橡卒論.PDF
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- えりか あいきょう
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1 1329
2 Booth Robertson Booth Wallace tree Booth
3 1 DSPDigital Signal Processor DSP 1
4 k 2k 2.1 Y k 2k 0 X 1 k k X k 2k k X 0 1 2k
5 Y LSB 2 3 Y 3 k
6 X 0 LSB X X X X k k k 2k 0 k 2k 4
7
8 k LSB Y 0 Y
9
10 Ripple Carry Adder n LSB n T c 1 T a T a = nt c 2.8 X3 Y3 X2 Y2 X1 Y1 X0 Y0 C-1 FA FA FA FA C3 C2 C1 C0 S3 S2 S1 S
11 2.8 Carry Look-ahead Adder XY 1 C i-1 C i XY 1 C i-1 C i Ci = Gi + Pi Ci - 1 G i = X i Y i P i = X i Å Y i G i P i C S C i C = G + P C C = G + P C = G + G P+ P P C C = G + G P + G P P + P P P C C = C + C P+ G P P+ G P P P+ P P P P C C = G + G P + G P P + + G P P P + P P P C n n n-1 n n-2 n-1 n n
12 2.9 4 Y 3 X 3 Y 2 X 2 Y 1 X 1 Y 0 X 0 C -1 G 3 P 3 G 2 P 2 G1 P 1 G 0 P 0 C 3 C 2 C 1 C CLA 4 1 CLA CLA 32 CLA 2 10
13 Carry Save Adder CSA n
14 D n i+ 1 i 1 n 1 SI n Q n-1: D n-1:0 2-to-1 L 0 n Q 0 12
15 Booth n n/2 Y 2 X 2 n n-1 n/2-1 i 2i å i å 2i+ 1 2i (2.1) i= 0 i= 0 XY = Xy 2 = X(2y + y )2 2y 2i+1 +y 2i X k 0 2 k -1 13
16 2 k Booth 2 X Booth 2 Y S V n-2 n-1 n-1 å i i i= 0 Y=- Y + Y =- y 2 + y2 n-2 n-2 n-1 i i n-1 å i å i i= 0 i= 0 =- y y2 - y2 n-1 n-2 i å i å i= 0 i= 0 =- y2 + 2 y2 i i n-1 n-1 n-1 i i i åy2 i åyi-12 å( yi y i-1)2 i= 0 i= 0 i= 0 =- + = - + (2.2) y º (2.2) 0 1 Y (2.2)2 n-1 å Y = (- y + y )2 i= 0 n/21 - å i= 0 n/21 - i= 0 i i-1 i = (2(- y + y ) +- ( y + y ))2 å 2i+ 1 2i 2i 2i-1 = (- 2y + y + y )2 2i+ 1 2i 2i-1 2i 2i (2.3) Y y = X 1 n/ Booth 14
17 2.12 Booth y i+2 y i+1 y i X X X X X X Booth k mx - 2 k-1 m 2 k-1 k (k+ 1) y8 y7 y6 y5 y4 y3 y2 y1 y (X) ) (Y) Booth 15
18 X Y P 3 P X Y Y P Y 3.1.1Robertson 3.1 Robertson 16
19 3.1 Robertson X MR Y MQ ACC 0 P ACC MQ MSB 2 Y MQ Y 1 ACC MR X ACC MR 2 Y ACC MQ P Y X > 0Y > 0 X Y -i i 0 X < 0Y > 0 X Y -i i 1 2 a X > 0Y < 0 X< 0Y < 0 2 X b a b ) )
20 3.1.2Booth 2 Booth Booth 3.2 Booth X MR Y MQ E ACCMQ E ACC E 0 Booth Q = 0E = 0 Q = 0E = 1 X ACC Q = 1E = 0 X ACC Q = 1E = 1 18
21 ACC P m n n 2 Y 0 0 Y X Y 4 P
22
23 3.2.2Wallace tree Wallace tree CSA CSA tree CSA CSA 3 2 log 3/2 n Y Wallace tree 16 log Wallace tree XY 7 XY 1 CSA 6 XY 5 XY 4 XY 2 3 CSA XY 2 XY1 3 CSA XY CSA CSA S C 21
24 3.6Wallace-tree 8 8 Wallace tree 3.6 CSA 1 2 CSA S CCSA S CSA XY 0 XY 1 CSA C CSA CSA m 1 (2/3)m 2 (2/3) 2 m 2 (2/m)m L 2/m = (2/3)L L = (log2m-1)/(log23-1) = 1.71(log2m- 1) 2 O(log m) 22
25 8 Y Y partial sum carry CSA sum/carry Wallace tree tree CSA Wallace tree 11 partial sum/carry 1 Wallace tree 6 1 Wallace tree Y k k Y X Wallace tree CSA 3.7 k CSA 6 Wallace tree 1 12 sum/carry Y m 1/m 2m X m CSA Wallace tree Wallace tree m 23
26 3.7 Wallace tree 24
27 3.3 1 n n n X Y 4 8 P D 3.4 AND
28 D 1 X 4 X7X4 2 X 4 X3X0 1 Y Y3Y0 P 4 P3P0 P 8 P11P4 L7L0 P11P8 L7L4 P7 1 L3 1 P6P4 L2L0 2 P 8 P11P4 X 4 X7X4 2 X 4 X3X0 1 Y Y7Y3 P 12 P11P0 26
29 k 4 é êk/2ù ú 8 é êk/3ù ú Y r P = (P + XY r)r P = 0 P = P (j+ 1) (j) k -1 (0) (k) j add shift right P = rp XY P = 0 P = P (j+ 1) (j) (0) (k) k-- j 1 shift left add r -1 r 2 XY j 4 X(Y i+1 Y i )
30 X Y X(Y Y) X(Y Y) P X 0X 1X 2X3X 0X1X2X 2X X 3X 3X = 2X + X 4 3X 2-way way X X 3X -X
31 [0, 4] X5X7X 3X X 6X 7X -3X -2X-X 1 X X Y (0) P X(Y Y) (1) 4P (1) P X(Y Y) (2) 4P (2) P X X 4X-X 29
32 Booth 2 Booth Booth X 2X Z i+1 Y i+1 Y i Z i Y i Y i-1 4 Z i/2 = (Z i+1 Z i ) 2 i Y i+1 Y i Y i Booth Y Y Z i i-1 i Booth Y Y Y Z Z Z' i+ 1 i i- 1 i+ 1 i i/
33 2 Booth 4 4 [0, 3] [-2, 2] 3.2 ( ) 4 = ( ) 2 = ( ) k 2 4 MSB 1 ê ëk/2ú+ û 1 = é(k+ 1)/2ù ê ú Y- 1 = Yk = Yk+ 1= ( ) 2 s-compl = ( ) 4 2 k 2 Booth 4 é êk/2ù ú k Yk = Yk - 1 Y -1 = 0 4 Booth [-2, 2] 4 Booth 3 Booth
34 X Y Z' 1 2 (0) P XZ' (1) 4P (1) P XZ' (2) 4P (2) P Booth Y = (1010) Z = (-1-2) 4 2 XZ 0 = -2X XZ 1 = -X P (1) P (1) Booth 32
35 3.14Booth 4 X nonzero 1 nonzero 1 nonzero 1 3 Y i+1, Y i, Y i-1 3 neg 0 1 non0 1 0 two 0 non0 1 1 non Booth Booth Wallace tree 33
36 4 34
37 35
38 Neil H. E. Weste, Kamran Eshraghian : PRINCIPLES OF CMOS VLSI DESIGN, ADDISON WESLEY (1992) Behrooz Parhami : Computer Arithmetic, Oxford University Press (1999) John L. Hennessy David A. Patterson 2 BP
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