1. ECED 4504 Digital Transmission Theory
Decoding of Convolutional Codes

2. Topics today
Revision of convolutional encoding: state diagram and generator matrix
ML decoding of convolutional codes and the free distance
Viterbi decoding
trellis diagram
surviving path
ending the decoding
Soft and hard decoding
Bounding error rate for convolutional codes
generating function and coding gain
weight spectrum
error-events and BER

3. Convolutional encoding
Figure shows how in memory depth L=v-1 k input bits are encoded to n output bits in a (n,k,L) code
This figure shows a general structure of a convolutional encoder

4. Example of using generator matrix
Verify that you can obtain the result shown!

5. State diagram of a convolutional code
Each new block of k bits causes a transition into new state (see -2 slides)
Hence there are 2k branches leaving each state
Assuming encoder zero initial state, encoded word for any input k bits can thus be obtained. For instance, below for u=( ) the encoded word v=(1 1, 1 0, 0 1, 0 1, 1 1, 1 0, 1 1, 1 1) is produced:
Verify that you have the same result!
Input state
Output state
Encoder state diagram for an (n,k,L)=(2,1,2) coder

6. Extracting the generating function by splitting and labeling the state diagram
The state diagram can be modified to yield information on code distance properties
Rules:
(1) Split S0 into initial and final state, remove self-loop
(2) Label each branch by the branch gain Xi. Here i is the weight of the n encoded bits on that branch
(3) Each path connecting the initial state and the final state represents a nonzero code word that diverges and re-emerges with S0 only once
The path gain is the product of the branch gains along a path, and the code weight is the power of X in the path gain
Code weigh distribution is obtained by using a weighted gain formula to compute its generating function (input-output equation) where Ai is the number of encoded words of weight i

7. Where these terms come from?
Example of splitting and labeling the state diagram
weight: 2
weight: 1
The path representing the state
sequence S0S1S3S7S6S5S2S4S0 has
path gain X2X1X1X1X2X1X2X2=X12
and the corresponding code word
has the weight 12. The generating function is:
Where these terms come from?

8. Distance properties of convolutional codes
Code strength is measured by the minimum free distance: where w(X) is weight of the entire encoded sequence X generated by a message sequence
The minimum free distance denotes:
The minimum weight of all the paths in the state diagram that diverge from and remerge with the all-zero state S0
The lowest power of the Generating Function T(X):
Coding gain:

9. Decoding convolutional codes
Maximum likelihood decoding of convolutional codes means finding the code branch in the code trellis that was most likely transmitted
Therefore maximum likelihood decoding is based on calculating code Hamming distances dfree for each branch forming encoded word
Assume that information symbols applied into a AWGN channel are equally alike and independent
Let’s denote by x the message bits (no errors) and by y the decoded bits:
Probability to decode the sequence y provided x was transmitted is then
The most likely path through the trellis will maximize this metric
Also, the following metric is maximized (prob.<1) that can alleviate computations:
received bits:
Decoder
non-erroneous bits:
bit
decisions

10. Example of exhaustive maximal likelihood detection
Assume a three bit message is to transmitted. To clear the encoder two zero-bits are appended after message. Thus 5 bits are inserted into encoder and 10 bits produced. Assume channel error probability is p=0.1. After the channel 10,01,10,11,00 is produced. What comes after decoder, e.g. what was most likely the transmitted sequence?

11. weight for prob. to
receive bit in-error
correct
errors

12. The largest metric, verify
that you get the same result!
Note also the Hamming distances!

13. Soft and hard decoding Transition for Pr[3|0] is indicated
Regardless whether the channel outputs hard or soft decisions the decoding rule remains the same: maximize the probability
However, in soft decoding decision region energies must be accounted for, and hence Euclidean metric dE, rather that Hamming metric dfree is used
Transition for Pr[3|0] is indicated
by the arrow

14. Decision regions
Coding can be realized by soft-decoding or hard-decoding principle
For soft-decoding reliability (measured by bit-energy) of decision region must be known
Example: decoding BPSK-signal: Matched filter output is a continuos number. In AWGN matched filter output is Gaussian
For soft-decoding several decision region partitions are used
Transition probability
for Pr[3|0], e.g. prob.
that transmitted ‘0’
falls into region no: 3

15. The Viterbi algorithm
Exhaustive maximum likelihood method must search all paths in phase trellis for 2k bits for a (n,k,L) code
By Viterbi-algorithm search depth can be decreased to comparing surviving paths where 2L is the number of nodes and 2k is the number of branches coming to each node (see the next slide!)
Problem of optimum decoding is to find the minimum distance path from the initial stage back to initial stage (below from S0 to S0). The minimum distance is the sum of all path metrics that is maximized by the correct path
The Viterbi algorithm gets its efficiency via concentrating into survivor paths of the trellis
Channel output sequence
at the RX
TX Encoder output sequence
for the m:th path

16. The survivor path
Assume for simplicity a convolutional code with k=1, and up to 2k = 2 branches can enter each stage in trellis diagram
Assume optimal path passes S. Metric comparison is done by adding the metric of S into S1 and S2. At the survivor path the accumulated metric is naturally smaller (otherwise it could not be the optimum path)
For this reason the non-survived path can be discarded -> all path alternatives need not to be considered
Note that in principle whole transmitted sequence must be received before decision. However, in practice storing of states for input length of 5L is quite adequate

17. Example of using the Viterbi algorithm
Assume received sequence is and the (n,k,L)=(2,1,2) encoder shown below. Determine the Viterbi decoded output sequence!
(Note that for this encoder code rate is 1/2 and memory depth L = 2)

18. The maximum likelihood path
Smaller accumulated
metric selected
After register length L+1=3
branch pattern begins to repeat 1 1
(Branch Hamming distance
in parenthesis)
First depth with two entries to the node
The decoded ML code sequence is whose Hamming
distance to the received sequence is 4 and the respective decoded
sequence is (why?). Note that this is the minimum distance path.
(Black circles denote the deleted branches, dashed lines: '1' was applied)

19. How to end-up decoding?
In the previous example it was assumed that the register was finally filled with zeros thus finding the minimum distance path
In practice with long code words zeroing requires feeding of long sequence of zeros to the end of the message bits: wastes channel capacity & introduces delay
To avoid this path memory truncation is applied:
Trace all the surviving paths to the depth where they merge
Figure right shows a common point at a memory depth J
J is a random variable whose magnitude shown in the figure (5L) has been experimentally tested for negligible error rate increase
Note that this also introduces the delay of 5L!

20. (the weight spectrum) at
Error rate of convolutional codes: Weight spectrum and error-event probability
Error rate depends on
channel SNR
input sequence length, number of errors is scaled to sequence length
code trellis topology
These determine which path in trellis was followed while decoding
An error event happens when an erroneous path is followed by the decoder
All the paths producing errors must have a distance that is larger than the path having distance dfree, e.g. there exists the upper bound for following all the erroneous paths (error-event probability):
Probability of the path at
the Hamming distance d
Number of paths
(the weight spectrum) at
the Hamming distance d

21. Selected convolutional code gains
Probability to select a path at the Hamming distance d depends on decoding method. For antipodal (polar) signaling in AWGN channel it is that can be further simplified for low error probability channels by remembering that then the following bound works well:
Here is a table of selected convolutional codes and their associative code gains Gc Gc=RCdf /2 (df = dfree)

22. The error-weighted distance spectrum and the bit-error rate
BER is obtained by multiplying the error-event probability by the number of data bit errors associated with the each error event
Therefore the BER is upper bounded (for instance for polar signaling) by where ed is the error-weighted distance spectrum where
ad is the number of paths (the weight spectrum) at the Hamming distance d
is the number of data-bit errors for the path at the Hamming distance d
Note: This bound is very loose for low SNR channels.
It has been found by simulations that partial bounds, eg taking terms of the summation of pb expression above yields good estimate to around BER<10-2 error rates