Dissertation Time - Frequency - Autoregressive - Moving - Average Modeling of Nonstationary Processes

This thesis introduces time-frequency-autoregressive-moving-average (TFARMA) models for underspread nonstationary stochastic processes (i.e., nonstationary processes with rapidly decaying TF correlations). TFARMAmodels are parsimonious as well as physically intuitive and meaningful because they are formulated in terms of time shifts (delays) and Doppler frequency shifts. They are a subclass of the well-known time-varying ARMA models with basis function expansion where the basis function set is given by the complex exponentials. The resulting mathematical structure allows for efficient estimation algorithms. We propose TFARMA models for nonstationary scalar processes and also two important special cases, namely, the TFAR and TFMA models. For all three model types, we derive several parameter and order estimation techniques based on the underspread property. From a practical viewpoint, the best TFAR parameter estimator is the TF-Yule-Walker method, which requires the inversion of a structured (Toeplitz-like) matrix. Such TF-Yule-Walker methods are available for all three discussed model types; however, for TFMA and TFARMA parameter estimation these fast methods are based on the results of a preceding TFAR analysis of the data. We also discuss other parameter estimators based on a novel TF cepstrum or on the least-squares or maximum likelihood principles. For order estimation, various information criteria are considered. Also, the issue of model stability is addressed. Using the concept of time-varying poles and zeros, an algorithm for model stabilization is developed. For the analysis of vector (multivariate) processes, we introduce vector-TFAR (VTFAR) models and a corresponding parameter estimation method that is also based on the TF-Yule-Walker approach. We furthermore discuss VTFAR models with banded parameter matrices, which are appropriate for vector processes exhibiting correlations only between some neighboring components. Finally, we verify the applicability of the discussed models in several fields of signal processing by analyzing some natural and artificial (vector) signals. Model-based spectral estimates are computed, and the application of the TFAR model to the prediction of nonstationary processes is discussed. Based on a recently proposed statistical description of mobile radio channels that do not fulfill the wide-sense stationary uncorrelated scatterers (WSSUS) assumption, we formulate a long-term model for non-WSSUS mobile radio channels based on VTFAR modeling. Finally, a stationarity test for vector signals is presented.