# Multiplication Algorithms for Radix-2 RN-Codings and Two s Complement Numbers

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3 x y + x + y x + y + x + y + x y x y x + y + x + y + 0 x y x 0 y x + y + x + y x y x 0 y 0 x 0 y + 0 x + 0 y 0 x + y + x x + y + x + y + y + x x y x y + + x y x y + y ψ (, ) ψ (0, ) ψ (0, ) ϕ (, ) ϕ (0, ) ϕ (0, ) (a) (4) (3) () (b) () (0) Fgure. (a) Computaton of X + Y + when n =3. (b) Computaton of X + Y + + X Y when n =3. Partal Product Generator Carry save adder tree (4,) compressors X + Y + X Y X + Y X Y + PPG PPG PPG PPG X + Y + Carry save adder X + Y + + X Y X Y Carry save subtracter X + Y X Y + n Carry save adder X + Y + X Y + partals products Carry save X + Y + + X + Y + X Y X + Y X Y + X Y PPG X + Y + X Y + Carry save subtracter PPG Carry save Two s complement to RN codng Two s complement Two s complement to RN codng Two s complement XY (RN codng) (a) XY (RN codng) (b) Fgure. Multplcaton of two RN-codngs. (a) Archtecture of the multpler descrbed n [3]. (b) Proposed mprovement. Example 3. Let us consder two 3-dgt RN-codngs X and Y. We could compute the sum of the sx partal products x + Y + and x Y, 0, by means of a carrysave adder tree. In ths example, Theorem allows to defne a sngle partal product (Fgure b), whose bt of weght j s denoted by λ (j) 0, 0 j 4. Accordng to Equaton (4), we replace an addton by an OR gate: λ (0) 0 = x + 0 y+ 0 x 0 y 0. Snce the computaton of λ (0) 0 does not generate a carry, we have λ () 0 = (x + 0 y+ + x+ y+ 0 + x 0 y + ) mod = ϕ(0, ) (Equaton (6). x y 0 We then apply twce Equaton (6) to compute ϕ(0, ), ψ(0, ), ϕ(, ), and ψ(, ). Fnally, Equatons (7), (8), and (5) respectvely allow to compute λ () 0, λ(3) 0, and λ(4) 0. Fgure b shows the general archtecture of the multpler. At the prce of a more complex partal product generaton, we save two carry-save adder trees and two carrysave adders based on (4, )-compressors. It s worth beng notced that the crtcal path of our new Partal Product Generator (two OR gates, one AND gate, and an XOR gate) s shorter than the one of a (4, )-compressor (three XOR gates). Let (U (c),u (s) ) denote the carry-save form of X + Y + + X Y.WehaveX + Y + + X Y =U (c) + U (s), where u (c) =0and u (s) = λ () 0 for {0,, n 3, n }. Algorthm 3 can also be appled to the computaton of (X + Y + X Y + ). Thus, we obtan a carrysave number (V (c),v (s) ) such that X + Y + X Y + = /05 \$ IEEE

6 c Y or Y + Z or Y y n z n x x X + X X (RN codng) (two s complement) R R PL 4 n bts c 0 0 n bts Sgn extenson Booth selector Sgn extenson Booth selector Carry save adder/subtracter n bts c 0 (two s complement) (RN codng) x + x c 3 x + + W + W W R R PL x + M 0 0 c 4 x + M Booth recodng Booth recodng P0 P Fgure 3. A multpler handlng two s complement numbers and radx- RN-codngs. W, X, Y, and W are n-bt two s complement numbers. X +, X, Y +, and Y are n-bt unsgned numbers. Table. Some operatons mplemented by the multpler shown n Fgure 3. Operaton c 4 c 3 c c c 0 P0 XY P WZ P0 XY + WZ P WZ P0 XY WZ P WZ P0 (X + X )Y P WZ P0 (X + X )Y P (X + X )Z P0 (X + X )(Y + Y ) P (X + X )Y 4. Concluson We have shown that very slghtly modfed arthmetc operators can effcently handle both conventonal bnary representatons and RN-codngs. Snce RN-codngs can also be effcently compressed for storage [7], we conclude that RN-codngs are a good canddate for numercal computatons. In a further study we wll desgn dedcated dvson and square root algorthms, as well as an ALU able to handle RN-codngs and conventonal bnary numbers. References [] J.-C. Bajard, J. Duprat, S. Kla, and J.-M. Muller. Some operators for on-lne radx- computatons. Journal of Parallel and Dstrbuted Computng, : , 994. [] J.-L. Beuchat and J.-M. Muller. Multplcaton algorthms for radx- RN-codngs and two s complement numbers. Techncal Report , Laboratore de l Informatque du Parallélsme, École Normale Supéreure de Lyon, 46 Allée d Itale, Lyon Cedex 07, Feb [3] J.-L. Beuchat and J.-M. Muller. RN-codes : algorthmes d addton, de multplcaton et d élévaton au carré. In SympA 005: 0ème édton du SYMPosum en Archtectures nouvelles de machnes, pages 73 84, Apr [4] A. D. Booth. A sgned bnary multplcaton technque. Quarterly Journal of Mechancs and Appled Mathematcs, 4():36 40, 95. [5] M. Daumas and D. W. Matula. Further reducng the redundancy of a notaton over a mnmally redundant dgt set. Journal of VLSI Sgnal Processng, 33:7 8, 003. [6] G. Goto, A. Inoue, R. Ohe, S. Kashwakura, S. Mtara, T. Tsuru, and T. Izawa. A 4.-ns compact b multpler utlzng sgn-select s. IEEE Journal of Sold-State Crcuts, 3():676 68, Nov [7] P. Kornerup and J.-M. Muller. RN-codng of numbers: defnton and some propertes. In Proceedngs of the 7th IMACS World Congress on Scentfc Computaton, Appled Mathematcs and Smulaton, Pars, July /05 \$ IEEE

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