"Detection of Zeros in the Short-Time Fourier Transform"Koliander, GüntherThe short-time Fourier transform (STFT) of a signal in white Gaussian noise can be interpreted as a weighted Gaussian analytic function. Without signal, the zero set of this random function has a well analyzed distribution. The zeros are quite uniformly dispersed and it is unlikely that two zeros are close to each other. Recently, several works analyzed the properties of these zeros in more detail and proposed to use them in the detection of the time-frequency support of the signal. However, a challenge that has hardly been addressed is the detection of the zeros in the first place. Note that with probability zero, any of the samples of the STFT given in practice is precisely at the position of any zero. One possible solution is thresholding, i.e., to consider all values as zeros that are below a certain absolute value. Unfortunately, this approach results in clusters around some zeros and missed detections of others. Another idea that has been proposed in the literature is to make use of the maximum modulus principle, which implies that any local minimum of a (non-constant) analytic function is a zero. Thus, the detection of zeros can be replaced by the detection of local minima, a task that more naturally extends to a discrete domain. In this talk, we present an algorithm based on this idea with probabilistic guarantees for the correct detection of zeros. Our result shows that the probability of correctly identifying all zeros within a given region increases to one as the grid size, i.e., the distance between samples, goes to zero. |
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