Thesis by: Shimon Peleg
Advisor:   Benjamin Friedlander
           Dept. Electrical and Computer Engineering
           University of California
           Davis, CA 95616
 
 
 
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%{\bf Shimon Peleg}\\
%{\bf September 1993}\\
%{\bf Electrical $\&$ Computer Engineering}
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{\bf Estimation and Detection with the Discrete Polynomial Transform} 
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{\underline {\bf Abstract}}
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Signals with continuous-phase and constant or slowly varying amplitude are 
common in engineering applications. This dissertation introduces
the {\em discrete polynomial transform} (DPT), and its applications in
estimating the
parameters of these kinds of signals. We show that the DPT-based
estimation algorithm offers a suboptimal but computationally efficient
alternative to the {\em maximum likelihood} algorithm.
 
The basic property of the DPT of order $M$ is that when it is applied to a
signal with constant-amplitude and polynomial-phase of order $M$ the result 
is a spectral line whose frequency is related to the highest-order 
coefficient of the polynomial. We use this property to estimate all of 
the signal parameters.
 
We derive in this dissertation theorems and properties of the DPT. Then
we introduce a DPT-based estimation algorithm for estimating the phase 
coefficients of constant-amplitude polynomial-phase signals. We give
a statistical analysis of the estimates of the highest order phase-coefficient 
and the amplitude. We consider a criterion for detection of these kinds 
of signals and determine the required SNR for the algorithm to
operate.  Also, we analyze the case of time-varying amplitude 
polynomial-phase signals. 
 
In some applications we are interested in the instantaneous phase (IP)
or instantaneous frequency (IF) of signals. In this dissertation we present
a method of obtaining estimates of the IP and IF of polynomial-phase
signals using a DPT-based estimation algorithm. Also,
using the fact that any continuous function on
a closed interval can be uniformly approximated by polynomial, 
we show how to use the DPT-based algorithm to estimate 
the IP and IF of the more general case of constant-amplitude and 
continuous-phase signals. 
 
We extend the algorithm to the case of multiple component signals. We
give a DPT-based algorithm for parameter estimation for this case
along with illustrative examples and Monte-Carlo simulations. We 
show that we can separate multiple-polynomial phase component
signals, and estimate the parameters of each component even though they
are observed at the same time and overlap in the frequency domain.
 
The DPT-based estimation algorithm is shown to be robust
to deviations from the constant-amplitude polynomial-phase
signal model, such as time-varying amplitude
and non-polynomial (but continuous) phase. 
 
In this dissertation we also develop tools for analyzing the algorithms. 
Using these tools we show that the accuracy with which the parameters are 
estimated by the algorithms are close to the CRB.
 
The use of the DPT appears to be a practical approach to the analysis
of an important class of non-stationary signals. In this dissertation
we discuss a few of the many possible applications of the
DPT-based estimation algorithm: two applications to radar and sonar and
detailed analysis of the signal transmitted by the brown bat.