Convolutional codes incomplete

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Convolutional codes(incomplete)

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

2
Source

• Communication Systems, IV Edition, A. B.
  Carlson and others, McGraw Hills.
• Digital Communication, IV edition, John G
  Proakis, McGraw Hills.
• Digital Communication, II edition,
• B Sklar, Prentice Hall.

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The Features

• It is a FEC technique.
• No need to divide the input stream in to message
  blocks.
• A coded stream is generated for entire input
  stream.
• The codewords are generated for each bit (may be
  double/triple/..) of input stream and they are
  transmitted sequentially as coded stream.

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• They are represented as (n,k,L) where
• n No. of bits in codeword,
• k No. of fresh input bits used in generation
  of code,
• L Encoders memory.
• There is no one to one correspondence like that
  of block/cyclic codes.
• Very simple encoder, thus suitable for satellite
  communication, mobile communication etc.
• High performance by sophisticated decoding.

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(2,1,2) Convolutional Encoder

• Constraint length n(L1) some author says its
  L.

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Code Tree, Code trellis and State Diagram
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The generator polynomials

• Continuing with same encoder
• G1(p) 1pp2 and G2(p) 1p2
• The message polynomial
• M(p) 1pp3p4p5p8 (see prvs slide)
• Xj M(p)G1(p) 1p5p7p8p9p10
• Xj M(p) G2(p) 1pp2p4p6p7p8p10
• Thus Xj XjXj 11 01 01 00 01 10 ...........
  (same result as obtained previously)

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• The transformation in p-domain simplifies the
  problem as it requires multiplication instead of
  convolution. (something similar to Fourier
  Transform of signals.)
• In normal operation Xj can be evaluated as
  following operation which is some what equivalent
  to convolution (refer slide no. 5)
• Xj ? mj-igi (mod-2) for 0ltiltL

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Free Distance

• Free Distance (df) w(X)min excluding all zero
  path.
• Termination of trellis (to ensure all zero state
  for next run and to find min weight non trivial
  path 5 in fig.)

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Free distance using generating function

• Draw modified state diagram (slide no. 6)
• Eliminate a-a loop.
• Draw c-a as final c-e transition.
• Assign a weight variable to each node.
• Label each branch with variables D and I where
  exponent of D equals branch weight and exponent
  of I equals corresponding number of non zero
  message bits.

DI
d
D
DI
D
D2I
D2
a
b
c
e
I
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• The modified state diagram is similar to signal
  flow graph.
• Masons Gain Formula is applicable to evaluate
  encoders generating function T(D,I).
• For the considered problem
• T(D,I) D5I/(1 - 2DI) D5I2D6I4D7I .
• (1-2DI)-1 has been expended in series.

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• T(D,I) represents all possible transmitted
  sequences that terminate with a c-e transition.
• Each term of T(D,I) has a specific physical
  interpretation.
• Exponent of I shows the number of non zero bits
  in input sequence
• Exponent of D shows the weight of corresponding
  valid coded sequence (path).
• The coefficient shows the number of valid paths
  for above statistics.

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• The free distance will be equal to the exponent
  of D of the first term.
• (WHY??)
• Unfortunately, explicit design formula for a
  given free distance does not exist, so good
  convolution codes must be discovered by hit and
  trial method (computer simulation).

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Coding Gain

• For a AWGN channel, the decoded error probability
  for a convolution code having code rate Rc k/n
  and free distance df can be derived as
• Pbe proportional to e-(Rcdf/2)?b
• Pbe improves when (Rcdf/2) gt1.
• By definition, Coding gain (Rcdf/2) usually
  expressed in dB.

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Catastrophic error

• If a finite number of channel errors causes an
  infinite number of decoding errors.
• If all generating polynomials Xj, Xj . of Xj
  has a common factor then it occurs.
• Systematic codes are free from this error but
  they have poor error correcting performance.

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Example-G1(p) 1p and G2(p) pp2 has a CF
(1p).
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Viterbi Decoding

• Maximum error correction, but max delay, high
  storage and computation requirements

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Other Decoding Methods

• Sequential Decoding (The output is available
  after each step. An increasing threshold is
  used.)
• (min delay, min error correction capability)
• Feedback Decoding (First output is available
  after predetermined number of steps (e.g. 3)
• (moderate delay, moderate error correction)

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Punctured Convolution codes

• Coding gain can be improved by increasing code
  rate.
• The decoding of high code rate convolution codes
  is very complicated due to very complex trellis
  diagram.
• Puncturing technique can be used to improve the
  effective code rate without increasing the
  complexity of decoder.

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Improving code rate to k/n from 1/n

• Choose a matrix P of dimension nxk.
• It should have n ones and all other elements as
  zero.
• Consecutive k codewords (corresponding to k
  input bits assuming 1/n encoder) should be
  compared with k columns of matrix P.
• Wherever there is a zero in matrix P
  corresponding bit (may be zero or one) of
  corresponding code word should be dropped.

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• Thus total number of bits that are transmitted
  are n for k input bits thus effective code rate
  will be k/n.
• Same matrix P is used at receiver. Either all
  zeros or all ones are inserted at the place of
  dropped bits.
• Now it can be decoded using a simple decoder
  which will treat the puncturing errors as
  transmission errors.