Scientific session DNIT of the RAS New optimization coding theory and its applied achievements...

Scientific session Scientific session DNIT of the RAS DNIT of the RAS

New optimization New optimization coding theory and coding theory and its applied its applied achievements achievements 24.04.2008.24.04.2008.

V.V.Zolotarev, SRI RASV.V.Zolotarev, SRI RAS

V.V.Zolotarev. The coding theory V.V.Zolotarev. The coding theory 22

The main fundamental The main fundamental scientific problem of transition scientific problem of transition

from analog from analog Communications(connections) Communications(connections)

and computer science and computer science To digital To digital

• Maintenance of a high Maintenance of a high level of reliability of level of reliability of formation, processing, formation, processing, transfer and storage of transfer and storage of figures.figures.

• Means of the decision Means of the decision of this problem - of this problem - methods of the theory methods of the theory of noiseproof codingof noiseproof coding

• Theoretical basis of unequivocal exact restoration of an analog signal is the theorem of readout of academician V.A.Kotelnikov.

• At transition of our technological civilization to transmission and storage of the information to a discrete form the main requirement to such systems, becomes their conformity to Shannon theorem.

• In this case it is possible to restore always in the receiver the digital message deformed in a channel, with as much as necessary small error probability if the length of the code block will grow.

This theorem has begun the modern This theorem has begun the modern theory of coding.theory of coding.

Coding - Coding - This introduction of This introduction of

redundancyredundancyK - the information

+

R - superfluous symbols

R=k/n <1 - code speed

n=k+r - length of the block

+

Coding - Coding - This is introduction of This is introduction of

redundancyredundancyk – the information

r - superfluous symbols

n=k+r - length of the blockn=k+r - length of the block

R=k/n <1 - code R=k/n <1 - code raterate


WITH

Channel capacity С is a function of energy in BCS

0

0,2

0,4

0,6

0,8

1

-8 -6 -4 -2 0 2 4

signal/noise ratio, dB

Cap

acit

y С

C <1!


The basic restriction The basic restriction of the Information of the Information

Theory Theory for coding for coding• Always condition Always condition

R <CR <C must satisfy!must satisfy!

• In this case there are systems of coding which In this case there are systems of coding which can provide transfer of the digital information can provide transfer of the digital information

with as much as small probability of a mistake if with as much as small probability of a mistake if the length of the block of the data will be great the length of the block of the data will be great

enough (K.Shannon, the theorem of coding enough (K.Shannon, the theorem of coding existence)existence)


What is necessary for What is necessary for codes in communication codes in communication networks?networks?• It is - a code gain, CG!, - a measure of It is - a code gain, CG!, - a measure of

signal energy effectiveness increase.signal energy effectiveness increase.• Now every one additional dB in CG gives Now every one additional dB in CG gives

in communication networks economic in communication networks economic benefit in many millions dollars!benefit in many millions dollars!

• Resource ЭВК can be realized at ERS for Resource ЭВК can be realized at ERS for decrease of aerials sizes, and also for decrease of aerials sizes, and also for increase in speed, reliability and distance increase in speed, reliability and distance of communication. It is extremely of communication. It is extremely important for satellite communication important for satellite communication systems such as VSAT, and also projects systems such as VSAT, and also projects micro- and nano- satellites or other high-micro- and nano- satellites or other high-speed communication systems. It is speed communication systems. It is achieved only by correct fast achieved only by correct fast mathematical processing of a digital mathematical processing of a digital stream!stream!

The low estimations of error probabilities The low estimations of error probabilities decoding for block codes with R=1/2 decoding for block codes with R=1/2

Even codes of length n=1000 are inefficient at Even codes of length n=1000 are inefficient at error probability in the channel Ro> 0.08. And the error probability in the channel Ro> 0.08. And the theory asserts it is possible to work successfully at theory asserts it is possible to work successfully at Ро <0.11 !!! Ро <0.11 !!!

And it occurs at 2 And it occurs at 2500500 variants of decisions! variants of decisions! Number of atoms in the Universe is Number of atoms in the Universe is

more little!more little!

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

0,11 0,1 0,09 0,08 0,07 0,06 0,05

P0 - вероятность ошибки в ДСК

Вер

оятн

ость

ош

ибки

на

блок

n=10000 n=3000

P0

n=1000

n=300

n=100

0 dB 1 dB -1 dB

234


Complexity of decoders for Complexity of decoders for different codes with length different codes with length

nn

1 2 3 4

Linear ~n Quadratic,

~n2

polinomial

~nm

Exponencial !!!

Asimptotic complexity of the decoding algoritms

minimal

maximum en

VA !!!

MTDDiscr. algebra


The block multithreshold decoder for The block multithreshold decoder for a code with R=1/2, d=5 and I iterations a code with R=1/2, d=5 and I iterations

MTD,complexity-N~d*I*n


The reasons of high efficiency new The reasons of high efficiency new MTD a methodMTD a method

• 1. Procedure is applied special very easy 1. Procedure is applied special very easy for realization iterative оптимизационная.for realization iterative оптимизационная.

• 2. Special codes with a minimum level of 2. Special codes with a minimum level of grouping of mistakes - too a method of grouping of mistakes - too a method of optimization are constructed.optimization are constructed.

• 3. Process of many hundreds parameters 3. Process of many hundreds parameters special optimization in the decoder was special optimization in the decoder was realized.realized.

• Problems 1 and 2 - “are very difficult"Problems 1 and 2 - “are very difficult"

• The problem 3 - has not appeared at allThe problem 3 - has not appeared at all

Minimum of calculations at Minimum of calculations at decoding - in MTD! decoding - in MTD!

(Number of operations per bit, (Number of operations per bit, soft realization)soft realization)

Usually: N1 ~ d*I, - product

and in MTD: only N2~d+I, - the

sum of key parameters d and I.It is in ~100 times easier and faster than, for example, at use of a turbo codes! It was realized in special TV-system.

Hardware realization MTD at Hardware realization MTD at VLSIVLSI

1. MTD will consist almost completely of store elements or shift registers. These are the fastest elements in VLSI. The share of other elements in MTD is less than 1 %.

2. MTD may be absolutely parallel one step algorithm. For this reason MTD for some values of parameters approximately in 1000 times faster, than for example, a turbo decoders. A delay - as for the elementary 2- input key - absolute minimum .

3. Realization: throughput: up to 1,6Gb/s, and CG = 7 - 9,5 dB


Chipset of the MTD decoder Chipset of the MTD decoder at Xilinx for channels with at Xilinx for channels with

speeds up to 150 Mb / sspeeds up to 150 Mb / s


Multithreshold decoder (MTD) for satellite and space channels, raises Multithreshold decoder (MTD) for satellite and space channels, raises efficiency of their use in 3-10 times, including for EDS. Simple MODEL efficiency of their use in 3-10 times, including for EDS. Simple MODEL MTD at Altera for channels up to 640 Mbit / s. The method can work MTD at Altera for channels up to 640 Mbit / s. The method can work

at information speeds up to 1,6 Gbit / sat information speeds up to 1,6 Gbit / s

MTD - for the Space!


New scientific and technological New scientific and technological revolution – data transmission revolution – data transmission

with the minimal powerwith the minimal powerЭффективность новых и старых методов кодирования

при кодовой скорости R=1/2

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

0 1 2 3 4 5 6 7 8 9 10

Ratio Eb/N0

BE

R,

ве

ро

ятн

ос

ть о

ши

бки

де

код

ер

а н

а б

ит

No codingVA

MTD - simple

MTD-CC

The first revolutionThe second revolution

WITH

, дБ


The symbolical multithreshold decoder for The symbolical multithreshold decoder for a code with R=1/2, d=5 and I iterations a code with R=1/2, d=5 and I iterations

QMTD-symbolic ?

-


Welcome! Visitors of web-site SRI RAS www.mtdbest.iki.rssi.ru in March, 2008.

More than 18000 visitors of our web-site from 56 countries have copied more than 7 Gbytes data about MTD algorithms during last year.

???

USA Network


Chipset MTD decoder Chipset MTD decoder at ALTERA basis at ALTERA basis


ConclusionConclusionss 1. We have opened iterative MTD algorithms 35 years ago.

2. Complexity of the soft versions MTD is an absolute known minimum of calculations. A difference with other codes is ~100 times! The rare case in the theory! We outstrip all countries ~ 7 ÷ 10 years. 3. Hard MTD are faster than a other codes ~1000 times! 4. Decisions MTD are almost always optimum even at the big noise level. 5. MTD - the absolute leader by criteria "complexity - efficiency". It is created in Russia! 6. Non-binary codes for MTD - the most unique discovery in the

coding theory during last 30 years. They are unknown abroad! The Russian scientific school - again in the group of world leaders in the

theory!


24.04.2008.24.04.2008.

SRI of the Russian Academy of Sciences SRI of the Russian Academy of Sciences in Moscow w.ph.: (495) 333 45 45, in Moscow w.ph.: (495) 333 45 45, www.mtdbest.iki.rssi.ru, www.mtdbest.iki.rssi.ru, e-mail: [email protected] e-mail: [email protected]

mobil: +7 916 518 86 28 mobil: +7 916 518 86 28 V.V.ZolotarevV.V.Zolotarev


Further there are help slides - appendices to the

report


Whenever possible - to code easier!!! An example of the coder for свёрточного a code

with code speed R=1/2


Differenr philosophy in coding theory

0

0,2

0,4

0,6

0,8

1

0 0,2 0,4 0,6 0,8 1

Algorithm throughput

Eff

ecti

ven

ess

Abroad

Russia, MTD


Complexity of various Complexity of various algorithms of decodingalgorithms of decoding

Codes and algorithmsCodes and algorithmsAlgorithm Algorithm ViterbiViterbiAlgebraic codesAlgebraic codesTurbo codesTurbo codesLow dencityLow dencity parity parity codescodesMajority codes-Majority codes-- only in Russia - - only in Russia - MTDMTD which which

almost always converges to almost always converges to optimum (optimum (total searchtotal search!) the !) the decision, but with linear decision, but with linear complexitycomplexity

ComplexityComplexity22nn

nn22

C1*nC1*nC2*nC2*nC3*n C3*n С3 С3

<C2 <C1<C2 <C1


Fig. 1. The multithreshold decoder сверточного JUICE with R=1/2, d=5 and 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

Convolutionale multithreshold decoder Convolutionale multithreshold decoder for a code with R=1/2, d=5 and 3 iterationsfor a code with R=1/2, d=5 and 3 iterations

The reasons of high efficiency new The reasons of high efficiency new МПД a methodМПД a method• 1. Procedure is applied special very easy 1. Procedure is applied special very easy

for realization iterative for realization iterative оптимизационная.оптимизационная.

• 2. Special codes with a minimum level of 2. Special codes with a minimum level of grouping of mistakes - a method of grouping of mistakes - a method of optimization are constructed.optimization are constructed.

• 3. Special optimization of many hundreds 3. Special optimization of many hundreds parameters of the decoder is carried out.parameters of the decoder is carried out.

• Problems(Tasks) 1 and 2 - "very difficult"Problems(Tasks) 1 and 2 - "very difficult"

• The problem(task) 3 - was not put at allThe problem(task) 3 - was not put at all


Multithreshold decoding (MTD) Multithreshold decoding (MTD)

– MTD repeatedly changes symbols of the MTD repeatedly changes symbols of the accepted message and can achieve at linear accepted message and can achieve at linear complexity of realization the decision of the complexity of realization the decision of the optimum decoder (OD). optimum decoder (OD).

– It is a result of the iterative approach It is a result of the iterative approach application to the error correction. It was application to the error correction. It was revealed at USSR at 22 years earlier, than in revealed at USSR at 22 years earlier, than in the West.the West.

– Usually "price" of optimum decodingUsually "price" of optimum decoding– (as for Viterbi algorithm, VA) - full search, and (as for Viterbi algorithm, VA) - full search, and

complexity MTD - only linear function of a code complexity MTD - only linear function of a code length!!!length!!!

–

What is necessary from What is necessary from codes for communication codes for communication networks?networks?• Prof. Berlecamp (USA) had said in 1980 in the Prof. Berlecamp (USA) had said in 1980 in the review :review :

• " It - a code gain! ", - a measure of effect of increase " It - a code gain! ", - a measure of effect of increase in energy of the signal, estimated as ~ $1 million in energy of the signal, estimated as ~ $1 million per 1 dB CG.per 1 dB CG.

• Now it is even more important as it is shown on Now it is even more important as it is shown on ours website SRI RAS ours website SRI RAS www.mtdbest.iki.rssi.ruwww.mtdbest.iki.rssi.ru

• Now every additional one dB CG gives in the big Now every additional one dB CG gives in the big networks economic benefit in hundred millions networks economic benefit in hundred millions dollars!dollars!

• Resource CG can be realized for decrease in the Resource CG can be realized for decrease in the sizes of aerials, and also for increase in speed, sizes of aerials, and also for increase in speed, reliability and distance of communication. It is reliability and distance of communication. It is extremely important for satellite systems of extremely important for satellite systems of communication such as VSAT, and also projects communication such as VSAT, and also projects micro- and nano- satellites or other high-speed micro- and nano- satellites or other high-speed systems of communication.systems of communication.

http://www.mtdbest.iki.rssi.ru/


The main fundamental The main fundamental scientific problem of transition scientific problem of transition

from analog from analog Communications(connections) Communications(connections)

and computer science and computer science To digital To digital

• Maintenance of a high Maintenance of a high level of reliability of level of reliability of formation, processing, formation, processing, transfer and storage of transfer and storage of figures.figures.

• Means of the decision Means of the decision of this problem - of this problem - methods of the theory methods of the theory of noiseproof codingof noiseproof coding

The main fundamental scientific The main fundamental scientific problem of transition from problem of transition from

analog analog Communications and computer Communications and computer

science to the digital one science to the digital one

• Maintenance on completely new Maintenance on completely new theoretical principles of a high level theoretical principles of a high level of reliability of transmission and of reliability of transmission and storage of data.storage of data.

• The most successful decisions of The most successful decisions of this problem are offered with the this problem are offered with the theory of noiseproof codingtheory of noiseproof coding

The main problems The main problems of coding technology of coding technology

• 1. To decode - it is easier!.1. To decode - it is easier!.• 2. Reliability - is higher!2. Reliability - is higher!• 3. Maximum to take into account 3. Maximum to take into account

Conditions of coding in real Conditions of coding in real systems of communication systems of communication

• 4. How it may be reached? 4. How it may be reached? Multithreshold Decoders!!!Multithreshold Decoders!!!

• Why? They are extremely simple Why? They are extremely simple and very affective!and very affective!


Limiting opportunities of Limiting opportunities of codescodes

Зависимость пропускной способности ДСК и скорости R1 от вероятности ошибки в канале 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 - вероятность ошибки в канале

Про

пуск

ная

спос

обно

сть

кана

ла и

ск

орос

ть R

1

C

R1

Capacity of the channelC

R1


Зависимость предельной энергетики канала Eb/N0

от кодовой скорости R

-1

0

1

2

3

4

5

6

7

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

R - кодовая скорость

Eb/N

0, д

Б

"жёсткий", М=2

"мягкий", М=16

. АВ


Limiting code cain (CG) in the case of the condition R <C

Scientific session DNIT of the RAS New optimization coding theory and its applied achievements...

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