Error Control Coding Techniques incomplete

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Error Control Coding Techniques(incomplete)

• Summarized by
• Neetesh Purohit
• Lecturer, IIIT,
• Allahabad, UP, India
• http//profile.iiita.ac.in/np/

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Source

• Communication Systems, IV Edition, A. B.
  Carlson and others, McGraw Hills.

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Why an Error occurs?

• Noise/ distortion/ interference/ fading.
• All the above parameters have random
  characteristics.
• One (more than one) factor(s) may change the
  shape of transmitted signal while it travels thru
  channel.
• Due to randomness sometimes the received signal
  may fall on the other side of the threshold level
  used for signal detection which introduces an
  error.

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Commercial advantages of Coding

• Coding gains can give as much as 10 dB
  performance improvement.
• VLSI and DSP chips are cheaper than higher power
  transmitters, bigger antennas

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Types of Channel Coding

• Waveform
• M-ary signaling
• Bipolar signaling
• Orthogonal
• Trellis-coded modulation

• Structured Sequences
• Block Codes, cyclic codes
• Convolutional Codes
• Turbo Codes

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Concept of redundancy

• Systematic generation and addition of redundant
  bits to the message bits, at the transmitter,
  such that transmission errors can be minimized by
  performing an inverse operation, at the receiver.
• Example triple repetition codes
• Logic 0 - 000
• Logic 1 - 111
• It has the ability to correct one bit error and
  detect two bit error.

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Code vectors, Hamming Distance

• n-bit codeword can be represented as a code
  vector in a n-dimensional vector space. Each unit
  weight vector represents a unique
  dimension/direction.

001
011
111
101
000
010
100
110
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• Only 2k are valid code vectors out of 2n total
  possible unique vectors.
• The distance between two code vectors is measured
  as the weight of the resultant when mod-2
  addition is done on these vectors.
• The minimum distance between all possible pairs
  of valid code vectors is called the hamming
  distance (dmin).

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6-D Code Space
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Error control capability , Code rate

• Detect up to t errors
• dmin gt t1
• Correct up to t errors
• dmin gt 2t1
• Code Rate (Rc)
• Rc k/n
• A good code should have
• High dmin and Rc should be close to 1.

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Triple repetition codes

• If probability of transmission error of a
  channel is x per bit and Triple repetition
  codes are used. Calculate the probability that
  the received word will be in error when
• (a) it is used for error correction
• (b) It is used for error detection

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• Pwe,uncoded x
• - With triple repetition code
• Pwe,detection P(3,3) x3
• Pwe,correction P(2,3) P(3,3) 3x2 -2x3
• - For any channel with xltlt1
• Pwe,uncoded gt Pwe,correction gt Pwe,detection
• - The cost is paid in terms of reduction in bit
  rate.

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FEC and BEC systems

• In FEC systems the receiver processes every
  received data packet. (sometimes it may introduce
  more errors instead of correcting an existing
  error).
• BEC systems uses error detecting codes and
  retransmission of packet is requested for error
  correction. Any one of the following ARQ
  protocols can be used for this purpose
• (a) Stop and wait (b) Go Back to n (c) Selective
  repeat

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ARQ Techniques
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Error Detecting codes

• Horizontal Redundancy Check (HRC)
• Vertical Redundancy check (VLR)
• Cyclic Redundancy Check (CRC)
• Checksum

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Classification of codes

• Error detecting/correcting.
• Systematic/nonsystematic
• Binary/nonbinary
• Block (includes cyclic codes)/convolutional
• Recursive/nonrecursive

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Linear Block codes (n,k)

• Linearity Property
• All zero vector must be a valid code vector
• Sum of any two valid code vectors produces
  another valid code vector.
• Advantage of linearity?

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• Requirement for Systematic codes
• Xkxn Mkxk Ckxq
• If G be the generator matrix such that
• XMG
• then G must be given as Ik P.
• What should be the dimension of P?

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• The contents of P will produce the redundant bits
  for a given message, thus governs the hamming
  distance and error correcting capability of
  generated code.
• Can we generate a block code for a specified
  hamming distance?
• Can we generate a block code for a specified
  hamming distance with a constraint on n and k??

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• Till date, no such method is proposed by any
  researcher. (surprised?? But there are reasons.)
• A large number of good block codes have been
  proposed using hit and trial method.
• Hamming codes have single bit error correction
  capability i.e. dmin 3, qgt3 and n2q-1 (why??
  It will be clear very soon).
• The parity matrix for generation of Hamming codes
  should contain all q-bit words with two or more
  than two 1s arranged in any order.

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(7,4) Systematic Hamming code

• The codewords for P 101 111 110 011

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Syndrome Decoding
HT P Iq
HT P Iq

• XiHT (0 0 0 . . . . .0)1xq
• If there is any error then Yi Xi Ei
• The operation YiHT will generate a non zero
  (1xq) vector called syndrome (S).

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• The Receiver already posses a syndrome table. The
  generated syndrome (from previous operation) is
  searched in it to find the corresponding error
  vector can be obtained.
• This error vector is added with Yi to get Xi. as
  Yi Ei Xi Ei Ei Xi .
• How does the receiver already has a syndrome
  table?

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• S YHT EHT
• All possible error patterns E are already known
  (depending upon error correcting capability of
  code) and HT is also a known quantity.
• Hence, If a code is to correct up to t errors
  per word then
• 2q 1 gt n nC2 nC3 . . . . . . nCt
• (Total no. of syndromes - 1) gt (possible error
  patterns)

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Syndrome table
To accommodate the syndrome table the decoder
needs to store (qn)2q bits. Larger is the table
size more time will be required to find error
vector thus decoding delay will be increased.
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Cyclic codes

• 2q 1 gt n nC2 nC3 . . . . . . nCt
• qLog (2) gt log? nCi 0ltiltt base 2
• n k n(1-k/n) log ? nCi
• 1 Rc (1/n)log ? nCi
• Rc should be close to unity in order to minimize
  overhead but it requires ngtgt1 which in turn
  requires more memory space and decoding time.

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• The solution lies in imposing some constraint on
  structure.
• cyclic codes are a subclass of linear block codes
  with a cyclic structure that leads to more
  practical implementation.
• Every cyclic shift of a valid code vector will
  produce another valid code vector.
• (see the table in slide no. 21)

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• A n bit codeword X can be represented in
  polynomial form as
• X(p) xn-1pn-1xn-2pn-2..x1px0
• Power of p ( i) will denote the position of
  codeword bit presented by its coefficient xi.
• Xi may be either zero or one. i.e. the code word
  is X (xn-1 xn-2 .. x1 x0)
• As per cyclic property X (xn-2 .. x1 x0
  xn-1) and all other iterations will also be valid
  codes.

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• How the first code should be generated?
• If a code word has n bits then at the most n
  different codewords can be generated by
  iteration.
• if the message size is k bits such that 2k gt n
  then how the other codewords should be generated?

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Generation of cyclic codes

• Mathematically the cyclic shift may also be
  represented as
• X(p) pX(p) xn-1(pn1)
• How? (The answer is mod-2 algebra)
• Let X(p)M(p)G(p). If G(p) is a factor of
  (pn1) and M(p) is a k-bit message then the
  aforesaid condition for cyclic codes will be
  satisfied.

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• If there are more than one factor of (pn1)?
• Choose any factor. It will generate a valid
  cyclic code but it may not possess optimum error
  correcting capability.
• How to find value of n or select a factor of
  (pn1) to maximize coding benefits?
• (my answer is hit and trial method)
• Hamming codes, BCH codes, Golay codes etc. are
  examples of cyclic codes.

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Systematic cyclic codes

• For (n,k) code (qn-k) the requirement is
• X(p) pqM(p) C(p)
• Combining with cyclic property
• M(p)G(p) pqM(p) C(p)
• M(p) pqM(p)/G(p) C(p)/G(p)
• pqM(p)/G(p) M(p) C(p)/G(p)

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• On performing the division pqM(p)/G(p) the
  q-bit remainder will be the required code bits
  C(p) for the given message M(p).
• Generate X(p) for a message using the above
  method. Then find other possible codewords by
  iteration.
• Complete the code table for all possible messages
  by repeating step 2.

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Advantages of Cyclic codes

• Simplified encoding.
• Easy syndrome calculation S(p) remY(p)/G(p)
• Ingenious error correcting decoding methods have
  been devised for specific cyclic codes. They
  eliminate the storage needed for table lookup.
• Cyclic Redundancy Codes (CRC), a class of cyclic
  codes, has ability to detect burst errors of
  length q, all single bit errors, any odd number
  of errors if (p1) is a factor of G(p), Double
  errors if G(p) has at least three 1s.