Error Control Coding3

8/2/2019 Error Control Coding3

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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=


==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


==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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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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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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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.

Error Control Coding3

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