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   -> Volume 2, Issue 17

Ph.D. abstract
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Author Message (Milos Doroslovacki)

PostPosted: Fri Nov 12, 1993 12:00 am    
Subject: Ph.D. abstract
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Ph.D. abstract


TITLE: Discrete-Time Signals: Uncertainty Relations, Wavelets, and Linear
System Modeling
AUTHOR: Milos Doroslovacki
ADVISOR: H. Howard Fan
GRANTING INSTITUTION: University of Cincinnati
ACCEPTANCE DATE: November 12, 1993
INFORMATION: Contact the author at Department of Electrical and Computer
Engineering, University of Cincinnati, Cincinnati,
Ohio 45221-0030, Tel: (513)556-6297, Fax: (513)556-7326,


A generalized uncertainty relation of the Heisenberg's type for signals is
derived and optimal signals, which reach the uncertainty limit are found. The
requirement for convolution invariance between optimal signals offers, as
unique solutions, the classical uncertainty relation for continuous-time
signals and a new one for the discrete-time case. Optimal discrete-time signals
are obtained and related in a limit case to Gaussian continuous-time signals.
The discrete-time uncertainty relation is given for any point in the
time-frequency plane. Second moments in time and frequency involved in the
relation, and whose product gives a measure of joint time-frequency
localization, are physically interpreted. A few more uncertainty relations
involving ordinary second moments are found.

Discrete-time wavelets are presented in a systematic and consistent way.
Sufficient and necessary conditions for different types of orthogonality,
relations to continuous-time wavelets, shift invariance of wavelet
representations, and relations to binary subband decomposition/reconstruction
of signals are addressed. Second moments characterizing localization in time
and frequency are found for a discrete-time scaling function generated by an
optimal signal. In a limit case, discrete-time scaling functions and wavelets
generated by optimal signals, tend towards continuous-time Gaussian signals. It
is possible to diminish the asymmetry of orthonormal continuous-time wavelets
by using the second moment in time as a criterion for the choice of
discrete-time wavelet generating filters.

Theoretical settings for the modeling of discrete-time linear systems by
wavelets are derived in time-invariant and time-varying cases. System
identification minimizing the mean square output error is studied. Optimal
coefficients and the corresponding minimum of the error are found. Least mean
square adaptive filtering algorithms are derived for on-line filtering and
system identification. Theoretically and by simulations the advantages of using
wavelet-based filtering are shown: faster convergence, smaller output error,
easy interpretation of modeling error. In the time-varying case adaptive
coefficients can tend to constants. A system-identification adaptive approach
is built for on-line and simultaneous detection and range and velocity
estimation of targets in radar and sonar applications.
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