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## Space vs. Speed: Binary Adders

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**Space vs. Speed: Binary Adders**11.3 Space vs. Speed**Binary Adders**• VHDL Adder • Carry Lookahead Adder**4-Bit Adder**C 1 1 1 0 A 0 1 0 1 B0 1 1 1 S 1 1 0 0**Adder in VHDL**entity adder is port ( a: in STD_LOGIC_VECTOR (3 downto 0); b: in STD_LOGIC_VECTOR (3 downto 0); sum: out STD_LOGIC_VECTOR (3 downto 0); carry: out STD_LOGIC ); end adder;**AiBi**00 01 11 10 Ci 0 1 1 1 1 1 Ci+1 Ci+1 = Ai & Bi # Ci & Bi # Ci & Ai**Binary Multiplier**2 bit by 2 bit Half Adders are Sufficient Since there is no Carry-in in addition to the two inputs to sum**Binary Multiplier**4 bit by 3 bit 4 bit by 3 bit yields 7 bit result**Binary Adders**• VHLD Adder • Carry Lookahead Adder**Carry Lookahead Adder**C2 = G1 + P1(G0 + P0C0) = G1 + P1G0 + P1P0C0 C3 = G2 + P2(G1 + P1 (G0 + P0C0)) = G2 + P2(G1 + P1 G0 + P0C0) = G2 + P2G1 + P2P1G0 + P2PlP0C0 G0-3 = G3 + P3G2 + P3P2G1 + P3P2PlG0 P0-3 = P3P2PlP0**Typically, longest delay path through n-bit ripple carry**adder is 2n + 2 • Tends to be one of the largest delays in a typical computer design Counts as 2 gate delays 0 2 2 4 0 1 3 4 1 0**4**4 0 2 6 0 5 6 4**4**6 6 4 0 2 8 0 7 8 6**8**4 6 8 6 4 0 2 10 0 9 10 8**8**10 4 6 10 8 6 4 • 10 Gate Delays • 16-bit Adder -- 34 Gate Delays • 64-bit Adder -- 130 Gate Delays**Carry Lookahead Adder**• Uses Propogate and Generate signals to “lookahead” for incoming carry signals • More complicated hardware configuration • Substantial decrease in gate delays**Ripple Carry**PFA: Partial Full Adders Carry Lookahead**Propagate P = A xor B If P = ‘1’ then the carry is**“propagated” through. If P = ‘0’ then the carry is not “propagated” through. • Generate G = A and B • If G = ‘1’ a carry is “generated” regardless • of the carry bit.**Cin A B**P G Cout S 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 1 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 For final carry determination, the Propagate signal is ANDed with the Carry Outand the Generate signal is ORed to the resulting signal. G P Cin Cout**Cin A B**Cin A B P G P G Cout S Cout S 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 1 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 Always Generate a Carry forA = 1, B = 0 Propagate the Carry in**2**4 1 2 4 3 Cout**2**4 PFA For Bit # 1 1 2 4 1 3 4 2 2 3 2 3 1 4 2 1 2 Cout 3 2 3 1 4 2**Bit #1**Bit #2 2 2 4 6 1 1 4 Bit #3 Bit #4 2 2 6 6 1 4 4**Significant Delay Reduction**• 4 - bit Ripple: 10 Delays CLA: 6 Delays 1 CLA level: 1*4 + 2 = 6 • 16 - bit Ripple: 34 Delays CLA: 10 Delays 2 CLA levels: 2*4 + 2 = 10 • 64 - bit Ripple: 130 Delays CLA: 14 Delays 3 CLA levels: 3*4 + 2 = 14 But at the expense of a significant increase in the number of gatesused by the circuit