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Oberlin, Thomas and Meignen, Sylvain and Perrier, Valerie
Second-Order synchrosqueezing transform or invertible
reassignment? Towards ideal time-frequency representations.
(2015) IEEE Transactions on Signal Processing, 63 (5). 1335-1344.
ISSN 1053-587X
Official URL:
https://doi.org/10.1109/TSP.2015.2391077
Second-Order Synchrosqueezing Transform
or Invertible Reassignment? Towards Ideal
Time-Frequency Representations
Thomas Oberlin, Sylvain Meignen, and Valérie Perrier
Abstract—This paper considers the analysis of multicomponent
signals
, deÞned as superpositions of real or complex modu-
lated waves. It introduces two new post-transformations for the
short-time Fourier transform that achieve a compact time-fre-
quenc
y representation while allowing for the separation and the
reconstruction of the modes. These two new transformations thus
beneÞt from both the synchrosqueezing transform (which allows
for
reconstruction) and the reassignment method (which achieves
a compact time-frequency representation). Numerical experi-
ments on real and synthetic signals demonstrate the efÞciency of
t
hese new transformations, and illustrate their differences.
Index Terms—Time-frequency, reassignment, synchrosqueez
-ing, AM/FM, multicomponent signals.
I. INTRODUCTION
M
OST re
al signals can be accurately modeled as su-
perpositions of locally bandlimited, amplitude and
frequency-modulated (AM-FM) waves. Containing several
modes
they
are often called multicomponent signals (MCS).
Many approaches have been built to decompose these signals
into their constituent modes either in time-frequency (TF)
doma
in, like the synchrosqueezing transform [1], or in time
domain, as for instance the empirical mode decomposition
(EMD) [2], [3]. Yet, very few results on the accuracy of the
mode
retrieval are available. In this regard, a new approxima-
tion result on the retrieval of weakly modulated modes of an
MCS has been shown using the synchrosqueezing transform
(SS
T) [4]. This major result inspired new developments of syn-
chrosqueezing or reassignment methods, and more generally of
local TF analysis and synthesis techniques [5]–[7]. To obtain
suc
h a result, very strong assumptions were made on the modes
making up the signal among which the most limiting one is the
weak modulation hypothesis on the modes.
T. Oberlin is with IRIT/INP-ENSEEIHT, University of Toulouse, Toulouse
31000, France (e-mail: thomas.oberlin@enseeiht.fr).
S. Meignen and V. Perrier are with the Jean Kuntzmann Laboratory, Uni-
versity of Grenoble-Alpes, and CNRS, Grenoble 38041,
France (e-mail: syl-
vain.meignen@imag.fr; valerie.perrier@imag.fr).
Indeed, most MCS contain strongly frequency modulated
modes, as for instance chirps involved in radar [8], speech
processi
ng [9], or gravitational waves [10]. Although many
TF transforms such as quadratic distributions [11], [12] or the
reassignment method (RM) [13]–[16] manage to accurately
represe
nt these kinds of signals, they do not allow for mode
separation and reconstruction. Recently, some work has been
done on how to deal with high FM with SST. For instance,
in [16]
, a two-steps algorithm is used, which Þrst computes
SST, estimates the FM, and Þnally recomputes SST on the
demodulated mode. In the same spirit, an iterative procedure is
propos
ed in [17], where at each iteration a SST is computed at
a Þner resolution. A different and earlier attempt [18] consisted
in building an invertible extension of RM. Unfortunately, the
propo
sed reconstruction was only an approximation, valid for
Gaussian windows with a small temporal width. We will see
that contrary to [16], [17], our approach is one-step: it only
consi
sts in computing a reassignment map and then rearranging
the coefÞcients of the STFT according to that map following
this idea of the original synchrosqueezing or reassignment
meth
ods. Thus, it can be seen as an extension of [18], with ad-
ditional theoretical results for representation and reconstruction
of linear chirps.
The ai
m of this paper is indeed to extend SST to MCS con-
taining strongly frequency modulated modes by generalizing
SST in two different ways. The Þrst attempt is inspired by RM,
which
is known to optimally sharpen the TF representation of a
linear chirp. However, in the original RM the sharpened repre-
sentation obtained from the spectrogram did not allow for mode
retr
ieval. We will see that by fully exploiting the information
contained in the so-called shape reassignment vector, one can
derive a new generation of SST that provides a sharpened TF
repr
esentation as good as the one given by RM, while enabling
exact retrieval of linear chirps. Our second approach relies on
a second-order local estimate of the instantaneous frequency,
whi
ch is used to improve modes localization and reconstruction
using SST.
The outline of the paper is as follows: in Section II, we recall
impo
rtant notations that are subsequently used in the paper and
we introduce the concept of weakly modulated modes. We then
deÞne SST and RM in Section III, putting the emphasis on the
dif
ferences in terms of reconstruction and representation. Then,
Section IV is devoted to the presentation of the new extensions
of SST. Section V Þnally delivers numerical results on both syn-
the
tic and real data, comparing the proposed methods with stan-
dard SST and RM.
I
I. TIME-FREQUENCY ANALYSIS OF
MULTICOMPONENT SIGNALS
A. Short-Time Fourier Transform
DeÞnition II.1: The Fourier transform of a function
is deÞned by
For a given represents the part of that oscillates at fre-
quency
on the whole time domain. The need for time-localized
frequency descriptors leads to the following deÞnition:
DeÞnition II.2: Let
a real-valued and even func-
tion, with unit norm. The short-time Fourier transform (STFT)
of
, with respect to the window , is deÞned by
(1)
The representation of the positive quantity in the TF
plane is called the spectrogram of
. Without ambiguity, will
be denoted by .
Remark: The usual deÞnition of STFT differs from (1) by
a factor and considers the conjugate of . Also, as we
assume to be real-valued, we do not need the conjugate in
(1).
It is well known from the properties of the Fourier transform
that a well-localized TF representation requires a window
well
localized both in time and frequency. We shall then mention
that STFT is invertible on
using either of the following
formulae:
(2)
(3)
Note that (3) is valid only if is non-zero and continuous at 0. If
the signal is analytic (i.e., then the integral
domain for
is restricted to .
B. MCS and Ridges
DeÞnition II.3: An AM-FM mode is an oscillating function
, with and positive and slow-varying.
is called the instantaneous amplitude of at time
being its instantaneous frequency. Note that the slow-varying
condition is not quantiÞed being model-dependent as explained
later. Thanks to that condition, a Þrst order expansion of the
phase combined with a zeroth order expansion of the amplitude
lead to the approximation, for close to a Þxed time :
The STFT of mode is then approximated by [4], [19]:
(4)
This shows that the STFT of an AM-FM mode is non-zero on
a TF strip, centered on the ridge corresponding to its instan-
taneous frequency and deÞned by
. The width of this
strip depends on that of the support of
. Let us now deÞne MCS
which are superpositions of AM-FM modes.
DeÞnition II.4: A MCS reads
(5)
where the modes
satisfy .
In order to retrieve each mode from the STFT of
, we assume
that the modes occupy non-overlapping regions in the TF plane.
Considering (4) and assuming
is compactly supported, this
requires that the distance between two ridges is always larger
than the width of the frequency support of window
[4], [19].
This will be detailed in Section III.C. Note that this separation
condition is of a linear type for STFT and can be adapted to
non-compactly supported, but rapidly decreasing windows in
the Fourier domain (e.g., Gaussian windows).
III. S
YNCHROSQUEEZING AND REASSIGNMENT IN A NUTSHELL
The so-called reassignment method (RM) [13], [14] is a gen-
eral way to sharpen a TF representation towards its ideal TF
representation which corresponds for MCS given by deÞnition
II.4 to:
(6)
being equal to 1 or 2 depending on whether a linear or a
quadratic representation is used (e.g., for the spectrogram
and for STFT). Note that keeps the phase informa-
tion, which enables the reconstruction of the mode in the case of
linear representations
. Quadratic representations often
remove the phase, and use the amplitude of instead of .
A. Reassignment of the Spectrogram
RM presented here in the spectrogram context consists of the
computation of so-called reassignment operators:
(7)
where can be viewed as an approximation of the instan-
taneous frequency and
as the group delay. Then, the reas-
signment step aims to retrieve the ideal TF representation by
moving the coefÞcients of the spectrogram according to the map
, which reads:
B. STFT-Based SST
The aim of SST based on STFT (FSST), introduced in [1] in
the CWT context, is to retrieve the ideal TF representation, i.e.,
with , from its STFT. Following [1], the authors in
[
20], [19] proposed to neglect operator
, and to reassign only
the complex coefÞcients according to the map
, which reads:
(8)
Knowing
, the th mode can then be retrieved by considering:
(9)
as explained in more details in Section III.C. In a nutshell, FSST
reassigns the information in the TF plane and then
makes use of
this sharpened representation to recover the modes. The ratio-
nale behind (8) and (9) is the reconstruction formula (3), where
the integration is restricted to a mode-rel
ated domain instead of
. This enables to retrieve the modes which was not possible
with RM based on the spectrogram, but the TF representation is
unfortunately less sharp as soon as the freq
uency modulation is
not negligible. This difference between FSST and RM is further
illustrated in Section III.D.
C. Perfect Localization and Reconstruction
We now recall the main properties of FSST and RM in terms
of TF localization and mode reconstruction for FSST. We Þrst
highlight for what type of signals they actually lead to the ideal
TF representation and then consider a more general case.
1) Synchrosqueezing for Perturbations of Pure Waves: One
can show that
actually matches the instantaneous frequency
only for a pure tone
. In this case, FSST provides
the ideal TF representation given by (6). More interesting i
s
the study of the behavior of FSST on perturbations of pure
waves, i.e., containing “small” amplitude and frequency mod-
ulations. This analysis was pioneered in [4] for the SS
T based
on wavelets, and was adapted for FSST in [19]. It starts by
deÞning precisely a MCS, originally referred to as ‘intrinsic
mode type function” [4].
DeÞnition III.1: Let
and . A MCS of type
(5) is in if:
• for all satisÞes:
and for all
and .
• the s are separated with resolution i.e., for all
and for all ,
One can then deÞne the synchrosqueezing operator, using a
function
, the space of compactly supported smooth
functions, such that
, a threshold and an accuracy
parameter
:
and Þnally state the main result.
Theorem III.1: Let
and . Consider a
window , the space of smooth functions with fast
decaying derivatives, such that
. Then, if is
small enough,
•
only if there exists such
that .
• For all
and all pair such that
, one has
• For all there exists a constant such that
for all ,
(10)
The proof of this result is available in [19]. Equation (10)
shows that the synchrosqueezing operator has to be summed up
around the ridge, the width of the interval being
. As in prac-
tice
is not known, one chooses a parameter instead. There-
fore we end up with reconstruction formula (9) for mode .
Note that a similar approximation result was stated before in
[20], but with stronger (global) assumptions on the mode sepa-
ration , instead of the local one in DeÞnition III.1.
To summarize, the FSST of a MCS offers both good localiza-
tion and reconstruction if two restrictive conditions are satisÞed:
• the frequency separation
, which is
however necessary to uniquely deÞne the decomposition,
• and the low-modulation assumption
, that we
want to relax in the sequel.
2) Reassignment of Perturbations of Linear Chirps: It was
shown in [14] that the reassignment operators perfectly localize
linear chirps, i.e., modes with a linear instantaneous frequency
and constant amplitude. More precisely, one has the following
result [14].
Theorem III.2: Consider a pure linear chirp
, where is a polynomial of degree 2 and .
Then, whenever the reassignment operators are deÞned one has
.
Proof: In spite this result is available in [14], we recall its
main lines for the sake of introducing some notation that will be
useful in the sequel.
Let us consider a linear chirp
with
. For such signal, one has for each and :
(11)
The STFT then writes
with . We can now derive the expression of the reas-
signment operator :
If we deÞne the function as
w
e Þnally get
(12)
Note that the nature of function is unknown, although in the
case of a Gaussian window
, one can show that it is linear
[7], [21]. But one can remark that it is still a smooth function
provided
is smooth and has fast decay.
Regarding the local time delay
deÞned in RM, in the case
of a linear chirp
, we can write
(13)
which again uses auxiliary function
. Remembering that
, we get from (12) and (13) that:
(14)
which means
is an exact estimate of the instantaneous
frequency
, but at a shifted time location given by .
As the operators are local, this suggests that whenever the
instantaneous frequency is quasi-linear and the amplitude slow-
varying, the reassignment provides an accurate approximation
of the ideal TF representation.
D. Illustrations
To illustrate the differences between FSST and RM, we use
two synthetic test signals throughout this paper. One (signal
1) is made of low-order polynomial chirps, that behave locally
as linear chirps, and the other one (signal 2) contains strongly
nonlinear sinusoidal frequency modulations and singularities
(points where
). Fig. 1 displays FSST and RM for these
signals. Note that wherever the slope of the ridge is strong,
RM gives a more concentrated representation than FSST.
To illustrate the speciÞcity of FSST regarding invertibility,
we display on Fig. 2 the reconstruction process of each test
signals from FSST. To compute an estimation of the ridges
, knowing the number of modes, we use the algo-
rithm introduced in [22] and used in [23] or [7], which computes
a local minimum of the functional
(15)
where
is an estimator of the ridge , and and
are two positive parameters tuning the level of regularization.
Other ridge detectors such as the one developed in [24] may of
course be used instead but this is beyond the scope of the present
paper.
Looking at the results of Fig. 2, it is worth noting that the
quality of the reconstruction of a mode is highly dependent on its
modulation. For instance mode 2 and 3 of signal 1 are properly
Fig. 1. Illustration of FSST and RM for test signal 1 (left) and 2 (right). From
top to bottom: the spectrogram, FSST and RM.
retrieved but so is not mode 3 since it contains stronger modu-
lations. Our goal is thus, in what follows, to improve mode re-
construction based on FSST when the modes are strongly mod-
ulated.
IV. T
OWARDS A SECOND-ORDER FSST
Recalling that FSST was designed to process superpositions
of perturbed pure waves, by second order FSST we mean that
we want to adapt FSST to superpositions of perturbed linear
chirps. This should enable us to obtain an invertible sharpened
TF representation of the same quality as the one provided by
RM. We are going to consider an approximation of the second
derivative of the phase of the modes which we will subsequently
use to build our extensions of FSST.
A. Local Instantaneous Modulation
This approximation requires the computation of second-order
derivatives of the phase of the STFT, that have been already used
to improve the reassignment method [6]. This is denoted by
and deÞned as follows:
DeÞnition IV.1: Let be in . Its modulation
operator is deÞned wherever and
by:
(16)
Operator
can be seen as some variation measure of the
reassignment vector. Note that there are many other possible
choices for this local estimate of the frequency modulation.
In particular, we could also use
, which amounts
