By David F. Findley
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Additional resources for Applied Time Series Analysis II. Proceedings of the Second Applied Time Series Symposium Held in Tulsa, Oklahoma, March 3–5, 1980
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. Here, then, is a challenge. recursive filter equation y(B) =§ff}u
Relatively little work has been done on the problem of finding optimal multidimensional windows, but Huang (Huang, 1972) has remarked that if w(x) is a good symmetric one-dimensional window, then the circularly symme tric window w2(x,y) = w(Vx2 + y 2) is probably not a bad two-dimensional window. } cess. 1 Pn 1 - T 1 1 0_ 0 it follows that the prediction error filter, A(z), is a (nor malized) least-squares inverse filter and so is guaranteed to be stable. Since this method is so easily applied in one dimension, and since we have the Levinson algorithm available for its solution, this is a highly attractive method for spectral estimation.
In this method, we define the (discrete) homomorphic transform, H, by the equation H = Z_1Log Z where Z stands for the Z-transform operator. The operator H has an inverse given by H"1 = Z"1Exp Z Now let b be any £-. sequence and let b be defined by b = Hr, where r is the autocorrelation of b. It follows from the Wiener-Levy theorem that if R(z) = |B(z)|2 > 0 for |z| = 1, then b is again in &1. If we now multiply b by the Heaviside function and halve its value at Higher Dimensional Signal Processing 31 the origin, we obtain a new function which we denote by b .
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