Wiener and Kalman filtering for self-similar processes

In this thesis, we formulate Wiener and Kalman filtering for the optimal estimation and prediction of self-similar processes. Our algorithms are based on the scale stationary processes framework which provides a mathematically tractable system theoretic model enabling the development of many signal processing techniques. In this framework, self-similar processes are modeled using scale stationary Auto Regressive Moving Average (ARMA) models. It has been shown that such continuous-time ARMA systems can be processed by generalized Mellin and Scale Transforms, analogous to the use of the Laplace and Fourier Transforms in the continuous-time processing of ordinary stationary ARMA systems. In this work, we first attempt to discretize these models by using exponential sampling and then formulate discrete time algorithms for discrete time processing. These discrete time techniques are used in the implementation of the optimal Wiener filtering algorithm obtained for self-similar processes using least squares techniques. Wiener filtering algorithm in Discrete Generalized Scale Transform (DGST) domain is defined in terms of scale power spectral densities. Hence, we develop a periodogram-like method for the estimation of the power spectrum of self-similar processes using DGST. Wiener filtering algorithm for the restoration of self-similar processes is tested on simulation examples. Investigation of the state space analysis is needed to fully develop the self-similar ARMA processes framework. First, a general state space representation of self-similar ARMA processes is obtained based on first order time varying ordinary differential equations. It is shown that the major difference of this representation is in the scale memory content. Here, we introduced new concepts, i.e., the "multivariate self-similarity" of the states which is captured in the "self-similarity matrix". In this state space representation, the self-similarity of the outputs is represented as a linear combination of self-similar states having different self-similarity parameters. Secondly, we develop a recursive predictive estimation algorithm in the form of a Kalman filter for self-similar processes using the proposed state space representation. Simulation examples suggest that the proposed algorithm is superior to the traditional Kalman filtering technique when the input and the output processes are self-similar.