FAST CONVERGENCE ADAPTIVE MMSE RECEIVER FOR ASYNCHRONOUS DS-CDMA SYSTEMS

Électronque et transmsson de l nformaton FAST CONVERGENCE ADAPTIVE MMSE RECEIVER FOR ASYNCHRONOUS DS-CDMA SYSTEMS CĂLIN VLĂDEANU, CONSTANTIN PALEOLOGU 1 Key words: DS-CDMA, MMSE adaptve recever, Least mean square (LMS), Recursve least square (RLS). Ths paper reconsders the Mnmum Mean-Squared Error (MMSE) sngle user adaptve recever for the asynchronous Drect Sequence Code Dvson Multple Access (DS-CDMA) system. Durng the tranng perod the classcal MMSE recever adapts the tap weghts once every data bt nterval. Our am s to reduce the overhead ntroduced durng the tranng perod. Ths wll be acheved by adaptng the tap weghts several tmes durng each bt nterval, n an teratve manner. Due to ths teratve process, the adaptve flter acheves a faster convergence rate. Hence, the tranng nterval or Multple-Access Interference (MAI) can be sgnfcantly reduced. Nevertheless, ths performance mprovement s computatonal expensve. 1. INTRODUCTION Many of the moble communcatons systems are employng the CDMA (Code Dvson Multple Access) technque, where the users are transmttng smultaneously wthn the same bandwdth by means of dfferent code sequences. CDMA technque has been found to be attractve because of such characterstcs as potental capacty ncreases over competng multple access methods, antmultpath capabltes, soft capacty, narrow-bandwdth ant-jammng, and soft handoff. In the DS-CDMA (Drect Sequence CDMA) system, each code sequence s used to spread the user data sgnal over a larger bandwdth, and to encode the nformaton nto a random waveform [1]. In ths case a smple multplcaton between the data sgnal and the code sequence waveform s needed, and the resulted sgnal nherts ts spectral characterstcs from the spreadng sequence. Hence, because of ts lnear sgnal processng functon ths scheme may be a 1 Faculty of Electroncs, Telecommuncatons and Informaton Theory / Telecommuncatons Department, Poltehnca Unversty of Bucharest, 1-3 Iulu Manu, ZIP 061071, Bucharest, Romana, E-mal: {caln.pale}@comm.pub.ro. Rev. Roum. Sc. Techn. Électrotechn. et Énerg., 52, 1, p. 51 60, Bucarest, 2007

52 Căln Vlădeanu, Constantn Paleologu 2 subject for possble performance mprovements by developng new sgnal processng technques for the recever. In DS-CDMA, the conventonal matched flter recever dstngushes each user s sgnal by correlatng the receved mult-user sgnal wth the correspondng sgnature waveform. Hence, the data symbol decson for each user s affected by Multple-Access Interference (MAI) from other users and by channel dstortons. The multple access nterference power depends also on the correlaton propertes of the chosen spreadng sequences [1-6]. Hence, the conventonal matched flter recever performances are lmted by ts orgnal purpose. It was desgned to be optmum only for a sngle user channel where no MAI s present, and to be optmum for a perfect power control, so t suffers from the near-far problem. These lmtatons have led to the ntroducton of an optmum mult-user detecton approach, based on Maxmum Lelhood Sequence Estmaton (MLSE) [2]. The computatonal complexty of such MLSE s nown to grow exponentally wth the actve number of users, a soluton that s not feasble. In ths context, many mult-user recevers have been proposed to overcome the nherent lmtatons of the conventonal matched flter recever. The use of these mult-user recevers has shown to mprove system s performance, and enhance ts capacty relatve to the conventonal matched flter detecton. Unfortunately, most of these mult-user detectors requre complete system nformaton on all users. Implementatons of adaptve Mnmum Mean-Squared Error (MMSE) recevers n DS-CDMA systems have been analyzed n [3] and [4]. The prncple of the adaptve MMSE recevers conssts of a sngle user detector that wors only wth the bt sequence of that user. In ths case the detecton process s done n a bt by bt manner, and the fnal decson s taen for a sngle bt nterval from the receved sgnal. The complexty of an adaptve MMSE recever s slghtly hgher than that of a conventonal recever, but wth superor performances [3-5, 11]. Besdes ts facle mplementaton the adaptve MMSE recever has the advantage that t needs no supplementary nformaton durng the detecton process, as compared to the conventonal matched-flter recever. Wthn the present paper we am to mprove adaptve MMSE recevers that reduce the MAI n a DS-CDMA system wth a faster rate or wthn a smaller tranng nterval. Ths wll be acheved by adaptng the tap weghts several tmes durng each bt nterval. Due to ths teratve process, the adaptve flter acheves a faster convergence rate. Ths performance s pad by an ncreased computatonal complexty. The paper s organzed as follows. In secton 2 we descrbe our asynchronous DS-CDMA system model, both the transmtter and adaptve recever parts of the scheme. The expermental results are presented n secton 3. Fnally, secton 4 concludes ths wor.

3 Adaptve MMSE recever for asynchronous DS-CDMA systems 53 2. SYSTEM MODEL In ths secton we present the asynchronous DS-CDMA system model used n our analyss. The adaptve MMSE teratve recever s ntroduced for the asynchronous DS-CDMA system. 2.1. TRANSMITTER MODEL In the transmtter part of the system, each user data symbol s modulated usng a unque sgnature waveform a (t) (wth a normalzed energy over a data bt T 2 nterval T, a () t dt = 1) gven by [1]: 0 a N () t a ( j) p ( t jt ), 1, K, = c c = j= 1 where the a ( j) represents the j-th chp of the th user s code sequence and are assumed to be elements of { 1, +1}, and p c (t) s the chp pulse waveform defned over the nterval [0, T c ), wth T c as the chp duraton whch s related to the bt duraton through the processng gan N (T c = T/N). K denotes the number of users n the system. In the followng analyss we consder bnary phase shft eyng modulaton (BPSK) for sgnal transmsson. Then, the -th user transmtted sgnal s gven by where P s the th user bt power, s ( t) = 2P b ( t) a ( t)cos( ω0t + θ ), 1, K, (2) = (1) N b b ( t) = b ( m) p( t mt ), m= 1 b ( m) { 1, + 1} (3) s the bnary data sequence for -th user, N b s the number of receved data bts, ω 0 = 2π f 0 and θ represent the common carrer pulsaton and phase, respectvely. 2.2. ADAPTIVE MMSE ITERATIVE RECEIVER A bloc dagram of the recever structure s shown n Fg. 1 [3]. After convertng the receved sgnal to ts baseband form usng a down converter, the receved sgnal s gven by:

54 Căln Vlădeanu, Constantn Paleologu 4 r( t) = P = 2 = 1 K K = 1 b s ( t τ ) + n( t) cos( ω t) ( t τ ) a ( t τ ) cos( θ ) + n( t) cos( ω t), 0 = 0 (4) where N b s the number of receved data bts, and n(t) s the two-sded PSD (Power Spectral Densty) N 0 /2 addtve whte Gaussan nose (AWGN). The asynchronous DS-CDMA system conssts of random ntal phases of the carrer 0 θ < 2π and random propagaton delays 0 τ < T for all the users = 1, K. There s no loss of generalty to assume that θ = 0 and τ = 0 for the desred user, and to consder only 0 τ < T and 0 θ < 2π for any [3]. Fg. 1 MMSE adaptve recever scheme. As shown n Fg. 1, the adaptve recever s modelled as a fnte mpulse response (FIR) flter wth flter taps delayed n steps of the chp duraton and total tme span equal to the transmtted bt duraton. Assumng perfect chp tmng at the recever, the receved sgnal n (4) s passed through a chp-matched flter followed by samplng at the end of each chp nterval to gve for the m-th data bt nterval: mt + ( l+ 1) Tc rm, l= r() t p( t ltc) d t, l = 0, 1,..., N 1, (5) mt + lt c

5 Adaptve MMSE recever for asynchronous DS-CDMA systems 55 where p(t) s the chp pulse shape, whch s taen to be a rectangular pulse wth ampltude 1 / N. Usng (5) and tang the th user as the desred one, the output of the chp matched flter after samplng for the mth data bt s gven by: where r K P 1 = Tcb ( m) a ( l) + P cosθb ( m) I, ( m, l) n( m, ), 2N 2N = 1 (6) m, l + l ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) b m 1 εa N 1 N + l + Tc ε a N N + l, 0 l N 1 I, ( m, l) = b m 1 εa N 1 + b m Tc ε a 0, l = N (7) b m εa l N 1 + Tc ε a l N, N + 1 l N 1, wth τ = N Tc + ε, 0 N N 1, 0 < ε < Tc. (8) Let us consder the followng vectors: ( ) r m = rm,0, rm,1... r m, N 1, (9) T wth r m,l gven by (6). The components of the nose n(m,l) vector n (6) conssts of ndependent zero-mean Gaussan random varables wth varance N0 /( 2N ). In the tranng mode, the recever attempts to cancel the MAI and adapts ts coeffcents usng a short tranng sequence employng an adaptve algorthm. After tranng s acqured, the recever swtches to the decson-drected mode and contnues to adapt and trac channel varatons [3-5, 11]. Durng the tranng mode, the flter tap weghts are adjusted teratvely G tmes every transmtted bt nterval. At each MMSE teraton g {1, 2, G} (.e., G tmes for every receved bt), the recever forms an error sgnal proportonal to the dfference between the flter output and the nown reference sgnal. Ths error sgnal s then used to adjust the flter tap weghts usng the adaptve algorthm. The error obtaned durng the Gth teraton of the m-th data bt s used by the algorthm n the 1st teraton of the (m + 1)-th data bt. Ths process s repeated for every receved bt untl convergence s reached. Now, let w (m) be the N coeffcent vector representng the adaptve flter weghts: w ( ) ( ) ( ) ( ) T m = w,0 m, w,1 m... w, N 1 m, (10) where the symbol m denotes the dscrete tme ndex of the data bt sequence. The dscrete output sgnal y (m) at the gth teraton s gven by:

56 Căln Vlădeanu, Constantn Paleologu 6 ( ) N 1 g ( 1 ( ) g y ) m = w, l ( m) rm, l. (11) l= 0 Usng vector notaton, (11) can be wrtten as: T g ( g) y m = w m r m (12) ( ) ( ) ( ) ( ). The soluton for the optmum flter tap weghts that acheve the MMSE are defned by the Wener-Hopf equaton as [7, 8]: w opt 1 = R P, (13) where R s the autocorrelaton matrx of the receved nput sgnal gven by: T ( m) ( m) R = E r r (14) and P s the crosscorrelaton vector of the nput sgnal wth the desred flter output gven by: P = E r ( m) d( m). (15) At the end of each teraton, the recever forms an error sgnal e (g) (m), e ( g ) ( g ( m) b ( m) y ) ( m), = (16) wth y (g) (m) gven by (12), and a new flter tap weght vector w (g) (m) s estmated accordng to the adaptve algorthm. When a new data bt s receved, the flter tap weghts are adapted n the same manner, wth the ntal tap weghts adapted gven by: ( 0 ) ( G w ( 1 ) ) m+ = w ( m), (17) ( 0 ) ( ) ( G ) ( ) where w m +1 and w m represent the ntal tap wegths at the (m+1)-th receved bt, and the fnal tap weghts at tme ndex m, respectvely. The adaptve algorthms used for MMSE recevers can be dvded nto two major categores [7, 8]. The frst one contans the algorthms based on mean square error mnmzaton, whose representatve member s the Least Mean Square (LMS) algorthm. The second category of algorthms uses an optmsaton procedure n the least squares (LS) sense, and ts representatve s the Recursve Least Squares (RLS) algorthm. The LMS algorthm wth ts smple mplementaton suffers from slow convergence, whch mples long tranng overhead wth low system throughput. On the other hand, LS algorthms such as the RLS offer faster convergence rate and tracng capablty than the LMS algorthm. Ths

7 Adaptve MMSE recever for asynchronous DS-CDMA systems 57 performance mprovement of the RLS over the LMS s acheved at the expense of the large computatonal complexty. In the case of LMS algorthm, the new flter tap weght vector w (g) (m) s gven by: ( ) ( ) ( 1 ) ( ) g g g ( ) ( ) ( ) w m = w m +µ e m r m. (18) The parameter µ n (18) s the flter adaptaton step sze chosen to optmze both the convergence rate and the mean squared error [7, 8]. For the RLS algorthm we have to ntroduce frst a new parameter, Q (g) (m). Ths parameter s teratve computed n the algorthm, startng wth an ntal value: ( 0 ) ( ) α, Q m = I N (19) where α s a postve constant (regularzaton parameter) and I s an N-by-N dentty matrx. A detaled study about choosng the value of α can be found n [9]. The RLS algorthm can be resumed as follows, w Q ( g ) ( g 1 ( m) = w ) ( m) 1 λ + λ + r ( g ) ( g 1 ( m) = Q ) ( m) Q ( g 1 Q ) ( m) r( m) ( ( ) ) ( ) ( ) ( g e ) ( m) ; T g 1 m Q m r m ( g 1 ) T ( g 1 ( m) ( m) ( m) ) r r Q ( m) T ( g 1 λ + r ( m) Q ) ( m) r( m) where λ s a postve constant, smaller than unt, whch s called the weghtng parameter (or forgettng factor). It controls the algorthm s memory [9, 10]., (20) 3. EXPERIMENTAL RESULTS The asynchronous DS-CDMA system usng MMSE teratve recever presented n the prevous secton was tested usng MATLAB programmng envronment. A bnary-phase shft eyng transmsson n a tranng mode scenaro was consdered. The bnary spreadng sequences are pseudo-random. The system smulaton parameters were fxed as followng: N = 16 and K = 8. The sgnal-tonose rato (SNR) s 15 db. The adaptve algorthm s terated G = 10 tmes for each data bt. The mean square error (MSE) was estmated by averagng over 100 ndependent trals usng LMS and RLS adaptve algorthms. The results are presented n Fg. 2 and Fg. 3.

58 Căln Vlădeanu, Constantn Paleologu 8 Fg. 2 Convergence of MMSE recevers usng LMS algorthm. Fg. 3 Convergence of MMSE recevers usng RLS algorthm. Comparng these results t can be concluded that the RLS algorthm outperforms LMS algorthm. Nevertheless, ths performance enhancement s pad by an ncreased computatonal complexty.

9 Adaptve MMSE recever for asynchronous DS-CDMA systems 59 Fnally, we nvestgated the steady-state BER (Bt Error Rate) performance as a functon of the energy per bt to nose power spectral densty rato (E b /N 0 ) of the teratve MMSE recever consdered above. These results are shown n Fg. 4, where we compared the performance of the teratve MMSE recever usng sx consecutve teratons wth the conventonal adaptve recever, usng LMS algorthm. The smulaton results were obtaned usng 2 000 bts tranng perod for each value of E b /N 0, n order to assure that the steady-state s reached. It s very mportant to note that under the same condtons the BER s mproved every new teraton. Nevertheless, one should mae a compromse between the computatonal complexty and the BER performances. Fg. 4 BER performance of MMSE recevers usng LMS algorthm. 4. CONCLUSIONS The MMSE teratve recever consdered above was shown to mprove the asynchronous DS-CDMA system performances. Thus, the MSE s decreased every new teraton by reducng MAI. Ths decrease offers a faster tranng mode for the recever. Hence, the desgnng procedure may consder one of these two enhancements: to shorten the tranng sequence for mantanng the same MAI n

60 Căln Vlădeanu, Constantn Paleologu 10 the system or to strongly reduce the MAI by eepng the same length of the tranng sequence. A very mportant result s that the BER s consderably mproved every new teraton. However, ths performance mprovement s computatonal expensve. Nevertheless, the systems performances are evaluated by means of smulatons. An analytcal estmaton of BER for ths MMSE teratve recever wll be consdered n perspectve. Receved on 16 July, 2006 REFERENCES 1. S. Glsc, B. Vucetc, CDMA Systems for Wreless Communcatons, Artech House, 1997. 2. S. Verdu, Optmum Multuser Asymptotc Effcency, IEEE Transactons on Communcatons, 34, 9, pp. 890-897, 1986. 3. S. Mller, An Adaptve Drect-Sequence Code-Dvson Multple-Access Recever for Mult-user Interference Rejecton, IEEE Transactons on Communcatons, 43, 4, pp. 1746-1755, 1995. 4. P. Rapajc, B. Vucetc, Adaptve Recever Structures for Asynchronous CDMA Systems, IEEE Journal on Selected Areas n Communcatons, 12, 5, pp. 685-697, 1994. 5. W. Hamouda, P. McLane, A Fast Adaptve Algorthm for MMSE Recevers n DS-CDMA Systems, IEEE Sgnal Processng Letters, 11, 2, pp. 86-89, 2004. 6. C. Vlădeanu, Optmum Chaotc Quantzed Sequences for Asynchronous DS-CDMA System, EUSIPCO 2005, Antalya, Turey, 4-8 Sept., 2005. 7. S. Cochnă, C. Negrescu, Ssteme adaptve, Ed. Tehncă, Bucharest, 1999. 8. S. Hayn, Adaptve Flter Theory, fourth edton, Prentce Hall Upper Saddle Rver, NJ, 2002. 9. G.V. Moustades, Study of the Transent Phase of the Forgettng Factor RLS, IEEE Transactons on Sgnal Processng, 45, 10, pp. 2468-2476, 1997. 10. S. Cochnă, C. Paleologu, A.A. Enescu, On the Behavour of RLS Adaptve Algorthm n Fxed- Pont Implementaton, SCS 2003, Iaş, Romana, 10-11 July, 2003. 11. C. Vlădeanu, C. Paleologu, MMSE Sngle User Iteratve Recever for Asynchronous DS-CDMA Systems, Communcatons 2006, Bucharest, Romana, 8-10 June, 2006.