# Gro?ebner bases in multidimensional systems and signal processing.

Abstract (Summary)

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The theory of Gröbner bases for ideals and modules over a multivariate polynomial
ring, K[z1, z2, . . . , zn], when K is an arbitrary but fixed field and z1, z2, . . . , zn
are independent complex variables, is applied to solve several problems of interest in
multidimensional systems and signal processing.
In 1979, Youla and Gnavi explained the implications of zero, minor and factor
coprimeness in matrix-fraction descriptions of multivariate rational transfer matrices. In
1982, Guiver and Bose showed how primitive factorization of bivariate matrices can be
implemented via computations only in the ground field, i.e., an extension field is not required.
Since then several publications in the area emerged, including, most recently, the
papers of Park-Kalker-Vetterli, Fornasini-Valcher, and Lin. The tests for multivariate
polynomial matrix zero coprimeness and minor coprimeness are implemented algorithmically
while progress towards the difficult problem of multivariate matrix factor extraction
is also reported. The multivariate polynomial matrix factorization algorithm when some
conditions are satisfied is developed. Some remarks and limitations of the algorithm are
also reported.
The multivariate polynomial matrix factorization algorithm developed is applied
to the multidimensional filter bank design problem. This algorithm and another algorithm
for computing a globally minimal generating matrix of the syzygy of solutions
associated with a polynomial matrix are both associated with a zero coprimeness constraint
that characterizes perfect reconstruction filter banks. Generalizations as well
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as limitations of recent results which incorporate the perfect reconstruction as well as
the linear phase constraints are discussed with several examples and counterexamples.
Specifically, a Gröbner basis-based proof for perfect reconstruction with linear phase is
given for the case of two-band multidimensional filter banks and the algorithm is illustrated
by a nontrivial design example. This detailed design example and simulation
results are based on an embedded zerotree wavelet encoding algorithm. Progress and
bottlenecks in the multidimensional multiband case are also reported.
The Gröbner basis theory is then applied to the minimax controller design using
rate feedback, which was initiated by Bucy-Namiri-Velman in 1990. The complete analytic
characterization and solution construction for this problem is given when the plant
consists of a known fixed set of coupled oscillators of cardinality three (or order six). The
case when the plant order is eight is also considered based on the use of Gröbner bases.
The results are given in the compact triangular form of polynomial equations so that the
numerical solutions are readily computed by simple recursive substitution of the values
of variables, already computed. In general, the problem is analytically intractable and
suboptimal solutions based on numerical techniques are then the only recourse.
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Bibliographical Information:

Advisor:

School:Pennsylvania State University

School Location:USA - Pennsylvania

Source Type:Master's Thesis

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