Discrete Stochastic Timefrequency Analysis and Cepstrum Estimation
Johan Sandberg
Centre for Mathematical Sciences
Mathematical Statistics
Lund University
2010
ISBN 9789162880804
LUTFMS10352010

Abstract:

The theory of stochastic timefrequency analysis of nonstationary 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 timelag 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 timefrequency theory, where we in Paper A compare three of the most common
definitions: the ClaasenMecklenbräuker, the Nuttall, and the JeongWilliams
ambiguity domain. We prove that amongst these three, only the JeongWilliams
ambiguity domain has the property that there exists a bijection between scaling
in this domain and convolution in the timelag 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 JeongWilliams
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 nonstationary 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
logperiodogram. 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."









