an algorithm for selecting compatible wavelet function in ...biorthogonal wavelet b splines bior....
Post on 24-Feb-2021
8 Views
Preview:
TRANSCRIPT
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
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 14 (2018) pp. 11440-11447
© Research India Publications. http://www.ripublication.com
11441
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.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 14 (2018) pp. 11440-11447
© Research India Publications. http://www.ripublication.com
11442
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
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 14 (2018) pp. 11440-11447
© Research India Publications. http://www.ripublication.com
11443
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.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 14 (2018) pp. 11440-11447
© Research India Publications. http://www.ripublication.com
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
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 14 (2018) pp. 11440-11447
© Research India Publications. http://www.ripublication.com
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
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 14 (2018) pp. 11440-11447
© Research India Publications. http://www.ripublication.com
11446
signal is minimum. Therefore, among 17 mother wavelets
‘bior3.5’ appears to be the most compatible function for
transient signals.
REFERENCES
[1] M.H.J. Bollen, Understanding power quality
problems: Voltage sags and Interruptions, New York:
IEEE Press, 2000.
[2] M.H.J. Bollen and I. Y. Huagu, “Signal Processing
of Power Quality Disturbances,” Wiley: IEEE Press,
2006.
[3] Douglas, J., 1993, “Solving problems of power
quality,” In EPRI Journal., 18, (8), pp. 6-15.
[4] Dugan, R. C., Mcgranaghan, M. F., Santoso, S., and
Beaty, H. W., 2003 “Electrical Power Systems
Quality,” New York: McGraw-Hili.
[5] Frunt, J., Kling, W. L., and Ribeiro, P. F., 2011
“Wavelet decomposition for power balancing
Analysis,” IEEE Transaction on Power Delivery, 26
(3), pp. 1608-1614.
[6] IEC, “IEC 6100-4-30 Standard”, 2007.
[7] Markiewiez, H., and Klajn, A., 2004, “Standard En
50160 voltage characteristics in public distribution
system, Wroclaw,” European Copper Institute,
Wroclaw University of Technology.
[8] Sundaram, P. K, and Neela, R., 2017, "Analysis and
classification of power quality events using Hilbert
transform and fuzzy system," In Sensing, Signal
Processing and Security (ICSSS), Third International
Conference on, pp. 269-274.
[9] Santoso S., Powers E. J., Lamoree, J., and Bhatt, S.
C., 2000, “Characterization of distribution power
quality event with fourier and wavelet transforms,”
IEEE Transaction on Powert Delivery, 15, (1), pp.
247-253.
[10] Santoso, S., Powers E. J., Grady, W. M., and
Hofmann, P., 1999, “Power quality assessments via
wavelet transform analysis,” IEEE Transaction on
Power Delivery, 11, pp. 924-930.
[11] Brito, N. S. D., Souza, B. A., and Pires, F. A. C.,
1998, “Daubechies Wavelets in Quality of Electrical
Power,” Processing of Harmonics and Quality of
Power, 1, pp. 511-515.
[12] Chen, S., and Zhu, H. Y., 2007, “Wavelet transform
for processing power quality disturbances,”
EURASIP journal Advance Signal Processing, 1, pp.
1-20.
[13] Poisson, O., Rioual, P., and Meunier, M., 1999,
“New signal processing tools applied to power
quality analysis,” IEEE Transaction on Power
Delivery, 14 (2), pp. 561–566.
[14] Peng, Z. K., and Chu, F. L., 2004, “Application of
the wavelet transform in machine condition
monitoring and fault diagnostics: a review with
bibliography,” Mechanical systems and signal
processing, 18 (2), pp. 199-221.
[15] Hamid, E. Y., and Kawasaki, Z., 2002, “Instrument
for the quality analysis of power systems based on
the wavelet packet transform,” IEEE Power
Engineering Review, 22 (3), pp. 52-54.
[16] Barros, J., and Diego, R. I., 2008, “Analysis of
harmonics in power systems using the wavelet packet
transform,” IEEE Transactions on Instrumentation
and Measurement., 57 (1), pp. 63-69.
[17] Parameswariah, C., and Cox, M., 2002, “Frequency
characteristics of wavelets,” IEEE Transaction on
Power Delivery., 17, (3), pp. 800-804.
[18] Domijan, A., Hari, A., and Lin, T., 2004, “On the
selection of appropriate filter bank for power quality
monitoring,” International Journal of Power Energy
System., 24, pp. 46-50, 2004.
[19] Barros, J., and Diego, R., 2006, “Application of the
wavelet-packet transform to the estimation of
harmonic groups in current and voltage waveforms,”
IEEE Transaction on Power Delivery., 21(1), pp.
533-535.
[20] Poisson, O., Rioual, P., and Meunier, M., 2000,
“Detection and measurement of power quality
disturbances using wavelet transform,” IEEE
Transaction on Power Delivery., 15 (3), pp. 1039–
1044.
[21] Ray, K., Prakash, B. K., Panigrahi, P. K., Rout, Asit
Mohanty., and Harishchandra Dubey., 2017
"Detection of Faults in Power System Using Wavelet
Transform and Independent Component Analysis,"
In Computer, Communication and Electrical
Technology: Proceedings of the International
Conference on Advancement of Computer
Communication and Electrical Technology (ACCET
2016)., pp. 227-231. CRC Press.
[22] Latran, M. B., and Teke, Ahmet., 2015, "A novel
wavelet transform based voltage sag/swell detection
algorithm," International Journal of Electrical Power
& Energy Systems., 71, pp. 131-139.
[23] Karthik, Thirumala., Umarikar, A. C., and Jain,
Trapti., 2015, "Estimation of single-phase and three-
phase power-quality indices using empirical wavelet
transform," IEEE Transactions on power
delivery., 30, (1), pp. 445-454.
[24] Masoum, M. A. S., Jamali, S., and Ghaffarzadeh, N.,
2010, “Detection and classification of power quality
disturbances using discrete wavelet transform and
wavelet networks,” IET Science, Measurement &
Technology., 4 (4), pp. 193-205.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 14 (2018) pp. 11440-11447
© Research India Publications. http://www.ripublication.com
11447
[25] Robertson, D. C., Camps, O. L., Mayer, J. S., and
Gish, W. B., 1996, “Wavelets and Electromegnetic
Power System Transients,” IEEE transaction on
power Delivery., 11(2), pp. 1050-1058.
[26] Ghartemani, M. K., and Iravani, M. R., 2003 “A
signal processing module for power system
applications,” IEEE Transection on Power Delivery.,
18 (4), pp. 1118–1126.
[27] Heydt, G. T. and Galli, A. W., 1997, “Transient
Powcr Quality Problems Analyzed Using Wavclcls,”
IEEE Transaction oil power Delivery., 12, (2), pp.
908-915.
[28] Averbuch, A., Lezar, D., and Israeli, M., 1996,
“Image compression using wavelet transform and
multiresolution decompossion,” IEEE Transaction
Image Processing., 14, (1), pp. 4-15.
[29] Mallat, S., and Zhong, S., 1992, “Characterization of
signals from multiscale edges,” IEEE Transaction on
Pattern Analysis and Machine Intelligence., 14 (7),
pp. 710-732.
[30] Daubechies, I., 1989, “Wavelets: A tool for time-
frequency analysis.” Sixth IEEE Workshop
in Multidimensional Signal Processing.
[31] Daubechies, I., 1992, “The Wavelet transform, time-
frequency localization and signal analysis,” IEEE
Transaction on Information Theory., 36 (5), pp. 961-
1005.
[32] I. Daubechies, Ten Lectures on Wavelets.
Philadelphia, PA: SIAM, 1992.
[33] Daubechies, I., 1995, “Wavelets: an overview, with
recent applications,” IEEE International Symposium
on Information Theory., pp. 17-22.
[34] Daubechies, I., 1996 “Where do wavelets come
from? A personal point of view,” in Processing Of
the IEEE., 84 (4), pp. 510-513.
top related