Time-limited signals and Gabor expansion

Gabor (1946) expansion suffers from the consequences of the zero theorem which states that all continuous functions have a zero on the unit square in the Zak space. In particular the Gaussian function routinely selected for a window in the Gaborian analysis has a zero at (1/2,1/2). As a result a zero-matching scheme between the signal and the window must be employed, which limits the class of signals that can be analyzed. We demonstrate that for a broad class of signals existence of zero of the window does not affect stability of the Gabor expansion and the zero-matching procedure can be avoided. It is shown that signal time-limited to the (-1/2,1/2) interval must have a zero at (1/2,1/2) in the Zak space, thus allowing a legal Gabor expansion based on a Gaussian window.<
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