LEAST‐SQUARES INVERSE FILTERING

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Abstract
Suppose we are given the autocorrelation function of a certain unknown sampled signal. Although a number of different signals might produce the given autocorrelation function, only one of these is minimum‐delay. Denoting this minimum‐delay unknown signal by the matrix K, the given autocorrelation may be written in the form K′K where K′ is the transpose of K. It is desired to determine approximately the inverse of this unknown signal K; that is, we wish to determine a vector X so that KX is as close to B′=(1, 0, 0, ⋯, 0) as possible in a least‐squares sense. If K were known, Rice shows that [Formula: see text] At first glance, the above formula appears useless as K′ is unknown. However, although K′ is indeed unknown, K′B has the form K′B=(c, 0, 0, ⋯, 0)′ where the scalar c simply plays the role of a scale factor. Thus, we determine X by simply selecting a convenient multiple of the first column of the inverse of the known matrix K′K. Although we do not present the details of the computer programming involved in the above calculation, we do present some simple examples to illustrate the process.
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