Convolutional Coding

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Convolutional Coding And The Viterbi Algorithm
Brian Carroll And Justin Ziniel
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Introduction to Convolutional Code

• Type of Error Correcting Code used to improve the
  reliability of communication systems
• Usually Specified by 3 parameters
• -the number of input bits (k)
• -the number of output bits (n)
• -the number of memory registers(m)
• Convolutional codes are said to have a code rate
  of k/n
• All convolutional codes can be generated by a
  shift register

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Covolutional Code Example

• ½ rate code
• Markov Series, where the state is (uk-1 ,uk-2)
• Probability of a transition given by
  P(xk1x0,x1,,xk)P(xk1xk)

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The Viterbi Algorithm

• Used to find the most likely sequence
  (x0,x1,,xK) given known states x0, xK, and K
  observations (zK)
• Want to find sequence where P(xz) is a maximum,
  or equivalently, a sequence which maximizes
• Can be viewed as finding the shortest path length
  in the trellis diagram
• Length of each transision branch defined by

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How to find the shortest length

• Do not have to consider every path
• Only consider paths originating from survivor
  paths
• The survivor path is the most likely path to a
  particular state at time index (k)
• The shortest (most likely) total path is
  guaranteed to go through one of the survior paths
• This cuts the complexity down from being
  exponential with K (the number of bits), to being
  linear with K

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Upper-Bounding the Probability of Error

• Strategy for calculating the probability of block
  and bit errors
• Introduce the notion of an error event
• Define a fundamental path
• Find an expression for all possible fundamental
  paths
• Determine the overall probability of an initial
  error event
• Use the probability of an initial error event to
  bound the total probability of block and bit
  errors

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Error Events

• Assume an ML decoder (i.e. Viterbi) is being used
• An error event is any path segment in a trellis
  that deviates from the correct path

• Notice that all error events must diverge and
  remerge to the same state, but this is not a
  problem

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Fundamental Paths

• Fundamental paths are paths that initiate and
  terminate in the same state, but do not return at
  any other time
• Helpful to visualize fundamental path sequences
  by modifying the state diagram
• Closed form expression exists

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Probability of Selection Error

• Selection error at node j, , is bounded
  by the sum of the error probabilities of
  selecting all potential remerging paths
• For an ML decoder on a discrete memoryless
  channel, the probability of a single error, ,
  is bounded by
• Consequently

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Computing block and bit error probabilities

• If is the number of input bits, then there
  are nodes at which the decoder must make a
  selection. Thus the block error probability,
  , is upper-bounded by
• By partially differentiating the complete path
  enumerator, it can be shown that the bit error
  probability, , is upper-bounded by
• In general, the free distance of a
  convolutional code has the greatest impact on the
  codes performance

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Plot of the Error Bounds
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Modifications to the Algorithm

• In some cases the initial state x0 may not be
  known
• If this is the case, the survivor paths should
  merge to the correct path after initial errors
• Also, it may be desired to shorten the length
  the survivor paths to a certain length (d)
• Then a firm decision has to made about the k- d
  states before the algorithm is complete
• Usually the survivor paths will agree about these
  bits, if not a majority vote can be taken