DC-Free Turbo Coding Scheme Using MAP/SOVA Algorithms

Transcription

1 Proceedngs of the 5th WSEAS Internatonal Conference on Telecommuncatons and Informatcs, Istanbul, Turkey, May 27-29, 26 (pp DC-Free Turbo Codng Scheme Usng MAP/SOVA Algorthms Prof. Dr. M. Amr Mokhtar Electrcal Engneerng Department, Alexandra Unversty, Egypt. Abstract :- A useful tool n the desgn of relable dgtal communcaton systems s channel codng. Turbo codes have been shown to yeld an outstandng codng gan close to theoretcal lmts n the Addtve Whte Gaussan Nose (AWGN channel. In ths paper, a novel DC-free turbo codng scheme usng Maxmum A Posteror algorthm (MAP/ Soft Output Vterb Algorthm (SOVA for the turbo decoder s presented. The proposed scheme acheves the DC-free codng and errorcorrectng capablty smultaneously. The scheme has a smple cascaded structure of the runnng dgtal sum (RDS control encoder and the turbo encoder. A gven sequence becomes DC-free f and only f the absolute RDS value of the sequence s bounded by a constant for any tme nstant. The RDS control encoder generates a sequence whch gves the turbo coded sequence wth a bounded RDS value. The structure allows us to explot effcent soft-decson decodng whch attans addtonal codng gans compared wth hard-decson decodng over an addtve whte Gaussan nose (AWGN channel. Key-Words: - Turbo Code DC-free Codes MAP algorthm SOVA algorthm Convolutonal Code 1 Introducton DC-free codes can be used n wred transmsson systems to decrease the degradaton due to the use of couplng components and/or solatng transformers [1],n magnetc-tape recodng to prevent wrte-sgnal dstorton caused by transformer couplng n the wrte electroncs [2], n optcal recodng to reduce nterference between servo sgnals and data [2], and n wreless systems to assst wth the nserton of plot tones [3]. In these applcatons, errorcontrol (EC codes are also employed to overcome mparments n the channel. The conventonal method of ncorporatng both EC codng and dc-free codng nto a dgtal communcaton system s through concatenaton of an EC code as the outer code and a dc-free code as the nner code [4]. In general, dc-free codes have lttle or no EC ablty, and ther decoders exhbt error extenson, expect bnary nput, and output hard decsons. To avod the mpact of error extenson of dc-free decodng on a subsequent EC decoder, and to enable the use of soft decson decodng and teratve algorthms durng EC decodng, t s desred that n the recever, dc-free decodng follow EC decodng [4]. Concatenaton schemes wth a partally reversed order of conventonal dc-free and EC codng have been proposed n [5], [6]. Constructons for some ntegrated bnary dc-free EC codes have been presented n [7] [8]. In [9], a method was ntroduced for ntegratng dc-free codes and EC block codes that fully reverses the conventonal order of dc-free decodng and EC decodng. There are several works on DC-free errorcorrectng codes based on convolutonal codes. Deng, L, and Herro [1] presented a DC-free error-correctng convolutonal codng technque. In ther method, the all 1 s vector n the generator matrx of a convolutonal code s exploted to control the runnng dgtal sum (RDS of encoded sequences. Nasr-Kenar and Rushforth [11] nvestgated DC-free subcodes of convolutonal codes. Recently, Chu [12] showed DC-free error-correctng codes based on convolutonal codes. In Chu s scheme, a codeword of a convolutonal codes wth a small RDS value s chosen wth the Vterb algorthm. These methods seem promsng and further nvestgaton on bnary DC-free codng schemes wth a smple trells structure, or equvalently, wth small decodng complexty s hoped for.

2 Proceedngs of the 5th WSEAS Internatonal Conference on Telecommuncatons and Informatcs, Istanbul, Turkey, May 27-29, 26 (pp In ths paper, we present a novel DC-free Turbo codng scheme wth an error correctng capablty. Fg. 1 represents the archtecture of our proposed scheme. Frst, the user message sequence (u u 1 u 2 s encoded to the ntermedate sequence by an RDS control encoder. The convolutonal encoder then converts an ntermedate sequence to the coded sequence (x x 1 x 2. After the ordnary bnary-bpolar converson, the coded sequence s transmtted over a nosy channel such as the addtve whte Gaussan nose (AWGN channel. The term RDS means the runnng dgtal sum of a (bpolar coded sequence. It s well known that the DC-free property s acheved f and only f the absolute value of the RDS s bounded by a constant value for any tme nstant [13]. The RDS control encoder must generate an ntermedate sequence whch gves a coded sequence wth a desred RDS constrant. In other words, an ntermedate sequence should be determned n such a way that t generates coded sequences wth a bounded RDS. Fg. 1 presents the cascaded structure of the RDS control encoder/ decoder and the turbo encoder/decoder. Wth ths archtecture, we are able to explot soft-decson decodng wth the Vterb algorthm. Moreover, the dashed box part n Fg. 1 s exactly dentcal to a turbo codng system. The RDS control encoder and decoder can be regarded as a front-end and a back-end of the turbo codng system. Thus, we can use a ready-made CODEC to mplement the proposed scheme. The proposed scheme s based on the followng three major deas: 1 addtve encodng usng a bnary lnear block code, 2 upper and lower bounds on the RDS for an addtve encoder, and 3 splttng a convolutonal code nto nfnte sequences of a lnear block code, whch s called a wndow code. In the followng, we shall explan these deas n order. 2. DC-Free Codng Scheme Based On An Addtve Encoder In ths secton, frstly, the necessary notatons and defntons wll be ntroduced. Then, a DCfree codng scheme based on an addtve encoder s presented. The scheme has a close relatonshp to the dea of addtve codng. 2.1 Notaton and Defnton n For v = (v,v 1,v 2,... v n-1 {,1}, the vector RDS of v s gven by S( v n 1 j = f ( v j (1 The bnary- bpolar converson mappng f s defned by 1 L f ( L = + 1 L = 1 (2 The upper and lower RDS of v are defned by U( v max f ( v t n 1 j j = t Lv ( mn f( v t n 1 j j = t (3 (4 For bnary lnear block code C ( n,k,d where n, k, and d denote the length, the dmenson, and the mnmum dstance, respectvely. If two bnary lnear codes C and C 1 satsfy and C = { c c : c, C, c C} C C = I 1, then the par of codes (Co, C 1 s called the drect sum decomposton of C. The code C s called the drect sum code based on C and C 1. Let k and k 1 be the dmensons of C and C 1, and G and C 1 be the generator matrces of C and C 1, respectvely. From the defnton above, t s obvous that the equalty k = k + k 1 holds. Assume one to one mappngs called encodng mappngs: ψ : F K 2 ψ : F C C k where F 2 s the Galos feld wth two elements {, 1} and the addton over F 2 s denoted by. 3 Addtve Encoder Assume an nfnte length bnary message sequence {a, a 1...}. Each vector a (=, 1, 2... F K 1 2 belongs to. An addtve encoder encodes a message block a to c c for each block ndex. The code C s a bnary lnear code of length n. The resultng sequence {c a, c 1...} s called a coded sequence. The addtve encoder appends redundancy k =k-k 1 bts per block and

3 Proceedngs of the 5th WSEAS Internatonal Conference on Telecommuncatons and Informatcs, Istanbul, Turkey, May 27-29, 26 (pp thus the codng rate becomes k 1 /n. After the bnary-bpolar converson, the bpolar sequence {f (C, f (c 1...} s transmtted over the-nosy channel. For achevng DC-free transmsson, the addtve encoder has to generate the coded sequence wth a RDS constrant. An addtve encoder encodes a message block a, nto c, n such a way: ψ ( b ψ1 ( a C = (5 F k 2 where b F s called control vector. In other words, the addtve encoder has freedom to select a control vector and should specfy a control vector so as to obtan a code sequence whch keeps the RDS value bounded. 4. Proposed DC-Free Error Correctng Codes 4.1. Introducton. Assume that a turbo code C together wth the parameterα, β, γ, and a decomposton matrx M are gven. The code C s called the base turbo code. The followng s the detal of the DC-free turbo codng such as encodng and decodng, 4.2. Encodng The message sequence (u u 1 u 2 s dvded nto blocks of length β. The -th (=, 1, 2 message block s denoted by The message sequences are encoded to the ntermedate sequences by the RDS control encoder. The ntermedate sequence (x x 1 x 2 s dvded nto the ntermedate block of length L. The -th ntermedate block s defned by u u, u, u, Κ, u = ( β β + 1 β + 2 ( + 1 β 1 where O s the frst pm-tuple of x and n s the last γ+ β-tuple of x such that x (x,x, x = (γ+ β (γ+ β+ 1 Κ (γ+ β+ L 1 x = ( \ n We obtan a coded sequence (y o y 1 by encodng the ntermedate sequence wth the turbo encoder. The coded sequence s dvded nto the coded blocks of length r. The -th ( =, 1, 2,Κ coded block has the form y = ( yr + qm, yr qm, Κ yr ( qm y = ( y qm, Κ, y r 1+ qm Note that the relaton between the message, ntermedate and coded sequences s shown n fgure 2 Notce that the ntermedate blocks x and x +1 are overlappng. The overlappng part corresponds to o. By applyng the addtve encoder to a wndow code, the overlappng s taken nto account. Wthn the ntermedate block x, only the vector n can be assgned freely wthout any nfluence of the prevous block. The overlappng part o s determned by the prevous ntermedate block x -1. As shown n fgure 2, the RDS control encoder adds redundancy (a control vector to the message sequence and thus the codng rate defned between the message and ntermedate sequence becomes β / ( γ + β. The turbo encoder appends redundancy to the ntermedate sequence. Consequently, the overall rate becomes R ( pβ /(q( γ + β. The rate loss can be consdered as a prce for obtanng a RDS constrant Decodng The decodng ssue for the proposed scheme s dscussed n ths secton. The receved sequence s frst decoded by the MAP/SOVA decoder for the base turbo code C. let the set of all allowable sequences generated by the proposed scheme be C RDS. The mnmum free Hammng d free dstance defned on C RDS s denoted by. From the cascaded structure of the proposed scheme, evdently, C s contaned n C and d free RDS the nequalty dfree holds. The symbol d free denotes the mnmum free hammng dstance of C. As a consequence of ths property, we can use the MAP decoder for the base Turbo code to decode C RDS. It can be consdered as a knd of a super code decodng. The decodng of the ntermedate sequence s straght forward from the defnton of the RDS control encoder. Let b *, n correspondng to u and b *, be the estmated blocks n u and respectvely. Multplyng the nverse matrx of M to left, we have 1 ( * n M = b / u (6 n / from

4 Proceedngs of the 5th WSEAS Internatonal Conference on Telecommuncatons and Informatcs, Istanbul, Turkey, May 27-29, 26 (pp u Where output block. s the -th estmated message 5. Smulaton Results As dscussed before, DC-free codes are usually requred n dgtal transmsson and recordng systems to reduce the effect of baselne wander and match spectra of the transmtted sgnals to frequency characterstcs of the transmsson meda. A gven sequence becomes DC-free f and only f the absolute Runnng Dgtal Sum (RDS value of the sequence s bounded for any tme nstant. Fg.3. shows degree of performance mprovement accordng to RDS value of the sequence. The smulaton results shows that turbo code whch has bounded runnng dsparty of code sequence gves better performance than turbo code whch has sequence of all ones or all zeros or any other sequence that does not have RDS=. In ths secton, smulaton results of BER versus E b /N o were plotted to show the nfluence of varous parameters. The channel model used s An (AWGN channel. The DC-Free Turbo decoder s used n an teratve fashon untl we acheved the 7 th teraton. The component decoder based on the MAP algorthm. Fg. 4 shows the performance mprovements when we apply the dc-free property to the turbo code usng MAP decoder algorthm and 3 rd teratons wth rate 1/3, frame sze 4 bts and 64-state DC-free codng scheme wth the overall rate 6/16 and the mnmum free dstance 1 has been obtaned compared wth turbo code usng decoder based on SOVA algorthm wth the same specfcatons. Fg. 5 shows the performance mprovements when we apply the dc-free property to the turbo code usng MAP decoder algorthm and 3 rd teratons wth rate 1/3, frame sze 4 bts and 64-state DC-free codng scheme wth the overall rate 6/16 and the mnmum free dstance 1 has been obtaned compared wth covolutonal code wth 8-state dc-free codng scheme wth the overall rate 3/7 and the mnmum free dstance 5 has been obtaned and ths scheme satsfes a bounded RDS constrant (from -9 to +9. Ths scheme satsfes a bounded RDS constrant (from 18 to Influence of number of teratons In fg 6. BER verses E b /N o curve s shown parameterzed by the number of decodng teratons. The results show that, the BER decreases as the number of teratons ncreases for the same frame sze and code rate Influence of Frame Sze In fg 7., the smulated performance results of DC-free turbo codes wth the same component code but dfferent frame sze s shown parameterzed by the frame sze. We can notce that at the same E b /N o DC-Free Turbo code gves the most effcent BER than the Turbo code. 6 Concluson A new constructon of DC-free codes based on turbo codes whch can smultaneously meet the dc constrant and error-correctng requrement s proposed. The presented scheme dvded nto two parts: the RDS control encoder/decoder and the turbo encoder/decoder. The RDS control encoder generates several codewords of a wndow code for selectng a control vector. The decodng requres smpler tasks than the encoder, also, BER performance for DC-free turbo code s nvestgated for many dfferent cases: performance of 7 decodng teratons for fxed code rates and constrant lengths but dfferent frame szes, performance of 7 decodng teratons for fxed frame szes but dfferent code rates, and performance mprovement between 1 decodng teraton and 7 decodng teratons for fxed code rates and constrant lengths but dfferent frame szes. References: [1] K. W. Cattermole, Prncples of dgtal lne codng, Int. J. Electron., pp. 3 33, July [2] K. A. S. Immnk, P. H. Segel, and J. K. Wolf, Codes for dgtal recorders, IEEE Trans. Inform. Theory, pp , Oct [3] A. Kokkos, A. Popplewell, and J. J. O Relly, A power effcent codng scheme for low-frequency spectral suppresson, IEEE Trans. Commun., pp , Nov [4] J. L. Fan and Calderbank, A modfed concatenated codng scheme wth applcatons to magnetc data storage, IEEE Trans. Inform.

5 Proceedngs of the 5th WSEAS Internatonal Conference on Telecommuncatons and Informatcs, Istanbul, Turkey, May 27-29, 26 (pp Theory, pp , July [5] W. G. Blss, Crcutry for performance error correcton calculatons on baseband encoded data to elmnate error propagaton, IBM Tech. Dscl. Bull., pp , [6] K. A. S. Immnk, A practcal method for approachng the channel capacty of constraned channels, IEEE Trans. Inform. Theory, pp , Sept [7] R. H. Deng and M. A. Herro, DC-free coset codes, IEEE Trans. Inform. Theory, pp , July [8] I. J. Far and D. R. Bull, DC-free error control codng through guded convolutonal codng, n Proc. 22 IEEE Int. Symp. Informaton Theory, 22, p [9] F. Zha, Y. Xn, and I. J. Far, DC-free multmode error control block codes, n Proc. IEEE Int. Symp. Informaton Theory, 23, p. 76. [1] R. H. Deng, Y. X. L, and M. A. Herro, DC-free error correctng convolutonal codes, Electron. Lett., vol. 29, pp , [11] M. Nasr-Kenar and C. K. Rushforth, A class of DC-free subcodes of convolutonal codes, IEEE Trans. Commun., vol. 44, pp , [12] M. C. Chu, DC-free error correctng codes based on convolutonal codes, n Proc. 2 IEEE Int. Symp. Informaton Theory, Sorrento, Italy, 2, pp [13] K. A. S. Immnk, Codng Technques for Dgtal Recorders. Englewood Clffs, NJ: Prentce-Hall, 1991.

6 Proceedngs of the 5th WSEAS Internatonal Conference on Telecommuncatons and Informatcs, Istanbul, Turkey, May 27-29, 26 (pp