Testing multiplexors $($muxes$).$

a$.$ Create a circuit to test a $4-$to$-1$ multiplexor $($mux$)$ using three $2-$to$-1$ multiplexors $($muxes$).$

b$.$ Design a testing procedure to make sure that the circuit does exactly the same thing as a $4-$to$-1$ multiplexor.

c$.$ Build and test.

$2.$

A communications company wants to encode all its data before transmitting anything. We will design a circuit to assist them with this encoding. For the unsigned $3-$bit binary numbers, the encoding requires an extra bit, called X$,$ and can be calculated using even and odd parity. Each number transmitted would have $4$ bits, the $3$ bits of the unsigned binary number and X$.$ The encoding works as follows: ~ If the number is divisible by $4$ it must be transmitted with odd parity. ~ All other numbers must be transmitted with even parity. Again, you can interpret this as the description of a truth table. Note $0$ is divisible by $4.$ A binary number has odd parity if it contains an odd number of $1$s$.$ So $111$ and $010$ have odd parity, and $011$ has even parity.

a$.$ Draw a truth table $($with the operands in this order: $,,,)$ showing the function the output bit $($You do not need to display $$ in the output$).$

b$.$ Write a Boolean expression as a sum of minterms for the output bit.

c$.$ Simplify the expression using a Kmap. $($Show the map and final expression$)$

d$.$ Draw the corresponding Circuit.

e$.$ Redesign the Circuit to use a $4-$to$-1$ multiplexor $($mux$),$ show how you determined the input value for each line of the mux using a table.

f$.$ Redesign the Circuit to use three $2-$to$-1$ multiplexors $($muxes$).$

g$.$ Draw the corresponding wiring diagram.

h$.$ Build and test. Your table of test values should contain many $$don$$t care$$ values.