/04/$ IEEE. 2

Transcription

1 Asymptotc Probablty xtracton or on-ormal Dstrbutons o Crcut Perormance Xn L, Jayong Le, Padmn Gopalarshnan and Lawrence. Plegg Dept. o C, Carnege ellon Unversty 5 Forbes Avenue, Pttsburgh, PA 53, USA {xnl, jayongl, pgopala, plegg}@ece.cmu.edu Abstract Whle process varatons are becomng more sgncant wth each new IC technology generaton, they are oten modeled va lnear regresson models so that the resultng perormance varatons can be captured va ormal dstrbutons. onlnear (e.g. quadratc) response surace models can be utlzed to capture larger scale process varatons; however, such models result n non-ormal dstrbutons or crcut perormance whch are dcult to capture snce the dstrbuton model s unnown. In ths paper we propose an asymptotc probablty extracton method, APX, or estmatng the unnown random dstrbuton when usng nonlnear response surace modelng. APX rst uses a novel bnomal moment evaluaton to ecently compute the hgh order moments o the unnown dstrbuton, and then apples moment matchng to approxmate the characterstc uncton o the random crcut perormance by an ecent ratonal uncton. A smple statstcal tmng example and an analog crcut example demonstrate that APX can provde better accuracy than onte Carlo smulaton wth 4 samples and acheve orders o magntude more ecency. We also show the error ncurred by the popular ormal modelng assumpton usng standard IC technologes.. Introducton As IC technologes are scaled to the deep sub-mcron regon, process varatons are becomng crtcal and sgncantly mpact the overall perormance o a crcut. able shows some typcal process parameters and ther 3 varatons as technologes are scaled rom.5 µm to 7 nm. hese large-scale varatons ntroduce uncertantes n crcut behavor, thereby mang IC desgn ncreasngly dcult. Low product yeld or unnecessary over-desgn cannot be avoded these process varatons are not accurately modeled and analyzed wthn the IC desgn low. able. echnology parameters and 3 varatons [] Le (nm) ox (nm) Vth (mv) W (µm) H (µm) Durng the past decade, varous statstcal analyss technques []-[7] have been proposed and utlzed n many applcatons such as statstcal tmng analyss, msmatch analyss, yeld optmzaton, etc. he objectve o these technques s to model the probablty dstrbuton o the crcut perormance under random process varatons. he author n [] apples lnear regresson to approxmate a gven crcut perormance (e.g /4/$. 4 I. delay, gan, etc.) as a uncton o the process varatons (e.g. Vth, ox, etc.), and assumes that all random varatons are normally dstrbuted. As such, the perormance s also a ormal dstrbuton, snce the lnear combnaton o normally dstrbuted random varables stll has a ormal dstrbuton [8]. he lnear regresson model s ecent and accurate when the process varatons are sucently small. However, the large-scale varatons n the deep sub-mcron technologes, whch can reach 35% as shown n able, suggest applyng hgh order regresson models n order to guarantee hgh approxmaton accuracy [4]-[7]. Usng a hgh order response surace model, however, brngs about new challenges due to the nonlnear mappng between the process varatons and the crcut perormance. he dstrbuton o s no longer ormal, as s the case or the lnear regresson model. he authors n [3]-[5] utlze onte Carlo smulaton to evaluate the probablty dstrbuton o, whch s computatonally expensve. ote that the computaton cost or ths probablty extracton s crucal, especally when the extracton procedure s an nner loop wthn the optmzaton low. In ths paper, we propose a novel Asymptotc Probablty Xtracton approach, APX, or estmatng the unnown PDF/CDF unctons usng nonlnear response surace modelng. Gven a crcut perormance (e.g. a dgtal crcut path delay or the perormance parameter o an analog/rf crcut), the response surace modelng approxmates as a polynomal uncton o the process parameters (e.g. Vth, ox, etc.). Snce the process parameters are modeled as random varables, the crcut perormance s a uncton o these random varables, whch s also a random varable. APX apples moment matchng to approxmate the characterstc uncton o (.e. the Fourer transorm o the probablty densty uncton [8]) by a ratonal uncton H. We conceptually consder H to be o the orm o the transer uncton o a lnear tme-nvarant (LI) system, and the probablty dstrbuton uncton (PDF) and the cumulatve dstrbuton uncton (CDF) o are approxmated by the mpulse response and the step response o the LI system H, respectvely. he resultng probablty dstrbuton uncton can then be used to characterze and/or optmze the statstcal perormance o analog and dgtal crcuts under process varatons. APX extends exstng moment matchng methods va three mportant new contrbutons whch sgncantly reduce the computaton cost and mprove the approxmaton accuracy or ths partcular applcaton. Frstly, a ey operaton requred by APX s to compute the hgh order moments, whch s extremely expensve when usng tradtonal technques. In APX, we propose a bnomal evaluaton scheme to recursvely compute the hgh order moments or a gven quadratc response surace model. he bnomnal moment evaluaton s derved rom statstcal ndependence theory and prncpal component analyss (PCA) methods. It can acheve more than 6 x speedup compared wth drect moment evaluaton. Secondly, APX approxmates the unnown probablty dstrbuton uncton by the mpulse response o an LI system. Drectly applyng such an approxmaton to any crcut perormance wth negatve values s neasble, snce t results n

2 an LI system that s non-causal. o overcome ths dculty, APX apples a generalzed Chebyshev nequalty or PDF/CDF shtng. Lastly, the best-case perormance (e.g. the % pont on CDF) and the worst-case perormance (e.g. the 99% pont on CDF) are two mportant metrcs to be evaluated. Drect moment matchng cannot capture the % pont value accurately snce the moment matchng approxmaton s most accurate or low requency components (correspondng to the nal values o CDF), and least accurate or hgh requency components (correspondng to the ntal values o CDF). o address ths problem, a reverse evaluaton technque s proposed n ths paper to produce an accurate estmaton o the % pont. he remander o the paper s organzed as ollows. In Secton we revew the bacground on response surace modelng. hen we propose our APX approach n Secton 3. We dscuss several mplementaton ssues, ncludng the hgh order moment evaluaton, PDF/CDF shtng and reverse PDF/CDF evaluaton, n Secton 4. he ecacy o APX s demonstrated by several crcut examples n Secton 5. Fnally, we conclude n Secton 6.. Bacground Gven a crcut topology, the crcut perormance (e.g. gan, delay) s a uncton o the desgn parameters (e.g. bas current, transstor szes) and the process parameters (e.g. V H, OX ). he desgn parameters are optmzed and xed durng the desgn process; however, the process parameters must be modeled as random varables to account or any uncertan varatons. Gven a set o xed desgn parameters, the crcut perormance can be approxmated by a lnear regresson model []: where X x, x,, X X B X ˆ () x denotes the process parameters, X s the mean value o X, X X X represents the process varatons, B ˆ R stands or the lnear model coecents and s the total number o these random varatons. he process varatons n (),.e. X, are oten approxmated by zero-mean ormal dstrbutons *. In addton, correlated process varatons can be expressed n terms o ndependent actors usng prncpal component analyss (PCA) [9]. Gven a set o normally dstrbuted random varables X and ther symmetrc, postve sem-dente correlaton matrx R, PCA decomposes R as: R V V () where dag,,, contans the egenvalues o R, and V V, V,, V contans the correspondng egenvectors that are orthonormal,.e. V V I (I s the dentty matrx). Based on and V, PCA denes a set o new random varables:. 5 Y V X (3) It s easy to very that the random varables Y are ndependent and satsy the ormal dstrbuton, (.e. zero mean and unt standard devaton). he actors extracted rom PCA can be nterpreted as coordnate rotatons o the space dened by the orgnal random varables. In addton, the magntude o the egenvalues deceases qucly, t s possble to use a small number o prncpal components to approxmate the orgnal -dmensonal space. * I a process parameter satses a log-ormal dstrbuton, t can also be transormed to a ormal dstrbuton by tang the logarthmc operator,.e. ln() s normally dstrbuted. 3 ore detals on PCA can be ound n [9]. Substtutng (3) nto () yelds: Y C B Y (4).5 where C X and B V Bˆ. he lnear regresson model n (4) s accurate when the process varatons are small. However, the large-scale varatons that are expected or nanoscale technologes suggest that applyng quadratc response surace models mght be requred to provde sucent accuracy: Y C B Y Y AY (5) In (5), C R s the constant term, B R represents the lnear coecents and A R denotes the quadratc coecents. Wthout loss o generalty, we assume that A s symmetrc n ths paper, snce any asymmetrc quadratc orm can be easly converted to an equvalent symmetrc orm []. 3. Asymptotc Probablty xtracton Gven the quadratc response surace model n (5), the objectve o probablty extracton s to estmate the unnown probablty densty uncton pd and cumulatve dstrbuton uncton cd or perormance. Instead o runnng expensve onte Carlo smulatons, APX tres to nd an -th order LI system H whose mpulse response h t and step response s t are the optmal approxmatons or the pd and cd respectvely. Here, the varable t n h t and s t corresponds to the varable n pd and cd. he optmal approxmaton s determned by matchng the rst moments between h t and pd or an -th order approxmaton. We rst descrbe the mathematcal ormulaton o APX n Secton 3.. hen, n Secton 3. we wll ln APX to tradtonal probablty theory and explan why t can be used to ecently approxmate PDF/CDF unctons. 3. athematcal Formulaton We dene the tme moments [] or a gven crcut perormance whose probablty densty uncton s pd as ollows: m pd d (6)! In (6), the denton o tme moments s dentcal to the tradtonal denton o moments n probablty theory except or the scalng actor!. Smlarly, tme moments can be dened or an LI system H []. Gven an -th order LI system whose transer uncton and mpulse response are: b a t a e t H s and h t (7) s b t he tme moments o H are expressed as []: m! a t h t dt (8) b b,,,, and resdues In (7), the poles,,, a, are the unnowns that need to be determned. atchng the rst moments n (6) and (8) yelds the ollowng nonlnear equatons:

3 a a a m b b b a a a m b b b (9) a a a m b b b he nonlnear equatons n (9) can be solved usng the algorthm proposed n []. Once the poles b and resdues a have been determned, the probablty densty uncton pd s optmally approxmated by h t n (7), and the cumulatve dstrbuton uncton cd s optmally approxmated by the step response: t a bt e t s t h d b () t It s worth notng that many mplementaton ssues must be consdered to mae our proposed approach, APX, easble and ecent. For example, the mpulse response o a causal LI system s only nonzero or t, but a PDF n practcal applcatons can be nonzero or. In secton 4, we wll propose several schemes to address these problems. he aorementoned moment-matchng method was prevously appled or IC nterconnect order reducton [], [] and s related to the Padé approxmaton n lnear control theory [3]. In the ollowng secton, we wll explan why ths momentmatchng approach s an ecent way to approxmate PDF/CDF unctons. 3. Connecton to Probablty heory In probablty theory, gven a random varable whose probablty densty uncton s pd, the characterstc uncton s dened as the Fourer transorm o pd [8]. j j pd e d pd d ()! Substtutng (6) nto () yelds: m j () hs mples an mportant act: the tme moments dened n (6) are related to the aylor expanson o the characterstc uncton at the expanson pont. atchng the rst moments n (9) s equvalent to matchng the rst aylor expanson coecents between the orgnal characterstc uncton and the approxmated ratonal uncton H s. o explan why the moment-matchng approach s ecent, we rst need to show two mportant propertes o the characterstc uncton [8]: Property : A characterstc uncton has maxmal magntude at.,.e. Property : A characterstc uncton when. Fg. shows the characterstc unctons or several typcal random dstrbutons. he above two propertes mply an nterestng act: namely, gven a random varable, ts 4 characterstc uncton decays as ncreases. hereore, the optmally approxmated H s n (7) s a low pass system. It s well-nown that a aylor expanson s accurate around the expanson pont. Snce a low-pass system s manly determned by ts behavor n the low-requency band (around ), t can be accurately approxmated by matchng the rst several aylor coecents at,.e. the moments. hs concluson has been vered n other applcatons (e.g. IC nterconnect order reducton [], []) and t provdes the theoretcal bacground to explan why moment-matchng wors well or the PDF/CDF evaluatons that we wll demonstrate n Secton 5. Fg.. () normal cauchy ch-square gamma Characterstc uncton or typcal dstrbutons. 4. Implementaton o APX Our proposed APX approach s made practcally easble by applyng several novel algorthms, ncludng: ) a bnomal scheme or hgh order moment computaton, ) a generalzed Chebyshev nequalty or PDF/CDF shtng and 3) a reverse evaluaton technque or best-case/worst-case analyss. In ths secton, we descrbe the mathematcal ormulaton o each o these algorthms. 4. Bnomal oment valuaton A ey operaton requred n APX s the computaton o the hgh order tme moments dened n (6) or a gven random varable. Such a moment evaluaton s equvalent to computng the expectaton o,,,,. Gven the quadratc response surace model n (5), s a hgh order polynomal n Y: Y c y y y (3) where y s the -th element n the vector Y, c s the coecent o the -th product term and j s the postve nteger exponent. Snce the random varables Y are ndependent ater PCA analyss, we have: c y y y (4) where stands or the expectaton. In addton, remember that each random varable y has a ormal dstrbuton,, whch yelds [8]: y,3,5, (5) 3,4,6, Substtutng (5) nto (4), the expectaton o can be determned. he above computaton scheme s called drect moment

4 evaluaton n ths paper. he ey dsadvantage o the drect moment evaluaton s that, as ncreases, the total number o the product terms n (4) wll ncrease exponentally, thereby qucly mang ther computaton neasble. o overcome ths dculty, we propose a novel bnomal moment evaluaton scheme that conssts o two steps: quadratc model dagonalzaton and moment evaluaton. he bnomal moment evaluaton scheme recursvely computes the hgh order moments, nstead o explctly constructng the hgh order polynomals n (4). A. Quadratc odel Dagonalzaton he rst step n bnomal moment evaluaton s to remove the cross product terms n the quadratc response surace model (5), thereby yeldng a much smpler, but equvalent, quadratc model. Accordng to matrx theory [], any symmetrc matrx A R can be dagonalzed as: A U U (6) where dag,,, contans the egenvalues o A and U U, U,, U s an orthogonal matrx (.e. U U I ) contanng the egenvectors. Dene new random varables Z as ollows: Z U Y (7) Substtutng (7) nto (5) yelds: where z Z C Q C q z z Z Z Z (8) s the -th element n the vector Z and q, q, q U B Q,. quaton (8) mples that there s no cross product term n the quadratc model ater dagonalzaton. In addton, the ollowng theorem guarantees that the random varables Z dened n (7) are stll ndependent and satsy the. ormal dstrbuton, heorem : Gven a set o ndependent random varables Y wth the ormal dstrbuton, and an orthogonal matrx U, the random varables Z dened n (7) are ndependent and satsy,. the ormal dstrbuton Proo: Snce the random varables Z are lnear combnatons o normally dstrbuted random varables Y, they are normally dstrbuted. he correlaton matrx or Z s gven by: Z Z U Y Y U U Y Y (9) U Remember that Y s a set o ndependent random varables wth a ormal dstrbuton,,.e. Y Y I orthogonal,.e. U U I. hus, we have: Z Z U Y Y U U I U I, and matrx U s () quaton () mples that the random varables n Z are uncorrelated. In addton, uncorrelated random varables wth ormal dstrbutons are also ndependent [8]. B. oment valuaton We now demonstrate the use o the smpled quadratc model (8) or ast moment evaluaton. Based on (8), we dene a set o new random varables: h C l g q l z g C z l q z z () Comparng () wth (8), t s obvous that when l, h. Instead o computng the hgh order moments o drectly, the proposed bnomal moment evaluaton scheme successvely computes the moments o h l, as shown n Fg... Start rom h C and compute h C,,,. Set l.. For each,,,, compute: or each gl ql zl l zl ql l z l () hl hl gl hl g l (3) 3. I l, then go to Step 4. Otherwse, l l and return Step., we have h 4. For each,,,. Fg.. Bnomal moment evaluaton algorthm. Step n Fg. s the ey operaton requred by the bnomal moment evaluaton algorthm. In Step, both () and (3) utlze the bnomal theorem to get the bnomal seres. hereore, we reer to ths algorthm as bnomal moment evaluaton n ths paper. In (), the expectaton can be easly evaluated usng the closed-orm expresson (5), snce z l s normally dstrbuted,. quaton (3) utlzes the property that h l and g l are ndependent, because h l s a uncton o z,,,, l, g l s a uncton o zl and all z are h g h g mutually ndependent. hereore, l l l l where the values o and h l z l, g l have already been computed n prevous steps. he man advantage o the bnomal moment evaluaton s that, unle the drect moment evaluaton n (4), t does not explctly construct the hgh order polynomal. hereore, unle drect moment evaluaton, where the total number o the product terms wll exponentally ncrease, both g l n () and h l n (3) contan at most product terms. Snce,,, and l,,, or an -th order APX approxmaton wth ndependent random varatons, the total number o and g l that need to be computed s h l O. In addton, the matrx dagonalzaton n (6) only 3 needs to be computed once and has a complexty o O. hereore, the computatonal complexty o the proposed 3 algorthm s O O. In most crcut applcatons, s small (around 5~) ater PCA analyss, and selectng 7 ~ provdes sucent accuracy or moment matchng. Wth these typcal values or and, the proposed bnomal moment evaluaton s extremely ast, as we wll demonstrate wth numercal examples n Secton 5. It should be noted that as long as the crcut perormance s represented by the quadratc model n (5) and the process varatons are normally dstrbuted, bnomal moment evaluaton provdes the exact hgh order moment values (except or numercal errors). here s no urther assumpton or 5

5 approxmaton made by the algorthm. In summary, bnomal moment evaluaton utlzes statstcal ndependence theory and prncpal component analyss (PCA) to ecently compute hgh order moment values, whch are requred n moment matchng or probablty extracton. 4. PDF/CDF Shtng Case ot Causal Fg. 3. pd ean µ Illustraton or PDF/CDF shtng. Case Large Delay pd as the APX approxmates the unnown PDF mpulse response h t o an LI system. he mpulse response o a causal system s only nonzero or t, but a PDF n practcal applcatons can be nonzero or. In such cases, we need to rght-sht the unnown response pd, as pd by and use the mpulse h t to approxmate the shted PDF shown n Fg. 3 (case ). In addton, even the unnown PDF pd s zero or all, t can be ar away rom the orgn, as shown n Fg. 3 (case ). As such, the correspondng mpulse response h t presents a large delay n tme doman, whch cannot be accurately captured by a low-order approxmaton. In such cases, we need to let-sht the unnown pd by and use the mpulse response h t to approxmate the shted PDF pd. he above analyss mples that t s crucal to determne the correct value o or PDF/CDF shtng. Over-shtng the unnown PDF to ether let or rght sde can ncrease the approxmaton error. In ths paper, process varatons are modeled as ormal dstrbutons, whch are unbounded and dstrbuted over,. hereore, any crcut perormance represented by the quadratc model n (5) s also unbounded. It s mpossble to completely sht pd to the postve axs. However, snce s a random varable, pd can be letshted by ( s negatve n case o rght-shtng) such that the probablty P s sucently small. As shown n Fg. 3, the PDF/CDF shtng problem can be stated as ollows: nd the value and let-sht pd by, where µ s the mean value o, such that the probablty P s not greater than a gven error tolerance. In addton, we want to select the value to be as small as possble,.e. nd the smallest satsyng P. A small results n a small tmedoman delay n h t and, thereore, hgh approxmaton accuracy or pd. o estmate, we need the ollowng theorem. heorem : Gven a random varable, or any and,4,6,, P (4) where µ s the mean value o. Proo: For any P,4,6,, we have pd d pd d pd d (5) ote that the above proo s not restrcted to any specal probablty dstrbuton. Based on (4), the unnown PDF, we have: P pd s let-shted by P P P (6) where,4,6,. hereore, one sucent condton or P s:,4,6, whch s equvalent to:,4,6, (7) (8) quaton (8) estmates usng hgh order central moments. In an -th order approxmaton, ater the hgh order expectatons,,, are computed by the bnomal moment evaluaton algorthm n Fg., the central moments can be easly calculated usng the bnomal theorem: (9) where. hen, usng (8), an estmated s computed or each,4,,, whch s denoted as. he mnmal value o all these values s utlzed as the nal or PDF/CDF shtng, snce we am to nd the smallest to acheve hgh approxmaton accuracy or pd. It s worth mentonng that when, equaton (4) s the well-nown Chebyshev nequalty [8]. We have generalzed the nd order Chebyshev nequalty to hgher orders and, thereore reer to (4) as the generalzed Chebyshev nequalty. In practcal applcatons we nd that hgh order moments provde a much tghter (.e. smaller) estmaton o, as s demonstrated by the numercal examples n Secton 5. In summary, the proposed generalzed Chebyshev nequalty (4) provdes an eectve way to estmate the boundary or PDF/CDF shtng. As such, the major part o the unnown PDF/CDF can be moved to the postve axs, whch s then accurately approxmated by the mpulse/step response o a causal LI system. 4.3 Reverse PDF/CDF valuaton In many practcal applcatons, such as robust crcut optmzaton [4], [4], the best-case perormance (e.g. the % pont on CDF) and the worst-case perormance (e.g. the 99% pont on CDF) are two mportant metrcs to be evaluated. As dscussed n Secton 3., APX matches the rst aylor expanson coecents between the orgnal characterstc uncton and the approxmated ratonal uncton. Remember that the 6

6 aylor expanson s most accurate around the expanson pont. Accordng to the nal value theorem o the Laplace transorm, accurately approxmatng at provdes an accurate pd at. hs, n turn, mples that the proposed approach can accurately estmate the 99% pont o the random dstrbuton, as shown n Fg. 4. pd Flp or Reverse valuaton pd Accurate or stmatng Accurate or stmatng the % Pont the 99% Pont Fg. 4. Illustraton or reverse PDF/CDF evaluaton. he above analyss motvates us to apply a reverse evaluaton scheme or accurately estmatng the % pont. As shown n Fg. 4, the reverse evaluaton algorthm lps the orgnal pd to pd. he % pont o the orgnal pd now becomes the 99% pont o the lpped pd whch can be accurately evaluated by APX. 4.4 Summary. Start rom the quadratc response surace model n (5) and a gven approxmaton order.. Dagonalze the quadratc model based on (6)~(8) and compute the hgh order expectatons usng the bnomal moment evaluaton algorthm n Fg.. 3. Compute the central moments based on (9). 4. Determne the value o usng (8) and, where µ s the mean value o. 5. Compute tme moments m!, where s smlarly evaluated by replacng µ by n (9). 6. Substtute m nto (9) and solve the problem unnowns a and b. 7. he shted response n (7) and the shted by the step response n (). pd s approxmated by the mpulse cd s approxmated 8. Sht pd and cd bac to cd. pd and Fg. 5. Overall mplementaton o APX. Fg. 5 summarzes the overall mplementaton o APX except or reverse evaluaton. I reverse evaluaton s requred to mprove estmaton accuracy or the % pont, we need to compute the hgh order expectatons n Step and repeat Step 3~8 or computng pd and cd. However, usng reverse evaluaton doesn t requre explctly computng the hgh order expectatons agan. ote that: (3) where the hgh order expectatons have already been calculated n prevous computatons. he algorthm n Fg. 5 s based on a gven approxmaton 7 order. he authors n [] and [] proposed several methods or teratvely determnng based on the approxmaton error. he approxmaton order should be ncreased the error s large. hese methods can also be appled here or APX. In addton, t s worth mentonng that usng an approxmaton order greater than can result n serous numercal problems [], []. In most practcal applcatons, we nd that selectng n the range o 7~ can acheve the best accuracy. 5. umercal xamples In ths secton we demonstrate the ecacy o APX usng several crcut examples. All experments are run on a SU Sparc GHz server. 5. ISCAS 89 S7 A. Response Surace odelng Fg. 6. Longest path n ISCAS 89 S7. We create a physcal mplementaton or the ISCAS 89 S7 benchmar crcut usng the S COS.3 µm process. hs benchmar crcut s smple, but t enables us to mae a ull comparson o APX wth varous PDF/CDF estmaton methods. Gven a set o xed gate szes, the longest path delay n the benchmar crcut (shown n Fg. 6) s a uncton o the process varatons (e.g. Vth, ox, L, etc.). Snce the crcut only conssts o sx gates whch can be put close to each other n the layout, nter-de varaton wll domnate ntra-de varaton, and gate delays wll domnate (local) nterconnect delays n ths example. hereore, or smplcty, we only consder nter-de varatons or COS transstors n ths example. he probablty dstrbutons and the correlaton normaton o the nter-de transstor varatons are obtaned rom the S desgn t. Ater PCA analyss, 6 prncpal random actors are dented to represent these process varatons. We should note, however, that nothng precludes us rom ncludng more detaled ntra-de and/or nterconnect varaton models n APX as well. We approxmate the longest path delay as a uncton o process varatons by a lnear regresson model and a quadratc response surace (.e. second order polynomal) model respectvely. he ttng error s 4.48% or the lnear model and.% or the quadratc model (4x derence). Whle t s worth notng that the lnear modelng error n ths example s not very large, as IC technologes are scaled to ner eature szes, the process varatons wll become relatvely larger, thereby mang the nonlnear terms n the quadratc model even more mportant. B. oment valuaton able compares the computaton tme or drect moment evaluaton and our proposed bnomal moment evaluaton. In drect moment evaluaton, the number o the total product terms ncreases exponentally, thereby mang the computaton tas qucly neasble. Bnomal moment evaluaton, however, s extremely ast and acheves more than 6 x speedup over drect moment evaluaton. In addton, we very that the moment values obtaned rom both approaches are dentcal except or numercal errors.

7 able. Computaton tme or moment evaluaton oment Drect Bnomal Order # o erms me (Sec.) me (Sec.) C. PDF/CDF Shtng As dscussed n Secton 4., PDF/CDF shtng s necessary to mae the proposed APX approach easble and ecent. A ey operaton or PDF/CDF shtng s determnng the value based on (8) (also see Fg. 3). We select an error tolerance 3 n (8). Fg. 7 shows the estmated value usng varous hgh order moments. From Fg. 7, we nd that the hgh order moments provde a much tghter (.e. smaller) estmaton o. However, ater the moment order, urther ncreases n do not have a sgncant mpact on reducng. (ns) Fg oment Order D. PDF/CDF valuaton stmated value usng hgh order moments. been noted n moment matchng o LI models o nterconnect crcuts [], [].. Comparson o Accuracy and Speed able 3 compares the accuracy and speed or three derent probablty extracton approaches: lnear regresson, onte Carlo analyss wth 4 samples, and the proposed APX approach. Several specc ponts on the cumulatve dstrbuton uncton are utlzed or comparng the accuracy. he % pont and the 99% pont, or example, denote the best-case delay and the worst-case delay respectvely. Ater the cumulatve dstrbuton uncton s explctly obtaned n the closed-orm expresson (), the bestcase delay, worst-case delay and any other specc ponts on CDF can be easly ound usng a bnary search algorthm. he error values n able 3 are calculated aganst the exact CDF obtaned by onte Carlo smulaton wth 6 samples. ote rom able 3 that the lnear regresson approach has the largest error. APX acheves more than x speedup over the onte Carlo analyss wth 4 samples, whle stll provdng better accuracy. In ths example, reverse evaluaton on pd reduces the % pont estmaton error by 4x, rom.% to.4%. hs observaton demonstrates the ecacy o the reverse evaluaton method proposed n Secton 4.3. able 3. stmaton error (compared to onte Carlo wth 6 samples) and computaton cost Lnear C ( 4 Runs) APX % Pont.43%.34%.4% * % Pont 4.63%.64%.% 5% Pont 5.76%.47%.3% 5% Pont 6.4%.3%.% 75% Pont 5.77%.5%.% 9% Pont 4.53%.66%.3% 99% Pont.8%.78%.9% Cost (Sec.) Low ose Ampler A. Response Surace odelng Cumulatve Dstrbuton Functon Approx Order = 4 Approx Order = 8 xact Delay (ns) Fg. 8. Cumulatve dstrbuton uncton or delay. Fg. 8 shows the cumulatve dstrbuton uncton usng varous approxmaton orders. In Fg. 8, the exact cumulatve dstrbuton uncton s evaluated by onte Carlo smulaton wth 6 samples. ote that, the CDF obtaned rom the low order approxmaton (Order = 4) s not accurate and contans numercal oscllatons. However, once the approxmaton order s ncreased to 8, these oscllatons are elmnated and the approxmated CDF asymptotcally approaches the exact CDF. Smlar behavor has Fg. 9. Crcut schematc or LA. As a second example we consder a low nose ampler desgned n the IB BCOS.5 µm process, as shown n Fg. 9. In ths example, the varatons on both OS transstors and passve components (capactance and nductance) are consdered. he probablty dstrbutons and the correlaton normaton o these varatons are provded n the IB desgn t. Ater PCA analyss, 8 prncpal actors are dented to represent the process varatons. he perormance o the LA s characterzed by 8 derent speccatons. Gven a set o determned crcut szes, each crcut perormance s a uncton o the process varatons. We approxmate these unnown unctons by lnear regresson models * hs % pont error s computed by usng reverse evaluaton. 8

8 and quadratc response (.e. second order polynomal) models respectvely. able 4 shows the modelng error or all these 8 perormances. In ths example, the quadratc modelng error s 7.5x smaller than the lnear modelng error on average. able 4. Regresson modelng error or LA Perormance Lnear Quadratc F.76%.4% S 6.4%.3% S 3.44%.6% S.94%.34% S 5.56% 3.47% F.38%.3% IIP3 4.49%.9% Power 3.79%.7% B. Comparson o Accuracy and Speed able 5. stmaton error or lower bound (% pont) Perormance Corner Lnear C ( 4 Runs) APX F 5.8%.%.4%.6% S 45.44% 5.78%.47%.9% S 38.87% 3.88%.59%.4% S 6.5%.9%.8%.7% S 3.8%.%.9%.7% F 5.9% 3.7%.%.6% IIP % 5.%.%.33% Power 6.56%.%.47%.9% able 6. stmaton error or upper bound (99% pont) Perormance Corner Lnear C ( 4 Runs) APX F.%.%.3%.5% S 5.53%.4%.74%.8% S 44.64%.6%.5%.8% S 5.63% 4.69%.%.9% S 36.% 5.6%.9%.9% F 7.8% 3.5%.8%.% IIP % 5.93%.%.6% Power 4.53%.4%.53%.% able 7. Computaton cost or statstcal analyss (Sec.) Perormance Lnear C ( 4 Runs) APX F S S S S F IIP Power able 5 and able 6 compare the estmaton accuracy or our derent statstcal analyss approaches: corner smulaton, lnear regresson, onte Carlo analyss wth 4 samples and the proposed APX approach. hese error values are calculated aganst the exact CDF obtaned by onte Carlo smulaton wth 6 samples. he corner smulaton approach computes the best-case and worst-case perormance by enumeratng all process corners,.e. combnng the extreme values o all process parameters. he corner smulaton approach s smple, but t can result n extremely large errors, as shown n able 5 and able 6. Lnear regresson provdes more accurate results than corner smulaton, but the errors are expected to ncrease as IC technologes contnue to scale. APX acheves better accuracy than the onte Carlo analyss wth 4 samples, and s more than x aster, as shown n able Concluson As IC technologes reach nanoscale, process varatons are becomng relatvely large and nonlnear (quadratc) response surace models mght be requred to accurately characterze the large-scale varatons. In ths paper we propose an asymptotc probablty extracton (APX) method or estmatng the non- ormal random dstrbuton resultng rom the nonlnear response surace modelng. hree novel algorthms,.e. bnomal moment evaluaton, CDF/PDF shtng and reverse PDF/CDF evaluaton, are proposed to reduce the computaton cost and mprove the estmaton accuracy. As s demonstrated by the numercal examples, applyng APX results n better accuracy than the onte Carlo analyss wth 4 samples, and acheves more than x speedup. APX can be ncorporated nto a yeld optmzaton loop or a tmng analyss envronment, or ecent probablty extracton and worst-case analyses. For example, the ecacy o applyng APX to robust analog desgn s urther dscussed n [4]. 7. Acnowledgements hs wor was unded n part by the ARCO Focus Center or Crcut & System Solutons (CS, under contract 3-C Reerence [] S. ass, odelng and analyss o manuacturng varatons, I CICC, pp. 3-8,. [] C. chael and. Ismal, Statstcal modelng o devce msmatch or analog OS ntegrated crcuts, I JSSC, Vol. 7, o., pp , Feb. 99. [3] Z. Wang and S. Drector, An ecent yeld optmzaton method usng a two step lnear approxmaton o crcut perormance, I DAC, pp , 994. [4] A. Dharchoudhury and S. Kang, Worse-case analyss and optmzaton o VLSI crcut peroormance, I rans. CAD, Vol. 4, o. 4, pp , Apr [5]. Felt, S. Zanella, C. Guardan and A. Sangovann-Vncentell, Herarchcal statstcal characterzaton o mxed-sgnal crcuts usng behavoral modelng, I ICCAD, pp , 996. [6] J. Swdzns,. Styblns and G. Xu, Statstcal behavoral modelng o ntegrated crcuts, I ISCAS, Vol. 6, pp. 98-, 998. [7] A. Graupner, W. Schwarz and R. Schüny, Statstcal analyss o analog structures through varance calculaton, I rans. CASI, Vol. 49, o. 8, pp. 7-78, Aug.. [8] A. Papouls and S. Plla, Probablty, Random Varables and Stochastc Processes, cgraw-hll,. [9] G. Seber, ultvarate Observatons, Wley Seres, 984. [] G. Golub and C. Loan, atrx Computatons, he Johns Hopns Unv. Press, 996. [] L. Pllage and R. Rohrer, Asymptotc waveorm evaluaton or tmng analyss, I rans. CAD, Vol. 9, o. 4, pp , Apr. 99. []. Cel, L. Plegg and A. Odabasoglu, IC Interconnect Analyss, Kluwer Academc Publshers,. [3]. Bosley and F. Lees, A survey o smple transer-uncton dervatons rom hgh-order state-varable models, Automatca, Vol. 8, pp , 97. [4] X. L, P. Gopalarshnan, Y. Xu and L. Plegg, Robust analog/rf crcut desgn wth projecton-based posynomal modelng, I ICCAD, 4. 9