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 27r 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. S INTRODUCTION HORT-TIME Fourier analysis [ l ] of discrete-time signals is of considerable interest in a number of signalprocessing applications. In order to study spectral properties of speech signals, for instance, the concept of a short-time Fourier transform of the signal is very convenient [ 11, [2]. Such a short-time Fourier transform is usually constructed by first multiplying the signal by a window function that is “slided” to a certain position, and then Fourier transforming the “windowed” signal. Therefore, we like to consider’the short-time Fourier transform as a sliding-window representation of the signal. There are other interpretations, of the short-time Fourier transform, including a well-knowPfilter bank interpretation [ 11. However, for the purpose of this paper, we find the slidingwindow interpretation to be the most appropriate, and, to emphasize this, we shall call the short-time Fourier transform the sliding-window spectrum of the signal. The sliding-window representation of a signal, which is a signal description in time and frequency simultaneously, is complete in the sense that the signal can be reconstructed from its sliding-window spectrum [ 11. However, to reconstruct the signal, we need not know the entire slidingwindow spectrum. In this paper we show that it suffices to know the values of the sliding-window spectrum only at the points of a certain rectangdar lattice in the time-frequency domain, and we describe how the signal Manuscript received January 17, 1984; revised January 2, 1985. The author i s with the Technische Hogeschool Eindhoven, Afdeling der Elektrotechniek, Postbus 513, 5600 MB Eindhoven, The Netherlands. BASTIAANS can be expressed directly in terms of the values of this sampled sliding-window spectrum. This will lead us in a natural way to Gabor’s representation [3] of a signal as a superposition of properly shifted and modulated versions of a function that is related to the window. We show a way to determine this function from the knowledge of the window, and we elucidate this with some simple examples of window functions. SLIDING-WINDOW REPRESENTATION F DISCRETE-TIME SIGNALS Let x(n) (n = + . , 1, 0, 1, * a ) denote a one-dimensional discrete-time signal and let w(n) represent a window sequence; the signal and the window may take complex values and they need not have a finite extent. We multiply the signal by a shifted and complex conjugated version of the window and take the Fourier transform of the product, thus constructing the function [cf. [l], (6.1)] f ( ~ ! , n) = C i (m> w*(m n) exp [ j ~ ! m ] . (1) Unlike (6.1) in [ l ] , (1) uses a complex conjugated version of the window; moreover, the window has not been timereversed. The only reason for doing this is to get more elegant formulas in the remainder of the paper. We shall callf(Q, n) the sliding-window spectrum [cf. [4], Section 4.11 of the discrete-time signal; it is clearly a function of two variables: the time index n , which is discrete and represents the position of the window, and the frequency variable L!, which is continuous. Of course, as in the case of normal Fourier transforms of discrete-time signals, the sliding-window spectrum f(Q, n) is periodic in L! with period 2n. Two choices of window sequences are of special interest. If w(n) vanishes for n # 0, then f ( 0 , n) is proportional to the signal x ( n ) ; the sliding-window spectrum thus reduces to a pure time representation of the signal. If, on the other hand, w(n) does not depend on n, thenf(Q, 0) is proportional to the Fourier transform of x ( n ) ; the sliding-window spectrum then reduces to a pure frequency representation of the signal. In general, however, the sliding-window spectrum is an intermediate signal description between the pure time and the pure frequency representation. We can reconstruct the signal x(n) from its sliding-window spectrum f(Q, n ) in the usual way [cf. [ l ] , (6.6)] by inverse Fourier transforming and taking m = n, which 03