Maximum Power Estimation Using the Limiting Distributions of Extreme Order Statistics

Transcription

1 Maxiu Power Estiatio Usig the Liitig Distributios of Extree Order Statistics Qiru Qiu Qig Wu ad Massoud Pedra Departet of Electrical Egieerig Systes Uiversity of Souther Califoria Los Ageles CA {qiru qigwu Abstract I this paper we preset a statistical ethod for estiatig the axiu power cosuptio i VLSI circuits. The ethod is based o the theory of extree order statistics applied to the probabilistic distributios of the cycle-by-cycle power cosuptio the axiu lielihood estiatio ad the Mote- Carlo siulatio. It ca predict the axiu power i the space of costraied iput vector pairs as well as the coplete space of all possible iput vector pairs. The siulatio-based ature of the proposed ethod allows it to avoid the liitatios of a gate-level delay odel ad a gate-level circuit structure. Last but ot least the proposed ethod ca produce axiu power estiates to satisfy user-specified error ad cofidece levels. Experietal results show that this ethod o average produces axiu power estiates withi 5% of the actual value ad with a 90% cofidece level by siulatig oly about 500 vector pairs. I. Itroductio Circuit reliability is a iportat issue i today s VLSI aufacturig. There are ay sources that ay cause circuit failure oe of the is excessive power dissipatio over a short period of tie. Uexpected high curret i short tie ay lead to peraet circuit daage sudde voltage chage o the supply source ad groud ets or teporary circuit failure. To desig a circuit with high reliability desigers have to rely o the efficiet ad accurate estiatio of axiu cycle-by-cycle power (or for short axiu power). Maxiu power estiatio i VLSI circuits is also essetial to deterie the IR drop ad bouce oise o supply lies ad to optiize the power ad groud routig etwors. I ost of the previous research wor axiu power estiatio refers to the proble of estiatig the axiu power (or curre that the circuit ay cosue withi ay cloc cycle. The proble is thus equivalet to looig for the axiu-power-cosuig vector pair aog all possible iput vector pairs. Therefore these techiques focus o fidig the lower boud ad upper boud of the axiu power. However differet desig requireets of today s VLSI chips ae thigs a little bit ore coplicated. I geeral we divide the scope of axiu power estiatio proble ito two categories: I. The axiu power for all possible vector pairs applied to the iputs of the circuit. We refer to this as the ucostraied axiu power. I. The axiu power for give trasitio/joit-trasitio probability specificatio for the circuit iputs. We refer to this as the costraied axiu power. A uber of techiques have bee developed to solve the probles i Category I. []-[8] ad Category I.. The ethod proposed i [] propagates the sigal ucertaity through the circuits to obtai a loose upper boud o the axiu power. The boud is the ade tighter by doig a detailed search o part of the priary iputs. The boud tighteig ethod teds to be tie cosuig whe the uber of the priary iputs is large. The Autoatic Test Patter Geeratio (ATPG) based techiques [3]-[4] try to geerate a iput vector pair that produces the largest switched capacitace i the circuit. The power cosuptio by the vector pair is the used as a lower boud o the axiu power of the circuit. The ATPG based techiques are very efficiet ad geerate a tighter lower boud tha that geerated by rado vector geeratio. The liitatios are however that the ATPG based techiques ca oly hadle siple delay odels such as the zero-delay ad uit-delay odels ad that the aalysis is doe at the gate-level. Cosequetly the estiatio accuracy is ot high. A cotiuous optiizatio ethod was proposed i [5] which treats the iput vector space as a cotiuous real-valued vector space ad the perfors a gradiet search to fid the fuctio axiu. Siilar to the ATPG based techiques the estiatio accuracy is ot high. The authors of [6] proposed a techique for fidig the axiu power-cosuig vector usig a geetic search algorith. The liitatio of this approach is that it requires siulatio of a lot of vectors i.e. its efficiecy is ot high. Statistical ethods have bee studied for axiu power estiatio. I [3] a Mote- Carlo based statistical techique for axiu curret estiatio was briefly discussed. The ethod radoly geerates highactivity vector pairs ad the axiu power is the estiated by siulatio. This ethod also suffers fro low efficiecy. The theory of order statistics has bee applied i [7][8] to estiate axiu power by estiatig the high quatile poit. Their efficiecy is as low as the rado vector geeratio techique. I this paper we preset a siulatio-based statistical ethod for axiu power estiatio for cobiatioal circuits. It is a ethod of estiatig the axiu power usig the theory of Asyptotic Extree Order Statistics. Copared to previous wor our approach aes the followig tagible cotributios:

2 . Our approach is the first approach that provides the cofidece iterval for the estiated axiu power for the user-specified cofidece level.. Our approach is the first approach which ca do axiu power estiatio for ay give error ad cofidece levels. 3. Our approach ca estiate the axiu power defied i both categories I. ad I.. 4. Because it is a siulatio-based techique the delay odel or the circuit structure do ot liit its accuracy. 5. By efficiet statistical estiatio of the extree distributios the estiatio efficiecy is largely iproved copared to existig statistical ethods (icludig siple rado saplig or quatile estiatio). O average the ethod ca do axiu power estiatio by siulatig oly about 500 vector pairs to achieve a 5% error at a cofidece level of 90%. This paper is orgaized as follows Sectio II itroduces the theory of asyptotic extree order statistics ad axiu lielihood estiatio. Sectio III describes our approaches for axiu power estiatio. Sectio IV presets our experietal results ad Sectio V gives the cocludig rears. II. Bacgroud. The asyptotic theory of extree order statistics The (cuulative) distributio fuctio (i short d.f.) of a rado variable (i short r.v.) x is defied as: F( = P{ x t} (.) The quatile fuctio (i short q.f.) of a d.f. F is defied as: F ( q) = if{ t : F( q} q [0] (.) rado where if(s) calculates the lower boud of set S. Notice that the q.f. F - is a real-valued fuctio ad F - (q) is the sallest q quatile of F that is if Z is a r.v. with d.f. F the F - (q) is the sallest value t such that P{Z < t} q P{Z t}. We rear that F(x)=sup{q [0]: F - (q) x}. Let z z... z be rado uits draw fro a coo distributio. If they are draw i a rado aer they are called idepedet idetically distributed (i short i.i.d.) r.v. s. If oe is ot iterested i the order i which z z...z are draw but iterested i the order of the agitude of their values oe has to exaie the ordered saple values X : X : X : which are the order statistics of a saple of size. X r: is called the rth order statistic ad the rado vector (X : X :... X : ) is the order statistic. Note that X : is the saple iiu ad X : is the saple axiu. X : is called the iia order statistic ad X : is called the axia order statistic or i geeral they are called the extree order statistics of a saple of size. The distributio fuctio of the saple axia X : is give by: P { X : = F( (.3) Three distributio fuctios are give for studyig the liitig d.f. of saple axia (i other words: extree value d.f. s): 0 0 ( ) = x G α x if exp( α FrFchet (.4) x ) x > 0 α exp( ( x) ) x 0 G α ( x) = if Weibull (.5) 0 x > 0 x G3( x) = exp( e ) for every x Gubel (.6) Defiitio [0] F is said to belog to the wea doai of attractio of liitig d.f. G if there exist serious of costats a > 0 ad reals b such that: F ( b + xa ) G( x) (.7) for every cotiuity poit of G. Let us defie the right edpoit of d.f. F as: ω ( = sup{ x : F( x) < } = F () (.8) Theore [0] A d.f. F belogs to the wea doai of attractio of a extree value d.f. G iα iff. oe of the followig coditios holds: F( tx) α ( α ) : ω( = ad li = x x > 0 t F( (.9) F( ω( + x α ( α ) : ω( < ad li = ( ) < 0 0 ( ( ) ) x x (.0) t F ω F t (3) : F( t + xg( ) li = e ω ( F ) F( t x - < x < (.) ad represets approachig decreasigly represets approachig icreasigly. Moreover costats a ad b ca be chose i the followig way: ( α ) : b = 0 a = F ( ) (.) ( α ) : b = ω( a = ω( F ( ) (.3) (3) : b = F ( ) a = g( b ) (.4) If a distributio F satisfies oe of the coditios i Theore we siply call the correspodig G iα the asyptotic distributio of the saple axia of distributio F. Theore ot oly gives the coditios of to which d.f. G the extree distributio will coverge but also the guidelies for us to choose a correct asyptotic extree distributio for a specific applicatio. Theore [0] The wea covergece to the liitig d.f. G holds for other choices of costats a ad b if ad oly if a a ad ( b b ) a 0 as (.5) Theore gives other possibilities of choosig a ad b i theore. I special cases whe F(x) has a fiite right edpoit by Theore the choice of b ( ) i Eq.(.3) is uique.. Maxiu-lielihood estiatio for paraeters of the Weibull distributio whe α > For reasos that will be ade clear i a later sectio we are iterested i developig a axiu-lielihood estiator for paraeters of a geeralized Weibull distributio defied as: α exp( β ( μ x) ) x μ G ( x; α β μ) = if (.6) 0 x > μ where μ is a locatio paraeter which deteries the right edpoit (i.e. axiu) of the distributio β > 0 is a scale paraeter ad α is the shape paraeter. The axiu-lielihood estiatio proble is defied as follows: Give idepedet rado saples x x x of G(x; α β μ) fid the values of α β μ which axiize the lielihood fuctio [9]: L = β μ) log[ G( x i i= ( α ; α β μ)] (.7) This axiu lielihood estiator whe it exists will be deoted by the vector αˆ βˆ ˆ μ ) ad satisfies: (

3 L ( αˆ ˆ ˆ β μ ) = 0 (.8) { α β μ} Let α 0 β0 μ0 deote the actual values of paraeters of the distributio G. It was proved i [9] that whe α > αˆ α βˆ β μˆ μ ) coverges i distributio ( ( ) to a oral rado distributio vector with ea 0. III. The estiatio Approach The proble of axiu power estiatio ca be stated as follows: Give a set V (called populatio) of iput vector pairs estiate the axiu power dissipatio that the circuit ay exhibit for ay vector pair i the populatio. A vector pair i V is called a uit of the populatio. I this paper the populatio ay iclude either all possible iput vector pairs applied to a circuit or all possible vector pairs uder soe iput trasitio probability costraits. Although there could oly be a fiite uber of distict vector pairs i the populatio but the size of V represeted by V is assued to be ifiite sice there is the possibility of repeatig the vector pairs. 3. The asyptotic distributio of the saple axiu power If we regard power cosuptio for a vector pair as a rado variable p a distributio of p is the fored by the power cosuptio values of vector pairs i set V. The average power is the ea value of the distributio. The axiu power is the the right edpoit of the distributio. Lie other papers o statistical power estiatio we assue the d.f. of power cosuptio i a large LSI circuit as a cotiuous distributio. Give populatio V the ith saple for ax power estiatio is fored by the power values of radoly selected uits: pi pi pi i = where is called the saple size ad the uber of saples. The axiu power i each saple is defied as: pi MAX = ax{ pi pi pi } i = (3.) Accordig to Eq.(.5) the d.f. of p imax ca be writte as: H ( p MAX ) = F ( p). As etioed i Theore H(b + p imax a ) asyptotically coverges to oe of the three distributios defied i Eq. s(.7) (.8) ad (.9). I the reaider of this paper we will use ω( deotig the actual axiu power of the populatio. We ow that power cosuptio i a LSI circuit is always a fiite value i.e. ω( <. Therefore the coditio i (.9) is ot et ad H(b + p imax a ) ca ot coverge to G α. Also because the upper boud of supportig doai for G 3 is ifiite while that of G α is fiite The coditio i (.0) is ore liely to hold tha that i (.). Therefore H(b + p imax a ) is ore liely to coverge to G α rather tha G 3. It is poited out i [0] that ost frequetly used cotiuous distributios with fiite right edpoit ( ω ( < ) satisfy the coditio i Eq.(.0). Therefore i ay egieerig applicatios of axia estiatio it is assued that the distributio uder study belogs to the wea covergece doai of G α. This stateet has also bee epirically proved to be true by our experiets (cf. later this sectio). Therefore we state that the distributio of p imax asyptotically follows the Weibull distributio G α. This eas that there exist a ad b such that: F( b + a p ) = F ( b + a p) G α ( p ) (3.) i MAX i MAX pi MAX b or F( pi MAX ) G α ( ) i = (3.3) a Fro Eq. s (.3) ad (.5) we get b = ω( where ω( is the axiu power cosuptio of the populatio. If we substitute the geeralized Weibull distributio defied i Eq.(.6) ito Eq.(3.3) we get F p ) G( p ; α β μ) i = ( i MAX i MAX where β=(/a ) α ad μ=b. Experiets have bee desiged to verify the asyptotic distributio of saple axia. The distributios of saple axia for differet saple size ( = ) was fored by 000 rado saples fro the populatio. Their closest Weibull distributios are obtaied by usig least-square fittig techiques. Figure shows the results for circuit C3540. = = 0 = 30 = 50 Figure The copariso betwee distributio of saple axia ad Weibull distributio Experiets are doe for other circuits ad populatios ad the siilar results are obtaied. Fro these results we cocluded that the differece betwee distributios of p imax ad Weibull distributio i the regio ear the axiu power is egligible whe 30. Sice we are oly iterested i estiatig the axiu power we fix the saple size to 30 ad assu that the distributio of p imax follows Weibull distributio whe 30. Cosequetly p imax (i= ) ( = 30) becoe the saples of the geeralized Weibull distributio i (.6). Most iportatly if previous assuptios hold we have: ω ( = μ. Therefore the proble of axiu power estiatio is equivalet to the proble of estiatig the locatio paraeter μ of a geeralized Weibull distributio fro rado saples. The siplest way of doig this is to curve-fit the saples to Eq.(.6) to get values of α β ad μ. However our study shows that the curve fittig approach is ustable sice the proble becoes difficult whe we tryig to costruct the distributio fro sall uber of saples. We choose aother estiatio ethod that is ore robust ad has a solid theoretical support. 3.3 A Maxiu-lielihood estiator of axiu power dissipatio The axiu-lielihood estiators for paraeters of geeralized Weibull distributio for α > have bee itroduced i Sectio II. I fact α is always large tha if the saple size is uch saller tha the populatio size V. Cosequetly let αˆ βˆ μˆ be the estiators that satisfy Eq. (.8) we ca prove the followig result. Theore 3 αˆ βˆ μˆ ( ) are the ubiased estiators of α β μ of the Weibull distributio which eas that

4 αˆ βˆ μˆ ( ) follow oral distributios with ea values of α 0 β0 μ0 ad covariace atrix VAR. The atrix VAR is defied as: σ α σ α β σ α μ VAR = σ β α σ β σ β μ (3.4) σ μ α σ μ β σ μ Fro Theore 3 we ow that the axiu power estiator μˆ coverges to a oral distributio with ea of μ 0 (which is the actual axiu power ω() ad variace of σ μ. Theore 4 μˆ is a ubiased estiator for axiu power ω(. Give cofidece level l (l (0)) the cofidece iterval of the estiated axiu power μˆ ( ) is give by: [ ω( ul σ μ ω( + ul σ μ ] (3.5) where ω( is the actual axiu power is the uber of saples σ μ is defied i Eq.(3.4) ad u l is defied as: ul e dx = l (3.6) ul π Theore 4 states that the probability that the estiated axiu power falls ito the iterval defied i Eq.(3.5) is l. For a give l saller cofidece iterval eas higher estiatio accuracy. Therefore the relative estiatio error is iversely proportioal to the square root of the variace of the estiator. I practice the theoretical cofidece iterval caot be calculated directly because σ μ is uow. Therefore we do ot ow a priori how ay saples are eeded to achieve certai cofidece iterval at give cofidece level. A iterative (Mote-Carlo) ethod has bee desiged to solve this proble. 3.4 The iterative estiatio procedure Experiets have bee desiged to study the distributio of the axiu lielihood estiator for axiu power i cases whe the uber of saples is fiite (we ow fro Theore 3 that whe this axiu lielihood estiator for ω( follows a oral distributio). The saple size is fixed at =30 ad differet uber of saples are used (=050). Durig each sigle experiet saples with saple size are radoly selected fro the populatio. Maxiu power is the estiated by usig the axiu lielihood estiator μˆ. The progra for axiu lielihood estiatio ca be foud i ay places. For each distict the saplig-estiatio procedure is repeated 00 ties to for the distributio of estiated value. The distributios of estiated axiu power for differet values of are the fored ad their earest oral distributios are obtaied by least-square curve fittig. The results for circuit C3540 are show i Figure. Figure The distributios of estiated axiu power copared with the earest oral distributio Siilar results are obtaied for other circuits. Fro the experietal results we ca coclude that the estiator for x =0 =50 axiu power is approxiately orally distributed whe the uber of saples is large eough ( 0). Therefore we assue oral distributio of estiator for axiu power whe 0. Before we itroduce our practical axiu power estiatio procedure we suarize our discussios i earlier part of this sectio as show i Figure 3. Saple MAX Populatio Maxiu- Lielihood Estiatio Hyper Saple Saple MAX The distributio of sapled uits follows the distributio of the origial populatio F(p) Distributio of saple axia follows the geeralized Weibull distributio G(p imax ; αβμ) whe 30 The estiated axiu power follows oral distributio whe 0 Figure 3 Syopsis of axiu power estiatio ethod I Figure 3 a hyper-saple is defied as the result of oe ru of axiu power estiatio for saples with size. We fix the value of to 30 ad value of to 0 the the uber of uits which is eeded to for a hyper-saple is 300. Theore 5 Let P ˆ i MAX (i=...) deote the ith hyper-saple for =30 ad =0 P ˆ i MAX follows the oral distributio with ea value of ω( ad variace of σ μ 0 where σ μ is defied i Eq.(3.4). By defie: PMAX = Pˆ i ad MAX s = ( Pˆ i MAX PMAX ) (3.7) i= i= Theore 6 P MAX ad s are ubiased estiators of the actual axiu power ω( ad σ μ 0 respectively. Give cofidece level l the cofidece iterval for the actual axiu power is give by: tl s tl s [ PMAX PMAX + ] (3.8) where t l- is the l 00% percetile poit of the t distributio with degree of freedo of -. Theore 6 gives us a guidelie for desigig a iterative procedure for axiu power estiatio subject to the required accuracy (relative error less tha or equal to ε) at give cofidece level l. The basic worflow is show i Figure 4. I Figure 4 the geeratio of a hyper-saple follows the procedure show i Figure 3. Cofidece iterval is calculated usig Eq.(3.8). The axiu relative error is calculated usig tl s the cofidece iterval as PMAX. If this quatity is

5 larger tha the required ε the the estiated value has ot coverged ad we add oe ore hyper-saple; otherwise the estiatio has coverged ad we report the estiatio result. START Add oe hyper-saple Copute the cofidece iterval for give Is the cofidece iterval saller tha the required value? YES Report Result Figure 4 Iterative flow of axiu power estiatio 3.5 Practical issue: fiite populatio versus ifiite populatio The approach discussed i the earlier part of this sectio is desiged for estiatig the axiu power of a ifiite populatio. However we ust deal with a fiite populatio i real applicatios. As a exaple our experietal setup i the ext sectio uses fiite populatios. Experietal results shows that if we use the sae approach for fiite populatio as for the ifiite populatio there will be bias i axiu lielihood estiatio i the sese that the ea of the estiated value is always larger tha the actual axiu power of the populatio. This happes because estiator μˆ is estiatig the axiu power of a ifiite populatio that should have (with soe probability) a eve tail after the actual axiu power of the populatio. While this tail does ot exist i the case of a fiite populatio. To solve this proble we ca regard the fiite populatio V as a saple of size V selected radoly fro the assued cotiuous distributio for the ifiite populatio. Assue there is oly oe uit i the fiite populatio which cosues the axiu power the the axiu power of the fiite populatio becoes the estiated (-/ V ) quatile poit of the assued cotiuous distributio. Accordig to the tail-equivalece property betwee a distributio ad the liitig distributio of its saple axia [0] estiatig the (-/ V ) quatile poit of the origial populatio is equivalet to estiatig the (-/ V ) quatile poit of the asyptotic Weibull distributio of the saple axia. Therefore whe estiatig the axiu power of a fiite populatio istead of usig the theoretical Add oe ore hyper-saple NO μˆ (which is the 00% quatile poit of the estiated geeralized Weibull distributio) we use the (-/ V ) quatile poit of the Weibull distributio (whose paraeters ca be estiated by usig the axiu lielihood estiator) as the estiator for the axiu power. We call this the odified estiator for the fiite populatio. Experietal results show that the odified estiator gives us a ubiased estiator for fiite populatios. IV. Experietal results Category I.. Estiatig the ucostraied axiu power. Proposed ethod: I this category the goal is to estiate the axiu power of the circuit for all possible iput vector pairs. Cosequetly the siple rado saplig procedure ca be realized by radoly geeratig vector pairs that is the two ethod (i.e. rado vector geeratio ad siple rado saplig) are equivalet i this case. Except for the fact that the saplig techique is replaced by the rado vector geeratio the reaiig part of our approach (cf. Figure 3 ad Figure 4) reais the sae. Experietal results ad discussio: Let us give a theoretical study o the efficiecy of the estiatio ethod of rado vector geeratio or siple rado saplig. Assue we wat to estiate axiu power of error less tha 5% at cofidece level 90% for a populatio. Let the size of the populatio be V. Defie the qualified uits as those uits whose values are withi 5% differece of the actual axiu. Assue the uber of the qualified uits is Z. The portio of the qualifies uits i the whole populatio is the Y=Z/ V. If we saple x uits fro the populatio the probability that there is at least oe qualified uit i these x uits is give by: x P = ( Y ). For P to be larger tha or equal to 90% we eed o average x = log(0.)/log(-y) sapled uits. Fro our experiets we have observed that Y is very sall (e.g. <0.000). This leads to very large x (e.g. >3000). The experietal setup is as follows. The populatio cotais radoly geerated high activity (average switchig activity larger that 0.3) vector pairs. Rado vector geeratio is equivalet to siple rado saplig vector pairs fro the populatio. The whole populatio is siulated usig Powerill [] to get the power cosuptio value for each uit ad i the process the actual axiu power. Our approach (=30 =0) ad siple rado saplig (SRS) have bee applied to do axiu power estiatio for relative error < 5% at cofidece level 90%. The experietal results are show i Table ad Table. Our approach has bee applied to do axiu power estiatio 00 ties for each circuit. Table shows the copariso of efficiecy ad accuracy of our approach versus siple rado saplig. The portio of the qualified uits i the whole populatio is give i the d colu. The axiu iiu ad average uber of uits eeded for our approach to coverge are reported i the 3 rd 4 th ad 5 th colus respectively. The 6 th colu gives the theoretically calculated (accordig to the discussio of the secod paragraph fro the botto of page 4) uber of uits eeded by siple rado saplig to achieve the sae error (5%) ad cofidece (90%) level. The 7 th ad 8 th colus give the absolute value of the axiu ad iiu estiatio error of our approach. We have ot give the relative error for SRS because the SRS techique is ot able to predict the axiu power subject to give error ad cofidece levels. Table shows the copariso for the estiatio quality. Siple rado saplig techiques usig 500 0K ad 0K uits are perfored 00 ties respectively. The d colu gives the actual axiu power of the populatio. Colus ad 6 give the results of largest-error estiates for differet techiques. Colu ad 0 give the results of the percetage of the tie whe the estiated value exceeds the error level. The experietal results shows that our approach is uch ore efficiet tha the siple rado saplig techique (about X speedup o average). More iportatly however siple rado saplig or siilar techiques are ot reliable because they caot provide cofidece iterval ad cofidece level for axiu power estiatio. Also the estiatio quality of our approach is obviously better tha siple rado saplig. Fro the results of Table if we copare our approach with siple

6 rado saplig with 0K uits the average largest error is 5.3% for our approach ad 0.4% for SRS. As to the average percetage of estiated value with error larger tha 5% it is 4.3% for our approach ad 3% for SRS. It ca be foresee that the advatage of our approach over SRS will be ore predoiat for ifiite populatio. Portio of the # of uits eeded Relative error Circuit qualified Our approach SRS Our approach uits MAX MIN AVE AVE MAX MIN C % 0.3% C %.4% C % 0.6% C %.% C %.7% C % 0.8% C % 0.05% C % 0.6% C %.9% Table Results for coparig the efficiecy ad accuracy Actual Largest estiatio error % of estiates with error > Circuits ax. 5% power Our SRS Our SRS (W) appr K 0K appr K 0K C % -3% -8.5% -6.3% 6% 80% 5% 5% C % -4% 7.5% -6.3% 3% 73% 8% 8% C % -8.6% -5.4% -.5% % 38% % 0% C % -4% -0% -8.9% 5% 80% 5% 33% C % -% -3% -4% 8% 89% 73% 57% C % -9.7% -7.7% -6.% % 73% 7% 3% C % -% -% -% 3% 76% 6% 5% C % -4% -0% -7.3% 7% 9% 69% 54% C % -0% -5% -% 4% 88% 4% 9% Table Results for coparig the estiatio quality Portio of the # of uits eeded Relative error Circuit qualified uits Our approach SRS Our approach MAX MIN AVE AVE MAX MIN C %.8% C %.0% C % 0.5% C % 4.0% C %.% C % 4.% C %.7% C % 0.9% C %.% Table 3 Results for coparig efficiecy ad accuracy Category I.. Estiatig the costraied axiu power. Proposed ethod: Siilar to Category I. except that vector pairs are geerated uder give costraits. Experietal setup: Siilar to Category I. this tie we geerate two populatios (each of size 80000) subject to the costrait that the average switchig activity per iput lie is 0.7 ad 0.3 respectively. Detailed copariso with siple rado saplig has bee perfored as well. However we give oly the tables for coparig efficiecy ad accuracy i order to save space. The experietal results for populatios of average switchig activity 0.7 ad 0.3 are show i Table 3 ad Table 4 respectively. The eaig of etries i differet colus is the sae as that i Table. The estiatio quality copariso ca be see fro the value of the portio of the qualified uits i the d colus of both tables. As expected whe he uber of qualified uits i the populatio decreases the uber of uits eeded to estiated the axiu power dissipatio i the circuit icreases. Portio of the # of uits eeded Relative error Circuit qualified Our approach SRS Our approach uits MAX MIN AVE AVE MAX MIN C %.% C % 3.5% C %.7% C %.9% C %.4% C % 3.4% C % 0.6% C % 3.3% C %.9% Table 4 Results for coparig efficiecy ad accuracy V. Coclusio I coclusio a statistical approach has bee proposed based o the asyptotic theory of extree order statistics. This is the first axiu power estiatio approach which ca provide cofidece iterval at give cofidece level. This is also the first approach which ca do axiu power estiatio for ay userspecified error ad cofidece levels. The proposed approach ca predict the axiu power i the space of costraied iput vector pairs as well as the coplete space of all possible iput vector pairs. It is a efficiet siulatio-based approach with high accuracy. The geerality of this approach aes it also applicable to other fields of VLSI desig autoatio for exaple axiu delay estiatio. REFERENCES [] H. Kriplai F. Naj ad I. Hajj Maxiu Curret Estiatio i CMOS Circuits Proceedigs of Desig Autoatio Coferece 99. [] S.Mae A.Pardo R.Bahar G.Hachtel F.Soezi E.Macii ad M.Pocio Coputig the Maxiu Power Cycles of a Sequetial Circuit Proceedigs of Desig Autoatio Coferece 995. [3] C.-Y. Wag ad K. Roy Maxiu Curret Estiatio i CMOS Circuits Usig Deteriistic ad Statistical Techiques Proceedigs of IEEE VLSI Desig Coferece Ja [4] C.-Y. Wag T.-L. Chou ad K. Roy Maxiu Power Estiatio Uder Arbitrary Delay Model Proceedigs of IEEE Iteratioal Syposiu o Circuits ad Systes 996. [5] C.-Y. Wag ad K. Roy COSMOS: A Cotiuous Optiizatio Approach for Maxiu Power Estiatio of CMOS Circuits Proceedigs of ACM/IEEE Iteratioal Coferece o Coputer- Aided Desig 997 to appear. [6] M.S. Hsiao E.M. Rudic ad J.H. Patel K: A Estiator for Pea Sustaiable Power of VLSI Circuits Proceedigs of Iteratioal Syposiu o Low Power Electroics ad Desig pp Aug [7] A. Hill C.-C. Teg ad S.M. Kag Siulatio-based axiu power estiatio Proceedigs of ISCAS [8] C.S. Dig Q. Wu C.T. Hsieh ad M. Pedra Statistical Estiatio of the Cuulative Distributio Fuctio for power dissipatio i VLSI Circuits Proceedigs of Desig Autoatio Coferece pp Ju [9] R.L. Sith Maxiu lielihood estiatio i a class of oregular cases Bioetria pp [0] J. Galabos The Asyptotic Theory of Extree Order Statistics Robert E. Krieger Publishig Copay 987 [] Aalog Cirsuit Egie Release Notes Syopsis 997.