ECE 274 Digital Logic Fall Digital Design. Optimization and Tradeoffs Introduction. Optimization and Tradeoffs Introduction

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1 ors) t r (t ECE 27 Digitl Logi ll 28 Optimitio Troffs Two-Lvl Miimitio, Krugh Mps, Et Huristi Miimitio, Multi-lvl Miimitio Digitl Dg..2 Digitl Dg Chptr : Optimitio Troffs Slis to omp th ttook Digitl Dg, irst Eitio, rk Vhi, Joh Wil Sos Pulishrs, Copright 27 rk Vhi strutors of ourss rquirig Vhi's Digitl Dg ttook (pulish Joh Wil Sos) hv prmiso to moif us ths slis for ustomr ours-rlt tivitis, sujt to kpig this opright oti i pl umoifi. Ths slis m post s uimt pf vros o pulil-sl ours wts.. PowrPoit sour (or pf with imtios) m ot post to pulil-sl wts, ut m post for stuts o itrl prott ts or istriut irtl to stuts othr ltroi ms. strutors m mk pritouts of th slis vill to stuts for rsol photoopig hrg, without iurrig roltis. A othr us rquirs pliit prmiso. strutors m oti PowrPoit sour or oti spil us prmisos from Wil s for iformtio. Optimitio Troffs troutio W ow kow how to uil igitl iruits How w uil ttr iruits? Lt s or two importt g ritri Dl th tim from iputs hgig to w orrt stl output Si th umr of trstors or quik stimtio, ssum Trsformig to rprsts Evr gt hs l of gt-l optimitio: Bttr i ll Evr gt iput rquirs 2 trstors ritri of itrst gor ivrtrs w w trstors 2 gt-ls trstors gt-l w = w + w = w = w(+ ) = w () (trstors) l (gt-ls) () Optimitio Troffs troutio Troff mprovs som, ut worss othr, ritri of itrst w w trstors 2 gt-ls w G G = w + w + G2 = w(+) + 2 trstors 3 gt-ls Trsformig G to G2 rprsts troff: Som ritri ttr, othrs wors. G2 (trstors) 2 G G2 2 3 l (gt-ls) 3

2 s Optimitio Troffs troutio Optimitios All ritri of itrst r improv (or t lst kpt th sm) l l Troffs Som ritri of itrst r improv, whil othrs r wors W oviousl prfr optimitios, ut oft must pt troffs You t uil r tht is th most omfortl, hs th st ful ffii, is th fstst ou hv to giv up somthig to gi othr thigs. Optimitio Troffs Comitiol Logi Optimitio Troffs Two-lvl optimitio ug lgri mthos Gol: iruit with ol two lvls (OR AND gts), with miimum trstors Though trstors gttig hpr (Moor s Lw), th still ost somthig Dfi prolm lgrill Sum-of-prouts ils two lvls = + is sum-of-prouts; G = w( + ) is ot. Trsform sum-of-prouts qutio to hv fwst litrls trms Eh litrl trm trslts to gt iput, h of whih trslts to out 2 trstors (s Ch. 2) gor ivrtrs for mpliit Not: Assumig -trstor 2-iput AND/OR iruits; i rlit, ol NAND/NOR r so ffiit..2 Empl = = ( + ) + ( + ) = * + * = + litrls + 2 trms = m gt iputs gt iputs = 2 trstors m m m Optimitio Troffs Algri Two-Lvl Si Miimitio Prvious mpl show ommo lgri miimitio mtho (Multipl out to sum-of-prouts, th) Appl followig s muh posl + = ( + ) = * = Comiig trms to limit vril (ormll ll th Uitig thorm ) Duplitig trm somtims hlps Not tht os t hg futio + = + + = Somtims ftr omiig trms, omi rsultig trms G = G = ( +) + (+ ) G = + (ow o gi) G = ( +) G = = = ( + ) + ( + ) = * + * = + = + + = = (+ ) + ( +) = + 7 Optimitio Troffs Krugh Mps for Two-Lvl Si Miimitio Es to miss g posl opportuitis to omi trms Krugh Mps (K-mps) Grphil mtho to hlp us fi opportuitis to omi trms Mitrms iffrig i o vril r jt i th mp C lrl s opportuitis to omi trms look for jt s or, lrl two opportuitis Top lft irl is shorth for + = ( +) = () = Drw irl, writ trm tht hs ll th litrls pt th o tht hgs i th irl Cirl, = & = i oth lls of th irl, ut hgs (= i o ll, i th othr) Miimi futio: OR th fil trms Er th ll tht lgr: = Noti ot i ir orr K-mp Trt lft & right s jt too = + = = ( + ) + ( + ) = * + * = + 8 2

3 Optimitio Troffs Krugh Mps for Two-Lvl Si Miimitio Optimitio Troffs Krugh Mps for Two-Lvl Si Miimitio our jt s ms two vrils limit Mks ituitiv ss thos two vrils ppr i ll omitios, so o must tru Drw o ig irl shorth for th lgri trsformtios ov G = G = ( ) (must tru) G = ( ( +) + (+ )) G = ( +) G = G G Drw th iggst irl posl, or ou ll hv mor trms th rll our jt lls i shp of squr OK to ovr twi Just lik uplitig trm Rmmr, + = + + No to ovr s mor th o Yils tr trms ot miimi J H H = ( pprs i ll omitios) Th two irls r shorth for: = = = ( + ) + ( ) = ( ) + () 9 Optimitio Troffs Krugh Mps for Two-Lvl Si Miimitio Cirls ross lft/right s Rmmr, gs r jt Mitrms iffr i o vril ol Cirls must hv, 2,, or 8 lls 3,, or 7 ot llow 3//7 os t orrspo to lgri trsformtios tht omi trms to limit vril Cirlig ll th lls is OK utio just quls K L E Optimitio Troffs Krugh Mps for Two-Lvl Si Miimitio our-vril K-mp follows sm priipl Ajt lls iffr i o vril Lft/right jt Top/ottom lso jt vril mps ist But hr to us Two-vril mps ist But ot vr usful s to o lgrill h w w G w =w + G= 2 3

4 Optimitio Troffs Krugh Mps for Two-Lvl Si Miimitio Optimitio Troffs Krugh Mps for Two-Lvl Si Miimitio Grl K-mp mtho. Covrt th futio s qutio ito sum-of-prouts form 2. Pl s i th pproprit K-mp lls for h trm 3. Covr ll s rwig th fwst lrgst irls, with vr ilu t lst o; writ th orrspoig trm for h irl. OR ll th rsultig trms to rt th miimi futio. Empl: Miimi: G = + + *( + ). Covrt to sum-of-prouts G = Pl s i pproprit lls G 3. Covr s G. OR trms: G = + Miimi: H = ( + ) Covrt to sum-of-prouts: H = Pl s i K-mp lls 3. Covr s. OR rsultig trms H H = + + u-lookig irl, ut rmmr tht lft/right jt, top/ottom jt 3 Optimitio Troffs Krugh Mps: Do t Cr put Comitios Optimitio Troffs Krugh Mps: Do t Cr put Comitios Wht if prtiulr iput omitios vr our?.g., Miimi =, giv tht (=) vr tru, tht (=) vr tru So it os t mttr wht outputs wh or is tru, us thos ss will vr our Thus, mk or for thos ss i w tht st miimis th qutio O K-mp Drw Xs for o t r omitios lu X i irl ONLY if miimis qutio Do t ilu othr Xs X X u Goo us of o t rs X X Ussr us of o t rs; rsults i tr trm Miimi: = + + Giv o t rs:, Not: Us o t rs with utio Must sur tht w rll o t r wht th futio outputs for tht iput omitio f w o r, v th slightst, th it s prol sfr to st th output to X X = +

5 Optimitio Troffs Krugh Mps: Do t Cr put Comitios Empl: Swith with potios 3-it vlu givs potio i ir Wt iruit tht Outputs wh swith is i potio 2, 3, or Outputs wh swith is i potio or Not tht th 3-it iput vr output ir,, or 7 Trt s o t r iput omitios 2 3 Withou t o t rs: = + G G 2,3,, ttor X X X G With o t rs: = + Optimitio Troffs Automtig Two-Lvl Logi Si Miimitio Miimiig h s hr for futios with or mor vrils M ot il miimum ovr pig o orr w hoos s rror pro Miimitio thus tpill o utomt tools Et lgorithm: fis optiml solutio Huristi: fis goo solutio, ut ot ssril optiml () trms Ol 3 trms 7 8 Optimitio Troffs B Copts Urlig Automt Two-Lvl Logi Miimitio Dfiitios O-st: All mitrms tht fi wh = Off-st: All mitrms tht fi wh = mplit: A prout trm (mitrm or othr) tht wh uss = O K-mp, lgl (ut ot ssril lrgst) irl Covr: mplit ovrs mitrms Epig trm: rmovig vril (lik lrgr K-mp irl) is po of implits of Not: W us K-mps hr just for ituitiv illustrtio of opts; utomt tools o ot us K-mps. Prim implit: Mimll p implit po woul ovr s ot i o-st,, ov But ot or th p Optimitio Troffs B Copts Urlig Automt Two-Lvl Logi Miimitio Dfiitios (ot) Esstil prim implit: Th ol prim implit tht ovrs prtiulr mitrm i futio s o-st mport: W must ilu ll sstil Ps i futio s ovr otrst, som, ut ot ll, osstil Ps will ilu G ot sstil sstil ot sstil sstil 9 2

6 Optimitio Troffs Automt Two-Lvl Logi Miimitio Mtho Optimitio Troffs Empl of Automt Two-Lvl Miimitio Stps 2: Et Stp 3: Hr. Chkig ll posilitis: t, ut omputtioll pv. Chkig som ut ot ll: huristi. 2. Dtrmi ll prim implits 2. A sstil Ps to ovr tlii s r thus lr ovr Ol o uovr rmis 3. Covr rmiig mitrms with osstil Ps Pik mog th two posl Ps () () 22 Optimitio Troffs Prolm with Mthos tht Eumrt ll Mitrms or Comput ll Prim mplits Too m mitrms for futios with m vrils utio with 32 vrils: 2 32 = illio posl mitrms. Too muh omput tim/mmor Too m omputtios to grt ll prim implits Comprig vr mitrm with vr othr mitrm, for 32 vrils, is ( illio) 2 = qurillio omputtios utios with m vrils oul rquirs s, moths, rs, or mor of omputtio ursol Optimitio Troffs Solutio to Computtio Prolm Solutio Do t grt ll mitrms or prim implits st, just tk iput qutio, tr to itrtivl improv it E: = fgh + fgh + jklmop Not: vrils, m hv thouss of mitrms But miimi just omiig first two trms: = fg(h+h ) + jklmop = fg + jklmop 23 2

7 ) ) ors t s r (t Optimitio Troffs Two-Lvl Miimitio ug trtiv Mtho Mtho: Roml ppl p oprtios, s if hlps Ep: rmov vril from trm Lik pig irl o K-mp.g., Epig to lgl, ut pig to ot lgl, i show futio Aftr p, rmov othr trms ovr wl p trm Kp trig (itrt) util os t hlp E: = fgh + fgh + jklmop = fg + fgh + jklmop = fg + jklmop () Optimitio Troffs Multi-Lvl Logi Optimitio Prform/Si Troffs W o t lws th sp of two lvl logi Multipl lvls m il fwr gts Empl = + + = + ( + ) = ( + ( + )) Grl thiqu: tor out litrls + = (+) 22 trstors 2 gt ls = + + () trstors gt-ls = (+(+)) (trstors) 2 () 2 3 l (gt-ls) 2 2 Optimitio Troffs Multi-Lvl Logi Optimitio Prform/Si Troffs Us multipl lvls to ru umr of trstors for = + f Solutio = + f = ( + f) Hs fwr gt iputs, thus fwr trstors f 22 trstors 8 trstors 2 gt ls 3 gt ls 2 8 f 2 3 l (gt-ls) = + f = ( + f) () () (trstors) 27 Optimitio Troffs Multi-Lvl Empl: No-Critil Pth Critil pth: logst l pth to output Optimitio: ru of logi o o-ritil pths ug multipl lvls g f 2 trstor s 3 gt-l s = (+) + fg + fg () f g 22 trstor s 3 gt-l s = (+) + (+)fg (trstors ) 2 2 () 2 3 l (gt-l s) 28 7