$(30$ points$)$

Two$-$Sequence Detector: Derive the state$-$diagram for a FSM sequence detector with a single input $w$ and single output $Z.$ The FSM generates an output $Z=1$ when the input sequence from $w$ is either $1110$ or $0110$ in four consecutive clock cycles; otherwise $Z=0.$ Overlapping input patterns are allowed.

a$)$ Complete the following example for a Moore FSM

$\backslash$table$[[w,0,1,0,0,1,1,0,1,1,1,0,0,1,1,1,1,0,1,1,0,0]]$

$\backslash$table$[[Z,,,,,,,,,,,,,,,,,,,,,,]]$

b$)$ Show the state$-$diagram for a Moore FSM sequence detector. Minimize the number of states.