research article fault diagnosis for analog circuits by...

Research ArticleFault Diagnosis for Analog Circuits by Using EEMDRelative Entropy and ELM

Jian Xiong12 Shulin Tian1 and Chenglin Yang1

1School of Automation Engineering University of Electronic Science and Technology of China Chengdu 611731 China2Department of Communication Engineering Chengdu Technological University Chengdu 611731 China

Correspondence should be addressed to Jian Xiong xiongjiancdtueducn

Received 21 May 2016 Revised 18 July 2016 Accepted 28 July 2016

Academic Editor Rodolfo Zunino

Copyright copy 2016 Jian Xiong et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a novel fault diagnosis method for analog circuits using ensemble empirical mode decomposition (EEMD)relative entropy and extreme learning machine (ELM) First nominal and faulty response waveforms of a circuit are measuredrespectively and then are decomposed into intrinsic mode functions (IMFs) with the EEMDmethod Second through comparingthe nominal IMFs with the faulty IMFs kurtosis and relative entropy are calculated for each IMF Next a feature vector is obtainedfor each faulty circuit Finally an ELM classifier is trained with these feature vectors for fault diagnosis Via validating with twobenchmark circuits results show that the proposedmethod is applicable for analog fault diagnosis with acceptable levels of accuracyand time cost

1 Introduction

Numerous researches have indicated that analog circuit faultdiagnosis is a significant fundamental for design validationand performance evaluation in the integrated circuit man-ufacturing fields [1ndash3] In contrast to the well-developeddiagnostic methods for digital circuits diagnosis for analogcircuits is an extremely difficult problem and an activeresearch due to the following reasons (1) there is lack of a reli-able and practical fault modeling method for analog circuitsbecause of the complexity and variability of analog circuitstructures (2) the parameter values of analog components arecontinuous (3) the impact of tolerance and nonlinear natureissues cannot be ignored (4) for actual analog circuits testpoints are limitations

The procedure of fault diagnosis for analog circuits canbe generally classified into four stages data acquisitionfeature extraction fault detection and fault identification andisolation As one of the foremost stages in fault diagnosisfeature extractionmethods are closely related to the efficiencyof fault diagnosis Many feature extraction methods havebeen proposed such as correlation function technique [4]information entropy approach [5] the fast Fourier transformtechnique [6] and the wavelet transform technique [7]

Zhang et al [8] directly used the output voltage as featuresfor fault diagnosis of analog circuits without preprocessingmethods and the results of fault diagnosis are not very goodM Aminian and F Aminian proposed a diagnostic methodof analog circuits using wavelet decomposition coefficientsprincipal component analysis (PCA) and data normalizationto construct fault feature vectors and then trained and testedneural network classifiers [3] The method can obtain higheraccuracy of diagnosis In [9] Long et al adopted conventionaltime-domain feature vectors to train and test least squaressupport vector machines (LS-SVM) for fault diagnosis ofanalog circuits which has better accuracy than that withtraditional wavelet feature vectors For information entropytechniques it is more sensitive to parameter variations ofcomponents in CUTs Therefore information entropy iswidely used with other techniques for fault diagnosis [5 10ndash12] Xie et al diagnosed soft faults of analog circuits usingRenyirsquos entropy and the result is effective [5] In [11] authorshave developed a new fault diagnosis approach by usingkurtosis and entropy of sampled signals as feature vectors totrain a neural network classifier

However there are some problems which should beconsidered and solved in feature extraction Firstly howto select features to train classifiers should be considered

Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2016 Article ID 7657054 9 pageshttpdxdoiorg10115520167657054

2 Computational Intelligence and Neuroscience

because different features with different classifiers for analogfault diagnosis have different results Secondly we find thatmost of the aforementioned methods were validated withsome discrete simulations data That is they only considereda CUT to be faulty when a component value is higher orlower than its nominal value by 50 It means this methodhas low fault coverageThirdly somemethods should take theinfluence of tolerance and the continuity of faulty parametersinto account

In our work therefore we use the techniques of EEMDkurtosis and relative entropy to construct new feature vectorsto train an ELM classifier to improve the diagnosability andreduce time cost As an adaptive time frequency data analysismethod ensemble empirical mode decomposition (EEMD)is suitable for linear nonlinear and no-stationary signals[13] Recently it has been successfully applied to extractsignificant fault features in many fields such as rotatingmachinery and locomotive roller bearings fault diagnosis [13ndash15] Relative entropy method is rarely used in the analogycircuit fault diagnosis field The difference between theprobability distributions of faulty and fault-free circuits canbe distinguished clearly by adopting relative entropy becausewhen a component is varied the energy distribution is alsochanged which leads to change in relative entropy Kurtosis isameasure of heavy tailed distribution of a real valued randomvariable It can clearly describe the difference from wave-forms As a result the combinationalmethods of kurtosis andrelative entropy are suitable as fault features for analog faultdiagnosis

As a consequence in this paper we decomposed impulseresponses of a CUT into IMFs using EEMD method andthen adopting kurtosis and relative entropy techniques toobtain feature vectors These features vectors can be usedfor diagnosis of faulty components among various variationpossibilities For this purpose a classifier is needed Weselected extreme learning machine (ELM) classifier becauseit is proven to have excellent generalization performanceand low computational cost [16 17] when it is fed to trainand test with fault features Utilizing the combination ofEEMD relative entropy and ELM algorithms for featureextraction and classification we can complete analog circuitfault diagnosis It demonstrates reliable and accurate faultdiagnosis with reduced test time

This paper is organized as follows Section 2 brieflypresents the principle of EEMD relative entropy and ELMalgorithms In Section 3 the diagnostic procedure of the pro-posed method is introduced Section 4 shows the simulationexperiment details and results for two benchmark analogcircuits And then the performance of the proposed methodis also discussed in the Section Finally the conclusions aredrawn in Section 5

2 A Review of Fundamental Theory

In the work we combined EEMD relative entropy and ELMto perform fault diagnosis of analog circuits Fundamentalsof EEMD relative entropy and ELM are introduced firstly asfollows

21 Ensemble Empirical Mode Decomposition (EEMD) En-semble empirical mode decomposition based on empiricalmode decomposition (EMD) is to solve the aliasing in timefrequency distribution with Gaussian white noise [13] Basedon simple assumption any signal consists of different simpleintrinsic modes of oscillations from low to high frequency[13 19] Thus original signal is defined as

119909 (119905) =

119899

sum

119894=1

119888119894(119905) + 119903

119899(119905) (1)

where 119888119894(119905) is the intrinsic mode functions (IMF) An IMF

is defined as a simple oscillatory function that satisfies twoconditions [18]

(1) It has the same number of extrema and zero crossingor has the difference nomore than one between them

(2) The mean value of the envelopes defined by the localmaxima and minima is zero

From (1) we can see that the original signal is decom-posed into 119899 IMFs and one residue 119903

119899(119905) The procedure of

decomposition with shifting method is described as follows

Step 1 Given a signal 119909(119905) all local maxima and minima ofit are gained firstly Then upper and lower envelopes of thegiven signal are determined from a cubic spline interpolationof the local maxima and minima Let 119898

1be the mean of the

two envelopes and the first component ℎ1(119905) is obtained as

ℎ1(119905) = 119909 (119905) minus 119898

1 (2)

Step 2 Let 11989811

be the mean of ℎ1(119905)rsquos upper and lower

envelopes and ℎ11(119905) is calculated as follows

ℎ11(119905) = ℎ

1(119905) minus 119898

11 (3)

Step 3 Repeat the above procedure 119899 times until ℎ1119899(119905)

satisfies IMF conditions The first IMF 1198881(119905) is obtained by

1198881(119905) = ℎ

1119899(119905)

Step 4 Subtract 1198881(119905) from 119909(119905) and a residue is obtained as

1199031(119905) = 119909 (119905) minus 119888

1(119905) (4)

Step 5 The residue which contains useful information isconsidered as main signal and Steps 1ndash4 are repeated to gainother IMFs Formula (4) is rewritten as

119903119894(119905) = 119903

119894minus1(119905) minus 119888

119894(119905) 119894 = 1 2 119899 (5)

Step 6 When the residue 119903119899(119905) becomes monotonic slope or

has only one extreme the whole procedure is stopped

From the procedure we can see that IMFs represent thedegree of oscillation of signal in amplitude and frequency Itmeans that these IMFs contain much time frequency infor-mation of the signal Thus the authors in [13] indicated thatthe algorithm is a new high-performance signal processingapproach which can deal with linear nonlinear and no-stationary signals More details about this technique can befound in [13 19]

Computational Intelligence and Neuroscience 3

1

1

Outputlayer

Hiddenlayer

Inputlayer

middot middot middot

middot middot middot middot middot middot


f (X)O1 Oj

1205731120573i

120573L

i L

d

X

g(X1)g(Xi) g(XL)

Figure 1 SLFN

22 Relative Entropy Let119883 be a continuous randomvariable119901(119909) and 119902(119909) are the probability distributions of 119883 Relativeentropy describes the distance between two probability dis-tributions of119883 The relative entropy is calculated as

119863(119901 119902) = int

119909isin119883

119901 (119909) log119901 (119909)

119902 (119909) (6)

where 119901(119909) denotes energy probability distribution function(PDF) of response voltages for faulty CUT and 119902(119909) indicatesnormal response voltage PDF of fault-free CUT Whenparameters of one or more components of CUT are changedthe PDF of corresponding output voltage will also varyThis means that it is more sensitive to parameter variationsof components in CUT By calculating the relative entropybetween faulty and fault-free circuit faults can be detectedConsequently for fault diagnosis relative entropy is suitableas fault feature

23 Extreme Learning Machine In order to accurately andquickly diagnose faults in our work extreme learningmachine (ELM) is adopted ELM is one kind of fast algo-rithm of single hidden-layer feedforward networks (SLFN)as shown in Figure 1 The hidden layer of SLFN need not betuned It is proven that it has excellent generalization per-formance and low computational cost in many applications[16 17] In the paper we utilize it to do fault diagnosis as aclassifier A brief of review of ELM is described as follows [16]

Suppose (119883119894 119905119894) isin 119877

119899

times 119877119898 are 119873 arbitrary distinct

samples 119883119894= [119909

1198941 1199091198942 119909

119894119899]119879 For a SLFN with 119871

hidden nodes taking one output node as example the outputfunction is defined as

119891119871(119883) = 119874

119895=

119871

sum

119894=1

120573119894119892 (119882119894sdot 119883119895+ 119887119894) = 119867120573

119895 = 1 2 119873

(7)

where 120573 = [1205731 1205732 120573

119871]119879 is the output weight between

hidden layer and output layer 119892(119883) is the activation functionwhich demonstrates the output vector of the hidden layerwith respect to the input 119883 119882

119894= [1199081198941 1199081198942 119908

119894119899]119879 is the

input weight of the 119894th hidden node and 119887119894denotes the bias

TestingTraining

Data processingData

acquisition EEMD Relativeentropy

ELMDiagnosisresult

Fault diagnosis

Figure 2 Block diagram of fault diagnosis

of hidden node 119894 And 119882119894sdot 119883119895represents the dot-product

between119882119894and 119883

119895 119867 is the hidden-layer output matrix It

is

119867 =

[[[[[[

[

119892 (1198821sdot 1198831+ 1198871) sdot sdot sdot 119892 (119882

119871sdot 1198831+ 119887119871)


119871sdot 1198832+ 119887119871)

sdot sdot sdot


119871sdot 119883119873+ 119887119871)

]]]]]]

]

(8)

The target of ELM is to minimize the output error hence theminimal norm least square method is adopted

Minimize 1003817100381710038171003817119867120573 minus 1198791003817100381710038171003817

2

(9)

where 119879 = [1199051 1199052 119905119873]119879 indicates the expected value of

output Once 119882119894and 119887119894are determined 119867 is also uniquely

confirmed According to formula (7) the output weight canbe calculated by

120573 = 119867+

119879 (10)

where119867+ is theMoorendashPenrose generalized inverse ofmatrix119867

3 Diagnostic Procedure

31 Diagnostic Procedure The diagnostic procedure basedon EEMD relative entropy and ELM is shown in Figure 2The procedure of the proposed method involves four majorstages data acquisition data processing training and faultdiagnosis Once the response voltage waveforms of fault-free circuit and fault circuits are recorded respectively theywill be decomposed into IMF components by using EEMDThrough utilizing the energy of each IMF then kurtosis andrelative entropy can be obtained between faulty IMFs andfault-free IMFs Kurtosis and relative entropy of some IMFsof each fault are selected to compose a fault feature vectorThe unique feature vector is extracted for each fault which isused for training and testing ELM classifier to complete faultdiagnosis

32 The Procedure of Feature Extraction The procedure offeature extraction of the proposed method is described asfollows

Step 1 Every fault (including fault-free status) of CUT issimulated in PSPICE And the relevant output waveforms areobtained


Seg1 Seg2 Seg3 Seg4 Seg5 Seg6

Signal

Figure 3 Segments of IMF

Step 2 Decompose each waveform with EEMD into 119899 IMFsaccording to the method in Section 21

Step 3 Calculate kurtosis and relative entropy of each IMF

Step 31 Obtain kurtosis from each IMF According to [11]kurtosis is a measure of the heaviness of the tails in adistribution of the signal 119909 [20] hence kurtosis could reactto the change of signals and be used as feature of signalsKurtosis is defined in the zero-mean case as follows [11]

kurt119894(119909) = 119864 119909

4

minus 3 [119864 1199092

]2

119894 = 1 2 119899 (11)

where 119864() is the expectation operator kurt119894() is the kurtosis

of IMF 119894 for a fault

Step 32 Calculate relative entropy of each IMF

(1) Calculate total energy of each IMF by

119864119899=

119870

sum

119894=1

(119862119899(119894))2

119894 = 1 2 119870 (12)

where 119862119899(119894) is the 119899th IMF and the length of 119862

119899(119894) is

equal to 119870(2) Calculate probability distribution According to [21]

the nonnegative energy distribution can be visualizedas probability distribution of signal Hence the pro-cess of calculating energy distribution is as followseach IMF is averagely divided into 119898 segments asshown in Figure 3 where 119898 is 6 The energy of eachsegment is equal to

119864119899119898=

1198992

sum

119894=1198991

(119862119899(119894))2

(13)

where 119898 = 1 2 3 6 is the number of segments1198991and 1198992are starting and stopping time points of the

segment The energy distribution of each segment inthe whole IMF can be expressed as

119901119899119898=119864119899119898

119864119899

(14)

C2

C1

R2

1k

5n

5nR3

R1

2k

R44k

R5

4k

0

Out

0

U1

+OutV1

minus+

518k

minus

Figure 4 Second-order Sallen-Key bandpass filter circuit

(3) According to the relative entropy theory the defini-tion of relative entropy of each IMFs is

119863119899=

6

sum

119898=1

119901119899119898

log119901119899119898

119901119900119898

(15)

where 119901119900119898

is the energy distribution of segment ofnominal IMF of fault-free circuit

Step 4 A feature vector for each fault can be given as

119879119896= [kurt

1 kurt2 kurt

119894 1198631 1198632 119863

119894]

119894 = 1 2 119899

(16)

where 119896 denotes the number of fault samples in a circuit and119899 is the number of IMFs of one fault Normalizing the featurevectors in formula (16) is reasonable to doHere we use partlynormalizedmethod to normalize some features in the featurevector which is defined as follows

119879norm 119896 = [kurt1

max119896(kurt1)

kurt2

max119896(kurt2)

kurt119894

max119896(kurt119894) 1198631 1198632 119863

119894] 119894 = 1 2 119899

(17)

Finally we could use the feature vectors to train and test anELM classifier for fault diagnosis

4 Experiment and Performance Results

41 A Sallen-Key Bandpass Filter To verify the capacity offault diagnosis with the proposed method the first examplecircuit is a second-order Sallen-Key bandpass filter circuitwhich is a benchmark circuit and is used as a CUT in[3 9 18] Figure 4 shows the schematic of the circuit withnominal parameter values From the figure we can see thatthe filter circuit consists of 5 resistors 2 capacitors and 1operational amplifier First the operational amplifiers in thecircuit are assumed to be fault-free Second we suppose eachpotential faulty componentrsquos nominal value is k and its faultyparameter range is [10minus4 lowast k 95 lowast k] and [105 lowast k104

lowast k] The nominal and faulty parameter ranges of thefilterrsquos components are shown in Table 1 In the table there are


Am

plitu

de (V

)

minus4minus2

024

times10minus41 2 30

Time (s)

(a)

Am

plitu

de (V

)

minus10

minus5

0

5

times10minus41 2 30

Time (s)

(b)

Am

plitu

de (V

)

minus4minus2

024

times10minus41 2 30

Time (s)

(c)

Am

plitu

de (V

)

minus10

minus5

0

5

times10minus41 2 30

Time (s)

(d)

Am

plitu

de (V

)

minus10minus5

05

10

times10minus41 2 30

Time (s)

(e)

Am

plitu

de (V

)

minus10minus5

05

10

times10minus41 2 30

Time (s)

(f)

Figure 5 Examples of waveforms of fault-free status and fault status for the Sallen-Key bandpass filter circuit (a) It is fault-free waveformand (b) (c) (d) (e) and (f) are fault waveforms for C1 uArr C1 dArr C2 dArr R2 dArr and R3 dArr respectively

Table 1 Fault configuration for Sallen-Key filter circuit

Fault ID Faults Nominal values Faulty parameter rangesF0 No-faultF1 R1 dArr 518k [05 492k]F2 R1 uArr 518k [544k 51meg]F3 R2 dArr 1k [01 095k]F4 R2 uArr 1k [105k 10meg]F5 R3 dArr 2k [02 19k]F6 R3 uArr 2k [21k 20meg]F7 R4 dArr 4k [04 38k]F8 R4 uArr 4k [42k 40meg]F9 R5 dArr 4k [04 38k]F10 R5 uArr 4k [42k 40meg]F11 C1 dArr 5n [05p 475n]F12 C1 uArr 5n [525n 50u]F13 C2 dArr 5n [05p 475n]F14 C2 uArr 5n [525n 50u]

total 15 faults including fault-free status where uArr and dArr standfor being higher and lower than nominal values respectively

According to the fault classes in Table 1 we useOrCADPSpice to simulate the circuit with time-domaintransient analysis and Monte Carlo analysis methods toobtain the simulation fault data First the Sallen-Key band-pass filter is stimulated by a excitation signal V1 which is asingle pulse of 10V with 10 120583s duration The run to time andmax step size are set as 300 120583s and 01 120583s respectively Theoutput voltage values are gained at the point ldquooutrdquo And toconsider the effects of the component tolerances the resistors

and capacitors are assumed to have tolerance limits of plusmn5When all the components are varying within their tolerancesthe circuit is considered no-fault Otherwise the parametervalue of any component is out of scope of its tolerance limitwith the other components varying within their toleranceswhich is regarded as a fault

In order to close to the actual circuit characteristic everyfault class will be simulated 150 times in faulty parameterranges using Monte Carlo analysis method in time domainand a total of 2250 corresponding impulse response wave-forms are obtained Some related waveforms are shownin Figure 5 In the figure (a) is the fault-free waveformand others are different impulse response waveforms aboutdifferent faulty circuits

The simulation data in PSpice are recorded and importedinto Matlab and then their feature vectors are constructedwith kurtosis and relative entropy to train an ELM classifierThe detail is described as follows

First to construct the feature vectors we decomposethese stored responses data into IMF components withEEMDmethod based on the discussion in Section 3 Figure 6displays the results of EEMD decomposition of no-faultcircuit and faulty C1 dArr In the figure each of the two responsesignals is decomposed into 10 IMF curves and a residuefrom high frequency to low frequency From the figure wecan clearly see that in same decomposition layer faulty IMFdiffers obviously from fault-free IMFTherefore here we onlytake C4ndashC9 IMF components into account to improve faultdistinguishability that can satisfy our work

Next kurtosis of each IMF for a certain fault circuitis calculated Meanwhile each IMF waveform (300 120583s 3000samples) is averagely divided into 6 segments The PDF of

6 Computational Intelligence and NeuroscienceC10

minus505

Inpu

t

minus010

01

C1

minus0050

005

C2

minus0050

005

C3

minus020

02

C4

minus505

C5

minus505

C6

minus050

05

C7

minus020

02

C8

minus010

01

C9

minus0050

005

minus050

05

R

500 1000 1500 2000 2500 3000 35000

Time(a)

C10

500 1000 1500 2000 2500 3000 35000

Time

minus0020

002

R

minus0010

001

minus0020

002

C9

minus0050

005C8

minus050

05

C7

minus101

C6

minus202

C5

minus020

02

C4

minus0050

005

C3

minus0050

005

C2

minus0050

005

C1

minus505

Inpu

t

(b)

Figure 6 EEMD decomposition results of two response signals In the decomposition noise for standard deviation 02 is added and theensemble number is 800 (a) The nominal response signal decomposition results (b) EEMD results of the response signal of C1 dArr

Table 2 PDF of the nominal IMFs of fault-free circuit

Fault name IMFs PDF1199011198991

1199011198992

1199011198993

1199011198994

1199011198995

1199011198996

Fault-free

C4 09710 00061 00047 00058 00063 00060C5 09683 00316 00000 00000 00000 00000C6 04853 04481 00657 00009 00000 00000C7 01316 00892 03222 03515 00982 00073C8 05588 03277 00419 00011 00228 00476C9 01302 02072 00276 02257 03032 01062

each IMF is calculated according to (12) (13) and (14) Table 2demonstrates PDFs of the nominal response IMFs for fault-free circuit Therefore relative entropy between each IMFof faulty components and corresponding nominal IMF offault-free circuit is achieved by adopting formula (15) Featurevector119879 for each fault classwill be built to feed directly into anELM classifier Take C1 dArr for example Table 3 shows a resultof feature extraction which is a feature vector of the faulty C1through calculating its kurtosis and relative entropy with (11)(15) and (16) Its fault feature vector is 119879 = [06162 0872909813 08713 09203 07989 00099 00109 02260 07400

01266 04903] As the same way a total of 2250 fault featurevectors of the circuit can be obtained

Finally for every fault class of the Sallen-Key circuit150 samples are split into two parts The first 100 faultfeature vectors are adopted to train an ELM classifier andthe remaining 50 fault feature vectors are used to test theELM Because the testing accuracy is sensitive to the selectionof activation functions the RBF function is proper for thediagnostic and the number of neurons is set as 250

In order to show the performance of the proposeddiagnostic method we compare our method with other


Table 3 Feature vector of C1 dArr

Fault name IMFs Relative entropy119863119894

Normalized kurtosis Feature vector 119879

C1 dArr

C4 00099 06162119879norm =[06162 08729 0981306713 09203 0798900099 00109 0266007400 01266 04903]

C5 00109 08729C6 02660 09813C7 07400 08713C8 01266 09203C9 04903 07989

Table 4 Results of fault classification of the Sallen-Key filter for single faults

Fault number Fault name Test accuracyProposed Wavelet [3] + ELM Lifting wavelet [18] + ELM [11] + ELM

F0 No-fault 10000 08600 10000 10000F1 R1 dArr 09700 09800 10000 09800F2 R1 uArr 10000 08700 10000 10000F3 R2 dArr 10000 1000 10000 09300F4 R2 uArr 10000 09800 10000 10000F5 R3 dArr 10000 08700 10000 10000F6 R3 uArr 09700 07800 09500 08200F7 R4 dArr 09900 09400 10000 10000F8 R4 uArr 10000 08400 10000 10000F9 R5 dArr 10000 09600 10000 10000F10 R5 uArr 10000 08200 10000 10000F11 C1 dArr 10000 08000 10000 10000F12 C1 uArr 09700 06400 09400 09600F13 C2 dArr 10000 10000 10000 10000F14 C2 uArr 10000 09800 10000 10000

Average accuracy 09940 08880 09930 09790

existing feature extraction methods which are presented in[3 11 18] to train anELMclassifierThe results of classificationare demonstrated in Table 4 For the single faults diagnosisof the Sallen-Key bandpass filter circuit the average testaccuracy of our method is 994 In contrast the wavelet andELM method (888) the lifting wavelet and ELM method(993) and the method in [11] (979) are lower than oursin test accuracy Thus we can see that the performance ofthe proposed method is superior to the combination methodof wavelet and ELM and the method of [11] Meanwhile ithas nearly the same accuracy as the lifting wavelet and ELMmethod Moreover these methods [3 11 18] only considereda CUT to be faulty when the value of potential faultycomponent is higher or lower than the nominal value by50 and did not take the continuity of faulty parameters andthe influence of tolerances into account If we use the samemethod considering only 50 variation as faulty parametervalues the test accuracy of our method could be up to 100in simulation

For reducing time cost we adopt an ELM algorithm as aclassifier because it is one of the best classification algorithmsand it also can provide higher performance in time costTable 5 shows ELM-based methodrsquos performance and SVM-based methodrsquos performance in time cost with the same fourtypes of original feature vectors respectivelyThe four feature

Table 5 Comparison of time cost and test accuracy between ELM-based method and SVM-based method for the Sallen-Key bandpassfilter

Feature extraction method Time (s)accuracy ()SVM classifier ELM classifier

Wavelet coefficients [3] 112931 00289888Kurtosis and entropy [11] 71986 00213979Lifting wavelet [18] 146992 00350993Proposed method 93998 00275994

vectors are based on different feature extraction methodsFrom the table it can be seen that the test accuracy ofELM-based method is approximate to SVM-based methodrsquosaccuracy For example the test accuracy of SVM-basedmethod is 986 for the kurtosis and entropy techniquewhich is similar to the ELM-basedmethod (979) Howeverthe time consumption of the ELM-based method is muchlower than the SVM-based method For instance usingwavelet coefficients as features to train SVM classifiersit takes 112 s on the contrary it takes 00289 s with anELM classifier For our proposed method its time cost isbetter than wavelet coefficients technique and lifting wavelet


U1

OPAMP

+Out

U2

OPAMP

+Out

U3

OPAMP

+Out

U4

OPAMP

+Out

U5

OPAMP

+Out

U6

OPAMP

+Out

R1

10k

R2

10k

R3

10k

R4

10k

R5

10k

R6

10k R7

10k R8

10k

R9

10k

R10

10k

R11

10k

R12

10k

R13

10k

C1

10n

C2

20n

C3

20n

C4

10n

Out0

0 0 00 0

0

V1

V

minus+minus

minus

minus

minus

minus

minus

V2 = 5V

TR = 1120583s

V1 = 0

PER = 20120583sPW = 10120583sTF = 10120583s

TD = 1120583s

Figure 7 Schematic of a leapfrog filter

technique As a result the proposed method in the paper canreduce time cost greatly

42 A Leapfrog Filter The second example circuit is aleapfrog filter which is used as a CUT in [9] The nominalvalues of the benchmark circuitrsquos components are shown inFigure 7 The input signal is also a single pulse with 5Vamplitude and 10120583s duration The ldquooutrdquo point of the circuitis the only test point As we all know several components of aCUT may cause faults simultaneously in practice Thereforein the experiment 10multifault cases are selected to verify ourproposed methodrsquos diagnostic performance for multifaultsin the CUT which is the same fault option as in [9] Thesefault classes are shown in Table 6 The experiment is alsocarried out through injecting these faults classes to the CUTrespectively according to the diagnostic procedure alreadydiscussed in Section 3 Diagnostic results of the circuit formultifaults are shown in Table 7 In the table the averagetest accuracy of our method is 989 whereas for thesemethods adopting these feature extraction methods in [3 1118] to diagnose these multifaults in the leapfrog filter circuittheir diagnostic accuracies are 881 905 and 868respectively Therefore we can see that the proposed methodis better than the other diagnostic methods for multifaults inthe leapfrog filter circuit

Through the two experiments the results of the proposedmethod can be summarized as follows

(1) The proposed method in the paper has better accu-racy than other methods such as the first waveletcoefficients technique and the lifting wavelet method

(2) For multifaults diagnosis the method adoptingEEMD kurtosis and relative entropy to constructfeature vectors has better classification accuracy thanthe traditional method used in [3 11 18]

(3) ELM classifiers with the techniques of EEMD kur-tosis and relative entropy sometimes get the samebetter classification results as SVM classifiers with

Table 6 Fault classes of the leapfrog filter for the multifaults

Fault ID Nominal value Fault valueF1 R2 10k R4 10k R2 20k R4 20kF2 R2 10k R4 10k R2 5k R4 5kF3 R3 10k R4 10k R3 20k R4 5kF4 R3 10k R4 10k R3 5k R4 5kF5 R4 10k R8 10k R4 15k R8 5kF6 R4 10k R8 10k R4 5k R8 20kF7 C2 20nf C3 20nf C2 30nf C3 30nfF8 C2 20nf C3 20nf C2 10nf C3 10nfF9 C2 20nf C4 10nf C2 10nf C4 20nfF10 C2 20nf C4 10nf C2 10nf C4 5nf

the same original feature vectors Meanwhile ELM-basedmethod hasmuch lower classification time thanSVM-based method

To sum up the proposed method in the paper is accept-able from two aspects test accuracy and time cost It hashigher test accuracy and fast classification capacity

5 Conclusions

In this paper a combinational diagnostic method for analogcircuit with EEMD relative entropy and ELM is proposedTheproposedmethodmakes gooduse of the EEMD kurtosisand relative entropy technique to construct fault featurevectors and then faults classification on CUTs are performedusing the ELM classifier The effectiveness of the proposedmethod has been validated with the classical two benchmarkcircuits for single and multifault diagnosis The results ofexperiments show that themethod can distinguish effectivelydifferent faults of circuit with the higher testing accuracy(994) and the lower testing time


Table 7 Diagnostic results of the leapfrog filter for the multiparametric faults

Diagnostic method Fault ID Average accuracy ()F1 F2 F3 F4 F5 F6 F7 F8 F9 F10

Reference [3] 100 100 51 46 86 100 100 100 98 100 881Reference [11] 100 70 100 98 70 74 100 100 95 98 905Reference [18] 100 100 52 34 82 100 100 100 100 100 868Our method 100 98 100 100 94 96 100 100 100 100 988

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the FundamentalResearch Funds for the Central Universities of China (Grantno ZYGX2015J074) Science and Technology Support Projectof Sichuan Province China (2014FZ0037 2015FZ0111) andSupport Project of CDTU (KY1311018B)

References

[1] C L Yang S Tian B Long and F Chen ldquoMethods of handlingthe tolerance and test-point selection problem for analog-circuit fault diagnosisrdquo IEEE Transactions on Instrumentationamp Measurement vol 60 no 1 pp 176ndash185 2011

[2] A S S Vasan B Long andMPecht ldquoDiagnostics andprognos-ticsmethod for analog electronic circuitsrdquo IEEETransactions onIndustrial Electronics vol 60 no 11 pp 5277ndash5291 2013

[3] M Aminian and F Aminian ldquoNeural-network based analog-circuit fault diagnosis using wavelet transform as preprocessorrdquoIEEE Transactions on Circuits and Systems II Analog andDigitalSignal Processing vol 47 no 2 pp 151ndash156 2000

[4] Z Hu Y Xie and T Li ldquoVLSI analog circuit fault diagnosticcorrelation coefficient analytical methodrdquo in Proceedings of theIEEE Circuits and Systems International Conference on Testingand Diagnosis (ICTD rsquo09) pp 1ndash4 IEEE Chengdu China April2009

[5] X Xie X Li D Bi Q Zhou S Xie and Y Xie ldquoAnalog circuitssoft fault diagnosis using renyirsquos entropyrdquo Journal of ElectronicTesting vol 31 no 2 pp 217ndash224 2015

[6] K Muhammad and K Roy ldquoFault detection and location usingIDD waveform analysisrdquo Design amp Test of Computers IEEE vol18 no 1 pp 42ndash49 2001

[7] B Long J Huang and S Tian ldquoLeast squares support vectormachine based analog-circuit fault diagnosis using wavelettransform as preprocessorrdquo in Proceedings of the InternationalConference on Communications Circuits and Systems (ICCCASrsquo08) pp 1026ndash1029 Fujian China May 2008

[8] Y Zhang X Y Wei and H F Jiang ldquoOne-class classifier basedon SBT for analog circuit fault diagnosisrdquoMeasurement vol 41no 4 pp 371ndash380 2008

[9] B LongM Li HWang and S Tian ldquoDiagnostics of analog cir-cuits based on LS-SVM using time-domain featuresrdquo CircuitsSystems and Signal Processing vol 32 no 6 pp 2683ndash27062013

[10] L Mingliang W Keqi S Laijun and Z Jianju ldquoApplyingempiricalmode decomposition (EMD) and entropy to diagnosecircuit breaker faultsrdquo Optik vol 126 pp 2338ndash2342 2015

[11] L Yuan Y He J Huang and Y Sun ldquoA new neural-network-based fault diagnosis approach for analog circuits by usingkurtosis and entropy as a preprocessorrdquo IEEE Transactions onInstrumentation and Measurement vol 59 no 3 pp 586ndash5952010

[12] R Krishnan W Wu F Gong and L He ldquoStochastic behav-ioral modeling of analogmixed-signal circuits by maximizingentropyrdquo in Proceedings of the 14th International Symposium onQuality Electronic Design (ISQED rsquo13) pp 572ndash579 Santa ClaraCalif USA March 2013

[13] Z Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[14] L Han C Li and H Liu ldquoFeature extraction method ofrolling bearing fault signal based on EEMD and cloud modelcharacteristic entropyrdquo Entropy vol 17 no 10 pp 6683ndash66972015

[15] T Wang X Wu T Liu and Z M Xiao ldquoGearbox faultdetection and diagnosis based on EEMDDe-noising and powerspectrumrdquo in Proceedings of the IEEE International Conferenceon Information and Automation (ICIA rsquo15) pp 1528ndash1531 IEEELijiang China August 2015

[16] G-B Huang Q-Y Zhu and C-K Siew ldquoExtreme learningmachine theory and applicationsrdquoNeurocomputing vol 70 no1ndash3 pp 489ndash501 2006

[17] A Lendasse V C Man Y Miche and G-B Huang ldquoAdvancesin extreme learning machines (ELM2014)rdquo Neurocomputingvol 174 part A pp 1ndash3 2015

[18] G Song Wang and S Jiang ldquoAnalog circuit fault diagnosisusing lifting wavelet transform and SVMrdquo Journal of ElectronicMeasurement amp Instrument vol 24 pp 17ndash22 2010 (Chinese)

[19] N E Huang ldquoNew method for nonlinear and nonstationarytime series analysis empiricalmode decomposition andHilbertspectral analysisrdquo in Proceedings of theWavelet Applications VIIvol 4056 of Proceedings of SPIE pp 197ndash209 April 2000

[20] CNikias andA PetropuluHigh-Order SpectraAnalysis ANon-linear Signal Processing Framework Prentice-Hall EnglewoodCliffs NJ USA 1993

[21] A Papoulis and H Saunders ldquoProbability random variablesand stochastic processes (2nd edition)rdquo Journal of Vibration ampAcoustics vol 111 1989

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



because different features with different classifiers for analogfault diagnosis have different results Secondly we find thatmost of the aforementioned methods were validated withsome discrete simulations data That is they only considereda CUT to be faulty when a component value is higher orlower than its nominal value by 50 It means this methodhas low fault coverageThirdly somemethods should take theinfluence of tolerance and the continuity of faulty parametersinto account

In our work therefore we use the techniques of EEMDkurtosis and relative entropy to construct new feature vectorsto train an ELM classifier to improve the diagnosability andreduce time cost As an adaptive time frequency data analysismethod ensemble empirical mode decomposition (EEMD)is suitable for linear nonlinear and no-stationary signals[13] Recently it has been successfully applied to extractsignificant fault features in many fields such as rotatingmachinery and locomotive roller bearings fault diagnosis [13ndash15] Relative entropy method is rarely used in the analogycircuit fault diagnosis field The difference between theprobability distributions of faulty and fault-free circuits canbe distinguished clearly by adopting relative entropy becausewhen a component is varied the energy distribution is alsochanged which leads to change in relative entropy Kurtosis isameasure of heavy tailed distribution of a real valued randomvariable It can clearly describe the difference from wave-forms As a result the combinationalmethods of kurtosis andrelative entropy are suitable as fault features for analog faultdiagnosis

As a consequence in this paper we decomposed impulseresponses of a CUT into IMFs using EEMD method andthen adopting kurtosis and relative entropy techniques toobtain feature vectors These features vectors can be usedfor diagnosis of faulty components among various variationpossibilities For this purpose a classifier is needed Weselected extreme learning machine (ELM) classifier becauseit is proven to have excellent generalization performanceand low computational cost [16 17] when it is fed to trainand test with fault features Utilizing the combination ofEEMD relative entropy and ELM algorithms for featureextraction and classification we can complete analog circuitfault diagnosis It demonstrates reliable and accurate faultdiagnosis with reduced test time

This paper is organized as follows Section 2 brieflypresents the principle of EEMD relative entropy and ELMalgorithms In Section 3 the diagnostic procedure of the pro-posed method is introduced Section 4 shows the simulationexperiment details and results for two benchmark analogcircuits And then the performance of the proposed methodis also discussed in the Section Finally the conclusions aredrawn in Section 5

2 A Review of Fundamental Theory

In the work we combined EEMD relative entropy and ELMto perform fault diagnosis of analog circuits Fundamentalsof EEMD relative entropy and ELM are introduced firstly asfollows

21 Ensemble Empirical Mode Decomposition (EEMD) En-semble empirical mode decomposition based on empiricalmode decomposition (EMD) is to solve the aliasing in timefrequency distribution with Gaussian white noise [13] Basedon simple assumption any signal consists of different simpleintrinsic modes of oscillations from low to high frequency[13 19] Thus original signal is defined as

119909 (119905) =

119899

sum

119894=1

119888119894(119905) + 119903

119899(119905) (1)

where 119888119894(119905) is the intrinsic mode functions (IMF) An IMF

is defined as a simple oscillatory function that satisfies twoconditions [18]

(1) It has the same number of extrema and zero crossingor has the difference nomore than one between them

(2) The mean value of the envelopes defined by the localmaxima and minima is zero

From (1) we can see that the original signal is decom-posed into 119899 IMFs and one residue 119903

119899(119905) The procedure of

decomposition with shifting method is described as follows

Step 1 Given a signal 119909(119905) all local maxima and minima ofit are gained firstly Then upper and lower envelopes of thegiven signal are determined from a cubic spline interpolationof the local maxima and minima Let 119898

1be the mean of the

two envelopes and the first component ℎ1(119905) is obtained as

ℎ1(119905) = 119909 (119905) minus 119898

1 (2)

Step 2 Let 11989811

be the mean of ℎ1(119905)rsquos upper and lower

envelopes and ℎ11(119905) is calculated as follows

ℎ11(119905) = ℎ

1(119905) minus 119898

11 (3)

Step 3 Repeat the above procedure 119899 times until ℎ1119899(119905)

satisfies IMF conditions The first IMF 1198881(119905) is obtained by

1198881(119905) = ℎ

1119899(119905)

Step 4 Subtract 1198881(119905) from 119909(119905) and a residue is obtained as

1199031(119905) = 119909 (119905) minus 119888

1(119905) (4)

Step 5 The residue which contains useful information isconsidered as main signal and Steps 1ndash4 are repeated to gainother IMFs Formula (4) is rewritten as

119903119894(119905) = 119903

119894minus1(119905) minus 119888

119894(119905) 119894 = 1 2 119899 (5)

Step 6 When the residue 119903119899(119905) becomes monotonic slope or

has only one extreme the whole procedure is stopped

From the procedure we can see that IMFs represent thedegree of oscillation of signal in amplitude and frequency Itmeans that these IMFs contain much time frequency infor-mation of the signal Thus the authors in [13] indicated thatthe algorithm is a new high-performance signal processingapproach which can deal with linear nonlinear and no-stationary signals More details about this technique can befound in [13 19]


1

1

Outputlayer

Hiddenlayer

Inputlayer




f (X)O1 Oj

1205731120573i

120573L

i L

d

X

g(X1)g(Xi) g(XL)

Figure 1 SLFN


119863(119901 119902) = int

119909isin119883

119901 (119909) log119901 (119909)

119902 (119909) (6)



Suppose (119883119894 119905119894) isin 119877

119899


samples 119883119894= [119909

1198941 1199091198942 119909

119894119899]119879 For a SLFN with 119871


119891119871(119883) = 119874

119895=

119871

sum

119894=1

120573119894119892 (119882119894sdot 119883119895+ 119887119894) = 119867120573

119895 = 1 2 119873

(7)

where 120573 = [1205731 1205732 120573



119894= [1199081198941 1199081198942 119908

119894119899]119879 is the


TestingTraining

Data processingData


ELMDiagnosisresult

Fault diagnosis



between119882119894and 119883


is

119867 =

[[[[[[

[


119871sdot 1198831+ 119887119871)


119871sdot 1198832+ 119887119871)

sdot sdot sdot


119871sdot 119883119873+ 119887119871)

]]]]]]

]

(8)


Minimize 1003817100381710038171003817119867120573 minus 1198791003817100381710038171003817

2

(9)




120573 = 119867+

119879 (10)








Signal





kurt119894(119909) = 119864 119909

4

minus 3 [119864 1199092

]2

119894 = 1 2 119899 (11)





119864119899=

119870

sum

119894=1

(119862119899(119894))2

119894 = 1 2 119870 (12)


119899(119894) is



119864119899119898=

1198992

sum

119894=1198991

(119862119899(119894))2

(13)



119901119899119898=119864119899119898

119864119899

(14)

C2

C1

R2

1k

5n

5nR3

R1

2k

R44k

R5

4k

0

Out

0

U1

+OutV1

minus+

518k

minus



119863119899=

6

sum

119898=1

119901119899119898

log119901119899119898

119901119900119898

(15)

where 119901119900119898



119879119896= [kurt

1 kurt2 kurt

119894 1198631 1198632 119863

119894]

119894 = 1 2 119899

(16)


119879norm 119896 = [kurt1

max119896(kurt1)

kurt2

max119896(kurt2)

kurt119894

max119896(kurt119894) 1198631 1198632 119863

119894] 119894 = 1 2 119899

(17)






Am

plitu

de (V

)

minus4minus2

024

times10minus41 2 30

Time (s)

(a)

Am

plitu

de (V

)

minus10

minus5

0

5

times10minus41 2 30

Time (s)

(b)

Am

plitu

de (V

)

minus4minus2

024

times10minus41 2 30

Time (s)

(c)

Am

plitu

de (V

)

minus10

minus5

0

5

times10minus41 2 30

Time (s)

(d)

Am

plitu

de (V

)

minus10minus5

05

10

times10minus41 2 30

Time (s)

(e)

Am

plitu

de (V

)

minus10minus5

05

10

times10minus41 2 30

Time (s)

(f)












minus505

Inpu

t

minus010

01

C1

minus0050

005

C2

minus0050

005

C3

minus020

02

C4

minus505

C5

minus505

C6

minus050

05

C7

minus020

02

C8

minus010

01

C9

minus0050

005

minus050

05

R

500 1000 1500 2000 2500 3000 35000

Time(a)

C10

500 1000 1500 2000 2500 3000 35000

Time

minus0020

002

R

minus0010

001

minus0020

002

C9

minus0050

005C8

minus050

05

C7

minus101

C6

minus202

C5

minus020

02

C4

minus0050

005

C3

minus0050

005

C2

minus0050

005

C1

minus505

Inpu

t

(b)




1199011198992

1199011198993

1199011198994

1199011198995

1199011198996

Fault-free

C4 09710 00061 00047 00058 00063 00060C5 09683 00316 00000 00000 00000 00000C6 04853 04481 00657 00009 00000 00000C7 01316 00892 03222 03515 00982 00073C8 05588 03277 00419 00011 00228 00476C9 01302 02072 00276 02257 03032 01062









C1 dArr

C4 00099 06162119879norm =[06162 08729 0981306713 09203 0798900099 00109 0266007400 01266 04903]

C5 00109 08729C6 02660 09813C7 07400 08713C8 01266 09203C9 04903 07989












U1

OPAMP

+Out

U2

OPAMP

+Out

U3

OPAMP

+Out

U4

OPAMP

+Out

U5

OPAMP

+Out

U6

OPAMP

+Out

R1

10k

R2

10k

R3

10k

R4

10k

R5

10k

R6

10k R7

10k R8

10k

R9

10k

R10

10k

R11

10k

R12

10k

R13

10k

C1

10n

C2

20n

C3

20n

C4

10n

Out0

0 0 00 0

0

V1

V

minus+minus

minus

minus

minus

minus

minus

V2 = 5V

TR = 1120583s

V1 = 0

PER = 20120583sPW = 10120583sTF = 10120583s

TD = 1120583s












5 Conclusions






Competing Interests


Acknowledgments


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




1

1

Outputlayer

Hiddenlayer

Inputlayer




f (X)O1 Oj

1205731120573i

120573L

i L

d

X

g(X1)g(Xi) g(XL)

Figure 1 SLFN


119863(119901 119902) = int

119909isin119883

119901 (119909) log119901 (119909)

119902 (119909) (6)



Suppose (119883119894 119905119894) isin 119877

119899


samples 119883119894= [119909

1198941 1199091198942 119909

119894119899]119879 For a SLFN with 119871


119891119871(119883) = 119874

119895=

119871

sum

119894=1

120573119894119892 (119882119894sdot 119883119895+ 119887119894) = 119867120573

119895 = 1 2 119873

(7)

where 120573 = [1205731 1205732 120573



119894= [1199081198941 1199081198942 119908

119894119899]119879 is the


TestingTraining

Data processingData


ELMDiagnosisresult

Fault diagnosis



between119882119894and 119883


is

119867 =

[[[[[[

[


119871sdot 1198831+ 119887119871)


119871sdot 1198832+ 119887119871)

sdot sdot sdot


119871sdot 119883119873+ 119887119871)

]]]]]]

]

(8)


Minimize 1003817100381710038171003817119867120573 minus 1198791003817100381710038171003817

2

(9)




120573 = 119867+

119879 (10)








Signal





kurt119894(119909) = 119864 119909

4

minus 3 [119864 1199092

]2

119894 = 1 2 119899 (11)





119864119899=

119870

sum

119894=1

(119862119899(119894))2

119894 = 1 2 119870 (12)


119899(119894) is



119864119899119898=

1198992

sum

119894=1198991

(119862119899(119894))2

(13)



119901119899119898=119864119899119898

119864119899

(14)

C2

C1

R2

1k

5n

5nR3

R1

2k

R44k

R5

4k

0

Out

0

U1

+OutV1

minus+

518k

minus



119863119899=

6

sum

119898=1

119901119899119898

log119901119899119898

119901119900119898

(15)

where 119901119900119898



119879119896= [kurt

1 kurt2 kurt

119894 1198631 1198632 119863

119894]

119894 = 1 2 119899

(16)


119879norm 119896 = [kurt1

max119896(kurt1)

kurt2

max119896(kurt2)

kurt119894

max119896(kurt119894) 1198631 1198632 119863

119894] 119894 = 1 2 119899

(17)






Am

plitu

de (V

)

minus4minus2

024

times10minus41 2 30

Time (s)

(a)

Am

plitu

de (V

)

minus10

minus5

0

5

times10minus41 2 30

Time (s)

(b)

Am

plitu

de (V

)

minus4minus2

024

times10minus41 2 30

Time (s)

(c)

Am

plitu

de (V

)

minus10

minus5

0

5

times10minus41 2 30

Time (s)

(d)

Am

plitu

de (V

)

minus10minus5

05

10

times10minus41 2 30

Time (s)

(e)

Am

plitu

de (V

)

minus10minus5

05

10

times10minus41 2 30

Time (s)

(f)












minus505

Inpu

t

minus010

01

C1

minus0050

005

C2

minus0050

005

C3

minus020

02

C4

minus505

C5

minus505

C6

minus050

05

C7

minus020

02

C8

minus010

01

C9

minus0050

005

minus050

05

R

500 1000 1500 2000 2500 3000 35000

Time(a)

C10

500 1000 1500 2000 2500 3000 35000

Time

minus0020

002

R

minus0010

001

minus0020

002

C9

minus0050

005C8

minus050

05

C7

minus101

C6

minus202

C5

minus020

02

C4

minus0050

005

C3

minus0050

005

C2

minus0050

005

C1

minus505

Inpu

t

(b)




1199011198992

1199011198993

1199011198994

1199011198995

1199011198996

Fault-free

C4 09710 00061 00047 00058 00063 00060C5 09683 00316 00000 00000 00000 00000C6 04853 04481 00657 00009 00000 00000C7 01316 00892 03222 03515 00982 00073C8 05588 03277 00419 00011 00228 00476C9 01302 02072 00276 02257 03032 01062









C1 dArr

C4 00099 06162119879norm =[06162 08729 0981306713 09203 0798900099 00109 0266007400 01266 04903]

C5 00109 08729C6 02660 09813C7 07400 08713C8 01266 09203C9 04903 07989












U1

OPAMP

+Out

U2

OPAMP

+Out

U3

OPAMP

+Out

U4

OPAMP

+Out

U5

OPAMP

+Out

U6

OPAMP

+Out

R1

10k

R2

10k

R3

10k

R4

10k

R5

10k

R6

10k R7

10k R8

10k

R9

10k

R10

10k

R11

10k

R12

10k

R13

10k

C1

10n

C2

20n

C3

20n

C4

10n

Out0

0 0 00 0

0

V1

V

minus+minus

minus

minus

minus

minus

minus

V2 = 5V

TR = 1120583s

V1 = 0

PER = 20120583sPW = 10120583sTF = 10120583s

TD = 1120583s












5 Conclusions






Competing Interests


Acknowledgments


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





Signal





kurt119894(119909) = 119864 119909

4

minus 3 [119864 1199092

]2

119894 = 1 2 119899 (11)





119864119899=

119870

sum

119894=1

(119862119899(119894))2

119894 = 1 2 119870 (12)


119899(119894) is



119864119899119898=

1198992

sum

119894=1198991

(119862119899(119894))2

(13)



119901119899119898=119864119899119898

119864119899

(14)

C2

C1

R2

1k

5n

5nR3

R1

2k

R44k

R5

4k

0

Out

0

U1

+OutV1

minus+

518k

minus



119863119899=

6

sum

119898=1

119901119899119898

log119901119899119898

119901119900119898

(15)

where 119901119900119898



119879119896= [kurt

1 kurt2 kurt

119894 1198631 1198632 119863

119894]

119894 = 1 2 119899

(16)


119879norm 119896 = [kurt1

max119896(kurt1)

kurt2

max119896(kurt2)

kurt119894

max119896(kurt119894) 1198631 1198632 119863

119894] 119894 = 1 2 119899

(17)






Am

plitu

de (V

)

minus4minus2

024

times10minus41 2 30

Time (s)

(a)

Am

plitu

de (V

)

minus10

minus5

0

5

times10minus41 2 30

Time (s)

(b)

Am

plitu

de (V

)

minus4minus2

024

times10minus41 2 30

Time (s)

(c)

Am

plitu

de (V

)

minus10

minus5

0

5

times10minus41 2 30

Time (s)

(d)

Am

plitu

de (V

)

minus10minus5

05

10

times10minus41 2 30

Time (s)

(e)

Am

plitu

de (V

)

minus10minus5

05

10

times10minus41 2 30

Time (s)

(f)












minus505

Inpu

t

minus010

01

C1

minus0050

005

C2

minus0050

005

C3

minus020

02

C4

minus505

C5

minus505

C6

minus050

05

C7

minus020

02

C8

minus010

01

C9

minus0050

005

minus050

05

R

500 1000 1500 2000 2500 3000 35000

Time(a)

C10

500 1000 1500 2000 2500 3000 35000

Time

minus0020

002

R

minus0010

001

minus0020

002

C9

minus0050

005C8

minus050

05

C7

minus101

C6

minus202

C5

minus020

02

C4

minus0050

005

C3

minus0050

005

C2

minus0050

005

C1

minus505

Inpu

t

(b)




1199011198992

1199011198993

1199011198994

1199011198995

1199011198996

Fault-free

C4 09710 00061 00047 00058 00063 00060C5 09683 00316 00000 00000 00000 00000C6 04853 04481 00657 00009 00000 00000C7 01316 00892 03222 03515 00982 00073C8 05588 03277 00419 00011 00228 00476C9 01302 02072 00276 02257 03032 01062









C1 dArr

C4 00099 06162119879norm =[06162 08729 0981306713 09203 0798900099 00109 0266007400 01266 04903]

C5 00109 08729C6 02660 09813C7 07400 08713C8 01266 09203C9 04903 07989












U1

OPAMP

+Out

U2

OPAMP

+Out

U3

OPAMP

+Out

U4

OPAMP

+Out

U5

OPAMP

+Out

U6

OPAMP

+Out

R1

10k

R2

10k

R3

10k

R4

10k

R5

10k

R6

10k R7

10k R8

10k

R9

10k

R10

10k

R11

10k

R12

10k

R13

10k

C1

10n

C2

20n

C3

20n

C4

10n

Out0

0 0 00 0

0

V1

V

minus+minus

minus

minus

minus

minus

minus

V2 = 5V

TR = 1120583s

V1 = 0

PER = 20120583sPW = 10120583sTF = 10120583s

TD = 1120583s












5 Conclusions






Competing Interests


Acknowledgments


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Am

plitu

de (V

)

minus4minus2

024

times10minus41 2 30

Time (s)

(a)

Am

plitu

de (V

)

minus10

minus5

0

5

times10minus41 2 30

Time (s)

(b)

Am

plitu

de (V

)

minus4minus2

024

times10minus41 2 30

Time (s)

(c)

Am

plitu

de (V

)

minus10

minus5

0

5

times10minus41 2 30

Time (s)

(d)

Am

plitu

de (V

)

minus10minus5

05

10

times10minus41 2 30

Time (s)

(e)

Am

plitu

de (V

)

minus10minus5

05

10

times10minus41 2 30

Time (s)

(f)












minus505

Inpu

t

minus010

01

C1

minus0050

005

C2

minus0050

005

C3

minus020

02

C4

minus505

C5

minus505

C6

minus050

05

C7

minus020

02

C8

minus010

01

C9

minus0050

005

minus050

05

R

500 1000 1500 2000 2500 3000 35000

Time(a)

C10

500 1000 1500 2000 2500 3000 35000

Time

minus0020

002

R

minus0010

001

minus0020

002

C9

minus0050

005C8

minus050

05

C7

minus101

C6

minus202

C5

minus020

02

C4

minus0050

005

C3

minus0050

005

C2

minus0050

005

C1

minus505

Inpu

t

(b)




1199011198992

1199011198993

1199011198994

1199011198995

1199011198996

Fault-free

C4 09710 00061 00047 00058 00063 00060C5 09683 00316 00000 00000 00000 00000C6 04853 04481 00657 00009 00000 00000C7 01316 00892 03222 03515 00982 00073C8 05588 03277 00419 00011 00228 00476C9 01302 02072 00276 02257 03032 01062









C1 dArr

C4 00099 06162119879norm =[06162 08729 0981306713 09203 0798900099 00109 0266007400 01266 04903]

C5 00109 08729C6 02660 09813C7 07400 08713C8 01266 09203C9 04903 07989












U1

OPAMP

+Out

U2

OPAMP

+Out

U3

OPAMP

+Out

U4

OPAMP

+Out

U5

OPAMP

+Out

U6

OPAMP

+Out

R1

10k

R2

10k

R3

10k

R4

10k

R5

10k

R6

10k R7

10k R8

10k

R9

10k

R10

10k

R11

10k

R12

10k

R13

10k

C1

10n

C2

20n

C3

20n

C4

10n

Out0

0 0 00 0

0

V1

V

minus+minus

minus

minus

minus

minus

minus

V2 = 5V

TR = 1120583s

V1 = 0

PER = 20120583sPW = 10120583sTF = 10120583s

TD = 1120583s












5 Conclusions






Competing Interests


Acknowledgments


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




minus505

Inpu

t

minus010

01

C1

minus0050

005

C2

minus0050

005

C3

minus020

02

C4

minus505

C5

minus505

C6

minus050

05

C7

minus020

02

C8

minus010

01

C9

minus0050

005

minus050

05

R

500 1000 1500 2000 2500 3000 35000

Time(a)

C10

500 1000 1500 2000 2500 3000 35000

Time

minus0020

002

R

minus0010

001

minus0020

002

C9

minus0050

005C8

minus050

05

C7

minus101

C6

minus202

C5

minus020

02

C4

minus0050

005

C3

minus0050

005

C2

minus0050

005

C1

minus505

Inpu

t

(b)




1199011198992

1199011198993

1199011198994

1199011198995

1199011198996

Fault-free

C4 09710 00061 00047 00058 00063 00060C5 09683 00316 00000 00000 00000 00000C6 04853 04481 00657 00009 00000 00000C7 01316 00892 03222 03515 00982 00073C8 05588 03277 00419 00011 00228 00476C9 01302 02072 00276 02257 03032 01062









C1 dArr

C4 00099 06162119879norm =[06162 08729 0981306713 09203 0798900099 00109 0266007400 01266 04903]

C5 00109 08729C6 02660 09813C7 07400 08713C8 01266 09203C9 04903 07989












U1

OPAMP

+Out

U2

OPAMP

+Out

U3

OPAMP

+Out

U4

OPAMP

+Out

U5

OPAMP

+Out

U6

OPAMP

+Out

R1

10k

R2

10k

R3

10k

R4

10k

R5

10k

R6

10k R7

10k R8

10k

R9

10k

R10

10k

R11

10k

R12

10k

R13

10k

C1

10n

C2

20n

C3

20n

C4

10n

Out0

0 0 00 0

0

V1

V

minus+minus

minus

minus

minus

minus

minus

V2 = 5V

TR = 1120583s

V1 = 0

PER = 20120583sPW = 10120583sTF = 10120583s

TD = 1120583s












5 Conclusions






Competing Interests


Acknowledgments


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







C1 dArr

C4 00099 06162119879norm =[06162 08729 0981306713 09203 0798900099 00109 0266007400 01266 04903]

C5 00109 08729C6 02660 09813C7 07400 08713C8 01266 09203C9 04903 07989












U1

OPAMP

+Out

U2

OPAMP

+Out

U3

OPAMP

+Out

U4

OPAMP

+Out

U5

OPAMP

+Out

U6

OPAMP

+Out

R1

10k

R2

10k

R3

10k

R4

10k

R5

10k

R6

10k R7

10k R8

10k

R9

10k

R10

10k

R11

10k

R12

10k

R13

10k

C1

10n

C2

20n

C3

20n

C4

10n

Out0

0 0 00 0

0

V1

V

minus+minus

minus

minus

minus

minus

minus

V2 = 5V

TR = 1120583s

V1 = 0

PER = 20120583sPW = 10120583sTF = 10120583s

TD = 1120583s












5 Conclusions






Competing Interests


Acknowledgments


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




U1

OPAMP

+Out

U2

OPAMP

+Out

U3

OPAMP

+Out

U4

OPAMP

+Out

U5

OPAMP

+Out

U6

OPAMP

+Out

R1

10k

R2

10k

R3

10k

R4

10k

R5

10k

R6

10k R7

10k R8

10k

R9

10k

R10

10k

R11

10k

R12

10k

R13

10k

C1

10n

C2

20n

C3

20n

C4

10n

Out0

0 0 00 0

0

V1

V

minus+minus

minus

minus

minus

minus

minus

V2 = 5V

TR = 1120583s

V1 = 0

PER = 20120583sPW = 10120583sTF = 10120583s

TD = 1120583s












5 Conclusions






Competing Interests


Acknowledgments


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







Competing Interests


Acknowledgments


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



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