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- pring 2003 Lecture 18 Adders Multipliers Today s lecture Adder design: Manchester carry chain, carry-bypass, carry-select, carrylookahead Multipliers 1 Adders Full-Adder A B in Full adder um out 2 Express

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- pring 2003 Lecture 18 Adders Multipliers Today s lecture Adder design: Manchester carry chain, carry-bypass, carry-select, carrylookahead Multipliers 1 Adders Full-Adder A B in Full adder um out 2 Express um and arry as a function of P, G, D Define 3 new variables that ONLY depend on A, B Generate (G) = AB Propagate (P) = A B Delete = A B an also derive expressions for and o based on D and P Transmission Gate Full Adder P A A A P i i P um Generation B A P B P A P o arry Generation i i etup A i P 3 Manchester arry hain P i φ P i o G i i i o G i P i D i φ Manchester arry hain φ P 0 P 1 P 2 P 3 3 i,0 G 0 G 1 G 2 G 3 φ Manchester arry hain tick Diagram Propagate/Generate Row P i G i φ P i +1 G i +1 φ i i -1 i +1 GND Inverter/um Row arry-bypass Adder i,0 P 0 G 1 P 0 G 1 P 2 G 2 P 3 G 3 o,0 o,1 o,2 o,3 Also called arry-kip P 0 G 1 P 0 G 1 P 2 G 2 P 3 G 3 BP=P o P 1 P 2 P 3 i,0 o,0 o,1 o,2 o,3 Idea:If(P0andP1andP2andP3=1) then o3 = 0, else kill or generate. 5 arry-bypass Adder (cont.) Bit 0-3 Bit 4-7 Bit 8-11 Bit etup etup etup etup i,0 arry Propagation arry Propagation arry Propagation arry Propagation um um um um arry Ripple versus arry Bypass t p ripple adder bypass adder 4..8 N 6 arry-elect Adder etup P,G arry Propagation arry Propagation o,k-1 o,k+3 um Generation arry Vector arry elect Adder: ritical Path Bit 0-3 Bit 4-7 Bit 8-11 Bit etup etup etup etup arry arry arry arry arry arry arry arry i,0 o,3 o,7 o,11 o,15 um Generation um Generation um Generation um Generation Linear arry elect Bit 0-3 Bit 4-7 Bit 8-11 Bit etup etup etup etup (1) (1) arry arry arry arry arry (5) (5) i,0 arry arry arry (5) (5) (5) (6) (7) (8) (9) um Generation um Generation um Generation um Generation (10) quare Root arry elect Bit 0-1 Bit 2-4 Bit 5-8 Bit 9-13 Bit etup etup etup etup (1) arry arry arry arry (1) arry arry arry arry (3) (3) (4) (5) (6) (4) (5) (6) (7) i,0 um Generation um Generation um Generation um Generation (7) Mux (8) um (9) 8 Adder Delays - omparison ripple adder 30.0 tp 20.0 linear select 10.0 square root select N LookAhead - Basic Idea A 0,B 0 A 1,B 1 A N-1,B N-1... i,0 P0 i,1 P 1 i,n-1 P N-1... o k, = fa ( k, B k, o k ) = G k + P k ok 1, 1, 9 Look-Ahead: Topology Expanding Lookahead equations: ok, = G k + P k ( G k 1 + P k 1 o, k 2 ) VDD G3 All the way: ok, = G k + P k ( G k 1 + P k 1 ( + P 1 ( G 0 + P 0 i, 0 ))) G2 G1 G0 i,0 o,3 P0 P1 P2 P3 Logarithmic Look-Ahead Adder A 0 F A 1 A 2 A 3 A 4 A 5 A 6 A 7 A 0 A 1 t p N A 2 A 3 A 4 A 5 A 6 A 7 F t p log 2 (N) 10 arry Lookahead Trees o0, = G 0 + P 0 i0, o1, = G 1 + P 1 G 0 + P 1 P 0 i0, o2 = G, 2 + P 2 G 1 + P 2 P 1 G 0 + P 2 P 1 P 0 i0, = ( G 2 + P 2 G 1 ) + ( P 2 P 1 )( G 0 + P 0 i 0 ) = G 2:1 + P 2:1 o0 an continue building the tree hierarchically.,, Tree Adders (A 0, B 0 ) (A 1, B 1 ) (A 2, B 2 ) (A 3, B 3 ) (A 4, B 4 ) (A 5, B 5 ) (A 6, B 6 ) (A 7, B 7 ) (A 8, B 8 ) (A 9, B 9 ) (A 10, B 10 ) (A 11, B 11 ) (A 12, B 12 ) (A 13, B 13 ) (A 14, B 14 ) (A 15, B 15 ) bit radix-2 Kogge-tone Tree 11 Tree Adders (a 0, b 0 ) (a 1, b 1 ) (a 2, b 2 ) (a 3, b 3 ) (a 4, b 4 ) (a 5, b 5 ) (a 6, b 6 ) (a 7, b 7 ) (a 8, b 8 ) (a 9, b 9 ) (a 10, b 10 ) (a 11, b 11 ) (a 12, b 12 ) (a 13, b 13 ) (a 14, b 14 ) (a 15, b 15 ) bit radix-4 Kogge-tone Tree Tree Adders (A 0, B 0 ) (A 1, B 1 ) (A 2, B 2 ) (A 3, B 3 ) (A 4, B 4 ) (A 5, B 5 ) (A 6, B 6 ) (A 7, B 7 ) (A 8, B 8 ) (A 9, B 9 ) (A 10, B 10 ) (A 11, B 11 ) (A 12, B 12 ) Brent-Kung Tree (A 13, B 13 ) (A 14, B 14 ) (A 15, B 15 ) 12 Brent-Kung Adder (G 0,P 0 ) (G 1,P 1 ) o,0 o,1 o,2 o,4 (G 2,P 2 ) o,3 o,5 (G 3,P 3 ) (G 4,P 4 ) (G 5,P 5 ) o,6 (G 6,P 6 ) (G 7,P 7 ) o,7 t add log 2 (N) Domino Adder lk G i =a i b i lk P i =a i +b i a i a i b i b i lk lk Propagate Generate 13 Domino Adder lk k P i:i-2k+1 lk k G i:i-2k+1 P i:i-k+1 P i:i-k+1 G i:i-k+1 P i-k:i-2k+1 G i-k:i-2k+1 Propagate Generate Domino um Keeper lk lkd um Gi lk i 0 lkd lk Gi i 1 lk 14 Multipliers The Binary Multiplication X Y M + N 1 Z k 2 k k = 0 M 1 X i 2 i N 1 Y j 2 j i = 0 j = 0 = = = = M 1 i = 0 N 1 X i Y j 2 i+ j j = 0 with Y = = M 1 X i 2 i i = 0 N 1 Y j 2 j j = 0 15 The Binary Multiplication AND operation Partial Products The Array Multiplier Z 0 X 3 X 2 X 1 X 0 Y 1 HA HA X 3 X 2 X 1 X 0 Y 2 Z 1 HA X 3 X 2 X 1 X 0 Y 3 Z 2 HA Z 7 Z 6 Z 5 Z 4 Z 3 16 The MxN Array Multiplier riticalpath HA HA HA ritical Path 1 ritical Path 2 ritical Path 1 & 2 HA arry-ave Multiplier HA HA HA HA HA HA HA HA Vector Merging Adder 17 Multiplier Floorplan X 3 X 2 X 1 X 0 Y 0 Y 1 Z 0 HA Multiplier ell Multiplier ell Y 2 Z 1 Vector Merging ell Y 3 Z 2 X and Y signals are broadcasted through the complete array. ( ) Z 7 Z 6 Z 5 Z 4 Z 3 Wallace-Tree Multiplier y 0 y 1 y2 i-1 y 0 y 1 y 2 y 3 y 4 y 5 y 3 i i-1 i i i-1 i-1 y 4 i i-1 i i-1 y 5 i 18 Multipliers ummary Optimization Goals Different Vs Binary Adder Once Again: Identify ritical Path Other possible techniques - Logarithmic versus Linear (Wallace Tree Mult) - Data encoding (Booth) - Pipelining FIRT GLIMPE AT YTEM LEVEL OPTIMIZATION The Binary hifter Right nop Left A i B i A i-1 B i-1 Bit-lice i... 19 The Barrel hifter A 3 B 3 A 2 h1 B 2 A 1 h2 B 1 :DataWire :ontrolwire A 0 h3 B 0 h0 h1 h2 h3 Area Dominated by Wiring 4x4 barrel shifter A 3 A 2 A 1 A 0 h0 h 1 h2 h3 Width barrel ~2p m M Buffer 20 Logarithmic hifter h1 h1 h2 h2 h4 h4 A 3 B 3 A 2 B 2 A 1 B 1 A 0 B bit Logarithmic hifter A 3 Out3 A 2 Out2 A 1 Out1 A 0 Out0 21

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