Applied Time Series Analysis II: Proceedings of the Second by David F. Findley

By David F. Findley

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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

We have already analyzed the extension of this method to higher dimensions in Section II. We have seen that the methods which are computationally tractable may result in unstable prediction error filters. On the other hand, if we follow Marzetta and guarantee stability of the predictor, 50 James H. Justice then we can no longer perform the (infinite) computations in most cases of interest. We are faced, then, with a rather severe problem in obtaining a multidimensional spectral esti­ mator by the least-squares approach.

FILTER REALIZATION Shanks (Shanks, 1967) first presented an algorithm for realizing a recursive filter. This algorithm (see below) has been widely used for the implementation of recursive filters, and for some time, no one really gave the problem much thought. As Jim Kaiser recently remarked, (Kaiser, 1980) (paraphrase) 'we used to be happy to get any solution. ' There is no doubt that processing multidimen­ sional data is a costly and time-consuming process. We would do well, then, to consider algorithms which are as cost effective as possible.

The Recursive Filter Stability Problem II. III. The Least-Squares Inverse Problem The Filter Realization Problem IV. The Spectral Estimation Problem II. THE RECURSIVE FILTER STABILITY PROBLEM A recursive digital filter is one whose (multivariable) Z-transform is of rational form, that is, the quotient of two (multivariable) polynomials. It has long been understood that if one passes a bounded input signal through a recursive 26 James H. Justice filter, the output may or may not be bounded. In practice, we require filters to have bounded outputs when excited by bounded inputs (Bounded Input-Bounded Output or BIBO sta­ bility).

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