The Analysis and the Performance Simulation of the Capacity of Bit-interleaved Coded Modulation System

Transcription

1 Sensors & Transdcers, Vol. 79, Isse 9, September 4, pp Sensors & Transdcers 4 by IFSA Pblshng, S. L. The Analyss and the Performance Smlaton of the Capacty of Bt-nterleaved Coded Modlaton System Hongwe ZHAO, Mengmeng Sh, Jnfang Wang School of Electroncs and Informaton, Northwestern Polytechncal Unversty, X an 79, Chna X an FengHo Electronc Technology Co., Ltd., X an 775, Chna Tel.: E-mal: Hongv_zhao@6.com Receved: 9 Agst 4 /Accepted: 9 September 4 /Pblshed: 3 September 4 Abstract: In ths paper, the capacty of the system over AWGN channels s frst analyzed; the crves of capacty verss SNR are also got by the Monte-Carlo smlatons, and compared wth the crves of the CM capacty. Based on the analyss reslts, we smlate the error performances of system wth LDPC codes. Smlaton reslts show that the capacty of system wth LDPC codes s enormosly nflenced by the mappng methods. Gven a certan modlaton method, the system can obtan abot -3 db gan wth Gray mappng compared wth Non-Gray mappng. Meanwhle, the smlaton reslts also demonstrate the correctness of the theory analyss. Copyrght 4 IFSA Pblshng, S. L. Keywords: Channel capacty, Bt-nterleaved coded modlaton, Coded modlaton, LDPC codes, Gray mappng.. Introdcton Bt-nterleaved coded modlaton (, whch composed of codng, bt-nterleaved, constellaton mappng, was orgnally proposed by Zehav n 99 [], and then, frther research was carred ot []. In the system, codng and modlaton s dvded by bt-nterleaved modles, so they can be desgned ndvdally, makng the overall desgn of the system easy and flexble. In the recevng termnal, demodlator gets the nformaton of bt based on the receved data, and otpts the fnal reslt by de-nterleavng and decodng [3, 4]. Coded modlaton (CM system s smlar to the. The trells coded modlaton (TCM, whch was proposed by Ungerboeck n 98, maxmze the Ecldean dstances between dfferent ways by desgn ontly the codng and modlaton and set partton of the constellaton ponts, whch mproved the performance. From the pont of vew of the capacty of channel, n fadng channels, the performance of s better than that of the trells coded modlaton (TCM and symbol-nterleaved coded modlaton (SICM. However, n the addtve whte Gassan nose (AWGN channel, the performance of s slghtly worse than that of CM [], the dfference can be gnored nder hgh sgnal-to-nose rado envronment. Owe to the great robstness, has become an ndstry standard n some felds of wreless commncaton, sch as HSPA, IEEE8.a/g and IEEE8.6e. Low-densty party-check code (LDPC s a knd of lnear block code whch has been wdely sed n the Date transmsson and storage Systems. LDPC 5

2 Sensors & Transdcers, Vol. 79, Isse 9, September 4, pp was frst proposed by Gallager n 963 n hs PHD dssertaton. Nevertheless, de to the lmt of hardware at that tme, an avalable LDPC decoder s hard to be obtaned, whch leads to that lttle mportance has been attached to LDPC for 3 years. In 993, the fndng of the Trbo codng arosed researchers nterest. And some researchers fond that for reglar LDPC, when adoptng sm-prodct algorthm (SPA, the decodng performance s smlar to that of Trbo code, and t even does better than Trbo n long code. The LDPC has low error floor and s able to realze parallel comptng, meanwhle, t s easer to decodng than Trbo, makng t be stable for the hardware desgn and havng the potental to decodng n hgh speed. The paper analyzes the nformaton between every bt and receved nmbers n mappng symbols frstly, then, the expresson of the capacty of s addressed, and fnally, the Monte Carlo Smlaton of based on the LDPC s gven and compares t wth the based on the RS nder the same condtons. The smlaton reslts show that the method of mappng of constellaton deeply has a great effect on the performance of the, the performance n Gray mappng s obvosly better than that n Non-Gray mappng. Meanwhle, wth the growth of the LDPC length, the performance of the wll get mproved closed to the lmt of the capacty of the. At the same tme, the reslts also demonstrate the valdty of theory analyss.. Model The model s st as shown n Fg.. Frstly, encoder codes the data flow U = (,,,, k prodced by nformaton sorce by sng the lnear block code whose parameter s (n, k, the length of the data flow U s k bts, and the otpt s V = ( v, v, v, vn, where, v {,}, k, n. v Π( s N û L(Λ Π ( Y Fg.. model. Then, nterweave and modlate the V and get the otpt S = ( s, s, s, s q -, where s and q are depended on the methods of modlaton and constellaton. Assme the constellaton, n whch the set of symbols s {A}, contans A = M symbols and each symbol can be expressed by M bts, that s, M f s = ( b, b, b, b, there wll be q = n/ M. Wthot loss of generalty, takng -dmensonal modlaton nto consderaton, the Fg. shows the constellatons correspondng to the Gray mappng and the Non-Gray mappng nder 8PSK and 6QAM modlaton. In Fg., the φ and φ denote a par of orthogonal bass. After the transmsson of symbol seqence S n AWGN channel, the recevng termnal gets a complex seqence Y = ( y, y, y, y q -, where the component y = s + n s the sperposed sgnal of the R I symbol s and nose, n = n + n s the component R of the nose vector N = ( n, n, n, nq, n and I n denote the real part and magnary part of the complex nose respectvely. And both of them are ndependent of each other and obey the Gass N dstrbton wth mean and σ = varance. Demodlaton modle detects the maxmm a posteror (MAP probablty for the sperposed sgnal, then gets the log-lkelhood rato (LLR vector L(/\ by de-nterleaver, fnally, the decoder complete the ntalzes and decodng. The otpt of the decoder s Û. It shold be noted that the nterleavng modle and de-nterleavng modle can be gnored when the LDPC s sed as forward error correcton code n system. 3. Capacty Analyss of For smplcty, we focs on the capacty of the nder the condtons where the otpttng of nformaton sorce s bnary and symbol eqprobablty. That s, the pror probablty px ( = = px ( = =.5 and for any ways of M modlaton, ps ( k =,( k -. M 5

3 Sensors & Transdcers, Vol. 79, Isse 9, September 4, pp ϕ ( ( 3 ( ( 5 ( ( ( ϕ x ( ( ϕ ( ( 3 ( 4 ( ϕ ( x ( 5 7 ( 6 ( ( ϕ (a (b ϕ ( 8 ( ( 4 ( ( 6 ( 7 ( 8 ( 9 ( 9 ( 3 ( 5 ( ϕ ( 5 ( ( ( ϕ ( ( 5 ( 7 ( 3 ( 4 ( 3 ( ( ( ( 4 ( 6 ( ( 5 ( 4 ( 3 ( (c (d Fg.. Dfferent methods of modlatons and mappng. In system, the bt nterleavng technqe was employed n transmttng termnal and the maxmm a posteror (MAP probablty of detecton was sed n recevng termnal for the transformaton between symbols and bt, so, t can be as the M parallel channels wth ndependent bnary nptcontnos otpt. The Capacty of the channels shold be M I ( X, Y = I( B, Y, B {,}, ( = where the symbol I( B, Y expresses the nformaton between receved vale Y and the th bt B n sendng symbol X, the calclaton process s as followed, I( B, Y = H( B- H( B Y = H (.5 + pb (, ylog pb ( y, b, py s ( k = { A} = - Eb, (log py s ( ' = { A' b k' ( where { Α' b denotes the set of symbols whch the vale of the th bt s k n constellaton, py ( s k expresses the lkelhood fncton, snce the sperposed nose s Complex Gassan Nose, the lkelhood fncton py ( s k can be expressed as follow, py ( = e πσ = e πσ R R I I ( y - + ( y - σ y- σ, (3 where y denotes the Ecldean dstance between receved complex sgnal y and the symbol s k n constellaton. Pt ( nto ( and we can get py s ( I ( X, Y = M E (log, (4 ( M k = { A} b, = p y ' ' = { A' b From the expresson of the capacty, we can fnd that nder the premse of gvng the way of modlaton, the mappng relatonshp between bt and symbol has greatly mportant nflence on the reslt. Meanwhle, from ( we also can fnd that the nformaton between Y and dfferent bt B ( M n mappng symbols s dfferent. Based on t, the system s able to adopt dfferent modlaton method to carry ot the neqal error protecton (UEP. Throgh pttng mportant bt nformaton nto the poston where the I ( B, Y s larger, t obtan relable nformaton wth hgh probablty after the demodlaton. The Fg. 3(a shows the relatonshp between the capacty I ( XY, of system and SNR( σ 53

4 Sensors & Transdcers, Vol. 79, Isse 9, September 4, pp correspondng to the Gray mappng and the Non- Gray mappng (Fg. nder the 8PSK and 6QAM modlaton. For comparson, Fg. 3(a shows the capacty of the CM nder 8PSK and 6QAM, and the expresson of the CM capacty s I CM py s ( k ( X, Y = E k, (log p( s p( y s, = { A} k k (5 From Fg. 3, we can see the followng ponts. Regardless of 8PSK or 6QAM, the capacty of correspondng to the Gray mappng s obvosly larger than Non- Gray mappng nder the same SNR; The capacty of and CM s almost the same when sng the Gray mappng nder the crcmstance wth hgh SNR; bt when the SNR s low, the capacty of the s slghtly smaller than that of CM. In order to reflect clearly the effect of the mappng methods on the capacty, the relatonshp between I ( XY, and Eb N of the s provded as Fg. 3(b. In the next sectons abot performance smlaton, we wll choose E N wll be chose as the parameter of smlaton. b (a capacty vs. SNR (b capacty vs. Eb N Fg. 3. The channel capacty crves. 4. Expresson of the LDPC and Decodng Algorthm Here we take the bnary LDPC code as an example to ntrodce the expresson method and the decodng algorthm. 4.. Expresson of the LDPC LDPC can be expressed by a matrx. Let H = ( be a sparse m n check matrx. When h, m n V satsfes HV T =, V s a code of LDP defned n check matrx H. Wheren, V F n. Defne the lne weght (or colmn weght of the -th lne (or -th colmn n matrx H as the nmber of ts non-zero elements, and call the LDPC code wth constant lne (colmn weght for H as reglar LDPC mode, otherwse, call them non-reglar LDPC mode. For reglar LDPC mode, the code rate m R. n A check matrx of LDPC corresponds to a Tanner graph [9]. Tanner graph s a bpartte graph, whch contans two types of nodes: varable nodes and check nodes. The nmber of varable nodes s n, whch s correspondng to n bt codes. There are m check nodes whch are correspondng to the m check eqaton. If the elements h, ( < m, < n n matrx H are not eqal to, an edge s ntrodced between -th check node and -th varable pont. Fg. 4 shows a check matrx of LDPC and ts correspondng Tanner graph. The se of Tanner graph can descrbe teratve decodng algorthm more clearly, ncldng the nformaton processng flow and the edge nformaton flow of dfferent types of node. 4.. SPA Decodng Algorthm of LDPC Code ( l Let, be the nformaton transmtted from check node c to varable node v n l-th teraton. ( l Let v represent the nformaton transmtted from, varable node v to check node c n l-th teraton. 54

5 Sensors & Transdcers, Vol. 79, Isse 9, September 4, pp H = v v v v3 v4 v5 c c c v v v v v 3 4 v5 c c c Fg. 4. Check matrx of LDPC and the correspondng Tanner graph. Demodlaton Usng the receved complex-vale y ( q, we can calclate the log-lkelhood rato nformaton correspondng to each bt n the symbol. py ( { Α b = } Λ = ln, ( M-, (6 py ( s ' { Α b = } where symbol { Αk b denotes the set of the symbol that the vale of -th symbol, -th bt s k n constellaton. The lkelhood fncton py ( s k can be compted accordng to eqaton (3. Based on eqaton (6, the ntal log-lkelhood rato nformaton L( Λ can be compted accordng to eqaton (6 for all bt, and L( Λ s looked as the ntal nformaton v ( ( n of decoder. Informaton process of check node Assmng that check node c receved the nformaton come from ts neghborng node n the l-th teraton. The nformaton ( l, transmtted from check node c to varable node v can be calclated as follow: k' = tanh ( tanh( v, (7 ( l ( l, ', ' N / where symbol N / denotes the varable nodes correspondng to the check node c besdes node v. The nformaton process of varable node ( l In the l-th teraton, the nformaton v +, transmtted from varable node v to check node c can be calclated as follow: v = v +, ( l+ ( ( l, ', ' M / (8 where symbol M / denotes the check nodes connected to the varable node v besdes c. When every teraton comes to the end, n order to get the estmaton of V, decoder need to make harddecson. The methods of nformaton pdate and hard-decson are as follow: Lv ( v = +, ( ( l, M (9 f L(v vˆ =, ( else Accordng to the above nformaton processng, we make the smmary for SPA algorthm and descrbe as follow, where J s the max nmbers of teraton n the process of decodng. Sm-prodct algorthm Intalzaton. For all varable nodes, calclate ( the ntal LLR nformaton v accordng to ( ( eqaton (6, meanwhle, let v, = v, and the nmber of teraton s set to. Iteraton. When l < J, carry ot the steps as follows:. Check nodes. For all check nodes, calclate the nformaton ( l, that transmtted to varable nodes from check nodes accordng to eqaton (7.. Varable nodes. For all varable nodes, calclate the nformaton ( l v, that transmtted from varable nodes to check nodes accordng to eqaton (8. 3. Decson: for all varable nodes, calclate Lv ( accordng to eqaton (9, and make harddecson accordng to eqaton (; 4. If H V ˆT =, let l = J, or else let l = l+. Otpt. Otpt ˆV as the reslt of decodng. 5. Smlatons In or smlatons, the performance of the system show as Fg. s nvestgated. When the LDPC code s sed as forward error correcton (FEC, the nterlaver / denterlaver can be gnored. The decodng algorthm for LDPC adopts the SPA 55

6 Sensors & Transdcers, Vol. 79, Isse 9, September 4, pp algorthm presented n secton 3., where the maxmm nmber of teratons s 5. Smlaton. Select LDPC codes of IEEE 8.6e standard as FEC code whose rate s.5 and length s 4. Modlaton and mappng method are shown as Fg., whch are 8 PSK/Gray mappng, 8PSK/Non- Gray mappng, 6 QAM/Gray mappng and 6 QAM/Non-Gray mappng respectvely. The stmlaton reslts are shown n Fg. 5. Meanwhle, For comparson easly, a performance crve of RS code s smlated, and the eqvalent length of the RS 8 code s 4, defnng n GF(. RS code are db and 9.45 db respectvely, the dfference s.38 db. That dfference s that the RS code are defned n fnte felds, the decodng process s to symbol, whle the decodng process of LDPC code s to bts. Under the same condton, LDPC code can redce the dstance to the theoretcal capacty lmt. Take the bt error rate -5 as crteron, the dstance to theoretcal capacty lmt of LDPC code and RS code are.96 db and db n Gray mappng, and there are.96 db and 4.89 db n Non-Gray mappng. When the modlaton method s 6 QAM, the conclson s 8 PSK. Smlaton. In ths smlaton, we choose LDPC codes of DVB S. standard as FEC code whose rate s.5, and code length s 64. In order to evalate the effect of dfferent code length on the performance, we draw the performance crve nder 8PSK modlaton and Gray/Non-Gray mappng. The smlaton reslts are shown n Fg. 6. In order to facltate comparson, the LDPC code performance crve based on IEEE 8.6e n smlaton s also drew n Fg. 6. (a 8PSK Fg. 6. Performance crve of based on (648, 34 LDPC code of DVB S.. (b 6QAM Fg. 5. Performance crve of based on (4, LDPC code of IEEE 8.6e. From Fg.5 we can see the followng ponts. When the modlaton s 8PSK. The effect of mappng method on LDPC code s far more than that on RS code. For example, take the bt error rate -5 as dge crteron, for Gray mappng and Non-Gray mappng, the SNR of LDPC code shold be 3.44 db and 6. db respectvely, whch the dfference s.78 db; whle the SNR of From Fg. 6 we can see that wth the ncrease of LDPC code length, the performance of system s approachng the capacty lmt. For example, take bt error rate -5 as crteron, the LDPC code of DVB S. reqred the SNR of.35 db and 5.5 db respectvely nder Gray and non-gray mappng. Compared to the LDPC code of IEEE 8,6e, the gan s.9 db and. db. 6. Conclsons In ths paper, we analyzed the capacty that system n AWGN channels frstly, then, by sng the Monte Carlo smlaton, we get the relatonshp crve between channel capacty and SNR nder dfferent modlaton and Gray/Non-Gray mappng, 56

7 Sensors & Transdcers, Vol. 79, Isse 9, September 4, pp frthermore, we compared the channel capacty to CM system nder the same condtons. Fnally, on the bass of the performance smlaton, the performance of the BER s evalated. Meanwhle, LDPC code and RS code s regard as FEC respectvely, and the dfference between them s compared. Smlaton reslts show that the mappng method of constellaton wll nflence the performance of system whch based on LDPC code n large extent, the performance wth Gray mappng s better than that wth Non-Gray mappng clearly, and wth the ncrease of LDPC code length, the performance of system s approachng the capacty lmt. Acknowledgements Ths work was spported by the Natonal Natral Scence Fondaton of Chna (No and Chna Postdoctoral Scence Fondaton (4M5549. References []. E. Zehav, 8-PSK trells codes for a Raylegh channel, IEEE Transacton on Commncaton, Vol. 4, Isse 3, 99, pp []. G. Care, G. Tarcco, and E. Bgler, Bt-nterleaved coded modlaton, IEEE Transacton on Informaton Theory, Vol. 44, Isse 3, 998, pp [3]. G. Ungerboeck, Channel codng wth mltlevel/phase sgnals, IEEE Transacton on Informaton Theory, Vol. 8, Isse, 98, pp [4]. R. G. Gallager, Low-densty party-check codes, MIT Press, Cambrdge, Mass, 963. [5]. C. Berro, A. Glavex, P. Thtmashma, Near Shannon lmt error-correctng codng and decodng: Trbo codes, n Proceedngs of the IEEE Internatonal Conference on Commncatons, 993, pp [6]. R. C. Bose, D. K. Ray-Chadhr, On a class of error correctng bnary grop codes, Informaton and Control, Vol. 3, 96, pp [7]. R. M. Tanner, A recrsve approach to low complexty codes, IEEE Transacton on Informaton Theory, Vol. 7, Isse 5, 98, pp Copyrght, Internatonal Freqency Sensor Assocaton (IFSA Pblshng, S. L. All rghts reserved. ( 57