1 Digital Logic Design Week 3 Logic gates (NOT, AND, OR, NAND, NOR, XOR, XNOR) Applications of...

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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)

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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

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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

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NOT gate

also known as an inverter

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

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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

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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

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AND gate

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

A

B

X

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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

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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

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Timing diagram

AND gateA

B

X

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AND gate

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OR gate

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

A

B

X

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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

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OR gates have 2 or more inputs

Write the truth table for a 3-input OR gate

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OR gate

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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

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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

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Name the mystery gate #1

?

Alarm to be activated if any door/window is open

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Name the mystery gate #2

?

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????

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

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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

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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

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Summary of logic gates

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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

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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.

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

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

?

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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

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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

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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

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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.

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

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