Time-Frequency Analysis for Biomedical Engineering

8/12/2019 Time-Frequency Analysis for Biomedical Engineering

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Time Frequency Analysis andWavelet Transform Tutorial

Time-Frequency Analysis for Biomedical Engineering

Chia-Jung Chang ()

National Taiwan University


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ABSTRACT

Biomedical related research requires lots of mathematical and engineering

techniques to analyze data. Among the subfields, electrophysiological research

plays the core role. In this tutorial, several tools are examined, includingelectrocardiogram, and electroencephalogram. These are the most common

tools used to diagnose our physiological activities since neural responses carry

information. Because biomedical signals are usually non stationary, Fourier

transform is not suitable to apply here. Besides, traditional signals are analyzed

in frequency domain, separately from time domain, such that extraordinary

conditions are hard to be observed. To solve such problem, time-frequency

analysis and wavelet transform provide both time and frequency information

simultaneously.

In the following tutorial, I would like to talk about the theoretical background

of both time-frequency analysis and wavelet transform methods, including

what properties they have, their common types, and how to operate them.

Secondly, I would briefly introduce three common physiological tools such as

ECG, and EEG. Where they can be applied, what they are targeting, and what

analysis methods can be used, and how they perform will be described.


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CONTENTS

Abstract

I. Theoretical Background1. Time-Frequency Analysis Methods

1.1 Cohens Class Distribution1.1.1 Four Derivative Distributions1.1.2 Electrophysiological Applications

2. Wavelet Transform2.1 From Fourier to Wavelet Transform

2.1.1 Fourier Transform2.1.2

Short Time Fourier Transform2.1.3 Wavelet Transform

2.2 Continuous Wavelet Transform2.2.1 Properties2.2.2 Representative Signals

2.3 Discrete Wavelet Transform2.3.1 Properties2.3.2 Representative Signals

2.4 Selection of Base Wavelet for Biomedical Signals2.4.1 Overview2.4.2 Selection Criteria

II. Biomedical Applications3. Electrocardiography (ECG)

3.1 Introduction3.2 Method3.3 Result

4. Electroencephalography (EEG)4.1 Introduction4.2 Method4.3 Result

Summary

References


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1. Time-Frequency Analysis MethodsGreat progress has been made in applying linear time-invariant techniques in

signal processing. In such cases the deterministic part of the signal is assumed

to be composed of complex exponentials, the solutions to linear time invariantdifferential equations. However, many biomedical signals do not meet these

assumptions. Thus, the emerging techniques of time-frequency analysis can

provide new insights into the nature of biological signals.

There are several different time frequency analysis methods such as short time

Fourier transform, S transform, Wigner transform, and Cohens distribution. I

would focus on Cohens class distribution since it is used in biomedical signals

analysis more commonly than S transform. As for short time Fourier transform,

it is also a popular analysis tool, which will be described in details in the latterchapters.

1.1 Cohens Class DistributionCohens class distribution is defined as follows

, , , +

where Axis called ambiguity function of x(t)

, Cohens class distribution can be regarded as a series of methods to eliminate

cross-term produced by Wigner distribution, and to keep clarity. However, the

trade-off is that it takes large computation, and thus losses some properties. As

there are different ways to filter out cross-term, there are several forms.

1.1.1 Four Derivative DistributionsThere are four different distributions being introduced in this section.

Choi-Williamns distribution (CWD) is defined with

, Born-Jordan distribution (BJ) is defined with

,


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Zhao-Atlas-Marks distribution (ZAM) is defined with

, || While Cohens class distribution can filter cross-term effectively, resolution onthe time-frequency domain is lost at the same time. Another distribution has

been brought out, which is called Reduced Interference distribution (RID). It

can be regarded as smoothed Wigner distribution. After we use a lowpass filter,

the cross term can be easily eliminated, and the resolution is enhanced as well.

Reduced Interference distribution (RID) has been defined as

,

,

, ||||

||( )

where h(t) is a time smooth window, g(v) is a frequency smooth window.

1.1.2 Electrophysiological ApplicationsA specific subset of time-frequency distribution is Cohen's class distribution.

For these distributions, a time shift in the signal is reflected as an equivalent

time shift in the time-frequency distribution, and a shift in the frequency of thesignal is reflected as an equivalent frequency shift in the time-frequency

distribution. The Wigner distribution, the RID, and ZAM all have this property.

Tracing back to history, Kawabata employed the instantaneous power spectrum,

a measure of the rate of change of the energy spectrum, to study dynamic

variations in the EEG during photic alpha blocking and performance of mental

tasks. DeWeerd compared time-frequency analysis of event related potentials

obtained by a non uniform filter and the complex energy distribution function.

The applicability of the Wigner distribution for the representation of event

related potentials was examined by Morgan and Gevins.

The RID with a product kernel exhibits the interesting and valuable property of

scale invariance. In addition, the RID has information invariance. Cohen's class

distribution does not vary with time or frequency automatically. On the other

hand, the continuous wavelet transform exhibits scale invariance and time-shift

invariance. In Figure 1a, a spectrogram with a 256-point window is used. In

Figure 1b, a spectrogram with a 32-point window is used. Finally, in Figure 1c,

an RID is used. Spectrogram and RID are compared, and the first is the original


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EEG segment. The second is the original EEG segment time-scaled to preserve

energy. The third segment is a frequency-shifted version of the original EEG

segment.

Figure 1

The RID provides a high-resolution representation of both time and frequency.

Furthermore, the RID preserves this structure well even when the original

signal is scaled or shifted in frequency. It is clear that these techniques can be

applied in neurophysiological research such as EEG analysis.


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2. Wavelet TransformsTo extract information from signals and reveal the underlying dynamics that

corresponds to the signals, proper signal processing technique is needed.

Typically, the process of signal processing transforms a time domain signal intoanother domain since the characteristic information embedded within the time

domain is not readily observable in its original form. Mathematically, this can

be achieved by representing the time domain signal as a series of coefficients,

based on a comparison between the signal x(t) and template functions {n(t)}.

The inner product between the two functions x(t) and n(t) is

, The inner product describes an operation of comparing the similarity between

the signal and the template function, i.e. the degree of closeness between the

two functions. This is realized by observing the similarities between the

wavelet transform and other commonly used techniques, in terms of the choice

of the template functions. A non stationary signal is shown in Figure 2 as an

example. The signal consists of four groups of impulsive signal trains. In these

groups, the signals are composed of two major frequencies, 650 and 1500 Hz.

Figure 2


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2.1 From Fourier to Wavelet Transform2.1.1 Fourier TransformUsing the notation of inner product, the Fourier transform of a signal can be

expressed as

, Assuming that the signal has finite energy

The Fourier transform is essentially a convolution between the time series x(t)and a series of sine and cosine functions that can be viewed as template

functions. The operation measures the similarity between x(t) and the template

functions, and expresses the average frequency information during the entire

period of the signal analyzed as shown in Figure 3.

Figure 3

However, it does not reveal how the signals frequency contents vary with time;

that is, it does not reveal if two frequency components are present continuously

throughout the time of observation or only at certain intervals, as is implicitly

shown in the time-domain representation. Because the temporal structure of the

signal is not revealed, the merit of the Fourier transform is limited; it is not

suited for analyzing non stationary signals.

2.1.2 Short Time Fourier TransformIn Figure 4, the STFT employs a sliding window function g(t). A time-localized

Fourier transform performed on the signal within the window. Subsequently,the window is removed along the time, and another transform is performed.


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The signal segment within the window function is assumed to be stationary. As

a result, the STFT decomposes a time signal into a 2D time-frequency domain,

and variations of the frequency within the window function are revealed.

Figure 4

STFT can be expressed as

, , , , According to the uncertainty principle, the time and frequency resolutions of

the STFT technique cannot be chosen arbitrarily at the same time.

||||

||||

As shown in Figure 5, the products of the time and frequency resolutions of the

window function (i.e., the area of f) are the same regardless of the window

size (or 0.5).

Figure 5


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2.1.3 Wavelet TransformWavelet transform is a tool that converts a signal into a different form. This

conversion reveals the characteristics hidden in the original signal. The wavelet

is a small wave that has an oscillating wavelike characteristic and has its energy

concentrated in time.

The first reference to the wavelet goes back to the early twentieth century.

Harrs research on orthogonal systems of functions led to the development of a

set of rectangular basis functions. The Haar wavelet (Figure 6) was named on

the basis of this set of functions, and it is also the simplest wavelet family

developed till this date.

Figure 6

In contrast to STFT, the wavelet transform enables variable window sizes in

analyzing different frequency components within a signal. By comparing the

signal with a set of functions obtained from the scaling and shift of a base

wavelet, it is realized as shown in Figure 7.

Figure 7


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Wavelet transform can be expressed as

. , , ( )

As in Figure 8, variations of the time and frequency resolutions of the Morlet

wavelet at two locations on the timefrequency plane, (1, s1) and (2, s2)

are illustrated.

Figure 8

Through variations of the scale and time shifts of the base wavelet function, thewavelet transform can extract the components within over its entire spectrum,

by using small scales for decomposing high frequency parts and large scales for

low frequency components analysis.

2.2 Continuous Wavelet Transform2.2.1 PropertiesThe definition of continuous wavelet transform

, ( ) where a shifts time, b modulates the width (not frequency), and (t) is mother

wavelt.

It has superpositionproperty.

If the continuous wavelet transform of x(t) is X(s,) and of y(t) is Y(s,), then

the continuous wavelet transform of z(t) = k1x(t) + k2y(t) can be expressed as

, , ,


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Moreover, it is covariantunder translation and dilation.

Suppose that the continuous wavelet transform of x(t) is X (s,), then the

transform of x(t-t0) is

,This means that the wavelet coefficients of x(t-t0) can be obtained bytranslating the wavelet coefficients of x(t) along the time with t0.

On the other hand, suppose that the continuous wavelet transform of x(t) is X

(s,), then the continuous wavelet transform of x(t/a) can be expressed as

, This indicates that, when a signal is dilated by a, its corresponding waveletcoefficients are also dilated by a along the scale and time axes.

2.2.2 Representative SignalsThere are several commonly used wavelets for performing the CWT.

One is Maxican hat, which is a normalized second derivative of a Gaussian

function, and frequently employed to model seismic data

Another is Morlet, which has been used to identify transient components

embedded in a signal, bearing defect-induced vibration.

Another is Frequency B-Spline Wavelet, which has been seen in biomedical

signal analysis.

[ ()] where fbis the bandwidth parameter, fcis the wavelet center frequency, and p is

an integer parameter.

There are also many other mother wavelets such as Shannon Wavelet (a special


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case of frequency b-spline wavelet), Gaussian Wavelet, Harmonic wavelet, etc.

2.3 Discrete Wavelet Transform2.3.1 PropertiesIt can be implemented if it is 1D as shown in Figure 9, and 2D case is shown inFigure 10. Similarly, it can be fitted into higher dimensionality.

Figure 9

Figure 10

Simply and clearly, we can tell from the implmentation structures that it has

higher computational speed than CWT, and the signals would be continuously

seperated into low frequencies and high frequencies as shown in Figure 11.


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Figure 11

2.3.2 Representative SignalsThere are several commonly used wavelets for performing the DWT.

One is Haar, which is orthogonal and symmetric. The property of symmetry

ensures that the Haar wavelet has linear phase characteristics, meaning that

when a wavelet filtering isperformed on a signal with this base wavelet, there

will be no phase distortion in the filtered signal. Furthermore, it is the simplest

base wavelet with the highest time resolution.

{ < . . <

However, the rectangular shape of the Haar wavelet makes its corresponding

spectrum with slow decay, leading to a low frequency resolution.

Another is Daubechies, is orthogonal and asymmetric, which introduces a

large phase distortion. This means that it cannot be used in applications where

a phase information needs to be kept. It is also a compact support base wavelet

with a given support width of 2N - 1, in which N is the order of the base

wavelet. 2 base and 4 base Daubechies transforms are illustrated in Figure 12.

Figure 12

The Daubechies wavelets have been widely investigated for fault diagnosis of

bearings and automatic gears.


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There are also Coiflet and Symlet, extended from Dauchechies families, but

are more symmetric and have vanishing moments in scaling functions.

2.4 Selection of Base Wavelet for Biomedical Signals2.4.1 OverviewOne of the advantages of wavelet transform for signal analysis is the abundance

of the base wavelets. From such abundance arises a natural question of how to

choose a base wavelet that is best suited for analyzing a specific signal. Since

the choice in the first place may affectthe result of wavelet transform at the end,

the question is valid. For example, as shown in Figure 13,

Therefore, in the following section, we first present a general strategy for base

wavelet selection. Then, we introduce several quantitative measures that can be

used as guidelines for wavelet selection. While Morlet wavelet is effective in

extracting the impulsive component, the Daubechies and Mexican hat wavelets

did not fully reveal the characteristics of impulsive component.

Figure 13

2.4.2 Selection CriteriaThere are two ways to measure wavelet performance, one is qualitative and the

other is quantitative.

Base wavelets are characterized by orthogonality, symmetry, and compact

support. Understanding these properties will help choose a candidate base

wavelet from the wavelet families for analyzing a specific signal. For example,


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the orthogonalityproperty indicates that the inner product of the base wavelet

is unity with itself, and zero with other scaled and shifted wavelets. As a result,

an orthogonal wavelet is efficientfor signal decomposition into nonoverlapping

subfrequency bands. The symmetricproperty ensures that a base wavelet can

serve as a linear phase filter. A compact supportwavelet is one whose basis

function is nonzero only within a finite interval. This allows the wavelet

transform to efficiently represent signals that have localizedfeatures.

In the area of biomedical engineering, the regularity and symmetry of base

wavelets were considered as essential features for auditory-evoked potentials

(AEP) signal analysis. The morphology and latency of peaks were preserved

with a symmetric base wavelet. By using the properties of compact support,

vanishing moment, and orthogonality, the Coiflet wavelet was selected to

separate burst andtonic components in the electromyogram (EMG). In addition

to orthogonality, the property of complex or real basis was used to guide the

choice of the basewavelet for electrocardiogram (ECG) signal analysis. The

Morlet wavelet, Gaussian wavelet, and quadratic B-Spline wavelet were

preselected as the candidates for ECG detection and segmentation.

Besides above properties, shape matching is alternative to wavelet selection.

For example, to measure the timing of multiunit bursts in surface EMG from

single trials, wavelets of different shapes, such as square, triangular, Gaussianand Mexican Hat, were investigated. The Daubechies wavelet was chosen for

its similarity to the shape of motor unit potentials hidden in the EMG signal.

Also, base wavelets of different shapes were compared with ECG signals to

determine their appropriateness for extracting a reference base from corrupted

ECG.

As far as shape matching is concerned, it is generally difficult to accurately

match the shape of a signal to that of a base wavelet through a visual

comparison. These deficiencies motivate the study of quantitative measuresforbase wavelet selection.

In the area of biomedical engineering, study on horse gait classification has

discussed an uncertainty model for wavelet selection. The model combines

the fuzzy uncertainty with the probabilistic uncertainty to provide a better

measure for choosing base wavelet to improve correct classification of different

horse gait signals.

In study on cardiovascular ailments in patients, experiments have revealed thesuitability of the Daubechies wavelet at order 8 for the ECG signal denoising,


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as it has the maximum cross correlation coefficientbetween the ECG signal

and the chosen base wavelets, (Daubechies, Symlet, and Coifletwavelets).

From above discussion and references, we know that

Coiflect 4 effectively separated burst and tonic components for EMG.(Wang et al. 2004)

Morlet, Gaussian, Paul 4, and quadratic B-Spline wavelets were selectedfor ECG detection and segmentation. (Bhatia et al. 2006)

Symlet 7, Coifect 3, Coifect 4, and Coifect 5 wavelets have betterdetection for ECG. (Abi-Abdallah et al. 2006)

Furthermore, there are two more quantitative creteria to determine optimal

wavelet base, which are maximum energyand minnimum Shannon entropy.

The energycontent of a signal x(t) can be calculated by

||It can also be calculated from its wavelet coefficients wt(s, )

|,|The above equation can be revised as

|,|If a major frequency component corresponding to a specific scaling s exists in

the signal, then the wavelet coefficients at that scale will have relatively high

magnitudes when this major frequency occurs. Thus, the energy related to that

component can be extracted from the signal when applying wavelet transform.

Therefore, we want a base wavelet that can extract the largest amount of energyfrom the signal.

The entropyhere refers to the entropy of the wavelet coefficients, instead of

the one of the signal itself. The energy distribution of the wavelet coefficients

can be described by Shannon entropy.

=

where piis the energy probability distribution of wavelet coefficients


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|,|From the probability principle, we find the bound of the entropy of the wavelet

coefficients.

If all the other wavelet coefficients are equal to zero except for one coefficient,

the entropy equals to zero. If the probability distribution is uniform, which

means all the wavelet coefficients are the same (i.e. 1/N), the entropy equals to

log2N. Therefore, the lower the entropy is, the higher energy concentration is.

3. Electrocardiography (ECG)3.1 IntroductionElectrocardiography (ECG) is a transthoracic interpretation of the electrical

activity of theheart over a period of time, as detected by electrodes attached to

the outer surface of the skin and recorded by a device external to the body. In

short, electrocardiogram is a test that records the electrical activity of the heart.

ECG has been used to measure the rate and regularity of heartbeats as well as

the size and position of the chambers, the presence of any damage to the heart,

and the effects of drugs or devices used to regulate the heart, as shown in

Figure 14 and 15.

Figure 14
http://en.wikipedia.org/wiki/Hearthttp://en.wikipedia.org/wiki/Heart


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Figure 15

A typical ECG tracing of the cardiac cycle consists of a P wave, a QRS wave, a

T wave, and a U wave. The baseline voltage of the electrocardiogram is known

as theisoelectric line. Typically the isoelectric line is measured as the portion of

the tracing following the T wave and preceding the next P wave, as shown in

Figure 16 and Table 1.

Figure 16

Feature Description

RP intervalThe interval between an R wave and the next R wave . Normal

resting heart rate is between 60 and 100 bpm

P waveDuring normal atrial depolarization, the main electrical vectoris directed from the SA node towards the AV node. This turns

into the P wave on the ECG.

PR interval

The PR interval is measured from the beginning of the P wave

to the beginning of the QRS wave. The PR interval reflects the

time the electrical impulse takes to travel from the sinus node

through the AV node and entering the ventricles. The PR

interval is therefore a good estimate of AV node function.

QRS wave

The QRS complex reflects the rapid depolarization of the right

and left ventricles. They have a large muscle mass compared to


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the atria and so the QRS wave usually has a larger amplitude

than the P wave.

T wave

The T wave represents the repolarization of the ventricles. The

interval from the beginning of the QRS wave to the apex of the

T wave is referred to as theabsolute refractory period. The last

half of the T wave is referred to refractory period.

ST interval The ST interval is from the J point to the end of the T wave.

Table 1

3.2 MethodWe need to select the QRS wave part from ECG to identify if the heart function

is normal or not. The first step is data acquisition with wavelet transform to getlowpass and hypass signals, as shown in Figure 17.

Figure 17

3.3 ResultThe different linear and quadratic approaches of time frequency representations,

such as the spectrogram, the wavelet transform, and the Wigner distribution,transform a 1D signal x(t) into a 2D function of time and frequency.

In the left panel of Figure 18, Wigner distribution of (a) a healthy control and

(b) a patient with ventricular tachycardia is compared; in the right panel of

Figure 18, schaologram is used.


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Figure 18

In Figure 19, ECG has been applied with Morlet Wavlet for the same control

and experimental data.

Figure 19

In short, these time-frequency analysis methods are mainly processed ECG

data as illustrated in Figure 20.


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Figure 20

4. Electroencephalography (EEG)4.1 IntroductionElectroencephalography (EEG) is the recording ofelectrical activity along

thescalp. It measures voltage fluctuations resulting from ionic current flows

within the neurons of thebrain. In clinical contexts, EEG refers to the

recording of the brain's spontaneous electrical activity over a short period of

time as recorded from multipleelectrodesplaced on the scalp. Diagnostic

applications generally focus on thespectral content of EEG, which means the

type of neural oscillations that can be observed in EEG signals. In neurology,

the maindiagnostic application of EEG is in the case of epilepsy, as epileptic

activity can create clear abnormalities on a standard EEG study. In figure 21,

several lines of EEG aree illustrated.
http://en.wikipedia.org/wiki/Electricalhttp://en.wikipedia.org/wiki/Scalphttp://en.wikipedia.org/wiki/Brainhttp://en.wikipedia.org/wiki/Electrodeshttp://en.wikipedia.org/wiki/Frequency_spectrumhttp://en.wikipedia.org/wiki/Diagnostichttp://en.wikipedia.org/wiki/Diagnostichttp://en.wikipedia.org/wiki/Frequency_spectrumhttp://en.wikipedia.org/wiki/Electrodeshttp://en.wikipedia.org/wiki/Brainhttp://en.wikipedia.org/wiki/Scalphttp://en.wikipedia.org/wiki/Electrical


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Figure 21

To get a closer look, we choose one EEG signal out within one second, as

shown in Figure 22.

Figure 22

It is typically described in terms of rhythmic activity and transients. Then, the

activity is divided into bands by frequency, which will have the following types,

from alpha wave, beta wave, delta wave, to theta wave, as shown in Figure 23.


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Figure 23

By analysizing the composition of EEG with wavelet transform, we categorize

and find out the combination percentage of each form.

4.2 MethodFourier transform, wavelet transforms are used. There is no much complex

algorithms here.

4.3 ResultSuppose there are two EEG data, one is from the patient and the other is from

healthy man, as shown in Figure 24.

Figure 24


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Among spectral analysis techniques, Fourier transform is considered to be the

best transformation between time and frequency domains because of it being

time-shift invariant, as shown in Figure 25.

Figure 26

If we decompose the EEG signals with wavelet transform, the effect would be

clearer if we want to categorize different subgroups and find the characteristics

of EEG signals between the patient and a healthy man. Also, by using wavelet

transform, one can view the shapes of the subspectral components of the EEG

signal in time domain to be different from those in Fourier transform, as shown

in Figure 27.

Figure 27


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SUMMARY

This tutorial starts from the introduction of theoretical backgrounds of how to

deduce time-frequency analysis methods such as Cohens class distribution,

short time Fourier transform, and wavelet transform. By describing how themathematical formulas come out, what properties they have, and how to select

optimal wavelets for our biomedical signal processing, we can gain help from

the brief categories and descriptions.

I didnt talk much on the second part, i.e. ECG and EEG applications, because

the main concept of how to operate these analysis tools have been mentioned in

the first part already, therefore, only brief method descriptions and the result

figures are shown.

There are still many other analysis tools with more complex algorithms that

have not been introduced here. Readers who are interested can find more

information in the references.

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Time-Frequency Analysis for Biomedical Engineering