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.

中文翻译：

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专为基于零的信号检测而设计的协变离散时频表示

最近在时频分析方面的工作建议将焦点从频谱图的最大值转移到零点，对于被高斯噪声破坏的信号，这会形成具有非常稳定结构的随机点模式，现代空间统计工具利用该模式来执行分量解缠结和信号检测。这种方法的主要瓶颈是短时傅里叶变换的离散化和时频观察窗口的有界性，恶化了信号处理过程所依赖的零点汇总统计的估计。为了规避这些限制，我们引入了Kravchuk 变换，这是一种适用于离散信号，为最近提出的离散变换提供协变和数值上易于处理的对应物，具有紧凑的相空间，特别适合空间统计。证明了 Kravchuk 变换的有趣性质，其中在$\文本{SO}(3)$和可逆性。我们进一步证明了高斯白噪声的Kravchuk变换的零点与球面高斯解析函数的零点一致，这意味着它在球体等距下的不变性。详细阐述这个定理，我们开发了一种基于 Kravchuk 频谱图零点空间统计的信号检测程序，其统计能力通过密集的数值模拟进行评估，并且与最先进的基于零点的检测相比具有优势程序。此外，它似乎对低信噪比和少量样本都特别稳健。