Point Processes and spatial statistics in time-frequency analysis (2402.19172v1)

Abstract: A finite-energy signal is represented by a square-integrable, complex-valued function $t\mapsto s(t)$ of a real variable $t$, interpreted as time. Similarly, a noisy signal is represented by a random process. Time-frequency analysis, a subfield of signal processing, amounts to describing the temporal evolution of the frequency content of a signal. Loosely speaking, if $s$ is the audio recording of a musical piece, time-frequency analysis somehow consists in writing the musical score of the piece. Mathematically, the operation is performed through a transform $\mathcal{V}$, mapping $s \in L^2(\mathbb{R})$ onto a complex-valued function $\mathcal{V}s \in L^2(\mathbb{R}^2)$ of time $t$ and angular frequency $\omega$. The squared modulus $(t, \omega) \mapsto \vert\mathcal{V}s(t,\omega)\vert^2$ of the time-frequency representation is known as the spectrogram of $s$; in the musical score analogy, a peaked spectrogram at $(t_0,\omega_0)$ corresponds to a musical note at angular frequency $\omega_0$ localized at time $t_0$. More generally, the intuition is that upper level sets of the spectrogram contain relevant information about in the original signal. Hence, many signal processing algorithms revolve around identifying maxima of the spectrogram. In contrast, zeros of the spectrogram indicate perfect silence, that is, a time at which a particular frequency is absent. Assimilating $\mathbb{R}^2$ to $\mathbb{C}$ through $z = \omega + \mathrm{i}t$, this chapter focuses on time-frequency transforms $\mathcal{V}$ that map signals to analytic functions. The zeros of the spectrogram of a noisy signal are then the zeros of a random analytic function, hence forming a Point Process in $\mathbb{C}$. This chapter is devoted to the study of these Point Processes, to their links with zeros of Gaussian Analytic Functions, and to designing signal detection and denoising algorithms using spatial statistics.

• The paper introduces a novel framework analyzing spectrogram zeros in Gaussian white noise to enhance signal detection and reconstruction.
• It employs Gaussian analytic functions to reveal non-random zero distributions, enabling statistically robust signal processing algorithms.
• The study provides numerical methods, such as the Minimal Grid Neighbor approach, offering practical tools for precise zero localization in discretized spectrograms.

Investigating the Zeros of Spectrogram Representations of White Noise

Introduction to Spectrogram Zeros in Signal Processing

The analysis of zeros in spectrograms generated from white noise provides an intriguing avenue of research within the field of signal processing. Traditionally, signal processing techniques have focused on the manipulation and analysis of signal amplitude and frequency components. However, recent studies suggest that examining the locations and patterns of zeros in time-frequency representations, such as spectrograms, of white noise signals can offer novel insights and methodologies for signal detection and reconstruction tasks.

The Mathematical Framework of Spectrogram Zeros

The paper of spectrogram zeros intersects areas of complex analysis, stochastic processes, and signal processing. Specifically, the representation of a finite-energy signal via the Short-Time Fourier Transform (STFT) under a Gaussian window is pivotal. This transformation maps a signal to a complex-valued function in the time-frequency plane, whose zeros signify points of silence or frequency absence in the signal. Crucially, for Gaussian white noise, these zeros do not distribute randomly but follow distinct spatial statistics governed by the underlying properties of the Gaussian Analytic Functions (GAFs).

Connection between Gaussian Analytic Functions and Time-Frequency Analysis

Gaussian Analytic Functions serve as a mathematical model to describe the distribution of zeros in the complex plane for specific analytic processes. Their relevance in signal processing emerges through the link between the STFT of white noise and particular types of GAFs. Such connections have been established, showing that for Gaussian windows, zeros of the Gaussian white noise spectrogram match those of the planar GAF. This relationship underscores the analytical tractability and statistical predictability of zeros in spectrograms, which are pivotal for algorithmic developments in signal processing.

Spatial Statistics and Signal Processing Applications

Spatial statistics of zeros, including their intensity functions and pair correlation functions, provide a quantitative measure to assess the regularity and repulsiveness among zeros. These properties are instrumental in designing signal processing algorithms. For instance, zero-based Monte Carlo envelope tests leverage the spatial distribution of zeros to distinguish between pure noise and signals corrupted by noise. These tests utilize computationally efficient estimations of spatial statistics, avoiding user-defined thresholds and offering robust detection capabilities even in low signal-to-noise ratio conditions.

Numerical Algorithms for Zero Identification

Practical application of zero-based signal processing techniques necessitates effective numerical algorithms to locate zeros in discretized spectrograms. Approaches such as the Minimal Grid Neighbor method, dealing with discretized data, have shown promise. Recent advancements propose algorithms with theoretical guarantees for precise zero detection, which are crucial for implementing zero-based signal processing methods in real-world scenarios.

Future Directions and Extensions

The investigation into spectrogram zeros is expanding into non-Gaussian windows and generalized signal representations, broadening the applicability and understanding of zero-based methodologies in signal processing. Efforts to characterize the zeros of non-Gaussian spectrogram representations through extended Weyl-Heisenberg groups illustrate the field's growth beyond its initial scope. Moreover, public benchmarks and contributions from the signal processing community further accelerate the development and evaluation of innovative zero-based approaches.

Conclusion

The zeros of time-frequency representations, particularly those arising from white noise signals, offer a rich mathematical structure and practical utility in signal processing. The intersectionality of complex analysis, probability theory, and algorithmic design in this research area not only enriches the theoretical foundation of signal processing but also propels the development of novel methods for signal analysis, detection, and reconstruction tasks. As the field progresses, exploring new mathematical models, extending applicability to various signal types, and enhancing computational methodologies will continue to drive innovation in signal processing research.