Soft In Soft Out Reed Solomon Decoding Anum Ali

• Communication Department, Lancaster University, Lancashire England

anum.ali@live.com

Abstract-The aim of this project was to investigate and implement Soft In Soft Out decoder based on Reed Solomon Codes. This type of error correction is widely used in data

communication applications, such as Digital Video Broadcast (DVB), optical carriers and Space Data Systems (CCSDS) primary choice is RS Codes by Consultative committee for deep space communications signal error correction.

I. INTRODUCTION

As demand for reliable digital communication is increasing new techniques are evolving and error prune coded algorithms are under development phase. This research was in context to Reed-Solomon decoding in areas where sensitivity to transmission errors are very high. Reed- Solomon codes are used to perform Forward­Error Correction Decoding (FEC) which is quite efficiently applied. Whatever the type of channel is in communication systems it requires error control. The technicalities are associated with the understanding of Soft In Soft Out decoding approach and using reference algorithm "Vardy and Be'ery Decomposition"[2] for developing effective Soft In Soft Out Simulation Decoder based on Reed Solomon Codes and BCH extended matrix.

II. MODERN ERA OF DIGITAL COMMUNICATION

Increasing demand for traditional services has been an important factor in the development of telecommunication technologies. Such development, combined with more general advancements in electronic and communication perspectives have been productive with reference to wireless communication services.

A. Noise Factor

Reliable transmission of information is one of the central topics in digital communications. While the advent of powerful digital computing and storage technologies have made the data traffic increasingly digital, the reality of inherently noisy communication channels persists and has thus made error control coding an important and necessary step in achieving redundancy in a message prior to transmission and using thus redundancy to recover the original message the one that erroneously received. There exists different type of errors. Following are the errors types:

• Random-Errors.

• Burst-Errors. • Compound Errors.

978-1-61284-941-6/11 /$26.00 ©20 11 IEEE

III. ERROR DECODING

In nearly four decades that have passed since the original work of Shannon, Hamming & Golay error control coding has matured into an important branch of communication system engineering. The field of coding have yielded many results on the mathematical structure of the codes as well as a number of very efficient coding techniques.

A. Error-Control Algorithms

There are three main classes of error correcting codes categorised in algorithms those are Block codes, Cyclic Redundancy Codes and Convolutional codes.

B. RS Codes Availability

What is special about Reed-Solomon codes? Reed­Solomon have higher error correcting capability that any other codes have. The parameters of RS code are:

m= the number of bits per symbol n= the block length k = the un coded message length in symbols (n-k) = the parity check symbols (check bytes) t= the number of correctable symbol errors.

RS codes operate on multi-bit symbol rather than on individual bits like binary codes. The users have the choice to set the number of correctable symbol errors (t), and the block length (n) as shown in Fig. 1.

In general, an RS decoder can detect and correct up to (t=r/2) incorrect symbols if there are (r=n-k) redundant symbols in the encoded message. One redundant symbol is used in identitying the precise value of that error.

This concept of using redundant symbols to either locate or correct errors is useful in the understanding of erasures. The term "erasures" is used for errors whose position is identified at the decoder by the external circuitry. If the RS decoder has been instructed that a specific message symbol is in error, it only has to use one redundant symbol to correct that error and does not have to use an additional redundant symbol to determine the location of the error.

If the locations of all the errors are given to the RS codec by the control logic of the system, 2t erasures can be corrected.

n /�-------------k --��--------------2 -t �

DATA PARITY

Fig. 1. RS codes

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C. Hard Decoding vs Soft Decoding

Channel noise is almost always a continuous phenomenon. What is transmitted may be selected from a discrete set, but what is received transfers from a continuum of values. This leads to the soft-decision decoder, which accepts a vector of real samples of the noisy channel output & estimates the vector of channel input symbols that was transmitted. Also Soft-decision takes advantage of "Side Information", i-e quality of received signal. If the outputs of the matched filter are unquantized or quantized in more than two levels, the demodulator makes soft-decisions.

By constract the hard decision decoder (lIDO) processes received sequence based on a specific decoding method by putting a matched filter in demodulator. Then, the decoder processes this hard-decision received sequence based on a specific decoding method. It requires that its input be from the same alphabet as the channel input.

A sequence of soft-decision outputs of the matched filter is referred to as a soft-decision received sequence. Because the decoder uses the additional information contained in the unquantized received samples to recover the transmitted codeword; soft-decision decoding provides better error performance than hard-decision decoding. Many soft-decision decoding algorithm have been devised.

These decoding algorithms can be classified into two major-categories.

1) Reliability- Based Decoding: Reliability-based decoding algorithms generally require generation of a list of candidate code words of a predetermined size to restrict the search space for finding maximum likelihood codeword.

2) Structure-Based Decoding: The most well known structure-based decoding algorithm is the Viterbi algorithm, which is devised based on the trellis representation of a code.

D. S1S0 Decoding

Soft In and Soft Out is different from soft-decision. Decoding as explained above. Generally the soft input and soft output module is a four port device that accept

1. A posterior / extrinsic information P(c;I) P(u;O). 2. Channel Information 3. A priori information 4. Information estimate P(c;O) P(u;I) as shown in the

Fig 2.

, ,

Fig. 2 Soft Input and Soft Output module model

All the error -correcting algorithm developed so far are based on hard decision in & out or soft decision in & hard decision out. The decoder processes the hard-decision received sequence based on a specific decoding method by putting a matched filter in demodulator.

If the outputs of the matched filter are unquantized or quantized in more than two levels, the demodulator makes soft-decisions. Further on in SISO L( "elk) is the

r------------I I I L(d)

I

t • priori l I value in I I

etector a posteriori LLR value

L'(d) = L.(x) + L(d)

Soft·in soft-out decoder

___ I L.(d)

extrinsic value out

L·(d) 8 posteriori

Output LLR value L(d) = L'(d) + L,(d

Fig. 3. Soft Input and Soft Output Decoder

soft-decision output at the decoder shown in Fig.3, and Lc(xk) is the LLR channel measurement, stemming from the ratio of likelihood functions p(xkldk = i) associated with the discrete memoryless channel model . Whole architecture is shown in Fig 3. As in this project decomposed factor graph is been used instead of iteration.

£. Simulation and Results

The simulation is based on Vardy-Be'ery decomposition [2] cycle free factor graph instead of typical trellis structure and thus shows an optimal SISO decoding algorithm. There are several methods of constructing such codes. The most effective method is the interleaving technique. The basic nature of RS codes can be defined by principle of interleaving.

1) Methodology

• Encoding phase is achieved by passing through BCH extended matrix and applying interleaving effect on information data bits before it travels on noise channel.

• Decoding phase is achieved by using Maximum likelihood criteria and distributed the information bits through factor graph.

Interleaving is another tool used by the code designer to match the error correcting capabilities of the code to the error characteristics of the channel. Interleaving in a digital communications systems enhances the random­error correcting capabilities of a code.

By interleaving, a t-random error correcting code (n,k) is redundantly expanded to degree A, obtain a (An, A.k) code capable of correcting any combination of t bursts of length A. or less. In that sense that if original input information vector after encoding with addition to parity bits is C={vo, . . . ,V�I) then after spreading effect it

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would be in such form C={v., . . .. ,Vm-l·, . . .. , . . ... ,vo', . . ... , Vm-l', . . ... , vo·', . . .... , Vm-l·'}. This is then further processed to noise channel and later passed into decoding stage. The idea behind interleaving is to separate the codeword symbol in the time. Separating the symbols in time effectively transforms a channel with memory to a memory less one, and thereby enables the random error correcting codes to be useful in a burst noise channel. The spreaded information bits are further encoded by using BCH extended matrix. The processing structure is shown in Fig-4 below.

: ...................... :

Vectors

Transmission oisc ___ �� Path

L- _

�_

�� __ �I�L_

D

_

C

_

coo

_

c

_

r

_�

R

Estimate of U Received Vector

(Analog Channel)

Coherent Demodulator

: ....................... :

Fig. 4 Architecture of BCH decoded digital communication system

I) Generator Matrix: Let R(N,K) be the Reed-Solomon code over GF(2m) of length N=2 m_1 and dimension K. Assume that R is used on a binary channel.

Hence, the encoder must employ some fixed linear mapping 6: GF(2 my -) GF(2) mN to convert a sequence of N elements of GF(2m) into string of mN binary digits as described earlier. Now let (l be a primitive element of GF(2m) and let as, (1"' • • • • • as+N-K-I and their cyclotomic conjugate over GF(2) employed for the linear mapping 6.

Defining the codes BI.B2, . . ... , Bm as Bi = {(Yib." '"fibl, '"fi bN-I) I b=(bl,bl, . . . ,bN-I) € B for j=l, 2, . . ... , m as shown in Fig. 5.

mn

�. 0

Ill]( '.

'-

0 [�� Olne Vectors I In(k'k'

Fig. 5 Structure of the Generator Matrix (Binary lmage- CRS where B generates CBCH)

Where, bi € GF(2) and product '"fib; is in GF(2-). B is a subfield subcode of R and, hence, the m codes defined (I) are also subcodes of R. Therefore if (Vii, . . . . . .. ,Vki) is a set of k generators for Bi and the set is

i'Il)- {o(w), O(V2' ), . • . . . . . , o(w)}

as the first mk rows of a binary generator matrix for R. By rearranging the columns the structure of Fig. 6 is obtained

Checks (In column.s

Fig. 6 BCH Extended Matrix Structure

F BCH Extended Matrix

(In ro�'S

The information vector is multiplied with BCH extended matrix and generating an encoded data with addition to parity bits. Rate is the square of the minimum hamming distance between two codewords. It specifies the duration or span of the channel memory, not its exact statistical characterization which is used as time diversity or interleaving.

G. Factor Graph vs Trellis

The generator matrix structure seen in Fig. 5 implies a cycle-free factor graph for RS codes. The RS factor graph consists of m parallel n-stage binary trellis and an additional glue node illustrated in Fig. 5, where variables are represented by circular vertices and states variables by double circles, and local constraints by using wolf method [ 3]. The final trellis stage is 2·" -ary variable node corresponding to the cosets, or equivalently the syndromes, of CSCH. The node connecting the final trellis stages corresponds to the glue code. The coded b�ts are labelled {cio)}i=O, . . . . n-l; j=I, . . ... m and uncoded bIts are similarly labelled {aPd=O, . . . . k-l; j=I, . . .. m. If there is no priori soft information on uncoded bits and if a systematic encoder is used then the equality constraints in the corresponding sections of the factor graph enforce ai"'= Ci'" for i=O, . . . . k-1 and j=I, . . . . . . . ,m. Using the factor graph shown in Fig.7 the following (nonunique) message­passing schedule ensures optimal SISO decoding.

H. Code Language

Visual C/C++ software package was installed for programming the C functions required to simulate the behaviour of factor graph decoding technique and plotted the results from different noise rates. Gaussian noise channel inputs were used to compare the performance of the coded with uncoded BPSK modulated data.

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1. Error Source

In comparing error performance, Additive White Gaussian Noise (A WGN) channel was used. A WGN channel is not always representative of the performance for real channel particular for radio transmission where fading and nonlinearities tend to dominate. Nevertheless A WGN channel is easy to derive and serves as a useful benchmark in performance comparisons. The noise source is assumed as Gaussian-distributed.

J Results

The error performance is a function of the signal-to­noise(SIN) ratio. Out of the many definition possible for SIN, the most suitable for comparison of decoding techniques is the ratio of energy Eb, to noise density (N.). To arrive at this definition and to obtain an appreciation for its usefulness let relate definitions for SIN and BIN •.

The effectiveness of the coding scheme is measured in terms of the Coding gain, which is the difference of the SNR levels of the uncoded and coded systems required to reach the same BER levels.

Comparisons are made between the code & uncoded coherent BPSK systems. At high EJN. ratio, it can be see that the performance of the coded BPSK system is better than the uncoded BPSK systems.

UtI(\)dt<! BIb W<llt' "1 S!.·r:t�fi:U TttUic; lIh'!CMCk 7lftJlllJ

= -0-

o

Fig. 7 Vardy & Be'ery Decomposition Wolf Trellis Factor Graph

At a bit error rate of 104 the (15, 13) double-error-

correcting Reed-Solomon coded BPSK systems give nearly approx 2.9db of coding gain over the uncoded BPSK systems respectively as shown in Fig 8.

'It' - � -_-.,;

! i

" .... , , '

, , ,

Fig. 8 Performance of RS code (15,13) under AWGN Channel

K. Further Enchantments

For future work, implementing the first phase of RS SISO with later section that includes LDPC codes which will accept soft values from RS SISO architecture and in output generates hard values for achieving efficient gain in respect for deep space communication. The system flow Fig 9 is shown below:

ChalUld Reed ' I non SO D«�qon I--� IS DecoOd Dlln

lDP DO<kf

Fig. 9 LDPC enchantments by RS SISO

L. Main Algorithmic Reference

HlldDeclf'1 I

Thomas R. Half ord, Vishankan Ponnampalam, Alex J. Grant and Keith M. Chugg, "Soft-In Soft-out Decoding of Reed-Solomon Codes Based on Vardy and Be'ery Decomposition Computing". IEEE Transactions on Information Theory, Vol. -51, No. 12, December 2005.

IV. CONCLUSION

The Results have established that Factor Graph decoding of Reed-Solomon codes can provide nearly all the predicted theoretical soft coding gain with respect to uncoded signal.

The soft in soft out front end can be implemented with a reasonable complexity using the proposed algorithm. Presented simulation results to characterize the coding gains of approx 2.7db to 3db is possible from the soft-decision Vardy & Be'ery algorithm for Reed-Solomon codes.

CODED ALGORITHM

For j=1,2, ........... m fInding the codeword

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b = I\bj which maximizes Mj (b): I\bj = arg 1£B max Mj(b) where Mj(b) = N-l\i=O log f(vij I bi+rij ) For each estimate I\bj , evaluate

M(c) = m\ j=O Mj (I\bj)= m\ j=l N-l\i=O log f(vij I bi+rij ) where cij = 1\ bij+Ij Selected the codeword c E R that maximizes above equations.

REFERENCES

[] TI.1lrnas R. Ha!�ord, Shr.1lEnt Mern�r, IEEE), Vishankan

Porma.li�alam, . {'Embel', IEEE , A.lex 1. Gram, (SEnior MEmber, IEEE.), .and Keiih � Chugg, (��mbEr, EEE). Soft·In Saft-cut Decoding Of Reed-SD.oJE"DD Codes B=:-a OD ard}·;.md Be'ery

Decomposmen Computi.�g.. IEEE T ransa.ccons en InfolVi"auon TI.'Emy, Vol. -51, No. 12, DE'temDer 2005.

[2] Alexanner ·axdy (Member, IEEE ;.md Yalr Be' Ery (:\1ember, IEEE. , Bit Le-vel SO:!-DECision d:coding Of Reed·Sol"Dmon Codes.. IEEE Trc.nsarnons on Lfo;rnanon Thee!)" "ol. 19, No.3, January 99 .

[3] Jerk KWoj, (Fe 0"', IEEE), Effi�iem Maximum Lik:lih1lod DeC1lding or linear Block Cod:-s Usil'Jg a nillis IEEE

T�ansa::Jlions en Inforrnali-on TbJ?l)ry, \ eol T - 4, No. 1, January 978.

[4] LH.O:;ules LeI? "frror-Conlrol Block Codes., For Cornmuniccti1lDS Engines-s", A.rtech H1)U5e, I�C, 20 . ISBN 1-58053--032-X

[5] J.L Massey, "Th.? h1)w and why of . rn'.Del coding, n in Proc Inl Zurich Seminar, � ar 1 84, pp. f 1(6 )-F 7( 3

[6] Turbo Code: ''BCH E.�llned Matrix Siructure". bt�rI""3"w-sC en<·-h1aVJf fribtr bi",! Date accmed 21-08-20 O.

[] St:phell B. Wicke.r. "Etror Control SYS""tns, For Digita CornmurUcati1)n and Sterage", Plenti��Hall, Inc, 9gS. \SB:\ 0-

3-200809-_ [8] l\1.atlab® 7.0 Documentation, "Signal Proce.;sing Toolbox",

b11j�d",,,,",,, m.Jiru.mw cQDllaccmlbelgd =s..1v'helgltQolboxlsi jin alln.i<>ol bIDlI Date �cessed 02- 1-_>D .

[9] Bemard S ar. Digital Commun:ca.tions: Fll1-.darnl?ntals and app ica.tions (2nd,ed)", P[entice-Hall, Inc, 2001. SB:\ 0-13-OM 88·75. � 1Iof:-tev and . P. eiko, Laser Assisted MicrorechoJogy, 21ld ed., R. �. Osgood, Jr., Ed. Berlin, Germany: S�inger- erlag, 999.

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