# Discrete Stochastic Time-Frequency Analysis and Cepstrum Estimation

Sammanfattning: The theory of stochastic time-frequency analysis of non-stationary random processes has mostly been developed for processes in continuous time. In practice however, random processes are observed, processed, and interpreted at a finite set of time points. For processes in continuous time, the ambiguity domain has interesting properties which makes it particularly useful. One such property is that there exists a certain relationship between scaling in the ambiguity domain and convolution in the time-lag domain. For processes in discrete time, several different definitions of the ambiguity domain have been proposed. Paper A and B of this thesis contributes to the discretization of time-frequency theory, where we in Paper A compare three of the most common definitions: the Claasen-Mecklenbräuker, the Nuttall, and the Jeong-Williams ambiguity domain. We prove that amongst these three, only the Jeong-Williams ambiguity domain has the property that there exists a bijection between scaling in this domain and convolution in the time-lag domain. For processes in continuous time, there is also a certain mapping between the mean square error (MSE) optimal smoothing covariance function estimator and the MSE optimal ambiguity function estimator. This mapping allows us to compute the MSE optimal smoothing estimator in a convenient way. In Paper B, we prove that a similar relationship is not valid between the scaling estimators in the Jeong-Williams ambiguity domain and the smoothing covariance function estimators for processes in discrete time. However, we show that the MSE optimal smoothing covariance function estimator for a non-stationary random process in discrete time can be found as the solution to a linear system of equations. It allows us to find the lower MSE bound of this family of estimators. In Paper C, we show that it is possible to compute a covariance function estimator which is MSE optimal to a set of processes in order to increase the robustness. The cepstrum of a stationary random process has a lot of interesting applications. It is usually estimated as the Fourier transform of the log-periodogram. In Paper D, we propose a multitaper based estimator and we derive approximations of its bias and variance. We demonstrate the performance of the multitaper based estimator in a speaker verification task. In Paper E we discuss four different families of cepstrum estimators based on smoothing. We find the MSE optimal smoother in each family and the lower MSE bound of each family of estimators. The robustness of the optimal estimators within each family is also considered.

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