2’s Complement Number System is widely implemented in computer architecture.
N-bit 2’s Complement number System can be able to represent Number from -2n-1 to 2n-1- 1
4 Bit can be able to represent numbers from ( -8 to 7 )
5 Bit can be able to represent numbers from ( -16 to 15 ) in 2’s Complementary System.
Overflow happens with respect to addition when 2 N-bit 2’s Complement Numbers are appended and the answer is too large to fit into that N-bit Group.
A computer contains N-Bit Fixed registers. Result of addition of two N-Bit Number will result maximum N+1 Bit number.
Carry Flag stores that Extra Bit. But Carry does not always specify overflow.
Overflow happens when −
Result of addition of two negative numbers is positive or
Result of addition of two negative numbers is negative.
So overflow can be detected by verifying Most Significant Bit(MSB) of two operands and result. But Instead of implementing 3-bit Comparator Overflow can also be detected implementing 2 Bit Comparator just by verifying Carry-in and Carry-Out from MSB’s. We consider N-Bit addition of 2’s Complement number.
Overflow happens when Carry-in not equal to Carry-out. Above expression for overflow can be discussed from below analysis.
In case of first Figure the MSB of two numbers are 0 which indicates they are positive. Here if Carry-in is 1 we get result’s MSB as 1 indicates result is negative (Overflow) and Carry-out as 0. Carry-in is not equal to Carry-out hence overflow.
In case of second Figure the MSB of two numbers are 1 which indicates they are negative. Here if Carry-in is 0 we get result MSB as 0 indicates result is positive(Overflow) and Carry-out as 1. Carry-in is not equal to Carry-out hence overflow.
So Carry-in and Carry-out at MSB’s are sufficient to detect Overflow.
Above XOR Gate can be implemented to detect overflow.