Parametric Modeling of Random Signals
This chapter deals with parametric modeling of random signals. Initially, we demonstrate the Paley-Wiener condition on the power spectral density of a signal. If this condition is verified, it is possible to factor the z power-density of a signal in the form of a product where appears the transfer function of a causal and stable system which has a causal and stable inverse. In that case, the noise is called regular. It appears that a regular random process can be seen as the output signal of a minimum phase filter driven by white noise. In the following, we study the filtering of white noise by an ARMA filter. We arrive at the Yule-Walker equations system connecting the values of the correlation function of the output signal to the filter coefficients. These equations make it possible to extrapolate the correlation function beyond the time interval used as the basis of the system of equations, or, in the case where the filter coefficients are unknown, to determine the coefficients of this filter. Calculating the coefficients of the MA part of the filter is delicate; one often seeks a more simple representation of a regular noise by an AR model. Then we arrive at a smoothed estimate of the power spectral density of the noise. The chapter concludes by modeling a regular noise by MA filtering of a white noise.