Robust Bind Mutiuser Detection in DS-CDMA Systems over Nakagami-m Fading Channes with Impusive Noise incuding MRC Receive Diversity Vinay Kumar Pamua, Member, IEEE, Department of ECE, MIC Coege of echnoogy, Kanchikachera, India 580. e-mai: pamuavk@ieee.org Habibua Khan, Department of ECE, K University, Vaddeswaram, India 550. e-mai: habibua@rediffmai.com Srinivasa Rao Vempati, Member, IEEE, Department of ECE, Anurag Engineering Coege, Kodad, India 50806. e-mai: vempatisr@ieee.org Ani Kumar ipparti, Member, IEEE, Department of ECE, Anurag Coege of Engineering, Aushapur, Ghatkesar, India 5030. e-mai: tvakumar000@yahoo.co.in Abstract his paper presents robust bind mutiuser detection in direct sequence-code division mutipe access systems (DS- CDMA) over Nakagami-m fading channes with impusive noise incuding maxima-ratio combining diversity. An M-estimator based structure for non-coherent demoduation of differentiay phase shift keyed (DPSK) signas transmitted over a synchronous CDMA fat-fading Nakagami channes with impusive noise is studied anayzed. A new M-estimator is proposed for robustifying the detector an approximate expression for average probabiity of error of non-coherent DPSK signas is derived using an approximation to generaized Marcum-Q function over singe-path Nakagami-m fading channes. Simuation resuts are presented to evauate the performance of the proposed robust bind mutiuser detector in comparison with the inear decorreating detector, Huber Hampe estimator based detectors. Keywords bind mutiuser detection; CDMA; impusive noise; infuence function; maxima ratio combiner; M-estimator; Nakagami-m distribution; probabiity of error. I. INRODUCION Recent research has expored the potentia benefits of optimum mutiuser detection (MUD) techniques to combat mutipe access interference (MAI) [], []. Compexity of optimum mutiuser detector, based on maximum ikeihood (M) criterion, grows exponentiay with number of active users suboptima mutiuser detectors are the aternatives with identica performance [3]. he decorreating detector presented in [4] is used to overcome the near-far probem as we as MAI. his decorreator requires no knowedge of received signa ampitudes but it requires matrix inversion, which increases the computationa compexity. he purpose of MUD is to detect the desired user symbos given the received signa assuming that the diagona matrix the signature matrix are known precisey at the receiver [5]. he Nakagami-m distribution has received considerabe attention of researchers as it can provide a good fit to measured data in different fading environments [6] ike Rayeigh (m = ), og-norma or Ricean (m > ) fading channes. It can aso accommodate fading conditions that are more or ess severe than that of the Rayeigh fading channe. Nakagami-m fading is often encountered in many practica appications such as ceuar mobie communications best fit to indoor mobie mutipath propagation as we as ionospheric radio inks [7]. Performance anaysis of MUD in synchronous DS-CDMA communication systems using decorreating decision feedback (DFB) detectors with singe-path Nakagami-m fading is presented in [8] by deriving the ower bounds of the probabiity of error. Recenty, in [9] decorreator receiver for DS-CDMA mobie radio system, empoying RAKE reception both MRC SC diversity, through Nakagami fading channe is anayzed. Performance of digita ceuar mobie radio systems in a frequency nonseective Rayeigh fading channe with ognorma shadowing, incuding the effects of white Gaussian noise, impusive noise interference co-channe interference is presented in [0]. here exist in the iterature a number of different approaches to the robust estimation probem, the M-estimator is one of the most sophisticated approaches to this probem. Recenty, the probem of robust mutiuser detection in non-gaussian channes has been addressed in the iterature [], which was deveoped based on the Huber the Hampe M-estimators, respectivey. Differentia non-coherent data detection in fat-fading CDMA impusive non-gaussian channes is presented in [0], []. Robust mutiuser detection in synchronous DS-CDMA systems with MRC receive diversity over Nakagami-m fading channe is presented in [3] by assuming that the moduation is binary PSK (BPSK). But, the differentia moduation must be used to overcome the phase ambiguity [], [4]. 978--4673-393-//$3.00 0 IEEE
Bind mutiuser detection can be used to reduce the intersymbo interference (ISI) aso to improve system throughput [5]. Subspace-based bind mutiuser detection technique is known for its better performance among a the other bind methods. In [4], 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 system operating under non-gaussian impusive noise is presented by [5]. Experimenta resuts have confirmed the presence of impusive (heavy-taied) noise in outdoor mobie communication channes, due to switching transients in power ines or automobie ignition, in radar sonar systems as a resut of natura or man-made eectromagnetic acoustic interference, in indoor wireess communication channes [], [], [6]. Hence, in this paper, the probem of differentia noncoherent data detection in synchronous CDMA fat-fading Nakagami-m channes with the assumption of impusive noise environment is considered. Pre-detection MRC receive diversity is incuded to combat the mutipath fading effects of the channe. An expression for average probabiity of error of an M-decorreator is derived using approximation to generaized Marcum-Q function for singe-path Nakagami-m channes incuding MRC receive diversity. his expression is used to compute the probabiity of error of inear decorreating detector, the Huber the Hampe estimator based detectors, the proposed M-estimator based detector. he remaining part of the paper is structured as: DS- CDMA systems over mutipath fading channe in impusive noise environment is presented in Section II. In Section III, M-estimation based regression the proposed M-estimator is presented. Subspace-based robust bind mutiuser detection is presented in Section IV. he asymptotic performance of proposed M-decorreator is discussed by deriving an expression for average probabiity of error in Section V. In Section VI, simuation resuts are presented. And finay, concusions are drawn in Section VII. II. SYSEM AND CHANNE MODE An -user synchronous CDMA system, where each user transmits information by moduating a signature sequence over a singe-path sowy fading Nakagami-m channe, is considered in this paper. he received signa over one symbo duration can be modeed as []: () () R M j t () ()e ( s ) + () = i= 0 rt = b iα t s t i -τ nt, () where R{} 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, () 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 nt ( ) is assumed as a zero-mean compex non-gaussian noise. he probabiity density function (PDF) of this noise mode has the form [], [] = ( ) ℵ (0, ) + ℵ (0, ) () f ε υ ε κυ with υ > 0, 0 ε, κ. Here ℵ(0, υ ) represents the nomina background noise the ℵ(0, κυ ) represents an impusive component, with ε representing the probabiity that impuses occur. For synchronous case (i.e., τ = τ =... = τ = 0 ), assuming that the fading process for each user varies at a sower rate that the magnitude phase can taken to be constant over the duration of a bit, the received signa can be expressed in matrix notation as [] where r() i = A () i + w() i (3) r( i) r ( i),..., r ( ) N i, (4) w() i w(),..., i w N () i, (5) () i () i (),..., i () i () i N b g b g (6) with w n () i is a sequence of independent identicay distributed (i.i.d.) compex rom variabes whose in-phase quadrature components are independent non-gaussian rom variabes with PDF given by (), g () i is the th channe fading coefficient with A [ a, a,..., a ] (7) a [ a, a,..., a N ]. (8) For the Nakagami-m fading channe, α () i are i.i.d. Nakagami rom variabes with PDF given by [7] m ( ) md md m α pα ( α ) = exp α, (9) Ω Γ( md) Ω where m is the Nakagami fading parameter that determines the severity of the fading, Ω = E [ α ] is the average channe power, E [] is statistica expectation Γ() is the compete Gamma function []. Consider a D - branch diversity receiver in Nakagami-m fading channes with SNR per bit on th branch as γ, =,,..., D. he overa instantaneous SNR per bit at the output of pre-detection MRC over Nakagami-m fading channes is given by [7]
the noise density, inear decorreating mutiuser detector is aso sensitive to the tai behavior of the noise distribution. Hence, a new M-estimator [8] is used to dea with heavy-taied noise. Suppose that ρ has a derivative with respect to the unknown parameters ( ψ = ρ ' ), caed the infuence function, since it describes the infuence of measurement errors on soutions. In this section, the infuence function of proposed M-estimator is presented (see Fig. ). he proposed M-estimator (modified Hampe) is determined by the penaty function its derivative, which is given, respectivey, by [0], [8], [9] where, Figure. Infuence function of the proposed estimator. E b D D b = α = γ, (0) No = = γ = ( b o) γ E / N α, E b is the transmitted signa energy per bit of th user N o / is the power spectra density of noise. he output SNR per bit γ is gamma distributed with PDF md md m γ γ pγ ( γ ) = exp m, m 0. 5, () γ Γ(mD) γ b o where γ =(E / N )E[ α ] 0is the average output received SNR per bit for a channe. III. M-ESIMAOR BASED REGRESSION An important cass of robust estimators is M-estimators they can resist outiers without preprocessing the data [6]. In M-estimates, unknown parameters,,... are soved by minimizing a sum of function ρ( ) of the residuas [] { N = argmin ρ R( rn() i [ A] n() i ) C n= = + ρ I( rn() i [ A] n() i ) } =, () where ρ 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 () i r () i respectivey, [ A] n is the n th eement of the matrix A, I ()denotes imaginary part. Since east squares (S) estimate is very sensitive to the tai behavior of ρ PRO where d is a constant, x a ab x exp b for x a ( x) = ax for a< x b, (3) sgn( ) x exp b b + d for x > b a x for a< x b ψ PRO ( x) =. (4) a x for x > b x for x a he choice of the constants a (= κυ ) b (= κυ ) depends on the robustness measures derived from the infuence function. A robust estimator shoud possess a finite vaue of rejection point. he proposed M-estimator is three-part with no sharp rejection point as in Hampe s three-part re-descending estimator. he proposed infuence function ψ PRO (x) is bounded has continuous derivatives [9]. he proposed detector has bounded infuence function, hence is robust. IV. SUBSPACE-BASED ROBUS BIND MUIUSER DEECION In this section, a subspace-based robust bind mutiuser detection technique for synchronous DS-CDMA systems is presented. et the signature waveforms { s} = of the users are ineary independent. he autocorreation matrix of the received signa, r is given by [4] C = E () () r = () () i r i W a i a i + υ I N =, (5) = AWA + υ I N A a a a, = ( ) where = [ ] W diag W W W. he eigen vaue decomposition (EVD) of the matrix C eads to
Λ C = Λ = s U s U U Us Un Λn, (6) U n where the vectors in U s, which are associated with the argest eigen vaues, span the signa subspace defined by the coumns of S the vectors U n are, associated with the (N-) noise eigen vaues, span the noise subspace. Using the resut deveoped in [], the received signa, r can be expressed as r () i = A () i + w () i = U ξ () i + w () i, (7) s where () i = [ () i () i () i ] ξ ξ ξ ξ. From the received signa, the signa subspace components U [ ] s ()= i u ˆ u ˆ u ˆ Λ ( ˆ ˆ ˆ ) s = diag λλ λ are estimated. Now the parameters ξ () i are robusty estimated using the estimated signa subspace coordinates U s () i. he parameters () i of the desired user can be estimated according to where uˆ j a () i = α ξ () i, =,,, (8) j= λj σ α = W = = j u j a σ j λ. (9) By using the modified residua method for updating the robust estimate of ξ () i according to ξ k ψ ( sξ ) k H () i = y U () i (0) () i = ξ () i + U s H () i, () μ k+ k k where µ is a step-size parameter chosen as =ν. μ 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 [] * ( R{ ˆ () ˆ ( ) } < 0 Δ φ() = 0,, ) P i i i x y x + y xy 4σ ˆ = Q M x, y I o e σ ˆ σ ˆ σ ˆ over the joint PDF of the rom variabes x y defined as where with () x () i ( i) N g g (3) y () i + ( i) N g g, (4) ν * σ ˆ ν R (5) N = ψ ( u) f ( u) du [ ψ ( u) f ( u) du] (6) * R is the cross-correation matrix of the rom infiniteength signature waveforms of the users, Q (, ) is the Marcum s Q-function Io () is the modified Besse function of the first kind with order zero. Assuming that the rom variabes x y are statisticay independent Nakagami-m rom variabes with PDF given in (9), the average probabiity of error for the th user can be expressed as P = e E QM x, y σ ˆ σ ˆ x + y 4σ ˆ ˆ xy E I o e σ Using an approximation to Marcum s Q-function given by [0] Q M ( a, b) P Γ P+ i P iai iγ M+ i b i= 0 Γ i + Γ P i + Γ M + i a M. (7) ( ) (, /) ( ) ( ) ( )exp( /) (8)
the reations [(6.455) (6.63), ], the terms of (7) can be expressed, respectivey, as i Γ ( P + i) P i i md σ m P ˆ = Ω Γ( md) i= 0 Γ ( i + ) Γ ( P i + ) Γ ( M + i) M+ ( i+ md)! Γ ( md+ M + i) 4σ ˆ ( i+ md) + M m md + Ω 4σ ˆ m/ Ω F, md+ M+ i; md+ ; m + Ω 4σ ˆ (9) Figure. Average probabiity of error versus SNR for user for inear mutiuser detector (S), minimax detector with Huber (HU), Hampe (HA) proposed (PRO) M-estimator in synchronous CDMA channe with impuse noise, N = 7, ε = 0.0. Ω = 0.5 F md, md;; 4 6σ m ˆ, (30) where F is the Gauss hypergeometric function []. hen the average probabiity of error of M-decorreator for the th user can be expressed as Pe. (3) Here, the approximation error is between.6 0-7 0-8, respectivey, for the truncating term P = 0 P = 500 [0]. Foowing the steps as in [], the average channe power Ω in (3) can be expressed as { } * R SNR ( ) Ω= + N ν σ ρ, (3) where σ = ( ε) υ + εκυ with υ κυ, respectivey, as the variance of in-phase quadrature components of noise sampes w n () i E SNR g () i σ. (33) Figure 3. Average probabiity of error versus SNR for user for inear mutiuser detector (S), minimax detector with Huber (HU), Hampe (HA) proposed (PRO) M-estimator in synchronous CDMA channe with impuse noise, N = 7, ε = 0.. VI. SIMUAION RESUS In this section, the performance of M-decorreator is presented by performing Monte Caro simuations by computing (3) for different vaues of Nakagami fading parameter different order of diversity. It is assumed that the channe mode is a ighty damped second-order autoregressive (AR) process given by [] with the bit rate, the poe radius, the spectra peak frequencies are b = 0 kbps, r d = 0.998 f p = 80 Hz, respectivey. Further, it is aso assumed that M = P = 50 in evauating average probabiity of error, (3). Performance of decorreating detector with different infuence functions is shown in Fig. Fig. 3. In Fig. ( ε = 0.0 & κ = 00), the average probabiity of error versus
the signa-to-noise ratio (SNR) corresponding to the user under perfect power contro of a synchronous DS-CDMA system with six users ( = 6) a arge processing gain, N = 7 is potted for m = D =. Simiary, in Fig. 3 ( ε = 0. & κ = 00), 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 SNR increases. It is aso cear that the proposed estimator performs we for very heavytaied noise with itte attendant increase in computationa compexity. VII. CONCUDING REMARKS Robust bind mutiuser detection in DS-CDMA systems with MRC receive diversity over Nakagami-m fading channes in an impusive noise environment is presented. An expression, using an approximation to Marcum-Q function, for average probabiity of error of the decorreating detector to detect DPSK signas is derived. An M-estimator based robust mutiuser detector is proposed its probabiity of error is computed using the expression derived. Simuation resuts show that the proposed robust bind mutiuser detector offers significant performance gain over the inear mutiuser detector the minimax decorreating detectors with Huber Hampe M-estimator, in impusive noise with itte attendant increase in the computationa compexity. Effect of fading parameter diversity order on the performance of decorreator is aso studied. REFERENCES [] A. Due-Haen, J. Hotzman, Z. Zvonar, "Mutiuser detection for CDMA systems," IEEE Pers. Commun., vo., no., pp.46-58, Apr. 995. [] S. Moshavi, "Muti-user detection for DS-CDMA communications," IEEE Commun. Mag., vo.34, no.0, pp.4-36, Oct. 996. [3] S. 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