Ch 2.7 Error Detection

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Ch 2.7Error Detection Correction

• CS-147
• Tu Hoang

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Error Detection and Correction

• No communication channel or storage device is
  completely error-free
• As the number of bits per area or the
  transmission rate increases, more errors occur.
• Impossible to detect or correct 100 of the
  errors

Ch 2.7 gt Introduction
3
Error Detection and Correction

• 3 Types of Error Detection/Correction Methods
• Cyclic Redundancy Check (CRC)
• Hamming Codes
• Reed-Solomon (RS)
• 10011001011 1001100 1011
• Code word information error-checking bits/
• bits parity bits/
• syndrome/
• redundant bits

Ch 2.7 gt Introduction
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Cyclic Redundancy Check (CRC)

• Mostly used in data communication
• Tells us whether an error has occurred, but does
  not correct the error.
• This is a type of systematic error detection
• The error-checking bits are appended to the
  information byte

Ch 2.7 gt Cyclic Redundancy Check
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Modulo 2 Arithmetic

• Addition Rules
• 0 0 0 Ex 1011
• 0 1 1 110
• 1 0 1 1101
• 1 1 0

Ch 2.7 gt Cyclic Redundancy Check gt Modulo 2
Arithmetic
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Modulo 2 Arithmetic

• Division
• ________
• Ex 1011 1001011
• divisor dividend

Ch 2.7 gt Cyclic Redundancy Check gt Modulo 2
Arithmetic
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Modulo 2 Arithmetic

• Division
• ____1___
• Ex 1011 1001011
• 1011

Ch 2.7 gt Cyclic Redundancy Check gt Modulo 2
Arithmetic
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Modulo 2 Arithmetic

• Division
• ____1___
• Ex 1011 1001011
• 1011
• 0010

Ch 2.7 gt Cyclic Redundancy Check gt Modulo 2
Arithmetic
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Modulo 2 Arithmetic

• Division
• ____10__
• Ex 1011 1001011
• 1011
• 00100

Ch 2.7 gt Cyclic Redundancy Check gt Modulo 2
Arithmetic
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Modulo 2 Arithmetic

• Division
• ____101_
• Ex 1011 1001011
• 1011
• 001001

Ch 2.7 gt Cyclic Redundancy Check gt Modulo 2
Arithmetic
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Modulo 2 Arithmetic

• Division
• ____101_
• Ex 1011 1001011
• 1011
• 001001
• 1011

Ch 2.7 gt Cyclic Redundancy Check gt Modulo 2
Arithmetic
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Modulo 2 Arithmetic

• Division
• ____101_
• Ex 1011 1001011
• 1011
• 001001
• 1011
• 0010

Ch 2.7 gt Cyclic Redundancy Check gt Modulo 2
Arithmetic
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Modulo 2 Arithmetic

• Division
• ____1010
• Ex 1011 1001011
• 1011
• 001001
• 1011
• 00101

Ch 2.7 gt Cyclic Redundancy Check gt Modulo 2
Arithmetic
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Modulo 2 Arithmetic

• Division
• ____1010 ? Quotient
• Ex 1011 1001011
• 1011
• 001001
• 1011
• 00101 ? Remainder

Ch 2.7 gt Cyclic Redundancy Check gt Modulo 2
Arithmetic
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Calculating and Using CRCs

• Let the information byte F 1001011
• The sender and receiver agree on an arbitrary
  binary pattern P. Let P 1011.
• Shift F to the left by 1 less than the number of
  bits in P. Now, F 1001011000.
• Let F be the dividend and P be the divisor.
  Perform modulo 2 division.
• After performing the division, we ignore the
  quotient. We got 100 for the remainder, which
  becomes the actual CRC checksum.
• Add the remainder to F, giving the message M
• 1001011 100 1001011100 M

Ch 2.7 gt Cyclic Redundancy Check gt Calculate CRCs
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Calculating and Using CRCs

• M is decoded and checked by the message receiver
  using the reverse process.
• ____1010100
  
• 1011 1001011100
• 1011
• 001001
• 1001
• 0010
• 001011
• 1011
• 0000 ? Remainder

Ch 2.7 gt Cyclic Redundancy Check gt Calculate CRCs
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Hamming Codes

• One of the most effective codes for
  error-recovery
• Used in situations where random errors are likely
  to occur
• Error detection and correction increases in
  proportion to the number of parity bits
  (error-checking bits) added to the end of the
  information bits
• code word information bits parity bits
• Hamming distance the number of bit positions in
  which two code words differ.
• 10001001
• 10110001
• Minimum Hamming distance or D(min) determines
  its error detecting and correcting capability.

Ch 2.7 gt Hamming Codes
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Hamming Codes

• Hamming codes can always detect D(min) 1
  errors, but can only correct half of those errors.

Ch 2.7 gt Hamming Codes
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Hamming Codes

• EX. Data Parity Code
• Bits Bit Word
• 00 0 000 01 1 011 10 1
  101 11 0 110
• 000 100 001 101 010 110
  011 111

Ch 2.7 gt Hamming Codes
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Hamming Codes

• Single parity bit can only detect error, not
  correct it
• Error-correcting codes require more than a single
  parity bit

Ch 2.7 gt Hamming Codes
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Hamming Codes

• EX. 0 0 0 0 0 0 1 0 1 1 1 0 1 1
  0 1 1 1 0 1
• Minimum Hamming distance 3
• Can detect up to 2 errors and correct 1 error

Ch 2.7 gt Hamming Codes
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Hamming Codes

• Hamming codes work well when we can reasonably
  expect errors to be rare events. (ex hard
  drives)
• Hamming codes are useless when multiple adjacent
  errors are likely to occur.
• These errors are called burst errors that
  result from mishandling removable media (ex
  magnetic tapes or CDs).

Ch 2.7 gt Hamming Codes
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Reed-Solomon (RS)

• Operates at block level instead of bit level
• - RS(n, k) codes are defined using the following
  parameters
• s the number of bits in a char (8 bits)
• k the number of s-bit characters comprising
  the data block
• n the number of bits in the code word
• RS(n, k) can correct (n k)/2 errors in the k
  information bytes
• EX RS(255, 223)
• can correct up to 16 erroneous bytes in the
  information block

Ch 2.7 gt Reed-Solomon
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Summary

• 3 Types of Error Detection/Correction Methods
• Cyclic Redundancy Check (CRC)
• Used primarily in communication
• Can only detect errors.
• Hamming Codes
• Can detect and correct errors.
• The more parity bits added, the more errors can
  be detected and corrected.
• Used to detect errors in memory bits or fixed
  magnetic disk drives, in which errors occur
  randomly.

Ch 2.7 gt Summary
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Summary

• Reed-Solomon (RS)
• Can detect errors that occur in blocks (adjacent
  errors)
• Used to detect errors in removable media such as
  magnetic tapes or compact disks, in which burst
  errors occur due to mishandling and
  environmental stress.

Ch 2.7 gt Summary