LOCAL MAXIMA OF WHITE NOISE SPECTROGRAMS AND GAUSSIAN ENTIRE FUNCTIONS LUIS DANIEL ABREU
2025-04-26
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LOCAL MAXIMA OF WHITE NOISE SPECTROGRAMS AND
GAUSSIAN ENTIRE FUNCTIONS
LUIS DANIEL ABREU
Abstract. We confirm Flandrin’s prediction for the expected average of local maxima of
spectrograms of complex white noise with Gaussian windows (Gaussian spectrograms or,
equivalently, modulus of weighted Gaussian Entire Functions), a consequence of the conjec-
tured double honeycomb mean model for the network of zeros and local maxima, where the
area of local maxima centered hexagons is three times larger than the area of zero centered
hexagons. More precisely, we show that Gaussian spectrograms, normalized such that their
expected density of zeros is 1, have an expected density of 5/3 critical points, among those
1/3 are local maxima, and 4/3 saddle points, and compute the distributions of ordinate
values (heights) for spectrogram local extrema. This is done by first writing the spectro-
grams in terms of Gaussian Entire Functions (GEFs). The extrema are considered under
the translation invariant derivative of the Fock space (which in this case coincides with the
Chern connection from complex differential geometry). We also observe that the critical
points of a GEF are precisely the zeros of a Gaussian random function in the first higher
Landau level. We discuss natural extensions of these Gaussian random functions: Gauss-
ian Weyl-Heisenberg functions and Gaussian bi-entire functions. The paper also reviews
recent results on the applications of white noise spectrograms, connections between several
developments, and is partially intended as a pedestrian introduction to the topic.
1. Introduction
The present paper is concerned with the critical points (divided in local maxima,local
minima and saddle points) of spectrogram transformed white noise. More precisely, we are
interested in statistical averages of critical points and local extrema of white noise spectro-
grams with Gaussian windows. As first observed in [14], such white noise spectrograms can
be written in terms of a Gaussian Entire Function (GEF), an observation which had the
virtue of connecting the traditionally applications-oriented field of time-frequency analysis,
to the traditionally more fundamental topic of Gaussian Analytic Functions.
Date: October 14, 2022.
Key words and phrases. Local maxima, White noise, Spectrograms, Gaussian Entire Functions, Landau
levels, polyanalytic functions.
L.D. Abreu was supported by FWF Project 31225-N32.
1
arXiv:2210.06721v1 [math.PR] 13 Oct 2022
2 LUIS DANIEL ABREU
We will both deal with the Gaussian Entire Function
F(z) =
∞
X
k=0
ak
zk
√k!
and its translation invariant derivative
F1(z) = F1(z, z)=(∂z−z)F(z) =
∞
X
k=0
ak
(k− |z|2)zk−1
√k!,
where akare i. i. d. Gaussian random variables. The zeros of F1(z) describe the criti-
cal points of H(z) = |F(z)|2e−|z|2, which can be divided in saddle points and local max-
ima/minima, while, due to the maximum modulus principle, the equation F0(z) = 0 can
only yield saddle points. As we will see below, e−|z|2/2F(z)
2and e−|z|2/2F1(z)
2can be
identified with white noise spectrograms windowed by the Gaussian and the first Hermite
function, respectively. This leads to a Gaussian functions companion to the determinantal
point processes in higher Landau levels [47, 3, 5, 45, 58, 48, 31], also defined using time-
frequency transforms. Observe that, while F(z) is analytic, F1(z) is just polyanalytic, since
∂2
zF1(z, z) = 0 (see [2] for an introduction to polyanalytic functions from the time-frequency
analysis perspective).
We now make an intermezzo in this introduction to point out two curious properties.
•Silence points (zeros) repel each other strongly (2-point intensity vanishes at zero
[17, Section 3]).
•Loud points (critical points) display weaker repulsion (positive 2-point intensity at
zero [11]).
Back to our work, we will first compute the expected number of critical, saddle and local
maxima coordinates, defined under constraints on the critical points equation,
F1(zc) = 0,
and then the corresponding average distribution of ordinate values (the expected average
values of F(z) at the critical, saddle and local maxima coordinates), where the ordinate
value xcassociated to the point zcis
xc=e−|z|2/2F(zc)=SpecgW(zc)1/2∈R+.
The description of the local extrema and corresponding ordinates of spectrogram transformed
white noise is likely to be useful in signal analysis applications involving local maxima of
spectrograms as in, for instance, [62], but also in reassignment [32] and syncrosqueezing [21]
techniques. This is particularly evident in differential reassignment [20], where local maxima
are attractors of the corresponding vector field.
Our results also complement the information used in the recent algorithms aimed at re-
covering a signal embedded in white noise from its spectrogram zeros [33, 34, 14, 15, 12,
LOCAL EXTREMA OF SPECTROGRAMS AND GAUSSIAN ANALYTIC FUNCTIONS 3
38, 16, 25] (these ‘zero-based’ methods have also been recently extended to wavelets, see
[49, 6, 15], to the sphere [55] and to the Stockwell transform [52]). Such algorithms have
found notable applications, for instance, in methods for anonymize motion sensor data and
enhance privacy in the Internet of Things (IoT) [57].
The analysis of local extrema of white noise spectrograms relies on the eigenvalues of the
Hessians of the Gaussian vectors, from which one can derive the Kac-Rice type formulas as
in [23]. The resulting expected number of local maxima is 1/3 times the expected number of
zeros. One can provide heuristics for the 1/3 proportion using a random honeycomb model,
suggested by Flandrin to describe zeros and local maxima of spectrograms of white noise
(see Figure 1 on page 6). By extrapolating such heuristics to the setting of holomorphic
sections of a Hermitian holomorphic line bundle over a complex manifold [23], the model
also suggests a hand-waving explanation for the occurrence of the factor 1/3 in the first
term of the statistics of supersymmetric vacua (modelled as fixed Morse index critical points
of random holomorphic sections). Moving to a different physical context, a correspondence
between the critical points of Gaussian entire functions and the zeros of a Gaussian bi-entire
function will be described. The Gaussian bi-entire function lives on the first higher Landau
level eigenspace, a space of paramount importance in condensed matter physics, used to
model electron dynamics on a energy level above the first layer of charged-like particles. The
Landau levels are the classical explanation for the step-like changes in conductivity under
the action of a constant magnetic field associated with the integer Quantum-Hall effect [50].
The results in this paper are novel to a certain extent, since their statement (both in the
GEF and time-frequency language) has not appeared before in the literature and given a
direct proof. But one should not go as far as calling them ‘genuinely new’, since general
results have been obtained for compact K¨ahler manifolds and the particular calculations
done for SU(2) random polynomials in [23] and [29]. It may come as a surprise that the
more simple (and also the most well-studied) case of the GEF has not been studied before,
since the cases treated in [23] and [29] are much more complicated. At least formally, our
main results could have been indirectly derived (one may say ‘guessed’, since actually some
definitions are a bit different for the non-compact case, most notably in the section on critical
values) by properly considering limit cases of the results for SU(2) polynomials in [23] and
[29]. Even if this argument could be made rigorous, it would be a long journey in comparison
to the direct and relatively elementary proofs we offer in this paper.
An outline of the paper follows. Since we will alternate between the spectrogram and
GEF formulations, to avoid confusing readers not familiar with both topics, we try to always
present both perspectives, sometimes at the cost of a little redundancy. In the next section
we present a short review of the required fundamental concepts about random Gaussian
functions and time-frequency analysis. In the third section we present the main results,
4 LUIS DANIEL ABREU
together with some remarks concerning the motivations, implications and connections to
the work of other authors. The proofs of the main results are gathered in section 4. In
section 5, we will consider Gaussian non-analytic functions motivated by the results of the
paper. First we observe that the nonzero critical points of H(z) are precisely the zeros of
a bi-analytic Gaussian function with correlation kernel given by the reproducing kernel of
the second Landau level. This suggests a more general class of Gaussian random functions
with correlation kernels given by the kernel of the Weyl-Heisenberg ensemble [5, 3]. We will
also consider Gaussian bi-analytic functions, with correlation kernel given by the sum of
the reproducing kernels of the first two Landau levels. We close the presentation with a
conclusion section, discussing the connections between the topics and results involved in the
paper, and a few considerations about their symbiotic nature.
2. Background
2.1. Gaussian entire functions and spectrograms of white noise. Given a window
function g∈L2(R), the short-time Fourier transform of f∈L2(R) is
(2.1) Vgf(x, ξ) = ZR
f(t)g(t−x)e−2πiξtdt, (x, ξ)∈R2.
when kgk2= 1, the map Vgis an isometry between L2(R) and a closed subspace of L2(R2):
kVgfkL2(R2)=kfkL2(R), f ∈L2(R).
If we choose the Gaussian function h0(t) = 21
4e−πt2,t∈R, as a window in (2.1), then a
simple calculation shows that, identifying z= (x, ξ)∈R2with z:= x+iξ ∈C,
(2.2) e−iπxξ+π
2|z|2Vh0f(x, −ξ)=21/4ZR
f(t)e2πtz−πt2−π
2z2dt =Bf(z),
where Bf(z) is the Bargmann transform of f[43]. The Bargmann transform Bis a uni-
tary isomorphism from L2(R) onto the Bargmann-Fock space F(C) consisting of all entire
functions satisfying
(2.3) kFk2
F(C)=ZC|F(z)|2e−π|z|2dz < ∞.
We will consider Hermite functions normalized as follows:
hr(t) = 21/4
√r!−1
2√πr
eπt2dr
dtre−2πt2, r ≥0,
Let g(t) := 21/4e−πt2,t∈R, be the one-dimensional, L2-normalized Gaussian. The short-
time Fourier transform of the Hermite functions with respect to gis
(2.4) Vghk(¯z
√π) = eixξ zk
√k!e−|z|2/2, k ≥0.
LOCAL EXTREMA OF SPECTROGRAMS AND GAUSSIAN ANALYTIC FUNCTIONS 5
One can formally define complex white noise Wby the series expansion
(2.5) W(t) =
∞
X
k=0
akhk(t)
where akare i. i. d. Gaussian random variables (the probability of such sequences being
square summable is zero, thus, with probability one, the series is not convergent in L2, but
with a little extra effort the argument can be made rigorous, by defining the sum (2.5) in a
larger Banach space, see [15, Section 3] for a detailed approach). Then, by linearity of the
Short-time Fourier transform Vg, we have formally from (2.5) and (2.4)
(2.6) VgW(¯z
√π) =
∞
X
k=0
akVghk(¯z
√π) = eixξe−|z|2/2
∞
X
k=0
ak
zk
√k!
2.2. Gaussian entire functions and Chern critical points. Gaussian entire functions
are defined, for z∈C, as
(2.7) F(z) =
∞
X
k=0
ak
zk
√k!,
where akare i. i. d. Gaussian random variables, and can be related to the Short-time
Fourier transform of white noise (see the formal manipulation (2.6) and [14, 15] for details)
as follows:
VgW(¯z
√π) = eixξe−|z|2/2
∞
X
k=0
ak
zk
√k!
Since F(z) is an entire function, all local minima are zeros (see Proposition 1 in Section
4 for a proof). Moreover, the maximum principle rules out the possibility of finding local
maxima for |F(z)|. However, this is possible to do if we look at the extrema of white
noise spectrograms. More precisely, at the critical, saddle and local maxima points and
corresponding ordinate values, of the following random function:
(2.8) SpecgW(z)=
VgW(¯z
√π)
2
=
e−|z|2/2
∞
X
k=0
ak
zk
√k!
2
=e−|z|2/2F(z)
2=H(z)
where Vgstands for the Short-Time Fourier Transform with a Gaussian window, or, equiva-
lently, of
log (SpecgW(z))1
2= log
F(z)e−|z|2
2
= log |F(z)| − |z|2
2=U(z).
The non-zero critical points of H(z) and U(z) are given by the solutions of the equation
(2.9) ∇0
zF(z) = 0,
where ∇0
zis the translation-invariant derivative (the symmetric part of the Chern connection,
see section 4 for more details):
∇0
z=∂z−z
摘要：
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LOCALMAXIMAOFWHITENOISESPECTROGRAMSANDGAUSSIANENTIREFUNCTIONSLUISDANIELABREUAbstract.Wecon rmFlandrin'spredictionfortheexpectedaverageoflocalmaximaofspectrogramsofcomplexwhitenoisewithGaussianwindows(Gaussianspectrogramsor,equivalently,modulusofweightedGaussianEntireFunctions),aconsequenceoftheconje...
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