Decoding of Convolutional Codes

1
Decoding of Convolutional Codes

• Let Cm be the set of allowable code sequences
  of length m.
• Not all sequences in 0,1m are allowable code
  sequences!
• Each code sequence can be
  represented by a unique path
• through the trellis diagram
• What is the probability that the code sequence
  is sent and the
• binary sequence is received?
• where p is the probability of bit error of BSC
  from modulation

2
Decoding Rule for Convolutional Codes

• Maximum Likelihood Decoding Rule
• Choose the code sequence through the trellis
  which has the
• smallest Hamming distance to the received
  sequence!

3
The Viterbi Algorithm

• The Viterbi Algorithm (Viterbi, 1967) is a
  clever way of
• implementing Maximum Likelihood Decoding.
• Computer Scientists will recognize the Viterbi
  Algorithm as an
• example of a CS technique called Dynamic
  Programming

• Reference G. D. Forney, The Viterbi
  Algorithm,
• Proceedings of the IEEE, 1973
• Chips are available from many manufacturers
  which
• implement the Viterbi Algorithm for K lt 10
• Can be used for either hard or soft decision
  decoding
• We consider hard decision decoding initially

4
Basic Idea of Viterbi Algorithm

• There are 2rm code sequences in Cm .
• This number of sequences approaches infinity as
  m
• becomes large
• Instead of searching through all possible
  sequences,
• find the best code sequence "one stage at a
  time"

5
The Viterbi Algorithm(Hamming Distance Metric)

• Initialization
• Let time i 0.
• We assign each state j a metric Z j (0) at time
  0.
• We know that the code must start in the state 0.
• Therefore we assign
• Z j (0) 0
• Z j (0) for all other states

6
The Viterbi Algorithm (continued)

• Consider decoding of the ith segment
• Let be the segment of n bits received
  between times i
• and i 1
• There are several code segments of n bits
  which lead
• into state j at time i1. We wish to
  find the most likely one.
• Let be the state from which the code
  segment emerged
• For each state j, we assume that is the
  path leading into
• state j if
• is the smallest of all the code
  segments leading into state j.

7
The Viterbi Algorithm (continued)

• Iteration
• Let
• Let ii1
• Repeat previous step
• Incorrect paths drop out as i approaches infinity.

8
Viterbi Algorithm Decoding Example

• r 1/2, K 3 code from previous example
• (0 0 1 1 0 1 0 0 10 10
  1 1) is sent
• (0 1 1 1 0 1 0 0 10 10
  1 1) is received.
• What path through the trellis does the Viterbi
  Algorithm choose?

9
Viterbi Algorithm Decoding Example(continued)
10
Viterbi Decoding Examples

• There is a company Alantro with a example
  Viterbi
• decoder on the web, made available to
  promote their
• website
• http//www.alantro.com/viterbi/workshop.html
• Your browser must have JAVA-enabled

11
Summary of Encoding and Decoding ofConvolutional
Codes

• Convolutional are encoded using a finite state
  machine.
• Optimal decoder for convolutional codes will
  find the path
• through the trellis which lies at the shortest
  distance to the
• received signal.
• Viterbi algorithm reduces the complexity of this
  search by
• finding the optimal path one stage at a time.
• The complexity of the Viterbi algorithm is
  proportional to the number of states
• exponential relationship to constraint length

12
Implementation of Viterbi Decoder

• Complexity is proportional to number of states
• increases exponentially with constrain length
  K 2K
• Very suited to parallel implementation
• Each state has two transitions into it
• Each state has two transitions out of it
• Each node must compute two path metrics, add
  them to previous metric and compare
• Much analysis as gone into optimizing
  implementation of this
• Butterfly calculation

13
Other Applications of Viterbi Algorithm

• Any problem that can be framed in terms of
  sequence detection can be
• solved with the Viterbi Algorithm
• MLSE Equalization
• Decoding of continuous phase modulation
• Multiuser detection

14
Continuous Operation

• When continuous operation is desired, decoder
  will automatically synchronize with transmitted
  signal without knowing state
• Optimal decoding requires waiting until all
  bits are received to trace back path.
• In practice, it is usually safe to assume that
  all paths have merged after approximately 5K time
  intervals
• diminishing returns after delay of 5K

15
Frame Operation of Convolutional Codes

• Frequently, we desire to transfer short (e.g.,
  192 bit)
• frames with convolutional codes.
• When we do this, we must find a way to terminate
• code trellis
• Truncation
• Zero-Padding
• Tail-biting
• Note that the trellis code is serving as a
  block code in
• this application

16
Trellis Termination Zero Padding

• Add K-1 0s to the end of the data sequence to
  force
• the trellis back to the all zeros state
• Performance is good
• Now both start and ending state are known by the
  decoder
• Wastes bits in short frame

17
Performance of Convolutional Codes

• When the decoder chooses a path through the
  trellis which diverges from the correct path,
  this is called an "error event
• The probability that an error event begins
  during the current time interval is the
  "first-event error probablity Pe
• The minimum Hamming distance separating any two
  distinct path through the trellis is called the
  free distance dfree.

18
Calculation of Error Event Probability

• What's the pairwise probability of choosing a
  path at distance d from the correct path?

19
Calculation of First Event Error Probability
20
Evaluating Error ProbabilityUsing the Transfer
Function Bound
21
Finding T(D) from State Diagram

• Break all 0s state in two, creating a starting
  state and a terminating state
• Re-label every output 1 as a D
• ad is the number of distinct paths leading from
  the starting state to the terminating state while
  generating the function Dd

22
Example of State Diagram
23
Performance Example for Convolutional Code
24
Performance of r1/2 Convolutional Codeswith
Hard Decisions
25
Performance of r1/3 Convolutional Codeswith
Hard Decisions
26
Punctured Convolutional Codes
27
Practical Examples of Convolutional Codes