UnidentifiedA Finite State Machine is designed to perform the function of $3-$bit binary counting.

Based on the state of the control variable A$,$ it counts forward as in an ordinary $3-$bit

binary counter or backwards as in a $3-$bit reverse binary counter. The Machine has

eight states: S$0,$ S$1,$ S$2,...$ S$6$ and S$7,$ corresponding to eight decimal numbers:

$0,1,2,3,4,5,6$ and $7.$ The states are coded as given in Table Q$3\mathrm{.}1.$

When input $A$ is false, i$.$e$.A=0,$ the Machine counts forward. When input $A$ is true,

i$.$e$.A=1,$ the Machine counts backwards.

When the Machine is at $S7$ state, it produces an output of $Z=1.$ At other states,

$Z=0.$ The output values are also shown in Table Q$3\mathrm{.}1.$

The current states of the Machine are denoted by the state bits $Q_2,Q_1$ and $Q_0$ and the

next states are denoted by $D_2,D_1$ and $D_0.$

Table Q$3\mathrm{.}1$

Question $3$ continues over the page.

Question $3$ continued.

$($a$)$ Draw the state diagram of the Finite State Machine and clearly indicate the

state names, input values and output values.

$[3$ marks$]$

$($b$)$ Derive the state transition table for this Finite State Machine. Show all

stages of work.

$[5$ marks$]$

$($c$)$ By using a Karnaugh map, derive the simplest expression for:

$($i$)D_2,$

$($ii$)D_1,$

$($iii$)D_0,$

as a function of $A,Q_2,Q_1$ and $Q_0.$ Clearly show all stages of work.

marks