# IF equation: a feature extractor for high-concentration time-frequency
representation

Xiangxiang Zhu<EMAIL_ADDRESS>Kunde Yang  Zhuosheng Zhang  School
of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an
710072, China  School of Marine Science and Technology, Northwestern
Polytechnical University, Xi’an 710072, China  School of Mathematics and
Statistics, Xi’an Jiaotong University, Xi’an 710049, China

###### Abstract

High-concentration time-frequency (TF) representation provides a valuable tool
for characterizing multi-component non-stationary signals. In our previous
work, we proposed using an instantaneous frequency (IF) equation to sharpen
the TF distribution and its effectiveness has been verified in experiments. In
this paper, we systematically discuss why the IF equation-based TF analysis
methods work and how to use the IF equation to improve the TF sharpness. By
theory analysis, many popular TF post-processing methods, such as the
synchroextracting transform, the multi-synchrosqueezing transform, and the
time extracting transform, fall into the IF equation-based category. A
comparison of the IF equation-based method with the popular synchrosqueezing
transform is made. Numerical examples of the theoretical derivations are
presented to illustrate the performance of the proposed IF equation-based TF
analysis method.

###### keywords:

Multi-component non-stationary signal , High-resolution time-frequency
analysis , Synchrosqueezing transform , IF equation

††journal: Time-frequency and its application

##  1 Introduction

The study and the interpretation of multi-component non-stationary signals
require a powerful analysis tool. Time-frequency (TF) analysis methods are
used to measure how a signal’s frequency components vary with time and provide
an effective tool in non-stationary signals analysis [1-7]. The most
frequently used TF representation is the linear TF analysis method, mainly
including the short-time Fourier transform (STFT) [8], the continuous wavelet
transform (CWT) [9], the Stockwell transform [10], and the chirplet transform
(CT) [11]. This kind of method characterizes the signal via inner products
with a pre-assigned basis and has invertibility and low computation cost.
However, restricted by the Heisenberg uncertainty principle [12], the linear
TF analysis can not simultaneously obtain optimal time and frequency
resolution, easily leading to a blurred TF representation.

To solve the above limitation, the post-processing methods of the linear TF
representations have been widely studied in the past decades. The reassignment
method (RM), proposed by Kodera et al. to the spectrogram [13] and generalized
by F. Auger and P. Flandrin to the time-scale distribution and others [14],
reassigns the TF energy from the original position to the gravity center of
the signal’s energy distribution such that a sharped TF representation is
yielded. F. Auger et al. also extended the standard RM to the Levenberg-
Marquardt reassignment [15], making the concentration of the signal energy
adjustable. J. Xiao and P. Flandrin [16] presented the multitaper TF
reassignment with the two-fold objective of a sharp localization for the chirp
components and a reduced level of statistical fluctuations for the noise. In
[17], Bruni et al. proposed a modified RM for analyzing multi-component
crossing signals with cross-over instantaneous frequencies (IFs). This method
corrects the TF result of the standard RM in the non-separable region partly.
Although RM and its variants improve the energy concentration of the linear TF
representation, they lack an explicit formula for signal reconstruction, which
limits its application [18,19].

Another popular TF post-processing method is the synchrosqueezing transform
(SST), initialized in [20] and further analyzed in [21]. This method was
developed in understanding the principle of the empirical mode decomposition
[22]. By squeezing the TF coefficients into the instantaneous frequency (IF)
trajectory in the frequency direction, SST not only enhances the TF
distribution but also allows for mode retrieval. Because of advantages to mode
representation and reconstruction, many new developments of SST have been
carried out in various directions. First, an extension to the STFT case was
proposed in [23], while a generalization of the wavelet approach by means of
wavelet packets decomposition for both one-dimensional and two-dimensional
cases was available in [24,25]. Extensions to other transform frameworks
included, but not limited to, the synchrosqueezing S-transform [26], the
synchrosqueezing three-parameter wavelet transform [27], and the fractional
synchrosqueezing transformation [28]. Second, the robust analysis of SST has
been studied in [29-31] and a new IF estimator within the framework of the
signal’s phase derivative and the linear canonical transform was introduced in
[32]. Moreover, a study of SST applied to multivariate data was done in [33].
An adaptive SST with a time-varying parameter was recently introduced in
[34,35], which obtains the well-separated condition for multi-component
signals using the linear frequency modulation to approximate a non-stationary
signal at every time instant. The theoretical analysis of adaptive SST was
studied in [36,37]. Finally, combining the SST with the ideal TF
representation theory, the synchroextracting transform (SET) [38] was proposed
to improve the TF concentration by extracting the ridges of the STFT.

SST works well for sinusoidal signals but suffers from low TF resolution when
dealing with strong modulation signals that consist of a fast-varying feature.
To address this drawback, many improvements have been presented, mainly
including the demodulated SST [39,40], the high-order SST [41,42], the multi-
synchrosqueezing transform [43,44], and the time-reassigned SST [45,46]. The
demodulated SSTs [39,40] aim to change broad-band signals to narrow-band
signals and the standard SST is followed then. The high-order SSTs [41,42] aim
to calculate an IF estimation in the TF or time-scale domain for more general
signal types. The multi-synchrosqueezing transformations [43,44] apply an
iterative reassignment technique to sharpen the blurred TF energy step-by-
step. The time-reassigned SSTs [45,46], in fact, introduce a dual operator by
considering time reassignment instead of frequency reassignment and such
methods have a good performance in addressing pulse-like signals. A
comprehensive review of the subject of reassigned TF representations is
provided in Ref. [5,19].

The mechanism of the SST and its variants introduced above is to derive an IF
or group delay (GD) estimator for different transform frameworks and
different-type signals, and further to reassign TF coefficients from the
original position to the estimated instantaneous trajectory. Indeed, the IF
(or GD) estimator is critical because it determines the accuracy of high-
concentration TF representations. Differing from the mechanism of the SST-
based methods, another high-concentration TF representation is based on the IF
equation [47-49], i.e., a non-linear function in the TF domain with the IFs as
its solution. This kind of method enhances the TF concentration by solving the
IF equation using the fixed point iterative algorithm [47] or the extractor
[48,49]. Such methods have been used in the sound signal analysis [47,48],
machine signal processing [48,50], and medical data analysis [51]. However,
several issues should be considered. First, its strengths and weaknesses over
the SST-based methods are not clear. The common goal of both methods is to
improve the TF readability, but we have to know which one is preferable in
various applications. Second, how to use the IF equation to concentrate the TF
distribution should be explained more comprehensively, involving the
uniqueness of the IF equation, and the convergence of its solution algorithm.
Last but not least, we should consider if there is a theoretical model to
cover the prevailing TF analysis methods, including the SET [38], the multi-
synchrosqueezing transformation [43,44], and the time synchroextracting
transform [52]. The relationship between these methods and the IF equation-
based methods should also be studied.

In this paper, we focus on the IF equation-based TF post-processing methods
and attempt to build a theoretical framework for them. We first review the
SST-based TF analysis methods and then define IF equation followed by two
concrete expressions under the STFT framework. We study the ways, that is, by
the extractor to detect signal information and by iterative algorithms for TF
reassignment, to concentrate the TF localization using the IF equation. The
uniqueness of the IF equation and the convergence of its solution algorithm
are also studied. The difference between the SST-based methods and the IF
equation-based methods is discussed. By comparison, we find that one of the
merits of the IF equation-based methods over the former is that IF equation
contains not only the IF information but also the GD information, which
unifies the IF and GD estimator of the SST-based methods. Finally, numerical
examples of the theoretical derivations are presented.

The remainder of the paper is organized as follows. In Section 2, we review
the SST-based TF analysis methods. In Section 3, we provide the unified
mathematical model for IF equation-based TF post-processing methods, and some
properties of IF equation are given. Comparative studies with the SST-based
methods are presented in Section 4. Finally, the conclusions are drawn in
Section 5.

##  2 SST-based TF post-processing methods

The following section presents the multi-component non-stationary signal model
and reviews the SST-based TF analysis methods.

###  2.1 Signal model

Multi-component non-stationary signals are comprised of several mono-
components and form the general case. In this paper we use the popular
amplitude-modulation and frequency-modulation (AM-FM) model to describe the
time-varying features of multi-component non-stationary signals, which is
defined as

$f(t)=\sum_{k=1}^{K}f_{k}(t)=\sum_{k=1}^{K}A_{k}(t)\mathrm{e}^{j\phi_{k}(t)},$
(1)

where $K$ is a positive integer representing the number of AM-FM components,
$\sqrt{-1}=j$ denotes the imaginary unit, $A_{k}(t)>0$ and $\phi_{k}(t)$ are
the instantaneous amplitude and instantaneous phase of the $k$-th component
(or mode), respectively. The first and the second derivatives of the phase,
i.e., $\phi^{\prime}_{k}(t)$, $\phi^{\prime\prime}_{k}(t)$, are referred to as
the instantaneous frequency (IF) and chirp rate (CR) of the $k$-th component.

###  2.2 SST-based TF post-processing method

We give the introduction of SST class under the STFT framework. Given signal
$f(t)\in L^{2}(\mathbb{R})$, its Fourier transform is defined as

$\hat{f}(\omega)=\int_{-\infty}^{+\infty}f(t)\mathrm{e}^{-j\omega
t}\mathrm{d}t.$ (2)

The STFT of $f(t)$, which is obtained through the use of a sliding window
$g(t)\in L^{2}(\mathbb{R})$, is given by

$\begin{split}S_{f}^{g}(t,\omega)=\int_{-\infty}^{+\infty}f(\mu)g^{*}(\mu-t)\mathrm{e}^{-j\omega(\mu-t)}\mathrm{d}\mu,\end{split}$
(3)

where the superscript $*$ denotes the complex conjugate. It is well known that
the STFT can not simultaneously localize a signal in both time and frequency
precisely due to the limitation of the Heisenberg uncertainty principle [12],
i.e., long windows provide good frequency resolution but poor temporal
resolution, whereas short windows increase temporal resolution at the expense
of frequency resolution.

To address this drawback, the SST-based TF post-processing methods
[21,23,24,26,41,42] are developed. Here the ”post-processing” means that the
SST methods make a further operation on the original TF representation, such
as STFT and CWT, to relocate the TF coefficients. With this regard, a key
problem is to know where are the blurred TF points rearranged. The ideal one
is to relocate the TF points to the true IF curves of the signal analyzed,
which leads to the development of IF estimations based on STFT.

Thanks to G. Thakur and H.T. Wu, they develop I. Daubechies’s work [21] and
give a two-dimensional IF estimator in the TF domain as follows [23]:

$\hat{\omega}_{1}(t,\omega)=\Re\left\\{\frac{\frac{\partial}{\partial
t}S^{g}_{f}(t,\omega)}{jS^{g}_{f}(t,\omega)}\right\\},$ (4)

here $|S^{g}_{f}(t,\omega)|>0$, and $\Re\\{\cdot\\}$ is the real part of
complex number. With $\hat{\omega}_{1}(t,\omega)$ defined by (4), the STFT-
based SST (FSST) [23] is to reassign the TF coefficients from the original
location to the estimated IF trajectory along the frequency direction:

$T_{f}^{1}(t,\omega)=\int_{|S^{g}_{f}(t,\eta)|>\lambda}S^{g}_{f}(t,\eta)\delta(\omega-\hat{\omega}_{1}(t,\eta))\mathrm{d}\eta,$
(5)

where $\lambda>0$, and $\delta(\cdot)$ is the Dirac delta function, a
generalized function formulated as $\delta(t)=0$ when $t\neq 0$, and
$\int_{-\infty}^{+\infty}\delta(t)\mathrm{d}t=1$.

The IF estimator (4) is accurate for the cosine-like signals with slow-varying
instantaneous information, i.e., the multi-component non-stationary signal
$f(t)$ satisfies $|A^{\prime}_{k}(t)|<\epsilon$,
$|\phi^{\prime\prime}_{k}(t)|<\epsilon$ for $k=1,2,\cdots,K$ (here $\epsilon$
is a small positive constant). For the fast-varying signals, such as chirp-
like signals, FSST suffers from a low TF resolution at the modulated part. To
address this shortcoming, paper [41] introduced another IF estimation as
follows:

$\begin{split}\hat{\omega}_{2}(t,\omega)=\begin{cases}\Re\left\\{\frac{\frac{\partial}{\partial
t}S^{g}_{f}(t,\omega)}{jS^{g}_{f}(t,\omega)}\right\\}+\Re\left\\{\tilde{q}(t,\omega)\frac{\frac{\partial}{\partial\omega}S^{g}_{f}(t,\omega)}{S^{g}_{f}(t,\omega)}\right\\},&~{}\text{if}~{}\frac{\partial}{\partial
t}\left(\frac{\frac{\partial}{\partial\omega}S^{g}_{f}(t,\omega)}{S^{g}_{f}(t,\omega)}\right)\neq
j,\text{and}~{}S^{g}_{f}(t,\omega)\neq 0,\\\
\hat{\omega}_{1}(t,\omega),&~{}\text{if}~{}\frac{\partial}{\partial
t}\left(\frac{\frac{\partial}{\partial\omega}S^{g}_{f}(t,\omega)}{S^{g}_{f}(t,\omega)}\right)=j,\text{and}~{}S^{g}_{f}(t,\omega)\neq
0,\end{cases}\end{split}$ (6)

where

$\tilde{q}(t,\omega)=\frac{\frac{\partial}{\partial
t}\left(\frac{\frac{\partial}{\partial
t}S^{g}_{f}(t,\omega)}{S^{g}_{f}(t,\omega)}\right)}{j-\frac{\partial}{\partial
t}\left(\frac{\frac{\partial}{\partial\omega}S^{g}_{f}(t,\omega)}{S^{g}_{f}(t,\omega)}\right)}.$
(7)

The quantity $\hat{\omega}_{2}(t,\omega)$ corresponds to the true IFs of the
Gaussian modulated linear chirp [41]. The second-order FSST then consists in
replacing $\hat{\omega}_{1}(t,\omega)$ by $\hat{\omega}_{2}(t,\omega)$ in the
standard FSST:

$T_{f}^{2}(t,\omega)=\int_{|S^{g}_{f}(t,\eta)|>\lambda}S^{g}_{f}(t,\eta)\delta(\omega-\hat{\omega}_{2}(t,\eta))\mathrm{d}\eta.$
(8)

Form (5) and (8), the SST-based TF post-processing methods can thus be unified
as

$T_{f}(t,\omega)=\int_{|S^{g}_{f}(t,\eta)|>\lambda}S^{g}_{f}(t,\eta)\delta(\omega-\hat{\omega}(t,\eta))\mathrm{d}\eta,$
(9)

where $\hat{\omega}(t,\eta)$ is an IF estimator calculated in a transform
domain. Similarly, based on the GD estimator $\hat{t}(t,\omega)$ of various TF
transformations, the time-reassignment SST [45,46] can be formulated as

$\tilde{T}_{f}(t,\omega)=\int_{|S^{g}_{f}(\mu,\omega)|>\lambda}S^{g}_{f}(\mu,\omega)\delta(t-\hat{t}(\mu,\omega))\mathrm{d}\mu.$
(10)

Differing from the squeezing manner of the SST that is based on an IF
estimation, in the following we will introduce the IF equation-based TF post-
processing methods and build an unified mathematical model for them.

##  3 IF equation-based TF post-processing method

The central idea of the IF equation-based TF post-processing methods is to
find an IF equation in the TF domain to characterize the instantaneous
features of the signal. By solving the IF equation, we want to output a
concentrated and accurate TF representation.

###  3.1 Definition of IF equation

######  Definition 1.

For signal $f(t)=\sum_{k=1}^{K}A_{k}(t)\mathrm{e}^{j\phi_{k}(t)}$, a binary
equation $h_{f}(t,\omega)=0$ is called the IF equation of $f(t)$, if
$h_{f}(t,\omega)$ is not identically zero and the set of points
$\omega=\phi^{\prime}_{k}(t)$ ($k=1,2,\cdots,K$) satisfies

$\displaystyle h_{f}(t,\phi^{\prime}_{k}(t))=0,~{}k=1,2,\cdots,K.$ (11)

From Definition 1, we can know that the IF curves
$\omega=\phi^{\prime}_{k}(t)$ of each model of $f(t)$ are the solutions of its
IF equation. Since the IF equation includes the IF information of the studied
signal, it can be considered as a feature extractor.

In order to show the IF equation more clearly, we now consider a specific
example of constant amplitude and linear frequency modulation signal (also
known as chirp), i.e., $f(t)=Ae^{j\phi(t)}$, where $A$ is a positive constant,
$\phi(t)=a+bt+\frac{c}{2}t^{2}$, $a,b,c$ are real numbers. In this case, the
STFT under the Gaussian window
$g(t)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{t^{2}}{2\sigma^{2}}}$ is calculated
as

$S_{f}^{g}(t,\omega)=f(t)\frac{1}{\sqrt{1-j\sigma^{2}c}}\exp\left(-\frac{\sigma^{2}}{2}\frac{(\omega-\phi^{\prime}(t))^{2}}{1-j\sigma^{2}c}\right),$
(12)

where $\phi^{\prime}(t)=b+ct$. From (12), we can obtain that

$\begin{split}\frac{\partial}{\partial\omega}S_{f}^{g}(t,\omega)=S_{f}^{g}(t,\omega)\left(-\frac{\sigma^{2}(\omega-\phi^{\prime}(t))}{1-j\sigma^{2}c}\right).\end{split}$
(13)

The set of points $\omega=\phi^{\prime}(t)$ satisfies
$\left|\frac{\frac{\partial}{\partial\omega}S_{f}^{g}(t,\omega)}{S_{f}^{g}(t,\omega)}\right|=0\text{~{}~{}for}~{}|S_{f}^{g}(t,\omega)|>\lambda$.
Therefore, we can define an IF equation of signal $f(t)=Ae^{j\phi(t)}$ as

$\begin{split}\left|\frac{\frac{\partial}{\partial\omega}S_{f}^{g}(t,\omega)}{S_{f}^{g}(t,\omega)}\right|=0,\text{~{}~{}for}~{}|S_{f}^{g}(t,\omega)|>\lambda.\end{split}$
(14)

This equation can be more efficiently computed by

$\begin{split}\left|\frac{S_{f}^{tg}(t,\omega)}{S_{f}^{g}(t,\omega)}\right|=0,\text{~{}~{}for}~{}|S_{f}^{g}(t,\omega)|>\lambda,\end{split}$
(15)

where $S_{f}^{tg}(t,\omega)$ denotes the STFT of $f(t)$ using $tg(t)$ as its
window.

It is easy to see that another IF equation of the chirp signal $f(t)$ is
formulated as

$\Re\left\\{\frac{\frac{\partial}{\partial\omega}S_{f}^{g}(t,\omega)}{S_{f}^{g}(t,\omega)}\right\\}=0,\text{~{}~{}for}~{}|S_{f}^{g}(t,\omega)|>\lambda.$
(16)

Therefore, the IF equation is not unique for signals of the same type. Finding
a powerful IF equation that covers as many signal types as possible and has
some good properties for its solutions is an important work that needs
exploration.

It is also noted that other window functions can be used to define the IF
equation. If an even one is employed, the IF equation of chirp signal can also
be defined as (14) (or (16)). In the following section, we discuss some
properties of IF equation (14).

###  3.2 Properties of IF equation (14)

1) Uniqueness of the solution: For each time $t$, $\omega=\phi^{\prime}(t)$ is
the unique solution of IF equation (14). This property denotes that the IF
equation (14) of chirps can characterize its IF information and excludes some
unwanted information, which is important for detecting the signal feature.

2) Impulse signals detection: The instantaneous information $t=t_{0}$ of
impulse signal $f(t)=A\delta(t-t_{0})$ is the unique solution of IF equation
(14).

The STFT of the transient signal $f(t)=A\delta(t-t_{0})$ with the Gaussian
window is computed as

$\displaystyle
S_{f}^{g}(t,\omega)=A{(\sqrt{2\pi}\sigma)}^{-1}\mathrm{e}^{-\frac{(t_{0}-t)^{2}}{2\sigma^{2}}}\mathrm{e}^{-j\omega(t_{0}-t)}.$
(17)

From (17), we can obtain that

$\begin{split}\left|\frac{\frac{\partial}{\partial\omega}S_{f}^{g}(t,\omega)}{S_{f}^{g}(t,\omega)}\right|=|t-t_{0}|.\end{split}$
(18)

Therefore, $t=t_{0}$ is the unique solution of IF equation (14).

Whereas IF equation (16) does not include the transient feature for the
impulse signal. Therefore, equation (14) is more preferred than (16) when
dealing with impulse signals.

3) Linear group delay detection: The GD of linear group delay signal
$\hat{f}(\omega)=Ae^{-j(a+b\omega+\frac{c}{2}\omega^{2})}$ is the unique
solution of IF equation (14) .

Since the STFT of $f(t)$ can be rewritten as

$\begin{split}S_{f}^{g}(t,\omega)&=\int_{-\infty}^{+\infty}f(\mu)g^{*}(\mu-t)\mathrm{e}^{-j\omega(\mu-t)}\mathrm{d}\mu=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\hat{f}(\nu)\hat{g}^{*}(\nu-\omega)\mathrm{e}^{jt\nu}\mathrm{d}\nu,\end{split}$
(19)

then the STFT of linear group delay model can be expressed as

$\begin{split}S_{f}^{g}(t,\omega)=\frac{1}{2\pi}\hat{f}(\omega)\mathrm{e}^{jt\omega}\int_{-\infty}^{+\infty}\mathrm{e}^{-j\frac{c}{2}\nu^{2}}\hat{g}^{*}(\nu)\mathrm{e}^{j(t-b-c\omega)\nu}\mathrm{d}\nu.\end{split}$
(20)

From (20), we have

$\begin{split}\frac{\partial}{\partial\omega}S_{f}^{g}(t,\omega)=j(t-b-c\omega)S_{f}^{g}(t,\omega)-\frac{jc}{2\pi}\hat{f}(\omega)\mathrm{e}^{jt\omega}\int_{-\infty}^{+\infty}\mathrm{e}^{-j\frac{c}{2}\nu^{2}}\nu\hat{g}^{*}(\nu)\mathrm{e}^{j(t-b-c\omega)\nu}\mathrm{d}\nu.\end{split}$
(21)

Due to the Fourier transform of Gaussian window
$g(t)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{t^{2}}{2\sigma^{2}}}$ is calculated
as

$\hat{g}(\nu)=\mathrm{e}^{-\frac{\sigma^{2}}{2}\nu^{2}},$ (22)

which yields that

$\int_{-\infty}^{+\infty}\mathrm{e}^{-j\frac{c}{2}\nu^{2}}\nu\hat{g}^{*}(\nu)\mathrm{d}\nu=0.$

Consequently, $t=b+c\omega$ satisfies

$\left|\frac{\frac{\partial}{\partial\omega}S_{f}^{g}(t,\omega)}{S_{f}^{g}(t,\omega)}\right|=0,\text{~{}~{}for}~{}|S_{f}^{g}(t,\omega)|>\lambda.$

This result shows that IF equation (14) not only covers the instantaneous
characteristic of chirp signals but also covers the instantaneous information
of the linear group delay model, which can unify the well-known IF and GD
estimators proposed in the SST [21,23] and the time-reassigned SST [45,46].

4) Characterizing a mixture of harmonic-like and pulse-like signals: Suppose
$f(t)=\sum_{k=1}^{K}A_{k}(t)\mathrm{e}^{j\phi_{k}(t)}$ be a multi-component
signal consisting of a harmonic and pulse signals, due to the analysis above,
we have
$\left|\frac{\frac{\partial}{\partial\omega}S_{f_{k}}^{g}(t,\omega)}{S_{f_{k}}^{g}(t,\omega)}\right|=0~{}(|S_{f_{k}}^{g}(t,\omega)|>\lambda$)
for each component $f_{k}(t)$. Since
$\left|\frac{\frac{\partial}{\partial\omega}S_{f}^{g}(t,\omega)}{S_{f}^{g}(t,\omega)}\right|\leq\sum_{k=1}^{K}\left|\frac{\frac{\partial}{\partial\omega}S_{f_{k}}^{g}(t,\omega)}{S_{f}^{g}(t,\omega)}\right|=\sum_{k=1}^{K}\left|\frac{\frac{\partial}{\partial\omega}S_{f_{k}}^{g}(t,\omega)}{S_{f_{k}}^{g}(t,\omega)}\frac{S_{f_{k}}^{g}(t,\omega)}{S_{f}^{g}(t,\omega)}\right|$,
then
$\left|\frac{\frac{\partial}{\partial\omega}S_{f}^{g}(t,\omega)}{S_{f}^{g}(t,\omega)}\right|=0$.
Furthermore, if $f_{k}(t)$ $(k=1,2,\cdots,K)$ are separable, leading to that
$|S_{f}^{g}(t,\omega)|=|S_{f_{k}}^{g}(t,\omega)|$ when $(t,\omega)$ located in
the energy spread band of $f_{k}(t)$, we can obtain that
$\left|\frac{\frac{\partial}{\partial\omega}S_{f}^{g}(t,\omega)}{S_{f}^{g}(t,\omega)}\right|=\sum_{k=1}^{K}\left|\frac{\frac{\partial}{\partial\omega}S_{f_{k}}^{g}(t,\omega)}{S_{f_{k}}^{g}(t,\omega)}\right|=0$.
This means that IF equation (14) extracts the instantaneous information of
signals consisting of a mixture of harmonic-like and pulse-like components.

5) Multi-resolution IF equation: It is known that a single-window is difficult
to balance time resolution and frequency resolution, the combination of
multiple spectrograms or wavelet transforms with short and long windows or a
set of wavelets has been shown to result in good joint TF resolution for
signals consisting of mixtures of tones and pulses [53,54]. Inspired by this
idea, we can also define a multi-resolution IF equation of chirp (impulse or
linear group delay) signal:

$\begin{split}MrIF(t,\omega):=\left|\left(\prod_{i=1}^{m}\frac{S_{f}^{g_{i}}(t,\omega)}{S_{f}^{g_{i}}(t,\omega)}\right)^{\frac{1}{m}}\right|=0,\text{~{}~{}for}~{}|S_{f}^{g_{i}}(t,\omega)|>\lambda,\end{split}$
(23)

where $S_{f}^{g_{i}}(t,\omega)$ denotes the STFT of $f(t)$ using the Gaussian
window with parameter $\sigma_{i}$. This equation corresponds to the geometric
mean of multiple IF equations and it is built on high-resolution TF
representation, with this regard, it can be used to characterize closely-
spaced instantaneous information.

###  3.3 TF post-processing method using IF equation

This section will introduce how to use the given IF equation $h_{f}(t,\omega)$
to concentrate the TF distribution. Generally, there are two classical ways to
achieve such goal: extraction and reassignment. In the following we will
discuss these two ways in detail.

####  3.3.1 Extracting transform

The main idea of extracting transform is to extract the solutions of the IF
equation because which characterize the instantaneous features of the signal.
Define an extractor as

$\begin{split}\delta(\omega)=\begin{cases}1,&~{}\text{if}~{}\omega=0,\\\
0,&~{}\text{if}~{}\omega\neq 0,\end{cases}\end{split}$ (24)

then a concentrated TF representation is obtained by keeping only the TF
points on the IF trajectories, which is given by

$\begin{split}T_{e}(t,\omega)=S_{f}^{g}(t,\omega)\delta(h_{f}(t,\omega)).\end{split}$
(25)

We refer to (25) as the extracting transform. This transform retains the TF
coefficients consistent with the solutions of the IF equation, making the TF
location match the IF or GD estimates. Therefore, the extracting transform
greatly sharpens the energy distribution of multi-component non-stationary
signals, even with slowing-varying and fast-varying components.

It should be pointed out that the popular synchroextracting transform (SET)
[38] can be classified in the IF equation-based extracting transform. Indeed,
from expression (12), it is easy to get:

$\hat{\omega}(t,\omega)=\Re\left\\{\frac{\frac{\partial}{\partial
t}S^{g}_{f}(t,\omega)}{jS^{g}_{f}(t,\omega)}\right\\}=\phi^{\prime}(t)+\frac{\sigma^{4}{c}^{2}(\omega-\phi^{\prime}(t))}{1+\sigma^{4}{c}^{2}}.$
(26)

Expression (26) indicates that the set of points $\omega=\phi^{\prime}(t)$
satisfies

$\omega-\hat{\omega}(t,\omega)=0.$ (27)

Thus, equation (27) is a special IF equation of constant amplitude and linear
frequency modulation signal. Based on equation (27), it is easy to define the
extracting transform as

$\begin{split}T_{e}(t,\omega)=S_{f}^{g}(t,\omega)\delta(\omega-\hat{\omega}(t,\omega)),\end{split}$
(28)

which corresponds to the SET given in [38].

It is known that the extracting transforms are not the type being energy
conservation because they only remain the signal energy on the TF curves
satisfying the IF equation. Differing from the extracting operation, we will
introduce an approach that maintains the signal energy.

####  3.3.2 TF reassignment based on iterative algorithm

Another way to sharpen the spread TF distribution is to move the TF values
towards the IF or GD curves. Due to the fact that the IFs or GDs of the
studied signal $f(t)$ are the solutions of a given IF equation
$h_{f}(t,\omega)=0$, i.e., the roots of $h_{f}(t,\omega)$, then for any TF
point ($t,\omega$) with TF energy larger than zero, we can consider it as an
initial point of a numerical algorithm to solve a root of $h_{f}(t,\omega)$.
When the numerical algorithm converges to the solutions of the IF equation
with the increasing of iteration times, the spreading TF energy can be
relocated to the instantaneous curves of the studied signal.

There are many algorithms, such as the fixed point algorithm, the Newton
algorithm, and the Levenberg-Marquardt (LM) algorithm [55] can be used to
obtain a solution of that

$\displaystyle h_{f}(t,\omega)=0.$ (29)

Specifically, starting with an arbitrary initial point ($t_{0},\omega_{0}$),
the fixed point algorithm yields the iteration that

$\displaystyle\omega_{k+1}=\omega_{k}+\tau h_{f}(t_{0},\omega_{k}),$ (30)

where $\tau\neq 0$ is hoped to converge to a root of $h_{f}$. Assume iteration
(30) is convergent, i.e., $\lim_{k\rightarrow\infty}\omega_{k}=\omega^{*}$
(here $\omega^{*}$ is a function of $t_{0}$ and $\omega_{0}$, and it can
rewritten as $\omega^{*}(t_{0},\omega_{0})$), we can obtain that

$\displaystyle h_{f}(t_{0},\omega^{*})=0.$ (31)

To show the instantaneous information of the signal more accurately and with
better readability for the TF plot, we reassign the TF value of
($t_{0},\omega_{0}$) to the location ($t_{0},\omega^{*}$) as follows:

$S_{f}(t_{0},\omega_{0})+S_{f}(t_{0},\omega^{*})\rightarrow
S_{f}(t_{0},\omega^{*}),$ (32)

where $S_{f}(t,\omega)$ is one of the TF representations, such as the STFT,
CWT, chirplet transform, and their modules. In fact, similar to the expression
(9) in the SST, this reassigment operation can be formulated as

${S}^{*}_{f}(t,\omega)=\int_{|S_{f}(t,\eta)|>\lambda}S_{f}(t,\eta)\delta(\omega-{\omega^{*}}(t,\eta))\mathrm{d}\eta.$
(33)

In the following, we utilize a concrete IF equation, i.e., the IF equation
(16) of chirp signals, to discuss the convergence of the fixed point algorithm
(30). In this case, the iteration can be expressed as

$\displaystyle\omega_{k+1}=\omega_{k}+\tau\Re\left\\{\frac{\frac{\partial}{\partial\omega}S_{f}^{g}(t,\omega)|_{\omega=\omega_{k}}}{S_{f}^{g}(t,\omega_{k})}\right\\}=\omega_{k}-\tau\frac{\sigma^{2}(\omega_{k}-\phi^{\prime}(t))}{\sqrt{1+\sigma^{4}c^{2}}}.$
(34)

From (34), we have

$\displaystyle|\omega_{k+1}-\omega_{k}|$
$\displaystyle=|\omega_{k}-\omega_{k-1}-\frac{\tau\sigma^{2}}{\sqrt{1+\sigma^{4}c^{2}}}(\omega_{k}-\phi^{\prime}(t)-(\omega_{k-1}-\phi^{\prime}(t)))|$
(35)
$\displaystyle\leq|1-\frac{\tau\sigma^{2}}{\sqrt{1+\sigma^{4}c^{2}}}||\omega_{k}-\omega_{k-1}|.$
(36)

Therefore, iteration (34) is convergent if $\tau$ is taken as a value such
that $|1-\frac{\tau\sigma^{2}}{\sqrt{1+\sigma^{4}c^{2}}}|<1$.

It should be pointed out that the multi-synchrosqueezing transform [43] is a
special case of the IF equation-based reassignment method. Indeed, when we
consider the IF equation (27), and take the value of $\tau$ as -1, then the
form of iteration (30) becomes

$\displaystyle\omega_{k+1}=\hat{\omega}(t,\omega_{k}),$ (37)

which corresponds to the multiple SST operation in [43].

Moreover, we also can use the Newton algorithm or the LM algorithm to get a
root of $h_{f}(t,\omega)$, which are expressed respectively as

$\displaystyle\omega_{k+1}$
$\displaystyle=\omega_{k}-\frac{h_{f}(t,\omega_{k})}{\frac{\partial}{\partial\omega}h_{f}(t,\omega)|_{\omega=\omega_{k}}},~{}\text{(Newton
algorithm)}$ (38) $\displaystyle\omega_{k+1}$
$\displaystyle=\omega_{k}-\frac{h_{f}(t,\omega_{k})}{\frac{\partial}{\partial\omega}h_{f}(t,\omega)|_{\omega=\omega_{k}}+\rho},~{}\text{(Levenberg-
Marquardt algorithm)}$ (39)

where $\rho>0$. The LM algorithm is generally considered to be more robust
than the Newton algorithm because it converges even if the starting point is
far from the final value [15,55].

##  4 Comparison

The setion above introduces the IF equation-based TF post-processing method to
obtain a concentrated and accurate TF output. In the following, we compare its
performance with the SST-based TF post-processing method.

First, the two methods are based on different feature extractors: the former
is based on IF equation, while the latter is based on IF estimation. For a
signal $f(t)$, its IF equation is not unique, and finding one that covers as
many different types of signals as possible is a challenging task.
Nevertheless, we can find an IF equation, such as equation (14), to
characterize the information of various signals, including the harmonic
signals, the chirps, and the impulse signals. By comparison, the IF estimators
presented in the SST and its extensions have limited generalization ability to
deal with a mixture of harmonic and impulsive signals.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 1:  Evaluation of TF results on a known signal structure. (a) Waveform
of the test signal, (b) TF plot by STFT, (c) TF plot by FSST, (d) TF plot by
Second FSST, (e) TF plot by TSST, (f) TF plot by ET_IF, (g) TF plot by RM_IF.

Let us consider an example to illustrate their performance. The waveform of
the test signal is illustrated in Figure 1(a), which contains two impulsive
signals, a chirp, and a cosine modulation component. We compute the TF
representations of the test signal using STFT, FSST, second-order FSST (Second
FSST for short), time-reassigned SST (TSST for short) [45], the extracting
transform based on IF equation (14) (ET_IF for short), and the reassignment
method based on IF equation (16) (RM_IF for short). From the results, we can
see that the TF result of STFT is blurred. The FSST and second FSST improve
the energy concentration of STFT for chirps and cosine modulated signals but
fail to characterize impulsive signals effectively. The TSST provides a better
TF localization for the transient signals while degrading the TF sharpness of
the stationary mode. For our proposed RM_IF approach, it provides a high-
concentration TF representation for the chirp and the cosine modulated signal,
but the operation of relocating the TF values along the frequency direction
limits its ability to handle the impulsive signals, which results in an
imprecise characterization, as shown in Figure 1(g). The ET_IF is clearly
superior to others and provides excellent performance in dealing with a
mixture of impulse-like signals, chirps, and cosine modulation signals.

Table 1:  The cost times of different analysis methods. Methods | STFT | FSST | Second FSST | TSST | ET_IF | RM_IF
---|---|---|---|---|---|---
Time (s) | 0.005 | 0.017 | 0.253 | 0.107 | 0.016 | 0.0672
|  |  |  |  |  |

The computational complexity of the two kinds of post-processing methods
mainly comes from the STFT calculations with different windows. We assume that
a signal of $N$ samples is utilized, then the STFT can be implemented by FFT
and it requires $O(N^{2}\log_{2}N)$ operations. Therefore, the total computing
complexity of the TF post-processing methods is $O(N^{2}\log_{2}N)$. Table 1
list the cost times of the above-mentioned TF analysis methods in addressing
the test signal (see Figure 1(a)). The tested computer configuration is as
follows: Intel Core i9-11900KF 3.50 GHz, 32.0 GB of RAM, and MATLAB version
R2021a. It can be seen from the results that these methods can finish the
analysis within one second. It takes less time to represent the signal by
ET_IF and RM_IF than by Second FSST and TSST.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 2:  Evaluation of TF results on the test signal with SNR=2 dB. (a)
Waveform of the test signal, (b) TF plot by STFT, (b) TF plot by FSST, (c) TF
plot by Second FSST, (d) TF plot by TSST, (e) TF plot by ET_IF, (f) TF plot by
RM_IF.

Moreover, Figure 2 provides TF representations of the test signal with strong
noise (see Figure 2(a)) using the six TF methods. From the results, we can see
that these post-processing methods are susceptible to noise, and ET_IF
produces discontinuous TF curves due to that the noise affects the precision
of the IF equation. Therefore, how to improve their noise robustness to obtain
an accurate TF plot is required.

##  5 Conclusion

In this paper, we discussed the IF equation-based TF analysis method to
generate a concentrated TF representation for multi-component non-stationary
signals. The IF equation we defined does serve as a feature extractor to
characterize the slow-varying and fast-varying modes. Several properties of
the IF equation were studied and proved theoretically. Based on the IF
equation, we introduced two classical ways, i.e., extraction and reassignment,
to obtain a highly-concentrated TF output. The IF equation-based TF post-
processing methods provide another way to improve the TF resolution and cover
many popular TF analysis methods, such as the synchroextracting transform, the
multi-synchrosqueezing transform, and the time extracting transform, etc.
Compared with the SST-based TF post-processing methods, our proposed method
showed better performance in dealing with a mixture of impulse-like signals,
chirps, and cosine modulation signals.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China
under Grant no. U20B2075, the Fundamental Research Funds for the Central
Universities under Grant no. G2021KY05103.

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# Unsupervised Pidgin Text Generation By Pivoting English Data and Self-
Training

Ernie Chang1, David Ifeoluwa Adelani2, Xiaoyu Shen2 & Vera Demberg1
1Department of Language Science and Technology, Saarland University, Germany
2Spoken Language Systems (LSV), Saarland Informatics Campus, Germany

<EMAIL_ADDRESS>

###### Abstract

West African Pidgin English is a language that is significantly spoken in West
Africa, consisting of at least 75 million speakers. Nevertheless, proper
machine translation systems and relevant NLP datasets for pidgin English are
virtually absent. In this work, we develop techniques targeted at bridging the
gap between Pidgin English and English in the context of natural language
generation. By building upon the previously released monolingual Pidgin
English text and parallel English data-to-text corpus, we hope to build a
system that can automatically generate Pidgin English descriptions from
structured data. We first train a data-to-English text generation system,
before employing techniques in unsupervised neural machine translation and
self-training to establish the Pidgin-to-English cross-lingual alignment. The
human evaluation performed on the generated Pidgin texts shows that, though
still far from being practically usable, the pivoting + self-training
technique improves both Pidgin text _fluency_ and _relevance_.

##  1 Introduction

Pidgin English is one of the the most widely spoken languages in West Africa
with roughly 75 million speakers estimated in Nigeria; and over 5 million
speakers estimated in Ghana (Ogueji & Ahia, 2019). 111Though variants of
Pidgin English are abound, the language is fairly uniform across the
continent. In this work, we directed our research to the most commonly spoken
variant of West African Pidgin English $-$ the Nigerian Pidgin English. While
there have been recent efforts in popularizing the monolingual Pidgin English
as seen in the BBC Pidgin222 https://www.bbc.com/pidgin , it remains under-
resourced in terms of the available parallel corpus for machine translation.
Similarly, this low-resource scenario extends to other domains in natural
language generation (NLG) such as summarization, data-to-text and so on
(Lebret et al., 2016; Hong et al., 2019; de Souza et al., 2018; Chang et al.,
2021b; a) $-$ where Pidgin English generation is largely under-explored. The
scarcity is further aggravated when the pipeline language generation system
includes other sub-modules that computes semantic textual similarity (Zhuang &
Chang, 2017), which exists solely in English.

Previous works on unsupervised neural machine translation for Pidgin English
constructed a monolingual corpus (Ogueji & Ahia, 2019), and achieved a BLEU
score of 5.18 from English to Pidgin. However, there is an issue of domain
mismatch between down-stream NLG tasks and the trained machine translation
system. This creates a caveat where the resulting English-to-Pidgin MT systems
(trained on the domain of news and the Bible) cannot be directly used to
translate out-domain English texts to Pidgin. An example of the English/pidgin
text in the restaurant domain (Novikova et al., 2017) is displayed in Table 1.

English | There is a pub Blue Spice located in the centre of the city that provides Chinese food.
---|---
Pidgin | Wan pub blue spice dey for centre of city wey dey give Chinese food.
Table 1:  Sample parallel English-Pidgin text from the restaurant domain.

Nevertheless, we argue that this domain-mismatch problem can be alleviated by
using English text in the target-domain as a pivot language (Guo et al.,
2019). To this end, we explore this idea on the task of neural data-to-text
generation which has been the subject of much recent research. Neural data-to-
Pidgin generation is essential in the African continent especially given the
fact that many existing data-to-text systems are English-based e.g. Weather
reporting systems (Sripada et al., 2002; Belz, 2008). This work aims at
bridging the gap between many of these English-based systems and Pidgin by
training an _in-domain_ English-to-pidgin MT system in an unsupervised way. By
this means, English-based NLG systems can be locally adapted by translating
the output English text into pidgin English. We employ the publicly available
parallel data-to-text corpus E2E (Novikova et al., 2017) consisting of
tabulated data and English descriptions in the restaurant domain. The training
of the in-domain MT system is done with a two-step process: (1) We use the
target-side English texts as the pivot, and train an unsupervised NMT
($model_{unsup}$) directly between in-domain English text and the available
monolingual Pidgin corpus. (2) Next, we employ self-training (He et al., 2019)
to create augmented parallel pairs to continue updating the system
($model_{self}$).

##  2 Approach

First phase of the approach requires training of an unsupervised NMT system
similar to Ogueji & Ahia (2019) (PidginUNMT). Similar to Ogueji & Ahia (2019),
we train the cross-lingual model using FastText Bojanowski et al. (2017) on
the combined Pidgin-English corpus. Next, we train an unsupervised NMT similar
to Lample et al. (2017); Artetxe et al. (2017); Ogueji & Ahia (2019) between
them to obtain $model_{unsup}$. Then we further utilize $model_{unsup}$ to
construct pseudo parallel corpus by predicting target Pidgin text given the
English input. We augment this dataset to the existing monolingual corpus. The
self-training step involves further updating $model_{unsup}$ on the pseudo
parallel corpus and non-parallel monolingual corpus to yield $model_{self}$.

##  3 Experiments and Results

We conduct experiments on the E2E corpus (Novikova et al., 2017) which amounts
to roughly 42k samples in the training set. The monolingual Pidgin corpus
contains 56,695 sentences and 32,925 unique words. The human evaluation was
performed on the test set (630 data instances for E2E) by averaging over
scores by 2 native Pidgin speakers on both Relevance (0 or 1 to indicate
relevant or not) and Fluency (0, 1, or 2 to indicate readability). Table 2
shows that $model_{self}$ outperforms direct translation (PidginUNMT) and
unsupervisedly-trained model $model_{unsup}$ on _relevance_ and performing on
par with PidginUNMT on _fluency_. We also display relevant sample outputs in
Table 3 at all levels of fluency.

System | Relevance | Fluency
---|---|---
PidginUNMT | 0.038 | 0.827
$model_{unsup}$ | 0.319 | 0.788
$model_{self}$ | 0.434 | 0.814
Table 2: PidginUNMT is trained on unparallel, out-domain English and pidgin
text. $model_{unsup}$ refers to unsupervised NMT trained on in-domain English
text and out-domain pidign text. $model_{self}$ further augments
$model_{unsup}$ with pseudo parallel pairs obtained from self-training.

Pidgin text | Fluency
---|---
Every money of money on food and at least 1 of 1 points. | 0
and na one na di best food for di world. | 1
People dey feel the good food but all of us no dey available. | 2

Table 3:  Sampled relevant (score of 1) Pidgin outputs from $model_{self}$
with various Fluency scores.

##  4 conclusion

In this paper, we have shown that it is possible to improve upon low-resource
Pidgin text generation in a demonstrated low-resource scenario. By using non-
parallel in-domain English and out-domain Pidgin text along with self-training
algorithm, we show that both _fluency_ and _relevance_ can be further
improved. This work serves as the starting point for future works on Pidgin
NLG in the absence of annotated data. For future works, we will also further
utilize phrase-based statistical machine translation to further improve upon
current work.

##  5 Acknowledgements

This research was funded in part by the SFB 248 “Foundations of Perspicuous
Software Systems”. We sincerely thank the anonymous reviewers for their
insightful comments that helped us to improve this paper.

## References

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