A Note on the Pseudo-Spectra and the Pseudo-Covariance Generating Functions of ARMA Processes

Although the spectral analysis of stationary stochastic processes has solid mathematical foundations, this is not the case for non-stationary stochastic processes. In this paper, the algebraic foundations of the spectral analysis of non-stationary ARMA processes are established. For this purpose the Fourier Transform is extended to the field of fractions of polynomials. Then, the Extended Fourier Transform pair pseudo-covariance generating function / pseudo-spectrum, analogous to the Fourier Transform pair covariance generating function / spectrum,is defined. The new transform pair is well defined for stationary and non-stationary ARMA processes. This new approach can be viewed as an extension of the classical spectral analysis. It is shown that the frequency domain has some additional algebraic advantages over the time domain.