A Frequency-domain Technique for Statistical Timing Analysis of Clock Meshes

Transcription

1 A Frequency-doman Technque for Statstcal Tmng Analyss of Clock Meshes Ruln Wang and Cheng-Kok Koh School of Electrcal and Computer Engneerng, Purdue Unversty, West Lafayette, IN47907 {rw, Abstract We propose a frequency-doman modelng technque wth applcatons on the statstcal tmng analyss of clock mesh/grd networks. Usng transmsson lnes to model clock mesh edges, we express the means and (co)varances of the snk arrval tmes as polynomal functons of the arrval tmes of the nput sgnals and the wre wdths of the mesh edges, wth up to second order accuracy. Expermental results show that the proposed frequency-doman statstcal tmng analyss technque s effcent and accurate. The relatve mean error s less than 1% and relatve varance error less than 3%. I. INTRODUCTION Clock meshes are wdely used n modern VLSI hghperformance clock networks to help reduce clock skew and jtter due to varatons n process, temperature and power supply voltages [1]. As random varatons become even more promnent n nanometer process technologes, statstcal tmng analyss (nstead of statc or worst-case tmng analyss) of clock meshes can be more effectve n combatng yeld loss and reducng the cost of over desgn. Whle gate-level statstcal tmng analyss tools can be easly appled to statstcally analyze clock trees, they cannot be drectly used to analyze clock meshes due to the exstence of large number of loops n them. Model order reducton (MOR) technques have been used n [2], [3], [4] to model varabltes of wre wdth wth frst order approxmaton. However, applyng MOR on clock meshes may not be effcent, due to the exstence of large number of nput/output termnals n clock meshes. Moreover, the mpact of the varatons on the nput-sgnal arrval tmes s not addressed. Of course, we could stll rely on numercal Monte Carlo smulatons for the tmng uncertanty analyss of clock meshes. However, the number of Monte Carlo smulatons requred to mprove accuracy grows quadratcally [5]. Thus, the run tme of Monte Carlo smulatons can be prohbtvely hgh for large nstances of clock meshes. Even when effcent technques, such as the sldng wndow scheme [6], [1], are used, the sze of the clock meshes that can be handled s stll too small (e.g., 16 16). In ths work, we propose a frequency-doman technque for statstcal tmng analyss of clock meshes. A frst, but farly ncremental contrbuton of ths work s that we extend the frequency-doman analyss technque n [7], whch was appled to clock trees, to handle clock meshes. The underlyng prncple of the frequency-doman analyss technque of clock Ths work was supported n part by the Natonal Scence Foundaton under Grant Number CCR meshes s based on the fact that clock sgnals are perodc and that the steady-state crcut response s of nterest n most cases. As any perodc sgnal can be decomposed nto the summaton of a seres of harmonc sgnals, analyzng the clock network (that are usually Lnear-Tme-Invarant) at each harmonc frequency separately wth the standard phasor analyss technque [8], [9] and combnng them together wll gve the steady state response. Usng a transmsson lne model and suffcent number of harmoncs, clock mesh analyss n frequency-doman can be very accurate. As the frequency-doman analyss technque s analytcal, the clock snk arrval tmes can be expressed analytcally n terms of the clock network parameters (wre wdth) and the nput sgnal arrval tmes. Our man contrbutons of ths work are: (1) We formulate a second-order tmng model of the snk sgnal arrval tmes wth respect to the wre wdth and nput sgnal arrval tmes. (2) Wth the lnear canoncal form used to model the varatons n the mesh wre wdth and the nput sgnal arrval tme, the mean and covarance expressons of the arrval tmes of the clock snks can be obtaned analytcally and computed effcently. (3) Compared wth Monte Carlo smulatons, the proposed statstcal tmng analyss technque s orders of magntude faster whle achevng hgh accuracy. The errors n mean and standard devaton are respectvely less than 1% and 3%. II. PRELIMINARIES A. Clock mesh structure Clock nput (a) I(t)=Vs(t)G (b) Fg. 1. A general clock mesh wth sze 8 8. The voltages of the I/O nodes (drve ponts and snk ponts) are of nterest /07/$ IEEE 334

2 In Fgure 1, we show a smple 8 8 general clock mesh. In general, we assume that the sze of the clock mesh s n x n y wth n = n x n y nodes and n e = 2n n x n y edges throughout the paper. We refer to the mesh nodes connected to the output of the nput drvers and flp-flops (or clock drver nputs of the next level clock network) as drve ponts and snks ponts, respectvely. Both the drve ponts and the snks are consdered as I/O nodes/termnals whose voltages are of nterest n ths paper. If a node s drven by a clock drver wth a drver resstor R and a voltage source Vs (t) n seres, we represent ts Norton equvalent crcut wth a conductor G = 1/R and a current source I = Vs G connected n parallel, as n Fgure 1. Otherwse, t s represented by a zero current source n parallel wth an nfnte resstor. These currents are consdered as the nputs. Wth ths formulaton, the number of currents nputs s equal to the number of I/O voltages. Consequently, the number of knowns and unknowns are equal, and thus the correspondng system (of lnear equatons) has a unque soluton. The mesh edges are modeled as transmsson lnes wth the reference pns grounded (not shown n the fgure). We assume that the wre wdth of each segment s unform for ease of presentaton, although the proposed model can be easly extended to analyze meshes wth non-unform wre wdths. B. Canoncal form of causal random varables In ths work, we consder two types of random varatons that drectly affect the tmng propertes of clock meshes: the varatons n the wre wdths of the mesh edges and the varatons n the arrval tmes of the nputs clock sgnals. These two sets of random varatons are affected by varatons on process, temperature, power supply, etc. For smplcty, we assume these two sets of random varables are ndependent from each other, normally dstrbuted and can be expressed as the lnear combnatons of a set of ndependently dstrbuted Gaussan random varables wth zero-mean and unt-varance. We denote vector s =[s 1,s 2,..., s n ] T as the nput clock sgnal arrval tmes. It can be expressed n the lnear canoncal form [10]: s = s 0 + PX, (1) where s 0 s the nomnal value of the nput clock sgnal arrval tme, P s the coeffcent matrx, and X s the random varable vector wth zero-mean and unt-varance. P can be obtaned by applyng PCA (prncpal component analyss) [11]. It s not dffcult to derve that the mean vector E(s) =s 0 and the covarance matrx cov(s) =PP T. Smlarly, we denote vector W =[w 1,w 2,..., w ne ] T as the mesh wre wdths wth W = W 0 + QY, (2) where Q s the matrx composed of egen vectors, and Y s the random prncpal components vector wth zero-mean and untvarance. The mean vector E(W )=W 0, and the covarance matrx cov(w )=QQ T. Fg V 1 Z (k) Y (k) /2 Y (k) /2 - - The π-model of transmsson lne III. CLOCK NETWORK FREQUENCY-DOMAIN ANALYSIS WITH TRANSMISSION LINE MODEL A. Frequency-doman analyss of clock mesh We assume that the nput clock sgnals are square-wave sgnals wth 50% duty, rsng/fallng tme τ and perod T. The physcal tme s represented by t, and the sgnal arrval tmes of the nput current stmulus sgnals are represented by vector s =[s 1,s 2,..., s n ] T. Let dag(v) denotes a dagonal matrx wth the dagonal beng the vector v. Then, wth ω 0 =2π/T beng the fundamental angular frequency of the perodc clock sgnals, the stmulus current vector can be expanded nto Fourer seres n complex exponental vector form [12] I(t )= dag[exp(jkω o (t s))] I (k), (3) k=0,±1,±2,±3... where kω o s the k-th order harmonc angular frequency; exp(jkω o t ) a coeffcent vector and exp(jkω o (t s)) = [exp(jkω o (t s 1 )), exp(jkω o (t s 2 )),..., exp(jkω o (t s n ))]; and I (k) = [I (k) 1,I(k) 2,..., I n (k) ] T the frequencydoman stmulus current vector. For k 0, I (k) = 2TG V dd sn(kω 0 τ/2)/(τk 2 π 2 )j [13] and I (0) = G V dd /2. As the nput stmulus current sgnals are perodc, the I/O node voltage sgnals are also perodc, and thus can also be expanded nto Fourer seres. The complex-exponental Fourer seres expanson of the node voltages n vector form s V (t )= dag[exp(jkω o t )] V (k), (4) k=0,±1,±2,±3... where V (k) =[V (k) 1,V (k) 2,..., V n (k) ] T s the frequency-doman node voltage vector. FortheclockmeshstructureshownnFg.1andforthe k-th harmonc, V (k) can be obtaned by solvng the complex lnear equaton: G (k) V (k) = I (k), (5) where G (k) s the admttance matrx. For clock mesh, G (k) s symmetrc and sparse, and has so-called block tr-dagonal structure. The arrval tmes of the clock snk sgnals can be obtaned by lettng V (t) = 0.5V DD and solvng for t. Typcally, only the frst few harmoncs are requred to acheve suffcent accuracy [7]. + V 2 335

3 B. The admttance matrx wth transmsson lne model To acheve hgh accuracy, we model the nterconnects on the clock meshes as transmsson lnes. We use the transmsson lne lumped π model (shown n Fgure (2)) that s often used n power system analyss [14] to model the clock mesh edges. Consder the -th mesh edge whose wrelength s l,forthe k-th harmonc frequency, the nductve mpedance across the edge s Z (k) = c, snh(γ(k) l ) (6) and the capactve admttance at each end of the edge s Y (k) /2 =1/ c, tanh(γ(k) l /2), (7) where c, = (R + jkωl )/(jkωc ) and γ (k) = (R + jkωl )(jkωc ) are respectvely the characterstc mpedance and propagaton functon of the -th mesh edge correspondng to the k-th harmonc frequency. We assume that R = R 0 /W, L = L 0 /W,andC = C 0 W, wth W beng the wre wdth of the -th mesh edge, and R 0, L 0,andC 0 beng the unt-wdth wre resstance, nductance and capactance, respectvely. The propagaton functon s then ndependent of wre wdth. The characterstc mpedance can be expressed as: c, = c,0 /W, (8) where c,0 = (R 0 + jkωl 0 )/(jkωc 0 ) s the unt-wdth characterstc mpedance of a wre at the k-th harmonc frequency. Let A be the adjacency matrx and G s the drver conductance vector. Defne A to be a matrx obtaned from A by takng the absolute value of each entry of A. (5) can then be wrtten n the followng form: A dag(g (k) E ) AT V (k) + dag(v (k) ) A G (k) D + dag(g s )V (k) = I s (k), (9) where G (k) E =[ W 1 c,0 snh(k) (γ (k) 1 l 1),..., W ne c,0 snh(k) (γ n (k) e l ne ) ]T (10) s the admttance vector for the edges and G (k) D =[W 1 tanh (k) (γ (k) 1 l 1/2),..., W n e tanh (k) (γ n (k) e l ne /2) c,0 c,0 (11) s the admttance vector for the end ponts of the edges. (9) mples that the snk voltage sgnal s a functon of the edge wre wdth, whch s useful for dervng the frst order approxmaton of the snk clock voltage sgnal as a functon of the edge wre wdth. Ths wll be dscussed n Secton IV. IV. MODELING OF THE CLOCK ARRIVAL TIME In ths work, we assume that the snk arrval tmes are lnearly dependent on the wre wdths. Ths assumpton s justfable because the wre wdth varatons are small as compared wth ther nomnal values. However, as the nput sgnal arrval tmes may have large random varatons, the ] T arrval tmes of the snks should be modeled as second-order polynomal functons of the nput arrval tmes. A. Frst order dervatves (Jacoban matrx) We denote v(t ) as the vector of clock snk voltage sgnals n tme doman (note: t s the physcal tme). For ease of explanaton, we defne a vector functon F (t ) v(t ) 0.5V dd. When the snk voltage sgnals cross the V dd /2 pont, we have F (t) =V (t) 0.5V dd =0. (12) Note that t =[t 1,t 2,..., t n ] T s the soluton of the above equaton, and s the vector of the arrval tmes of the snk sgnals. Snce V (t) s the functon of both t and nput sgnal arrval tme s, t s clear from (12) that vector t s an mplct functon of vector s. Accordng to the mplct functon theorem [15], [16], the Jacoban matrx D s t can be obtaned by takng the dervatve of (12) wth respect to s and t, respectvely. We then have D s t = (D t F ) 1 D s F, (13) where D t F = D t V (t) s the Jacoban matrx that corresponds to the frst-order dervatves of F wth respect to t and D t F = dag{ jkω 0 dag[exp(jkω 0 t)]v (k) }, (14) wth exp(jkω 0 t)=[exp(jkω 0 t 1 ),..., exp(jkω 0 t n )] T.Obvously, D t F s a dagonal matrx, and ts nverse can be easly obtaned. D s F s the Jacoban matrx that corresponds to the frst-order dervatves of F wth respect to s. From (3), (4) and (5), we can easly obtan D s F = jkω 0 dag[exp(jkω 0 t)] (G (k) ) 1 dag[i (k) ]. (15) Smlar to (13), D W t, the Jacoban matrx of snk sgnal arrval tmes v.s. the wre wdths, can also be obtaned. B. Second order dervatves: Smlarly, we can obtan the second-order dervatves of t wth respect to s. From (13), takng dervatve of D s t wth respect to s (the nput sgnal arrval tme of the -th node) requres takng dervatves of D t F and D s F. The dervatve of the Jacoban matrx D t F wth respect to s s: D t F = (jkω 0 ) 2 dag[exp(jkω 0 t)]{i (k) dag[(g (k) ) 1 (:,)] + dag[v (k) ] dag[d s t(:,)]}, (16) where D s t(:,) and G (k) ) 1 (:,) denote the -th columns of the matrx D s t and G (k) ) 1, respectvely. The dervatve of the Jacoban matrx D s F wth respect to s s: 336

4 D s F = (jkω 0 ) 2 dag[exp(jkω 0 t)]{(g (k) ) 1 dag[0, 0,..., I (k),..., 0] T + dag[d s t(:,)](g (k) ) 1 dag[i (k) ]}. (17) Apply chan-rule of dervatves, the dervatve of the Jacoban matrx D s t wth respect to s s D s t = (D t F ) 1 [ D s F (D t F ) 1 D t F D s F ]. s s s (18) C. Tmng model of the arrval tme of the snk sgnals Wth both the arrval tmes vector of the nput sgnals s and the mesh wre wdths vector W consdered as varables, the fnal second-order tmng model of the arrval tmes vector of the snk sgnals t s: t = t 0 + D W t Q Y + D s t P X + 1/2 (s s,0 ) D s t (s s 0 ). (19) s Note that s s,0 s a scalar and s s 0 =[s 1 s 1,0,..., s n s n,0 ] s a vector. The statstcal tmng analyss of clock meshes s based on ths second-order tmng model. V. STATISTICAL ANALYSIS OF THE CLOCK SIGNAL ARRIVAL TIMES Wth the second-order tmng model (19) obtaned n the secton IV, the mean and covarance of the arrval tmes vector t of the snks sgnals can be explctly expressed n terms of the means and (co)varances of W (the clock mesh wre wdth) and s (the arrval tmes of the nput sgnals). A. The mean vector By takng expectaton of the second-order tmng model (19), usng (18) and the fact that E(X) =0and E(Y )=0, the mean vector of the arrval tmes of snk sgnals t s E(t) = 1/2(D t F ) 1 [ D s F cov(s,s) =1 (D t F ) 1 D t F D s F cov(s,s)] + t 0, (20) where cov(s,s) = [E[(s s,0 )(s 1 s 1,0 )],..., E[(s s,0 )(s n s n,0 )]] T, = =1 s D s F cov(s,s) (jkω) 2 dag[exp(jkω 0 t)]{(g (k) ) 1 (I (k) σ 2 s ) + {D s t [(G (k) ) 1 dag[i (k) ] PP T ]}[1, 1,..., 1] T },(21) and = =1 s D t F D s F cov(s,s) (jkω) 2 dag[exp(jkω 0 t)] {{[(G (k) ) 1 (D s F P P T )] I s (k) } + dag[v (k) ] [D s t (D s F P P T )][1, 1,..., 1] T }. (22) Note that σs 2 = E((s 1 s 1,0 ) 2, (s 2 s 2,0 ) 2,..., (s n s n,0 ) 2 ) T s the varance vector of s, and denotes the Hadamard product. B. The covarance matrx The accuracy requrement for the covarance values s usually not very hgh. Moreover, the second-order dervatve terms n the tmng model (19), as compared wth the frst order dervatve, s always very small. In ths work, we therefore choose to gnore the second-order terms of the tmng model (19) and keep only the frst-order terms. In other words, we use the followng lnear canoncal form of the tmng model to estmate the covarance: t = t 0 + D W t Q Y + D s t P X. (23) As X and Y are assumed mutually ndependent and have zero-mean and unt-varance, the covarance of the arrval tme of the snk sgnal s: cov(t) = [t 0 E(t)][t 0 E(t)] T + D s tp P T D s t T + D W tqq T D W t T, (24) where E(t) s determned by (20). Note that ths covarance expresson s a theoretcal expresson. In practce, t s not necessary to compute all of the n 2 entres of the covarance matrx. VI. IMPLEMENTATION AND COMPLEXITIES The major steps of computng the mean and varance of the snk sgnal arrval tme vector t are outlned below: 1) Generate G (k) for k =1to K. 2) Calculate the nomnal V (k) and t from (5). 3) Calculate D s t P (and D W t Q) based on (13). 4) Calculate the varance of t based on (24). 5) Calculate the mean of t from (20), (21) and (22). As (G (k) ) 1 s a full matrx wth O(n 2 ) entres, t s not effcent to calculate (G (k) ) 1 drectly. In ths work, (G (k) ) 1 s mostly used to left-multply wth another vector or matrx. Ths s equvalent to solve one or a few matrx equatons wth the same matrx G (k). We therefore use PLU (permuted LU decomposton) to obtan the L and U factorzatons and apply back-substtuton to solve these matrx equatons as they share the same L and U. Let the tme complexty of PLU factorzaton and back substtuton are O(n α ) and O(n β ), respectvely, then the tme complexty of performng steps 2,3 and4so(k n α +K n p n β ) where n p s the summaton of the sze of the vector X and the sze of vector Y. Typcally, 1 <β<α<2, α s close to 1.5 and β s close to

5 In step 5, Hadamard products of matrces (G (k) ) 1 and D s t are nvolved n (21) and (22). Because (G (k) ) 1 and D s t are full matrces, the tme complexty of each Hadamard products s O(n 2 ). However, due to the RC low-pass flter effect, whch nduces the exponental attenuaton [6], the entres n the matrx (G (k) ) 1 and D s t decrease very quckly from the man dagonal. Both (G (k) ) 1 and D s t can be approxmated wth sparse matrces wth O(n) entres. In ths work, we use SPAI (Sparse Approxmate Inverson) algorthm [17] to obtan the sparse approxmate nverse of G (k). The sparse approxmaton of D s t can be computed drectly from (13). Whle the error due to ths approxmaton s less than 0.1%, the tme complexty of the Hadamard products can be reduced to O(n) from O(n 2 ). The tme complexty for performng step 5 s therefore O(Kn β n p ). VII. EXPERIMENTAL RESULTS The proposed frequency-doman statstcal tmng analyss technque for the calculaton of the mean vector and the varance vector of the snk arrval tmes t s mplemented n a MATLAB (R2006b) scrpt and performed on an Intel Core 2 Duo E6600 (3.0GHz) computer wth 2GB memory. For smplcty, the I/O nodes (the drve ponts and the snks) of the meshes are always on the grd ponts, although t s very easy to extend the proposed methods to more general meshes. For nstances of clock meshes larger than 20 20, the matrces P and Q used n (1) are generated wth PCA based on the lnear correlaton assumpton [18]. 200 prncpal components for both nput sgnal arrval tme varatons and the wre wdth varatons are used. The clock frequency s 5GHz, wth rse/fall tme of 10ps. In order to verfy the accuracy of the frequency-doman modelng technque, we create a clock mesh, and measure the arrval tmes at some snk nodes. We model the edge segments between snk nodes wth one and 10 π-type RLC crcuts respectvely, and perform HSPICE smulatons to obtan the snk arrval tmes. The arrval tme results obtaned wth 10 π-type RLC models are treated as accurate. The nterconnect parameters used for the edge segments between snk nodes are R=23Ω, L=0.1nH, and C=0.25pF. The frequency-doman modelng technque s then appled wth the number of harmoncs varyng from 1 to 99. Fgure (3) shows the arrval tme comparson wth dfferent methods used. As shown n ths fgure, the delay result of the frequency-doman technque converges to the accurate value very quckly wth the ncrease of the number of harmoncs. Wth 11 harmoncs used, the delay obtaned by the frequency-doman modelng technque s as accurate as the one π-type model, whch gves less than 1.3% of relatve error. Wth more than 33 harmoncs used, the error s close to 0. Note that the frequencydoman Monte Carlo smulatons can also be easly performed by repeatedly solvng (5) for samples of t. Because of the hgh accuracy of the frequency-doman modelng technque and the long run tme of HSPICE transent smulatons, we use the frequency-doman Monte Carlo smulaton results as the golden reference nstead of the HSPICE Monte Carlo smulaton results. Fg. 3. Delay accuracy comparson wth number of harmoncs used Fg. 4. Trend of mean and standard devaton of snk sgnal arrval tme v.s. # of Monte Carlo runs In order to fnd the approprate number of Monte Carlo smulatons needed for golden reference, we also run some frequency-doman Monte Carlo smulatons wth small nstances of meshes used (as the accuracy of the Monte Carlo smulatons s affected by the number of samples and not the sze of the nstances [5]). σ s s assgned to be 3ps. Fgure 4 shows the trend of the mean and varance of the snk arrval tme wth nput sgnal arrval tme varatons and wre wdth varatons. As the tradeoff between accuracy and speed, we choose 10K MC runs as the golden reference. Wth mesh sze varyng from to and 33 harmoncs used, we compute the mean and the varance of the snk sgnal arrval tmes. The run tme ncludes the tme requred to compute the mean vector and the coeffcent matrx that s used for the calculaton of the varance vector. We dd not compute the covarance matrx, as t s not requred. As 1K Monte Carlo smulatons are commonly used n the lteratures, we lst ts run tme results. For meshes smaller than , we use 10K Monte Carlo results as the reference; for meshes larger than , we use the results of 1K Monte Carlo smulatons as reference. As shown n table I and Fgure 5, the proposed frequency-doman modelng technque for SSTA 338

6 REFERENCES Fg. 5. Maxmum relatve error of the mean and varaton of the proposed method s 50 tmes faster than 1K (500 tmes faster than 10K runs) frequency-doman Monte Carlo smulatons wth relatve mean error of only 1% and relatve standard devaton error of only 3%. Note that the error for meshes larger than s slghtly hgher because we use 1K Monte Carlo smulatons results as the reference, whch may not be accurate. Due to prohbtve computaton cost, we perform HSPICE-based and SWS-based (sldng wndow scheme) Monte Carlo smulatons wth only 100 repettons used, and lst ther run tmes n table I as well. The nterconnect model used s 4 segment π RLC model as a tradeoff between speed and accuracy. The proposed statstcal tmng analyss technque s more than thousands of tmes faster than 100 repettons of HSPICEbased Monte Carlo smulatons. For large nstances of clock meshes, HSPICE fals to smulate the clock meshes, and SWS takes more than a few days to complete the smulatons. mesh sze Proposed 1K M.C. Hspce 100 SWS fal >3 days fal >5 days fal >5 days >5days fal >5 days TABLE I RUN TIME COMPARISON.TIME UNIT: SECOND(S) [1] Subodh M. Reddy, Gustavo R. Wlke, and Rajeev Murga. Analyzng tmng uncertanty n mesh-based clock archtectures. In DATE 06: Proceedngs of the conference on Desgn, automaton and test n Europe, pages , 3001 Leuven, Belgum, Belgum, European Desgn and Automaton Assocaton. [2] Yng Lu, Lawrence T. Plegg, and Andrzej J. Strojwas. Model order-reducton of rc(l) nterconnect ncludng varatonal analyss. In DAC 99: Proceedngs of the 36th ACM/IEEE conference on Desgn automaton, pages , New York, NY, USA, ACM Press. [3] Janet M. Wang, Omar A. Hafz, and Jun L. A lnear fractonal transform (lft) based model for nterconnect parametrc uncertanty. In DAC 04: Proceedngs of the 41st annual conference on Desgn automaton, pages , New York, NY, USA, ACM Press. [4] Peng L, Frank Lu, Xn L, Lawrence T. Plegg, and San R. Nassf. Modelng nterconnect varablty usng effcent parametrc model order reducton. In DATE 05: Proceedngs of the conference on Desgn, Automaton and Test n Europe, pages , Washngton, DC, USA, IEEE Computer Socety. [5] Ashsh Srvastava, Denns Sylvester, and Davd Blaauw. Statstcal Analyss and Optmzaton for VLSI: Tmng and Power, pages Sprnger-Verlag, [6] H. Chen, C. Yeh, G. Wlke, S. Reddy, H. Nguyen, W. Walker, and R. Murga. A sldng wndow scheme for accurate clock mesh analyss. In ICCAD 05: Proceedngs of the 2005 IEEE/ACM Internatonal conference on Computer-aded desgn, pages , Washngton, DC, USA, IEEE Computer Socety. [7] Guoqng Chen and Eby G. Fredman. An RLC nterconnect model based on fourer analyss. IEEE Transactons on Computer-Aded Desgn, 24(2): , February [8] Charles Proteus Stenmetz. De anwendung complexer grssen n der elektrotechnk. In Elektrotechnsche Zetschrft, pages , [9] Davd E. Johnson, Johnny R. Johnson, John L. Hlburn, and Peter D. Scott. Electrc Crcut analyss, pages Prentce Hall, 3rd ed edton, [10] Lzheng Zhang, Wejen Chen, Yuhen Hu, and Charle Chung png Chen. Statstcal statc tmng analyss wth condtonal lnear max/mn approxmaton and extended canoncal tmng model. IEEE Transactons on Computer-Aded Desgn, 25(6): , June [11] Honglang Chang and Sachn S. Sapatnekar. Statstcal tmng analyss consderng spatal correlatons usng a sngle pert-lke traversal. In ICCAD 03: Proceedngs of the 2003 IEEE/ACM nternatonal conference on Computer-aded desgn, page 621, Washngton, DC, USA, IEEE Computer Socety. [12] Alan V. Oppenhem, Alan S. Wllsky, S. Hamd, and S. Hamd Nawab. Sgnals and Systems. Prentce Hall, 2nd ed edton, [13] T. Tang and E. G. Fredman. Lumped versus dstrbuted rc and rlc nterconnect mpedance. In Proc. IEEE Mdwest Symp. Crcuts System, pages , [14] A. R. Bergen and V. Vttal. Power system analyss. Prentce Hall, [15] Frank Warner. Foundatons of Dfferentable Manfolds and Le Groups. Sprnger Scence+Busness Meda, [16] James Munkres. Analyss on manfolds. Addson-Wesley, [17] Marcus J. Grote and Thomas Huckle. Parallel precondtonng wth sparse approxmate nverses. SIAM Journal on Scentfc Computng, 18(3): , [18] Paul Fredberg, Yu Cao, Jason Can, Ruth Wang, Jan Rabaey, and Costas Spanos. Modelng wthn-de spatal correlaton effects for process-desgn co-optmzaton. In ISQED 05: Proceedngs of the 6th Internatonal Symposum on Qualty of Electronc Desgn, pages , Washngton, DC, USA, IEEE Computer Socety. VIII. CONCLUSION By usng frequency-doman modelng technque, we establsh a second-order tmng model to analyze the tmng of clock meshes. Based on ths tmng model, we perform statstcal tmng analyss on clock meshes and compute the mean and varance of the arrval tmes of the snk sgnals. Experments result shows that the proposed technque s accurate and effcent. 339