Given the two Boolean functions F and G below. You are asked to compare the "gate input

cost$"$ of their implementation using optimised sum$-$of$-$products form and hierarchical

design approach.

F$($A$,$B$,$C$,$D$)=$AB$(+)/($b$)$ar $($A$)((()/($bar$)($C$)(+)/($b$)$ar $($D$))($A$+$D$))$

G$($A$,$B$,$C$,$D$)=($CD$(+)/($b$)$ar $($A$)/($b$)$ar $($D$))$B$(+)/($b$)$ar $($B$)/($b$)$ar $($D$)$

$($a$)$ By using Boolean algebraic identities, expand F and G into sum$-$of$-$products form $($no

simplification is required, but terms like $($A$)/($b$)$ar $($A$)$ should be eliminated$).$ Hence, write

down the sum$-$of$-$minterms expressions $($using the notation $\backslash$sum m$($dots$))$ of both F and G$.$

$($b$)$ By using K$-$maps, find the optimised sum$-$of$-$products forms of both F and G$.$ Show

your steps clearly.

$($c$)$ Find the gate input cost $($not counting the cost for inverters$)$ for the implementation of

F and G respectively using your results in part $($b$)$ above.

$($d$)$ In hierarchical approach, Boolean functions F and G are implemented using THREE

copies of a suitable hierarchical component which is represented by Boolean function

H$($x$,$Y$,$Z$).$

$($i$)$ Write down the Boolean expression for the hierarchical function H$,$ and draw

the logic circuit diagram of such hierarchical component.

$($ii$)$ Rewrite the functions F and G in a form that the hierarchical function H is used

twice for each function. Put your answers in the format F$=$H$($dots$)$ and G$=$

H$($dots$).$

$($iii$)$ Draw a diagram to show the hierarchical implementation of F and G$.$ Use

graphical symbols to represent the hierarchical component in your answer.

$($iv$)$ Find the gate input cost $($not counting the cost for inverters$)$ for the

implementation of F and G respectively using your results in part $($iii$)$ above.