  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.
