Performance of Turbo Coded 64-QAM with Joint Source Channel Decoding, Adaptive Scaling and Prioritised Constellation Mapping

Transcription

1 CTRQ 03 : The Sixh Inernaional Conference on Communicaion Theory, Reliabiliy, and Qualiy of Service Performance of Turbo Coded 64-QAM wih Join Source Channel Decoding, Adapive Scaling and Prioriised Consellaion Mapping Tulsi Pawan Fowdur, Yogesh Beeharry and K.M. Sunjiv Soyjaudah Dep of Elecrical and Elecronic Engineering Universiy of Mauriius Redui, Mauriius p.fowdur@uom.ac.mu, yogesh536@homail.com, ssoyjaudah@uom.ac.mu Absrac Turbo coded 64-QAM sysems have been adoped by sandards such as CDMA-000 and Long Term Evoluion (LTE o achieve high daa raes. Alhough several echniques have been developed o improve he performance of Turbo coded QAM sysems, combinaions of hese echniques o produce hybrids wih beer performances, have no been fully exploied. This paper proposes a combinaion of Join Source Channel Decoding (JSCD, adapive Sign Division Raio (SDR based scaling and prioriised consellaion mapping, o improve he performance of Turbo coded 64-QAM. JSCD explois a-priori source saisics a he decoder side and SDR based scaling provides a scale facor for he exrinsic informaion as well as a sopping crierion. Addiionally, prioriised consellaion mapping explois he inheren Unequal Error Proecion (UEP characerisic of he 64-QAM consellaion and provides greaer proecion o he sysemaic bis of he Turbo encoder. Simulaion resuls show ha a Bi Error Raes (BERs above 0 -, he combinaion of hese hree echniques achieves an average gain of.5 db over a convenional Turbo coded 64-QAM sysem. However, a BERs below 0 -, he combinaion of only JSCD and SDR scaling provides an average gain of db. Keywords- Turbo Code; QAM; JSCD; SDR; Priorised Mapping. I. ITRODUCTIO Since he inspecion of Turbo codes by Berrou e.al in 993 [], several communicaion sandards have adoped his powerful near Shannon limi error correcing code. For example, Turbo coded 64-QAM sysems have been widely exploied o achieve reliable ransmission a high daa raes in several sandards such as Long Term Evoluion (LTE [],[3], CDMA 000 [4] and HomePlug Green PHY [5]. These sysems have also been repored o be promising for IEEE 80.a [6]. The major impac of Turbo codes has led o he emergence of several echniques such as Join Source Channel Decoding (JSCD [7], [8], [9], [0], exrinsic informaion scaling and ieraive deecion [], [], [3], [4], o improve is error performance and lower is decoding complexiy. Moreover, cerain characerisics of he 64- QAM consellaion have also been exploied o improve he performance of Turbo coded QAM [5]. An overview of hese echniques is given below. JSCD essenially involves he use of a-priori source saisics and he exploiaion of residual redundancy o enhance he channel decoding process. For example, Murad and Fuja [7] proposed a composie rellis, made up of a Markov source, a Variable Lengh Code (VLC, and a channel decoder s sae ransiions, o exploi a priori source saisics. A low complexiy version of he echnique in [7] was developed by Jeanne e.al [8] and more recenly Xiang and Lu [9] proposed a JSCD scheme for Huffman encoded muliple sources, which could exploi he a-priori bi probabiliies in muliple sources. Also, Fowdur and Soyjaudah [0] proposed a JSCD scheme wih ieraive bi combining, which incorporaed wo ypes of a-priori informaion, leading o significan performance gains. On he oher hand, exrinsic informaion scaling aims a improving he Turbo decoder s performance by scaling is exrinsic informaion wih a scale facor. For example, Vog and Finger [] used a fixed scale facor o improve he Max- Log-MAP Turbo decoding algorihm, while Gnanasekaran and Duraiswamy [] proposed a modified MAP algorihm using a fixed scale facor. Ineresingly hough Lin e.al [3] proposed a scaling scheme ha exended he Sign Division Raio (SDR echnique of Wu e.al [4] o adapively deermine a scaling facor for each daa block a every ieraion. Finally, he Turbo decoding process can be furher enhanced by exploiing he UEP characerisic of he 64- QAM consellaion o give more proecion o he sysemaic bis of he Turbo encoder. This echnique has been applied o LTE Turbo codes by Lüders e.al [5]. In conras wih previous works, which have mosly considered he schemes developed o improve he performance of Turbo codes independenly, his paper analyses he performance of a Turbo coded 64-QAM scheme, which inegraes hree differen echniques. Firsly, a he encoder side, prioriized consellaion mapping [5] is performed so ha he sysemaic bis oupu by he Turbo encoder are given he highes proecion when hey are mapped ono he 64-QAM consellaion. The second echnique employed is JSCD [7], [0], which explois a- priori source saisics during Turbo decoding. The final echnique used is adapive exrinsic scaling based on he SDR crierion [3]. Significan performance gains are obained for boh ieraive and non-ieraive decoding wih he combinaion of hese hree echniques. The organizaion of his paper is as follows. Secion II describes he complee sysem model. Secion III presens he simulaion resuls and analysis. Secion IV concludes he paper and liss some possible fuure works. Copyrigh (c IARIA, 03. ISB:

2 CTRQ 03 : The Sixh Inernaional Conference on Communicaion Theory, Reliabiliy, and Qualiy of Service II. SYSTEM MODEL The complee ransmission sysem is shown in Fig.. Random Alphabe Source S 0,S 0,P,P,P,P 64-QAM RVLC π Bi Reordering AWG RSC RSC Fig. Transmier wih prioriised consellaion mapping. A random alphabe source is firs generaed wih a nonencoded ino bis uniform probabiliy disribuion and hen wih he Reversible Variable Lengh Code (RVLC of [6]. The coded bis are fed o a Turbo encoder, which consiss of a parallel concaenaion of wo Recursive Sysemaic Convoluional (RSC encoders, RSC and RSC, separaed by an inerleaver,. The Turbo encoder generaes a sysemaic sream, S 0 and wo pariy sreams P and P. To achieve prioriized consellaion mapping, such ha he sysemaic bis, S 0, are placed a he mos srongly proeced poins on he 64-QAM consellaion, bi reordering [5] mus be performed afer he muliplexing process. The bi reordering is performed on a group of six bis a a ime since six bis are mapped ono one complex 64-QAM symbol. From Fig., i is observed ha afer bi re-ordering, he pariy bis S 0 occupy he firs wo posiions of he six bis ha are mapped on one symbol of he 64-QAM consellaion shown in Fig.. In his consellaion, he bis found in he firs wo posiions are mos proeced, while he bis found in he las wo posiions receive he lowes proecion. R S 0 P P M U X S 0,P,P,S 0,P,P This can be explained by considering he four major and 6 minor quadrans of his consellaion. The major quadrans are disinguished by he firs wo bis of he consellaion poin, for example, in he upper righ major quadran, he firs wo bis are 00. Hence, if he de-mapper only disinguishes beween he four quadrans correcly, he firs wo bis are correcly de-mapped. Each major quadran is divided ino four minor quadrans, which are disinguished using he 3 rd and 4 h bis of he consellaion poins. Therefore, wih bi ordering [5], he sysemaic bis S 0 receive he highes proecion while he second pariy bis, P, receive he lowes. Since he sysemaic bis of a Turbo encoder have he greaes impac on is performance, he re- of he Turbo ordering scheme improves he performance decoder. The modulaed 64-QAM symbols are hen ransmied over a complex Addiive Whie Gaussian oise (AWG channel and he corresponding received sequence is denoed by R. The complee sysem model for he receiver is shown in Fig. 3. The received symbols R are fed o a sof-oupu 64- QAM de-mapper o produce sof bis. These sof bis are hen de-muliplexed and sen for Turbo decoding. The firs Turbo decoder is modified so ha i can incorporae a-priori source saisics by combining he rellis of he Turbo decoder wih he rellis of he RVLC decoder as described in [7] and [0]. This resuls ino a composie rellis srucure wih which JSCD can be performed. Wih JSCD he compuaion of he branch ransiion probabiliy is modified. r0 Demapper Demuliplexer π r r0 r Pr{C i } π: inerleave π - : de-inerleave R ( JOIT DEC (Dec DEC ( e ( e ( e ( π - SDR SCALIG e ( Fig. 3 Turbo decoding sysem wih JSCD and SDR scaling. SDR SCALIG T T ( x x HD S r π S r π - Fig. 64-QAM consellaion wih major and minor quadrans. Copyrigh (c IARIA, 03. ISB:

3 CTRQ 03 : The Sixh Inernaional Conference on Communicaion Theory, Reliabiliy, and Qualiy of Service Assuming ha he Max-Log-MAP algorihm [7] is used, he branch meric probabiliy for he join decoder is compued as follows: ( i γ ( l', l where, ( i [ r0 0 ] [ ] log (.exp x r x p i + =.Pr{ C } i σ = log [ ] [ r0 0 ] [ ] ( x r x p i + + log[ Pr{ C }] σ γ ( l', l is he branch ransiion probabiliy from sae l o l of bi i ( i = 0 or a ime insan, p ( i is he a-priori probabiliy of bi i derived from he channel exrinsic informaion and inpu o he join (firs decoder, Pr{C i }is he a-priori probabiliy of bi i obained from source saisics, r0 and r are he de-mapped sof bis corresponding o he bipolar equivalen of he ransmied sysemaic bis, x0 and firs pariy bis, x. σ is he noise variance [0]. Wih he join decoder, he a-priori saisics, Pr{C i } can be incorporaed ino he Turbo decoding process. The derivaion of he a-priori source saisics for he RVLC source given in Table I is now explained. The RVLC decoder s bi-level rellis is shown in Fig. 4 [0]. From he bi level rellis, he probabiliy of he ransiion from sae M - = l o M = l where l, l Є (F,IA,IB,IC,ID,IE,IF, given an inpu bi i a ime insan, can be derived for all possible sae ransiions. For example, he probabiliy of he ransiion from he final sae F o he inermediae sae IA, is given by [0]: ( M IA, i = 0 M = F = PA + PB 0 ( Pr = = P For simpliciy, he sae ransiion probabiliy for any sae corresponding o bi i is denoed as Pr{C i } and he join decoder explois his probabiliy in compuing he branch meric probabiliy as per equaion ( [0]. The forward recursive variable, α ( l, a ime and sae l is compued as follows for a join decoder wih M j saes: ( max ( i α ( ' ( ', l = l + γ l l for 0 l' M j (3 TABLE I. RVLC CODEWORDS Symbol Probabily RVLC [6] A 0.33 (PA 00 B 0.30 (PB 0 C 0.8 (PC D 0.0 (PD 00 E 0.09 (PE 000 i ( Fig. 4. Bi level rellis of RVLC decoder [0]. The number of saes of he join decoder, M j is greaer han he number of saes, M s of he second decoder (DEC, because he join decoder is obained by merging he saes of he RVLC decoder wih he saes of he Turbo decoder as described in [0]. The backward recursive variable, β ( l, is compued as follows: β ( l max β ( l, l' for ( i = ( l' + γ 0 l' M j The Log-Likelihood Raio (LLR, a ieraion r and ime for he join decoder is compued as follows: = max ( l' + γ ( ( l', l + β ( l (0 max ( ' ( ', ( l + γ l l + β l for 0 l' M j The exrinsic informaion e a ieraion r and ime for he join decoder is compued as follows: ( r e = r0 e (6 σ ( r where, e is he de-inerleaved exrinsic informaion obained from he second decoder a ieraion r-. The exrinsic informaion, e and he LLR, are hen sen o a SDR scaling mechanism, which compues a scale facor S r as follows: (, ( S r = f e = (4 (5 (7 Copyrigh (c IARIA, 03. ISB:

4 CTRQ 03 : The Sixh Inernaional Conference on Communicaion Theory, Reliabiliy, and Qualiy of Service where, f ( e, = if e same sign, oherwise f (, ( and have he e =0. is he frame size in bis. When S r akes is maximum value of.0, he swich T is opened, he ieraive decoding process is sopped and a hard decision is made on. However, when Sr is less han one, T remains closed and he exrinsic informaion e is scaled wih S r and inerleaved o obain e. Hence, he SDR scaling mechanism acs boh as a sopping crierion and a scale facor generaor. The mechanism is derived from he one proposed in [3], bu, in his work e and are used o compue he scale ( r facor and no e and e. Anoher difference is ha in his work only he exrinsic informaion has been scaled and no he sof channel inpus, as was he case in p i [3]. The a-priori probabiliy, (, is compued as follows and sen o decoder : ( r exp( e ( r + exp( ( ( = p i e ( r + exp( e for i = for i = 0 The branch meric probabiliy for he second decoder is compued as follows: [ ] [ r0 x0 ] + [ r x ] p ( i σ ( i γ ( l ', l = log (8 where, r is he de-mapped sof bis corresponding o he bipolar version of he ransmied second pariy bis x and r0 is he inerleaved counerpar of r0. The forward and backward recursive variable, α ( l and β ( l a ime and sae l are compued as follows: ( max ( i α ( ' ( ', l = l + γ l l for 0 l' M s (9 ( max ( i β ( ' (, ' l = β l + γ l l for 0 l' M s (0 The LLR, and exrinsic informaion, e a ieraion r and ime are compued as follows: (7 ( = max ( ' ( ', ( l + γ l l + β l (0 max ( ' ( ', ( l + γ l l + β l for e = r0 e σ The scale facor S r is compued as follows: (, ( S r = f e = where, f (, ( 0 l' M j ( ( (3 e = if e and have he same sign. Finally, he a-priori probabiliy, p ( i, is compued as per equaion (7 bu using e. If S r =.0, T is opened o sop he ieraive decoding process and a hard decision, (HD is made on. The combinaion of prioriized consellaion mapping, JSCD and adapive scaling cerainly lead o an enhanced Turbo coded 64-QAM sysem, bu a he cos of greaer compuaional complexiy and delay. The complexiy increase due o he bi re-ordering scheme is negligible and may even be inegraed wih he muliplexer. JSCD on he oher hand leads o he greaes increase in complexiy and delay because as menioned previously he join decoder is obained by merging he saes of he RVLC decoder wih he saes of he Turbo decoder. The number of compuaions involved in compuing S r and S r o perform adapive scaling also increase he delay. However, his is compensaed by he faser convergence achieved wih he use of he scale facor and he possibiliy of sopping he ieraive decoding process once convergence is achieved. This prevens he decoder from performing unnecessary ieraions. III. SIMULATIO RESULTS AD AALYSIS The performances of he following four Turbo coded 64- QAM schemes are compared: Scheme The Turbo coded 64-QAM sysem wih JSCD, adapive scaling and prioriised consellaion mapping. The encoding and decoding frameworks are given in Fig. and Fig. 3, respecively. Scheme - This scheme only uses prioriised consellaion mapping. The encoding is as per Fig., bu he decoding does no include JSCD or adapive scaling. Scheme 3 This scheme uses JSCD and adapive scaling and is decoder is similar o Scheme. However, prioriised consellaion mapping is no performed, as such, he bi reordering block of he encoder shown in Fig. is omied. Copyrigh (c IARIA, 03. ISB:

5 CTRQ 03 : The Sixh Inernaional Conference on Communicaion Theory, Reliabiliy, and Qualiy of Service Scheme 4 This scheme is a convenional Turbo coded 64-QAM sysem wihou prioriised consellaion mapping, JSCD and SDR scaling. In all simulaions, a random alphabe source wih he probabiliy disribuion given in Table I has been used. Afer generaing he alphabes, hey are grouped ino packes of size P = 64 symbols. The packes are hen Reversible Variable Lengh Coded o obain an RVLC bi-sream as shown in Fig.. ormally, he lengh in bis, L, of each packe is ransmied as side-informaion because L is differen for each packe. The packeizaion is imporan o preven error propagaion. The RVLC bi-sreams of all packes are grouped ino blocks of 4056 bis since an inerleaver size of 4056 bis has been used in he simulaions. The parameers for he Turbo code used are as follows: Generaor: G = [, g/g], where g0 = 7 and g = 5 in Ocal. Inerleaver size, = 4056 bis. Maximum number of ieraions, I =. Code-rae = /3 and channel model: Complex AWG. The graphs of Bi Error Rae (BER as a funcion of Eb/o have been ploed separaely over a low Eb/o range: 0 db Eb/o 3 db and a high Eb/o range: 3.5 db 0 Eb/o 6.5 db in seps of 0.5 db. Eb/o is he raio of he bi energy, Eb o he noise power specral densiy, o. I is o be noed ha he ransiion from he low Eb/o range o he high Eb/o range is essenially a coninuiy from 3 db o 3.5 db and up o 6.5 db. The performance analysis has also been made for boh ieraive and non-ieraive decoding. Fig. 5 shows he graph of BER agains Eb/o for ieraive decoding over he low Eb/o range. I is observed ha he Turbo coded 64-QAM sysem wih JSCD, adapive scaling and prioriised consellaion mapping (Scheme provides he bes performance wih an average gain of.5 db for BER > 0 - over he convenional Turbo coded sysem (Scheme 4. A an Eb/o of db, Scheme also provides a gain of abou.5 db over Scheme 3, which does no employ prioriised consellaion mapping. Moreover, Scheme, which uses only prioriised consellaion mapping ouperforms Scheme 3 by db a an Eb/o = db. I is o be noed from a heoreical poin of view he performance of he sysem for a BER > 0 - is imporan because i is revealing a new characerisic of he sysem whereby i is seen ha significan gains can be obained in his BER region using he proposed echnique. However, from a pracical poin of view he performance of he sysem for BER > 0 - is no really relevan. Fig. 6 shows he graph of BER agains Eb/o for ieraive decoding over he high Eb/o range. In his range i is observed ha prioriised consellaion mapping is no beneficial. For example, Scheme provides he wors performance while he performance of Scheme is comparable o ha of Scheme 4. A possible explanaion is ha wih ieraive decoding, in he high Eb/o range, convergence akes place. As such, giving more proecion o he sysemaic bis does no provide furher gains. Also, since lower proecion has been given o he pariy bis, his can lead o performance degradaion. Over his Eb/o range, Scheme 3 which uses only JSCD and adapive scaling provides he bes performance wih an average gain of db in Eb/o over Scheme and Scheme 4. I is o be noed ha Scheme 3 ouperforms Scheme over his high Eb/o range because Scheme suffers from a performance loss, which resuls from he use of prioriised consellaion mapping afer convergence. Fig. 7 shows he graph of average number of ieraions versus Eb/o over he range 3 db Eb/o 6.5 db. Scheme and Scheme 3, which employ SDR based scaling wih a sopping crierion, show a progressive decrease in he number of ieraions required as he Eb/o increases. For example a an Eb/o of 5.5 db, Scheme 3 consumes six ieraions less han Scheme and Scheme 4. However, Scheme consumes on average.5 ieraions more han Scheme 3 due o performance loss as a resul of using prioriised mapping afer convergence. The number of ieraions required by Schemes and 4 remains fixed a. Fig. 5. Ieraive low Eb/o performance wih = Fig. 6. Ieraive high Eb/o performance wih = Copyrigh (c IARIA, 03. ISB:

6 CTRQ 03 : The Sixh Inernaional Conference on Communicaion Theory, Reliabiliy, and Qualiy of Service Fig. 7. Average number of ieraions vs Eb/o for = Fig. 8 shows he graph of BER agains Eb/o for nonieraive decoding over he low Eb/o range. I is observed ha Scheme ouperforms all he oher schemes wih an average gain of.5 db over Scheme 4 and db over Scheme 3. However, wih non-ieraive decoding, convergence does no ake place and he use of prioriized consellaion mapping does no lead o degradaion a BERs below 0 -. This is observed in Fig. 9 whereby Scheme ouperforms Scheme 3 by 0.5 db on average and Scheme 4 by almos.5 db. I is o be noed ha in [5], whereby only bi-reordering was used, i was also observed ha wih nonieraive decoding a performance gain is obained hroughou he Eb/o range whereas wih ieraive decoding convergence akes place a a cerain poin. Hence when prioriized consellaion mapping is used he ieraive scheme does no presen a similar relaion as he non-ieraive. Fig. 8. on-ieraive low Eb/o performance wih = Fig. 9. on-ieraive high Eb/o performance wih = IV. COCLUSIO AD FUTURE WORK This paper proposed an efficien Turbo coded 64-QAM scheme wih JSCD, adapive scaling and prioriised consellaion mapping. A he encoder side a re-ordering mechanism is used o map he sysemaic bis of he Turbo encoder on he mos srongly proeced poins of he 64- QAM consellaion. To enhance he decoding process, JSCD is used o incorporae a-priori source saisics and adapive SDR based scaling is also performed. A BERs above 0 -, he proposed scheme provides a significan performance gain of.5 db wih ieraive decoding over a convenional Turbo coded scheme. For BERs below 0 -, he use of prioriised consellaion mapping degrades performance as a resul of convergence. Hence, for BERs below 0 -, i is preferable o use only JSCD and SDR scaling, which achieves a gain of db on average over a convenional Turbo coded scheme. However, wih nonieraive decoding, he proposed scheme, ouperforms a convenional Turbo coded scheme a all BERs because here is no performance degradaion due o prioriised consellaion mapping. Overall, he combinaion of prioriised consellaion mapping wih JSCD and SDR based scaling appears promising for Turbo coded 64-QAM sysems. Several ineresing fuure works can be envisaged from he scheme proposed in his work. A sraighforward exension would be o assess is suiabiliy for communicaion sysems such as LTE. A more challenging fuure work would be o use JSCD schemes, which are less complex and hence do no incur significan delays while sill allowing he exploiaion of source saisics. The prioriised consellaion mapping scheme could also be opimised so ha performance gains could be obained in he high Eb/o range also. Finally, invesigaions could be made on how o exend he scheme o block Turbo codes and also on he possibiliy of using bi inerleaved coded modulaion. Copyrigh (c IARIA, 03. ISB:

7 CTRQ 03 : The Sixh Inernaional Conference on Communicaion Theory, Reliabiliy, and Qualiy of Service ACKOWLEDGMET The auhors would like o hank he Universiy of Mauriius for providing he necessary faciliies for conducing his research as well as he Teriary Educaion Commission of Mauriius. REFERECES [] C. Berrou, A. Glavieux and P. Thiimajshima, ear Shannon limi error-correcing coding and decoding: Turbo-codes, IEEE Inernaional Conference on Communicaions, ICC 93. Geneva, vol., 3-6 May 993, pp [] S. Sesia, I. Toufik and M. Baker, LTE The UMTS Long Term Evoluion: From Theory o Pracice, John Wiley & Sons, Ld, 009, ISB: [3] 3GPP, Technical Specificaions Rel.8., 009. [4] 3GPP C.S004-B, CDMA 000 High Rae Packe Daa Air Inerface Specificaion Version.0, May 006. Available Online: hp:// B_v.0_0605.pdf [Accessed: ovember 0]. [5] J. Zyren, Home Plug Green PHY Overview, Technical Paper, Aheros Communicaions, 00. [6] I. Lee, C.E.W. Sundberg, S. Choi and W. Lee, A modified medium access conrol algorihm for sysems wih ieraive decoding, IEEE Transacions on Wireless Communicaions, vol. 5(, 006, pp [7] A.H. Murad and T.E. Fuja, Join source channel decoding of variable lengh encoded sources, Proceedings of he Informaion Theory Workshop (ITW. Killarney, Ireland, June. 998, pp [8] M. Jeanne, J.C. Carlach and P. Siohan, Join source channel decoding of variable lengh codes for convoluional codes and urbo codes, IEEE Trans Commun vol. 53(, 005, pp.0 5. [9] W. Xiang and P. Lu, Bi-Based Join Source-Channel Decoding of Huffman Encoded Markov Muliple Sources, Journal of eworks, vol. 5(4, 00, pp [0] T.P. Fowdur and K.M.S. Soyjaudah Performance of join source channel decoding wih ieraive bi combining and deecion, Ann. Telecommun. vol. 63, 008, pp [] J. Vog and A. Finger, Improving he MAX-Log-MAP Turbo decoder, Elecr. Le., vol. 36, no. 3, ov. 000, pp [] T. Gnanasekaran and K. Duraiswamy, Performance of Unequal Error Proecion Using MAP Algorihm and Modified MAP in AWG and Fading Channel, Journal of Compuer Science, vol. 4 (7,008, pp [3] Y. Lin, W. Hung, W. Lin, T. Chen, E. Lu, "An Efficien Sof- Inpu Scaling Scheme for Turbo Decoding," IEEE Inernaional Conference on Sensor eworks, Ubiquious, and Trusworhy Compuing Workshops, vol., 006, pp [4] Y. Wu, B. Woerner, and J. Ebel, A Simple Sopping Crierion for Turbo Decoding, IEEE Commun. Le., vol. 4, no. 8, Aug. 000, pp [5] H. Lüders, A. Minwegen, and P. Vary, Improving UMTS LTE Performance by UEP in High Order Modulaion, 7h Inernaional Workshop on Muli-Carrier Sysems & Soluions (MC-SS 009, Herrsching, Germany, 009, pp [6] Y. Takishima, M. Wada, and H. Murakami, Reversible variable lengh codes, IEEE Trans. on Commun., vol. 43, 995, pp [7] B. Vuceic and J. Yuan, Turbo Codes: Principles and Applicaions, Kluwer Academic Publishers, 000. Copyrigh (c IARIA, 03. ISB: