The Pole Behaviour of the Phase Derivative of the Short-Time Fourier   Transform
  The short-time Fourier transform (STFT) is a time-frequency representation widely used in applications, for example 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. 
Peter Balazs, Dominik Bayer, Florent Jaillet and Peter S{\o}ndergaard
math.FA