$($n$-$bit full adder$)n=42^k$ with integer $k.$

a$.$ Design a $4-$bit adder in a way you like. Assuming a delay of $1$ time unit

$($TU$)$ for any gate you may use, determine the delay $($in TU$)$

corresponding to each output.

b$.$ Utilizing your $4-$bit adder, design an $8-$bit adder using the ripple carry

approach discussed in class. Determine the delay for the carry$-$out bit

and the maximum delay for the sum bits.

c$.$ Big$-$O notation ${?}^2$ is very useful in inspecting the asymptotic behavior of

algorithms and devices. Generalize your solution in part $b$ to an $n-$bit

adder and find out the delays in big$-$O notation.

d$.$ We now work on a carry look ahead idea to speed up the operation of an

$8-$bit adder. A ripple carry adder waits for the carry out bit's generation.

Alternatively, we can have two $4-$bit adders for the more significant $4$

bits, one with a carry$-$in of $0$ and the other of $1,$ in order to account for

two carry$-$in possibilities that would come from the sum of the lower $4$

bits. After carry$-$in is determined, we can simply select the correct one.

Design such an $8-$bit adder by making use of the $4-$bit adder from part $b$

and a $2-1$ MUX.

e$.$ Determine the delays in TU for your solution in part $d.$

f$.$ Generalize your solution to an n$-$bit adder and find out the delays in big$-$

O notation. $($Hint: The answer should not be $O(n)$ and think about why

there is a $2^k$ term in $n.)$

g$.$ Nothing should be free in real life. What is the tradeoff?