Some Modular Adders and Multipliers for Field Programmable Gate Arrays

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(1)Some Modular Adders and Multipliers for Field Programmable Gate Arrays Jean-Luc Beuchat. To cite this version: Jean-Luc Beuchat. Some Modular Adders and Multipliers for Field Programmable Gate Arrays. [Research Report] LIP RR-2002-37, Laboratoire de l’informatique du parallélisme. 2002, 2+10p. �hal02101822�. HAL Id: hal-02101822 https://hal-lara.archives-ouvertes.fr/hal-02101822 Submitted on 17 Apr 2019. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) Laboratoire de l’Informatique du Parallélisme École Normale Supérieure de Lyon Unité Mixte de Recherche CNRS-INRIA-ENS LYON no 5668.

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(4) Ó . École Normale Supérieure de Lyon 46 Allée d’Italie, 69364 Lyon Cedex 07, France Téléphone : +33(0)4.72.72.80.37 Télécopieur : +33(0)4.72.72.80.80 Adresse électronique :

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(509)

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(606)

(607)

(608)

(609) (xy) (2n + 1)! K

(610)

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(617) (2n + 1) 5 7:! &

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(621)

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(623)

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(630) 639! .

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(633)

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(639) m "

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(641) 0 . 3FG #% "

(642)

(643) &

(644) ? m!

(645) k = log2 m + 1

(646) 0 m! +

(647) m ≤ 2k % B0 5: . (xy) m = ((xy) 2k + ((2k − m) · (xy) 2k ) m.

(648) .

(649) k " (xy) 2k k ! , 0%

(650) # 4 ) &

(651) 3G×3F 5! 3 . 3F :

(652) m " 3! $ ) "

(653) 0 <,>78 ! B

(654) 0 3FG #! ! ? @ 1!' ?@. m=5. m = 13. m = 29. m = 61. m = 125. m = 253. m = 509. m = 1021.

(655) .

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(663) F.

(664) . 7 D

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(668) . ? % "

(669) "

(670) .

(671) 5 7 K:

(672) & 1"=

(673)

(674) 639!

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(678) x = 0 y = 0% 5 K:! ?

(679) $ •

(680)

(681) (2n + 1)

(682) ) *

(683) ( !

(684) $

(685) (2n + 1) .

(686) " (n + 1) ! , 0%

(687) " (2n + 1) . 0 2 · n/2 5 7: n/2 + 2 5 K: .

(688) ! •

(689) & 1"=

(690)

(691) "

(692)

(693) (2n + 1) !

(694) n × n n Pi = 2ixi y% i ∈ {0, . . . , n − 1} "

(695) !

(696)

(697) #

(698) ; . "

(699)

(700) 1C

(701) !

(702)

(703) ) *

(704)

(705) %

(706)

(707)

(708) " 51C ? :! • ="% "

(709)

(710) %

(711) 7 n ≤ 28!

(712) ?

(713)

(714)

(715)

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(718)

(719) ? ! n. 7$ ) (2n + 1) Z∗2. n +1. % *" #% 0. % *" 0. % *" #% 0. % *" 0. % *" #% 0. % *" 0. <,>B8 !. n=4. n=8. n = 12. n = 16. n = 20. n = 24. n = 28. n = 32.

(720)

(721)

(722)

(723)

(724)

(725) .

(726)

(727)

(728)

(729)

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(731) .

(732)

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(738)

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(744)

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(749) .

(750)

(751)

(752)

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(756)

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(762)

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(769) # 5 % % :!

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(776)

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(795) 2n + 1

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(797) B0 58: (x + y + 1) (2n + 1) "

(798) 0 ≤ x, y ≤ 2n ! %

(799) 0 ≤ x + y ≤ 2n+1 x+y+1 x + y < 2n , n (x + y + 1) (2 + 1) = x + y − 2n x + y ≥ 2n . +

(800)

(801)

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(805) .

(806) $.

(807) E.

(808) • • •. x + y = 2n+1 5!! x = y = 2n :% "

(809) (x + y) 2n = 0% sn = 0% (x + y + 1) (2n + 1) = 2n = x + y − 2n % "

(810)

(811)

(812) !. sn+1 = 1!

(813) . 2n ≤ x+y < 2n+1 % " #"

(814) sn+1 = 0 sn = 1! , 0% (x+y+1) (2n +1) = (x+y) 2n = x + y − 2n ! % 0 ≤ x+y < 2n % sn+1 = sn = 0 (x+y) 2n = x+y! + (x+y +1) (2n +1) = x+y +1!.

(815)

(816) ! "

(817)

(818) −2ψ2i + ψ2i+1 + ψ2i−1 ∈ {−2, −1, 0, 1, 2}! B

(819)

(820)

(821)

(822) (±2j x− 1) (2n + 1)! 1 ξ = (x − 1) (2n + 1)

(823)

(824) x! +

(825)

(826) " ξ! 1

(827) "

(828) x = 0 5!! ξ = 2n :! 4 2n+q −2q 5 (2n + 1):% "