Lect 6_Error Control Coding III

8/2/2019 Lect 6_Error Control Coding III

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SET 358 3 DIG ITAL CO MMUNICATIO N

Dr. Nurul Muazzah Abdul Latif f

Faculty of Electrical Engineer ing, UTM

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Content

O ptimum decod ing

Maximum Likelihood decoder

Soft decisions and hard decisions?

Viterbi algorithm

2


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O ptimum decod ing

If the input seq uence messages a re e qually likely, the op timumde cod er which minimizes the prob ab ility of error is theMaximum likelihooddecoder.

ML decoder, selects a codeword among all the possiblecodewor ds which max imizes the likelihood f unctionwhere is the received seq uence a nd is one of thepossible codewords.

)( )(m

p

U|Z

Z )(mU

)(max)(ifChoose )(allover

)()( mmm pp(m)

U|ZU|ZUU

ML decoding rule:

2Lcodewords

to search!!!

3

ML decod ing for memory -less channels

Due to the indep endent channel sta tistics for memory lesschannels, the likelihood function becomes

and equivalently, the log-likelihood function becomes

The pa th metric up to time index , is called the pa rtia l pa th metric.

1 1

)(

1

)()(

21,...,...,,

)( )|()|()|,...,...,,()(21

i

n

j

m

jiji

i

m

ii

m

izzz

muzpUZpUZZZpp

iU|Z

1 1

)(

1

)()()|(log)|(log)(log)(

i

n

j

m

jiji

i

m

ii

muzpUZppm U|Z

U

Path metric Branch metric Bit metric

ML decoding rule:Choose the path with maximum metric among

all the paths in the trellis.

This path is the closest path to the transmitted sequence.

""i


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Binary symmetric channels (BSC)

If is the Hamming d ista nce b etween Z and U,then

Modulator

input

1-p

p

p

1

0 0

1

)( )(mm

dd UZ,

)1log(1

log)(

)1()( )(

pL

p

pdm

ppp

nm

dLdm mnm

U

U|Z

Demodulator

output)0|0()1|1(1

)1|0()0|1(

ppp

ppp

ML decoding rule:Choose the path with minimum Hamming distance

from the received sequence.

Size of coded sequence

Sof t and ha rd decisions

In hard decision:

The demodulator makes a firm or hard decision whetherone or zero is transmitted and provides no otherinfor mat ion for the decoder such tha t how relia ble t hedecision is.

Hence, its output is only ze ro or one (the output is

quantized only to two level) which are called hard-bits.

Decod ing based on ha rd -b its is ca lled the ha rd -

decision decoding.


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Soft and hard decision-contd

In Sof t decision: The demodulator provides the decoder with some side

information together with the decision.

The side information provides the decoder with a measure ofconfid ence for t he decision.

The demodulator outputs which a re ca lled sof t-b its, arequantized to more than two levels.

Decoding based on soft-bits, is called the soft-decisiondecoding.

O n AW G N channels, 2 dB and on fa ding channels 6 dBgain are obtained by using soft-decoding over hard-decoding.

The Viterb i a lgorithm The Viterb i alg orithm pe rf orms Ma ximum likelihood

decoding.

It find a p at h through trellis with the larg est metric(ma ximum corr ela tion or minimum d ista nce).

It processes the demodulator outputs in an iterative manner.

At ea ch step in the trellis, it compa res the metric of all pa thsentering ea ch state, a nd keep s only the pa th with the smallestmetric, called the survivor, together with its metric.

It proceeds in the tre llis by eliminating the least likely p aths.

It red uces the de cod ing complexity to !12K

L


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The Viterb i a lgorithm - contd

Viterbi alg orithm:

A. Do the fo llowing set up :

For a d ata block of L b its, form the tr ellis. The trellis hasL+K-1 sections or levels and sta rt s a t time and endsup at time .

La bel a ll the b ra nches in the trellis with theircorresponding branch metric.

For ea ch sta te in the tre llis a t the time which is

denoted by , def ine a pa rameterB. Then, do the f ollow ing:

it

}2,...,1,0{)(1

K

itS

ii

ttS ),(

1t

KLt

1t

The Viterb i a lgorithm - contd

1 . Set and

2 . At time , comp ute the p ar tia l p at h metrics for a ll thepaths entering each state.

3 . Set eq ua l to the b est pa rtia l pa th metricentering each sta te at time .

Keep the survivor pa th a nd delete the dea d p at hs f romthe tr ellis.

1 . If , increa se by 1 and return to step 2.

C. Start at state zero at t ime . Follow thesurviving b ra nches ba ckwards through the trellis.The pa th thus def ined is unique a nd correspond tothe M L codeword .

0),0( 1 t.2i

it

ii ttS ),(it

KLi i

KLt


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Examp le of Hard decision Viterb idecoding

1/11

0/00

0/10

1/11

1/01

0/00

0/11

0/10

0/01

1/11

1/01

1/00

0/00

0/11

0/10

0/01

0/00

0/11

0/00

6t1t 2t 3t 4t 5t

)101(m)1110001011(U)0110111011(Z

Examp le of Hard decision Viterb i

decoding-contd

Label all the branches with the branch metric (Hammingdistance)

0

2

0

1

2

1

0

1

1

0

1

2

2

1

0

2

1

1

1

6t1t 2t 3t 4t 5t

1

0

ii ttS ),(


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Examp le of Hard decision Viterb idecoding-contd

i= 2

0

2

0

1

2

1

0

1

1

0

1

2

2

1

0

2

1

1

1

6t1t 2t 3t 4t 5t

1

0 2

0


decoding-contd

i= 3

0

2

0

1

2

1

0

1

1

0

1

2

2

1

0

2

1

1

1

6t1t 2t 3t 4t 5t

1

0 2 3

0

2

30


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i= 4

0

2

0

1

2

1

01

1

0

1

2

2

1

0

2

1

1

1

6t1t 2t 3t 4t 5t

1

0 2 3 0

3

2

3

0

2

30


decoding-contd

i= 5

0

2

0

1

2

1

01

1

0

1

2

2

1

0

2

1

1

1

6t1t 2t 3t 4t 5t

1

0 2 3 0 1

3

2

3

20

2

30


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i= 6

0

2

0

1

2

1

01

1

0

1

2

2

1

0

2

1

1

1

6t1t 2t 3t 4t 5t

1

0 2 3 0 1 2

3

2

3

20

2

30

Exa mple of Hard decision Viterbi decoding-

contd

Trace back and then:

0

2

0

1

2

1

01

1

0

1

2

2

1

0

2

1

1

1

6t1t 2t 3t 4t 5t

1

0 2 3 0 1 2

3

2

3

20

2

30

)100( m)0000111011( U


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Example of soft-decision Viterbi decoding

)101( m)1110001011( U

)101(m)1110001011(U

)1,3

2,1,

3

2,1,

3

2,

3

2,

3

2,

3

2,1( Z

5/3

-5/3

4/3

0

0

1/3

1/3

-1/3

-1/3

5/3

-5/3

1/3

1/3

-1/3

6t1t 2t 3t 4t 5t

-5/3

0 -5/3 -5/3 10/3 1/3 14/3

2

8/3

10/3

13/33

1/3

5/35/3

ii ttS ),(

Branch metric

Partial metric

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-4/35/3

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Lect 6_Error Control Coding III