Root tracking using time-varying autoregressive moving average models and sigma-point Kalman filters

Root tracking is a powerful technique that provides insight into the mechanisms of various time-varying processes. The poles and the zeros of a signal-generating system determine the spectral characteristics of the signal under consideration. In this work, time-frequency analysis is achieved by tracking the roots of time-varying processes using autoregressive moving average (ARMA) models in cascade form. A cascade ARMA model is essentially a high-order infinite impulse response (IIR) filter decomposed into a series of first- and second-order sections. Each section is characterized by real or conjugate pole/zero pairs. This filter topology allows individual root tracking as well as immediate stability monitoring and correction. Also, it does not suffer from high round-off error sensitivity, as is the case with the filter coefficients of the direct-form ARMA structure. Instead of using conventional gradient-based recursive methods, we investigate the performance of derivative-free sigma-point Kalman filters for root trajectory tracking over time. Based on simulations, the sigma-point estimators provide more accurate estimates, especially in the case of tightly clustered poles and zeros. The proposed framework is applied to real data, and more specifically, it is used to examine the time-frequency characteristics of raw ultrasonic signals from medical ultrasound images.