A Covariant, Discrete Time-Frequency Representation Tailored for Zero-Based Signal Detection

Recent work in time-frequency analysis proposed to switch the focus from the maxima of the spectrogram toward its zeros, which, for signals corrupted by Gaussian noise, form a random point pattern with a very stable structure leveraged by modern spatial statistics tools to perform component disentan...

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Bibliographic Details
Published in:	IEEE transactions on signal processing 2022, Vol.70, p.2950-2961
Main Authors:	Pascal, Barbara, Bardenet, Remi
Format:	Article
Language:	eng
Subjects:	Analytic functions coherent states Computer Science Covariant representations Fourier transforms gaussian analytic functions Gaussian noise Methodology monte carlo envelope tests Random noise Representations Robustness (mathematics) Signal and Image Processing Signal detection Signal processing Signal to noise ratio spatial statistics Spectrogram Statistical methods Statistics Time-frequency analysis Transforms
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Summary:	Recent work in time-frequency analysis proposed to switch the focus from the maxima of the spectrogram toward its zeros, which, for signals corrupted by Gaussian noise, form a random point pattern with a very stable structure leveraged by modern spatial statistics tools to perform component disentanglement and signal detection. The major bottlenecks of this approach are the discretization of the Short-Time Fourier Transform and the boundedness of the time-frequency observation window deteriorating the estimation of summary statistics of the zeros, on which signal processing procedures rely. To circumvent these limitations, we introduce the Kravchuk transform , a generalized time-frequency representation suited to discrete signals, providing a covariant and numerically tractable counterpart to a recently proposed discrete transform, with a compact phase space, particularly amenable to spatial statistics. Interesting properties of the Kravchuk transform are demonstrated, among which covariance under the action of \text{SO}(3) and invertibility. We further show that the point process of the zeros of the Kravchuk transform of white Gaussian noise coincides with those of the spherical Gaussian Analytic Function, implying its invariance under isometries of the sphere. Elaborating on this theorem, we develop a procedure for signal detection based on the spatial statistics of the zeros of the Kravchuk spectrogram, whose statistical power is assessed by intensive numerical simulations, and compares favorably to state-of-the-art zeros-based detection procedures. Furthermore it appears to be particularly robust to both low signal-to-noise ratio and small number of samples.
ISSN:	1053-587X 1941-0476