Convolutional Codes 1

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Convolutional Codes
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Basic Definitions

• k 1, n 2 , (2,1) Rate-1/2 convolutional code
• Two-stage register ( M2 )
• Each input bit influences the output for 3
  intervals (K3)
• K constraint length of the code M 1

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Generator Polynomial

• A convolutional code may be defined by a set of n
  generating polynomials for each input bit.
• For the circuit under consideration
• g1(D) 1 D D2
• g2(D) 1 D2
• The set gi(D) defines the code completely. The
  length of the shift register is equal to the
  highest-degree generator polynomial.

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State Diagram Representation

• The output depends on the current input and the
  state of the encoder ( i. e. the contents of the
  shift register).

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Trellis Diagram Representation

• Expansion of state diagram in time.

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Decoding

• A message m is encoded into the code sequence c.
• Each code sequence represents a path in the
  trellis diagram.
• Minimum Distance Decoding
• Upon receiving the received sequence r, search
  for the path that is closest ( in Hamming
  distance) to r .

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The Viterbi Algorithm

• Walk through the trellis and compute the Hamming
  distance between that branch of r and those in
  the trellis.
• At each level, consider the two paths entering
  the same node and are identical from this node
  onwards. From these two paths, the one that is
  closer to r at this stage will still be so at any
  time in the future. This path is retained, and
  the other path is discarded.
• Proceeding this way, at each stage one path will
  be saved for each node. These paths are called
  the survivors. The decoded sequence (based on
  MDD) is guaranteed to be one of these survivors.

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The Viterbi Algorithm (contd)

• Each survivor is associated with a metric of the
  accumulated Hamming distance (the Hamming
  distance up to this stage).
• Carry out this process until the received
  sequence is considered completely. Choose the
  survivor with the smallest metric.

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• 6.3 The Viterbi Algorithm
• The viterbi algorithm is used to decode
  convolutional codes and any structure or system
  that can be described by a trellis.
• It is a maximum likelihood decoding algorithm
  that selects the most probable path that
  maximizes the likelihood function.
• The algorithm is based on add-compare-select the
  best path each time at each state.

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• Example For the convolutional code example in
  the previous lecture, starting from state zero,
  Decode the following received sequence.

At the end of the trellis, select the path
with the minimum cumulative Hamming weight
This is the survival path in this example
Decoded sequence is m10 1110
Compute the two possible paths at each state and
select the one with less cumulative Hamming
weight
Add the weight of the path at each state
This is called the survival path
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Distance Properties of Conv. Codes

• Def The free distance, dfree, is the minimum
  Hamming distance between any two code sequences.
• Criteria for good convolutional codes
• Large free distance, dfree.
• Small Hamming distance (i.e. as few differences
  as possible ) between the input information
  sequences that produce the minimally separated
  code sequences. dinf
• There is no known constructive way of designing a
  conv. code of given distance properties. However,
  a given code can be analyzed to find its distance
  properties.

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Distance Prop. of Conv. Codes (contd)

• Convolutional codes are linear. Therefore, the
  Hamming distance between any pair of code
  sequences corresponds to the Hamming distance
  between the all-zero code sequence and some
  nonzero code sequence. Thus for a study of the
  distance properties it is possible to focus on
  the Hamming distance between the all-zero code
  sequence and all nonzero code sequences.
• The nonzero sequence of minimum Hamming weight
  diverges from the all-zero path at some point and
  remerges with the all-zero path at some later
  point.

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Distance Properties Illustration

• sequence 2 Hamming weight 5, dinf 1
• sequence 3 Hamming weight 7, dinf 3.

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Modified State Diagram

• The span of interest to us of a nonzero path
  starts from the 00 state and ends when the path
  first returns to the 00 state. Split the 00 state
  (state a) to two states a0 and a1.
• The branches are labeled with the dummy variables
  D, L and N, where
• The power of D is the Hamming weight ( of 1s)
  of the
• output corresponding to that branch.
• The power of N is the Hamming weight ( of 1s)
  of the
• information bit(s) corresponding to that branch.
• The power of L is the length of the branch
  (always 1).

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Modified State Diagram (contd)
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Properties of the Path

• Sequence 2
• code sequence .. 00 11 10 11 00 ..
• state sequence a0 b c a1
• Labeled (D2LN)(DL)(D2L) D5L3N
• Prop. w 5, dinf 1, diverges from the allzero
  path by 3 branches.
• Sequence 3
• code sequence .. 00 11 01 01 00 10 11 00 ..
• state sequence a0 b d c b c
  a1
• Labeled (D2LN)(DLN)(DL)(DL)(LN)(D2L)
  D7L6N3
• Prop. w 7, dinf 3, diverges from the allzero
  path by 6 branches.

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Transfer Function

• Input-Output relations
• a0 1
• b D2LN a0 LNc
• c DLb DLNd
• d DLNb DLNd
• a1 D2Lc
• The transfer function T(D,L,N) a1 /a0

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Transfer Function (contd)

• Performing long division
• T D5L3N D6L4N2 D6L5N2 D7L5N3 .
• If interested in the Hamming distance property of
  the code only, set N 1 and L 1 to get the
  distance transfer function
• T (D) D5 2D6 4D7
• There is one code sequence of weight 5. Therefore
  dfree5.
• There are two code sequences of weight 6,
• four code sequences of weight 7, .