Q1

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Q1. (6 points) A full-adder is a combinational circuit that forms the arithmetic sum of three input bits. It consists of three inputs, x, y, z, and two outputs, C and S. Two of the input, that is, x and y, represent the two significant bits to be added. The third input, z, represents the carry from the previous lower significant position. The output S denotes the sum of two bits and C denotes carry. Answer the following sub-questions.

1) Construct a truth table for the Full-Adder

2) Based on the truth table, construct a K-map for the output S and derive a Boolean equation using K-map. Make the equation as simple as possible.

3) Based on the truth table, construct a K-map for the output C and derive a Boolean equation using K-map. Make the equation as simple as possible.

4) By algebraic manipulation, show that S can be expressed as the exclusive-OR of the three input variables. That is, show that,

S = x XOR y XOR z.      

You can prove by deriving from S = x XOR y XOR z to the answer you got in the question 2)

5) By algebraic manipulation, show that C can be expressed as the following term.

            C = xy + (x XOR y)z.

You can prove by deriving from  C = xy + (x XOR y)z to the answer you got in the question 3)

6) Based on 4) and 5), draw a circuit for the full-adder in Logisim simulator, attach the image file and submit the circuit file!

Q2. (4 points) A sequential circuit has one D flip-flop and one JK flip-flop, two inputs x and y, and one output z. A is the output of D flip-flop, and B is the output of JK-flip-flop; A and B together form the "output state" of the circuit. The flip-flop input equations and the circuit output are as follows. Here DA is the D input of the D-flip flop of A, and JB, KB is the J and K input of the JK-flip flop of B.

DA = ~xy + yB

JB = ~yB + xy

KB = xB + ~yA

z = x+~xy

1) Draw the logic diagram of the circuit and test it with Logisim.

Please attach the circuit image and the generated table!

Please also submit the circuit file!
2) Construct a state diagram of this circuit.

Q3. (10 points) Design a system with the following state changes: This is a sequential circuit with three flip-flops. The state sequence is changed with a clock as in the order of, 111, 010, 100,110, 001, 011, 101, 000, 111 and repeat. Use JK flip-flops.

1) Draw a state diagram.

2) Construct an excitation table .

3) Draw K-maps and derive Boolean equations using K-maps. Make the equations as simple as possible.

4) Draw the system in Logisim simulator, attach the circuit image and submit the circuit file.

5) Test the system and attach the generated table.