Chalmers Publication Library

Transcription

1 Chalmers Publicatio Library Copyright Notice 200 IEEE Persoal use of this material is permitted However, permissio to reprit/republish this material for advertisig or promotioal purposes or for creatig ew collective works for resale or redistributio to servers or lists, or to reuse ay copyrighted compoet of this work i other works must be obtaied from the IEEE This documet was dowloaded from Chalmers Publicatio Library ( where it is available i accordace with the IEEE PSPB Operatios Maual, ameded 9 Nov 200, Sec 89 ( (Article begis o ext page)

2 Exploitig UEP i QAM-based BICM: Iterleaver ad Code Desig Alex Alvarado, Studet Member, IEEE, Erik Agrell, Leszek Szczeciski, Seior Member, IEEE, ad Are Svesso, Fellow, IEEE Abstract I this paper we formally aalyze the iterleaver ad code desig for QAM-based BICM trasmissios usig the biary reflected Gray code We develop aalytical bouds o the bit error rate ad we use them to predict the performace of BICM whe uequal error protectio (UEP) is itroduced by the costellatio labelig Based o these bouds the optimum desig of iterleaver ad code is foud, ad umerical results for represetative cofiguratios are preseted Whe the ew desig is used, the improvemets may reach 2 db, ad they are obtaied without ay icrease o the trasceiver s complexity We also itroduce the cocept of geeralized optimum distace spectrum covolutioal codes, which are the optimum codes for QAM-based BICM trasmissios Idex Terms BICM, iterleaver desig, multiple iterleavers, optimum distace spectrum codes, QAM, UEP I INTRODUCTION I bit-iterleaved coded modulatio (BICM) [] with highorder costellatios, the bit mappig causes the so-called uequal error protectio (UEP) [2], ie, depedig o the bits positio withi the symbol, the bits experiece differet protectio, which may be iterpreted i terms of ucoded error probability or average mutual iformatio I this paper we formally aalyze the problem of the iterleaver ad code desig for uequally protected BICM trasmissios BICM, first itroduced by Zehavi [] ad later aalyzed i detail by Caire et al i [2], owes its popularity to the fact that the chael ecoder ad the modulator are separated by a bit-level iterleaver Because of this separatio, the code rate ad the costellatio ca be chose idepedetly allowig for a simple ad flexible desig [2, Sec V] At the receiver s side, the reliability metrics are calculated for the coded bits i the form of logarithmic likelihood ratios, also kow as L- values These metrics are the deiterleaved ad further used by the soft-iput chael decoder From a capacity poit of Paper approved by XXYY, the Editor for ZZZ of the IEEE Commuicatios Society Mauscript received Moth Day, Year; revised Moth Day, Year Research supported by the Swedish Research Coucil, Swede (uder research grat # ), ad by NSERC, Caada (uder research grat # ) Differet parts of this work were preseted at the Europea Wireless Coferece EW2008, Prague, Czech Republic, Jue 2008, ad at the Iteratioal Coferece o Commuicatios ICC2009, Dresde, Germay, Jue 2009 A Alvarado, E Agrell, ad A Svesso are with the Departmet of Sigals ad Systems, Commuicatio Systems Group, Chalmers Uiversity of Techology, Gotheburg, Swede ( {alexalvarado,agrell,ares}@chalmersse) L Szczeciski is with the Istitut Natioal de la Recherche Scietifique, INRS-EMT, 800, Gauchetiere W Suite 6900 Motreal, H5A K6, Caada ( leszek@emtirsca) Digital Object Idetifier 009/TCOMM2009XXXXX view, BICM with appropriately desiged mappig itroduces oly a small pealty whe compared to a coded modulatio scheme (CM) where the chael ecoder ad mapper are joitly desiged [2] Ugerboeck s trellis coded modulatio (TCM) [3] is oe of the most popular CM schemes, ad it maximizes the miimum Euclidea distace betwee trellis paths correspodig to differet code sequeces O the other had, BICM maximizes the code diversity, ad therefore, it outperforms TCM i fadig chaels Whe compared to TCM, BICM decreases the miimum Euclidea distace, ad cosequetly, it is suboptimal for the AWGN chael Nevertheless, sice this decrease is oly margial, BICM is very robust to variatios of the chael characteristics [4, Sec 46] BICM is owadays a de facto stadard, ad it is used i most of the existig wireless systems, eg, HSPA, IEEE 802a/g, IEEE 8026, etc Whe BICM is used with Gray-mapped 4-QAM, all the bits are equally treated by the modulator O the other had, if UEP is produced by the modulator, it ca be exploited to improve the receiver s performace I this paper we are iterested i UEP caused by the biary labelig of highorder costellatios, however, we ote that UEP ca also be itetioally imposed This ca be doe by usig uequal power allocatio for systematic/parity bits, a idea first used for turbo-ecoded BICM (TC-BICM) i [5] ad later aalyzed i [6] [0], or by simply deletig some bits (pucturig) The coclusios available i the literature about the best strategy to exploit the UEP are somehow cotradictory Accordig to [5], [] the performace of turbo-ecoded trasmissios ca be improved if the parity bits are more protected, while i [6], [8], [0] it is show that systematic bits must receive sroger protectio The ifluece of the block legth ad code rate for optimal power allocatio was aalyzed i [6], [2] It has bee show i [3] ad i [4, Sec 932] that to improve the performace of TC-BICM, the systematic bits must be assiged to the most protected positios Accordig to [5], i the waterfall regio, pucturig systematic bits (strog protectio for parity bits) improves the performace, while i [6] the opposite is claimed Iterleaver desig aimig to assig the coded bits to differet bit positios for high-order modulatio schemes was aalyzed i [7] UEP has bee studied for LDPC codes i [8] [22], ad for turbo coded modulatio schemes i [23], where the bits were grouped ito differet classes of importace To take advatage of the UEP caused by the modulator, ad for a give chael code, the desig of the iterleaver coectig both etities becomes crucial Followig the frame-

3 2 work set i [2], for the aalysis of BICM, a sigle iterleaver (S-iterleaver) is most ofte cosidered This simplifies the aalysis of the resultig system, but leads to sub-optimality already oted i the literature [24] I fact, the origial BICM paper of Zehavi [] postulated the applicatio of multiple iterleavers (M-iterleavers) betwee each of the ecoder s output ad the correspodig modulator s iput (eg, usig three iterleavers for a 2/3-rate ecoder, each of them feedig bits to oe of the bits positios i the 8-PSK symbol) Similar M-iterleavers have bee used for BICM [24], [28], for BICM with iterative demappig ad decodig (BICM-ID) [29], for serially cocateated systems [25], ad for BICM-OFDM [26] M-iterleavers have also bee proposed i the 3GPP/HSPA stadard [4], [30] with 6-QAM or 64-QAM Their use i that cotext is relevat from a implemetatio poit of view sice two parallel iterleavers i HSPA with 6-QAM (or three for 64-QAM) are costructed re-usig the iterleaver already implemeted for 4-QAM Whe such M-iterleavers are used, the performace gais will strogly deped o the bit assigmet betwee the ecoder s output ad the bit positios i the complex symbol Although previous works we cite oted the ifluece of the iterleaver desig ad the UEP, to the best of our kowledge, this paper is the first to aalyze formally this problem for BICM trasmissios More particularly, we preset a methodology for the iterleaver ad code desig for QAM-based BICM trasmissios (BICM-QAM) To obtai simple desig rules, we use the Gaussia model for the distributio of the L-values i QAM trasmissios preseted i [3] ad the geeralized trasfer fuctio of a code [27], [32], [33], which allows us to develop uio bouds for the coded bit error rate (BER) of the system Usig these bouds, the optimum desig of iterleaver ad code is preseted, provig for example that the aswer about the protectio of systematic/parity bits caot be give i abstractio of the code ad the modulatio As aother applicatio of the developed bouds, we itroduce the geeralized optimum distace spectrum (GODS) codes as the aswer to the problem of selectig good covolutioal codes i BICM-QAM II SYSTEM MODEL Hereafter we use lowercase letters x to deote a scalar, ad boldface letters x to deote a vector of scalars Capital letters X deote radom variables, P( ) deotes probability, ad f X (x) deotes the probability desity fuctio (pdf) of the radom variable X Blackboard bold letters X represet matrices or vectors We cosider the BICM system show i Fig The k c vectors of N iformatio bits b l = [b l (),, b l (N)] are ecoded by a rate R = k c / chael ecoder, where l =,,k c The vectors of coded bits c,,c are the fed to the iterleaver uits where the pth output vector of the ecoder is give by c p = [c p (),, c p (N)] We emphasize here that the proposed scheme is differet from the so-called Differet ames have bee give to this iterleaver: for example, i-lie [25], itralevel [26], M [24], dual [4], or modular [27] iterleavers Its formal defiitio will be preseted i Sec II-A b b kc ˆb Ecoder c c L π π π π c c L MUX DEMUX u u m M-PAM Decoder Demapper ˆb kc L L M 2 -QAM BICM Chael U U m M-PAM Mapper Fig Model of BICM-QAM trasmissio: a chael ecoder followed by the iterleavers (π,, π ), a multiplexig uit (MUX), the M-PAM mapper, the chael, ad the iverse processes at the receiver s side multi-level codig [34] through the fact that oly oe ecoder is preset i the system A The iterleavers ad the multiplexig uit The iterleavers (π,,π ) i Fig are assumed to be ifiite ad idepedet (ideal), yieldig radomly permuted sequeces of the coded bits c p = π p {c p } This idealizig assumptio lets us focus o the essetial features of the desig ad is also justified by the fact that the resultig desig s optimality does ot seem to be affected by fiitelegth iterleavers used durig the simulatios We ote that a more realistic aalysis would cosider fiite-legth (ie, oideal) iterleavers, however, this requires a differet ad more complex approach The multiplexig uit (MUX) assigs the coded ad iterleaved bits to the differet bit positios i the M 2 -QAM symbol The mappig cosidered here is based o the socalled biary reflected Gray code (BRGC) 2 [37], [38], so each symbol is a superpositio of idepedetly modulated real/imagiary parts [39] Cosequetly, we focus o the equivalet M-PAM costellatio (cf Fig ) where M = 2 m For a fully geeral approach, we defie the multiplexig uit usig a matrix K m K of dimesios m, whose elemets, 0 κ p,q, deote the fractio of bits c p that will be assiged to the qth output u q As all the vectors u q for q =,,m have the same legth, so the costrait p= κ p,q = m must be satisfied, ad sice all the bits i the vector c p must be assiged to oe of the m outputs, the coditio m q= κ p,q = must also be fulfilled The matrix K ca be the writte as show i (), where the last row ad the last colum of K take ito accout the costraits imposed o κ p,q, ad cosequetly, whe desigig K, oly κ p,q for p =,, ad q =,,m may be freely set (cosiderig also 0 κ p,q p, q) We emphasize that K i () represets the multiplexig uit, ie, it defies how the coded bits are assiged to the iputs of the modulator This matrix ad the multiple (parallel) iterleavers i Fig model the whole iterleavig, ad allow us to cosider its differet cofiguratios For this reaso we 2 The BRGC is selected for our aalysis due to its relevace i practical systems, its optimality i terms of BER i ucoded trasmissios [35], ad also because it maximizes the BICM capacity for a wide rage of SNRs ad costellatio sizes [36] x y z

4 3 K = κ, κ,m m q= κ,q κ 2, κ 2,m m q= κ 2,q κ, κ,m m m p= κ p, m p= κ p,m + m + p= q= κ,q m q= κ p,q () will refer to iterleaver desig as the process of selectig the elemets κ p,q defiig K For example, for = m, if K = I (I beig the idetity matrix), the system is trasformed ito the Zehavi s cofiguratio where all the bits from the same ecoder s output are assiged to the same modulator s iput Exchagig the rows of this matrix allows us to cosider differet ways of coectig the ecoder to the modulator If we cosider κ p,q = m for all p ad q, a uiform distributio of the coded bits at the iputs of the modulator is achieved Whe comparig our model to the S- iterleaver (sigle iterleaver) i [2] we ote that due to the ifiite iterleaver assumptio, the S-iterleaver also results i a uiform distributio, ad therefore our model ad the iterleaver i [2] become equivalet At ay time istat t, the coded ad iterleaved bits [u (t),, u m (t)] are mapped to a M-PAM symbol x(t) X usig a biary memoryless mappig M : {0, } m X, where X = {( M), (3 M),, (M ) } is the set of M-PAM symbols 3, ad where 2 is the miimum distace betwee them The costellatio is ormalized to The result of the trasmissio of N s symbols is give by y = x + z, where x = [x(),, x(n s )], ad z R Ns is a vector with samples of zero-mea ad idepedet Gaussia radom variables with variace N 0 /2 The sigal-to-oise ratio (SNR) per complex symbol is give by γ = N 0 At the receiver s side, the reliability metrics of the trasmitted bits are calculated i the form of logarithmic likelihood ratios (L-values) 4 for each bit positio as [], [2], [40], [4] uit average eergy so = 3 2(M 2 ) ( U q (t) = γ { mi (x(t) a) 2 } { mi (x(t) a) 2 } a X q,0 a X q, ), where X q,b is the set of symbols labelled with the qth bit equal to b Sice the mappig is memoryless, from ow o we drop the time idex t, eg, U q (t) U q It is worth to metio that (2) is a suboptimal metric sice it is based o the max-log approximatio This simplificatio, proposed i the early works of Zehavi ad Caire et al, is recommeded by the 3GPP workig groups [4] as it has small impact o the receiver s performace whe Graymapped costellatios are used [42] [44] The vector of soft iformatio U q is demultiplexed (L p ), deiterleaved (L p ) ad the passed to a chael decoder which 3 The M 2 -QAM costellatio is formed by the direct product of two M- PAM costellatios, ie, X jx 4 L-values covey iformatio about the bits probabilities ad are ofte used i practice Alterative implemetatios ca use differet metrics or the actual probabilities (2) produces a estimate of the trasmitted bits ˆb B Equivalet Chael Model Usig the results preseted i [3] it is possible to build a equivalet model for the M 2 -QAM BICM chael show i Fig I this model each bit u q after the MUX ca be see as beig set over a virtual chael whose output L-value U q has a distributio that depeds o the bit s positio q ad the symbol set, ie, the value of the other bits u v, v q We explai it briefly below while more details may be foud i [3] Let d q (x) deote the Euclidea distace betwee the symbol x ad the closest symbol i the costellatio with the opposite value of the bit labelig x at positio q, ie, if x X q,b, b {0, }, d q (x) = mi a Xq, b x a Due to the properties of the BRGC, symbols with the qth bit set to 0 or are clustered so that d q (x) may be at a distace that varies from 2 to 2 M 2 q That is, whe q = m, there is always a adjacet symbol (at distace 2 ) with the opposite value of the bit O the other had, for q =, the umber of possible distaces is Sice d q (x) determies the protectio experieced by the bit, differet values of d q (x) cause UEP For q = m the bits have always the same weak protectio but for q =, depedig o the value of other bits i the modulatig codeword, the protectio may be relatively strog I Fig 2 we show the 8-PAM costellatio with BRGC ad also the distaces d q (x) for some symbols All the distaces are listed i Table I Accordig to [3], there are differet Gaussia distributios that ca be used to model the L-values A bit trasmitted at positio q passes through the virtual chael Θ j whe it is set usig a symbol x such that d j (x) = 2 j The, the L-value U q has a distributio that may be approximated as Gaussia with mea µ j ad variace σ 2, where (µ j, σ 2 ) = (4γ 2 (2j ), 8γ 2 ), (3) with j =,, It is worth to metio that the equivalet model preseted i this sectio is slightly differet from the oe preseted i [2] While both of them cosider m parallel biary-iput soft-ouput chaels, i our model we use the kowledge of the desities of the L-values These desities were previously calculated i [3] ad are based o the use of the max-log approximatio Moreover, i order to make the aalysis tractable, we use the simplified Gaussia model for these desities as proposed i [3] The probability that a L-value at bit positio q is distributed

5 d ( ) d ( 3 ) d ( 5 ) d ( 7 ) d 2 ( 5 ) d 2 ( 7 ) d 3 ( 7 ) q = q = 2 q = 3 Fig 2 8-PAM costellatio with BRGC The biary labeligs per positio are show together with the distaces d q(x) for some symbols The weaker protectio of the bit positio q = 3 is evidet due to the smaller (o average) values of d 3 (x) TABLE I UEP CAUSED BY THE BRGC: MODULATING CODEWORDS, 8-PAM SYMBOLS, DISTANCES d q(x), AND VIRTUAL CHANNELS Θ j [u u m] x d (x) Θ j Θ 4 Θ 3 Θ 2 Θ Θ Θ 2 Θ 3 Θ 4 d 2 (x) Θ j Θ 2 Θ Θ Θ 2 Θ 2 Θ Θ Θ 2 d 3 (x) Θ j Θ Θ Θ Θ Θ Θ Θ Θ with parameters (µ j, σ 2 ) is give by ω q,j = 2 m q if j =,, 2 m q 0 if j = 2 m q +,, M, (4) 2 that is, the virtual chael Θ ca be used by the bit for all positios q, Θ 2 oly for q m, Θ 3 ad Θ 4 oly for q m 2, Θ 5,, Θ 8 for q m 3, ad so o It is worth to metio that for the BRGC, all the poits i the costellatio have oly oe closest eighbor with the opposite bit label at the same distace (cf Fig 2) This is a property of the mappig aalyzed i this paper, ad it does ot hold i geeral To fully characterize the equivalet M 2 -QAM BICM chael we defie the matrix O m M O of dimesios m 2 where each elemet ω q,j i O is the probability that a trasmitted bit at positio q is trasmitted usig the chael Θ j The resultig equivalet chael model is schematically show i Fig 3 Based o the previous discussio, the M 2 -QAM BICM chael of Fig ca be replaced by a compoud chael completely defied by the matrices K (iterleaver) ad O (mappig) If we defie the matrix X as m m κ,q ω q, κ,q ω q, q= q= X KO =, (5) m m κ,q ω q, κ,q ω q, q= q= the the pth output L p R of this chael is associated with the pth biary iput c p, where L p is a Gaussia mixture with desity give by f Lp (λ) = ξ p,j Φ(µ j, σ 2 ; λ), (6) j= ( where Φ(µ j, σ 2 ; λ) = exp 2πσ (λ µj)2 is a Gaussia 2 fuctio, ad ξ p,j is the (p, j)th elemet of X which deotes the probability that the pth bit passes through the chael Θ j Example : Cosider a rate R = /3 ( = 3) code ad a 8-PAM costellatio (m = 3, M = 8) preseted i Table I I this table the virtual chaels associated with the differet symbols ad bit positios are show For this case, we cosider two matrices K 0 0 /3 /3 /3 K = , K = 2σ 2 ) /3 /3 /3 /3 /3 /3, (7) ad the matrix O is give by /4 /4 /4 /4 O = /2 /2 0 0 (8) While the matrix K represets Zehavi s cofiguratio, we ote that the etries of the matrix K are equal to /m, which meas that thaks to the ifiite iterleavig the ecoder output bits are uiformly distributed over all m iputs of the modulator, ad therefore, the M-iterleaver represeted by K is equivalet to the S-iterleaver postulated by Caire et al i [2] III INTERLEAVER AND CODE DESIGN I this sectio, based o the model itroduced i Sec II ad usig a geeralized trasfer fuctio of a code, we develop uio bouds o the BER of BICM-QAM Based o these bouds the optimum desig of iterleaver ad code is foud ad later used i Sec IV to aswer simple questios such as: What are the attaiable gais obtaied by usig M-iterleavers? Which bits (systematic/parity) should receive stroger protectio? What are the optimum covolutioal codes i this sceario?

6 5 Θ 2 m q coded BICM is give by u q Θ 2 m q Θ 2 m q + Θ 2 m q Fig 3 Equivalet chael model: the virtual chaels Θ j, j =,2 m q are selected with equal probability, while the chaels Θ j, j = 2 m q +,, are ot available for the bit at positio q A Geeralized weight distributio spectrum For ay covolutioal code (CC) it is possible to defie a geeralized trasfer fuctio (GTF) which eumerates ot oly the umber of o-zero output bits over a path, but the locatio of those bits, ie, it idicates which brach the ozero outputs are associated with [], [27] For a rate-k c / CC we defie the GTF of the code as T(W, I, L) = w t w,i,l I i L l i l p= U q W wp p, (9) where the geeralized weight w = (w,,w ) gathers the weight w p of the pth output of the ecoder, ad W = (W,, W ), I, ad L are dummy variables The coefficiet t w,i,l eumerates the umber of paths divergig from the zero state ad mergig with the zero state after l steps, associated with a iput sequece of weight i, ad a output sequece of geeralized weight w The coefficiets t w,i,l ca be calculated usig stadard techiques [45, Ch 4] Efficiet methods for this calculatio iclude the recursive algorithm of Divsalar et al [46], or a breadth first search algorithm [47] Usig the GTF, it is possible to obtai a geeralized weight distributio spectrum (GWDS) of the code [], [45, Ch 4] β(w) = k c p= w p! [ w L)] W w I W=0,I=L= T(W, I,, where w W = w w w W w ad w = w W w + + w If a turbo code (TC) is cosidered, the cocept of uiform iterleaver itroduced by Beedetto et al [48] ca be used to calculate the spectrum of the code The extesio to a GWDS is straightforward; more details ca be foud i [46], [48], [49] BER UB = l=w free w W (l) β(w)pep(w), (0) where w free is the free distace of the code, ad PEP(w) is the pairwise error probability which represets the probability of detectig a codeword with geeralized weight w istead of the trasmitted all-oe codeword 5 Obviously, ad for practical reasos, the boud i (0) is calculated usig oly a limited umber of terms i the first sum This meas that (0) is ot a UB aymore, but rather its approximatio Nevertheless, throughout this paper we will use the ame UB to refer to approximatios of the true boud To calculate the PEP we eed to calculate the probability that the decoder selects a codeword with geeralized weight w istead of the trasmitted all-oe codeword To this ed, we ote that the decisio is made based o the sum of w + +w L-values i the diverget path Let Z be the decisio variable where w Z = i= w L (i) + + i= L (i) w p = p= i= L (i) p, () ie, a sum of l idepedet radom variables, where the radom variable associated with the ith output is a sum of iid Gaussia mixtures give by (6) Cosequetly, for a give value of w, the PEP ca be calculated as the tail itegral of the pdf of Z, ie, PEP(w) = P(Z < 0) = 0 f Z (λ)dλ (2) To calculate f Z (λ) we first defie the j-fold self covolutio operator as follows Let L be a radom variable with desity f L (λ), its j-fold self covolutio is deoted by [f L (λ)] (j) f L (λ) f L (λ), (3) }{{} j times which correspods to the PDF of the sum of j iid radom variables L Usig the above otatio ad (6), we ca calculate the PDF of the decisio variable Z i () as f Z (λ) =[f L (λ)] (w) [f L (λ)] (w), (4) where the pth term i (4) ca be approximated 6 by B Uio bouds for BICM-QAM I order to use the GWDS of the code to calculate uio bouds for the BER, we defie the set W i (l) as all the combiatios of i oegative itegers such that the sum of the elemets is l, ie, W i (l) {(w,, w i ) (Z + ) i : w + + w i = l} Usig the GWDS of the code, the uio boud (UB) o the BER for both covolutioally ad turbo 5 We ote that the costellatio labelig produces a o-symmetric chael, ie, the coditioal chael trasitio probability for a bit b = 0 is ot the same that for b = Cosequetly, the exact value of the PEP i (0) depeds o both w ad the trasmitted codeword However, the symmetry coditio ca be easily fulfilled if the bits at the ecoder output are radomly egated ad the sig of the L-values at the decoder iput chaged afterwards Moreover, umerical results showed that this symmetrizatio causes egligible impact o the performace of QAM-based BICM trasmissios 6 The approximatio refers to the fact that the Gaussia model for the L- values is used istead of the exact desities

7 6 [ flp (λ) ] (w p) ξ p,j Φ(µ j, σ 2 ; λ) = = j = r W (w p) j= ( wp Φ j wp = ( wp µ ji, w p σ 2 ; λ i= (w p) ) wp (5) ξ p,ji (6) i= ) Φ r j µ j, w p σ 2 ; λ r j= j= ξ rj p,j (7) To pass from (5) to (6) we have expaded the covolutio of sums as sums of covolutios ad the applied Φ(µ i, σi 2; λ) Φ(µ j, σj 2; λ) = Φ(µ i + µ j, σi 2 + σj 2 ; λ) To pass from (6) to (7) we ote that a Gaussia fuctio with parameters (r µ + + r µ, w p σ 2 ) ca be geerated by differet combiatios of (j,, j wp ) Furthermore, the umber of combiatios (multiplicities) for a give value of r = (r,, r ) are the multiomial coefficiets give by ( ) wp w p! r r! r! (20) Usig (7) i (4) we get the fial ad exact expressio for the desity of Z show i (8) ad (9), where ) g(r,,r ) = ( wp ξ rp,j p,j (2) r p p= j= Based o the previous discussio, we preset three propositios which are the mai results of this sectio They will help us to simplify the desig of the system (cf Sec IV) Propositio : The UB o the BER for BICM-QAM ca be approximated as UB β(w) g(r,,r ) r,,r where l=w free w W (l) Q ( h(r,,r ) ), (22) p= j= h(r,,r ) = r p,jµ j, (23) lσ 2 g(r,,r ) is give by (2), Q(x) = 2π x e t2 /2 dt, ad r p W (w p ) for p =,, Proof: From (0), (2), ad (9) Aalyzig the expressio i (22), it is possible to see that it is composed of three terms: β(w) which depeds oly o the code, Q (h(r,,r )) which depeds oly o the chael [cf (23)], ad g(r,,r ) which depeds o the iterleaver [cf (5)] Expressig the UB i this way shows how to optimize the BICM-QAM trasmissios I particular, we ote that the chael properties defied by O are fixed for a give value of M, ad that the optimum performace of the system will be achieved by a joit desig of the iterleaver ad the code We also ote that all combiatios i (8) are i geeral tedious to evaluate (especially for large values of ad/or m), thus we seek further approximatios The simplificatio preseted i the followig propositio is based o cosiderig, for each l, oly the Gaussia desity with the smallest mea-to-stadard deviatio ratio The ituitio behid this approximatio is that the error coefficiets geerated by other Gaussia desities are less importat Propositio 2: The UB i (22) ca be further approximated by UB = l=w free Q ( 2lγ 2) w W (l) β(w) p= ξ wp p, (24) Proof: Approximate W (w p ) i the third sum of (22) by its leadig elemet r p = (w p, 0,,0) The g(r,,r ) = p, from (2) ad h(r,,r ) = p= ξwp lµ /σ = 2lγ from (23) ad (3) Now (24) follows from (22) We emphasize here that (24) is quite simple to evaluate compared with the origial expressio i (22), ad it still takes ito accout the parameters to optimize the trasmissio (iterleaver ad code) The followig propositio presets a eve simpler asymptotic approximatio of the origial expressio i (22), ie, whe the SNR goes to ifiity This result will provide us with the ew criteria to select the optimum code ad iterleaver desig (cf Sec IV-B) Propositio 3: The asymptotic performace of BICM- QAM is give by UB = Q ( 2γ 2 w free ) w W (w free ) β(w) p= ξ wp p, (25) Proof: The boud (22) is a sum of weighted Q-fuctios, whose argumet h(r,,r ) depeds o the umber of bits that were trasmitted usig the differet virtual chaels If γ, oly oe of those Q-fuctios will domiate the boud, ie, the Q-fuctio with the smallest argumet For a give value of w we eed to choose the combiatio of (r,,r ) that miimizes h(r,,r ), ie, mi r,,r { } h(r,,r ) mi r,,r = mi r,,r = mi r,,r r p,j µ j p= j= j= { p= j= r p,jµ j lσ 2 r,j µ j + + j= } r,j µ j (26) Sice µ j > 0, j =,, ad µ j > µ, j = 2,,, it is clear that r p = (w p, 0,,0) p miimizes (26) Usig the previous result ad the defiitios of µ j ad σ 2 i (3), it ca be see that the fuctio h(r,,r ) has a miimum value of 2γ 2 l Moreover, if l is icreased, the argumet of the domiat Q-fuctio will icrease ad cosequetly, the miimum is obtaied whe l = w free, ie, whe all the w free bits were trasmitted usig the least

8 7 f Z (λ) = = r W (w ) r W (w ) ( w r ) Φ r,j µ j, w σ 2 ; λ j= r W (w ) j= ξ r,j,j r W (w ) ( w r ) Φ r,j µ j, w σ 2 ; λ (8) g(r,,r )Φ r p,j µ j, σ 2 w p ; λ (9) p= j= p= j= j= ξ r,j,j protected chael Θ The weightig coefficiet i (25) ca be obtaied usig the defiitio of X i (5) By combiig the results preseted above, (25) ca be obtaied For the umerical evaluatio of (22) ad (24), l will be be limited betwee w free ad l max K W Aalytical K S Aalytical K B Aalytical K W Simulatio K S Simulatio K B Simulatio A UB for BICM-QAM IV NUMERICAL RESULTS I this sectio we cotrast the boud i (22) with the results obtaied based o umerical simulatios With these results we aim to quatify the potetial gais whe M-iterleavers are used istead of S-iterleavers, ad also to cofirm the aalytical developmets preseted i Sec III For a spectral efficiecy of bit/s/hz, two cases are aalyzed A rate-/2 TC or CC is used i cojuctio with 6- QAM ( = 2 ad m = 2), ad a rate-/3 TC or CC is used with 64-QAM ( = 3 ad m = 3) For the CC we use ODS codes from [50] with polyomials give i octal otatio ad where the pth polyomial geerator is associated with the pth ecoder s output For the TC, two idetical rate-/2 recursive systematic covolutioal (RSC) ecoders are cocateated i parallel separated by a sigle iterleaver of legth N Eve if formally the rate-/2 TC has three outputs (systematic bits, parity bits from the RSC ad from the RSC2), here we make o distictio betwee the parity bits, ad we cosider them to be oe output For = m = 2 we see from () that there is oly oe degree of freedom whe selectig K (κ, ) I Fig 4 the boud (22) is compared with the simulatio results 7 for the values of κ, that yield the two M-iterleavers (κ, {0, }) ad the S- iterleaver (κ, = /2) Let us first aalyze the CC case From Fig 4 we ote that the simulatio results perfectly match the aalytical bouds For this particular code, the best iterleaver desig deoted by K B is obtaied whe κ, =, ie, whe the bits comig from the first ecoder s output (geerator polyomial 23) are more protected by the chael tha the secod ecoder s output The worst iterleaver desig deoted by K W is obtaied whe κ, = 0, while the S-iterleaver deoted by K S gives a performace betwee K B ad K W From the two-dimesioal GWDS of this particular code, we observed that the o-zero elemets w = (w, w 2 ) W (w free ) are ot balaced, ie, the weigths w are o average larger tha the weigths w 2 Usig this code property i Propositio 3, oe 7 To calculate the boud i (22) umerically, we used l max = 00 for the TC ad l max = 50 for the CC The iterleaver size for the TC is N = 000 BER TC SNR γ [db] Fig 4 UB (22) (Propositio ) ad simulated BER for BICM-QAM for TC ad CC: = 2, m = 2 (R = /2 ad 6-QAM) ad differet iterleaver cofiguratios The CC is the ODS code with K = 5 ad polyomial geerators (23, 35) The TC is a parallel cocateatio of two idetical RSCs defied by their polyomial geerators (, 5/7) Alterate pucturig of the the parity bits is performed to reach R = /2 The iterleaver size is N = 000 ad 0 iteratios are performed by the turbo decoder ca easily demostrate that protectig more the bits from the first output will decrease the UB The differece betwee the two cofiguratios is relatively small (03 db at BER = 0 6, cf Fig 4), however, we will see i the followig that for other codes, or code rates, the gais ca be much more importat If the rate-/2 TC is used istead, the optimum iterleaver K B is achieved settig κ, = 0, ie, whe the parity bits are more protected tha the systematic bits (ad K W if κ, = ) This cotradicts [4, Sec 932] ad [3], where it is claimed that systematic bits should always be set to the more reliable positios However, usig the developed bouds, we see that the optimum assigmet depeds o the code defied by its GWDS I Fig 4 these results are preseted, where the boud (22) perfectly predicts the error floor of the TC We emphasize that for this code, ad for a target BER of 0 6, the differece betwee K B ad K W is db, which is obtaied without complexity icrease but oly by properly assigig the coded bits to the bit positios i the QAM symbol If we aalyze the asymptotic behaviour of this code usig Propositio 3, we discover that the boud (25) is tight oly for very high SNR values (BER 0 2 ) The reaso behid this is the so-called spectral thiig property of the TCs, ie, CC

9 8 BER SNR γ [db] K W Aalytical K S Aalytical K B Aalytical K W Simulatio K S Simulatio K B Simulatio Asymptotic Fig 5 UB (22) (Propositio ) ad simulated BER for BICM-QAM for TC ad CC: = 3, m = 3 (R = /3 ad 64-QAM) ad differet iterleaver cofiguratios The CC is the ODS code with K = 5 ad polyomial geerators (25, 33, 37) The TC is a parallel cocateatio of two idetical RSCs defied by their polyomial geerators (, 5/7) The iterleaver size is N = 500 ad 0 iteratios are performed by the turbo decoder The asymptotic bouds based o (24) (Propositio 2) for the TC ad o (25) (Propositio 3) for the CC are also show the values of the GWDS for w W (w free ) are quite small To aalyze the TC i the error floor regio, we will thus use Propositio 2 sice it cosiders more terms i the spectrum, cf (24) I Fig 5 we preset the bouds ad the result of umerical simulatios for a rate-/3 TC or CC used i cojuctio with 64-QAM ( = 3 ad m = 3) I this case, the optimizatio space is formed by the variables κ,, κ,2, κ 2,, ad κ 2,2, uder the costraits preseted i Sec II-A The variables of the optimizatio space are i geeral cotiuous, however, we oly aalyze the six possible M-iterleavers (κ p,q {0, }) ad the S-iterleaver (κ p,q = /3) The results preseted i Fig 5 are for the best ad worst M-iterleaver foud, ad also the S-iterleaver The best (or worst) M-iterleaver was foud by selectig the matrix K that miimizes (resp maximizes) the UB at a give target BER The selected target BER was 0 6, however, we oted that chagig the target BER to ay other value of practical iterest (betwee 0 4 ad 0 7 ) does ot chage the coclusio about the best (or worst) M-iterleaver For this particular code, the matrices foud are K B = K W = (27) For this cofiguratio we used N = 500 i order to double check the correct computatio of the GWDS of the TC ad the bouds I this figure we ca see agai that the boud (22) match the simulatio results, ad that for a target BER of 0 6 there is differece of approximately 2 db betwee K W ad K B I order to calculate the boud (22) for = m = 3 (cf Fig 5), we used l max = 50 for the TC ad l max = 25 for the CC As metioed before, whe m ad/or icrease, coutig all the combiatios i (22) becomes tedious, ad cosequetly, the maximum value of l cosidered must be relatively small I Fig 5 we also preset results for the (asymptotic) simplificatios preseted i Sec III-B For the CC we calculate UB usig (25) ad l max = 50, ad for the TC we calculate UB usig (24) ad l max = 00 The computatios for these simplificatios are very simple compared with (22), ad yet they predict the asymptotic performace of the system as show i Fig 5 From the results preseted i Fig 4 ad Fig 5, we ca draw the followig iterestig coclusios: For a give target BER of 0 6, the SNR gais betwee the best ad the worst iterleaver cofiguratio are betwee some teths of db ad up to 2 db (cf TC i Fig 5) The boud (22) is tight for BER values less tha 0 3 for the CC ad for the error floor regio of the TC, while (24) ad (25) ca be used to predict the asymptotic performace of a TC ad a CC respectively Optimized M-iterleavers were always better tha S- iterleavers for the aalyzed cases Improperly desiged M-iterleavers (K W ) ca degrade the system performace compared to K S Thus, whe usig M-iterleavers, the optimizatio of K becomes a madatory step K S ca be worse tha K W (cf for example the CC i Fig 5), so S-iterleavers caot, i geeral, be cosidered as a coservative solutio betwee K B ad K W The assigmet of the coded bits to the positios we preseted ca be see as a code-depedet iterleaver desig that does ot modify the flexibility of BICM which allows the desiger to choose the ecoder idepedetly of the mappig The proposed scheme should ot be cofused for example with TCM where code ad mappig are joitly desiged The oly differece with previous BICM desigs is that here we propose a optimum way of coectig the ecoder ad mapper Also ote that for give values of ad m, the problem of selectig the optimum iterleaver cofiguratio (selectio of K) is a multidimesioal optimizatio problem, however, the optimizatio was performed over oly a limited umber of poits B Optimum Iterleaver ad Code Desig for BICM with Covolutioal Codes It is well kow that ODS codes tabulated for example i [50] are the optimum covolutioal codes for biary trasmissios However, accordig to (25), whe UEP is itroduced by the chael, the optimizatio criterio is differet to [50, Sec II], amely, the iterleaver ad the GWDS of the code must be take ito accout I this sectio we defie the geeralized optimum distace spectrum (GODS) codes, which are the optimum codes for this sceario For a give costrait legth K, code rate R, costellatio size m, ad assumig that the optimum free distace w free for that family of codes is kow (cf for example [50, Table I, II

10 9 ODS GODS Other 0 6-QAM 0 0 w W(wfree) β(w) p= ξwp p, 0 0 w W(wfree) β(w) p= ξwp p, QAM 256-QAM κ, Fig 6 Cost fuctio i (28) for all possible codes with optimum w free for R = /2, 6-QAM, ad K = 9 as a fuctio of the iterleaver parameter The thick solid lie represets the ODS code (56, 753), ad the thick dashed lie the ew code (55, 677) ODS Codes GODS Codes C i Fig 7 Weightig coefficiet of the UB i (25) for the best (K B ) ad worst (K W ) iterleaver desig, K = 9, ad the 2 possible codes with w free = 2 for k c =, = 2 (R = /2) ad m = 2 ( ), m = 3 ( ), ad m = 4 ( + ) The dashed lies represet the rage of variatio betwee the best ad the worst iterleaver desig or III]), ay combiatio of code ad iterleaver will produce a asymptotic BER give by (25) Defiitio : A GODS covolutioal code (C GODS ) is a code that usig a optimized iterleaver cofiguratio (K GODS ) produces a asymptotic BER which is a miimum compared to the values that ay other ecoder ad iterleaver combiatio ca geerate, ie, { } [C GODS, K GODS ] = argmi C,K w W (w free ) β(w) p= ξ wp p,, (28) where C belogs to the set of all codes with optimum w free Usig the previous defiitio, a exhaustive search for pairs [C GODS, K GODS ] with costrait legth up to K = 0 was performed Three differet cofiguratios were tested: code rate R = /2 ( = 2) ad 6-QAM (m = 2), 64-QAM (m = 3) or 256-QAM (m = 4) The optimizatio space for K i these cases was κ, {0, /2, } for m = 2, κ,, κ,2 {0, /3, 2/3} for m = 3, ad κ,, κ,2, κ,3 {0, /2, } for m = 4 The results are preseted i Table II, where the asterisks deote codes foud that are differet from the ODS codes listed i [50] Amog the 24 combiatios studied, 7 resulted i ew optimal codes Extesio to ay other combiatio of code rate ad modulatio order is straightforward I Fig 6 the cost fuctio i (28), which is the iterleaverdepedet factor of UB, for R = /2, 6-QAM, ad K = 9 is preseted as a fuctio of the iterleaver parameter κ, The ODS code (56, 753) is marked with a black thick lie Aalyzig this curve, it is clear that the performace of this code ca be optimized by settig κ, =, ad that the curve has a maximum for κ, = 04 which will result i the worst iterleaver desig for this particular code The cost fuctio obtaied for the code (55,677) (thick dashed lie) is the smallest amog all other codes (icludig the ODS oe) Cosequetly, if the multiplexig uit is adequately desiged settig κ, = 0 (best M-iterleaver), this code is the optimal code for this particular trasmissio with o icrease of complexity However, if the iterleaver is ot optimized, for example settig κ, = /2 (S-iterleaver), the ew code is ot optimal aymore Fially, i Fig 7 the performace of the optimum desig [C GODS, K GODS ] ca be compared with all codes with K = 9 (ad w free = 2) usig the best ad the worst iterleaver desig (K B ad K W ) The dashed lies represet the rage of variatio betwee the best ad the worst iterleaver desig, ie, ay other iterleaver cofiguratio will have a coefficiet betwee the correspodig pair of markers We ote that the optimum desig may sigificatly outperform other codes, eg, 256-QAM ad C 5 i Fig 7 The improvemet with respect to ODS codes is less evidet but clear Thus, the results preseted i this sectio idicate that fidig the iterleaver ad code should be a madatory step i the desig of BICM- QAM V CONCLUSIONS I this paper we developed aalytical bouds to predict the performace of BICM with QAM schemes whe UEP is itroduced by the costellatio labelig Together with the origial uio boud, two asymptotic expressios which are simple to evaluate were developed The aalytical developmets were supported by simulatio results yieldig accurate results We quatified the attaiable gais whe usig optimized M-iterleavers over S-iterleavers for covolutioally-ecoded ad turbo-ecoded schemes These improvemets ca be up to 2 db for the aalyzed cases, ad they ca be obtaied without

11 0 TABLE II OPTIMUM INTERLEAVERS AND CODES FOR R = /2 AND 6, 64, AND 256-QAM ASTERISK ( ) DENOTES A NEW CODE, BETTER THAN THE ODS CODES 6-QAM (m = 2) 64-QAM (m = 3) 256-QAM (m = 4) K w free C GODS κ C GODS κ κ 2 C GODS κ κ 2 κ (5, 7) 0 (5, 7) 0 /3 (5, 7) /2 / (5, 7) (5, 7) 2/3 /3 (5, 7) /2 / (23, 35) (27, 3) 0 /3 (23, 35) /2 / (53, 75) 0 (53, 75) 0 /3 (53, 75) /2 / (33, 7) (35, 47) 0 /3 (35, 47) 0 0 /2 8 0 (247, 37) (225, 373) 0 /3 (247, 37) /2 / (55, 677) 0 (557, 75) 0 /3 (457, 755) /2 / (5, 753) 0 (5, 753) 0 /3 (5, 753) /2 /2 0 complexity icrease but oly if the assigmet of the coded bits to the bit positios i the complex symbol is optimized We also itroduced the cocept of GODS codes, which are the optimum codes for the aalyzed sceario REFERENCES [] E Zehavi, 8-PSK trellis codes for a Rayleigh chael, IEEE Tras Commu, vol 40, o 3, pp , May 992 [2] G Caire, G Taricco, ad E Biglieri, Bit-iterleaved coded modulatio, IEEE Tras If Theory, vol 44, o 3, pp , May 998 [3] G Ugerboeck, Chael codig with multilevel/phase sigals, IEEE Tras If Theory, vol 28, o, pp 55 67, Ja 982 [4] J G Proakis ad M Salehi, Digital Commuicatios, 5th ed McGraw- Hill, 2008 [5] J Hokfelt ad T Maseg, Optimizig the eergy of differet bitstreams of turbo code, i Iteratioal Turbo Codig Semiar, Lud, Swede, Aug 996, pp [6] M M Salah, R A Raies, M A Temple, ad T G Bailey, Eergy allocatio strategies for turbo codes with short frames, i The Iteratioal Coferece o Iformatio Techology: Codig ad Computig, ITCC 2000, Las Vegas, USA, Mar 2000, p 408 [7] W Zhag ad X Wag, Optimal eergy allocatios for turbo codes based o distributios of low weight codewords, Electroics Letters, vol 20, o 9, pp , Sep 2004 [8] T Duma ad M Salehi, O optimal power allocatio for turbo codes, i Iteratioal Symposium o Iformatio Theory, ISIT 997, Ulm, Germay, Jue 997, p 04 [9] A H S Mohammadi ad A K Khadai, Uequal error protectio o the turbo-ecoder output bits, i IEEE Iteratioal Coferece o Commuicatios, ICC 997, vol 2, Motreal, Caada, Jue 997, pp [0], Uequal power allocatio to the turbo-ecoder output bits with applicatios to CDMA systems, IEEE Tras Commu, vol 47, o, pp , Nov 999 [] M A Kousa ad A H Mugaibel, Pucturig effects o turbo codes, Proc IEE, vol 49, o 3, pp 32 38, Jue 2002 [2] Y M Choi ad P J Lee, Aalysis of turbo codes with symmetric modulatio, Electroics Letters, vol 35, o, pp 35 36, Ja 999 [3] S Le Goff, Badwidth-efficiet turbo codig over Rayleigh fadig chaels, It J Commu Syst, vol 5, pp , July 2002 [4] E Dahlma, S Parkvall, J Sköld, ad P Bemig, 3G Evolutio: HSPA ad LTE for Mobile Broadbad, st ed Academic Press, 2007 [5] I Lad ad P Hoeher, Partially systematic rate /2 turbo codes, i Iteratioal Symposium o Turbo Codes ad Related Topics, Brest, Frace, Sep 2000, pp [6] C Wag ad Q Vo, Uequal protectio with turbo decodig for high-order modulated sigalig, i IEEE Wireless Commuicatios ad Networkig Coferece, WCNC 2005, vol 2, New Orleas, USA, Mar 2005, pp [7] E Roses ad Ø Ytrehus, O the desig of bit-iterleaved turbo-coded modulatio with low error floors, IEEE Tras Commu, vol 54, o 9, pp , Sep 2006 [8] H Sakar, N Sidhushayaa, ad K R Narayaa, Desig of lowdesity parity-check (LDPC) codes for high order costellatios, i Global Telecommuicatios Coferece, GLOBECOM 2004, vol 5, Dallas, USA, Nov 2004, pp [9] N vo Deetze ad S Sadberg, Desig of uequal error protectio LDPC codes for higher order costellatios, i IEEE Iteratioal Coferece o Commuicatios, ICC 2007, Washigto, DC, USA, Jue 2007, pp [20] Y Li, C K Ho, Y Wu, ad S Su, Bit-to-symbol mappig i LDPC coded modulatio, i IEEE Vehicular Techology Coferece, VTC Sprig, vol, Stockholm, Swede, Jue 2005, pp [2] R D Maddock ad A H Baihashemi, Reliability-based coded modulatio with low-desity parity-check codes, IEEE Tras Commu, vol 54, o 3, pp , Mar 2006 [22] Y Li ad W E Rya, Bit-reliability mappig i LDPC-coded modulatio systems, IEEE Commu Lett, vol 9, o, pp 3, Ja 2005 [23] M Aydilik ad M Salehi, Turbo coded modulatio for uequal error protectio, IEEE Tras Commu, vol 56, o 4, pp , Apr 2008 [24] I Abramovici ad S Shamai, O turbo ecoded BICM, Aales des Telecommuicatios, vol 54, o 3-4, pp , Mar - Apr 999 [25] A Nilsso ad T M Auli, O i-lie bit iterleavig for serially cocateated systems, i Iteratioal Coferece o Commuicatios ICC 2005, vol, May 2005, pp [26] C Stierstorfer ad R Fischer, Itralevel iterleavig for BICM i OFDM scearios, i 2th Iteratioal OFDM Workshop, Hamburg, Germay, Aug 2007 [27] X Li, A Chidapol, ad J A Ritcey, Bit-iterlaved coded modulatio with iterative decodig ad 8PSK sigalig, IEEE Tras Commu, vol 50, o 6, pp , Aug 2002 [28] U Hasso ad T Auli, Chael symbol expasio diversity improved coded modulatio for the Rayleigh fadig chael, i IEEE Iteratioal Coferece o Commuicatios, ICC 996, vol 2, Dallas, USA, Jue 996, pp [29] X Li ad J A Ritcey, Bit-iterleaved coded modulatio with iterative decodig, IEEE Commu Lett, vol, o 6, pp 69 7, Nov 997 [30] 3GPP, Uiversal mobile telecommuicatios system (UMTS); multiplexig ad chael codig (FDD), 3GPP, Tech Rep TS 2522, V850 Release 8, Mar 2009 [3] A Alvarado, L Szczeciski, R Feick, ad L Ahumada, Distributio of L-values i Gray-mapped M 2 -QAM: Closed-form approximatios ad applicatios, IEEE Tras Commu, vol 57, o 7, pp , July 2009 [32] Y S Leug, S G Wilso, ad J W Ketchum, Multifrequecy trellis codig with low delay for fadig chaels, IEEE Tras Commu, vol 4, o 0, pp , Oct 993 [33] D N Rowitch ad L B Milstei, Covolutioal codig for direct sequece multicarrier CDMA, i Military Commuicatios Coferece 995, MILCOM 95, vol, Sa Diego, CA, USA, 995, pp [34] U Wachsma, R Fischer, ad J B Huber, Multilevel codes: Theoretical cocepts ad practical desig rules, IEEE Tras If Theory, vol 45, o 5, pp 36 39, July 999 [35] E Agrell, J Lassig, E G Ström, ad T Ottosso, Gray codig for multilevel costellatios i Gaussia oise, IEEE Tras If Theory, vol 53, o, pp , Ja 2007 [36] C Stierstorfer ad R Fischer, (Gray) Mappigs for bit-iterleaved coded modulatio, i IEEE Vehicular Techology Coferece 2007, VTC-2007 Sprig, Dubli, Irelad, Apr 2007 [37] F Gray, Pulse code commuicatios, U S Patet , Mar 953

12 [38] E Agrell, J Lassig, E G Ström, ad T Ottosso, O the optimality of the biary reflected Gray code, IEEE Tras If Theory, vol 50, o 2, pp , Dec 2004 [39] K Hyu ad D Yoo, Bit metric geeratio for Gray coded QAM sigals, IEE Proc-Commu, vol 52, o 6, pp 34 38, Dec 2005 [40] A J Viterbi, A ituitive justificatio ad a simplified implemetatio of the MAP decoder for covolutioal codes, IEEE J Sel Areas Commu, vol 6, o 2, pp , Feb 998 [4] Ericsso, Motorola, ad Nokia, Lik evaluatio methods for high speed dowlik packet access (HSDPA), TSG-RAN Workig Group Meetig #5, TSGR#5(00)093, Tech Rep, Aug 2000 [42] M K Simo ad R Aavajjala, O the optimality of bit detectio of certai digital modulatios, IEEE Tras Commu, vol 53, o 2, pp , Feb 2005 [43] B Classo, K Blakeship, ad V Desai, Chael codig for 4G systems with adaptive modulatio ad codig, IEEE Wireless Commu Mag, vol 9, o 2, pp 8 3, Apr 2002 [44] A Alvarado, H Carrasco, ad R Feick, O adaptive BICM with fiite block-legth ad simplified metrics calculatio, i IEEE Vehicular Techology Coferece 2006, VTC-2006 Fall, Motreal, Caada, Sep 2006 [45] A J Viterbi ad J K Omura, Priciples of Digital Commuicatios ad Codig McGraw-Hill, 979 [46] D Divsalar, S Doliar, R J McEliece, ad F Pollara, Trasfer fuctio bouds o the performace of turbo codes, JPL, Cal Tech, TDA Progr Rep 42-2, Aug 995 [47] J Belzile ad D Haccou, Bidirectioal breadth-first algorithms for the decodig of covolutioal codes, IEEE Tras Commu, vol 4, o 2, pp , Feb 993 [48] S Beedetto ad G Motorsi, Uveilig turbo codes: Some results o parallel cocateated codig schemes, IEEE Tras If Theory, vol 42, o 2, pp , Mar 996 [49] L C Perez, J Seghers, ad D J Costello, Jr, A distace spectrum iterpretatio of turbo codes, IEEE Tras If Theory, vol 42, o 6, pp , Nov 996 [50] P Freger, P Orte, ad T Ottosso, Covolutioal codes with optimum distace spectrum, IEEE Tras Commu, vol 3, o, pp 37 39, Nov 999