P$2.$ Design a circuit based on JK flip$-$flops that will step through the $3-$bit Gray code cycle.

$($What$\text{'}$s Gray code? See below.$)$ Each of the JKs will hold one bit, together they make the

$3-$bit value $Q_3{Q}_2{Q}_1$ of the code.

$($a$)$ First remind yourself of the $Q$ output transitions of a JK flip$-$

flop by writing down a chart such as this one $($recall that it is

possible to have "don't care" inputs use $x$ to show these$).$

$($b$)$ Make a table showing the flip$-$flop outputs $Q_3{Q}_2{Q}_1$ of the $3-$bit Gray code cycle. Add

six columns for $J_1,K_1,dots{K}_3,$ and fill in these columns with the values necessary to

generate the cycle.

$($c$)$ Pick your favorite two columns from the six $J_1,K_1,dots{K}_3$ and determine the logic gates

necessary to generate those columns from the $Q$ outputs.

Gray Code: How does a machine know when it is in the right location? This problem is often solved using

some version of "optical encoder": a set of photodiodes reads the light$/$dark pattern on a moving part and

determines where the part is$.$ For example, to determine the rotational position of a disc, three photodiodes

could be used $($at three different radii$)$ to sense light$/$dark marks, these can be converted into $2^3=8$

possible locations. You might think it would be a good idea to mark the pattern using binary counting,

but that can lead to serious problems if either the marks or the light sensors are slightly misaligned. In

the 'binary counting' example below, going from sector $011100$ involves all three bits changing. If the

markings aren't quite lined up it is possible that the state briefly goes through some other value $($such as

$011111100),$ in which case the machine might malfunction $($for example, if it were trying to move to

the sector actually marked $111).$ But if you look carefully at the eight positions using Gray code below, you

will see that for each transition only one bit ever changes this means there is no danger of accidentally

passing through a false position state.