MODULE_2_Ch4_01082014.pptx .

1. Department of Electronics and Communication Engineering, MIT, Manipal
PART II
DIGITAL ELECTRONICS
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Reference:
1. Malvino and Leach, Digital Principles & applications, 7th
edition, TMH, 2010
2. Morris Mano, “Digital design”, Prentice Hall of India, Third
Edition.
Chapter 4 : Number systems and Codes

2. Department of Electronics and Communication Engineering, MIT, Manipal
Module 2: Codes
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Learning outcomes
At the end of this module, students will be able to:
• Discuss different binary combinations to represent code
characters.
• Explain error detection using parity bit.
• Describe error correction using hamming code.

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Binary coded decimal codes
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Weighted Codes

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Binary coded decimal codes
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• Weighted binary codes are those binary codes which
obey the positional weight principle.
• Each position of the number represents a specific weight.
• There exists a fixed weight associated with each bit
position in the binary representation of the code
character.
Weighted codes

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Binary coded decimal codes
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Decimal number (A)10
Encoded in the binary form as a3 a2 a1 a0.
w3, w2, w1 and w0 are the weights selected for a given
code
(A)10 = w3a3 + w2a2 + w1a1 +w0a0
The more popularly used codes have these weights as
W3 W2 W1 W0
8 4 2 1
2 4 2 1
8 4 -2 -1

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Binary coded decimal codes
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Binary Coded Decimal code (BCD)
Consider the number (16.85)10
(16.85)10 = (0001 0110 . 1000 0101)

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Binary coded decimal codes
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Non-Weighted Code is one in which the positions in the code
do not have a specific weight. Examples are Excess-3. And
Gray.
• EXCESS-3 CODE
• GRAY CODE

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Binary coded decimal codes
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Decimal BCD = 8421 Excess-3 Gray
0 0000 0011 0000
1 0001 0100 0001
2 0010 0101 0011
3 0011 0110 0010
4 0100 0111 0110
5 0101 1000 0111
6 0110 1001 0101
7 0111 1010 0100
8 1000 1011 1100
9 1001 1100 1101
Decimal to BCD, Excess-3 and Gray code

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Self complementing codes
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Example: Consider the 2421 code.
The 2421 code of (4)10 is 0100.
Its complement is 1011 which is 2421 code for (5)10 = (9 - 4)10.
The complement of a code (by inverting each bit of code word)
the new code word formed represents the complement of the
number then such codes are called self-complementing code.

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Self Test
1. What do you mean by a code word?
2. What are the two types of BCD codes?
3. What do you mean by self – complementing code?
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Error Detection and Correction
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• Introduction
• Single bit Error detection using parity bit
• Single bit error correction using (7,4) Hamming code

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Introduction
• When data is transmitted in digital form from one place to
another place through a transmission channel some data bits
may be modified.
• In a communication system data integrity is extremely
important.
• Ability to identify the error is called the error detection. It
would be preferred even well if we can correct the error besides
detection.
• A code which has the ability to correct the is called error
correction code.

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Error Detection Codes
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• Parity: Number of ones in the given code word.
• Even & Odd parity:
Example: 0000 (1)odd-parity (0)even-parity
Example: 0100 (1)odd-parity (0)even-parity

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Error Correction code
 Principle of error correction
 Consider a (7,4) Hamming code
 Let i1 i2 i3 i4 be information symbols
 Let p1p2 p4 be check symbols
 The parity equations:
p1 = i1  i2  i4
p2 = i1  i3  i4
p4 = i2  i3  i4
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Hamming Code
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Can write the equations as follows (easy to remember)
p1 p2 i1 p4 i2 i3 i4
1 0 1 0 1 0 1
0 1 1 0 0 1 1
0 0 0 1 1 1 1
1 2 3 4 5 6 7
This encodes a 4-bit information word into a 7-bit code
word

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Structure of Encoder and Decoder of
Hamming Code
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Hamming Code
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There is 10 different messages for BCD and EXCESS-3.
Decimal Digit Hamming Code bits
For BCD For EXCESS 3
P1 P2 D3 P4 D5 D6 D7 P1 P2 D3 P4 D5 D6 D7
0 0 0 0 0 0 0 0 1 0 0 0 0 1 1
1 1 1 0 1 0 0 1 1 0 0 1 1 0 0
2 0 1 0 1 0 1 1 0 1 0 0 1 0 1
3 1 0 0 0 0 1 1 1 1 0 0 1 1 0
4 1 0 0 1 1 0 0 0 0 0 1 1 1 1
5 0 1 0 0 1 0 1 1 1 1 0 0 0 0
6 1 1 0 0 1 1 0 0 0 1 1 0 0 1
7 0 0 0 1 1 1 1 1 0 1 1 0 1 0
8 1 1 1 0 0 0 0 0 1 1 0 0 1 1
9 0 0 1 1 0 0 1 0 1 1 1 1 0 0

18. Department of Electronics and Communication Engineering, MIT, Manipal
Self Test
1. What is parity bit? How many types are there?
2. What do you mean by distance between two words?
3. What is the minimum distance of 7- bit Hamming code?
4. What must be the minimum distance of a code for it to be
single – bit error correcting code?
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Summary
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 Different forms of BCD representation
 Weighted and non weighted codes
 Error detection using parity bit.
 Error correction using Hamming code.