Error Control Code

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Error Control Code
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Goal

• Goal of this lecture
• Review. CDMA, OFDM, MIMO
• Introduction to error control codes

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Review

• Still remember the analogy of TDMA, FDMA, and
  CDMA?
• The cell phone to base station model. Also, in
  wireless LAN, its PC to access point model.
• What does the chip sequence achieve in CDMA?

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Error Control Code

• Widely used in many areas, like communications,
  DVD, data storage
• In communications, because of noise, you can
  never be sure that a received bit is right
• In physical layer, what you do is, given k data
  bits, add n-k redundant bits and make it into a
  n-bit codeword. You send the codeword to the
  receiver. If some bits in the codeword is wrong,
  the receiver should be able to do some
  calculation and find out
• There is something wrong
• Or, these things are wrong (for binary codes,
  this is enough)
• Or, these things should be corrected as this for
  non-binary codes
• (this is called Block Code)

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Error Control Codes

• You want a code to
• Use as few redundant bits as possible
• Can detect or correct as many error bits as
  possible

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Error Control Code

• Repetition code is the simplest, but requires a
  lot of redundant bits, and the error correction
  power is questionable for the amount of extra
  bits used
• Checksum does not require a lot of redundant
  bits, but can only tell you something is wrong
  and cannot tell you what is wrong

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(7,4) Hamming Code

• The best example for introductory purpose and is
  also used in many applications
• (7,4) Hamming code. Given 4 information bits,
  (i0,i1,i2,i3), code it into 7 bits
  C(c0,c1,c2,c3,c4,c5,c6). The first four bits are
  just copies of the information bits, e.g., c0i0.
  Then produce three parity checking bits c4, c5,
  and c6 as (additions are in the binary field)
• c4i0i1i2
• c5i1i2i3
• c6i0i1i3
• For example, (1,0,1,1) coded to (1,0,1,1,0,0,0).
• Is capable of correcting one bit error

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Generator matrix

• Matrix representation. CIG where

• G is called the generator matrix

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Parity check matrix

• It can be verified that any CH(0,0,0) for all
  codeword C

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Error Correction

• What you receive is RCE. You multiply R with H
  SRH(CE)HCHEHEH. S is called the syndrome.
  If there is only one 1 in E, S will be one of
  the rows of H. Because each row is unique, you
  know which bit in E is 1.
• The decoding scheme is
• Compute the syndrome
• If S(0,0,0), do nothing. If S!(0,0,0), output
  one error bit.

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How G is chosen

• How G is chosen such that it can correct one
  error?
• Any combinations of the row vectors of G has
  weight at least 3 (having at least three 1s)
  and codeword has weight at least 3.
• The sum of any two codeword is still a codeword,
  so the distance (number of bits that differ) is
  also at least 3.
• So if one bit is wrong, wont confuse it with
  other codewords

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The existence of H

• We didnt compare a received vector with all
  codewords. We used H.
• The existence of H is no coincidence (need some
  basic linear algebra!) Let \Omega be the space of
  all 7-bit vectors. The codeword space is a
  subspace of \Omega spanned by the row vectors of
  G. There must be a subspace orthogonal to the
  codeword space spanned by 3 vectors which is the
  column vectors of H.

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Linear Block Code

• Hamming Code is a Linear Block Code. Linear Block
  Code means that the codeword is generated by
  multiplying the message vector with the generator
  matrix.
• Minimum weight as large as possible. If minimum
  weight is 2t1, capable of detecting 2t error
  bits and correcting t error bits.

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Cyclic Codes

• Hamming code is useful but there exist codes that
  offers same (if not larger) error control
  capabilities while can be implemented much
  simpler.
• Cyclic code is a linear code that any cyclic
  shift of a codeword is still a codeword.
• Makes encoding/decoding much simpler, no need of
  matrix multiplication.

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Cyclic code

• Polynomial representation of cyclic codes. C(x)
  C_n-1xn-1 C_n-2xn-2 C_1x1
  C_0x0.
• Division of polynomials. Try divide x33x23x1
  by x1.
• Verify that xC(x) \mod xn-1 is the cyclic shift
  of C(x).
• A (n,k) cyclic code can be generated with a
  polynomial g(x) which has degree n-k and is a
  factor of xn-1. Call it the generator
  polynomial.

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Cyclic Code

• One way of thinking it is to write it out as the
  generator matrix

• Another equivalent way is to think it as

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Cyclic Code

• Given a code polynomial

• We have

• C1(x) is the cyclic shift of C(x) and (1) has a
  degree of no more than n-1 and (2) an divide g(x)
  (why?) hence is a code polynomial (why?).

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Cyclic Code

• So, to generate a cyclic code is to find a
  polynomial that (1) has degree n-k and (2) is a
  factor of xn-1.
• So, each row of the generator matrix is just a
  shifted version of the first row. Unlike Hamming
  Code.

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Generating Systematic Cyclic Code

• A systematic code means that the first k bits are
  the data bits and the rest n-k bits are parity
  checking bits.
• To generate it, you let

where

• The claim is that C(x) must divide g(x) hence is
  a code polynomial. 33 mod 7 6. Hence 33-628
  can be divided by 7.

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Division Circuit

• Division of polynomials can be done efficiently
  by the division circuit. (just to know there
  exists such a thing, no need to understand it)

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Remaining Questions for Those Really Interested

• Decoding. Divide the received polynomial by g(x).
  If there is no error you should get a 0 (why?).
  Make sure that the error polynomial you have in
  mind does not divide g(x).
• How to make sure to choose a good g(x) to make
  the minimum degree larger? Turns out to learn
  this you have to study more its the BCH code.

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Cyclic code used in IEEE 802

• g(x) x32 x26 x23 x22 x16 x12 x11
  x10 x8 x7 x5 x4 x2 x 1
• all single and double bit errors
• all errors with an odd number of bits
• all burst errors of length 32 or less

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Other codes

• RS code. Block code. Used in CD, DVD, HDTV
  transmission.
• LDPC code. Also block code. Reinvented after
  first proposed 40 some years ago. Proposed to be
  used in 802.11n. Achieve close-to-Shannon bound
• Trellis code. Not block code. More closely
  coupled with modulation.
• Turbo code. Achieve close-to-Shannon bound.