Error Probability of RS Code Over Wireless Channel

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1 Internatonal Journal of Electroncs Engneerng, 3 (), 11, pp Serals Publcatons, ISS : Error Probablty of RS Code Over Wreless Channel Mohammad Aftab Alam Khan 1 & Mehwash Farooq 1 1 Department of ECE, IIMT Engneerng College, Meerut, Inda E-mal: er.aftab@ymal.com, er.mehwash@ymal.com Abstract: Reed-Solomon codes are an mportant class of error correctng codes used n many applcatons related to communcatons and dgtal storage. The fundamental operatons n Reed-Solomon encodng and decodng nvolve Galos feld arthmetc. Computer smulaton tool, MATLAB s used to create and run extensvely the entre smulaton model for performance evaluaton. It s dscovered that the performance of RS codes can be assess through the functon of ts bloc sze, redundancy and code rate and t s observed usng Bt-Error Rate (BER) performance curve. Because RS codes wor at byte level, thus t s also apparent that RS codes can perform well aganst burst nose. The results show that code gan wth hgh code rate s better than that of low code rate and t s found that RS coded QAM sgnal performs better than the coded D-PSK sgnal wth the same modulaton sze. Keywords: Reed-Solomon Codes, RS encoder and decoder, M-PSK, Bt Error Rate. 1. ITRODUCTIO In real world communcaton, errors are ntroduced n messages sent from one pont to another as shown n Fgure 1. Reed-Solomon s an error-correctng codng system that was devsed to address the ssue of correctng multple errors especally burst-type errors n mass storage devces (hard ds drves, DVD, barcode tags), wreless and moble communcatons unts, satellte lns, dgtal TV, dgtal vdeo broadcastng (DVB), and modem technologes le xdsl ( x referrng to all the exstng DSL solutons, whether ADSL, VDSL, SDSL, or HDSL). of code s also nown as a systematc code. A well-nown example of a RS code s RS (55, 3) wth 8-bt symbols. For ths specfc Reed-Solomon code, each code word has 55 total bytes, wth 3 bytes of data and 3 bytes for party. Ths code has: n = 55, = 3, s = 8, t = 3, t = 16. Fgure : Reed Solomon Code Word Ths means that the decoder can automatcally correct 16 symbol errors up to 16 bytes anywhere n the code word. Fgure 1: Two Ponts Exchange Informaton In order for the transmtted data to be corrected n the event that t acqures errors, t has to be encoded. The recever uses the appended encoded bts to determne and correct the errors upon recepton of the transmtted sgnal. The number and type of errors that are correctable depend on the specfc Reed-Solomon codng scheme used. A Reed- Solomon code s specfed as RS (n, ) wth s-bt symbols, where n s the total number of bytes the code word contans and s the number of data bytes. The number of party bytes s equal to n, where n s rased to the power of s mnus one (s 1). A Reed-Solomon decoder can correct up to t number of bytes, where t = n. Fgure shows a Reed-Solomon code word n whch the data s left unaltered whle the party bts are suffxed to the data bts. Ths type 1.1. Reed-Solomon Termnology Symbol Wdth s the number of bts per symbol Code Word s the bloc of n symbols RS (n, ) code: - n s the total number of symbols per code word - s the number of nformaton symbols per code word Code Rate s equal to / n r = (n ) s the number of chec symbols. t = (n ) / s the maxmum number of Symbols wth errors that can be corrected.. RS SYSTEM MODEL In ths subsecton the testng s appled to sngle carrer system wth (AWG) channel model usng (QAM) technque. AWG channel s havng a frequency spectrum that s contnuous and unform over a specfed frequency

2 3 Internatonal Journal of Electroncs Engneerng band or t has equal power per hertz over the specfed frequency band QAM s a modulaton technque where ts ampltude s allowed to vary wth phase, also can be vewed as a combnaton of ampltude shft eyng (ASK) as well as phase shft eyng (PSK). It can be vewed as ASK n two dmenson. Fgure 3 demonstrates the smulaton model employed by ths secton. 55/39 assocated wth codng, whch s to ncrease the number of symbol states n the modulaton scheme. For ths reason, spectrally effcent multlevel modulaton schemes such as M-PSK and M-QAM were developed. However, ncreasng the number of symbol states may ncur an mplementaton penalty as well as a large energy effcency penalty, requrng hgher phase and ampltude accuracy n both transmtter and recever systems. Fgure 3: Sngle Carrer Transcever 3. RS ECODIG & DECODIG PROCESS Reed-Solomon (RS) code s a cyclc symbol error-correctng code that operates at the bloc level rather than the bt level. For bloc codes, the ncomng data stream s frst pacaged nto small blocs. These blocs are then treated as ew set of symbols to be pacaged nto a super-coded bloc of n symbols, by appendng the calculated redundancy. Such symbols can ether be comprsed of one bt (bnary code) or, of several bts (symbol codes). Therefore, the nformaton transfer rate s reduced by a factor called code rate R= /n, and the bandwdth of the sgnal produced by the modulator s expanded by the rato 1/R= n/, relatve to a system usng the same modulator wthout codng [1]. The Reed-Solomon encodng and decodng requre a consderable amount of computaton and arthmetcal operatons over a fnte number system wth certan propertes,.e. algebrac systems, whch n ths case s called felds. RS s ntal defnton focused on the evaluaton of polynomals over the elements n a fnte feld (Galos feld) []. Suppose we are dealng wth a 56-level RS code of a natural bloc length 55 n conjuncton wth a modulaton/ demodulaton scheme. Here the feld sze s 56, and the nformaton and code symbols can be regarded as 8-bt symbols. Let that we see a d mn = 17, producng t = 8 or fewer symbol-error-correctng capablty. Ths mples that n - =16, or the number of nformaton symbols s 39. The generator polynomal for (55,39) RS code s a 16 degree polynomal over GF (56) wth coeffcents gven n an ascendng order as α 136 α 4 α 8 α 195 α 181 α 158 α 1 α 1 α 11 α 83 α 167 α 17 α 113 α 11 α 16 α 11. The feld generator polynomal for ths code over GF () s x 8 + x 4 + x 3 + x + 1. In such applcatons, there s one obvous method to avod the tradtonal bandwdth expanson by a factor Fgure 4: Bloc Dagram of Coded Sngle Carrer System Model The performance of RS code s tested by combnng channel and modulaton codng n sngle carrer system, through the smulaton of such a system as depcted n Fgure 4. The RS coded data are nterleaved to provde addtonal error correcton. Ths process spreads the data from several RS blocs over a much longer perod of tme so that long burst of nose s requred to overcome the capablty of the RS code. 4. RS ECODER AD DECODER ALGORITHMS 4.1.Encoder Algorthm Cyclc codes, such as Reed-Solomon codes, are descrbed n numerous codng theory boos [3] [4] [5]. Gven a data polynomal a(x) of degree < n, n = m, n Galos feld GF (m) and a code generatng polynomal g(x) of degree p, where p n- and p g ( x) = ( ), x + α a ( x) = a x = = wth α successve unty roots n GF ( m ) and a elements of the same feld, the systematc encodng of a(x) s gven by n ( ) ( ) ( ) C x = a x x R x () where R(x) s the remander of the dvson a(x)x n- by g(x). Gven two polynomals P(x) P() x and G(x) wth coeffcents n the GF ( m ) feld, M (1) ( ) =, ( ) = + α (3) P x a x G x x = = the remander R(x) of P(x) dvded by G(x)can be expressed as:

3 Error Probablty of RS Code Over Wreless Channel 31 ( ) R x () x (4) = = = β + α Where = n, α are the roots of the polynomal G(x), and β () are GF polynomals generated by successve Horner reductons of P(x) by α. Successve Horner reductons of the polynomal by are llustrated n Table 1. [6]. The entres n the frst row are the coeffcents of the polynomal to be reduced and the entres n the rghtmost column are the unty roots. If the remander ( j) s zero, then j s a root of P(x). The entres of the second row are the coeffcents of the reduced polynomal wth, and so on. Each row represents the coeffcents of the polynomal to be reduced wth the root of the g(x) polynomal n the correspondng poston of the rghtmost column. The result of the reducton s the next row. Table 1 Horner Reducton Table -1 - a α α + -1 ( α + -1 )α + - β α 1 α 1 + α + -1 ( α 1 + α + -1 )α 1 + α ( α + -1 )α + - β (-1) α -1 The encodng algorthm s performed as follows: Perform the successve Horner reductons on the polynomal a(x)x n- to obtan the coeffcents β (), Calculate R(x) as descrbed n Equaton 4, Calculate C(x) as descrbed n Equaton. 4.. Decoder Algorthm Suppose the receved code word s ( ) C x n = c x (5) = Instead of checng the valdty of the code word by encodng the data porton of the receved code word and then comparng the computed party bts to the receved party bts, the decoder starts by drectly computng the syndromes of the receved codeword usng Equaton 6. S T = C (6) where C s a vector formed from the coeffcents of the data polynomal and Σ are the syndrome vectors computed usng, n = ( ) = x α (7) If all the syndromes are zeroes, there are no errors. If any syndrome dffers from zero, the followng syndrome matrx equaton s solved, where λ are the coeffcents of the error locator polynomal. s s1. s λ s s s. s λ s 1 s s3. s λ s..... = s s. s.. + λ1 s s s. s λ s The above system of equatons s solved usng Gaussan elmnaton to obtan Equaton 9. 1 Ξ1. Ξ λ s 1 1. Ξ λ s + 1. s Ξ + 1 λ 3 + = Ξ λ 1 s. 1 s λ Ths system of equatons s then solved to produce, 1... (8) (9) λ = Ξ λ = Ξ λ + Ξ (1) The error locator polynomal s solved usng Chen search. The nverse of the soluton of the error locator polynomal represents the poston of the bt error. 5. SIMULATED RESULTS Dfferental M-DPSK and coherent M-QAM modulaton/ demodulaton technque are adopted n ths wor. Fgure 5 smulates the transmsson of encoded and modulated sngle carrer sgnal n the presence of addtve whte Gaussan nose, where the encoded data are 15 symbols depth nterleaved. BER remans at hgh value untl SR exceeds a certan pont dependent on the constellaton sze. As sequence of ponts from a bandwdth-effcent constellaton, we can have a few postons where code words dffer by symbols havng large nter-sgnal dstance, or we can have a relatvely larger number of postons where the symbol

4 3 Internatonal Journal of Electroncs Engneerng dstance s small. Ultmately, mnmum dstance between constellaton ponts results n hgher requred SR to acheve error-free recepton, for example, a change from 64-DPSK to 18-DPSK results n ecessary ncrease n SR by a maxmum of 6dB. Moreover, t s clear that the coherent modulaton performs better than dfferental modulated sgnal wth the same constellaton sze of about 1 db. For ncluson of ISI effect on system performance under consderaton due to mult-path propagaton, Fgure 6 llustrates the BER as a functon of SR over Rcan fadng channel wth two path and τ MED (Mean Excess Delay) of approxmately.4 sample and τ max (Maxmum Excess Delay) of 1 sample. It s apparent that the ISI so nduced, degrades the performance of coded sngle carrer n terms of the requred SR, for example, of about 1 db at BER = 1 - n 16-QAM system. bnary modulaton. Fgure 7 graphed the decoder output BER versus channel SR for two dfferent t values 1 and 4 and fxed n = 55 RS code over Rcan channel of τ max =3 sample and τ MED =.5 sample. Fgure 7: BER Performance Comparson of Coded 16-QAM Sgnal at Dfferent t Values Over Rcan Channel of max =3 Sample Fgure 5: BER Performance of Sngle Carrer System n AWG Channel wth RS (55,39) Code Fgure 8: BER Performance Comparson of Dfferent RS Coded 16-QAM Sgnal Over Rcan Channel of max =3 Sample Fgure 6: BER Performance of Sngle Carrer System wth RS (55,39) Code n Rcan Channel of MED =.4 Sample However, the transton to 64-QAM or 16-DPSK results n an rreducble error floor, even over smaller delay spread. Varous studes have been carred out to deal wth optmzaton of the structure of Reed-Solomon codes usng Recall that the relaton between code redundancy and t s n- = t. The error rates are related to the code error correcton capablty, because the channel generates errors randomly wth numbers mght exceed t wthn some of the messages. Ths effect gets more obvous for lower t. In the fgure dashed lne s conducted for -error correcton capablty wth 17-code length, where lower error rates are resulted under the same channel condton. Because of that a longer code word s more susceptble to random channel errors. The encodng process, to ensure enough error protecton aganst channel degradaton, s based on a mother code. In order to guarantee more system flexblty a shortenng procedure s nserted, as shown n Fgure 8. RS

5 Error Probablty of RS Code Over Wreless Channel 33 (45,35) and (3,) shortened code performs better than the mother RS code (55,45) by a maxmum of 3dB n channel SR to acheve the error-free recepton. 6. COCLUSIO From the smulated error rates of Reed-Solomon coded sngle carrer system over AWG channel usng QAM and DPSK modulated types, t s found that QAM sgnal performance s better than of DPSK sgnal by about 1dB for 64-pont constellaton sze. Also, 16-QAM coded sgnal over fadng channel exhbts a more robust performance than DPSK even of lower order constellaton. Lower error rates can be acheved usng more error code correcton capablty and/or shorter code word snce t s less susceptble to random channel errors. REFERECES [1] M. J. Rley and E. G. Rchardson, Dgtal Vdeo Communcatons, Artech House, Inc., [] Y. Xu and T. Zhang, Varable Shortened-and-Punctured Reed-Solomon Codes for Pacet Loss Protecton, IEEE Transactons on Broadcastng, Vol. 48, o. 3, pp , September. [3] E. R. Berleamp, Algebrac Codng Theory, Aegean Par Press, [4] R. E. Blahut, Algebrac Codes for Data Transmsson, Cambrdge Unversty Press, 3. [5] S. B. Wcer, Error Control Systems for Dgtal Communcaton and Storage, Prentce Hall Inc., [6] C. Sdney Burrus, James W. Fox, Gary A. Stton, and Sven Tretel, Horner s Method for Evaluatng and Deûatng Polynomals, Rce Unversty ovember 6, 3.

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