1. Reliability and Channel Coding
Chapter 8
Reliability and Channel Coding
© Bobby Hoggard, Department of Computer Science, East Carolina University
These slides may not be used or duplicated without permission

2. Causes of Transmission Errors
Cause of Problem
How to Avoid the Problem
Device failure
Careful monitoring of the network to detect device failures
Equipment that does not meet engineering standards
Extensive testing of the equipment when it's manufactured
Physics of the universe
Algorithms to detect and correct errors

3. Categories of Transmission Errors
Category
Description
Results In
Interference
Electromagnetic radiation emitted from devices, and from background cosmic radiation causing noise; disturbs signals traveling across wires
Spike in voltage, causing
single-bit errors
Distortion
Properties of wires that block certain frequencies while allowing others
Longer duration interference causing Burst errors or Erasure
Attenuation
Weakening signal;
The further distance a signal travels, the weaker it gets
Erasure
Type of Error
Description
Single-bit error
A single bit (out of a block of bits) gets changed – a 1 changes to 0 or 0 to 1
Burst error
Multiple bits (in a block of bits) get changed
Erasure
The signal is ambiguous (can't determine if it is a 0 or 1)

4. Strategies for Handling Channel Errors
FEC – Forward Error Correction
Add extra bits to the data to help determine if an error has occurred & possibily to help rebuild the data
ARQ – Automatic Repeat Request
Sender and receiver communicate with messages to determine if an error has occurred, and if so, to resend the data

5. Forward Error Correction Techniques
Block Error Codes
Divide the data into blocks
Attach extra information to each block (called redundancy or check bits)
Blocks are independent of each other
Convolutional Error Codes
Treats data as a series of bits
Compute a code over a continuous series
The code for a series of bits depends on the current input and some of the previous bits

6. Single Parity Check
Sender and receiver decide in advance whether to use even parity or odd parity
How it works:
Divide the data in to blocks of 8-bit units
Add a parity bit (also called a check bit) to each block
The parity bit value is determined such that the total number of 1's in the block (including the parity bit) is even or odd (depending on what was previously agreed upon)
This algorithm is designed to detect errors, but not correct them
This algorithm cannot detect all errors

7. Example: Single Parity Check
Original Data
# of 1's
Even Parity
Odd Parity 1 5 1
Sent (using even parity)
Received
odd number of 1's
even number of 1's
This algorithm is guaranteed to detect only single-bit errors
Cannot detect the error if an even number of bits get changed (burst error)
Technically, it can detect if an odd number of bits get changed

8. Hamming Distance for Words
The number of bits that must be changed in order to turn one piece of data into another
Example vs
Hamming Distance = 3
3 bits must change to turn one of these values into the other
Can be computed by calculating the XOR of the two words and counting the bits in the result:
XOR =

9. Codewords 2n codewords 2n codewords 2n - 2m 2m 2m valid data words
The data bits the extra check bits added for error detection
2n codewords
2n codewords
2n - 2m
invalid
codewords 2m
valid
codewords
2m valid data words

10. Codeword Example 512 codewords 29 codewords 512 codewords
Simple Parity Check – consists of 9-bit codewords
512 codewords
29 codewords
512 codewords
= 256
invalid
codewords
256
valid
codewords
28 valid data words
256 valid data words
8 bits of data, 1 check bit

11. Hamming Distance for Algorithm
Given an algorithm to produce check bits, we can produce a list of all valid codewords
Feed all possible data words into the algorithm
From this list, find the 2 codewords which have the minimum Hamming distance
this is the Hamming Distance for the Algorithm
this defines the weakest point in the algorithm
this defines the guarantee for the algorithm
All possible 2-bit data values 00 01 10 11
Single Parity
Check Algorithm
(with odd parity)
001
010
100
111
All of the valid codewords

12. Hamming Distance for Single Parity
Data
Codeword
0 0
0 0 1
0 1
0 1 0
1 0
1 0 0
1 1
1 1 1
d (0 0 1, )
= 2
d (0 0 1, )
= 2
d (0 0 1, )
= 2
d (0 1 0, )
= 2
d (0 1 0, )
= 2
minimum
codeword
distance
d (1 0 0, )
= 2
data
parity
Hamming distance for the
simple parity check algorithm is 2

13. What Does Hamming Distance Determine?
To detect d single-bit errors:
Need an algorithm with a distance of: d + 1
To correct d single-bit errors:
Need an algorithm with a distance of: 2d + 1
Key point: with d number of errors, the received codeword is closer to the original codeword than to any other codeword in the list

14. Detection/Correction Distance Example
Algorithm with Distance = 12
Detects: <= 11 errors
12 =
Corrects: <= 5 errors
11 = 2 * 5 + 1
correcting 6 errors requires
distance of 13 6
valid
codeword
distance = 12
valid
codeword
send
valid
codeword
invalid
codeword
received
distance = 5
distance = 7

15. Hamming Distance for Simple Parity
Data
Codeword
0 0
0 0 1
0 1
0 1 0
1 0
1 0 0
1 1
1 1 1
d (0 0 1, )
= 2
This algorithm can detect: 1 error
d (0 0 1, )
= 2
2 = 1 + 1
d (0 0 1, )
= 2
algorithm distance
number of errors
d (0 1 0, )
= 2
d (0 1 0, )
= 2
This algorithm can correct: None
d (1 0 0, )
= 2
3 = 2 * 1 + 1
distance needed
minimum number of
errors
Hamming distance for the
simple parity check algorithm is 2

16. Correction Algorithm Example
We have an algorithm which produces 4 valid codewords of 10-bits each: 1) 2) 3) 4) 5
What is the algorithm's distance?
d (1, 2)
= 5
d (1, 3)
= 5
<=9
How many single-bit errors can be detected?
d (1, 4)
= 10
<=2
How many single-bit errors can be corrected?
d (2, 3)
= 10
d (2, 4)
= 5
Algorithm overhead (data bit –to– check bit ratio):
d (3, 4)
= 5
2 data bits
8 check bits
10 bits total
400%

17. Correction Algorithm Example
We have an algorithm which produces 4 valid codewords of 10-bits each: 1) 2) 3) 4)
Transmitted:
Received:
d (r, 1)
= 3
d (r, 2)
= 2
Assume data was:
d (r, 3)
= 8
d (r, 5)
= 7

18. Correction Algorithm Example
We have an algorithm which produces 4 valid codewords of 10-bits each: 1) 2) 3) 4)
Transmitted:
Received:
d (r, 1)
= 2
Assume data was:
How many single-bit errors can be corrected? <=2
d (r, 2)
= 3
How many single-bit errors occurred? 3
d (r, 3)
= 7
Number of errors exceeded the algorithm's limit
d (r, 5)
= 8

19. Hamming Algorithm
An algorithm with low overhead, that can correct one single-bit error
Data Bits
Check Bits
Overhead 1 2
200%
2 - 4 3 %
5 - 11 4 % 5 % 6 % 7
12 - 6% 8
7 - 3%