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Convolutional encoding
Figure shows how in
memory depthL=v-1kinput bits are
encoded to
n output bits in a
(n,k,L) code
This figure
shows a general
structure of a
convolutionalencoder
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Example of using generator matrix
1
2
[1 0 11]
[111 1]
= =
g
g
Verify that you can obtain the result shown!
11 10
01
11 00 01 11 01 =
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State diagram of a convolutional code
Each new block ofkbits causes a transition into new state (see -2 slides)
Hence there are 2kbranches leaving each state
Assuming encoder zero initial state, encoded word for any input kbits
can thus be obtained. For instance, below for u=(1 1 1 0 1) the encoded
word v=(1 1, 1 0, 0 1, 0 1, 1 1, 1 0, 1 1, 1 1) is produced:
Encoder state diagram for an (n,k,L)=(2,1,3) coder
Verify that you have the same result!
Input state
Output state
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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 gainXi. 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 ofXin the path gain
Code weight distribution is obtained by using a weighted gain formula
to compute its generating function (input-output equation)
whereAiis the number of encoded words of weight i
( ) ii
i
T X A X =
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The path representing the statesequence S0S1S3S7S6S5S2S4S0 has
path gain X2X1X1X1X2X1X2X2=X12
and the corresponding code word
has the weight 12. The generating
function is:
6 7 8
9 10
( )
3 5
11 25 ....
i
ii
T X A X
X X X
X X
=
= + +
+ + +
Where these terms come from?
weight: 1weight: 2
Example of splitting
and labeling the
state diagram
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Distance properties of convolutional codes
Code strength is measured by the minimum free distance:
where w(X) is weight of the entire encodedsequenceXgenerated by a
message sequence
The minimum free distance denotes:
The minimum weight of all the paths in the state diagram thatdiverge from and remerge with the all-zero state S0
The lowest power of the Generating Function T(X):
{ }min ( )free
d w X=
6 7 8
9 10
( )
3 511 25 ....
i
ii
T X A X
X X X X X
=
= + ++ + +
6free
d =
( )/ 2 1
c free
G kd n= >Coding gain:
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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
Lets denote by x the message bits (no errors) and by y the received 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.
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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 intoencoder 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?
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0
( , ) ( | )m j mj
j
p p y x
=
=y x
0ln ( , ) ln ( | )jm j mjp p y x
==y x
errors correct
weight for prob. to
receive bit in-error
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Note also the Hamming distances!
correct:1+1+2+2+2=8;8 ( 0.11) 0.88
false:1+1+0+0+0=2;2 ( 2.30) 4.6
total path metric: 5.48
=
=
The largest metric, verify
that you get the same result!
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Soft and hard decoding
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
E
fre be Cd d E R=
0ln ( , ) ln ( | )jm j mjp p y x
==y x
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Decision regions
Coding can be realized by soft-decoding or hard-decoding principle
For soft-decoding reliability (measured by bit-energy) of decision regionmust be known
Example: decoding BPSK-signal: Matched filter output is a continuos
number. In AWGN matched filter output is Gaussian
For soft-decoding
several decisionregion partitions
are used
Transition probability
for Pr[3|0], e.g. prob.that transmitted 0
falls into region no: 3
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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 2kis 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 S0to S0). Theminimum 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
0ln ( , ) ln ( | )jm j mjp p y x
==y x
Channel output sequence
at the RX
TX Encoder output sequence
for the m:th path
2 2k L
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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 transmittedsequence must be received before decision.
However, in practice storing of states for
input length of 5L is quite adequate
2 branches enter each nodek
2 nodesL
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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!
01101111010001y =
(Note that for this encoder code rate is 1/2 and memory depthL = 2)
states
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The maximum likelihood path
The decoded ML code sequence is 11 10 10 11 00 00 00 whose Hamming
distance to the received sequence is 4 and the respective decoded
sequence is 1 1 0 0 0 0 0 (why?). Note that this is the minimum distance path.
(Black circles denote the deleted branches, dashed lines: '1' was applied)
(1)
(1)
(0)
(2)
(1)
(1)
1
1
Smaller accumulated
metric selected
First depth with two entries to the node
After register length L+1=3
branch pattern begins to repeat
(Branch Hamming distance
in parenthesis)
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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 thispath memory truncation is applied:
Trace all the surviving paths to thedepth where they merge
Figure right shows a common point
at a memory depth J
Jis 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! 5 stages of the trellisJ L>
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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 eventhappens 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):
2( )
free
e dd d
p a p d
=
Number of paths
(the weight spectrum) at
the Hamming distance d
Probability of the path at
the Hamming distance d
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Selected convolutional code gains
Probability to select a path at the Hamming distance ddepends 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)
2
0
2( ) b
C
Ep d Q R d
N
=
( )21
( ) exp / 22
Q x x
( 0)x
2( )
free
e dd d
p a p d
=
21( ) exp( / 2)2 x
Q x d
=
/C
R k n=
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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 edis the error-weighted distance spectrum
where
adis 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 3 - 10 terms of
the summation ofpb expression above yields good estimate to around
BER
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Punctured Convolutional Codes
Puncturing is the process of systematically deleting, or not sending,
some output bits of a low-rate encoder. Since the trellis structure of the low-rate encoder remains the same, the
number of information bits per sequence does not change the outputsequences belong to a higher-rate punctured convolutional (PC) code.
Apuncturing matrix P specifies the rules of deletion of output bits. P isa knp binary matrix, with binary symbolspijthat indicate whether the
corresponding output bit is transmitted (pij= 1) or not (pij= 0).
A rate k/np PC encoder based on a rate-l/n encoder, has a puncturing
matrix P that contains l zero entries, where np = kn - l,0 l< kn
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Punctured Convolutional Codes
Ex: A rate-2/3 memory-2 convolutional code can be constructed by
puncturing the output bits of the rate-1/2 memory-2 convolutional encoder,according to the puncturing matrix
1 1
1 0P
=
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Punctured Convolutional Codes
One of the goals of puncturing is that the same decoder can be used for a
variety of high-rate codes. One way to achieve decoding of a PC code using the Viterbi decoder of
the low-rate code, is by the insertion of "deleted" symbols in the positions
that were not sent.
The "deleted" symbols are marked by a special flag (i.e., bit 1).
If a position is flagged, then the corresponding received symbol is nottaken into account in the branch metric computation.
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Punctured Convolutional Codes
Puncturing matrices are employed with the memory-6 rate-1/2
convolutional code with generators (g0, g1) = (171,133): vi(m)
indicates theoutput, at time m, associated with generatorgi, i = 0, 1.
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