论文标题
针对零基信号检测量身定制的协变，离散的时频表示
A covariant, discrete time-frequency representation tailored for zero-based signal detection
论文作者
论文摘要
时频分析中的最新工作提议将焦点从频谱图的最大值转换为其零，该零件对于因高斯噪声而损坏的信号，形成了一个随机点模式，并具有由现代空间统计工具利用的非常稳定的结构，以执行组件解散和信号检测。这种方法的主要瓶颈是短期傅立叶变换的离散化以及时间频观测窗口的界限，从而导致信号处理程序依赖的零汇总统计数据的估计。为了避免这些局限性，我们引入了Kravchuk Transform，这是一种适合离散信号的广义时频表示，为最近提出的分散转换提供了协方差且在数值上可牵引的对应物，具有紧凑的相位空间，尤其是可与空间统计相关的。证明了Kravchuk变换的有趣特性，其中在SO（3）和可逆性的作用下协方差。我们进一步表明，白色高斯噪声的kravchuk变换的零点与球形高斯分析功能的零相吻合，这意味着它在球体的异构体下的不变性。详细阐述了该定理，我们根据Kravchuk频谱图的空间统计数据开发了一个信号检测的程序，Kravchuk频谱图的空间统计数据是通过密集的数值模拟评估的，其统计能力与基于最新的Zeros Zeros检测程序进行了比较。此外，对于低信噪比和少量样品，它似乎特别健壮。
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.
