lect17-adders.ppt - Week 8 Adders 1 Outline Single-bit Addition half or full Carry-Ripple Adder basic or PG Carry-Skip Adder Carry-Lookahead Adder

lect17-adders.ppt - Week 8 Adders 1 Outline Single-bit...

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Week 8: Adders 1
2 Outline Single-bit Addition half or full Carry-Ripple Adder basic or PG Carry-Skip Adder Carry-Lookahead Adder Carry-Select Adder Carry-Increment Adder Tree Adder
3 Single-Bit Addition Half Adder Full Adder A B C out S 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 A B C C out S 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 A B S C out A B C S C out out S A B C A B out ( , , ) S A B C C MAJ A B C
4 PGK For a full adder, define what happens to carries (in terms of A and B) B
5 Full Adder Design I Brute force implementation from eqns out ( , , ) S A B C C MAJ A B C A B C S C out MAJ A B C A B B B A C S C C C B B B A A A B C B A C B A A B C C out C A A B B
6 Full Adder Design II Factor S in terms of C out S = ABC + (A + B + C)(~C out ) Critical path is usually C to C out in ripple adder S S C out A B C C out MINORITY
7 Layout Clever layout circumvents usual line of diffusion Use wide transistors on critical path Eliminate output inverters
8 Full Adder Design III Complementary Pass Transistor Logic (CPL) Slightly faster, but more area A C S S B B C C C B B C out C out C C C C B B B B B B B B A A A
9 Full Adder Design IV Dual-rail domino Very fast, but large and power hungry Used in very fast multipliers C out _h A_h B_h C_h B_h A_h C out _l A_l B_l C_l B_l A_l S_h S_l A_h B_h B_h B_l A_l C_l C_h C_h
10 Carry Propagate Adders N-bit adder called CPA Each sum bit depends on all previous carries How do we compute all these carries quickly? + B N...1 A N...1 S N...1 C in C out 11111 1111 +0000 0000 A 4...1 carries B 4...1 S 4...1 C in C out 00000 1111 +0000 1111 C in C out
11 Carry-Ripple Adder Simplest design: cascade full adders Critical path goes from C in to C out Design full adder to have fast carry delay C in C out B 1 A 1 B 2 A 2 B 3 A 3 B 4 A 4 S 1 S 2 S 3 S 4 C 1 C 2 C 3
12 Inversions Critical path passes through majority gate Built from minority + inverter Eliminate inverter and use inverting full adder C out C in B 1 A 1 B 2 A 2 B 3 A 3 B 4 A 4 S 1 S 2 S 3 S 4 C 1 C 2 C 3
13 Generate / Propagate Equations often factored into G and P Generate and propagate for groups spanning i:j Base case Sum: : : : 1: : : 1: i j i k i k k j i j i k k j G G P G P P P : : i i i i i i i i i i G G A B P P A B

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