Chapter 10
Part III: Data Link Layer
Error Detection and Correction
© 2012 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
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.
10.2
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
10.3 Other Cyclic Codes
Data Link Layer Duties
10.4
10-1 INTRODUCTION
Let us first discuss some issues related, directly or
indirectly, to error detection and correction.
10.5
10.1.1 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.
10.6
Figure 10.1: Single-bit and burst error
NOTE:
Burst error is more likely to occur than a
single-bit error
10.7
10.1.2 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.
10.8
10.1.3 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.
10.9
10.1.4 Coding




10.10
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.
10.1.4 Coding

10.11
We can divide coding schemes into two broad
categories:

Block coding

Convolution coding
10.1.5 Modulo 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=0
0+1=1
0-0=0
0-1=1
1-1=0
1+0=1
1+1=0
1-0=1
addition and subtraction give the same results. In
this arithmetic we use the XOR (exclusive OR)
operation for both addition and subtraction
10.12
10-2 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.
10.13
10.2.1 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.
10.14
Figure 10.6: Process of error detection in block coding
10.15
10.15
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
10.16
Error Detection Methods
10.17
1. Parity Check

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 1011. Since the number of
10.18
1's is odd then the parity bit is equal to 1 and the codeword for
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)
10.19
10.4.1. 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 1011000 is a codeword and we cyclically left-shift,
then 0110001 is also a codeword.
1
10.20
0
1
1
0
0
0
Table 10.6: A CRC code with C(7, 4)
10.21
Figure 10.14: CRC encoder and decoder
10.22
The Process of CRC
At the Sender:
1.
2.
3.
4.
5.
10.23
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.
The Process of CRC
At the Receiver:
1.
2.
3.
4.
10.24
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.
Figure 10.15: Division in CRC encoder
10.25
Figure 10.16: Division in the CRC decoder for two cases
10.26
10.4.2 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.
10.27
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?
10.28
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
X6
1
0
0
0
0
X1
X0
1
1
Input Data
( dataword /
Rc codeword )
X6
+
+
X1
X0
The feedback taps + are determind by the coefficent 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
10.29
Polynomial Arithmetic for CRC
Adding and Subtracting Polynomials
+
x5 + x 4 + x2
x6 + x 4 + x2
+
x6 + x 5
x5 + x4 + x2
x6 + x4 + x2
x5 + x2 +
xx16 + x2 +
x1
Multiplying or Dividing Terms
It is the same as the binary division.
Shifting
Shifting left 3 bits (multiplying by x3):
10011 becomes 10011000
or
x4 + x +1 becomes x7 + x4 +x3
Shifting right 3 bits (dividing by x3):
10011 becomes 10
becomes x
10.30
or
x4 + x + 1
10.4.3 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 10.6. We can see that the process is
shorter.
10.31
Figure 10.22: CRC division using polynomials
10.32
10.4.4 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.
10.33
Example 10.15
Which of the following g(x) values guarantees that a singlebit error is caught? x + 1, x3 and 1
Solution
10.34
Figure 10.23: Representation of isolated single-bit errors
10.35
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 + x7 + 10.
Solution
10.36
Table 10.7: Standard polynomials
10.37
10.4.5 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.
10.38
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, 99.97 percent of the time,
burst errors with a length of 12 or more.
10.39
10-5 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.
10.40
Figure 10.15: Checksum
10.41
Checksum Concept
10.42
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.
10.43
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.
10.44
Example 10.22
Let us use the idea of the checksum in Example 10.19. 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).
10.45
Figure 10.16: Example 10.24
100100 36
10
_________
0110
6
1001
9
10.46
101101 45
10
_________
1111
15
0000
0
Table 10.5: Procedure to calculate the traditional checksum
10.47
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.

10.48