# On the Intersection of Signal Processing and Machine Learning: A Use Case-
Driven Analysis Approach

Sulaiman Aburakhia, Abdallah Shami, and George K. Karagiannidis Sulaiman
Aburakhia and Abdallah Shami are with the Department of Electrical and
Computer Engineering, Western University, N6A 3K7, Canada (e-mail:
<EMAIL_ADDRESS>abdallah.shami@uwo.ca).
George K. Karagiannidis is with Department of Electrical and Computer
Engineering, Aristotle University of Thessaloniki, Greece and also with
Artificial Intelligence and Cyber Systems Research Center, Lebanese American
University (LAU), Lebanon (e-mail: geokarag@auth.gr).

###### Abstract

Recent advancements in sensing, measurement, and computing technologies have
significantly expanded the potential for signal-based applications, leveraging
the synergy between signal processing and Machine Learning (ML) to improve
both performance and reliability. This fusion represents a critical point in
the evolution of signal-based systems, highlighting the need to bridge the
existing knowledge gap between these two interdisciplinary fields. Despite
many attempts in the existing literature to bridge this gap, most are limited
to specific applications and focus mainly on feature extraction, often
assuming extensive prior knowledge in signal processing. This assumption
creates a significant obstacle for a wide range of readers. To address these
challenges, this paper takes an integrated article approach. It begins with a
detailed tutorial on the fundamentals of signal processing, providing the
reader with the necessary background knowledge. Following this, it explores
the key stages of a standard signal processing-based ML pipeline, offering an
in-depth review of feature extraction techniques, their inherent challenges,
and solutions. Differing from existing literature, this work offers an
application-independent review and introduces a novel classification taxonomy
for feature extraction techniques. Furthermore, it aims at linking theoretical
concepts with practical applications, and demonstrates this through two
specific use cases: a spectral-based method for condition monitoring of
rolling bearings and a wavelet energy analysis for epilepsy detection using
EEG signals. In addition to theoretical contributions, this work promotes a
collaborative research culture by providing a public repository of relevant
Python and MATLAB signal processing codes. This effort is intended to support
collaborative research efforts and ensure the reproducibility of the results
presented.

###### Index Terms:

Signal processing, signal preprocessing, signal denoising, feature extraction,
signal transforms

##  I Introduction

The rapid advancements in sensing and measurement represent a paradigm shift
in how data is collected, processed, and interpreted. This opens the door for
a wide range of signal-based applications, marking a transformative phase
across various fields. Moreover, the development of computing technologies and
the rise of the Internet of Things (IoT) have paved the way to leverage
machine learning (ML) within signal-based applications, offering new insights
and achieving unprecedented levels of accuracy and efficiency. This merge
between signal processing and ML is expected to play a major role in the next
generations of sensor-enabled systems across various areas [1]. The
integration of signal processing pipelines into ML models forms the
fundamental core of these systems. Further, it represents a critical
intersection in their advancement, motivating the research community to
address the role of signal processing in ML. However, the diverse landscape of
signal types and application requirements shapes the scope of the existing
body of work to be application-centric, limiting their scopes to specific
applications. For instance, The role of feature extraction in ML has been
extensively reviewed within the context of vibration-based predictive
maintenance (PdM). The work in [2] presents a review that focuses on
transforming traditional methods to ML techniques in applying vibration-based
damage detection in civil structures. The article highlights traditional
methods and presents a comprehensive review of the latest applications of ML
algorithms used for this purpose. In [3], a systematic review is conducted on
adopting machine learning for failure prediction in industrial maintenance.
The review covers the used datasets, preprocessing, and the training and
evaluation of prediction models. In [4], a comprehensive review is presented
on signal processing techniques for vibration-based feature extraction in
structural health monitoring (SHM). The work in [5] addresses the application
of vibration-based condition monitoring techniques for the PdM of rotating
Machinery. It provides a comprehensive review of vibration data acquisition
and analysis, as well as the methods used for fault interpretation and
diagnosis, including data acquisition, data transmission, signal processing,
and fault detection. The work in [6] introduces a tutorial on the same topic
that describes relevant signal processing methods in this field. Furthermore,
the tutorial provides Python and MATLAB code examples to demonstrate these
methods alongside explanatory videos.

In the biomedical field, the topic of signal processing in ML is an active
area of research. For instance, the work in [7] provides an end-to-end review
of biomedical signal processing for health monitoring applications. It
introduces a flow for developing biomedical signal processing systems.
Further, it covers recent applications, types of low-cost, non-invasive
biomedical sensors, signal processing techniques, and future perspectives for
building reliable systems. The application of MI-based brain-computer
interfaces (BCIs) in controlling external devices through EEG signal
processing is addressed in [8]. The article reviews recent ML models,
identifies major challenges, and suggests potential solutions by focusing on
feature extraction and classification methods. The work in [9] reviews various
feature extraction techniques for electrocardiogram (ECG) Signal analysis. In
[10], the main steps in detecting and classifying EEG epileptic seizure
activities are addressed, along with a review of related feature extraction
techniques. The studies in [11] and [12] deal with the application of emotion
recognition using EEG signals, where different existing feature extraction
methods are analyzed and compared in terms of classification performance. A
comprehensive review of methods and techniques that covers the entire process
of EEG signal processing is presented in [13]. The study analyzed numerous
articles related to EEG signal processing, identified limitations, and
analyzed future development trends. Besides biomedical and BdM fields, the
role of signal processing in ML has been addressed in various fields such as
audio analysis and recognition [14, 15, 16, 17, 18, 19, 20], seismic signal
analysis [20, 21, 22], and telecommunications[23, 24, 25, 26, 27, 28, 29, 30].

Despite this diverse spectrum of articles addressing the role of signal
processing within ML, there remains a noticeable gap in the literature,
characterized by the following four main limitations in the existing studies:

  * •

Application-Specific Focus: The analysis are often confined to specific
applications, which limits the coverage and depth in which the topic is
addressed.

  * •

Limited Audience Reach: Many articles assume a substantial background in
signal processing, restricting accessibility to a limited audience.

  * •

Task-Centric Approach: The exploration of signal processing in ML tends to
focus on particular tasks, such as signal preprocessing or feature extraction.
This neglects other critical tasks such as signal segmentation, smoothing, and
denoising, which are essential components of a typical signal processing ML
pipeline.

  * •

Theory-to-Practice Gap: There is a notable lack of practical demonstrations
through use-case applications. Such demonstrations are critical to connecting
theoretical concepts with real-world applications, and thus bridging the gap
between theory and practice.

This paper attempts to bridge this gap by implementing an integrated-article
approach that addresses the aforementioned shortcomings through the following
contributions:

  * •

Comprehensive Tutorial for Diverse Readership: The paper starts with a
comprehensive tutorial on the fundamentals of signal processing that is aimed
at readers from different domains, allowing the interested reader to develop a
proper background before delving into the review.

  * •

Application-Independent Approach: The paper adopts a broad, application-
independent review, providing a comprehensive overview of signal processing in
ML that is not shaped to a specific use case.

  * •

End-to-End Overview of Signal Processing Workflow: The article thoroughly
discusses the key tasks in a typical signal processing pipeline, grouping them
under three main categories: preprocessing, processing, and application.

  * •

Exhaustive Review with Novel Taxonomy: The paper conducts a detailed review of
feature extraction techniques, presented and categorized through a new
taxonomy that presents new insights and enriches the reader’s understanding.

  * •

Addressing Major Challenges and Potential Solutions: The paper identifies the
primary challenges faced in implementing signal processing-based ML
applications and discusses both existing and potential solutions.

  * •

Bridging Theory with Practice: The paper addresses the practical application
of signal processing in ML through two use cases. In the first use case, a
spectral-based method is introduced for vibration-based condition monitoring
of rolling bearings. In the second case, wavelet-energy analysis is utilized
for epilepsy detection using EEG signals.

  * •

Public Repository of Signal Processing: The paper introduces a public
repository of the Python and MATLAB codes used throughout the article, along
with additional codes relevant to signal processing, thereby fostering a
collaborative research environment and ensuring reproducibility of work.

Figure 1: Visual representation of the paper’s contents.

With these contributions, the paper aims to become a foundational reference on
the topic of signal processing in ML, serving a wide range of readers and
offering new perspectives on the intersection of signal processing and ML.
Each of these contributions has its own section in the paper. The sections are
logically grouped under four parts. For reader convenience, a visual
representation that shows the various sections grouped under each part is
depicted in Fig. 1. Although these parts are organized to build knowledge
progressively, they are written in a self-contained manner, allowing for
selective reading based on interest or need.

The article is structured as follows: The tutorial is carried out in Section
II and Section III. A typical signal-processing pipeline for signal-based ML
applications is presented in Section IV. The review of feature extraction
techniques is introduced in Section V. Section VI discusses the main
challenges faced in implementing signal processing-based ML applications and
addresses potential solutions. The practical application use cases are
introduced in Section VII. The paper is finally concluded in Section VIII.

##  II Introduction to Signals

This tutorial provides an in-depth- introduction to signal processing,
highlighting concepts, mathematical formulation, applications, advantages, and
limitations of common signal processing tools. Additionally, the tutorial
offers insights into implementation considerations and highlights programming
libraries that offer functionalities for implementing these tools. Throughout
the tutorial, illustrative examples are generated using Python and MATLAB
codes. These codes are publicly available on the Github site of the Optimized
Computing and Communications (OC2) Laboratory111https://github.com/Western-
OC2-Lab/Signal-Processing-for-Machine-Learning

###  II-A Signal Characteristics

A signal is a function of time; it represents the value of a physical entity
or phenomenon as it evolves over time, such as voltage, current,
acceleration…etc. Depending on the field, there are more definite definitions
of the term ”signal”. For instance, within a signal processing context, a
signal can be defined as a function that conveys information about the
behavior of a system or attributes of some phenomenon [31]. In manufacturing,
the term “signal” refers to a physical quantity that carries a certain type of
information and serves as a means for communication [32]. Examples of such
signals include a vibration signal generated by an accelerometer attached to
the rolling-bearing element of a rotating machinery and a torque signal
generated by a torque Sensor in a computer numerical control (CNC) milling
machine. The changes in these signals are directly related to the operation of
the machine and can therefore be used to communicate the operating status of
the machine to the machine operator [32].

Signals are typically represented by their time waveforms; Fig. 2. displays
the waveform of a given $x(t)$ as it evolves over time $t$.

Figure 2: Waveform of a signal, $x(t)$, as it evolves over time $t$.

Amplitude, frequency, and phase are the main characteristics of a signal.
Amplitude measures the amount and direction of change in the signal with
respect to a reference value, as shown in Fig. 2. Frequency is the number of
cycles or oscillations that occur in a given unit of time. It represents the
rate at which the signal oscillates. Frequency is measured in Hertz (Hz) where
$1$ Hz is equal to $1$ cycle per second. The phase of a signal refers to the
position of a point (time instant) on the signal’s waveform cycle. The phase
of a wave refers to its position within its cycle and is measured in either
degrees ($0-360$) or radians ($0-2\pi$). It helps describe the relative
position and timing of two signals at a given moment. The concept of frequency
of a signal is closely related to the rate at which its phase changes.
Specifically, the frequency at a particular moment in time indicates the speed
at which the phase is changing at that moment. A higher frequency indicates a
faster rate of phase change. Therefore, frequency and phase are mathematically
linked through the concept of the derivative. By knowing the phase information
of a signal, the frequency can be calculated as the first derivative of the
phase with respect to time. The period of a waveform refers to the time it
takes to complete one cycle. A periodic signal is a signal that repeats its
pattern or the sequence of values exactly after a fixed duration time, known
as the period $T$. This can be expressed mathematically as:

$x(t)=x(t+T)$ (1)

Since the period is the time duration a signal takes to complete one cycle, it
is related to the signal’s frequency, which represents the number of cycles
per second, through the following relation:

$T=\frac{1}{f}\;\;(seconds)$ (2)

While purely periodic signals do not exist in practice [33], they represent an
essential theoretical concept in signals theory [34]. A sinusoidal wave,
$s(t)$, represents the basic form of a periodic signal; it is expressed
mathematically as:

$s(t)=A\sin(2\pi ft+\phi)$ (3)

Where $A$ is the signal’s peak amplitude, $f$ is frequency, and $\phi$ is the
phase. With respect to zero, signal amplitude can be either positive or
negative, as shown in Fig.3.

Figure 3: Sinusoidal waveform $s(t)$.

On the other hand, the magnitude of the signal $|s(t)|$, shown in Fig. 4,

Figure 4: Magnitude of the sinusoidal waveform $s(t)$.

is the absolute value of the amplitude, it shows the change in the signal
regardless of its direction since it is always a positive quantity. Amplitude
values indicate the strength of the signal, reflecting the intensity of the
measured physical phenomenon.

As explained earlier, a signal represents a physical entity or phenomenon that
changes over time. A signal that remains unchanged over time when measured
again under the same conditions is referred to as a deterministic signal. Such
a signal has no uncertainty about its value at any given point in time and can
be modeled using mathematical expressions. The sinusoidal function is an
example of a deterministic function. However, signals that are completely free
of unknown and uncontrollable factors and have true deterministic properties
are extremely rare [35]. In practical scenarios, signals are non-deterministic
as they are random in nature and exhibit various degrees of uncertainty. Since
non-deterministic signals are stochastic, they are described in terms of their
statistical properties. From this perspective, random signals can be
classified into two categories: stationary and nonstationary. Stationary
signals generally maintain constant statistical properties over time, while
nonstationary signals exhibit changing statistical properties over time.

From the preceding discussion, it is apparent that signal variation describes
the changes in the measured physical entity or phenomenon as it evolves over
time. This variation is subject to the influence of uncontrollable factors
during the measurement period, which manifest themselves as noise or outliers
in the signal. Other influencing factors include subsequent processing and
operations imposed on the signal, commonly referred to as ”systems” in the
context of signal processing. Specifically, a system in signal processing is
any process or operation that produces an output signal in response to an
input signal. In practical situations, real-world systems exhibit mostly
nonlinear behaviors222A linear system in signal theory is defined by two key
properties: homogeneity and additivity. Homogeneity means if an input signal
$x(t)$ results in an output $y(t)$, then any scaled input $ax(t)$ leads to a
scaled output $ay(t)$. Additivity implies that the response to a sum of input
signals $x_{1}(t)+x_{2}(t)$ is the sum of their responses,
$y_{1}(t)+y_{2}(t)$. and operate under transient, nonstationary conditions
[36]. Therefore, real-world signals in practical situations exhibit nonlinear,
time-varying, and nonstationary characteristics. Understanding the unique
characteristics of a signal holds significant importance, as it directly
influences the selection of the proper signal processing tool, ensuring
precise and effective analysis of the signal and, consequently, extracting
discriminative features.

Another fundamental aspect of signal processing is signal acquisition and the
time representation of the signal. From a time perspective, signals are
classified as continuous or discrete-time signals. The continuous-time signal
$x(t)$ is the signal that has a continuous value over the observation time
range, as depicted in Fig. 2. A discrete-time version of the signal $x(t)$ is
displayed in Fig. 5, denoted by $x[n]$. In the discrete-time representation,
the independent variable $n$ is discrete in type and takes integer values
only. That’s it; in contrast to a continuous-time signal, the discrete signal
is defined only over discrete time intervals.

Figure 5: $x[n]$, a discrete-time version of signal $x(t)$.

The discrete-time representation $x[n]$ is obtained by sampling the
continuous-time signal $x(t)$ at time instants separated by $T_{s}$. i.e.

$x[n]=x(nT_{s}),\;n=0,1,2,\dots$ (4)

where $T_{s}$ represents the sampling interval or sampling period in seconds.
Accordingly, the sampling frequency or sampling rate $f_{s}$ is the number of
samples obtained in $1$ second; it is given by:

$f_{s}=\frac{1}{T_{s}}$ (5)

The unit of $f_{s}$ is samples per second or hertz. In practice, a measured
signal is a sampled version of the actual physical signal since it is acquired
by recording a specific number of measurements (samples) every second. The
sampling rate plays a crucial role in signal measurement and processing. A
higher sampling rate means more measured samples per second and, hence, a
higher signal resolution. However, processing time and memory requirements
increase as the sampling rate increases. A fundamental concept in signal
sampling is the Nyquist theorem [37], which states that a signal can be
correctly reconstructed from its sampled discrete-time sequence provided that
the sampling rate is equal to or greater than twice the highest frequency
(effective bandwidth) of the signal. This implies that the highest frequency,
represented as $f_{max}$, that can be represented accurately in an acquired
signal is one-half of its sampling rate $f_{s}$; i.e.

$f_{s}=2f_{max}$ (6)

This formula is called the Nyquist rate; it provides a lower bound on the
sampling rate that is required to reconstruct a signal from its sampled
version accurately. Accordingly, a sampled signal is categorized as either
undersampled, critically sampled, or oversampled:

  * •

Undersampled signal: is one where the sampling rate is below the Nyquist
sampling rate. Undersampling leads to a distortion in the signal known as
aliasing, causing signal components at higher frequencies than the sampling
frequency to appear at a lower, aliased frequency [38].

  * •

Critically sampled signal: is one where the sampling rate equals the Nyquist
sampling rate. In this case, the signal is sampled at the lowest rate, which
still allows for a complete reconstruction of the continuous-time signal,
assuming the signal bandwidth is limited to half of the sampling frequency.
While perfect reconstruction is theoretically still possible, it is less
common in practical applications.

  * •

Oversampled signal: is one where the sampling rate exceeds the Nyquist rate.
Oversampling is widely used in signal measurements because it improves the
resolution of the acquired signal. Additionally, with oversampling, the
quantization noise (an error that represents the difference between the value
of the measured sample and the closest mapped digital value) is spread over a
broader frequency range compared to the effective bandwidth of the signal.
Accordingly, filtering out high frequencies outside the effective bandwidth
effectively reduces noise in the signal, improving the signal-to-noise ratio
(SNR).

###  II-B Signal Energy and Power

As noted above, signal amplitude reflects the intensity of the measured
physical phenomenon, making it significant in identifying critical features of
the signal, such as energy and power. Energy quantifies the total ”work” done
by the signal; mathematically, the energy of a signal is the area under the
squared magnitude of the signal over a given time interval. for a continuous-
time signal $x(t)$, its energy $E$ is defined as:

$E=\int_{-\infty}^{\infty}|x(t)|^{2}\,dt$ (7)

For a discrete-time signal $x[n]$, the energy is given by:

$E=\sum_{n=-\infty}^{\infty}|x[n]|^{2}$ (8)

This definition assumes the signal is energy-limited, meaning that its energy
is finite over the entire time domain. Signal power measures the average rate
at which the signal transmits energy. It is calculated as the average power
over a period of time. For a continuous-time signal, its average power $P$ is
defined as:

$P=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|x(t)|^{2}\,dt$ (9)

Similarly, for discrete signals:

$P=\lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}|x[n]|^{2}$ (10)

If the signals’ energy $E$ converges to a finite but non-zero value, the
signal is classified as an energy signal. Examples include non-zero signals
over a limited period but zero elsewhere, such as pulse signals. The average
power of an energy signal is zero because the calculation of power involves
averaging the finite energy over an infinite period of time. Power signals, on
the other hand, are signals that have finite, non-zero average power. A common
example is a periodic sinusoidal signal that continues indefinitely; such
signals have infinite energy. For signals of finite duration, energy and power
can be evaluated over their active time intervals. For periodic signals, due
to the repetitive nature of the signal, the computation is simplified by
considering one single period $T$ as follows:

$P=\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}|x(t)|^{2}dt$ (11)

For a discrete-time periodic signal $x[n]$ with period $N$, the power is:

$P=\frac{1}{N}\sum_{n=0}^{N-1}|x[n]|^{2}$ (12)

Instantaneous power refers to the power of a signal at a specific moment in
time. Unlike average power, which is calculated over a period of time,
instantaneous power gives a moment-to-moment view of the power level of the
signal. For a continuous-time signal $x(t)$, the instantaneous power $P(t)$ at
time $t$ is defined as the square of the signal’s magnitude at that particular
time. Mathematically, it is expressed as:

$P(t)=|x(t)|^{2}$ (13)

For a discrete signal $x[n]$, where $n$ represents discrete time indices, the
instantaneous power $P[n]$ at point $n$ is defined as the square of the
magnitude of the signal at that point. Mathematically, this is expressed as:

$P[n]=|x[n]|^{2}$ (14)

For a given discrete time signal $x[n]$, represented by $N$ samples, the total
energy $E$ is the sum of its instantaneous power values over its discrete time
interval:

$E=\sum_{n=0}^{N-1}|x[n]|^{2}$ (15)

Accordingly, the average power $P$ is obtained by dividing the energy $E$ by
the number of samples $N$:

$P=\frac{\sum_{n=0}^{N-1}|x[n]|^{2}}{N}$ (16)

along with power and energy, statistical properties of the signal, such as
mean and variance, represent important aspects of signal characteristics and
play a major role in signal processing. For the discrete time signal $x[n]$,
its mean $\mu$ is equal to:

$\mu=\frac{1}{N}\sum_{n=0}^{N-1}x[n]$ (17)

The mean represents the central tendency or the average value of the signal
over its time period. The signal’s variance $\sigma^{2}$ is given by:

$\sigma^{2}=\frac{1}{N}\sum_{n=0}^{N-1}(x[n]-\mu)^{2}$ (18)

The variance measures the spread of the signal’s amplitude around its mean
value, quantifying the deviation of the signal’s samples from its mean. Note
that for a zero-mean signal, The variance equals the signal’s average power.
For such signals, the root-mean-square (RMS) is a useful measure that
represents the effective value of the signal because it provides a meaningful
measure of the average magnitude of the signal, even though its mean is zero.
The RMS is expressed as:

$x_{RMS}=\sqrt{\frac{1}{N}\sum_{n=0}^{N-1}[x[n]]^{2}}$ (19)

###  II-C Signal Correlation

Correlation can be viewed as a similarity measure; it quantifies the
similarity or the degree of dependence between two signals, providing insights
into their common characteristics and patterns. Auto-correlation refers to the
correlation of a signal with a delayed version of itself. It represents the
degree of similarity between a signal and a time-shifted (lagged) version of
itself over a given time interval. Auto-correlation helps to detect repeating
patterns in the signal and assess its stability or predictability over time.
For a continuous-time signal $x(t)$, the auto-correlation function is
expressed as:

$R_{xx}(\tau)=\int_{-\infty}^{\infty}x(t)x(t-\tau)dt$ (20)

where $\tau$ represents time shift. For a discrete-time signal $x[n]$,
autocorrelation $R_{xx}[k]$, at a lag $k$ is given by defined as:

$R_{xx}[k]=\sum_{n=-\infty}^{+\infty}x[n]x[n-k],\;k=0,\pm 1,\pm 2,\dots$ (21)

Note that for zero lag, there is no time shift; hence, the autocorrelation
$R_{xx}$ will be at its maximum value,— which equals the signal’s total
energy— representing the maximum similarity as depicted in Fig. 6 which shows
the autocorrelation of the signal $x(t)$.

Figure 6: Signal $x(t)$ and its autocorrelation.

Cross-correlation extends the concept of auto-correlation to two different
signals, measuring the degree of similarity or dependence between them. It
quantifies the degree of similarity between one signal and a time-shifted
(lagged) version of another signal over a given time interval. For two
continuous-time signals $x(t)$ and $y(t)$, their cross-correlation is
expressed as:

$R_{xy}(\tau)=\int_{-\infty}^{\infty}x(t)y(t-\tau)dt$ (22)

For discrete-time signals $x[n]$ and $y[n]$, the cross-correlation $R_{xy}$ at
a given time lag $k$ is defined as:

$R_{xy}[k]=\sum_{n=-\infty}^{+\infty}x[n]y[n-k],\;k=0,\pm 1,\pm 2,\dots$ (23)

The value of $R_{xy}[k]$ indicates how much of one signal is present in the
other signal at a given delay $k$. Hence, it describes the mutual relationship
between the two signals as a function of the displacement of one relative to
the other. Cross-correlation is often used to search for desired features or
patterns within a signal.

The correlation coefficient between two signals is a statistical measure that
quantifies the degree to which the two signals are linearly related. It is a
normalized form of the cross-correlation function and is widely used in signal
processing and statistics to assess the strength of the linear relationship
between two signals [39]. For discrete-time signals $x[n]$ and $y[n]$, each of
length $N$, the correlation coefficient $\rho_{xy}$ is defined as:

$\rho_{xy}=\frac{\sum_{n=0}^{N-1}(x[n]-\mu_{x})(y[n]-\mu_{y})}{\sqrt{\sum_{n=0}^{N-1}(x[n]-\mu_{x})^{2}\sum_{n=0}^{N-1}(y[n]-\mu_{y})^{2}}}$
(24)

where $\mu_{x}$ and $\mu_{y}$ are the means of the signals $x[n]$ and $y[n]$,
respectively.
The correlation coefficient lies between $-1$ and $+1$. Values close to $+1$
indicate a strong positive linear relationship, i.e. as one signal increases,
so does the other. Values close to $-1$ indicate a strong negative linear
relationship, where one signal increases as the other decreases. Values close
to $0$ suggest a weak or no linear relationship between the signals.

###  II-D Signal Convolution

Convolution is an integral operation on two signals that produces a third
signal; it expresses the amount of overlap of one signal as it is shifted over
another signal. The convolution of two continuous-time signals $x(t)$ and
$h(t)$ is given by:

$x(t)\ast h(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau$ (25)

where $\ast$ denotes the convolution operator, which is defined as the
integral of the product of $x(t)$ with a time-reversed and shifted version of
$h(t)$. The time reversal of $h(\tau)$, represented by $h(-\tau)$, involves
reflecting the signal about the vertical axis that represents time origin
($\tau=0$). Accordingly, the integral is evaluated for all shift values $t$,
producing the convolution signal $x(t)\ast h(t)$. For discrete-time signals
$x[n]$ and $h[n]$ the convolution is expressed as:

$x[n]\ast h[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k]$ (26)

The convolution process can be visualized as one signal sliding over the other
signal, computing the area of overlap as a function of the degree of overlap.
In contrast to correlation, which measures the similarity between two signals
as a function of the displacement of one relative to the other, the
convolution represents the amount of overlap between two signals as one is
inverted and shifted over the other signal. Therefore, while correlation
measures the similarity between two signals, convolution can be viewed as a
measure of the effect of one signal on the other.

Convolution is particularly useful for describing how a linear time-invariant
(LTI) system or filter responds over time to an input signal since the output
of an LTI system or a filter for any arbitrary input can be determined by
convolving the input signal with the system’s impulse response. LTI Systems
are systems that are characterized by linearity and time-invariance.
Linearity, as explained earlier, implies that the response (output) to a
weighted sum of inputs is the weighted sum of the responses to each input.
Time-invariance means the system’s response to a given input does not change
over time. In signal theory, the impulse response of an LTI system or a
filter, denoted as $h(t)$ for continuous-time representation or $h[n]$ in
discrete-time case, represents the system’s output when subjected to an
impulse function. In continuous-time representation, the impulse function is
denoted as $\delta(t)$ and is defined as a function that is zero at all times
except at $t=0$, where it is infinitely high in such a way that its integral
over time is $1$. In discrete time, the impulse function is denoted as
$\delta[n]$ and is defined as a sequence that is zero at all samples except at
$n=0$, where its value is $1$. The impulse response fully characterizes the
system’s or filter’s behavior. Hence, given the impulse response of an LTI
system or a filter for any arbitrary input, its output can be determined by
convolving the input signal with the impulse response.

###  II-E Signal Entropy

Entropy is an important concept that originated in information theory as a
measure of ”information” in a random variable. In signal processing, entropy
quantifies the degree of unpredictability or randomness in a signal, thereby
reflecting its structure and offering a quantitative measure of various signal
characteristics. The fundamental concept of entropy-based applications in
signal processing is to treat the signal as a random stochastic process. In
this context, the signal samples represent a collection of possible outcomes
of the process; each outcome is associated with a probability of occurrence.
Hence, for a discrete signal where its samples are represented as a set of
possible outcomes $x_{1},x_{2},\dots,x_{n}$ with associated probabilities
$p(x_{1}),p(x_{2}),\dots,p(x_{n})$, the Shannon entropy is calculated as
follows:

$H(X)=-\sum_{i=1}^{n}p(x_{i})\log p(x_{i})$ (27)

This formula calculates the average ”information,” ”surprise,” or
”uncertainty” inherent in the signal’s possible outcomes. In information
theory, high entropy suggests that a signal contains a wide range of
information. In signal processing, high entropy values are associated with
high degrees of unpredictability, uncertainty, irregularity, and randomness,
indicating a complex signal with more varying structures compared to
predictable or repetitive patterns within the signal. On the other hand, low
entropy values often imply a high degree of uniformity and predictability
within a signal. This is illustrated in Fig. 7, which compares the entropy
values of two signals: the random vibration signal $v(t)$ and a square wave.
Each signal contains $12800$ samples. The signals are segmented into $128$
segments so that each segment contains $100$ samples. Accordingly, the Shannon
entropy ($H$) is calculated for each segment to quantify uncertainty within
each segment and to assess signal irregularity in terms of the variation in
entropy values between signal segments.

Figure 7: A comparison between Shannon entropy values of a random vibration
$v(t)$, and a square wave.

The comparison shows that segments of the random vibration have higher entropy
values than those of the square wave, indicating a higher degree of
uncertainty. Moreover, the high variation of entropy values among different
segments of the random vibration reflects the complex and irregular patterns
in the signal. These concepts find useful applications in signal processing,
including but not limited to:

  * •

Measurement of complexity: Entropy serves as a metric for signal complexity,
with higher entropy values indicating more complex signals. Such signals are
typically characterized by their low compressibility.

  * •

Uncertainty Quantification: Entropy is a direct measure of the uncertainty in
a signal. A signal with high entropy suggests less predictability and greater
uncertainty about its current state or future values.

  * •

Regularity Analysis: Lower entropy values indicate a higher degree of
regularity in the signal, which could indicate a fault or abnormal behavior in
applications such as condition monitoring and fault detection.

  * •

Randomness Evaluation: The degree of randomness in a signal can be assessed
using entropy. High entropy signals have a high degree of randomness, whereas
low entropy signals involve minimal randomness.

The Shannon entropy formula in (27), introduced by Claude Shannon in 1948
[40], is the fundamental form of entropy that is widely used in signal
processing. Spectral Entropy is another type of entropy that applies Shannon
entropy to the power spectrum of a signal, providing insight into its spectral
characteristics. Other types of entropy include approximate entropy and sample
entropy, commonly used in time-series analysis to measure the regularity and
unpredictability of fluctuations in data series. More information on the
different types of entropy can be found in [41].

##  III Signal Transformation and Analysis

Signal transformation is a fundamental concept in signal theory; it plays an
important role in many signal processing applications, such as signal
analysis, filtering, denoising, compression, and feature extraction.
Transforming signals from their time-domain representations into other-domain
representations allows the signal to be analyzed in a new paradigm, revealing
new aspects of the signal that are not apparent in the time domain. The
process of signal transformation involves mapping a signal from its original
domain (usually time) to a new domain. This is achieved by computing the inner
product of the signal with a set of basic functions (kernels) that serve as
the building blocks for representing the signal in the new domain. These
functions are chosen based on the properties desired in the transformed
domain, for example, sinusoidals in Fourier transforms and wavelets in wavelet
transforms. The inner product measures how much of the base function is
present in the signal. A large value of the inner product indicates a high
degree of similarity or a strong presence of the characteristics represented
by the base function in the signal. This section reviews common frequency-
domain and time-frequency domain transformations widely used in signal
processing.

###  III-A Fourier Transform

The Fourier transform (FT) is one of the most widely used transforms in signal
processing. It provides a frequency-domain representation of the signal,
revealing its spectral contents and enabling frequency-based analysis. The
basic principle underlying the FT is that any waveform can be expressed as a
combination of sinusoidal components with different amplitudes, frequencies,
and phases. Accordingly, FT attempts to decompose signals into their
constituent sinusoidal components. FT is mathematically expressed as:

$F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-j\omega t}\,dt$ (28)

where,
$F(\omega)$ represents the FT the function f(t),
$f(t)$ is the original time-domain signal,
and $e^{-j\omega t}$ is the complex exponential function used in the
transform.
The discrete Fourier transform (DFT) is the discrete version of the continuous
FT, which is used for analyzing discrete signals; it is defined as:

$X[k]=\sum_{n=0}^{N-1}x[n]e^{-\frac{j2\pi}{N}kn}$ (29)

where,
$X[k]$ represents the DFT of the discrete time-domain signal, $x[n]$,
$N$ is the total number of samples in $x[n]$,
$k=0,1,...,N-1$, is the index in the frequency domain,
and $n=0,1,...,N-1$, is the index in the time domain.
Both the FT and the DFT are reversible operations where the frequency domain
representation of a signal can be transformed back into its original time-
domain form by applying the inverse transform operation. This property is
fundamental in signal processing and has various applications, such as signal
compression, filtering, and noise reduction.

The fast Fourier transform (FFT) and its inverse, IFFT, are computational
algorithms that efficiently implement the DFT and its inverse, IDFT. The key
advantage of FFT and IFFT is their high computational efficiency. NumPy and
SciPy Python libraries offer reliable functions for FFT and IFFT computation.
When computing the FFT of a signal, two critical parameters need to be
considered: the sampling rate, $f_{s}$, and the FFT length, $n$. The values of
these parameters determine the frequency range and frequency resolution of the
resulting spectrum. It is essential to choose an appropriate FFT length,
usually a power of two (e.g., 256, 512, 1024), as the FFT algorithm performs
efficiently when the number of points is a power of two. The FFT length also
determines the number of frequency bins in the spectrum, where each bin
corresponds to a specific range of frequencies within the total frequency
spectrum analyzed by the FFT. The number of unique frequency bins in the FFT
output is:

$\frac{n}{2}+1$ (30)

The bin width defines the frequency resolution, $\Delta f$, of the FFT
representing the smallest distinguishable frequency in the spectrum. It is
defined as:

$\Delta f=\frac{f_{s}}{N}$ (31)

Frequency resolution is critical in frequency analysis because it affects the
accuracy with which closely spaced frequency components can be resolved in the
resulting spectrum. If the FFT length is set equal to the number of samples
$N$ in the input signal, the FFT operates directly on the signal as it is. If
the FFT length is set greater than $N$, the signal is typically zero-padded to
the desired length. Zero padding does not alter the actual frequency content
of the signal but increases the number of frequency bins in the FFT result,
leading to a finer frequency resolution in the spectrum. On the other hand,
setting the FFT length to less than $N$ truncates the signal, resulting in
loss of information and reduced frequency resolution. The maximum frequency
that can be accurately represented in the spectrum is the Nyquist frequency of
the signal ($\frac{f_{s}}{2}$, which equals half of the sampling rate.
Frequencies higher than the Nyquist frequency will not be correctly resolved
in the spectrum. The output of the FFT function is an array that contains
complex Fourier coefficients representing the amplitude and phase information
of each frequency bin. The function fftfreq in Python can be used to obtain
the frequency bins; it takes two inputs, the FFT length $n$ and the sampling
period, $T_{s}=\frac{1}{f_{s}}$, and returns an array containing frequency
bins. Accordingly, the frequency spectrum is constructed by mapping the
coefficients to the $x$-axis and the corresponding frequency bins to the
$y$-axis. Since Fourier coefficients are complex numbers, each bin is
expressed either by the magnitude or power of the corresponding Fourier
coefficients, resulting in amplitude spectrum or power spectrum, respectively.
This gives a two-sided frequency spectrum symmetric around the $y$ axis,
showing positive and ”negative” frequencies. The negative frequency spectrum
is inherent to the Fourier analysis of signals; it is a mirror of the positive
”real” spectrum around the $y$-axis. Thus, it represents redundant frequency
information. Reconstructing a positive spectrum involves discarding the second
half of coefficients and frequency bins arrays. Thus, to accurately represent
the signal’s energy in the frequency domain, it’s necessary to multiply the
magnitudes of Fourier coefficients by $2$ to restore the energy distribution
in the positive frequency since the signal’s energy was equally divided
between the positive and negative frequencies. It is worth noting that the
number of Fourier coefficients is directly proportional to the length of the
input signal, as stated in (30). Consequently, the magnitude of the FFT
coefficients increases with the length of the signal. Therefore, when longer
signal segments are used, the frequency spectrum displays larger values as
more components contribute to the sum. Therefore, to reflect the true
amplitudes of the signal components, the magnitudes of the Fourier
coefficients are usually normalised by the signal length.

To demonstrate the concept of FT in resolving various frequency components, a
sinusoidal signal, denoted as $s_{3}(t)$ composed of three components, is
created as shown in Fig. 8.

Figure 8:  Composite sinusoidal signal $s_{3}(t)$.

The used sampling rate is $1600$ Hz, and the length of the signal is $1536$
data points; accordingly, this corresponds to $0.96$ seconds of the signal’s
time duration. The signal consists of three sinusoidal components with
different durations, amplitudes and frequencies. The first component consists
of $960$ data points and oscillates at $100$ Hz with an amplitude of $10$. The
second component contains $64$ data points with a frequency of $400$ Hz and an
amplitude of $5$. The third component consists of $512$ data points
oscillating at a dominant frequency of $200$ Hz and an amplitude of $20$.
Consequently, the time duration which corresponds to each of the components
are $0.6$ seconds, $0.04$ seconds, and $0.32$ seconds, respectively. Compared
to the first and third components, the second component represents weak, rapid
oscillations of very short duration, simulating a transient component of
particular importance in the composite sinusoidal. The frequency spectrum of
the signal is displayed in Fig. 9, which clearly shows that the three
components in the composite signal at $100$ HZ, $200$ Hz, and $400$ Hz are
accurately resolved and represented in the spectrum. The peaks correspond to
the amplitudes of the components and therefore convey how much energy is being
dissipated within each component. In practical terms, the presence of a high
peak in the spectrum indicates the presence of a dominant (energetic)
oscillatory component in the time domain signal. The frequency value
associated with this peak reflects the oscillation rate of the component.

Figure 9:  Frequency spectrum of $s_{3}(t)$.

While the FT can identify the three frequency components within the signal,
its analysis spans the signal’s entire duration, which is evident from the
infinite integral in the transform equation (28). Thus, although it accurately
distinguishes between different frequency components, it cannot provide
temporal information about the timing of the occurrence of these frequencies
within the signal. This lack of time resolution in the FT limits its
applications to pure-frequency analysis only. Additionally, the FT is not
ideal for analyzing signals with highly time-localized components, like short
bursts with high energy concentration. Such components produce a broad range
of frequencies in the frequency spectrum of the signal due to the inherent
uncertainty principle associated with Fourier analysis [42], which states that
a signal cannot simultaneously have an arbitrarily small duration in time and
an arbitrarily narrow bandwidth in frequency. This means that if a signal is
highly localized in time, it must spread widely in the frequency domain and
vice versa. This concept is demonstrated in Fig. 10 and Fig. 11, which show a
composite sinusoidal signal, denoted as $s_{m3}(t)$, and its frequency
spectrum, respectively. The signal $s_{m3}(t)$ is a modified version of the
sinusoidal signal $s_{3}(t)$ where the duration of the second component is
reduced to $0.05$ (The duration of the first component is increased
accordingly), and its amplitude is increased to $30$ to simulate localized
oscillations of high energy concentration.

Figure 10: Composite sinusoidal signal $s_{m3}(t)$.

Figure 11: Frequency spectrum of $s_{m3}(t)$.

As depicted in Fig. 11, the frequency of the localized component in
$s_{m3}(t)$ is broadly spread across the frequency spectrum, which poses a
challenge in distinguishing subtle frequency components in practical
applications.

###  III-B Short-Time Fourier Transform

The short-time Fourier transform (STFT) bridges the gap between time-domain
and frequency-domain analysis, making it particularly useful for analyzing
nonstationary signals whose spectral characteristics evolve over time. Unlike
FT, which spans the signal’s entire duration for frequency analysis, STFT
analyzes the signal over finite, overlapping time windows, thereby preserving
temporal information. The computation of the STFT involves dividing the time-
domain signal into segments of equal time length, which is achieved by
multiplying the signal with a sliding window function. The FFT is then
computed for each segment, providing frequency bins and corresponding Fourier
coefficients for each segment. For a continuous signal, $x(t)$, the STFT is
mathematically defined as:

$STFT\\{x(t)\\}(\tau,\omega)=\int x(t)w(t-\tau)e^{-j\omega t}\,dt$ (32)

where $w(t-\tau)$ is the window function centered at time $\tau$, and $\omega$
is the angular frequency. For discrete signal $x[n]$, the discrete STFT is
defined as:

$DSTFT[n,k]=\sum_{m=0}^{N-1}x[n+m]w[m]e^{-j2\pi\frac{km}{N}}$ (33)

where,
$N$ is the size of the window, defining the number of signal samples included
in each signal’s segment for analysis,
$n$ is the time index around which the window is centered,
$k$ is the frequency bin index,
$w[n+m]$ represents the signal samples within the window,
$w[m]$ is the window function of size $N$,
and $e^{-j2\pi\frac{km}{N}}$ is is the DFT kernel.
The SciPy Python library provides an efficient implementation of SFTF and its
inverse transformation. The output of the STFT function contains three sets:

  * •

The temporal set contains the duration of each segment in seconds. Its size
equals the number of segments.

  * •

The frequency set that contains groups of frequency bins corresponding to each
time segment. The size of the set is equal to $\frac{n}{2}+1$.

  * •

Coefficients set: A matrix containing Fourier coefficients in each frequency
bin for every segment. Thus, its size is $d_{1}\times d_{2}$, where $d1$ and
$d2$ are sizes of frequency and temporal sets, respectively. STFT features are
usually obtained by examining the coefficients matrix. However, since Fourier
coefficients are complex numbers, extracting features in the STFT commonly
involves analyzing the magnitudes of these coefficients.

Heatmaps serve as an effective tool for visualizing the STFT output, providing
a clear representation of the variations in the frequency content of the
signal over time. These heatmaps, commonly referred to as spectrograms, are
generated by mapping the temporal set to the $x$-axis and frequency set to the
$y$-axis. The ”intensity” of each time-frequency pair in the $x-y$ plane is
represented by the magnitude of the corresponding Fourier coefficients. Fig.
12 shows the STFT of the composite signal $s_{3}(t)$ introduced earlier. The
signal is segmented into ten segments using a Hamming window with an overlap
of $50\%$ of the segment length. The Fourier coefficients are computed for
each segment using an FFT length of $2048$. Compared to the signal’s frequency
spectrum depicted in 9, the STFT spectrogram shows the spectral contents and
reveals their temporal characteristics simultaneously, providing time-
frequency analysis of various frequency components in the signal.
Specifically, The duration of each frequency component in the signal is
clearly shown over the timeline ($X$-axis). Furthermore, the temporal edges,
represented by the start and end times of each component, are well-defined,
thereby reflecting high temporal resolution that allows the identification of
the transient component at $400$ Hz.

Figure 12: STFT spectrogram of the composite sinusoidal signal $s_{3}(t)$.

In real-world scenarios, the efficiency of the STFT in time-frequency analysis
is significantly influenced by the choice of its parameters, particularly the
segment length and the percentage of overlap between adjacent segments.
Segment overlap provides continuity and reduces information loss between
adjacent segments. A high degree of overlap introduces more redundancy into
the segments and improves time resolution, which is particularly useful when a
large segment length is used. The segment length determines the resolution of
the time-frequency analysis. A smaller segment length enables better time
resolution, which helps identify rapid transient patterns in the signal, as
demonstrated in Figure 12. On the other hand, a larger segment length enhances
frequency resolution by allowing more frequency cycles to fit within the
segment. However, this comes at the cost of reduced temporal resolution, as
transient changes in the signal over short time durations may be overlooked.
Therefore, there is a trade-off between time and frequency resolutions in the
STFT analysis. This concept is demonstrated in Fig. 13, which shows the STFT
spectrogram of $s_{3}(t)$ re-computed by dividing the signal into three
segments only instead of ten segments. The frequency resolution is represented
by the ”thickness” of the timeline. A thinner timeline— reflecting a finer
frequency resolution— allows precise identification of the corresponding
frequency bin on the $y$-axis compared to a thick timeline that spans several
frequency bins. A comparison of the two spectrograms shows that the use of a
longer segment improves the frequency resolution but reduces the temporal
resolution, resulting in overlapping temporal edges that obscure the
identification of the transient component.

Figure 13: STFT spectrogram of the composite sinusoidal signal $s_{3}(t)$.

The fixed segment length is a major limitation in STFT that results in a
uniform resolution analysis of the signal and, hence, leads to an inherent
compromise between time and frequency resolutions when applying STFT for time-
frequency analysis.

###  III-C Power Sepctral Density

Power spectral density (PSD) describes how the power of a signal is
distributed across different frequencies, providing a measure of the power per
unit frequency. This is achieved through the normalization of the power
spectrum by frequency resolution (frequency bin width), converting the power
spectrum into a power spectral density that shows the power distribution per
unit frequency. Mathematically, the PSD $S_{xx}(f)$ of a continuous-time
signal $x(t)$ is defined as the FT of the autocorrelation function
$R_{xx}(\tau)$; it is expressed as:

$S_{xx}(f)=\int_{-\infty}^{\infty}R_{xx}(\tau)e^{-j2\pi f\tau}\,d\tau$ (34)

where,

$R_{xx}(\tau)=E[x(t)x(t+\tau)]$ (35)

and $E[]$ denotes the expected value. For discrete signals, the discrete PSD
(DPSD) is utilized, involving the DFT of the autocorrelation sequence. The
DPSD $S_{xx}[k]$ of a discrete-time signal $x[n]$ is given by:

$S_{xx}[k]=\sum_{n=-\infty}^{\infty}r_{xx}[n]e^{-j\frac{2\pi}{N}kn}$ (36)

where $r_{xx}[n]$ is the autocorrelation sequence of $x[n]$.

The SciPy Python library provides two methods for PSD estimation: the
periodogram method (scipy.signal.periodogram) and the Welsh method
scipy.signal.welch [43]. The periodogram estimation involves obtaining the
spectral power of the signal by applying FFT and squaring the magnitude of the
resultant FFT coefficients. The PSD estimate is obtained by normalizing the
spectral power by the frequency resolution of the FFT. One of the limitations
of the periodogram method is its high variance, especially when the number of
samples in the signal is limited. The Welsh method overcomes this limitation
by segmenting the signal into overlapping segments using a sliding window,
computing the periodogram for each segment, and averaging the periodograms to
obtain the PSD estimate. This effectively reduces noise and provides a
smoother estimate than the periodogram method. Fig. 14 compares the two
methods in estimating PSD of the random vibration signal $v(t)$ introduced
earlier. In the Welsh method, the signal is divided into four segments with
$50\%$ of overlap. It is evident that averaging the estimate over several
segments greatly reduces the variance of the estimate, resulting in a smoother
PSD. The smoothness of the PSD is of significant importance in power spectral
analysis and feature extraction since high estimation variance can hinder
major characteristics in the PSD, such as peaks. A smooth PSD allows precise
identification of such characteristics, improving the sensitivity of the
extracted features.

Figure 14: Periodogram and Welsh’s PSD estimations of vibration signal $v(t)$.

Similar to the STFT, the choice of parameters in the Welch method, such as
window type, segment length and overlap, has a significant effect on the PSD
estimate. While STFT and the PSD possess similar aspects, they serve different
purposes and are based on distinct theoretical foundations. Understanding
their differences is crucial, especially in machine learning and feature
extraction applications. Table I compares STFT and PSD and summarises their
main characteristics.

Table I: Comparison between STFT and PSD Properties |  STFT |  PSD
---|---|---
Purpose and Concept |  STFT is used for time-frequency analysis of signals, especially nonstationary signals where frequency components vary over time |  PSD is used to describe the power distribution of a signal across frequency. It is particularly useful in understanding the energy content of different frequency components of a signal
Representation |  The result of STFT is a two-dimensional representation of the signal, showing both time and frequency dimensions. This is crucial in analyzing signals whose spectral properties evolve over time |  PSD provides a one-dimensional frequency domain representation, indicating how power is distributed across frequencies. Unlike STFT, it does not offer time-domain information
Characteristics |  STFT involves a trade-off between time and frequency resolution controlled by the window size. A larger window offers better frequency resolution but poorer time resolution, and vice versa. |  PSD assumes signal stationary, making it less suitable for analyzing signals with time-varying characteristics
Application |  STFT is used for analyzing signals with time-varying spectral properties. |  PSD is more suited for stationary signals or for assessing overall frequency content

###  III-D Wavelet Transform

The accuracy of time-frequency analysis depends heavily on the localization of
the base function in time and frequency domains. In Fourier analysis, the
basis functions are sinusoidals (complex exponentials) that are perfectly
localized in the frequency domain since each corresponds to a single frequency
component with no frequency spread. In the time domain, these sinusoidal
functions extend infinitely, lacking time localization due to their infinite
duration. In the context of wavelets, signal analysis is based on a family of
wavelets that are localized in both time and frequency domains. These wavelets
are generated from a single mother wavelet through scaling and translation,
facilitating signal analysis at different frequencies and time intervals. A
wavelet is a mathematical function that satisfies certain conditions, such as
finite energy and a zero mean. The term ”wavelet” itself means a small wave,
which captures the essence of these functions and how they behave. The main
characteristics of a wavelet function include the following:

  * •

Zero Mean: This ensures the wavelet oscillates around the zero level; this is
crucial for detecting changes in the signal since it allows it to capture both
positive and negative variations in the signal effectively.

  * •

Finite Energy: This implies that the total energy of a wavelet is finite,
ensuring integrability and existence of the inner product with the signal. The
energy of a given wavelet function is calculated by integrating the squared
amplitude of the wavelet over its entire duration. Mathematically, the total
energy of a wavelet function, $\psi(t)$, is expressed as:

$\int_{-\infty}^{\infty}|\psi(t)|^{2}\,dt<\infty$ (37)

A wavelet with finite energy is inherently localized in time and has compact
support, meaning it is non-zero only over a limited range, and its
oscillations are confined to a small region. This time localization allows
wavelets to effectively represent signals that have nonstationary or transient
components. A wavelet’s integrability implies that it won’t produce infinite
results when integrated over its domain. This is crucial for signal analysis
since it is essentially carried out by computing the inner product of the
signal with scaled and transformed versions of the wavelet.

Wavelet analysis is conducted through two operations, scaling and translation:

  * •

Scaling (Dilation): This involves stretching or compressing the wavelet.
Scaling changes the frequency content of the wavelet, allowing the signal to
be analysed in different frequency bands. Specifically, the oscillation
frequency of a wavelet is directly affected by the scale. At lower scales, the
wavelet compresses and oscillates more rapidly, which allows it to detect
high-frequency components or rapid changes in the signal. On the other hand,
at higher scales, the wavelet is stretched, resulting in fewer oscillations
for the same amount of time, making it more sensitive to low-frequency
components or slow-changing features in the signal.

  * •

Translation (Shifting): This refers to moving the wavelet along the time axis,
enabling the wavelet to analyze different signal parts at a given scale.

These aspects allow high time-frequency localization and provide a multi-
resolution analysis through varying scales and translations. The choice of the
mother wavelet and scale values is derived by signal characteristics and
application-specific requirements. Attributes such as orthogonality, compact
support, vanishing moments, and the ability to capture signal discontinuities
and singularities play a crucial role in this choice. The wavelet family
includes various mother wavelets; each with unique properties that cater to
various signal types and application requirements. Fig. 15 displays the time-
domain waveforms of some of the commonly used wavelet functions.

Figure 15: Some of the commonly used wavelet functions.

####  III-D1 The Continuous Wavelet Transform (CWT)

The CWT of a signal $x(t)$ can be expressed mathematically as follows:

$CWT_{x}(a,b)=\frac{1}{\sqrt{|a|}}\int_{-\infty}^{\infty}x(t)\psi^{*}\left(\frac{t-b}{a}\right)dt$
(38)

where,
$a$ is the scale parameter, $b$ is the translation parameter,
$CWT_{x}(a,b)$ represents the wavelet coefficient at scale $a$ and translation
$b$,
$\psi(t)$ is is the mother wavelet, and $\psi^{*}(t)$ is its complex
conjugate.
The output of CWT is a matrix of wavelet coefficients, where each row contains
the coefficients of the corresponding scale. These coefficients represent the
correlation between the signal and a scaled and translated wavelet, indicating
how closely the signal matches the wavelet at those specific scales and
translations. This emphasizes the importance of selecting a proper wavelet
function for signal analysis. The CWT output is commonly visualized through 2D
heatmaps, known as scalograms, where the $x$-axis and $y$-axis represent time
and scale, respectively. The intensity of the heatmap is represented by the
magnitude of the wavelet coefficients, which reflect the signal wavelet energy
at different scales and translations. The scale parameter inversely relates to
the signal’s frequency content. High scales correspond to low frequencies,
capturing the slower-varying components of the signal. On the other hand, low
scales, corresponding to high frequencies, capture rapidly changing components
within the signals. Hence, scalograms show how the spectral content varies
over time, enabling the identification of transient or nonstationary
components within the signal. Fig.16 displays the scalogram of the composite
signal $s_{3}(t)$ using a scale range from 1 to 135. The fbsp, displayed in
the bottom right corner of Fig. 15, is employed in the analysis; fbsp is a
spline wavelet constructed using a spline function. The selection of this
wavelet is based on its sinusoidal-like waveform, resembling the waveform of
the composite signal $s_{3}(t)$. The scalogram demonstrates a multi-resolution
signal analysis at various scale values. In particular, scales greater than
$120$ effectively capture the slowly varying components (components $1$ and
$3$) of the signal. On the other hand, the raid change in the signal,
represented by component $2$, can only be detected at small-scale values
(scales between $30$ and $40$).

Figure 16:  CWT (scalogram) of the composite signal $s_{3}(t)$.

####  III-D2 Discrete Wavelet Transform (DWT)

DWT is a wavelet-based signal decompression tool that uses discrete values for
scale and translation, reducing redundancy and computational complexity. In
DWT, wavelet multi-resolution analysis is achieved by decomposing the signal
into high and low-frequency components, commonly known as elementary modes,
using digital filter banks and down-sampling operations. The Key parameters
include the type of mother wavelet and the degree of decomposition.
Mathematically, the DWT of a discrete signal $x[n]$ can be expressed as:

$DWT_{x}[j,k]=\sum_{n=-\infty}^{\infty}x[n]\psi_{jk}[n]$ (39)

where,
$DWT_{x}[j,k]$ represents the wavelet coefficient at the $j$-th level of
decomposition and the $k$-th position,
$\psi_{jk}[n]$ are the discrete wavelets, defined as scaled and shifted
versions of the mother wavelet.
In the decomposition process, the signal undergoes convolution with
consecutive low-pass and high-pass filters followed by down-sampling. The main
aspects and steps of the decomposition process include:
Mother Wavelet: The first step involves selecting an appropriate wavelet base
function, often called the ”mother wavelet.”
Scaling and Translation: The decomposition process starts by scaling and
translating the mother wavelet to generate a family of wavelet functions that
serve as bases to decompose the signal. In the DWT, the mother wavelet is used
in a discretized form where scaling and translation are done in discrete
steps, typically in powers of two for scaling and integer multiples of that
scale for translations. For each level of decomposition, the wavelet is scaled
by a factor of $2^{j}$, where $j$ indicates the level of decomposition. This
scaling effectively compresses the wavelet to capture signal features at
different frequencies. Consequently, the wavelet is translated along the
signal in steps that are multiples of the current scale factor $2^{j}$. This
stepwise translation allows the wavelet to cover the entire signal, ensuring
that all signal parts are analyzed.
High-Pass Filter: This filter represents the mother wavelet since its
coefficients are essentially a discrete representation of the mother wavelet.
As a high-pass filter, it extracts high-frequency detail components from the
signal, capturing rapid changes within the signal.
Low-Pass Filter: This filter represents the scaling function, as its
coefficients are derived from the scaled wavelets. As a low-pass frequency
filter, it extracts low-frequency approximate components that vary slowly over
time.
Down-Sampling: After separating the signal into high and low-frequency
components, each component is down-sampled by a factor of two. Down-sampling
involves keeping every second sample of the filtered signals. This process
reduces the sample rate by half, effectively decreasing the data size and
ensuring the transform is computationally efficient.
Approximate and Detail Coefficients: Approximation and detail coefficients are
outputs of down-sampling operations performed on approximate and detail
components, respectively. The approximate coefficients represent a smoothed
and down-scaled version of the original signal, while the detail coefficients
describe high-frequency content in the signal.
Iterative Analysis: The process of high and low pass filtering followed by
down-sampling is performed iteratively on the resultant approximation
coefficients only until reaching the specified number of levels ($j$) or until
reaching maximum decomposition constrained by signal length. This process
creates a $j$-level tree structure of wavelet coefficients, shown in Fig.17.
Signal reconstruction: The original signal can be perfectly reconstructed by
applying the inverse DWT, which processes the resultant coefficients through
reconstruction filters in an inverse manner to the decomposition process.
Elementary modes: These modes are obtained through the individual
reconstruction of the resultant coefficients (coefficients cD1 to cDj and cAj
as shown in Fig. 17), allowing the processing of the composite signal at the
level of its constituent modes. Since these modes form the underlying
structure that constitutes the original signal, perfect reconstruction of the
original signal is possible through direct summation of these modes.

Figure 17: $7$-level DWT decomposition tree.

Fig. 18 displays elementary modes of the random vibration signal $v(t)$ along
with their corresponding spectra. These modes are obtained through $7$-level
DWT decomposition of the signal using Daubechies db4 wavelet (shown in the top
right corner of Fig. 15).

Figure 18: $7$-level DWT decomposition of random vibration signal $v(t)$:
Elementary modes and their corresponding frequency spectra.

The Daubechies family of wavelets, especially those with higher orders (e.g.,
db4, db5), are compactly supported and well-localized in time, making them
highly effective for analyzing signals that contain transient patterns or
nonstationary components. This is particularly useful in vibration analysis
[44], where detecting sudden changes or faults is essential for condition
monitoring and diagnostics. The DWT decomposition of $v(t)$ iteratively
extracts slowly varying modes of the signal, corresponding to lower
frequencies in the signal at each level. This progressively reduces the
resolution of high-frequency details with each level of decomposition, making
DWT well-suited for signals where the information of interest is concentrated
in lower frequencies.

####  III-D3 Stationary Wavelet Transform (SWT)

A major limitation of the DWT is the lack of time invariance, which stems
primarily from the down-sampling process. The down-sampling performed at each
level makes the output of DWT sensitive to shifts in the input signal. Even
small shifts can cause significant changes in the resulting wavelet
coefficients, which is problematic in applications where the precise timing of
signal features is essential. Furthermore, down-sampling reduces the signal
length by half at each level of decomposition, which may lead to the loss of
specific signal details, especially at higher decomposition levels. SWT is an
extension of the DWT that addresses these limitations by omitting the down-
sampling step, thus maintaining the original size of the signal throughout the
decomposition levels. Unlike DWT, which provides a critically sampled (non-
redundant) representation, SWT provides a more detailed but redundant
representation of the signal. SWT is implemented by applying high-pass and
low-pass filters without downsampling the filtered signals. Instead, the
filters are up-sampled at each level to match the size of the original signal.
The up-sampling of filters is done by inserting zeros between the filter
coefficients, effectively increasing the size of the filters as the
decomposition level increases. This leads to an overcomplete signal
representation and increases computational burden compared to DWT.

####  III-D4 Wavelet Packet Transform (WPT)

The WPT extends the capabilities of classical wavelet-based decomposition.
Unlike the DWT, which limits its decomposition to approximation coefficients,
the WPT performs a full decomposition at each level by processing detail and
approximation coefficients, as depicted in Fig. 19, thereby capturing both
low-frequency and high-frequency components. While this process is more
computationally intensive, it enhances frequency resolution in the analysis,
enabling the detection of subtle signal features that DWT may overlook.

Figure 19: $3$-level WPT decomposition tree.

Fig. 20 displays elementary modes of the vibration signal $v(t)$, obtained
through $3$-level WPT decomposition using the db4 wavelet, along with their
corresponding spectral contents.

Figure 20: $3$-level WPT decomposition of random vibration signal $v(t)$:
Elementary modes and their corresponding frequency spectra.

In contrast to DWT, which offers a hierarchical frequency decomposition with
good frequency resolution at lower frequencies, WPT provides a more uniform
time-frequency localization across the spectrum due to its symmetric
decomposition of detail and approximation coefficients at each level. Hence,
it maintains a balance between time and frequency resolution throughout the
frequency spectrum, allowing for a more precise localization across a wider
range of frequencies. Additionally, the spectrum in DWT is skewed towards
lower frequencies with each successive level of decomposition. While this can
be advantageous for analyzing signals characterized by low-frequency
components, it inadequately represents higher-frequency details. WPT, on the
other hand, provides a balanced spectrum with a more equitable representation
of both high and low-frequency components.

Wavelet decomposition offers an effective tool for analyzing complex real-
world signals that involve transit and nonstationary components, opening the
door for a wide range of applications such as:

  * •

Multi-resolution analysis: Wavelet decomposition methods attempt to decompose
the signal into elementary modes characterized by their high time-frequency
localization, offering the capacity for conducting a multi-resolution analysis
of the original signal.

  * •

Feature Extraction: The obtained modes encompass meaningful information as
they exhibit distinct amplitude and frequency characteristics. Accordingly,
distinctive features directly related to the inherent structure of the signal
can be extracted from these modes or their corresponding coefficients.

  * •

Noise Reduction: Noise and signal components often have different
characteristics in the wavelet domain. By thresholding certain wavelet
coefficients before signal reconstruction [45], noise can be effectively
reduced or eliminated.

  * •

Signal Enhancement: Wavelet reconstruction can be utilized to enhance signals,
where certain characteristics can be amplified or smoothed out as required,
improving the quality of the reconstructed signal.

  * •

Signal Compression: By keeping only the most significant wavelet coefficients
for reconstruction, signals can be compactly represented with minimal loss of
quality.

As previously stated, selecting the appropriate mother wavelet and
decomposition level are crucial steps in wavelet analysis. Different wavelets
have different shapes and properties, such as smoothness, symmetry, and the
number of vanishing moments, making the selection mainly dependent on
application requirements and the shape similarity between the wavelet and the
signal. The decomposition level determines the resolution at which the signal
is analyzed. Higher decomposition levels allow for the analysis of lower-
frequency components but also increase computational complexity and could
dilute the significance of higher-frequency components. The maximum level of
decomposition in DWT and WPT is typically limited by the length of the signal,
$N$. Generally, each level of decomposition reduces the number of data points
by half. Thus, the number of maximum possible decomposition levels equals
$log_{2}(N)$. In SWT, the maximum decomposition level is determined by the
signal’s length and the filters’ length at each decomposition level, which
increases with each level due to the up-sampling operation. Further
decomposition becomes impractical when the filters become too long relative to
the signal.

The Python PyWavelets package provides comprehensive modules for wavelet
analysis. It supports custom wavelets and provides various wavelet functions
such as CWT, DWT, SWT, WPT and over 100 built-in wavelet filters. Other
libraries supporting wavelet analysis include PyCWT, a Python module for
continuous wavelet spectral analysis, and scipy.signal, which provides CWT and
other wavelet functions, particularly for signal filtering.

###  III-E Hilbert Transfrom

The Hilbert transform (HT) is a fundamental operator in signal theory; it is
particularly useful in obtaining the analytic signal representation of real-
valued signals. An analytic signal is a complex-valued representation of the
signal that provides a comprehensive way to describe both its amplitude and
phase characteristics. The HT of a signal $x(t)$ is defined as:

$H\\{x(t)\\}=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}d\tau$
(40)

The transform essentially modifies the phase of each frequency component of
the signal by $\pm 90^{\circ}$. The $X_{a}(t)$ of $x(t)$ is formed by
augmenting the signal with its HT $H\\{x(t)\\}$ as the imaginary part.
Mathematically, it is expressed as:

$X_{a}(t)=x(t)+jH\\{x(t)\\}$ (41)

Represented in its polar form, the analytical is expressed as:

$x_{a}(t)=A(t)e^{j\theta(t)}$ (42)

where, $A(t)$ is the instantaneous amplitude (also known as the amplitude
envelope), it is given by:

$A(t)=|x_{a}(t)|=\sqrt{x(t)^{2}+H\\{x(t)\\}^{2}},$ (43)

and $\theta(t)$ is the instantaneous phase; given by:

$\theta(t)=\arctan\left(\frac{H\\{x(t)\\}}{x(t)}\right)$ (44)

Accordingly, the instantaneous frequency $f(t)$ can be obtained by taking the
derivative of the instantaneous phase $\theta(t)$:

$f(t)=\frac{1}{2\pi}\frac{d\theta(t)}{dt}$ (45)

The instantaneous amplitude of a signal shows how the signal strength varies
with time. The instantaneous phase provides the phase angle of the signal as a
function of time. The instantaneous frequency reveals how the frequency
content of a signal evolves over time, providing a dynamic view of the
signal’s spectral properties. By obtaining instantaneous amplitude, phase, and
frequency information, the HT serves as an effective tool to identify distinct
characteristics, such as common patterns and sudden changes in phase and
frequency where relevant features can be extracted accordingly. For instance,
HT can be used to obtain instantaneous amplitudes of amplitude-modulated
signals such as fault vibration signals. In these signals, vibrations of
damaged bearings are manifested as modulations in the amplitude of the
generated vibration signal. Hence, obtaining the signal envelope provides an
efficient approach for extracting fault signature frequencies from the
envelope’s spectrum. Fig. 21 displays a composite signal, $c(t)$, that has a
time duration of $1$ second. The signal is composed of various components so
that it exhibits four sudden changes as follows:

  * •

Rapid oscillations of $50$ Hz from $0.3$ time instant to $0.5$ time instant.

  * •

Rapid oscillations of $100$ Hz from $0.7$ time instant to $0.9$ time instant.

  * •

Abrupt phase change of $180$ degrees at $0.4$ time instant.

  * •

Abrupt phase change of $90$ degrees at $0.6$ time instant.

Figure 21: Composite signal $c(t)$.

The function scipy.signal.hilbert in the scipy.signal library is used to
compute the HT of $c(t)$. The function returns the analytical signal
$c_{a}(t)$ as an output. The instantaneous amplitude, phase, and frequency are
calculated accordingly and displayed in Fig. 22.

Figure 22: Hilbert Transform of the signal $c(t)$.

As shown in the plots, the obtained instantaneous information provides
critical insights into the signal’s behavior, highlighting common patterns and
sudden changes. Specifically, the envelope shows the main pattern of the
signal, reflecting its energy content over time. The abrupt phase changes at
the $0.4$ and $0.6$ time instants are manifested as small step changes at
these time instants in the instantaneous phase. Moreover, the phase is
directly impacted by temporal changes in the signal frequency, as shown in the
plot. The instantaneous frequency provides vital information about frequency
variations and sudden changes in the signal. Particularly, the instantaneous
frequency plot shows the rapid oscillations of $50$ Hz and $100$ Hz between
$0.3-0.5$ and $0.7-0.9$ time instants, respectively. Additionally, the sudden
changes in frequency and phase are manifested as a spike at the corresponding
time instant in the plot. In real-world applications, such spikes represent
significant features in the signal that would be of particular interest.

###  III-F Hilbert–Huang transform

The Hilbert–Huang transform (HHT) [46] is a well-known adaptive method for
analyzing nonlinear and nonstationary signals. Unlike traditional transforms,
HHT does not impose a fixed basis function in signal analysis. Instead, it
utilizes an adaptive approach to analyze the signal, making it highly
responsive to variations in signal. The HHT is a two-stage process that
involves empirical mode decomposition (EMD) of the signal followed by the
Hilbert Spectral Analysis (HSA).

####  III-F1 Empirical Mode Decomposition (EMD)

EMD is an adaptive decomposition method that decomposes the signal into a set
of simpler functions known as intrinsic mode functions (IMFs). EMD forms the
basis of adaptive mode decomposition (AMD) methods introduced over the last
three decades. In contrast to wavelet-based decomposition, AMD uses adaptive
approaches rather than priori basis functions to decompose the signal. Hence,
the obtained IMFs are not influenced by a priori basis function [47]. This
makes AMD particularly effective when dealing with nonlinear and nonstationary
signals due to its ability to adapt to varying signal characteristics.
However, this adaptability comes at the cost of higher computational
complexity. The EMD algorithm, introduced in 1998 as part of the HHT,
represents the core of all AMD methods. The process of EMD, commonly known as
sifting, involves the following steps:

  1. 1.

Identification of Extrema: Start by identifying all the local maxima and
minima of the original signal.

  2. 2.

Envelope Creation: Construct the upper and lower envelopes of the signal by
interpolation between the local maxima and minima, respectively. These
envelopes essentially outline the signal’s oscillatory amplitude.

  3. 3.

Mean Envelope Calculation: Compute the mean of the upper and lower envelopes.

  4. 4.

Extraction of Detail: Subtract the mean envelope from the original signal.
This step isolates a component of the signal that potentially qualifies as an
IMF.

  5. 5.

IMF Check: Verify if the extracted component meets the criteria for being an
IMF:

     * •

The number of extrema (local maxima and minima) and the number of zero
crossings in the component must either be equal or differ at most by one. This
condition ensures that the IMF captures a well-defined oscillatory mode
without bias toward upward or downward trends.

     * •

The mean value of the envelope defined by the local maxima and the envelope
defined by the local minima must be zero at any point in the component. This
condition guarantees that the IMFs have well-balanced oscillations around
zero, reflecting true oscillatory modes rather than trends or biases in the
signal.

If the component doesn’t qualify for an IMF, return to step 2 and use this
component as the new signal.

  6. 6.

Completion of one IMF: Once an IMF is identified, subtract it from the
original signal. This leaves a residue signal.

  7. 7.

Repetition: Repeat steps 1-6 on the residue signal. This process is iterated,
yielding a new IMF at each iteration, and the residual becomes the input for
the next iteration.

  8. 8.

Termination (Stopping Criterion): The process terminates when the residual
signal becomes a monotonic function from which no more IMFs can be extracted,
or it becomes sufficiently small in amplitude (based on a predefined
threshold)

The output of the EMD process of the random vibration signal $v(t)$ is
displayed in Fig. 23, including IMFs, residual, and their respective spectra.

Figure 23: EMD of the random vibration signal $v(t)$: IMFs, residual, and
their respective frequency spectra.

As shown, the EMD adaptively decomposed the signal into ten IMFs of distinct
frequency spectra where the frequency contents of IMFs decrease with each
successive IMF. While EMD is well-known to be effective in analyzing complex
signals, the decomposition process is sensitive to noise since it works
directly on the signal’s extrema and minima. Additionally, the process
requires a complete envelope to decompose the signal into IMFs accurately,
which can be difficult to define at the boundaries due to the absence of
neighboring data points. This can lead to distortions of IMFs near the ends of
the signal, commonly known as the end effect [48], impacting the accuracy and
reliability of EMD for signal analysis, particularly when the signal’s
behavior at the boundaries is critical. Another challenge in EMD is mode
mixing [49]; ideally, each IMF should represent a unique frequency component.
However, in signals with close spectral proximity or exhibit intermittency
characteristics such as abrupt amplitude and frequency changes, the standard
EMD algorithm may not be able to separate these features accurately. As a
result, a single IMF may contain information from multiple modes, which can
lead to a loss of significant physical interpretation of the IMFs. Further
improvements to the standard EMD algorithm have been introduced to address
these limitations. For example, the ensemble empirical mode decomposition
(EEMD) algorithm [50] was developed by incorporating an ensemble approach and
adding finite amplitude white noise to alleviate problems of mode mixing and
end effects. Accordingly, the natural oscillatory modes are obtained by
averaging the corresponding IMFs obtained from an ensemble of the signal and
the added noise. Other common AMD algorithms include variational mode
decomposition (VMD)[51], local mean decomposition [52], and empirical wavelet
transform (EWT) [53]. More details on AMD techniques and their applications
can be found in [54][55][56][57].

####  III-F2 Hilbert Spectral Analysis

The second step of HHT involves applying the HT to each IMF to obtain its
analytic signal and, accordingly, its insinuations amplitude, phase, and
frequency. An IMF, due to its defining conditions, ensures that its HT is
well-behaved and meaningful for signal analysis [58]. Specifically, due to the
equal number of extrema and zero crossings, an IMF exhibits a consistent
oscillatory pattern, which is essential for a coherent HT. Further, the
symmetry of the IMF’s envelope about the zero line guarantees that the IMF
doesn’t have a bias towards positive or negative values, facilitating a more
accurate HT.
Mathematically, the HSA can be expressed as follows: Given a signal $x(t)$,
decomposed into $n$ IMFs $c_{i}(t)$, where $i=1,2,,n$, the analytic signal
$Z_{i}(t)$ for each IMF is formed as follows:

$Z_{i}(t)=c_{i}(t)+jH\\{c_{i}(t)\\}$ (46)

where $H\\{c_{i}(t)\\}$ is the HT of $c_{i}(t)$. The analytical signal in the
polar form is expressed as:

$Z_{i}(t)=A_{i}(t)e^{j\theta_{i}(t)}$ (47)

Accordingly, the instantaneous amplitude $A_{i}(t)$ is:

$A_{i}(t)=|Z_{i}(t)|=\sqrt{c_{i}(t)^{2}+H\\{c_{i}(t)\\}^{2}},$ (48)

and the instantaneous phase $\theta_{i}(t)$ is:

$\theta_{i}(t)=\arctan\left(\frac{H\\{c_{i}(t)\\}}{c_{i}(t)}\right)$ (49)

Accordingly, the instantaneous frequency $f_{i}(t)$ is obtained by taking the
derivative of $\theta_{i}(t)$:

$f_{i}(t)=\frac{1}{2\pi}\frac{d\theta_{i}(t)}{dt}$ (50)

The instantaneous frequencies and amplitudes obtained from the HT, can be used
to construct a time-frequency distribution of the signal. This is accomplished
by plotting the instantaneous amplitude or energy against the instantaneous
frequencies of the IMFs for each point in time. The result is a distribution
that shows how the signal’s frequencies evolve over time, highlighting the
signal’s nonlinear and nonstationary characteristics. The time-frequency-
energy distribution is commonly visualized as a 2D heatmap, where the $x$-axis
represents time, the $y$-axis represents frequency, and the color intensity
represents the amplitude or energy level of the signal at each time-frequency
point. Considering the composite sinusoidal signal $s_{3}(t)$, Fig. 24
displays the resulting IMFs and residuals from the EMD process, as well as
their spectral contents. Consequently, Fig. 25 shows the 2D heatmap
visualization of the output from the HTT. These results are obtained using
textitemd, vmd, and hht built-in MATLAB functions, which, in contrast to
Python, provide a mature and user-friendly environment for EMD, VMD, and HHT
computations and visualization.

Figure 24: EMD of the composite sinusoidal signal $s_{3}(t)$: IMFs and
residual alongside with their respective frequency spectra.

Figure 25: 2D heatmap visualization of the HTT of $s_{3}(t)$.

It is obvious that EMD and, consequently, HHT effectively capture all
components in the signal, including the transit osculations of $400$ Hz at
$0.6$ seconds, and effectively shows the temporal energy distribution of the
signal at different frequencies. This demonstrates the advantage of the HHT
that lies in its adaptability to nonlinear and nonstationary signals.

However, the performance of the HHT is tied to the choice of AMD method, which
is fundamental to the accurate decomposition of the signal into its IMFs.
Furthermore, the employed stopping criterion for the sifting process [59] and
the used interpolation method for envelope estimation [60], have direct
impacts on the effectiveness of the HHT. A proper stopping criterion ensures
that the decomposition process neither overfits nor underfits the signal,
thereby preserving the signal’s essential characteristics without introducing
artifacts. The interpolation method for envelope estimation influences the
HHT’s performance since it determines how well the upper and lower envelopes
capture the true oscillatory modes of the signal.

###  III-G Wigner–Ville Distribution

The Wigner–Ville distribution (WVD), as a time-frequency analysis tool is
characterized by its ability to provide highly resolved energy-time-frequency
representations of signals. The WVD of a continuous-time signal $x(t)$ is
expressed as:

$WVD(t,f)=\int_{-\infty}^{\infty}x(t+\frac{\tau}{2})x^{*}(t-\frac{\tau}{2})e^{-j2\pi
f\tau}\,d\tau$ (51)

where, $x^{*}(t)$ is the complex conjugate of $x(t)$, $\tau$ and $f$ are
variables representing time-shift and frequency, respectively.
According to this expression, the kernel function of WVD is given by:

$x(t+\frac{\tau}{2})x^{*}(t-\frac{\tau}{2}),$ (52)

which is essentially the instantaneous autocorrelation function (IACF) of
$x(t)$, commonly known as Wigner auto-correlation function [61]. The use of
the complex conjugate of the signal in the autocorrelation accounts for the
magnitude and phase parts of the signal. Note that the traditional
autocorrelation function of (20) integrates the autocorrelation over time $t$,
providing a global measure of the correlation. In contrast, the IACF provides
a local measure of the correlation, which is better suited to reflect local
and time-varying features of the signal. Hence, facilitating a more accurate
representation of the signal’s characteristics [62]. The Fourier kernel
$e^{-j2\pi f\tau}$ in the WVD expression transforms the IACF into the
frequency domain with respect to the time-shift variable $\tau$, thereby
providing a joint time-frequency representation of the signal. For a discrete
signal $x[n]$ with $N$ samples,The discrete WVD is given by:

$WVD[n,k]=\sum_{m=-N}^{N}x[n+\frac{m}{2}]x^{*}[n-\frac{m}{2}]e^{-j2\pi\frac{km}{N}}$
(53)

Here, $x^{*}[n]$is the complex conjugate of $x[n]$, and $k$ represents the
discrete frequency index. The term,

$x[n+\frac{m}{2}]x^{*}[n-\frac{m}{2}],$ (54)

captures the auto-correlation of the signal at different lags.

The output of WVD, $WVD(t,f)$, is essentially a 2D function of time and
frequency that is commonly visualized as a 2D heatmap where the value of
$WVD(t,f)$ at any point $(t,f)$ reflects signal’s energy at that particular
time and frequency. The high energy concentration of WVD results from its
inherent quadratic (nonlinear) structure since the energy itself is a
quadratic representation of the signal [63]. In contrast to FT, STFT, and CWT,
which are liner transforms333In signal theory, a linear transform satisfies
the superposition or linearity condition, which states that if the input
signal $x(t)$ is a linear combination of some signal components, then, the
transom of $x(t)$ is also a linear combination of the transforms of each
signal component., the WVD is nonlinear since it involves a product of the
signal with a time-shifted version of itself (correlation). This energetic and
correlative nature of the WVD [63] makes it a unique tool for energy-time-
frequency analysis compared to other transforms, however the quadratic nature
of the WVD presents a major challenge when analysing multi-component signals
due to the high level of cross-term interference [64] that appears in the
time-frequency representations. Cross-terms are caused by undesired cross-
correlation between various signal components, leading to a false indication
of signal components between the desired auto-correlation terms in the WVD
representation of the signal. To overcome this problem, window-based
approaches [65, 66] and kernel-based approaches [67, 68], such as the pseudo-
Wigner distribution (PWVD), are commonly utilized to suppress cross-terms.
PWVD is a windowed version of the WVD that applies a smoothing kernel to WVD
in either time-domain or frequency domain to reduce the cross-terms at the
cost of a certain loss in resolution. The PWVD of a signal $x(t)$ can be
expressed as:

$PWVD(t,f)=\int_{-\infty}^{\infty}h(\tau)x\left(t+\frac{\tau}{2}\right)x^{*}\left(t-\frac{\tau}{2}\right)e^{-j2\pi
f\tau}d\tau$ (55)

where $h(\tau)$ is the smoothing kernel in the time domain. In the frequency
domain, the smoothing operation becomes the convolution operation between the
smoothing kernel and the WVD. The smoothing operation essentially averages the
signal’s energy over the time and/or frequency domains, reducing the presence
of cross-terms arising from the quadratic nature of the WVD. However, this
comes at the cost of blurring the signal’s time-frequency representation. The
limitation of WVD resulting from undesired cross-correlation and the advantage
of PWVD are demonstrated by comparing the WVD and PWVD of the composite
signal, $c(t)$, which, as illustrated earlier, comprises multiple time-varying
components. The MATLAB function wvd is used to compute WVD and PWVD and
visualize the outputs as 2D heatmaps. This built-in function provides
straightforward computations and visualizations of various WVD types. The WVD
and PWVD results are displayed in Fig.26 and Fig.27, respectively.

Figure 26: WVD of the composite sinusoidal signal $c(t)$.

Figure 27: PWVD of the composite sinusoidal signal $c(t)$.

The comparison reveals that the WVD representation is impaired by false
indications of non-existent signal components due to the cross-correlation
between the actual signal components. In contrast, the PWVD, through its
smoothing operation, eliminated the interference of cross-terms and accurately
resolved all the time-varying frequency components. However, it can be seen
that the improvement achieved with the PWVD led to a reduction in frequency
resolution, manifested as wider lines in the PWVD’s heatmap compared to the
WVD’s heatmap, thereby affecting the precise estimation of the corresponding
frequency values.

###  III-H Conclusive Comparison

Aimed at providing interested readers a thorough understanding of signal
processing fundamentals, this tutorial offered a comprehensive introduction to
signal characteristics with an in-depth discussion of prevalent signal
transformation and analysis tools. The discussion focused on elucidating the
main concepts, mathematical foundations, essential characteristics,
advantages, and limitations. Additionally, the tutorial addressed
implementation considerations by underscoring the use of programming libraries
and built-in functions to facilitate efficient implementations of these tools.
Furthermore, the codes employed for generating the illustrative plots
throughout this section have been made publicly accessible as previously
mentioned.

Throughout the section, it became evident that selecting the proper tool
depends entirely on signal nature, application-specific requirements, and
available computational resources. For stationary signals, where the primary
interest lies in frequency analysis, The FT efficiently obtains the frequency
spectrum, thereby enabling the identification of predominant frequency
components in the signal. The PSD is particularly advantageous in scenarios
involving comparative spectral analysis of multiple signals varying in length
and/or bandwidth. It provides a normalised spectral density measure that
facilitates effective comparison. The HT is particularly effective in
applications where the signal envelope is of primary importance. Moreover,
instantaneous phase and frequency information can be efficiently obtained from
the analytical signal of the HT.

For signals with time-varying spectral characteristics, the STFT provides a
computationally efficient tool for the time-frequency representation of the
signal. However, its fixed window size results in a uniform resolution across
all frequencies, reducing its effectiveness for applications requiring
variable-resolution analysis. In such contexts, Wavelet analysis presents a
viable solution due to its ability to perform multi-resolution analysis
through scaled and shifted versions of a wavelet base function.

For the analysis of nonstationary and nonlinear signals characterized by
rapidly changing components, the HHT provides an adaptive and more robust
approach than CWT but with more computational requirements. The WVD offers a
powerful method for providing a joint energy-time-frequency representation of
the signal, although it is more computationally intensive than the other
tools. It is particularly useful in high-resolution analysis scenarios or
where energy concentration is a focal requirement.

Signal decomposition techniques allow the analysis of the elementary
components of a signal. This makes it possible to extract unique manageable-
sized features closely related to the signal’s inherent structure. Regarding
wavelet decomposition, its performance is highly dependent on the base wavelet
function and the level of decomposition. In contrast, AMD methods use an
adaptive mechanism rather than an a priori basis function to decompose the
signal, thereby generating elementary modes that are not influenced by an a
priori basis function. Moreover, AMD methods dynamically adapt to varying
signal characteristics, making them effective for analyzing nonlinear and
nonstationary signals. However, AMD methods are more computationally intensive
than wavelet decomposition. Additionally, AMD methods are sensitive to noise
and their efficiency highly depends on the employed stopping criterion and the
used interpolation method for envelope estimation. Table II summarizes the
comparison, highlighting the main aspects of each tool regarding signal
nature, application requirements, and computational complexity.

Table II: Comparison between common signal analysis and transformation tools Tool | Signal Nature | Application | Computational Complexity
---|---|---|---
Fourier Transform (FT) | Stationary | Frequency analysis | Low
Power Spectral Density (PSD) | Stationary | Spectral analysis | low
Hilbert Transform (HT) | Stationary/nonstationary | Envelope and instantaneous phase/frequency analysis | low
Short-Time-Frequency Transform (STFT) | nonstationary | Fixed-resolution time-frequency analysis | moderate
Wavelet Transform (WT) | nonstationary | Multi-resolution time-frequency analysis | moderate
Hilbert-Huang Transform (HHT) | nonstationary and nonlinear | Adaptive time-frequency analysis | High
Wigner-Ville Distribution (WVD) | nonstationary | High-resolution energy-time-frequency analysis | High
Wavelet Decomposition (WD) | nonstationary | Signal decomposition | moderate
Adaptive Mode Decomposition (AMD) | nonstationary and nonlinear | Signal decomposition | High

##  IV A Typical Signal-Based ML Pipeline

In signal-based ML applications, the pipeline of signal processing is
typically divided into three core stages: preprocessing, processing, and
application, as illustrated in Fig. 28. This section addresses each stage,
highlighting its purpose and main aspects.

Figure 28: Overview of a typical signal-based ML framework.

###  IV-A Preprocessing

The first stage encompasses the preprocessing of acquired signals. Here, it is
essential to distinguish between data preprocessing techniques[69], commonly
used in ML models, and signal preprocessing, which involves the tasks of
signal smoothing, signal denoising, and signal segmentation. Signal
preprocessing is performed on the raw signals, whereas data preprocessing
techniques, such as normalization and scaling, are applied to the extracted
features.

####  IV-A1 Signal Smoothing

The acquired signal may exhibit rapid amplitude changes between successive
samples. Such random changes in amplitude can negatively impact application-
level performance. Furthermore, during measurements, the signal is susceptible
to outlier samples caused by external factors such as instrument malfunction
[70]. Signal smoothing attempts to remove such impairments in the signal by
adjusting the amplitudes of individual samples with respect to the amplitudes
of adjacent samples. Smoothing acts as an approximation filter that works on
$N$-successive samples at a time and outputs $N$ smoothed samples. Common
smoothing methods involve the well-known moving average (MA) filter and the
Savitzky–Golay (SG) filter [71]. MA filter offers a simple method to smooth
the signal by averaging neighboring data points while preserving the signal’s
shape. It operates by averaging a set of signal data points (window ) and
sliding this window across the signal to smooth out short-term fluctuations.
The SG filter uses a different approach to smoothing the signal by performing
a local polynomial regression on the signal data points present in the moving
window to provide more nuanced smoothing that preserves the original shape and
characteristics of the signal. The design of an SG filter involves properly
selecting the window size and the order of the polynomial [72, 73, 74, 75].
Signal smoothing helps refine the signal’s waveform shape and captures its
main patterns, particularly useful in many signals-based applications.
However, as a lossy operation, smoothing could cause a notable distortion in
applications where peak-related features are of main interest. More
information on signal smoothing can be found in [76, 77, 78, 79].

####  IV-A2 Signal Denoising

Denoising refers to the process of removing noise from the acquired signal or,
in other words, reconstructing the desired signal from its acquired noisy
representation. This can be expressed mathematically as follows:

$w\left(t\right)=s\left(t\right)+n\left(t\right)$ (56)

where $w\left(t\right)$ denotes the noisy signal, $s\left(t\right)$ is the
desired signal, and $n\left(t\right)$ is the noise component, Noise can be
defined as any unwanted signal present in the measurement process other than
the desired signal [80]. Signal denoising and signal smoothing are often
treated synonymously in the literature [81], as both aim to refine the
signal’s waveform while preserving meaningful patterns within the signal.
However, they have different impacts on the noise and frequency contents of
the signal. Smoothing is typically applied in the time domain on a sample-by-
sample basis while denoising attempts to remove the whole noise component,
$n\left(t\right)$.

Furthermore, in smoothing, the rapid fluctuations can be viewed as highly
oscillating components of small amplitudes compared to the whole signal over a
relatively long duration. From a frequency point of view, such oscillations
represent high-frequency components of low amplitudes in the signal’s
spectrum. Smoothing, therefore, acts as a low-pass filter that removes these
high-frequency components from the signal. On the other hand, signal denoising
typically involves the use of various signal processing methods to develop
more advanced noise filtering techniques. Signal denoising is addressed with
more details in Section VI-A.

####  IV-A3 Signal Segmentation

Depending on the length of the acquired signal, expressed in terms of the
number of samples, it might be necessary to divide the signal into smaller
segments for the following practical considerations:

  * •

Online deployment: Online applications require working on a pre-defined length
of the input signal.

  * •

Real-time and delay-sensitive applications: Given the sampling frequency,
$f_{s}$ in samples per second, the time duration, $t_{s}$, of the segment,
expressed in seconds equals:

$t_{s}=\frac{N_{o}}{f_{s}}$ (57)

Hence, in real-time and delay-sensitive applications, the number of samples,
$N_{o}$, in the input segment is of paramount importance since time delay
increases as a function of $N_{o}$ [44].

  * •

Computation burden: The computational complexity of signal-based ML systems
increases as a function of $N_{o}$ [44].

  * •

Effectiveness of extracted features: The number of samples, $N_{o}$, in the
input segment significantly shapes the discriminative characteristics of the
extracted features. Shorter segments may not provide an adequate
representation of the signal’s characteristics, reducing the reliability of
extracted features. Conversely, very long segments could incorporate a high
redundancy level, weakening the discriminative capability of the extracted
features.

Constant segmentation is commonly used in various applications where the
segment length, $N_{o}$, is kept constant. Consequently, signals are divided
into smaller segments of equal lengths. Determining the value of $N_{o}$ is
application-specific and typically relies on achieving good trade-offs among
the aforementioned practical considerations. Constant segmentation is commonly
implemented using a sliding window of size $N_{o}$. Another aspect of constant
segmentation is the percentage of overlap between successive segments.
Introducing overlap between segments increases the number of segments
extracted from the signal. The more overlap, the more segments are extracted.
However, increasing the overlap increases correlation or dependence among
adjunct segments. In general, the extent to which overlap can be used and its
percentage depends on the nature of the application and the signal processing
techniques used [82, 83, 84]

In some applications, the acquired signal exhibits random intervals of
activity and inactivity, as observed in various sound and biomedical signals.
In such scenarios, constant segmentation becomes unreliable, as it does not
count for these inherent intervals. In such situations, adaptive segmentation
[85, 86, 87, 88, 89, 90, 91] is commonly utilized to dynamically adjust the
segment length, $N_{o}$. Adaptive segmentation exploits the underlying
characteristics of the signal and adapts $N_{o}$ accordingly. Unlike fixed
length segments, adaptive length segments are more discriminative because the
signal’s characteristics and inherent patterns change from segment to segment,
allowing more discriminative features to be extracted. However, adaptive
segmentation involves more computational burden compared to constant
segmentation.

###  IV-B Processing

The processing stage involves processing the resultant segments to extract
appropriate features that serve as inputs to ML models. The methods and
algorithms available in the literature can be broadly categorized into two
main approaches: feature extraction and signal-to-image conversion. In feature
extraction, signal processing or deep learning (DL) can be used to extract
relevant features from the input segment. The next section comprehensively
covers the topic of feature extraction. Signal-to-image conversion approaches
[92, 93, 94, 95, 96, 97, 98, 99, 100, 101] transform 1-dimensional (1D) signal
segments into 2-dimensional (2D) spatial representations, allowing to leverage
DL models in signal-based applications. Gramian Angular Field (GAF) and Markov
Transition Field (MTF) [92] are two well-known techniques that are commonly
used to transform time series into images. Advanced techniques involve the use
of signal processing to generate 2D time-frequency representations of the
signal segments. The essence of these techniques is to represent time-
frequency-energy characteristics of the signal and features as width-height-
color intensity, mimicking image features. However, to serve training and
testing purposes of DL models, a large number of transformed images should be
available. Nevertheless, signal-to-image conversion is especially feasible in
integrated systems such as unmanned aerial vehicles (UAVs) and autonomous
vehicles (AVs) [102]. In these systems, computer vision applications, such as
object detection and navigation, and signal-based applications are deployed
within the same environment. Therefore, it is deemed computationally efficient
to convert signals into images and use transfer learning approaches that
utilize the trained deep-learning infrastructure for inference tasks of
signal-based applications. Moreover, signal-to-image conversion and transfer
learning offer a practical solution when the size and type of available signal
data are insufficient to fulfill training and testing requirements.

###  IV-C Application

The application is the final stage of a typical signal-based ML pipeline. As a
task-specific stage, it uses appropriate ML models [103, 104, 105] to address
tasks, including classification, clustering, and anomaly detection. Feature
preprocessing, hyperparameter tuning, and relevant metrics [106, 107] are
often used to improve and evaluate model performance. Table III serves as a
high-level overview of a typical signal-based ML framework, highlighting each
stage’s purpose, inputs, functions, and outputs.

Table III: Inputs, functions, and outputs of each stage of typical signal-based ML framework Stage | Inputs | Functions | Outputs
---|---|---|---
Preprocessing | Acquired signal+impairments (noise / outliers) | Smoothing / Denoising / Segmentation | Segmented signal
Processing | Segmented signal | Feature extraction / Signal-to-image conversion | Features/ Images
Application | Features/ Images | Classification / Clustering / Anomaly detection | Results

Figure 29: Taxonomy of feature extraction techniques.

##  V Feature Extraction Methods

This section provides a comprehensive review of existing feature extraction
methods. The review utilizes a new hierarchical taxonomy, depicted in Fig. 29,
to categorize and structure various methods that exist in the literature. The
taxonomy starts by grouping the existing methods under two main categories:
DL-based and signal processing-based. This categorization distinguishes
between two mechanisms that exist for feature extraction: feature learning and
feature engineering. The first mechanism relies on DL models— such as
convolutions neural networks (CNN), recurrent neural networks (RNN), and auto-
encoders— to learn high-level feature representations from signal data [108,
109, 110, 111, 112, 113, 114, 115, 116, 117]. The second mechanism, on the
other hand, employs signal processing to extract distinctive features from the
signal, deliberately engineered to emphasize specific attributes and reveal
certain characteristics. This deliberate engineering allows meaningful
connections to be made between the resulting features and various conditions
or classes that exist within the signals. In contrast, DL features are not
interpretable since DL models are black-box models, making establishing such
meaningful connections impossible. In addition, features extracted by DL
models are model-dependent, whereas features based on signal processing are
solely signal-dependent. This implies that variations in the model’s
architecture, parameters, training, or tuning lead to different feature
representations and, consequently, inconsistent performance results. Thus, it
can be deduced that in DL-based approaches, performance is model-dependent,
while in signal processing-based approaches, performance is mainly feature-
dependent. Moreover, DL features would exhibit higher dimensionality and
redundancy than signal processing-based features. This requires the use of
appropriate dimensionality reduction and feature selection techniques[111,
118, 119], thereby increasing the online processing time and computational
burden. Table IV summarizes the aforementioned comparison between signal
processing-based and DL-based approaches.

Table IV: A comparison between deep learning-based and signal processing-based feature extraction Aspect | Deep Learning Approaches | Signal Processing Approaches
---|---|---
Feature Extraction Mechanism | Feature learning | Feature engineering
Interpretability of Extracted Features | Not interpretabile | Interpretabile
Size of Required Dataset | Large | Small-to-medium
Data Dimensionality and Redundancy in Extracted Features | High, depends on DL model | Low-to-medium
Computational Cost | High | Moderate

A considerable amount of work in the literature uses hybrid approaches where
signal processing-based features are fed into DL models to further extract
highly distinctive representations from these features [120, 121, 122, 123,
124, 125, 126, 127, 128, 129, 130, 131, 132, 133]. Compared to DL-based and
signal processing-based features, hybrid features have higher discriminative
power, which generally leads to better performance. However, employing hybrid
approaches involves increased online processing time and higher computational
complexity, making practical deployment subject to trade-offs between system
complexity and performance requirements. In general, hybrid approaches prove
feasible in applications that involve complex signals, such as radio
modulation recognition [128, 129, 130, 131, 132, 133]. In these applications,
signals are randomly modulated, corrupted with high noise levels, and subject
to various types of interference.

Focusing on signal processing-based feature extraction, the taxonomy of Fig.
29, from a computational point-of-view, divides the existing methods into four
main categories: time domain-based methods, transform-based methods,
decomposition-based methods, and envelope-based methods. Time domain-based
methods extract features directly from time domain waveforms of the signal
without further processing. On the other hand, transform-based, decomposition-
based, and envelope-based methods use additional steps to process the signal
and, consequently, additional computations to extract features. The remainder
of this section reviews common methods that exist within each category.

###  V-A Time Domain-Based Methods

In time-domain-based methods, features are calculated from the signal’s
amplitude, representing specific aspects of the signal’s dynamics over its
time period. This allows for the quantification of amplitude variations from
one period to another. Conventional methods [134, 135, 136, 137, 138, 139,
140, 141, 142, 143, 144, 145, 146, 147, 148, 149] involve extricating shape
characteristics— such as maximum, peak-to-peak value, and crest factor— and
statistical properties of the input signal. Statistical features describe
characteristics of the probability distribution of the input signal; common
statistical features involve mean, root-mean-square (RMS) value, variance, and
high-order statistics (HOS), such as skewness and kurtosis. Compared to first
and second-order statistics (mean and variance), HOS features have more
distinguishable characteristics as they can identify more complex aspects of
the input signal, such as the shape of its distribution. Furthermore, HOS are
less sensitive to noise. Besides shape-based and statistical methods, time-
domain entropy methods [150, 151, 152, 41] are commonly utilized for feature
extraction. Entropy methods rely on the observation that signals belonging to
different classes or conditions typically exhibit different levels of
irregularity or dynamic changes depending on the underlying state. Hence,
different entropy measures can be utilized to quantify the irregularity of the
input signal. In contrast to statistical features, entropy-based features are
more sensitive to changes and complexities that arise within the signal. Thus,
they can capture variations and irregularities in the signal that may be
difficult to detect using traditional statistical measures. Additionally,
entropy-based features are resilient to noise and outliers. In general, time-
domain feature extraction approaches work directly on input signals, making
them conceptually straightforward and relatively easy to implement. Further,
they are computationally efficient since no further processing steps are
required, making them advantageous in applications where real-time processing
is crucial, especially with limited computational resources. However, time-
domain analysis is highly susceptible to noise since noise in the signal would
alter its amplitude and mask the dynamic characteristics of the signal.
Furthermore, to achieve reliable performance, input segments of relatively
long duration are usually required to capture the changes and complexities
that evolve within the signal over time.

###  V-B Transform-Based Methods

Transform-based approaches use advanced techniques to convert signals from the
time domain to the frequency domain or time-frequency domain. This allows for
the identification of frequency and frequency-temporal information,
facilitating robust signal analysis and feature extraction.

####  V-B1 Frequency-Based Methods

Frequency-based approaches represent signals in terms of their frequency
content, revealing details that are not apparent in the time-domain waveform.
Further, in contrast to time-domain analysis, frequency-based analysis allows
the isolation and removal of noise or unwanted components by applying
appropriate frequency filtering mechanisms to improve the signal-to-noise
ratio and improve reliability of the extracted features. Frequency-based
approaches involve the application of the Fourier transform (FT) to the input
signal to compute its frequency content, which can be broadly categorized into
two main categories: spectrum-based [153, 154, 155, 156, 157, 135, 158, 12,
159, 160, 161, 162, 163, 164, 165, 166] and spectral density-based [163, 167,
168, 169, 170, 171]. Spectrum-based approaches use the frequency spectrum of
the signal for feature extraction. Specifically, Some approaches directly use
resultant Fourier coefficients of the input segment as input features [153,
154, 155, 156, 157] since these coefficients describe the distribution of the
signal’s energy over the range of frequencies contained in the signal. Here,
dimensionality reduction and feature selection techniques are often used to
reduce the size of coefficients and select the most discriminative
coefficients.

Other approaches involve the extraction of various frequency-domain features
from the frequency spectrum, such as statistical properties, energy, entropy,
and correlation coefficients[135, 158, 12, 159, 160, 161, 162, 163, 164, 165,
166]. Spectral density-based approaches, on the other hand, utilize the power
spectral density (PSD) of the input signal to extract the features [163, 167,
168, 169, 170, 171] since it provides a normalized measure of the power per
unit frequency compared to the frequency spectrum. Entropy is frequently
utilized in PSD analysis to calculate Spectral Entropy (SE) [168, 169]. SE is
a very useful spectral feature that measures the irregularity or randomness of
power distribution across signal frequencies and, hence, can be used to
quantify the spectral complexity of the input signal. Other methods utilize
Mel-frequency cepstral coefficients (MFCCs) as input features [172, 173, 174,
175, 176, 177, 178, 179, 180, 181] because they effectively capture the shape
of the power spectrum. MFCCs are originally developed for audio signals [182]
and are gaining attraction in other various applications as well. A thorough
explanation of MFCCs, including computational steps and usage scenarios, can
be found in [182]. Higher-order spectra [183, 184, 185, 186] are commonly used
in frequency-domain analysis to conduct a more comprehensive spectral analysis
compared to second-order spectral analysis. Common higher-order spectra
include bispectrum and trispectrum [187, 188, 189, 190, 191, 192, 193, 194,
184, 195, 196], which correspond to the FT of the signal’s third-order
cumulant and the fourth-order cumulant, respectively.

A key advantage of higher-order spectra over power spectrum is their ability
to retain phase information [197]. Specifically, the power spectrum,
representing the second-order spectrum or the FT of the autocorrelation
function, captures the magnitude of frequencies present in a signal but
discards phase information. This limitation makes it challenging to identify
phase-related properties of the signal, such as frequency coupling or
interactions between different frequency components. In contrast, higher-order
spectra provide insight into the phase relationships among frequency
components by examining moments or cumulants of the signal beyond the second
order, enabling the detection of nonlinearities, phase coupling, and other
characteristics that are invisible in the power spectrum. However, higher
order spectra are more complex to calculate and interpret than traditional
power spectral analysis because they require more data for reliable estimation
and involve more sophisticated mathematical and computational methods.
Furthermore, due to their multi-dimensional nature, visualisation and
interpretation of higher order spectra can be challenging. Because of this
complexity, specific cross sections or slices of the bispectrum or trispectrum
are often analysed to extract meaningful information. This approach simplifies
the analysis by reducing the dimensionality of the data, making it easier to
visualise and interpret.

While frequency analysis accurately distinguishes between different frequency
components in the signal, the analysis spans the signal’s entire duration,
lacking the ability to provide temporal information about the timing of
occurrence of these frequencies within the signal. Additionally, the FT is not
ideal for analyzing signals that contain highly time-localized components,
such as short bursts with high energy concentrations, because the components
produce a wide range of frequencies in the frequency spectrum due to the
inherent uncertainty principle associated with Fourier analysis [42].

####  V-B2 Time-Frequency-Based Methods

In contrast to frequency-domain analysis, which lacks time resolution, time-
frequency transforms produce representations that map the signal’s energy
across both time and frequency, allowing for a localized analysis in the time-
frequency domain. This approach allows the identification of transient
oscillatory components within the signal, providing a more complete
understanding of its dynamics. Time-frequency transforms that are commonly
used in the literature for feature extraction include short-time Fourier
transform (STFT), [198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209,
210] Hilbert-Huang transform (HHT) [211, 212, 213, 214, 215, 216, 217, 218,
219, 220, 221], Wigner-Ville distribution (WVD) [222, 223, 224, 225, 226, 227,
228, 229, 230], and continuous wavelet transform (CWT) [231, 202, 232, 233,
234, 235, 236, 237, 228, 238, 239, 240, 241]. STFT is efficient in analyzing
nonstationary signals whose spectral properties vary over time. However, it
uses a fixed segment length in analyzing the signal, imposing a trade-off
between time-domain and frequency-domain resolutions. In CWT, on the other
hand, time-frequency analysis is based on a family of wavelets generated from
a single mother wavelet through scaling and translation, offering a multi-
resolution analysis of the signal at different frequencies (scales) and time
intervals (translations). Moreover, in contrast to STFT, which uses complex
exponentials as basis functions— that extend infinitely in the time domain—,
wavelet base functions are localized in time and frequency, making them more
suitable for signals with transient or highly localized components. The HHT
uses an adaptive approach for time-frequency analysis. Unlike STF and CWT, HHT
does not impose a fixed basis function on the signal analysis, but instead
uses an adaptive approach to analyze the signal.. This adaptivity makes it
highly responsive to variations in the signal, facilitating very effective
analysis of nonlinear and nonstationary signals. However, as an adaptive
analysis method, its performance depends heavily on two factors: the stopping
criterion for the sifting process [59] and the choice of the interpolation
method for envelope estimation [60].

A proper stopping criterion ensures that the decomposition process neither
overfits nor underfits the signal, thereby preserving the signal’s essential
characteristics without introducing artifacts. On the other hand, the
interpolation method for envelope estimation determines how well the upper and
lower envelopes capture the actual oscillatory modes of the signal, heavily
influencing the HHT’s performance. The WVD offers a time-frequency
representation with high energy concentration, facilitating detailed analysis
of a signal’s energy distribution over both time and frequency domains. The
high concentration of energy arises from the quadratic nonlinear structure of
WVD compared to STFt and CWT. This energetic and correlative nature of the WVD
[63] makes it very useful in analyzing single-component signals. However, the
quadratic nature of the WVD represents a major challenge when analyzing multi-
component signals due to high cross-term interference [64] caused by undesired
cross-correlation between various signal components, leading to a false
indication of non-existent signal components in the resultant time-frequency
representations of the signal. Window-based WVD [66, 65] and kernel-based
approaches [67, 68] are commonly used to reduce the cross-terms at the cost of
some loss in resolution.

From a feature extraction perspective, time-frequency-based methods can be
grouped under three main approaches in which the generated energy-time-
frequency mappings are treated differently:

  1. 1.

Transformation coefficients as input features: In these approaches, the
generated transform energy coefficients, representing signals’ energy
distribution across time-frequency instants, are used directly as input
features for the ML model [198, 199, 211, 231, 216, 219, 220, 240]. Feature
selection techniques are commonly used here to select the most distinctive
representations, which helps reduce redundancy and relaxes computational
requirements for training and inference.

  2. 2.

Energy-time-frequency 2D heatmaps as imagery data: The generated energy-time-
frequency representations are used to visualize 2D heatmaps that map the
signal’s energy across time-frequency instants. Accordingly, computer vision
and DL techniques are leveraged where these heatmaps serve as input images
[200, 201, 202, 203, 204, 205, 206, 207, 208, 212, 232, 233, 222, 223, 217,