Convolutional Codes Presentation17!12!2014

7/24/2019 Convolutional Codes Presentation17!12!2014

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Overview

• Channel Coding• Convolutional Codes

• Convolutional Encoding• Convolutional Decoding

• Merits and Demerits• Applications• Conclusions



What is Channel Coding?• Channel Coding refers to the class of signal transmission

designed to – Improve communication performance

– etter withstand the effects of various channal

impairments! such as noise! interference and fading



• "tructured se#uences deals with transforming datase#uences into $etter se#uences!% having structuredredundanc&' (he redundant its can then e used for thedetection and correction of errors

"tructured "e#uences



Convolutional Codes

• Convolutional codes offer an approach to error controlcoding

• "e#uentiall& convolves the se#uence of information itswith generator se#uences according to some rule

• It is a channal coding scheme with memor&



• Convolutional code is specified & ) parameters n! * and +

where!

* , inputs n , outputs

+ , constraint length of the convolutinal code

• n-tuple emitted & convolutional encoding procedure is not onl&

a function of an input *-tuple! ut is also a function of the

previous *-. input *-tuple

• Code /atecode rate , ratio *0n determine the numer of data its per

coded it'

Convolutional Codes "pecifications



• (he performance of a convolutional code depends on

– coding rate

– constraint length

• "maller coding rate *0n

– More powerful code due to e1tra redundanc&

– 2ess andwidth efficienc&

• 2onger constraint length +

– More powerful code

– More coding gain

Convolutional Codes 3erformance

Criteria



Convolutional Encoding



A 4eneral Convolutional Encoder



Convolutional Encoder /epresentation

Methods• Connection /epresentation

– Impulse /esponse of Encoder

– 3ol&nomial /epresentation

• (ree Diagram

• "tate Diagram

• (he (rellis Diagram



Connection /epresentation• Convolutional encoder

– 5rate 6! +,)7

– ) shift-registers! where the first one ta*es the incoming

data it and the rest form the memor& of the encoder



• 3eriodic (runcation – Adds numer of 8ero its to the input data se#uence forthe purpose of clearing or flushing the encoding shiftregister

• Connection 9ectors – Descries the connection of the encoding shift register to

that of module : adder

– If connection e1ists , connection vector is . – If no connection , connection vector is ;

Connection /epresentation Contd'



Connection /epresentation E1ample

• Connection vector g. for the upper connection , ...

• Connection vector g: for the lower connection , .;.



E1ample Contd'



Impulse /esponse of Encoder • /esponse of the encoder to a single $one% it that moves through

register

• (he output se#uence for the Input $one% is called impulse responseof the encoder

• E1ample – Consider input se#uence m , .;.' We generate output & the

convolution of the input se#uence with the impulse response of

the encoder! that we drive the name convolutional encoder



3ol&nomial /epresentation

• /epresent a convolutional encoder with a set of n generator

pol&nomial! one for each of the n module-: adder anddescries the connection of the encoding shift register to that

module-: adder li*e connection vector

• 4enerator pol&nomial for upper and lower connection

• E1press m,.;. as a pol&nomial

• (he output se#uence is found as



3ol&nomial /epresentation Contd'



"tate Diagram

• A state diagram is simpl& a graph of the possile states

of the encoder and the possile transitions from one

state to another

• It is not possile in a single transition to move form a

given state to an aritrar& state

• It can e used to show the relationship etween the

encoder state! input! and output

• <odes are connected & ranches



"tate Diagram Contd'''



(ree Diagram

• (ree diagram adds the dimension of time to the statediagram

• If the input it is ;! its associated ranch word is found &

moving to the ne1t rightmost ranch in the upward direction

• If the input it is .! its associated ranch word is found &

moving to the ne1t rightmost ranch in the downward

direction



• If the input it is ; , output

ranch word is ;;

• If the input it is . , output

ranch word is ..

•If the first input it is . =

:nd input it is . , output

ranch word is ;.

• If the first input it is . =

:nd input it is ; , output

ranch word is .;

(ree Diagram Contd'• (he ranching rule for finding a codeword se#uence is



(rellis Diagram

• (rellis diagram is an e1tension of state diagram which

e1plicitl& shows the passage of time

• All the possile states are shown for each instant of time

• (ime is indicated & a movement to the right

• (he input data its and output code its are represented& a uni#ue path through the trellis

• >lue line indicates that the input it is .' /ed lineindicates that the input it is ;



• Assume that the state start with@a,;; ! m , ..;..

• "tep .Current status B a , ;;

Encoding /ule B ; -- ;;! . -- ..

Input to the status B . 5. it7

Output it 5according to Econding

/ule7 B .. 5: its7Output "tatus B , .;

• "tep :Current status B , .; 5(he result

of 3revious "tep7Encoding /ule B ; -- .;! . -- ;.

Input to the status B . 5. it7

Output 5according to Econding/ule7 B ;. 5: its7

Output "tatus B d , ..

(rellis Diagram Contd'



• Step 3Current status B d , .. 5(he result of 3revious

"tep7

Encoding /ule B ; -- ;.! . -- .;Input to the status B ; 5. it7

Output 5according to Econding /ule7 B ;. 5: its7

Output "tatus B c , ;.

• Step 4Current status B c , ;. 5(he result of 3revious

"tep7

Encoding /ule B ; -- ..! . -- ;;Input to the status B . 5. it7

Output 5according to Econding /ule7 B ;; 5: its7

Output "tatus B , .;

• Step 5Current status B c , .; 5(he result of 3revious

"tep7

Encoding /ule B ; -- .;! . -- ;.Input to the status B . 5. it7

Output 5according to Econding /ule7 B ;.

Output "tatus B d , ;.

(rellis Diagram Contd'



Convolutional Decoding

•"everal decoding algorithm – 9iteri decoding

– "e#uential decoding

• 9iteri decoding algorithm – sed for relativel& small values of *

– niversall& adopted

– 3rovides ma1imum li*elihood – Easil& implemented in 92"I



E1ample of 9iteri Algorithm

• 2et@s assume that we have received a it stream

@..;.;.;;;.@ and want to decode this to get the original

data 5the data efore encoding7

• Assuming that we have used : it encoder! we split the

received it stream into multiple groups of two its

• (hen we have .. ;. ;. ;; ;.

• <ow we have to do is to ac* trac* the trellis diagram to

figure out the input it stream that would have produced

.. ;. ;. ;; ;.



"teps in Decoding



"teps in Decoding



"teps in Decoding



"teps in Decoding



"teps in Decoding



Merits and Demerits of Convolutional

Codes

• Merits – (o correct the errors in nois& channels these convolutional codes are

fre#uentl& used

– (hese convolutional codes perform well on ver& ad conditions also

– (hese convolutional codes are generall& used often in satellite

communications – Convolutional codes generall& gives good results in lower noise

environments

– Convolutional codes are also used in the voice and modems andalso in the 4"M moile phones

• Demerits – Even though the convolutional encoder has the simplest procedure!

decoding of it is ver& comple1 tas*'

– Convolutional codes cannot provide more protection against noise'



Applications

• Digital video

• /adio

• Moile communication

• "atellite communication



Conclusions

• Convolution codes outperform loc* codes for the sameimplementation comple1it& of encoder -decoder'

• Convolutional encoding can e used to improve the

performance of wireless s&stems'

• 9iteri algorithm is an optimum decoding algorithm in thesense that it alwa&s finds the nearest path to the nois&modification of the encoder output se#uence and it is

#uite useful when the code has a short memor&'







• <ot much differernce' As an operation convolutionalcodes are called loc* codes ut traditional& calledconvolutional codes ecause of impulse responseehaviour and memor&

• Encoder of convolutional codes has memor&

– Output n-tuple is function of oth input *-tuple andprevios *-. input *-tuple

• Encoder of loc* codes has no memor&

– Output n-tuple is function of input *-tuple onl&

• Convolution codes perform etter than loc* codes for

the same implementation of encoding and decoding

Comparison of >loc* = Convolutional

Codes?



9iteri Decoding with Error

• Decoding algorithms with errors uses amming distancemetric to find the most li*el& 5minimum distance7 path

through the trellis

• amming distance is path etween : nodes

• 3ath etween nodes having shorter hamming distance is

chosen as correct path

• "tate ;; to ;; transition has hamming distance :

• "tate ;; to .; transition has hamming distance ;



Decoding with Error

. "tep



: "tep

Decoding with Error



) "tep

Decoding with Error



F "tep

Decoding with Error



G "tep

Decoding with Error

D di i h E



H step

Decoding with Error



"tep

Decoding with Error

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Convolutional Codes Presentation17!12!2014

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