Computer algebra and signal processing

A new area in mathematics called computer algebra is rapidly evolving to implement old, but very useful, algorithms. In this thesis, we will focus on Buchberger's algorithms, which have a wide variety of applications, including stability of two-dimensional (2-D) digital filters.; Since a filter cannot be used in practice unless it is stable, determining its stability is a very important aspect of signal processing. However, stability tests for such filters often involve a great deal of computation.; The criteria developed in this thesis apply directly to the transfer function of the filter, which is a ratio of two relatively prime polynomials. The first criterion locates the zeros of the denominator polynomial when the polynomial has real coefficients. The second criterion generalizes the first one to include the case of complex coefficients in the denominator polynomial. The computation required to use these tests can be carried out with computer algebra systems such as Maple and Mathematica. The use of these criteria is demonstrated by examples.