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

A learning algorithm is proposed for fully recurrent convolutive blind source separation. Let s_i(n) and x_j(n) be the signal sources and the observations. H_ji(z) expresses a transfer function from s_i(n) to x_j(n). It is assumed that the transmission delay time of H_ji(z), j≠i is longer than that of H_ii(z). In many practical applications, this assumption is acceptable. Based on this assumption, s_i(n) in the output y_j(n), j≠i of an unmixing block is cancelled through the feedback C_ji(z) from the ith output to the jth observation. However, s_i(n) in the output y _i(n) cannot be cancelled, because a noncausal C_ij(z) is required. A cost function E[q(y_j(n))] can be used, where q is an even function with a single minimum point. The coefficients of C_ji(z), i.e. c_ji(l) are updated following a gradient descent method. The correction term is expressed uq˙[y_j(n)]y_i(n-l). q˙ 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 C₁₂(z) and C₂₁(z), respectively. The power ratio of the main signals and the cross-components is about 15 dB