Error Detection and Correction

1
Error Detection and Correction

• Trac D. Tran
• ECE Department
• The Johns Hopkins University
• Baltimore, MD 21218

2
Outline

• Motivation and background
• General communication system
• Objectives
• Assumptions
• Error detection
• Repetition
• Parity checking
• Examples
• Error correction
• Repetition
• Redundant-check code word
• Hamming error correction code

3
Communication System
received signal
received message
message
signal
source
transmitter
receiver
destination


m
x
x
m
noise
communication channel
Examples of noisy communication channels
Robust communication over noisy channels

• 56K modem
• Radio link
• Cellular phone
• Disk drive or CD drive

• Physical solutions
• System solutions

4
Main Concept

• System solution to robust communication
• Accept the given noisy channel, try to find
  solutions to the detection and correction of
  transmission errors
• Additional computational load at the transmitter
  and the receiver
• Redundancy helps!
• Objectives
• Detection and correction of error introduced by
  noisy channels
• Efficient transmission of data (bandwidth
  efficiency)
• Fast simple encoding/decoding algorithms
• Error detection codes
• Add redundant check bits to detect bit error in
  the transmission
• Error correction codes
• More powerful not only detecting errors, but
  also capable of correcting errors as well

5
Assumptions

• All length-n code-words (consisting of n bits
  0,1) sent over the channel will be received as
  blocks of binary digits of length n no erasure
  error
• Probability of error for any single bit is
  independent of the probability of error of any of
  its neighbors
• The channel is a binary symmetric channel (BSC)

p
0
0
error
1p
1p
1
1
p
Reliability of BSC
6
Error Detection Codes Repetition

• Repetition
• Another example where is the error?

• Double bandwidth
• Cannot correct errors 00 or 11?

1 1 0 0 1 1 0 0 0 1 1 1
0 0 1 1 0 0 1 0 0 0 1 1
7
Error Detection Codes Parity Check

• Parity-Check
• One redundant bit, called parity bit, per
  code-word
• Parity bit is based on the number of 1s and
  whether the count should be even or odd
• More bandwidth-efficient than the repetition
  method
• (N, K) code
• length-K source bits, length-N transmitted bits,
  NgtK

(3,2) code
0 0 0 0 1 1 1 0 1 1 1 0

• Is error correction possible? 000 or 110 or 011?

0 1 0 0 1 1 1 0 1 1 1 0 1 0 0
(4,3) Parity error detection code 0 0 1 1 0 0
1 0 0 0 1 1
8
Error Correction Codes Repetition

• Error correction
• More sophisticated then error detection can
  correct bit error as well
• Working principle relying on the redundant bits
  to check and correct errors
• Repetition codes
• (3,1) error correction code via repetition

1 1 1 0 0 0 1 1 1 0 0 0

• Triple the bandwidth!
• Majority-vote in correction
• No absolute immunity!

1 1 1 0 1 0 0 0 0 0 1 1
9
Hamming Distance

• Number of differing digits between 2 code-words
• Examples d(01101,01100)1 d(01001,01100)2

d(000,111)3
d(011,110)d(011,101)d(101,110)2
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Error Detection Correction Capability
?
?
Maximum Likelihood Decoding

• minimum Hamming distance between valid
  code-words
• Number of bit errors that ECC can detect
• Number of bit errors that ECC can correct
• For an (N,K) block code, the code rate is
• Example (5,1) repetition code

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Single Parity-Check Code Revisit

• Single Parity-check Code (SPC)
• (N, N1) block code, appending a single parity
  bit at the end to force even-parity overall
• Construction
• Properties
• Rate
• Minimum Hamming distance
• Detection capability
• Correction capability

1 bit error
NO!
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Hamming Error Correction Code

• An (N, K) Hamming code has NK parity checking
  bits

message bits
parity bits
check the odd parity of these 3 bits
K
NK
Inputs
Outputs
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Hamming ECC
Message Code-words 0000 0000000 0001
0001011 0010 0010111 0011
0011100 0100 0100101 0101
0101110 0110 0110010 0111
0111001 1000 1000110 1001
1001101 1010 1010001 1011
1011010 1100 1100011 1101
1101000 1110 1110100 1111
1111111

• Rate 4/7
• Additional cost of 3 bits per 4-bit message
• Minimum Hamming distance 3
• Can detect 2 bits of error
• Can correct a single bit error

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Hamming Encoder
inputs
outputs
Modulo-2 arithmetic
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Hamming ECC Decoding

• Key to decoding

• Suppose that we receive

• Check

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Hamming ECC Decoding

• Suppose that we only have at most 1 bit error per
  block

• Parity check

Action
no error

• 0 0 0
• 0 0 1
• 0 1 0
• 0 0
• 0 1 1
• 0 1
• 1 1 0
• 1 1 1

• What if there are 2 bit errors?

17
Hamming ECC Decoding Example
Send
Received
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Redundancy-Check Code

• Redundancy-check code-word
• Appends to the end of a data sequence
• Check code-word is computed from the data
  code-words hence redundant!
• Data code-words contains error detection to
  identify the error bits that the check code-word
  can then correct
• Built on top of error detection
• Problem error correction fails when there are
  more than one error occurring during
  transmission. To correct multiple errors, we need
  additional redundant bits (more check code-words
  for example)

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Redundancy-Check Example

• Can only protect a short message
• More redundancy check code-words are needed for
  better correction

Even 1s check
0 0 0 0 1 1 1 0 1 1 1 0
1 1 0
Even 1s check
redundancy-check code-word
Transmitted
0 0 0 0 1 1 1 0 1 1 1 0 1 1 0
Even 1s check
Received
0 1 0 0 1 1 1 0 1 1 1 0 1 1 0
compare
1 0
Even 1s check
Error occurs in 2nd bit position
20
Reed-Solomon Code

• A generalization of Hamming codes
• Reed-Solomon error correction code
• has a different (larger) alphabet
• uses Modulo-N arithmetic
• used on CDs, in the transmission of information
  from outer space (for instance, the Voyager
  images)