Available via license: CC BY 4.0
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Received April 28, 2022, accepted May 23, 2022, date of publication June 1, 2022, date of current version June 8, 2022.
Digital Object Identifier 10.1109/ACCESS.2022.3179517
A Review of Wavelet Analysis and Its
Applications: Challenges and Opportunities
TIANTIAN GUO 1,2, TONGPO ZHANG 1,2, ENGGEE LIM 1, (Senior Member, IEEE),
MIGUEL LÓPEZ-BENÍTEZ 2,3, (Senior Member, IEEE), FEI MA 4,
AND LIMIN YU 1, (Member, IEEE)
1Department of Communications and Networking, School of Advanced Technology, Xi’an Jiaotong-Liverpool University (XJTLU), Suzhou, Jiangsu 215123,
China
2Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool, Merseyside L69 3BX, U.K.
3ARIES Research Centre, Universidad Antonio de Nebrija, 28040 Madrid, Spain
4Department of Applied Mathematics, School of Science, Xi’an Jiaotong-Liverpool University (XJTLU), Suzhou, Jiangsu 215123, China
Corresponding author: Limin Yu (limin.yu@xjtlu.edu.cn)
This work was supported in part by the Research Enhancement Fund of Xi’an Jiaotong-Liverpool University (XJTLU) under Grant
REF-19-01-04, in part by the National Natural Science Foundation of China (NSFC) under Grant 61501380, in part by the AI University
Research Center (AI-URC), in part by the XJTLU Laboratory for Intelligent Computation and Financial Technology through the XJTLU
Key Program Special Fund under Grant KSF-P-02, in part by the Jiangsu Data Science and Cognitive Computational Engineering Research
Centre, and in part by the ARIES Research Centre.
ABSTRACT As a general and rigid mathematical tool, wavelet theory has found many applications and is
constantly developing. This article reviews the development history of wavelet theory, from the construction
method to the discussion of wavelet properties. Then it focuses on the design and expansion of wavelet
transform. The main models and algorithms of wavelet transform are discussed. The construction of rational
wavelet transform (RWT) is provided by examples emphasizing the advantages of RWT over traditional
wavelet transform through a review of the literature. The combination of wavelet theory and neural networks
is one of the key points of the review. The review covers the evolution of Wavelet Neural Network (WNN),
the system architecture and algorithm implementation. The review of the literature indicates the advantages
and a clear trend of fast development in WNN that can be combined with existing neural network algorithms.
This article also introduces the categories of wavelet-based applications. The advantages of wavelet analysis
are summarized in terms of application scenarios with a comparison of results. Through the review, new
research challenges and gaps have been clarified, which will serve as a guide for potential wavelet-based
applications and new system designs.
INDEX TERMS Wavelets, multiresolution analysis, wavelet transform, rational wavelets, wavelet neural
network.
ABBREVIATIONS
The acronym words in this paper and their full names are
listed below.
ACRONYM FULL NAME
BER Bit Error Ratio.
BP Back Propagation.
complex WT complex Wavelet Transform.
CWT Continuous Wavelet Transform.
CROW Complex Rational Orthogonal Wavelet.
CO-OFDM Coherent Optical OFDM.
CDF Cohen Daubechies Feauveau.
The associate editor coordinating the review of this manuscript and
approving it for publication was Ramakrishnan Srinivasan .
ACRONYM FULL NAME
DT-CWT Dual-Tree Complex Wavelet Transform.
DWT Discrete Wavelet Transform.
Dmey Discrete meyer wavelet.
DWPT Discrete Wavelet Packet Transform.
ECG ElectroCardioGrams.
EEG ElectroEncephaloGraphy.
EWT Empirical Wavelet Transform.
FIR Finite Impulse Response.
FrWF Fractional Wavelet Filter.
VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 58869
T. Guo et al.: Review of Wavelet Analysis and Its Applications: Challenges and Opportunities
FrWT Fractional Wavelet Transform.
fBm fractional Brownian motion.
MRA Multi-Resolution Analysis.
MS Multiple Sclerosis.
NOMA Non-Orthogonal Multiple Access.
OFDM Orthogonal Frequency Division Multiplex.
ON Optic Neuritis.
OWDM Orthogonal Wavelet Division Multiplex.
PSD Power Spectral Density.
PSNR Peak Signal to Noise Ratio.
PNN Probabilistic Neural Network.
PAPR Peak to Average Power Ratio.
PSO Particle Swarm Optimization.
QAM Quadrature Amplitude Modulation.
QPSK Quadrature Phase Shift Keying.
RWT Rational Wavelet Transform.
RADWT Rational Dilation Wavelet Transform.
RMS Root Mean Square.
RNN Recurrent Neural Network.
RWNN Recurrent Wavelet Neural Network.
RMSE Root Mean Square Error.
STFT Short-Time Fourier Transform.
SWT Stationary Wavelet Transform.
SAF Sigmoid Activation Function.
SNR Signal to Noise Ratio.
sEMG surface ElectroMyoGraphy.
SVM Support Vector Machines.
SSIM Structural Similarity.
UWT Undecimated Wavelet Transform.
WNN Wavelet Neural Network.
WP Wavelet Packet.
WAF Wavelet Activation Function.
WK Wavelet Kernel.
WFNN Wavelet Fuzzy Neural Network.
WSS Weighted Spectral Slope.
I. INTRODUCTION
In signal processing, much attention has been paid to multi-
resolution analysis and data feature extraction. As a powerful
mathematical tool for analyzing time-varying non-stationary
signals, the time-frequency analysis method offers informa-
tion on joint distribution in both the time and frequency
domains. This method clearly describes the relationship
between time and signal frequency. Standard time-frequency
distribution functions include short-time Fourier transform
(STFT, including Gabor transformation), Cohen distribu-
tion function (including Wegener distribution), improved
Wegener distribution, Gabor-Wigner distribution function
and S transform [1]. The advantage of STFT is that its
physical meaning, which represents the energy contained
in each frequency component of a signal over a specified
time interval, is clear. Many actual test signals provide a
time-frequency structure consistent with people’s intuitive
perception, which has become the most used time-frequency
analysis. Nevertheless, the time or frequency resolution of the
STFT is limited by the window’s width function and cannot
be optimized at the same time [2]. Such limitations can be
overcome with wavelets.
Similar to the Fourier transform, the wavelet transform
can be seen as the projection of a signal into a set of basis
functions that provide localization in the frequency domain.
However, in contrast to the Fourier transform, which pro-
vides constant, equally spaced time-frequency localization,
the wavelet transform provides high-frequency resolution at
low frequencies and high time resolution at high frequen-
cies. Thus, different from the Fourier transform, the wavelet
transform utilizes a series of orthogonal bases with differ-
ent resolutions to represent or approximate a signal through
the expansion and translation of the wavelet basis function.
Wavelet transform is considered to be a significant break-
through in mathematical analysis. It can be applied to various
fields. For example, signal processing, image processing,
pattern recognition, speech analysis and many applications
could introduce wavelet analysis.
The wavelet research developed rapidly in the 1980s.
In 1981, Stromberg proved the existence of wavelet functions.
From 1984 to 1988, Meyer, Battle and Lemarie designed
different wavelet basis functions with fast decay character-
istics [3]. Mallat proposed a fast wavelet transform algorithm
for signal analysis and reconstruction, namely Mallat algo-
rithm [4]. Based on the concept of multi-resolution analysis,
the Mallat algorithm is expressed as a two-channel filter.
Whether 2-dimension images or 1-dimension signals, signals
could be approximated by a set of sub-signals with different
resolutions. It is widely applied in signal decomposition and
reconstruction. In 1992, Soman and Vaidyanathan proposed
wavelet packet theory [5]. Compared with wavelet transform,
wavelet packet can divide the time-frequency plane more
finely, and the resolution of the high-frequency part of the
signal is better than that of the wavelet analysis.
In 1992, Zou and Tewfik [6] proposed the M-band
Wavelet theory, which extended people’s research on
wavelet transform from ‘‘two-band’’ to ‘‘multi-band’’.
In 1994, Goodman et al. established a multi-wavelet theo-
retical framework based on R-order multi-scaling functions
and multi-resolution analysis [7]. The multi-resolution space
generated by multiple-scale functions in a single wavelet
is expanded to be generated by multiple-scale functions to
obtain greater degrees of freedom. Geronimo and his team
in 1994 [8] designed a multi-scale wavelet transform. The
construction of the wavelet function is completed by multiple
scaling functions. It can have the characteristics of tight
support, orthogonality, symmetry and interpolation at the
same time. In 1995, Sweldens et al. proposed a new wavelet
construction algorithm-Lifting Scheme. First, the original
discrete sample signal is divided into odd and even, and
then the odd and even sample points are filtered. All first-
generation wavelets can be constructed using lifting schemes.
It is characterized by fast computing speed, small memory
requirements, and the ability to implement integer-to-integral
conversion [9]. In recent years, with the development of mod-
ern communication systems and image processing, wavelet
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design has also evolved to adapt to new types of signals and
image analysis.
The development of wavelet transform has never stopped.
Numerous new wavelet systems have been created and
applied to improve the limitations of classic wavelets. Classi-
cal discrete wavelets are translation sensitive, which means
a slight shift of signals will cause an extensive range of
wavelet coefficients to fluctuate. Complex wavelet transform
(complex WT) can overcome the above problems, but it has
another problem [10]. Since the input form of the complex
WT with more than one level of decomposition is complex,
it is challenging to construct its complete reconstruction
filter. Kingsbury [11] in 1998 constructed Dual-Tree Com-
plex Wavelet Transform (DT-CWT). Compared with classi-
cal wavelet transform, DT-CWT could also provide phase
information. It not only satisfies the condition of complete
reconstruction but also retains other advantages of complex
wavelets. It has been applied to many image processing fields.
Combining the advantages of wavelet transform and neural
network is another research hotspot. One method is to use
wavelet analysis to preprocess the signal. The feature extrac-
tion of the signal is achieved through wavelet transform,
and the extracted feature vector is then sent to the neural
network for processing. The other is a compact combination
of wavelet transform and neural network, fully integrating
the advantages of the two. Pati and Krishnaprasad [12] first
studied the relationship between neural network and wavelet
transform and proposed a discrete affine wavelet network
model. The idea is to introduce the discrete wavelet transform
into the neural network model. In 1992, Zhang and Ben-
veniste [13] formally proposed the concept and algorithm of
wavelet networks. The idea is to replace neurons with wavelet
elements, that is, to use wavelet functions instead of Sigmod
functions as activation functions. In wavelet neural network
(WNN), the hidden layer of the neural network is replaced
by a wavelet function. Meanwhile, the corresponding weights
from the input layer to the hidden layer and the threshold of
the hidden layer are replaced by the scaling factor and the
time shift factor of the wavelet function [14].
Then Szu and Harold [15] proposed two adaptive WNN
models based on continuous wavelet transform. One is used
for signal representation, focusing on function approxima-
tion; the other focuses on selecting appropriate wavelets for
feature extraction. Because it does not involve reconstruction
problems, the orthogonality requirements of wavelets are not
very strict. However, because the orthogonal wavelet base has
good time-frequency resolution performance when the signal
changes drastically, the network can increase the resolution
scale to ensure the accuracy of the approximation. In addition,
due to the orthogonality of the function bases, adding or delet-
ing network nodes during the training process does not affect
the trained network weights, which can greatly shorten the
network learning time. Bhavik et al. [16] used the orthogonal
wavelet function as the activation function of neurons and
proposed an orthogonal multi-resolution WNN. According to
the theory of multi-resolution analysis, the scale function and
wavelet function are included in the network together, and the
network is trained by the step-by-step learning method.
In this review, a clear track of the wavelet development
in history is sorted out and reviewed systematically. The
contribution of this paper could be concluded as follows:
1) Wavelet theory, including the wavelet construction
method and properties definition, is briefly summarized.
The development of the wavelet base is discussed, and
the research direction, which is rationalization, has been
pointed out.
2) Signal decomposition methods are discussed, including
widely used DWT (Discrete wavelet transform) and
its extensions: wavelet packet (WP), complex WT, and
primarily, rational wavelet transform (RWT). RWT is
a more powerful signal processing tool with a more
satisfactory frequency resolution, which can be applied
to a wide range of fields by adjusting the rational factor.
3) The advantages of WNNs that use wavelet analysis as
preprocessing are further clarified. More satisfactory
scale domain resolution is more convenient for signal
processing, whether in denoising or feature extraction.
Applying the wavelet function to the neural network,
combined with some interdisciplinary algorithms, can
significantly increase the neural network’s performance.
4) The application of wavelets in signal processing in the
emerging fields, image processing, and optimization
algorithms is introduced to broaden the application sce-
narios of wavelets.
5) Some current challenges and research gaps are discussed
in the review, and some future research directions are
suggested.
The paper is organized as follows. Section II is the
description of review method. Wavelet theory is described
in Section III. The properties of different wavelet bases are
discussed. Wavelet transform, especially RWT, which can
provide finer resolution analysis, is reviewed in Section IV.
WNN is introduced in Section V. It is divided into two
research directions: wavelet as signal preprocessing and
wavelet as activation function. The advantages of wavelet
in signal processing (traditional and emerging fields), image
processing, and application of optimization algorithms are
also reviewed in Section VI. Through the review, some
research challenges and research gaps are clarified in
Section VII. In the end, the conclusion and future work are
presented in Section VIII.
II. REVIEW METHOD
A. THE ORGANIZATION OF REVIEW
The development of wavelets in four main categories is
reviewed, covering wavelet theory, applications of wavelet
theory in general signal processing problems and practical
application scenarios. WNN, one of the fast-developing hot
spots, is also covered to reflect its theoretical development
and rich applications. The organization of the review is visu-
alized in Fig. 1and the detailed organization of the wavelet
application is in Section VI. The historical background and
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FIGURE 1. The orgnization structure of this review paper.
families of wavelet basis functions are first reviewed. Various
wavelet decomposition architectures and time-scale analy-
sis tools for general signal processing are summarized. The
review of WNN is divided into two major categories: one is
to preprocess the signal with wavelets and input them into the
neural network. The other is the deep fusion of the wavelet
function and the neural network. The most relevant, recent
and representative practical implementations using wavelets
are summarized to reflect the current status of wavelet appli-
cations. Through the comprehensive review, research gaps
and challenges are identified.
B. SELECTION CRITERIA
We first used the filter provided by the database to screen
the publication date, publication platform, and fields of the
documents. For example, we choose the latest papers, which
are mostly from the past ten years and divide them into journal
articles and conference papers. Furthermore, we browse the
abstracts and keywords of the literature to determine whether
it meets the requirements of the review paper. If the abstract
and keywords cannot be accurately identified, we need to read
the introduction part and find the objective statement of the
literature to determine its relevance. Through the extensive
and intensive reading of the literature, the main contribution
is extracted and presented in a concise way. The main aspects
for the evaluation of cited papers are novelty, contribution,
relevance to the field and timelines. The survey includes
papers that represent recent advances in the field, as well
as relevant contributions to the field, regardless of their pub-
lication date. Through the literature, it is judged whether the
literature focuses on the theoretical proposal or application
design and divided into two categories. In each category, the
literature is subdivided according to the collected keywords.
We rank the articles by publication date for each subcategory.
We intensively read the literature and extracted the novelty
and achievements of the literature for further comparison and
analysis.
III. WAVELET THEORY
Current review papers related to wavelets focus on one nar-
row field in many cases. For example, Paul S. Addison’s
article focused on near-infrared spectroscopy and defined
cross-wavelet transform [17]. The author suggests that
wavelets can be used for real-time local transformation phase
monitoring to obtain valuable new high-resolution views.
Moreover, low-oscillation wavelets may be worth consider-
ing when the time resolution needs to be increased in the
transform domain. Huang et al. [18] pays particular attention
to the influence of different wavelets chosen (i.e., wavelet
packet and Gabor wavelet) and proposes to use the recently
introduced empirical wavelets. In order to pay attention to
the texture existing in the image, experiments are conducted
based on an extensive selection of wavelets. The wavelet
families in different directions were used to represent tex-
ture more effectively. Reference [19] introduced continuous
wavelet transform (CWT) and DWT in ECG (ElectroCar-
dioGrams) signal denoising and data storage reduction. It is
analyzed that the most suitable denoising technology method
is the bionic wavelet transform method. It shows high selec-
tivity and sensitivity with high noise reduction.
In comparison with the previous reviews, this paper con-
tains the scope of wavelets and various fields of wavelet
applications as detailed in the following sections.
A. WAVELET CONSTRUCTION METHOD
There are two methods to construct wavelets. The wavelet’s
concept was initially praised in 1981 [20], [21]. After that,
multi-resolution analysis (MRA) was constructed as a tool-
box for constructing standard wavelet bases. Meyer and some
other researchers perfected the details of the MRA. Man-
souri Jam and Sadjedi proposed an orthogonal MRA and
designed a matched wavelet to satisfy orthogonal MRA con-
ditions [22]. Assume {Vj}j∈Zis a subspace of L2(R) space,
consider {Vj}j∈Zis an MRA of L2(R) space when [23]:
•Vj−1⊂Vj
The space expands with the increase of jand is strictly
contained (subspaces are contained layer by layer), indi-
cating that the information of Vj−1is completely con-
tained in Vjand contains less information than Vj.
•T
j∈Z
Vj= {0},S
j∈Z
Vj=L2(R)
The subspaces have and only have the intersection of 0.
All of them could form a L2(R) space.
•f(t)∈Vj⇔f(2t)∈Vj+1
The subspaces have dyadic scalability.
•f(t)∈Vj⇒f(t−k)∈Vjfor all k∈Z
The function still belongs to this subspace after
translation.
The other method is based on the lifting scheme, which
was raised by Sweldens in 1995 [9]. The method has a
faster calculation speed, and takes up less memory. It retains
wavelet characteristics while overcoming the original limita-
tions. In [24], Ansari and Gupta extended the lifting frame-
work from dyadic wavelet to rational wavelet. It inherits
all the advantages of the lifting framework. These so-called
second-generation wavelets are easy to implement and can
process signals of any size in the spatial domain and show
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perfect reconstruction [25]. It has two main applications. The
first is the acceleration of the fast wavelet transform algo-
rithm. The boundary processing has also been simplified. The
second application is to design wavelets suitable for multi-
dimensional bounded domains and curved surfaces, which
cannot be achieved by Fourier transform [26].
Haouam et al. [27] presented the lifting scheme with a
biorthogonal wavelet and applied it in Magnetic Resonance
Imaging (MRI) images compression. Their method gives bet-
ter compression results than traditional methods. In [28], they
also designed a rational wavelet filter bank based on a lifting
scheme and achieved better sparsifying property than other
wavelets. Chen et al. [29] proposed an undecimated lifting
scheme to exact shift-invariance and show superiority over
other methods. A biorthogonal wavelet with shape control
is constructed under the lifting scheme in [30], which gives
better performance in data compression and noise reduction.
Guptha and his team introduced transistor technology based
on a lifted wavelet transform architecture. The complete
boosting step is processed as a continuous stream of samples.
Compared with the existing architecture, the architecture
is optimized by integrating forward and backward lifting
schemes [31].
B. WAVELET PROPERTIES
The properties of the wavelet are an important reference
for constructing wavelets or choosing suitable wavelets to
process various signals. Farge [20] has discussed the fac-
tors that need to be considered when choosing the mother
wavelet, such as orthogonal and non-orthogonal, negative
and real values, and the width and shape of the mother
wavelet. There are several basic properties or standards for
wavelets: vanishing moment, support length, regularity, sym-
metry and orthogonality. Orthogonality wavelet has good
time-frequency localization characteristics. Selesnick gave
the necessary conditions for an orthogonal wavelet system to
form a Hilbert transform pair [32] and proposed a construc-
tion algorithm based on a delay filter in [33], [34]. Selesnick
further pointed out that the necessary conditions given by
Selesnick in [32] are still sufficient [35]. Vanishing moment
is defined as [36]. If the wavelet has Nvanishing moments,
it should satisfy that:
Ztpψ(t)dt =0 (1)
where 0 ≤p<N, ψ (t) is the wavelet function, and tis
the time variable of the wavelet function. Higher vanishing
moments of wavelets are required for signal compression,
denoising, fast calculation. The larger the vanishing moment
is, the more wavelet coefficients are zero. Fig. 2shows
the wavelet coefficients with different vanishing moments.
We apply DWT with Daubechies wavelets on an example
sine function. The wavelets are db2, db5, and db10, with
vanishing moments being 2, 5 and 10, respectively. The larger
the vanishing moment of the wavelet, the smaller the high-
frequency coefficients after wavelet decomposition, the more
concentrated the signal energy, and the higher the signal
compression ratio.
The support interval of the wavelet function is the length
at which the function converges from a finite value to 0 when
the time or frequency tends to infinity. The longer the support
length, the larger cost of computation is required, and more
high-amplitude wavelet coefficients are generated. Wavelets
often have a compact support requirement, which means that
the wavelet function is zero except for a small value range
near 0. The compact support and the vanishing moment are
contradictory. The support length represents the length of
the filter. If the vanishing moment increases, the wavelet
coefficients of the high-frequency sub-band decrease and a
larger amount of coefficients are close to zero, so that the
support length is shorter. Therefore, the support length and
the vanishing moment must be compromised. Compactly
support and vanishing moments can be better balanced under
the multi-wavelet construction [26]. In actual situations, they
can be weighed according to the singularity of the signal.
High vanishing moments are more suitable if the signal singu-
larities are few, and if singularities are trivial, shorter supports
interval are required.
Regularity is generally used to describe the smoothness of a
function. The higher the regularity, the better the smoothness
of the function. The Lipschitz exponent k usually character-
izes the regularity of the function. Given a positive integer
n, if there is a positive integer Aand a polynomial of degree
n(Pn(t)), so that the function f(t) has the characteristics in
Equation (2) at t∈(t0−h,t0+h), then f(t) has Lips-
chitz exponent αat the point t0.his a sufficiently small
amount. The Lipschitz exponent characterizes the approxi-
mation degree of the function and the local polynomial, which
is related to the differentiability of the function. The regularity
of the wavelet base affects the stability of the reconstruction
of the wavelet coefficients.
A certain regularity (smoothness) is usually required for
wavelet analysis to obtain a better-reconstructed signal. The
wavelet function has the same regularity as the scale function
because the wavelet function is composed of the linear combi-
nation of the corresponding scale function translation. When
quantizing the wavelet coefficients, to reduce the influence
of the reconstruction error on the human eye, the smooth-
ness or continuous differentiability of the wavelet must be
increased as much as possible. Wavelets with reasonable
regularity can achieve a better smoothing effect in signal
or image reconstruction. However, if the regularity is rea-
sonable, the support length will be longer, and the cost of
computation will be larger. There is an excellent relationship
between vanishing moments and regularity. For many intrin-
sic wavelets (e.g., spline wavelets, Daubechies wavelets), the
regularity of the wavelet becomes larger as the vanishing
moment increases [26]. However, this does not mean that as
the vanishing moment of the wavelet increases, the regularity
of the wavelet also increases.
|f(t)−Pn(t−t0)| ≤ A|t−t0|α,n< α < n+1 (2)
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FIGURE 2. The wavelet coefficients with different vanishing moment of an example signal: (a) Original example signal; (b) Low fequency
coefficients with vanishing moment =2; (c) High fequency coefficients with vanishing moment =2; (d) Low fequency coefficients with
vanishing moment =5; (e) High fequency coefficients with vanishing moment =5; (f) Low fequency coefficients with vanishing moment =
10; (g) High fequency coefficients with vanishing moment =10.
Daubechies has rigorously proved that for compactly sup-
ported 2-band orthogonal wavelets, there is no symmet-
ric (antisymmetric) wavelet except Haar wavelet [37]. The
wavelet with symmetry can effectively avoid phase distortion
in image processing. Therefore, researchers generalized the
2-band wavelet and obtained several significant branches,
such as the biorthogonal wavelet [38], the vector wavelet [39],
and the M-band wavelet [40]. The properties of biorthogonal
wavelets are similar to those of orthogonal wavelets, but
they can be completely symmetric. The properties discussed
above are still applicable to biorthogonal wavelets. Generally,
a wavelet with a high vanishing moment is used for decom-
position, and then another wavelet is used for reconstruction.
In the continuous wavelet transform (CWT), the wavelet
family is obtained from the base wavelet through expansion
and translation. At the same time, CWT has the characteris-
tics of being unchanged after translation and the characteristic
of changing together after expansion, so the CWT coefficients
have a certain degree of correlation. In other words, the
wavelet transform coefficients corresponding to two adjacent
points in the time-scale plane are correlated. The closer the
two points are, the stronger the correlation is. As the distance
between the two points increases, their correlation weakens
rapidly. It means that there is data redundancy in the CWT
of the signal, which increases the difficulty of analyzing and
interpreting the results of the wavelet transform [26].
Although the discrete wavelet transform (DWT) can effec-
tively capture the singularity of one-dimensional signals, it is
not the case in two-dimensional cases. The two-dimensional
orthogonal wavelet base is formed by the tensor product
of two one-dimensional orthogonal wavelet bases, and its
direction selectivity is very limited. Only the horizontal, ver-
tical, and diagonal two-dimensional DWT can not effectively
represent the contour and edge information of the image; that
is, DWT is not a sparse representation of the contour and edge
of the image. Its performance in image denoising, texture
classification, and image retrieval is lower than that of multi-
directional wavelet [41].
C. WAVELET BASES
1) COMMON BASIC WAVELET BASES
Haar wavelet is a step function and the earliest discovered
wavelet with the simplest form. Its expression is shown in
Equation (3). Numerous wavelet theory books also start from
the introduction of Haar wavelet [42]–[46] for the reader to
study. Due to its simple and convenient nature, a large number
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of works have adopted the Haar wavelet to achieve their goals.
9(t)=
1,0≤t<1
2
−1,1
2≤t<1
0,Else
(3)
Daubechies wavelet was constructed in 1987. The most
significant advantage of Daubechies wavelet is that it can be
realized by finite impulse response conjugate mirror filter.
Both Haar and Daubechies wavelets are orthogonal wavelets.
Daubechies wavelets are generally abbreviated as dbN, where
Nrepresents the order of the wavelet. dbN has no closed-form
expression. Daubechies wavelet has better regularity. As the
order Nincreases, the vanishing moment of the wavelet
is larger, and localization ability in the frequency domain
is stronger; but at the same time, the amount of calcula-
tion is larger. In order to have better symmetry, Daubechies
improved the wavelet system and constructed the Symlets
wavelet, and Coiflets wavelet [25]. Coiflet has better sym-
metry than dbN. Symlet wavelet function is an approximately
symmetric wavelet function, which improves the dbN func-
tion. Compared with dbN wavelet, they are consistent with
dbN wavelet in terms of continuity, support length, and filter
length, but Symlet and Coiflet have better symmetry; they can
reduce the phase of signal analysis and reconstruction to a
certain extent distortion.
Biorthogonal wavelet is conducive to signal reconstruction
and can accurately reconstruct the signal through a finite
impulse response filter (FIR). Biorthogonal wavelets are able
to have tight support, high vanishing moments and symmetry
in the meantime. Its construction methods are able to be
roughly separated into two categories: spectral decomposi-
tion and lifting schemes. Many researchers have proposed
different methods of constructing biorthogonal wavelets.
Bhatnagar [47] explained the biorthogonal representation
of functions. Biorthogonal wavelets are a generalization of
orthogonal wavelets. Therefore, there are more degrees of
freedom in designing biorthogonal wavelets. Reference [30]
introduced Catmull-Clark subdivision surface and combined
it with biorthogonal wavelets. They applied this novel wavelet
to noise suppression, data compression, and other applica-
tions and achieved better results. In [48], researchers used the
homotropy method instead of Newton’s method to construct
a biorthogonal wavelet, which enlarged the selective range of
the biorthogonal wavelet.
Coffey and Etter introduced internalized MRA and con-
structed biorthogonal wavelet based on the bounded domain
efficiently [49]. Tay and Lin proposed a technique for con-
structing biorthogonal wavelets with rational coefficients.
This wavelet has a linear phase and also has very similar
properties to quadrature filters [50]. In [51], an algorithm
for estimating rotor displacement of a magnetic bearing
motor based on a multi-resolution filter bank biorthogonal
spline wavelet is proposed. The algorithm utilizes biorthog-
onal spline wavelets with generalized linear phase and tight
support characteristics, which can accurately demodulate the
ripple current in the coil and extract the displacement infor-
mation. Nagare et al. [52] proposed a new half-band polyno-
mial with rational coefficients using Bernoulli polynomials to
design biorthogonal filter banks. Singh and Pathak construct
biorthogonal wavelet packets in the Sobolev space Hs(K) on
the local positive eigenfield and derive their biorthogonality
at each layer by Fourier transform [53].
Meyer wavelet is different from the previous wavelets.
It is defined in the frequency domain [26]. Although it has
an analytical form, it does not have a compact support set.
Meyer wavelet, therefore, has no fast discrete wavelet trans-
form algorithm. An FIR filter can be used to construct a
filter matrix to approximate and simulate Meyer wavelet
transform. In [54], the potential problem of contaminated
data is handled by a regularization scheme based on Meyer
wavelets. The regularization solution is recovered by Meyer
wavelet projection of the Meyer MRA elements. Lee and
Ryu also suggested that the OFDM system using Dmey is
the most similar to traditional OFDM but solves the disad-
vantages of the traditional Discrete Fourier transform-OFDM
system [55], [56]. A novel fractional Meyer neuroevolution-
based intelligent computational solver is proposed in [57] for
numerical processing of bi-singular multi-fractional Lane-
Emden systems using a combination of Meyer WNNs.
Regimanu et al. [58] used a multi-resolution wavelet trans-
form technique to remove dithered signals. The five-level
multi-resolution analysis uses various wavelet types such as
discrete Meyer wavelets (Dmey) and Daubechies wavelets.
The dithered signal is attenuated by 107.0 dB, and the phase
characteristic is found to be linear in the passband, with lower
computational complexity. In [59], Sabir and his team pro-
pose a novel stochastic computational framework based on
fractional Meyer wavelet artificial neural networks, designed
for nonlinear singular fractional Lane-Emden differential
equations. The statistical results verify the model’s superi-
ority in solving singular nonlinear fractional-order systems.
Fig. 3shows some example wavelet functions about Haar
wavelet, Daubechies wavelet, Biorthogonal wavelet, Meyer
wavelet, Symlets wavelet and Coiflets wavelet.
2) THE DEVELOPMENT OF WAVELET BASE
Classical wavelet has the advantages of multi-resolution anal-
ysis structure and time-frequency localization. However, this
advantage is only applied in signal processing and cannot be
generalized to two-dimensional or even higher dimensions.
In order to make up for this shortcoming while retaining the
advantages of wavelet analysis, Daubechies and Mallat have
constructed numerous new wavelet systems based on classic
wavelets, each with its own characteristics.
In order to analyze high-quality audio and speech, a non-
uniform frequency domain representation is required [28].
Rational wavelets can provide non-uniform frequency parti-
tions of the signal spectrum and further improve flexibility.
They can also provide greater flexibility and higher time-
frequency analysis accuracy for WNN design. Chertov and
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FIGURE 3. Various Wavelet functions and their frequency responses: (a) Haar wavelet; (b) Daubechies wavelet family; (c) Biorthogonal
wavelet family; (d) Meyer wavelet; (e) Symlets wavelet family; (f) Coiflets wavelet family.
Malchykov verified that the perfect reconstruction condition
is satisfied for the reducible fraction as the dilation factor by
an example [60]. Ansari and Gupta used lifting scheme to
design a rational learning wavelet, which also extends dyadic
to rational wavelet [24]. It owns all the lifting framework’s
advantages and has better results when applied in compressed
sensing reconstruction of signals.
Another novel wavelet is the fractional wavelet. Fractional
wavelets extend classical wavelets and are suitable for higher
dimensions due to their low memory. Tausif, Jain, Khan and
Hasan designed two-type architectures of Fractional wavelet
filter (FrWF) with 5/3 filter bank: with multiplier and without
a multiplier, and it required less memory than existing archi-
tecture [61], [62]. Tausif et al. [63] proposed a segmented
modified FrWF to reduce the high time complexities of DWT
and FrWF, about 16.8% and 53.6%, respectively, and has
about 65% lower energy consumption than traditional FrWF
for high-resolution images. In [64], Liu et al. combined frac-
tional wavelets and a scattering network and constructed a
fractional scattering network to obtain improved signal and
image classification performance. Shi et al. studied the sam-
pling theorem for fractional wavelet transform and discussed
sampling and aliasing errors estimating [65]. A hybrid fractal
wavelet coder is proposed in [66]. It has the advantages
of wavelet transform and achieves significantly improved
image quality without obviously blur. In [67], the authors
combined fractional wavelet and biorthogonal wavelet and
defined fractional MRA. They constructed the necessary and
sufficient conditions for translation of the wavelet to form
a fractional Riesz basis. Reference [68] discussed fractional
spline wavelets and verified that it is more effective than
traditional wavelets in the texture recognition of the surface
texture of machine parts.
When the selected wavelet function has a complex domain
instead of a real one, the wavelet is defined as a com-
plex wavelet. Complex wavelet has the form ψc(t)=
ψreal(t)+jψimage (t). The real part ψreal (t) and imaginary
part ψimage(t) of most complex wavelets is a Hilbert trans-
form pair. Fernandes et al. [69] combined a mapping filter
and an inverse mapping filter with a complex wavelet and
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constructed a mapping-based complex wavelet transform.
This wavelet transform has both directivity and is non-
redundant. Toda and Zhang and some other researchers [70],
[71] proposed a series of complex wavelet transforms hav-
ing perfect translation invariance based on different meth-
ods: Hilbert transform pair [70], and 3-dilation orthogonal
basis with perfect translation invariance [71]. They also con-
structed a tight wavelet frame based on a designed complex
wavelet [72] in the frequency domain. In [73], the com-
plex wavelet packet energy moment entropy is defined as a
new monitoring index to characterize bearing performance
degradation.
There are some special wavelets. The widely applied non-
orthogonal wavelets are Gaussian Wavelet, Morlet Wavelet,
and Mexican Hat Wavelet [74]. Orthogonal wavelet func-
tion is generally used for discrete wavelet transform; non-
orthogonal wavelet function can be used for discrete wavelet
transform or continuous wavelet transform [75]. Gaussian
wavelet is the first derivative of Gaussian function; its expres-
sion is ψ(t)= −te−0.5t2. Mexican hat wavelet is the sec-
ond derivative of Gaussian function; express as ψ(t)=
(1 −t2)e−0.5t2detector for finding a Gaussian noise is the
Mexican Hat wavelet [76]. Mexican hat is real-valued and
captures both the positive and negative oscillations of the time
series as separate peaks in wavelet power [20]. In addition,
to obtain information on both the amplitude and phase of
the time series, it is necessary to choose a complex wavelet
because the complex wavelet has an imaginary part, which
can express the phase well. Morlet wavelet (express as ψ(t)=
cos(5t)e−0.5t2) is simply a complex wave within a Gaus-
sian envelope. Reference [25] considered several different
test signals, such as noise, phase shift, bump and a slight
spike, to test the performance of Morlet wavelet with differ-
ent parameters. Morlet wavelet has a good balance between
the localization of time and frequency. Fig. 4is the wave-
form of the three non-orthogonal wavelets. Table 1shows
the properties of several standard wavelet bases introduced
above.
Novel wavelets could also be constructed by combin-
ing different properties of different wavelets. Wen et al. [77]
studied the decomposition and reconstruction orthogonal
rational wavelet filter bank with dilation factor M=3
2.
They constructed high-pass filter banks from low-pass filter
banks and gave a perfect reconstruction method. In [78],
Li also proved the condition of the perfect reconstruction for
the orthonormal wavelet bases with rational dilation factor
M=p
qand gave two examples of orthogonal wavelet
bases to verify the perfect reconstruction. Yu and her team
constructed a wavelet that combined complex wavelet, ratio-
nal wavelet and orthogonal wavelet [79]. It achieves better
robustness in broadband sonar pulse than linear frequency
modulated pulse-based system [80], [81], and has well sys-
tem performance against Doppler effect and inter-symbol
interference caused by multipath while reducing channel
noise [82].
3) EXPANSION OF WAVELET BASE IN DIMENSIONS AND
SCALES
One of the wavelet extension directions is higher-dimensional
wavelets. At present, two-dimensional wavelet analysis has
made significant progress both in theory and application
(mostly in image processing). In [83], Rinoshika designed a
three-dimensional orthogonal wavelet based on Daubechies
wavelet and analyzed instantaneous 3-D velocity fields
of a high-resolution tomographic particle image velocime-
try. At different wavelet decomposition levels, different
vortexes could be extracted. Reference [84] proposed a
three-dimensional discrete wavelet transform for hyperspec-
tral faces feature extraction compared with three existing
hyperspectral face recognition methods and achieved higher
accuracy.
The wavelets discussed previously are all single-scaling
wavelets. Whether it is a classic wavelet or a newly designed
wavelet, the wavelet function is constructed by a single scal-
ing function. In signal processing, whether the wavelet has
properties such as compact support, symmetry, orthogonality,
and the vanishing moment is essential. However, it is not
easy for a single-scaling wavelet to have these properties
at the same time. Multi-wavelet means that multiple scaling
functions complete the construction of wavelet functions. The
construction of multi-wavelets can usually be transformed
into the solution of vector filter matrix coefficients. Com-
pared to single-scaling wavelets, multi-wavelets have supe-
rior properties such as symmetry, regularity, and vanishing
moments in the compactly supported range, so they have
received extensive attention in the field of signal processing.
Reference [85] uses the Optimized multi-wavelet trans-
form of electroencephalography (EEG) signals for the clas-
sification of eye movements of humans and achieves higher
accuracy when there are different movements and blinking.
In [86], multi-wavelet transform is applied on mechanical
features extraction of on-load tap-changer and achieve a
better result in fault detection. Both the authors in [87]
and [88] combined multi-wavelet and neural networks and
then achieved better results in their research fields. In [88], its
approximate performance is far better than that of some clas-
sical algorithms, even in algorithms that use mother wavelets.
In [87], this method can effectively expand the data set and
build a CNN model through experiments and has good robust-
ness to noise, misalignment, and different numbers of training
samples of the same type.
IV. CLASSIFICATION OF WAVELET-BASED SIGNAL SPACE
DECOMPOSITION
A. DISCRETE WAVELET TRANSFORM
Wavelet transform has the characteristics of multi-resolution
analysis and can characterize the local characteristics of
the signal in both the time and frequency domains. This
method performs multiscale analysis on the signals through
calculation functions such as expansion and translation.
Compared with the Fourier transform, it is able to provide
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FIGURE 4. Non-orthogonal wavelet functions and their spectrums: (a) Gaussian wavelet; (b) Mexican Hat wavelet; (c) Morlet wavelet.
a ‘‘time-frequency’’ window that changes with frequency.
It can also fully highlight certain aspects of the signals. DWT
is the most basic and most widely used wavelet transform,
which is implemented by a two-channel filter bank with
different levels. DWT is obtained by discretizing the scale
and displacement of continuous wavelet transform according
to the power of 2, so it is also called dyadic wavelet transform.
For many signals, the low-frequency component is essential,
it contains the characteristics of the signal in many cases,
and the high-frequency component gives the details or differ-
ences of the signal. In DWT decomposition, low-frequency
information represents the high-scale of the signal, which is
an approximation of the signal; high-frequency information
represents the high-scale of the signal, which is the detail of
the signal. Therefore, the original signal passes through two
mutual filters to produce two signals. The approximate signal
is continuously decomposed through the continuous decom-
position process, and the signal can be decomposed into
many low-resolution components. Theoretically, the decom-
position can proceed without limit. In practical applications,
the appropriate number of decomposition layers is gener-
ally selected according to the characteristics of the signal or
appropriate standards.
The DWT of the signal is not directly realized by the inner
product between signals and ψ(t) (the wavelet function) and
φ(t) (the scaling function), but by using high-pass filter h[n]
and low-pass filter g[n]. It regards the wavelet coefficients
cj[k] and dj[k] of the signal as discrete signals, and h[n]
and g[n] as digital filters, thereby establishing the wavelet
transform and filter bank. The filter bank theory realizes
the relationship between the signal wavelet analysis. Most
research involving wavelets will introduce wavelet analysis
into the design of filter banks. A particular wavelet filter
bank can be designed according to the processing object. The
structure of DWT is shown in Fig. 5(a). Fig. 5introduces
the wavelet decomposition structure of different wavelet
transforms, which are DWT, discrete wavelet packet trans-
form (DWPT), dual-tree complex WT (DT-CWT), stationary
wavelet transform (SWT). DWT also use a downsampling
filter after the high-pass filter and low-pass filter. Assume
the original signal x[n], the ith level coefficients could be
calculated as:
xi,h[n]=
K−1
X
k=0
xi−1,h[2n−k]h[k] (4)
xi,g[n]=
K−1
X
k=0
xi−1,g[2n−k]g[k] (5)
where Kis the length of the filters, h[n] and g[n] are high-pass
filter and low-pass filter, respectively.
SWT (stationary wavelet transform) is a DWT with no
down-sampling. The structure of SWT is shown in Fig. 5(b)
Zheng and his team [89] chose the basic wavelet as Haar
wavelet and applied SWT on heart-rate monitoring. They
achieved very high accuracy when estimating heart rate in
the driving scenario. The results of different wavelets and
different decomposition levels are compared from the three
aspects of accuracy, sensitivity, and specificity of EEG signal
classification and found that deeper levels may have bet-
ter accuracy in EEG data classification using SWT in [90].
In [91], undecimated wavelet transform (UWT) is used in
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TABLE 1. The properties, advantages, disadvantages and applications of common wavelet bases.
order to ensure the shift insensitivity property of the coeffi-
cients for time series prediction. The core idea of UWT is to
remove the down-sampling in the sampling wavelet transform
and replace it with the up-sampling of the filter.
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FIGURE 5. Different structure of wavelet transform: (a) DWT; (b) SWT; (c) DWPT; (d) DT-CWT.
FIGURE 6. Wavelet decomposition tree: (a) DWT; (b) WP.
B. EXTENSION OF DISCRETE WAVELET TRANSFORM
1) WAVELET PACKET
In the process of decomposition, wavelet analysis only re-
decomposes low-frequency signals and does not decompose
high-frequency signals. Therefore, its frequency resolution
decreases as the frequency increases. Wickerhauser and other
researchers proposed the concept of wavelet packet [92].
The wavelet packet (WP) decomposes the low-pass and
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high-pass components of the signal frequency band simulta-
neously to locate any frequency band. Fig. 6is the wavelet
decomposition tree of DWT and WP. It presents the con-
cept of optimal basis selection based on wavelet analysis
theory. Many researchers designed different wavelet packet
bases. [73] combined complex wavelet and wavelet packet
energy moment entropy and defined it as a new monitor-
ing index to characterize bearing performance degradation.
In [93], the combination of wavelet packets and genetic pro-
gramming significantly improves prediction accuracy.
Islam et al. [94] introduced a particular wavelet packet
called perceptual wavelet packet to enhance speech signals.
This method resulted in better spectrogram output and higher
scores in subjective listening tests. In [95], defects are char-
acterized by the power of low-frequency components and
wavelet packet energy after wavelet decomposition, complex
signals are analyzed, and defect features are extracted. [96]
combining wavelet packets and CNNs to classify sound sig-
nals from excitation-induced extraction of wavelet packet
decomposition (WPD) features. Liu and his team proposed an
improved wavelet packet denoising algorithm, which deter-
mines the optimal decomposition layer according to the dif-
ference in the correlation function values of the wavelet
packet coefficients [97]. In addition, the wavelet packet
coefficients are divided into the approximate part, blur part
and detail part. Singular spectrum analysis, fuzzy threshold
and correlation analysis are carried out on the selection of
these three different types of coefficients to preserve the
dynamic performance of chaotic signals to the greatest extent.
An energy analysis method based on wavelet packet is pro-
posed in [98]. This method is used to calculate the wavelet
packet energy index of the ground-penetrating radar signal of
clay samples with water content. The results show that there
is a highly correlated linear relationship between WPEI and
soil water content, and the relationship between the two fits a
linear fitting function.
Fig. 5(c) shows the discrete wavelet packet transform
(DWPT). 2D-DWT has three priority directions: horizontal,
vertical and diagonal. Due to the supplementary decomposi-
tion of the output of the high-pass filter, 2D-DWPT has higher
directional selectivity [99]. The decorrelation property is
closely related to the shape of the Fourier transform that sup-
ports the width and wavelet packet function [100] addressed
the DWPT for continuous-time fBm and considered station-
arization and asymptotic decorrelation. They also studied the
influence of fBm with or without independent white Gaussian
noise on selecting the best wavelet packet basis. Khaleel and
his team proposed an adaptive neuro-fuzzy method based on
a discrete packet wavelet transform-Kalman filter for power
quality identification and classification [101]. [102] proposed
a flexible architecture that computes generalized wavelet
packet trees with the help of boost-based bypass wavelet
filters and bit-swapping circuits. An enhanced fault detec-
tion method combining maximum overlap discrete wavelet
packet transform and Teager energy adaptive spectral kurtosis
denoising algorithm to identify weak periodic pulses is pro-
posed in [103].
2) COMPLEX WAVELET TRANSFORM
Complex Wavelet Transform (complex WT) is a complex
extension of DWT. Remenyi et al. [104] defined the complex
maximal overlap scale mixing 2D complex WT and applied
it to image denoising. Their method achieved excellent visual
performance. Fernandes et al. constructed a new framework
of complex WT and provided a mapping-based and non-
redundant complex WT [69]. The new framework based on
the mapping of complex WT overcomes the serious short-
comings of DWT and has the benefits of controllable redun-
dancy and flexibility. In [105], Xu et al. applied complex WT
to mitigate noise in gas-insulated switchgear signals. They
achieved better noise filtering results by extracting two kinds
of the information-the real part of the wavelet analysis and the
imaginary part of the wavelet analysis. One important design
of complex WT is dual-tree complex WT (DT-CWT). Its
implementation uses two real-valued DWTs, one giving the
real part of the transform coefficient and one giving the imag-
inary part. Its advantage is that it has better directionality in
two dimensions or even higher dimensions, low redundancy,
and is an effective, fast calculation algorithm. Kingsbury [11]
first proposed the DT-CWT structure in 1998. Fig. 5(d) is the
filter bank structure of DT-CWT.
As introduced before, DT-CWT has advantages in two
dimensions or even higher dimensions signals, so it is widely
used in image processing. Fahmy et al. [106] used DT-CWT
on the video magnification techniques and introduced a new
and accurate method of orthogonal filter design for construct-
ing the DT-CWT system. They modify the phase differences
between the wavelet coefficients and achieve better video
quality with less calculation cost. Farhadiani et al. [107] pro-
posed a new method to reduce the speckles on synthetic
aperture radar images based on an undecimated DT-CWT
and achieve better performance. However, this method con-
sumes more computational cost. [108] obtained a neural net-
work dataset using chest X-ray images and subband images
obtained by applying a DT-CWT to the above images.
Prashar et al. [109] evaluated in detail the impact of thresh-
old, threshold algorithm and distribution function choice on
the performance of ECG denoising with DT-CWT. [110]
proposed a method using global-based DT-CWT for kinship
recognition on similar full-face images. Then, the researchers
proposed novel patch-based kinship recognition methods for
DT-CWT: local patch-based DT-CWT and selective patch-
based DT-CWT. The former extracts the coefficients of
smaller face patches for kinship identification. The latter
extends the former, only extracting the coefficients of rep-
resentative blocks with similarity scores above the normal-
ized accumulation threshold. All the references above focus
on image processing, which is 2-D DT-CWT. In [111], the
authors extended DT-CWT to higher dimensional (e.g., 3-D)
and studied the power spectral density of the real and
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FIGURE 7. Rational branch.
imaginary parts of the complex coefficients of the DT-CWT.
They achieve more accurate results in the wavelet noise fil-
tering area.
C. RATIONAL WAVELET TRANSFORM
The traditional wavelet transforms are dyadic wavelet trans-
forms. Its iterative decomposition process repeatedly divides
the frequency domain space at the input into two parts with
equal bandwidth. The concept of rational multi-resolution
analysis was first proposed by Auscher [112] and then sys-
tematically introduced by Mallat. Auscher [112] proved in
1992 that real rational orthogonal wavelets were derived
under the framework of a rational MRA. Assume M=
p
q(p,q∈ZandM >1),{Vj}j∈Zis a subspace of L2(R) space,
consider {Vj}j∈Zis a rational MRA of L2(R) space when:
•Vj⊂Vj−1
•T
j∈Z
Vj= {0},S
j∈Z
Vj=L2(R)
•f(t)∈Vj⇔f(M−1t)∈Vj+1
•f(t)∈Vj⇒f(t−k)∈Vjfor all k∈Z
The explanation for the expression is similar to MRA
above, but the expansion factor is different. Original MRA
is dyadic, and rational MRA contains rational factor M.
The orthogonal basis of Vjis constructed by extending and
translating the mother wavelet function ψ(t)∈L2(R). It is
called the scaling function. The basis function of Vjis given
by [113]:
ψj,n(t)=M−j/2(M−jt−n),t,n∈Z(6)
In [114], Kovacevic constructed perfect reconstruction
filter banks with rational sampling factors. The perfect
reconstruction filter bank theory is generalized to a ratio-
nal situation, thereby allowing non-uniform division of the
frequency spectrum. This feature may be helpful in speech
and music analysis. Reference [113] reviewed the theory of
rational MRA, proposed a pyramid algorithm for calculating
fast orthogonal wavelet transform, and explained the analysis
process and synthesis part in detail. It proposes the applica-
tion of signal denoising through a rational wavelet to show
that the scale factor matches the signal information better.
Fig. 7is the rational part of the filter bank.
The Q factor (Q factor is the Quality Factor, defined as
the filter centre frequency to bandwidth ratio) of the wavelet
transform should be selected reasonably according to the
oscillation behaviour of the signal [115]. For example, the
wavelet transform should have a relatively high Q factor
when using wavelets to process and analyze oscillating sig-
nals (such as speech and EEG signals). However, in addition
to continuous wavelet transforms, most wavelet transforms
FIGURE 8. Rational filter bank where p0=q−p.
have poor tuning capabilities for wavelet Q-factors. The Q
factor is constant and low in the dyadic wavelet [37], [115].
In this transformation model, the bandwidth of the band-
pass filter in the higher frequency domain space is wider,
resulting in a sparse partition of the higher frequency domain
space. Therefore, this conversion mode is suitable for signals
with fewer oscillation characteristics but not for signals with
significant oscillation characteristics [115]. Compared with
the traditional dyadic and integer wavelet transform, the Q
factor of the rational wavelet transform (RWT) is adjustable,
which can realize more free and fine frequency domain seg-
mentation [37], [115]. However, the local performance in the
time domain is relatively weak.
In [116], Bayram and Selesnick introduced a filter
bank with a rational q/psampling factor based on [117].
Reference [117] designed an orthogonal rational filter bank,
and it is close to wavelet transform. Fig. 8is a two-
band rational filter bank example, which is defined as
rational-dilation discrete wavelet transform (RADWT) [118].
In [118], Bayram and Selesnick also designed overcomplete
RWT, which is composed of a self-reverse HIR filter based on
the rational sampling factor, so it realizes the reconstruction
of the decomposed signal and has translation invariance.
The Q-factor can be controlled by changing pand qin
Fig. 8. Han et al. [119] design rational coefficients biorthog-
onal wavelet filters by the thought of complete reconstruc-
tion filter idea and adding vanishing moment characteristics.
By reducing the vanishing moment of the wavelet filter, more
high-frequency information can be retained in the wavelet
transform domain, which is suitable for edge detection. The
simulation results show that image edge detection under a
noisy environment has achieved some significant effects.
Fig. 9shows the structure of DT-RADWT. The structure is
based on Fig. 5(d) and Fig. 8. In [120], Canditiis and his
team use a complete filter bank (i.e. RADWT) to guarantee a
perfect reconstruction property and a tunable Q-factor.
To illustrate the difference between DWT filter bank and
RWT filter bank, the example complex rational orthogonal
wavelet (CROW) constructed in [79] is shown below. Dilation
factor a=1+1
q. The wavelet basis function is defined in the
frequency domain by (7):
9(ω)=
(2π)−1
2ejω
2
×sin(π
2β(q
ω1
|ω| −q)), ω1≤| ω|≤ ω2
(2π)−1
2ejω
2
×cos(π
2β(q
ω2
|ω| −q)), ω2≤| ω|≤ ω3
0,|ω|/∈[ω1, ω3]
(7)
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FIGURE 9. The structure of DT-RADWT.
FIGURE 10. Wavelet function ψ(t) with dilation factor of a=3/2,q=2.
And the rational scaling function is given also in the fre-
quency domain by:
8(ω)=
(2π)−1
2,|ω|< ω1
(2π)−1
2
×cos(π
2β(q
ω1
|ω| −q)), ω1≤| ω|≤ ω2
0,|ω|< ω2
(8)
where
ω1=(q−q
2q+1)π, ω2=aω1, ω3=aω2=a2ω1(9)
β(t) is the construction function and has the form in (10).
It is not unique.
β(t)=t4(35 −84t+70t2−20t3) (10)
Based on these definitions above, the CROW function is:
ψ+(t)=ψ(t)+jˆ
ψ(t) (11)
where ˆ
ψ(t) is the Hilbert transform pair of ψ(t) and defined
in the frequency domain.
ˆ
8(ω)= −jsign(ω)8(ω) (12)
Fig. 10 shows the time and frequency response of the
wavelet ψ(t) with an example dilation factor a=3/2,q=2.
Fig. 11 shows the frequency response of DWT filter bank and
RWT filter bank. RWT has a better frequency resolution than
DWT. As qincreases, the dilation factor gradually decreases
and is close to 1, and the frequency resolution of the filter
bank is better.
Reference [121] proposed a high-accuracy general rational
approximation model of Gaussian wavelet series in the time
domain. The proposed wavelet basis approximation model
can be extended to any order and wavelet function without
explicit formulation. In 2009, Selesnick and Bayram [115],
[118] constructed an over-complete fractional wavelet trans-
form method, which is different from the early critical sam-
pling mode. It allows a small amount of redundancy to
improve local performance in both the time and frequency
domains. If the signal and noise have strong time-frequency
coupling, that is, the distribution of signal and noise overlaps
on the time axis or frequency axis, it is challenging to design
a reasonable filter.
Fractional wavelet transform (FrWT) has flourished with
the further development of wavelet technology. As intro-
duced above, fractional wavelet is a novel wavelet system.
FrWT extends the wavelet transform to the time domain-
generalized frequency domain (fractional Fourier domain),
which has greater signal analysis and processing flexibility.
Mendlovic et al. [122] first introduced FrWT and suggested
that the FrWT may be used for image compression since
it improved the reconstruction performance of the wavelet
transformation. Reference [61] proposed a fractional wavelet
filter and compared it to state-of-the-art low memory DWT,
which showed that it has better performance. The architecture
proposed by FrWF, which uses filter banks to calculate the
two-dimensional DWT coefficients of images, requires less
memory and fewer hardware components.
In [64], Liu’s team introduced FrWT and designed a
scattering network based on FrWT. They extended the tra-
ditional scattering network with fractional coefficients and
achieved higher image classification accuracy. The authors
in [123] also considered combining FrWT and neural net-
work. FrWT is treated as a set of linear translation variable
multiscale filters. They defined fractional wavelet scatter-
ing transform based on it and validated it with computer
simulations. In [124], Fan’team detailed the construction of
FrWT and applied it to signal denoising. They proposed
a two-dimensional search method to determine the optimal
order of the fractional wavelet transform and verified the
effectiveness and superiority of the method. For an exam-
ple of denoising in sine signal, SNR can be increased by
about 40%, and RMSE can be reduced by about 50% when
applying FrWT. Shi et al. [65] extended the sampling the-
orem based on the FrWT subspace and discussed some
applications of exporting results. Kumar and Naik combined
compressive sensing and FrWT and ensured the security of
picture transmission [125]. In [126], the authors proposed
the definitions and properties of a novel designed FrWT to
overcome the limitations of some existing wavelet transform
and FrWT.
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FIGURE 11. Frequency response of DWT filter bank with: (a) Dyadic DWT; (b) RWT with dilation factor q=2; (c) RWT with dilation factor q=5; (d) RWT
with dilation factor q=50.
In recent years, attention to wavelet analysis has increased
day by day, and many researchers have published literature on
wavelet analysis. Searching for literature on wavelet-related
topics on the ’Web of Science’ website (the statistics are as
of the end of 2021), a total of 5972 samples were obtained.
Related keywords in these documents include fractional
wavelet (1734 articles), dyadic wavelet (702 articles), orthog-
onal wavelet (3239 articles) and rational wavelet (297 arti-
cles). The time periods include the period from 1990 to 2021.
For the convenience of statistics, the periods are divided into
five parts: 1990-1996, 1997-2003, 2004-2009, 2010-2015,
and 2016 to the present. Statistics may contain duplicate
documents because designing wavelets with multiple charac-
teristics is more in line with research needs to apply wavelet
analysis better. Fig. 12 shows the statistics about publications
in recent years. It can be seen from the figure that orthogonal
wavelets are always published with the largest number. The
reason is that orthogonal wavelets reduce the correlation of
sub-band data and reduce redundancy. When wavelet theory
first developed, there were more dyadic wavelets; however,
the proportion of research on rational wavelets and fractional
wavelets has increased with the development of wavelet
analysis. In recent years, the number of publications has far
exceeded dyadic wavelets.
V. WAVELET NEURAL NETWORK
Wavelet Neural Network (WNN) integrates the advantages
of artificial neural networks and wavelet analysis, which
makes the network converge fast and has the characteristics
of time-frequency local analysis. Searching for literature on
WNN-related topics on the ‘Web of Science’ website (the
statistics are as of the end of 2021), a total of 10990 samples
were obtained. The statistic is shown in Fig. 13. In recent
years, the research on WNN has shown a blowout type
development.
FIGURE 12. Research statistics of different wavelets.
There are two primary forms of WNN. In the first one,
wavelet analysis performs preliminary processing on the
input of the neural network, making the information input
to the neural network more effortless for the neural network
to process. The decomposed signal obtained by the original
signal through different wavelet decomposition levels will be
used as the input of the neural network. Furthermore, the
features of the decomposed signal could also be extracted
as input. These obtained features could be the maximum,
minimum, average, and deviation value of the decomposed
signals and the amplitude, slope (or gradient) of amplitude,
time of occurrence, mean, standard deviation, and energy of
the signals.
The second approach is the deep fusion of wavelets and
neural networks. There are two ways to integrate. One is
to replace neurons with wavelet elements, replace the acti-
vation function with the positioned wavelet function, and
establish the connection between the wavelet function and
the neural network coefficients through affine transformation.
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FIGURE 13. Research statistics of WNN.
The corresponding weights from the input layer to the hidden
layer and the threshold of the hidden layer are replaced by
the scaling factor and the time shift factor of the wavelet
function [14]. The other is recently proposed to replace the
convolution kernel with wavelet in CNN because the kernel
of CNNs seems like a filter. It provides a very efficient way
to obtain custom filter banks [127].
A. SIGNALS PREPROCESSED BY WAVELET ANALYSIS
The signals are preprocessed by wavelet analysis, which
means the wavelet space is used as the feature space for
pattern recognition. The inner product of the wavelet base
and the signal is weighted to realize the feature extraction of
the signal, and then the extracted feature vector is sent to the
neural network for processing. Reference [128] uses Coiflet
wavelet as an envelope extraction and then chooses the mean
value, standard deviation, peak value, and RMS (root mean
square) value as the features which are the input of PNN
(Probabilistic neural network). They also compared it with
a traditional back-propagation neural network and achieved
better classification accuracy. In [129], wavelet decomposi-
tion is first applied to the signal and then obtains the energy
and the PSD value of the detailed divided signal as the
extracted features for the input of the neural network. Wavelet
decomposition architecture is shown in Fig. 14, which uses
DWT as an example. X[n] is the input signal, G[n] and H[n]
are the lowpass filter and highpass filter. When the wavelet
decomposition level is different, the number of signals after
decomposition is also different. These decomposed signals
have different energy and other characteristics.
Appropriate level numbers need to be selected according
to the specific situation in actual applications. Reference [90]
compared several decomposition levels and found that deeper
levels may have better accuracy in EEG data classification
using SWT. Sun et al. [130] use wavelet packet to decompose
the signal into seven layers and extract the energy of wavelet
coefficients in the seventh layer as the input of the PNN
classifier. They test several different wavelets and finally, the
most effective wavelet is db3. The energy of the 128 nodes
of the seventh layer wavelet coefficients is normalized into
less dimensional eigenvectors to speed up the classification
process. [131] uses wavelet transform to remove noise effects
on images and perform feature extraction for recognition.
On a limited dataset, the algorithm was still able to iden-
tify COVID-19 cases. In [132], the authors introduce syn-
chronous compression wavelet transform to more clearly
represent the intrinsic properties of AE waves in the time-
frequency domain and find that AE waves caused by different
mechanisms exhibit different energy distribution patterns.
Then, a multi-branch convolutional neural network model
with two branches is developed to automatically classify
three types of acoustic emission waves by considering their
simultaneous compressed wavelet transform maps at differ-
ent time-frequency scales.
The authors in [133] and [134] calculated the wavelet
energy spectrum of the signal treated with wavelet
decomposition and a separate reconstruction algorithm.
Shao et al. [133] applied DWT for wavelet decomposition
and calculated wavelet energy of each wavelet coefficient
as wavelet energy spectrum Eas the feature vector, which
includes all Ej. Use Fig. 14(a) as an example:
E=[EA3,ED1,ED2,ED3] (13)
where Ecould be calculated as:
E=
N
X
i=1
|f(i)|2(14)
In [134], Zhang et al. use WP for wavelet decomposition
and strike the energy distribution of the wavelet packet as the
feature vector. Fig. 14(b) is a three-level WP decomposition,
as an example. The total energy of the third level is:
E3=
7
X
n=0
E3,n(15)
and the energy distribution vector is:
E=[E3,0
E3
,E3,1
E3
,E3,2
E3
,E3,3
E3
,E3,4
E3
,E3,5
E3
,E3,6
E3
,E3,7
E3
] (16)
Another possible feature vector x, which represents the
WNN inputs, is shown in Equation (17). These features are
from the original signals and the decomposed signals at dif-
ferent decomposition levels. The most widely used features
are the energy of decomposed signals at different decompo-
sition levels.
x=[M(f),Std(f),RMS(f),Pk(f),E(f)]T(17)
where
Mean Value M(f)=f,(18)
Standard Deviation Std(f)=sPN
i=1(f(i)−f)2
N,(19)
RMS Value RMS (f)=sPN
i=1(f(i))2
N,(20)
Peak Value Pk (f)=max (f(i)),(21)
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FIGURE 14. The structure of wavelet decomposition: (a) DWT; (b) WP.
The wavelet as the preprocessing method of the input
signal of the neural network is similar to most current wavelet
analysis methods for feature extraction. Wavelet transform is
a local transform of time and frequency. It has the character-
istics of multi-resolution analysis, and it can characterize the
local characteristics of the signal in the time and frequency
domains. Since the wavelet transform can concentrate the
energy of the original signal on a small part of the wavelet
coefficients, and the decomposed wavelet coefficients have
a high degree of local correlation in the detail components,
this provides a decisive condition for feature extraction. The
use of wavelet transform for feature extraction has been
widely used in texture analysis, image compression, and
defect detection. The neural network has the characteristics of
self-learning, self-adaptation and fault tolerance. Then use the
neural network to classify or predict the extracted features,
and better results can be obtained.
B. COMBINATION OF WAVELET FUNCTION AND NEURAL
NETWORK
1) WAVELET KERNEL-BASED NEURAL NETWORK
Wavelet kernels (WK) are a strong contender for initializ-
ing convolutional neural network kernels because the use of
these kernels produces useful approximations of the signal
after convolution operations [135]. The initialization of ker-
nels in a CNN plays a crucial role in network performance.
Better initialization provides better performance with fewer
training iterations/epochs. The kernel of CNNs seems like
a filter. Therefore, wavelet kernels may be good candidates
for initializing CNN kernels, which are hardly reported in the
existing literature [136]. Wavelet kernels are usually used in
convolutional layers, similar to filters. The kth kernel of the
lth layer before the nonlinear activation has the feature value
hl
kcan be denoted as [127]:
hl
k=wl
k∗x+bl
k(22)
where wl
kis the weight of kth convolutional kernel of the lth
layer and bl
kdenotes the bias. xis the input signal, ∗is the
convolutional operator. The proposed WK performs the con-
volution operation with a predefined wavelet function ψu,v(t)
that depends on tranfer parameters uand scale parameter s
only, the feature value hcan be denoted as:
h=ψu,v(t)∗x(23)
In [137], researchers combined CNN, genetic algorithm
and Extreme Learning Machine with WK to increase the
performance of classification. They investigated several state-
of-art CNN architectures like AlexNet and VGG-19 and
achieved more than 95% accuracy even in 10 classes. [127]
proposed a novel wavelet deep neural network called WKNet,
where a continuous wavelet convolutional layer was designed
to replace the first convolutional layer of standard CNN.
This enables the layer to discover more meaningful filters.
Furthermore, the raw data are directly learned from the
scale and translation parameters. It provides a very effi-
cient way to obtain custom filter banks. Mo et al. [138]
refer to [127] and designed their variational kernel. They
compared their designed kernel with WK, which uses three
different wavelets: the Laplace wavelet, Morlet wavelet, and
Mexican hat wavelet and concluded that if a wrong type of
wavelet kernel (Mexican hat wavelet kernel in this reference)
is selected, it may even reduce the network performance.
In [136], a WK-based CNN is designed for acoustic sensor
data analysis. The proposed network has less training time
than other designed CNN and achieves higher accuracy than
standard CNN. [135] also used the WK-based CNN in [136]
for fault identification and classification and achieved better
performance than some other designed CNN.
2) WAVELET FUNCTION AS ACTIVATION FUNCTION
The basic idea was formally put forward by Zhang et al. [13],
that is, the wavelet function is used to replace the hidden
layer function of the conventional neural network, and the
corresponding input layer to the hidden layer weight and
hidden layer threshold are respectively determined by the
wavelet basis function. The scale parameter and translation
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parameter are used instead [14]. Its basic structure is shown
in Fig. 15, where Xi,(i=1,2,...,L) is the input sample,
9j,(j=1,2,...,M) is the wavelet basis function, Fk,(k=
1,2,...,N) is the output of the network, and Ui,jrepresents
the connection weight between the ith neuron in the input
layer and the jth neuron in the hidden layer, and ωj,krepre-
sents the jth neuron in the hidden layer and the kth neuron in
the output layer.
According to the continuity of the selected wavelet basis
function, the connection weight between the neurons can be
divided into two types: WNN with continuous parameters and
WNN based on wavelet framework [139]. For WNN with
continuous parameters, wavelet function is ψ(t) and bj,aj
are the scale parameter and translation parameter. It comes
from the definition of the continuous wavelet transform. Its
characteristic is that the positioning of the basis function is
not limited to the finite discrete value, the redundancy is high,
the expansion is not unique, and the correspondence between
the wavelet parameters and the function is not fixed. It has
a nonlinear optimization problem similar to the BP network.
However, wavelet analysis theory helps the initialization of
the network and guides the learning process to have a faster
convergence speed. The wavelet function of the hidden layer
is:
ψj(t)=ψt−bj
aj(24)
The output of the simple three-layer WNN in Fig. 15 could
be written as:
Fk=
M
X
j=1
ωj,kψj=
M
X
j=1
ωj,kψ PL
i=1Ui,jXi−bj
aj!,
k=1,2,...,N(25)
For WNN based on a wavelet framework, the theoretical
basis is the wavelet frame (detailed information is in [140]).
However, the wavelet basis under the tight frame is not
necessarily orthogonal and may not have tightly supported
characteristics, representing a certain degree of redundancy in
the estimation. Since the wavelet frame can represent smooth
signals and signals with singular characteristics, the wavelet
frame method has been widely used in signal, image process-
ing, and other fields. The wavelet function in the hidden layer
could be written as:
ψj(t)=ψ(a−mj
0t−njb0) (26)
where a0,b0are the basic units of scaling and translation.
So the output of WNN in Fig. 15 is:
Fk=
M
X
j=1
ωj,kψj
=
M
X
j=1
ωj,kψ(
L
X
i=1
Ui,ja−mj
0Xi−njb0),
k=1,2,...,N(27)
FIGURE 15. A three-layer structure of WNN.
The construction of WNN is a critical issue. Zhang [141]
used regression analysis to give a method for constructing
wavelet networks. He constructed a feed-forward neural net-
work based on WNN structure and discussed that it is suit-
able for neural network construction methods development.
Pati and Krishnaprasad [142] gave two methods of wavelet
network synthesis, which systematically defined the structure
of the network and determined some weight values in the
network in advance, thus simplifying the network training
problem. Reference [143] also proposed a ‘‘decomposition-
synthesis’’ method of wavelet basis function network struc-
ture design, which effectively reduces the wavelet primitives
required to construct wavelet networks.
For the feed-forward network, Stepanov in [74] detailed the
construction of activation functions of WNN and provided the
procedure of choosing proper wavelet models. He concluded
that polynomials, neural networks and spline wavelet models
could be used when constructing the activation function of
WNNs. The spline wavelet model provides the guaranteed
accuracy of the wavelet approximation to the sample, but the
model has a high degree of complexity. In [144], researchers
use a single hidden layer feed-forward WNN. The results
demonstrate the effectiveness and feasibility of the proposed
observer in detecting nonlinear system faults. An example
is shown in Fig. 16.ωi,jare the weights from input to the
wavelet neurons in the hidden layer, and ωψjare the weights
from the wavelet hidden layer to the output layer. ωxiare the
weights of the input connected to the output directly, and θ
is used for nonzero mean functions on finite domains [145].
The output of the feed-forward could be written in (28), where
wavelet function is ψ(x) and aj,bjare the scale parameter and
translation parameter.
F=θ+
M
X
i=1
ωxixi+
N
X
j=1
ωψjψj
=θ+
M
X
i=1
ωxixi+
N
X
j=1
ωψjψ(PM
i=1ωi,jxi−bj
aj
) (28)
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FIGURE 16. The structure of feed-forward WNN.
FIGURE 17. The structure of WNN in [146]: ψkis the kth wavelet function
and Skis the sigmoid function.
Banakar and Azeem [146] combined the feed-forward
network with wavelet functions, where the sigmoid activa-
tion function (SAF) and the Morlet wavelet activation func-
tion (WAF) are paralleled in each neuron model. After the
introduction of the wavelet function, the performance of
the performance calculation method adopted by [146] has
increased by about 20% on different examples. An example
structure of the Sigmoid and Wavelet network is shown in
Fig. 17. Different from previous WNN structure, Wωiand WSi
on Fig. 17 represent the weight values. In order to represent
two sets of parallel activation functions, there should be two
sets from the input layer to the hidden layer. After being
summed separately, they will go through the sigmoid func-
tion and wavelet function of the hidden layer. Fig.18 is the
sigmoid function and Morlet wavelet function. Equation (29)
is the sigmoid function. Different from traditional WNN,
each neuron has two parallel sets of weights. Then the val-
ues calculated by the two sets of weights are respectively
passed through the wavelet function and the sigmoid func-
tion. Finally, they are multiplied to obtain the output of the
neuron of the neural network. The output of kth neuron is
shown in (30).
S(t)=1
1+e−t(29)
Fk=FψkFsk=ψk(
L
X
i=1
WωiXi)×S(
L
X
i=1
WsiXi) (30)
FIGURE 18. Activation functions: (a) Sigmoid function; (b) Morlet wavelet
function.
FIGURE 19. Wavelet layer with self-feedback loop.
Some researchers also combined recurrent neural net-
work (RNN) with WNN. Simple recurrent WNN (RWNN) is
similar to traditional RNN. The value of the hidden layer of
the RNN depends not only on the current input but also on the
last value of the hidden layer. A novel Type-2 Fuzzy RWNN
is proposed to estimate nonlinear systems [147]. This novel
structure has been shown to outperform other conventional
techniques in nonlinear system modelling, with better conver-
gence, lower error, and faster response. Fig. 19 isthe input and
hidden layer of the designed RWNN. Reference [146] also
considered the idea of a recurrent neural network. Since there
are two parallel lines in one neuron, the feedbacks, which
are the outputs of wavelet function and sigmoid function,
could be fed back to themselves and the parallel part. Fig. 20
are the designed RWNN with different feedback positions.
The output from the sigmoid function could be sent to both
the wavelet part and sigmoid part, and the output from the
wavelet function could also be feedback to the wavelet part
and sigmoid part. In addition, based on the design of RNN,
the neural network’s final output could also be sent back to
the input layer.
In [148], it designed a four-layer WNN. Wavelet function
is used in the second hidden layer. It is similar to the structure
in Fig. 15 but still adds a sigmoid function as the first hidden
layer between the wavelet hidden layer. Reference [149] not
only used WP as a signal preprocessed tool but also used a
three-layer WNN for prediction and achieved high accuracy.
In [150]–[152], they all used a four-layer WNN; the two
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FIGURE 20. The structure of RWNN designed in [146]: (a) Wavelet output to wavelet part; (b) Sigmoid output to sigmoid part; (c) Sigmoid output to
wavelet part; (d) Wavelet output to sigmoid part.
layers between the input and output layer are the mother
wavelet layer and wavelet layer. Fig. 21 is an example of this
four-layer WNN. Another important network structure is the
BP network, which is currently one of the most widely used
and most successful neural network models.
In [153], a variable translation WNN is proposed and
compared with other neural networks, which shows a bet-
ter learning probability. The translation parameter of the
mother wavelet in the hidden layer are setting depends on
the input variable and is controlled by a non-linear function.
The combination of wavelet network and fuzzy logic uses
the membership function to express the weight value. The
fuzzy wavelet network model with fuzzy weights and output
is constructed. The authors in [154] and [155] combined
fuzzy Neural Network and WNN and designed a wavelet
fuzzy neural network (WFNN). In [155], each node in the
fifth layer is with a wavelet function. WFNN proved to be
a convergent network. The effectiveness of the proposed
control system has been verified by computer simulation and
experimental results. Huang et al. [154] extended WFNN to
Hybrid WFNN, which is based on PNN. Compared with the
FIGURE 21. A four-layer WNN structure example.
results produced by some well-known and commonly used
fuzzy neural network models, experimental studies involving
three commonly used data sets show some better results. The
RMSE of the best-performing method among their proposed
methods is about 65% higher than the previously proposed
method.
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FIGURE 22. Three widely used wavelet activation function: (a) Gaussian
wavelet; (b) Mexican Hat wavelet; (c) Morlet wavelet.
Similar to the artificial neural network, not only the struc-
ture design of the neural network is a problem, but the
choice of the activation function (mother wavelet) is also a
hot research topic. As introduced above, the hidden layer of
the WNN structure is a scaled and shaped mother wavelet.
Both orthogonal and non-orthogonal wavelet functions could
be applied to the hidden layer. The widely applied non-
orthogonal wavelets are Gaussian Wavelet, Morlet Wavelet,
and Mexican Hat Wavelet [74], which are introduced above.
[150], [152], [155]–[158] all choose the first derivative of the
Gaussian function as the mother wavelet, which is usually
called Gaussian wavelet. [146], [149], [151], [156], [159],
[160] applied Morlet Wavelet function in the hidden layer.
In [144], [147], [153], [156], Mexican Hat Wavelet is chosen
as the mother wavelet. Reference [156] compared differ-
ent mother wavelet activation functions, including Gaus-
sian wavelet, Mexican Hat wavelet and Morlet wavelet. The
experiment results showed that the Gaussian and Morlet
wavelets have better classification accuracy. Fig. 22 is the
waveform in the time domain of the three wavelet activa-
tion functions. Orthogonal wavelet network is more effective
for function approximation due to the orthogonality of its
basic function, but the orthogonal basis structure and network
learning algorithm are more complicated, and the network’s
anti-interference ability is poor.
In [148], Rajankar and Talbar compared several
Daubechies orthogonal wavelets, such as Coiflet Wavelet and
Symlet Wavelet, as the mother wavelet function and found
that db6 has the best MSE (Mean square error) performance.
In [161], Lemarie Meyer wavelet is chosen, which is an
orthonormal function. Their model converges quickly and
obtains low RMS errors, which is a simple three-layer WNN.
Chun and his team [162] used Meyer scaling function as
the activation function. The orthogonal wavelet network
training method is used to determine the number of hidden
layer neurons and the weight of the hidden layer and the
output layer, and the Gray system can compensate for the
characteristics of the ambiguity problem of the orthogonal
wavelet network model. An orthogonal WNN is proposed
in [163]. Both orthogonal scaling functions and the corre-
sponding mother wavelets are used and extended WNN to
the multi-dimensional cases. They designed WNN based on
wavelet framework theory and verified the better function
approximation performance. Table 2shows a brief conclusion
of signal preprocessed by wavelet analysis and different
WNN structures with various WAF (i.e., different mother
wavelets) of reviewed references.
WNN was initially used in function approximation and
speech recognition and then gradually extended to prediction,
classification, image compression and other aspects. WNN
is a neural network constructed based on wavelet transform
theory, which makesfull use of the localized nature of wavelet
transform and the large-scale data-parallel processing and
self-learning capabilities of neural networks. Therefore, it can
accurately identify signals with local singularities, has a
strong approximation ability, faster convergence speed, and
fault tolerance, and its realization process is relatively simple.
Usually, in signal approximation and estimation, the choice
of wavelet function should match the characteristics of the
signal, and the wavelet waveform, supporting length, and the
number of vanishing moments should be considered. The sys-
tem established by WNN identification can approximate the
system’s dynamic characteristics well on the linear model.
WNN has a strong non-linear mapping ability. Its low-
pass filtering effect is good because the wavelet function
has limited support in the time-frequency domain. Therefore,
in terms of function approximation and signal processing,
WNN has received more and more attention from experts.
Generally speaking, the theoretical research of wavelet net-
works is still in the initial stage, and there are still many prob-
lems to be solved so far. For example, the research combines
existing models like emerging neural network models or opti-
mization algorithms, theoretical research on the convergence,
robustness, generalization ability, computational complexity
of wavelet networks and the selection and design criteria of
wavelet base.
VI. REVIEW OF PRACTICAL APPLICATION OF WAVELET
Wavelets can be used in communication, image processing,
signal processing and many other areas. BER and PAPR
(Peak to Average Power Ratio) are two important crite-
ria to judge the performance of wavelet applications like
[164]–[167]. How to reduce or avoid the effect of BER
during the application of wavelets has been discussed and
analyzed in various research. Methods reducing PAPR dur-
ing wavelet’s application are also designed and discussed in
many ways. Lower PAPR is an important index to ensure
higher efficiency of wavelet application. Joint methods,
pilot symbols, wavelet transform and approaches based on
Wavelet Networks are valuable ways to achieve the reduc-
tion of PAPR, and recent studies have focused on these
ways [166], [167]. Krishna et al. [166] introduced DWT in
channel estimation of OFDM system and achieved better
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TABLE 2. Brief conclusion of recent WNN researches.
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PAPR performance than DFT. In [167], Anoh and his team
investigated several mother wavelets in a wavelet-based
OFDM system for PAPR reduction. They found that their
performance of it is often better than traditional wavelets,
especially when adding pilot symbols. Fig. 23 shows the
structure of wavelet applications reviewed in this section.
A. COMMUNICATION SYSTEM
1) TRADITIONAL SIGNAL PROCESSING
There are many typical applications for signal processing.
Classification is a significant branch of signal processing.
Applying wavelet analysis to signal processing can achieve
a better classification effect, which focuses on feature extrac-
tion. The characteristics of the signal can be in the time
domain, such as zero-crossing rate, and short-term energy.
The characteristics of the signal can also be in the frequency
domain, such as the characteristics that contain energy, mean
square frequency and frequency variance.
Reference [168] designed fractional wavelet packet
decomposition for energy entropy calculation to obtain more
information. They investigated several fractional orders to
increase the identification accuracy and compared the pro-
posed method with other existing classifiers, showing their
method’s superiority. The authors in [169], [170] use orthog-
onal wavelets to achieve feature extraction of fault signals
in the power system. In [169], frequency response analysis
signals use orthogonal wavelet filter banks to detect winding
faults. They extracted log energy features after Daubechies
wavelet decomposition and increased classification accuracy.
In [170], Aggarwal and Saini used the criterion of energy-
to-Shannon-entropy ratio to choose the best mother wavelet
to decompose the voltage sag signals. They compared it
with the classic classifier and showed superiority. Wang and
Zhang [171] analyzed wavelet entropy characteristics by
extracting the Shannon entropy of wavelet coefficients and
the correlation dimension of signals as the feature vector
of signal and designed a new method for feature extraction.
The classification results showed that the combined features
have better performance. The authors in [172] focus on the
spectrum characteristics of healthy and faulty parts of signals
and are then reconstructed with RADWT to analyze. The
energy possessed by the RADWT processed signal is used
to estimate the torque.
For identification, in [173], system identification of linear
time-invariant systems is studied and has better performance
with wavelets. They focused on Basis Pursuit identification
using a rational wavelet basis and compared it with the
existing method of adaptive Fourier decomposition; the per-
formance is comparable. In [174], Ma and his team studied
the spectral identification of Fusarium head blight by apply-
ing continuous wavelet analysis to the reflectance spectra
of wheat ears. This model performance suggests that spec-
tral signatures obtained using CWA can potentially reflect
Fusarium head blight infestation in winter wheat ears. The
researchers in [175] proposed a DTCWT-based method to
extract sensor pattern noise from a given image, which
achieved better performance in regions around strong edges.
Authors in [151], [155] tested the identification ability of dif-
ferent designed WNNs. Reference [155] designed a wavelet
fuzzy neural network for identifying and controlling non-
linear dynamic systems. Computer simulations have verified
the effectiveness of the proposed control system. Khan’s team
attempted to use a new kind of wavelet-based self-tuned
wavelet controller for IPM motor drives which has already
been implemented using the MATLAB/Simulink software
and the dSPACE digital signal processor hardware and shows
better performance than traditional controllers [151].
Wavelet transform provides an efficient way for noise
suppression/mitigation [176]. Huang and his team designed a
Gaussian wavelet basis expansion [177], and a pseudo-pilot-
aided complex Gaussian wavelet basis expansion base [178]
and compared the BER performance of it with some other
phase noise compensation methods. The proposed method
is more efficient than other existing methods. In [124],
Fan et al. compared FrWT and DWT denoising performance
based on SNR (Signal-Noise ratio) and RMSE (Root Mean
Squared Error). Reference [30] compared the noise-filtering
effects of different wavelet construction and found that
biorthogonal wavelet transforms with shape control has the
best performance. Chien and Yu [179] focused on impulse
noise mitigation in the wavelet-OFDM system for power-
line communication. The BER performance shows that the
proposed method mitigates the impulse noise much more
effectively, especially by adding ideal channel estimation.
Denoising ECG signals can also be realized by using suitable
wavelet methods [148], [180].
Arvinti and Costache [180] propose a robust and easy-
to-implement algorithm and achieve high SNR, low RMSE
and MSE for ECG signals. They also investigated different
choices of mother wavelets and found that reverse biorthog-
onal wavelet 2.4 is the best mother wavelet for ECG signals.
Reference [148] applies WNN as the ECG signal denoising
method and investigates the BER and RMSE performance of
different wavelet functions as activation functions. They con-
cluded that WNN is a better alternative to the traditional DWT
based noise mitigation method and db6 is more suitable for
ECG signal denoising. In [94], the authors applied perceptual
wavelet packet transform on speech enhancement. Segmental
SNR, Perceptual Evaluation of Speech Quality and Weighted
Spectral Slope (WSS) are used to evaluate the efficiency
of their method compared with some of the state-of-the-art
speech enhancement methods. The simulation results show
higher segmental SNR, higher output perceptual evaluation of
speech quality, and lower WSS values than existing methods.
2) SIGNAL PROCESSING IN EMERGING FIELDS
Wavelet transform is widely used in traditional signal pro-
cessing. It expands many emerging applications, for example,
in electrical signal processing in power systems, biomedical
signal processing, IoT (Internet of Things) mobility predic-
tion, and even quantum image processing. Reference [181]
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FIGURE 23. The structure of practical applications of wavelets.
replaces the sampling in the DWT with compressed sens-
ing and reconstructs the high-frequency characteristics of
the voltage and current to estimate the equivalent series
resistance (ESR) of the aluminium electrolytic capacitor.
The cost of data sampling, transmission, and storage is
reduced and suitable for various environments. In [182], Gao
and his team introduced the empirical wavelet transform
(EWT), which has superior time-frequency resolution abil-
ity. They compared it with other feature extraction methods
like WP, which verified that EWT is more suitable for the
extraction of High-impedance faults signals. They consid-
ered permutation entropy, which denotes the similarity, the
cross-correlation coefficient, the tracking original signal abil-
ity and energy ratio, and the energy loss for feature extraction
measurements.
Compared to the traditional empirical mode decomposi-
tion method, the three criteria improved by about 2%, 260%
and 44%, respectively. Reference [183] uses complex WT
to detect the phase and duration of voltage sags accurately.
Compared with the db4 real wavelet detection voltage sag,
it verifies the effectiveness of combining the DQ trans-
form method and the complex WT for voltage sag detection.
Wavelet transform can also be applied to the texture feature
analysis of microscope images [184]. They extracted detailed
information from the wavelet decomposition coefficients and
analyzed these features to evaluate the changes in artificially
aged power transformer winding insulation paper samples.
Biological signals can also be used to extract features
through wavelet transform. Reference [185] tracks the user
gait phase and identifies relevant biomechanical gait events.
DWT method can robustly adapt to different walking speeds
and reduce the RMS of the phase reset error by 64% and
21% in assistive mode and transparent mode, respectively.
In [186], WP has been used for feature extraction in electroen-
cephalography (EEG) signals, and the recognition accuracy
achieved 68%. Zhang et al. [187] applied DWT to analyze
retinal ganglion cell inner plexiform layer (GCIPL) topo-
graphic thickness map to extract useful features and used
three machine learning methods for further analysis. The per-
formance of traditional thickness analysis in discrimination
ability in patients with multiple sclerosis (MS) and a history
of optic neuritis (ON) is improved. Machine learning methods
may be expected to facilitate the diagnosis of MS patients
and ON patients. Wang’s team used an improved wavelet
threshold method to denoise measured surface electromyog-
raphy (sEMG) signals [188]. Compared with the traditional
wavelet threshold denoising algorithm, it has better SNR
and RMS error performance for sEMG signal denoising,
improved by about 5%. The features are extracted from the
denoised sEMG signal and used as the input of the neural
network algorithm to achieve accurate fatigue state recog-
nition. In [189], researchers used ECG signals to predict
sudden cardiac death with high accuracy. Use DWT for signal
preprocessing, extract features as the classifier’s input, and
achieve the highest accuracy compared with other research.
Reference [190] applied DWT decomposition to construct
an adaptive mobility sampling algorithm, which can reduce
wasting computational resources in IoT network mobility
prediction. In [191], quantum wavelet transform is used in
embedding watermark information in the quantum image.
The simulation results show that the watermarked image
is not significantly different from the original image for
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different images. After watermarking, the image distortion
is smaller than the quantum image watermarking algorithm
using a quantum Fourier transform. The authors in [192]
focused on the vibration signal of rolling bearing, and
Daubechies wavelet is selected for 3-level wavelet packet
decomposition. The proposed method achieved higher classi-
fication accuracy than existing classifiers like SVM (support
vector machines). Bărbulescu et al. modelled the signals that
ultrasonic waves propagate in diesel [193]. Through statisti-
cal verification, the method combined with the wavelet better
describes the experimental data, which can be used to predict
or control the evolution of the cavitation process.
Reference [194] introduced orthogonal wavelet division
multiplex (OWDM) as a more flexible alternative for OFDM.
It replaces the fast Fourier transform and Inverse Fast Fourier
Transform parts in the OFDM structure with DWT and
IDWT. Fig. 24 shows the block diagram of DWT-OFDM.
In [195], the BER performance shows the superiority of
DWT-OFDM over traditional FFT-OFDM in a hybrid power-
line communication (PLC)-visible light communication-
based system. Lokesh and his team [196] compared several
wavelets in DWT-OFDM and showed that the biorthogonal
wavelet transform provides a lower BER in all wavelets by its
characteristics. Sarowa et al. [197] compared in more detail.
They designed a mitigation technique and compared the
wavelet-OFDM system based on this technique with the tra-
ditional OFDM system based on self-cancellation and maxi-
mum likelihood. In [198], a new wavelet-based multi-carrier
modulation technique, namely filtered orthogonal wavelet
division multiplexing, is proposed as an effective alterna-
tive to traditional OFDM to reduce PAPR. In this model,
the system does not require a cyclic prefix, which exhibits
higher bandwidth efficiency. Avcı and his team proposed a
new asymmetrically clipped optical-OFDM method based
on lifting wavelet transform to restore spectral efficiency
and improve the performance of the system [199]. In order
to improve the spectral efficiency of multi-carrier modula-
tion in sonar image transmission, reference [200] proposes a
sparse non-OWDM scheme based on sparse representation.
The results show that compared with OFDM, the proposed
scheme requires fewer frequency resources and has higher
PSNR and lower PAPR.
The coherent optical OFDM (CO-OFDM) system
has unique advantages in optical fibre transmission
and utilization, which can effectively solve the disper-
sion and interference problems generated in the sys-
tem. Reference [201] combined DWT and CO-OFDM and
reduced the disadvantages of CO-OFDM. The BER per-
formance of DWT-CO-OFDM is better than CO-OFDM
in QPSK (Quadrature Phase Shift Keying) and 16-QAM
(Quadrature Amplitude Modulation) modulation. Non-
orthogonal multiple access (NOMA) is a currently emerging
technology adopted by 5G as a new multiple access tech-
nology. Bringing wavelet analysis to NOMA could achieve
better results. The authors in [164], [202] both studied
wavelet transform-based with pulse-shaped data for downlink
FIGURE 24. The structure of DWT-OFDM.
NOMA. Baig’s team [202] compared the noise variance
and BER performance between FFT-NOMA and wavelet-
NOMA; wavelet-NOMA outperforms FFT-NOMA in all
simulation scenarios. In [164], Haar, Daubechies and coiflet
wavelet are applied in the NOMA system and compared with
conventional FFT-NOMA. Both BER and PAPR performance
showed that wavelet-NOMA is usually superior to traditional
FFT-NOMA, and the Haar wavelet has the best PAPR perfor-
mance. In [165], the wavelet-OFDM system is also applied
on precoded NOMA, and the BER and PAPR performance
are better than OFDM-based precoded NOMA.
B. IMAGE PROCESSING
Wavelets can be used in image processing areas. While real-
izing pattern matching and recognition applications, DWT is
used in a wide variety of areas [203]. By creating a rational
biorthogonal wavelet filter bank, it is possible to optimally
extract features in different sizes [204]. They compared the
proposed RWT with a biorthogonal wavelet with a standard
wavelet filter bank and achieved higher image classification
accuracy. Wavelet’s applications involve feature extraction
and texture. Approaches used to solve challenges of feature
extraction in image processing contain optimally extracting
features in different sizes. In [64], Liu et al. designed a
scattering network based on FrWT. They extended the tra-
ditional scattering network with fractional coefficients and
improved image classification accuracy. Reference [205]
focused on 2-D palm-print images. The investigation of palm-
print images after two-level wavelet decomposition shows
that the extracted feature values can maintain the uniqueness
of each palmprint image and can be used for palmprint image
classification.
Furthermore, wavelets are helpful in denoise, enhance-
ment and compression in the image processing area [107].
Reference [107] combines complex wavelet shrinkage and
non-local filtering. Experimental results show that the pro-
posed method effectively reduces speckle in Synthetic Aper-
ture Radar images and ensures detail preservation in uniform
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areas. However, the proposed method is relatively time-
consuming. Norbert Remenyi’s team presents an image
denoising procedure [104]. They compared the existing
image denoising methods and achieved a better denoising
effect through the performance of PSNR. The PSNR value is
improved by about 5% compared to Hidden Markov Model.
Inspired by the powerful learning ability of GAN and the
structural information extraction ability of wavelet transform,
Su and his team [206] propose a combination of extracting
structure and noise information through wavelet transform
and generating high-quality images through GAN. Experi-
mental results show that excellent performance is achieved,
and noise can be effectively extracted while preserving
texture details.
Image enhancement technology is a method that recon-
structs a higher-resolution image. It is widely used in satel-
lite image resolution. Many researchers used wavelets to
obtain higher resolution images, such as using discrete frac-
tional wavelet transform and fractional fast Fourier transform
and combining level set method, biorthogonal CDF (Cohen
Daubechies Feauveau) wavelet-based on lifting scheme
and complex WT [27], [207]. In [207], Choudhury and
Dahake compared traditional DWT and FrWT decomposi-
tion, and interpolation is performed in these high-frequency
bands using interpolation methods in order to obtain super-
resolution images. In terms of PSNR, MSE and structural
similarity (SSIM) performance, FrWT achieves better results.
A medical image compression algorithm combining geomet-
ric active contour model and biorthogonal wavelet transform
is proposed in [27]. This algorithm is superior to traditional
MRI image methods and provides better PSNR and Mean-
SSIM values.
Image compression is also an essential part of image pro-
cessing, reducing data storage and bandwidth limitations.
Research related to using wavelets in image compression is
also a hot spot. One trend is focused on different kinds of
designed wavelets, such as using Daubechies and biorthog-
onal wavelets with the fusion of Spatial-orientation tree
wavelets [30], [208], [209]. In [30], the theoretical analy-
sis and numerical experiments of the proposed biorthogonal
wavelet transform are based on the unified Catmull-Clark
subdivision with shape control parameters. The proposed
wavelet transform achieves a higher compression ratio and
a more stable noise filtering effect than the most advanced
lifting-based solutions. They improve the PSNR of the recon-
struction model and reduce the time cost of encoding and
decoding.
Reference [208] proposed a multimedia image compres-
sion method based on biorthogonal wavelet packets. The
methods include the establishment of linear phase biorthogo-
nal wavelet basis, the selection of 3 or 4 level wavelet decom-
position and reconstruction stages, and the combination of
improved frequency band division. PSNR was used as the
reconstructed image quality evaluation index and achieved
a better compression effect, improving about 3%. Bharati
and his team compared several Daubechies wavelet and
Biorthogonal wavelets at different decomposition levels, and
the PSNR, MSE and compression ratio are used to indi-
cate the efficiency of the wavelet-based image compression
method [209].
C. OPTIMIZATION PROBLEM
There are many ways to solve optimization problems, but
they all have some shortcomings. On one hand, the traditional
mathematical optimization method takes the gradient descent
direction as the forward direction of the optimization, which
can easily fall into the local minimum solution and cannot
get an optimal global solution of the problem with a high
degree of nonlinearity [210]. On the other hand, the optimal
global solution of some optimization problems is often near
the pole of the feasible region, and these places correspond to
the discontinuity of the derivative of the function mathemati-
cally, which makes the traditional mathematical optimization
method invalid here.
Some non-traditional optimization methods developed
since the 1970s are designed based on the inspiration
of certain physical or biological phenomena. These meth-
ods include Genetic Algorithms, Simulated Annealing, Ant
Colony Optimization Algorithm, Tabu Search and Particle
Swarm Optimization (PSO). Although they can theoretically
obtain the optimal global solution, the calculation time is
theoretically infinite, which is not conducive to practical
applications. Wavelet theory has a special function in describ-
ing the singularities of functions because many engineering
optimization problems can approximate linear objective func-
tions. It transforms the optimization of functions into a finite
number of singular points in the feasible region, regardless of
the optimization content and constraints [211]. As long as the
singularity is determined, the optimal global solution is also
obtained.
In [212]–[216], researchers combined PSO and wavelet
analysis and achieved more exemplary optimization methods.
The idea of the particle swarm algorithm originates from
the study of predation behaviour of birds/fish schools [217].
It simulates the behaviour of bird swarms flying for food.
The cooperation between birds makes the group achieve
the optimal goal. It is an optimization method based on
Swarm Intelligence. It finds the global optimum by follow-
ing the optimal value currently searched. Compared with
other modern optimization methods, the obvious feature of
particle swarm optimization is that few parameters need to
be adjusted, it is simple and easy to implement, and the
convergence speed is fast. Like WNN, combining wavelet
analysis and optimization methods is also divided into two
directions. On the one hand, References [212], [216] are new
methods proposed after mixing particle swarm and wavelet
analysis.
Zhang and Min [212] designed an improved particle swarm
with a wavelet threshold and also used a WNN using the
Morlet wavelet as the activation function for classification.
The improved PSO algorithm achieved higher classification
accuracy, and different wavelet functions were applied for
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better noise filtering. The subjective visual effects, mean
square error, peak signal-to-noise ratio, and structural sim-
ilarity of the images after noise reduction are better than
traditional noise filtering algorithms. The proposed method
to classify the data set reduces the number of features and
classification error rate. 21.543% reduces the maximum clas-
sification error rate, and the number of features is reduced
by 12; 29.243% reduces the minimum classification error
rate, and the number of features is reduced by 9. In [216],
an improved particle swarm optimization scheme using lift-
ing wavelet transform proposes dynamic range enhance-
ment for optical time-domain reflectometry. This scheme
enables the design of custom lifting wavelet filters to increase
the signal-to-noise ratio and thus improve the dynamic
range.
On the other hand, the authors in [213]–[215] first pro-
cess signals with wavelet analysis and then apply PSO or
Enhanced-PSO for optimization. Reference [213] proposed
a hybrid prediction model combining wavelet transform,
particle swarm optimization and support vector machine for
short-term power generation prediction of practical micro-
grid photovoltaic systems. The prediction accuracy of the
proposed model has been compared with seven other pre-
diction strategies and shows excellent performance in terms
of prediction accuracy improvement. In [214], Djaghloul and
his team performed segmentation and tracking of deformable
structures during intervention through an improved PSO
scheme. The reconstructed 3D models are analyzed using
wavelet-based methods to perform registration tasks. The
system can thus track surgical instruments through updates
of the colour model guided by prior anatomical knowledge.
The researchers in [215] extracted seven wavelet features
for Fusarium head blight detection based on continuous
wavelet analysis of wheat spike hyperspectral reflectance.
They constructed a Fusarium wilt detection model, taking
wavelet features and traditional spectral features as input
features and combining them with the PSO-SVM algo-
rithm. The accuracy of random forest (RF), backpropaga-
tion neural network, and PSO-SVM detection models with
wavelet features are improved by 3.7%, 2.9%, and 8.3%,
respectively.
For other optimization algorithms, in [218], Yin et al. pro-
posed wavelet transform subspace-based optimization meth-
ods and investigated various wavelet functions for the minor
part of induced current in the inverse problem. The proposed
method increased the resolution of a specific area and signif-
icantly accelerated the convergence speed of the algorithm.
Temel and his team modified the Cat Swarm Optimization
algorithm with wavelet transform to seek the best position-
ing sensor to cover the specified area in the 3D environ-
ment as effectively as possible [219]. Compared with the
random deployment and the Delaunay Triangulation based
deployment approaches, when covering 90% of the specified
area, their method needs the least number of sensors. It has
the best QoC (Quality Of Conformance) performance with
96 sensors.
VII. CHALLENGES AND RESEARCH GAP
Although wavelet analysis has achieved certain results in
many application fields, it still faces many problems.
1) Except for mature one-dimensional wavelet theory, the
theory of high-dimensional wavelet is not well devel-
oped. Multiwavelet theory is not extensively developed
either. There is no general construction formula for high-
dimensional wavelets and multi-wavelet. In practical
applications, the two-dimensional and high-dimensional
wavelet bases currently used are separable; the low-
dimensional wavelet base is constructed as a tensor
product. However, using separable wavelet bases con-
structed from tensor products to analyze signals may
lose their anisotropic properties. Designing the scaling
factor value of multi-wavelet also needs to be further
studied according to actual applications.
2) Selecting the most suitable wavelet basis for a spe-
cific application or data source has been a challenge in
wavelet analysis all the time, both in wavelet transform
and WNN. Although there has been researching on opti-
mal basis selection methods in the literature as presented
in the review, a systematic way of optimal wavelet basis
selection and performance evaluation is still a significant
research gap. The current selection of wavelet basis has
the following problems:
- Considering that some desirable properties of
wavelets, such as symmetry and orthogonality, are
not easy to obtain at the same time, it is a huge
challenge to select or design suitable wavelets to
deal with various problems in reality. At present,
there are not many qualitative studies in this area.
- The RWT is more suitable for oscillating signals
because of its more satisfactory frequency reso-
lution. However, the choice of rational factors is
worth studying. The rational wavelet preferably
includes the characteristics of the analyzed signal
in the construction process. In order to make the
rational wavelet transform better match the signal
characteristics, it is necessary to adjust the param-
eters to obtain the rational wavelet basis with dif-
ferent time-frequency distribution characteristics.
Among them, adjusting the parameters can change
the frequency division method and change the time-
domain oscillation properties of the wavelet func-
tion. For a given signal, achieving the adaptability
of the wavelet base to the signal needs to be studied.
- Most of the literature uses simple non-orthogonal
wavelets, such as Mexican hat wavelets and Morlet
wavelets, as activation functions because they are
simple and easy to implement. Similar to multi-
wavelets, the new wavelet network can use multi-
ple mother wavelets to select the best wavelet to
the greatest extent. However, if a complex wavelet
function is used, the calculation time of WNN will
be significantly increased. Moreover, if the initial
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settings of the wavelet function’s scale parame-
ter and translation parameter are unreasonable, the
entire network will be difficult to converge. In the
aspect of high-dimensional data processing, there is
little research on WNN, which is determined by the
complicated structure of multi-dimensional wavelet
theory. Therefore, the development of wavelet net-
works also depends on further research of wavelet
theory.
3) Traditional signal processing has applied a variety of
RWT, and most of them have achieved better results than
DWT. However, in emerging fields, DWT still occupies
the majority of studies. In the review of wavelet signal
processing in the power system voltage and current sig-
nals, biological signals and quantum fields, few applica-
tions use RWT. The application direction of RWT could
be expanded. It can also be seen that rational wavelet
is used less for applying wavelets in the optimization
algorithm. The type of optimization algorithm combined
with wavelet is also relatively less, most of them consid-
ering PSO algorithm.
4) For WNN, the neural network structure is still under
extensive research. Numerous studies in the literature
use WNN based on the Radial Basis Function neural
network, which is only a three-layer neural network.
Choosing the proper WNN structure to deal with dif-
ferent problems is a great challenge in this field. The
wavelet network only uses the expanded and translated
version of a mother wavelet to construct the network.
It is unrealistic to rely solely on a particular theory
and technology. Therefore, attention should be paid to
combining interdisciplinary research on fuzzy, fractal
and genetic algorithms. Wavelet Kernel in convolution
layer is a developing research field. Similar to using
the wavelet function as the activation function, selecting
a suitable wavelet kernel is also a direction worthy of
further study.
VIII. CONCLUSION
The key highlights and concluding points of this paper are
summarized as follows:
1) Wavelet theory is briefly summarised, including the
construction method and properties of different wavelet
bases. There are currently two main wavelet design
methods: MRA-based and lifting schemes. The second-
generation wavelet constructed using the lifting scheme
contains the multi-resolution characteristics of the first-
generation wavelet. They have fast calculation speed
and low memory consumption. For various practical
problems, rational wavelets, high-dimensional wavelets
and multi-wavelets are worthy of further study.
2) Related algorithms using wavelet analysis were also dis-
cussed. For example, wavelet packet theory and wavelet
transform are constructed by filter banks. DWT is the
most basic and most widely used wavelet transform.
RWT can achieve finer frequency domain segmentation.
It enhances the signal frequency domain localization
and is a very powerful signal processing tool. RWT is
more suitable for oscillating signals, and its application
in Doppler analysis and radar or sonar detection is very
promising. In terms of denoising, FrWT achieves better
results than DWT. SNR can be increased by about 40%,
and RMSE can be reduced by about 50% [124]. In the
case of in-depth analysis of the signal, many particular
wavelet transforms, such as DT-CWT and RADWT, are
also designed to achieve better or more adaptable effects
to special situations.
3) With the development of neural networks, combining
neural networks and wavelet analysis has also flour-
ished. The WNN, whose signal is preprocessed by
wavelet analysis, combines the advantages of artificial
neural networks and wavelet analysis. After wavelet
analysis preprocesses the signal, the performance of
WNN can reduce the prediction error by about 50%
in [220]. Some commonly used WNN structures are
summarised for WNN, where wavelet cells replace
neurons, and examples of WNN structures combined
with RNN are given. Applying WNN to more complex
structures or combining it with some interdisciplinary
algorithms can improve the performance of neural net-
works, such as in [154], which increases the perfor-
mance by about 65%. Wavelet Kernel in convolution
layer is a developing research field. Selecting a suit-
able wavelet kernel is also a direction worthy of fur-
ther study. In summary, WNN can avoid the blindness
of traditional neural network design. It has more vital
learning ability, higher accuracy, simple structure and
fast convergence speed. It is also the focus of future
research.
4) Wavelet analysis has a wide range of applications in
signal processing, and it has more advantages than
traditional methods in signal analysis in terms of
enhancement, denoising, compression and classifica-
tion. In emerging fields like power systems and biologi-
cal signals, wavelet signal processing has also achieved
better results than traditional FFT. The performance of
RMSE after noise reduction and feature extraction has
been improved to varying degrees. The application of
wavelet analysis in image processing includes image
compression, classification and denoising. It deals with
the low-frequency and high-frequency parts of wavelet
images. In most cases, wavelet transform can achieve
better performance in image processing, such as com-
pression rate and denoising effect. The PSNR after
noise filtering could be improved by about 5% in [104].
Wavelet analysis is superior to traditional methods in
image quality reconstruction under the same compres-
sion ratio. PSNR is about 3% higher than existing meth-
ods in [208]. Combining wavelets with optimization
algorithms can often get better optimization results. The
combination of the optimization algorithm and wavelets
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method reduces the number of features and reduces the
classification error rate.
5) The main challenges and research gaps in wavelet
research have been discussed. When applied to sig-
nal processing, it is necessary to study the selection
or design of the optimal wavelet basis. Multiwavelet
and high-dimensional wavelet theories are still under
development. Although RWT can flexibly adjust the
time-frequency distribution characteristics, the amount
of calculation and memory consumption has increased.
The application areas of RWT also need to be further
expanded, not only in traditional signal processing. The
wavelet basis function required by the hidden layer of
WNN is inconsistent with the wavelet basis selection
criteria of signal processing, and it is necessary to intro-
duce advanced wavelet theory further. WNN can be
combined with multi-interdisciplinary algorithms such
as fuzzy, fractal and genetic algorithms to obtain broad
application prospects.
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TIANTIAN GUO received the B.Eng. degree in
spatial information and digital technology from the
University of Electronic Science and Technology
of China (UESTC), in 2017, and the M.Sc. degree
in multimedia telecommunications from the Uni-
versity of Liverpool, in 2019. She is currently pur-
suing the Ph.D. degree in electronic and electrical
engineering with the School of Advanced Technol-
ogy, Xi’an Jiaotong-Liverpool University, Suzhou,
China. Her research interests include wavelet anal-
ysis, wavelet neural networks, signal processing, and communications.
TONGPO ZHANG received the B.Eng. degree in
computer science and electronic engineering from
the University of Liverpool, in 2018, and the M.Sc.
degree in computer systems engineering from the
University of Glasgow, in 2019. He is currently
pursuing the Ph.D. degree in electronic and elec-
trical engineering with the School of Advanced
Technology, Xi’an Jiaotong-Liverpool University,
Suzhou, China. His research interests include inte-
grated navigation, scheduling optimization, het-
erogeneous networks, machine learning, and robotics.
ENGGEE LIM (Senior Member, IEEE) received
the Ph.D. degree from Northumbria University,
in 2002. He has been with Xi’an Jiaotong-
Liverpool University (XJTLU) as an Associate
Professor, since 2007; and a Full Professor, since
2015. His research interests include RF and
microwave applications, antennas, filters, diplex-
ers, couplers, RFID, UWB, WIMAX, 3G/4G/5G
mobile communication networks, wireless capsule
endoscopy, EM measurement and simulation, co-
operative and cognitive wireless communication networks, smart-grid com-
munication, robotic networking technology, and wireless communication
networks for smart and green cities (e.g., mobile APP and public transporta-
tion information).
MIGUEL LÓPEZ-BENÍTEZ (Senior Member,
IEEE) received the B.Sc. and M.Sc. degrees
(Hons.) in telecommunication engineering from
Miguel Hernández University, Elche, Spain, in
2003 and 2006, respectively, and the Ph.D. degree
(summa cum laude) in telecommunication engi-
neering from the Technical University of Catalo-
nia, Barcelona, Spain, in 2011. From 2011 to 2013,
he was a Research Fellow with the Centre for
Communication Systems Research, University of
Surrey, Guildford, U.K. In 2013, he became a Lecturer (Assistant Professor)
with the Department of Electrical Engineering and Electronics, University of
Liverpool, U.K., where he has been a Senior Lecturer (Associate Professor),
since 2018. His research interests include wireless communications and
networking, with special emphasis on mobile communications, dynamic
spectrum access, and the Internet of Things.
FEI MA received the M.Sc. degree from Xiamen
University, China, in 2002, and the Ph.D. degree
in applied mathematics from Flinders University,
Australia, in 2008. He has been with the Depart-
ment of Mathematical Sciences, Xi’an Jiaotong-
Liverpool University, since 2012; and is currently a
Full Professor. His research interests include non-
negative matrix, medical image analysis, forecast-
ing methods, graph theory, and AGV scheduling.
LIMIN YU (Member, IEEE) received the B.Eng.
degree in telecommunications engineering and the
M.Sc. degree in radio physics/underwater acoustic
communications from Xiamen University, China,
in 1999 and 2002, respectively, and the Ph.D.
degree in telecommunications engineering from
The University of Adelaide, Australia, in 2007.
She worked with ZTE Telecommunications Com-
pany, Shenzhen, China, as a Software Engineer.
She also worked with South Australia University
and The University of Adelaide as a Research Fellow and a Research Asso-
ciate. She has been with Xi’an Jiaotong-Liverpool University(XJTLU), since
2012; and is currently an Associate Professor. Her research interests include
sonar detection, wavelet analysis, sensor networks, coordinated multi-AGV
systems design, and medical image analysis.
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