On the penalty factor for autoregressive order selection in finite samples

Fig. 1. Absolute value of the crosstalk with respect to the number of samples (NS) used to estimate the cross cumulants. Each point is the average of 10 experiments. to estimate the cross-cumulants. Each point in Fig. 1 corresponds to the average over 10 experiments, in which the mixing matrix is randomly chosen: The matrix entries mC3 (i # j) are random numbers in the range [-1, +1]. With 500 samples, a residual crosstalk of about-20 dB is obtained. In the case of nonstationary signals, cross-cumulant estimation must be done on few samples and has a larger variance. Consequently, it can lead to more inaccurate estimation of the mixing matrix. We still obtained an interesting performance: a residual crosstalk of about-15 to-20 dB, with various signals (colored noise, speech) and statistics estimated over 500 samples. In this correspondence, we proved that the mixing matrix can be. estimated using fourth-ordercross-cumulants, for two mixtures of two non-Gaussian sources. Solutions are obtained by rooting a fourth-order polynomial equation. Using second-order cross-cumulants allows us to simplify the method; the solution is then obtained by rooting two second-order polynomial equations and gives the result if one source is Gaussian. The methods are then quite simple, but its roots are very sensitive to the accuracy of the estimated cumulants. In fact, this direct solution is less accurate than indirect methods, especially adaptive a l g o r i b s. Moreover, we restricted the study to the separation of two sources, and theoretical solutions for three sources or more seems not easily tractable. However, in the case of two mixtures of two sources, it may give a good starting point with a small computation cost for any adaptive algorithm. REFERENCES J.-F. Cardoso, " Blind identification of independent signals, " in Proc.