1. Part III: Data Link Layer Error Detection and Correction1
Chapter 10
Part III: Data Link Layer
Error Detection and Correction
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2. Error Detection and Correction
10.#
Error Detection and Correction
What does accuracy means?
What about corrupted message?
Which layer is responsible for error detection and correction?
Does all application has the same sensitivity to errors?
Data can be corrupted during transmission.
Some applications require that errors be detected and corrected.
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3. Error Detection and Correction
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Error Detection and Correction
By the End of this Chapter you will be able to:
Understand different Types of Errors & Redundancy
Detection Versus Correction
Forward Error Correction Versus Retransmission
Coding
10.4 CYCLIC CODES
CRC Cyclic Redundancy Check
Polynomials Cyclic Code Analysis
Advantages of Cyclic Codes Other Cyclic Codes
10.5 CHECKSUM
Idea /One's Complement /Internet Checksum
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4. Data Link Layer Duties
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5. 10.#
INTRODUCTION
Let us first discuss some issues related, directly or indirectly, to error detection and correction.
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6. 10.#
Types of Errors
Whenever bits flow from one point to another, they are subject to unpredictable changes because of interference.
There are two types of errors:
Single-Bit Error
means that only 1 bit of a given data unit is changed from 1 to 0 or from 0 to 1.
Burst Error
means that 2 or more bits in the data unit have changed from 1 to 0 or from 0 to 1.
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7. 10.# Figure 10.1: Single-bit and burst error NOTE:
Burst error is more likely to occur than a
single-bit error
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8. 10.#
Redundancy
The central concept in detecting or correcting errors is redundancy. To be able to detect or correct errors, we need to send some extra bits with our data. These redundant bits are added by the sender and removed by the receiver. Their presence allows the receiver to detect or correct corrupted bits.
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9. 10.1.3 Detection versus Correction
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Detection versus Correction
The correction of errors is more difficult than the detection. WHY?
In error correction, we need to know the exact number of bits that are corrupted and, more importantly, their location in the message. However, in error detection we need only to know is there an error or not.
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10. 10.#
Coding
Redundancy is achieved through various coding schemes.
The sender adds redundant bits through a process that creates a relationship between the redundant bits and the actual data bits.
The receiver checks the relationships between the two sets of bits to detect errors.
The ratio of redundant bits to data bits and the robustness of the process are important factors in any coding scheme.
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11. 10.#
Coding
We can divide coding schemes into two broad categories:
Block coding
Convolution coding
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12. 10.1.5 Modulo Arithmetic 10.# Modulo-2 Arithmetic
In modulo-N arithmetic, we use only the integers in the range 0 to N - 1, inclusive.
Example: clock system (12 numbers but we use 0 to 11)
Modulo-2 Arithmetic
0+0= =1 1+0=1 1+1=0
0-0= = = =0
addition and subtraction give the same results. In this arithmetic we use the XOR (exclusive OR) operation for both addition and subtraction
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13. 10.#
BLOCK CODING
In block coding, we divide our message into blocks, each of k bits, called datawords.
We add r redundant bits to each block to make the length n = k + r. The resulting n-bit blocks are called codewords.
How the extra r bits are chosen or calculated is something we will discuss later.
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14. 10.#
Error Detection
How can errors be detected by using block coding? If the following two conditions are met, the receiver can detect a change in the original codeword.
1. The receiver has (or can find) a list of valid codewords.
2. The original codeword has changed to an invalid one.
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15. 10.# Figure 10.6: Process of error detection in block coding 10.15

16. 1616
Example 10.2
Let us assume that k = 2 and n = 3. Table 10.1 shows the list of datawords and codewords. Later, we will see how to derive a codeword from a dataword.
Table 10.1: A code for error detection in Example 10.1
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17. Error Detection Methods
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18. 1. Parity Check How does it work?
The most common and least expensive.
Simple or Two Dimensional.
How does it work?
A k-bit dataword is changed to an n-bit codeword where n = k + 1. The extra bit,called the parity bit, is selected to make the total number of 1s in the codeword even.
Example:
Assume the sender sends the dataword Since the number of 1's is odd then the parity bit is equal to 1 and the codeword for this dataword is
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19. Parity Check
NOTE:
Simple parity check can detect all single-bit errors. It can detect burst errors only if the total number of errors in each data unit is odd.
A better approach is the two-dimensional parity check which organizes the data units into table (rows and columns). The parity bit for each column is Calculated and added to the table as a new row (column parity)
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20. 10.4.1. Cyclic Redundancy Check
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Cyclic Redundancy Check
In this section, we simply discuss a subset of cyclic codes called the cyclic redundancy check (CRC), which is used in networks such as LANs and WANs.
In a cyclic code, if a codeword is cyclically shifted (rotated), the result is another codeword.
Example:
if is a codeword and we cyclically left-shift, then is also a codeword. 1 1
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21. 2121
Table 10.6: A CRC code with C(7, 4)
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22. 10.#
Figure : CRC encoder and decoder
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23. The Process of CRC
At the Sender:
the encoder, the dataword has k bits (4 here).The codeword has n bits ( 7).
The size of the dataword is augmented by adding n-k (3 here) 0s to the right-hand size of the word.
The generator uses a divisor of size n-k-1 (4 here), predefined and agreed upon.
The generator divides the augmented dataword by the divisor ( modulo-2 division).
The reminder is appended to the dataword to create the codeword.
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24. The decoder does the same division process as the encoder.
The Process of CRC
At the Receiver:
The decoder does the same division process as the encoder.
The reminder of the division is the syndrome.
If the syndrome is all 0s, there is no error, the dataword separated from the received codeword and accepted.
Otherwise, everything is discarded.
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25. 10.#
Figure 10.15: Division in CRC encoder
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26. 10.#
Figure 10.16: Division in the CRC decoder for two cases
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27. 10.#
Polynomials
A better way to understand cyclic codes and how they can be analyzed is to represent them as polynomials.
A pattern of 0s and 1s can be represented as a polynomial with coefficients of 0 and 1.
The power of each term shows the position of the bit; the coefficient shows the value of the bit.
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28. 10.# Figure 10.21: A polynomial to represent a binary word
Note that: the degree of a polynomial is 1 less that the number of bits in the pattern.
What is the polynomial degree and bit pattern for x6+ x+ 1?
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29. 10.# Hardware Implementation for a Divisor in CRC +
Although CRC division may be performed in software, it is usually performed using a feedback shift register and XOR gates.
Let us take the following polynomial as the divisor x6+ x+ 1
The circuit has n-k registers, initially has 0's and at the end it has the remainder.
The feedback taps are determined by the coefficient of the generated polynomial.
The divisor line and XOR are missing if the corresponding bit in the divisor is O. No lines for X2, X3, X4 and X5 +
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30. 10.# 10.30 We only shift to the right since feedback is 0.
Not affect for XOR
Feedback=1
It will go to each XOR connected line
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31. 10.# x5 + x4 + x2 x5 + x4 + x2 x6 + x4 + x2 x6 + x4 + x2 x5 + x2 + x1
Polynomial Arithmetic for CRC
Adding and Subtracting Polynomials
x5 + x4 + x2
x5 + x4 + x2
x6 + x4 + x2
x6 + x4 + x2 +
x5 + x2 + x1 +
x6 + x5
x6 + x2 + x1
Multiplying or Dividing Terms
It is the same as the binary division.
Shifting
Shifting left 3 bits (multiplying by x3):
10011 becomes or x4 + x +1 becomes x7 + x4 +x3
Shifting right 3 bits (dividing by x3):
10011 becomes or x4 + x +1 becomes x
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32. 10.4.3 Encoder Using Polynomials
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Encoder Using Polynomials
Now that we have discussed operations on polynomials, we show the creation of a codeword from a dataword. Figure 10.9 is the polynomial version of Figure We can see that the process is shorter.
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33. 10.#
Figure 10.22: CRC division using polynomials
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34. 10.#
Cyclic Code Analysis
The divisor in a cyclic code is normally called the generator polynomial or simply the generator.
We can analyze a cyclic code to find its capabilities by using polynomials. We define the following, where f(x) is a polynomial with binary coefficients.
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35. 3535
Example 10.15
Which of the following g(x) values guarantees that a single-bit error is caught? x + 1, x3 and 1
Solution
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36. 10.#
Figure 10.23: Representation of isolated single-bit errors
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37. 3737
Example 10.16
Find the suitability of the following generators in relation to burst errors of different lengths: x6 + 1, x18 + x7 + x + 1, and x32 + x23 + x
Solution
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38. 3838
Table 10.7: Standard polynomials
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39. 10.4.5 Advantages of Cyclic Codes
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Advantages of Cyclic Codes
Cyclic codes have a very good performance in detecting single-bit errors, double errors, an odd number of errors, and burst errors.
In burst error CRC can detect all burst errors that affect an odd number of bits, errors of length less than or equal to the degree of the polynomial, and it can detect, with very high probability, burst errors of length greater than the degree of the polynomial.
They can easily be implemented in hardware and software. They are especially fast when implemented in hardware. This has made cyclic codes a good candidate for many networks.
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40. Example
The CRC-12 x12 + x11 + x3 + x + 1 which has a degree of 12, will detect:
1. all burst errors affecting an odd number of bits,
2. will detect all burst errors with a length less than or equal to 12, and
3. will detect, percent of the time, burst errors with a length of 12 or more.
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41. 10.#
CHECKSUM
Checksum is an error-detecting technique that can be applied to a message of any length.
In the Internet, the checksum technique is mostly used at the network and transport layer rather than the data-link layer.
Like CRC, checksum based on the concept of redundancy.
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42. 10.#
Figure 10.15: Checksum
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43. Checksum Concept
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44. 4444
Example 10.18
Suppose the message is a list of five 4-bit numbers that we want to send to a destination.
In addition to sending these numbers, we send the sum of the numbers.
For example, if the set of numbers is (7, 11, 12, 0, 6), we send (7, 11, 12, 0, 6, 36), where 36 is the sum of the original numbers.
The receiver adds the five numbers and compares the result with the sum. If the two are the same, the receiver assumes no error, accepts the five numbers, and discards the sum.
Otherwise, there is an error somewhere and the message not accepted.
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45. 4545
Example 10.20
In the previous example, the decimal number 36 in binary is (100100)2. To change it to a 4-bit number we add the extra leftmost bit to the right four bits as shown below.
Instead of sending 36 as the sum, we can send 6 as the sum (7, 11, 12, 0, 6, 6). The receiver can add the first five numbers in one’s complement arithmetic. If the result is 6, the numbers are accepted; otherwise, they are rejected.
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46. 4646
Example 10.22
Let us use the idea of the checksum in Example The sender adds all five numbers in one’s complement to get the sum = 6. The sender then complements the result to get the checksum = 9, which is 15 − 6. Note that 6 = (0110)2 and
9 = (1001)2; they are complements of each other. The sender sends the five data numbers and the checksum (7, 11, 12, 0, 6, 9). If there is no corruption in transmission, the receiver receives (7, 11, 12, 0, 6, 9) and adds them in one’s complement to get 15 (See Figure 10.24).
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47. 10.# Figure 10.16: Example 10.24 100100 36 101101 45 10 10 _________ 10
_________ 10
_________
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48. 4848
Table 10.5: Procedure to calculate the traditional checksum
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49. Checksum Performance Checksum can detect errors in a message of
any size.
Almost detects all errors involving odd numbers
of bits or even.
It is not as strong as CRC. WHY?
Because if the value of one word is incremented
and the value of another word is decremented by
the same amount, the two errors are not detected.
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