Accuracy of the estimation of the transfer function of the linear filter

Abstract

The identification problem of linear regression parameters is studied. Unknown impulse response is es- timated by Least Squares and Maximum Likelihood methods. The main application considered is the acoustic echo cancellation problem with about two thousand parameters to be estimated. The number of measurements is ten times more only and the problem is ill conditioned due to specific speech spectrum. The superfast Toeplitz inversion numerical algorithm is applied for estimation of the echo impulse response and echo transfer function. As a rule, speech signal has a poor spectrum that leads to singularity in matrix inversion. Accuracy of the transfer function estimate is explicitly expressed by initial data. Complexity of calculation of the accuracy is shown to be the same as complexity of FFT for the length of impulse response. Regularization of the matrix inversion problem is studied. An explicit formula for the loss of the cost function after regularization is presented. The underlying mathematical technique is based on theory of Szego and Schur polynomials. Application results of echo cancellation are shown. Refs 8. Figs 3.

Keywords:

parameters identification, computational complexity, superfast Schur algorithm, Szego polynomials