Scientia Iranica, Vol. 13, No. 4, pp 337{347 c Sharif University of Technology, October 2006 Performance Evaluations and Comparisons of Several LDPC C

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1 Scenta Iranca, Vo. 13, No. 4, pp 337{347 c Sharf Unversty of Technoogy, Octoer 006 Performance Evauatons and Comparsons of Severa LDPC Coded MC-FH-CDMA Systems H. Behrooz, J. Haghghat 1, M. Nasr-Kenar and S.H. Jama 3 In ths paper, the appcaton of reguar Low-Densty Party-Check (LDPC) codes n Mut- Carrer Frequency-Hoppng (MC-FH) CDMA systems s studed. To ths end, derent weknown constructons of reguar LDPC codes are consdered and the performance of LDPC coded MC-FH-CDMA systems, ased on these constructons, are evauated and compared n a frequency-seectve sowy Rayegh fadng channe. These resuts are compared wth those prevousy reported for super-orthogona convoutonay coded MC-FH-CDMA systems. The smuaton resuts ndcate that the LDPC coded MC-FH-CDMA system sgncanty outperforms the uncoded and super-orthogona convoutonay coded schemes. To aevate the restrctons mposed y we known LDPC constructon methods when apped to the coded MC-FH-CDMA system consdered, a new sem-random constructon s proposed and ts performance s evauated n the coded scheme. The smuaton resuts ndcate that ths new constructon sustantay outperforms other we-known constructon methods n the appcaton consdered. INTRODUCTION Recenty, a mutcarrer frequency-hoppng CDMA (MC-FH-CDMA) system, usng the concept of frequency dversty on a phase-coherent Frequency- Hoppng Spread Spectrum (FHSS) system [1], was proposed [] as a canddate for future hgh-rate mutmeda wreess mutpe access communcaton systems [3,4]. In ths system, the tota gven andwdth s equay dvded nto N s suands, each contanng N h derent orthogona carrer frequences. At each t nterva,, for each user, k, N s carrers are chosen from N s dstnct suands. These N s carrers are moduated wth the th data t of the user, usng nary phase shftkeyng (BPSK) moduaton. These moduated carrers are then added together and transmtted through the channe. For the next sgnang nterva, each of these N s carrers ndependenty hops n ts suand and *. Correspondng Author, Department of Eectrca and Computer Engneerng, Concorda Unversty, Montrea, Queec, Canada. 1. Department of Eectrca and Computer Engneerng, Concorda Unversty, Montrea, Queec, Canada.. Department of Eectrca Engneerng, Sharf Unversty of Technoogy, Tehran, I.R. Iran. 3. Department of Eectrca and Computer Engneerng, Unversty of Tehran, Tehran, I.R. Iran. another frequency n the suand s chosen. The frequency-hoppng pattern, whch s determned y the dedcated sgnature sequence of the user, s assumed to e known n the recever. After dehoppng at the recever sde, N s suand correators separate the sgna transmtted n derent suands. These N s detected sgnas are used to make a decson on the transmtted data t. To ths end, the correators' weghted outputs are smpy comned and the resut s compared to threshod zero to make a decson on the transmtted data t. Note that the moduatng and demoduatng of the N s carrers smutaneousy can e mpemented easy y usng the Inverse Fast Fourer Transform (IFFT) and the Fast Fourer Transform (FFT), respectvey. In the MC-FH-CDMA system descred aove, the dversty s otaned va oth mutcarrer transmsson and frequency-hoppng. The conventona Fast Frequency-Hoppng (FFH) systems, whch transmt one carrer at a tme and change t n a fracton of a t duraton, make coherent demoduaton reatvey dcut. However, the MC-FH-CDMA system aows sower carrer frequency hoppng and mposes each carrer to hop soey n a fracton of the tota gven andwdth. Ths system s thoroughy studed and anayzed n [5]. Thus, wth sow frequency-hoppng, wth a perod at east equa to the t duraton, a

2 338 H. Behrooz, J. Haghghat, M. Nasr-Kenar and S.H. Jama coherent recepton w e fease n a sowy fadng channe []. (Note that n some appcatons, FFH systems are more pause and desrae, such as mtary appcatons. Wth fast hoppng, t s much more dcut for the transmsson to e ntercepted y an undesred user.) To vsuaze the derence, the tradtona sow and fast frequency hoppng schemes are depcted n Fgure 1. Even though coherent detecton gves a somewhat superor performance compared to the non-coherent case, n a case of fast fadng, a non-coherent recepton s nevtae. Snge user performance of MC-FH and FFH systems usng non-coherent detecton has een evauated n [6], n whch t has een shown that the non-coherent MC-FH system outperforms the FFH system when the channe deay spread s severe, whe the FFH system s superor to the MC-FH system for a fast fadng channe. The mutuser performance of a MC-FH system, wth coherent and non-coherent detectons n addtve whte Gaussan nose (AWGN) and a frequency-seectve sowy Rayegh fadng channe, s evauated n [7]. The resuts n [7] have shown that the coherent detecton sustantay outperforms the non-coherent detecton. In addton, t has een shown that [8] the mutuser performance of MC-FH and FFH systems s amost dentca, when utzng non-coherent detecton. In ths paper, t s assumed that the coherent recepton s fease and the focus s on coherent detecton. To expot the gven andwdth more ecenty, n [5], the authors have proposed to use a practca ow rate error correctng code n the MC-FH- CDMA system, whch does not requre any addtona andwdth to that needed n an uncoded MC-FH- CDMA system. The dea s as foows: Instead of sendng the N s carrers at each t nterva wth an dentca phase (0 or 180 ), whch s determned y the correspondng nput data t, these carrers can Fgure 1. Sow and fast frequency hoppng schemes. e sent wth derent phases, the vaues of whch are determned wth output symos of an encoder. In fact, wth the aove nsght, the uncoded scheme s consdered as a coded scheme wth a repetton ock code of rate 1=N s. Snce a repetton code s not a good code, t s expected that appyng a more powerfu code wth the same rate, 1=N s, sustantay mproves the system performance wthout requrng any andwdth expanson n addton to that needed y the uncoded scheme. The error correctng code consdered n [5] s a super orthogona convoutona code [9], as ts path generatng functon s avaae for performance evauaton. The authors have shown [5] that the super orthogona convoutonay coded MC- FH-CDMA system sustantay outperforms the uncoded scheme. On the other hand, LDPC codes, orgnay ntroduced y Gaager [10] and redscovered y MacKay and Nea [11], have receved consderae attenton recenty and ther appcatons n varous communcaton systems, ncudng OFDMA, Moe Satete and CDMA systems, have een consdered. The nterest n these codes s due to ther near Shannon mt performance and ther smpe descrptons, mpementatons and decodng agorthms. In ths paper, the appcaton of a LDPC code n a MC-FH-CDMA system s consdered and the performance of the LDPC coded system s evauated, usng the coded scheme ntroduced n [5] and descred aove, n a Rayegh fadng channe. Derent constructons of reguar LDPC codes, are examned, namey Gaager [10], MacKay [11] and semrandom [1] constructons. It w e shown that for the appcaton consdered, these constructons have some practca mtatons. To aevate those mtatons, a moded sem-random constructon s proposed. The smuaton resuts ndcate that the LDPC coded systems n the cases consdered perform etter than the super orthogona coded systems. Furthermore, the resuts show that the proposed moded sem-random constructon dspays a etter performance than the other we-known reguar constructon methods for the appcaton consdered, despte ts very ow encodng compexty. The outne of the paper s as foows. In the foowng secton, the LDPC coded MC-FH-CDMA system and ts conventona snge user recever structure are descred. Then, a ref descrpton of LDPC codes and ther derent constructons are presented. After that, the performance of a LDPC coded MC-FH- CDMA system n Rayegh fadng channes s evauated and some smuaton resuts are provded. Then, the proposed moded sem-random constructon of LDPC codes s descred and ts performance n the MC- FH-CDMA system s evauated. Fnay, the paper s concuded.

3 LDPC Coded MC-FH-CDMA Systems 339 SYSTEM DESCRIPTION In ths secton, rst, a ref descrpton of the MC- FH-CDMA system s provded, as ntroduced n [1,5]. In ths system, every transmtter sends N s carrers for each data t usng BPSK moduaton. The carrers are spaced apart n sequenta suands. The tota gven frequency andwdth s equay parttoned nto N s suands, where each suand contans N h derent frequency carrers spaced apart y f d. f d s chosen such that every par of carrers s orthogona,.e., f d = 1=T s, where T s denotes a t tme duraton. Thus, the tota frequency carrers avaae for mutpeaccess communcatons are N s N h. In ths system, the equvaent aseand sgna transmtted y user k can e wrtten as foows: s (t) = X s 1 =0 p Pw d ej(f +c f d)(t T s) P Ts (t T s ); where ndex ndcates the suand numer, P w s the transmt power of each carrer and fd g s the transmtted nary sequence of user k. Ths sequence moduates the dedcated carrers. fc g s the pseudorandom sequence of user k, whch determnes the carrer frequency seected from each suand,, durng the th t nterva. The eements of ths sequence are ndependent and dentcay dstruted (..d) random varaes, whch take on nteger vaues n the nterva [0;N h 1]. P (t) s a rectanguar puse over the nterva [0;] wth amptude equa to 1. f s the rst carrer frequency n suand and s equa to N h f d, =0; 1; ; ;N s 1. In an uncoded scheme, sequence d s N s repettons of the user nformaton data sequence,.e., d = D, for = 0; 1; ;N s 1, where D s the th nformaton t of user k. Thus, the uncoded scheme can e consdered as a coded scheme wth a smpe repetton ock code of rate 1=N s. Snce the repetton code s not a good code, a more powerfu code can e used to mprove system performance. Use of a LDPC code s suggested wth rate 1=N s. In the coded scheme consdered, d n Equaton 1 s the th code t of user k at nformaton t nterva. Thus, n ths coded scheme, the phase of each carrer s determned y the correspondng coded symo. Fgure presents a ock dagram of the transmtter structure. In ths gure, the LDPC encoder s mpemented n the same way as any near ock code, y knowng ts generator or party-check matrx. The channe s consdered as a frequency-seectve sowy Rayegh fadng channe wth an AWGN. However, each carrer s assumed to experence a frequencynon seectve fadng channe,.e., f d << (f) c, where (f) c s the coherence andwdth of the channe. Snce, wth very hgh proaty, the dstance among adjacent carrers s arge as they are ocated n derent suands, note that the average frequency dstance etween adjacent carrers s N h f d.) t can we e assumed that the dedcated carrers experence ndependent at fadng. In addton, the Dopper shft s assumed to e sma enough to have a sow fadng. The aove assumptons are reasonae for most practca appcatons. Propagaton measurements n wreess envronments, ncudng ndoor, open rura, suuran and uran areas, show that the deay spread s typcay dstruted over the range of [0.1 s, 3 s] [13]. So, for nstance, at a t rate equa to 1Mps for ndoor Fgure. Transmtter ock dagram of user k.

4 340 H. Behrooz, J. Haghghat, M. Nasr-Kenar and S.H. Jama envronments, whch has the deay spread of 0.1 s, the coherence andwdth s arger than the symo rate. As a resut, each carrer experences a at fadng. Furthermore, for N h greater than 0, the average frequency dstance etween adjacent carrers (.e., N h f d ) s greater than the coherence andwdth and, as a resut, each carrer experences ndependent fadng. Note that for future very hgh rate wreess communcaton networks, to make the carrers experence ndependent fadng, a sera-to-parae converter (ke n OFDMA and MC-CDMA systems) mght e requred efore MC-FH-CDMA moduaton, n order to reduce the symo rate n each parae ranch to much ess than the coherence andwdth of the channe. Due to the random constructon and very sparse property of the party-check matrx of a LDPC code, the decoder of a LDPC code has a ut-n \ntereaver" [14]. So, one can we assume that the successve symos transmtted y the same carrer n a suand aso experence ndependent fadng. Let g = ej e the channe compex gan oserved y the th carrer of user k at t nterva. Then, g s for derent vaues of k, and are ndependent. Furthermore, snce the channe s assumed to e sowyfadng,.e., the compex channe gan correspondng to each carrer does not change sgncanty durng severa t ntervas, the channe gan can e we estmated at the recever. However, n the current work, t s assumed that perfect estmatons of the channe parameters are avaae and the eects of parameter estmaton errors are not consdered n the performance evauatons. Aso, the eects of the synchronzaton errors are not consdered. For some recent work on channe estmatons and synchronzaton proems n a MC-FH CDMA system and ther error eects on the system performance, pease see [4,15]. As the channe s assumed to have Rayegh dstruton, 's are ndependent from 's, wth,. Aso, 's w have a unform dstruton over the nterva [0; ]. The receved sgna, due to user k, s equa to: X s 1 p s rec(t) = g Pw d ej(f +c f d)(t T s) =0 P Ts (t T s ); () For smpcty of presentaton, a synchronous system wth perfect power contro s assumed. Then, the tota receved sgna can e wrtten as: r(t) = XN u k=1 s rec(t)+(t); (3) where N u s the numer of actve users and (t) s AWGN wth a two sded power spectra densty of N 0 =. By susttutng Equaton n Equaton 3, one has: r(t) = XN s k=1 X X N s 1 =0 p Pw g d ej(f +c f d)(t T s) P Ts (t T s ) + (t): (4) In the foowng, the recever structures for uncoded and coded schemes are descred. Wthout any oss of generaty, t s assumed that the desred user s user 1. Uncoded Scheme For ths system, a snge user Maxma Rato Comnng (MRC) recever s consdered. Let the desred user e user 1. The we-known MRC recever makes a decson at each t nterva,, accordng to the foowng rue: Z z } m; { Suand correator output ( Ns 1 z X Z } { Re g 1 (+1)Ts ) m; r(t)e j(fm+c m; fd)t dt T m=0 s T s {z } Coded Scheme Decson varae,z 1? 0: 0 (5) The ock dagram of the recever for user 1 s shown n Fgure 3. In ths recever, at each nformaton t nterva,, the N s receved coded symo sgnas carred y the N s carrers of that nterva are rst demoduated and, then, are gven to the dehopper foowed y a suand correator. The suand correator outputs, as dened n Reaton 5, are mutped y the conjugate of the channe gans g 1) m; 's. The rea part of these resuts,.e., Re(Z m; )'s n Reaton 5, are gven to the parae to sera converter and then passed to the LDPC decoder. In fact, Re(Z m; ) denotes the receved sampe (the decoder soft nput) correspondng to the mth transmtted code t at the th nformaton t nterva. LDPC CODES LDPC codes, orgnay nvented and nvestgated y Gaager [10] n 196, are near ock error-correctng codes ased on very sparse party check matrces. A(N;d v ; d c )-reguar LDPC code s a near nary code determned y the condton that each code t partcpates n exacty d v party-check equatons and that each such check equaton conssts of exacty d c code ts. In other words, the correspondng partycheck matrx, H, has d v ones n each coumn and

5 LDPC Coded MC-FH-CDMA Systems 341 Fgure 3. Recever ock dagram (coded scheme) of user 1. d c ones n each row. Thus, the numer of party ts and nformaton ts are, respectvey, equa to M = N:d v =d c and K = M N, where N s the ength of the code. In other words, as oth N:d v and M:d c represent the numer of edges n the correspondng partte graph of the party check matrx, they are equa. Therefore, the rate, R, of the code s gven y R =1 (M=N) =1 (d v =d c ). LDPC Encodng In the foowng, the three we-known constructon methods of reguar LDPC codes are rey descred, namey, Gaager, MacKay and sem-random constructons, whch are apped to the coded MC-FH-CDMA systems n the next sectons. Gaager Constructon In ths constructon, the party-check matrx s dvded horzontay nto d v equa sze sumatrces, each contanng a snge `1' n each coumn. The rst sumatrx s constructed as foows: the th row contans 1's n coumns ( 1)d c + 1 to d c. The susequent sumatrces are merey random coumn permutatons of the rst one. MacKay Constructon MacKay has presented [11] severa constructon methods for reguar LDPC codes. The est one s used, for whch the party-check matrx, H, s generated randomy, wth coumn Hammng weght d v, row Hammng weght as unform as posse, and overap etween any two coumns s no greater than one. Sem-Random Constructon The major proem wth reguar LDPC codes s ther very hgh encodng compexty. Snce the party-check matrx of the code s sparse, the assocated generator matrx w e dense and, as a resut, the encodng process requres a hgh numer of computatons. To overcome ths proem, the concept of sem-random LDPC codes has een ntroduced [1]. In sem-random constructon, to produce a (N;K) code, the foowng party check matrx, H MN = [h j ], s used: H MN = M 0 K d M 0 K {z } M where M = N K and M 0 = M=d v. Each component matrx, M 0 K, has a random constructon wth coumn weght equa to one and row weght equa to d 0 c, where d 0 c = Kd =M. As can e oserved, n ths constructon, the party check matrx, H, conssts of two parts, ony one part of whch s generated randomy. For the systematc codeword,.e., V = (p 1 ; ;p M ; u 1 ; ;u k ), party ts are computed as: 8 >< >: P K p 1 = u j=1 jh P 1;M+j (mod ) K p m = p m 1 + u j=1 jh m;m+j (mod ) m M (6) where h j s the eement n the th row and jth coumn of the party-check matrx and u 1 ; ;u k are nformaton ts. Ecent encodng s acheved y drecty computng the party-check ts from Equaton 6 wthout any requrement to compute the generator matrx. ;

6 34 H. Behrooz, J. Haghghat, M. Nasr-Kenar and S.H. Jama One of the advantages of the sem-random constructon s ts Unequa Error Protecton (UEP). In [16], Davey has shown that n an rreguar LDPC code, n whch symos do not a partcpate n the same numer of party-checks, those symos partcpatng n the hghest numer of party-check equatons receve the most nformaton durng the decodng process. As a resut, the vaue of those symos s determned more qucky and more accuratey than the \ess prveged" symos. Ths property s caed as Unequa Error Protecton (UEP). In the sem-random constructon, nformaton ts have a degree of d v (d v > ) and party ts have a degree of, except the party t, p M, whch s of degree one. Therefore, the nformaton ts partcpate n the hghest numer of party-check equatons and w e decoded more qucky wth hgher reaty. Because of ths property, as seen ater, the sem-random code gves a etter performance n comparson wth the other two reguar LDPC codes when apped to MC-FH-CDMA systems, despte ts ower encodng compexty. LDPC Decodng Iteratve soft decodng of LDPC codes can e mpemented usng the Beef Propagaton (BP) agorthm [11]. In ths agorthm, nformaton s exchanged etween neghorng nodes n the graph y passng messages aong the edges. At the nta stage, ths agorthm requres the knowedge of the a pror proaty of the transmtted code t. Let P (d n = z) e the a pror proaty of code t d n. The vaue of ths proaty s fuy reated to the moduaton used and the channe characterstcs. In the foowng, n the Bnary Phase Shft Keyng (BPSK) moduaton, t s assumed that 0 s mapped to -1 and 1 s mapped to 1. Let y n e the nth receved sampe at the LDPC decoder nput,.e., the rea part of Z m;, as dened n Reaton 5, correspondng to code t d n. Then, the Log Lkehood Rato (LLR) of y n s computed as foows: LLR = og p(y n=d n = 0) p(y n =d n = 1) : (7) Then, from Equaton 7, the a pror proaty of the correspondng code t, d n, s easy otaned as: P (d n = z) =1=(1 + exp((z 1) LLR)): (8) In the foowng secton, for performance evauaton, the a pror proaty of the code t s rst computed. Then, y usng these vaues at the nta stage of the BP agorthm, the performance of the coded system s evauated y smuaton. PERFORMANCE EVALUATION (RAYLEIGH FADING CHANNEL) A Pror Proaty Evauaton for the LDPC Decoder As mentoned efore, a pror proates of the code ts are requred y the LDPC decoder mpemented y the BP agorthm. To ths end, one must rst compute the dstruton of the nterference term n the receved sampes at the LDPC decoder nput. Through ths secton and susequent sectons, smar to [5], the dscrete moment generatng functon (mgf) approach s used for computng the dstruton. Wthout any oss of generaty, the receved sgna at the nformaton t nterva s consdered to e zero. So, the sundex zero s dropped from Z m;0, dened n Reaton 5. From Equaton 4, the receved sgna at t nterva zero s equa to: r(t) = XN u k=1 s 1 =0 g d p Pw e j(f +c f d )t + (t); (9) where, for the sake of smpcty, g, ;0 d ;0 and c have ;0 een repaced wth g, d and c, respectvey. For the desred user (user 1), from Equatons 4 and 5, the suand correator output,.e., Z m, s smpy otaned as foows: Z m = g m T s = g m T s Z Ts 0 Z Ts ( Nu 0 r(t)e j(fm+c m f d)t dt X k=1 s 1 =0 g p d Pw e j(f f m+(c c m )f d)t +(t)e j(fm+c m f d)t ) dt: (10) As the carrers are orthogona, the ntegra n Equaton 10 s nonzero ony for = m. Consequenty, one has: Z m = jg m j d m p Pw XN u g m g m d p m Pw Z T s + T s k= 0 e j(c m c m )f dt dt+ m ; (11) where m s the compex nose component. In Equaton 11, the rst term s due to the desred user sgna and the second and thrd terms are the mutpe-access nterference and whte Gaussan nose components, respectvey. The rea part of the suand correator's

7 LDPC Coded MC-FH-CDMA Systems 343 output,.e., Re(Z m ), s smpy otaned as: y m = Re(Z m ) = d m ( m ) p P w + I z m } { XN u m m d p m Pw Z cos( m m ) T s e j(c m c m )fdt dt T s k= 0 {z } I m + m ; where g m = m e j m, I m s the nterference caused y nterferng user k at mth suand correator's output, and I m s the tota nterference due to a nterferng users at the mth suand correator output of user 1. In Equaton 1, m, the nose component, s a Gaussan random varae wth zero mean and varance equa to N 0 ( m ) =T s (or, equvaenty, equa to N s P w ( m ) = ), where s the receved sgna to nose rato per t. It s necessary to otan the mgf of I m condtoned on g m. To ths end, the mgf of the nterference caused y each nterferng user k must e determned. Then, as the nterferng user components are ndependent, the mgf of the tota nterference s computed y mutpyng the mgfs of derent nterferng users' components. Under the assumpton of fu power contro, the mgfs of the nterference, due to derent users, are dentca. Therefore, t s sucent to determne the mgf of the nterference caused y ony one user. Snce random varae m m has unform dstruton on nterva [ m ; m ] and m has a Rayegh dstruton wth m = = 1 and, aso, m and m m are ndependent, t can e concuded that m cos( m m ) has a Gaussan dstruton wth zero mean and varance equa to 1/ [17]. As a resut, t can easy e shown that random varae m d p m Pw m cos( m m ) has aso a Gaussan dstruton wth zero mean and varance equa to P w ( m ). On the other hand, the ntegra n Equaton 1 s equa to T s for c m = c m and zero for other vaues of c m. Consderng the dstruton of fc m g's, the proaty that c m s equa to c m s = 1=N h and the proaty that c m s not equa to c m s = 1 = 1 1=N h. Consequenty, the moment generatng functon of I m, condtoned on m = Re(g m ), s computed as: I m j m (s) = + exp s P w m ; (13) where m has een repaced wth m. Then, from Equatons 1 and 13, the mgf of I m s smpy otaned as: Imjm (s) = = YN u k= I m j m (s) + exp s P w m Nu 1 : (14) Now, from Equaton 1, one can easy compute the condtona mgfs of the soft receved sampes,.e., y m, condtoned on m and the correspondng transmtted code symo,.e., d m, as foows: ymjm;d m (s) = y mjm;1(s) = exp(s mp Pw ) : = + exp s P w m u 1 =0 Nu 1 + (P w + P w : N s ) s Nu 1 : exp Nu 1 exp! s P w :N s : m s p P w ( m); (15) where the second equaty smpy foows from poynoma expanson of the second mutpcaton term. Wthout oss of generaty, p P w = 1 s set. Snce m has a Rayegh dstruton, m have a ch-square dstruton of order. Takng the expectaton on m 's and usng some agerac manpuatons ead to: Nu 1 ymj1 (s) = = u 1 =0 u 1 =0 1 s p ;1 + 1 s Nu 1 Nu 1 + Ns s 4 Nu 1 q Ns 1 p ; s where p ;1 and p ; are gven, as foows: p ;1 = q ; Ns ; (16) p ; = q : (17) Ns p ;1 s are a negatve and p ; s are a postve. Aso, the Regon Of Convergence (ROC) of an mgf aways

8 344 H. Behrooz, J. Haghghat, M. Nasr-Kenar and S.H. Jama ncudes the axs, s = j!. Therefore, from Equaton 16, the condtoned proaty densty functon (pdf) of soft receved sampes,.e., y m s, can easy e otaned usng an nverse Lapace transform, as foows: f ym=1 = : exp u 1 =0 Nu 1 Nu 1 q Ns y + Ns m jy m j: s 1++ N s!! : (18) Now, one can easy use Equatons 7, 8 and 18 to compute the a pror proaty of a transmtted code t, whch s requred y the LDPC decoder. Computaton of a Pror Proates Under the Gaussan Assumpton If one assumes that the dstruton of nterference at the output of the suand correator s Gaussan, the mean and varance of ths random varae shoud e determned, n order to compute the a pror proates. In terms of mgf, the mean and varance of an artrary random varae, X, can e expressed as: X = EfXg = d X(s) j S=0 ; ds and: X = EfX g (EfXg) = d X (s) ds j S=0 dx (s) j S=0 : ds Usng Equaton 16, t can e shown that: and: d Imjm (s) j S=0 = 0; ds d Imjm (s) ds j S=0 = P W (N u 1) m: So, at the weghted output of each suand correator, m, one has a Gaussan sgna, wth the foowng mean and varance: ( p ym = m Pw y m = Therefore: (N u 1) + Ns P w m: Imjm;d (s)=exp s+ s m 4 (N u 1)+ N s m (19) ; where p P w = 1. After takng expectaton on m s and dong some manpuatons, the condtona pdf of y m can e otaned, as foows: 1 f ym=1 = q 1 + Nu 1 N h + Ns :exp N u 1 y N h + Ns m jy m j: s 1+ N u 1 + N s N h!! : (0) Ths formua smpes the computatona compexty of the a pror proaty formua n Equaton 18. In the next susecton, the performance of LDPC codes n a MC-FH-CDMA system s evauated, ased on oth exact and Gaussan approxmated evauatons of the a pror proates. Smuaton Resuts In ths secton, some smuaton resuts are presented to evauate the performance of LDPC coded MC-FH- CDMA systems. MacKay, Gaager and sem-random constructons of the LDPC codes are consdered. The eef propagaton agorthm s used for decodng of the LDPC codes, n whch Equatons 7, 8 and 18 are rst used to compute the a pror proaty of the transmtted code t. Under Gaussan assumpton, the a pror proaty s cacuated, usng Equaton 0 nstead of Equaton 18. Then, the BP agorthm uses ths a pror proaty for ts nta stage, to decode the transmtted code ts. A processng gan of N h N s = 30 and sgna to nose rato per t ( ) of 1 db are assumed. Fgures 4a and 4 present the pots of the Bt Error Rate (BER) versus the numer of users for uncoded and coded schemes n a Rayegh fadng channe at a numer of suands,.e., N s, equa to and 4, respectvey. The Rayegh parameter,, s set to one and the maxmum numer of decodng teratons ( m ) for LDPC codes s set to To evauate the t error proaty, ocks are transmtted. In these gures, the performance of LDPC coded schemes, wth a data ock ength of 500 and d v = 3, s compared wth the ower ound of BER of the superorthogona convoutona coded schemes, as reported n [5]. (Note that the performance evauaton of the super-orthogona coded schemes n [3] s ased on the anaytca resuts n whch ony the ower and upper ound of BER can e computed usng the path generatng functon of the super-orthogona code.) Even though the super-orthogona codes show a good performance at BER=10 3, whch s sutae for voces at ower BER, requred for hgh-quaty servces such as data, LDPC codes perform sustantay etter. For nstance, at BER=10 5, the numer of users supported

9 LDPC Coded MC-FH-CDMA Systems 345 Fgure 5. Bt error rate versus the numer of users n LDPC coded MC-FH-CDMA system (MacKay constructon). Fgure 4. Bt error rate versus the numer of users n MC-FH-CDMA system. s set to 1dB. The average numer of decodng teratons s for (a) and for (). y the sem-random code s aout 5 (for N s = ) and 60 (for N s = 4), whereas, y the super-orthogona codes, t s, at most, (for N s = ) and 30 (for N s = 4). From these gures, t can aso e reazed that the semrandom code, despte ts encodng smpcty, performs etter than the more compex Gaager and MacKay codes. The performance of the Gaussan dstruton assumpton for mutuser nterference at the output of the suand correator s ustrated n Fgure 5, where the t error rate versus the numer of users for N s = 4 under MacKay constructon s evauated. As can e reazed, the Gaussan assumpton, whe smpfyng the computaton of a pror proates, Fgure 6. BER versus the numer of users n LDPC coded MC-FH-CDMA system (MacKay constructon) parameterzed y the maxmum numer of teratons (m). The average numers of decodng teratons for m = 1000, 00 and 50 are , 5.673, and , respectvey. performs we n the LDPC coded MC-FH-CDMA system. To consder the eect of a maxmum numer of decodng teratons n the performance of a LDPC decoder, the BER versus the numer of users parameterzed y the maxmum numer of teratons s shown n Fgure 6. It can e reazed that the performance of a LDPC decoder wth m = 00 s cose to the performance of the LDPC decoder wth m = 1000.

10 346 H. Behrooz, J. Haghghat, M. Nasr-Kenar and S.H. Jama MODIFIED SEMI-RANDOM CONSTRUCTION The we-known constructons of reguar LDPC codes, rey descred n the prevous secton, mpose some restrctons. That s that M (the numer of party ts) must e dvse y d v. The sem-random constructon aso requres the dvsty of K d v y M. Furthermore, n the appcaton consdered (MC- FH-CDMA), the code rate must e equa to 1=N s, n order not to have any andwdth expanson, due to the codng apped. Thus, K d v M = R d v 1 R = d v N s 1 ; where the rst equaty foows from denton of the code rate, R = K N = K K+M, and the second equaty foows from the requrement of the coded MC-FH- CDMA,.e., R = 1 N s. As a resut, ths requrement mposes the mtaton of the dvsty of d v y N s 1. To overcome these restrctons for the appcaton consdered, a new constructon w e presented, ased on the sem-random code, whch w e caed \moded sem-random" constructon. In ths method, an approprate vaue s chosen for d v (wthout any restrcton) and d 0 v s computed as foows: K d 0 dv v = : () M The determnstc part of the party check matrx, H, s smar to that of the sem-random constructon and the random part s generated y matrx HKM 0 wth weght d 0 v per coumn, usng the MacKay constructon. So, the matrx, H, for ths constructon s gven as foows: H MN = j j j HMK 0T j j Note that, n ths constructon, the vaue of d v does not depend on the vaue of N s. As a resut, for the appcaton consdered, ts vaue can e seected sma enough for good performance [10], even for hgh vaues of N s. The performance of the LDPC coded MC-FH- CDMA system has een evauated usng the proposed constructon y smuaton. Fgures 7a and 7 present the pots of BER versus the numer of users for derent constructons of LDPC codes. As can e reazed, the smpe moded sem-random constructon outperforms the other constructons, especay at hgh vaues of N s. For nstance, at N s = 4 and BER : Fgure 7. Performance of derent constructon methods of reguar LDPC codes for MC-FH-CDMA system. equa to 10 5, the numer of users supported y MacKay, Gaager, sem-random and moded semrandom constructons are aout 46, 47, 57 and 65, respectvey. CONCLUSION In ths paper, a LDPC coded mut-carrer frequencyhoppng CDMA scheme was rst consdered, whch does not requre any addtona andwdth to that needed y an uncoded spread spectrum MC-FH- CDMA system. Then, the performance of the coded system was evauated, usng derent constructons of LDPC codes n a Rayegh fadng channe. The smuaton resuts have ndcated that the coded schemes sustantay outperform the uncoded scheme. Furthermore, t has een reazed that the LDPC coded scheme performs sustantay etter than the super-

11 LDPC Coded MC-FH-CDMA Systems 347 orthogona convoutonay coded scheme (prevousy reported n [5]) at ow BERs. It has aso een oserved that the sem-random LDPC codes perform very we, despte ther smpe encodng structures. Then, to overcome the restrctons mposed y the we-known constructons of LDPC codes for the appcaton consdered, a moded sem-random constructon has een proposed. Despte ts encodng smpcty, whch s the same as that of the sem-random constructon, the proposed moded constructon outperforms prevous constructon methods. REFERENCES 1. Lance, E. and Kaeh, G.K. \A dversty scheme for a phase-coherent frequency-hoppng spread-spectrum system", IEEE Trans. Commun., 45(9), pp (Sept. 1997).. Shn, O.S. and Lee, K.B. \Performance comparson of FFH and MCFH spread-spectrum systems wth optmum dversty comnng n frequency-seectve Rayegh fadng channes", IEEE Trans. Commun., 49(3), pp (March 001). 3. Nkjah, R. and Nasr-Kenar, M. \Uned performance anayss of severa mutcarrer spread-spectrum mutpe-access systems n correated Rayegh fadng channes", n Proceedngs of Fourth IEEE Conference on Moe and Wreess Communcatons Networks, MWCN, Stockhom, Sweden (00). 4. Nkjah, R. and Nasr-Kenar, M. \Uned mutpeaccess performance anayss of severa mut-rate mutcarrer spread-spectrum systems", n Proceedngs of IEEE Vehcuar Technoogy Conference (VTC) Fa 004, Los Angees, CA, USA (004). 5. Erahm, T.M. and Nasr-Kenar, M. \Performance anayss of mutcarrer frequency-hoppng (MC-FH) code dvson mutpe-access systems: Uncoded and coded schemes", 55th IEEE Vehcuar Technoogy Conference, 00,, pp Aso appeared n IEEE Trans. Vehcuar Technoogy (Juy 004). 6. Shn, O.S. and Lee, K.B. \Performance comparson of FFH and MCFH spread-spectrum systems wth optmum dversty comnng n frequency-seectve Rayegh fadng channes", IEEE Transactons on Communcatons, 3, pp (March 001). 7. Zenapour Yazd, Z. and Nasr-Kenar, M. \Performance comparson of coherent and non-coherent mutcarrer frequency-hoppng code dvson mutpeaccess systems", n Proceedngs of IEEE Internatona Symposum on Persona, Indoor and Moe Rado Communcatons, PIMRC 004, Span (004). 8. Zenapour Yazd, Z. and Nasr-Kenar, M. \Mutuser performance comparsons of fast frequency hoppng and mutcarrer sow frequency hoppng systems: Uncoded and coded schemes", n Proceedngs of IEEE Vehcuar Technoogy Conference (VTC) Fa 004, Los Angees, CA, USA (004). 9. Vter, A.J., CDMA Prncpes of Spread Spectrum Communcaton, MA, Addson-Wesey (1995). 10. Gaager, R.G., Low-Densty Party-Check Codes, Camrdge, MA: MIT Press, USA (1963). 11. MacKay, D.J.C. \Good error correctng codes ased on very sparse matrces", IEEE Trans. Inform. Theory, 45, pp (Mar. 1999). 1. Png, L., Leung, W.K. and Phamdo, N. \Low densty party check codes wth sem-random party check matrx", Eectroncs. Letter, 35(1,7), pp (Jan. 1999). 13. Lee, W.C.Y., Moe Communcatons Engneerng, New York, McGraw-H, nd Ed. (1998). 14. Je, H., Sege, P.H. and Msten, L.B. \Performance anayss and code optmzaton of ow densty partycheck codes on Rayegh fadng channes", IEEE J. Seect. Areas Commun., 19(5), pp (May 001). 15. Taghav, Z. and Nasr-Kenar, M. \Iteratve mutuser recever for coded MC-FH mutpe access systems n the presence of parta-and nterference," n Proceedngs of IEEE Vehcuar Technoogy Conference (VTC) Fa 004, Los Angees, CA, USA (004). 16. Davey, M.C., Error Correctng Usng Low-Densty Party Check Codes, Ph.D. Dssertaton, Camrdge Unversty (Dec. 1999). 17. Papous, A., Proaty, Random Varaes and Stochastc Processes, McGraw-H, 3rd Ed. (1991).