Decoding of Convolutional Codes
 Let Cm be the set of allowable code sequences of length m.
 Not all sequences in {0,1}m are allowable code sequences!
 Each code sequence c  Cm can be represented by a unique path
through the trellis diagram
 What is the probability that the code sequence c is sent and the
binary sequence y is received?


 
dH y,
c
mdH y,
c


Pr y | 
c p
.(1  p)

















where p is the probability of bit error of BSC from modulation
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Decoding Rule for Convolutional
Codes
 Maximum Likelihood Decoding Rule:
max
cCm

Pry, c  min cCm dH y, c 
 Choose the code sequence through the trellis which
has the
smallest Hamming distance to the received
sequence!
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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 < 10
 Can be used for either hard or soft decision decoding
We consider hard decision decoding initially
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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"
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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
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The Viterbi Algorithm (continued)
Consider decoding of the ith segment:
 Let y i be the segment of n bits received between times i
and i + 1
 There are several code segments
c i
of n bits which lead
into state j at time i+1. We wish to find the most likely one.
 Let sc i  be the state from which the code segment
c i
emerged
 For each state j, we assume that
c i
into
state j if:
Z s 
c i i   d
 
is the path leading
c ,y
H i 
is the smallest of all the code segments leading into state
j.
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The Viterbi Algorithm (continued)
 Iteration:
• Let
 
i   d c i , y
Z i  1  Z
s
ci
j
H






• Let i=i+1
• Repeat previous step
•Incorrect paths drop out as i approaches infinity.
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Viterbi Algorithm Decoding Example
 r =1/2, K = 3 code from previous example

c

= (0 0 1 1
01
00
10
10
1 1) is sent
 y = (0 1 1 1
received.
01
00
10
10
1 1) is
What path through the trellis does the Viterbi Algorithm
choose?
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Viterbi Algorithm Decoding Example
(continued)
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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
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Summary of Encoding and Decoding of
Convolutional 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
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to the number
of states
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
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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
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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
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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
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Trellis Termination: Zero Padding
Add K-1 0’s 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
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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.
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Calculation of Error Event Probability
 What's the pairwise probability of choosing a path at distance
d from the correct path?
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Calculation of First Event Error Probability
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Evaluating Error Probability
Using the Transfer Function Bound
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Finding T(D) from State Diagram
 Break all 0’s 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
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Example of State Diagram
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Performance Example for Convolutional
Code
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Performance of r=1/2 Convolutional Codes
with Hard Decisions
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Performance of r=1/3 Convolutional Codes
with Hard Decisions
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Punctured Convolutional Codes
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Practical Examples of Convolutional
Codes
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