7

2.3.3 Finite impulse response (FIR) Filter identify the abnormalities and sudden changes in heartbeats
FIR filter is used to implement almost any type of digital [77]. WT can analyze the short-duration signal changes;
frequency response. The FIR filter design can be done in due to this property, it is more popular for examining the
two ways, such as frequency sampling and windowing electrophysiological signals of humans as compared to the
FIR filters. In the case of frequency sampling method, fil- other signal processing methods. Especially investigate the
ter design uses Inverse Discrete Fourier Transform (IDFT) pulsed electromagnetic field (PEMF) in case of extremely
for desired frequency response for filter coefficient. In the low frequency (ELF) of ECG, EEG, and PPG (Photoplethys-
windowing method, FIR filters are determined by Inverse mography) analysis its plays major role. This analysis was
Discrete Time-Frequency Transform (IDTFT).FIR filter with done with the help of multi-resolution decomposition[78].
frequency method is similar to the FFT. Both use the Discrete In wavelet transmission, a wavelet is an oscillation function
Fourier Transform, and representation is also easy. The whose energy is concentrated in time to represent tran-
problem of this design is the length of the filter is long sient, non-stationary signals better. WT working is done
because the IDFT of the sampled frequency response with by two operations such as scaling and translation to ob-
the number of samples is equal to the number of filter serving the correlation between observed signal & the basis
coefficients if we using frequency sampling method [70]. function.[79]. As with FFT analysis, WT also expands the
The IDTFT would be a continuous frequency function, and function in terms of basic function but not in terms of
it calculates the result of the impulse response (which is trigonometric polynomials (in FFT) but in terms of wavelets
generated by IDFT) for the windowing filter. The IDTFT generated by mother wavelet for translate & dilations [80].
process is an estimation of the multiplication of impulse Digital signal processing provides two types of wavelet
response by a rectangular window. Due to this reason, for transform such as Continuous Wavelet Transform (CWT) &
extracting sub-bands from EEG signal prefer windowing Discrete Wavelet Transform (DWT) [80], [81]. Depending on
based FIR filter, it is time-domain representation. Here for the application, one may be preferred to the others.
all the passband frequencies ideal frequency response of a
filter is one, and for stopband frequency, the ideal frequency 2.3.6 Continuous Wavelet Transform (CWT)
is zero. There are many windowing functions for a Window- The input signal is continuous, and it has continuous pa-
based FIR filter to extract the required frequencies from EEG rameters like time and scale, then CWT is best for extracting
signals. Those are the Rectangular window, Bartlett window, required features. CWT can be expressed as
Hanning window, Hamming window, Blackman window,
and Kaiser window [71]. In the case of EEG signal analysis Z inf
among Blackman, Hamming, Hann, and Rectangular win- CW T (a, b) = ∗
x(t)ψa,b (t)dt (4)
dows, Blackman window is better if we consider stopping −inf

band attenuation and transitivity bandwidth [72], if you

consider all five windows, Kaiser window is best according 1 t−b
ψa,b (t) = √ ψ( ) (5)
to main and side lobes, and SNR [73]. The difficulty of FIR a a
is EEG signal sampled at 1000Hz to mine the Delta, Theta,
alpha, and beta waves. ψ(t) = motherW avelet (6)

2.3.4 Infinite Impulse Response (IIR) Filter The CWT parameters such as ’a’ and ’b’ (scaling or
These filters are recursive, means it uses current input reciprocal of frequency and translation or time localization
and previous output samples for computing current output or shifting)are continuous. [82], [81]. The main drawback of
sample. IIR filters are considerably more effective than FIR the CWT is these parameters change continuously and leads
filters [74]. There are many IIR filters: Butter-worth, Cheby- to redundancy and impracticality. Due to this, the wavelet
shev (Type I & II), and Elliptic IIR filters. To analyze EEG, will waste its energy to generate a lot of new information.
commonly used IIR filters are Butter-worth, and elliptic IIR To reduce wasting of effort in the wavelet, the DWT method
filters [75]. With the help of IIR, we can extract five main is considered.
sub-bands of EEG signal with the help of relating to low-
pass and band-pass IIR. This process is opposite to the Finite 2.3.7 Discrete Wavelet Transform (DWT)
impulse response Filter. The generated sub-bands of EEG According to Daubechies [83] there are two types of DWT,
signals are shown in the next section. one is based on frames (redundant discrete systems), and
another one is based on wavelets. In the case of frames, pa-
2.3.5 Wavelet Transform rameters (dilation and translation) are discrete and chooses
WT is a mostly used Digital signal processing method to any positive or negative integer of powers of one. Due to
perform time-frequency analysis of signals by providing its redundant description of the original function doesn’t
a unified framework for different techniques for various support the multi-resolution system, so it is not useful for
applications such as telecommunication and biology. Be- analyzing physiological signals such as EEG. The other
cause of their non-stationary signal analysis property, this DWT system, called the wavelet-based system, supports
method is a powerful method and alternative to the Fourier multi-resolution analysis. Here, it uses dyadic scales and
methods for analyzing EEG signal [76]. The appropriate Positions as replacements of scaling and shift parameters of
possessions of wavelets have been utilized as a part of the CWT (shown in Table 4). These are selected based on powers
investigation of medicinal signs such as EEG and ECG to of two. As compared to the CWT, its analysis is efficient[84].
 8

inf
1 t − 2j k
Z
DW T (j, k) =) p x(t)ψ( )dt (7)
|2j | −inf 2j

TABLE 4: Scaling and Shifting parameters of CWT and

DWT.

CWT DWT
Scaling a 2j
Shifting b k2j
Fig. 9: Sample DWT Decomposition model for EEG signal
DWT decomposes only discrete signals for gathering of 64Hz sample frequency.
different frequency bands, and this can be done with the
help of low and high pass filters of the time-domain signal.
Here it follows multi-resolution decomposition for getting we are using Multi-Resolution (MR) algorithm because each
the required frequency band signal of the time domain. A stage of decomposition uses high pass filter H[.] and low-
signal can be decomposed into many sub-bands, where each pass filter G[.]. Again we apply these two filters to the
sub-band contains a part of the whole signal. The output of output of the high pass filter. This process is done until we
the high pass filter is called a complex coefficient, and the get the required frequency ranges. DWT on EEG signal pro-
output of the low-pass filter is the approximation coefficient vides multi-resolution analysis for multiple non-statistical
[85]. The operational flow of DWT is shown in Fig. 8 to parameters through time, frequency and time-frequency.
generating the sub-bands. In this process, we must specify By this, the subsets of the decomposition coefficients (both
the decomposing levels and wavelet function. approximate and detailed) were used to analyze and pre-
dict the EEG subject’s behavior. Choosing the best mother
wavelet function is another important issue in DWT for
generating delta, theta, alpha, beta, and gamma sub-bands.
There are multiple wavelet families such as Daubechies, Bi-
orthogonal Coif-let, etc., and each wavelet family contains
multiple wavelet functions. The choice of the wavelet trans-
form function depends upon the application, and it must be
closely matched to the signal [87].
There are many wavelet families offered in wavelet
Transformation showed in Table 5. For knowing the number
of wavelet families available in wavelet Transformation
in MATLAB, there are some commands such as "wavem-
Fig. 8: DWT working flow diagram.
nger(’read’)" or "wavelet families(’f’)."
In [85], [86] chooses the decomposition levels based on
TABLE 5: Wavelet Transformation and support Wavelets.
dominant frequency means; if EEG signals useful frequency
bands are below the 30Hz, then the decomposition levels Wavelet Family function
were 5. This paper chooses the decomposition level de- Haar haar
pending on the sampling frequency used to gather the EEG Daubechies db
signal. For example, if the sampling frequency is 256 Hz, Symlets sym
then the decomposition levels are six. At Decomposition Coiflets coif
level CA6 (approximate coefficient), we get delta, and at BiorSplines bior
CD6, CD5, CD4, CD3 (Detailed coefficients) levels, we get ReverseBior rbio
Theta, Alpha, Beta, and Gamma. The remaining decomposi- Meyer meyr
tion considers as noise that is at CD2 and CD1. DMeyer dmey
The decomposition of raw EEG signal for generating Gaussian gaus
sub-bands is shown in Fig. 9. Each step contains two digital Mexican–hat mexh
filters g[n] is the high pass filter, and h[n] is the low pass fil- Morlet morl
Complex Gaussian cgau
ter. The outputs CD, CA are the detailed and approximation
Shannon shan
coefficients of the EEG signal. The relation between the WT Frequency B-Spline fbsp
and H and G is Complex Morlet cmor
Fejer-Korovkin fk
H(z)H(z −1 ) + H(−z)H(−z −1 ) = 1 (8)
There are many wavelets families, and each wavelet
−1 family has different wavelet functions. There are some
G(z) = zH(−z ) (9) commands for MATLAB editor for finding the number
Here z is H’s Z-transform. And G is complementary of of wavelet functions in each wavelet family. Those are
z-transform displayed in the above equation. In this process, "wavemngr(’read’,1)" or "wavelet families(’a’)" or "wavelet
 9

families(’n’)". All wavelet families available in wavelet because analysis of such sensitive signals must be useful
transformation are not supported by the DWT. A particular to predicting human behavior and finding the diseases even
wavelet family that supports a DWT operation or not can be analysis taken more time.
determined with the help of one MATLAB command such The help of appropriate decomposition levels based
as "waveinfo(’Wavelet function name’)," example, "wave- on sample frequency and reconstruction of the signal af-
info(’db’)." In Table 6 we visualized all wavelet family ter decomposition finds the best mother wavelet function
details that support the DWT procedure. among all available wavelets till now in the Matlab–2017.
The decision of wavelet function fundamentally influ- For decomposing the signal into a small set of signals,
ences the result and selection depending on how smooth the there are many wavelet families explained in Table 6. All
signal and computation involved [88]. However, there are these wavelet functions are support by DWT to analyze
no known methods for choosing a practical wavelet func- the biomedical signal such as EEG. To determine the best
tion other than experience means; in many cases, selecting wavelet function for generating EEG sub-bands, first find
the appropriate Wavelet function is based on experimental the number of levels (L) based on the sampling frequency
result only, means trial and error method. Epileptic seizure (Fs ) of EEG signal with the help of the following equation.
detection by analyzing the EEG by considering the multi-  
domain features uses the "db4" Daubechies family wavelet Fs
L = log2 ( ) (10)
function [89]. They didn’t specify the reason for choosing 4
this wavelet function.
Decompose the EEG signals up to L levels using every
In [90], [91] and [88] they uses the wavelet transforma-
available mother wavelet function as shown in Fig. 8. After
tion for epileptic seizure detection, but they didn’t specify
getting sub-brands such as Delta, Theta, Alpha, Beta, and
the which wavelet function they used for decomposing the
gamma along with unwanted data (greater than 50Hz),
EEG signal to extract the features. For the same application,
reconstruct the signal and compare the reconstructed signal
select "db4" for the mother wavelet function, which was
with the original EEG signal, which has less difference that
selected based on classification accuracy comparing with
is the best mother wavelet function for generating EEG
few other mother wavelet functions such as Morlet, bior,
sub-bands. To find the difference between the reconstructed
orthogonal cubic spline, Mexican hat, complex Gaussian,
signal and the original signal, we have many quantitative
etc. [92]. But all these mother wavelet family functions are
measurement techniques in signal processing; among them,
not supported by the DWT.
we choose any method that performs well in less time in
In [93] provide a comparative analysis of wavelet fam-
the case of both single-dimensional and multidimensional
ilies for EEG. Here they use coiflets (coif1) for analyzing
signals. We are choosing Mean Squared Error (M SE ). M SE
EEG signal, which was selected by compared four mother
is a quantitative measurement technique used to calculate
wavelet functions such as Haar(db1), Daubechies (db2),
the degree of similarity between the two signals [97].
coiflets (coif1), and Bi-orthogonal (Bior1.1). In this work,
MSE of single dimensional signal done with help of the
they find the best wavelet function by using statistical
following equation
analysis of classification done through Probabilistic Neural
Network (PNN) classifier were equated with Support Vector 1 X
N

Machine (SVM) with the assistance of three features such M SE = (Xi − Yi )2 (11)
N i=1
entropy, standard deviation, and energy of different sub-
bands was subtracted using all wavelet functions. MSE of multidimensional signal done with help of the
In all the above cases, classification was done for epilep- following equation
tic, so it may be limited to analyzing epileptic EEG signals. It
is also a time-consuming process for checking accuracy for sum(D(:))
M SE = (12)
every mother wavelet function. In [94] and [95] select the T otalSamples
wavelet function using cross-correlation. Hear they speci- here
fied that "sym9" is best among the forty-five mother wavelet D = abs(X − Y )2 (13)
functions of three wavelet families: Daubechies, coiflets,
and Symlets. The Same author uses the "sym9" wavelet In this work, we compare the total of one hundred thirty-
function in [96] for removing artifacts from EEG signal with one (131) wavelet functions of eight wavelet families, which
the help of Independent Component Analysis with wavelet DWT supports (shown in Table 6). Here we use two types of
transform (ICA-WT). Here they didn’t check all available datasets. Standard datasets gathered online from Physionet
mother wavelet functions and computing cross-correlation [98] and BCI competition [99]. Another type of data set
for the multidimensional signal is the time taken to process, is gathered through our device. Here we considered both
and its value depends on the overall signal, not on the single-dimensional and multidimensional data sets in two
individual samples. formats such as "edf " and "mat" formats. In edf format
The Wavelet Transmission supports many mother the multidimensional datasets are available at [100]. In this
wavelets functions. All these wavelet functions have differ- work, we used twenty-one datasets; among them, eleven are
ent filter lengths; long filter length wavelet function required ".edf " format which was gathered through the online and
a higher computational cost in case of both time and space local hospitals. The remaining ten are ".mat" format; among
compared to the small width filter. Finding the best mother them, five are gathered online, such as BCI computation
wavelet function through this filter length is not useful websites, and five are gathered directly from the human
for analyzing biomedical signals such as EEG and ECG brain with the help of the available device in our institute.
 10

TABLE 6: DWT supported Wavelets

Wavelet–family Short Name Orthogonal Biorthogonal Order N Example
Fejer–Korovkin fk YES NO 4,6,8,14,18,22 fk4, fk6
Daubechies db YES YES 1 to 45 db1 or haar, db2, db25
Symlets sym YES YES 1 to 45 sym1, sym2,sym6, sym8
Coiflets coif YES YES 1 to 5 coif2, coif4
Biorthogonal bior NO YES 1.1, 1.3, 1.5, 2.2, 2.4, 2.6, 2.8, 3.1, bior 1.1, bior 2.2
3.3, 3.5, 3.7, 3.9, 4.4, 5.5, 6.8
Reverse Biorthogonal rbio NO YES 1.1, 1.3, 1.5, 2.2, 2.4, 2.6, 2.8, 3.1, rbio 1.1,rbio 2.2
3.3, 3.5, 3.7, 3.9, 4.4, 5.5, 6.8
DMeyer dmey YES YES - dmey

In this experiment, we use each wavelet function to

decompose the twenty-one datasets and reconstruct the new
signal with the help of the decomposed components. After
getting the reconstructed signal, we calculate the individ-
ual MSE of all datasets. After calculating MSE for every
dataset with the same wavelet function, we calculate the
mean value of all these MSE’s. This is the final MSE value
of that particular wavelet function. Initially, we compare
all the average MSE values of individual wavelet fami-
lies. Daubechies W avelets: Is belongs to the orthogonal
wavelets family used in many problems solving techniques
related to the signal analysis and it has infinitely supported
function [80] and supports vanishing wavelet moments and
computed with finite impulse response conjugate mirror Fig. 11: Evaluation of Coiflets Wavelet functions.
filter [101].
These Wavelets are written as "dbN" where N is from
1 to 45. db1 is also called haar, and it is related to the
Haar transform mathematical operation. In db family haar 2.3.9 Symlet Wavelets
is only symmetric wavelet, and its advantages are fast It is similar to the Daubechies family wavelets. "db" and
and memory-efficient, and simple[102]. The epilepsy EEG "sym" properties are similar. As like "dbN" symlets also
analysis was done with the help of extracting EEG sub- have "symN" Hear N is from 1 to 45. These wavelets and
bands by using Daubechies (db4) wavelet function [103]. their MSE values are shown in Fig. 12. Many researchers
We find all forty-five wavelet functions MES’s of Daubechies use these mother wavelet function for extracting the sub–
wavelet family. Among them, we plot the graph for the first bands from EEG signal. Some researchers specifies sym9
eleven wavelet functions in ascending order, as shown in is best among sym1 to sym20 [95], some other researcher
Fig. 10. specifies that sym7 is best as compared to the sym of order
5,7,8,and 9 for extracting the features from EEG to identify
the dyslexia [106]. In our observation up to "sym35" DWT
perform decomposition in less time. After on wards, means
from sym36–sym45 Matlab takes more time. First, eleven
best sym wavelet functions for analyzing EEG signal is
showed in Fig.12.

2.3.10 Bi-orthogonal Wavelets

It has the capability to reconstruct and decompose both sig-
nal and image. These wavelets support symmetric wavelets.
Orthogonal property of wavelet has flexibility for devel-
opment of wavelet bases. In case of Bi-orthogonal, there
Fig. 10: Evaluation of DAUBECHIES Wavelet functions. is more freedom as compared to orthogonal wavelets by
providing two scaling functions useful for multiple multi-
resolution analysis of two wavelet functions. This feature
makes decomposition and reconstruction with help of two
2.3.8 Coiflets Wavelets different wavelet functions in bi-orthogonal mother wavelet
This is constructed by Daubechies and R.Coifman. It has two [107]. To decompose EEG signal for extracting the sub–
functions such as wavelet function has 2N movements and bands use biorthogonal mother wavelet function for classi-
scaling function has 2N-1 to support the length 6N-1 [104]. fying the normal and epileptic human [108], but they didn’t
Some researchers using coif 1 mother wavelet function ana- specify which wavelet function was used in biorthogonal
lyze the EEG and detect the seizure [105]. Determination of wavelet family. With the help of bior4.4 analyze the EEG
best Coiflets wavelet function is shown in Fig. 11. signal for predicting the epileptic seizure [109], here also
 11

Fig. 12: Evaluation of Symlets Wavelet functions.

Fig. 14: Evaluation of Reverse–Bi–orthogonal Wavelet func-
tions.
they didn’t provide any reason to choose bior4.4 even
we have multiple bior wavelet functions available in Bi-
orthogonal mother wavelet family (shown in Table 6). These among remaining. In Fig. 15 we compare the all its family
wavelets and their M SE values are shown in Fig.13. wavelet function.

Fig. 13: Evaluation of Bi-orthogonal Wavelet functions. Fig. 15: Evaluation of Fejer–Korovkin Wavelet functions.

2.3.11 Reverse Bi-orthogonal Wavelets 2.3.13 DMeyer Wavelet

By using this wavelet functions we can perform decomposi- In this wavelet family we have only one wavelet function.
tion and reconstruction of the signal as, like bi-orthogonal, It supports both orthogonal and bi–orthogonal operations.
the difference is reverse the role of the two available wavelet Main advantages of this mother wavelet function are sym-
functions for decomposing and reconstruction [110]. In metry and speed of computation. But it is not suitable for
some cases, decomposition done with bi-orthogonal and time domain signal [110].In our experiment, it performs
reconstruction did with reverse–bi–orthogonal. In our obser- very poor for decomposing the signal. In Fig.16 average
vation MSE’s of these two families almost similar for mul- MSE’s of "dmey" wavelet function is displayed.
tidimensional signal and same for the single dimensional
signal. These wavelets and their MSE values are shown in
Fig. 14. 3 I LLUSTRATE THE OVERVIEW OF THE RESULT
For generating these sub-bands, we use some techniques
2.3.12 Fejer Korovkin Wavelets such as Fourier Transform (FFT & STFT), Digital filters (FIR
It is also an orthogonal wavelet family. This family contains & IIR) Wavelet Transform (CWT & DWT). In the case of
only six wavelet functions. Till now no one can use this Fourier Transformation, problems of FFT are suffer from
wavelet function for decomposing the EEG signal. This huge noise sensitivity, and if a signal is linear means station-
wavelet function available in latest MATLAB versions only ary, then it performs the operation well, but in the case of the
such as 2016a on wards. In our experiment, it performs best non-stationary signal, its performance is low as compared