An Extension of Wiener Filter Theory to Partly Sampled Systems

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 function , a sampler with sampling interval , and an output filter with transfer function . Given the power spectra of the input signal and the noise, the object is to find a realizable and which in combination minimize the mean square difference between the output and "desired output" . is defined by a "desired transfer function" , not necessarily realizable, which would produce from the input signal if the noise were absent. will in general contain factors periodic in with period , and such factors may be moved to either side of the sampler without changing the final output, thus introducing a considerable arbitrariness in and . 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 of , the nonperiodic part of , and the remaining (periodic) factor of . 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.