Review of modern noise proof coding methods D. Sc. Valeri V. Zolotarev

V. Zolotarev - Review of modern coding methods2 The large volume of transmitting data demands to provide their high veracity w 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 methods3 Principles of noise proof coding w 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. w The relative part of initial information characters in such enlarged block is called as code rate R.

V. Zolotarev - Review of modern coding methods4 The main concepts of information theory w Channel capacity С - w characterizes a maximum mean information quantity, which one can be transferred to the receiver during the period of one usage of a channel. w С - function of a channel noise level, i. e. of mean transmission error probability for binary digits.

V. Zolotarev - Review of modern coding methods5 The main limitation in information theory for coding w The condition should be always satisfied: w R < C ! w 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. w.w.

V. Zolotarev - Review of modern coding methods6 How to fulfill the indicated condition in communication engineering? Is it difficult or not? w 1. The introducing w of redundancy conforming to a given value of code rate R is very simply. So R<C - w 3. So R<C - understandable for the specialists condition w 2. For given error probabilities of transmitting binary digits in Gaussian channel its capacity C also is easily calculated w

V. Zolotarev - Review of modern coding methods7 The elementary encoder for a block code with 2 correcting errors! It is way to form redundancy (code rate): R=1/2

V. Zolotarev - Review of modern coding methods8 Whenever possible - it is else easier!!! An example of the encoder for a convolutional code with the same code rate R=1/2.

V. Zolotarev - Review of modern coding methods9 Limit possibilities of coding

V. Zolotarev - Review of modern coding methods10 What quality of codes is main? w - C ode distance d ! w It determines minimum number of symbol positions, in which the code words (permissible data) are different. w 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 !

V. Zolotarev - Review of modern coding methods11 What for it is necessary to take codes with large d values? w The more d, then the greater number of errors appeared in the transmitted code block by, can be corrected. w In this case portion of blocks grows, which one after decoding can be error-free. w And then what maximum d values are possible?

V. Zolotarev - Review of modern coding methods12 Limits of correcting properties for two code classes

V. Zolotarev - Review of modern coding methods13 One of main questions: // What may the code length be? w 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.

V. Zolotarev - Review of modern coding methods14 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 > But the theory affirms, that it is possible to work successfully at Po 1/2. And it is true for total searching methods!

V. Zolotarev - Review of modern coding methods15 The main «jokes» of the Nature The main «jokes» of the Nature w 1. Almost all codes are "good". If decoder is optimal then resulting error probability will be close to the best ones. w 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 (!!!) versions of the possible code words. But it exceeds number of atoms in the Universe! w So what must we do? PROBLEM!!!

V. Zolotarev - Review of modern coding methods16 The Main Problem of the noise proof coding theory w 1. To find and to investigate simple non exhaustive search decoding methods in noisy channel. w 2. To ensure such decoding quality with these methods, that they were more close to efficiency of optimal procedures. w 3. To remember needs and conditions of coding usage in communication systems.

V. Zolotarev - Review of modern coding methods17 everything is simple ! Threshold decoders: everything is simple ! Let's pay attention: It is truly the elementary errors correcting scheme!

V. Zolotarev - Review of modern coding methods18 But TD efficiency - is paltry! It is extremely far from Ро=0.11.

V. Zolotarev - Review of modern coding methods19 Multithreshold decoders (MTD) for Gaussian channels They 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!

V. Zolotarev - Review of modern coding methods20 The main consequence from MTD properties w 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. w Usually solutions OD - are the outcomes of exponential growing with code length exhaustive search....., w but here we get linear complexity?!! w

V. Zolotarev - Review of modern coding methods21 It is multithreshold decoder!!! 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?

V. Zolotarev - Review of modern coding methods22 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 n A =14

V. Zolotarev - Review of modern coding methods23 The resolved MTD problems w 1. The very complicated problem of an error propagation effect (EP) estimation in TD is completely resolved w 2. The codes with minimum EP were successfully constructed ! w 3. Four generations of MTD coding equipment have been built. w 4. Most important: the minimum possible complexity of decoding, referenced for customary TD is saved. w 5. Consequent. MTD works at high noise levels almost as OD. w 6. TOTAL. Creation of the effective decoder near channel capacity C w - generally resolved problem.

V. Zolotarev - Review of modern coding methods24 The estimations of convolutional code error probability decoding for Viterbi algorithm and MTD in BSC with R=1/2. 2.0

V. Zolotarev - Review of modern coding methods25 And what is necessary for communication engineering? w “The energy decrease in communication channel at 1 dB gives an economic efficiency $1’000’000,” - E.R.Berlecamp, IEEE, 1980, vol.68, №5. w Now at enormous growth of communication network cost the price of signal power decrease has increased (!!!) multiply. w But how to fasten probabilistic channel parameters to its signal energy?

V. Zolotarev - Review of modern coding methods26 The coding considerably reduces signal power in transmission channel! w The value of a decrease is called code gain (CG) G = 10*Lg(R*d) dB w w The signalmen for a long time know how to change the receiver for increase code gain. w And where are limits of signal power decrease? w

V. Zolotarev - Review of modern coding methods27 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. « - » « + »

V. Zolotarev - Review of modern coding methods28 The minimally possible ratio of energy per bit of the transmitted information to a noise power density E b /N o in binary channel for different code rate R can be submitted for “hard” and “soft” modems so:

V. Zolotarev - Review of modern coding methods29

V. Zolotarev - Review of modern coding methods30 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.

V. Zolotarev - Review of modern coding methods31 BUT! MTD in 100 times more simple!!!

V. Zolotarev - Review of modern coding methods32 What shall we use ? - Most simple and effective methods !!! MTD-K

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