Identification of non-minimum phase systems using evolutionary spectral theory

Abstract

In this paper we present an application of the Wold-Cramer representation of non-stationary processes to the identification of non-minimum phase linear time-invariant (LTI) systems and modeling of non-stationary signals. According to the Wold-Cramer representation, a non-stationary signal can be expressed as an infinite sum of sinusoids with time-varying random magnitudes and phases. For the identification of LTI systems with non-stationary input and output, we relate the Wold-Cramer representations of the system's input and output to obtain estimates of the magnitude and phase frequency responses of the system. Our procedure uses estimators from the evolutionary periodogram recently introduced, permits the identification of non-minimum phase systems, and provides also rational estimates. Furthermore, we show that when the system output is corrupted by additive stationary noise it is possible to avoid the effects of the noise in the identification. If the noise is non-stationary and Gaussian, we show that it can be removed by considering the evolutionary bispectrum, recently introduced. The analysis also provides a model for a non-stationary signal, as the output of the cascade of a linear time-varying (LTV) and an LTI systems with white noise as input. To illustrate the performance of our procedure, we present simulations of the identification of non-minimum phase LTI systems with noisy outputs.