Revision Mid 2

Prof. Sin-Min Lee

Department of Computer Science

Example Problem A hall light is controlled by two light switches, one

at each end. Find (a) a truth function, (b) a Boolean expression, and (c) a logic network that allows the light to be switched on or off by either switch.

x y f(x,y)

0 0 0

0 1 1

1 0 1

1 1 0

(What kind of gate has this truth table?

Let Let xx and and yy be the switches: be the switches:

Example (cont)

One possible equation is the complete sum-of-products form:

f(X,Y) = XY* + X*Y Use The Most Complex Machine xLogicCircuit

Module to implement the

equation.

x y f(x,y)

0 0 0

0 1 1

1 0 1

1 1 0

How to use NAND gates to build an OR gate?

Truth TableA B C D Q

0 0 1 1 0

0 1 1 0 1

1 0 0 1 1

1 1 0 0 1Hint 1 : Use 3 NAND gates

Hint 2 : Use 2 NAND gates to build 2 NOT gates

Hint 3 : Put the 3rd NAND gate after the 2 “NOT” gates

A

B

C

DQ

How to use NAND gates to build a NOR gate?

Truth TableA B C D E Q

0 0 1 1 0 1

0 1 1 0 1 0

1 0 0 1 1 0

1 1 0 0 1 0

A

B

C

DQ

E

Hint 4 : Put the “NOT” gate after “OR” gate

Hint 3 : Use a NOR gate to build a NOT gate

Hint 2 : Use 3 NAND gates to build an OR gate

Hint 1 : Use 4 NAND gates

Multiplexers

A combinational circuit that selects info from one of many input lines and directs it to the output line.

The selection of the input line is controlled by input variables called selection inputs.

They are commonly abbreviated as “MUX”.

These pictures have errors.

T

CLK

Q1

T

CLK

Q1

How to use D to implement T Flip-Flop

Q

QSET

CLR

D

D = TQ’ + T’Q

T

How to use T to implement D Flip-Flop

T Q+

0

1

Q

Q’

D Q+

0

1

0

1

T = DQ’ + D’Q

Q+ 0 1

D

0

1

0 1

1 0

How to use T to implement D Flip-Flop

Q

QSET

CLR

DT

T = DQ’ + D’Q

D

Implementing with a D AND a T flip-flop

Using this FSM with three states, an operating only on inputs and transitions from one state to another, we will be using both D and T flip-flops.

Implementing with a D AND a T flip-flop

Since we have no state “11”, our Q(t+1) is “don't care” = “XX” for both of these transitions.

Consider the first column of the Q(t+1) values to be “D” and the second to be “T” and then we derive two corresponding charts.

D T

Implementing with a D AND a T flip-flop

Then we need to derive the corresponding equations.

Implementing with a D AND a T flip-flop

We assume that Q(t) is actually a pair of Q

DQ

T.

Now, with these equations, we can graph the results.

Implementing with a D AND a T flip-flop