GATE CSE 2017 Set 1 | Question: 9

Question

When two $8\text{-bit}$ numbers $A_{7}\cdots A_{0}$ and $B_{7}\cdots B_{0}$ in $2$'s complement representation (with $A_{0}$ and $B_{0}$ as the least significant bits) are added using a ripple-carry adder, the sum bits obtained are $S_{7}\cdots S_{0}$ and the carry bits are $C_{7}\cdots C_{0}$. An overflow is said to have occurred if

• A. the carry bit $C_{7}$ is $1$
• B. all the carry bits $\left ( C_{7},\cdots ,C_{0} \right )$ are $1$
• C. $\left ( A_{7} \cdot B_{7} \cdot \overline{S_{7}}+\overline{A_{7}} \cdot \overline{B_{7}} \cdot S_{7} \right )$ is $1$
• D. $\left ( A_{0} \cdot B_{0} \cdot \overline{S_{0}}+\overline{A_{0}} \cdot \overline{B_{0}} \cdot S_{0} \right )$ is $1$

Comments

repeated concept.

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Overflow condition can be written in any of the three ways – 

1. $C_{n} \oplus C_{n-1}=1$ which says that either 1st MSB or 2nd MSB should give carry but not both.
2. $A_{n}B_{n}\bar{C_{n-1}} + \bar{A_{n}}\bar{B_{n}}C_{n-1}=1$ where $A_n$ and $B_n$ are the MSB (which are sign bits) and $C_{n-1}$ is the carry from the 2nd MSB.
3. $A_{n}B_{n}\bar{R_{n}} + \bar{A_{n}}\bar{B_{n}}R_{n}=1$ where $R_{n}$ is the MSB (sign bit) of result of the addition.

From the third condition above, we can see that overflow will happen when:

 1. $A_n,B_n$ is 0 (which means positive numbers) and $R_n$ is 1 (which means a negative number).

2. $A_n,B_n$ is 1 (which means both are negative numbers) and $R_n$ 0 (which means a positive number).

Conclusion: For overflow to happen, two positive numbers are added and the result is negative OR two negative numbers are added and the result is positive.

PS: 2nd result can be derived from 1st result by replacing the value of $C_n$ with $A_nB_n + C_{n-1}(A_n \oplus B_n)$ and 3rd result can be derived from the 2nd result by thinking a bit. Let me know if you find it difficult to derive it.

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ALL QUESTION ONE SOLUTION NICE  jatinmittal199510

Answer 1

Answer: (C) 

Explanation: Overflow indicates that the result was too large or too small to fit in the original data type.
Overflow flag indicates an overflow condition for a signed operation. Signed numbers are represented in two’s complement representation. 
The overflow occurs only when two positive number are added and the result is negative or two negative number are added and the result is positive. Otherwise, the sum has not overflowed.

Therefore, a XOR operation can quickly determine if an overflow condition exists. i.e., 
(A7 . B7 )⊕(S7) = (A7 . B7 . S7‘ + A7‘ . B7‘ . S7 = 1

http://www.geeksforgeeks.org/gate-gate-cs-2017-set-1-question-35/

Answer 2

Option (c) is correct