H8: Adders & Multipliers

Time Estimate1-2 hours   Grade Impact1%   DueSep 14 @ 12pm  

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Notes

Show your work. You must show all steps of your work for full credit. While calculators can perform these calculations, demonstrating your process ensures understanding and mastery of the concepts.

Notation. To encourage flexibility between notations, the problems in this assignment use a variety of notations. We have discussed each style already; review the textbook or your notes if you need help.

Exercises

1. Derive the sum and carry outputs for a half adder. Construct the full truth table and write the Boolean expressions for each output in both sum-of-products and XOR/AND form.

2. Design a full adder using only AND, OR, and NOT gates. Show how the carry-out is formed from the three input bits and simplify your expressions as much as possible.

3. Add the two 5-bit binary numbers 10110 and 01111 using a ripple-carry adder. Show each intermediate carry and sum bit, and give the final result in binary and decimal.

4. For signed 4-bit two’s complement addition, determine whether overflow occurs for each case below. Explain your reasoning using carry into and out of the sign bit.
   a. 0101 + 0011
   b. 0111 + 0001
   c. 1100 + 1011

5. A full adder can be written as $S = A \oplus B \oplus C_{in}$ and $C_{out} = AB + AC_{in} + BC_{in}$. Show that the carry expression can also be written as $C_{out} = AB + C_{in}(A \oplus B)$. Verify the equivalence algebraically.

6. Use an adder to compute 9 - 6 in 4-bit unsigned binary and -3 + 5 in 4-bit two’s complement. Show how subtraction is converted into addition and identify any carry-out or overflow.

7. For a 4-bit adder, define the propagate and generate signals for each bit. Using these signals, derive expressions for C1, C2, C3, and C4 in a carry lookahead adder.

8. Explain how a subtraction-based adder can be used to compare two unsigned 4-bit numbers A and B. Describe how to determine whether A > B, A = B, or A < B from the result and carry-out.

9. Multiply 1011₂ by 110₂ using the partial-products method. Show each shifted partial product and add them to obtain the final product.

10. Multiply the 4-bit two’s complement numbers 1101 and 0011. Check your work by converting each number to decimal, perform the multiplication, and express the result in binary using enough bits to avoid overflow.