An Extension of Wiener Filter Theory to Partly Sampled Systems

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Abstract
The growing use of digital computers as components of control systems has given great importance to the study of linear systems which are partly sampled and partly continuous. This paper treats the problem of optimizing the simplest possible mixed system consisting of an input filter with transfer functionK(s), a sampler with sampling intervalT, and an output filter with transfer functionL(s). Given the power spectra of the input signal and the noise, the object is to find a realizableKandLwhich in combination minimize the mean square difference between the outputhanda"desired output"h_d.h_dis defined by a "desired transfer function"{G_d}(s), not necessarily realizable, which would produceh_dfrom the input signal if the noise were absent.KLwill in general contain factors periodic inswith period2 \pi j/T, and such factors may be moved to either side of the sampler without changing the final output, thus introducing a considerable arbitrariness inKandL. However, since these periodic factors represent linear operations on discrete data (such as might be performed inside a digital computer), it is appropriate to separate them out. There are then three functions to be determined: the nonperiodic part ofK, the nonperiodic part ofL, and the remaining (periodic) factor ofKL. Methods for determining these three functions are given. The interesting theoretical point is that the determination is not always unique. In general, there will be a finite number of distinct but equivalent solutions.

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