Digital Systems

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 Is. So 1110 and 0100 have odd parity, and 0110 has even parity. Draw a truth table (with the operands in this order A, B, C, X) showing the function the output bit X (You do not need to display ABC in the output). Write a Boolean expression as a sum of minterms for each output bit. Simplify each expression using a Kmap. (Show the map and final expression) Draw the corresponding logic diagram