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.

翻译：近期时频分析领域的研究提出将关注点从谱图的极大值转向其零点，对于受高斯噪声污染的信号，这些零点形成具有高度稳定结构的随机点模式，现代空间统计工具可据此进行分量分离和信号检测。该方法的主要瓶颈在于短时傅里叶变换的离散化以及时频观测窗口的有界性，这会破坏用于信号处理过程的零点汇总统计量估计。为克服这些限制，我们引入克拉夫丘克变换——一种适用于离散信号的广义时频表示，为近期提出的离散变换提供了协变且数值可计算的对应形式，其相空间紧致且特别适合空间统计分析。我们证明了克拉夫丘克变换的若干重要性质，包括在SO(3)作用下的协变性及可逆性。进一步证明，白高斯噪声克拉夫丘克变换的零点点过程与球面高斯解析函数的零点一致，这意味着其在球面等距变换下具有不变性。基于该定理，我们开发了一套基于克拉夫丘克谱图零点空间统计的信号检测程序，通过大量数值模拟评估其统计效能，并证明该方法优于现有最先进的基于零点的检测程序。此外，该方法在低信噪比和小样本量条件下均表现出极强的鲁棒性。