Review

1

• Review

2

• Error correction and error detection.
• What is an error? Bits missed or altered.
• How to find an error?
• Make sure that when errors occur, the resulting
  message is invalid.
• E.g two bits message 00, 01, 10, 11 or 00, 11?

3

• How to find an error?
• Introducing redundancy -- using 2 bits message to
  send 1 bit information in the previous example.
• Message information bits redundant bits
  (checksum).
• How to design codes that have error
  correction/detection capability?
• Hamming distance between two code words the
  number of different bits between the two code
  words.
• E.g 010101 and 111000? Hamming distance ?
• Hamming distance of a complete code the minimum
  Hamming distance any of the two codewords in the
  code.
• E.g 010101, 111000, 000111, 111111
  Hamming_distance ?

4

• Relation between Hamming distance of a code and
  the codes error detection/correction capability
• Hamming distance N
• How many bits of errors can be detected?
• How many bits of errors can be corrected?
• Example 00000000, 00001111, 11110000, 11111111.
• Hamming distance ? How many bits of errors can
  be detected? How many bits of errors can be
  corrected?
• Example 2 Parity code (Even parity code)?
• Let number of information bits 2, How to
  construct the even parity code? Hamming distance
  ?

5

• Error correction code
• How many (r) redundant bits do we need to correct
  a single error for the m information bits?
• A message contains mr bits
• total number of possible codewords 2(mr)
• total number of valid codewords 2(m)
• To correct single error, each single error must
  results in a different (invalid) codeword.
• Total number of (invalid) codewords for one bit
  error (mr)2m
• total number of valid codewords plus the total
  number of (invalid) codewords for single bit
  error must be less than the total number of
  possible codewords.
• 2m(mr)2m lt 2(rm)
• mr1 lt 2r

6

• Error correction code
• How many (r) redundant bits do we need to correct
  single error for the m information bits?
• mr1 lt 2r
• m 1, r 2
• m 2, r 3
• m 3, r 3
• m 1000, r 10
• This formula gives the lower bound of the
  redundant bits to correct a single error. Hamming
  code achieves this lower bound (Chapter 3.2.1).
• How many redundant bits are needed to detect a
  single error?

7

• Error correction code/error detection code which
  code is more efficient?
• Consider only the single bit error
• Assumption error rate 10(-6), one error in 1
  million bits
• packet size 1000 bits for information, total 1000
  packets.
• Error correction 1000 (100010)
• Error detection 1000(10001) (10001)
• Error correction code is in general NOT
  efficient!!!

8

• Error detection code
• parity code detect single error, not good
  enough.
• To detect N errors, we need to have a code whose
  hamming distance gtN1. How to get this kind of
  code?
• The most commonly used error detection code is
  called polynomial code, or cyclic redundancy code
  or CRC code.
• A bit stream is treated as a polynomial with
  coefficients 0s and 1s
• k bits gt k-terms polynomial
• 11001 ---gt x4x30x20x1
  x0
• The sender and the receiver have a generator
  polynomial G(X) with degree r
• Add the checksum bits to make sure that the final
  message is divisible by G(X).

9

• How to compute the checksum?
• Let r be the degree of G(x). Append r zero to the
  low-order end of data bits (m bits) so that the
  frame contains mr bits corresponding to xrM(x)
• Using modulo 2 division to divide the bit stream
  corresponding to xrM(x) by the bit stream
  string corresponding to G(x).
• Add the remainder back to the frame xrM(x).
• Example Data 1101011011, Generator 10011
• What would be the final data frame?

10

• The power of the CRC code
• Depends on the selection of the generator, G(x)
• T(x) frame polynomial,
• E(x) error polynomial
• G(x) generator polynomial
• T(x) E(x) / G(x) E(x) / G(x)
• Single error E(x) xj
• G(x) has more than two terms, guarantee to detect
  single error.
• Double error E(x) xjxkxk(x(j-k)1)
• xk1 (klt32768) cannot be divisiable by
  x15x141.
• x1 cannot divide any polynomial with odd number
  of terms (detect all odd number of errors).
• xr can detect all less than r number of burst
  errors.

11

• Three standard polynomials
• CRC-12 x12x11x3x2x1
• CRC-16 x16x15x21
• CRC-CCITT x16x12x51
• What is the hamming distance for the code
  generated by CRC-16?