Modified Latin Hypercube Sampling Monte Carlo (MLHSMC) Estimation for Average Quality Index

Transcription

1 Analog Integrated Circuits and Signal Processing, vol. 9, no., April 999. Abstract Modified Latin Hypercube Sapling Monte Carlo (MLHSMC) Estiation for Average Quality Index Mansour Keraat and Richard Kielbasa Departent of Electrical Engineering, Texas A&M University, College Station, Texas Ecole Supérieure d'electricité (SUPELEC), Service des Mesures, Plateau de Moulon, F-99 Gif-sur-Yvette Cedex France The Monte Carlo (MC) ethod exhibits generality and insensitivity to the nuber of stochastic variables, but it is expensive for accurate Average Quality Index (AQI) or Paraetric Yield estiation of MOS VLSI circuits or discrete coponent circuits. In this paper a variant of the Latin Hypercube Sapling MC ethod is presented which is an efficient variance reduction technique in MC estiation. Theoretical and practical aspects of its statistical properties are also given. Finally, a nuerical and a CMOS clock driver circuit exaple are given. Encouraging results and good agreeent between theory and siulation results have thus far been obtained. Key Words: Monte Carlo estiation, Latin hypercube sapling, average quality index. I. Introduction During the past decade, the feature sizes of VLSI devices have been scaled down rapidly. Despite the technological progress in patterning fine-line features, the fluctuations in etch rate, gate oxide thickness, doping profiles, and other fabrication steps that are critical to device perforances have not been scaled down in proportion. Consequently, the Average Quality Index (AQI) [] or its special case Paraetric Yield is becoing increasingly critical in VLSI design. Circuit designers ust ensure that their chips will have an acceptable AQI or paraetric yield under all anufacturing process variations. The Monte Carlo (MC) ethod [] is the ost reliable technique in AQI and yield estiation of electrical circuits. evertheless, it requires a large nuber of circuit siulations to have a valuable estiation, i.e., to have a low variance estiator. The well-known variance reduction techniques in MC yield estiation are [3]: ) iportance sapling, ) control variates, and 3) stratified sapling. The two first ethods require soe knowledge about the circuit responses, and for the latter a partitioning schee ust be realized []. Furtherore, soe of these techniques are based on the acceptability region [3], which is not defined in AQI. Therefore, these techniques are not applicable to AQI estiation. The efficiency of Latin Hypercube Sapling (LHS) [4] in MC (LHSMC) yield and AQI estiation has been shown in our earlier work [5], [6]. In LHSMC, instead of rando sapling, - -

2 the LHS approach is used. In this contribution, the odified LHS (MLHS) which is ore efficient than the standard LHS (SLHS), is presented. This paper is organized as follows. The following Section reviews the AQI definition. In Section III, the general aspect of LHS generation is presented. The Modified LHS Monte Carlo (MLHSMC) estiators will be discussed in Section IV. Section V describes the successful application of the MLHSMC ethod in soe nuerical and circuit exaples. Finally, concluding rearks will be given in Section VI. II. Average Quality Index The quality of a circuit can be defined in various ways. In a fabrication line, the circuit quality changes fro one to another. This is why one needs to define an AQI for a circuit production. One of the iportant quantities of AQI is the anufacturing paraetric yield of a circuit. Assue that we have circuit perforances y = [ y, y,..., y ]. For each perforance y i, a ebership function ηi = ηi ( yi ) can be defined by using fuzzy sets [7]. η i can be interpreted as a quality index easuring the goodness of perforance y i. A good circuit should have a high value of the quality index η i for every corresponding perforance. The circuit quality index can be defined as [ y y y ] where ϕ [.] is an appropriate intersection operator [4]. η( y ) = ϕ η ( ), η ( ),..., η( ), () In the case of integrated circuits, the circuit paraeters can be odeled as functions of their deterinistic noinal values, x, and a set of process disturbances, ξ, i.e.,p = p( x, ξ ). In addition, the coponents of ξ can be considered utually independent [8]. One can forulate AQI in disturbance space as ( y x ) Q( x) = η = η (, ξ) fξ ( ξ) dξ, () R n where f ξ (.) is the joint probability density function (pdf) of the process disturbances, and R n disturbance space. Furtherore, the paraetric yield is a special case of AQI where the quality index takes on only or value (pass/fail). For statistical circuit design, we need to calculate AQI. It can be evaluated nuerically using either the quadrature-based, or MC-based ethods [], [3]. The quadrature-based ethods have high coputational costs that grow exponentially with the diensionality of the statistical space (curse of diensionality). The MC ethod is the ost reliable technique for statistical analysis of electrical circuits. The unbiased Priitive MC (PMC) based estiator of AQI can be expressed as is - -

3 Q MC i ( x) = η( x, ξ ) (3) i= where η( x, ξ i ) denotes η( y( x, ξ i )), ξ i s are independently drawn rando saples fro f ξ ( ξ), and is the saple size. III. Latin Hypercube Sapling Monte Carlo (LHSMC) The Latin Hypercube Sapling (LHS) [4] is an extension of quota sapling [9], and can be considered as an n-diensional extension of Latin square sapling []. The sapling region is partitioned in a specific way by dividing the range of each coponent of the disturbance vector ξ. We will only consider the case where the coponents of ξ are independent or can be obtained fro an independent basis by linear transforation. However, the saple generation for correlated coponents with Gaussian distribution can easily be achieved. A Sapling Schee The LHS ethod operates in the following anner to generate a saple size fro the n variables ξ = [ ξ ξ ξ ],,..., n with joint pdf f ξ ( ξ). The range of each variable is partitioned into non overlapping intervals of equal probability. One value fro each interval is selected at rando according to the probability density in the interval. The values thus obtained for ξ are randoly paired with the values of ξ. These pairs are cobined in a rando anner with the values of ξ 3 to for triplets, and so on, until a set of n-tuples is fored. This set of n- tuples is a Latin hypercube saple. Fig. illustrates an LHS saple in a two-diensional space with saple size = 3. In order to have a atheatical notation, LHS can be described as follows. The ranges of each of the n coponents of ξ are partitioned into intervals of probability. The Cartesian product of these intervals partitions the disturbance space S into n cells each of probability n. Each cell can be labeled by a set of n cell coordinates i = [ i, i,..., in ] where ij is the interval nuber of coponent ξ j represented in cell i. A Latin hypercube saple of size is obtained fro a rando selection of the cells,,...,, with the condition that for each j the set [ ij ] i = be a perutation of the integers,,...,. Then one saple in each selected cell is generated by using its conditional joint pdf. Let C denote cell, which is deterined by = [,,..., n ]. Then the conditional joint pdf of ξ, given ξ belongs to C, is f C n fξ ( ξ) ( ξ) = ξ C. (4) otherwise - 3 -

4 B Practical Ipleentation of LHS Generator In order to generate an -point LHS saple, one should first create correlated saple for each variable in the way pointed out in the previous subsection. Generally, the saple generator can be based on the following ethods []: - inverse transfor ethod, - coposition ethod, 3- acceptance-rejection ethod. Here, the Inverse Transfor Method is used. In this ethod the inverse Cuulative Distribution Function (cdf) ust be known analytically or nuerically. The saple generation for each variable is as follow. The interval [, ] is divided into equal intervals. Then in each interval a saple is drawn with unifor pdf. Then these values are apped to the ξ axis by the corresponding inverse cdf ( F ( ξ ) ). Fig. illustrates the saple generation for =3. The saples of each variable are randoly ordered. By juxtaposing these vectors, the LHS saple is achieved. IV. Modified Latin Hypercube Sapling (MLHS) The LHS generation ethod pointed out in the previous section is called Standard LHS (SLHS). In this section, a variant of the SLHS ethod is presented. First the saple generation will be described, and then the statistical properties of its estiator are discussed. A MLHS Generator In MLHS, after partitioning the range of each variable into intervals of equal probability, the ean value of the conditional rando variable of each interval is chosen as a saple of the interval. The rest are the sae as in the LHS procedure (Fig. ). After deterining the upper and lower liits of each partition, we can calculate the ean values. For instance, consider the truncated Gaussian partition shown in Fig. 3. It can be shown that the ean value is given by (see Appendix A) i ξ ( a µ ) ( b µ ) = e σ e σ µ π +. (5) Suppose that the Inverse Transfor Method is used for saple generation, in which the inverse of the cdf F ( ξ ) ust be nuerically evaluated. The coputational coplexities of SLHS and MLHS for an -saple are given in Table I. Generally, the inverse cdf evaluation is the doinant part of the saple generation tie. It is seen that if MLHS is used in an optiization procedure with the nuber of iterations = and n=7 with Gaussian pdf, (which is case of integrated circuits), then MLHS is 7 ties as fast as the LHS generator. Therefore, the MLHS approach is ore adapted for AQI or yield optiization procedure. If an analytical for of the ean value of each interval is not available, one can use the edian of the interval conditional pdf instead of its ean value. The saple generation can be - 4 -

5 stated as follows. First, the interval of [,] is divided into equal intervals. Then the idpoint of each interval is deterined. The coponents of this vector are apped to the variable axis by the corresponding inverse cdf. The eleents of this vector are randoly disordered. By repeating this procedure for all variables and juxtaposing the resulting vector, the MLHS saple with edian type is generated. This type of MLHS takes less coputational tie than that of the ean value type. The saple value differences between edian and ean types for a Gaussian pdf with (, / 9 ) is shown in Fig. 4(a). The axiu saple value differences vs. saple size is illustrated in Fig. 4(b). It is seen that for large values of saple size the differences between the two types of MLHS can be ignored. In what follows the ean type MLHS is considered. B Statistical Properties of MLHSMC The MLHSMC estiator with saple size is given by T ML = i= η ξ i ( ), (6) where ξ i s are generated by the MLHS ethod. Since the probability of selecting ξ i C is n, and is the sae for all cells, we have fro cell i i E[ η( ξ )] = η( ξ ) Pr( ξ C ) = n η( ξ ), (7) R where E[.] stands for the expectation, Pr(.) is the probability, R denotes the space of all cells, and ξ is the expectation of ξ in cell C, expressed as ξ [ ξ ξ ] R = E C. (8) Furtherore, the expectation of the MLHSMC estiator can be described as i [ ML ] E[ η ξ ] E T Therefore, we have = ( ) = n η( ξ ) i= R { E[ η ξ C ] E[ η ξ C ] η( ξ )} = n + = R n R C [ [ ]] n η( ξ) f ξ dξ EC E C + η ξ η ξ ξ ( ) ( ) [ ] [ ML ] = [ η( ξ) ] + C η( ξ ) [ η ξ ]. (9) E T E E E C. () - 5 -

6 Fro (), it is seen that the MLHSMC estiator is not generally an unbiased estiator. It will be an unbiased estiator if the second ter of the right hand side of Eq. () is equal to zero. In fact, the difference between the standard LHSMC (SLHSMC) and MLHSMC estiators is the use of the following approxiation [ ] ( [ ]) E η( ξ) ξ C η E ξ ξ C = η( ξ ). () ow consider the following proposition concerning the bias of the MLHSMC estiator. Proposition : Assue that in each cell C the function η can be expressed as a ultilinear function of ξ (linear in each ξ i while other ξ j s are fixed). Then the MLHSMC estiator is an unbiased estiator. The proof of this proposition is given in Appendix B. It is iportant to note that the class of ultilinear functions is uch larger than the class of linear functions. Furtherore, the ultilinear property ust be satisfied in each sall cell, but does not need to be satisfied all over the disturbance space. The behavior of the MLHSMC estiator bias in the general case is described in the following theore. Theore : It is assued that the set of discontinuity points of the quality index function η over disturbance space is of the zero Lebesgue [] easure. Then the bias of the MLHSMC estiator converges to zero as the saple size approaches infinity ( ). The proof of Theore is given in Appendix B. It should be ephasized that the assuption of Theore holds for all realistic quality index functions and is not a restrictive condition. In practical circuit applications, it is found that the bias of the MLHSMC estiator is negligible for >. Thus the bias of the estiator is not a restriction for this ethod. One of the iportant properties of an estiator is its variance. The variance of the MLHSMC estiator (6) can be described as (see Appendix B) Var( TML ) = Var( η ) + Cov( ηl, η ), () where ηl = η( ξl ), η = η( ξ ), and the two cells C l and C have no cell coordinates in coon. Consider the following proposition. Proposition : If the quality index function η is a ultilinear function in each cell C, then the variance of the MLHSMC estiator is less than that of the SLHSMC estiator. The proof of Proposition is given in Appendix C. It should be noted that the ultilinear property does not iply the onotonicity condition. We now consider the following theore about the variance of the MLHSMC estiator in the general case

7 Theore : Suppose that the set of discontinuity points of the quality index function η over disturbance space is of the zero Lebesgue easure. Then the variance of the MLHSMC estiator approaches the variance of the SLHSMC estiator when. The proof of this theore is described in Appendix C. Siulation results have shown that variance efficiency gain with respect to SLHSMC can be obtained for < <. This is a very interesting property for the AQI or yield optiization by Stochastic Approxiation Approach [9], where AQI or yield is estiated by a sall saple size. V. uerical and Circuit Exaples Here we present a CMOS clock driver circuit and a 3-diensional quadratic perforance function to show the advantages of the MLHSMC estiator over those of PMC. In order to copare two different estiation ethods, an efficiency easure is introduced which is the product of the ratio of the respective variances and the ratio of the respective coputation ties [3] γ σ τ, (3) σ τ = R R L where τ R and σ R denote the coputation tie and the variance of the PMC estiator, and τ L and σ L are the coputation tie and the variance of the LHSMC estiator, respectively. Exaple : CMOS Clock Driver Circuit [] A CMOS clock driver is shown in Fig. 5. The clock driver provides two outputs V out and V out in opposite phase. The noinal response of the circuit is shown in Fig. 6. The perforance of interest is the clock skew as shown in Fig. 6. The specifications are that the skew belongs to the interval [-,] ns. OMEGA [5] is an open electric siulator which was developed at Ecole Supérieur d Electricité (SUPELEC). OMEGA was used as the circuit siulator with BSIM transistor odels, and Matlab [6] was as our prograing environent. The interactions between OMEGA and Matlab are done through Interprocess Counications [7]. The odel paraeters used to characterize CMOS anufacturing process disturbances are listed in Table II. These variables are considered independent, and to be of Gaussian probability distribution. The bias of the MLHSMC yield estiator of this circuit is depicted in Fig. 7. One can see that the bias of the estiator can be ignored for practical values of saple size and it converges to zero as saple size approaches infinity. The results confir the theoretical bases described in the previous section. It should be noted that the bias is estiated and it contains soe uncertainty. That is why there are soe fluctuations in the presented curves. The standard deviation (SD) of the MLHSMC and PMC estiators, as well as the efficiency of the MLHSMC yield estiator over the PMC estiator for this circuit are shown in Fig. 8. It is seen that the MLHSMC estiator is alost 3 ties as fast as the PMC estiator. L - 7 -

8 Exaple : Quadratic Perforance Function Suppose that the behavior of a circuit perforance can be expressed as a 3-diensional quadratic function. The function for this exaple is taken as where a = 3, and the atrices J and H are as follows: J = T y( ξ) = a + Jξ + ξ H ξ, (4) 4 5 H = (5) 5 4 The disturbances are considered to be independent with Gaussian pdf over the following region of tolerance where ξ = [. 5,. 5,. 5] T { ξ ξ ξ } 3 R = t i = T i i i and t = [,, ] T,,, (6) is the tolerance vector of disturbances. For quality index definition, a sigoidal fuzzy ebership function [] is chosen. The bias of MLHSMC for this exaple is given in Fig. 9. It is seen that the bias practically can be ignored for >. In addition, the efficiency of MLHSMC and SLHSMC with respect to PMC are shown in Figs. and, respectively. Fig. shows the efficiency of the MLHSMC estiator over the SLHSMC estiator. It is seen that the MLHSMC ethod is ore efficient than that of SLHSMC for < <. This property can be used in estiators with sall saple size, such as the yield gradient estiator in the Stochastic Approxiation approach [9]. It should be noted that the efficiency here is related to the ratio of two variances, and the efficiency of saple generation was not taken into account. In order to estiate the variance of each estiator, we repeated the estiation process ties. In Figs. (a) and (c), one can see that the standard deviation of the AQI estiator using MLHSMC is less than that of the PMC estiator with respect to saple size and AQI value, respectively. The dashed line is the theoretical standard deviation of the PMC yield estiator with the sae value of AQI. Also, the efficiency of SLHSMC is shown in Figs. (b) and (d). In Fig. (b), it is seen that the MLHSMC ethod is alost 7.5 ties as fast as PMC with the sae degree of confidence on results. Furtherore, the results of several applications have shown that the efficiency of the MLHSMC approach in AQI estiation is greater than in yield estiation. VI. Conclusions In this paper, a variant of the Latin hypercube sapling, called MLHS, is presented. The MLHS approach is a variance reduction technique in MC yield or AQI estiation ethods. It has - 8 -

9 the following advantages over the SLHSMC estiator in practical probles: ) fast saple generation, particularly in an optiization procedure, and ) ore precise estiator (saller estiation variance) with the sae coputational tie. The theoretical properties of MLHSMC were also described. Finally, siulation results of a CMOS clock driver circuit and a 3- diensional quadratic perforance function were given to show the efficiency of the MLHSMC approach. Good agreeent between theory and siulation was achieved. Furtherore, it is felt that the MLHSMC estiator is a well-adapted estiator in AQI or yield optiization by the Stochastic Approxiation approach [9] or the Centers of Gravity algorith [8]. Further research in this area is in progress. Appendix A Mean Value of Truncated Gaussian PDF For the sake of siplicity, we first consider a standard Gaussian pdf with zero ean and σ =. We suppose that this pdf is partitioned into intervals of equal probability. An interval with liits a and b is considered (Fig. 3). The conditional pdf of this interval is given by The ean value is obtained by f ξ e ξ [ a, b ] ξ = π. (A) otherwise ab( ) and after soe anipulations, we have b ξ ξ = ξ ξ ξ ξ = ξ ξ f d e ( ) d, (A) π a a b ξ = e e. (A3) π Furtherore, for a Gaussian pdf with ean µ and variance σ, the ean value is described as ξ ( µ ) ( µ ) σ σ = a b µ π e e +. (A4) - 9 -

10 Appendix B Bias of MLHSMC Estiators In this section, the bias property of the MLHSMC estiator is atheatically described. The η = E η( ξ), is expressed as MLHSMC estiator, which is an estiator of [ ] T ML = i= η ξ i ( ), (B) where ξ i s are generated by the MLHS ethod. After soe algebra, the expectation of this estiator can be written as [ ] [ ML ] = η + C η ξ [ η ξ ] E T E ( ) E C, (B) where C denotes a cell in the disturbance space, and ξ is the ean value of ξ in the cell C, ow, consider the following proposition. ξ [ ξ ξ ] = E C. (B3) Proposition : Assue that in each cell C the function η can be expressed as a ultilinear function of ξ (linear in each ξ i while other ξ j s are fixed). Then the MLHSMC estiator is an unbiased estiator. Proof: Fro the ultilinear assuption, the quality index function η in each cell C can be written as j = j = j = n j j jn η( ξ) =... a j j... j ξ ξ... ξn. (B4) In addition, in SLHS and MLHS saple generation it is supposed that the disturbance rando variables are independent. Thus by using (B4), the following result is straightforward [ ] ( [ ]) n E η( ξ) ξ C η E ξ ξ C = = η( ξ ). (B5) By substituting (B5) into (B), the proposition is proved. ow we consider the following lea before describing the asyptotic behavior of the estiator bias in the general case. Lea : Suppose that the cell C s are obtained fro the LHS ethod. Then for all cells in which the function quality index η is continuous, we have - -

11 [ η ξ ξ C ] li E ( ) = li η( ξ ). (B6) Proof: Let ξ Min and ξ Max denote the point where η has its axiu and iniu in the cell C, respectively. Then it is obvious that [ C ] η( ξmin ) E η( ξ) ξ η( ξmax ). (B7) η( ξ ) η( ξ ) η( ξ ) Min The shape of each cell is hyperbox with diensions approaching zero when saple size approaches infinity. Therefore, if ξ and ξ are two arbitrary points within the cell C, then Max li d( ξ, ξ ) =, (B8) where d stands for Euclidean distance. Fro (B8), and the continuity property of η over cell C, one can conclude that li η( ξ ) = li η( ξ ). (B9) Min By taking the liit of (B7) and using (B9), the proof is copleted. In the general case, the behavior of the MLHSMC estiator bias is described as the following theore. Theore : It is assued that the set of discontinuity points of the quality index function η over disturbance space is of the zero Lebesgue [] easure. Then the bias of the MLHSMC estiator converges to zero when the saple size approaches infinity ( ). Proof: Fro (B), the bias of the MLHSMC estiator ( B lhs ) can be expressed as where is defined as B lhs [ ] Max = EC, (B) [ η ξ ] = η ξ ( ) E C. (B) Suppose that approaches infinity. Then the right hand side of (B) can be expressed as an integral. According to Lea, one can conclude that the liit of is equal to zero over the points where η is continuous. To the contrary, can be non-zero over the point where η has discontinuity points ( [ ], ). Fro the assuption of zero Lebesgue easure over the discontinuity points of η, it can be concluded that the set of non-zero is of the zero Lebesgue easure. Therefore, the related integral (right hand side of (B)) is equal to zero. Hence the MLHSMC estiator approaches an unbiased estiator when. - -

12 Appendix C Variance of MLHSMC Estiators In this section the variance of the MLHSMC estiator is atheatically discussed in coparison with the SLHSMC estiator. Let CP, CP,... C P represent the cells fro which ξ, ξ,... ξ let [ ] are sapled, respectively, and U = C C C P, P,... P, (C) represent the ordered -tuple of disjoint cells. There are M = (!) n such ordered -tuples. We will index U and the corresponding cells with superscripts, such as [ P ] P P Each of these -tuples are equally likely, that is U i = C i C i C i,,... i =,,..., M. (C) Using the well-known forula [4], we have It can easily be shown that and then i Pr( U = U ) =. (C3) M Var( T ) E [ Var( T U i )] Var( E [ T U i LM LM LM ]) = +. (C4) i Var( T U ) = i =,,..., M, (C5) LM i [ LM ] E Var( T U ) =. (C6) After soe algebra, in the sae way as in the Result 5 of [3], the second ter of (C4) can be expressed as Var( TML ) = Var( η ) + Cov( ηl, η ), (C7) where ηl = η( ξl ), η = η( ξ ), and the two cells C l and C have no cell coordinates in coon. In addition, the variance of the SLHSMC estiator can be written as [3] Var( TSL ) = Var( η ) + Cov( µ l, µ ), (C8) - -

13 where T SL denotes the SLHSMC estiator, µ is defined as and the pairs ( µ l, µ ) [ C ] µ = E η( ξ) ξ, (C9) correspond to cells having no cell coordinates in coon. Proposition : If the quality index function η is a ultilinear function in each cell C, then the variance of the MLHSMC estiator is less than that of the SLHSMC estiator. Proof: According to the ultilinear assuption, Eq. (B5) holds and one can write [ C ] By inserting (C) into (C7) and (C8), we have and by using the well-known forula, µ = E η( ξ) ξ = η( ξ ) = η. (C) Var( TML ) Var( TSL ) = [ Var( η ) Var( η) ], (C) [ ] ( [ ]) Var( η ) = E Var( η ( ξ ) ξ C ) + Var η ( ξ ) E ξ C. (C) Furtherore, by substituting (C) into (C), the difference between the variance of the two estiators is given by [ C ] Var( T ) Var( T ) Var( ( ) ) E ML SL = η ξ ξ. (C3) Therefore, the proof of Proposition is copleted. The following theore states the asyptotic behavior of the variance of the MLHSMC estiator in the general case. Theore : Suppose that the set of discontinuity points of the quality index function η over disturbance space is of the zero Lebesgue easure. Then the variance of the MLHSMC estiator approaches the variance of the SLHSMC estiator when. Proof: Suppose that. Then fro Lea, over the point where η is continuous, we have li η = li µ. (C4) In the theore it is assued that the set of discontinuity points of η is of the zero Lebesgue easure. In addition, the quality index η belongs to [, ], so η, and µ [, ] properties, it can be shown that. Fro these - 3 -

14 li Var( η ) = li Var( µ ). (C5) li Cov( η, η ) = li Cov( µ, µ ) By substituting (C5) into (C7) and (C8), we have l l [ TML TSL ] = [ C ] li Var( ) Var( ) li Var( ( ) ) E η ξ ξ. (C6) In [], it is shown that [ C ] ax Var( η( ξ) ξ ) =. 5, and then { E[ C ]} ax Var( η ( ξ ) ξ ) =. 5. (C7) By using (C7) in (C6), it is seen that the right hand side of Eq. (C6) is equal to zero. Therefore, the result of Theore follows iediately. Moreover, in [] it is shown that [ C ] li E Var( η( ξ) ξ ) =. (C8) Consequently, the difference between the variance of the MLHSMC and SLHSMC estiators converges to zero with a convergence rate greater than

15 Acknowledgents The authors wish to thank F. Trelin for helpful discussions in prograing aspects. They would also like to thank J. Ostensen for reading the anuscript. References [] J. C. Zhang and M. A. Styblinski, Yield and Variability Optiization of Integrated Circuits. Kluwer Acadeic Publisher, 995. [] R. Y. Rubinstein, Siulation and the Monte Carlo Method. John Wiley & Sons, Inc., 98. [3] D. E. Hocevar, M. R. Lightner, and T.. Trick, A study of variance reduction techniques for estiating circuit yields, IEEE Trans. Coputer-Aided Design, vol. CAD-, no. 3, pp. 8-9, July 983. [4] M. D. McKay, R. J. Beckan, and W. J. Conover, A coparison of three ethods for selecting values of input variables in analysis of output fro a coputer code, Technoetrics, vol., no., pp , May 979. [5] M. Keraat and R. Kielbasa, Latin hypercube sapling Monte Carlo estiation of average quality index for integrated circuits, Analog Integrated Circuits and Signal Processing, vol. 4, no. /, pp. 3-4, 997. [6] M. Keraat and R. Kielbasa, Worst case efficiency of Latin hypercube sapling Monte Carlo (LHSMC) yield estiator of electrical circuits, in Proc. IEEE Int. Syp. Circuits Syst., Hong Kong, June 997, pp [7] A. Torralba, J. Chavez, and L. G. Franquelo, Circuit perforance odeling by eans of fuzzy logic, IEEE Trans. Coputer-Aided Design of Integrated Circuits Syst., vol. 5, no., pp , oveber 996. [8] Cox, P. Yang, S. S. Mahant-Shetti, and P. Chatterjee, Statistical odeling for efficient paraetric yield estiation of MOS VLSI circuits, IEEE Trans. Electron Devices, vol. ED-3, no., pp , February 985. [9] M. A. Styblinski and A. Ruszczynski, Stochastic approxiation approach to statistical circuit design, Electronics Letters, vol. 9, no. 8, pp. 3-3, April 983. [] M. Keraat and R. Kielbasa, A study of stratified sapling in variance reduction techniques for paraetric yield estiation, IEEE Trans. Circuits and Systes-II: Analog and Digital Signal Processing, vol. 45, no. 5, pp , May 998. [] A. Papoulis, Probability, Rando Variables, and Stochastic Process, 3rd edition. McGraw-Hill, Inc., 99. [] M. Keraat, Properties of average quality index in statistical circuit design, Ecole Superieure d Electricite (SUPELEC), Paris, France, Tech. Rep. o. SUP-496-, October 996. [3] R. L. Ian and W. J. Conover, Sall saple sensitivity analysis technique for coputer odels, with an application to risk assessent, Counications in Statistics A9-(7), pp , 98. [4] A. M. Mood, F. A. Graybill, and D. C. Boes, Introduction to Theory of Statistics, 3rd ed., ew York: McGrawHill,

16 [5] P. Aldebert and R. Klielbasa, OMEGA: syste of electric siulator, user's anual, Version 6., Ecole Superieure d Electricite (SUPELEC), Paris, France, Tech. Rep., Septeber 993. [6] The Mathworks, Inc., MATLAB: External Interface Guide, 993. [7] M. Keraat, Using Matlab in interprocess counications, Ecole Superieure d Electricite (SUPELEC), Paris, France, Tech. Rep. o. SUP-496-7, April 996. [8] R. S. Soin and R. Spence, Statistical exploration approach to design centring, IEE Proc., vol. 7, Pt. G, no. 6, pp. 6-69, Deceber 98. [9] H. A. Steinberg, Generalized quota sapling, uc. Sci and Eng., vol. 5, pp. 4-45, 963. [] Raj and Des, Sapling Theory. ew York: McGraw-Hill,

17 ξ SLHS MLHS ξ F( ξ ) Fig.. SLHS saple generation for = 3. ξ ξ 3 ξ ξ Fig.. Saple generation by the Inverse Transfor Method. ξ σ a ξ i b µ ξ Fig. 3. A truncated Gaussian partition.

18 6 x 3 Median/Mean MLHS generators (=) 4 error ξ (a).4 Median/Mean MLHS generators.35.3 Maxiu Error (b) Fig. 4. Differences between ean and edian types MLHS.

19 V DD V DD V DD PMOS_ PMOS_ PMOS_3 V out V in MOS_ C_ MOS_ MOS_3 V DD V DD PMOS_4 PMOS_5 V out MOS_4 C_ MOS_5 Fig. 5. Scheatic circuit of a CMOS clock driver. 4.5 CMOS clock driver transient response Vout Vout 3 Vout & Vout (v).5.5 Skew tie (ns) Fig. 6. oinal response of the CMOS clock driver.

20 Bias of MLHS for Y=43.8 Bias of MLHS saple size Fig. 7. Bias of the MLHSMC yield estiator for the CMOS clock driver. SD of estiators 6 4 SD forula PMC MLHSMC efficiency Y=43.3% (a) saple size (b) saple size SD of estiators efficiency 4 3 =3 5 (c) Y % 5 (d) Y % Fig. 8. Siulation results of the CMOS clock driver circuit.

21 .4 Bias of MLHS for Q=5.6. Bias of MLHS saple size Fig. 9. Bias of MLHSMC in the case of the quadratic function. SD of estiators 6 4 SD forula PMC LHSMC 4 (a) saple size 8 efficiency 8 6 Q=5.% 4 4 (b) saple size SD of estiators 6 4 efficiency =4 5 (c) Q % 5 (d) Q % Fig.. Efficiency of MLHSMC in the case of the quadratic function.

22 SD of estiators 6 4 SD forula PMC SLHSMC 4 (a) saple size 8 efficiency Q=5.% 4 (b) saple size SD of estiators (c) Q % efficiency =4 5 (d) Q % Fig.. Efficiency of SLHSMC for the quadratic function. 3.5 efficiency saple size Fig.. Efficiency of MLHSMC over SLHSMC for the quadratic function.

23 Table I. Coplexities of the SLHS and MLHS Methods Type # Unifor R.V. # F evaluation SLHS n n MLHS dif. pdf n n sae pdf n Table II. Process oise Factors i ξ i E(ξ i ) siga σ i Description ξ.74 µ. µ PMOS Width Reduction ξ. µ. µ PMOS Length Reduction 3 ξ 3.4 V. V PMOS Flat Band Voltage 4 ξ 4 5 n n Oxide Thickness 5 ξ 5.98 µ. µ MOS Width Reduction 6 ξ 6.5 µ. µ MOS Length Reduction 7 ξ V. V MOS Flat Band Voltage

24 Figure Captions Fig.. SLHS saple generation for = 3. Fig.. Saple generation by the Inverse Transfor Method. Fig. 3. A truncated Gaussian partition. Fig. 4. Differences between ean and edian types MLHS. Fig. 5. Scheatic circuit of a CMOS clock driver. Fig. 6. oinal response of the CMOS clock driver. Fig. 7. Bias of the MLHSMC yield estiator for the CMOS clock driver. Fig. 8. Siulation results of the CMOS clock driver circuit. Fig. 9. Bias of MLHSMC in the case of the quadratic function. Fig.. Efficiency of MLHSMC in the case of the quadratic function. Fig.. Efficiency of SLHSMC for the quadratic function. Fig.. Efficiency of MLHSMC over SLHSMC for the quadratic function. Table Captions TABLE I. COMPLEXITIES OF THE SLHS AD MLHS METHODS TABLE II. PROCESS OISE FACTORS