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.