The pole behavior of the phase derivative of the short-time Fourier transform

• We investigate the peculiar behavior of the derivative of the phase around the zeros of the STFT.
• Numerical evidence shows two characteristically shaped peaks, one negative, one positive.
• We give a complete analytic explanation of the observed phenomenon. It is closely related to the smoothness of the window function.
• Any sufficiently often continuously differentiable window function will exhibit the observed pattern.

The short-time Fourier transform (STFT) is a time–frequency representation widely used in audio signal processing. Recently it has been shown that not only the amplitude, but also the phase of this representation can be successfully exploited for improved analysis and processing. In this paper we describe a rather peculiar pole phenomenon in the phase derivative, a recurring pattern that appears in a characteristic way in the neighborhood around any of the zeros of the STFT, a negative peak followed by a positive one. We describe this phenomenon numerically and provide a complete analytical explanation.