Estimation of the instantaneous amplitude and frequency of non-stationary short-time signals

We consider the modeling of non-stationary discrete signals whose amplitude and frequency are assumed to be nonlinearly modulated over very short-time duration. We investigate the case where both instantaneous amplitude and frequency can be approximated by orthonormal polynomials. Previous works dealing with polynomial approximations refer to orthonormal bases built from a discretization of continuous-time orthonormal polynomials. As this leads to a loss of the orthonormal property, we propose to use discrete orthonormal polynomial bases: the discrete orthonormal Legendre polynomials and a discrete base we have derived using Gram-Schmidt procedure. We show that in the context of short-time signals the use of these discrete bases leads to a significant improvement in the estimation accuracy. We manage the model parameter estimation by applying two approaches. The first is maximization of the likelihood function. This function being highly nonlinear, Preprint submitted to Elsevier Science 21 September 2007 ha l-0 02 00 11 5, v er si on 1 20 D ec 2 00 7 Author manuscript, published in "Signal Processing 88, 7 (2008) 1636-1655"