Fault detection in analog and mixed-signal circuits by using Hilbert–Huang transform and coherence analysis

Microelectronics Journal 46 (2015) 893–899

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Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo

Fault detection in analog and mixed-signal circuits by using Hilbert–Huang transform and coherence analysis Shengxue Tang n, Zhigang Li, Li Chen Province-Ministry Joint Key Laboratory of Electromagnetic Field and Electrical Apparatus Reliability, Hebei University of Technology, Tianjin 300130, China

art ic l e i nf o

a b s t r a c t

Article history: Received 22 April 2014 Received in revised form 19 May 2015 Accepted 22 July 2015

Using Hilbert–Huang transform (HHT) and coherence analysis, a signature extraction method for testing analog and mixed-signal circuits is proposed in this paper. The instantaneous time–frequency signatures extracted with HHT technique from the measured signal of circuits under test (CUT) are used for faults detection that is implemented through comparing the signatures of faulty circuits with that of the faultfree circuit. The coherence functions of the instantaneous time–frequency signatures and its integral help to test faults in the faulty dictionary according to the minimum distance criterion. The superior capability of HHT-based technique, compared to traditional linear techniques such as the wavelet transform and the fast Fourier transform, is to obtain the subtle time-varying signatures, i.e., the instantaneous time–frequency signatures, and is demonstrated by applying to Leapfrog filter, a benchmark circuit for analog and mixed-signal testing, with 100% of F.D.R (fault detection rate) in the best cases and with the least 24.2% of F.L.R. (fault localization rate) with one signature. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Analog and mixed-signal circuit Circuit test Hilbert–Huang transform Coherence function

1. Introduction Fault test of analog and mixed-signal circuits are fundamental important processes for design validation and performance evaluation, and are getting more attentions in the integrated circuit manufacturing fields [1–3]. However, in many cases faults test in analog and mixed-signal circuits is still a complex and complicated task, and the efficiency based on oriented-fault test is closely related to the signature preprocessing techniques for extracting the signature from the signal measured on out terminal of CUT. For fault detection, test method must be sensitive enough to show the signature variations, especial for the parametric fault detection, which helps to deal with the tasks of test method, such as compressing the response of CUT for data storage, reducing the influence of noises and magnifying the difference of response data to show faults in analog and mixed-signal circuits. Three spaces, in which the tasks of signature extraction on the response of CUT are implemented, can be used to classify the test method, that is, the time domain method, the frequency domain method and the time– frequency domain method. For example, correlation function technique [2] and Kurtosis and Entropy technique [4] are the time domain method, and fast Fourier transform (FFT) technique [5] is the typical frequency method. The wavelet transform (WT) [6]

n

Corresponding author.

http://dx.doi.org/10.1016/j.mejo.2015.07.004 0026-2692/& 2015 Elsevier Ltd. All rights reserved.

technique and sub-band filtering technique [7] are two typical time–frequency method. It has been verified in many cases that the time–frequency method has better performance than two other methods and is more sensitive to signature variations because the signatures include both of time and frequency information [7,8]. However, there are needs for an a priori knowledge on selection of mother wavelet and time scales for the WT technique [9], and needs the band-width and band-center location ahead for the subband filtering technique [7], which affect the signature extraction accuracy in the time–frequency domain, especially for the parametric faults. Although the input and output signals in the CUT change, the signature of the CUT will not change at sometimes or changes small so that it leads to detect fault difficultly in applications, due to the inappropriate a priori knowledge. Recently, an instantaneous time–frequency decomposition technique, i.e., the Hilbert–Huang transform (HHT), has been proposed, and does not require an a priori functional basis as the WT technique, which has been used to analyze non-stationary and nonlinear signal [9–11]. Unlike the WT and FFT, the basis functions of HHT are derived adaptively from the data by the empirical mode decomposition (EMD) with sifting procedures and the instantaneous frequencies are computed from derivatives of the phase functions of the Hilbert transform of the basis functions, and hence the final result is presented in the time–frequency space. Our motivation in this research is to apply the HHT technique combined with the coherence analysis technique to fault test in analog and mixed-signal circuits.

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2. Previous work For fault detection and localization in analog and mixed-signal circuits, there have been some investigations focused on fault signature analysis. Some of these works are reviewed in this section, and then our method by HHT technique and coherence analysis technique are presented in detail. 2.1. Previous work To detect the faults, Chatterjee et al. proposed a low cost DC built-in self-test scheme [12], in which the DC voltage is for the fault signature and is only effective to the catastrophic faults of linear analog circuits. In [13], the slope voltage increment in two nodes was presented for the fault signature to detect the parametric faults of the linear resistive circuit. These signatures are extracted from time domain. Through the FFT technique, Muhammad et al. presented the frequency signature extracted from the IDD signal to locate the faults in CMOS circuits [5]. In [14], Luo proposed an optimal fractional Fourier transform method to extract the fault signature. However, the FFT-based methods assume that the signal measured is the linearity and stationary processes, and has low accuracy for nonlinear circuits. AT the same time, the signature extraction techniques in time– frequency domain are studied by researchers. Aminian proposed the principal wavelet coefficients as signatures from the impulse response of CUT preprocessed by wavelet decomposition and principal component analysis [6,15]. In [16], the energy in the wavelet band is applied to fault signatures through the fast WT technique for diagnosis the analog circuits. In [7], Roh and Abraham proposed the integral of signal processed by sub-band filters implemented with the wavelet decomposition to test analog circuits. However, once the sampling frequency is selected, the dyadic frequency decompositions of WT technique, carried out by the Mallat algorithm, cause that the bands associated with the wavelet signals are fixed and imply loose flexibility mainly when studying possible frequency components introduced by the fault, that is, there needs an a priori knowledge on selection of mother wavelet and time scales. Deng et al. in [2] applied the Volterra series of sub-band timedomain for fault signature to detect and locate the parametric faults in the nonlinear circuits. However, calculation of the discrete time-domain Volterra series is complex. 2.2. Works in this study Here, we focused on how to detect and locate the parametric faults occurred in analog and mixed-signal circuits through the HHT technique and coherence analysis technique. The instantaneous amplitude, the instantaneous frequency, and their respective coherence function and their integrals are selected for fault testing data. Simulation on testing Leapfrog filter, a benchmark circuit for analog and mixed-signal test, has been done to show the accuracy of fault detection and the results are compared with the WT technique and sub-band technique. The remainder of this paper is organized as follows: the HHT and coherence analysis are reviewed and the test strategy is described in Section 3, and simulation is given in Section 4 and conclusions in Section 5.

3. Test method Here, the signal x(t) denotes current or output voltage of the accessible out-terminals of CUT, which is assumed to be sampled from the nonlinearity and non-stationary processes. The HHT is a magnifying glass to analyze the data from nonlinear and

non-stationary processes and is developed to extract signatures as following. 3.1. Overview of HHT Considering the real-valued signal xðtÞ for signature extraction and the Hilbert transform is mathematically defined as Z 1 1 xðτÞ yðtÞ ¼ P Udτ ð1Þ π 1 t τ where P is the Cauchy principal value of the singular integral. Combining the real-valued signal xðtÞ with its Hilbert transform yðtÞ, analytic signal is obtained and can be expressed as zðtÞ ¼ xðtÞ þ yðtÞi ¼ aðtÞeiθðtÞ pffiffiffiffiffiffiffiffi where i ¼  1, and  1=2 aðtÞ ¼ x2 ðtÞ þ y2 ðtÞ

ð2Þ

ð3Þ

And

θðtÞ ¼ arctan

yðtÞ xðtÞ

ð4Þ

where aðtÞ is the instantaneous amplitude of the analytic signal zðtÞ, and θðtÞ is the instantaneous phase. Then the instantaneous frequency ωðtÞ is defined as

ωðtÞ ¼ dθðtÞ=dt:

ð5Þ

In fact, the Hilbert transform of signal generates the spurious amplitudes at negative frequencies due to the multiple components of the signal xðtÞ, which is originated from nonlinear and non-stationary processes. To avoid the unphysical results, Huang proposed a method to transform the original signal into a set of band limited signals by projecting the original signal onto a set of basis functions by a preprocess termed empirical mode decomposition (EMD) and the projections termed intrinsic mode functions (IMFs), which are usually implemented with the sifting algorithm. The sifting algorithm decomposes the original signal xðtÞ into n IMFs cj ðtÞ and the residue r n ðtÞ, the original signal xðtÞ can be expressed as xðtÞ ¼

n X

cj ðtÞ þr n ðtÞ

ð6Þ

j¼1

where the IMFs cj ðtÞ, also called basis functions and localized in time and frequency, are obtained adaptively by the sifting algorithm. The issues of decomposition criterion, convergence and uniqueness are given in [8,17]. Calculate the instantaneous amplitude aj ðtÞ and instantaneous frequencyωj ðtÞ for each IMFs cj ðtÞ with Eqs. (3) and (4), respectively, and hence the original signal can expressed xðtÞ as 2 3 n X iωj ðtÞdt 5 4 ð7Þ xðtÞ ¼ Re aj ðtÞe j¼1

where Re[] represents the real part of terms within brackets. The amplitudes of both aj ðtÞ and ωj ðtÞ in terms of time–frequency space are called Hilbert–Huang spectrum Hðω; tÞ. The decomposition expression (7) can be viewed as the expansion of the Fourier representation where both aj ðtÞ and ωj ðtÞ are constants. Therefore, the Hilbert–Huang spectrum Hðω; tÞ, serving as an alternative of the traditional Fourier or wavelet spectrum analysis, gives a way to obtain the subtle time–frequency information in the time–frequency space and it can be used for signature extraction.

S. Tang et al. / Microelectronics Journal 46 (2015) 893–899

data storage of fault dictionaries, the integrals of Hilbert–Huang spectrum and their coherence values of data measured from the accessible out-terminals of CUT are applied to extract the fault signatures. For example, the integral of C x;y ðωÞ is calculated as below Z T C x;y dω ð13Þ I C x;y ¼

3.2. Coherence analysis Coherence analysis is a basic technique of the information, which gives a basic measurement of relationship between two signals. Here, coherence analysis is introduced to locate faults with the instantaneous amplitude aj ðtÞ and frequency ωj ðtÞ. Some basic description of coherence analysis is given as following. The cross-correlation function of two stochastic process of variables xðtÞ and yðtÞ is defined as Z 1 T Rx;y ðτÞ ¼ xðtÞyðτ tÞ U dt ð8Þ T 0

0

3.4. Testing strategy The first task of fault test is fault detection. The fault detection is implemented by comparing the measured integral signature of faulty circuit with that of fault-free circuit. In the process one of integrals which is not in the tolerance level mean fault occurs. The instantaneous data for integral, includes the instantaneous amplitude aj ðtÞ and the instantaneous frequency ωj ðtÞ of each IMFs cj ðtÞ, and their self-coherence values C x;x ðωÞ. After the fault is detected, the second task is to locate fault position in the circuits. According to the minimum distance criterion, the fault can be identified by comparing integral signatures of faulty circuit with those correspondents in the fault dictionary which is associated with predefined fault values. The dictionary data can be achieved previously with HHT technique by the Hspice simulation from the predefined set of faulty circuit. The main signature used for localization is cross-coherence values C x;y ðωÞ, where x represents the data measured from the CUTs and y represent one faulty data in the dictionary. The least values C ðx;yÞmin ðωÞ, one member of C x;y ðωÞ, means the fault component lies in y position.

where T is the time length of stochastic process and τ is the middle variable. Then, the self-correlation function of x(t) is obtained as Z 1 T xðtÞxðτ  tÞ Udt ð9Þ Rx ðτÞ ¼ T 0 Applying the Fourier transformation to the correlation function Rx ðτÞ, the cross-power spectrum of variables xðtÞ and yðtÞ is obtained as Z 1 Rx;y ðτÞe  iωτ U dτ ð10Þ P x;y ðωÞ ¼ 1

Similarly, the power spectrum of variables xðtÞ is calculated as Z 1 Rx ðτÞe  iωτ U dτ ð11Þ P x ðωÞ ¼ 1

Then, the coherence function C x;y ðωÞ of variables xðtÞ and yðtÞ is obtained as C x;y ðωÞ ¼

P x;y ðωÞ P x ð ωÞ U P y ð ωÞ

ð12Þ

4. Experiments and results

The values of C x;y ðωÞ are between 0 and 1, which describe a relationship measurement between variables xðtÞ and yðtÞ in frequency domain.

Simulations are executed to evaluate the efficiencies of the proposed methods. The data, which is obtained from results of HSPICE simulation with HSPICE Toolbox for MATLAB, are processed in MATLAB to obtain the signatures, in which the EMD codes with the sifting algorithm available at Ref. [17] are utilized to implement the HHT technique. The benchmark circuit of the low-pass leapfrog filter is shown in Fig. 1. All values of the components are shown in Fig. 1(a) and each

3.3. Integral of variable In this paper, testing strategy is based on the fault dictionaries that is generated offline with simulation approach. To reduce the

C1 0.01u 2 R1 20

10k

R3 10k R5

10k

OA1

R4 4 10k

10k

5

R6

0.02u 0.02u

C3

R7

OA2 10k

R12 10k

11

C2 R2

895

OA3

10

Input

10k

C4 0.01u 14 R8

R9 10k

R11

OA4 4 10k

R13 10k

17

10k R10

OA5 16

10k

OA6 19

Vdd m5

m9

0.1V m7

-

m1

Input

+

output

m2 m6 m8

m4

m3

Vss

Fig. 1. (a) Leapfrog circuit for test and (b) OA circuit.

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S. Tang et al. / Microelectronics Journal 46 (2015) 893–899

one of the operational amplifier (OA) circuit is shown in Fig. 1(b). The benchmark circuit is based on MITEL Semiconductor's complementary metal-oxide-semiconductor (CMOS) technologies [7]. In the fault-free case, the component values vary within their tolerance of nominal values (σ is the standard deviation from the nominal value for each component by Monte Carlo analysis) [18]. For the catastrophic fault, open fault and short fault are implemented by injecting a resistor of 100 M Ω in series and short fault 100 Ω in parallel, respectively. In all simulations, the stimulus source is a 1-kHz sinusoidal signal, as in [7], and the sampling frequency was 100 kHz. To be universal, twenty-three circuit cases, including the fault free circuit, the catastrophic faults, the parametric faults, and multiple faults, are list in Table 1. For comparison, cases 1–4 (short faults) and cases 5–8 (open faults) are randomly selected from the Ref. [2,7] and cases 9–13 (parametric faults) are selected from the Ref. [7]. Cases 1–13 are these passive faults, in which the OP Amps are fault-free, and the CUTs are linear circuit. Cases 14–20 are the nonlinear fault circuits, in which the transistors are faulty elements. Table 2 presents results of fault detection and location. The data are calculated from the first three of IMFs cj for abbreviation. The % error means the relative error of signature values in percentage between faulty circuits and fault-free circuit, F.D.R is the fault detection rate, and F.L.R is the fault localization rate. The results indicate that the signatures of some IMFs have good fault detection for all kinds of faults. The minimal difference of frequency integral I ω2 of the second IMF is 36.6%, and I ω2 has clearly 100% of the detectable rate. Obviously, the signatures I C a2;a2 and I C ω3;ω3 have the same detectable rate as I ω2 . The frequency integral I ω3 has the least detectable rate, 36.4%, within the HHT signatures and I C a3;a3 has the least detectable rate, 27.3%, within the Coherence analysis techniques. For catastrophic cases, cases 1–8 can be detected effectively. In cases 1–3, 5–6, I ω1 is close to that of fault-free, and the I ω3 is close to that of cases 2, 4, 8. For parametric cases, reactive components are more detectable than active components. The faults of reactive components cannot be detected by I a1 and I C a1;a1 . For example, the maximal error of I a1 is 0.94% in case 14. The signature in some cases with HHT technique is not able to detect fault, however its coherence signature has the help to solve

the problem. For example, case 17 is difficult to detect with the signatures I a1 , I a2 , I a3 and I ω1 , I ω3 because of the maximal error 2.89%, however their coherence signature is much smaller than that of fault-free 257. Therefore, the proposed method has more detection capability. The results in Table 2 also indicate that the localization rate is much lower than the detection rate. For catastrophic faults cases 1–8, the fault location is easily implemented by the signatures. For example, case 2 is located by the signature I a3 because of the difference of I a3 between case 2(88.18) and case 3 (74.78) is 13.4, which is more close to that of case 2. However, for the parametric faults of active components, especial for the cases 17–20, many of the signatures is not suitable for fault location. For example, the difference between cases 18–20 lies within 2%, and it is difficult to distinguish from each other. For these active parametric faults, the linearity performances of OP Amps are all same with the values of faulty transistor M4, and hence the responses of CUT in such cases have less variation. It should be noted that although the signatures I ω2 and I C ω3;ω3 have powerful detectable capability, their locating efficiency is lower. Fig. 2 shows the time series, the IMFs, and the Hilbert spectrum of fault-free circuit. It is obvious that the noises lay mainly in the first and second IMF component, as shown in Fig. 2(b). In time domain, these components have low capability for the signatures because they have waveform like white noises. However, the integral of their HHT spectrum can remove the noise with average filtering and has the fault detection capability, as shown in Table 2. For the third IMF component, its Hilbert spectrum is almost a constant and can be used for the signature for fault detection. This is another advantage comparing with wavelet transform technique. The results of wavelet package decomposition are demonstrated in Fig. 3. Comparing Fig. 3(a) with Fig. 2(b), it is seen that each component of wavelet decomposition still has a portion of noises, which is verified by their FFT spectrum, shown in Fig. 3(b). The remaining noises result in accuracy reduction for signatures, duo to the frequency full-band feature of white noises and the fixed-band filtering technique of wavelet transform. The proposed method based on sifting algorithm is an adaptive decomposition technique.

Table 1 Fault configuration for leapfrog filter. Fault type

Fault no.

Fault

Faulty values

Short faults

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Node 10&4 short Node 20&10 short Node 5&2 short Node 16&13 short R2 open R3 open R9 open C1 open R2 variation R3 variation R7 variation R12 variation C3 variation M2 of OP Amp1 variation M4 of OP Amp1 variation M2 of OP Amp2 variation M4 of OP Amp3 variation M4 of OP Amp4 variation M4 of OP Amp5 variation M4 of OP Amp6 variation R9 between 14&11 and R10 variation

22

R5 variation and M2 of OP Amp4 variation

100 Ω 100 Ω 100 Ω 100 Ω 100 MΩ 100 MΩ 100 MΩ 100 MΩ 10k-5 kΩ 10k-5 kΩ 10k-5 kΩ 10k-15 kΩ 0.02μ-0.01 μF Width 20μ-16μ Width 36μ-40μ Width 20μ-16μ Width 36μ-40μ Width 36μ-40μ Width 36μ-40μ Width 36μ-40μ R9: 10k-15 kΩ R10: 10k-8 kΩ R5: 10k-12 kΩ Width 20μ-16μ

Open faults

Parametric faults

Nonlinear faults

Multiple faults

S. Tang et al. / Microelectronics Journal 46 (2015) 893–899

897

Table 2 Results of fault detection and location. Fault no.

Fault -free 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 F.D.R F.L.R

Integral Ix of HHT

Integral I x of coherence function

a1

a2

a3

ω1

ω2

ω3

Ca1,a1

Ca2,a2

Ca3,a3

Cω1,ω1

Cω2,ω2

Cω3,ω3

1583.1 668.04 % 57.8 2063.8 %30.4 452.86 % 71.3 1151.6 %  27.2 3104.4 %96.0 2202.4 %39.1 2572.8 %62.5 20.029 %  98.7 1003.1 %  36.6 1061.2 %  32.9 2221.9 %40.3 1841 %16.2 1551.4 %  2.00 1568.3 %  0.94 1573.1 %  0.63 1570.7 %  0.78 1584.6 %0.090 1571.4 %  0.74 1572.5 % 0.67 1573.1 %  0.63 976.07 %  38.3 1637.7 %3.44 %59.6 %24.1

27.851 8.8417 %  68.2 93.823 %236 116.84 %319 186.58 %569 208.43 %648 131.27 %371 123.88 %344 1.6060 % 94.2 13.493 %  51.5 66.2630 %137 138.23 %396 108.58 %289 93.07 %234 93.804 %236 94.672 %239 94.41 %238.7 28.227 %1.34 94.686 %239 94.524 %239 94.681 %239 50.652 %81.8 98.195 %252 %95.5 %62.8

19.210 4.7590 %  75.2 88.178 %359 74.775 %289 163.49 %751 197.09 %925 176.12 %816 130.98 %581 1.4513 %  92.4 8.1470 %  57.5 47.138 %145 133.53 %595 64.541 %235 63.21 %229 50.526 %163 55.707 %189 55.918 %191.8 18.654 %  2.89 55.903 %191 55.441 %188 55.709 %190 35.715 %85.9 55.174 %187 %96.7 %78.9

4.7565 5.0828 %6.86 4.8853 %2.70 5.0507 %6.18 11.017 %131 5.0462 %6.08 5.0517 %6.20 6.477 %36.1 11.246 %136 4.7520 %  0.09 6.4499 %35.6 6.4469 %35.5 5.4903 %15.4 6.4683 %35.9 5.9567 %25.2 5.9568 %25.2 5.9571 %25.21 4.7567 %0.002 5.957 %25.2 5.9567 %25.2 5.9568 %25.2 6.4274 %35.1 6.4379 %35.3 %68.2 %41.9

31.091 1.5253 % 95.1 2.0155 %  93.5 0.92178 %  97.0 2.0999 %  93.2 3.1181 %  89.9 3.0323 %  90.2 2.3223 %  92.5 2.0809 %  93.3 1.5549 %  94.9 2.9191 %  90.6 3.2116 %  89.6 3.015 %  90.3 2.9412 %  90.5 2.9312 %  90.5 2.9468 %  90.5 2.9468 %  90.5 2.0682 %  93.3 2.9468 %  90.5 2.9473 %  90.5 2.947 %  90.5 19.705 %  36.6 2.9334 %  90.5 %100 %20.1

1.1387 0.7156 %  37.1 1.1608 %1.93 0.65404 %  42.5 1.1116 %  2.38 0.9954 % 12.5 0.8696 %  23.6 1.5391 %35.1 1.1844 %4.00 0.94624 % 16.9 1.1312 %  0.66 0.83765 %  26.4 1.0921 %  4.09 1.0536 %  7.47 1.0881 %  4.45 1.0817 % 5.01 1.0797 % 5.18 1.1413 %0.228 1.082 %  4.98 1.0816 % 5.01 1.0817 %  5.00 1.0067 %  11.5 1.0936 %  3.96 %36.4 %22.4

257 200.91

257 106.07

257 246.10

257 200.07

257 126.85

257 110.99

153.9

115.93

251.77

155.03

178.36

150.94

217.44

87.301

236.1

220.06

234.41

108.05

164.71

95.443

246.02

184.09

175.58

114.59

145.78

135.49

235.23

147.59

230.34

105.07

131.57

88.35

204.4

134.86

88.937

106.51

212.29

85.116

233.7

205.84

146.43

107.12

146.28

91.655

223.48

183.6

156.63

178.52

209.99

103.68

248.37

211.23

233.21

163.68

224.34

105.63

238.56

212.04

109.57

163.7

145.32

84.243

191.76

157.9

82.192

100.19

226.03

105.5

237.56

223.98

200.93

130.27

184.77

130.82

232.67

204.02

231.81

175.87

144.16

118.21

113.97

161.44

234.13

163.69

252.49

109.64

239

226.85

231.7

158.93

253.54

109.97

239.04

227.72

231.72

161.4

231.13

172.8

186.59

234.21

233.45

163.68

252.12

110.07

238.98

226.4

231.72

159.16

253.05

109.24

239.01

226.91

231.66

158.5

252.37

109.51

239.01

226.94

231.69

158.76

146.91

81.717

241.32

149.09

89.121

117.46

133.04

112.89

238.28

155.79

231.84

179.04

%54.8 %54.7

%100 %55.5

%27.3 %19.6

%72.8 %68.8

%77.8 %45.9

%100 %64.1

Fig. 4 shows the Hilbert spectrum of fault circuit of the case 9. It is obvious that these waveforms distinct from those of Fig. 2(c) and (d). Therefore, fault is detected with the Hilbert spectrum. The other components of IMFs in some cases have more powerful detection capability, such as the frequency HHT spectrum of IMF4. Table 3 presents the comparison of detection rate with the maximal signature differences for some cases which are difficult to be detected. For the node 5&2 short fault (case 3), the Nagi's integrator scheme has 5.4% error and Roh and Abrahm's integrator scheme has the 76.1% error on HP3 (case 4 in [7]). In our scheme, the results of the second IMFs are the 319% increment error on the instantaneous amplitude and the  97% decrement error on the instantaneous frequency. In our proposed scheme, the capacitive faults can be detected effectively. For example, the case 8 has the maximal percentage 98.7% (decrement error) and the case 13 has the maximal percentage  90.3% (decrement error) or 234% (increment error). For Roh and Abrahm's integral scheme, the case 8 (C1 open) has only the maximal percentage 125% of error in the eighth sub-band, and the case 13 (C3 variation, 0.02tegral μF) has only the maximal

percentage  8.5%. When the capacitive faults on capacitors occur, the phase of CUT changes, and hence leads to the instantaneous frequency of response varies. The HHT technique has powerful capability to extract the instantaneous frequency features. Therefore, our scheme has more effectiveness for detection of capacitive faults.

5. Conclusion In this paper, the HHT and coherence analysis techniques are applied for faults test in analog and mixed-signal circuits. The subtle features, i.e., the instantaneous time–frequency signatures, are obtained with EMD decomposition and Hilbert transform, and coherence analysis and integral scheme are used to help the fault detection for reducing noise interference. Through comparing the signatures of faulty circuits with that of the fault-free circuit, the faults detection is implemented, and the faults location is done by using the minimum normalized distance. The Leapfrog filter, as an example, is investigated, in which different faults are setted for test

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4 3

0

0

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0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009

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S. Tang et al. / Microelectronics Journal 46 (2015) 893–899

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Table 3 Comparison of differences scheme. Fault no.

Nagi's scheme

Roh's scheme[7]

Deng's scheme[2]

Our scheme

Case Case Case Case Case

 0.1% 5.4%  7.7%  5.6%  8.5%

663% 76.1% 12.5%  8.5% 16.6%

77.9% 63.6% 84.1% 215.5% 55.5%

 95.1% 319%  98.7% 234% 239%

1 3 8 13 18

and for comparison with other methods. The experiment results demonstrate that the technique has more advantages to the traditional linear techniques such as the wavelet transform and the fast Fourier transform. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant no. 51477040, no. 51377044. References [1] Alexios D. Spyronasios, Michael G. Dimopoulos, Alkis A. Hatzopoulos, Wavelet analysis for the detection of parametric and catastrophic faults in mixed-signal circuits, IEEE Trans. Instrum. Meas. 60 (6) (2011) 2025–2038. [2] Yong Deng, Yibing Shi, Wei Zhang, An approach to locate parametric faults in nonlinear analog circuits, IEEE Trans. Instrum. Meas. 61 (February( 2)) (2012) 358–367. [3] Arvind Sai Sarathi Vasan, Bing Long Michael, PechtDiagnostics and prognostics method for analog electronic circuits., IEEE Trans. Ind. Electron. 60 (11) (2013) 5277–5291. [4] Lifen Yuan, Yigang He, Jiaoying Huang, Yichuang Sun, A new neural-networkbased fault diagnosis approach for analog circuits by using kurtosis and entropy as a preprocessor, IEEE Trans. Instrum. Meas. 59 (March(3)) (2009) 586–595.

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