A covariant, discrete time-frequency representation tailored for zero-based signal detection

TL;DR

This paper introduces the Kravchuk transform, a covariant, discrete time-frequency representation that improves zero-based signal detection by addressing discretization and boundedness issues, demonstrating robustness and superior detection performance.

Contribution

The paper presents the Kravchuk transform as a novel, covariant, and numerically tractable discrete time-frequency representation tailored for zero-based signal detection, with demonstrated invariance and robustness.

Findings

Kravchuk transform is covariant under SO(3) and invertible.

Zeros of the Kravchuk transform of Gaussian noise match spherical Gaussian Analytic Function.

Proposed detection method outperforms existing zeros-based procedures, especially at low SNR and small sample sizes.

Abstract

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 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.