review of modern noise proof coding methods d. sc. valeri v. zolotarev

Review of modern noise proof coding methods

D. Sc. Valeri V. Zolotarev

2V. Zolotarev - Review of modern coding methods

The large volume of transmitting data demands to provide their high veracity

One of major ways for transmission error probability decrease in noisy digital channels is usage of noise proof coding methods

V. Zolotarev - Review of modern coding methods

3

Principles of noise proof coding

The information is broken into blocks, for example, binary digits, to which one the check bits being by a function from an information part of the transmitting data are added.

The relative part of initial information characters in such enlarged block is called as code rate R.


4

The main concepts of information theory

Channel capacity С - characterizes a maximum mean

information quantity, which one can be transferred to the receiver during the period of one usage of a channel.

С - function of a channel noise level, i. e. of mean transmission error probability for binary digits.


The main limitation in information theory for coding

The condition should be always satisfied:

R < C !

In this case there are coders, which one can ensure a digital transmission with an arbitrary small probability of an error, if the block length will be great enough.

.


How to fulfill the indicated condition in communication engineering? Is it difficult or not? 1. The introducing of redundancy

conforming to a given value of code rate R is very simply.

3. SoSo R<CR<C - - understandable for the specialists condition

2. For given error probabilities of

transmitting binary digits in Gaussian channel its capacity C also is easily calculated


7

The elementary encoder for a block code with 2 correcting errors!It is way to form redundancy (code rate): R=1/2


8

Whenever possible - it is else easier!!!

An example of the encoder for a convolutional code with the same code rate R=1/2.


Limit possibilities of codingInterconnection between channel capacity C

and computational rate R1 for BSC with channel error probability Po

0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0,90

1,00

0,000 0,050 0,100 0,150 0,200 0,250 0,300 0,350 0,400 0,450 0,500

Po - channel error probability

Cap

acity

C a

nd

com

puta

tiona

l rat

e R

1

C

R1


What quality of codes is main? - Code distance d ! It determines minimum number of

symbol positions, in which the code words (permissible data) are different.

For example, in parity checking codes all permissible words - are only ones with an even number of «ones». So its code distance is d=2 !


11

What for it is necessary to take codes with large d values?

The more d, then the greater number of errors appeared in the transmitted code block by, can be corrected.

In this case portion of blocks grows, which one after decoding can be error-free.

And then what maximum d values are possible?


12

Limits of correcting properties for two code classes

0,000 0,050 0,100 0,150 0,200 0,250 0,300 0,350 0,400 0,450 0,500R block0,00

0,10

0,200,30

0,40

0,500,60

0,70

0,800,90

1,00

Code

rate

R

Code distance d to code length n ratio: d/n

Interconnection between C and R1 for different values of d/n ratio

R block

R conv

d freed min

Codes exist !

There are no codes !


One of main

questions: // What may

the code length be? As at R<C the theory guarantees good outcomes of the coded data transmission, let's see, as far as lengthy should be the code block in different cases.


The lower estimations of error probabilities of optimum block code decoding with R=1/2 in BSC. Even the codes with length n=1000 are ineffective at channel error probability Po > 0.07.

But the theory affirms, that it is possible to work successfully at Po < 0.11, in accordance to main condition C > 1/2.

And it is true for total searching methods!


15

The main «jokes» of the NatureThe main «jokes» of the Nature 1. Almost all codes are "good". If decoder is

optimal then resulting error probability will be close to the best ones.

2. Almost all codes can be decoded only by total searching methods. For a code length n=1000 exhaustive search at R=1/2 requires to look through 2500(!!!) versions of the possible code words. But it exceeds number of atoms in the Universe!

So what must we do? PROBLEM!!!


16

The Main Problem of the noise proof coding theory

1. To find and to investigate simple non exhaustive search decoding methods in noisy channel.

2. To ensure such decoding quality with these methods, that they were more close to efficiency of optimal procedures.

3. To remember needs and conditions of coding usage in communication systems.


Threshold decoders: everything is simpleeverything is simple!! Let's pay attention: It is truly the elementary errors correcting scheme!


But TD efficiency - is paltry! It is extremely far from Ро=0.11.


19

Multithreshold decoders (MTD) Multithreshold decoders (MTD) for Gaussian channelsfor Gaussian channelsThey are designed and deeply investigated

during last 30 years multithreshold decoders very poorly distinguished from customary extremely simple classic threshold procedures, offered by J.L.Massey.

The main property MTD - at each change of symbols new decoder

decision becomes more close to the optimum one!


20

The main consequence from MTD properties

If MTD for a long time changes characters of the received data, it can achieve the solution of the optimum decoder (OD) at linear complexity of decoding.

Usually solutions OD - are the outcomes of exponential growing with code length exhaustive search .....,

but here we get linear complexity?!!


It is multithreshold decoder!!!0 1 2 3 4 5 6 7 8 9 10 11 12

0 1 2 3 4 5 6 7 8 9 10 11 12

12 11 10 9 8 7 6 5 4 3 2 1 0

T

v

u

Декодер блокового СОК с R = 1/2, d = 5 и n = 26

It is a view of block MTD. The new register contains a difference between the MTD solutions and values of information bits of a channel.

Why?


This is convolutional MTD with R=1/2, d=5 and 3 iterations

Рис. 1. Многопороговый декодер сверточного СОК с R=1/2, d=5 и nA=14

0 1 2 3 4 5 6

0 1 2 3 4 5 6

6 5 4 3 2 1

T1

v

u

0

0 1 2 3 4 5 6

0 1 2 3 4 5 6

6 5 4 3 2 1

T2

0

0 1 2 3 4 5 6

0 1 2 3 4 5 6

6 5 4 3 2 1

T3

0


23

The resolved MTD problems 1. The very complicated problem of an error propagation effect

(EP) estimation in TD is completely resolved 2. The codes with minimum EP were successfully constructed ! 3. Four generations of MTD coding equipment have been built. 4. Most important: the minimum possible complexity of

decoding, referenced for customary TD is saved. 5. Consequent. MTD works at high noise levels almost as OD.

6. TOTAL. Creation of the effective decoder near channel capacity C

- generally resolved problem.


The estimations of convolutional code error probability decoding for Viterbi algorithm and MTD in BSC with R=1/2.

2.0

Decoder's error probabilities in BSC

1,0E-09

1,0E-08

1,0E-07

1,0E-06

1,0E-05

1,0E-04

1,0E-03

1,0E-02

0,1170 0,1038 0,0909 0,0787 0,0671 0,0563 0,0464 0,0375 0,0297 0,0229

Signal-to-noise ratio in a Guassian channel, dB and channel probability in BSC

Bit

er

ror

pro

bab

ility

, P

b(e

)

MTD1MTD2MTD3av7hav11hav15hn1000n=10000n=3000G4G6G8

1

23

47

1115

СBSCn10000- -0.5

n1000

Po

Es/No-1.5 -1.0 0.0 0.5 1.0 1.5 2.5 3.0n3000

G=4

G=6G=8

CC

2.0


25

And what is necessary for communication engineering?

“The energy decrease in communication channel at 1 dB gives an economic efficiency $1’000’000 ,” - E.R.Berlecamp, IEEE, 1980, vol.68, №5.

Now at enormous growth of communication network cost the price of signal power decrease has increased (!!!) multiply.

But how to fasten probabilistic channel parameters to its signal energy?


26

The coding considerably reduces signal power in transmission channel!

The value of a decrease is called code gain (CG) G = 10*Lg(R*d) dB

The signalmen for a long time know how to change the receiver for increase code gain.

And where are limits of signal power decrease?


27

The “soft” modem estimating reliability of a signal reception instead of "hard", which one only makes a decision about value of received bit, allows to diminish signal power approximately at 2 dB.

Distribution of voltage output of a binary signal in the modemРаспределение выходного напряжения двоичного сигнала в модеме

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0,16

0,18

№п/п

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Номера областей

Зн

ач

ен

ия

Ноль

Один

1 0

« - »

« + »


28

The minimally possible ratio of energy per bit of the transmitted information to a noise power density Eb/No in binary channel for different code rate R can be submitted for “hard” and “soft” modems so:

-1

0

1

2

3

4

5

6

7

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

R - code rate

Eb/N

0,

dB

'hard' М=2

"soft" М=16


29

1,0E-08

1,0E-07

1,0E-06

1,0E-05

1,0E-04

1,0E-03

1,0E-02

0,117 0,104 0,091 0,079 0,067 0,056 0,046 0,038 0,030 0,023

Signal to noise ratio in Gausssian channel, dB and Po

Bit

err

or r

ate,

Pb

(e)

av7

av11

av15

м11

m7

m9

э4

э6

э8

7

11

m11

-0.5

Po

Es/No-1.5 -1.0 0.0 0.5 1.0 1.5 2.5 3.0

m9

G=4

G=6G=8

м7

2.0

15CC

Error probabilities of the main base decoding algorithms in Gaussian channel with "soft" demodulation


30

Concatenation - it’s the best!Concatenation - it’s the best!In this case the coding implements two and more codes, which ones in the

receiver are decoded in the return order and at definite interplay of decoders.

On the chart - best known outcomes on efficiency in Gaussian channel: Viterbi algorithm (VAk), MTD usual and cascaded (MTDK), VA+RS-code,

best of turbo (T1 and T2), and woven code (W1) too.

Decoder bit error probability as function of bit energy to noise power density ratio

1,0E-06

1,0E-05

1,0E-04

1,0E-03

1,0E-02

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0

Eb/No, dB

Bit

err

or

rate

C=1/2

T1 VA7

VA-RS

VA15

VA20

MTD

MTDK

VA11W1

T2


31

BUT! BUT! MTD in 100 times more simple!!!MTD in 100 times more simple!!!

Decoder operation number comparision

1

10

100

1000

10000

100000

0 0,25 0,5 0,75 1 1,25 1,5 1,75 2 2,25 2,5

Channel bit energy/noise density ratio, Eb/No, dB

Op

erat

ion

nu

mb

er

MTD-S

MTD-a

MTD-b

turbo


32

What shall we use ?What shall we use ? - Most simple and effective methods !!!- Most simple and effective methods !!!

History of achivements - increase in code gain

0

1

2

3

4

5

6

7

8

9

10

1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

Years

Co

de

ga

in,

dB

TD VA

CC: АВ+RSMTD

Turbo

CC: turbo? MTD?…….??

MTD-K


0055.0.099.2003.2003

E-mail: [email protected]: [email protected] tel. +7 - 095-333-23-56,Work tel. +7 - 095-333-23-56,

+7 -+7 - 095-095-261261-5-544--4444моb.: +7-916-518-86-28моb.: +7-916-518-86-28

www.mtdbest.iki.rssi.ru

review of modern noise proof coding methods d. sc. valeri v. zolotarev

Documents

zolotarev review

modern coding methods10

modern coding methodshow

redundancy code rate

block code

given value of code

convolutional code

channel capacity c