Research Article Fault Diagnosis for Analog Circuits by ...downloads.hindawi.com/journals/cin/2016/7657054.pdf · Research Article Fault Diagnosis for Analog Circuits by Using ...
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
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
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
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
4 Computational Intelligence and Neuroscience
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
(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
Computational Intelligence and Neuroscience 5
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
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
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
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
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
8 Computational Intelligence and Neuroscience
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
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
Computational Intelligence and Neuroscience 9
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
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
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
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
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
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
4 Computational Intelligence and Neuroscience
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
(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
Computational Intelligence and Neuroscience 5
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
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
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
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
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
8 Computational Intelligence and Neuroscience
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
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
Computational Intelligence and Neuroscience 9
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
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
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
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
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
4 Computational Intelligence and Neuroscience
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
(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
Computational Intelligence and Neuroscience 5
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
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
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
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
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
8 Computational Intelligence and Neuroscience
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
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
Computational Intelligence and Neuroscience 9
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
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
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
(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
Computational Intelligence and Neuroscience 5
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
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
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
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
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
8 Computational Intelligence and Neuroscience
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
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
Computational Intelligence and Neuroscience 9
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
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
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
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
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
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
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
8 Computational Intelligence and Neuroscience
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
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
Computational Intelligence and Neuroscience 9
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
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
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
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
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
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
8 Computational Intelligence and Neuroscience
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
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
Computational Intelligence and Neuroscience 9
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
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
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
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
8 Computational Intelligence and Neuroscience
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
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
Computational Intelligence and Neuroscience 9
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
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
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
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
Computational Intelligence and Neuroscience 9
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
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
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