An Algorithm for Selecting Compatible Wavelet Function in ...Biorthogonal Wavelet B splines Bior....

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 14 (2018) pp. 11440-11447

© Research India Publications. http://www.ripublication.com

11440

An Algorithm for Selecting Compatible Wavelet Function in Electrical

Signals to Detect and Localize Disturbances

Jitendra Kumar Sharma1, Yatindra Gopal2, Dinesh Birla3, Mahendra Lalwani4

1Research Scholar, Department of Electrical Engineering, Rajasthan Technical University Kota, India. 2Research Scholar, Department of Electrical Engineering, Rajasthan Technical University Kota, India.

3Professor, Department of Electrical Engineering, Rajasthan Technical University Kota, India. 4Associate Professor, Department of Electrical Engineering, Rajasthan Technical University Kota, India.

Abstract

In power system, power quality (PQ) distortion is a topical

issue. This event has spurred to develop sophisticated methods

for processing and analysing the signal to detect and classify

the power quality distortion. Fourier transform (FT) technique

was the core of many conventional techniques However, FT

being replaced by the newer method notably Wavelet

Transform (WT) technique. In WT spectral leakage problem

was introduced when suitable wavelet family was not

available to analyse the signal. This paper presents an

algorithm to select most compatible wavelet function with

transient signal to detect and localized disturbances. In this

paper, the transient signal is analyzed using seventeen

different mother wavelets i.e. Daubechie, Coiflet,

Biorthogonal, Symlet family members etc. and then selecting

most compatible mother wavelet for transient signal by

calculating Percentage Root Means Square Difference (PRD)

and Mean Square Error (MSE) values. Here the threshold

value is used to calculate PRD and MSE values as global

threshold, this work is performed using MATLAB software.

Keywords: Wavelet Transform, Mother Wavelet, Percentage

Root Mean Square Difference (PRD), Mean Square Error

(MSE), Power Signal, Power Quality ,Transient Signal.

INTRODUCTION

Now a days, the electricity demand is growing up due to many

reasons therefore, the use of power electronic equipment is

gradually increasing. These power electronics devices

introduce harmonics and other power quality distortions in the

system [1, 2]. Basically, quality of power is introduced by the

shape of power signal waveform and due to distortion;

electricity waveform is deregulated from the sinusoidal wave

shape. Many types of equipment used in industries are very

sensitive to PQ disturbances and small amount of PQ

disturbance can decrease the efficiency of the industries [3, 4].

In summary PQ is compatible issue between load and

generation because variation in load and generation profile

affects each other [5]. Healthy power supplied to the

consumers is one of the big challenges towards the electricity

utilities consequently new standards for power quality criteria

[6, 7] have been fixed to establish the maximum allowed

disturbance level. Several power quality mitigation devices

(PQMD) are used to overcome these problems and PQMDs

require knowing the type of disturbance before mitigating the

problems. Many other technologies are evaluated to detect and

localize the disturbances as reported by [8].

Fourier transform is one of the conventional method to detect

disturbances and convert the time domain into frequency

domain [9]. In FT at a time only one domain information is

extracted either in frequency or time domain or in discreate

fourier transform (DFT), the frequency coefficient is a function

of sin and cosine. If the signal is predominantly sinusoidal,

periodic, and stationary than the DFT method is useful for

analysis and for the transient or non-periodic disturbances DFT

is inefficient method for analysis. Fourier transform have

constant time-frequency spectrum window hence, FT

technique now replaced by new approach remarkably WT.

Wavelet transforms are the mathematical functions which

convert time domain into time scale domain rather than

frequency domain [10, 11]. The important features of wavelet

are translation (shift in time) and dilation (stretches or shrinks

in time). WT have flexible time-frequency window which

resolve the difficulty of signal cutting by variable resolution

[12, 13], consequently wavelet transform can analysis the

transient signal, which is advantageous over FT. Wavelet

transform uses long time interval for low frequency and short

time interval for high frequency information [14].

Wavelet Packet Transform (WPT) and Discreate Wavelet

Transform (DWT) are the basic types of wavelet transform

techniques. WPT uses full signal decomposition tree i.e.

“approximation” as well as “detailed signal” can be splits thus

large number of nodes are generated that increase the

calculation time, WPT also has uniform frequency band [15,

16]. On the other hand, DWT uses only basic branches of

decomposition tree and the frequency band is also flexible

hence, the analyzing time is low.

In WT, signal is analyzed using mother wavelet and in doing

so an appropriate mother wavelet can increase the efficiency of

analysis. If compatible mother wavelet is not used for a

particular signal than some information is lost, which is known

as ‘spectral leakage’ and this is the main drawback in wavelet

transform technique [17-19] that can be overcome by the use

of suitable mother wavelet for the particular signal.

Wavelet Transforms have different kind of mother wavelets

including complex wavelet function to analyze a given signal

[20]. In WT technique, the signal is analyzed by mother



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wavelet to detect, localize the disturbances [21, 22] and to

extract the information about disturbance time and frequency.

Wavelet transform is also used in many power system

applications [23-27] and mostly utilized for image processing

[28, 29].

This paper introduces an algorithm to select the appropriate

mother wavelet by calculating Percentage Root Means Square

Difference (PRD) and Mean Square Error (MSE) values and

thus the mother wavelet having minimum PRD and MSE value

is the best compatible mother wavelet for the signal. Section 2

of the paper provides a synoptic introduction of WT and

elaborates a WT property which explains the flexibility and

procedure of wavelet decomposition. In section 3,

classification of wavelet family according to their properties is

presented and provides information about some extremely used

wavelet family members. Section 4 provides details of the

procedure to calculate PRD and MSE values and to select most

appropriate mother wavelet for the signal. Power quality

disturbance detection and localization by wavelet transform

analysis with the use of most compatible mother wavelet is

presented in section 5.

DISCREATE WAVELET TRANSFORM

WT is a mathematical tool for signal processing and it was

introduced around 1980s. The wavelet transforms decompose

the signal into different scales with different level of resolution

by dilation of the mother wavelet.

To construct the scale analysis of a signal it is important to

consider two basic functions which fulfils the properties of

Lebesgue vector space L2(R). These two functions are scale

function [(t)] and wavelet function [(t)]:

(t)= 2 φ(2t - k)k

k

g (1)

( ) 2 (2 )k

k

t h t k (2)

Scaling and the Wavelet function are the two scale difference

equations which are based on selecting of scaling function

with properties that fulfil certain criteria, discrete filters hk

and gk are used to solve the equation. where gk= (-1)k hN-1-k

Let x(t) is the signal to process which is defined in L2(R) space

(vector space for finite energy signal) and R is the real number.

/2 2

, ( )2

2m

m

m

t nDWT x m n x t dt

(3)

where m and n are the scale and time dilation parameters,

respectively and (t) denotes the mother wavelet. The mother

wavelet used to analysis the signal must satisfy following

equation:

( ) dt = 0t

(4)

As in eq. (3) the mother wavelet is dilating by a factor two for

the transformation performed, thus the resulting wavelet family

becomes an orthonormal basis function by which there is no

data (information) lost during the compression of the signal.

By the wavelet filters g(n) (denoted by ‘H’ in figure 1) and

h(n) (denoted by ‘L’ in figure 1) the sampled version of x(t)

denoted by c0(n) is decomposed into two signals detailed d1(n)

and smoothed c1(n) signals respectively. Detailed value d1(n)

of original signal c0(n) extracted by the band pass filter g(n)

contains higher frequency component and by low pass filter

h(n) smooth signal provides the approximation value of c0(n).

d1(n) and c1(n) are the first level decomposition coefficients of

c0(n), which are mathematically denoted as:

1 0( ) ( 2 ) ( )k

c n h k n c k (5)

1 0( ) ( 2 ) ( )k

d n g k n c k (6)

In the wavelet transform, if c0(n) have ‘N’ number of samples

then coefficients c1(n) and d1(n) have approximate N/2

samples. Further these samples will decompose as the level

increases.

In multi-level decomposition processes, signal is decomposed

in two parts by low-pass filter and high pass filter. The

coefficients from low pass filtering process are the

“approximation coefficients” that can be processed again to

decompose the data by bank of filters to generate another

group of “approximation” and “detail coefficients”. This

process has repeated until selected levels are reached. The two

levels decomposition tree by DWT process is shown in Figure

1.

x(t)

2

2

H

L

H 2

2 L

d1(n)

d2(n)

c1(n)

c2(n)

Level 1

Level 2

Figure 1. Two level signal decomposition algorithm

The number of filters used to analyse the signal depend on type

of family members and all the family members have different

kind of properties. Some type of family members present in

WT are introduced in section 3.

WAVELET FAMILY

WT have several types of wavelet families to analyse the

signal and all these families are further sub-divided in many

family members according to their properties and application

to use. Figure 2 shows the classification of different kind of

wavelet families according to their properties.



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Wavelet

Family

Crude Wavelet

Gaussian

waveletMorlet Maxican

Hat

Infinity Regular

Wavelet

Mayer

Orthogonal

Wavelet

Daubechies Symlets Coiflets

Biorthogonal

Wavelet

B splines Bior.

Wavelet

Complex Wavelet

Complex

Gaussians

Complex

Morlet

Figure 2. Classification of Wavelet Family

Mother wavelet is the core of WT technology to analyse the

signal and different type of mother wavelets are presents in

WT tool i.e. Haar, Daubechies, Biorthogonal, Morlet,

Symlets, Maxican-hat, Coiflets, Mayer etc. These wavelets are

also classified according to the number of coefficients like -

db2 & da4 (Daubechies wavelet with 2 & 4 coefficient,

respectively).

In this paper work with some most potential wavelet family

members is presented and the functions and properties of

these mother wavelets are discussed in following section.

Daubechies Wavelet Family

Ingrid Daubechie was firstly introduced daubechies mother

wavelet and piloted much research in the wavelet transform

domain [30-34]. The daubechies family members represented

as ‘dbN’, where surname of daubechies family is ‘db’ and the

order of family is represented by ‘N’ (N=1, 2, 4, 6, 8,10). Db1

is the basic daubechies wavelet, which is also known as ‘Haar

wavelet’. Figure 3 presents the scaling and wavelet function of

haar mother wavelet.

Figure 3. Haar Mother Wavelet Scaling and Wavelet Function

Almost all the Daubechies members are unsymmetrical

therefore; this wavelet is mostly used to the analysis of non-

periodic signal. Figure 4 shows the wave-shape of daubechies

wavelet family members db2, db4, db6, db8 with their scaling

function.

Figure 4. db2, db4, db6 and db8 Mother Wavelet Scaling and

Wavelet Function.

Coiflet Wavelet family

Ingrid daubechies invented coiflet mother wavelet, on the

request of R. Coifman. ‘Coif’ represents the surname of

coiflet family member and five members (coif1 to coif5) are

present in this family. This wavelet function and scaling

function has 2N and 2N-1 moments equal to zero respectively.

Orthogonal, compactly supported but nearly symmetry are the

general properties of the coiflet family members. Figure 5

shows waveform of some basically used wavelet family

members along with their scaling function.

Figure 5. coif1, coif2 and coif4 Mother Wavelet Scaling And

Wavelet Function



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Symlet Wavelet family

Sysmlet mother wavelet is modified form of daubechies

family. Due to unsymmetrical property of ‘db’ mother

wavelet, symlet mother wavelet was invented which have

modified symmetry property of ‘db’ family wavelets. It is also

known as Daubechies least asymmetric wavelet and

represented as ’symN’ and there are seven members (sym2 to

sym8) are belonging to this family. “Orthogonal” and

“compactly supported in time” are the most important

properties of symlet family members. The scaling and wavelet

function waveforms for some mostly used symlet family

members are shown in Figure 6.

Figure 6. Scaling and Wavelet Function of Symlet Family

Members sym4, sym6, sym8

Biorthogonal Wavelet family

These wavelet family members have biorthogonal property.

The surname of these family members is represented by ‘bior’

and their order is denoted by Nd and Nr. The general

characteristics of this family are compactly supported but do

not have orthogonal property. Figure 7 present the scaling and

wavelet function waveforms of bior3.3 and bior3.5 mother

wavelet.

Figure 7. Scaling and wavelet function of bior3.5

and bior3.3 wavelet

All These ‘wavelet 1 D’ family members are used to analyze

the signal and to calculate their PRD values. The processer to

calculate PRD and MSE values is described in next section.

PERFORMANCE EVALUATION CRITERIA

The main intention of wavelet transform analysis is extract the

important information from the signal in the form of energy

and remove the redundancy, unwanted and irrelevant

information. Error criterion is the one of the most difficult

problems in signal compression and reconstruction

applications. Consequently, to measure the ability of the

reconstructed signal to preserve the relevant information, error

criterion has to be defined and the distortion is defined as the

difference between the original and the reconstructed signal.

The PRD and MSE are the most prominently used distortion

measurement techniques and defined as below:

12 2

0

2

0

( ) ( )*100

( )

rn

n

x n x nPRD

x n

(7)

2

0

1( ) ( )rn

MSE x n x nN

(8)

Here original signal is denoted by ‘x0’ and reconstruction

signal is denoted by ‘xr’, ‘N’ introduces the number of

samples in both the signals.

In WT analysis, most appropriate mother wavelet for the

signal is selected by PRD and MSE values in three steps.

Firstly, the wavelet transformed to be applied on the signal is

analyzed. Secondly, by the wavelet coefficients which are

extracted in step1, the threshold values are calculated. Finally,

inverse wavelet transform is applied to decomposed signal

and then by eq. (7) PRD values and by eq. (8) MSE values are

calculated for the signal. All the process covered in these steps

is known as decomposition and reconstruction sequence. The

appropriate level for decomposition and reconstruction

process is selected by considering the dominant frequency

components of the signal. In wavelet transform analysis, the

signal is decomposed into P+1 sub-bands with detailed and

approximation coefficients by selecting the sampling

frequency. Here ‘P’ denotes the level of the wavelet

transform. In this paper, level 5 is chosen for wavelet

decomposition & reconstruction and thus Global threshold

value is used.



11444

Figure 8 shows the algorithm to find the most suitable mother

wavelet among various mother wavelets for analyzing the

electric signal by calculating PRD and MSE value.

Wavelet Transform Power SignalCalculate Global

Threshold Value

Reconstract SignalCalculate PRD

Value

PRD1=0

MSE1=0

K=number

of mother

wavelet

PRD<=PRD1

MSE<=MSE1

All mother

wavelet used

PRD1=PRD

MSE1=MSE

Yes

No

Evaluate PRD &

MSE values and

their corresponding

wavelet is best

wavelet

Next wavelet used

(K+1)

Yes

No

Figure 8. Flow chart of performance evaluation criteria

By this algorithm the family member which has lowest value

of PRD and MSE is considered the most compatible mother

wavelet for the signal analysis with high efficiency.

RESULTS AND ANALYSIS

In this paper, the transient signal is analyzed by most

compatible wavelet function to detect and localize the

distortion. For calculating most compatible mother wavelet,

transient signal with 1500 data samples is analyzed using

wavelet transform and decomposed signal up to 5 levels. The

global threshold value is calculated using the detailed

coefficients and then the signal is reconstructed. The original

signal data and reconstructed signal data are used to calculate

the PRD and MSE values for various mother wavelets. This

process is repeated till all the mother wavelets are used. The

minimum PRD and MSE values give the better way to select

compatible mother wavelet function for signal i.e. the mother

wavelet with minimum PRD and MSE value is best wavelet

for the signal.

Figure 9 and Figure 10 shows the transient signal with

approximate and detailed signals up to 5 levels, analyzed by

Bior3.5 mother wavelet.

Figure 9. Transient and 5th Level Approximation Signal

Figure 10. 5 Level Wavelet Decomposition by Bior3.5

Mother Wavelet detailed signal

Figure 11. Original signal, reconstructed signal and error

signal using Bior3.5 mother wavelet

where ‘S’ represents original signal, ‘a5’ is 5th level

approximate coefficient and d1 to d5 are detail coefficient of



11445

corresponding level. Considering level one detailed coefficient

(D1) signal, it can be concluded that when the distortion occurs

in the power signal then spikes are introduced in detailed

signal which represent the distortion energy by which the

disturbance is detected and localized in time format.

Figure 11 shows the original signal which is to be analyzed by

the wavelet transform and reconstructed signal after the

decomposition up to five levels by bior3.5 mother wavelet,

with error signal is the difference between these two signals.

From the Table 1 and Figure 12 to Figure 15 it can be

observed that bior3.5 mother wavelet has lowest value of PRD

and MSE among all the 17 mother wavelets therefore, bior3.5

mother wavelet is best compatible mother wavelet for this

transient signal.

Table 1. PRD and MSE Values for Various Mother Wavelets

Wavelet Family PRD Values MSE Values

Haar 0.7338 2.8604

Db2 0.4367 1.0130

Db4 0.2628 0.3669

Db6 0.2628 0.4167

Db8 0.2888 0.4429

Db10 0.3149 0.5269

Sym4 0.2601 0.3594

Sym6 0.3144 0.5249

Sym8 0.2967 0.4676

Coif1 0.4620 1.1339

Coif2 0.2558 0.3475

Coif4 0.2467 0.3232

Bior 1.3 0.7263 2.8022

Bior 2.2 0.2841 0.4287

Bior 3.3 0.2073 0.2282

Bior 3.5 0.1959 0.2038

Bior 3.9 0.2413 0.3092

Figure 12. PRD values of seventeen mother wavelets

Figure 13. MSE values of seventeen mother wavelets

Figure 14. PRD value variation in 17 mother wavelets

Figure 15. MSE value variation in 17 mother wavelets

The information lost from original signal to reconstructed

signal by bior3.5 is low in comparison to other 16 mother

wavelets. In other words, haar wavelet leaks more

informations in comparison to other mother wavelet due to

highest PRD and MSE value.

CONCLUSION

This paper presents a method to select the most compatible

mother wavelet for the transient signal to minimize the

leakage problem and to detect the distortion in signal. The

main advantage of wavelet transform technique is step by step

analysis of the signal and possibly provides good quality of

resultant signal with all the information about the

disturbances. In term of PRD and MSE choice of appropriate

mother wavelet for particular signal at any level is found

optimal.

In summary, for the transient signal biorthogonal family

member ‘bior3.5’ has minimum PRD and MSE values and it

means information lost between original and reconstructed



11446

signal is minimum. Therefore, among 17 mother wavelets

‘bior3.5’ appears to be the most compatible function for

transient signals.

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