On the existence of a class of invertible FIR filters for spectral shaping

This paper presents a proof for the existence of a class of LTI causal IFIRF converting spectrum of an arbitrary stationary input process to an output with prescribed set of normalized autocorrelation samples. The input is an MA, AR, or ARMA process of finite order, with or without additive white noise, whose stable model may or may not be minimum phase. The prescribed values of finite number of output autocorrelation lags (except lag zero) may or may not all be equal to zero. The FIR filter is minimum phase and of finite order. It is derived from the predefined input and output autocorrelation samples directly using no intermediate filtering stage or minimization of a cost function. The filters' output autocorrelation lags match the prescribed values precisely. Such a filter provides an alternative solution to the problem of finding a causal IFIRF for spectral shaping in statistical signal processing applications.