BER Performance for Downlink MC-CDMA Systems over Rician Fading Channels

Transcription

1 EURASIP Journal on Applied Signal Processing 2005:5, c 2005 Hindawi Publishing Corporation BER Performance for Downlink MC-CDMA Systems oer Rician Fading Channels Zhihua Hou Positioning & Wireless Technology Centre (PWTC, School of Electrical & Electronic Engineering, anyang Technological Uniersity, 50 anyang Drie, Singapore houzhihua@pmail.ntu.edu.sg Vimal K. Dubey Schoolof Electrical & ElectronicEngineering, anyang Technological Uniersity, 50 anyang Aenue, Singapore ekdubey@ntu.edu.sg Receied 30 July 2003; Reised 1 April 2004 We consider downlink multicarrier code-diision multiple-access (MC-CDMA systems using binary phase-shift keying (BPSK modulation scheme and maximal ratio combining (MRC in frequency-selectie Rician fading channels. A time-domain method to obtain bit error rate (BER by calculating moment generating function (MGF of the decision ariable for a tapped-delayline channel model is proposed. This method does not require any assumption regarding the statistical or spectral distribution of multiple access interference (MAI, and it is also not necessary to assume that the fading encountered by the subcarriers is independent of each other. The analytical formula is also erified by simulations. Keywords and phrases: frequency-selectie Rician fading channels, MC-CDMA, moment generating function, performance analysis, BER for downlink MC-CDMA systems oer Rician fading channels. 1. ITRODUCTIO MC-CDMA systems, based on the combination of code-diision multiple-access (CDMA and orthogonal frequency-diision multiplexing (OFDM techniques, were proposed in 1993 [1]. The multicarrier CDMA schemes can be categorized into two groups: MC-CDMA and MC- DS-CDMA [2]. Due to the attractie features like efficient frequency diersity and high bandwidth efficiency [3], MC-CDMA has receied greater attention. Furthermore, MC-CDMA outperforms direct sequence CDMA (DS- CDMA and MC-DS-CDMA in terms of BER performance oer the downlink. Hence, MC-CDMA appears to be a suitable candidate for supporting multimedia serices in mobile radio communications for the downlink. Most of the preious papers [1, 4, 5], which inestigated the performance of the MC-CDMA systems, assumed that the fading in different subcarriers is independent of each other, so that the ariance of the interference can be approximated by using the central limit theorem. eertheless, the assumption is not guaranteed in practice, as the fading of the subcarriers is usually correlated due to insufficient frequency separation between the subcarriers. Also, the assumption of independent fading characteristic implies a frequency-selectie fading channel at each subcarrier, since it requires sufficient independent paths uniformly distributed oer the symbol duration [3], which contradicts the assumption of flat fading at each subcarrier. An exact error floor without taking into account the noise term is obtained in [6] under the assumption of exponential multipath intensity profile. In [7], a closed-form BER expression for a synchronous MC-CDMA in the uplink has been obtained assuming independent fading among subcarriers. In [8], a performance ealuation using characteristic method is proposed for MC-DS-CDMA systems. All the aboe-mentioned papers consider Rayleigh fading channels, and the results for other more general channel models are not aailable. In this paper, we propose a time-domain approach instead of usual frequency-domain approach to obtain the error performance of downlink MC-CDMA with maximal ratio combining (MRC in correlated Rician fading channels. We canobtainanexactberperformancewithouthaingto make any assumption about the MAI distribution by calculating the moment generating function (MGF. It is also not necessary to assume that the fading of the subcarriers is independent of each other. In addition, it is not necessary to make assumption regarding the correlation property of the spreading sequence. The closed-form error performance may proide helpful insights releant to designing the spreading sequences for MC-CDMA systems and may

2 710 EURASIP Journal on Applied Signal Processing lead to an improed performance for MRC which has been shown to suffer seere MAI when the number of users is large. otation (,( T,and( H denote complex conjugate, transpose, and conjugate transpose operation, respectiely. Vectors and matrices are represented in bold. E[ ] denotes expectation. stands for the norm of a complex ariable or a ector. Re[ ] andim[ ] correspond to the real and imaginary parts of a complex number. Matlab s notation FFT(, n and IFFT(, n denoten-point fast Fourier transform (FFT and inerse FFT (IFFT. x y stands for point multiplication between two ectors. represents circular conolution. (m mod(m,. (µ, σ 2 denotes normal distributed random ariable with µ as mean and σ 2 as ariance, respectiely. det( denotes the determinant of a matrix. 2. SYSTEM MODEL 2.1. Transmitter We consider the system proposed in [1] and assume that there are Ku actie users in a downlink. At the transmitter side, the spreading binary data stream is serial-to-parallel conerted to parallel substreams. All the data in subcarriers are modulated in baseband using binary phase-shift keying (BPSK by the means of the inerse discrete Fourier transform (IDFT. The resultant signals are then conerted back into serial data. The guard interal is inserted between symbols to aoid intersymbol interference (ISI caused by multipath fading, and finally the signal is transmitted after radio frequency (RF upconersion. Let a k (q and{c k,i } i1 denote the qth data bit and the spreading sequence of user k, respectiely. The equialent lowpass transmitted signal for user k can be expressed as [3] S k (t + q a k(qc k,i p b ( t qtb e j2πi f (t qt b, E b is the power of the data bit and assumed to be the same for all users, p b (t is a rectangular pulse defined in [0, T b ]witht b denoting the bit duration, and f 1/T b is the minimum subcarrier separation Channel model We consider a slowly arying frequency-selectie Rician fading channel. The channel is modeled as a tapped delay line (TDL haing the following baseband equialent impulse response [9]: (1 L 1 1 c(t b(lδ ( t lt c, (2 l0 δ( is the Dirac delta function, L 1 is the number of resolable paths of the channel, T c is the chip duration, b(l is the lth path gain which is a complex Gaussian random process with zero mean (Rayleigh or nonzero mean (Rician and are mutually independent for different l. The Rician probability density function (PDF is obtained as the PDF of b(l B 2 lx + B 2 ly,b lx (µ lx, σ 2 l, B ly (µ ly, σ 2 l, and B lx, B ly are independent. The Rician K-factor is defined as the ratio of signal power in dominant component oer the (local-mean scattered power. We hae K l u2 lx + u2 ly 2σl 2. (3 Since only the nonzero paths need to be considered, we define { l } L 1 l0 as the propagation delay for the nonzero paths normalized by the sample interal, and σl 2 as the ariance, L is the number of the nonzero paths. In the frequency domain, all subcarriers are assumed to experience flat but correlated fading. The channel gain for the ith subcarrier is h i ρ i e jϕi,ρ i is Rician distributed with E[ρi 2 ] 1. It has been shown in [10] that L 1 1 ( j2πil h i b(lexp (i 0, 1,..., 1. (4 l0 The receied signal can be written as r(t q Ku k1 a k (qp b ( t qtb ρ i c k,i e j[2πi f (t qtb+ϕi] + n(t, n(t is additie white Gaussian noise (AWG haing a double-sided power spectrum density of 0 /2forbothreal and imaginary components Receier At the receier, the RF signal is first conerted to baseband signal. After the portion of the signal corresponding to the prefix is remoed, DFT is performed on the signal samples. A coherent correlation receier with MRC is then used. As the focus of this paper is on the ealuation of BER, we assume perfect subcarrier synchronization with no frequency offset and no nonlinear distortion and also assume perfect subcarrier amplitude/phase estimation. Assuming user u is the desired user, the decision ariable of the zeroth data bit is gien by Z u (0 Re { 3. PERFORMACE AALYSIS (5 } 1 Tb ρ i c u,i r(te j(2πi ft+ϕi dt. (6 T b 0 We aim at obtaining a concise form of the decision ariable for further processing. With the knowledge that there are usually only a few actie taps compared to the number

3 BER Downlink MC-CDMA Systems oer Rician Fading Channels 711 of subcarriers, the idea is to transform the calculation of the decision ariable from the frequency domain to the time domain by applying discrete Parseal s theorem, so that the MGF of the decision ariable can be deried. The exact BER is then obtained from the MGF by the Laplace inersion integral or numerical methods. Assuming the length of the cyclic prefix is longer than the length of the channel impulse response, so that there is no ISI. Here we discuss real spreading codes, which are commonly used in practice, howeer, extension to complexalued codes is straightforward. The power loss due to the cyclic prefix is not considered in the analysis. The decision ariable of the zeroth data bit in (6 with proper sampling time can be written as Z u (0 Re { 1 β ui h i h i + c u,i n i h i }, (7 β ui a u (0+ Ku a k (0c k,i c u,i (i 0, 1,..., 1, n i corresponds to the complex additie Gaussian noise at the ith subcarrier. In order to normalize the AWG noise, a factor equal to 2/0 is multiplied to (7. We obtain Z u (0 Re { 2γb β ui h i h i + c u,i ñ i h i }. (8 After normalization, the ñ i in (8 iszero-mean,complex Gaussian random ariable with a ariance of σn 2 1 for the real and imaginary components, respectiely, and the ratio γ b E b / 0 represents the signal-to-noise ratio (SR per bit. Since the noise is AWG, the multiplication of the noise by spreading codes of the desired user will not change the distribution of the noise. The AWG noise in the frequency domain is the DFT of the normalized AWG noise in the time domain η(i multiplied by 1/,η(iisactually n(t sampled at the rate of 1/T c : ñ i 1 ( j2πin η(nexp n0 (i 0, 1,..., 1. Assume y is the corresponding signal of β u in the time domain gien the alue y IFFT(β u,, x is the signal corresponding to β u h in the time domain, that is x IFFT( β u h,. The multiplication of the DFT of two sequences is equialent to the circular conolution of the two sequences in the time domain [11], the relationship between x and y can be thus expressed as x(i y(i b(i b(my(i m m0 m0 b ( ( m y i m, i 0, 1,..., 1. (9 (10 Applying discrete Parseal s theorem [11], the summation in the frequency domain in (8 can be conerted to the summation in the time domain. Omitting a factor equal to, the decision ariable in (8 can be rewritten in the time domain as Z u (0 Re { 2γb x(ib(i + η(ib(i }. (11 From (10 and(11, the decision ariable in the time domain can be obtained as Z u (0 Re { 2γb + m0 η ( i b ( i }. b ( m y ( i m b( i (12 For a u (0 1, an error occurs if Z u (0 < 0. For a u (0 1, an error occurs if Z u (0 > 0. The decision ariable Z u (0 is conditioned on the channel coefficients {b( i } L 1 and the user data {a k (0} Ku k1.we compute the BER by first aeraging oer {b( i } L 1 and then oer {a k (0} Ku k1. In order to obtain an expression of exact error performance, we calculate the Laplace transform of Z u (0 conditioned on {a k (0} Ku k1, Φ(s E(e sz, that is, the MGF of the decision ariable Z u (0. Usually it is not an easy task to calculate the MGF for a random ector since it is by definition the mean of an exponential function of the random ariables inoled. This generally requires the calculation of multidimensional integral and the knowledge of the joint probability density of the random ariables. In this case, since the calculation is conerted to the time domain, the channel coefficients {b( i } L 1 for different paths are statistically independent Gaussian random ariables [9] and the exponential of the function turns out to be a noncentral Gaussian quadratic form. The MGF can be calculated as (see the appendix for details Φ( s exp ( M m1 bmsλ 2 m / ( 1 sλ m Mm1 ( wm, (13 1 sλm λ m,form 1,..., M, are distinct eigenalues of A defined in (A.10 M is the total number of distinct eigenalues, w m is the multiplicity of the eigenalue λ m, b m is gien by (A.14. It is clear that ( 1/λ 1,..., 1/λ M are the M distinct poles of the MGF Φ(s ofz u (0 conditioned on {a k (0} Ku k Closed-form expression If an exact calculation of P(Z u (0 < 0 is sought, a common starting point is the Laplace inersion formula [9] P e,u au(01,{a k(0} Ku P( Z u (0 au(01,{a k(0} Ku 0 1 2πj c+j c j Φ(s ds s, (14

4 712 EURASIP Journal on Applied Signal Processing c is a small positie constant between zero and the smallest positie pole of Φ(s/s. This complex contour integral might be ealuated exactly by calculating the residues of Φ(s/s [12] oer the poles in the right-hand side of the complex s-plane. Since Φ(s contains essential singularities which make the residue ealuation ery difficult, power series expansion is applied to sole the problem instead. Assuming that among the M distinct poles of Φ(s(13, M 1 of them are positie. Following the method described in [13], we obtain a closed-form expression for (14 as P e,u au(01,{a k(0} Ku M 1 ( M ( ( λ wm ( k b 2 exp m λ m λ k1 m1,m k k λ m λ k λ m exp ( ( bk 2 n+w k 1 b 2n k G(rk k (λ n!r k! ( rk, λ k G (r k (λ [ r 1 r 10 ( r 1 r 1 g (r 1 r1 k In (16, g (n k (λisgienby g (n k (λ M m1,m k n0 r k0 r 1 1 ( r1 1 (λ r 20 r 2 (15 ] g (r1 1 r2 k (λ. (16 [ ( λk λ n+1 ( n! m w m +(n+1 b2 mλ ] k. λ k λ m λ k λ m ( umerical method Equation (15 gies a general closed-form BER expression for MC-CDMA systems oer arbitrary Rician multipath fading channels. Since (15 is deried using the fast-conergent power series expansion of the MGF about its positie poles, it is expected to conerge rapidly with respect to the index n. Howeer, as shown in [13], the series (15conergesmore slowly as the Rician K-factor increases, therefore it is more suitable for analysis of Rician fading channels with a relatiely small K-factor, for example, K 7dB. To achiee a more conenient solution to this problem, a different approach towards the exact calculation of (14 is used instead [14]. Since the left-hand side of (14 is a real quantity, we can write P e,u au(01,{a k(0} Ku 1 2π 1 2π Φ(c + jω c + jω dω c Re { Φ(c + jω } + ω Im { Φ(c + jω } c 2 + ω 2 dω. (18 Thechangeofariableω c 1 x 2 /x yields P e,u au(01,{a k(0} Ku 1 { 1 [ ( 1 x 2 ] Re Φ c + jc 2π 1 x 1 x 2 + x [ ( Im Φ c + jc 1 x 2 x ] } dx 1 x 2. (19 Finally, using a Gauss-Chebyshe quadrature rule with an een number of nodes, we hae P e,u au(01,{a k(0} Ku 1 /2 k1 { Re [ Φ ( c + jcτk ] + τk Im [ Φ ( c + jcτ k ]} + E, (20 τ k tan((2k 1π/(2 and the error term E anishes as. To achiee the desired degree of accuracy, it is practical increasing alues of and accepting the result when it does not change significantly. In general, 32 to 64 nodes are sufficienttoobtainanaccuracybetterthan. The alue of c affects the number of nodes necessary to achiee a preassigned accuracy. The detailed discussion concerning the selection of c can be found in [14]. In our numerical results, we assume 64, and c is set equal to one half the smallest real part of the poles of Φ(s. Howeer, een with a suboptimum choice of c, the alue of does not grow large enough as to become unmanageable. In the aboe section, a closed-form expression and a numerical method are gien to obtain the error probability conditioned on the user data based on the MGF. For coherent detection, we hae P e,u au(01,{a k(0} Ku P e,u au(0 1,{a k(0} Ku. (21 By aeraging the conditioned BER (15 or(20 oer {a k (0} Ku which are independent and identically distributed binary random ariables, we obtain the BER of the user u: P e,u 1 2 Ku 1 a 1(0 {1, 1} a k u (0 {1, 1} P e,u au(01,{a k(0} Ku. a Ku(0 {1, 1} (22 TheaerageBERofallusersisgienby P e 1 Ku P e,i. (23 Ku i1

5 BER Downlink MC-CDMA Systems oer Rician Fading Channels 713 As the computational complexity to ealuate (22 increases exponentially with the number of users, the proposed approach is appropriate for systems with relatiely small number of users, say Ku < 20. When the number of users is large, we may resort to traditional Gaussian approximation. In the following section, we propose a simple ealuation method to remedy this problem Gaussian approximation method for large number of users From (7, the decision ariable of the desired user can be rewritten as I u Z u (0 D u + I u + u, (24 D u a u(0 u Re h i 2 [ K u h i 2, (25 a k (0c u,i c k,i, (26 c u,i h i n i ]. (27 E b / 0 (db Closed-form method, 2-ray Closed-form method, 3 p umerical method, K 0.2, 1 u umerical method, K 0.2, 10 u umerical method, K 1, 1 u umerical method, K 1, 10 u Figure 1: Theoretical BER of MC-CDMA for the 2-ray (K 0 K 1 K and 3-path channel models (K 0 K 1 K 2 KusingMRCby the closed-form and the numerical method. The term D u is the desired output. I u corresponds to the MAI component. u is the noise term contributed by AWG. Due to the equal power, equally likely, antipodal data modulation a k (0 in (26, we apply the central limit theorem and approximate g u,i K u a k(0c u,i c k,i (i 0, 1,..., 1 as a zero mean Gaussian random ariable. ote that g u,i is correlated between subcarriers. The coariance matrix M u is an matrix with each element gien by M u (i1, i2 Ku c u,i1c k,i1 c u,i2 c k,i2. The decision ariable conditioned on {h i } 1 is a Gaussian random ariable with E ( D u a u(0 ar ( 1 E I b u i10 E b i10 ar ( u 0 2 i20 i20 h i 2 a u(0 h i1 2 h i2 2 Mu (i1, i2 ρ 2 i1ρ 2 i2m u (i1, i2, h i ρ 2 i, (28 (29 ρ 2 i. (30 Equation (29 isderiedfrom[15] for the ariance of the linear combination of correlated random ariables. The probability of error conditioned on {ρ i } 1 is simply gien by P [ e ρ ] 1 2 erfc E ( D u ar ( (. (31 I u +ar u The BER for the zeroth data stream is obtained ia aeraging P[e ρ]oerρ: P e 0 0 p[e ρ]p 0 ( ρ 0, ρ 1,..., ρ 1 dρ0 dρ 1 dρ 1. (32 Equation (32 can be ealuated by Monte Carlo integration [4, 16]. 4. UMERICAL RESULTS In this section, we present numerical results. To calculate the BER, it is assumed that the number of subcarriers is 32, and two channel delay profiles are considered, a simple tworay multipath delay profile often encountered in urban and hilly areas [14], and a three-path channel with linear delay power profile [17]. For the two-ray channel, we assume 0 0, Furthermore, Walsh Hadamard codes are used as the spreading codes. For the channels and the signals described aboe, we can compute the BER based on the closed-form expression gien by (15 and(22 and the numerical method (20. Shown in Figure 1, the results obtained by the two methods match ery well as long as the alue of n is sufficiently large. To check how quickly the infinite series in (15 conergeswithrespect to n, we gie the BER alues based on the truncated series for the index n summing up from 0 to n max in Tables 1 and 2 for two-ray and three-path channels. The BER for n max is obtained by setting n max large enough in (15, so that the first

6 714 EURASIP Journal on Applied Signal Processing Table 1: BER alues with truncated series for the 2-ray channel: Ku 1, SR 0dB. n max K-factor K 0.2 K 1 K umerical Table 2: BER alues with truncated series for the 3-path channel: Ku 1, SR 0dB. n max K-factor K 0.2 K 1 K umerical nine significant digits for each BER alue conerge. It is also shown in the tables that the results of the closed-form expression conform exactly with those computed by the numerical method. The results in Tables 1 and 2 also show that the series in (15 conerges ery rapidly for a relatiely small K-factor, and n max 5 10 may be accurate enough for practical interest. For large K-factors, it requires n max to get an accurate approximation. Since the numerical method appears more computationally efficient compared with the closed-form expression especially when the K-factoris large, in Figures 2 7, the analytical results are obtained using the numerical method. The results of our proposed time-domain method described herein are erified by comparing with the Monte Carlo simulations. In Figures 2 and 3, the K-factors for different taps are equal, while in Figure 4, the K-factors for each tap are different. In all these cases, the simulation results agree ery well with the theoretical results obtained by the technique proposed in this paper. The simulation results are based on the calculation of the decision ariable for each bit, and aeraging the results oer large number of bits (say bits. Figure 5 shows the accuracy of our lowcomplexity method proposed in Section 3.3. The results indicate that our low-complexity method gies quite a good approximation as compared with the accurate result, and the accuracy improes when the number of users is high. ext, the effect of the Rician K-factor on the BER performance is inestigated using the analytical formula deried in Section 3. TheRicianK-factor is defined as the ratio of signal power in dominant component oer the (local-mean scattered power which corresponds to a deterministic strong wae receied. It is natural to expect a better performance for larger alues of K. From the analytical results shown in (Figure 6, it can be seen that the performance improement due to the increase in K is eident when the number of user is small (i.e., MAI is not dominant, while for 20 users E b / 0 (db A, 2-ray, 1 u S, 2-ray, 1 u A, 2-ray, 10 u S, 2-ray, 10 u A, 2-ray, 20 u S, 2-ray, 20 u A, 3 p, 1 u S, 3 p, 1 u A,3p,10u S, 3 p, 10 u A,3p,20u S, 3 p, 20 u Figure 2: BER of MC-CDMA for the 2-ray (K 0 K 1 K 0.2 and 3-path channel models (K 0 K 1 K 2 K 0.2 using MRC. (A: theoretical results; S: simulation results. E b / 0 (db A, 2-ray, 1 u S, 2-ray, 1 u A, 2-ray, 10 u S, 2-ray, 10 u A, 2-ray, 20 u S, 2-ray, 20 u A, 3 p, 1 u S, 3 p, 1 u A, 3 p, 10 u S,3p,10u A, 3 p, 20 u S,3p,20u Figure 3: BER of MC-CDMA for the 2-ray (K 0 K 1 K 1 and 3-path channel models (K 0 K 1 K 2 K 1 using MRC. (A: theoretical results; S: simulation results. ( MAI dominates, there is no difference between the BER performance for different alues of K (i.e., K 0.2, K 1, and K 10. For this case, MAI trends to be so seere and it becomes the dominant factor in determining the

7 BER Downlink MC-CDMA Systems oer Rician Fading Channels E b / 0 (db 10 8 E b / 0 (db A, 2-ray, 1 u S, 2-ray, 1 u A, 2-ray, 10 u S, 2-ray, 10 u A, 2-ray, 20 u S, 2-ray, 20 u A, 3 p, 1 u S, 3 p, 1 u A, 3 p, 10 u S,3p,10u A, 3 p, 20 u S,3p,20u Figure 4: BER of MC-CDMA using MRC for the 2-ray channel model (K 0 10, K 1 4 and for the 3-path channel model (K 0 10, K 1 2.5, K 2 0. (A: theoretical results; S: simulation results. A, 1 u A, 10 u A, 20 u S, K 0.2 S, K 1 S, K 10 Figure 6: BER of MC-CDMA system ersus SR with arious number of users and K-factor for the 2-ray channel model: 0 0, 1 14, K 0 K 1 K. (A: theoretical results; S: simulation results. 20 users 10 users 15 users E b / 0 (db K Exact Approx. A, SR 10, 1 u S, SR 10, 1 u A, SR 10, 10 u S, SR 10, 10 u A, SR 10, 20 u S, SR 10, 20 u Figure 5: BER of MC-CDMA system with MRC ersus SR for the 2-ray channel model: 0 0, 1 14, and K 0 K 1 1. Figure 7: BER of MC-CDMA system ersus K-factor, SR 10 db for the 2-ray channel model: 0 0, 1 14, K 0 K 1 K. (A: theoretical results; S: simulation results. system performance. Simulation results are also gien to alidate the analytical results. Figure 7 shows the BER comparison for arious K when SR equals 10 db for 1, 10, and 20 users. A distinct ariation can be seen when the number of users is small, while for 20 users, there is hardly any ariation in the BER as MAI dominates the performance in this case. 5. COCLUSIO In this paper, we hae proposed an accurate and simple method to calculate the performance of MC-CDMA systems with MRC combining scheme oer downlink correlated Rician fading channels. A general algorithm is proposed to

8 716 EURASIP Journal on Applied Signal Processing calculate the moment generating function by expressing the decision ariables in Gaussian quadratic forms. Based on the moment generating function, an exact BER is obtained in the time domain. Howeer, the complexity of this method increases exponentially with the number of users. To alleiate this problem, we hae also proposed a low-complexity approximate BER ealuation method by using the Monte Carlo integration. The results obtained by the analytical formula hae been thoroughly erified by computer simulations. APPEDIX DERIVATIO OF MGF In this appendix, we show the detailed deriation of (13 starting from (12. The MGF of the decision ariable for frequency-selectie channels under Rician distribution is deried. The decision ariable Z u (0 in (12 canberewritten as { [ 2γb Z u (0 Re b ( i 2 y(0 + + m0, m i η ( i b ( i }, b ( i b ( m yi,m ] (A.1 with y( i m denoted by y i,m for simplicity. If the decision ariable in (A.1 is written in ector form, the deriation of the MGF can be expressed more concisely. The ector form of the decision ariable for the desired user is gien by Z u (0 H (0Q(0(0, (A.2 (0 [ b ( ( ( ( ( ( ] T 0 b 1 b L 1 η 0 η 1 η L 1. (A.3 Q(0 is a 2L 2L matrix defined as [ ] Q1 0.5I Q(0 L, (A.4 0.5I L 0 L y(0 y 0,1 y 0,2 y 0,L 1 y0,1 y(0 y 1,2 y 1,L 1. Q 1 2γb y0,2 y1, (A yl 2,L 1 y0,l 1 y1,l 1 yl 2,L 1 y(0 In (A.4, 0 L and I L are the L L zero and identity matrices, respectiely. With the assumption that the channel coefficients b(l for different paths are statistically independent of each other and the AWG noise term is independent of the channel impulse response, a simple expression of the coariance matrix of (0 is achieed: P E { (0 H (0 } 2 diag σ0 2, σ1 2,..., σl 1,1,...,1 2. }{{} L (A.6 The mean ector of (0 is gien by µ E ( (0 ( ( µ 0x + jµ 0y, µ1x + jµ 1y,..., T ( µ(l 1x + jµ (L 1y,0,...,0. }{{} L (A.7 Employing a result for the distribution of a noncentral Gaussian quadratic form [13], the MGF of Z u (0 in (A.2can be obtained as Φ( s exp ( µ H F(sµ det(i sp 1/2, (A.8 F(s P 1/2 (( I sp 1/2 QP 1/2 QP 1/2 1 I P 1/2, (A.9 P 1/2 is the inerse of the symmetric square root of P. To represent the conditioned MGF Φ( s in a more compact form, we define a matrix A P 1/2 QP 1/2 (A.10 and represent its eigendecomposition as A U Λ U H, Λ diag ( λ 1, λ 2,..., λ 2L is the eigenalue matrix of A.Also,wedefine (A.11 µ [ µ 1, µ 2,..., µ 2L ] T U H P 1/2 µ. (A.12 Equation (A.8 can now be simplified to Φ( s exp ( µ H Λ(sũ det(i sλ exp ( 2L k1 µ k 2 sλk / ( 1 sλ k 2L (, k1 1 sλk Λ(s (I sλ 1 I is a diagonal matrix. (A.13

9 BER Downlink MC-CDMA Systems oer Rician Fading Channels 717 In the case of repeated eigenalues, the MGF of Z u (0 is gien by (13. In (13 bm 2 µ k 2, (A.14 k κ m κ m denotes the set of k indices associated with the mth distinct eigenalue. Here, the MGF of the decision ariable for the two-ray channel model is gien as an example: Φ(s e (K0+K1 e 2(σ2 1 K 0M 1+σ0 2 K 1M 0 W/R, R the coefficients are gien by R c 4 s 4 + c 3 s 3 + c 2 s 2 + c 1 s +1, c 4 σ0 2 σ1 2, c 3 4 2γ b σ0 2 σ1 2 y(0, c 2 ( σ0 2 + σ1 2 +8γb σ0 2 σ1 2 y(0 2 8γ b σ0 2 σ1 2 y 0,1 2, c 1 2 ( 2γ b σ σ1 2 y(0, W 2γb s [ µ 0x µ 1x Re ( y 0,1 + µ0y µ 1x Im ( y 0,1 (A.15 +µ 0y µ 1y Re ( y 0,1 µ0x µ 1y Im ( y 0,1 ]. (A.16 The region of conergence is the ertical strip enclosing the jω axis bounded by the closest poles on either side. The BER performance can be obtained through the method described aboe. REFERECES [1]. Yee, J. P. Linnartz, and G. Fettweis, Multi-carrier CDMA in indoor wireless radio networks, IEICE Trans. Commun., ol. E77-B, no. 7, pp , [2] S. Hara and R. Prasad, Oeriew of multicarrier CDMA, IEEE Communications Magazine, ol. 35, no. 12, pp , [3] S. Hara and R. Prasad, Design and performance of multicarrier CDMA system in frequency-selectie Rayleigh fading channels, IEEE Trans. Vehicular Technology,ol.48,no.5,pp , [4] E. A. Sourour and M. akagawa, Performance of orthogonal multicarrier CDMA in a multipath fading channel, IEEE Trans. Communications, ol. 44, no. 3, pp , [5]. Yee and J. P. Linnartz, BER of multi-carrier CDMA in an indoor Rician fading channel, in Proc. 27th Asilomar Conference on Signals, Systems and Computers, ol. 1, pp , Pacific Groe, Calif, USA, oember [6]Q.H.ShiandM.Lata-aho, Anexacterrorfloorfor downlink MC-CDMA in correlated Rayleigh fading channels, IEEE Communications Letters, ol. 6, no. 5, pp , [7] Q. H. Shi and M. Lata-aho, Exact bit error rate calculations for synchronous MC-CDMA oer a Rayleigh fading channel, IEEE Communications Letters, ol. 6, no. 7, pp , [8] B. Smida, C. L. Despins, and G. Y. Delisle, MC-CDMA performance ealuation oer a multipath fading channel using the characteristic function method, IEEE Trans. Communications, ol. 49, no. 8, pp , [9] J. G. Proakis, Digital Communications, McGraw-Hill, ew York, Y, USA, 4th edition, [10] Y. Li, L. J. Cimini Jr., and. R. Sollenberger, Robust channel estimation for OFDM systems with rapid dispersie fading channels, IEEE Trans. Communications,ol.46,no.7,pp , [11] J. G. Proakis and D. G. Manolakis, Digital Signal Processing, Prentice-Hall, Englewood Cliffs, J, USA, 3rd edition, [12] J. K. Caers and P. Ho, Analysis of the error performance of trellis-coded modulations in Rayleigh-fading channels, IEEE Trans. Communications, ol. 40, no. 1, pp , [13] Y. Ma, T. L. Lim, and S. Pasupathy, Error probability for coherent and differential PSK oer arbitrary Rician fading channels with multiple cochannel interferers, IEEE Trans. Communications, ol. 50, no. 3, pp , [14] E. Biglieri, G. Caire, G. Taricco, and J. Ventura-Traeset, Simple method for ealuating error probabilities, Electronics Letters, ol. 32, no. 3, pp , [15] C. W. Helstrom, Probability and Stochastic Processes for Engineers, Macmillan Publishing, ew York, Y, USA, [16] X. Gui and T. S. g, Performance of asynchronous orthogonal multicarrier CDMA system in frequency selectie fading channel, IEEE Trans. Communications,ol.47,no.7,pp , [17] J. K. Caers, Mobile Channel Characteristics, Kluwer Academic, Boston, Mass, USA, Zhihua Hou receied the B.S. and M.S. degrees from Xi an Jiaotong Uniersity, China, in 1993 and 1996, respectiely, both in electrical engineering and information science. She is now pursuing the Ph.D. degree at the School of Electrical and Electronic Engineering, anyang Technological Uniersity, Singapore. Her research interests include wireless communications, with emphasis on the analysis of wireless digital communications oer fading channels, multicarrier and CDMA techniques, channel estimation, and MIMO communication systems. Vimal K. Dubey receied the B.S. (honors degree in mathematics from the Uniersity of Rajasthan, India, the B.E. and M.E. degrees in electrical communication engineering from the Indian Institute of Science, Bangalore, India, and the Ph.D. degree in electrical engineering from McMaster Uniersity, Hamilton, Ontario, Canada. He has worked in arious research and deelopment laboratories in India for more than ten years. From 1972 to 1976, he was a Research Scientist at DLRL, Hyderabad, India. From 1976 to 1982, he was with DEAL, Dehradun, India, he conducted research on spread-spectrum systems for satellite communications and transportable troposcatter communication system. From 1982 to 1986, he was a Commonwealth Research Scholar at McMaster Uniersity. He joined the School of Electrical and Electronic Engineering, anyang Technological Uniersity, Singapore, in 1988, he is now an Associate Professor. His main research interests are in the areas of digital communications, specializing in coding, modulation, and spread-spectrum systems for satellite and wireless communications.