1

Digital Logic Design

Week 3

Logic gates (NOT, AND, OR, NAND, NOR, XOR, XNOR)

Applications of Binary Codes (MOTIVATION)(ASCII and UNICODE codes, Seven-segment Display Code, Gray code-rotation counter, Error Detection and Correction code)

2

Logic gates

Logic gates– NOT gate– AND gate– OR gate– NAND gate– NOR gate– Exclusive-OR (XOR) and exclusive-NOR (XNOR) gates

Fixed-function logic & programmable logic

3

True only if all input conditions are true

True only if one or more input conditions are true

Indicates the opposite condition

Basic logic operations and symbols

True/false conditions are represented by voltages:HIGH = true LOW = false

X

4

NOT gate

also known as an inverter

performs inversion or complementation• HIGH ↔ LOW• 1 ↔ 0

5

NOT gate Truth Table

A X = A’

0 1

1 0

A truth table shows the output corresponding to each possible input

Three equivalent ways to write the NOT condition:X = NOT A

X = A’

A X

A X

We’ll use truth tables extensively in the course

6

Timing diagram

A

X

A X

Application: A group of inverters can be used to form the 1’s complement of a binary number:

Binary number

1’s complement

1 0 0 0 1 1 0 1

0 1 1 1 0 0 1 0

NOT gate

7

AND gate

Produces a HIGH output when all inputs are HIGH; otherwise, the output is LOW

A

B

X

8

Truth table for a 2-input AND gate

AND operation is sometimes shown with a dot between the variables, but it may be implied (no dot):

AND operation is written as X = A·B or X = AB

InputsA B X

Output

0 00 11 01 1

00 01

AND gateA

B

X

9

AND gates have 2 or more inputs

Write the truth table for a 3-input AND gate

#inputs = n #input combinations (= #rows in truth table) = 2n

2 inputs → 22 = 4 input combinations 3 inputs → 23 = 8 input combinations 4 inputs → 24 = 16 input combinations

10

Timing diagram

AND gateA

B

X

11

AND gate

12

OR gate

Produces a HIGH output if any input is HIGH; if all inputs are LOW, the output is LOW

A

B

X

13

Truth table for a 2-input OR gate

The OR operation is shown with a plus sign (+) between the variables:

OR operation is written as X = A + B

InputsA B X

Output

0 00 11 01 1

01 11

OR gateA

B

X

14

OR gates have 2 or more inputs

Write the truth table for a 3-input OR gate

15

OR gate

16

NAND gate

NAND = NOT-AND

Truth table for a 2-input NAND gate

NAND operation is written as or

InputsA B X

Output

0 00 11 01 1

11 10

BA X BA X

17

NOR gate

NOR = NOT-OR

Truth table for a 2-input NOR gate

NOR operation is written as or

InputsA B X

Output

0 00 11 01 1

10 00

BA X BA X

18

Name the mystery gate #1

?

Alarm to be activated if any door/window is open

19

Name the mystery gate #2

?

20

????

Name the mystery gate #3

Want green light ON if both tanks are at least 25% full

Level sensor output is HIGH if tank is at least 25% full

Light turns on if this signal is

LOW

21

XOR gate 2-input exclusive-OR (XOR) gate produces a HIGH

output if the inputs are at opposite logic levels– one HIGH & one LOW

Truth table for a 2-input XOR gateInputsA B X

Output

0 00 11 01 1

01 10

B A X XOR operation is written 3-input XOR defined:

– extends to N-input XOR

CBACB A CB A X

22

XNOR gate 2-input exclusive-NOR (XNOR) gate produces a HIGH

output if the inputs are at same logic levels– both HIGH, or both LOW

Truth table for a 2-input XNOR gateInputsA B X

Output

0 00 11 01 1

10 01

BA X XNOR operation is written

A

B

X

23

Summary of logic gates

24

1. The binary number 100011010100011011112 expressed in hexadecimal is:

(a) AD46716

(b) 8C46F16

(c) 8D46F16

(d) AE46F16

2. Convert the decimal number 28410 to hexadecimal

25

3. Express the decimal number 246910 in binary-coded decimal.

4. A communication system transmits 9-bit blocks of information using an even parity scheme. The following byte of information is to be transmitted: 01110101

Calculate the value of the parity bit to be attached to this byte.

26

5. For the set of input waveforms shown below, draw the timing diagram showing the output X in relation to the inputs.

27

6. By examination of the timing diagram below, identify the type of 3-input gate which generates the output X.

?

28

7. A 2-input gate with inputs A and B generates output X shown in the timing diagram below. The gate is:

(a) an OR gate(b) an AND gate(c) a NOR gate(d) a NAND gate

A

X

B

29

9. The truth table for a 2-input NOR gate is:

© 2008 Pearson Education

0 00 11 01 1

InputsA B X

Output

0 00 11 01 1

InputsA B X

Output

0 00 11 01 1

InputsA B X

Output

InputsA B X

Output

0 00 11 01 1

a. b.

c. d.

0110

00 01

10 00

01 11

30

10. Compute the 2’s complement of 011011002

11.The decimal number +12210 is expressed in 8-bit 2’s complement form as:

(a) 01111010(b) 11111010(c) 01000101(d) 10000101

31

12. Write −3410 as a binary number in 8-bit 2’s complement form

13. Convert each of the following decimal numbers to 8-bit 2’s complement form, and add them: −52 and 25.

Check your answer by converting the result back to decimal.

32

14. Convert the hexadecimal number 6B16 to octal (base-8)

15. Perform the following addition of hexadecimal numbers: DF16 + AC16