On the sliding-window representation in digital signal processing

The short-time Fourier transform of a discrete-time signal, which is the Fourier transform of a "windowed" version of the signal, is interpreted as a sliding-window spectrum. This sliding-window spectrum is a function of two variables: a discrete time index, which represents the position of the window, and a continuous frequency variable. It is shown that the signal can be reconstructed from the sampled sliding-window spectrum, i.e., from the values at the points of a certain time-frequency lattice. This sampling lattice is rectangular, and the rectangular cells occupy an area of 2π in the time-frequency domain. It is shown that an elegant way to represent the signal directly in terms of the sample values of the sliding-window spectrum, is in the form of Gabor's signal representation. Therefore, a reciprocal window is introduced, and it is shown how the window and the reciprocal window are related. Gabor's signal representation then expands the signal in terms of properly shifted and modulated versions of the reciprocal window, and the expansion coefficients are just the values of the sampled sliding-window spectrum.