The work described by this report was supported by NSF under contract

Transcription

1 BINAR ULTIPLICATION UING PARTIALL REDUNDANT ULTIPLE Gary Bewck chae J. Fynn Technca Report No. CL-TR June 992 The work descrbed by ths report was supported by NF under contract IP

2 BINAR ULTIPLICATION UING PARTIALL REDUNDANT ULTIPLE by Gary Bewck chae J. Fynn Technca Report No. CL-TR June 992 Computer ystems Laboratory Departments of Eectrca Engneerng and Computer cence tanford Unversty tanford, Caforna Abstract Ths report presents an extenson to Booth's agorthm for bnary mutpcaton. ost mpementatons that utze Booth's agorthm use the 2 bt verson, whch reduces the number of parta products requred to haf that requred by a smpe add and shft method. Further reducton n the number of parta products can be obtaned by usng hgher order versons of Booth's agorthm, but t s necessary to generate mutpes of one of the operands (such as 3 tmes an operand) by the use of a carry propagate adder. Ths carry propagate addton ntroduces sgncant deay and addtona hardware. The agorthm descrbed n ths report produces such dcut mutpes n a partay redundant form, usng a seres of sma ength adders. These adders operate n parae wth no carres propagatng between them. As a resut, the deay ntroduced by mutpe generaton s mnmzed and the hardware needed for the mutpe generaton s aso reduced, due to the emnaton of expensve carry ookahead ogc. Key Words and Phrases: arthmetc mutpcaton, Booth's agorthm, redundant mutpes, computer

3 Copyrght c 992 by Gary Bewck chae J. Fynn

4 Contents Introducton 2 Background 2. Add and hft : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2.. Dot Dagrams : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Booth's Agorthm : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Booth 3 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Booth 4 and hgher : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 3 Redundant Booth 5 3. Booth 3 wth fuy redundant parta products : : : : : : : : : : : : : : : : : : : : : : Booth 3 wth partay redundant parta products : : : : : : : : : : : : : : : : : : : Deang wth negatve parta products : : : : : : : : : : : : : : : : : : : : : : 3.3 Booth wth bas : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3.3. Choosng the rght constant : : : : : : : : : : : : : : : : : : : : : : : : : : : : Producng the mutpes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Redundant Booth 3 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Redundant Booth 4 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Choosng the adder ength : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 4 ummary 2

5 Lst of Fgures 6 bt add and shft mutpy : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 6 bt add and shft exampe : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 3 Parta product seecton ogc for add and shft : : : : : : : : : : : : : : : : : : : : : bt Booth 2 mutpy : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : bt Booth 2 Exampe : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : bt Booth 2 Parta Product eector Logc : : : : : : : : : : : : : : : : : : : : : : bt Booth 3 mutpy : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : bt Booth 3 Exampe : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : bt Booth 3 Parta Product eector Logc : : : : : : : : : : : : : : : : : : : : : : 7 Booth 4 Parta Product eecton Tabe : : : : : : : : : : : : : : : : : : : : : : : : : 8 6 x 6 Booth 3 mutpy wth fuy redundant parta products : : : : : : : : : : : : bt fuy redundant Booth 3 exampe : : : : : : : : : : : : : : : : : : : : : : : : : 9 3 Computng 3 n a partay redundant form : : : : : : : : : : : : : : : : : : : : : : 4 Negatng a number n partay redundant form : : : : : : : : : : : : : : : : : : : : : 5 Booth 3 wth bas : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 6 Transformng the smpe redundant form : : : : : : : : : : : : : : : : : : : : : : : : : 3 7 ummng K utpe and Z : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 8 Producng K 3 n partay redundant form : : : : : : : : : : : : : : : : : : : : : 4 9 Producng other mutpes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : x 6 Redundant Booth 3 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : bt partay redundant Booth 3 mutpy : : : : : : : : : : : : : : : : : : : : : : : 6 22 Parta product seector for redundant Booth 3 : : : : : : : : : : : : : : : : : : : : : 7 23 Producng K 6 from K 3 : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 24 A derent bas constant for 6 and 3 : : : : : : : : : : : : : : : : : : : : : : : : : 9 25 Redundant Booth 3 wth 6 bt adders : : : : : : : : : : : : : : : : : : : : : : : : : : 2 v

6 Introducton utpcaton s a basc arthmetc operaton that s mportant n mcroprocessors, dgta sgna processng, and other modern eectronc machnes. VLI Desgners have recognzed ths mportance by dedcatng sgncant resources and area to nteger and oatng pont mutpers. As a resut, t s desrabe to reduce the cost of these mutpers by usng ecent agorthms that do not compromse performance. Ths report descrbes an extenson to Booth's agorthm for bnary mutpcaton that reduces the cost of such hgh performance mutpers. To expan ths scheme, background matera on exstng agorthms s presented. These agorthms are then extended to produce the new method. nce the method s somewhat compex, ths report w attempt to stay away from mpementaton detas, but nstead concentrate on the agorthms n a hardware ndependent manner. In order that the basc agorthms are not obscured wth sma detas, unsgned mutpcaton ony w be consdered here, but the agorthms presented are easy generazed to dea wth sgned numbers. 2 Background 2. Add and hft The rst and smpest method of mutpcaton to be addressed s the add and shft mutpcaton agorthm. Ths agorthm condtonay adds together copes of the mutpcand (parta products) to produce the na product, and s ustrated n Fgure, for a 6 x 6 mutpy. Each dot s Parta Product eecton Tabe utper Bt eecton utpcand Parta Products u t p e r sb sb Product Fgure : 6 bt add and shft mutpy

7 a pacehoder for a snge bt whch can be a zero or one. Each horzonta row of dots represents a snge copy of the mutpcand,, (.e. one parta product), whch s condtoned upon a partcuar bt of the mutper. The condtonng agorthm s shown n the seecton tabe n the upper eft hand corner of the gure. The mutper tsef s represented on the rght edge of the gure, wth the east sgncant bt at the top. The na product s represented by the row of 32 horzonta dots at the bottom of the gure. An exampe of the add and shft agorthm usng actua numbers s shown n Fgure 2. utper = = utpcand () = 49 = = = Product Fgure 2: 6 bt add and shft exampe u t p e r sb 2.. Dot Dagrams The dot dagram, as Fgure s referred to, can provde nformaton whch can be used as a gude for examnng varous mutpcaton agorthms. Roughy speakng, the number of dots (256 for Fgure ) n the parta product secton of the dot dagram s proportona to the amount of hardware requred (tme mutpexng can reduce the hardware requrement, at the cost of sower operaton [4]) to sum the parta products and form the na product. The atency of an mpementaton of a partcuar agorthm s aso reated to the heght of the parta product secton (.e the maxmum number of dots n any vertca coumn) of the dot dagram. Ths reatonshp can vary from ogarthmc (tree mpementaton where nterconnect deays are nsgncant) to near (array mpementaton where nterconnect deays are constant) to somethng n between (tree mpementatons where nterconnect deays are sgncant). Impementatons are not beng consdered here, so the ony concuson s that a smaer heght ought to be faster. Fnay, the ogc whch seects the parta products can be deduced from the parta product seecton tabe. For the add and shft agorthm, the ogc s partcuary smpe and s shown n 2

8 Fgure 3. Ths gure shows the seecton ogc for a snge parta product (a snge row of sb utpcand utper bt sb Parta Product Fgure 3: Parta product seecton ogc for add and shft dots). Frequenty ths ogc can be merged drecty nto whatever hardware s beng used to sum the parta products. Ths can reduce the deay of the ogc eements to the pont where the extra tme due to the seecton eements can be gnored. However, a rea mpementaton w st have nterconnect deay due to the physca separaton of the common nputs of each AND gate, and dstrbuton of the mutpcand to the seecton eements. 2.2 Booth's Agorthm nce fewer dots can be faster and requre ess hardware, how can the number (and heght) of dots be reduced A common method s to use Booth's Agorthm []. Hardware mpementatons commony use a sghty moded verson of Booth's agorthm, referred to appropratey as oded Booth's Agorthm [2]. Fgure 4 shows the dot dagram for a 6 x 6 mutpy usng the 2 bt verson of ths agorthm (Booth 2). The mutper s parttoned nto overappng groups of 3 bts, and each group s decoded to seect a snge parta product as per the seecton tabe. Each parta product s shfted 2 bt postons wth respect to t's neghbors. j kthe number of parta products has been n2 reduced from 6 to 9. In genera the there w be parta products, where n s the operand 2 ength. The varous requred mutpes can be obtaned by a smpe shft of the mutpcand (these are easy mutpes). Negatve mutpes, n 2's compement form, can be obtaned usng a bt by bt compement of the correspondng postve mutpe, wth a added n at the east sgncant poston of the parta product (the bts aong the rght sde of the parta products). An exampe mutpy s shown n Fgure 5. The number of parta products to be added has been reduced from 6 to 9 vs. the add and shft agorthm. In addton, the number of dots has decreased from 256 to 77 (ths ncudes sgn extenson and constants). Ths reducton n dot count s not a compete savng { the parta product seecton ogc s more compex (Fgure 6). In fact, dependng on actua mpementaton detas, the extra cost and deay due to the more compex parta product seecton ogc may overwhem the savngs due to the reducton n the number of dots [3]. 2.3 Booth 3 hft amounts between adjacent parta products of greater than 2 are aso possbe [2], wth a correspondng reducton n the heght and number of dots n the dot dagram. A 3 bt Booth dot dagram s shown n Fgure 7, and an exampe s shown n Fgure 8. Each parta product coud be from the set f,, 2, 3, 4 g. A mutpes wth the excepton of 3 are 3

9 Parta Product eecton Tabe utper Bts eecton utpcand utpcand 2 x utpcand -2 x utpcand - utpcand - utpcand - = f parta product s postve (top 4 entres from tabe) = f parta product s negatve (bottom 4 entres from tabe) u t p e r sb Fgure 4: 6 bt Booth 2 mutpy easy obtaned by smpe shftng and compementng of the mutpcand. The number of dots, constants, and sgn bts to be added s now 26 (for the 6 x 6 exampe) and the heght of the parta product secton s now 6. Generaton of the mutpe 3 (referred to as a hard mutpe, snce t cannot be obtaned va smpe shftng and compementng of the mutpcand) generay requres some knd of carry propagate adder to produce. Ths carry propagate adder may ncrease the atency, many due to the ong wres that are requred for propagatng carres from the ess sgncant to more sgncant bts. ometmes the generaton of ths mutpe can be overapped wth an operaton whch sets up the mutpy (for exampe the fetchng of the mutper). Another drawback to ths agorthm s the compexty of the parta product seecton ogc, an exampe of whch s shown n Fgure 9, aong wth the extra wrng needed for routng the 3 mutpe. 2.4 Booth 4 and hgher A further reducton n the number and heght n the dot dagram can be made, but the number of hard mutpes requred goes up exponentay wth the amount of reducton. For exampe the Booth 4 agorthm (Fgure ) requres the generaton of the mutpes f,, 2, 3, 4,5,6,7,8g. The hard mutpes are 3 (6 can be obtaned by shftng 3), 5 and 7. The formaton of the mutpes can take pace n parae, so the extra cost many nvoves the adders for producng the mutpes, and the addtona wres that are needed to route the varous mutpes around. 4

10 utper = = utpcand () = 49 = sb u t p e r Fgure 5: 6 bt Booth 2 Exampe 3 Redundant Booth Ths secton presents a new varaton on the Booth 3 agorthm, whch emnates much of the deay and part of the hardware assocated wth the mutpe generaton, yet produces a dot dagram whch can be made to approach that of the conventona Booth 3 agorthm. Before ntroducng ths varaton, a smpe and smar method (but s not partcuary hardware ecent) s expaned. Ths method s then extended to produce the new varaton. ethods of further generazng to a Booth 4 agorthm are then dscussed. 3. Booth 3 wth fuy redundant parta products The tme consumng carry propagate addton that s requred for the hgher Booth agorthms can be emnated by representng the parta products n a fuy redundant form. Ths method s ustrated by examnng the Booth 3 agorthm, snce t requres the fewest mutpes. A fuy redundant form represents an n bt number by two n bt numbers whose sum equas the number t s desred to represent (there are other possbe redundant forms see [5]). For exampe the decma number 4568 can be represented n redundant form as the par (4568,), or (4567,), etc. Usng ths representaton, t s trva to generate the 3 mutpe requred by the Booth 3 agorthm, snce 3 = 2, and 2 and are easy mutpes. The dot dagram for a 6 bt Booth 3 mutpy usng ths redundant form for the parta products s shown n Fgure (an exampe appears n Fgure 2). The dot dagram s the same as that of the conventona Booth 3 dot dagram, but each of the parta products s twce as hgh, gvng roughy twce the number of dots and twce the heght. Negatve mutpes are obtaned by the same method as the prevous Booth agorthms { bt by bt compementaton of the correspondng postve mutpe wth a added at the sb. nce every parta product now conssts of two numbers, there w be two s added at the 5

11 sb utpcand eect eect 2 utper Group Booth Decoder sb 2 more And/Or/Excusve- Or bocks sb Parta Product Fgure 6: 6 bt Booth 2 Parta Product eector Logc utper Bts Parta Product eecton Tabe eecton utper Bts utpcand utpcand 2 x utpcand 2 x utpcand 3 x utpcand 3 x utpcand 4 x utpcand eecton -4 x utpcand -3 x utpcand -3 x utpcand -2 x utpcand -2 x utpcand - utpcand - utpcand - = f parta product s postve (eft-hand sde of tabe) = f parta product s negatve (rght-hand sde of tabe) u t p e r sb Fgure 7: 6 bt Booth 3 mutpy 6

12 utper = = utpcand () = 49 = 3 x utpcand (3) = 2357 = sb u t p e r Fgure 8: 6 bt Booth 3 Exampe Bts of utpcand and 3 x utpcand utpcand Bt k 3 x utpcand Bt k utpcand Bt k- utpcand Bt k-2 eect eect 3 eect 2 utper Group eect 4 Booth Decoder sb of 8 mutpexer bocks Bt k of Parta Product Fgure 9: 6 bt Booth 3 Parta Product eector Logc 7

13 utper Bts eecton utpcand utpcand 2 x utpcand 2 x utpcand 3 x utpcand 3 x utpcand 4 x utpcand utper Bts Parta Product eecton Tabe eecton utper Bts 4 x utpcand 5 x utpcand 5 x utpcand 6 x utpcand 6 x utpcand 7 x utpcand 7 x utpcand 8 x utpcand eecton -8 x utpcand -7 x utpcand -7 x utpcand -6 x utpcand -6 x utpcand -5 x utpcand -5 x utpcand -4 x utpcand utper Bts eecton -4 x utpcand -3 x utpcand -3 x utpcand -2 x utpcand -2 x utpcand - utpcand - utpcand - Fgure : Booth 4 Parta Product eecton Tabe u t p e r sb Fgure : 6 x 6 Booth 3 mutpy wth fuy redundant parta products 8

14 utper = = 3 = 2357 = utpcand () = 49 = 4 = 6476 = sb u t p e r Fgure 2: 6 bt fuy redundant Booth 3 exampe sb, whch can be combned nto a snge whch s shfted to the eft one poston. Athough ths agorthm s not partcuary attractve, due to the doubng of the number of dots n each parta product, t provdes a steppng stone to a reated, ecent agorthm. 3.2 Booth 3 wth partay redundant parta products The conventona Booth 3 agorthm assumes that the 3 mutpe s avaabe n non-redundant form. Before the parta products can be summed, a tme consumng carry propagate addton s needed to produce ths mutpe. The Booth 3 agorthm wth fuy redundant parta products avods the carry propagate addton, but has the equvaent of twce the number of parta products to sum. The new scheme tres to combne the smaer dot dagram of the conventona Booth 3 agorthm, wth the ease of the hard mutpe generaton of the fuy redundant Booth 3 agorthm. The dea s to form the 3 mutpe n a partay redundant form by usng a seres of sma ength adders, wth no carry propagaton between the adders (Fgure 3). If the adders are of sucent ength, the number of dots per parta product can approach the number n the nonredundant representaton. Ths reduces the number of dots needng summaton. If the adders are sma enough, then carres are not propagated across arge dstances, and are faster than a fu carry propagate adder. Aso, ess hardware s requred due to the emnaton of the ogc whch propagates carres between the sma adders. There s a desgn tradeo whch must be resoved here. 9

15 Fuy redundant form bt adder 4 bt adder 4 bt adder 4 bt adder Carry Carry Carry Carry C C C C Partay redundant form Fgure 3: Computng 3 n a partay redundant form

16 3.2. Deang wth negatve parta products There s a dcuty wth the partay redundant representaton descrbed by Fgure 3. Reca that Booth's agorthm requres the negatve of a mutpes. The negatve (2's compement) can normay be produced by a bt by bt compement, wth a added n at the sb of the parta product. If ths procedure s done to a mutpe n partay redundant form, then the arge gaps of zeros n the postve mutpe become arge gaps of ones n the negatve mutpe (see Fgure 4). In the worst case (a parta products negatve), summng the partay redundant parta products C C C C Negate C C C C Gaps fed wth s Fgure 4: Negatng a number n partay redundant form requres as much hardware as representng them n the fuy redundant form. The probem then s to nd a partay redundant representaton whch has the same form for both postve and negatve mutpes, and aows easy generaton of the negatve mutpe from the postve mutpe (or vce versa). The smpe form used n Fgure 3 cannot meet both of these condtons smutaneousy. 3.3 Booth wth bas In order to produce mutpes n the proper form, Booth's agorthm needs to be moded sghty. Ths modcaton s shown n Fgure 5. Each parta product has a bas constant added to t before beng summed to form the na product. The bas constant (K) s the same for both postve and negatve mutpes. The bas constants may be derent for each parta product. The ony restrcton s that K, for a gven parta product, cannot depend on the whch partcuar mutpe

17 Compensaton constant Parta Product eecton Tabe utper Bts eecton utper Bts eecton K K-4 x utpcand K utpcand K-3 x utpcand K utpcand K2 x utpcand K2 x utpcand K3 x utpcand K3 x utpcand K4 x utpcand K-3 x utpcand K-2 x utpcand K-2 x utpcand K- utpcand K- utpcand K- u t p e r sb Fgure 5: Booth 3 wth bas s seected for use n producng the parta product. Wth ths assumpton, the constants for each parta product can be added (at desgn tme!) and the negatve of ths sum added to the parta products (the Compensaton constant). The net resut s that zero has been added to the parta products, so the na product s unchanged Choosng the rght constant Now consder a mutpe n the partay redundant form of Fgure 3 and choose a vaue for K such that there s a n the postons where a "C" dot appears and zero esewhere, as shown n the top part of Fgure 6. Notce the topmost crced secton encosng 3 vertca tems (two dots and the constant ). These tems can be summed as per the mdde part of the gure, producng the dots "" and "". The three tems so summed can be repaced by the equvaent two dots, shown n the bottom part of the gure, to produce a redundant form for the sum of K and the mutpe. Ths s very smar to the smpe redundant form descrbed earer, n that there are arge gaps of zeros n the mutpe. The key advantage of ths form s that the vaue for K utpe can be obtaned very smpy from the vaue of K utpe. Fgure 7 shows the sum of Kutpe wth a vaue Z whch s formed by the bt by bt compement of the non-bank portons of K utpe and the constant n the sb. When these two vaues are summed together, the resut s 2K (ths assumes proper sgn extenson to however many bts are desred). That s : K utpe Z = 2K Z = K utpe In short, K utpe can be obtaned from K utpe by compementng a of the non-bank 2

18 C C C C Combne these bts by summng K utpe C OR = = EOR C C C K utpe Fgure 6: Transformng the smpe redundant form C C K utpe Z (the bt by bt compement of the non-bank components of Kutpe, wth a added n at the sb) 2K Fgure 7: ummng K utpe and Z 3

19 bts of K utpe and addng. Ths s exacty the same procedure used to obtan the negatve of a number when t s represented n ts non-redundant form. Ths partay redundant form satses the two condtons presented earer, that s t has the same representaton for both postve and negatve mutpes, and aso t s easy to generate the negatve gven the postve form (the entres from the rght sde of the tabe n Fgure 5 w contnue to be consdered as negatve mutpes) Producng the mutpes Fgure 8 shows n deta how the based mutpe K 3 s produced from and 2 usng 4 bt adders and some smpe ogc gates. The smpe ogc gates w not ncrease the tme needed to Carry 4 bt adder Carry 4 bt adder Carry 4 bt adder Carry 4 bt adder C K 3, where K = Fgure 8: Producng K 3 n partay redundant form produce the based mutpe f the carry-out and the east sgncant bt from the sma adder are avaabe eary. Ths s usuay easy to assure. The other requred based mutpes are produced by smpe shftng and nvertng of the mutpcand as shown n Fgure 9. In ths gure the bts of the mutpcand () are numbered (sb = ) so that the source of each bt n each mutpe can be easy seen. 4

20 K K K K K4 3.4 Redundant Booth 3 Fgure 9: Producng other mutpes Combnng the partay redundant representaton for the mutpes wth the based Booth 3 agorthm provdes a workabe redundant Booth 3 agorthm. The dot dagram for the compete redundant Booth 3 agorthm s shown n Fgure 2 for a 6 x 6 mutpy. The compensaton constant has been computed gven the sze of the adders used to compute the K 3 mutpe (4 bts n ths case). There are paces where more than a snge constant s to be added (on the eft hand dagona). These constants coud be merged nto a snge constant to save hardware. Ignorng ths mergng, the number of dots, constants and sgn bts n the dot dagram s 55, whch s sghty more than that for the non-redundant Booth 3 agorthm (prevousy gven as 26). The heght s 7, whch s one more than that for the Booth 3 agorthm. Each of these measures are ess than that for the Booth 2 agorthm (athough the cost of the sma adders s not reected n ths count). A detaed exampe for the redundant Booth 3 agorthm s shown n Fgure 2. Ths exampe uses 4 bt adders as per Fgure 8 to produce the mutpe K 3. A of the mutpes are shown n deta at the top of the gure. The parta product seectors can be but out of a snge mutpexer bock, as shown n Fgure 22. Ths gure shows how a snge parta product s but out of the mutpcand and K3 generated by ogc n Fgure 8. The dagram ndcates a snge coumn (2) wth heght 8, but ths can be reduced to 7 by manpuaton of the bts and the compensaton constant. 5

21 Compensaton constant C C C C C Fgure 2: 6 x 6 Redundant Booth 3 u t p e r sb utper = = K = K3 = K = utpes (n redundant form) K = utpcand () = 49 = K2 = K4 = Compensaton constant K-3 K- K3 K-4 K- 2 sb u t p e r Fgure 2: 6 bt partay redundant Booth 3 mutpy 6

22 K 3 utpcand 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D 3D D 2D 4D ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock ux Bock eects from Booth decoder. A correspondng seect and nvert nputs are wred together Parta Product D D 2D 4D eect 3 eect eect 2 eect 4 Note : A unwred D,2D, or 4D nputs on uxbocks shoud be ted to Invert ux Bock Fgure 22: Parta product seector for redundant Booth 3

23 3.5 Redundant Booth 4 At ths pont, a possbe queston s "Can ths scheme be adapted to the Booth 4 agorthm". The answer s yes, but t s not partcuary ecent and probaby s not vabe. The dcuty s outned n Fgure 23 and s concerned wth the based mutpes 3 and 6. The eft sde of the gure C C C C 3 K C K 3 Left hft C 2K 6 K 6 Fgure 23: Producng K 6 from K 3 shows the format of K 3. The probem arses when the based mutpe K 6 s requred. The norma (unbased) Booth agorthms obtan 6 by a snge eft shft of 3. If ths s tred usng the partay redundant based representaton, then the resut s not K 6, but 2K 6. Ths voates one of the orgna premses, that the bas constant for each parta product s ndependent of the mutpe beng seected. In addton to ths probem, the actua postons of the bts has shfted. These probems can be overcome by choosng a derent bas constant, as ustrated n Fgure 24. The bas constant s seected to be non-zero ony n bt postons correspondng to carres after shftng to create the 6 mutpe. The three bts n the area of the non-zero part of K (crced n the gure) can be summed, but the summaton s not the same for 3 (eft sde of the gure) as for 6 (rght sde of the gure). Extra sgnas must be routed to the Booth mutpexers, to smpfy them as much as possbe (there may be many of them f the mutpy s fary arge). For exampe, to fuy form the 3 dots abeed "", "", and "Z" requres the routng of 5 sgna wres. Creatve use of hardware dependent crcut desgn (for exampe creatng OR gates at the nputs of 8

24 2 C 3 C 2 C C C C C C K 6 Z 2 C C = EOR = 2 EOR ( AND C ) Z = 2 OR ( AND C ) Z C = = EOR C Z = OR C 9 C Z Z Z K 3 C Z Z Z K 6 Fgure 24: A derent bas constant for 6 and 3

25 the mutpexers) can reduce ths to 4, but ths st means that there are more routng wres for a mutpe than there are dots n the mutpe. Of course snce there are now 3 mutpes that must be routed (3, 5, and 7), these few extra wres may not be sgncant. There are many other probems, whch are nherted from the non-redundant Booth 4 agorthm. Larger mutpexers { each mutpexer must choose from 8 possbtes, twce as many as for the Booth 3 agorthm { are requred. There s aso a smaer hardware reducton n gong from Booth 3 to Booth 4 then there was n gong from Booth 2 to Booth 3. Optmzatons are aso possbe for generaton of the 3 mutpe. These optmzatons are not possbe for the 5 and 7 mutpes, so the sma adders that generate these mutpes must be of a smaer ength (for a gven deay). Ths means more dots n the parta product secton to be summed. Thus a redundant Booth 4 agorthm s possbe to construct, but probaby has tte speed or mpementaton advantage over the redundant Booth 3 agorthm. The hardware savngs due to the reduced number of parta products s exceeded by the cost of the adders needed to produce the three hard mutpes, and the ncreased compexty of the mutpexers requred to seect the parta products. 3.6 Choosng the adder ength By and arge, the rue for choosng the ength of the sma adders necessary for s straghtforward - The argest possbe adder shoud be chosen. Ths w mnmze the amount of hardware needed for summng the parta products. nce the mutpe generaton occurs n parae wth the Booth decodng, there s tte pont n reducng the adder engths to the pont where they are faster than the Booth decoder. The exact ength s dependent on the actua technoogy used n the mpementaton, and must be determned emprcay. Certan engths shoud be avoded, as ustrated n Fgure 25. Ths gure assumes a redundant Fgure 25: Redundant Booth 3 wth 6 bt adders u t p e r sb Booth 3 agorthm, wth a carry nterva of 6 bts. Note the accumuaton of dots at certan postons n the dot dagram. In partcuar, the coumn formng bt 5 of the product s now 8 hgh (vs 7 2

26 for a 4 bt carry nterva). Ths accumuaton can be avoded by choosng adder engths whch are reatvey prme to the shft amount between neghborng parta products (n ths case, 3). Ths spreads the bts out so that accumuaton won't occur n any partcuar coumn. 4 ummary Ths report has descrbed a new varaton on conventona Booth mutpcaton agorthms. By representng parta products n a partay redundant form, hard mutpes can be computed wthout a sow, fu ength carry propagate addton. Wth such hard mutpes avaabe, a reducton n the amount of hardware needed for summng parta products s then possbe usng the Booth 3 mutpcaton method. A detaed evauaton of mpementatons usng ths agorthm s currenty n progress, whch w gve precse answers to performance, area, and power mprovements. References [] A. D. Booth. A sgned bnary mutpcaton technque. Quartery Journa of echancs and Apped athematcs, 4(2):236{24, June 95. [2] O. L. acorey. Hgh-speed arthmetc n bnary computers. Proceedngs of the IRE, 49():67{ 9, Jan 96. [3] ark antoro. Desgn and Cockng of VLI utpers. PhD thess, tanford Unversty, Oct 989. [4] ark antoro and ark Horowtz. pm: A ppened 64x64b teratve array mutper. IEEE Internatona od tate Crcuts Conference, pages 35{36, February 988. [5] Naofum Takag, Hroto asuura, and huzo ajma. Hgh-speed VLI mutpcaton agorthm wth a redundant bnary addton tree. IEEE Transactons on Computers, C{34(9), ept 985. [6]. Waser and. J. Fynn. Introducton to Arthmetc for Dgta ystems Desgners. Hot, Rnehart and Wnston,