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Several recent works [1,2,3) have discussed the role of the Gabor transform as a tool for signal detection and feature extraction in the presence of noise. Gabor, or more generally Weyl—Heisenbrerg, expansions of signals provide a greater degree of sensitivity to local phenomena such as local frequency changes as compared to classical Fourier methods. To be an effective tool, such expansions should highlight relatively few components and not spread signal energy throughout an unreasonably large number of components. An important first step in applying the Gabor transform to detect a signal is to maximally exploit a priori signal knowledge to design an appropriate weighting function as a window on input data. The effect of such a window in practice is to reduce the dimension of the signal search space. If complete signal information is known at the outset, then the optimal signal processing strategy is the matched filter, which reduces the search—space to one dimension. In this work, we fix a window and provide tools for measuring how well various subspaces of signals can be analyzed relative to the window. In a Fourier signal processing strategy, signal decay rate (along with that of its Fourier transform) and signal smoothness usually serve as the essential a priori information necessary to set sampling rates and establish error estimates. This information will not be sufficient for an effective application of the Gabor transform. The Zak transform plays a key role in several works [4,5,6,7] dealing with the Gabor transform. The function formed from the ratio of the Zak transform of a signal to the Zak transform of the window will provide essential information about the window's effectiveness in compactly representing signal information. Various subspaces of signals will be defined by functional properties of the quotient. In general, the quotient is doubly periodic but is not necessarily continuous. However, under rather general conditions, Fourier coefficients can be defined which in some cases determine the coefficients of a Weyl-Heisenberg expansion. It is natural to ask whether these coefficients can be used to provide a (discrete) time—frequency representation of the signal relative to the fixed window independent of their role in a Weyl—Heisenberg expansion.
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