Linearly time-varying systems and their fast implementation

Abstract

Linear time-invariant systems can be implemented in O(N log N) time, whereas the most general family of linear systems can be implemented as a vector-matrix product in O(N2) time. However, there are time-variant systems that can be implemented in O(N log N) time. In this paper, we introduce a particular family of such systems, which we refer to as the class of linearly time varying (LTV) systems. These systems interpolate between multiplicative systems and convolutive systems, and are characterized by their chirp-type eigenfunctions and their relationship to fractional Fourier domain filtering. We derive expressions for the linear transform kernel of LTV systems, and illustrate their use with examples. Recognizing LTV systems, or approximating linear systems with LTV systems when possible, can reduce the time of computation from O(N2) to O(N log N).