Problem $5(15$ points$)$:

This problem involves the design of a circuit that finds the square root of an $8-$bit unsigned binary number $N$ using

the method of subtracting out odd integers. To find the square root of $N,$ we subtract $1,$ then $3,$ then $5,$ etc.,

until we can no longer subtract without the result going negative. The number of times we subtract is equal

to the square root of $N.$ For example, to find $\sqrt[\phantom{2}]{?}27$ : $27-1=26$;$26-3=23$;$23-5=18$;$18-7=11$;$11-$

$9=2$;$2-11($can$\text{'}$t subtract$).$ Since we subtracted $5$ times, $\sqrt[\phantom{2}]{?}27=5.$ Note that the final odd integer is ${11}_{10}=$

${1011}_2,$ and this consists of the square $\sqrt[{101}_2]{?}=5_{10}$ followed by a $1.$

$($a$)$ Draw a block diagram of the square rooter that includes a register to hold $N,$ a subtracter, a register

to hold the odd integers, and a control circuit. Indicate where to read the final square root. Define

the control signals used on the diagram.

$($b$)$ Draw a state graph for the control circuit using a minimum number of states. The $N$ register should

be loaded when $St=1.$ When the square root is complete, the control circuit should output a done

signal and wait until $St=0$ before resetting.