Sommatore 1 0 0 0 0 1 1 1 0 1 1 0 C i+1 = G i + C i P i - - PowerPoint PPT Presentation

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Jun 02, 2023 345 likes •399 views

Metodologia della sintesi logica Carry Look-Ahead A i B i C i C i+1 G i P i G = AB 0 0 0 0 0 0 P = A B 0 1 0 0 0 1 Sommatore 1 0 0 0 0 1 1 1 0 1 1 0 C i+1 = G i + C i P i 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1

SLIDE 1

Metodologia della sintesi logica

Sommatore Carry Look-Ahead

Reti logiche Carry Look-Ahead

Ai Bi Ci Ci+1 Gi Pi 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

G = AB P = A⊕B Ci+1 = Gi + CiPi

C1 = (P0 C + G0) = P0 C + G0 = CG-1 (C, P0, G0) C2 = (P1 C1 + G1) = P1 P0 C + P1 G0 + G1 = CG-2 (C, P0, P1, G0, G1) C3 = (P2 C2 + G2) = P2 P1 P0 C + P2 P1 G0 + P2 G1 + G2 = CG-3 (C, P0, P1, P2, G0, G1, G2)

Carry generator: CG-1

C0 P0 G0 VDD C2

C1 = G0 + P0 C0

P0 C0 G0 C1

Carry generator: CG-2

C0 P0 P1 G0 G1 VDD C2

C2 = G1 + P1 C1

P0 P1 C0 G0 G1 C2

SLIDE 2

Carry generator: CG-3

C0 P0 P1 P2 G0 G1 G2 VDD C3

C3 = G2 + P2 C2

P0 P1 P2 C0 G0 G1 G2 C3

Carry Look-Ahead: 4-bit

SUM

A0 B0 S0 G0 P0 C0

CG-1

G0,G1 P0,P1

CG-2

G0,G1,G2 P0,P1,P2

CG-3 SUM

A1 B1 S1

SUM

A2 B2 S2

SUM

A3 B3 S3

2 2 3 3

C1 C2 C3 A0,A1,A2,A3 B0,B1,B2,B3 P0,P1,P2,P3 G0,G1,G2,G3

Setup

4 4 4 4

Carry Look-Ahead: 4-bit

SUM

A0 B0 S0 G0 P0 C0 G0,G1 P0,P1 G0,G1,G2 P0,P1,P2

SUM

A1 B1 S1

SUM

A2 B2 S2

SUM

A3 B3 S3

2 2 3 3

C1 C2 C3

Carry Generator

A0,A1,A2,A3 B0,B1,B2,B3 P0,P1,P2,P3 G0,G1,G2,G3

Setup

4 4 4 4

C4 = (P3 C3 + G3) = P3 P2 P1 P0 C + P3 P2 P1 G0 + P3 P2 G1 + P3 G2 + G3 = P[0,3] C + G[0,3] P[0,3] = P3 P2 P1 P0 G[0,3] = P3 P2 P1 G0 + P3 P2 G1 + P3 G2 + G3 = CG-3 (G0, P1, P2, P3, G1, G2, G3) C4 = CG-1 (C, P[0,3], G[0,3]) C5 = (P4 C4 + G4) = CG-1 (C4, P4, G4) C6 = (P5 C5 + G5) = P5 P4 C4 + P5 G4 + G5 = CG-2 (C4, P4, P5, G4, G5) C7 = (P6 C6 + G6) = P6 P5 P4 C4 + P6 P5 G4 + P6 G5 + G6 = CG-3 (C4, P4, P5, P6, G4, G5, G6)

Carry Look-Ahead: 16-bit

SLIDE 3

C8 = (P7 C7 + G7) = P7 P6 P5 P4 C4 + P7 P6 P5 G4 + P7 P6 G5 + P7 G6 + G7 = = P7 P6 P5 P4 P[0,3] C + P7 P6 P5 P4 G[0,3] + P7 P6 P5 G4 + P7 P6 G5 + P7 G6 + G7 = = P[4,7] P[0,3] C + P[4,7] G[0,3] + G[4,7] P[4,7] = P7 P6 P5 P4 G[4,7] = P7 P6 P5 G4 + P6 G5 + P7 G6 + G7 = CG-3(G4, P5, P6, P7, G5, G6, G7) C8 = CG-2 (C, P[0,3] , P[4,7] , G[0,3] , G[4,7]) C9 = CG-1 (C8, P8, G9) C10 = CG-2 (C8, P8, P9, G8, G9) C11 = CG-3 (C8, P8, P9, P10, G8, G9, G10)

Carry Look-Ahead: 16-bit

C12 = (P11 C11 + G11) = P11 P10 P9 P8 P[4,7] P[0,3] C + P11 P10 P9 P8 P[4,7] G[0,3] + P11 P10 P9 P8 G[4,7] + P11 P10 P9 G8 + P11 P10 G9 + P11 G10 + G11 = = P[8,11] P[4,7] P[0,3] C + P[8,11] P[4,7] G[0,3] + P[8,11] G[4,7] + G[8,11] P[8,11] = P11 P10 P9 P8 G[8,11] = P11 P10 P9 G8 + P11 P10 G9 + P11 G10 + G11 = CG-3(G8, P9, P10, P11, G9, G10, G11)

Carry Look-Ahead: 16-bit

C13 = CG-1 (C12, P12, G12) C14 = CG-2 (C12, P12, P13, G12, G13) C15 = CG-3 (C12, P12, P13, P14, G12, G13, G14)

Carry Look-Ahead: 16-bit

P0 P1 P2 G0 G1 G2 C1 C2 C3 C0 P4 P5 P6 G4 G5 G6 C5 C6 C7 P8 P9 P10 G8 G9 G10 C9 C10 C11 P12 P13 P14 G12 G13 G14 C13 C14 C15 P[0,3] P[4,7] P[8,11] G[0,3] G[4,7] G[8,11] C4 C8 C12 C0

Carry Generator Carry Generator Carry Generator Carry Generator Carry Generator

Setup

P[0,3] = P3 P2 P1 P0 G[0,3] = CG-3(G0, P1, P2, P3, G1, G2, G3) P[4,7] = P7 P6 P5 P4 G[4,7] = CG-3(G4, P5, P6, P7, G5, G6, G7) P[8,11] = P11 P10 P9 P8 G[8,11] = CG-3(G8, P9, P10, P11, G9, G10, G11) P[12,15] = ... G[12,15] = ...

Setup (P , G)

Pi = Ai Bi Gi = Ai · Bi

SUM

C16 = CG-1 ( C, P[0,15], G[0,15] )

P[0,15] = P[12,15] P[8,11] P[4,7] P[0,3] G[0,15] = P[12,15] P[8,11] P[4,7] G[0,3] + P[12,15] P[8,11] G[4,7] + P[12,15] G[8,11] + G[12,15] = = CG-3 ( G[0,3], P[4,7], P[8,11], P[12,15], G[4,7], G[8,11], G[12,15] )

C32 = CG-2 ( C, P[0,15], P[16,31] , G[0,15] , G[16,31] )

P[16,31] = P[28,31] P[24,27] P[20,23] P[16,19] G[16,31] = CG-3 ( G[16,19], P[20,23], P[24,27], P[28,31], G[20,23], G[24,27], G[28,31] )

C48 = CG-3 ( C, P[0,15], P[16,31], P[31,47], G[0,15] , G[16,31] , G[31,47] )

P[32,47] = P[44,47] P[40,43] P[36,39] P[32,35] G[32,47] = CG-3 ( G[32,35], P[36,39], P[40,43], P[44,47], G[36,39], G[40,43], G[44,47] )

Carry Look-Ahead: 64-bit

SLIDE 4

Carry Look-Ahead: 64-bit

Setup (P , G)

Pi = Ai Bi Gi = Ai · Bi Setup Setup Setup Setup

Setup

P[16,31] = ... G[16,31] = ... P[32,47] = ... G[32,47] = ... G[0,3] , G[4,7] , G[8,11] , G[12,15] P[0,15] P[16,31] P[32,47] G[0,15] G[16,31] G[32,47] C0 C0 P[0,3] , P[4,7] , P[8,11] , P[12,15] 8 P[48,63] = ... G[48,63] = ... P[0,15] = P[12,15] P[8,11] P[4,7] P[0,3] G[0,15] = CL-3(G[0,3], P[4,7], P[8,11], P[12,15], G[4,7], G[8,11], G[12,15]) Carry generators (2 levels)

SUM

Carry generators (2 levels) Carry generators (2 levels) Carry generators (2 levels)

Carry Generator

Carry Look-Ahead: delay

P[...] and G[...] signals

– do not depend on C0 – propagate from lower levels to higher levels of hierarchy

Ci signals

– propagate from higher levels of hierarchy to final adders

Carry Look-Ahead: delay

Ts = delay of Setup P, G L = hierarchy levels

– L = log4 n

Tsum = delay of Ai Bi Ci TG3 = delay of CG-3

– standard cells: TG ~ 2 · TG

Carry Look-Ahead: delay

Latency for generation of G[] = (L-1) · TG3 Latency for generation of last C = L · TG3 TCLA(n) = Ts + (L-1) · TG3 + L · TG3 + Tsum TCLA(n) log4 n

– standard cells: TCLA(n) ~ TG + (L-1)·2·TG + L·2·TG + TG

TCLA(n) ~ (1 + 4 · log4 n) · TG