Time-frequency localization and the spectrogram

Abstract

One method by which the time-frequency content of a signal can be measured is by the Gabor (or windowed Fourier) transform. It is defined as the Fourier transform of the product of the signal against a translate of a fixed window function. Multiplication by the window localizes the signal in time, and hence the Fourier transform computes the spectrum of the signal localized in time. The magnitude squared of the Gabor transform is known as the spectrogram of the signal. In this paper, we are concerned with the problem of finding signals that are maximally concentrated within a given region of phase space. It is well known that such problems give rise to an integral eigenvalue problem for an associated concentration operator. It is easy to see that such an operator is positive self-adjoint, with eigenvalues bounded above by one. In this paper, we show that for a bounded measurable domain and for the gaussian window function, the eigenfunctions of such a concentration operator with nonzero eigenvalue are analytic and of exponential decay, while the eigenvalues decay exponentially in the index. Furthermore, the number of eigenvalues close to unity is asymptotically equal to the measure of the domain, upon dilation. With some regularity assumptions on the domain, we also prove that the number of eigenvalues strictly between zero and one grows at most linearly in the dilation factor. These results are sensitive to the regularity of the window function. © 1994 Academic Press, Inc.