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Zeros of the Spectrogram of White Noise with Hermite Windows
Charaf Bellourak, Sylvain Meignen, Kévin Polisano, Mickaël Nahon
TL;DR
Intrinsic zero statistics for a single Hermite-window white-noise spectrogram have received comparatively little attention. The paper uses the polyanalytic Bargmann transform to classify zeros by charge and characterize their statistics; unlike the Gaussian-window case, pair correlation tends to a positive constant at the origin.
Problem
Intrinsic statistics for zeros of a single Hermite-window spectrogram have received comparatively little attention, beyond cross-correlations between different Hermite orders.
Source: §1 · p. 2Method
The paper uses the polyanalytic Bargmann transform to classify zeros by Jacobian-sign charge and derive one- and two-point intensities, alongside a winding-number detection method.
Source: §1 · p. 2Results
For Hermite windows, the pair correlation converges at the origin to a strictly positive constant, unlike the total repulsion observed with a Gaussian window.
Source: §7 · p. 20Takeaways & Limitations
The proposed practical detection method identifies spectrogram zeros using winding numbers on small circular contours.
Source: §7 · p. 20For n≥1, the local-maxima and saddle-point characterization available at n=0 does not extend because f_n is no longer holomorphic.
Source: §4.2 · p. 10
Abstract
from arXiv · showhide
We study the zeros of the spectrogram of complex white noise analyzed with Hermite windows of order n, extending the classical Gaussian-window correspondence with Gaussian Analytic Functions to the polyanalytic setting. Using the polyanalytic Bargmann transform, these zeros split into two families according to the sign of the local Jacobian, giving rise to a natural notion of charge, and we show that these zeros are almost surely simple, isolated, and form a translation and rotation-invariant point process. We derive closed-form expressions for the one-point intensity, decomposed by charge, and also for the unsigned, signed, and hybrid two-point intensities. Besides, we prove that unlike the total repulsion observed with a Gaussian window, the Hermite window of order n generates a pair correlation function that converges at the origin to a strictly positive constant depending on n. Finally, we propose a numerical method for detecting spectrogram zeros via the winding number around candidate local minima.
1. Introduction
Prior work connected Gaussian-window white-noise spectrogram zeros to Gaussian analytic functions, while Hermite windows move the problem into a comparatively understudied polyanalytic setting. This article develops single-window zero statistics and a winding-number detection method for Hermite windows.
1. Introduction
- Gaussian-window white-noise spectrogram zeros coincide with zeros of a Bargmann–Fock Gaussian Analytic Function, grounding earlier observations in a probabilistic framework.Earlier work also extended related correspondences to orthogonal-polynomial generating functions and wavelet transforms.Source: §1 · pp. 1–2
- Hermite windows yield polyanalytic random functions, but intrinsic statistics for one spectrogram received comparatively little attention beyond cross-correlations between orders.Hermite windows are also used in multitaper, reassignment, ConceFT, and Gabor-frame methods.Source: §1 · p. 2
- The article studies the single-window case, classifies zeros by Jacobian-sign charge, derives one- and two-point statistics, and analyzes near-origin pair correlation.It also proposes zero detection using winding numbers around local minima and discusses contour-radius choice.Source: §1 · p. 2
- The paper proceeds from transform and function-space background to white-noise stationarity, zero properties, pair correlations, and numerical detection.Technical proofs are deferred to the appendix.Source: §1 · pp. 2–3
2. Background
The background sets up the STFT and its spectrogram, then relates Gaussian and Hermite windows to analytic and polyanalytic function spaces. It introduces the Gabor-space and transform framework used for the paper’s Hermite-window analysis.
2.1. The Short-Time Fourier Transform
- The STFT evaluates a signal against a translated and modulated window, and its squared modulus is the spectrogram.For a unit-norm window, the STFT is an isometry onto its Gabor space, which is a reproducing-kernel Hilbert space.Source: §2.1 · p. 3
2.2. The Bargmann Transform and Fock Space
- With the unitary Gaussian window, the STFT has a Bargmann representation and maps into the Fock space of entire functions.The Bargmann transform is described as an isometry from L2(R) onto that Fock space.Source: §2.1 · p. 3, §2.2 · pp. 3–4
2.3. Gabor space, Hermite functions and polyanalytic functions
- For Hermite windows, the order-n Bargmann transform is polyanalytic and relates to the STFT; the corresponding true polyanalytic Fock space is a reproducing-kernel Hilbert space.The background also introduces Hermite functions, complex Hermite polynomials, and the STFT of one Hermite function analyzed with another Hermite window.Source: §2.3 · pp. 4–6
3. Polyharmonic Bargmann transform of white Gaussian noise and stationarity properties
The paper represents complex white noise through its Hermite coefficients and the polyharmonic Bargmann transform. A translation law for the associated kernel implies translation invariance of the transform’s zeros.
3.1. Polyharmonic Bargmann transform of white Gaussian noise
- Complex white noise has i.i.d. standard complex Gaussian Hermite coefficients, which are used to express its order-n Bargmann transform as a polyanalytic function.The paper then abbreviates this function as f_n(z).Source: §3.1 · p. 6
3.2. Stationarity with respect to translation
- Translating both kernel arguments changes the kernel only by a nonzero prefactor, so the zeros of B_n are translation invariant.Source: §3.2 · p. 7
4. Zeros of White Noise with Hermite Functions
For Hermite windows, the zeros are almost surely simple and isolated, and for order 1 their Jacobian charge distinguishes maxima from saddle points of the order-0 spectrogram.
4.1. Zeros of White Noise as a Point Process
- The zeros are almost surely simple and isolated, so they define a point process.Source: §4.1 · pp. 8–9
- For n≥1, the sign of the Jacobian assigns positive or negative charge to a zero; degenerate zeros occur with probability zero.Source: §4.1 · pp. 8–9
4.2. On Maxima and Zeros of |Vh0ξ|2 and |Vh1ξ|2
- For order 1, negatively charged zeros of f1 are local maxima of |Vh0ξ|2, while positively charged zeros are saddle points.Source: §4.2 · p. 10
- This maxima-and-zero correspondence does not extend to n≥1 because fn is no longer holomorphic.Source: §4.2 · p. 10
5. Computation of the one and two points intensities
Using Kac–Rice and stationarity, the section derives Hermite-zero intensities and examines how they vary by charge, separation, and contour radius.
5.1. Computation of the First Intensity of Hermite Zeros
- The one-point intensity is constant, and the section derives separate intensities for zeros with positive and negative Jacobian signs.Stationarity permits evaluation at zero using Kac–Rice.Source: §5.1 · pp. 11–12, §5 · p. 11
5.2. Computation of the 2-point intensity of Hermite zeros
- For Hermite order n, the pair correlation has a strictly positive limit at zero separation, unlike the holomorphic case; the sequence l_n increases toward 1/4.The figure illustrates the curves for n = 1, 2, and 3 and a zoom highlighting l_1, l_2, and l_3.Source: §5.2 · p. 14
5.3. Signed 2-point intensity
- The signed two-point intensity is given by an expression involving the covariance matrix Γ defined earlier.The proof is deferred to Appendix 8.6.Source: §5.3 · p. 15
5.4. Hybrid 2-point intensity
- The hybrid two-point intensity is computed using transformations that exploit the zeros’ translation and rotation invariance.The proposition’s proof is deferred to Appendix 8.7.Source: §5.4 · p. 15
5.5. On the repulsivity of charged zeros
- Same-charge zeros show strong short-range repulsion that weakens with distance, while opposite-charge zeros have weaker repulsion near the origin.The opposite-charge pair correlation remains larger near the origin.Source: §5.5 · p. 16
6.1. Detection based on winding number
- For n ≥ 1, local spectrogram minima serve as candidates, and a contour winding number of magnitude 1 accepts a zero and assigns its charge.The method evaluates the phase around a small contour; not every local minimum is a zero.Source: §6.1 · pp. 17, 19, §5.5 · p. 17
6.2. The optimal radius of the contour of the winding numbers for zeros detection
- The contour radius trades interpolation sensitivity at small radii against the risk of enclosing multiple zeros at large radii.The estimated error decreases to a minimum around R*_n, then increases when additional zeros can fall inside the contour.Source: §6.2 · pp. 19–20
7. Conclusion
The conclusion summarizes the charged zero populations and intensity formulas for Hermite-window spectrograms, highlights their nonzero pair-correlation limit, and proposes winding-number detection. Figure 4 plots detection error against contour radius for three window orders.
7. Conclusion
- The Hermite-window zero patterns split into positive- and negative-charge populations, with closed-form one-point and unsigned, signed, and hybrid two-point intensities.The pair correlation converges at the origin to a strictly positive constant, unlike the total repulsion reported for the Gaussian window.Source: §7 · p. 20
- A winding-number method is proposed for detecting spectrogram zeros on small circular contours.Figure 4 plots error E_n against radius r for n = 1, 2, and 3; the conclusion also proposes using charged-zero structure with Delaunay triangulation to discriminate signal from noise.Source: §7 · p. 20, Figure 4 · p. 21
8. Appendix
The appendix supplies proofs for propositions used in the paper and explains numerical implementation details, including a change of variables to address instability in an integral calculation.
8.1. Proof of Proposition 2.2
- The coefficient expansion is re-indexed to identify the standard generalized Laguerre polynomial form.The proof uses the orthonormal basis and coefficient extraction in the Fock space, with separate cases for k < n and k > n.Source: §8.1 · pp. 21–22
8.2. Proof of Proposition 5.1
- The proof applies the Dirichlet integral to J0 and Jr to obtain the ensuing two-variable integral expression.The appendix then states that this completes the proof.Source: §8.2 · p. 23
8.3. Implementation details
- A change of variables rescales the integral so its internal variations no longer depend on x, addressing rapid oscillations that challenge numerical discretization.The logarithmic plots of a0 and ar against r motivate the changes of variable used in the implementation.Source: §8.3 · pp. 24, 26, Figure 5 · p. 25
8.4. Proof of the monotonicity of ln
- The monotonicity proof reduces showing that l_n increases to showing that x_n increases.The appendix notes that ϕ(n) is strictly increasing before making this reduction.Source: §8.4 · p. 26
8.5. Proof of Proposition 5.2
- For the small-r asymptotics, the proof shows D0 and D1 converge to the same distribution and that each is asymptotically independent of its corresponding sum variable.It also develops expansions in x = r^2 and uses them in the asymptotic calculation.Source: §8.5 · pp. 26, 27, 29, 30
8.6. Proof of Proposition 5.3
- The proof constructs the covariance matrix of a zero-mean Gaussian vector from the kernel and uses Isserlis’ theorem in the expectation calculation.The resulting expressions are reinjected into the conditional expectation.Source: §8.6 · pp. 31–32
8.7. Proof of Proposition 5.4
- The proof computes E[Jr cos(sJ0)] through the function ϕ(s, t), using Jacobi’s formula for the matrix M.The appendix then gives the resulting expression to conclude the proof.Source: §8.7 · p. 33