Fast and Precise Power Prediction for Combinational Circuits

Transcription

1 Fast and recse ower redcton for obnatona rcuts Hongng L, John K. ntono, and Sudarshan K. Dha S-TR-0-00 Noveber 00 Technca Reort Schoo of outer Scence Unverst of Oahoa 00 Fegar Street Noran OK

2 Fast and recse ower redcton for obnatona rcuts Hongng L, John K. ntono, and Sudarshan K. Dha Schoo of outer Scence, Unverst of Oahoa 00 Fegar Street, Noran OK bstract The ower consued b a cobnatona crcut s dctated b the swtchng actvtes of a sgnas assocated wth the crcut. n anatca aroach s roosed for cacuatng sgna actvtes for cobnatona crcuts. The aroach s based on a Marov chan sgna ode, and drect accounts for correatons resent aong the sgnas. The accurac of the aroach s verfed b coarng sgna actvt vaues cacuated usng the roosed aroach wth corresondng vaues roduced through suaton studes. It s aso deonstrated that the roosed aroach s accurate and coutatona effcent. Inde Ters: obnatona rcut, robabt, ctvt, Marov han, orreaton Factor.. Introducton ower consuton of ntegrated crcuts Is s of growng concern as ore eectronc devces are beng deoed n obe and ortabe acatons, e.g., Ds, obe teehones, and other batterowered eectronc devces. s the functonat of such devces ncreases, so does the coet and sohstcaton of the underng crcuts. More coet and faster coc rates genera transate nto hgher ower consuton for a gven hardware eentaton technoog. ecause batter

3 technoog has not roved at the sae rate as I technoog, there s strong otvaton to desgn crcuts that are as ower effcent as ossbe to etend batter fe for ortabe devces. Iroveents n I technooges e.g., reducton n feature sze can reduce ower requreents of a gven crcut desgn. However, functonat and coet of coerca devces genera ncrease fro one generaton to the net. So, the net generaton devce eented wth the net generaton I technoog w genera have ore functonat and coet than the revous generaton, and thus the ssue of archtectura desgn of the underng crcuts to be ower effcent reans ortant. Severa sar and reated aroaches to ths robe have been roosed n the ast, ncudng suaton-based [] and anatca aroaches [,, 4]. good surve of ast aroaches can be found n [5]. Genera, suaton-based aroaches acheve hgh accurac but requre ong eecuton tes; n contrast, the anatca aroaches are faster but are genera ess accurate. In ths aer a new anatca aroach s roosed that acheves fast eecuton te and accurac that s coarabe wth suaton-based ethods. s eaned beow, the artcuar focus s on ower consuton of crcuts eented n MOS, but the roosed aroach a be acabe for other technooges as we. ower consuton n a MOS crcut s rar due to three tes of current fow: eaage current, swtchng transent current, and oad caactance chargng current [9]. The eaage current s assocated wth the erfecton of fed effect transstors FETs that are used n MOS devces. Ths te of current fow n MOS technoog s genera ver sa. The swtchng transent current wthn MOS gates s caused b a bref short crcut that can occur when the state of the coentar gates change fro on-to-off and off-to-on. Ths short crcut occurs when the coentar MOSFETs are concurrent on for a bref transent erod of te.

4 The ower oss due to swtchng transent current s deendent on the swtchng actvt of the gate and s genera greater than eaage current. The fna te of current fow s oad caactance chargng current. Ths s the current fow that s requred to charge the caactve oad that s assocated wth a transstor gate, and occurs when the state of a gate changes. Ths s the donant coonent of ower consuton n MOS devces, and s strong deendent on sgna swtchng actvt. Let Φ denote the set of a sgnas assocated wth a crcut. For each s Φ, et s denote the caactve oad assocated wth sgna s. so, et s denote the actvt of sgna s, whch has a vaue between zero and one, and reresents the sgna s norazed average frequenc reatve to the frequenc of a sste coc, f. Thus, fs gves the average frequenc of sgna s. ased on these assutons and notaton, the average ower for a MOS crcut oeratng at a votage eve of V can be eressed as [4, 5]: ower V f s s avg. s Φ The robe addressed n ths reort s to deterne the actvt of a sgnas of a cobnatona crcut gven an arorate robabstc ode for the rar nut sgnas that drve the crcut. The sgna ode roosed n ths reort s based on a Marov chan. The sgna actvt s eas couted fro the araeters assocated wth the roosed sgna ode. In the roosed aroach, sgnas wth nown Marov chan reresentatons are roagated through a ode of the crcut to roduce a Marov chan reresentaton for the outut of each gate n the crcut. ccurac of the aroach s verfed b coarng sgna actvtes roduced b the roosed ethod wth corresondng actvtes roduced through suaton studes. When coared wth other reated aroaches, a e asect of 4

5 the roosed aroach s that correatons resent aong the sgnas due to re-convergent fan-out [6] are accounted for drect. The rest of ths reort s organzed as foows. In Secton, an overvew of reated aroaches s rovded. The roosed aroach, whch utzes a Marov chan sgna ode, s gven n Secton. The transforatons and agorth for ang ths aroach to a crcut s descrbed and anazed n Secton 4. In Secton 5, the accurac of the roosed aroach s verfed through Sce crcut suatons, foowed b concusons n the fna secton.. revous Reated roaches. asc robabstc Sgna araeter Defntons Sgnas n a cobnatona ogc crcut can be treated n a robabstc sense [],.e., for sgna, the robabt that has ogc vaue s defned b. Let t, t,, be a stochastc rocess that taes the vaues of ogca 0 or ogca, transtonng fro one to the other at rando tes. Genera, a stochastc rocess s sad to be strct-sense statonar SSS f ts statstca roertes are nvarabe to a shft of te orgn. ased on the assutons of a SSS 0- ean-ergodc rocess t, the foowng defntons are derved fro []. Defnton. Sgna robabt: The robabt of a ogc sgna t s the average fracton of te that the sgna s hgh and s gven b T t dt. T T Defnton. Sgna ctvt: ssue the average nuber of transtons n a te nterva of ength T s gven b nt. The sgna actvt of the corresondng sgna t s gven b T 5

6 n T. T T. Sgna robabt acuaton In [], the concet of usng robabstc sgna odeng for anass of cobnatona crcuts was frst ntroduced. In ths wor, each sgna s odeed wth a snge robabstc araeter,, defnng the robabt of a sgna havng a ogca vaue of one. The urose s to cacuate the robabt araeter for a sgnas, gven the robabt araeters of the crcut s rar nuts. The otvaton for ths wor orgnated fro the area of seudorando testng, n whch faut coverage and dentfcaton s acheved wthout resortng to ehaustve testng. Instead, b subectng a crcut to a arge nuber of rando generated nut sgna vectors, one can deduce fauts n the crcut b easurng the fracton of te that an gven sgna has ogc vaue one. If an of the easured sgna robabtes do not atch cacuated sgna robabtes, then the ossbt of a faut s resent. For sgna, the robabt that has ogc vaue s defned b. Two agorths for cacuatng sgna robabtes are ntroduced n []. These aroaches requre that a ooean functon eresson assocated wth each sgna be derved n ters of the rar nuts. ecause the nuber of ters n these eressons can grow eonenta wth the nuber of nuts, the coet of these aroaches can be rohbtve for ractca crcuts. coutatona effcent agorth for cacuatng sgna robabtes s ntroduced n [7], naed gorth, whch oerates b roagatng robabt vaues through the gates of crcut, thereb drastca reducng the sze of the ooean functons that ust be evauated. Secfca, the robabt of the outut of a gate s eressed n ters of the robabt vaues for the nuts to that 6

7 gate nstead of the rar nuts of the entre crcut, as requred b the aroach n []. Ths agorth s se and fast t has a near coet n the nuber of gates but s not accurate for a casses of crcuts. 4 Fgure. n eae cobnatona crcut used to ustrate sgna robabt cacuatons derved fro [7]. To ustrate the naccuraces of gorth, assue n Fgure that the robabtes of rar nuts and are both 0.5. ang gorth of [7], the couted robabtes of the crcut s sgnas were cacuated and are rovded n Tabe. Tabe. oarson of actua sgna robabtes and those cacuated usng gorth n [7] for the crcut of Fgure wth. 4 ctua 4 4 gorth n [7] The robe wth the accurac of gorth arses n crcuts n whch re-convergent fan-out sgnas are resent. Re-convergent fan-out ntroduces functona deendences and statstca correatons aong the sgnas; however, gorth assues statstca ndeendence aong the nuts to each gate. For eae, sgnas and n Fgure both deend on sgna due to reconvergent fan-out. Thus, ang the agorth to cacuate 4 under the assuton that sgnas 7

8 and are ndeendent resuts n an error n the vaue cacuated for 4, as shown n Tabe. Sar, the vaues cacuated for and are aso n error. nother agorth s roosed n [7] caed the Weghted veragng gorth W, whch genera acheves better accurac than gorth and has a coarabe te coet. However, the W st does not awas roduce correct vaues. ethod for accountng for sgna robabt correatons was deveoed n [6] naed the correaton coeffcent ethod M. usng ths aroach, the robabt of the outut of a twonut gate can be ore accurate cacuated, gven the robabtes of the two nuts and an assocated correaton factor assocated wth the two sgnas. In ths agorth, the correaton factor can aso be cacuated anatca b eans of a set of basc roagaton rues. ang ths M agorth to the crcut shown n Fgure, the vaues of,,, and 4 are roer cacuated and corresond to the actua vaues shown n Tabe. The te coet of the M agorth s O N for a crcut wth N gates... Sgna ctvt acuaton The above-descrbed aroaches of [], [6], and [7] are concerned wth deternng the robabtes of sgna vaues, not the robabtes of sgna transtons,.e., actvtes, whch are necessar for estatng ower consuton, refer to Eq.. n ear aroach for estatng sgna actvtes was deveoed n [], n whch sgnas of a crcut are odeed to be utua ndeendent strct-sense-statonar SSS ean-ergodc 0- rocesses. Under these assutons, the actvt of a sgna fro a crcut wth n-rar nuts can be eressed as Sharer te coet resuts can be obtaned; for eae, t can be shown that a crcut wth N eves has a coet of O N 8

9 9 n, where s the ooean dfference of functon wth resect to and s defned b.,,,0,,,,,,,,, 0 n n L L L L Intutve, the ooean dfference defnes whether a transton of sgna w cause a transton n outut sgna. Secfca, f the ooean dfference functon evauates to one, then a transton of sgna causes a transton n ; f the ooean dfference functon evauates to zero, then a transton of sgna does not cause a transton n. So, the robabt of the ooean dfference functon,, defnes the robabt that a change n w occur gven that there s a change n. s an eae of how to evauate Eq., consder a se case of a three-nut ND functon n whch., 4 0 0, 5 and sar, 6. 7

10 Thus,. ecause,, and are utua ndeendent, we can further sf the robabt ters as foows: 8 The above eresson s read evauated usng the vaues of and, whch are the nown robabtes and actvtes of the rar nut sgnas. though the cacuaton of the robabt of the ooean dfference ters,.e.,, for the above eae was reatve straghtforward, ths cacuaton can be cocated for arge and coe crcuts. In [], the cacuaton of these ters s accoshed b frst reresentng the nodes of the crcut wth a bnar decson dagra DD [, 5]. In ractce, the DD aroach often acheves near or near near te coet; however, n the worst case the coet can grow eonenta wth the nuber of gates. It s noted n [4] that Eq.,.e., the aroach descrbed n [], fas to consder the effect of sutaneous swtchng of gate nuts. Fgure shows an eae of how sutaneous swtchng of nuts to a ogc gate affects the actvt of the outut node. s shown n the fgure, f the two nut sgnas awas swtch sutaneous, then the outut sgna of the XOR gate w have an actvt of zero, even though the robabt and actvt ters n Eq. are nonzero [4]. Ths eae s an etree case, but s gven to ustrate the ortance of consderng sutaneous swtchng. 0

11 Fgure. Eae to ustrate the effect of sutaneous swtchng derved fro [4]. Each ooean dfference ter assocated wth Eq. descrbes an nut-swtchng event n whch eact one of the nuts aes a transton. Thus, Eq. does not account for events nvovng sutaneous swtchng of two or ore of the nut sgnas. The concet of the generazed ooean dfference was ntroduced n [4] to account for sutaneous swtchng, and s denoted as foows:,,...,,,...,...,...,, b b b b b b b b b 9 where s a ostve nteger,,,...,,, are dstnct utua ndeendent rar nuts of, and b are bnar vaues of 0 or. Note that f the generazed ooean dfference evauates to one, then the sutaneous transtons of sgnas,...,, fro,...,, b b b to,...,, b b b or fro,...,, b b b to,...,, b b b w cause a transton at. Eq. s adated n [4] usng the generazed ooean dfference concet to account for sutaneous swtchng, resutng n: [ ] [ ] }, { } {,,..., 0 00 n n n n n n n n n n n c c c c c c 0

12 where c, c 00,, n n c L 0... are condtona robabtes of the generazed ooean dfferences under the condton that on the ndcated nuts sutaneous swtch, and the rest do not. Detas on how to cacuate these condtona robabtes can be found n [4]. ng Eq. 0 to the sae three-nut ND functon used earer resuts n the foowng: 4 Observe that a of the ters of Eq. 8 aso aear as ters n Eq.. The rest of the ters n Eq. arse due to the generazed ooean dfference factors that account for sutaneous swtchng. In genera, the aroach of Eq. 0 eds ore accurate resuts than Eq.. However, the overa coet assocated wth evauatng Eq. 0 s genera uch arger than that of Eq.. Ths hgh coet s due to a otenta arge nuber of ters eonenta n the nuber of nuts and the coet assocated wth evauatng the condtona robabtes. For ore dscusson about the coet and technques for cacuatng the condtona robabtes, refer to [4]..4. Suar of revous Reated roaches The sgna ode for the aroaches overvewed n Subsecton. s based on a snge robabt araeter [, 6, 7]. though ths robabt araeter s not drect used n cacuatng a crcut s ower consuton, refer to Eq., t s a necessar coonent for the sgna ode coon to the

13 aroaches of Subsecton., whch utze both sgna robabt and sgna actvt araeters [, 4]. The aroaches of [], [], and [4] can have hgh coutatona coetes because the nuber of ters n the underng equatonstransforatons can grow eonenta wth the nuber of rar nuts to the crcut. In [7], a trade-off between coutatona coet and resutng accurac s ustrated n the contet of the underng equatonstransforatons ntroduced n []. In artcuar, an aroate aroach s defned n [7] n whch the transforatons of [] are aed n a gate-bgate fashon. Thus, nstead of dervng the transforaton for a sgna s robabt araeter n ters of the crcut s rar nuts, t s derved n ters of the edate nuts to the ogc gate assocated wth the sgna. Ths aroach great reduces the coutatona coet, but ntroduces error n the cacuated robabt araeters for crcuts wth re-convergent fan-out. Sar trade-offs between coutatona coet and accurac are ossbe reatve to the evauaton of Eq. and Eq. 0 assocated wth [] and [4], resectve. Instead of dervng a sgna s ogc functon n ters of the crcut s rar nuts, the araeters to the edate nuts the sgna s ogc gate can be used. gan, ths te of gate-b-gate technque w genera ntroduce error because t does not account for correatons resent aong the nterna sgnas that drve the gates wthn the crcut. The aroach of [6] s a fast and accurate gate-b-gate technque for cacuatng a sgna s robabt araeter. It ntroduces the concet of a correaton factor to account for and arorate adust the transforaton for correated nuts to a gate.. Marov han Sgna Mode

14 .. renares In ths secton we ntroduce a sgna ode that s based on a Marov chan havng three event araeters. It s shown that the roosed Marov chan ode s equvaent to the two-araeter robabtactvt sgna ode of [] and [4]. The advantage of odeng sgnas wth Marov chans s that t aes t ossbe to coute correatons between sgnas reated to both robabt and actvt. The aroach derved here can be vewed as a generazaton of the aroach n [6]. Instead of tracng a correaton factor for the snge robabt araeter ode, transforatons for correaton factors assocated wth the three araeters of the Marov ode are derved. Ths utate eads to a fast and accurate gate-b-gate agorth for cacuatng sgna robabtes and actvtes. s ustrated n Fgure, the roosed Marov chan sgna ode has three event araeters for sgna. The event denoted b reresents the sgna beng n state, and and reresent the events that there s a transton fro state 0 to and fro state to 0, resectve. Note that the robabt of event s denoted b, and s equvaent to the sgna robabt defned n the revous secton. 0 Fgure. roosed Marov chan sgna ode. For notatona convenence and cart, we w denote the vaue of as for the vaue of the robabt of sgna and the vaue of the actvt as for the vaue of the actvt of sgna 4

15 throughout the rest of the aer. Usng these notatons and ang basc roertes of Marov chans aong wth the defnton of sgna actvt, the foowng eressons can be derved for, and :,,. 4 Thus, f the vaues of both the robabt and actvt araeters of a sgna are nown.e., and, then the robabtes of the three events assocated wth the roosed Marov ode for the sgna are coete deterned. Lewse, nowng the robabt vaues of the three araeters of the Marov ode fu deternes the robabt and actvt araeters of the sgna. In order to defne correatons between two sgnas odeed wth Marov chans, soe basc defntons are needed. Let and denote two events and et denote the robabt of both and occurrng. Fro basc robabt theor [8],, where reresents the robabt of gven. so, the correaton coeffcent of two events and s defned as σ ρ, 5 σ σ where σ s the covarance and σ and σ are the ostve square roots of the varances of and. It can be shown that ρ. 6 In order to sf ater dervatons, t s convenent to defne the correaton factor of two events and as 5

16 . 7 ang Eq. 7 to Eq. 6, the foowng reatonsh can be derved: ρ. 8 Thus, s reated to ρ through scang and shftng. The vaue of ρ, b defnton [8], s a rea nuber n the nterva [-, ]; therefore, accordng to Eq. 8, taes on rea non-negatve vaues. so, ρ 0 corresonds to, and ndcates that the events and are utua ndeendent. Sar, ρ < 0.e., and are negatve correated corresonds to 0 <, and ρ > 0.e., and are ostve correated corresonds to >... Marov han Mode for asc Logc Gates The focus n ths subsecton s on dervng the Marov chan ode for the outut of a basc ogc gate n whch the Marov chan odes of the nut sgnas are nown. The se case of a NOT gate s consdered frst foowed b the anass of two-nut basc ogc gates. For a NOT gate wth nut, the ooean outut functon s gven b Y. Fro Fgure, t s cear that the Marov ode for Y s gven b Y, Y, Y. 9 onsder now the case of a two-nut basc ogc gate, as shown n Fgure 4. ssung the Marov chan odes of and are nown, the obectve s to derve the Marov chan ode for outut sgna Y. 6

17 Two-Inut Logc Gate Y Fgure 4. Generc two-nut ogc gate. e to dervng the Marov chan ode for sgna Y of Fgure 4 s to reresent the state transton dagra assocated wth the gate s two nuts, as shown n Fgure 5. The four states n the fgure corresond to the four nut cobnatons for the two nuts. The frst dgt of each state abe corresonds to the vaue of, and the second to the vaue of, e.g., the state abeed 0 corresonds to 0 and. though not abeed on the fgure, the drected edges reresent transton events. To ustrate the notaton to abe transton events, 00 0 w be used to reresent the event that nut sgna transtons fro 0 to and sgna stas n state Fgure 5. State transton dagra for nuts and of Fgure 4. The nown araeters of the Marov chan odes for sgnas and are gven b,,,,, and. so assued to be nown are the correaton factors for ars of 7

18 events assocated wth the Marov chan odes for the nuts. Fro Eq. 7 note that, where s the correaton factor assocated wth events and. Sar, the correaton factor enabes the cacuaton of usng the fact that. Reca fro Eq. 8 that ndeendent events corresond to a correaton factor of unt. Gven the Marov chan odes for sgnas and and the corresondng correaton factors t s ossbe to derve the robabt assocated wth ever event shown n the state transton dagra of Fgure 5. To ustrate, consder the robabt of event 00 0: 00 0 [ ] 0 Eressons for the robabtes of a events assocated wth the state transton dagra of Fgure 5 can be derved sar; a coete tabuaton of these eressons are gven n Tabe. Tabe. robabtes of events assocated wth Fgure 5. Event robabt state state 0 0 state 0 0 state Dervng transforatons to deterne correatons factors assocated wth ars of sgnas w be dscussed n Subsecton.; for uroses of the resent subsecton the are assued to be nown. 8

19 Dervng a Marov chan ode for Y of Fgure 4 deends on the artcuar functon of the gate. To ustrate how to deterne the Marov chan ode for Y, consder the secfc eae of an ND gate,.e., Y. For an ND gate, the outut taes on ogc vaue f and on f both nuts are. Thus, 9

20 Y. The event Y s assocated wth three events fro Fgure 5, nae: 00, 0, and 0. Thus, equat can be estabshed as foows: Y Y Sovng Eq. for Y and usng Eqs. through 4 resuts n the foowng eresson: Y 4 where, and. Dervaton for Y foows n a sar fashon and can be eressed as. 4 Y Dervatons of Y, Y, and Y for two-nut OR and XOR gates,.e. Y and Y resectve, are sar to the above dervaton for the ND gate and the resuts are shown n Tabe. To reduce the notatona burden, the foruas n Tabe are eressed n ters of sgna robabtes and actvtes nstead of the Marov chan araeters.e., Eqs. to 4 were aed. Tabe. Foruas for coutng Marov chan araeters for the outut of basc gates. Gate Y Y Y NOT Y 0

21 ND Y 4 OR Y 4 XOR Y ng Eqs. to 4, and usng the araeter resuts sted n Tabe, the robabt and actvt vaues of the outut sgna Y of these two-nut ND, OR and XOR gates and the NOT gate can be derved and the resuts are shown n Tabe 4. Tabe 4. robabt and actvt vaues of outut sgnas of basc gates. Gate Y Y NOT Y ND Y OR Y

22 Y XOR.. acuaton of orreaton Factors The urose of ths subsecton s to rovde ethods for cacuatngroagatng correaton factors through basc eeents of a crcut. For two sgnas and, there are two nds of correatons that need to be estabshed: robabt correaton factor donated as corresondng to correaton factor between event and event and transton correaton donated as corresondng to correaton factor between event and event, where, {, } and and are transton events corresondng to sgna and sgna resectve as shown n Fgure. The frst rue to be estabshed s the fan-out rue assocated wth the crcut dagra n Fgure 6. Fgure 6. The crcut dagra assocated wth the fan-out rue. ecause sgna s the sae sgna as,, Q and 5

23 Q 6 Sar, The second rue s naed ND rue and s assocated wth the crcut dagra n Fgure 7. Fgure 7. The crcut dagra assocated wth the ND rue. Gven correaton factors between nut sgnas, and, the correaton factors between outut sgnas and can be derved b foows: ecause and usng the resuts n Tabe 4,, 8

24 4 so So. Sovng Eg. b ang Eqs. 0 and, 4

25 5 Other correaton factors.e.,,, and can be obtaned sar: Dervatons of correaton factors for OR and XOR gates foow n a sar fashon.

26 6 Fgure 8. The crcut dagra assocated wth the OR rue. Fgure 8 shows the crcut dagra assocated wth the OR rue, and the correaton cofactors of the OR rue can be derved as foows: so

27 7 Y Y

28 Fgure 9 shows the rcut dagra assocated wth the XOR rue, foowed b the dervaton of the correaton factors. Fgure 9. The crcut dagra assocated wth the XOR rue. 4

29 a a a a 4 a a a

30 Fna, Fgure 0 s the rcut dagra assocated wth the NOT rue, foowed b the correaton factor dervaton. Fgure 0. The crcut dagra assocated wth the NOT rue

31 The resuts of these basc rues used to roagate correaton factors fro the nuts to the outut are sted n Tabe 5. These basc rues aong wth the transforatons for deternng the Marov chan araeters for the outut of a ogc functon Tabe are the foundatona coonents for the agorth deveoed n the net secton. Tabe 5. Set of basc rues used to cacuate the outut correaton factors. Rues robabt orreaton Factors Transton orreaton Factors Indeendent rue Fan-out rue 0 0 ND rue

32 OR rue NOT rue XOR rue a a a

33 a a a 4. Marov han roagaton gorth Ths secton descrbes a roosed Marov han roagaton M agorth for deternng the Marov chan odes for a sgnas of a gven cobnatona crcut. The Marov chan sgna ode of Secton s eoed, and t s assued that the araeters of the ode are nown for the crcut s rar nuts. The overa aroach s to roagate sgna nforaton assocated wth the Marov chan ode through the crcut n a gate-b-gate fashon. Reca that once the Marov chan ode s deterned for a sgnas, the sgna actvtes and crcut ower estate are deterned usng Eq. and Eq., resectve. It s assued that the gven crcut s secfed at the eve of basc ogc gates.

34 M gorth Ste : Reresent the gven cobnatona crcut as a drected accc grah DG. Vertces of the DG corresond to basc gates and edges reresent sgnas. Two etra vertces a source and a sn are ncuded n the DG to accoodate the rar nuts and oututs of the crcut. n eae of how to reresent a crcut wth the DG ode s ustrated b Fgures a and b. Ste : erfor a tooogca sort [0] on the DG to obtan an orderng of the gates. See Fgure c. Ste : Transforaton to two-nut basc ogc gates. s shown n Fgure d, reace a basc gates havng ore than two nuts wth an equvaent sequence of two-nut basc gates. Ste 4 artton the crcut nto eves. s shown n Fgure e, eves are defned at the nut and outut of each basc gate. Note that there s at ost one gate between an two consecutve eves. Ste 5: Successve a roagaton rues at each eve. the roagaton rues fro Tabes and 5 for cacuatng the araeters of the Marov ode for the basc gate oututs and the assocated correaton factors a 4

35 b c d L L L L 4 L 5 L 6 L 7 L 8 L 9 L 0 L L L L e Fgure. Iustraton of the basc stes of the M gorth. 5

36 In dervng the te coet of the M agorth, et N denote the nuber of basc gates, M be the nuber of fan-outs, and S the nuber of hsca sgnas. Fan-out s assocated wth a sgna that s broadcast.e., ducated. To ustrate, for the crcut of Fgure e, N7, M7, S7. ecause two eves are assocated wth each gate one s aced before the gate and the other after, there are N eves for a crcut wth N gates, whch s 4 eves for the eae shown n Fgure e. onstructng the DG Ste fro the gven crcut requres ONS oeratons and t s shown n [0] that tooogca sort Ste aso requres ONS oeratons. Ste can be fnshed wth no ore than S oeratons and at ost N oeratons are needed for Ste 4. For Ste 5, there are two cases: fro eve L to eve L and fro eve L to eve L,where,,, N-. For the frst case, because there s on one gate e.g., gate when 5 as shown n Fgure e between eve L and eve L, the cacuaton needed s to roagate the nuts of the snge gate to the outut of that gate. s shown n Fgure e, when 5, the three araeters of the outut sgna of gate can be obtaned n a constant nuber of oeratons, denoted b. The correaton factors between ths outut sgna and other sgnas need to be cacuated and nserted to the correaton factor tabe durng ths ste. ecause of the foowng three facts, t foows that the nuber of oeratons needed for ths case of Ste 5 can be eressed as S : on those sgnas havng correatons wth the nut sgnas of the gate w have correatons wth the outut sgna of the gate need to be cacuated; the au ength of the correaton tabe of ever entr s no ore than S; and the correaton factors between two sgnas can be done n a constant nuber of oeratons assued to be usng basc rues shown n Tabe 5. 6

37 For the other case there sn t a gate between eve L to eve L e.g., as shown n Fgure e, when 5, ths corresonds to L 6 to L 7. The on cacuaton needed n ths case s to cacuate the correaton factors due to recovergent fan-outs. ssue there are fan-outs fro eve L to eve L. The needed nuber of oeratons s bounded b. So the tota nuber of oeratons n Ste 5 s thereb N S M N NS O NS obnng the derved coet resuts of Ste to Ste 5, the te coet of ths M gorth s ONS. 5. Eerenta Resuts The M gorth has been eented and evauated usng severa test crcuts. To verf the accurac of the resuts roduced b the M agorth, Sce crcut suatons were erfored on the sae test crcuts. In the suaton studes, te-seres reazatons fro the assued Marov chan ode for each rar nut were used to drve the crcut suaton. Estates of sgna robabtes were derved fro the suatons b countng the fracton of te each sgna too on a vaue of unt. Estates of sgna actvtes were derved fro the suatons b countng sgna transtons. Fgure shows a s-gate crcut used for nta testng and evauaton. The coarson between robabt and actvt vaues roduced b the M gorth and those roduced through suaton are rovded n Tabe 6. 7

38 Fgure. se test crcut. Tabe 6. Resuts fro M gorth and Suaton Studes for the crcut of Fgure. Sgna Sgna robabt Sgna ctvt No. M Suaton M Suaton The M gorth was aso evauated usng a crcut naed 4 fro the ISS-85 enchar Set. For ths crcut there are a tota of 45 dstnct sgnas, not ncudng the rar nuts. Note that there are a tota of 4 hsca sgnas, whch ncudes fan-out sgnas. Tabe 7 show the dstrbuton of absoute dfferences between actvt vaues couted b the M gorth and those derved through suaton. These resuts ndcate that the M gorth roduces ver accurate redctons of sgna actvtes. Tabe 7. Resuts fro M gorth and Suaton Studes for rcut 4 fro the ISS-85 enchar Set. Range of Dfference n ctvt Vaues Nuber of Sgnas [0, 0.0] , 0.0] 5 8

39 0.0, 0.0] 9 0.0, 0.04] , 0.05] , 0.06] 0.06, ] 0 6. Suar and Future Wor The robe of deternng the actvtes of a sgnas of a cobnatona crcut s addressed n ths aer. new sgna ode s roosed based on a Marov chan. Sgna actvt s eas couted fro the araeters assocated wth the roosed sgna ode. In the roosed aroach, sgnas wth nown Marov chan reresentatons are roagated through the crcut to roduce a Marov chan reresentaton for the outut of each gate n the crcut. ccurac of the aroach s verfed b coarng sgna actvtes roduced b the roosed ethod wth corresondng actvtes roduced through suaton studes. These nta testng resuts w be etended n future wor b testng ore and arger crcuts. The current crcut ode assues zero roagaton dea through each gate. In reat, gates have non-zero deas, whch resuts n sgna gtchng. To ustrate how non-zero deas cause gtches, consder an eae crcut as shown n Fgure a. Under the assuton of zero dea, the sae nut sgnas, and resut n the outut sgnas and as shown n Fgure b. Notce that outut sgna eerences no transtons. For non-zero deas assue the dea of each gate s d the outut sgna for the sae nuts s derved and shown n Fgure c, whch has severa gtchng transtons. ower consuton s acted b these sgna gtches; thus, future wor s underwa to etend the wor resented n ths reort to consder the effect of gtches due to non-zero roagaton deas. 9

40 a b Te d d Te c Fgure. n eae used to show how non-zero deas cause gtches. 40

41 cnowedgents Ths research was suorted b DR under ontract F The authors woud e to than Dr. S. Lashvarahan for hs contrbutons to ths wor. References [] R. urch, F. N. Na,. Yang, and T. Trc, Monte aro roach for ower Estaton, IEEE Trans. VLSI Sstes, Vo., No., Mar. 99, [] K.. arer and E. J. Mcuse, robabstc Treatent of Genera obnatona Networs, IEEE Trans. outers, Vo. -4, No. 6, June 975, [] F. N. Na, Transton Denst: New Measure of ctvt n Dgta rcuts, IEEE Trans. outer-ded Desgn of Integrated rcuts and Sstes, Vo., No., Feb. 99,. 0-. [4] T.-L. hou and K. Ro, Estaton of ctvt for Statc and Dono MOS rcuts onsderng Sgna orreatons and Sutaneous Swtchng, IEEE Trans. outer-ded Desgn of Integrated rcuts and Sstes, Vo. 5, No. 0, Oct. 996, [5] F. N. Na, Surve of ower Estaton Technques n VLSI rcuts, IEEE Trans. on VLSI Sstes, Vo., No. 4, Dec. 994, [6] S. Ercoan, M. Fava, M. Daan,. Oovo, and. Rcco, Estate of Sgna robabt n obnatona Logc Networs, roc. IEEE Euroean Test onference, r 989,

42 [7]. Krshnaurth and I. G. Tos, Iroved Technques for Estatng Sgna robabtes, IEEE Trans. outers, Vo. 8, No. 7, Ju 989, [8] J.. Thoas, n Introducton to ed robabt and Rando rocesses, Kreger ubshng, Huntngton, NY, 98. [9] M. J. M. Sth, caton-secfc Integrated rcuts, ddson Wese, Readng, M, 997. [0] T. H. oren,. E. Leserson, R. L. Rvest, and. Sten, Introducton to gorths, McGraw- H New Yor, NY, 00. 4