A learning algorithm for convolutive blind source separation with transmission delay constraint

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Abstract
A learning algorithm is proposed for fully recurrent convolutive blind source separation. Let s/sub i/(n) and x/sub j/(n) be the signal sources and the observations. H/sub ji/(z) expresses a transfer function from s/sub i/(n) to x/sub j/(n). It is assumed that the transmission delay time of H/sub ji/(z), j/spl ne/i is longer than that of H/sub ii/(z). In many practical applications, this assumption is acceptable. Based on this assumption, s/sub i/(n) in the output y/sub j/(n), j/spl ne/i of an unmixing block is cancelled through the feedback C/sub ji/(z) from the ith output to the jth observation. However, s/sub i/(n) in the output y/sub i/(n) cannot be cancelled, because a noncausal C/sub ij/(z) is required. A cost function E[q(y/sub j/(n))] can be used, where q is an even function with a single minimum point. The coefficients of C/sub ji/(z), i.e. c/sub ji/(l) are updated following a gradient descent method. The correction term is expressed uq/spl dot/[y/sub j/(n)]y/sub i/(n-l). q/spl dot/ is a partial derivative of q. Two-channel blind source separation has been simulated using speech signals. 100th- and 70th-order FIR filters are used for C/sub 12/(z) and C/sub 21/(z), respectively. The power ratio of the main signals and the cross-components is about 15 dB.

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