Bind Mutiuser Detection in Asynchronous DS-CDMA Systems over akagami-m Fading Channes Vinay Kumar Pamua JU Kakinada, Andhra Pradesh, India 533 003 pamuavk@yahoo.com ABSRAC his paper presents a technique for bind mutiuser detection in asynchronous direct sequence-code division mutipe access (DS-CDMA) systems over akagami-m fading channes with impusive noise. A new M- estimator is proposed anayzed for robustifying the detector. A cosed-from expression for average error rate of DPSK signas is derived. Average probabiity of error is computed to evauate the performance of the new M-estimator based bind mutiuser detector in comparison with the inear decorreating detector, Huber Hampe estimator based detectors. Simuation resuts show that the new M-estimator based detector performs we. Proceedings of Internationa Conference on Emerging rends in Eectronics & eecommunications (ICEE) 3th - 4th December -04 Karimnagar, eangana, India (DS0EC030) ISS (onine): 349-000 ISS (print): 394-4544 Keywords: CDMA; impusive noise; M-estimator; mutiuser detection; akagami-m distribution; probabiity of error. I. IRODUCIO Bind mutiuser detection can be used to reduce the inter-symbo interference (ISI) aso to improve system performance []. Subspace-based bind mutiuser detection technique is known for its better performance among a the other bind methods. In [], an improved subspace-based robust bind mutiuser detection technique for synchronous asynchronous CDMA systems over non-gaussian channes is presented. A new adaptive agorithm for bind mutiuser detection of coherent BPSK signas in DS-CDMA systems operating under non-gaussian impusive noise is presented in []. he probem of robust mutiuser detection in non-gaussian channes has been addressed in the iterature [3], which was deveoped based on the Huber the Hampe M- estimators, respectivey. Differentia non-coherent data detection for CDMA over fat-fading non- Gaussian channes with impusive noise is presented in [], [4]. Robust mutiuser detection in synchronous DS-CDMA system with MRC receive diversity over akagami-m fading channe is presented in [5] by assuming that the moduation is binay PSK (BPSK). But, the differentia moduation must be used to overcome the phase ambiguity [], [4]. Recenty, [6] considered the robust bind mutiuser detection for synchronous DS-CDMA systems over akagami-m fading channes. his paper extends the work of [6] to asynchronous DS-CDMA systems over akagami-m channes in the presence of heavy-taied impusive noise by empoying subpace-based bind mutiuser detection technique. An approximate expression for average probabiity of error of an M-decorreator derived in [6] by using an approximation to Marcum-Q function. In this paper, a cosed-form exact expression for average error rate is derived using [Eq. 3, 7]. his expression is used to compute the probabiity of error of inear decorreating detector, ICEE 04 DS0EC030 P a g e 93
the Huber the Hampe estimator based detectors, the newy (modified-hampe) proposed M- estimator based detector. Simuation resuts shows that the new M-estimator based detector outperforms the inear decorreating detector, Huber Hampe estimator based detectors. he remaining part of the paper is organized as: DS- CDMA system over mutipath fading channe in impusive noise environment is presented in Section II. In Section III, new M-estimator proposed to robustify the detector is presented. Section IV presents the subspace-based robust bind mutiuser detection technique. An expression for average probabiity of error of M-decorreator is derived in Section V. Section VI, presents the simuation resuts finay, concusions are drawn in Section VII. II. SYSEM MODE An asynchronous DS-CDMA system shared by - users, each user transmits information by moduating a signature sequence over a singe-path akagami-m fading channe, is considered. he received signa over one symbo duration can be modeed, when the users are symbo chip asynchronous, as [4]: () ( ) M jθ t r t = b ( ) ( )e i α t s ( t is - τ ) n( t ) () = i0 denotes the rea part, M is the number of data symbos per user in the data frame of interest, s is the symbo interva, α () t is the time-varying fading gain of the th user s channe, θ () t is the time-varying phase of the th user s channe, b () i is the i th bit of the th user, s () t is the normaized signaing waveform of the th user, is the propagation deay nt () is assumed as a zero-mean compex non-gaussian noise given by [3] f ( ) (0, ) (0, ) () with > 0, 0,. Here (0, ) represents the nomina background noise the (0, ) represents an impusive component, with representing the probabiity that impuses occur. At the receiver, the received signa rt () is first fitered by a chip-matched fiter then samped at the chip rate, / c. he resuting discretetime signa sampe corresponding to the n th chip of the i th symbo is given by [4] is( n) jct rn ( i) r( t) e dt, n.... (3) isn Assuming that the fading process for each user varies at a sower rate that the phase ampitude can taken to be constant over the duration of a bit, (3) simpifies to [4] () r ( ) ( ) (0) ( ) ( ) () j i g ( ) i b i hj g i b i hj w j i (4) 0, j m h j (0) ( ) a, j m jm (5) a ( ), j m a j jm m ( ), a a j m j m j m h j () a, j m jm (6) 0, j m with m, m c is( j) jct w j ( i) n( t) e dt. (7) is j et r() i r 0( i),..., r ( i), w() i w 0( i),..., w( i) ( i) ( i) ( i),..., ( i) ( i). b g b g By stacking k successive data sampes, the foowing quantities can be defined [], [4], ICEE 04 DS0EC030 P a g e 94
r( i) ( i) () i r ( ), ( ) k i k i ( ) ( ) r i k k i k ( k ) (8) wi () w () k i ( ) w i k k r k () i can be written as r ( i) H ( i) ( i) w ( i), (9) k k k k H() H(0) 0 0 0 0 H k () i 0 H() H(0) with (0) (0) h h H (0) h (0) (0) h () () h h H () h () () h (). () (0) he parameter k is caed the smoothing factor is chosen such that the matrix III. M- ESIMAOR H k has fu coumn rank. here exist in the iterature a number of different approaches to the robust estimation probem, the M-estimation is one of the most sophisticated approaches to this probem [8]. An important cass of robust estimators is M-estimators they can resist outiers without preprocessing the data [9]. In M-estimates, unknown parameters,,... are soved by minimizing a sum of function () of the residuas [4] θ arg min r ( ) [ ] ( ) n i A n i θc n r ( ) [ ] ( ) n i A n i (3) is a symmetric positive-definite function with a unique minimum at zero, is chosen to be ess increasing than square, rn () i () i are the n th th eements of the vectors r() i () i respectivey, [ A] n is the n th eement of the matrix A [see 4 for detais], ()denotes imaginary part. Since east squares (S) estimate is very sensitive to the tai behavior of the noise density, inear decorreating mutiuser detector is aso sensitive to the tai behavior of the noise distribution. Hence, a new M-estimator [0] is used to dea with heavy-taied noise. he infuence function of new M-estimator is given by [0], [] PRO ( ) sgn( ) a exp b b for a a for a b. (4) for b he choice of the constants a (= ) b (= ) depends on the robustness measures derived from the infuence function. IV. ROBUS BID MUIUSER DEECIO In robust bind mutiuser detection for asynchronous channes the signa subspace components of the covariance matrix of the signa r k () i are computed as [], [4] ICEE 04 DS0EC030 P a g e 95
( ) ( ) E i i k () iwk k. (5) U s su s U nu n C r r H H I Here, the signa subspace has dimension of (m+). hen, the robust estimation in the signa subspace foows as k k r ( i ) r Us ( i ) (6) ( u) f ( u) du ( u) f ( u) du () * R is the cross-correation matrix of the rom infinite-ength signature waveforms of the users, Q (, ) is the Marcum s Q-function I () is the M modified Besse function of the first kind with order o k k U r k s ( i) ( i) ( i) (7) zero. Here, it is assumed that the rom variabes x y are statisticay independent akagami-m µ is a step-size parameter chosen as. he estimate of the product of the ampitude the data bit () i W b () i of the th user is given by [4] k ( ) ujh ( i) ( i),,,..., (8) j j 0 h ( k) finay, the th user h data bit is demoduated as * b ( i ) sgn k, ( i) k, ( i). (9) V. AVERAGE PROBABIIY OF ERROR OF M- DECORREAOR he asymptotic probabiity of error of non-coherent DPSK demoduator can be evauated by averaging the conditiona probabiity [4] ˆ* P ˆ ( i) ( i ) 0 ( i) 0, x, y xy Q M x, y I o e ˆ ˆ ˆ x y 4 ˆ (0) over the joint probabiity density function (PDF) of the rom variabes x y [4]. Here * ˆ R () with distributed with PDF []. By foowing the steps as in [6] using [Eq. 3, 7], the average probabiity of error of DPSK signas can be derived as (3) shown at the end of the paper. In (3), (/4 ˆ )/(4 m/ / ˆ ) H ( m/ /4 ˆ ), F () is the Gauss hyper- geometric function () is the gamma function [3], * R SR ( ), (3) ( ) (with, respectivey, as the variance of in-phase quadrature components of noise sampes) E SR g () i. (4) VI. SIMUAIO RESUS In this section, the performance of M-decorreator is presented by computing (3) for different vaues of akagami fading parameter different order of diversity. It is assumed that the channe mode is a ighty damped second-order autoregressive (AR) process given by [4] with the bit rate, the poe radius, the spectra peak frequencies are b = 0 kbps, rd = 0.998 fp = 80 Hz, respectivey. Further, it is aso assumed that M = in evauating average probabiity of error, (3). ICEE 04 DS0EC030 P a g e 96
Performance of decorreating detector with different infuence functions is shown in Fig. Fig.. In Fig., the average probabiity of error versus the signa-to-noise ratio (SR) corresponding to the user under perfect power contro of an asynchronous DS-CDMA system with six users ( = 6) a arge processing gain, = 7 is potted for m = D =. Simiary, in Fig., average probabiity of error is potted for m = D =. he simuation resuts revea that the increase in diversity order (from to ) improves the detector performance. Simuation resuts aso reveas that the proposed M-estimator outperforms the inear decorreating detector minimax decorreating detector (both with Huber Hampe estimators), even in highy impusive noise. Moreover, this performance gain increases as the SR increases. Fig.. Average probabiity of error versus SR for user for inear mutiuser detector (east Squares), minimax detector with Huber, Hampe proposed M-estimator in asynchronous DS-CDMA channe with moderate ( = 0.0) impusive noise. VII. COCUDIG REMARKS Bind mutiuser detection in asynchronous DS- CDMA systems over akagami-m fading channes in an impusive noise environment is presented. A new M-estimator (modified-hampe) based bind mutiuser detection technique is presented its performance is anayzed by deriving a cosed-form expression for average error rate. his expression is used to compute the average probabiity of error of the M-decorreator with east-squares, Huber, Hampe proposed M-estimators. Simuation resuts show that the bind mutiuser detector offers significant performance gain over the inear mutiuser detector the minimax decorreating detectors with Huber Hampe M-estimators, in impusive noise. Effect of fading parameter diversity order on the performance of decorreator is aso studied. Fig.. Average probabiity of error versus SR for user for inear mutiuser detector (east Squares), minimax detector with Huber, Hampe proposed M-estimator in asynchronous DS-CDMA channe with highy ( = 0.) impusive noise. REFERECES [] R. irmaa Devi,. Ani Kumar, K. Kishan Rao, "Improved adaptive agorithm for bind mutiuser detection in non-gaussian noise," Information Sciences Signa Processing their Appications (ISSPA), 00 0th Internationa Conference on, vo., no., pp.70-73, 0-3 May 00. ICEE 04 DS0EC030 P a g e 97
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