2. Background Material Wavelets are transform methods that has received great deal of attention over the past several years. The wavelet transform is a time-scale representation method that decomposes signals into basis functions of time and scale, which makes it useful in Proceedings of the 5th WSEAS International Conference on Signal Processing, Istanbul, Turkey, May 27-29, 2006 (pp145-150)