Geometrical Interpretation of the PCA Subspace Approach for Overdetermined Blind Source Separation
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Geometrical Interpretation of the PCA Subspace Approach for Overdetermined Blind Source Separation S. Winter,1, 2 H. Sawada,1 and S. Makino1 1 NTT
Communication Science Laboratories, NTT Corporation, 2-4 Hikaridai Seika-cho, Soraku-gun, Kyoto 619-0237, Japan of Multimedia Communication and Signal Processing, University of Erlangen-Nuremberg, 91058 Erlangen, Germany
2 Department
Received 25 January 2005; Revised 24 May 2005; Accepted 26 August 2005 We discuss approaches for blind source separation where we can use more sensors than sources to obtain a better performance. The discussion focuses mainly on reducing the dimensions of mixed signals before applying independent component analysis. We compare two previously proposed methods. The first is based on principal component analysis, where noise reduction is achieved. The second is based on geometric considerations and selects a subset of sensors in accordance with the fact that a low frequency prefers a wide spacing, and a high frequency prefers a narrow spacing. We found that the PCA-based method behaves similarly to the geometry-based method for low frequencies in the way that it emphasizes the outer sensors and yields superior results for high frequencies. These results provide a better understanding of the former method. Copyright © 2006 S. Winter 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.
1.
INTRODUCTION
Blind source separation (BSS) is a technique for estimating original source signals using only sensor observations that are mixtures of the original signals. If source signals are mutually independent and non-Gaussian, we can employ independent component analysis (ICA) to solve a BSS problem. Although in many cases equal numbers of source signals and sensors are assumed [1], the use of more sensors than source signals (overdetermined systems) often yields better results [2–4]. Different techniques are employed to map the mixture signal space to the output signal space with reduced dimensions. In this paper we present results for overdetermined BSS based on two different methods of subspace selection. Each provides better separation results than when the number of sensors and sources is the same. The first method utilizes the principal components obtained by principal component analysis (PCA) as described in [5]. The second method is based on geometrical selection, which depends on the frequency and sensor spacing as described in [6]. We compared the two methods by performing experiments with real world data in a reverberant environment. We found that for low frequencies the PCA-based method behaves similarly to the geometry-based method, and support this result analytically. For high frequencies the former method yields better results, since it normally removes the noise subspace more efficiently than the geometry-based
method. These results provide a better understanding of the PCA
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