The document discusses error detection and correction techniques. It explains that bit errors can occur during data transmission due to noise. Various error detection strategies are described, including parity schemes, checksum, and CRC. Parity schemes like even and odd parity can detect some errors but not all. Checksum adds up data words and transmits the sum. CRC generates a polynomial for data and divisor, allowing it to detect more error types than checksum. For error correction, Hamming codes are discussed which can detect and correct single bit errors by using redundant parity bits and calculating separate parities.

2. ERROR DETECTION
 Bit errors are sometimes introduced into frames
 Bit error simply means change from “ 0 to 1 ” or “ 1 to 0 ”
This most of the time happens because of:
 NOISE

Where noise is defined as the induction of unwanted signal into
data

3. ERROR DETECTION
Data is transmitted in form of signals
Process of translating binary data into signal is called
 ENCONDING SCHEME
Most commonly the signal travelling on a wire is an
electrical or electromagnetic signal
 Usually one bit is represented by one voltage value
 And second bit by another voltage value

E.g.
1 = +5 Volt
0 = -5 Volt

4. ERROR DETECTION
Data 101010
10
10
10
1 01010
 In simple words electricity is flowing between the two PCs
 Whenever electricity or electrical signal is flowing through a wire
 It creates electromagnetic field around it
 Simply the area around the wire in which current has its affect is
called electromagnetic field

5. Data 101010
10
10
10
1 01010
10
10
10
1
0 01010
Noise

6. ERROR DETECTION
This type of noise is called electrical interference
There are many other types of noise:
 Thermal noise, Shot noise, Burst Noise, Flicker Noise
Bit errors are introduced into frames because of noise
Some strategy is needed to detect these errors
 Parity
 Checksum
 CRC ( Cyclic Redundancy Check )

7. ERROR DETECTION
General steps in every error detection strategy:
 Send extra information with data
 This information is calculated from data
 This information is generally called Redundant Information
 Errors are detected in data by using Redundant Information
 If error is detected in a frame


The frame is dropped
Sender retransmits the frame

8. ERROR DETECTION
Every strategy has its own way of calculating redundant
information and has its own name for it
Even
Parity
Odd Parity
Redundant
Information
Internet
Checksum
CRC
Number

9. ERROR DETECTION

10. ERROR DETECTION
Lets design a simple error detection strategy
In simplest case we can think of sending two frames
 Original data frame
 Copy of data frame ( Redundant Information )
 Receiver compares the two frames


If the two match there is no error
 Frame is accepted and copy is discarded
If the two do not match there is an error
 Frame is rejected and both data and copy are discarded

11. ERROR DETECTION
10 10 10
101010
Data
Copy
10
10
Data
10
00
10
10
10
10
10
Copy
 Receiver compares the two frames


If the two match there is no error
 Frame is accepted and copy is discarded
If the two do not match there is an error
 Frame is rejected and both data and copy are discarded

12. ERROR DETECTION
Is this a good Error Detection Strategy?
Too much redundant information

13. ERROR DETECTION
Parity Schemes
 Even Parity

Check digit (1 or 0) is added to make sure there are an even
number of 1s
 Odd Parity

Check digit (1 or 0) is added to make sure there are an odd
number of 1s
 Two-Dimensional Parity

Add parity bits in both dimension i.e. row-wise and column-wise

14. ERROR DETECTION
1 0 0 0 0 1 1 1
0 1 1 0 0 1 1 0
0
 Even Parity
 With every 7 bits of data one parity bit is added to keep number of 1s even
 When receiver receives the data, it checks the number of 1s
 If number of 1s are even then, data is correct
 If number of 1s are odd then there is an error

15. ERROR DETECTION
1 0 0 0 0 1 1 1
1 1
 Even Parity
 If more then one bits are corrupted in such a way that number of 1s stay
even, the error goes undetected

16. ERROR DETECTION
Is this a good Error Detection Strategy?
Cannot detect all types of errors

17. ERROR DETECTION
1 0 0 0 0 1 1 1
0 0 1 0 0 0 1 0
1 1 0 0 0 1 0 1
1 0 0 1 0 1 1 0
0 0 0 1 0 0 0 1
0 1 1 0 0 1 1 0
0 1 0 0 1 1 0 1
1 1 0 0 1 1 0 0
 Two-Dimensional Parity
 Divides data into blocks, where every block contains 7 groups of 7 bits
 Adds parity bits both column-wise and row-wise

18. ERROR DETECTION
Is this a good Error Detection Strategy?
Can detect more types of errors as compared to simple
parity schemes
Sends more Redundant Information as compared to
simple parity scheme

19. ERROR DETECTION
 When the logic for detecting errors is based on addition, such
strategies may be called checksum
 Internet Checksum Algorithm
 Add up all the words of sending data and then transmit the
result of that sum ( R.I ) with data

A word can be 1,2 or 4 bytes of data
 The receiver performs the same calculation on the received data
and compares the result with the received checksum
 If answers match there is no error otherwise error detected

20. ERROR DETECTION
1
1 1 1 1 1 1 1
Sum
1 0 1 0 0 1 1 1
• Add
• Words
0 1 0 0 1 1 0 0
1 0 1 1 0 1 1 0
• Data
• Sum
Transmitted
1 0 1 0 1 0 0 1
Compare
1
1 0 1 0 1 0 1 0
• Recalculate
• Sum
1 0 1 0 1 0 1 0

21. ERROR DETECTION
 This strategy is implemented at network layer
 Sends less redundant information
 A very weak strategy for Error Detection
 If bits are corrupted in such a way that sum does not change
error will go undetected

22. ERROR DETECTION
CRC ( Cyclic Redundancy Check)
 Before communication sender and receiver agree on a divisor
 Before sending the data CRC number is combined with data
 It makes the data completely divisible by divisor
On Receiver Side
 If data is again completely divisible, no Error detected
 If data is not completely divisible, Error is detected

24. ERROR DETECTION
CRC process is usually carried in polynomial form
 Polynomial Form is used for two reasons


Polynomial mathematics is easy and efficient
Long binary messages can be represented by short polynomials
 Conversion Process




Start from right side and write “x” under each binary digit with
powers in sequence
Multiply them with binary digits
Add all the terms
Highest power in a polynomial is the DEGREE of polynomial

25. 1
5 bits
0
Polynomial
1
1
Of
0
Degree 4

26. 1
0
6 bits
1
Polynomial
0
Of
1
1
Degree 5

27. 1
0
7 bits
0
Polynomial
0
0
Of
0
Degree 6
1

28. ERROR DETECTION
CRC Process:
 n + 1 bits of data (Message) M(x) is represented by a polynomial of degree n
For Calculating CRC:
 Both sender and receiver choose a common divisor C(x) with degree k
 Any number can be chosen as a divisor as long as k < n
 Multiply M(x) into xk to obtain T(x), can also be called zero-extended
message. T(x) = M(x) * xk
 Divide T(x) by C(x) and find remainder.
 Subtract Remainder from T(x).
T(x) / C(x) => Remainder
T(x) - Remainder
 At this point we get data combined with CRC , which is completely
divisible by divisor

29. C(x) = 1101
M(x) = 10011010
x7 + x4 + x3 + x1
x 3 + x2 + 1
10011010
M(x) Where
n=7
C (x) k=3
agreed ,k < n
101
(x10 + x7 + x6 + x4) / ( x3 + x2 + 1 ) = Remainder
T(x) = M(x) * xk5
6
7
x3 + x2 + 1
x - x + x – x4 + x3 +1
x8 + x7 + x5
7 + x 4 + x 3 + x 1 ) * x3
T(x) = (x
x3 + x2 + 1
CRC 101
T(x) – Remainder 3 43 Completely7 5 4
7.x3 + x4.x3 + x .x + x1.x3
T(x) = x x10 7 + x6 + x Divisible– x – T(x) x
On x + Zero
2–
Extended – x2 – 1
(x10 + x7 x10 6 + x4) x73+3 (– x1+3 1)
+ x + 4+3 + –
Divisor – x7 – x6 – x4
T(x) = x7+3 + x x9 + x + x
Message
x10 + x7 + x6 7 x4 6 x2 4 1 Data + CRC
+ + +
Remainder = 101
T(x) = x10 + 9 + x + x
x 6 4
6 – x5 + 2x4
–x +x +x
x
0
6
– 9 x8 000
10011010x –101– x
T(x) = 10011010
x8 + 2x6 + x4
0
x6 + x 5 + x3
– 0 5 + x3
2x
C(x) = 1101
Complete Division =
No Error
Incomplete Division =
Error Detected

30. ERROR DETECTION

31. ERROR DETECTION
BISYNC protocol uses two dimensional parity
IP uses Internet Checksum
HDLC, DDCMP, use CRC
Ethernet and 802.5 networks use CRC-32, while HDLC
uses CRC-CCITT. ATM uses CRC-8, CRC-10, and CRC-32.

32. ERROR CORRECTION
 Error correction are more complex schemes then detection
 We need to know the exact position of corrupted bits
 Hamming Codes:
 Developed by Richard Hamming
 Can correct up to one bit-error ( one we will study )
 Takes help from simple parity
 Redundant bits are placed at every 2K position
 Data is placed in between redundant bits
 Every redundant bit is responsible for checking bits at some
position
 CORRECTION comes after DETECTION

33. ERROR CORRECTION
R parity bits are required for 2R-R-1 data bits
Parity bits are placed at position 2K i.e. 1,2,4,8,16,…..
Parity bit at position k is responsible for every k bit(s)
starting from position k and at a gap of k bits. e.g.
 Parity bit 1 is responsible for bits at position [1], [3], [5], [7]……
 Parity bit 2 is responsible for bits at position [2,3], [6,7], [10,11]……
 Parity bit 4 is responsible for bits at position [4,5,6,7], [12,13,14,15]…..
Data is placed in between the Redundant Bits. i.e. any
position ≠ 2k
 e.g. 3,5,6,7,9,10,11,12,13,14,15,17…….

34. ERROR CORRECTION
Sender:
 Separate parity checking is done for every redundant bit
and the bits that it is responsible for
 By using even parity, number of 1s are kept even
Receiver:
 Recalculates parity bit in received data for every redundant
bit and its responsible bits
 By combining these recalculated parity bits we get the
position of corrupted bit
 The bit at that position is simply flipped

35. ERROR CORRECTION
 R parity bits are required for 2R-R-1 data bits
 E.g. for 7 data bits.
 For R = 1, 21 – 1 – 1 = 0
 For R = 2, 22 – 2 – 1 = 1
 For R = 3, 23 – 3 – 1 = 4
 For R = 4, 24 – 4 – 1 = 11
 So total of 11 bits with 7 data bits and 4 parity bits
 Parity bits are placed at position 2k and data bits between
them

36. ERROR CORRECTION
 Parity bit at position k is responsible for every k bit(s) starting from position
k and at a gap of k bits.

37. Insert
Data at
position
≠ 2k
1 0 0 1 1 0 1 Data
1 0
0 1
1 1
0 0
Separate
Parity
Checking
Flip Bit
Receiver
Sender
1
1 0
Combine
Bits =
Position
Transmission
Recalculate
Parity Bits
0
1
0
1
1
0
11 10 9
8
7
6
5
1
0
0
4
1
3
0
2
1
1
0
0 1 1 1
Position =7

38. Insert
Data at
position
≠ 2k
1 0 0 1 1 0 1 Data
1 0
0 1
1 1
0 0
Separate
Parity
Checking
Flip Bit
Receiver
Sender
1
1 0
Combine
Bits =
Position
Transmission
Recalculate
Parity Bits
1
0
0
1
1
1
0
11 10 9
8
7
6
5
0
4
1
3
0
2
1
1
0 0 0 0
Position = 0

39. HAMMING DISTANCE
To find the hamming distance we exclusively or every
code word with the code word below it
The minimum distance is then called
 dmin = Hamming Distance
Any error detection scheme having a minimum hamming
distance of dmin is guaranteed to:
 Detect errors up to : dmin – 1
 Correct errors up to: (dmin – 1) / 2