# Signal Analysis via the Stochastic Geometry of Spectrogram Level Sets
111Digital Object Identifier 10.1109/TSP.2022.3153596. S. Ghosh was partially
supported by the MOE grants R-146-000-250-133, R-146-000-312-114 and
MOE-T2EP20121-0013.

Subhroshekhar Ghosh222Department of Mathematics, National University of
Singapore, Singapore (email<EMAIL_ADDRESS>webpage: https://subhro-
ghosh.github.io/)., Meixia Lin333(Corresponding author) Institute of
Operations Research and Analytics, National University of Singapore, Singapore
(email<EMAIL_ADDRESS>webpage: https://linmeixia.github.io/).,
Dongfang Sun444Department of Mathematics, National University of Singapore,
Singapore (email: sun_dongfang@u.nus.edu).

###### Abstract

Spectrograms are fundamental tools in time-frequency analysis, being the
squared magnitude of the so-called short time Fourier transform (STFT). Signal
analysis via spectrograms has traditionally explored their peaks, i.e. their
maxima. This is complemented by a recent interest in their zeros or minima,
following seminal work by Flandrin and others, which exploits connections with
Gaussian analytic functions (GAFs). However, the zero sets (or extrema) of
GAFs have a complicated stochastic structure, complicating any direct
theoretical analysis. Standard techniques largely rely on statistical
observables from the analysis of spatial data, whose distributional properties
for spectrograms are mostly understood only at an empirical level. In this
work, we investigate spectrogram analysis via an examination of the stochastic
geometric properties of their level sets. We obtain rigorous theorems
demonstrating the efficacy of a spectrogram level sets based approach to the
detection and estimation of signals, framed in a concrete inferential set-up.
Exploiting these ideas as theoretical underpinnings, we propose a level sets
based algorithm for signal analysis that is intrinsic to given spectrogram
data, and substantiate its effectiveness via extensive empirical studies. Our
results also have theoretical implications for spectrogram zero based
approaches to signal analysis. To our knowledge, these results are arguably
among the first to provide a rigorous statistical understanding of signal
detection and reconstruction in this set up, complemented with provable
guarantees on detection thresholds and rates of convergence.

Keywords: Gaussian analytic function, level sets, stochastic geometry, Gabor
spectrogram, statistical inference, signal analysis

##  1 Introduction

Short Time Fourier Transforms (abbrv. STFTs) are foundational objects in time-
frequency analysis [1, 2, 3]. By introducing a window function in the
traditional Fourier transform, STFTs lead to a two-dimensional representation
of a signal on the time-frequency plane.

In a bit more detail, if $f\in L^{2}(\mathbb{R})$ is a signal and $\phi\in
L^{2}(\mathbb{R})$ is a (real-valued) window function, then the STFT of $f$
with respect to the window function $\phi$, denoted by $V_{\phi}f$, is a
function on $\mathbb{R}^{2}$ defined as

$V_{\phi}f(\tau,\omega)=\int_{R}f(t)\phi(t-\tau)e^{-2i\pi tw}\mathrm{d}t.$

Informally, the variable $\tau$ is the time variable, whereas $\omega$ is the
frequency variable in the time-frequency plane. Popular choices for the window
function include the Gaussian density function
$\phi(t)=\frac{1}{\sqrt{2\pi}\sigma}e^{-t^{2}/2\sigma^{2}}$. A key effect of
such window functions is to localise the signal $f$ in time (e.g., at the
scale $\sigma$ for the Gaussian window above).

A fundamental quantity that is a functional of the STFT of a signal is the
spectrogram, which is going to be germane to the investigations carried out in
this paper. The spectrogram of a signal is the square of the modulus of the
STFT of the signal. To be more precise, if $f\in L^{2}(\mathbb{R})$ is a
signal and $\phi\in L^{2}(\mathbb{R})$ is a (real-valued) window function,
then the spectrogram of $f$ with respect to the window function $\phi$,
denoted by $S_{\phi}f$, is a function on $\mathbb{R}^{2}$ defined as

$S_{\phi}f(\tau,\omega)=\left|\int_{R}f(t)\overline{\phi(t-\tau)}e^{-2i\pi
tw}\mathrm{d}t\right|^{2}=|V_{\phi}f(\tau,\omega)|^{2}.$ (1)

The specific setting of STFT with a Gaussian window function is referred to as
the Gabor transform [1]. Informally speaking, the spectrogram of a signal
associating with each point in the time-frequency plane is a localised measure
of the energy of the signal at that time and that frequency [4].

For a more detailed technical discussion on STFTs and their implications for
white noise, we refer the reader to Section 2. A comprehensive exposition with
complete theory and applications is available in [1].

Spectrogram analysis has been demonstrated to be highly effective in many
applications. A particularly significant domain of application is in the field
of acoustics. For instance, AM-FM-type signals with a small number of
components admit spectrogram representations that are sparse, in the sense
that locations with high spectral energy are few and far between on the time-
frequency plane, and allow for a cogent explanation in terms of the structure
of the original signal [5]. We refer the interested reader to [1], and the
references therein, for a more exhaustive overview of the various application
domains and their specific problems.

Classically, considerable attention has been focused on the maxima of the
spectrogram. This is related to the understanding that these capture greater
energy of the spectrogram, and therefore greater information about the signal.
Techniques such as synchrosqueezing, reassignment and ridge extraction have
gained prominence in the context of identifying and processing the maxima of
the spectrogram. For details on the classical methods, we refer the reader to
the excellent monographs [1, 2, 3, 6].

In recent years, a somewhat different strain of investigations has gained
prominence. This pertains to examining the zeros, as opposed to the maxima of
the spectrogram. In other words, instead of the high peaks, the attention
shifts to the deep valleys of the spectrogram. This line of investigations
originates from seminal work of Flandrin [5]. In essence, the central idea in
this vein of works revolves around the empirical observation of [5] that the
Gabor transform (i.e., STFT with a Gaussian window) of white noise is, in
fact, an analytic function on the time-frequency plane (in the single complex
variable $\tau+i\omega$), and as such, has a well-structured set of zeros that
form a discrete set without any accumulation points. Furthermore, Flandrin
observed that these zeros (which are also the zeros of the Gabor spectrogram
of white noise) exhibit a spatial distribution that is highly uniform (i.e.,
grid-like) on the time-frequency plane, with highly regular Voronoi
tessellation patterns. On the other hand, the presence of a non-zero signal
creates distortions in the highly uniform spatial distribution of points,
which can possibly be exploited for separating signal from noise, and
estimating the signal if one is present.

In a related direction, there has been a recent line of work connecting the
STFT of white noise to the so-called Gaussian Analytic Functions (abbrv.,
GAFs). In particular, it was established in [7] that the zeros of the
spectrogram of white noise have the same statistical distribution as the zeros
of the planar GAF, which is the most common model of GAFs defined in the
setting of the complex plane. GAFs and related models of Determinantal Point
Processes (abbrv., DPPs) have attracted great interest in probability theory
and statistical physics over the past two decades; see [8, 9] for a
comprehensive overview. Such processes are characterised by the so-called
hyperuniformity of the spatial distribution of points, which is of independent
interest in the statistical physics and condensed matter physics communities
[10, 11, 12].

In subsequent work, the relationship between spectrogram zeros of white noise
and zeros of GAFs has been extended to other geometries (apart from the planar
case, such as spherical and hyperbolic geometries) [4]. There is an extensive
theory of GAFs and their zeros on these spaces (see, e.g., [8, 13]), which
have been connected to zeros of spectrograms of white noise with non-Gaussian
windows on the time-frequency analysis side. In particular, the zeros of the
hyperbolic GAF have been demonstrated to correspond to the so-called analytic
wavelet transform of Daubechies and Paul [14]; in this context we also refer
the reader to the work [15] that appeared in parallel to and independent of
[4]. Further connections to Gaussian analytic functions were investigated in
[16].

In the context of Gabor spectrogram zeros of white noise, the hyperuniformity
of the GAF zero sets provides a cogent explanation for the empirical
observation in [5] that these zeros have a highly homogeneous spatial
distribution. However, the procedures to exploit this property in order to
perform signal detection and inference are largely empirical in nature. The
mathematically rigorous understanding of why and how such procedures are
effective, in terms of theoretical guarantees or quantitative rates, is rather
limited.

Fig. 1: Gabor spectrogram and spectrogram level set of linear combination of
fundamental modes corrupted by noise with $k_{1}=10$, $k_{2}=40$, $k_{3}=95$
in (15).

In this work, we investigate the stochastic geometry of the level sets of the
Gabor spectrogram of white noise. A random field $X$ on a set $E$, roughly
speaking, is a collection of (usually real or complex valued) random variables
$\\{X_{v}\\}_{v\in\mathbb{E}}$ indexed by the elements of $E$ that have a
joint statistical distribution. Generally, random fields are studied under
regularity conditions with respect to some intrinsic geometry on the set $E$
(such as almost sure continuity or smoothness or H$\ddot{\rm o}$lder
regularity, etc.). A typical example of such a background set $E$ would be a
Euclidean space $\mathbb{R}^{d}$ with its canonical Euclidean metric geometry
and Lebesgue measure. For any random field $X$ on a set $E$, and a threshold
$u\in\mathbb{R}$, the level set $\Lambda_{X}(u)$ is the set

$\Lambda_{X}(u):=\\{v\in E:|X_{v}|\geq u\\}.$ (2)

In other words, $\Lambda_{X}(u)$ is the random subset of $E$ where the random
field $X$ exceeds the threshold $u$. To be more precise, $\Lambda_{X}(u)$ as
defined above is sometimes referred to as the upper level set for the
threshold $u$; whereas the random subset $\Lambda_{X}(u)^{\complement}\subset
E$ is referred to as the lower level set. However, since the understanding of
these two sets is more or less interchangeable for our purposes, we will
adhere to the concept of the level set as outlined in (2).

In this work, we undertake a comparative investigation of the level sets of
the Gabor spectrogram of Gaussian white noise and those of a signal corrupted
by Gaussian white noise. In particular, we obtain a mathematically rigorous
understanding of certain (random) geometric and analytical properties of such
level sets. This analysis is carried out in Section 3. Utilising this analysis
as a cornerstone, we establish the theoretical foundations of a spectrogram
level sets based approach to signal analysis in the time-frequency domain.
This is explored in Section 4.

To be precise, we frame our result in terms of a natural hypothesis testing
problem to decide between pure white noise and the presence of a fundamental
signal. We provide an efficient test of hypothesis for this problem, and
provide theoretical guarantees for its effectiveness via non-asymptotic bounds
on the power of the test. Based on stochastic geometric considerations of
spectrogram level sets, we also demonstrate an estimation procedure for a
fundamental signal if it is present, and provide error bounds for the accuracy
of such estimation in terms of the quantity of data available.

Our analysis of level sets also has bearing on the spectrogram zero based
approach to filtering (on the lines of [5]), by rigorously demonstrating that
the presence of a non-trivial signal creates zero-free regions for the
spectrogram on the time-frequency plane, whose geometry is related to the
structure of the signal in a systematic manner. The zero sets (or the maxima
or minima) of GAFs have a complicated stochastic structure, which makes a
direct theoretical analysis of usual spectrogram analysis techniques via GAFs
a difficult proposition.

For inferential purposes, these techniques largely rely on statistical
observables from the analysis of spatial data, whose distributional properties
for spectrogram extrema are mostly understood empirically. Such statistics
include, e.g., the celebrated Ripley’s K function (cf. [7]). Since the (upper)
level sets of the Gabor spectrogram of signals cannot contain zeros of the
spectrogram, our results provide theoretical validation for spectrogram zero
based techniques, by demonstrating the appearance of canonical zero-free
regions on the time-frequency plane when a signal is present along with noise.

Motivated by our theoretical analysis, we propose an algorithm for signal
analysis that is intrinsic to the spectrogram data (and, in particular, does
not require any additional inputs regarding signal strength and such). This
procedure is able to effectively perform detection and estimation for linear
combinations of fundamental signals. The signal estimation turns out to be
highly accurate as long as the fundamental signals being combined are
reasonably well separated. We demonstrate the effectiveness of our algorithm
via empirical evidence in the form of numerical experiments. The algorithm and
its various attributes are analysed in Section 5.

At this point, a few words would be in order regarding the comparison and
contrast of our approach vis-a-vis thresholding-based techniques, which are
among the most common denoising methods in signal processing and statistics,
see e.g. [17, 18, 19]. It may be noted in this vein that the spectrogram level
sets we consider are spatial objects that live on the time-frequency plane
while the standard thresholding-based procedures generally truncate the small
coefficients in some basis expansion of an estimator of the signal.
Furthermore, the two approaches are conceptually different from an algorithmic
point of view. Our approach considers the level set as an input and provides
an estimate of the signal as an output (if it detects that one exists). In
contrast, generic thresholding-based methods first find a rough estimate of
the signal and then truncate small coefficients to zero. That is to say, such
techniques use thresholding as a step in their algorithm. Last but not least,
the coefficients in the standard thresholding methods are finite in number, so
the cost of an inadvertent error in thresholding can be substantial. On the
other hand, spectrogram level sets being continuous objects, we believe that
our approach would be more robust to procedural noise. Our approach sheds
light on the limiting performance of standard thresholding-based techniques
when the discretization parameters tend to zero, which is a common and useful
idealisation of spectrogram-based methods in practice.

To the best of our knowledge, our results are arguably among the first to
provide a rigorous statistical understanding of signal detection and
reconstruction in this set up, complemented with provable guarantees on
detection thresholds and rates of convergence. We therefore focus our
attention on the most fundamental setting for a signal, namely Hermite
functions, which form an orthonormal basis for $L^{2}(\mathbb{R})$ and is
central to Gabor analysis; our results appear to be novel even in such a set
up. Consideration of more refined signal classes, with characteristic
analytical or geometric properties, suggests a natural avenue for future
research.

##  2 STFT and white noise

###  2.1 The short-time Fourier transform

Given a signal $f$ and a window function $\phi$, the short-time Fourier
transform $V_{\phi}f(\cdot,\cdot)$ is defined as the inner product between $f$
and shifted $\phi$ (in the sense of time and frequency). The detailed
definition is shown as follows, which could be found in [1, Definition 3.1.1].

######  Definition 1.

Fix a window function $\phi\in L^{2}(\mathbb{R})$. The short-time Fourier
transform (STFT) of a function $f\in L^{2}(\mathbb{R})$ with respect to $\phi$
is defined as

$\displaystyle V_{\phi}f(u,v):=\langle f,M_{v}T_{u}\phi\rangle,\quad
u,v\in\mathbb{R},$ (3)

where $M_{v}f=e^{2i\pi v\cdot}f(\cdot)$ and $T_{u}f=f(\cdot-u)$. Here
$\langle\cdot,\cdot\rangle$ denotes the inner product in $L^{2}(\mathbb{R})$
with respect to the Lebesgue measure, which is defined as

$\displaystyle\langle f,g\rangle=\int f(t)\overline{g(t)}\mathrm{d}t,\quad
f,g\in L^{2}(\mathbb{R}).$

It needs to be mentioned that the above definition is not unique, as there is
some flexibility in the choice of the phase term in the integral. For example,
the symmetric form $\int f(t+\tau/2)\overline{\phi(t-\tau/2)}e^{-2i\pi
tw}\mathrm{d}t$, which is often called the cross-ambiguity function, plays an
important role in radar and in optics [20, 21]. Introducing a pure-phase term
in (3) has no impact on the corresponding spectrogram but it can help in
getting symmetric forms of the STFT, see Proposition 1.

######  Remark 1.

Note that the definition of the STFT in (3) could be extended to more general
functions, such as tempered distributions. The detailed discussion is given in
Section 2.2. A typical example of tempered distributions is Gaussian white
noise.

It is of special interest to study the STFT with respect to a Gaussian window
function $g(t)=2^{1/4}e^{-\pi t^{2}}$ such that $\|g\|_{L^{2}(\mathbb{R})}=1$.
This special setting of the STFT is referred to as the Gabor transform. The
following proposition provides a way to compute the Gabor transform using the
Bargmann transform, which is taken from [1, Proposition 3.4.1].

######  Proposition 1.

For $f\in L^{2}(\mathbb{R})$, it holds that

$\displaystyle V_{g}f(u,v)=\exp(-\pi
iuv-\frac{\pi}{2}|z|^{2})Bf(\bar{z}),\quad u,v\in\mathbb{R},$

where the complex number $z$ is taken as $z=u+iv$, and the Bargmann transform
[22] of a function $f$ on $\mathbb{R}$ is the function $Bf$ on $\mathbb{C}$
defined by

$\displaystyle Bf(z)=2^{1/4}\int_{\mathbb{R}}f(t)\exp(2\pi tz-\pi
t^{2}-\frac{\pi}{2}z^{2})\mathrm{d}t,\quad z\in\mathbb{C}.$

Some functions are known to have a simple closed-form Bargmann transform, such
as the Hermite functions. Let $\\{H_{k}\\}$ be Hermite polynomials [1, 23]
defined as

$\displaystyle H_{k}(x)=\frac{(-1)^{k}}{2^{2k-1/4}\pi^{k}k!}e^{\frac{3}{2}\pi
x^{2}}\frac{\mathrm{d}^{k}}{\mathrm{d}x^{k}}\left(e^{-2\pi x^{2}}\right),\quad
k=0,1,\cdots,$

which are orthonormal in $L^{2}(\mathbb{R})$ with respect to the window
function $g$, i.e.,

$\displaystyle\int H_{k}(x)H_{l}(x)g(x)\mathrm{d}x=\delta_{kl},\quad k,l\geq
0,$

where $\delta_{kl}$ is the Kronecker delta. The Hermite functions are then
defined as: for each $k=0,1,\cdots$,

$\displaystyle h_{k}(x)=\frac{(-1)^{k}}{2^{2k-1/2}\pi^{k}k!}e^{\pi
x^{2}}\frac{\mathrm{d}^{k}}{\mathrm{d}x^{k}}\left(e^{-2\pi x^{2}}\right).$ (4)

It can be proved that $\\{h_{k}\\}$ form an orthonormal basis of
$L^{2}(\mathbb{R})$. According to [1, Theorem 3.4.2], the Bargmann transform
of the Hermite function $h_{k}$ is in a quite simple form as

$\displaystyle Bh_{k}(z)=\frac{\pi^{k/2}z^{k}}{\sqrt{k!}},\quad k=0,1,\cdots.$

As a side note, we consider the form of Hermite functions in (4) instead of
the canonical form is to help us get a clean formula of its Bargmann
transform.

Based on the above discussions and Proposition 1, we can see that the Gabor
transform of the Hermite function $h_{k}$ could be computed via the following
proposition.

######  Proposition 2.

For any $k\in\mathbb{N}$, it holds that

$\displaystyle V_{g}h_{k}(u,v)=\exp(-\pi
iuv-\frac{\pi}{2}|z|^{2})\frac{\pi^{k/2}\overline{z}^{k}}{\sqrt{k!}},\quad
u,v\in\mathbb{R},$

where the complex number $z$ is taken as $z=u+iv$.

###  2.2 Gaussian white noise

In order to define white noise in signal processing, we need the definitions
of Schwartz space and tempered distributions [24, 23].

######  Definition 2.

Schwartz space ${\cal S}(\mathbb{R})$ is the function space consisting of
rapidly decreasing smooth functions from $\mathbb{R}$ to $\mathbb{C}$, i.e.,

$\displaystyle{\cal S}(\mathbb{R})=\left\\{\phi\in
C^{\infty}(\mathbb{R},\mathbb{C})\Big{|}\sup_{x\in\mathbb{R}}|x^{\alpha}\phi^{(\beta)}(x)|<\infty,\
\forall\alpha,\beta\in\mathbb{N}\right\\}.$

######  Definition 3.

The space of tempered distributions on $\mathbb{R}$, denoted as ${\cal
S}^{\prime}(\mathbb{R})$, is the continuous dual of ${\cal S}(\mathbb{R})$,
which consists of all linear and continuous functions from ${\cal
S}(\mathbb{R})$ to $\mathbb{C}$.

From the definition of Schwartz space, we can see that ${\cal S}(\mathbb{R})$
is a dense subspace of $L^{2}(\mathbb{R})$. Furthermore, we have that ${\cal
S}(\mathbb{R})\subseteq L^{2}(\mathbb{R})\subseteq{\cal
S}^{\prime}(\mathbb{R})$. For any $\psi\in S^{\prime}(\mathbb{R})$, $\phi\in
S(\mathbb{R})$, we define the action $\langle\psi,\phi\rangle:=\psi(\phi)$.
Based on this notation, the definition of STFT in (3) could be naturally
extended. That is, the STFT of $f\in{\cal S}^{\prime}(\mathbb{R})$ with
respect to a window function $\phi\in{\cal S}(\mathbb{R})$ is

$\displaystyle V_{\phi}f(u,v):=\langle f,M_{v}T_{u}\phi\rangle,\quad
u,v\in\mathbb{R}.$

Now we could define the white noise in a classical approach. We use the space
of tempered distributions ${\cal S}^{\prime}(\mathbb{R})$ as the basic
probability space, and its Borel subsets, denoted as ${\cal B}({\cal
S}^{\prime}(\mathbb{R}))$, as the event space. According to the Bochner-Minlos
theorem [23, Theorem 2.1.1], there exists a unique probability $\mu_{1}$ on
${\cal B}({\cal S}^{\prime}(\mathbb{R}))$ with the following property:

$\displaystyle\mathbb{E}\left[e^{i\langle\cdot,\phi\rangle}\right]:=\int_{S^{\prime}(\mathbb{R})}e^{i\langle\xi,\phi\rangle}\mathrm{d}\mu_{1}(\xi)=e^{-\frac{1}{2}\|\phi\|^{2}},\quad\phi\in{\cal
S}(\mathbb{R}),$

where $\|\phi\|^{2}=\|\phi\|^{2}_{L^{2}(\mathbb{R})}$,
$\langle\xi,\phi\rangle=\xi(\phi)$ is the action of $\xi\in
S^{\prime}(\mathbb{R})$ on $\phi\in S(\mathbb{R})$ and
$\mathbb{E}=\mathbb{E}_{\mu_{1}}$ denotes the expectation with respect to
$\mu_{1}$. The triplet $(S^{\prime}(\mathbb{R}),{\cal
B}(S^{\prime}(\mathbb{R})),\mu_{1})$ is called the while noise probability
space, and $\mu_{1}$ is called the white noise measure. The measure $\mu_{1}$
is also called the normalized Gaussian measure on ${\cal
S}^{\prime}(\mathbb{R})$, since according to [23, Lemma 2.1.2], for any random
variable $\xi$ with distribution $\mu_{1}$ and
$\phi_{1},\cdots,\phi_{n}\in{\cal S}(\mathbb{R})$ that are orthonormal in
$L^{2}(\mathbb{R})$, the vector
$(\langle\xi,\phi_{1}\rangle,\cdots,\langle\xi,\phi_{n}\rangle)$ is a real
standard Gaussian random vector.

Let $\xi$ be a random variable with distribution $\mu_{1}$. The following
proposition identifies each value of the random function ${\cal
F}[\xi](z):=V_{g}\xi(u,v)$, i.e., the Gabor transform of $\xi$, as a limit in
$L^{2}(\mu_{1})$, which comes from [7, Proposition 2 and Proposition 3].

######  Proposition 3.

Let $u,v\in\mathbb{R}$ and write $z=u+iv\in\mathbb{C}$. Then

$\displaystyle{\cal F}[\xi](z)=\sqrt{\pi}\exp(i\pi
uv-\frac{\pi}{2}|z|^{2})\sum_{k=0}^{\infty}\langle\xi,h_{k}\rangle\frac{\pi^{k/2}z^{k}}{\sqrt{k!}},$
(5)

where $\\{h_{k}\\}$ are the orthonormal Hermite functions defined in (4), and
convergence is in $L^{2}(\mu_{1})$. In addition, the random series

$\displaystyle\sum_{k=0}^{\infty}\langle\xi,h_{k}\rangle\frac{\pi^{k/2}z^{k}}{\sqrt{k!}}$

$\mu_{1}$-almost surely defines an entire function.

It should be noted that in the remaining part, we use the notation
$V_{g}\xi(\cdot,\cdot)$ when we treat the function in (5) as a function on
$\mathbb{R}^{2}$, and use the notation ${\cal F}[\xi](\cdot)$ when we treat
the function in (5) as a function on $\mathbb{C}$.

##  3 The spectrogram of white noise

The squared modulus of the STFT is called the spectrogram, which is commonly
interpreted as a measure of the signal around time and frequency. In order to
identify fundamental modes, we need the boundedness of the Gabor spectrogram
(i.e., the spectrogram of the STFT with a Gaussian window) of white noise,
within a bounded set, which enables us to obtain a mathematically rigorous
understanding of certain (random) geometric and analytical properties of
spectrogram level sets.

###  3.1 Gaussian random fields

Note that $\\{F[\xi](z)\\}_{z\in\mathbb{C}}$ is a centered Gaussian random
field indexed by $\mathbb{C}$, which is determined by its covariance function.
The covariance function of $\\{F[\xi](z)\\}_{z\in\mathbb{C}}$ could be
computed as in the following proposition, whose proof is in Section 7.1.

######  Proposition 4.

For any $z,w\in\mathbb{C}$, we write them in the form as $z=u_{1}+iv_{1}$,
$w=u_{2}+iv_{2}$. The covariance function of
$\\{F[\xi](z)\\}_{z\in\mathbb{C}}$ is

$\displaystyle K(z,w)$ $\displaystyle=\mathbb{E}\left[\left({\cal
F}[\xi](z)-\mathbb{E}\left[{\cal F}[\xi](z)\right]\right)\overline{\left({\cal
F}[\xi](w)-\mathbb{E}\left[{\cal F}[\xi](w)\right]\right)}\right]$
$\displaystyle=\pi\exp(i\pi u_{1}v_{1}-i\pi
u_{2}v_{2}-\frac{\pi}{2}|z|^{2}-\frac{\pi}{2}|w|^{2}+\pi z\overline{w}).$

Moreover, the variance function of $\\{F[\xi](z)\\}_{z\in\mathbb{C}}$ is

$\displaystyle\sigma^{2}(z):=\mathbb{E}\left[|{\cal
F}[\xi](z)-\mathbb{E}\left[{\cal F}[\xi](z)\right]|^{2}\right]=\pi.$

Before describing our main result, we need a metric $d$ on $\mathbb{C}$, known
as the canonical metric for ${\cal F}[\xi](\cdot)$, by setting

$\displaystyle
d(z,w)=\left\\{\mathbb{E}\left[|F[\xi](z)-F[\xi](w)|^{2}\right]\right\\}^{1/2},\quad
z,w\in\mathbb{C}.$

The following proposition provides the computation of $d(\cdot,\cdot)$ in an
explicit formula, and the proof could be referred to Section 7.2.

######  Proposition 5.

For any $z=u_{1}+iv_{1}$, $w=u_{2}+iv_{2}\in\mathbb{C}$, it holds that

$\displaystyle
d^{2}(z,w)=2\pi\left[1-\cos\left(\pi(u_{1}+u_{2})(v_{1}-v_{2})\right)\exp(-\frac{\pi}{2}|z-w|^{2})\right].$

###  3.2 The Gabor spectrogram of white noise

In this subsection, we focus on the boundedness of the Gabor spectrogram of
white noise $\\{F[\xi](z)\\}_{z\in\mathbb{B}_{L}}$, where the parameter set
$\mathbb{B}_{L}$ is the ball in $\mathbb{C}$ with size $L>0$ defined as

$\displaystyle\mathbb{B}_{L}:=\\{u+iv\in\mathbb{C}:\max\\{|u|,|v|\\}\leq
L,u,v\in\mathbb{R}\\}.$ (6)

It can be seen later that the boundedness depends on how large the parameter
set is when the size is measured by a metric that comes from the process
itself. We measure the size of the parameter set $\mathbb{B}_{L}$ via the
metric entropy, whose definition is given as follows.

######  Definition 4.

Suppose $\\{f(t)\\}_{t\in T}$ is a centered real valued Gaussian random field
with a parameter set $T\subseteq\mathbb{R}$, and $d_{f}$ is the associated
canonical metric defined as
$d_{f}(t,w)=\left\\{\mathbb{E}\left[|f(t)-f(w)|^{2}\right]\right\\}^{1/2}$.
Assume that $T$ is $d_{f}$-compact. Denote the $d_{f}$ ball centered on $t\in
T$ and of radius $\varepsilon$ as

$\displaystyle B_{d_{f}}(t,\varepsilon):=\\{w\in T\mid
d_{f}(t,w)\leq\varepsilon\\}.$

The metric entropy function for $\\{f(t)\\}_{t\in T}$ is defined as
$N(T,d_{f},\varepsilon)$, which is the smallest number of such balls that
cover $T$.

Note that in the above definition, the real valued Gaussian random field is
considered, while our Gaussian random field $\\{F[\xi](z)\\}_{z\in\mathbb{C}}$
is complex valued. Thus the boundedness of
$\\{F[\xi](z)\\}_{z\in\mathbb{B}_{L}}$ should be analysed through its real
part and imaginary part. The following theorem states that the Gabor
spectrogram of white noise is of order $\log{L}$ over the bounded set
$\mathbb{B}_{L}$ with high probability.

######  Theorem 1.

For $L\geq\pi$, we have that for any $\tau>0$,

$\displaystyle\mathbb{P}\left[\sup_{z\in\mathbb{B}_{L}}|{\cal
F}[\xi](z)|\leq\sqrt{2}(14K+\tau)\sqrt{\log L}\right]\geq
1-4\exp\left(-\frac{\tau^{2}}{2\pi}\cdot\log L\right),$

where $K>0$ is a constant.

We defer the proof of Theorem 1 to Section 7.3. Theorem 1 plays an important
role in investigating the random geometric and analytical property of the
spectrogram level sets of fundamental modes corrupted with white noise.

###  3.3 Spectrogram level sets of a noisy fundamental mode

Before investigating the property of the spectrogram level sets of a noisy
fundamental mode, we need the following proposition, which shows the maximum
value of the Gabor spectrogram of the Hermite function $h_{k}$ defined in (4).
The proof could be found in Section 7.4.

######  Proposition 6.

Given $k\in\mathbb{N}$, let $h_{k}$ be the Hermite function defined in (4).
Then it holds that

$\displaystyle\max_{u,v\in\mathbb{R}}|V_{g}h_{k}(u,v)|=\prod_{t=1}^{k}\sqrt{\frac{k}{et}},$

where the maximum value is obtained when $\sqrt{u^{2}+v^{2}}=\sqrt{k/\pi}$.

Furthermore, for any $u,v\in\mathbb{R}$ such that
$\sqrt{u^{2}+v^{2}}-\sqrt{k/\pi}=r\sqrt{k/\pi}$ with $r\in\mathbb{R}$, we have

$\displaystyle\frac{|V_{g}h_{k}(u,v)|}{\max_{u,v\in\mathbb{R}}|V_{g}h_{k}(u,v)|}=\frac{(1+r)^{k}}{e^{k(r+r^{2}/2)}}.$
(7)

Denote the spectrogram level set of a signal $y$, restricted to the ball
$\mathbb{B}_{L}\subset\mathbb{C}$ defined in (6), with threshold
$\gamma\in\mathbb{R}$ as:

$\displaystyle\Lambda(\gamma):=\\{(u,v)\in\mathbb{R}^{2}:u+iv\in\mathbb{B}_{L},\
|V_{g}y(u,v)|\geq\gamma\\}.$ (8)

We are now ready to investigate the structure of the level sets of the Gabor
spectrogram of a fundamental mode corrupted by additive white noise. The
following theorem gives us a mathematically rigorous understanding of
geometric and analytical properties of such level sets, with its proof given
in Section 7.5.

######  Theorem 2.

Assume the signal $y$ is generated as

$\displaystyle y=\lambda h_{k}+\xi,$

where $h_{k}$ is an Hermite function defined in (4), white noise $\xi$ is a
random variable with distribution $\mu_{1}$, and $\lambda\in\mathbb{R}$, is a
parameter. Then it can be seen that $y\in{\cal S}(\mathbb{R})$. Consider the
Gabor spectrogram of $y$ restricted to $\mathbb{B}_{L}$ as in (6). Let

$|\lambda|\geq\frac{5\sqrt{2}(14K+\tau)\sqrt{\log
L}}{\prod_{t=1}^{k}\sqrt{k/(et)}},$

where $K>0$ is the constant in Theorem 1 and $\tau>0$ is a parameter. Then,
for $L\geq\max\\{\sqrt{k/\pi},\pi\\}$, we have

$\displaystyle\emptyset\neq\Lambda\left(3\sqrt{2}(14K+\tau)\sqrt{\log
L}\right)\subseteq\left\\{(u,v)\in\mathbb{R}^{2}:\frac{|V_{g}h_{k}(u,v)|}{\prod_{t=1}^{k}\sqrt{k/(et)}}>\alpha\right\\},$

with probability $\geq 1-4\exp\left(-\frac{\tau^{2}}{2\pi}\cdot\log L\right)$,
where

$\alpha:=\frac{\sqrt{2}(14K+\tau)\sqrt{\log
L}}{|\lambda|\prod_{t=1}^{k}\sqrt{k/(et)}}\in(0,\frac{1}{5}].$

Up to now, we have proved that a well-chosen level set of the Gabor
spectrogram of a fundamental mode corrupted by Gaussian white noise is
nonempty with high probability, which makes it possible to conduct signal
detection via the level sets of the spectrogram of data.

##  4 Signal analysis via spectrogram level sets

In this section, we examine the implications of the geometric and analytic
properties of spectrograms, as studied in Section 3, to the problem of signal
analysis. We will focus on the setting of a single fundamental signal
corrupted by Gaussian white noise, which offers the requisite structural
clarity for obtaining a complete theoretical understanding of the signal
analysis problem, at the same time capturing foundational features of the
spectrogram level sets perspective on this question.

###  4.1 The generative and observational models

Herein, we clarify the generative and observation models under which we will
work for the rest of Section 4.

  * •

(Generative model): The observation $y$ is generated as

$\displaystyle y=\lambda h_{k}+\xi,$

where the signal $h_{k}$ is an Hermite function defined in (4), white noise
$\xi$ is a random variable with distribution $\mu_{1}$, and $\lambda$ is the
signal strength. We further assume that $1\leq k\leq k_{0}$, where
$k_{0}\in\mathbb{N}$ is given.

  * •

(Observational model): The level sets of the spectrogram at any prescribed
level $\theta$ may be observed, restricted to the $L_{\infty}$ ball
$\mathbb{B}_{L}\subset\mathbb{C}$ of radius $L$ defined in (6), i.e., the set
$\Lambda(\theta)$ defined in (8).

###  4.2 Signal Detection

We will structure our result for signal detection in the form of a hypothesis
testing problem as discussed below. The signal detection problem consists in
investigating the signal strength $|\lambda|$, relative to the noise (of
strength $1$), that is necessary for detecting the presence of the signal.

####  4.2.1 A hypothesis testing perspective

We frame the signal detection problem in terms of a natural simple versus
composite hypothesis testing question. To be precise, we desire to test the
following null vs. alternative hypotheses:

  * $H_{0}$:

The observation $y=\xi$, i.e., there is no signal but only pure noise

  * vs.

  * $H_{1}$:

The observation $y=\lambda h_{k}+\xi$ with $\lambda\neq 0$ and some integer
$k\in[0,k_{0}]$.

A test $\psi_{\theta}$ is a measurable function of the observed level set
$\Lambda(\theta)$ that maps $\Lambda(\theta)$ to the (two-element) set
$\\{0,1\\}$ (with the understanding that, the value $0$ corresponds to
acceptance of $H_{0}$ whereas the value $1$ pertains to rejecting the null and
accepting the alternative $H_{1}$). Let $\mathbb{P}^{0}$ denote the
distribution of $\Lambda(\theta)$ under $H_{0}$, whereas $\mathbb{P}_{m}$
denote the distribution of $\Lambda(\theta)$ under $H_{1}$ with $k=m$. We say
that we detect the signal at strength $\lambda$ if, for any $\delta>0$, there
exists a test $\psi_{\theta(\delta)}$ such that

$\mathbb{P}^{0}[\psi_{\theta(\delta)}=1]\vee\max_{1\leq k\leq
k_{0}}\mathbb{P}_{k}[\psi_{\theta(\delta)}=0]\leq\delta.$

It may be noted that the quantities $\mathbb{P}^{0}[\psi_{\theta(\delta)}=1]$
and $\max_{1\leq k\leq k_{0}}\mathbb{P}_{k}[\psi_{\theta(\delta)}=0]$ pertain
to the probabilities of Type I and Type II errors in this model, and we desire
to detect the signal such that the probabilities of there two types of errors
are less than $\delta$.

####  4.2.2 The test $\psi_{\theta}$

We consider the test

$\psi_{\theta}=\mathbbm{1}\\{\Lambda(\theta)\text{ is non-empty}\\}.$ (9)

For any $l\geq 0$, we define the minimax quantity

$\mathfrak{M}(l)=\min_{1\leq k\leq
l}\max_{u,v\in\mathbb{R}}|V_{g}h_{k}(u,v)|=\min_{1\leq k\leq
l}\prod_{t=1}^{k}\sqrt{k/(et)}.$ (10)

We are now ready to state the following theorem with its proof given in
Section 7.6, which provides the theoretical guarantees for the proposed test
of hypothesis via non-asymptotic bounds on the power of test.

######  Theorem 3.

In the generative and observational models as in Section 4.1, we consider the
hypothesis testing problem in Section 4.2.1. For any given $\delta>0$,
consider the test $\psi_{\theta(\delta)}$ as in (9) obtained by setting

$\theta(\delta)=3\sqrt{2}\left(14K\sqrt{\log
L}+\sqrt{2\pi\cdot\log(4/\delta)}\right),$

where $K>0$ is the constant in Theorem 1. Then, for
$L\geq\max\\{\sqrt{k_{0}/\pi},\pi\\}$, this test $\psi_{\theta(\delta)}$
performs signal detection as defined in Section 4.2.1 at signal strength

$\displaystyle|\lambda|\geq 5\sqrt{2}\
\mathfrak{M}(k_{0})^{-1}\left(14K\sqrt{\log
L}+\sqrt{2\pi\cdot\log(4/\delta)}\right).$ (11)

In addition to the hypothesis testing procedure stated above, we also
investigate an estimation procedure for a fundamental mode if it is present,
with comprehensive theoretical understandings.

###  4.3 Signal estimation

In this section, we demonstrate that signal estimation is possible with high
probability (as the observation size $L\to\infty$) in the set up of Section
4.1. To this end, we define the following statistics:

######  Definition 5.

We define

  * •

$\hat{\theta}:=\max\\{\theta:\psi_{\theta}=1;\theta\geq 0\\}$, where
$\psi_{\theta}$ is as in (9);

  * •

$\hat{k}:=[[\pi\cdot(\min\\{|z|:z\in\Lambda(\hat{\theta})\\})^{2}]]$, where
$[[x]]$ denotes the nearest integer to $x\in\mathbb{R}$;

  * •

$\hat{\lambda}:=\hat{\theta}/\prod_{t=1}^{\hat{k}}\sqrt{\hat{k}/(et)}$.

We further introduce a few notations. First, we recall (7) and define

$C(k,r):=\frac{(1+r)^{k}}{e^{k(r+r^{2}/2)}},\quad k\in\mathbb{N},\ r>-1.$

For every $k\in\mathbb{N}_{+}$, we continue with

$\beta(k):=\max\left\\{C(k,\frac{1}{4k+2}),C(k,-\frac{1}{4k+2})\right\\}.$
(12)

Observe that $\beta(k)<1$ for every $k\in\mathbb{N}_{+}$. Finally, for any
given $k_{0}\in\mathbb{N}_{+}$, we define

$C(k_{0})=\frac{2}{5\left(1-\max_{1\leq k\leq k_{0}}\beta(k)\right)}+1.$ (13)

We then state our mode estimation theorem as follows, whose proof is given in
Section 7.7. Note that this theorem provides an error bound for the accuracy
of the mode obtained by our estimation in terms of the quality of data
available.

######  Theorem 4.

In the set-up of Section 4.1 and parameter $\tau>0$, for
$L\geq\max\\{\sqrt{k_{0}/\pi},\pi\\}$, and signal strength

$|\lambda|\geq
t_{\mathrm{mode}}(\tau):=5C(k_{0})\sqrt{2}(14K+\tau)\mathfrak{M}(k_{0})^{-1}\sqrt{\log
L},$

where $K>0$ is the constant in Theorem 1, we have

$\inf_{1\leq k\leq k_{0}}\mathbb{P}_{k}[\hat{k}=k]\geq
1-4\exp\left(-\frac{\tau^{2}}{2\pi}\cdot\log L\right).$

In addition to the error bound for the estimated mode, we also provide an
estimation theorem for signal strength, which demonstrates that our estimation
procedure could give a highly accurate estimation of signal strength with high
probability. The proof is deferred to Section 7.8.

######  Theorem 5.

In the set-up of Section 4.1 and parameter $1\geq\delta>0$, for
$L>\max\\{\sqrt{k_{0}/\pi},\pi,\exp(14K/\delta)^{2}\\}$, and signal strength

$|\lambda|\geq
t_{\mathrm{strength}}:=5C(k_{0})\sqrt{2}\quad\mathfrak{M}(k_{0})^{-1}\log L,$

where $K>0$ is the constant in Theorem 1, we have

$\displaystyle\inf_{1\leq k\leq
k_{0}}\mathbb{P}_{k}\left[\left|\frac{\hat{\lambda}}{|\lambda|}-1\right|\leq\delta\right]\geq
1-4\exp\left(-\frac{\delta^{2}\log^{2}L}{2\pi}\cdot\left(1-\frac{14K}{\delta\sqrt{\log
L}}\right)^{2}\right).$

##  5 Algorithm for signal learning via spectrogram level sets

In this section, we propose a spectrogram level sets based algorithm for
signal detection and estimation. The algorithm is based on the analysis of
geometric and analytic properties of the white noise spectrogram in Section 3,
and the theoretical investigation of their implications for signal analysis
done in Section 4.

Algorithm 1 : Signal learning via spectrogram level sets

Step 1. Consider the setting as stated in Section 4.1. Compute the Gabor
spectrogram of a given signal $y$ with respect to the Gaussian window function
$g$, which is denoted as $V_{g}y(u,v)$, $u,v\in\mathbb{R}$;

Step 2. Given $L>0$ large enough, compute  $\displaystyle
m_{L}:=\max_{u+iv\in\mathbb{B}_{L}}|V_{g}y(u,v)|;$

Step 3. Compute the spectrogram level set
$\displaystyle\Lambda(0.2m_{L})=\left\\{(u,v)\in\mathbb{R}^{2}\mid
u+iv\in\mathbb{B}_{L},|V_{g}y(u,v)|\geq 0.2m_{L}\right\\};$

Step 4. Detect whether these exists an annulus in $\Lambda(0.2m_{L})$ centered
at the origin.

  * 4.1

If so, we say that there exists a fundamental mode.

  * 4.2

Moreover, suppose the average of the radii of the large circle and the smaller
one is $\eta$, the unknown index $k$ could be estimated as
$\lfloor{\pi\eta^{2}}\rfloor$, where $\lfloor{x}\rfloor$ denotes the greatest
integer less than or equal to $x\in\mathbb{R}$;

A key feature of the algorithm is that its operation is intrinsic to the
spectrogram data, and in particular, does not depend on the prior knowledge of
additional information (e.g., regarding the choice of threshold for the level
sets, or otherwise). It also utilizes a single level set of the spectrogram,
and does not entail the possibility of having to compare multiple spectrograms
(which might be necessary, e.g., to determine
$\max\\{\theta:\psi_{\theta}=1\\}$).

A few words are in order regarding the choice of the factor “0.2”.
Intuitively, we are going to pick a factor that is intrinsic to the
spectrogram data, which can not be close to neither 1 nor 0 as we would like
to observe an annulus if there exists a fundamental mode. Empirical evidence
shows that “0.2” is a quite good choice, which provides satisfactory
performance for most of the cases.

As for the procedure to detect the annulus and then estimate the average
radii, we give a bit more implementation details in the following remark.

######  Remark 2.

In practice, we always work on a set of grid points
$\\{(u_{i},v_{j})\\}_{i,j=1}^{2N+1}$ where $u_{i}=(i-1-N)L/N$,
$v_{j}=(j-1-N)L/N$ and $N>0$ is a given integer. We could construct a $0$-$1$
matrix $S\in\mathbb{R}^{(2N+1)\times(2N+1)}$ as $S_{ij}=1$ if
$(u_{i},v_{j})\in\Lambda(0.2m_{L})$, and $S_{ij}=0$ otherwise.

Choose a set of slopes $\\{\kappa_{m}\\}_{m=1}^{M}$ with
$-\infty<\kappa_{1}\leq\cdots\leq\kappa_{M}<\infty$. For each $\kappa_{m}$,
the line $v=\kappa_{m}u$ approximately intersects the grid at
$\\{(u_{i},v_{j_{i}^{m}})\\}_{i=1}^{2N+1}$ with
$j_{i}^{m}:=[[k_{m}u_{i}N/L]]+1+N$. Define $b^{m}\in\mathbb{R}^{2N+1}$ as
$b_{i}^{m}=S_{u_{i},v_{j_{i}^{m}}}$, which is a $0$-$1$ vector. Let $D^{m}$ be
the set of distances between grid points indexed by the start points and the
end points of the subvector of $b_{i}^{m}$ with consecutive elements to be
$1$. Set $D=\cup_{i=1}^{M}D^{m}$. We then apply histogram bin counts on $D$ to
see whether there exists a count in one bin exceeds $3M/2$. If so, we say that
there exists a fundamental mode, and $\eta$ in the algorithm could be
calculated as the median of the bin.

As a side note, the procedure in Algorithm 1 could be extended to learn linear
combinations of fundamental modes as long as the signals being combined are
reasonably well separated.

The effectiveness of Algorithm 1 for learning linear combinations of
fundamental modes, can be demonstrated via detailed empirical studies, which
we undertake in the following section.

##  6 Empirical investigations

We conduct empirical studies in this section to demonstrate the performance of
our algorithm on detection and estimation for linear combinations of
fundamental signals.

In order to measure the performance of Algorithm 1 for learning linear
combinations of fundamental signals $\sum_{i=1}^{m}{\lambda_{i}}h_{k_{i}}$
corrupted with noise, where $1\leq k_{i}\leq k_{0}$, we need to provide a
reasonable metric. It is known that the fundamental modes corresponding to $k$
and $k+1$ are quite similar as functions when $k$ becomes large, but this is
no longer very applicable if $k$ is very small, like $1$ or $2$. In order to
take this into consideration, we define modified accuracy (mACC) of the
estimation $\\{\hat{k_{j}}\\}_{j=1}^{\hat{m}}$ as

$\displaystyle\mbox{mACC}=\left\\{\begin{aligned} &0,&&\mbox{if $\hat{m}\neq
m$}\\\
&\max\left\\{0,1-\sum_{i=1}^{m}\left|\frac{\hat{k_{i}}}{k_{i}}-1\right|\right\\}&&\mbox{if
$\hat{m}=m$, $\max_{i}|\hat{k_{i}}-k_{i}|\leq 1$},\\\ &0,&&\mbox{if
$\hat{m}=m$, $\max_{i}|\hat{k_{i}}-k_{i}|>1$}.\end{aligned}\right.$

By definition, we can see that $\mbox{mACC}=1$ means the perfect estimation,
and $\mbox{mACC}=0$ represents the worst.

One fundamental mode. We first consider the case of one fundamental mode as
stated in Section 4.1. That is the generative model is given as

$\displaystyle y=\lambda h_{k}+\sigma\xi,$ (14)

where the signal $h_{k}$ is an Hermite function defined in (4), white noise
$\xi$ is a random variable with distribution $\mu_{1}$, $\lambda$ is the
signal strength and $\sigma$ is the noise strength. In each trail, we

  * •

take the observation radius $L=8$;

  * •

randomly generate the integer $k$ uniformly from $\\{1,2,\cdots,112\\}$;

  * •

randomly generate the parameter $\lambda$ uniformly in $[1,2]$;

  * •

take the parameter $\sigma$ as $\frac{1}{10\sqrt{\log L}}$.

Based on the Gabor spectrogram of the data $y$, we define the spectrogram
level set with threshold $\gamma m_{L}$ as

$\displaystyle\Lambda(\gamma m_{L})=\left\\{(u,v)\in\mathbb{R}^{2}\mid
u+iv\in\mathbb{B}_{L},\ |V_{g}y(u,v)|\geq\gamma m_{L}\right\\},$

where $m_{L}:=\max_{u+iv\in\mathbb{B}_{L}}|V_{g}y(u,v)|$. The parameter
$\gamma$ is typically taken to be $0.2$ as in our algorithm design.

Figure 2 and Figure 3 show two examples to illustrate the Gabor spectrogram of
data $y$ and its corresponding spectrogram level sets, where we fix $k=10$ and
$k=100$, respectively. As we can see, it is promising to detect the
fundamental mode through the annulus in the spectrogram level set.

Fig. 2: Gabor spectrogram and spectrogram level set of one fundamental mode
corrupted by noise with $k=10$ in (14).

Fig. 3: Gabor spectrogram and spectrogram level set of one fundamental mode
corrupted by noise with $k=100$ in (14).

The first row of Table 1 shows the empirical performance of Algorithm 1 for
learning one fundamental mode over $10000$ trails. One can see that the
average mACC is 99.32%, which implies that almost all trials could get the
estimation in $\\{k-1,k,k+1\\}$ where $k$ is the true mode index.

Table 1: Empirical performance of Algorithm 1 for learning the linear combination of fundamental modes over $10000$ trails with different parameters. Number of modes $(m)$ | The parameter $w$ | Average mACC
---|---|---
1 | – | 99.32%
2 | 1.5 | 94.96%
2 | 2 | 99.85%
3 | 1.5 | 91.93%
3 | 2 | 97.11%

Two or more fundamental modes. Note that Algorithm 1 could be extended to
estimate the linear combination of fundamental modes, as long as these modes
are reasonably well separated. We then test the case of $m\geq 2$ fundamental
modes, that is the generative model is given as

$\displaystyle y=\sum_{i=1}^{m}\lambda_{i}h_{k_{i}}+\sigma\xi,$ (15)

where signals $h_{k_{i}}$’s are Hermite functions defined in (4), white noise
$\xi$ is a random variable with distribution $\mu_{1}$, $\lambda_{i}$’s are
the signal strength and $\sigma$ is the noise strength. In each trail, we

  * •

take the observation radius $L=8$;

  * •

randomly generate the integers $k_{i}$’s uniformly from $\\{1,2,\cdots,112\\}$
such that each $k_{i}$ are well separated as:

$\displaystyle\sqrt{\frac{k_{i}}{\pi}}\notin[\sqrt{\frac{k_{j}}{\pi}}-w,\sqrt{\frac{k_{j}}{\pi}}+w],\quad
1\leq i\neq j\leq m,$

where $w>0$ is a preset parameter;

  * •

randomly generate each parameter $\lambda_{i}$ uniformly in $[1,2]$;

  * •

take the parameter $\sigma$ as $\frac{1}{10\sqrt{\log L}}$.

Note that in the generative model above, larger $w$ will give us a signal with
more separate peaks, which makes it easier to detect and estimate the
fundamental modes.

Figure 4 and Figure 1 show the Gabor spectrogram and spectrogram level set of
data generated by two and three fundamental modes, respectively. It can be
seen that, in these two cases, the annuli in the spectrogram level sets are
well separated, which enables us to detect and estimate the signals via
analysing the spectrogram level sets.

Fig. 4: Gabor spectrogram and spectrogram level set of linear combination of
fundamental modes corrupted by noise with $k_{1}=8$, $k_{2}=90$ in (15).

Table 1 also shows the empirical performance of Algorithm 1 for learning two
or more fundamental modes over $10000$ trails with different choices of $w$.
As we can see from the table, when the parameter $w$ is taken as the value of
$2$, our algorithm gives more accurate estimation due to the fact that in this
case, the fundamental modes are more separated. Overall, our algorithm is
quite effective for learning linear combinations of fundamental modes as long
as these modes are reasonably well separated.

##  7 Proofs of main results

In this section, we present the proofs of the main theoretical results in this
paper. The proofs will be divided into different subsections, one pertaining
to each result.

###  7.1 Proof of Proposition 4

###### Proof.

Notice that $\mathbb{E}\left[{\cal F}[\xi](z)\right]=0$ for all
$z\in\mathbb{C}$. Denote $\xi_{k}:=\langle\xi,h_{k}\rangle$, which is real
valued as $\xi$ follows the white noise measure $\mu_{1}$. Then we can see
that

$\displaystyle{\cal F}[\xi](z)\overline{{\cal F}[\xi](w)}$ $\displaystyle=$
$\displaystyle\pi\exp(i\pi u_{1}v_{1}-i\pi
u_{2}v_{2}-\frac{\pi}{2}|z|^{2}-\frac{\pi}{2}|w|^{2})\times\sum_{k=0}^{\infty}\left[\sum_{l=0}^{k}\left(\xi_{l}\frac{\pi^{l/2}z^{l}}{\sqrt{l!}}\right)\left(\xi_{k-l}\frac{\pi^{(k-l)/2}\overline{w}^{k-l}}{\sqrt{(k-l)!}}\right)\right].$

Therefore, due to the fact that $\\{\xi_{k}\\}$ are i.i.d. real standard
Gaussian random variables, it holds that

$\displaystyle\mathbb{E}\left[{\cal F}[\xi](z)\overline{{\cal
F}[\xi](w)}\right]$ $\displaystyle=\pi\exp(i\pi u_{1}v_{1}-i\pi
u_{2}v_{2}-\frac{\pi}{2}|z|^{2}-\frac{\pi}{2}|w|^{2})\times\sum_{k=0}^{\infty}\mathbb{E}\left[\sum_{l=0}^{k}\left(\xi_{l}\frac{\pi^{l/2}z^{l}}{\sqrt{l!}}\right)\left(\xi_{k-l}\frac{\pi^{(k-l)/2}\overline{w}^{k-l}}{\sqrt{(k-l)!}}\right)\right]$
$\displaystyle=\pi\exp(i\pi u_{1}v_{1}-i\pi
u_{2}v_{2}-\frac{\pi}{2}|z|^{2}-\frac{\pi}{2}|w|^{2})\sum_{k=0}^{\infty}\frac{\pi^{k}z^{k}\overline{w}^{k}}{k!}$
$\displaystyle=\pi\exp(i\pi u_{1}v_{1}-i\pi
u_{2}v_{2}-\frac{\pi}{2}|z|^{2}-\frac{\pi}{2}|w|^{2}+\pi z\overline{w}).$

Thus by definition, the covariance function could be computed as

$\displaystyle K(z,w)$ $\displaystyle=\mathbb{E}\left[{\cal
F}[\xi](z)\overline{{\cal F}[\xi](w)}\right]$ $\displaystyle=\pi\exp(i\pi
u_{1}v_{1}-i\pi u_{2}v_{2}-\frac{\pi}{2}|z|^{2}-\frac{\pi}{2}|w|^{2}+\pi
z\overline{w}).$

Taking $w=z$ in the above formula, we have that

$\displaystyle\sigma^{2}(z)=\mathbb{E}\left[|{\cal F}[\xi](z)|^{2}\right]=\pi
e^{-\pi|z|^{2}}e^{\pi|z|^{2}}=\pi,$

which completes the proof. ∎

###  7.2 Proof of Proposition 5

###### Proof.

According to the definition of $d(\cdot,\cdot)$, we have that

$\displaystyle d^{2}(z,w)$
$\displaystyle=\mathbb{E}\left[\left(F[\xi](z)-F[\xi](w)\right)\overline{\left(F[\xi](z)-F[\xi](w)\right)}\right]$
$\displaystyle=\sigma^{2}(z)+\sigma^{2}(w)-K(z,w)-K(w,z)$
$\displaystyle=2\pi-K(z,w)-\overline{K(z,w)}.$

Writing $z=u_{1}+iv_{1}$ and $w=u_{2}+iv_{2}$, we have that

$\displaystyle K(z,w)+\overline{K(z,w)}$ $\displaystyle=2{\rm
Re}\left(K(z,w)\right)$
$\displaystyle=2\pi\exp(-\frac{\pi}{2}|z|^{2}-\frac{\pi}{2}|w|^{2}+{\rm
Re}\left(\pi z\overline{w}\right))\times\cos\left(\pi u_{1}v_{1}-\pi
u_{2}v_{2}+{\rm Im}\left(\pi z\overline{w}\right)\right)$
$\displaystyle=2\pi\exp(-\frac{\pi}{2}|z-w|^{2})\cos\left(\pi u_{1}v_{1}-\pi
u_{2}v_{2}-\pi u_{1}v_{2}+\pi v_{1}u_{2}\right)$
$\displaystyle=2\pi\cos\left(\pi(u_{1}+u_{2})(v_{1}-v_{2})\right)\exp(-\frac{\pi}{2}|z-w|^{2}).$

Therefore,

$\displaystyle
d^{2}(z,w)=2\pi\left[1-\cos\left(\pi(u_{1}+u_{2})(v_{1}-v_{2})\right)\exp(-\frac{\pi}{2}|z-w|^{2})\right],$

which completes the proof. ∎

###  7.3 Proof of Theorem 1

###### Proof.

For any $z\in\mathbb{C}$, we could write

$\displaystyle{\cal F}[\xi](z)={\cal F}_{\rm R}[\xi](z)+i{\cal F}_{\rm
I}[\xi](z),$

where ${\cal F}_{\rm R}[\xi](z)$, ${\cal F}_{\rm I}[\xi](z)$ are real valued.
Thus $\\{{\cal F}_{\rm R}[\xi](z)\\}_{z\in\mathbb{C}}$, $\\{{\cal F}_{\rm
I}[\xi](z)\\}_{z\in\mathbb{C}}$ are (real) centered Gaussian random fields.

Due to Borell-TIS inequality [25, Theorem 2.1.1], we have that for all
$\rho>0$,

$\displaystyle\mathbb{P}\left[\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm
R}[\xi](z)-\mathbb{E}\left[\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm
R}[\xi](z)\right]>\rho\right]\leq$
$\displaystyle\exp(-\frac{\rho^{2}}{2\sup_{z\in\mathbb{B}_{L}}\mathbb{E}\left[{\cal
F}_{\rm R}[\xi](z)^{2}\right]})$ $\displaystyle\leq$
$\displaystyle\exp(-\frac{\rho^{2}}{2\sup_{z\in\mathbb{B}_{L}}\mathbb{E}\left[|{\cal
F}[\xi](z)|^{2}\right]})=\exp(-\frac{\rho^{2}}{2\pi}),$

where the last equality follows from Proposition 4. As a result, with
probability $\geq 1-\exp(-\frac{\rho^{2}}{2\pi})$, we have

$\displaystyle\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm R}[\xi](z)$
$\displaystyle=\mathbb{E}\left[\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm
R}[\xi](z)\right]+\left(\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm
R}[\xi](z)-\mathbb{E}\left[\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm
R}[\xi](z)\right]\right)$
$\displaystyle\leq\mathbb{E}\left[\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm
R}[\xi](z)\right]+\rho.$

According to [25, Theorem 1.3.3], we know that there exists a universal
constant $K_{\rm R}>0$ such that

$\displaystyle\mathbb{E}\left[\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm
R}[\xi](z)\right]\leq K_{\rm
R}\int_{0}^{\infty}\sqrt{\log\left(N(\mathbb{B}_{L},d_{\rm
R},\varepsilon)\right)}\mathrm{d}\varepsilon,$

where $N(\mathbb{B}_{L},d_{\rm R},\varepsilon)$ denotes the metric entropy for
$\\{{\cal F}_{\rm R}[\xi](z)\\}_{z\in\mathbb{B}_{L}}$ with the associated
canonical metric defined as $d_{\rm R}(z,w)=\left\\{\mathbb{E}\left[|F_{\rm
R}[\xi](z)-F_{\rm R}[\xi](w)|^{2}\right]\right\\}^{1/2}$. Thus it can be seen
that for any $z,w\in\mathbb{B}_{L}$, $d_{\rm R}(z,w)\leq d(z,w)$, which
indicates that for any $\varepsilon>0$,

$\displaystyle N(\mathbb{B}_{L},d_{\rm R},\varepsilon)\leq
N(\mathbb{B}_{L},d,\varepsilon).$

Therefore, with probability $\geq 1-\exp(-\frac{\rho^{2}}{2\pi})$, we have

$\displaystyle\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm R}[\xi](z)\leq K_{\rm
R}\int_{0}^{\infty}\sqrt{\log\left(N(\mathbb{B}_{L},d,\varepsilon)\right)}\mathrm{d}\varepsilon+\rho.$
(16)

Similarly, we could prove that with probability $\geq
1-\exp(-\frac{\rho^{2}}{2\pi})$,

$\displaystyle\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm I}[\xi](z)\leq K_{\rm
I}\int_{0}^{\infty}\sqrt{\log\left(N(\mathbb{B}_{L},d,\varepsilon)\right)}\mathrm{d}\varepsilon+\rho,$
(17)

where $K_{\rm I}>0$ is a constant.

Note that by the definition of $N(\mathbb{B}_{L},d,\varepsilon)$, when

$\displaystyle\varepsilon\geq{\rm
diam}(\mathbb{B}_{L}):=\sup_{z,w\in\mathbb{B}_{L}}d(z,w),$

we have $N(\mathbb{B}_{L},d,\varepsilon)=1$. Moreover, by Proposition 5, we
can see that $d^{2}(z,w)\leq 4\pi$ for any $z,w\in\mathbb{C}$, which implies
that ${\rm diam}(\mathbb{B}_{L})\leq 2\sqrt{\pi}$. Thus

$\displaystyle\int_{0}^{\infty}\sqrt{\log\left(N(\mathbb{B}_{L},d,\varepsilon)\right)}\mathrm{d}\varepsilon$
$\displaystyle=\int_{0}^{2\sqrt{\pi}}\sqrt{\log\left(N(\mathbb{B}_{L},d,\varepsilon)\right)}\mathrm{d}\varepsilon$
$\displaystyle=\int_{0}^{L^{-2}}\sqrt{\log\left(N(\mathbb{B}_{L},d,\varepsilon)\right)}\mathrm{d}\varepsilon+\int_{L^{-2}}^{2\sqrt{\pi}}\sqrt{\log\left(N(\mathbb{B}_{L},d,\varepsilon)\right)}\mathrm{d}\varepsilon.$

Next we will prove that when $0\leq\varepsilon\leq L^{-2}$, given
$z,w\in\mathbb{B}_{L}$ with $z=u_{1}+iv_{1}$, $w=u_{2}+iv_{2}$, if
$\max\\{|u_{1}-u_{2}|,|v_{1}-v_{2}|\\}\leq\frac{\varepsilon}{3\sqrt{2\pi^{3}}L}$,
it holds that $d(z,w)\leq\varepsilon$. Suppose $z,w\in\mathbb{B}_{L}$
satisfies
$\max\\{|u_{1}-u_{2}|,|v_{1}-v_{2}|\\}\leq\frac{\varepsilon}{3\sqrt{2\pi^{3}}L}$.
It can be observed that for any $x\in\mathbb{R}$, $e^{-x}\geq 1-x$ and
$\cos(x)\geq 1-x^{2}$. Based on these, we can see that

$\displaystyle|\pi(u_{1}+u_{2})(v_{1}-v_{2})|\leq\frac{2\pi
L\varepsilon}{3\sqrt{2\pi^{3}}L}\leq\frac{2\varepsilon}{3\sqrt{2\pi}}\leq\frac{2}{3\sqrt{2\pi}L^{2}}<1,$

which further implies

$\displaystyle\cos\left(\pi(u_{1}+u_{2})(v_{1}-v_{2})\right)>0.$

Thus, together with Proposition 5, it holds that

$\displaystyle d^{2}(z,w)$ $\displaystyle\leq
2\pi\left[1-\left(1-\pi^{2}(u_{1}+u_{2})^{2}(v_{1}-v_{2})^{2}\right)\left(1-\frac{\pi}{2}|z-w|^{2}\right)\right]$
$\displaystyle\leq
2\pi\left[\pi^{2}(u_{1}+u_{2})^{2}(v_{1}-v_{2})^{2}+\frac{\pi}{2}|z-w|^{2}\right]$
$\displaystyle\leq(8\pi^{3}L^{2}+\pi^{2})|z-w|^{2}\leq
9\pi^{3}L^{2}\frac{\varepsilon^{2}}{9\pi^{3}L^{2}}=\varepsilon^{2},$

where the last inequality follows from the fact that $L\geq\pi$, which means
that $d(z,w)\leq\varepsilon$.

Due to the above discussion, when $0\leq\varepsilon\leq L^{-2}$, if
$\\{B_{d_{\rm E}}(z_{i},\frac{\varepsilon}{3\sqrt{2\pi^{3}}L})\\}_{i=1}^{M}$
is a covering of $\mathbb{B}_{L}$, where $d_{\rm
E}(z,w):=\max\\{|u_{1}-u_{2}|,|v_{1}-v_{2}|\\}$,
$\\{B_{d}(z_{i},\varepsilon)\\}_{i=1}^{M}$ is also a covering of
$\mathbb{B}_{L}$. Therefore, we have

$\displaystyle
N(\mathbb{B}_{L},d,\varepsilon)\leq\left(\frac{2L}{\frac{2\varepsilon}{3\sqrt{2\pi^{3}}L}}\right)^{2}=18\pi^{3}\frac{L^{4}}{\varepsilon^{2}},\quad
0\leq\varepsilon\leq L^{-2}.$

As a result, it holds that

$\displaystyle\int_{L^{-2}}^{2\sqrt{\pi}}\sqrt{\log\left(N(\mathbb{B}_{L},d,\varepsilon)\right)}\mathrm{d}\varepsilon$
$\displaystyle\leq
2\sqrt{\pi}\sqrt{\log\left(N(\mathbb{B}_{L},d,L^{-2})\right)}$
$\displaystyle\leq 2\sqrt{\pi}\sqrt{\log\left(18\pi^{3}L^{8}\right)}$
$\displaystyle\leq 2\sqrt{\pi}\sqrt{\log 2+2\log 3+3\log\pi+8\log L}\leq
2\sqrt{14\pi}\sqrt{\log L},$

and

$\displaystyle\int_{0}^{L^{-2}}\sqrt{\log\left(N(\mathbb{B}_{L},d,\varepsilon)\right)}\mathrm{d}\varepsilon$
$\displaystyle\leq\int_{0}^{L^{-2}}\sqrt{\log\left(18\pi^{3}\frac{L^{4}}{\varepsilon^{2}}\right)}\mathrm{d}\varepsilon\leq\int_{0}^{L^{-2}}\sqrt{\log\left(\frac{L^{10}}{\varepsilon^{2}}\right)}\mathrm{d}\varepsilon$
$\displaystyle\leq\sqrt{7}\int_{0}^{L^{-2}}\sqrt{\log\left(\frac{1}{\varepsilon}\right)}\mathrm{d}\varepsilon=\sqrt{7}\left(\frac{\sqrt{\log
L}}{L^{2}}+\int_{\sqrt{2\log L}}^{\infty}e^{-t^{2}}\mathrm{d}t\right).$

According to Mills’ ratio inequality [26], we know that

$\displaystyle
e^{\frac{x^{2}}{2}}\int_{x}^{\infty}e^{-\frac{t^{2}}{2}}\mathrm{d}t<\frac{1}{x},$

which means

$\displaystyle\int_{\sqrt{2\log
L}}^{\infty}e^{-t^{2}}\mathrm{d}t<\frac{1}{2\sqrt{2}L^{2}\sqrt{\log L}}.$

Therefore, it holds that

$\displaystyle\int_{0}^{L^{-2}}\sqrt{\log\left(N(\mathbb{B}_{L},d,\varepsilon)\right)}\mathrm{d}\varepsilon$
$\displaystyle\leq\sqrt{7}\left(\frac{\sqrt{\log
L}}{L^{2}}+\frac{1}{2\sqrt{2}L^{2}\sqrt{\log L}}\right)$
$\displaystyle<\frac{4\sqrt{7}+\sqrt{14}}{4\pi^{2}}\sqrt{\log L},$

which further implies

$\displaystyle\int_{0}^{\infty}\sqrt{\log\left(N(\mathbb{B}_{L},d,\varepsilon)\right)}\mathrm{d}\varepsilon$
$\displaystyle\leq(2\sqrt{14\pi}+\frac{4\sqrt{7}+\sqrt{14}}{4\pi^{2}})\sqrt{\log
L}$ $\displaystyle<14\sqrt{\log L}.$

Take $\rho=\tau\sqrt{\log L}$ in (16), we have with probability $\geq
1-\exp\left(-\frac{\tau^{2}}{2\pi}\cdot\log L\right)$,

$\displaystyle\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm R}[\xi](z)$
$\displaystyle\leq K_{\rm
R}\int_{0}^{\infty}\sqrt{\log\left(N(\mathbb{B}_{L},d,\varepsilon)\right)}\mathrm{d}\varepsilon+\tau\sqrt{\log
L}$ $\displaystyle\leq(14K_{\rm R}+\tau)\sqrt{\log L}.$

Similarly, according to (17), we have with probability $\geq
1-\exp\left(-\frac{\tau^{2}}{2\pi}\cdot\log L\right)$,

$\displaystyle\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm I}[\xi](z)\leq(14K_{\rm
I}+\tau)\sqrt{\log L}.$

Denote $K=\max\\{K_{R},K_{I}\\}$, then it holds that

$\displaystyle\mathbb{P}\left[\sup_{z\in\mathbb{B}_{L}}|{\cal
F}[\xi](z)|>\sqrt{2}(14K+\tau)\sqrt{\log L}\right]$
$\displaystyle\leq\mathbb{P}\left[\sup_{z\in\mathbb{B}_{L}}|{\cal F}_{\rm
R}[\xi](z)|>(14K+\tau)\sqrt{\log
L}\right]+\mathbb{P}\left[\sup_{z\in\mathbb{B}_{L}}|{\cal F}_{\rm
I}[\xi](z)|>(14K+\tau)\sqrt{\log L}\right]$ $\displaystyle\leq
2\mathbb{P}\left[\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm
R}[\xi](z)>(14K+\tau)\sqrt{\log
L}\right]+2\mathbb{P}\left[\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm
I}[\xi](z)>(14K+\tau)\sqrt{\log L}\right],$

where the last inequality follows from symmetry. As a result, we have

$\displaystyle\mathbb{P}\left[\sup_{z\in\mathbb{B}_{L}}|{\cal
F}[\xi](z)|\leq\sqrt{2}(14K+\tau)\sqrt{\log L}\right]$ $\displaystyle\geq
2\mathbb{P}\left[\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm
R}[\xi](z)\leq(14K+\tau)\sqrt{\log
L}\right]+2\mathbb{P}\left[\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm
I}[\xi](z)\leq(14K+\tau)\sqrt{\log L}\right]-3$ $\displaystyle\geq
2\mathbb{P}\left[\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm
R}[\xi](z)\leq(14K_{\rm R}+\tau)\sqrt{\log
L}\right]+2\mathbb{P}\left[\sup_{z\in\mathbb{B}_{L}}{\cal F}_{\rm
I}[\xi](z)\leq(14K_{\rm I}+\tau)\sqrt{\log L}\right]-3$ $\displaystyle\geq
1-4\exp\left(-\frac{\tau^{2}}{2\pi}\cdot\log L\right),$

which completes the proof. ∎

###  7.4 Proof of Proposition 6

###### Proof.

According to Proposition 2, we have for any $u,v\in\mathbb{R}$,

$\displaystyle|V_{g}h_{k}(u,v)|=\exp(-\frac{\pi}{2}|z|^{2})\frac{\pi^{k/2}|z|^{k}}{\sqrt{k!}},$
(18)

where $z\in\mathbb{C}$ is taken as $z=u+iv$. Define
$f:\mathbb{R}\rightarrow\mathbb{R}$ as
$f(x)=e^{-\frac{\pi}{2}x^{2}}\frac{\pi^{k/2}x^{k}}{\sqrt{k!}}.$ Due to the
fact that

$\displaystyle
f^{\prime}(x)=e^{-\frac{\pi}{2}x^{2}}\frac{\pi^{k/2}x^{k-1}}{\sqrt{k!}}\left(-\pi
x^{2}+k\right),$

we know for any $x\geq 0$,

$\displaystyle f(x)\leq
f(\sqrt{\frac{k}{\pi}})=\prod_{t=1}^{k}\sqrt{\frac{k}{et}}.$

Therefore, together with the equation (18), we have that

$\displaystyle\max_{u,v\in\mathbb{R}}|V_{g}h_{k}(u,v)|=\prod_{t=1}^{k}\sqrt{\frac{k}{et}},$

where the maximum is obtained when $\sqrt{u^{2}+v^{2}}=\sqrt{k/\pi}$.

In addition, for any $u,v\in\mathbb{R}$ such that
$\sqrt{u^{2}+v^{2}}=(1+r)\sqrt{k/\pi}$ with $r\in\mathbb{R}$, it holds that

$\displaystyle\frac{|V_{g}h_{k}(u,v)|}{\max_{u,v\in\mathbb{R}}|V_{g}h_{k}(u,v)|}$
$\displaystyle=\frac{e^{-\frac{k}{2}(1+r)^{2}}(1+r)^{k}(\sqrt{k/\pi})^{k}}{e^{-\frac{k}{2}}(\sqrt{k/\pi})^{k}}=\frac{(1+r)^{k}}{e^{k(r+r^{2}/2)}},$

which completes the proof. ∎

###  7.5 Proof of Theorem 2

###### Proof.

By the definition of the STFT in (3), we can see that

$\displaystyle V_{g}y(u,v)=\lambda V_{g}h_{k}(u,v)+V_{g}\xi(u,v),\quad
u,v\in\mathbb{R}.$

Thus it holds that for any $u,v\in\mathbb{R}$,

$\displaystyle|\lambda||V_{g}h_{k}(u,v)|+|{\cal F}[\xi](z)|$
$\displaystyle\geq|V_{g}y(u,v)|\geq|\lambda||V_{g}h_{k}(u,v)|-|{\cal
F}[\xi](z)|,$

where $z=u+iv\in\mathbb{C}$ and ${\cal F}[\xi](z)=V_{g}\xi(u,v)$ as in (5).
According to Theorem 1, for $L\geq\pi$ and $\tau>0$, we have

$\displaystyle\mathbb{P}\left[\sup_{z\in\mathbb{B}_{L}}|{\cal
F}[\xi](z)|\leq\sqrt{2}(14K+\tau)\sqrt{\log L}\right]\geq
1-4\exp\left(-\frac{\tau^{2}}{2\pi}\cdot\log L\right),$

which means

$\displaystyle\sup_{z\in\mathbb{B}_{L}}\bigg{|}|V_{g}y(u,v)|-|\lambda||V_{g}h_{k}(u,v)|\bigg{|}\leq\sqrt{2}(14K+\tau)\sqrt{\log
L}$

with probability $\geq 1-4\exp\left(-\frac{\tau^{2}}{2\pi}\cdot\log L\right)$.
We then consider the case when the above inequality holds.

We first prove that the level set $\Lambda\left(3\sqrt{2}(14K+\tau)\sqrt{\log
L}\right)\neq\emptyset$. Consider $u_{0}+iv_{0}\in\mathbb{B}_{L}$ such that
$\sqrt{u_{0}^{2}+v_{0}^{2}}=\sqrt{k/\pi}$, we have

$\displaystyle|V_{g}y(u_{0},v_{0})|$
$\displaystyle\geq|\lambda||V_{g}h_{k}(u_{0},v_{0})|-\bigg{|}|V_{g}y(u_{0},v_{0})|-|\lambda||V_{g}h_{k}(u_{0},v_{0})|\bigg{|}$
$\displaystyle\geq|\lambda|\prod_{t=1}^{k}\sqrt{\frac{k}{et}}-\sup_{u+iv\in\mathbb{B}_{L}}\bigg{|}|V_{g}y(u,v)|-|\lambda||V_{g}h_{k}(u,v)|\bigg{|}$
$\displaystyle\geq 5\sqrt{2}(14K+\tau)\sqrt{\log
L}-\sqrt{2}(14K+\tau)\sqrt{\log L}$
$\displaystyle=4\sqrt{2}(14K+\tau)\sqrt{\log L}.$

That is to say,

$\displaystyle(u_{0},v_{0})\in\Lambda\left(3\sqrt{2}(14K+\tau)\sqrt{\log
L}\right).$

Next we prove that

$\displaystyle\Lambda\left(3\sqrt{2}(14K+c)\sqrt{\log
L}\right)\subseteq\left\\{(u,v):\frac{|V_{g}h_{k}(u,v)|}{\prod_{t=1}^{k}\sqrt{k/(et)}}>\alpha\right\\}.$

For any $u+iv\in\mathbb{B}_{L}$ such that
$|V_{g}h_{k}(u,v)|\leq\alpha\prod_{t=1}^{k}\sqrt{k/(et)}$, we can see that

$\displaystyle|V_{g}y(u,v)|$
$\displaystyle\leq|\lambda||V_{g}h_{k}(u,v)|+\sup_{u+iv\in\mathbb{B}_{L}}\bigg{|}|V_{g}y(u,v)|-|\lambda||V_{g}h_{k}(u,v)|\bigg{|}$
$\displaystyle\leq|\lambda|\alpha\prod_{t=1}^{k}\sqrt{k/(et)}+\sqrt{2}(14K+\tau)\sqrt{\log
L}$ $\displaystyle=2\sqrt{2}(14K+\tau)\sqrt{\log L}.$

which indicates that

$\displaystyle(u,v)\notin\Lambda\left(3\sqrt{2}(14K+\tau)\sqrt{\log
L}\right).$

Therefore, from the above discussion, we have that

$\displaystyle\emptyset\neq\Lambda\left(3\sqrt{2}(14K+\tau)\sqrt{\log
L}\right)\subseteq\left\\{(u,v):\frac{|V_{g}h_{k}(u,v)|}{\prod_{t=1}^{k}\sqrt{k/(et)}}>\alpha\right\\}$

with probability $\geq 1-4\exp\left(-\frac{\tau^{2}}{2\pi}\cdot\log L\right)$.
∎

###  7.6 Proof of Theorem 3

###### Proof.

We observe that if we set $\tau=\sqrt{2\pi\cdot\log(4/\delta)/\log L}$, then
the conclusion of Theorem 2 applies with probability $\geq 1-\delta$, the
level set

$\Lambda\left(3\sqrt{2}\left(14K\sqrt{\log
L}+\sqrt{2\pi\cdot\log(4/\delta)}\right)\right)$

and, for the fundamental signal $h_{k}$, the required signal strength

$\displaystyle|\lambda_{k}|\geq
5\sqrt{2}\left(\prod_{t=1}^{k}\sqrt{k/(et)}\right)^{-1}\left(14K\sqrt{\log
L}+\sqrt{2\pi\log(4/\delta)}\right).$

In addition, according to (10) and (11), we observe that the choice of signal
strength in the statement of this theorem satisfies

$\displaystyle|\lambda|\geq
5\sqrt{2}\left(\prod_{t=1}^{k}\sqrt{k/(et)}\right)^{-1}\left(14K\sqrt{\log
L}+\sqrt{2\pi\cdot\log(4/\delta)}\right)$

for all $1\leq k\leq k_{0}$.

In view of the above discussion, we can conclude that, for the given choices
of level set $\Lambda(\theta(\delta))$ and signal strength $|\lambda|$ as in
the statement of the present theorem, the conclusion of Theorem 2 holds with
probability $\geq 1-\delta$. Thus, if a fundamental signal $h_{k}$ is present
(under the generative model as in Section 4.1) with the prescribed signal
strength, then $\mathbb{P}_{k}[\psi_{\theta(\delta)}=0]\leq\delta$.

Further, with the same choices, the conclusion of Theorem 1 can also be
verified to hold with probability $\geq 1-\delta$. Thus, if no signal is
present (under the generative model as in Section 4.1), then
$\mathbb{P}^{0}[\psi_{\theta(\delta)}=1]\leq\delta$.

In consideration of this, and the definition of signal detection as laid out
in Section 4.2, we could conclude that the prescribed test as in Section 4.2
performs signal detection at the signal strength in the statement of the
present theorem. ∎

###  7.7 Proof of Theorem 4

###### Proof.

We consider the setting where $y=\lambda h_{k}+\xi$ for a fixed
$k\in[1,\ldots,k_{0}]$. Let $\Omega_{\tau}$ denote the event

$\Omega_{\tau}:=\\{\max_{u+iv\in\mathbb{B}_{L}}|V_{g}\xi(u,v)|\leq\sqrt{2}(14K+\tau)\sqrt{\log
L}\\}.$

From Theorem 1, we have $\mathbb{P}[\Omega_{\tau}]\geq
1-4\exp\left(-\frac{\tau^{2}}{2\pi}\cdot\log L\right)$. Denote

$\displaystyle{\cal M}_{k}$
$\displaystyle=\left\\{u+iv\in\mathbb{B}_{L}:(u,v)\in\arg\max_{u+iv\in\mathbb{B}_{L}}|V_{g}y(u,v)|^{2}\right\\}.$

We claim that on the event $\Omega_{\tau}$, for any $z\in\mathcal{M}_{k}$, the
quantity $\pi\cdot|z|^{2}$ satisfies

$k-\frac{1}{2}<\pi\cdot|z|^{2}<k+\frac{1}{2},$

where the proof of the claim is presented later. Thus, on the event
$\Omega_{\tau}$, for any $z\in\mathcal{M}_{k}$ we have
$[[\pi\cdot|z|^{2}]]=k$. Since $\mathbb{P}[\Omega_{\tau}]\geq
1-4\exp\left(-\frac{\tau^{2}}{2\pi}\cdot\log L\right)$, and the above analysis
holds for every $1\leq k\leq k_{0}$, we may deduce that

$\inf_{1\leq k\leq k_{0}}\mathbb{P}_{k}[\hat{k}=k]\geq
1-4\exp\left(-\frac{\tau^{2}}{2\pi}\cdot\log L\right),$

as desired.

We now prove the claim. We are going to prove that on the event
$\Omega_{\tau}$, if $z\in\mathcal{M}_{k}$, then it must hold that
$\left|\pi\cdot|z|^{2}-k\right|<1/2$. This condition can be elaborated as

$\left(1-\frac{1}{2k}\right)^{1/2}<\frac{|z|}{\sqrt{{k}/{\pi}}}<\left(1+\frac{1}{2k}\right)^{1/2}.$

This can further be expressed as

$-\frac{\sqrt{2k}-\sqrt{2k-1}}{\sqrt{2k}}<\frac{|z|}{\sqrt{{k}/{\pi}}}-1<\frac{\sqrt{2k+1}-\sqrt{2k}}{\sqrt{2k}};$

equivalently,

$\displaystyle-\frac{1}{\sqrt{2k}(\sqrt{2k}+\sqrt{2k-1})}<\frac{|z|}{\sqrt{{k}/{\pi}}}-1<\frac{1}{\sqrt{2k}(\sqrt{2k+1}+\sqrt{2k})}.$
(19)

We next define the set ${\cal A}_{k}$ as

$\displaystyle{\cal
A}_{k}:=\left\\{z\in\mathbb{B}_{L}:-\frac{1}{4k+2}<\frac{|z|}{\sqrt{k/\pi}}-1<\frac{1}{4k+2}\right\\}.$

Observe that any $z\in{\cal A}_{k}$ satisfies (19). We will demonstrate that,
in fact, on the event $\Omega_{\tau}$, $\mathcal{M}_{k}\subset{\cal A}_{k}$.
We prove this by deducing that
$A_{k}^{\complement}\subset\mathcal{M}_{k}^{\complement}$ on the event
$\Omega_{\tau}$. Note that the complement of the set is taken with respect to
the universal set $\mathbb{B}_{L}$. The proof is divided into three steps.

Step 1. Suppose $z_{0}=u_{0}+iv_{0}\in{\cal A}_{k}^{\complement}$, which
implies

$\displaystyle\left|\frac{|z_{0}|}{\sqrt{k/\pi}}-1\right|\geq\frac{1}{4k+2}.$

We may invoke (7) with
$|r_{0}|=\left|\frac{|z_{0}|}{\sqrt{{k}/{\pi}}}-1\right|$ and the definition
of $\beta(k)$ in (12) to conclude that

$\frac{|V_{g}h_{k}(u_{0},v_{0})|}{\max_{u,v\in\mathbb{R}}|V_{g}h_{k}(u,v)|}=C(k,r_{0})\leq\beta(k).$
(20)

For $L\geq\max\\{\sqrt{k_{0}/\pi},\pi\\}$, according to Proposition 6, we have

$\displaystyle\max_{u,v\in\mathbb{R}}|V_{g}h_{k}(u,v)|=\max_{u+iv\in\mathbb{B}_{L}}|V_{g}h_{k}(u,v)|.$
(21)

Now, on the event $\Omega_{\tau}$ we have

$\displaystyle|V_{g}y(u_{0},v_{0})|\leq|\lambda||V_{g}h_{k}(u_{0},v_{0})|+\max_{u+iv\in\mathbb{B}_{L}}|V_{g}\xi(u,v)|$
$\displaystyle\leq$
$\displaystyle|\lambda|\max_{u+iv\in\mathbb{B}_{L}}|V_{g}h_{k}(u,v)|\
\beta(k)+\max_{u+iv\in\mathbb{B}_{L}}|V_{g}\xi(u,v)|\quad\text{ [using
\eqref{eq:decay}, \eqref{eq: max_max}]}$ $\displaystyle=$
$\displaystyle|\lambda|\max_{u+iv\in\mathbb{B}_{L}}|V_{g}h_{k}(u,v)|\
\beta(k)\left(1+\frac{\max_{u+iv\in\mathbb{B}_{L}}|V_{g}\xi(u,v)|}{|\lambda|\max_{u+iv\in\mathbb{B}_{L}}|V_{g}h_{k}(u,v)|\beta(k)}\right)$
$\displaystyle\leq$
$\displaystyle|\lambda|\max_{u+iv\in\mathbb{B}_{L}}|V_{g}h_{k}(u,v)|\
\beta(k)\left(1+\frac{1}{5C(k_{0})\beta(k)}\right),$

where, in the last step, we have use the threshold for $|\lambda|$ as in the
statement of the present theorem, the definition of $\mathfrak{M}(k_{0})$ and
the event $\Omega_{\tau}$.

Step 2. Let ${\cal T}_{k}$ be the circle of radius $\sqrt{k/\pi}$ as: ${\cal
T}_{k}:=\\{z\in\mathbb{B}_{L}:|z|=\sqrt{k/\pi}\\}$. For
$z_{1}=u_{1}+iv_{1}\in{\cal T}_{k}$, on the event $\Omega_{\tau}$ we have

$\displaystyle|V_{g}y(u_{1},v_{1})|\geq$
$\displaystyle|\lambda||V_{g}h_{k}(u_{1},v_{1})|\left(1-\frac{\max_{u+iv\in\mathbb{B}_{L}}|V_{g}\xi(u,v)|}{|\lambda||V_{g}h_{k}(u_{1},v_{1})|}\right)$
$\displaystyle=$
$\displaystyle|\lambda|\max_{u+iv\in\mathbb{B}_{L}}|V_{g}h_{k}(u,v)|\left(1-\frac{\max_{u+iv\in\mathbb{B}_{L}}|V_{g}\xi(u,v)|}{|\lambda|\max_{u+iv\in\mathbb{B}_{L}}|V_{g}h_{k}(u,v)|}\right)$
$\displaystyle\geq$
$\displaystyle|\lambda|\max_{u+iv\in\mathbb{B}_{L}}|V_{g}h_{k}(u,v)|\left(1-\frac{1}{5C(k_{0})}\right),$

where, the equality in the second last step is due to Proposition 6 and in the
last step, we have use the threshold for $|\lambda|$, the definition of
$\mathfrak{M}(k_{0})$ and the event $\Omega_{\tau}$.

Step 3. From Step 1 and Step 2, we have that

$\displaystyle\sup_{u+iv\in{\cal A}_{k}^{\complement}}|V_{g}y(u,v)|$
$\displaystyle\leq|\lambda|\max_{u+iv\in\mathbb{B}_{L}}|V_{g}h_{k}(u,v)|\beta(k)\left(1+\frac{1}{5C(k_{0})\beta(k)}\right)$
$\displaystyle\inf_{u+iv\in{\cal T}_{k}}|V_{g}y(u,v)|$
$\displaystyle\geq|\lambda|\max_{u+iv\in\mathbb{B}_{L}}|V_{g}h_{k}(u,v)|\left(1-\frac{1}{5C(k_{0})}\right)$

Now, we can invoke the definition of $C(k_{0})$ as in (13) and observe that

$\max_{1\leq k\leq
k_{0}}\beta(k)\cdot\left(1+\frac{1}{5C(k_{0})\beta(k)}\right)<\left(1-\frac{1}{5C(k_{0})}\right).$

This implies that $\inf_{u+iv\in{\cal T}_{k}}|V_{g}y(u,v)|>\sup_{u+iv\in{\cal
A}_{k}^{\complement}}|V_{g}y(u,v)|$. Recalling the definition of
$\mathcal{M}_{k}$, we therefore deduce that
$A_{k}^{\complement}\subset\mathcal{M}_{k}^{\complement}$ on the event
$\Omega_{\tau}$. This completes the proof. ∎

###  7.8 Proof of Theorem 5

###### Proof.

We consider the setting where $y=\lambda h_{k}+\xi$ for a fixed
$k\in[1,\ldots,k_{0}]$. Define $\tau(\delta)=\delta\sqrt{\log L}-14K>0$. Let
$\Omega_{\tau(\delta)}$ denote the event

$\Omega_{\tau(\delta)}:=\\{\max_{u+iv\in\mathbb{B}_{L}}|V_{g}\xi(u,v)|\leq\sqrt{2}(14K+\tau(\delta))\sqrt{\log
L}\\}.$

Using Theorem 1, we may conclude that

$\displaystyle\mathbb{P}[\Omega_{\tau(\delta)}]$ $\displaystyle\geq
1-4\exp\left(-\frac{\tau(\delta)^{2}}{2\pi}\cdot\log
L\right)=1-4\exp\left(-\frac{\delta^{2}\log^{2}L}{2\pi}\cdot\left(1-\frac{14K}{\delta\sqrt{\log
L}}\right)^{2}\right).$

Using the fact that $0<\delta\leq 1$, we may deduce that the threshold
$t_{\mathrm{strength}}$ as in the statement of the present theorem satisfies

$\displaystyle t_{\mathrm{strength}}\geq
5C(k_{0})\sqrt{2}(14K+\tau(\delta))\mathfrak{M}(k_{0})^{-1}\sqrt{\log
L}=t_{\mathrm{mode}}(\tau(\delta)).$

Thus we may invoke Theorem 4 on the event $\Omega_{\tau(\delta)}$ above and
the signal strength $t_{\mathrm{strength}}$ in order to obtain $\hat{k}=k$.

For the rest of this proof, we will work on the event $\Omega_{\tau(\delta)}$,
whence $\hat{k}=k$ as discussed above. For any
$u_{0}+iv_{0}\in\mathbb{B}_{L}$, we have

$\displaystyle|\lambda||V_{g}h_{k}(u_{0},v_{0})|$
$\displaystyle-\max_{u+iv\in\mathbb{B}_{L}}|V_{g}\xi(u,v)|\leq|V_{g}y(u_{0},v_{0})|\leq|\lambda||V_{g}h_{k}(u_{0},v_{0})|+\max_{u+iv\in\mathbb{B}_{L}}|V_{g}\xi(u,v)|,$

which implies that

$\displaystyle|\lambda|\max_{u+iv\in\mathbb{B}_{L}}$
$\displaystyle|V_{g}h_{k}(u,v)|-\max_{u+iv\in\mathbb{B}_{L}}|V_{g}\xi(u,v)|\leq\max_{u+iv\in\mathbb{B}_{L}}|V_{g}y(u,v)|$
$\displaystyle\leq|\lambda|\max_{u+iv\in\mathbb{B}_{L}}|V_{g}h_{k}(u,v)|+\max_{u+iv\in\mathbb{B}_{L}}|V_{g}\xi(u,v)|.$

The above inequality could in turn be rewritten as:

$\displaystyle\left|\frac{\max_{u+iv\in\mathbb{B}_{L}}|V_{g}y(u,v)|}{|\lambda|\max_{u+iv\in\mathbb{B}_{L}}|V_{g}h_{k}(u,v)|}-1\right|\leq\frac{\max_{u+iv\in\mathbb{B}_{L}}|V_{g}\xi(u,v)|}{|\lambda|\max_{u+iv\in\mathbb{B}_{L}}|V_{g}h_{k}(u,v)|}.$

Note that by the definition of $\hat{\lambda}$, we obtain

$\left|\frac{\hat{\lambda}}{|\lambda|}-1\right|\leq\frac{\max_{u+iv\in\mathbb{B}_{L}}|V_{g}\xi(u,v)|}{|\lambda|\max_{u+iv\in\mathbb{B}_{L}}|V_{g}h_{k}(u,v)|}.$

Using the threshold $|\lambda|\geq 5C(k_{0})\sqrt{2}\
\mathfrak{M}(k_{0})^{-1}\log L$ as in the statement of the present theorem, we
can see that on the event $\Omega_{\tau(\delta)}$,

$\left|\frac{\hat{\lambda}}{|\lambda|}-1\right|\leq\frac{14K+\tau(\delta)}{5C(k_{0})\sqrt{\log
L}}\leq\frac{14K+\tau(\delta)}{\sqrt{\log L}}.$

Using $\tau(\delta)=\delta\sqrt{\log L}-14K$, we obtain
$\frac{14K+\tau(\delta)}{\sqrt{\log L}}=\delta$, implying
$\left|\frac{\hat{\lambda}}{|\lambda|}-1\right|\leq\delta$ on the event
$O_{\tau(\delta)}$. Finally, we recall that

$\mathbb{P}[\Omega_{\tau(\delta)}]\geq
1-4\exp\left(-\frac{\delta^{2}\log^{2}L}{2\pi}\cdot\left(1-\frac{14K}{\delta\sqrt{\log
L}}\right)^{2}\right).$

This completes the proof of the theorem. ∎

##  8 Conclusion and future work

In this paper, we have investigated signal analysis via an examination of the
stochastic geometric properties of spectrogram level sets. We have obtained
rigorous theorems demonstrating the efficacy of a spectrogram level sets based
approach to the detection and estimation of signals, and further proposed a
level sets based algorithm for signal analysis that is intrinsic to given
spectrogram data. We hope that the present work, to our knowledge the first to
establish rigorous statistical guarantees for spectrogram based methods for
signal analysis, will serve as a prototype and provide a roadmap for similar
investigations with regard to broader classes of signals. While the ambit of
our results is substantiated by the fact that a general signal can in
principle be written as a linear combination of Hermite functions, other bases
might allow for more parsimonious signal representations, and thus be of
practical interest as well. This includes the setting of complex-valued
(analytic) chirps, which is a natural avenue for future investigation.

## Acknowledgment

We are grateful to Rémi Bardenet for illuminating discussions. We thank the
anonymous referees for their careful reading of the paper, and for their
helpful comments and suggestions.

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