Consider the design of a Moore type sequential circuit for the channel search part of a hand$-$held remote controller for the TV set with three channels: CH$1,$ CHC and CH$3.$ There is the CS $($Channel Search$)$ button on the controller. As long as this button is pressed, next channel on the TV is accessed, ie$,$ if the TV is at CH$1$ before the button is pressed, it will change to CH$2$ as soon as CS is pressed. Otherwise, it will stay at the current

channel Requirements:

$($ The channel selection unit of the remote controller has one input namely CS and three outputs CH$1,$ CH$2$ and CH$3.$

$(1)$ The CS input is LOW $($Logic $0)$ as long as the CS button is not pressed and goes HIGH $($Logic $(1)$ when the button is pressed.

$($i$)$ Only one of the outputs CH$1,$ CH$2$ and CH$3$ will be HIGH at the same time and it depends which channel is to be selected.

$($iv$)$ The state representing CH$1$ will be encoded as $01,$ the state representing CH$2$ will be encoded as $10$ and CH$3$ as $11$

$($V$)$ There is one state that is not used. The unused state will directly move to the state of CHT but all outputs must be zero so that no channel will be selected during that transition

a$)(10$ points$)$ Derive the state diagram of the circuit

b$)(10$ points$)$ Derive the state table of the circuit

c$)(10$ points$)$ If T type flip$-$flops are used in the circuit, derive the simplified flip$-$flop input and the outeut equations of the circuit