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Statistical Modelling of Analog Circuits for Test Metrics Computation
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Statistical Modelling of Analog Circuits for Test Metrics Computation

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Analog Built-In Test (BIT) techniques should be evaluated at the design stage, before the real production, by estimating the analog test metrics, namely Test Escapes (TE) and Yield Loss (YL). Due to ...

Analog Built-In Test (BIT) techniques should be evaluated at the design stage, before the real production, by estimating the analog test metrics, namely Test Escapes (TE) and Yield Loss (YL). Due to the lack of comprehensive fault models, these test metrics are estimated under process variations. In this paper, we estimate the joint cumulative distribution function (CDF) of the output parameters of a Circuit Under Test (CUT) from an initial small sample of devices obtained from Monte Carlo circuit simulation. We next compute the test metrics in ppm (parts-per-million) directly from this model, without sampling the density as in previous works. The test metrics are obtained very fast since the computation does not depend on the size of the output parameter space and there is no need for density sampling. An RF LNA modeled with a Gaussian copula is used to compare the results with past approaches.

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    Statistical Modelling of Analog Circuits for Test Metrics Computation Statistical Modelling of Analog Circuits for Test Metrics Computation Document Transcript

    • Statistical Modelling of Analog Circuits for Test Metrics Computation Kamel Beznia† , Ahcène Bounceur† , Salvador Mir‡ , Reinhardt Euler† † Lab-STICC Laboratory - European University of Britanny - University of Brest 20, Avenue Victor Le Gorgeu 29238, Brest, France Email: {Kamel.Beznia, Ahcene.Bounceur, Reinhardt.Euler}@univ-brest.fr ‡ TIMA Laboratory 46, Avenue Félix Viallet 38031, Grenoble Cedex, France Email: Salvador.Mir@imag.fr Abstract—Analog Built-In Test (BIT) techniques should be evaluated at the design stage, before the real production, by estimating the analog test metrics, namely Test Escapes (TE) and Yield Loss (YL). Due to the lack of comprehensive fault models, these test metrics are estimated under process variations. In this paper, we estimate the joint cumulative distribution function (CDF) of the output parameters of a Circuit Under Test (CUT) from an initial small sample of devices obtained from Monte Carlo circuit simulation. We next compute the test metrics in ppm (parts-per-million) directly from this model, without sampling the density as in previous works. The test metrics are obtained very fast since the computation does not depend on the size of the output parameter space and there is no need for density sampling. An RF LNA modeled with a Gaussian copula is used to compare the results with past approaches. Index Terms—Analog test, mixed-signal test, RF test, test metrics estimation, theory of Copulas, statistical model. I. INTRODUCTION AND PREVIOUS WORK Analog BIT techniques must be evaluated at the design stage before the real production in order to estimate test errors. These test errors include the Test Escapes TE, or proportion of faulty circuits that pass the test, and the Yield Loss YL, or proportion of circuits that fail the test within those that are functional. The approaches used to evaluate analog BIT techniques at the design stage are based on simulation. The approach based on fault simulation allows the injection of typically a few hundred catastrophic or parametric faults [1][2] and they require the use of analog circuit-level fault simulators [3]. This approach is extremely time consuming and often not realistic because of the lack of relevant fault models. A second approach is based on statistical simulation (typically Monte Carlo) of the circuit process variations. The inconvenient of this technique is that potential structural defects are not considered. Instead, it is assumed that BIT techniques that are efficient for detecting parametric deviations will also be efficient for the detection of real defects. In [4], it is proposed to extract the joint probability density function (PDF) of the output parameters (performances and test measures) of a CUT from an initial sample of about one thousand circuits obtained from Monte Carlo circuit simulation (e.g. Spice or Spectre simulation). Next, a larger sample of millions of circuits having the same statistical model can be generated by sampling the PDF model. From this new large sample, the estimation of the test metrics can be done accurately by computing relative frequencies. Different multivariate probability density estimation tech- niques have been considered to generate a statistical model of the CUT, including a multivariate Gaussian approach [4], the use of Copulas Theory [5] or the use of a non-parametric method [6]. Once this model has been obtained, all these previous techniques generate a large sample of several mil- lion instances. For example, for circuits having 20 output parameters, a set of at least 20 million circuits must be generated. In a more recent work, [7] considered the use of Extreme Value Theory. Only circuits having extreme values of their output parameters are simulated by making use of the statistical blockade technique [8] in order to discriminate the process parameters that generate extreme circuits from those that generate the non-extreme ones. However, the approach is not multivariate. In this paper, we consider the estimation of the joint CDF of the output parameters which will allow us to directly compute the test metrics without the need of sampling a PDF. To see how, let us assume that we have a sample of a circuit that has one performance P and one test measure T. Let s be a specification and l a test limit. By definition, the functional circuits are the ones for which (P ≤ s) and the circuits that pass the test are the ones for which (T ≤ l). Thus, the Test Escapes can be calculated as Pr(P > s/T ≤ l) = 1−Pr(P ≤ s/T ≤ l) = 1 − FP T (s, l), where FP T (.) is the joint CDF of the performance and the test measure. We can directly calculate the test metric if the mathematical form of the CDF is known. The paper is structured as follows. Section II presents the test metrics and their computation based on their probabilistic definition. The statistical model using copula distributions is presented in Section III. Section IV shows how to estimate the test metrics using a known statistical model. As a case study, in
    • Section V we evaluate a BIT technique for a 1.9 GHz cascode LNA designed in a 0.25 µm BiCMOS ST Microelectronics technology. The obtained results are compared with the ones presented in [5]. Finally, Section VI concludes the paper. II. TEST METRICS In this section we will present the definitions of the test metrics and their computation. First, we introduce the termi- nology used and then we explain how to estimate these test metrics theoretically. A. Terminology • Performance: is an output parameter used to decide if the circuit is functional or not. • Specification: is the interval of the acceptable values of the corresponding performance. • Test measure: is an output parameter used to decide if the circuit passes the test or not. • Test limit: is the interval of the acceptable values of the corresponding test measure. • Functional circuit: a circuit is functional if all its perfor- mances are verified (i.e. inside the specifications). • Faulty circuit: a circuit is faulty if at least one of its performances is not verified. • Circuit that passes the test: is the one for which all its test measures are verified (i.e. inside the test limits). • Circuit that fails the test: is the one for which at least one of its test measures is not verified. B. Computation of Test Metrics Analog test metrics are presented in [1]. In this paper we will consider Test Escapes TE (also called Defect Level) and Yield Loss YL since they are used to set the test limits of a BIT technique. In terms of probability, TE is the probability that a circuit is faulty knowing that it passes the test and YL is the probability that a circuit fails the test knowing that it is functional. By applying Bayes’ theorem we can then define TE as the ratio of the probability that a circuit is functional and passes the test over the probability that a circuit passes the test, and the YL as the ratio of the probability that a circuit is functional and passes the test over the probability that a circuit is functional. Let us consider a circuit with n performances P = (P1, P2, . . . , Pn) and n specifications s = (s1, s2, . . . , sn). This circuit is designed such that each performance Pi satisfies the specification si (i.e. Pi ≤ si). For this circuit we consider a set of m test measures T = (T1, T2, . . . , Tm) and m test limits l = (l1, l2, . . . , lm). The circuit passes the test if each test measure Ti satisfies the test limit li (i.e. Ti ≤ li). For simplicity, henceforth we will write P ≤ s to specify P1 ≤ s1, . . . , Pn ≤ sn and T ≤ l to specify T1 ≤ l1, . . . , Tm ≤ lm. Then, the test metrics can be written in a probabilistic form as follows: TE = 1 − Pr(P ≤ s, T ≤ l) Pr(T ≤ l) (1) YL = 1 − Pr(P ≤ s, T ≤ l) Pr(P ≤ s) (2) ⇒ TE = 1 − FP T (s, l) FT (l) (3) YL = 1 − FP T (s, l) FP (s) (4) ⇒ TE = 1 − FP T (s1, s2, . . . , sn, l1, l2, . . . , lm) FT (l1, l2, . . . , lm) (5) YL = 1 − FP T (s1, s2, . . . , sn, l1, l2, . . . , lm) FP (s1, s1, . . . , sn) (6) where FT (.) is the joint CDF (Cumulative Distribution Func- tion) of the test measures, FP (.) is the joint CDF of the per- formances, and FP T (.) is the joint CDF of the performances and the test measures. III. CDF ESTIMATION USING COPULAS In the previous section, we have shown how to write the test metrics using the CDFs of the performances and the test measures. In this section we will present a mathematical description of these CDFs. In particular we will show that Sklar’s theorem [9] allows to describe any joint CDF if the marginal distributions and the dependence structure, called Copula, of the output parameters are known. First, let us recall the concept of Copula. A. Copulas A copula is a multivariate distribution with uniform marginal distributions on [0, 1]. For an n-dimensional random vector U = (U1, U2, . . . , Un) on the unit cube, a copula C is a multivariate CDF such that: C(u1, u2, . . . , un) = Pr(U1 ≤ u1, . . . , Un ≤ un). (7) By applying Sklar’s theorem [9], we can easily derive the expression of the joint CDF F(x1, . . . , xn) associated with a copula C. Let X1, X2, . . . , Xn be n random variables with CDFs F1(x1), F2(x2), . . . , Fn(xn), respectively. Sklar’s theorem states that there exists a copula C such that ∀x = (x1, x2, . . . , xn) ∈ Rn F(x1, . . . , xn) = C(F1(x1), . . . , Fn(xn)). (8) B. The Gaussian Copula The Gaussian copula is given by: C(u1, u2, . . . , un) = u1 −∞ u2 −∞ · · · un −∞ 1 √ det Σ × exp − 1 2 wT · (Σ−1 − I) · w ∂w1∂w2 . . . ∂wn (9)
    • where Σ is the correlation matrix and w = (w1, . . . , wn). Figure 1 shows an example of the CDF of a bivariate Gaussian Copula C(u1, u2). 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 u1 C(u1,u2) u2 Fig. 1. Bivariate Gaussian Copula distribution. As we can see, the Gaussian copula given by Equation (9) has not an analytical form because of the presence of multiple integrals in the formula. In [4][5] it is proposed to estimate it using Monte Carlo techniques by generating a large sample of this copula from which it will be estimated. In this paper, we use an efficient algorithm presented in [10][11] to calculate accurately this copula function. This algorithm is implemented under the R software ([12]) which is a powerful open source statistical software. IV. PROCEDURE FOR TEST METRICS COMPUTATION The procedure for computing test metrics, illustrated in Figure 2, is as follows: Initial Population (Monte Carlo circuit simulation) Statistical model estimation Estimation of the marginal distributions Estimation of the Copulas Test metrics calculation Fig. 2. Approach to estimate test metrics. 1) Start with a small sample of circuits generated using Monte Carlo circuit simulation, Fig. 3. LNA schematic. 2) Estimate the marginal distribution FPi (xi) = Pr(Pi ≤ xi) of each performance Pi where i = 1, . . . , n, 3) Estimate the marginal distribution FTj (yj) = Pr(Tj ≤ yj) of each test measure Tj where, j = 1, . . . , m, 4) Estimate the copulas: CP of the performances, CT of the test measures, and CP T of the performances and the test measures, 5) Then, the statistical model is written as follows : a) FP (x1, . . . , xn) = CP (FP1 (x1), . . . , FPn (xn)) b) FT (y1, . . . , ym) = CT (FT1 (y1), . . . , FTm (ym)) c) FP T (x1, . . . , xn, y1, . . . , ym) = CP T (FP1 (x1), . . . , FPn (xn), FT1 (y1), . . . , FTm (ym)) 6) Based on the theory of copulas presented in the previous section (Section III-B) and using Equations (5) and (6) the Test Escapes and the Yield Loss, respectively, can be calculated as follows : TE = 1 − CP T (FP (s), FT (l)) CT (FT1 (l1), . . . , FTm (lm)) (10) YL = 1 − CP T (FP (s), FT (l)) CP (FP1 (s1), . . . , FPn (sn)) (11) where FP (s) = FP1 (s1), . . . , FPn (sn) and FT (l) = FT1 (l1), . . . , FTm (lm)). The theory of Copulas is an efficient method to estimate joint CDFs, which can then be used to estimate the statistical model in order to calculate the test metrics. However, if the joint CDFs are estimated with other methods, the procedure to compute test metrics is of course still the same. V. CASE STUDY In this section we will evaluate a BIT technique for an RF LNA and we will compare the obtained test metrics with the previous approach in [5]. A. Test Vehicle The case-study is a 1.9 GHz cascode LNA designed in a 0.25 µm BiCMOS ST Microelectronics technology. The schematic of the LNA is shown in Figure 3. Table I sum- marizes the performances of the LNA and its specifications.
    • The considered test measure is the DC signature obtained by computing the RMS value of the cross-correlation between the output voltage and the power supply current of the LNA when a 1.9 GHz sinusoidal stimulus with magnitude −30 dBm is applied at its input [13]. The output parameter includes the five circuit performances (e.g. P1=NF, P2 = S11, P3=Gain, P4=1-dB CP, and P5=IIP3) and the test measures (T1=BIT). TABLE I LIST OF THE PERFORMANCES AND THE SPECIFICATIONS OF THE LNA. i Performance Specifications si 1ower limit upper limit 1 P1= NF −∞ 1.3dB 2 P2=S11 −∞ −9dB 3 P3=Gain 17dB +∞ 4 P4=1-dB CP −11.3dBm +∞ 5 P5=IIP3 −5.1dBm +∞ B. BIT Evaluation To evaluate the BIT technique for the LNA we use the procedure described in Section IV. 1) Initial circuit sample: We have performed a Monte Carlo circuit-level simulation of 1000 instances. The matrix plot of Figure 4 shows in the diagonal the histograms of each output parameter. The rest of the matrix shows the bivariate distributions of each pair of output parameters. Fig. 4. Initial sample of LNA circuits obtained from the Monte Carlo circuit level simulation. 2) CDFs of the performances: Using the classical univari- ate goodness of fit test1 , we have estimated the CDF of each performance (P1=NF, P2 = S11, P3=Gain, P4=1-dB CP, and P5=IIP3) of the LNA. The obtained distributions and their statistical parameters are shown in Figure 5(a) to (e) and in Table II. 3) CDFs of the Test Measures: In the same way as the previous step, we have estimated by use of the classical univariate goodness of fit test the CDF of the test measure (T1=BIT). The obtained values are presented in Figure 5(f) and in Table II. 1The Kolmogorov-Smirnov test. TABLE II PARAMETERS OF THE FITTED MARGINAL DISTRIBUTIONS. CDF Fitted Parameters CDF µ σ Bandwidth ξ FP1 (x1) Gauss 1.15 0.03 — — FP2 (x2) NP — — 0.4 — FP3 (x3) GEV 17.46 0.12 — -0.34 FP4 (x4) NP — — 0.089 — FP5 (x5) Gauss -1.12 1.014 — — FT1 (y1) GEV 0.36 0.034 — -0.049 4) Performances and Test Measure Copulas: In this step, by transforming each sample in Figure 4 via the CDF ˆFPi (xi) and the CDF ˆFTi (yi) of each marginal we obtain the empirical copula of Figure 6. The resulting distribution has an elliptical form typical of a gaussian dependence. Using a goodness of fit test of copulas [5] we conclude that the empirical copula of the performances and the test measures is Gaussian. Fig. 6. Empirical copula of the initial data of the LNA. Therefore, the copulas CP and CP T are Gaussian. They have the same form as Equation (9). Note that any univari- ate copula C(F(x)) is by definition equal to the univariate CDF F(x) (i.e. C(F(x)) = F(x)). In our case study the copula CT (FT1 (y1)) distribution of the test measure T1 is the same as the CDF of this test measure. That is to say that CT (FT1 (y1)) = FT1 (y1) which follows, from Table II, the generalized extreme value distribution. 5) The Statistical Model: The statistical model is given by: FP (x1, . . . , x5) = CP (FP1 (x1), . . . , FP5 (x5)) FT1 (y1) = exp − 1 + ξ y1 − µ σ −1/ξ FP T (x1, . . . , x5, y1) = CP T (FP1 (x1), . . . , FP5 (x5), FT1 (y1)) where µ ∈ R is the location parameter, σ > 0 the scale parameter and ξ ∈ R the shape parameter. 6) Test Metrics Estimation: Based on these previous steps and equations (5) and (6), the test metrics of our case study are written as follows: TE = 1 − CP T (FP1 (s1), . . . , FP5 (s5), FT1 (l1)) FT1 (l1) (12) YL = 1 − CP T (FP1 (s1), . . . , FP5 (s5), FT1 (l1)) FP1 (s1), . . . , FP5 (s5) (13)
    • (a) (b) (c) (d) (e) (f) Fig. 5. Fitted CDFs of the performances (a) NF, (b) S11, (c) Gain, (d) 1-dB CP and (e) IIP3 and (f) the test measure (BIT). C. Setting test limits In order to fix the values of the test limits, we vary the value of the test limit l1 from 0.40V to 0.80V with a step of 0.01V and then we use equations (12) and (13) to calculate the test metrics. The black curves with triangle symbols of Figure 7 show the obtained values for the test metrics. Test limits l1 Test limits l1 Test Escapes(TE) Yield Loss (YL) Fig. 7. The Test Metrics YL and TE obtained (1) from the Copula-based estimation [5] in red circles and (2) from the proposed method in black triangles. The test limits can be fixed, for example, to the values where the YL is equal to TE. For this case, the obtained test limit is YL = TE = 198ppm. Using the proposed statistical model we have then compared the results of the directly estimated test metrics with those estimated from a large sample of 1 million generated by copula-based simulation [5]. The resulting test metrics are shown in Figure 7. As we can see, the values obtained by each method are very close. Both approaches start from estimating the statistical model from the Monte Carlo circuit simulation. The main difference between them is that in the first one, which is based on density simulation, another large sample of circuits is generated with the same statistical model as the one obtained by Monte Carlo. The problem with this approach is that it is time consuming: the number of samples that need to be generated to guarantee ppm precision for the test metrics grows with the dimensionality, and the samples must be examined to compute relative frequencies. However, in the new approach, once the statistical model is determined, the test metrics are computed directly using the model. Even if the number of performances and test measures is very important, the computation of test metrics is simple. Indeed, in this paper the model of the case study is based on a Gaussian copula which does not have an analytical form. However, an algorithm has been developed by [12], which allows a fast and accurate computation.
    • VI. CONCLUSION The estimation of analog test metrics (Test Escapes and Yield Loss) at the design stage is important to evaluate BIT techniques. At the design stage, there are no fabricated circuits, and test metrics must be estimated from a statistical model of the CUT. This model is obtained from a small sample of circuits obtained by Monte Carlo circuit-level simulation. In this paper, we have described an approach to directly calculate the test metrics from such a model. This approach is faster than previous ones and the complexity does not grow with the dimensionality of the output parameter space of the CUT. The obtained results using an RF LNA of ST Microelectronics show good accuracy in comparison with previous approaches. REFERENCES [1] S. Sunter and N. Nagi. Test metrics for analog parametric faults. In Proc. VLSI Test Symposium (VTS), 1999, pp. 226-234. [2] Y. Lechuga, A. Bounceur, R. Mozuelos, M. Martínez, S. Bracho and S. Mir. Test limits evaluation for an ADC Design-for-Test approach by using a CAT platform. 23rd International Conference on Design of Circuits and Integrated Systems, Grenoble, France, November 2008. [3] A. Bounceur, S. Mir, L. Rolíndez and E. Simeu. CAT platform for analogue and mixed-signal test evaluation and optimization, 2007, in IFIP International Federation for Information Processing, Volume 249, VLSI-SoC : Research trends in VLSI and Systems on Chip, eds. De Micheli, G., Mir, S., Reis, R., (Boston : Springer), pp. 281-300. [4] A. Bounceur, S. Mir, E. Simeu, and L. Rolíndez, Estimation of test metrics for the optimisation of analogue circuit testing, Journal of Electronic Testing: Theory and Applications, vol.23, no. 6, pp. 471-484, 2007. [5] A. Bounceur, S. Mir and H-G. Stratigopoulos. Estimation of analog parametric test metrics using copulas. IEEE Transactions Computer-Aided Design of Integrated Circuits and Systems, September 2011, volume 30, number 09, ITCSDI, ISSN 0278-0070 [6] H. Stratigopoulos, S. Mir, and A. Bounceur. Evaluation of analog/RF test measurements at the design stage. IEEE Transactions on Computer- Aided Design of Integrated Circuits and Systems, 28(4), April 2009, pp. 582-590. [7] H. Stratigopoulos and S. Mir. Analog test metrics estimates with PPM accuracy. IEEE International Conference on Computer-Aided Design (ICCAD), San Jose, USA, November 2010, pp. 241 - 247. [8] A. Singhee and R. A. Rutenbar, "Statistical blockade: Very fast statistical simulation and modeling of rare circuit events and its application to memory design," IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 2009. [9] A. Sklar. Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Université de Paris 8, 1959, pp. 229-231. [10] Alan Genz. Numerical Computation of Multivariate Normal Probabili- ties. Journal of Computational and Graphical Statistics, 1992, 1: 141-150. [11] Miwa, Tetsuhisa and Hayter, A. J. and Kuriki, Satosh. The evaluation of general non-centred orthant probabilities. Journal of the Royal Statistical Society Series B, 2003, 65: 223-234. [12] Xuefei Mi, Tetsuhisa Miwa and Torsten Hothorn, mvtnorm: New Nu- merical Algorithm for Multivariate Normal Probabilities , The R Journal, May 2009, vol. 1/1, 37-39. [13] J. Machado da Silva, Low-power in-circuit testing of an LNA, In IEEE International Mixed-Signals Testing Workshop, 2005, pp. 206-210.