On two fundamental properties of the zeros of spectrograms of noisy signals

The spatial distribution of the zeros of the spectrogram is significantly altered when a signal is added to white Gaussian noise. The zeros tend to delineate the support of the signal, and deterministic structures form in the presence of interference, as if the zeros were trapped. While sophisticated methods have been proposed to detect signals as holes in the pattern of spectrogram zeros, few formal arguments have been made to support the delineation and trapping effects. Through detailed computations for simple toy signals, we show that two basic mathematical arguments –the intensity of zeros and Rouché’s theorem– allow discussing delineation and trapping, and the influence of parameters like the signal-to-noise ratio. In particular, interfering chirps, even nearly superimposed, yield an easy-to-detect deterministic structure among zeros.