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
The spectrum is then estimated in any of the usual ways from the extrapolated data. Various models have been assumed for the extrapolation including linear prediction (Frost, 1980) and band-limited extrapolation (Abend, 1980). The techniques for linear pre dictive data extrapolation are based on deriving a linear predictor and then "running it off the edge of the data" to 54 James H. Justice obtain the extrapolated data values. These methods are, as pointed out earlier, sensitive to the geometry of the filter support.
Efficient recursive algorithms have been devised for Burg's method (Burg, 1975) and various extensions to the mul tichannel case have been proposed (Jones, 1978; Gambotto, 1979). However, when we consider the extension of these algorithms to the multidimensional case, we again encounter serious problems. So serious, in fact, that no one has found a good way around them. Following Newman (Newman, 1977), we may pose the two-di mensional problem as follows. Let us define the average entropy of the process whose power spectrum, S, is assumed to be bandlimited to the rectangle [-v,v] x [-|J,H] by the equa tion H w = 8uV ; J" ^ n [s(",v)]dudv + £n(2*e) 1/2 The problem is to maximize H subject to the constraint that the observed (estimated) autocorrelation values should agree with the power spectrum S.
22 James H. Justice processing as a discipline worthy of study in its own right, however, is a recent phenomenon, and it seems safe to say that the important developments in multidimensional digital signal processing have occurred within the last decade. Indeed, it has only been within the last few years that this relatively new discipline has attracted a sufficient number of investigators to have become somewhat visible in the pub lished literature and in the agenda of professional socie ties.