Question $4(8$ points$)$

Design the finite state machine with input x and output z described by the state diagrams shown below: input x and output z values are shown at the state transitions as numerator and denominator, respectively. The initial state S $0$ is obtained by flip$-$flop clearing so that the flip$-$flop outputs in state S $0$ are $\backslash (\backslash$mathrm$\{$Q$\}1\backslash$mathrm$\{$Q$\}0=00\backslash ).$

a$)$ Assign state codes Q$1$Q$0$ to states S$1,$ S$2($your choice$).$

b$)$ Obtain D$-$flip$-$flop input excitation equations and equations for the output forming logic in terms of input x and state variables Q$1$Q$0.$

c$)$ Implement the circuit with two D$-$flip$-$flops, any number of $2-$to$-1$ multiplexers, inverters. Sketch the schematic.

d$)$ Obtain the waveforms for the flip$-$flop outputs Q$1$Q$0$ and output z for input sequence $\backslash (\backslash$mathrm$\{$x$\}=100101\backslash )$ for $6$ clock periods. The flip$-$flops have positive edge triggering. Input x changes at negative clock edges.