By David F. Findley

ISBN-10: 0122564200

ISBN-13: 9780122564208

**Read or Download Applied Time Series Analysis II. Proceedings of the Second Applied Time Series Symposium Held in Tulsa, Oklahoma, March 3–5, 1980 PDF**

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**Extra resources for Applied Time Series Analysis II. Proceedings of the Second Applied Time Series Symposium Held in Tulsa, Oklahoma, March 3–5, 1980**

**Example text**

It is also possible for stability to fail under the same conditions. 29 Higher Dimensional Signal Processing Surely the last word has not been written on the filter stability problem for specified numerators (non-unity). How ever, it has been pointed out, that in any event, it would be unwise to try to implement any digital filter with a nonessential singularity of the second kind on |z| = |w| = 1 for obvious reasons. It is generally true that the filter must be represented (convolutionally) by an £-.

1979) have pointed out that essentially all known stability tests for multidimensional filters follow from a result of W. Rudin (Rudin, 1969), which we state here for reference. Theorem (Rudin) A polynomial p(z,w) is non-zero for |z| <_ 1, |w| <_ 1 if and only if (1) P(z,w) / 0 for |z| = | w| = 1 (2) P(z,z) ? 0 for |z| <_ 1 . We see that condition (2) reduces the problem to a standard one-dimensional problem over much of the bidisk and standard tests such as Schur-Cohn can then be invoked. Higher Dimensional Signal Processing 35 Delsarte, Genin, and Kamp have given the multidimensional extension of this result, which retains its form exactly, and have shown how standard stability tests are derivable from this result.

In this final section I thought that it would be of interest to consider the physical ramifications of dimension ality by examining the wave equation in different numbers of dimensions. Whatever the underlying mathematical properties may be, the effect of dimensionality shows up most clearly in the (scalar) wave equation. In order to motivate our discussion, let us consider the solutions of the initial value problem for the scalar wave equation in one, two, and three (spatial) dimensions. A. The One-Dimensional Wave Equation Consider the equation '-2 Utt = °: 56 James H.