Part IV$.$ Sequential Circuits $(45\%)$

$6.[15\%]$ From equation to state graph.

A sequential circuit has three D flip$-$flops, A$,$ B$,$ C$,$ one input X$,$ and one output $2.$ The flip$-$flop input equations are described as follows:

$A^{**}=D_A=A^{\text{'}}B{C}^{\text{'}}+x{A}^{\text{'}}{B}^{\text{'}}C+x{B}^{\text{'}}A{B}^{\text{'}}$

$B^{**}=D_B=x^{\text{'}}{A}^{\text{'}}C+x{A}^{\text{'}}{C}^{\text{'}}$

$C^{**}=D_C=x^{\text{'}}{A}^{\text{'}}{B}^{\text{'}}+x{A}^{\text{'}}C+A^{\text{'}}{B}^{\text{'}}C$

$($a$)3\%$ Please derive the state table for the design. $($assume $S_0=$

$000,S_1=001,S_2=010,S_3=011,S_4=100,S_5=101,S_6=$

$110,S_7=111$

$($b$)4\%$ Please draw the state graph if the output equation is: $Z=$

$x^{\text{'}}{A}^{\text{'}}BC+x^{\text{'}}A{B}^{\text{'}}{C}^{\text{'}}+x{A}^{\text{'}}BC+x{B}^{\text{'}}{C}^{\text{'}}$

$($c$)4\%$ Please draw the state graph if the output equation is: $Z=$

$A{B}^{\text{'}}{C}^{\text{'}}+A^{\text{'}}BC$

$($d$)4\%$ Continue on $($c$),$ can you use only $2$ state variables to implement a state machine which have the same function? If so$,$ please show how you eliminate the redundant states, and draw the state graph with minimum number of states.