Performance Comparison of RS Code and Turbo Codes for Optical Communication

Transcription

1 Internatona Journa of Eectroncs Engneerng, (), 0, pp Seras Pubcatons, ISSN : Performance Comparson of RS Code and Turbo Codes for Optca Communcaton Reena Tyag, Puneet Chandra Srvastava, Mahendra Kumar & R. K. Sngh ECE Dept., Raj Kumar Goe Engg.Coege,Pakhua (Ghazabad), Mahamaya Technca Unversty, Noda,(UP), Inda Assocate Professor,ECE Dept.,BIET, Jhans (UP) Professor, ECE Dept., Kumaon Engg. Coege, Dwarhat (Amora), Uttarakhand Ema: reenatyag94@gma.com Abstract: RS codes can be consdered as serous compettors to turbo codes n terms of performance and compexty. They are based on a smar phosophy.e. constraned random code ensembes and teratve decodng agorthms. In ths paper, we present the performance comparson of RS code and bock turbo codes. The RS code couped wth receve dversty technques are empoyed as the error correcton scheme over optca fber communcaton Channes by empoyng bnary puse poston moduaton (BPPM) moduaton scheme for optca fber communcaton. The performance of codes s evauated n term of bt error rate (BER) for a gven vaue of Eb/No. Smuaton resuts demonstrate that the bt error rate (BER) performance of the Turbo codes (concatenated RS codes) provde better random as we as burst error correcton capabty as compared to RS codes. Keywords: Channe codng, RS Code, Turbo Code, Optca Communcaton Channe, Recever Dversty, BER, BPPM, NECG.. INTRODUCTION FEC s wdey used n wred and mobe communcaton, deep space communcaton as we as data storage systems. In the recent past, t has begun to fnd appcatons of FEC n optca nks [0]. Error correctng codes are broady cassfed n two categores, vz., Bock Codes and Convoutona Codes. We use bock codes n optca communcaton systems snce they operate at very hgh data rate and by usng bock codes we can fnd ow overhead codes that are capabe of correctng random errors due to nose, and burst errors due to dsperson and nter-channe cross tak wth speca emphass on compexty and cost. It s dffcut to mpement convoutona codes that operate at hgh code rate requred for fber-optc systems [9]. Agebrac bock codes, such as Bose-Chaudhur- Hocqueaghem (BCH) and Reed-Soomon (RS) codes are capabe of correctng mutpe bt-errors wth the ow overhead constrant. In case of fber-optc communcaton systems operatng at very hgh data rate (Rc>0.8). Whe seectng an error correctng code one shoud take nto account the practca mtaton mposed by the hardware to make t feasbe to ntroduce an overhead of (n-k) symbos. Thus, ow overhead constrant becomes an mportant parameter whe seectng FEC for optca communcaton appcaton. In ths paper, we have done a comparatve anayss of the performance of dfferent concatenated code usng RS code consderng the ow overhead requrement as a prme desgn crteron.. ALGEBRIC DECODING ALGORITHMS FOR RS CODES.. Berekamp Massey Agorthm for RS Codes Decodng of the nonbnary RS codes nvoves not ony determnng the ocaton of errors, but aso ther magntudes. The error poynoma s e() X e X e X... e X () j j j j j j For the RS case e j, e j,... e j are fed eements of GF ( m ). The t syndrome equatons are j j j t j j j j j j t t t t j j j The s and s j s are reated by the Newton Identtes and shown n [4] that the syndrome Sj can be expressed n recursve form as a functon of σ s and the earer syndromes s j-..., s j- such that s j ( σ s j σ s j... σ s j σs j () for j,,..., t

2 5 Internatona Journa of Eectroncs Engneerng The error ocator poynoma σ(x) w be determned after t teratons nstead of t teratons. As the roots of the error ocator poynoma σ(x) are determned, the task of the decoder s to fnd the error magntude at the error ocaton numbers. Now, we w utze the foowng error evauator poynoma Ω(X) havng the degree equa to the degree of σ(x). [ ] Ω () X ()() s X σ X ( s )( X s X... X X...) σ σ ()() s σ X s sσ σ X () s s... s X σ σ σ Ω Ω X X... Usng t syndromes and coeffcents of σ(x), the error magntudes are computed. The Forney agorthm [7] used to derve the error magntudes. Equaton can be rewrtten as t t [ ()()( s X ] X...) X X mod t X X () σ Ω Ω Ω (4) The equaton can be rearranged n terms of the known poynoma σ(x), the error ocator poynoma Ω(X) and s(x) the syndrome poynoma t [ ] Ω () X ()() mod s X σ X X (5) Ω(X) s naturay reated to the error ocatons and error vaues by the t reatons Ω()() β : e,,..., β β for t (6) j The error magntudes are computed usng the expresson βω() β e for,,..., t j σ () β where σ (X ) denotes the forma dervatve of σ(x ) wth respect to X. The forma dervatve s smar to the usua dervatve, but does not have the same nterpretaton. If f (X) f 0 f X f X... f n X n...s a poynoma over GF(q), then the forma dervatve f (X) s defned as f (X) f f X... nf n X n-....the and quotent rues are apped to forma dervatves. Snce f(x) GF( m ), then f (X) has no odd power term. Appyng the defnton of forma dervatve to σ(x) equaton 7 can be wrtten as e j Ω() β (7) t () ββ (8), where e j s the error vaue at the coordnate n r specfed by β The teratve procedure based on whch the Berekamp- Massey agorthm work s descrbed n [, 8,, 5]. After the t teratons we have σ(x) whose roots are determned by the Chen search. The nverse of the roots are the error ocaton numbers β s. Now we know where the errors are n r but not ther vaues. The decoder has to fnd the error magntudes at these ocatons, whch s accompshed by the Forney agorthm. The decodng s accompshed by addng the error pattern to r over GF (q)... Eucd s Agorthm for BCH and RS Codes The Eucd s agorthm s a recursve technque to fnd the greatest common dvsor (GCD) of two poynomas [6]. If f (X) and g(x) are two poynomas where deg (f(x )) deg(g(x)) then the GCD s computed dvdng f (X) by g(x) recursvey so that the agorthm aways converges to a remander poynoma d(x) 0 and the ast nonzero poynoma d(x) s the GCD. The recursve reaton between f (X) and g(x) s obtaned by wrtng the ntazaton equaton [6] such that m(x) f (X) n(x)g(x) d(x), (9) where m(x ) and n(x ) are ntermedate poynomas obtaned durng the dvson process. The dvson process s expaned n short At the th teraton m()()()()() X f X n X g X d X For the th teraton f ( ) (X) g () (x) and g () (X) d () (X ) The agorthm termnates when d () ( X) 0 and g () (X) s the GCD of f (X) and g(x) In the case of decodng BCH and RS codes we are not nterested n the GCD of f (X ) and g(x) but the ntermedate poynomas m(x) and n(x) at each teraton. We defne a quotent poynoma [6]. q () X d () X d () X 0 where [ ] 0 then d ()()()() X d X q X d X denotes non negatve powers of X (0) m ()()()() X m X q X m X and () n ()()()() X n X q X m X The nta condtons for the agorthm are m ()() X ;()() n X 0 m X n X 0 0 d X f X and d X g X ()()()() 0 At the th teraton m ()()()()() X f X n X g X d X.

3 Performance Comparson of RS Code and Turbo Codes for Optca Communcaton 5 Comng back to the decodng of BCH/RS codes we re-wrte the key equaton (5) for our anayss [ ] t σ () X()() mod s X Ω X X From equaton (9), we can wrte n ()()() X gmod() X d X f X () Compare equaton (9) wth the key equaton n ()(),()()() X σ X d X Ω X and f X X t To observe that the approach produces the desred souton to the key equaton, we need to utze the property of Eucd s agorthm [] that states [ ] [ ] [ ] deg() n Xdeg() deg() d X f X t and [ ] [ ] deg() n Xdeg() d X < t of systems utzng such soft decsons compared to hard decsons are typcay approxmated as db n optca AWGN. Such a decoder coud be caed a soft nput/hard output decoder, because the fna decodng process out of the decoder must termnate n bts (hard decsons) []. Wth turbo codes, where two or more component codes are used, and decodng nvoves feedng outputs from one decoder to the nputs of other decoders n an teratve fashon, a hard-output decoder woud not be sutabe. Because hard decsons nto a decoder degrade system performance (compared to soft decsons). Hence, decodng of turbo codes n optca channes needs a soft nput/soft output decoder. For the frst decodng teraton of such a soft nput/ soft output decoder n optca communcaton, ustrated n Fgure, we generay assume the bnary data to be equay key, yedng an nta a pror LLR vaue of L(d) 0. For t errors the souton of nterest has [ ] [ ] deg() Ω Xdeg() σ X t There exsts ony one poynoma σ(x) wth degree not greater than t, whch satsfes the key equaton (5). Ths means that the ntermedate resut at the th teraton provdes the souton of our nterest to the key equaton. Thus, smpy appyng the Eucd s agorthm unt deg [d (X)] t gves souton to the key equaton. Rest of the decodng nvoves fndng the roots of σ(x) by the Chen search. The nverses of the roots are the error ocaton numbers n a usua manner.. BLOCK TURBO CODE Concatenated codng schemes were frst proposed by Forney [5] as a method for achevng arge codng gans by combnng two or more reatvey smpe budng bock or component codes (sometmes caed consttuent codes). Turbo codes were frst ntroduced n 99 by Berrou, Gaveux, and Thtmajshma [,], where a scheme s descrbed to acheves a bt-error probabty of 0-5, usng a rate / code over an addtve whte Gaussan nose (AWGN) channe and BPSK moduaton at an Eb/N0 of 0.7 db. The codes are constructed by usng two or more component codes on dfferent ntereaved versons of the same nformaton sequence. Whereas, for conventona codes, the fna step at the decoder yeds hard-decson decoded bts (or, more generay, decoded symbos), for a concatenated scheme such as a turbo code to work propery, the decodng agorthm shoud not mt tsef to passng hard decsons among the decoders... Prncpes of Iteratve (Turbo) Decodng In a typca communcatons recever, a demoduator s often desgned to produce soft decsons, whch are then transferred to a decoder. The error-performance mprovements Fgure : Soft Input/Soft Output Decoder (for a Systematc Code) The channe LLR vaue, Lc(x), s measured by formng the ogarthm of the rato of the vaues of and for a partcuar observaton of x as shown n Fgure. The output L(dˆ) of the decoder n Fg. s made up of the LLR from the detector, L (dˆ),and the extrnsc LLR output, Le(dˆ), representng knowedge geaned from the decodng process. As ustrated n Fgure, for teratve decodng, the extrnsc kehood s fed back to the decoder nput, to serve as a refnement of the a pror probabty of the data for the next teraton. 4. PRODUCT CODE Fgure : Lkehood Functons Product codes (or terated codes) [4, ] are seray concatenated codes usng two or more short bock codes to form ong bock codes. If C (n, k, d mn ) and C (n, k, d mn )

4 54 Internatona Journa of Eectroncs Engneerng are two systematc near bock codes, then the product code P C C s obtaned by pacng (k * k ) nformaton symbos n a matrx of k rows and k coumns codng the k rows usng code C codng the n coumns usng code C The resutant product code [] P (n, k, d mn ) has n n * n, k k * k, d mn d mn * d mn and code rate s gven by R R * R. 5. SIMULATION RESULTS Fgure 4 depcts the seray concatenated RS (55, 9) and RS (55, 9) code wth a row-coumn ntereaver of ength 55 bytes and depth 9 bytes on a mage. The seray concatenated RS (55, 9) and RS (55, ) wth a row-coumn ntereaver of ength 55 bytes and depth bytes on same mage are shown n Fgure 5. The output of the nner encoder s transposed to show that SCBC wth row-coumn ntereaver s apparenty the product code. 4.. RS Product Code In our work we have used the cassca method for the constructon of RS product codes, where the nformaton symbo matrx contans k *k o q-ary nformaton symbos. The codes C and C have the same code ength n n o. The resutant code desgn scheme s easy to understand by Fgure where the span N of the ntereaver s equa to the code ength n o of the outer code and the depth D s equa to the nformaton ength k of the nner code. Fgure 4: Sera Concatenaton of RS Codes Fgure : Constructon of Product Code 4.. Decodng of RS Product Code In the present paper, the product codes based on RS component codes are decoded by sequentay decodng the rows and coumns of P by hard-decson agebrac bounded dstance decoder for optca communcaton, whch s sub-optma but has sgnfcant reduced decodng compexty compared to the optmum ML decoder. We have smuated the dfferent RS product codes on a Gaussan channe wth sequenta row by coumn hardnput/hard-output (HIHO) component decoders usng the Berekamp-Massey agorthm. Soft-decson decodng of the component RS codes [] wth SISO decoders w defntey provde addtona codng gan but the decodng compexty w be extremey hgh. Another mtng factor s the hgh data rate at whch optca communcaton systems operate. Impementng the SISO decoders at such a hgh data rate s mpractcabe. Further terated decodng of RS product codes s performed usng the component HIHO decoders. Fgure 5: RS Product Code The dfferent RS code parameters are presented n Tabe. From Tabe we observe as the code rate decreases the redundancy n the code, the mnmum dstance of the code and the asymptotc codng gan ncreases. The code rate and redundancy are nversey proportona to each other. The anaytca resuts for the canddate RS codes for BER are presented n Fgure 6. A smuatons are performed on Matab pece-wse for an output BER of 0-8 for the canddate RS codes. Tabe RS Code Parameters Canddate Code Redundancy d mn G a (db) RS(n, k) Codes Rate (k/n) (n-k)/k (%) (*t) Asymptotc RS(55, 47) RS(55, 9) RS(55, )

5 Performance Comparson of RS Code and Turbo Codes for Optca Communcaton 55 The smuated resuts presented n Fgure 7 are confrmed wth the anaytca resuts n Fgure 6. of the concatenated code es n proper seecton of the nner and the outer code to meet the ow overhead requrement. We extend the assumptons used to compute the output BER of RS codes to evauate the approxmate anaytca output BER of the concatenated RS codes. The RS product code parameters are presented n Tabe. Tabe RS Product Code Parameters Canddate RS Code Rate Redundancy (%) d mn Product Codes RR *R (n *n -k *k )/ k *k d mn * d mn Fgure 6: Theoretca Performance of the RS Codes RS (55, 9) RS (55, 9) RS (55, 9) RS (55,)mn Fgure 7: Smuated Performance of the RS Codes The approxmate anaytca output BER for RS product codes are presented n Fgure 8. The smuatons were performed pece-wse for an output BER of 0-8 on Matab. The smuated resuts for same RS product codes are presented n Fgure 9. The smuated performance w be defntey poor than the approxmate anaytca performance. Snce the decodng s a two-step decodng procedure based on HIHO component decoders, whch s sub-optma. Hence, the approxmated anaytca performance n Fgure 8 can be consdered as a ower bound on the output BER performance for the canddate RS product codes. Net eectrca codng gan (NECG) s commony used to quantfy FEC performance and ndcates an mprovement n the SNR or Q factor at the recever due to FEC. The comparson of the performance of the canddate RS codes n terms of codng gan s presented n Tabe. Tabe Codng Gan Comparson of the RS Codes at Output BER of 0-8 Canddate RS Redundancy NECG BER 0-8 (n,k) Codes (n-k)/k (%) (Theoretca) (Smuated) RS (55,47) RS (55,9) RS (55,) It can be notced from the Tabe, the ncreased redundancy n the code pays n terms of NECG. The RS (55, 9) code wth amost twce the redundancy compared to the RS (55, 47) code offers roughy more than db addtona codng gan. Approxmatey db addtona codng gan s offered by the RS (55, ) code wth amost 4.5 tmes more redundancy than the RS (55, 47) code. Concatenated codes provde an ncreased codng gan but wth ony a near ncrease n hardware cost. Good desgn Fgure 8: Approxmate Output BER Performance of RS Product Codes Tabe 4 Codng Gan Comparson of Codes wth 4% Redundancy Canddate Code Redundancy Code Rate NECG (%) (db)@ 0-8 RS (55, ) RS (55, 9) RS (55, 9)

6 56 Internatona Journa of Eectroncs Engneerng Fgure 9: Smuated Output BER Performance of RS Product Codes Further, we present the comparson n terms of net codng gan performance offered by the dfferent codng schemes wth comparabe redundancy and code rate n Tabe CONCLUSION It can be observed from Tabe 4, even though the canddate codes offer comparabe performance for the same amount of redundancy, the computatona compexty of the encoder and decoder for these canddate codes are not the same. The RS (55, ) encoder has sghty ess computatona compexty compared to RS (55, 9) RS (55, 9) encoder. Athough, the RS (55, ) decoder has hgher computatona compexty compared to the RS (55, 9) decoder, the decoder for RS (55, 9) RS (55, 9) code nvoves two component RS (55, 9) decoders and a memory eement for the dentereaver. However, the overa encoder/decoder compexty for these two codes s comparabe. From the resuts presented n Fgure 6,7,8 and 9, we concude that whe desgnng an error correctng scheme for reabe communcaton we have to tradeoff not ony redundancy aganst NECG but aso compexty aganst NECG. So, Smuaton resuts demonstrate that the bt error rate (BER) performance of the Turbo codes (concatenated RS codes) provde better random as we as burst error correcton capabty as compared to RS codes. REFERENCES [] Berrou, C. and Gaveux, A., Near Optmum Error Correctng Codng and Decodng: Turbo-Codes, IEEE Trans. on Communcatons, Vo. 44, No. 0, October 996, pp [] Berrou, C., Gaveux, A. and Thtmajshma, P., Near Shannon Lmt Error-Correctng Codng and Decodng: Turbo Codes, IEEE Proceedngs of the Int. Conf. on Communcatons, Geneva, Swtzerand, May 99 (ICC 9), pp [] E. R. Berekamp, Agebrac Codng Theory, New York, McGraw-H, 968. [4] F. J. MacWams and N. J. A. Soane, The Theory of Error Correctng Codes, Amsterdam: North Hoand, 977. [5] Forney, G. D. Jr., Concatenated Codes (Cambrdge, MA: MIT Press, 966). [6] G. Cark, J.Can, Error-Correctng Codng for Dgta Communcatons, New York, Penum Press, 98. [7] G. D. Forney, On Decodng BCH Codes, IEEE Transactons on Informaton Theory, Vo. IT-, pp , October 965. [8] I. S. Reed, X. Chen, Error Contro Codng For Data Networks, Kuwer Academc, Cop [9] ITU-T G.975, Forward Error Correcton for Submarne Appcatons, October 000 [0] N.Ramanujam, A.B. Puc, G.Lenner, H.D. Kdorf, C. R. Davdson, I. Hayee, J-X, Ca, M. Nssov, A. Ppetsk, C.Rvers, N. S. Bergano, Forward Error Correcton (FEC) Technques n Long-hau Optca Transmsson Systems, Lasers and Eectro-Optcs Socety 000 Annua Meetng. LEOS 000. th Annua Meetng. IEEE, Vo., pp [] O. A. Sab, R. Pyndah, Performance of Reed-Soomon Bock Turbo Code, Proceedng of IEEE GLOBECOM Conference, Vo. /, pp. -5, Nov [] P. Eas, Error-Free Codng, IRE Transacton on Informaton Theory, Vo. IT-4, pp. 9-7, Sept [] S.B.Wcker, Error Contro Systems for Dgta Communcaton and Storage, Upper Sadde Rver, NJ, Prentce Ha Inc., 995. [4] S.B.Wcker, Error Contro Systems for Dgta Communcaton and Storage, Upper Sadde Rver, NJ, Prentce Ha Inc., 995. [5] S. Ln, D. J. Costeo, Jr., Error Contro Codng, Engewood Cffs, NJ, Prentce Ha Inc., 98. [6] Y. Sugyama, M. Kasahara, S.Hrasawa, T.Namekawa, A Method for Sovng Key Equatons for Decodng Goppa Codes, Informaton and Contro, Vo. 7, pp , January 975.