How can a 4-bit two's complement operation be implemented using only boolean logic gates (AND, OR, NOR, NOT, NAND, XOR, and XNOR)?
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Sign up to join this communityHow can a 4-bit two's complement operation be implemented using only boolean logic gates (AND, OR, NOR, NOT, NAND, XOR, and XNOR)?
(This question was redirected to CS from Stack Overflow)
A two's complement operation is simply a one's complement operation followed by the addition of 1
to the result. One's complement is easy: simply invert all of the input bits.
The addition of 1
must be done with a 4-bit adder. A 4-bit adder is constructed using four stages of a 1-bit full adder. The 1-bit full adder accepts two bits, plus a Carry input, and generates the sum of the two bits, plus a Carry output. The following diagram is a 1-bit full adder:
We can cascade four of the 1-bit full adder stages together, feeding the Carry output of each stage to the Carry input of the next stage. The inverted (one's complement) inputs are applied to the B
inputs of the four stages. To perform an addition of 1
, we apply the 4-bit binary value 0001
to the A
inputs. The complete boolean circuit is shown below:
The above circuit can be reduced by noting that each XOR operation on the input of each adder stage can be replaced either with an inverter if the A
input is a 0
, or a NOP (no operation) if the A
input is a 1
. On further analysis, further reductions may be made to the circuit, as well.