Optimum Nonlinear Filters

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Abstract
The theory of optimum nonlinear filters outlined in this paper is based on the consideration of a sequence of classes of nonlinear filters, designated as 𝔑1, 𝔑2, 𝔑3, …, such that each class in the sequence includes all the preceding classes and, furthermore, the class of linear filters is a subclass of every class in the sequence. A filter of class 𝔑m is described in terms of a characteristic function which involves m age variables and m values of the input time-function. The input-output relationship for a filter of class 𝔑m has the form of an m-fold integral of the characteristic function with respect to the m age variables. It is shown that the characteristic function of the optimum filter (in the least squares sense) within the class 𝔑m satisfies a linear integral equation of 2mth order. The optimization of filters of class 𝔑1 is treated in detail, and methods of approximate realization of such filters in the form of nonlinear delay line filters and power series filters are indicated. The results are extended to the case of nonstationary time series.

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