EURASIP Journal on Applied Signal Processing 2005:5, 709–717 c(cid:1) 2005 Hindawi Publishing Corporation
BER Performance for Downlink MC-CDMA Systems over Rician Fading Channels
Zhihua Hou Positioning & Wireless Technology Centre (PWTC), School of Electrical & Electronic Engineering, Nanyang Technological University, 50 Nanyang Drive, Singapore 637553 Email: houzhihua@pmail.ntu.edu.sg
Vimal K. Dubey School of Electrical & Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 Email: evkdubey@ntu.edu.sg
Received 30 July 2003; Revised 1 April 2004
We consider downlink multicarrier code-division multiple-access (MC-CDMA) systems using binary phase-shift keying (BPSK) modulation scheme and maximal ratio combining (MRC) in frequency-selective Rician fading channels. A time-domain method to obtain bit error rate (BER) by calculating moment generating function (MGF) of the decision variable for a tapped-delay- line 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 verified by simulations.
Keywords and phrases: frequency-selective Rician fading channels, MC-CDMA, moment generating function, performance anal- ysis, BER for downlink MC-CDMA systems over Rician fading channels.
1. INTRODUCTION
combination of MC-CDMA systems, based on the code-division multiple-access (CDMA) and orthogonal frequency-division 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 attractive features like efficient frequency diversity and high bandwidth efficiency [3], MC-CDMA has received greater attention. Furthermore, MC-CDMA outperforms direct sequence CDMA (DS- CDMA) and MC-DS-CDMA in terms of BER performance over the downlink. Hence, MC-CDMA appears to be a suitable candidate for supporting multimedia services in mobile radio communications for the downlink.
requires sufficient independent paths uniformly distributed over the symbol duration [3], which contradicts the assump- tion 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 inten- sity profile. In [7], a closed-form BER expression for a syn- chronous MC-CDMA in the uplink has been obtained as- suming independent fading among subcarriers. In [8], a performance evaluation using characteristic method is pro- posed for MC-DS-CDMA systems. All the above-mentioned papers consider Rayleigh fading channels, and the results for other more general channel models are not available. 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 can obtain an exact BER performance without having to make any assumption about the MAI distribution by cal- culating 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 neces- sary to make assumption regarding the correlation prop- erty of the spreading sequence. The closed-form error per- formance may provide helpful insights relevant to designing the spreading sequences for MC-CDMA systems and may Most of the previous papers [1, 4, 5], which investigated the performance of the MC-CDMA systems, assumed that the fading in different subcarriers is independent of each other, so that the variance of the interference can be ap- proximated by using the central limit theorem. Neverthe- less, the assumption is not guaranteed in practice, as the fading of the subcarriers is usually correlated due to insuffi- cient frequency separation between the subcarriers. Also, the assumption of independent fading characteristic implies a frequency-selective fading channel at each subcarrier, since it
710 EURASIP Journal on Applied Signal Processing
(cid:5)
lx + B2 B2
lead to an improved performance for MRC which has been shown to suffer severe MAI when the number of users is large.
is the lth path gain which is a complex Gaussian random pro- cess 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 l ), Bly ∼ N(µly, σ 2 ly, where Blx ∼ N(µlx, σ 2 |b(l)| = l ), and Blx, Bly are independent. The Rician K-factor is defined as the ratio of signal power in dominant component over the (local-mean) scattered power. We have
lx + u2 u2 ly 2σ 2 l
(3) . Kl =
Since only the nonzero paths need to be considered, we l=0 as the propagation delay for the nonzero paths l as the variance, L define {vl}L−1 normalized by the sample interval, and σ 2 is the number of the nonzero paths. Notation (·)∗, (·)T , and (·)H denote complex conjugate, transpose, and conjugate transpose operation, respectively. Vectors and matrices are represented in bold. E[·] denotes expectation. | · | stands for the norm of a complex variable or a vec- tor. Re[·] and Im[·] correspond to the real and imaginary parts of a complex number. Matlab’s notation FFT(·, n) and IFFT(·, n) denote n-point fast Fourier transform (FFT) and inverse FFT (IFFT). (cid:2)x · y(cid:3) stands for point multiplica- tion between two vectors. ⊗ represents circular convolution. (m)N = mod(m, N). N(µ, σ 2) denotes normal distributed random variable with µ as mean and σ 2 as variance, respec- tively. det(·) denotes the determinant of a matrix.
i ] = 1. It has been shown in [10] that
(cid:6)
(cid:7)
L1−1(cid:1)
2. SYSTEM MODEL In the frequency domain, all subcarriers are assumed to experience flat but correlated fading. The channel gain for the ith subcarrier is hi = ρie jϕi, where ρi is Rician distributed with E[ρ2 2.1. Transmitter
− j2πil N
l=0
(4) b(l) exp (i = 0, 1, . . . , N − 1). hi =
(cid:2)
∞(cid:1)
Ku(cid:1)
(cid:3)
(cid:4)
The received signal can be written as
q=−∞
k=1
N −1(cid:1)
×
r(t) = ak(q)pb t − qTb Eb N
i=0
ρick,ie j[2πi∆ f (t−qTb)+ϕi] + n(t),
(5)
(cid:2)
+∞(cid:1)
N −1(cid:1)
(cid:4)
We consider the system proposed in [1] and assume that there are Ku active users in a downlink. At the transmitter side, the spreading binary data stream is serial-to-parallel converted to N parallel substreams. All the data in N sub- carriers are modulated in baseband using binary phase-shift keying (BPSK) by the means of the inverse discrete Fourier transform (IDFT). The resultant signals are then converted back into serial data. The guard interval is inserted between symbols to avoid intersymbol interference (ISI) caused by multipath fading, and finally the signal is transmitted after radio frequency (RF) upconversion. Let ak(q) and {ck,i}N i=1 denote the qth data bit and the spreading sequence of user k, respectively. The equivalent lowpass transmitted signal for user k can be expressed as [3] where n(t) is additive white Gaussian noise (AWGN) having a double-sided power spectrum density of N0/2 for both real and imaginary components.
(cid:3) t − qTb
q=−∞
i=0
2.3. Receiver Sk(t) = e j2πi∆ f (t−qTb), ak(q)ck,i pb Eb N (1)
where Eb is the power of the data bit and assumed to be the same for all users, pb(t) is a rectangular pulse defined in [0, Tb] with Tb denoting the bit duration, and ∆ f = 1/Tb is the minimum subcarrier separation.
2.2. Channel model
(cid:8)
(cid:10)
(cid:9)
N −1(cid:1)
Tb
At the receiver, the RF signal is first converted to baseband signal. After the portion of the signal corresponding to the prefix is removed, DFT is performed on the signal samples. A coherent correlation receiver with MRC is then used. As the focus of this paper is on the evaluation of BER, we as- sume 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 variable of the zeroth data bit is given by
0
i=0
L1−1(cid:1)
(cid:4) ,
(6) r(t)e− j(2πi∆ f t+ϕi)dt . Zu(0) = Re ρicu,i We consider a slowly varying frequency-selective Rician fad- ing channel. The channel is modeled as a tapped delay line (TDL) having the following baseband equivalent impulse re- sponse [9]: 1 Tb
(cid:3) t − lTc
l=0
(2) c(t) = b(l)δ 3. PERFORMANCE ANALYSIS
We aim at obtaining a concise form of the decision vari- able for further processing. With the knowledge that there are usually only a few active taps compared to the number where δ(·) is the Dirac delta function, L1 is the number of resolvable paths of the channel, Tc is the chip duration, b(l)
BER Downlink MC-CDMA Systems over Rician Fading Channels 711
(cid:8)
(cid:10)
(cid:5)
N −1(cid:1)
N −1(cid:1)
Applying discrete Parseval’s theorem [11], the summa- tion in the frequency domain in (8) can be converted to the summation in the time domain. Omitting a factor equal to √ N, the decision variable in (8) can be rewritten in the time domain as of subcarriers, the idea is to transform the calculation of the decision variable from the frequency domain to the time domain by applying discrete Parseval’s theorem, so that the MGF of the decision variable can be derived. The exact BER is then obtained from the MGF by the Laplace inversion in- tegral or numerical methods.
i=0
i=0
(11) x(i)b(i)∗ + η(i)b(i)∗ . Zu(0) = Re 2γb
(cid:8)
(cid:5)
L−1(cid:1)
L−1(cid:1)
(cid:4)
(cid:4)
(cid:4)∗
From (10) and (11), the decision variable in the time do- main can be obtained as
(cid:3) vm
(cid:3) vi − vm
(cid:3) vi
N b
m=0
i=0
(cid:10)
b y Zu(0) = Re 2γb
(cid:10)
L−1(cid:1)
N −1(cid:1)
N −1(cid:1)
(cid:3)
(cid:4)∗
(12) 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, however, extension to complex- valued codes is straightforward. The power loss due to the cyclic prefix is not considered in the analysis. The decision variable of the zeroth data bit in (6) with proper sampling time can be written as (cid:8)(cid:2)
(cid:4) b
(cid:3) vi
i +
i=0
i=0
i=0
(cid:11)
+ η . vi , (7) Zu(0) = Re βuihih∗ cu,inih∗ i Eb N
= −1, an error occurs if Zu(0) > 0.
i=0 and the user data {ak(0)}Ku
For au(0) = 1, an error occurs if Zu(0) < 0. For au(0)
Ku where βui = au(0)+ k=1, k(cid:7)=u ak(0)ck,icu,i (i = 0, 1, . . . , N −1), ni corresponds to the complex additive Gaussian noise at the ith subcarrier. (cid:12)
In order to normalize the AWGN noise, a factor equal to
(cid:10)
N −1(cid:1)
N −1(cid:1)
2/N0 is multiplied to (7). We obtain (cid:8)(cid:2)
i +
i=0
i=0
(8) . Zu(0) = Re βuihih∗ cu,i ˜nih∗ i 2γb N
√
After normalization, the ˜ni in (8) is zero-mean, complex = 1 for the Gaussian random variable with a variance of σ 2 n real and imaginary components, respectively, and the ratio γb = Eb/N0 represents the signal-to-noise ratio (SNR) per bit.
(cid:3)
(cid:4)(cid:4)
(cid:3) (cid:11)
(cid:6)
(cid:7)
(cid:13)
The decision variable Zu(0) is conditioned on the chan- nel coefficients {b(vi)}L−1 k=1. We compute the BER by first averaging over {b(vi)}L−1 i=0 and then over {ak(0)}Ku k=1. In order to obtain an expression of exact er- ror performance, we calculate the Laplace transform of Zu(0) k=1, Φ(s) = E(e−sZ), that is, the MGF conditioned on {ak(0)}Ku of the decision variable Zu(0). Usually it is not an easy task to calculate the MGF for a random vector since it is by defini- tion the mean of an exponential function of the random vari- ables involved. This generally requires the calculation of mul- tidimensional integral and the knowledge of the joint prob- ability density of the random variables. In this case, since the calculation is converted to the time domain, the chan- nel coefficients {b(vi)}L−1 i=0 for different paths are statistically independent Gaussian random variables [9] and the expo- nential of the function turns out to be a noncentral Gaussian quadratic form. The MGF can be calculated as (see the ap- pendix for details) Since the noise is AWGN, the multiplication of the noise by spreading codes of the desired user will not change the distribution of the noise. The AWGN noise in the frequency domain is the DFT of the normalized AWGN noise in the time domain η(i) multiplied by 1/ N, where η(i) is actually n(t) sampled at the rate of 1/Tc:
N −1(cid:1)
M m=1 b2 msλm/ (cid:3) M 1 − sλm m=1
− j2πin N
n=0
, (13) Φ(−s) = exp 1 − sλm (cid:4) wm η(n) exp (i = 0, 1, . . . , N − 1). ˜ni = 1√ N (9)
u
Assume y is the corresponding signal of β
u
where λm, for m = 1, . . . , M, are distinct eigenvalues of A defined in (A.10) where M is the total number of distinct eigenvalues, wm is the multiplicity of the eigenvalue λm, bm is given by (A.14). It is clear that (−1/λ1, . . . , −1/λM) are the M distinct poles of the MGF Φ(s) of Zu(0) conditioned on {ak(0)}Ku k=1.
3.1. Closed-form expression
u in the time domain given the value y = IFFT(β u, N), x is the signal corresponding to (cid:2)β · h(cid:3) in the time domain, that is x = IFFT((cid:2)β · h(cid:3), N). The multiplication of the DFT of two se- quences is equivalent to the circular convolution of the two sequences in the time domain [11], the relationship between x and y can be thus expressed as
N −1(cid:1)
(cid:14)
If an exact calculation of P(Zu(0) < 0) is sought, a common starting point is the Laplace inversion formula [9]
(cid:15) ≤ 0
= P
au(0)=1,{ak(0)}Ku
au(0)=1,{ak(0)}Ku
k=1, k(cid:7)=u
k=1, k(cid:7)=u
m=0
c+ j∞
L−1(cid:1)
(cid:4)
=
(cid:3) vm
(cid:3) i − vm
= 1 2π j
c− j∞
(cid:4) N ,
m=0
x(i) = y(i) ⊗ b(i) = b(m)y(i − m)N Pe,u| Zu(0)| (cid:9) (10) , Φ(s) b y i = 0, 1, . . . , N − 1. ds s (14)
√
712 EURASIP Journal on Applied Signal Processing
k=1, k(cid:7)=u
au(0)=1,{ak(0)}Ku (cid:8)
√
(cid:9)
(cid:23)
(cid:6)
(cid:7)(cid:24)
1
The change of variable ω = c 1 − x2/x yields where c is a small positive constant between zero and the smallest positive pole of Φ(s)/s. Pe,u|
−1
= 1 2π
√
√
(cid:23)
(cid:6)
(cid:7)(cid:24)(cid:10)
Φ Re c + jc 1 − x2 x
Φ + Im c + jc 1 − x2 x 1 − x2 x dx√ 1 − x2 . (19) This complex contour integral might be evaluated exactly by calculating the residues of Φ(s)/s [12] over the poles in the right-hand side of the complex s-plane. Since Φ(s) contains essential singularities which make the residue evaluation very difficult, power series expansion is applied to solve the prob- lem instead. Assuming that among the M distinct poles of Φ(s) (13), M1 of them are positive. Following the method de- scribed in [13], we obtain a closed-form expression for (14) as
(cid:16)
(cid:16)(cid:6)
(cid:6)
(cid:7)(cid:18)(cid:18)
au(0)=1,{ak(0)}Ku k=1, k(cid:7)=u M(cid:17)
M1(cid:1)
(cid:7)wm
=
au(0)=1,{ak(0)}Ku
k=1
k=1, k(cid:7)=u
m=1,m(cid:7)=k
(cid:18)
∞(cid:1)
n+wk −1(cid:1)
(cid:3)
(cid:4)
v/2(cid:1)
(cid:21)
(cid:4)(cid:26)
(cid:25)
(cid:4)(cid:26)(cid:22)
·
Pe,u| Finally, using a Gauss-Chebyshev quadrature rule with an even number v of nodes, we have exp Pe,u| λk λk − λm (cid:16)
· exp
(cid:25) Φ
− b2 k
(cid:3) c + jcτk
(cid:3) c + jcτk
n=0
rk =0
= 1 v
k=1
, Φ Re + τk Im + Ev, b2 mλm λk − λm k G(rk) b2n (λ) k (cid:4) (cid:3) rk − λk n!rk! (15) (20)
where
(cid:16)
(cid:18)
(cid:16)
(cid:18)
(cid:20)
r−1(cid:1)
r1−1(cid:1)
=
G(r) k (λ) (cid:19)
r1=0
r2=0
(λ)· (λ) · · · . g (r−1−r1) k g (r1−1−r2) k r −1 r1 r1 −1 r2
(16)
(cid:19)
k (λ) is given by (cid:6)
(cid:7)(cid:20)
(cid:7)n+1(cid:6)
M(cid:1)
In (16), g (n)
m=1,m(cid:7)=k
n! . wm+(n+1) g (n) k (λ) = λkλm λk −λm b2 mλk λk −λm where τk = tan((2k − 1)π/(2v)) and the error term Ev van- ishes as v → ∞. To achieve the desired degree of accuracy, it is practical increasing values of v and accepting the result when it does not change significantly. In general, 32 to 64 nodes are sufficient to obtain an accuracy better than 10−5. The value of c affects the number of nodes necessary to achieve a pre- assigned accuracy. The detailed discussion concerning the se- lection of c can be found in [14]. In our numerical results, we assume v = 64, and c is set equal to one half the smallest real part of the poles of Φ(s). However, even with a suboptimum choice of c, the value of v does not grow large enough as to become unmanageable. (17)
In the above section, a closed-form expression and a nu- merical method are given to obtain the error probability con- ditioned on the user data based on the MGF. For coherent detection, we have
= Pe,u|
au(0)=1,{ak(0)}Ku
au(0)=−1,{ak(0)}Ku
k=1, k(cid:7)=u
k=1, k(cid:7)=u
(cid:1)
(cid:1)
· · ·
· · ·
(21) . Pe,u|
a1(0)∈{1,−1}
ak(cid:7)=u(0)∈{1,−1} (cid:1)
×
au(0)=1,{ak(0)}Ku
k=1, k(cid:7)=u
aKu(0)∈{1,−1}
By averaging the conditioned BER (15) or (20) over {ak(0)}Ku k=1, k(cid:7)=u which are independent and identically dis- tributed binary random variables, we obtain the BER of the user u: Pe,u = 1 2Ku−1 3.2. Numerical method Equation (15) gives a general closed-form BER expression for MC-CDMA systems over arbitrary Rician multipath fad- ing channels. Since (15) is derived using the fast-convergent power series expansion of the MGF about its positive poles, it is expected to converge rapidly with respect to the index n. However, as shown in [13], the series (15) converges more slowly as the Rician K-factor increases, therefore it is more suitable for analysis of Rician fading channels with a rela- tively small K-factor, for example, K ≤ 7 dB. To achieve a more convenient solution to this problem, a different ap- proach towards the exact calculation of (14) is used instead [14]. Since the left-hand side of (14) is a real quantity, we can . Pe,u| write
k=1, k(cid:7)=u
(cid:9) ∞
(22) Pe,u|
(cid:22)
(cid:21)
(cid:22)
−∞ (cid:9) ∞
Ku(cid:1)
−∞
au(0)=1,{ak(0)}Ku = 1 2π = 1 2π
i=1
The average BER of all users is given by dω Φ(c + jω) c + jω (cid:21) c Re Φ(c + jω) + ω Im Φ(c + jω) dω. (23) Pe,i. c2 + ω2 Pe = 1 Ku (18)
100
10−1
10−2
e t a r
BER Downlink MC-CDMA Systems over Rician Fading Channels 713
r o r r e
10−3
As the computational complexity to evaluate (22) in- creases exponentially with the number of users, the pro- posed approach is appropriate for systems with relatively small number of users, say Ku < 20. When the number of users is large, we may resort to traditional Gaussian approxi- mation. In the following section, we propose a simple evalu- ation method to remedy this problem.
t i B
10−4
3.3. Gaussian approximation method for large number of users
10−5
From (7), the decision variable of the desired user can be rewritten as
0
2
4
6
10
12
14
16
8 Eb/N0 (dB)
(24) Zu(0) = Du + Iu + Nu,
(cid:2)
N −1(cid:1)
(cid:27) (cid:27)2,
where
(cid:27) (cid:27)hi
i=0 Ku(cid:1)
N −1(cid:1)
(cid:27) (cid:27)2
(25) au(0) Eb N Du = (cid:2)
(cid:27) (cid:27)hi
Closed-form method, 2-ray Closed-form method, 3 p Numerical method, K = 0.2, 1 u Numerical method, K = 0.2, 10 u Numerical method, K = 1, 1 u Numerical method, K = 1, 10 u
i=0
k=1, k(cid:7)=u
(cid:20)
(cid:19)
N −1(cid:1)
(26) Iu = ak(0)cu,ick,i, Eb N
i ni
(27) . cu,ih∗ Nu = Re
Figure 1: Theoretical BER of MC-CDMA for the 2-ray (K0 = K1 = K) and 3-path channel models (K0 = K1 = K2 = K) using MRC by the closed-form and the numerical method.
i=0
The BER for the zeroth data stream is obtained via
(cid:9) ∞
(cid:9) ∞
(cid:11)
· · ·
0
0
0
(cid:4)
×
averaging P[e|ρ] over ρ: (cid:9) ∞ Pe = p[e|ρ]p (cid:3) ρ0, ρ1, . . . , ρN −1 dρ0dρ1 · · · dρN −1. (32) The term Du is the desired output. Iu corresponds to the MAI component. Nu is the noise term contributed by AWGN. Due to the equal power, equally likely, antipodal data modu- lation ak(0) in (26), we apply the central limit theorem and Ku k=1, k(cid:7)=u ak(0)cu,ick,i (i = 0, 1, . . . , N − 1) approximate gu,i = as a zero mean Gaussian random variable. Note that gu,i is correlated between subcarriers. The covariance matrix Mu is an N × N matrix with each element given by Mu(i1, i2) = (cid:11)
Ku k=1, k(cid:7)=u cu,i1ck,i1cu,i2ck,i2. The decision variable conditioned on {hi}N −1 i=0
Equation (32) can be evaluated by Monte Carlo integra- is a Gaus- tion [4, 16].
(cid:2)
N −1(cid:1)
N −1(cid:1)
(cid:4)
(cid:27) (cid:27)2 =
=
(cid:27) (cid:27)hi
(cid:3) Du
i=0
i=0
N −1(cid:1)
N −1(cid:1)
(cid:4)
(cid:27) (cid:27)2
(cid:27) (cid:27)2
sian random variable with (cid:2) 4. NUMERICAL RESULTS (28) E au(0) au(0) ρ2 i , Eb N Eb N
(cid:3) Iu
(cid:27) (cid:27)hi1
(cid:27) (cid:27)hi2
≈ Eb N
var Mu(i1, i2)
i1=0 N −1(cid:1)
i2=0 N −1(cid:1)
i1ρ2 ρ2
i2Mu(i1, i2),
= Eb N
i1=0
i2=0
N −1(cid:1)
N −1(cid:1)
(29)
(cid:3)
(cid:4)
(cid:27) (cid:27)hi
= N0 2
(cid:27) (cid:27)2 = N0 2
i=0
i=0
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 two- ray multipath delay profile often encountered in urban and hilly areas [14], and a three-path channel with linear de- lay power profile [17]. For the two-ray channel, we assume v0 = 0, v1 = 14. Furthermore, Walsh Hadamard codes are used as the spreading codes. var (30) Nu ρ2 i .
(cid:4)
(cid:26)
(cid:25)
(cid:30) (cid:31) (cid:31)
(cid:4)
Equation (29) is derived from [15] for the variance of the linear combination of correlated random variables. The probability of error conditioned on {ρi}N −1 is simply given i=0 by
.
= 1 2
(cid:3) E Du (cid:4) (cid:3) + var Iu
(cid:3) Nu
erfc (31) P e|ρ var For the channels and the signals described above, we can compute the BER based on the closed-form expression given by (15) and (22) and the numerical method (20). Shown in Figure 1, the results obtained by the two methods match very well as long as the value of n is sufficiently large. To check how quickly the infinite series in (15) converges with respect to n, we give the BER values based on the truncated series for the index n summing up from 0 to nmax in Tables 1 and 2 for two-ray and three-path channels. The BER for nmax is obtained by setting nmax large enough in (15), so that the first
100
EURASIP Journal on Applied Signal Processing 714
Table 1: BER values with truncated series for the 2-ray channel: Ku = 1, SNR = 0 dB.
10−1
nmax
10−2
e t a r
5 10 ∞
r o r r e
10−3
Numerical
K = 0.2 0.114438389 0.114438390 0.114438390 0.114438390
K-factor K = 1 0.107710909 0.107723090 0.107723090 0.107723090
K = 4.5 0.0670963590 0.0915495818 0.0917931623 0.0917931623
t i B
10−4
Table 2: BER values with truncated series for the 3-path channel: Ku = 1, SNR = 0 dB.
10−5
nmax
0
2
4
10
12
14
16
6
8 Eb/N0 (dB)
10 15 20 ∞
Numerical
K = 0.2 0.108401585 0.108401585 0.108401585 0.108401585 0.108401585
K-factor K = 1 0.117951253 0.102827722 0.102830484 0.102830484 0.102830484
K = 2.8 0.146651024 0.128790431 0.103149956 0.0939229789 0.0939229789
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, 3 p, 10 u A, 3 p, 20 u S, 3 p, 20 u
Figure 2: BER of MC-CDMA for the 2-ray (K0 = K1 = K = 0.2) and 3-path channel models (K0 = K1 = K2 = K = 0.2) using MRC. (A: theoretical results; S: simulation results.)
100
10−1
10−2
e t a r
r o r r e
10−3
t i B
10−4
10−5
0
2
4
10
12
14
16
6
8 Eb/N0 (dB)
nine significant digits for each BER value converge. It is also shown in the tables that the results of the closed-form ex- pression conform exactly with those computed by the nu- merical method. The results in Tables 1 and 2 also show that the series in (15) converges very rapidly for a relatively small K-factor, and nmax = 5–10 may be accurate enough for prac- tical interest. For large K-factors, it requires nmax = 15–30 to get an accurate approximation. Since the numerical method appears more computationally efficient compared with the closed-form expression especially when the K-factor is large, in Figures 2–7, the analytical results are obtained using the numerical method.
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, 3 p, 10 u A, 3 p, 20 u S, 3 p, 20 u
The results of our proposed time-domain method de- scribed herein are verified by comparing with the Monte Carlo simulations. In Figures 2 and 3, the K-factors for dif- ferent taps are equal, while in Figure 4, the K-factors for each tap are different. In all these cases, the simulation re- sults agree very well with the theoretical results obtained by the technique proposed in this paper. The simulation re- sults are based on the calculation of the decision variable for each bit, and averaging the results over large number of bits (say 100 000 bits). Figure 5 shows the accuracy of our low- complexity method proposed in Section 3.3. The results in- dicate that our low-complexity method gives quite a good approximation as compared with the accurate result, and the accuracy improves when the number of users is high.
Figure 3: BER of MC-CDMA for the 2-ray (K0 = K1 = K = 1) and 3-path channel models (K0 = K1 = K2 = K = 1) using MRC. (A: theoretical results; S: simulation results.)
(where MAI dominates), there is no difference between the BER performance for different values of K (i.e., K = 0.2, K = 1, and K = 10). For this case, MAI trends to be so se- vere and it becomes the dominant factor in determining the Next, the effect of the Rician K-factor on the BER per- formance is investigated using the analytical formula derived in Section 3. The Rician K-factor is defined as the ratio of signal power in dominant component over the (local-mean) scattered power which corresponds to a deterministic strong wave received. It is natural to expect a better performance for larger values of K. From the analytical results shown in (Figure 6), it can be seen that the performance improve- ment due to the increase in K is evident when the number of user is small (i.e., MAI is not dominant), while for 20 users
100
100
10−1
10−1
10−2
10−2
10−3
e t a r
e t a r
10−3
10−4
r o r r e
r o r r e
10−4
t i B
t i B
10−5
10−5
10−6
10−6
10−7
10−7
10−8
0
2
4
6
12
14
16
10
0
2
4
6
10
12
14
16
8 Eb/N0 (dB)
8 Eb/N0 (dB)
A, 1 u A, 10 u A, 20 u
S, K = 0.2 S, K = 1 S, K = 10
A, 3 p, 1 u S, 3 p, 1 u A, 3 p, 10 u S, 3 p, 10 u A, 3 p, 20 u S, 3 p, 20 u
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
BER Downlink MC-CDMA Systems over Rician Fading Channels 715
Figure 6: BER of MC-CDMA system versus SNR with various number of users and K-factor for the 2-ray channel model: v0 = 0, v1 = 14, K0 = K1 = K. (A: theoretical results; S: simulation results.)
Figure 4: BER of MC-CDMA using MRC for the 2-ray chan- nel model (K0 = 10, K1 = 4) and for the 3-path channel model (K0 = 10, K1 = 2.5, K2 = 0). (A: theoretical results; S: simulation results.)
100
10−1
20 users
10−1
10−2
e t a r
e t a r
15 users
r o r r e
r o r r e
t i B
t i B
10−2
10−3
10 users
10−3
10−4
0
2
4
6
10
12
14
16
0
1
2
3
4
6
7
8
9
10
5
8 Eb/N0 (dB)
K
Exact Approx.
A, SNR = 10, 1 u S, SNR = 10, 1 u A, SNR = 10, 10 u
S, SNR = 10, 10 u A, SNR = 10, 20 u S, SNR = 10, 20 u
Figure 5: BER of MC-CDMA system with MRC versus SNR for the 2-ray channel model: v0 = 0, v1 = 14, and K0 = K1 = 1.
Figure 7: BER of MC-CDMA system versus K-factor, SNR = 10 dB for the 2-ray channel model: v0 = 0, v1 = 14, K0 = K1 = K. (A: theoretical results; S: simulation results.)
5. CONCLUSION
system performance. Simulation results are also given to val- idate the analytical results. Figure 7 shows the BER compar- ison for various K when SNR equals 10 dB for 1, 10, and 20 users. A distinct variation can be seen when the number of users is small, while for 20 users, there is hardly any variation in the BER as MAI dominates the performance in this case. In this paper, we have proposed an accurate and simple method to calculate the performance of MC-CDMA systems with MRC combining scheme over downlink correlated Ri- cian fading channels. A general algorithm is proposed to
716 EURASIP Journal on Applied Signal Processing
(cid:22)
With the assumption that the channel coefficients b(l) for different paths are statistically independent of each other and the AWGN noise term is independent of the channel im- pulse response, a simple expression of the covariance matrix of v(0) is achieved:
(cid:21) v(0)vH (0)
= 2 diag
σ 2
0 , σ 2
1 , . . . , σ 2
. L−1, 1, . . . , 1 / ,
-. L
(cid:4)
Pv = E calculate the moment generating function by expressing the decision variables in Gaussian quadratic forms. Based on the moment generating function, an exact BER is obtained in the time domain. However, the complexity of this method in- creases exponentially with the number of users. To alleviate this problem, we have also proposed a low-complexity ap- proximate BER evaluation method by using the Monte Carlo integration. The results obtained by the analytical formula have been thoroughly verified by computer simulations. (A.6) APPENDIX The mean vector of v(0) is given by DERIVATION OF MGF
(cid:3) v(0)
= E
(cid:4)
=
µ v
(cid:4) , . . . ,
(cid:3) µ0x + jµ0y
(cid:3) µ1x + jµ1y
(cid:8)
(cid:19)
(cid:5)
L−1(cid:1)
(cid:3)
(cid:4)
(cid:4)(cid:27) (cid:27)2
, (A.7) In this appendix, we show the detailed derivation of (13) starting from (12). The MGF of the decision variable for frequency-selective channels under Rician distribution is de- rived. The decision variable Zu(0) in (12) can be rewritten as
(cid:27) (cid:27)b
(cid:3) vi
i=0
T , 0, . . . , 0 / -. , L
(cid:20)
L−1(cid:1)
L−1(cid:1)
(cid:4)
(cid:4)∗
. y(0) µ(L−1)x + jµ(L−1)y Zu(0) = Re 2γb
(cid:3) vi
(cid:3) vm
i=0
(cid:10)
L−1(cid:1)
(cid:3)
+ b b (A.1) yi,m
m=0, m(cid:7)=i (cid:4)∗
(cid:4) b
(cid:3) vi
(cid:3)
(cid:4)
i=0
v
(cid:4) ,
v
(cid:3)(cid:3)
Employing a result for the distribution of a noncentral Gaussian quadratic form [13], the MGF of Zu(0) in (A.2) can be obtained as , + η vi exp (A.8) Φ(−s) = det(I − sP1/2 with y(vi − vm)N denoted by yi,m for simplicity.
, (A.9) F(s) = P−1/2 I − sP1/2
v F(s)µ µH v QP1/2 (cid:4) (cid:4)−1 − I P−1/2 v
v
v QP1/2 v
v
where P−1/2 If the decision variable in (A.1) is written in vector form, the derivation of the MGF can be expressed more concisely. The vector form of the decision variable for the desired user is given by is the inverse of the symmetric square root of Pv. To represent the conditioned MGF Φ(−s) in a more com- (A.2) Zu(0) = vH (0)Q(0)v(0), pact form, we define a matrix
v QP1/2 v
where A = P1/2 (A.10)
(cid:4)
(cid:4)
(cid:4)
(cid:4)
(cid:4)
(cid:4)$
T
=
· · · b
· · · η
(cid:3) v0
(cid:3) v1
(cid:3) v0
(cid:3) v1
(cid:3) vL−1
(cid:3) vL−1
(cid:4)
v(0) # b b η η and represent its eigendecomposition as A = UvΛvUH v , where . (A.3)
(cid:3) λ1, λ2, . . . , λ2L
(cid:19)
(cid:20)
(A.11) Q(0) is a 2L × 2L matrix defined as Λv = diag
(cid:26)
=
T = UH
is the eigenvalue matrix of A. Also, we define Q(0) = , (A.4) Q1 0.5IL 0.5IL 0L
(cid:25) ˜µ1, ˜µ2, . . . , ˜µ2L
v P−1/2
v
· · ·
(A.12) where ˜µ v µ v.
(cid:3)
(cid:4)
(cid:5)
Equation (A.8) can now be simplified to
· · · ... ...
exp Λ(s) ˜uv (cid:4) . (A.5) Q1 = 2γb Φ(−s) =
(cid:4)(cid:4)
1,L−1
L−2,L−1
= exp
(cid:3) 1 − sλk (cid:4)
2L k=1 (cid:13)
(A.13) y0,2 y1,2 ... ... · · · y∗ y0,L−1 y1,L−1 ... yL−2,L−1 y(0) y(0) y0,1 y∗ y(0) 0,1 y∗ y∗ 0,2 1,2 ... ... 0,L−1 y∗ y∗ , ˜µH v det(I − sΛv (cid:27) (cid:27) (cid:3) (cid:11) (cid:27)2 (cid:27)˜µk sλk/ (cid:3) 2L 1 − sλk k=1 In (A.4), 0L and IL are the L×L zero and identity matrices, respectively. where Λ(s) = (I − sΛv)−1 − I is a diagonal matrix.
BER Downlink MC-CDMA Systems over Rician Fading Channels 717
[9] J. G. Proakis, Digital Communications, McGraw-Hill, New
In the case of repeated eigenvalues, the MGF of Zu(0) is
York, NY, USA, 4th edition, 2001.
(cid:1)
=
(cid:27) (cid:27)2,
given by (13). In (13)
(cid:27) (cid:27)˜µk
k∈κm
(A.14) b2 m
[10] Y. Li, L. J. Cimini Jr., and N. R. Sollenberger, “Robust chan- nel estimation for OFDM systems with rapid dispersive fad- ing channels,” IEEE Trans. Communications, vol. 46, no. 7, pp. 902–915, 1998.
1 K0M1+σ 2
0 K1M0−W)/R,
where κm denotes the set of k indices associated with the mth distinct eigenvalue. Here, the MGF of the decision variable for the two-ray channel model is given as an example:
(A.15) Φ(s) = e−(K0+K1) e2(σ 2 R
[11] J. G. Proakis and D. G. Manolakis, Digital Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, USA, 3rd edition, 1996. [12] J. K. Cavers and P. Ho, “Analysis of the error performance of trellis-coded modulations in Rayleigh-fading channels,” IEEE Trans. Communications, vol. 40, no. 1, pp. 74–83, 1992. [13] Y. Ma, T. L. Lim, and S. Pasupathy, “Error probability for co- herent and differential PSK over arbitrary Rician fading chan- nels with multiple cochannel interferers,” IEEE Trans. Com- munications, vol. 50, no. 3, pp. 429–441, 2002.
where the coefficients are given by
R = c4s4 + c3s3 + c2s2 + c1s + 1,
[14] E. Biglieri, G. Caire, G. Taricco, and J. Ventura-Traveset, “Sim- ple method for evaluating error probabilities,” Electronics Let- ters, vol. 32, no. 3, pp. 191–192, 1996.
c4 = σ 2 (cid:5)
(cid:27) (cid:27)2,
(cid:3) c2 = −
(cid:27) (cid:27)y0,1
0 σ 2 1
c3 = −4 (cid:4)
[15] C. W. Helstrom, Probability and Stochastic Processes for Engi- neers, Macmillan Publishing, New York, NY, USA, 1991. [16] X. Gui and T. S. Ng, “Performance of asynchronous orthog- onal multicarrier CDMA system in frequency selective fad- ing channel,” IEEE Trans. Communications, vol. 47, no. 7, pp. 1084–1091, 1999.
y(0), 2γb
[17] J. K. Cavers, Mobile Channel Characteristics, Kluwer Aca-
(cid:5)
(cid:3)
(cid:4)
0 σ 2 1 , 2γbσ 2 0 σ 2 1 y(0), 1 y(0)2 − 8γbσ 2 0 σ 2 + 8γbσ 2 (cid:5) (cid:4) (cid:3) 0 + σ 2 σ 2 1 (cid:4) (cid:3)
demic, Boston, Mass, USA, 2000.
0 + σ 2 σ 2 1 c1 = 2 (cid:25) µ0xµ1x Re 2γbs
(cid:4)(cid:26)
− µ0xµ1y Im
W = y0,1 (cid:3) + µ0yµ1x Im (cid:4) y0,1 (cid:3) y0,1 y0,1 +µ0yµ1y Re . (A.16)
The region of convergence is the vertical strip enclosing the jω axis bounded by the closest poles on either side. The BER performance can be obtained through the method de- scribed above.
REFERENCES
