1 Block Coding Messages are made up of k bits. Transmitted packets have n bits, n > k: k-data bits and r-redundant bits. n = k + r
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1
Block Coding
Messages are made up of k bits.
Transmitted packets have n bits, n > k:k-data bits and r-redundant bits.
n = k + r
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Modulo-2 Arithmetic
Addition and subtraction are described by the logical exclusive-or operation.
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Modulo-2 Arithmetic (logical xor)
4
Chapter 10
Error Detection and
Correction
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Interference
Example causes of interference:HeatNoise due to interference (EM fields)AttenuationDistortion
Interference causes bit errors.
Bit Error Classifications
Single bit error
Burst error
Single Bit Error
Single bit errorA one is interpreted as a zero (or vice-versa)Refers to only one bit modified in a specified
transmission unit of dataAn uncommon type of error for serial data due to
the duration of a bit being much less than the duration of interference.
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Figure 10.1 Single-bit error
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Burst Error
More than one bit is damaged by interference.
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Figure 10.2 Burst error of length 8
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Error Detection
Detection of errors is necessary to determine if data should be rejected.
Possible responses to error detection include:Retransmission request Forward error correction (corrected on the receiving
end)Forward correction saves bandwidth & time
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Redundancy
Extra bits can be included with the data transmission to assist in the detection and correction of errors
– For digital transmissions that implies using block-coding
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Redundant Bits
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Detection vs Correction
Detection is easier than correction
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Data Coding Schemes
Block coding – described in this course
or
– Convolution coding – this is covered in advanced Math or EE signal processing courses.
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Block Coding
Data-words are made up of k bits.
Codewords are made up of k data bits (a data-word) and r redundant bits.
n = k + r
Where n is the number of bits in a codeword
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Figure 10.5 Datawords and codewords in block coding
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Examples
No block coding for 10base-T Ethernet, Manchester coding.
4B/5B block coding (Fast Ethernet), 8B6T or MLT-3 line coding.
8B/10B block coding (Gigabit Ethernet), PAM-5 line coding.
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Error Detection With Block Coding
• The receiver can detect an error if
– The receiver has a list of valid code words
– A received codeword is not a valid code word
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Table 10.2 A code for error correction (Example 10.3)
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Possible Transmission Outcomes
• A codeword is sent and received without incident
• A codeword is sent but is modified in transmission. The error is detected if the codeword is not in the valid codeword list.
• A codeword is sent but is modified in transmission. The error is not detected if the resulting new codeword is in the list of valid codewords.
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Error-Detecting Code
• Error detecting code can only detect the types of errors it was designed to detect. Other types of errors may go undetected.
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The Hamming Distance
• The Hamming Distance, d(x,y), between two words of the same size is the number of differences between corresponding bits.
• Examples
d(000,011) = 2
d(10101,11110) = 3
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Let us find the Hamming distance between two pairs of words.
1. The Hamming distance d(000, 011) is 2 because
Example 10.4
2. The Hamming distance d(10101, 11110) is 3 because
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The minimum Hamming distance is the smallest Hamming distance between all possible pairs in a set of words.
Note
10.27
Table 10.1 A code for error detection (Example 10.2)
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Find the minimum Hamming distance of the coding scheme in Table 10.1.
SolutionWe first find all Hamming distances.
Example 10.5
The dmin in this case is 2.
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Table 10.2 A code for error correction (Example 10.3)
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Find the minimum Hamming distance of the coding scheme in Table 10.2.
SolutionWe first find all the Hamming distances.
The dmin in this case is 3.
Example 10.6
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To guarantee the detection of up to s errors in all cases, the minimum
Hamming distance in a block code must be dmin = s + 1.
Note
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Hamming Code Notation
What is the maximum number of detectable errors for each of the two previous coding schemes?
– d-min = 2, s =
– d-min = 3, s =
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Figure 10.8 Geometric concept for finding dmin in error detection
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To guarantee correction of up to t errors in all cases, the minimum Hamming distance in
a block code must be dmin = 2t + 1.
Note
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Figure 10.9 Geometric concept for finding dmin in error correction
How many errors can be corrected for the two example coding schemes:
d-min = 2, t =
d-min = 3, t =
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To guarantee correction of up to t errors in all cases, the minimum Hamming distance in
a block code must be dmin = 2t + 1.
Note
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Example
• A code scheme has dmin = 5.
– What is the maximum number of detectable errors?
– What is the maximum number of correctable errors?
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Linear Block Codes
• A linear block code is a code where the logical exclusive-or of any two valid codewords creates another valid codeword.
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Linear Block Codes
• The min Hamming distance for a LBC is the minimum number of ones in a non-zero valid codeword.
Exercise:
For each of the next two examples,
Compute the • Hamming distance,
• the number of detectable errors,
• number of correctable errors
• Show that the code is linear
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Table 10.1 A code for error detection C(3,2)
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Table 10.2 A code for error correction C(5,2)
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Parity Check
• Count the number of ones in a data word.
• If the count is odd, the redundant bit is one
• If the count is even, the redundant bit is zero
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A simple parity-check code is a single-bit error-detecting
code in which n = k + 1 with dmin = 2.
Note
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A simple parity-check code can detect an odd number of errors.
Note
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Table 10.3 Simple parity-check code C(5, 4)
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How Useful is a Parity Check?
• Detecting any odd number of errors is pretty good, can we do better?
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2-Dimensional Parity Check
• It is possible to create a 2-D parity check code that detects and corrects errors.
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Figure 10.11 Two-dimensional parity-check code
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Figure 10.11 Two-dimensional parity-check code
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Figure 10.11 Two-dimensional parity-check code
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Figure 10.11 Two-dimensional parity-check code
Interleaving
● By interleaving the columns into slots, it becomes possible to
● detect up to n-row errors.● The example is 70%efficient. The efficiency can
be improved by adding more rows.
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Cyclic Codes
• If a codeword is shifted cyclically, the result is another codeword.
– (highest order bit becomes the lowest order bit)
– Cyclic codes are linear codes
C(7,4)
Assume d-min = 3.
Answer the following about table 10.6:
● What is the codeword size?● What is the data-word size?
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Table 10.6 is this C(7, 4) cyclic?
C(7,4)
Is the table 10.6 code cyclic?
Is it linear?
What is d-min?
How many detectible errors?
How many correctible errors?
Hamming Codes
d-min >= 3
Minimum number of detectable errors: 2
Minimum number of correctable errors: 1
For C(n,k), • n = 2^r – 1
• r = n – k (number of redundant bits)
• r >= 3 (author uses m = r)
Which are Hamming Codes?
C(7,4)
C(7,3)
Error Correction
1. Using CRC codes computing bit syndromes
2. Using interleaving with multiplexing.
– Use a parity bit in each frame
– Check for invalid code words
(see example exercises)
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Checksum
• Adding the codewords together at the source and destination.
• If the sum at the source and destination match, there is a good chance that no errors occurred.
• Checksums are not as reliable as the CRC.
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Checksum Examples
• 1's complement
• 16 bit checksum used by the Internet
Sender: Checksum Calculation
1. Divide into 16 bit unsigned words
2. Add the 16 bit words using 1’s complement addition.
3. Complement the total
4. Send all the above words.
Receiver: Checksum Calculation
1. Divide into 16 bit words
1.Add the 16 bit words and the checksum value using 1’s complement arithmetic.
2.The complement of the total should be zero.
Example: Sender
0x466f
0x726f
0x757a
+_______
0x12e58 partial sum
0x2e59 sum
0xd1a6 complement
Example: Receiver
0x466f
0x726f
0x757a
0xd1a6 (sender’s checksum)
+_______
0x1fffe (partial sum)
0xffff (sum)
0x0000 (complement)
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