1 Unsupervised classiﬁcation of the spectrogram zeros

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Unsupervised classiﬁcation of the spectrogram zeros

Juan M. Miramont, Franc¸ois Auger, Marcelo A. Colominas, Nils Laurent, and Sylvain Meignen

Abstract—The zeros of the spectrogram have proven to be a

relevant feature to describe the time-frequency structure of a

signal, originated by the destructive interference between com-

ponents in the time-frequency plane. In this work, a classiﬁcation

of these zeros in three types is introduced, based on the nature of

the components that interfere to produce them. Echoing noise-

assisted methods, a classiﬁcation algorithm is proposed based

on the addition of independent noise realizations to build a 2D

histogram describing the stability of zeros. Features extracted

from this histogram are later used to classify the zeros using a

non-supervised clusterization algorithm. A denoising approach

based on the classiﬁcation of the spectrogram zeros is also

introduced. Examples of the classiﬁcation of zeros are given for

synthetic and real signals, as well as a performance comparison

of the proposed denoising algorithm with another zero-based

approach.

Index Terms—Zeros of the spectrogram, time-frequency

analysis, non-stationary signals, noise-assisted methods.

I. INTRODUCTION

Time-frequency (TF) representations, such as the widely-

known spectrogram [1], are powerful tools to reveal the time-

varying frequency structure of a signal. A common interpre-

tation of the spectrogram is that of a TF distribution of the

energy, a conception that naturally leads to consider that more

information is located where the energy of the signal is more

concentrated in the TF plane [2]–[6].

More recently, a paradigm-shifting idea was proposed in

the work of Flandrin [7], where the zeros of the spectrogram

have a central role instead of its larger values. Indeed, it can

be shown that zeros or silent points of the spectrogram are

relevant for the characterization of the TF structure of a signal

[7]–[9]. In addition, Bardenet et al. [10], [11] showed that

the distribution of the zeros of the spectrogram of complex

circular white noise corresponds to that of the zeros of a planar

Gaussian analytic function, establishing promising connections

between signal processing and the study of spatial point

processes.

The zeros of the spectrogram are created by destructive

interference between components in the short-time Fourier

transform (STFT) [7], [12], hence they are intimately related

Juan M. Miramont and Franc¸ois Auger are with Nantes Universit´

e, Institut

de Recherche en ´

Energie ´

Electrique de Nantes Atlantique (IREENA, UR

4642), F-44600 Saint-Nazaire, France (email: juan.miramont@univ-nantes.fr

and francois.auger@univ-nantes.fr).

Marcelo A. Colominas is with the Institute of Research and Development

in Bioengineering and Bioinformatics (IBB, UNER - CONICET), and with

the Faculty of Engineering (UNER) Oro Verde, Entre Rios, Argentina (email:

macolominas@conicet.gov.ar).

Nils Laurent and Sylvain Meignen are with Jean Kuntzmann Lab-

oratory, University of Grenoble-Alpes and CNRS UMR 5224, F-

38401 Grenoble, France (email: nils.laurent1@univ-grenoble-alpes.fr and

sylvain.meignen@univ-grenoble-alpes.fr).

This work was supported by the ANR ASCETE project with grant number

ANR-19-CE48-0001-01.

to the distribution of the signal energy in the plane. This will

be a core idea throughout this work. We shall see that, when

two signal components interfere, the position of the resulting

zeros of the spectrogram seems to be robust to the addition of

noise, for some low relative noise amplitudes with respect to

the signal amplitude.

This leads to the main three contributions of this paper. The

ﬁrst one is a criterion to classify the zeros of the spectrogram

depending on the nature of the components that interfere to

create them. We will show that it is possible to discriminate

the zeros of the spectrogram in three categories: 1) the

zeros created by the interference between deterministic signal

components, 2) the zeros only related to noise components and

3) the zeros created through the interaction between signal and

noise components.

The second main contribution of this article is the intro-

duction of 2D histograms that describe the spectrogram zeros

variability when several independent noise realizations are

added to the signal. We shall show that this representation

can be used to correctly discriminate the zeros in the three

categories described before by computing ﬁrst a number of

features from the 2D histogram and using a clusterization

technique to classify them in a non-supervised, automatic way.

Finally, a third contribution is a denoising method based on

the classiﬁcation of the spectrogram zeros. We will describe

a simple multicomponent signal for which other zero-based

strategies face limitations, and illustrate how the here proposed

approach can be more efﬁcient for disentangling noise and

signal in this case.

The rest of the paper is organized as follows. In Sec. II

we deﬁne the STFT, the spectrogram and give other relevant

elements to set the context of the article. In Sec. III we detail a

criterion to classify the zeros in three types, whereas in Sec. IV

we introduce the 2D histograms of the position of zeros. Later,

in Sec. V we show how to automatically classify the zeros of

the spectrogram using an unsupervised approach. In Sec. VI

we illustrate the performance of the classiﬁcation algorithm

with simulated and real signals, and we give an example of a

denoising strategy derived from the previous ﬁndings. Finally,

we comment on the obtained results in Sec. VII and draw

some conclusions in Sec. VIII.

II. CONTEXT

Given a signal x∈L1(R)∩L2(R), we deﬁne its Fourier

transform (FT) as

ˆx(f):=Z+∞

−∞

x(t)e−i2πf tdt, (1)

where t, f ∈Rare the time and frequency variables. The

STFT, which can be considered as a time-localized version of

arXiv:2210.05459v1 [eess.SP] 11 Oct 2022

the Fourier transform, is a fundamental tool for time-frequency

signal analysis. We deﬁne the STFT of x(t)as

x(t, f):=Z+∞

−∞

x(u)g(u−t)∗e−i2πf udu, (2)

where g∈L1(R)∩L2(R)a and g(t)∗is complex conjugate

of g(t). Henceforth, we consider the analysis window g(t)as

a unitary Gaussian window given by

g(t) = 21/4

√Te−πt2

T2,(3)

where Tdetermines its width. We shall consider T= 1 s,

unless stated otherwise, so that the window has the same

essential support in time and frequency.

The spectrogram of x(t), deﬁned as the squared modulus

of the STFT, will be given then by

x(t, f):=|Vg

x(t, f)|2.(4)

We deﬁne the discrete-time counterpart of Eq. (2) as

x[n, q]:=

n+L

m=n−L

x[m]g[m−n]e−i2πqm

N,(5)

where n, q ∈Zare the discrete time and frequency variables,

Lis the half-width of the discretized analysis window g[n]

and N∈Nis the number of frequency bins of the discrete

frequency axis.

In the following, we shall consider signal and noise mix-

tures. What one means by signal and noise depends on the

application, but usually it can be considered that the signal

is a quantity of interest that exhibits some organization and,

in contrast, noise is usually regarded as a random ﬂuctuation

including all the other inﬂuences on the observed phenomena

that are not of interest [12].

We will consider ξ(t)a real white Gaussian noise realiza-

tion, where ξ(t)is distributed as N(0, γ2

0)for all values of t,

satisfying:

E{ξ(t)ξ(t−τ)}=γ2

0δ(τ),(6)

where γ2

0is the noise variance and δ(t)is the Dirac distri-

bution. We shall express the Signal-to-Noise Ratio between a

deterministic signal x(t)and ξ(t)as

SNR(x, ξ) = 10 log10 kxk2

γ2

0!(dB),(7)

where kxk2is the usual 2-norm of xand its square is the

energy of the signal.

III. THREE TYPES OF ZEROS OF THE SPECTROGRAM

The zeros of the spectrogram are a consequence of the

destructive interference between components in the time-

frequency plane [7], [12]. Indeed, the spectrogram of the sum

of two signals x1(t)and x2(t)can be written as [13], [14]

x1+x2(t, f) = Sg

x1(t, f) + Sg

x2(t, f)

+ 2<{Vg

x2(t, f)Vg

x1(t, f)∗}.(8)

(a) (b)

Fig. 1. (a) Logarithm of the spectrogram of real white Gaussian noise.

(b) Spectrogram of the mixture of the same noise realization with a signal.

The dashed line indicates the curve Γgiven by Eq. (10). In both cases, the

superimposed white dots mark the positions of the spectrogram zeros.

Expressing Vg

x1(t, f) = Mx1(t, f)eiΦx1(t,f)and Vg

x2(t, f) =

Mx2(t, f)eiΦx2(t,f), one obtains:

x1+x2(t, f) = Mx1(t, f)2+Mx2(t, f)2

+ 2Mx1(t, f)Mx2(t, f) cos (Φx2(t, f)−Φx1(t, f)) ,(9)

from which the following proposition is derived:

Proposition III.1: Destructive Interference. For Sg

x1+x2(t, f)

to be equal to zero at a point (t, f), the following necessary

and sufﬁcient conditions must be satisﬁed:

1) The phases Φx1(t, f)and Φx2(t, f)must differ from an

integer odd multiple of π.

2) The modulus of Vg

x2(t, f)and Vg

x1(t, f)must be equal.

A proof of this simple proposition is given in the Appendix.

In the noiseless case, those conditions are fulﬁlled for some

(t, f)provided that the signal is multicomponent. One can

then consider x1(t)and x2(t)as two signal components that

are linearly combined to produce the signal. Thus, the zeros

of the spectrogram would be generated by the interference

between x1(t)and x2(t), and therefore they will be completely

deterministic. We will denote them as zeros of the ﬁrst kind.

In the noise only case, shown in Fig. 1a, the zeros are

uniformly distributed in the TF plane, generated by the in-

terference of multiple randomly located logons (albeit not

at arbitrary positions because of the restrictions posed by

the reproducing kernel of the STFT [7], [12]). Hence, x1(t)

and x2(t)can be thought of as two adjacent logons, the

interference of which would produce zeros in the spectrogram.

This case has been by far the most studied one, and the

characterization of the zeros of noise components and the

lattice-like structure they generate in the TF plane is the basis

of some recently developed denoising methods [7], [10], [15].

In the sequel, we will term the zeros created by the interference

between noise logons, zeros of the second kind.

Regarding the mixture of signal and noise, the situation is

more complex. Comparing Fig. 1a and 1b, it is possible to see

that the presence of the signal only affects the spectrogram

zeros locally [7], [10], since zeros far away from the signal

energy are not modiﬁed. Considering this, one can conclude

that in the regions of the TF plane where the noise dominates,

the zeros of the spectrogram will correspond to zeros of the

second kind. In contrast, zeros of the ﬁrst kind will be found

in the regions where the signal components are more energetic

than the noise, as described before, provided that the signal

components are close enough.

It remains to analyze the zeros created by the interfer-

ence between the deterministic signal components and the

noise where neither of these inﬂuences can be neglected.

As expected, these zeros will be located close to the border

of the signal domain. Considering now x1(t)as a signal

component and x2(t)as noise, it is not possible to give an

exact description of the position of the zeros created by the

interference between them because of the random nature of

the noise. But despite this, one can approximate where the

zeros will be located by replacing the spectrogram of noise

in Condition 2 of Prop. III.1 by its expected value (constant

and proportional to the noise variance in the case of white

Gaussian noise and a unitary energy window) [1], [12]. Then,

Condition 2 can be written as

x1(t, f) = γ2

0.(10)

Equation (10) deﬁnes a level curve Γgiven by

Γ = (t, f)|Sg

x1(t, f) = γ2

0,(11)

that well approximates the location of the zeros that surround

the signal domain, as illustrated by Fig. 1b, where the dashed

line indicates the level curve Γ. This poses a restriction on

the zeros created by the interference of signal and noise, in

the sense that they are created in the neighborhood of Γ,

making these zeros of a very different nature from the ﬁrst

two categories introduced before. Hence we will consider these

zeros as belonging to a third kind: those that are created by

the interference between signal and noise components.

For lower (resp. higher) values of SNR(x, ξ), the domain

determined by the interior of Γprogressively shrinks (resp.

expands), and constitutes a good approximation to the signal

domain [15]. Notice that, however, one usually does not have

access to Sg

x1(t, f)and an estimation of γ0is commonly used

for denoising by thresholding Sx+ξ(t, f)[16]–[18].

IV. A 2D HISTOGRAM OF THE POSITION OF ZEROS

A. A noise-assisted Study of Zeros

Consider a signal composed of two parallel linear chirps and

contaminated with real white Gaussian noise, the spectrogram

of which is shown in Fig. 2a.

Hypothetically, if one changed the noise realization of the

signal shown in Fig. 2a for another one with the same variance,

and then proceeded to compare the position of the zeros of the

spectrogram of the original noisy signal with the zeros of the

modiﬁed one, it would appear as if the zeros have moved.

This effect can be seen in Fig. 2b, where the circles and dots

indicate, respectively, the position of the spectrogram zeros

with the original noise realization and a new one.

One can notice that the zeros between the chirps, i.e. zeros

of the ﬁrst kind, are not affected by this change. This is

somehow expected, because of the predominantly determin-

istic nature of these zeros. However, this is not true for the

(a)

(b)

Fig. 2. (a) Logarithm of the spectrogram of a signal with two parallel

linear chirps contaminated by real white Gaussian noise. (b) Position of the

spectrogram zeros. Circles and dots indicate the position of the spectrogram

zeros corresponding to two different noise realizations. Dashed lines denote

the instantaneous frequency associated with each chirp.

zeros of the second and third kind. As one can observe in

Fig. 2b, the new zeros are created at different locations in the

remaining regions of the TF plane when the noise realization

is modiﬁed.

Because the new noise realization has the same variance as

the original one, the signal domain determined by Γin Eq. (10)

is not affected. This may explain why Fig. 2b also shows that

the new zeros created near the border of the signal domain

seem to be located closer to the original zeros. This effect

is not observed in the noise-dominated regions of the plane,

where the locations of the new zeros look less correlated to

that of the original ones. These observations lead to a principle

that could be used to differentiate between the different types

of zeros by studying how far the new zeros are created from

the original ones after modifying the noise.

Nevertheless, changing only the noise component of a signal

for an equivalent one is not feasible in practice. Drawing

inspiration from the noise-assisted methods [19], [20], one can

rather add a new realization of noise that will then modify

the position of all the zeros of the spectrogram. To see why

this occurs, let us ﬁrst call y(t) = x(t) + ξ(t)the original

signal mixture, where ξ(t)is the noise already present in the

signal, and yj(t) = x(t) + ξ(t) + ηj(t)anew mixture, where

ηj(t)j∈Nis a noise realization (here real white Gaussian

noise) with zero mean and variance γ2

Then, the STFT of yj(t)can be written as the following

series (see [21], chapter 3, and [10] for more details):

yj(t, f) = eiπfte−π|z|2/2∞

k=0 hyj, hkiπk/2zk

√k!(12)

where z=t+if,{hk}∞

k=0 is the set of Hermite functions and

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1UnsupervisedclassicationofthespectrogramzerosJuanM.Miramont,Franc¸oisAuger,MarceloA.Colominas,NilsLaurent,andSylvainMeignenAbstractThezerosofthespectrogramhaveproventobearelevantfeaturetodescribethetime-frequencystructureofasignal,originatedbythedestructiveinterferencebetweencom-ponentsinthetime-...

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