By David F. Findley
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Extra info for Applied Time Series Analysis II. Proceedings of the Second Applied Time Series Symposium Held in Tulsa, Oklahoma, March 3–5, 1980
For simplicity, we shall consider a two-dimensional case, and refer to the literature for the extensions to other cases. (S) where S is the cross product of the set of non-negative integers with itself. (S). (S). Having thus converted the problem to an equivalent one of £-,(S) invertibility, we may take advantage of the fact that L ( S ) is a Banach algebra with convolution Higher Dimensional Signal Processing 27 product and a generalized form of the Wiener Tauberian theorem can be used to obtain our result.
0 for |z| £ 1, |w| £ 1 . One then utilizes appropriate algorithms when applying such filters in order to guarantee stability. While this method does extend to the multidimensional case, we note some significant departures from the character istics of the single variable case. If we consider (unstable) filters whose support is a quarter plane, for example, we see that the spectral factorization may yield factors whose support is not in the same quarter plane as the original filter. Murray (Murray, 1978) considered this problem and gave quite severe necessary conditions on the amplitude response of a (unstable) quarter plane filter in order that it have a stable polynomial representation with support in the same quarter plane as the original filter.
Since 30 James H. Justice R(z) is the Z-transform of an autocorrelation function, z 0 is a root of zNR(z) if an only if l/z0 is also a root. We construct B'(z) by re-combining all the factors whose roots lie outside the unit circle. Clearly this method does not extend to higher dimensions since it relies on polynomial factorization. A second technique for stability is based on the leastsquares inverse problem and will be discussed in the next section. This method has to date found no computationally useful extension to higher dimensions.