Nonlinear Matched Filters via Chen–Fliess Series

A class of nonlinear matched filters is introduced suitable for detection problems using Chen-Fliess functional series. Such series can be viewed as a noncommutative analogue of Taylor series. They are written in terms of a weighted sum of iterated integrals indexed by words over a noncommuting set of symbols. The primary goal is to identify within this class the set of filters which maximizes the signal-to-noise ratio at a given time instant in order to provide a statistic for detecting a known signal.