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
1
Modulo-2 Arithmetic

Addition and subtraction are described by the
logical exclusive-or operation.
2
Modulo-2 Arithmetic (logical xor)
3
Chapter 10
Error Detection
and
Correction
4
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Interference

Example causes of interference:

Heat

Noise due to interference (EM fields)

Attenuation

Distortion
Interference causes bit errors.
Bit Error Classifications

Single bit error

Burst error
Single Bit Error

Single bit error



A one is interpreted as a zero (or vice-versa)
Refers to only one bit modified in a specified
transmission unit of data
An uncommon type of error for serial data due to
the duration of a bit being much less than the
duration of interference.
Figure 10.1
Single-bit error
8
Burst Error

More than one bit is damaged by interference.
9
Figure 10.2
Burst error of length 8
10
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
11
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
12
Redundant Bits
13
Detection vs Correction

Detection is easier than correction
14
Data Coding Schemes

Block coding – described in this course
or
– Convolution coding – this is covered in
advanced Math or EE signal processing
courses.
15
Block Coding


Data-words are made up of k bits.
Codewords are made up of k data bits (a dataword) and r redundant bits.
n=k+r
Where n is the number of bits in a codeword
16
Figure 10.5
Datawords and codewords in block coding
17
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.
18
19
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
20
Table 10.2
A code for error correction (Example 10.3)
21
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.
22
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.
23
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
24
Example 10.4
Let us find the Hamming distance between two pairs of
words.
1. The Hamming distance d(000, 011) is 2 because
2. The Hamming distance d(10101, 11110) is 3 because
25
Note
The minimum Hamming distance is the
smallest Hamming distance between
all possible pairs in a set of words.
26
Table 10.1 A code for error detection (Example 10.2)
10.27
Example 10.5
Find the minimum Hamming distance of the coding
scheme in Table 10.1.
Solution
We first find all Hamming distances.
The dmin in this case is 2.
28
Table 10.2
A code for error correction (Example 10.3)
29
Example 10.6
Find the minimum Hamming distance of the coding
scheme in Table 10.2.
Solution
We first find all the Hamming distances.
The dmin in this case is 3.
30
Note
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.
31
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 =
32
Figure 10.8 Geometric concept for finding dmin in error detection
33
Note
To guarantee correction of up to t errors in
all cases, the minimum Hamming distance in
a block code
must be dmin = 2t + 1.
34
Figure 10.9 Geometric concept for finding dmin in error correction
35
How many errors can be corrected for the two
example coding schemes:
d-min = 2, t =
d-min = 3, t =
Note
To guarantee correction of up to t errors in
all cases, the minimum Hamming distance in
a block code
must be dmin = 2t + 1.
37
Example
•
A code scheme has dmin = 5.
–
What is the maximum number of detectable
errors?
–
What is the maximum number of correctable
errors?
38
Linear Block Codes
•
A linear block code is a code where the logical
exclusive-or of any two valid codewords
creates another valid codeword.
39
Linear Block Codes
•
The min Hamming distance for a LBC is the
minimum number of ones in a non-zero valid
codeword.
40
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
Table 10.1 A code for error detection C(3,2)
42
Table 10.2 A code for error correction C(5,2)
43
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
44
Note
A simple parity-check code is a
single-bit error-detecting
code in which
n = k + 1 with dmin = 2.
45
Note
A simple parity-check code can detect an
odd number of errors.
46
Table 10.3 Simple parity-check code C(5, 4)
47
How Useful is a Parity Check?
•
Detecting any odd number of errors is pretty
good, can we do better?
48
2-Dimensional Parity Check
•
It is possible to create a 2-D parity check code
that detects and corrects errors.
49
Figure 10.11 Two-dimensional parity-check code
50
Figure 10.11 Two-dimensional parity-check code
51
Figure 10.11 Two-dimensional parity-check code
52
Figure 10.11 Two-dimensional parity-check code
53
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.
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
55
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?
Table 10.6 is this C(7, 4) cyclic?
57
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)
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.
62
Checksum Examples
•
1's complement
•
16 bit checksum used by the Internet
63
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)