University of Illinois at Urbana-Champaign, Urbana, IL 61801, /11/$ IEEE 162

Transcription

1 oward Efficient Spatia Variation Decomposition via Sparse Regression Wangyang Zhang, Karthik Baakrishnan, Xin Li, Duane Boning and Rob Rutenbar 3 Carnegie Meon University, Pittsburgh, PA 53, wangyan@ece.cmu.edu, xini@ece.cmu.edu Massachusetts Institute of echnoogy, Cambridge, MA 39, karthikb@mit.edu, boning@mt.mit.edu 3 University of Iinois at Urbana-Champaign, Urbana, IL 68, rutenbar@iinois.edu ABSRAC In this paper, we propose a new technique to accuratey decompose process variation into two different components: () spatiay correated variation, and () uncorreated random variation. Such variation decomposition is important to identify systematic variation patterns at wafer and/or chip eve for process modeing, contro and diagnosis. We demonstrate that spatiay correated variation carries a unique sparse signature in frequency domain. Based upon this observation, an efficient sparse regression agorithm is appied to accuratey separate spatiay correated variation from uncorreated random variation. An important contribution of this paper is to deveop a fast numerica agorithm that reduces the computationa time of sparse regression by severa orders of magnitude over the traditiona impementation. Our experimenta resuts based on siicon measurement data demonstrate that the proposed sparse regression technique can capture spatiay correated variation patterns with high accuracy. he estimation error is reduced by more than 3.5 compared to other traditiona methods.. INRODUCION With the continued scaing of CMOS technoogy, process variation has become a critica issue for design and manufacture of integrated circuits []-[4]. Large-scae performance variabiity has been observed for integrated circuits fabricated at advanced technoogy nodes, resuting in significant parametric yied oss. For this reason, accurate process characterization and modeing is required in order to fuy understand the variation sources and, hence, faciitate robust circuit design to achieve high parametric yied [5]. owards this goa, identifying and modeing systematic variation patterns is of great importance. Once the systematic variation sources are found, it is possibe to optimize the manufacturing process and/or modify the circuit design to improve yied. raditionay, systematic variation patterns are often determined by cacuating the averaged variation from a arge number of wafers and/or chips [3]-[4]. As such, uncorreated random variation can be statisticay removed. hese traditiona approaches, however, suffer from severa major imitations. First, it requires a arge number of measured wafers/chips to accuratey eiminate the impact of uncorreated random variation. In practice, the number of avaiabe wafers/chips can be imited (e.g., in owvoume production). Second, the systematic variation patterns must be identica for a tested wafers/chips. If a number of wafers/chips carry a different systematic variation pattern (e.g., due to manufacturing equipment drift) or contain a ot of missing data (e.g., due to measurement error), they can substantiay bias the estimation resut. It has been demonstrated in the iterature that systematic variation often presents a unique spatia pattern [3]. Namey, systematic variation is spatiay correated. For exampe, it has been observed in [5] that the spatia correation in gate ength is partiay caused by the systematic variation due to ithography. Motivated by these observations, we propose a new technique to uncover systematic variation patterns by decomposing process variation into two different components: () spatiay correated variation, and () uncorreated random variation. In other words, by removing the uncorreated random variation component, the remaining spatiay correated variation wi accuratey represent the systematic variation of interest. Our proposed technique is based upon an important fact that spatiay correated variation and uncorreated random variation present competey different signatures in frequency domain. Namey, spatiay correated variation typicay carries a unique sparse structure in frequency domain [6]-[8], impying that it can be accuratey represented by a sma number of dominant DC (i.e., discrete cosine transform) coefficients. On the other hand, uncorreated random variation has a white frequency spectrum and the corresponding DC coefficients are eveny distributed over a frequencies. By exporing the unique sparsity in frequency domain, we derive a sparse regression formuation to identify the dominant frequency-domain components and, hence, approximate the spatiay correated systematic variation. Another important contribution of this paper is to borrow the Simutaneous Orthogona Matching Pursuit (S-OMP) method from the statistics community [] to sove the aforementioned sparse regression probem. A number of impementation detais are carefuy considered in order to further tune the S-OMP method to fit the need of our specific appication. In particuar, severa new numerica agorithms are deveoped and integrated with S-OMP to substantiay reduce the computationa time for arge-scae wafer/chip-eve data anaysis. Our key idea is to expore the specia properties of DC (e.g., orthogonaity of DC basis functions) [6] to simpify the numerica operations that are required by S-OMP. he proposed variation decomposition technique has been vaidated by using the measurement data of contact pug resistance coected from 4 test chips in a 9 nm CMOS process. As wi be demonstrated by the experimenta resuts in Section 5, the proposed sparse regression approach reduces the estimation error by more than 3.5 compared to other traditiona methods. In addition, our improved S-OMP agorithm achieves more than 6 speed-up over the traditiona impementation. he remainder of this paper is organized as foows. In Section, we first derive the mathematica formuation for the proposed variation decomposition probem and then describe the S-OMP agorithm in Section 3. Next, we deveop severa fast numerica agorithms to impement S-OMP in Section 4. he efficacy of the proposed method is demonstrated by severa exampes in Section 5. Finay, we concude in Section 6.. VARIAION DECOMPOSIION Let g(x, y) be a two-dimensiona function representing the spatia variation of interest, where x and y denote the coordinate of //$6. IEEE 6

2 a spatia ocation within the two-dimensiona pane. he spatia variation g can be the device-eve threshod votage variation within a chip, chip-eve eakage current variation on a wafer, etc. In practice, the spatia variation g is measured at a finite number of spatia ocations. herefore, without oss of generaity, the spatia coordinates x and y can be abeed as integer numbers: x {,,...,P} and y {,,...,}, as shown in [6]-[8]. If the spatia variation g is measured for mutipe chips and/or wafers, it can be represented by a set of two-dimensiona functions: {g () (x, y); =,, L}, where L denotes the tota number of wafers/chips. In this paper, we aim to decompose each spatia variation function g () (x, y) into two different components: g()( x, y) = s()( x, y) + r()( x, y) ( =,,, L) () where {s () (x, y); x =,,...,P, y =,,...,} and {r () (x, y); x =,,...,P, y =,,...,} stand for the spatiay correated variation and the uncorreated random variation, respectivey. As demonstrated in [6]-[8], the spatia variation {g () (x, y); x =,,...,P, y =,,...,} can be mapped to frequency domain by a two-dimensiona inear transform such as discrete cosine transform (DC) [6]: G where 3 P ()( u, α u βv g( ) ( x, y) = x= y= π ( y ) ( v ) π ( u = ) ( u P) ( x ) ( u ) P ( =,,, L) () P α u = (3) P ( v = ) ( v ) 4 β v =. (4) In (), {G () (u, ; u =,,...,P, v =,,...,} represents the DC coefficients (i.e., the frequency-domain components) of the spatia variation function g () (x, y). Equivaenty, the function {g () (x, y); x =,,...,P, y =,,...,} can be represented as the inear combinations of {G () (u, ; u =,,...,P, v =,,...,} by inverse discrete cosine transform (IDC): 5 P π ()( ) ( ) ( x )( u ) x, y = αu β v G( ) u, v u= v= P π ( y )( v ) ( =,,, L) g. (5) Due to the inearity of DC [6], the DC coefficients {G () (u, ; u =,,...,P, v =,,...,} can be further decomposed into two different components: 6 G() S() + R() ( =,, L) = (6), where {S () (u, ; u =,,...,P, v =,,...,} and {R () (u, ; u =,,...,P, v =,,...,} denote the DC coefficients of the spatiay correated variation s () (x, y) and the uncorreated random variation r () (x, y) defined in (). Once S () (u, and R () (u, are found, s () (x, y) and r () (x, y) can be determined by IDC, simiar to the case in (5). o accuratey sove the decomposition probem in (6), we first need to anayze the signatures of S () (u, and R () (u, in the DC domain. As is demonstrated in [6]-[8], the DC coefficients S () (u, (corresponding to spatiay correated variation) are typicay sparse, i.e., many of these coefficients are cose to. In other words, there exist a sma number of (say, λ () where λ () << P) dominant DC coefficients to satisfy: 7 S() S() D( ) u = v = P where D () denotes the set of the indices of the dominant DC coefficients for S () (u,. Eq. (7) simpy impies that the tota energy of a DC coefficients {S () (u, ; u =,,...,P, v =,,...,} are amost equa to the energy of the dominant DC coefficients {S () (u, ; (u, D () }. On the other hand, uncorreated random variation can be characterized as white noise [7] and eveny distributed among a frequencies. herefore, given the set of indices D (), the foowing equation hods: P λ() 8 R()( u, R()( u,. (8) D P () u = v= Because of the inequaity λ () << P, we have λ () /P << in (8). If the vaue of λ () is sufficienty sma (i.e., the DC coefficients of spatiay correated variation are sufficienty sparse), the efthand side of (8) is approximatey zero and the foowing inequaity hods: 9 R()( u, << S()( u,. (9) D() D() Based on these assumptions, an accurate approximation of the DC coefficients S () (u, (corresponding to spatiay correated variation) can be expressed as: G( ) D( ) ) S( ) =. () ( otherwise ) In other words, we simpy approximate S () (u, by the dominant DC coefficients {G () (u, ; (u, D () }.Comparing (6) and (), it can be further proven that the approximation error of () is given by: P [ S ] ()( u, S()( u, u = v=. () = R()( u, + S()( u, D() D() Given the assumptions in (7) and (9), the error terms in () are amost negigibe. In practice, however, Eq. () cannot be directy used for variation decomposition because of two reasons. First, the index set of dominant DC coefficients D () is not known in advance and it must be estimated from the measurement data. Second, if the spatia variation g () (x, y) is not measured at a ocations, it is not possibe to directy cacuate G () (u, from (). In many practica appications, measurement error and manufacturing defect can resut in missing data at a number of spatia ocations, as is demonstrated in the iterature [3], [8]. hese observations, hence, motivate us to derive an efficient Simutaneous Orthogona Matching Pursuit (S-OMP) agorithm [] to determine S () (u, in () for accurate variation decomposition. 3. S-OMP ALGORIHM In this section, we describe the S-OMP agorithm in detai. o this end, we first show a simpified version of S-OMP, referred to as Orthogona Matching Pursuit (OMP) [], where ony the measurement data from a singe wafer/chip are considered. Next, we derive the fu S-OMP agorithm [] that expores the correation among mutipe wafers/chips to further improve the accuracy for variation decomposition. (7) 63

3 3. Orthogona Matching Pursuit he objective of OMP is to determine the index set D () so that a sma number of dominant DC coefficients can be identified to approximate the spatiay correated variation in (). Mathematicay, our variation decomposition probem can be formuated as the foowing optimization: minimize A() η() B() η ( ) () subject to η() λ () where and stand for the L -norm (i.e., the square root of the summation of the squares of a eements) and the L -norm (i.e., the number of non-zero eements) of a vector respectivey, and: A(),,, A(),,, A(),, P, () () () 3 A,,, A,,, A,, P, A = ( ) (3) A() () () () () (), M A,,, M A,,, M, P, π ( x(), m ) ( u ) A(), m, u, v = α u β v P 4 (4) π ( y(), m ) ( v ) η = S, S, S P, (5) [ ] [ g ( )] x,, y, g x, M, y M 5 () ()( ) ()( ) ()( ) ( ) () () () () () 6 B () = () () (),. (6) In ()-(6), the vector B () represents the measurement data coected from M () different spatia ocations {(x (),m, y (),m ); m =,,,M () } of the th wafer/chip, the vector η () contains the unknown DC coefficients for the spatiay correated variation that we want to extract, and the matrix A () defines the inear transform to map the DC coefficients from the frequency domain to the spatia domain. he optimization in () attempts to use a sma number of (i.e., λ () ) dominant DC coefficients to approximate the measurement data B () with east-squares error. Studying (), we woud have two important observations. First, if the number of measured sampes (i.e., M () ) is equa to the tota number of DC coefficients (i.e., P), the matrix A () represents the IDC matrix and it is a fu-rank square matrix. On the other hand, if M () is ess than P (e.g., due to missing data), the matrix A () contains M () rows taken from the IDC matrix and it is not simpy a square matrix. In genera, soving the optimization in () is not trivia, since the probem is NP-hard. OMP [] is an efficient greedy agorithm to approximate the soution of (). It was recenty adopted by the CAD community for arge-scae performance modeing [9]. In this paper, we further extend the OMP agorithm to our appication of variation decomposition. In what foows, we briefy review the major steps of the OMP agorithm. More detais on OMP can be found in the iterature [9], []. he key idea of OMP is to iterativey use the inner product to identify a sma number of important DC coefficients. owards this goa, we re-write the matrix A () by its coumn vectors: 7 A () = [ A(), A(), A(), P ] (7) where each coumn vector A (),i can be conceptuay viewed as a basis vector associated with the DC coefficient η (),i. he inner product <B (), A (),i > measures the correation between the measurement data B () and the basis vector A (),i. A strong correation between B () and A (),i impies that the basis vector A (),i (hence, the DC coefficient η (),i ) is an important component to approximate B (). Based on this idea, OMP appies an iterative process to find a set of important basis vectors, as summarized in Agorithm. At each iteration, OMP performs two major operations. First, it seects the basis vector A (),s that is most correated to the residua Res (). Second, the DC coefficients associated with a seected basis vectors are soved by east-squares fitting. It shoud be noted that Agorithm reies on a given input parameter λ (). In practice, the vaue of λ () is not known in advance. However, it can be accuratey estimated by cross-vaidation, as wi be discussed in detai in Section 3.3. Agorithm : Orthogona Matching Pursuit (OMP). Start from the optimization probem in () with a given integer λ () specifying the tota number of basis vectors.. Initiaize the residua Res () = B (), the set Ω () = {}, and the iteration index p =. 3. Seect the new basis vector A (),s according to the foowing criterion: 8 maximize Res (), A()s,. (8) s 4. Update Ω () by Ω () = Ω () {s}. 5. Sove the east-squares fitting: 9 minimize A(), i η(), i B() η i ( ), i, Ω( ) i Ω ( ). (9) 6. Cacuate the residua: Res () = B() A(), i η (), i. () i Ω () 7. If p < λ (), p = p + and go to Step For any i Ω (), set η (),i =. 3. Simutaneous OMP Whie the spatiay correated variation for mutipe wafers/chips can be extracted by independenty performing OMP, this method is ceary not optima since it ignores the strong correation among different wafers/chips. Such strong correation exists, if these wafers/chips are produced by the same manufacturing ine and, hence, a significant portion of systematic variation can be shared [8]. In this sub-section, we further extend the OMP agorithm and derive an efficient Simutaneous Orthogona Matching Pursuit (S-OMP) agorithm [] so that the aforementioned correation information can be used to improve the accuracy of variation decomposition. As demonstrated in [8], if mutipe wafers share simiar spatia variation patterns, the corresponding DC coefficients are strongy correated. In this case, dominant DC coefficients can be found at a number of common frequencies shared by a wafers. A simiar observation can be made at chip eve, where the systematic variation of a chip is often characterized by ayoutdependent patterns. Since mutipe chips share the same ayout design, their systematic variation is expected to share simiar spatia patterns. Hence, the dominant DC coefficients associated with chip-eve systematic variation shoud be distributed over a set of common frequencies shared by mutipe chips. Based upon these observations, we propose to mode the spatia variation of mutipe wafers/chips by a shared index set D for dominant DC coefficients. Namey, we assume: D() = D( ) = = D( L ) = D. () 64

4 Consequenty, the sizes of the sets {D () ; =,,,L}, i.e., {λ () ; =,,,L}, are identica and can be modeed by a singe parameter λ: λ () = λ( ) = = λ( L) = λ. () With ()-() in mind, we re-visit the OMP agorithm (i.e., Agorithm ) where a set of dominant DC coefficients are seected to approximate the spatiay correated systematic variation. At each iteration of Agorithm, a singe DC basis vector is chosen according to the inner product in (8). For S- OMP, since the index set of dominant DC coefficients is shared for L different wafers/chips as shown in (), we use the inear combination of mutipe inner products as a quantitative criterion for basis vector seection: L 3 maximize Res(), A(), s. (3) s = Eq. (3) is expected to be more accurate than (8), since it is ess sensitive to the random noise caused by uncorreated random variation and/or measurement error. In other words, by adding the inner products over L wafers/chips, the impact of random noise is reduced and the spatia pattern associated with systematic variation can be accuratey detected. his is the fundamenta reason why S-OMP is preferred over OMP, if the spatiay correated systematic variation shares simiar patterns across mutipe wafers/chips. Agorithm summarizes the major steps of the aforementioned S-OMP agorithm. Note that S-OMP is an extended version of OMP (i.e. Agorithm ). If there is ony one wafer/chip (i.e., L = ), S-OMP is exacty equivaent to OMP. Agorithm : Simutaneous OMP (S-OMP). Start from the optimization probem in () for L wafers/chips {,,,L} with a given integer λ specifying the tota number of basis vectors.. Initiaize the set Ω = {}, and the iteration index p =. 3. For each {,,,L}, set the residua Res () = B (). 4. Seect the new basis vector s according to (3). 5. Update Ω by Ω = Ω {s}. 6. For each {,,,L}, sove the east-squares fitting in (9). 7. Cacuate the residua for {,,,L} by using (). 8. If p < λ, p = p + and go to Step For any i Ω, set η (),i =. 3.3 Cross-Vaidation he S-OMP agorithm (i.e., Agorithm ) reies on a user defined parameter λ to contro the number of dominant DC coefficients that shoud be seected. In practice, λ is not known in advance. he appropriate vaue of λ must be determined by considering the foowing two important issues. First, if λ is too sma, S-OMP cannot seect a sufficient number of basis vectors to represent the spatiay correated variation, thereby eading to arge modeing error. On the other hand, if λ is too arge, S-OMP can incorrecty seect too many DC coefficients and some of these coefficients are associated with uncorreated random variation, instead of spatiay correated systematic variation. It, again, resuts in arge modeing error due to over-fitting. In order to achieve the best accuracy, we must accuratey estimate the modeing error for different λ vaues and then find the optima λ with minimum error. In this paper, we adopt the cross-vaidation method [8] to estimate the modeing error for our variation decomposition appication. An F-fod cross-vaidation partitions the entire data set into F groups. Modeing error is estimated according to the cost function in () from F independent runs. In each run, one of the F groups is used to estimate the modeing error and a other groups are used to cacuate the DC coefficients. Note that the training data for coefficient estimation and testing data for error estimation are not overapped. Hence, over-fitting can be easiy detected. In addition, different groups shoud be seected for error estimation in different runs. As such, each run resuts in an error vaue ε f (f =,,...,F) that is measured from a unique group of data points. he fina modeing error is computed as the average of {ε f ; f =,,...,F}, i.e., ε = (ε + ε ε F )/F. 4. IMPLEMENAION DEAILS Whie Agorithm summarizes the major steps of S-OMP for variation decomposition, a number of impementation detais must be carefuy considered in order to make the S-OMP agorithm computationay efficient for arge-scae probems. In this section, we derive severa efficient numerica agorithms to address the aforementioned issue reated to computationa cost. 4. Inner Product Computation It can be easiy observed from Agorithm that the computationa cost is dominated by two steps: the inner product computation in Step 4 and the east-squares fitting in Step 6. In this sub-section, we first derive an efficient numerica agorithm to cacuate the inner product vaues in (3). We wi discuss the numerica agorithm for east-squares fitting in the next subsection. In order to appropriatey seect the basis vectors by (3), the inner product <Res (), A (),i > must be cacuated for a basis vectors i {,,,P} and a wafers/chips {,,,L}. If the inner product vaues are simpy cacuated by vector-vector mutipications, the computationa cost is in the order of O(LP ). Note that the computationa cost quadraticay increases with the probem size P. Hence, the aforementioned impementation can quicky become computationay intractabe, as the probem size increases. For this reason, an efficient numerica agorithm for inner product computation is needed in order to reduce the computationa cost. owards this goa, we first re-write the inner product <Res (), A (),i > as: 4 Res() A(), i = A(), i Res(),. (4) For each {,,,L}, we need to cacuate (4) for each basis vector, i.e., i {,,,P}. he resuts can be expressed by the foowing matrix-vector mutipication: Res(), A(), Res 5 (), A(), = A() Res(). (5) Res(), A(), P In other words, by cacuating the matrix-vector mutipication in (5), we are abe to obtain the inner product vaues for a (i.e., P) basis vectors. If the measurement of the th wafer/chip does not contain any missing data, the matrix A () in (5) represents the IDC matrix and it is a fu-rank square matrix, as defined in (3). In this case, since DC/IDC is an orthogona transform [6], A () = A () is exacty the DC matrix. Namey, cacuating the inner product vaues in (5) is equivaent to performing DC on the residua Res (). Simiar to fast Fourier transform (FF), there exists a number of fast agorithms for DC/IDC. he computationa cost 65

5 of these fast agorithms is in the order of O(P og(p)) [6]. herefore, by using a fast DC agorithm, the computationa cost for Step 4 of Agorithm is reduced from O(LP ) to O(LP og(p)). he aforementioned fast DC agorithm is appicabe, if and ony if there is no missing data and, hence, the matrix A () is a furank square matrix. If a number of missing data exist (e.g., due to measurement error), we can construct an augmented vector Res * () R P where the eements corresponding to missing data are simpy fied with zeros. Mathematicay, the augmented vector Res * () can be represented as: 6 () * Res() Res = () W (6) where W () is a permutation matrix to map the residua Res () and the zero vector to the appropriate eements in Res * (). Appying DC to the augmented vector Res * () yieds: * * * Res() 7 A Res() = A W( ) (7) where A * represents the IDC matrix and, hence, A * is the DC matrix. Remember that the matrix A () in (3) contains M () rows taken from the IDC matrix A *. Hence, the matrix A * W () in (7) can be re-written as: [ A A ] * A W = 8 () () ( ) (8) where the matrix A ( ) contains the P M () rows of A * that are not incuded in A () due to missing data. Substituting (8) into (7), we have: * * Res() 9 A Res() = [ A() A ( )] = A () Res(). (9) Note that the DC resuts in (9) are exacty equa to the inner product vaues in (5). It, in turn, demonstrates that by fiing the missing data with zeros, we can efficienty cacuate the inner product vaues by using a fast DC agorithm. In this case, the computationa cost for Step 4 of Agorithm is again reduced from O(LP ) to O(LP og(p)). In addition to the reduction in computationa cost, the aforementioned fast agorithm based on DC can aso efficienty reduce the memory consumption. Note that the direct matrixvector mutipication in (5) requires to expicity form a dense matrix A () with about P entries. Whie it is possibe to cacuate each inner product in (4) one by one without forming the matrix A (), such an approach eads to arge computationa time since each coumn of A () must be repeatedy formed during the iterations of Agorithm. For these reasons, the direct approach based on matrix-vector mutipication or vector-vector mutipication is expensive in either memory consumption or computationa time. On the other hand, our proposed method ony needs to form the augmented vector Res * () in (6) with P entries. A fast DC agorithm can be appied to Res * () without expicity buiding the DC matrix in memory, thereby significanty reducing the memory consumption for arge-scae probems. 4. Least-Squares Fitting In addition to inner product computation, east-squares fitting is another computationay expensive operation that is required by Step 6 of Agorithm. he goa is to sove the optimization probem in (9). In this sub-section, we wi deveop an efficient numerica agorithm to reduce the computationa cost of (9). We first re-write (9) for the th wafer/chip at the pth iteration step: 3 minimize η()(, p ) A()(, p) η()(, p) B() (3) where the matrix A (),(p) contains p coumn vectors seected from A () and the vector η (),(p) contains the DC coefficients corresponding to these seected basis vectors. he reation between A (),(p) and A () can be further expressed as: 3 A () W( p) = [ A()(, p) A()(, p )] (3) where W (p) is a permutation matrix, and the matrix A (),(p ) contains the basis vectors that are not incuded in A (),(p). he east-squares soution η (),(p) of (3) satisfies the foowing norma equation [9]: 3 A()(, p) A()(, p) η ()(, p) = A()(, p) B( ). (3) raditionay, the soution η (),(p) of (3) is soved by R decomposition [9]: 33 A ()(, p) = ()(, p) R()(, p) (33) where (),(p) is an M () -by-p matrix with orthonorma coumns and R (),(p) is a p-by-p upper trianguar matrix. Substituting (33) into (3) yieds: 34 R( ),( p) η ( ),( p) = ( ), ( p) B( ). (34) In (34), since R (),(p) is upper trianguar, η (),(p) can be soved by back substitution. he computationa cost of the aforementioned east-squares fitting is dominated by the R decomposition step and it is in the order of O(M () p ). he traditiona east-squares sover based on R decomposition is not computationay efficient for arge-scae probems. An aternative way to sove (3) is based on an iterative agorithm that is referred to as the LSR method [3]. LSR reies on the bi-diagonaization process of the matrix A (),(p). During its iterations, LSR generates a sequence of soutions to approximate η (),(p). hese soutions are exacty identica to the resuts cacuated by the conjugate gradient method [9] for the norma equation in (3). However, unike the conjugate gradient method that suffers from numerica issues when soving (3), LSR aims to directy sove (3) in order to improve numerica stabiity. he detais of LSR can be found in [3]. When appying LSR, it is not necessary to expicity form the matrix A (),(p). Instead, ony the matrix-vector mutipications A (),(p) α and A (),(p) β, where α is a p-by- vector and β is an M () - by- vector, are required. hese matrix-vector mutipications can be efficienty cacuated by appying a fast numerica agorithm. In what foows, we wi show the mathematica formuation of our proposed fast agorithm. First, to efficienty compute A (),(p) α, we construct an augmented vector α * R P : * α 35 α = W ( p) (35) where W (p) is the permutation matrix defined in (3). We conceptuay consider the augmented vector α * as a set of DC coefficients and appy IDC to it: * * * α 36 A α = A W( p) (36) where A * denotes the IDC matrix as defined in (7). On the other hand, we can derive the foowing equation from (8): A() 37 () ( ) * A = W. (37) A Substituting (37) into (36) yieds: 66

6 A() ( ) * * W p α 38 A α = W(). (38) A ( ) W ( p) In (38), A () W (p) can be represented as two sub-matrices as shown in (3). If we simiary re-write A ( ) W (p) as two sub-matrices: 39 A ( ) W( p) = [ A ( )( ) A ( )( )], p, (39) p Eq. (38) becomes: A()( p) A()( p ) α A()( ) α 4 () () ( )( ) ( )( ) ( )( ) * *,,, p A α = W = W. (4) A A A α, p, p, p Since W () is a permutation matrix, Eq. (4) is equivaent to: A()(, p) α * * 4 = W () A α. (4) A( )(, ) α p Eq. (4) reveas an important fact that the matrix-vector mutipication A (),(p) α can be efficienty computed by appying IDC to the augmented vector α *. he vaue of A (),(p) α is determined by seecting the appropriate eements from the IDC resut A * α *. If a fast IDC agorithm is appied [6], the computationa cost of the aforementioned matrix-vector cacuation is in the order of O(P og(p)). Next, we consider the other matrix-vector mutipication A (),(p) β that is required by the LSR agorithm. Simiary, we first construct an augmented vector β * R P : * β 4 β = W () (4) where W () is the permutation matrix defined in (6). We appy DC to the augmented vector β * : * * * β 43 A β = A W() (43) where A * is the DC matrix as defined in (7). Substituting (37) into (43) yieds: * * β 44 A β = [ A() A ( )] W() W() = A() β. (44) Based on (3), Eq. (44) can be further re-written as: A 45 ()(, ) β p * * = W( p) A β. (45) ()(, A p ) β Hence, the matrix-vector mutipication A (),(p) β can be cacuated by appying DC to the augmented vector β *. he vaue of A (),(p) β is determined by seecting the appropriate eements from the DC resut A * β *. he computationa cost is in the order of O(P og(p)). Finay, it is worth mentioning that simiar to other iterative sovers, a good initia guess shoud be provided to LSR to achieve fast convergence. If the initia guess is cose to the actua soution, LSR can reach convergence in a few iterations [3]. In this paper, LSR is required at each iteration step of the S-OMP agorithm (i.e., Agorithm ). When Agorithm is appied, the soution from the previous iteration step can serve as a good initia guess for the current iteration step. By adopting such a heuristic, LSR typicay converges in 3 iterations in our tested exampes. 5. NUMERICAL EXAMPLES In this section, we demonstrate the efficacy of our proposed variation decomposition agorithm using severa exampes. A numerica experiments are performed on a.8ghz Linux server. 5. Measurement Data for Contact Pug Resistance We consider the contact pug resistance measurement data coected from 4 test chips in a 9 nm CMOS process. Each chip contains 36,864 test structures (i.e., contacts) arranged as a array, as described in []. Among these 4 test chips, three of them contain missing data due to externa measurement error. he number of faied measurements are 936, 864 and 8 for these three chips, respectivey DC Coeff(Mag) (a) (b) Figure. (a) Measured contact pug resistance (normaized) of a array for one of the 4 test chips. (b) Discrete cosine transform (DC) coefficients (magnitude) of the measured contact pug resistance for the same test chip. Figure (a) shows the measured contact pug resistance (normaized) from one of the 4 test chips. Studying Figure (a), we woud notice that there is a unique spatia pattern due to ayout dependency. However, the spatia pattern is not ceary visibe because of the arge-scae uncorreated random variation found in this exampe. Figure (b) further shows the DC coefficients (magnitude) of the measured contact pug resistance for the same test chip. Note that there ony exist a sma number of dominant DC coefficients with arge magnitude. hese DC coefficients are distributed over a sma number of frequencies, representing a unique signature of the ayout-dependent systematic variation in frequency domain. A other DC coefficients are sma in magnitude and have a white frequency spectrum (i.e., eveny distributed over a frequencies). hey correspond to the uncorreated random variation that we observe from Figure (a). hese observations demonstrate the important fact that the spatiay correated systematic variation can be extracted by identifying the dominant DC coefficients in frequency domain. A. Variation Decomposition We appy the proposed S-OMP agorithm (i.e. Agorithm ) to extract the ayout-dependent systematic variation of a test chips. he extracted systematic variation of the chip in Figure (a) is shown in Figure (a). Comparing Figure (a) with Figure (a), we woud notice that the spatia pattern of systematic variation becomes cear, after S-OMP is appied. Such a spatia variation pattern can serve as an important basis for diagnosing the sources of systematic variation. In this exampe, the systematic variation is mainy caused by different ayout patterns reguary distributed over the entire chip. o verify the ayout dependency, we pot the spatia distribution of different ayout patterns in Figure (b) where there exist 55 ayout patterns in tota and different ayout patterns are shown in different coors. Note that Figure (b) perfecty matches Figure (a). It, in turn, demonstrates that the aforementioned ayout dependency is the dominant source for the extracted systematic variation in Figure (a). 67

7 (a) (b) Figure. (a) Extracted ayout-dependent systematic variation (normaized) of contact pug resistance. (b) Spatia distribution of different contact ayout patterns in the test chip. B. Runtime Comparison o demonstrate the efficiency of the fast numerica agorithms proposed in Section 4, we impement three different versions of OMP/S-OMP where the inner product and the east-squares fitting are cacuated by different methods. In the first impementation, the inner product is directy computed by (4) and the eastsquares fitting is directy computed by the R decomposition in (33)-(34). In the second impementation, the traditiona inner product cacuation is repaced by the fast agorithm proposed in Section 4.. Finay, in the third impementation, both the inner product and the east-squares fitting are cacuated by the fast agorithms proposed in Section 4. For testing and comparison purposes, we first run the OMP agorithm (i.e., Agorithm ) with the aforementioned three impementations. abe shows the computationa time for the proposed variation decomposition of a singe test chip. Note that the fast agorithm for inner product computation achieves 73 speed-up and the fast east-squares fitting further brings 8.8 speed-up. he overa speed-up achieved by our proposed fast agorithms is 647, compared to the traditiona direct impementation. abe. Computationa time of variation decomposition for a singe chip by OMP Inner product Least-squares fitting CPU time (Sec.) Direct Direct.88 6 Fast Direct Fast Fast abe. Computationa time of variation decomposition for 4 chips by S-OMP Inner Product Least-squares fitting CPU time (Sec.) Fast Direct 5. 6 Fast Fast.97 5 Next, we run S-OMP for a 4 test chips and abe compares the computationa time for two different impementations. Once S-OMP is appied to a test chips, the computationa time increases significanty. he simpe impementation with direct inner product cacuation and eastsquares fitting is not computationay feasibe. Hence, its resut is not shown in abe. In this exampe, the proposed fast agorithm for east-squares fitting achieves 6.3 speed-up over the direct impementation. 5. Synthetic Data for Contact Pug Resistance o further vaidate the accuracy of the proposed S-OMP agorithm, we create a set of synthetic data for contact pug resistance. Simiar to the siicon measurement data shown in Section 5., the synthetic data set aso contains 4 test chips, with a array of test structures in each chip. hree different variation sources are modeed for the synthetic data set: () die-to-die variation (σ = 5%), () ayout-dependent systematic variation (σ = 3.5%), (3) uncorreated random variation (σ = 3.5%). he standard deviation of these variation sources is approximatey equa to what is observed from the siicon measurement in Section 5.. Since we exacty know the systematic variation for the synthetic data set, it enabes us to quantitativey compare the accuracy of the proposed variation decomposition agorithm with severa traditiona techniques DC Coeff(Mag) (a) (b) Figure 3. (a) Layout-dependent systematic variation (normaized) of contact pug resistance for one of the 4 test chips in the synthetic data set. (b) Discrete cosine transform (DC) coefficients (magnitude) of the systematic variation for the same synthetic test chip. Figure 3(a) shows the ayout-dependent systematic variation (normaized) of contact pug resistance for one of the 4 test chips in the synthetic data set. Figure 3(b) further shows the DC coefficients (magnitude) of the systematic variation for the same synthetic test chip. Note that the DC coefficients are sparse in frequency domain where most DC coefficients are cose to. A. Accuracy Comparison For testing and comparison purposes, we appy severa different agorithms to extract the ayout-dependent systematic variation in this exampe: () the proposed S-OMP agorithm (i.e., Agorithm ), () the OMP agorithm (i.e., Agorithm ), (3) the non-oca means method [4], (4) the moving average method [6], (5) the Wiener fiter method [6], (6) the Gaussian fiter method [6], and (7) the waveet threshoding method [6]. Except S-OMP and OMP, other methods are borrowed from the image processing community. o compare the accuracy of these different techniques, we define the average estimation error of systematic variation as: [ s()( ) x, y s()( x, y) ] 4 x= y= Error =. (46) = [ s()( x, y) ] x= y= where s () (x, y) and s ()(x, y) denote the exact systematic variation and the estimated systematic variation for the th chip, respectivey. abe 3 shows the average estimation error for seven different agorithms. Studying the resuts in abe 3, we woud have two important observations. First, our proposed S-OMP agorithm is more accurate than the simpe OMP agorithm. Compared to OMP, S-OMP improves the accuracy by exporing the correation information among different chips, as discussed in Section

8 Second, the proposed S-OMP agorithm achieves more than 3.5 error reduction over the traditiona image processing techniques that have been widey appied for noise remova. Most image processing methods are particuary deveoped to capture the owfrequency components of a -D image (e.g., by oca smoothing). hey cannot accuratey capture the high-frequency DC coefficients shown in Figure 3(b), thereby resuting in arge error. abe 3. Average estimation error of ayout-dependent systematic variation for seven different agorithms Agorithm Error S-OMP (Agorithm ).67% OMP (Agorithm ).% Non-oca means [4].44% Moving average [6] 3.% Wiener fiter [6] 3.% Gaussian fiter [6].55% Waveet threshoding [6] 3.6% B. Missing Data Finay, to further study the impact of missing data on our proposed S-OMP agorithm, we purposey introduce a number of missing data into the synthetic data set. he ocations of these missing data are made identica to the siicon measurement data in Section 5.. Namey, three chips contain missing data where the number of faied measurements are 936, 864 and 8 respectivey. With these missing data, the estimation error of S-OMP ony increases by.% for the chip with 936 missing sampes, and no notabe change in estimation error is observed for other chips. Consequenty, the change of average estimation error is amost negigibe. In addition, we further randomy inject % and % missing sampes to each synthetic chip. he average error of S- OMP is.68% and.7% in these two cases, respectivey. hese observations, in turn, demonstrate an important fact that the proposed S-OMP agorithm is extremey robust to the missing data caused by measurement error. 6. CONCLUSIONS In this paper, we propose a new technique to efficienty separate spatiay correated systematic variation from uncorreated random variation. he proposed method is based upon the fact that spatiay correated variation typicay carries a unique sparse signature in frequency domain and it can be accuratey represented by a sma number of dominant DC coefficients. An efficient S-OMP agorithm is borrowed from the statistics community to accuratey find these dominant DC coefficients corresponding to systematic variation. In addition, a number of fast numerica agorithms are deveoped to make the computationa cost tractabe for arge-scae data anaysis probems. Our experimenta resuts for contact pug resistance demonstrate that the proposed S-OMP agorithm achieves more than 3.5 error reduction compared to other traditiona methods. he variation decomposition technique deveoped by this paper can be appied to a number of practica appications, incuding manufacturing process modeing, contro and diagnosis. 7. ACKNOWLEDGEMENS he authors acknowedge the support of the CS Focus Center and the Interconnect Focus Center, two of six research centers funded under the Focus Center Research Program (FCRP), a Semiconductor Research Corporation entity. his work is aso supported in part by the Nationa Science Foundation under contract CCF REFERENCES [] S. Nassif, Deay variabiity: sources, impacts and trends, IEEE ISSCC, pp ,. [] Semiconductor Industry Associate, Internationa echnoogy Roadmap for Semiconductors, 9. [3] A. Gattiker, Unraveing variabiity for process/product improvement, IEEE IC, pp. -9, 8. [4] S. Reda and S. Nassif, Accurate spatia estimation and decomposition techniques for variabiity characterization, IEEE rans. on Semiconductor Manufacturing, vo. 3, no. 3, pp , Aug.. [5] P. Friedberg, Y. Cao, J. Cain, R. Wang, J. Rabaey, and C. Spanos, Modeing within-fied gate ength spatia variation for process-design co-optimization, Proceedings of SPIE, vo. 5756, pp , May. 5. [6] X. Li, R. Rutenbar and R. Banton, Virtua probe: a statisticay optima framework for minimum-cost siicon characterization of nanoscae integrated circuits, IEEE ICCAD, pp , 9. [7] W. Zhang, X. Li and R. Rutenbar, Bayesian virtua probe: minimizing variation characterization cost for nanoscae IC technoogies via Bayesian inference, IEEE DAC, pp. 6-67,. [8] W. Zhang, X. Li, E. Acar, F. Liu and R. Rutenbar, Muti-wafer virtua probe: minimum-cost variation characterization by exporing wafer-to-wafer correation, IEEE ICCAD, pp ,. [9] X. Li, Finding deterministic soution from underdetermined equation: arge-scae performance modeing of anaog/rf circuits, IEEE rans. on CAD, vo. 9, no., pp , Nov.. [] K. Baakrishnan and D. Boning, Measurement and anaysis of contact pug resistance variabiity, IEEE CICC, pp. 46-4, 9. [] J. ropp, A. Gibert, and M. Strauss, Agorithms for simutaneous sparse approximation. Part I: Greedy pursuit, Signa Processing, vo. 86, pp , Mar. 6. [] J. ropp and A. Gibert, Signa recovery from random measurements via orthogona matching pursuit, IEEE rans. Information heory, vo. 53, no., pp , Dec. 7 [3] C. Paige and M. Saunders, LSR: An agorithm for sparse inear equations and sparse east squares, ACM rans. on Mathematica Software, vo. 8, no., pp. 43-7, Mar. 98. [4] A. Buades, B. Co, and J. More, A nonoca agorithm for image denoising, IEEE CVPR, vo., 5, pp [5] M. Orshansky, S. Nassif, and D. Boning, Design for Manufacturabiity and Statistica Design: A Constructive Approach, Springer, 7. [6] R. Gonzaez and R. Woods, Digita Image Processing, Prentice Ha, 7. [7] A. Oppenheim, Signas and Systems, Prentice Ha, 996. [8] C. Bishop, Pattern Recognition and Machine Learning, Prentice Ha, 7. [9] W. Press, S. eukosky, W. Vettering and B. Fannery, Numerica Recipes: he Art of Scientific Computing, Cambridge University Press, 7. 69