The Phase Derivative Around Zeros of the Short-Time Fourier Transform

Details The Phase Derivative Around Zeros of the Short-Time Fourier Transform The Phase Derivative Around Zeros of the Short-Time Fourier Transform

2011-03-02

arxiv.org/abs/1103.0409v1

Mathematics

Functional Analysis

22 pages, 4 figures

Scientific paper

The phase of the short-time Fourier transform (STFT) is sometimes considered difficult to interpret. However, the phase information is important for improved analysis and processing. The phase derivative, in particular, is essential for the reassignment method or the phase vocoder algorithm. In order to understand the phase derivative of the STFT more thoroughly, we describe an interesting phenomenon, a recurring pattern in the neighborhood of zeros. Contrary to the possible expectation of an arbitrary behavior, the phase derivative always shows a singularity with the same characteristic shape with a negative and a positive peak of infinite height at these points. We show this behavior in a numerical investigation, present a simple explicit analytic example and then do a complete analytical treatment. For this we present several intermediate results about the regularity of the STFT for Schwartz windows, which are of independent interest.

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