Blind Deconvolution in Nonminimum Phase Systems Using Cascade Structure
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Research Article Blind Deconvolution in Nonminimum Phase Systems Using Cascade Structure Bin Xia and Liqing Zhang Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai 200030, China Received 27 September 2005; Revised 11 June 2006; Accepted 16 July 2006 Recommended by Andrzej Cichocki We introduce a novel cascade demixing structure for multichannel blind deconvolution in nonminimum phase systems. To simplify the learning process, we decompose the demixing model into a causal finite impulse response (FIR) filter and an anticausal scalar FIR filter. A permutable cascade structure is constructed by two subfilters. After discussing geometrical structure of FIR filter manifold, we develop the natural gradient algorithms for both FIR subfilters. Furthermore, we derive the stability conditions of algorithms using the permutable characteristic of the cascade structure. Finally, computer simulations are provided to show good learning performance of the proposed method. Copyright © 2007 Hindawi Publishing Corporation. All rights reserved.
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INTRODUCTION
Recently, blind deconvolution has attracted considerable attention in various fields, such as neural network, wireless telecommunication, speech and image enhancement, biomedical signal processing (EEG/MEG signals) [1–4]. Blind deconvolution is to retrieve the independent source signals from sensor outputs using only sensor signals and certain knowledge on statistics of source signals. A number of methods [2, 5–13] have been developed for the blind deconvolution problem. For blind deconvolution problem in minimum phase systems, causal filters are used as demixing models. Many algorithms work well in learning the coefficients of causal filters, such as the second-order statistical (SOS) approaches [2, 5–11, 13], higher-order statistical (HOS) approaches [2, 5, 9, 10], and the Bussgang algorithms [6–8, 14]. In the real world, the mixing systems are usually nonminimum phase. To deal with the blind deconvolution problem in nonminimum phase systems, Amari et al. [15] used doubly sided infinite impulse response (IIR) filters as demixing model. To our knowledge, it is still a difficult task to develop a practical algorithm for doubly sided IIR filters. To simplify the problem of blind deconvolution, some researchers introduced the cascade structure for demixing filter. In [16], Douglas discussed a cascade structure for multichannel system. The main idea of cascade structure is to divide the difficult task into several easy subtasks. By intro-
ducing this idea in blind deconvolution, we can decompose the demixing filter into subfilters to recover the counterparts in mixing system. Labat et al. [17] presented a cascade structure for single channel blind equalization. Zhang et al. [18] provided a cascade structure to multichannel blind deconvolution. Waheed and Salam [19] discussed several cascade structures for blind deconvolution problem. Theoretically, a nonminimum phase system can be decomposed into a minimum phase subsystem and a corresponding maxi
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