Blind deconvolution, information maximization and recursive filters

Starting from maximizing information ow through a nonlinear neuron Bell and Sejnowski derived adaptation equations for blind deconvolution using an FIR lter [1]. In this paper we will derive a simpler form of the adaptation and we will apply it to more complex lter structures, such as recursive lters. As an application, we study blind echo cancellation for speech signals. We will also present a method that avoids whitening the signals in the procedure. 1. BLIND DECONVOLUTION Assume an unknown signal s convolved with an unknown lter with impulse response a (which can be any kind of a lter, for example, a causal FIR lter ak; k = 0; :::; La). The resulting corrupted signal x is a convolution x = a s. The task is to recover s by learning a lter w which reverses the e ect of lter a so that u = w x would be equal to the original signal s upto a delay and a constant. The corrupting lter spreads information from one sample st to all the samples xt; :::; xt+La . The task of blind deconvolution is now to remove these redundancies assuming that the samples of the original signal st are statistically independent. Some practical applications include blind acoustic echo cancellation, (where only the echo-corrupted signal is available) and suppression of intersymbol interference in communications (blind equalization) [3]. Several methods for blind deconvolution are based on the fact that if a source signal having a non-Gaussian PDF (probability density function) is convolved with a lter, the PDF of the resulting signal is closer to a Gaussian PDF due to the central limit theorem. Deconvolution can then be achieved by nding a lter which drives the output PDF away from a Gaussian. Functions of higher-order statistics, for example, kurtosis, can be used as a cost function to minimize/maximize [6, 5, 2, 3, 4]. Bell and Sejnowski formulated blind deconvolution as redundancy reduction between samples of data [1]. We will rst review their information maximization approach. By viewing their approach rather as shaping of the output PDF, we will show that the same learning rule can be achieved via a slightly simpler path. We will show how this facilitates learning more complicated lter structures for blind deconvolution. Finally, some experiments with blind acoustic echo cancellation will be presented. 2. INFORMATION MAXIMIZATION Bell and Sejnowski proposed to learn the restoring lter w by using an information theoretic measure [1]. In their con guration, w is a causal FIR lter.