Multipliers. Overview. Introduction. Reading. Computer Systems Laboratory. Stanford University. Copyright 2001 by Mark Horowitz

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1 Lctr : ltplrs ptr Systs Laratry Stafrd Uvrsty hrwtz@stafrd.d pyrght 00 y ark Hrwtz H/JZ EE 7 Lctr Ovrvw adg Itrdct Thr ar ts f paprs wrtt ltplcat. Ufrtatly t s rar that th papr talks at th lgc ad crct sss tgthr. Thr ar ay lgc paprs wth dscr clvr ways f addg th partal prdcts, t thy rarly addrss th cst f th wrg. Othr crct paprs talk at fast ways t ld th asc addr clls. W wll start wth th ltplr rgazat, ad lk at th th lgc ad crct sss. H/JZ EE 7 Lctr

2 ltplcat asc da s prtty spl * = D y gratg a r f partal prdcts, ad addg th tgthr: w f dts s a partal prdct: slt f ltplyg y t f ( s th ltplcad) slt s ts Fr tgr prats wat LS ts, fr F wat S F.XXXX *.XXX H/JZ EE 7 Lctr Spl ltplr Taks cycls, ad grats ad adds partal prdct ach cycls: ltplcad artal rdct Grat ltplr ddr Shft rght ach cycl slt Shft Shft rght ach cycl H/JZ EE 7 Lctr 4

3 . rat th partal prdcts. S th tgthr a Isss Fr a fast ltplr dc th r f partal prdcts Fast ddr clls Tr addrs Talk at ths stff t H/JZ EE 7 Lctr 5 artal rdct Grat I th spl fr t s vry asy: 0 * = 0 * = Ths s a spl ND gat at ach t t grat a fr ach ltplr t W ca grat a fr vry tw ltplr ts * 0 * (ths s asy t grat y a shft) * (ych -- t ca ak asy 4 -, ad dlay th 4 t th t grp H/JZ EE 7 Lctr 6

4 dfd th dg ak t asy Wh th S f th ts s add 4 t t grp p ( = 4 -) th rcdr lks at ts Th tw ts t s rcdg, ad th S f prvs grp Tal S LS rv Ot H/JZ EE 7 Lctr 7 th Ecdr Fr d crcts Otpts ar ht a d gat Nd larg tpt ffrs t drv lg wr t ctrl all th that ar cctd t t. lk Zr Dr 4 lck Dlay.4 6 Tw_ Dr lk 4/ 4/ 4/ 4/4 O_ OTw Dr Dr Dr _ Otpt Drvr (Dr) 9/ /0 4/5 0/60 00/0 lk 4 lk 5 _ ac ZrOO_TwTw_ Ipt Ivs 9/ _ H/JZ EE 7 Lctr 8

5 .4 0/4 th lk _ 8 Zr 6 O 6 Tw O_ Tw lk 0 ltplcad s cvrtd t dal ral partal prdcts aga y sg lcal vrtrs, ad s st dw t sgfcac hghr th, gratg. Th addrs gt tc dal ral partal prdcts fr th th. H/JZ EE 7 Lctr 9 t th a rcd ltplr ts at a t Grats / f th partal prdcts t y d p wth dg *ltplcad Ths taks a addr S ths s grally t d Tal S LS Ot Nd t lk at S f prvs grp ad add f t s. Ths stll kps th rs tw -4 ad 4. Th ly hard t grat s H/JZ EE 7 Lctr 0

6 ddg artal rdcts If y wat t add r partal prdcts a cycl y cld s ltpl addrs Ths slt has tw prls: Ev fast addrs ar prtty slw 7 FO4 dlays Fast addrs ar prtty larg Epsv ara artal rdct Grat ddr artal rdct Grat ddr artal rdct Grat ddr H/JZ EE 7 Lctr arry Sav ddr T gt a fastr addr w wll s a trck: prst th rslt f addr (s) a rddat fr S = S S That s wh y add S ad S tgthr y gt th rght rslt. Ths s c sc w w d t d t prpagat th carry th addr. t Σ Istad f rpplg th carry, shft t lft y, ad s t as th thr part f th s. Each stag f th addr taks pts, ad prdcs tw tpts. H/JZ EE 7 Lctr

7 arry Sav ddr Each addr ca add partal prdct t a rddat s ddat s taks tw pts, partal prdct taks th thrd Each addr s sall ad fast D S s ard.5 FO S s prtty spl, t s jst a fll addr Th prl s th tpt s rally a sal fr t th d f th array y d t add tw parts f rddat r tgthr Ths tak a fast addr, t y ly d at th d f ltplr, t fr ach partal prdct H/JZ EE 7 Lctr ltplr Ovrvw lck dagra f ltplr: N/N N ltplr ltplcad OOTH UXES Σ pp Σ N t N Ecdrs Th Σ array s carry sav addrs, ad fal s s a ral addr H/JZ EE 7 Lctr 4

8 rray ltplr Fr flatg pt rs y hav 5 t rs, whch gv 7 = S 4 S 7 S 5 5 Ss datapath, all crtcal path (Frst addr adds ) 56 ts tall rglar datapath Shrt wrs H/JZ EE 7 Lctr 5 Tr ltplrs a ld fastr Σ arrays t sg tr addr Nd t carfl sc: Trs hav lgr wrs (ach addr s a rw f addrs) Draw a ary tr, t S ar pt tpt dvcs a ld a 4: cr sg S H/JZ EE 7 Lctr 6

9 4- ddr t Σ If y gr th sd rachs y gt 4 pts ach t pst, ad tw tpt at ach t pst. Ths allws y t ld ary trs. ca ld : trs, Wallac Trs, (ad t s a lttl fastr) t th wrg s ch r cpl H/JZ EE 7 Lctr 7 Tr Layt Fll (4:) Trs: t-slc: 4 Datapath Hght: Ss datapath, 8 crtcal path Icrasd datapath hght Lg wrs, ay crss-vrs, wrg chals H/JZ EE 7 Lctr 8

10 prs slt Itrlavd rrays 4: r d d v 4: Fll-array crtcal path rdcd y half fr sall ara vrhad Lgr wrs wth th array a w d v ttr? H/JZ EE 7 Lctr 9 rray-f-rrays Gral Ida: p p p p r partal arrays: hghr parallls, wrg cplty, largr cg twrk hs dpdg prl sz H/JZ EE 7 Lctr 0

11 EE 7 Lctr H/JZ Tw artal rrays glar wrg, sall r f crss-vrs a ffctv wr ladg ~.5S dlays Fr ptal parallls, p= p=4. ~4.5 S dlays crtcal path 4: 4: r EE 7 Lctr H/JZ Fr artal rrays Tr cr: Wrg, ara vrhad atchg rray Dlays: 5570 pp.0 Ss Optal arallls: 448 pp. Ss r artal arrays?: argal rtrs 4 4: r

12 Dtals Thr ar tw dtals that w stll d t cvr Dalg wth th ts that fall ff th d f th array I th array rgazat th datapath wdth ra cstat Th tra (LS) ts fall ff th tp f th array t ths ts ar stll rddat fr Fr rdg ths ts d t addd t Dalg wth gatv partal prdcts Th prl s wth sg ts Nd t td th sg ts th fll N ts Ths s N addtal ts fr th frst Lk at ths sss t H/JZ EE 7 Lctr LSs Fallg Off th rray Spd thrgh th array s S dlay = S 4 S 7 S 5 t sc th ar th cdd, tw t psts fall ff ach stag Nd t d a tw t carry prpagat add S dlay ld a sall carry slct addr arry t carry t s ly a dlay H/JZ EE 7 Lctr 4

13 Fastr Strctrs D r adds paralll, akg dalg wth th d ts hardr I tr addr, all th ts fall ff th d at rghly th sa t rray f arrays has a slar prl Nd t ld a cplt carry prpagat addr fr ths ts Nral Fast ddr 4: r H/JZ EE 7 Lctr 5 4 rtsy D. Fty Sg Ets Wh th s gatv, d t sg td Ths s ad sc th fat ca larg Us a trck stad r-add th tragl f s 000 T clar t s add t th rw S H/JZ EE 7 Lctr 6

14 Sg Ets S y ly d t add fw ts: S SS S S S 000 ddg ths ts tgthr s th sa as cplt sg ts H/JZ EE 7 Lctr 7

Complex Numbers. Prepared by: Prof. Sunil Department of Mathematics NIT Hamirpur (HP)

Complex Numbers. Prepared by: Prof. Sunil Department of Mathematics NIT Hamirpur (HP) th Topc Compl Nmbrs Hyprbolc fctos ad Ivrs hyprbolc fctos, Rlato btw hyprbolc ad crclar fctos, Formla of hyprbolc fctos, Ivrs hyprbolc fctos Prpard by: Prof Sl Dpartmt of Mathmatcs NIT Hamrpr (HP) Hyprbolc