Source separation techniques applied to speech linear prediction

The prediction filters are well known models for speech signal, in communications, control and many others areas. The classical method for deriving linear prediction coding (LPC) filters is often based on the minimization of a mean square error (MSE). Consequently, second order statistics are only required, but the estimation is only optimal if the residue is independent and identically distributed (iid) Gaussian. However, if the residue is not Gaussian, the estimation is no longer optimal. If one knows the theoretical statistics, it is possible to improve the estimation by using optimal (odd value higher order) statistics. Otherwise, i.e. if the statistics is not known, one can wonder how to implementing a quasi-optimal estimation. In this paper, we derive the ML estimate of the prediction filter. Relationships with robust estimation of auto-regressive (AR) processes, with blind deconvolution and with source separation based on mutual information minimization are shown. The algorithm, based on the minimization of a high-order statistics criterion, uses on-line estimation of the residue statistics. Improvements in the experimental results with speech signals emphasize on the interest of this approach. 1. CLASSICAL LPC The classical LPC methods are based on the minimization of a mean square error, defined as the difference between the input signal x k ( ) and the predicted signal ) 1 ( )] ( [ ) ( − = k x z w k y , where w z ( ) is a L -th order causal finite impulse response filter, i.e. a filter whose entries wi = 0 for 1 , , 0 − ∉ L i Κ . The block diagram of a linear predictor is shown in Fig. 1. Denoting E x k x k l R l xx [ ( ) ( )] ( ) − = , the cost function reduces to: ( ) [ ] ( ) ( ) ( ) ∑ ∑ ∑