Correlated Bayesian Model Fusion: Efficient Performance Modeling of Large-Scale Tunable Analog/RF Integrated Circuits

Transcription

1 Correlated Bayesian odel Fusion: Efficient Perforance odeling of Large-Scale unable Analog/RF Integrated Circuits Fa Wang and Xin Li ECE Departent, Carnegie ellon University, Pittsburgh, PA 53 {fwang, ABSRAC unable circuit has eerged as a proising ethodology to address the grand challenge posed by process variations. Efficient high-diensional perforance odeling of tunable analog/rf circuits is an iportant yet challenging tas. In this paper, we propose a novel perforance odeling approach for tunable circuits, referred to as Correlated Bayesian odel Fusion (C- BF). he ey idea is to encode the correlation inforation for both odel teplate and coefficient agnitude aong different nob configurations by using a unified prior distribution. he prior distribution is then cobined with a few siulation saples via Bayesian inference to efficiently deterine the unnown odel coefficients. wo circuit exaples designed in a coercial 3n SOI COS process deonstrate that C-BF achieves ore than cost reduction over the traditional state-ofthe-art odeling technique without surrendering any accuracy.. INRODUCION he advent of nanoscale technology aes analog/rf integrated circuit (IC) increasingly susceptible to process variations. Process variations, including inter-die variations and local isatches, significantly ipact the paraetric yield of analog/rf circuit, and ust be properly handled at all levels of design hierarchy []-[]. raditional approaches based on overdesign [3]-[4] are not sufficient to aintain high paraetric yield [5]-[6], due to the large-scale process variations and aggressive design specifications at advanced technology nodes. In this context, tunable circuit has eerged as a proising ethodology to address the variability issue [6]-[]. It eploys a set of tuning nobs (e.g., bias current) to facilitate post-silicon tuning. In particular, a tunable circuit possesses ultiple states associated with tuning nobs that can be adaptively selected after anufacturing. For each particular process corner and/or environental condition, the tunable circuit is able to adaptively select the optial state and, hence, achieve high paraetric yield. Since tunable circuit closely interacts with process variations, the variability associated with tunable circuit ust be appropriately odeled. oward this goal, perforance odeling is an iportant tas circuit-level perforance (e.g. phase noise) is approxiated as an analytical function (e.g. polynoial) of device-level variations (e.g. ΔVH). he perforance odel, once built, can be applied to various iportant applications, such Perission to ae digital or hard copies of all or part of this wor for personal or classroo use is granted without fee provided that copies are not ade or distributed for profit or coercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for coponents of this wor owned by others than AC ust be honored. Abstracting with credit is peritted. o copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific perission and/or a fee. Request perissions fro Perissions@ac.org. DAC '6, June 5-9, 6, Austin, X, USA 6 AC. ISBN /6/6$5. DOI: as yield estiation []-[3], corner extraction [4], design optiization [5], etc. Historically, the perforance odeling proble of non-tunable circuit has been thoroughly studied [6]- [8]. For tunable circuit, however, this fundaental odeling proble can be extreely difficult with the following characteristics: High-diensional variation space: A large nuber of devicelevel rando variables ust be used to odel the process variations associated with a tunable analog/rf syste at advanced technology nodes. Large nuber of nob configurations: o axiize tuning range and resolution, a large nuber of nob configurations (i.e. states ) should be designed. his, however, significantly increases the coplexity of perforance odeling, since a unique perforance odel is required to accurately capture the variability at each state. Correlated nob configurations: Since different states are associated with the sae tunable circuit, the perforance odels of the different states are correlated. In particular, the odel teplate and odel coefficients are expected to be siilar aong these states. It is iportant to note that ost traditional approaches [6]- [] are not fully equipped for perforance odeling of tunable circuit. aing sparse regression [6]-[7] as an exaple, the sparsity of odel coefficients is explored to iprove efficiency. However, no correlation is considered between different states. In other words, perforance odeling is done for each state independently and the perforance odeling cost would quicly becoe intractable as the nuber of states increases. Several other perforance odeling ethods (e.g., siultaneous orthogonal atching pursuit [9], ulti-ode ulti-corner sparse regression [], group lasso [], etc.) have been proposed to exploit the aforeentioned correlation inforation. However, these ethods only tae into account the shared odel teplate aong different states and they ignore the correlation of coefficient agnitude. In Bayesian odel fusion (BF) [8], a prior distribution is defined to encode the prior nowledge. However, the prior distribution in [8] is not able to appropriately encode the correlation inforation for tunable circuit. In this paper, we propose a new Correlated Bayesian odel fusion (C-BF) technique to facilitate large-scale perforance odeling of tunable analog/rf circuit. he proposed C-BF ethod is otived by the fact that perforance odels associated with different states are correlated. Our ey idea is to encode the correlation inforation for both odel teplate and coefficient agnitude aong different states by using a unified prior distribution. he prior distribution is then cobined with a few siulation saples via Bayesian inference []-[3] to efficiently deterine the unnown odel coefficients. In addition, efficient and robust hyper-paraeter inference is proposed to solve a large nuber of hyper-paraeters associated with C-BF. It consists of two ajor steps: (i) estiating the hyper-paraeters to obtain an initial guess using a heuristic approach, and (ii) optiizing the

2 hyper-paraeter values using an expectation-axiization algorith []. As will be deonstrated by our nuerical exaples in Section 4, the proposed C-BF approach is able to achieve runtie speedup over the state-of-the-art odeling technique without surrendering any accuracy. he reainder of this paper is organized as follows. We briefly review the bacground of perforance odeling for tunable circuit in Section, and then describe the proposed C- BF approach in Section 3. wo circuit exaples are presented to deonstrate the efficacy of C-BF in Section 4. Finally, we conclude in Section 5.. BACKGROUND Given a tunable analog/rf circuit (e.g. a down-conversion ixer) with totally K states (i.e. nob configurations), the perforance odel of the -th state can be represented by an analytical function (e.g., polynoial) of device-level variations (e.g., ΔVH): y f( x) = α, b( x) ( =,,, K), () = {y; =,,, K} contains the perforance of interest (PoIs) associated with the K states, x is a vector representing the device-level variations, f(x) denotes the perforance odel of interest which establishes a apping fro x to y, {,; =,,..., } contains the odel coefficients of f(x), {b(x); =,,..., } contains the basis functions (e.g., linear or quadratic polynoials), and denotes the total nuber of basis functions. Without loss of generality, we assue that the sae set of basis functions {b(x); =,,..., } are shared aong all the states. In order to deterine the perforance odels in (), we need to find the odel coefficients {,; =,,, K; =,,..., }. oward this goal, the traditional least-squares fitting ethod first collects a nuber of sapling points of x and {y; =,,, K} for all K states, and then solves the odel coefficients fro the following optiization probles: in y B ( =,,, K), () () () () b( x ) b( x ) b ( x ) ( ) ( ) ( ) ( ) ( ) ( ) 3 b x b x b x B = ( =,,, K) (3) ( N) ( N) ( N) b( ) b( ) b x x ( x ) 4 = α, α, α, ( =,,, K) (4) () ( ) ( ) 5 y N = y y y ( =,,, K ). (5) In (3)-(5), x (n) is the value of x at the n-th sapling point of the - th state, y (n) is the value of y at the n-th sapling point, stands for the L-nor of a vector, and N represents the total nuber of sapling points collected for each state. o avoid overfitting, the nuber of sapling points ust be substantially greater than the nuber of unnown coefficients. o reduce the nuber of saples required for perforance odeling, the siultaneous orthogonal atching pursuit (S-OP) algorith [9] has been proposed in the literature. he ey idea of S-OP is to explore the sparse property and the correlation of odel teplate between different states. As an extension of the orthogonal atching pursuit (OP) algorith [6], S-OP naturally inherits the assuption of sparse regression. In other words, S-OP exploits the fact that only a sall nuber of basis functions in {b(x); =,,..., } are iportant and, hence, should be used to approxiate {y; =,,, K}. ore iportantly, S-OP also explores the correlation of odel teplate aong {y; =,,, K}. In particular, S-OP assues that different states possess the sae set of iportant basis functions. his is usually a valid assuption in practice, because different states are associated with the sae tunable circuit. While S-OP has been successfully applied to any practical applications (e.g. wafer spatial pattern analysis [9]), it is possible to explore other correlation inforation between states to further iprove the efficiency of perforance odeling for tunable circuits. otivated by this observation, we will propose a novel C-BF approach that taes advantage of the correlation of coefficient agnitude in Section PROPOSED APPROACH Siilar to S-OP, the proposed C-BF ethod relies on two assuptions: (i) the sparsity of odel teplate, and (ii) the correlation of odel teplate. However, unlie S-OP which ignores the correlation of coefficient agnitude aong different states, C-BF also encodes such agnitude correlation to iprove odeling accuracy. In what follows, we will discuss the atheatical forulation of C-BF in detail. 3. Prior Knowledge Definition o start with, we consider the perforance odels of K states in a tunable circuit as defined in (). We rewrite their odel coefficients in a concatenated for as: 6 =, (6) 7 = α, α, αk, ( =,, ), (7) {,; =,,..., K; =,,..., } are the odel coefficients defined in (), and is the cobined colun vector with size of K. In (6), the odel coefficients are organized such that the coefficients associated with the -th basis function are grouped together in ( =,,..., ). Since the odel coefficients in are related to the sae basis function, we assue that they are statistically correlated and define their prior distribution as: 8 pdf ( ) ~ N (, λ R) ( =,,, ). (8) In (8), {; =,,..., } are the hyper-paraeters that control the sparsity of the basis functions, and {R; =,,..., } are the positive definite atrices that quantify the correlation aong odel coefficients. Studying (8) yields several iportant observations. First, the sparse property is naturally encoded in {; =,,..., }. o understand this, we consider the -th basis function is zero. In this case, the variance of the zero-ean Gaussian distribution in (8) is also zero. As such, the coefficients can only tae zero values based on their prior distribution, thereby iplying the sparsity of. Second, the correlation of odel teplate is also encoded by the atheatical representation in (8). In particular, all the odel coefficients in share the sae sparse pattern, because their prior distribution is controlled by the sae hyper-paraeter. hird, but ost iportantly, the correlation of coefficient agnitude is also encoded by this prior definition, as long as the covariance etrics {R; =,,..., }

3 are not diagonal. In this paper, we further assue that: 9 R = R = R = R, (9) in order to reduce the nuber of hyper-paraeters and avoid overfitting. As shown in Figure, the prior nowledge of sparsity and correlation is encoded by the hyper-paraeters. o coplete the definition of prior distribution for all odel coefficients in, we further assue that i and j are statistically independent for any i j. herefore, the joint distribution can be represented as: pdf ( ) ~ N ( A, ), () λ R A =. () λ R he independence assuption in ()-() siply indicates that we do not now any correlation inforation between different basis functions as our prior nowledge. Such correlation will be learned fro sapling points, when the posterior distribution is calculated fro Bayesian inference in Section 3.. Correlation atrix: R Sparsity paraeters:,,,,,, K, K, K, Figure. he prior nowledge for odel coefficients is illustrated. Each row (shown by the red box) represents the odel coefficients of a particular state, while each colun (shown by the blue box) represents the odel coefficients associated with the sae basis function. he hyper-paraeters {; =,,..., } and R are defined to encode the sparsity and correlation respectively. 3. axiu-a-posteriori Estiation Once the prior distribution pdf() is defined in ()-(), we will cobine pdf() with a nuber of saples collected fro K states {(x (n), y (n) ); n =,,, N; =,,, K} to solve the odel coefficients by axiu-a-posteriori (AP) estiation. According to Bayes theore, the posterior distribution is proportional to the prior distribution pdf() ultiplied by the lielihood function, which can be represented as []: pdf ( y) pdf ( ) pdf ( y ). () In (), pdf(y ) is the lielihood function, () () ( ) ( ) ( N) ( N) 3 y = y yk y yk y y K (3) is a vector containing N K sapling points collected fro K states. he lielihood function pdf(y ) represents the probability of observing these saples given the odel coefficients. o derive the lielihood function, we assue that the error associated with the perforance odel f(x) follows a zero-ean Gaussian distribution. We therefore rewrite () as: 4 y = α, b( x) + ε ( =,,, K), (4) = represents the odeling error with the distribution: ε 5 pdf ( ε) = exp ~ N (, σ ). (5) π σ σ In (5) the standard deviation σ controls the agnitude of the odeling error. Cobining (4)-(5) yields the probability of observing a particular sapling point (x (n), y (n) ): ( ) ( ) ( ) 6 ( ) ( ) pdf y ~ exp n n n, y b α πσ σ x.(6) = Given that all sapling points are independently generated, the joint lielihood function pdf(y ) can be written as: 7 pdf ( y, σ) ~ N ( D, σi ), (7) D is a atrix with N K rows and K coluns. he atrix D can be obtained by peruting the rows and coluns of the following bloc diagonal atrix: B 8, (8) B K {B; =,,, K} are defined in (3). Cobining (), () and (7), the posterior distribution can be calculated as []: 9 pdf ( ) ~ N ( P, p ) ( σ ) p y, (9) = A AD I+ DAD DA () P = σ pdy. () Since the Gaussian distribution reaches its axiu at the ean value, the AP solution of the odel coefficients is: = P =σ pdy. () While the proposed Bayesian inference has been illustrated, the hyper-paraeters in the prior distribution () and the lielihood function (7) ust be appropriately deterined in order to solve the odel coefficients. In the next section, we will further discuss a statistical inference to find the optial hyperparaeter values. 3.3 Hyper-Paraeter Inference he hyper-paraeters of our proposed Bayesian inference are defined in Section 3. and Section 3.: = {,,,, R, σ}. o deterine these hyper-paraeters, one possible approach is to apply cross-validation [] which has been successfully used for various applications [6]-[8]. However, the cross-validation approach is not applicable to our Bayesian inference in this paper. It is well-nown that the coputational coplexity of crossvalidation exponentially grows with the nuber of hyperparaeters, and thus is only suitable to handle sall-size probles. In this paper, our Bayesian inference involves a large nuber of hyper-paraeters, since the nuber of basis functions (i.e., ) can easily reach several thousand due to the large nuber of device-level process paraeters at an advanced technology node. Hence, an alternative approach ust be developed to efficiently deterine these hyper-paraeters in. Statistically, the hyper-paraeters can be deterined by axiizing the conditional probability of observing the data set, i.e., the arginal lielihood:

4 3 ( Ω ) = ( Ω ) = (, Ω) ( Ω) l pdf y pdf y pdf d. (3) he integration in (3) indicates that the effect of odel coefficients is averaged out in the arginal lielihood. Given that the two ters in the integration depend on different sets of hyper-paraeters in, we can rewrite (3) as: 4 l ( Ω ) = pdf ( y, σ) pdf ( λ,..., λ, R) d. (4) By cobining () and (7) and taing the negative logarith for (4), we have: 5 L( ) ( ) Ω = y σ I+ DAD y + log σ I+ DAD. (5) he hyper-paraeters can be found by iniizing the cost function in (5). However, iniizing (5) is not trivial, because the cost function is not convex. In this paper, we adopt an expectation-axiization (E) algorith to address this challenge. In particular, instead of directly iniizing (5), the E algorith approaches a local optiu of (5) by iteratively perforing two operations []-[3], nown as the expectation step and the axiization step respectively. In the expectation step, we first calculate the posterior distribution pdf( y, (old) ) according to (9)-() based on (old), (old) denotes the hyper-paraeters calculated in the last iteration. Next, we define the following quantity []: 6 Q( Ω Ω ) = E, log p( y, Ω ), (6) y Ω the operator E represents the expectation of the log probability function with respect to the posterior distribution pdf( y, (old) ). Note that the expected value in (6) depends on the ean vector p and the covariance atrix p of the posterior distribution in (9). In the axiization step, we try to find such that: 7 Ω= arg ax Q Ω Ω. (7) Ω ( ) he optial solution of corresponds to: 8 Q( Ω Ω ) =. (8) Ω Cobining (6) and (8), we can find the optial solution of (7) as a function of (old), resulting in the following rule for updating [3]: r R ( + ( ) ) 9 λ p p p ( =,,, ) (9) K 3 = ( ) p + p p R ( =,,, ) (3) λ p ( p ) y D + r DD 3 σ ( =,,, ), (3) N p corresponds to the -th sub-vector in p with size of K, p corresponds to the -th diagonal bloc in p with size of K K. he expectation step (9)-() and the axiization step (9)-(3) are perfored iteratively until convergence is reached. Given that the E algorith only converges to a local optiu, the initial guess of ust be carefully chosen. Here we propose a odified S-OP algorith to achieve this goal. he ey idea of the proposed algorith is to (i) reduce the hyperparaeter space by posing additional constraints on R, (ii) apply a greedy algorith for basis function selection, and (iii) apply cross-validation to solve this siplified proble within the reduced hyper-paraeter space. Following this idea, we first consider the paraeterized covariance atrix R defined as: K r r r 3 R =, (3) r K r r r < is a non-negative hyper-paraeter. he atrix in (3) has several iplications. First, the correlation becoes sall when two states (i.e., their indexes) are substantially different. Second, the sae decaying rate r is used for all the states. his assuption often provides a good approxiation, even though it is not highly accurate. However, since our goal here is to find a good initial guess of, a reasonably accurate approxiation should suffice. As the covariance atrix R depends on a single paraeter r in (3), we can now apply cross-validation to estiate its value with low coputational cost. On the other hand, a greedy basis selection algorith based on S-OP [9] is used to infer the sparsity inforation that is encoded by the hyperparaeters {,,, }. In this way, we do not have to directly estiate {,,, }, which is coputationally expensive. At this point, the hyper-paraeter space has been reduced to three variables: r, σ and, denotes the nuber of selected basis functions. hese three hyper-paraeters can be efficiently deterined using cross-validation []. Naely, we start fro a candidate set {(r (q), σ (q), (q) ); q =,,, }, denotes the size of the set. At the q-th iteration step, we choose the hyper-paraeter values (r (q), σ (q), (q) ), estiate the odel coefficients, and evaluate the odeling error. o solve the odel coefficients with given (r (q), σ (q), (q) ), we adopt a odified S- OP algorith to iteratively identify the iportant basis functions. Siilar to S-OP [9], a single ost iportant basis function is greedily chosen at each iteration step by axiizing the correlation between the selected basis function and the odeling residual over all states. he odel coefficients are then calculated fro the Bayesian inference in (), ()-() and (3) based on r (q), σ (q) and the currently selected basis functions. Different fro the traditional S-OP ethod, we tae into account the correlation of coefficient agnitude when solving the unnown odel coefficients. he aforeentioned iteration continues until the nuber of chosen basis functions reaches (q). he odeling error can now be evaluated by cross-validation for the hyper-paraeters (r (q), σ (q), (q) ). Eventually, the optial values of (r, σ, ) are chosen to iniize the odeling error, and they are used to initialize the hyper-paraeters for the iterative E algorith. 3.4 Suary Algorith suarizes the ajor steps of the correlated Bayesian odel fusion (C-BF) algorith. It consists of two ain coponents: (i) a odified S-OP algorith fro Step to Step 7 to find the initial guess of the hyper-paraeters, and (ii) the E algorith fro Step 8 to Step to iteratively solve the optial values of the odel coefficients are updated at each iteration step. Algorith : Correlated Bayesian odel Fusion Algorith. Start fro a candidate set of hyper-paraeters {(r (q), σ (q), (q) ); q =,,, }. Partition the sapling points into C groups with the sae size for cross-validation.. Set the candidate index q =. 3. Set the cross-validation index c =. 4. Assign the c-th group as the testing set and all other groups as

5 the training set. 5. Initialize the residual Res = y, the basis vector set = {}, and the iteration index p =. 6. Calculate the inner product values {,; =,,, K; =,,, } between Res and all basis vectors b, for each state, b, represents the -th colun of B in (3). 7. Select the index s based on the following criterion: 33 s = arg ax ξ,. (33) K = 8. Update = s. 9. By considering the basis functions in and =, solve the odel coefficients {,; =,,, K; } using (), ()-() with r (q), σ (q) and R defined in (3).. Calculate the residual: 34 Res = y α, B, ( =,,, K ). (34) Θ. If p < (q), p = p + and go to Step 6.. Calculate the odeling error eq,c using the testing set. 3. If c < C, c = c + and go to Step Calculate eq = (eq, + eq, eq,c)/c. 5. If q <, q = q + and go to Step Find the optial r, σ, and aong the set {(r (q), σ (q), (q) ); q =,,, } such that the odeling error eq is iniized. 7. Initialize the atrix R using (3), =, and = Calculate p and p based on the expectation step in (9)-(). 9. Update {; =,,, }, R and σ based on the axiization step using (9)-(3).. If convergence is not reached, go to Step 8. Otherwise, stop iteration and calculate the odel coefficients by using (). 4. NUERICAL EXAPLES In this section, two circuit exaples designed in a coercial 3n SOI COS process are used to deonstrate the efficacy of the proposed C-BF algorith. Our objective is to build the perforance odels for tunable circuits. For testing and coparison purposes, two different odeling algoriths are ipleented: (i) S-OP [9], and (ii) C-BF. Here, S-OP is chosen for coparison, as it is one of the state-of-the-art techniques in the literature. In each exaple, a set of rando saples are generated by transistor-level onte Carlo siulations. he data set is partitioned into two groups, referred to as the training and testing sets respectively. he training set is used for coefficient fitting, while the testing set is used for odel validation. All nuerical experients are run on a.53ghz Linux server with 64GB eory. 4. Low Noise Aplifier Figure (a) shows the siplified circuit scheatic of a tunable.4ghz low noise aplifier (LNA) designed in a coercial 3n SOI COS process. In this exaple, there are totally 64 independent rando variables to odel the device-level process variations, including both the inter-die variations and the rando isatches. he LNA is designed with 3 different nob configurations (i.e., states) controlled by a tunable current source. Our goal is to odel three perforance etrics, noise figure (NF), voltage gain (VG) and third-order intercept point (IIP3), as linear functions of all rando variables for 3 states. he odeling error is estiated by using a testing set with 5 saples per state. Figure (b)-(d) show the perforance odeling error for NF, VG and IIP3 respectively. wo iportant observations can be ade here. First, for both S-OP and C-BF, the odeling error decreases when the nuber of saples increases. Second, with the sae nuber of saples, C-BF achieves significantly higher accuracy than S-OP, because C-BF taes into account the additional correlation inforation of coefficient agnitude. able further copares the perforance odeling error and cost for S-OP and C-BF. he overall odeling cost consists of two ajor coponents: (i) the circuit siulation cost for collecting training saples, and (ii) the odel fitting cost for solving all odel coefficients. As shown in able, the overall odeling cost is doinated by the siulation cost in this exaple. C-BF achieves ore than cost reduction over S-OP without surrendering any accuracy. IN VDD VDD to tunable current irror (a) (c) OU Figure. A tunable.4ghz LNA is used as an exaple for perforance odeling: (a) the siplified circuit scheatic, and (b)-(d) the perforance odeling error for NF, VG and IIP3 respectively. able. Perforance odeling error and cost for LNA S-OP C-BF Nuber of training saples 48 odeling error for NF.36%.85% odeling error for VG.577%.566% odeling error for IIP3.738%.497% Siulation cost (Hours).7.6 Fitting cost (Sec.) Overall odeling cost (Hours) Down-conversion ixer Shown in Figure 3(a) is the siplified circuit scheatic of a tunable.4ghz down-conversion ixer designed in a coercial 3n SOI COS process. In this exaple, there are totally 33 independent rando variables to odel the device-level process variations, including both the inter-die variations and the rando isatches. he ixer is designed with 3 different states controlled by two tunable load resistors. he perforances of interest include NF, VG, and input referred db copression point (IdBCP). he odeling error is estiated by using a test set with 5 saples per state. Figure 3(b)-(d) show the perforance odeling error for NF, VG and IdBCP respectively. Note that C-BF requires substantially less training saples than S-OP to achieve the (b) (d)

6 sae odeling accuracy. In this exaple, C-BF achieves ore than cost reduction over S-OP, as shown in able. LO+ IN+ (a) (c) VDD LO+ OU+/- LO- IN- odeling error (%) Figure 3. A tunable.4ghz down-conversion ixer is used as an exaple for perforance odeling: (a) the siplified circuit scheatic, (b)-(d) the odeling error for NF, VG and IdBCP respectively. able. Perforance odeling error and cost for ixer S-OP C-BF Nuber of saples 48 odeling error for NF.73%.66% odeling error for VG.758%.569% odeling error for IdBCP.4%.34% Siulation cost (Hours) Fitting cost (Sec.) Overall odeling cost (Hours) CONCLUSIONS In this paper, a novel C-BF algorith is proposed for efficient perforance odeling of tunable analog/rf circuits. C- BF encodes the correlation inforation for both odel teplate and coefficient agnitude aong different states by a prior distribution. Next, the prior distribution is cobined with very few siulation saples to accurately solve the odel coefficients. An efficient and robust hyper-paraeter inference is proposed to deterine the large nuber of hyper-paraeters associated with our proposed Bayesian inference. As is deonstrated by two circuit exaples designed in a coercial 3n SOI COS process, the proposed C-BF ethod achieves ore than cost reduction copared to the state-of-the-art odeling technique. Finally, it is worth entioning that C-BF assues a unified correlation odel across all states. If the states are utually different, such an assuption will no longer hold. In this case, a clustering algorith is needed to group siilar states into clusters before applying the proposed C-BF algorith. 6. ACKNOWLEDGEN his wor is sponsored in part by the DARPA HEALICS progra under Air Force Research Laboratory (AFRL) contract FA865-9-C-794 and the National Science Foundation under contract CCF he views expressed are those of the author and do not reflect the official policy or position of the (b) (d) Departent of Defense or the U.S. Governent. he authors would than B. Sadhu, J.-O. Plouchart and A. Valdes-Garcia for valuable discussions and D. Friedan for anageent support. 7. REFERENCES [] Seiconductor Industry Associate, International echnology Roadap for Seiconductors,. [] X. Li et al., Statistical Perforance odeling and Optiization, Now Publishers, 7. [3] X. Li et al., Robust analog/rf circuit design with projectionbased perforance odeling, IEEE rans. on CAD, vol. 6, no., pp. -5, Jan. 7. [4] G. Debyser et al., Efficient analog circuit synthesis with siultaneous yield and robustness optiization, IEEE ICCAD, pp. 38-3, 998. [5]. 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