Point Processes and spatial statistics in time-frequency analysis
CoRR(2024)
摘要
A finite-energy signal is represented by a square-integrable, complex-valued
function t↦ 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 𝒱, mapping s ∈ L^2(ℝ)
onto a complex-valued function 𝒱s ∈ L^2(ℝ^2) of time t
and angular frequency ω. The squared modulus (t, ω) ↦|𝒱s(t,ω)|^2 of the time-frequency representation is
known as the spectrogram of s; in the musical score analogy, a peaked
spectrogram at (t_0,ω_0) corresponds to a musical note at angular
frequency ω_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 ℝ^2 to ℂ through z = ω +
it, this chapter focuses on time-frequency transforms 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 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.
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