Time-frequency representation of digital signals and systems based on short-time Fourier analysis

This paper develops a representation for discrete-time signals and systems based on short-time Fourier analysis. The short-time Fourier transform and the time-varying frequency response are reviewed as representations for signals and linear time-varying systems. The problems of representing a signal by its short-time Fourier transform and synthesizing a signal from its transform are considered. A new synthesis equation is introduced that is sufficiently general to describe apparently different synthesis methods reported in the literature. It is shown that a class of linear-filtering problems can be represented as the product of the time-varying frequency response of the filter multiplied by the short-time Fourier transform of the input signal. The representation of a signal by samples of its short-time Fourier transform is applied to the linear filtering problem. This representation is of practical significance because there exists a computationally efficient algorithm for implementing such systems. Finally, the methods of fast convolution age considered as special cases of this representation.