Performance of Asynchronous MC-CDMA Systems with Maximal Ratio Combining in Frequency-Selective Fading Channels

The bit error rate (BER) performance of the asynchronous uplink channel of multicarrier code division multiple access (MC-CDMA) systems with maximal ratio combining (MRC) is analyzed. The study takes into account the e ﬀ ects of channel path correlations in generalized frequency-selective fading channels. Closed-form BER expressions are developed for correlated Nakagami fading channels with arbitrary fading parameters. For channels with correlated Rician fading paths, the BER formula developed is in one-dimensional integration form with ﬁnite integration limits, which is also easy to evaluate. The accuracy of the derived BER formulas are veriﬁed by computer simulations. The derived BER formulas are also useful in terms of computing other system performance measures such as error ﬂoor and user capacity.


INTRODUCTION
Multicarrier code division multiple access (MC-CDMA) is a technique that combines direct sequence (DS) CDMA with orthogonal frequency division multiplexing (OFDM) modulation. It is one of the candidate technologies considered for the 4th-generation wireless communication systems [1]. MC-CDMA transmits every data symbol on multiple narrowband subcarriers and utilizes cyclic prefix to absorb and remove intersymbol interference (ISI) arising from frequency-selective fading. As it is unlikely for all subcarriers to experience deep fade simultaneously, frequency diversity is achieved when the subcarriers are appropriately combined at the receiver. In [2,3], it is shown that MC-CDMA outperforms the conventional DS-CDMA and two other forms of CDMA with OFDM modulation, namely MC-DS-CDMA and multitone CDMA.
Several combining techniques have been proposed for MC-CDMA systems. Maximal ratio combining (MRC) offers maximum improvement in the presence of spectrally white Gaussian noise [4,5]. It is shown to achieve better performance for MC-CDMA uplink than equal gain combining (EGC) in [3], and the resultant system has lower error floor than DS-CDMA and MC-DS-CDMA.
The bit error rate (BER) performance of MC-CDMA systems is not easy to analyze as the receiver operations involve coherent combining of a large number of independent or correlated fading subcarriers with possibly different fading statistics. Signal analysis is further complicated by the presence of multiuser interference (MUI) in the received signal. In the literature, simulations are often used to study the performance of MC-CDMA systems [2,6,7,8]. For the downlink performance of MC-CDMA with MRC, performance lower bounds are given in [2,9]. For the MC-CDMA uplink with MRC, to the authors' knowledge, the most general performance analysis is given in [3], where Monte Carlo integration is used to evaluate the BER expressions which involve multidimensional integration of dimensions equal to the number of subcarriers. Although simplified performance formulations are given in [10,11,12], they are based on the assumptions of independent and identically distributed (i.i.d.) fading among the channel paths [10], or independent fading among the subcarriers [11,12]. Furthermore, all the works reported in [2,3,9,10,11,12] only consider Rayleigh fading channels.
In this paper, we conduct a BER analysis for MC-CDMA uplink with MRC in Rayleigh, Rician, and Nakagami fading channels with arbitrary fading parameters and correlations between the channel paths or subcarriers. A simplified signal model for the MRC output in uplink channel is first developed. By employing the time-frequency equivalence for multicarrier signals, the generic BER expression of the MC-CDMA system with MRC is expressed in terms of the channel impulse response (CIR) information, instead of the subcarrier fading information. Then, by using the technique of Cholesky decomposition, a closed-form BER formula that does not require integration is obtained for channels with correlated Nakagami fading paths. For channels with correlated Rician fading paths, the BER formula is reduced to a form of one-dimensional integration by employing an alternative form of the Gaussian Q-function.
The organization of the paper is as follows. The generic BER analysis is given in Section 2, while specific BER formulations for a variety of fading channels are given in Section 3. Section 4 presents the verification results and Section 5 concludes the paper.

Signal model
Considering an MC-CDMA system with N u users, each of whom employs N c subcarriers modulated with BPSK, the transmitted signal corresponding to the kth user can be expressed as follows: where E b and T s are the bit energy and symbol duration respectively; u Ts (t) represents a rectangular pulse waveform with amplitude 1 and duration T s ; b k (v) is the vth transmitted data bit, c k,n is the random spreading code; w n is the frequency of the nth subcarrier; and θ k,n is the random phase at transmitter.
For uplink transmission, the base station receives signals from different users through different propagation channels. This leads to different channel amplitudes and phases to be associated with different users. More importantly, asynchronous transmission between users results in misalignment in the signal arriving times among different users. Hence the received MC-CDMA uplink signal from a quasistatic frequency-selective fading channel is of the form where φ k,n = θ k,n +ϕ k,n −w n ξ k , ξ k is the time misalignment of user k with respect to user 1 (the reference user), h k,n and ϕ k,n are the frequency-domain subcarrier fading gain and phase for the nth subcarrier, respectively, and η(t) is the additive white Gaussian noise (AWGN). The decision variable U of user 1 is Nc n=1 c 1,n cos w n t + φ 1,n α 1,n dt where D and η are the desired signal and noise components, respectively, I and J are the MUI components, and α 1,n is the combiner coefficient for the nth subcarrier of user 1. Without loss of generality, we abbreviate h 1,n as h n and α 1,n as α n . For MRC, the combiner coefficient α n = h n . Then the desired signal component D = E b T s /2N c Nc n=1 h 2 n , and the noise component η has zero mean and variance (N 0 T s /4) Nc n=1 h 2 n . For simplicity in analysis, the uplink MUI is divided into two parts: I is the MUI from the same subcarrier of other users while J is the MUI from other subcarriers of other users [3], that is, where b k (0) and b k (−1) represent the current and previous data bits of the kth user, respectively. Since the user data and fading parameters of different users are uncorrelated, the summands in (4) are uncorrelated too. Even though the subcarrier fading gain h k,n of the same user may be correlated to some extent, the summands in (4) in this case are still uncorrelated due to the presence of other uncorrelated variables such as phase φ k,n in the equations. Moreover, since the number of summands in (4) are very large (e.g., N c can be at least 64 and N u can be as large as N c ), both I and J are the summations of large number of uncorrelated terms. Hence, central limit theorem (CLT) can be applied to approximate I and J as Gaussian random variables (RVs) [13]. It is shown in [3] that I and J have zero mean and variance given by where σ 2 is the subcarrier fading power of other users. Thus the BER of an MC-CDMA uplink channel conditioned on the set of subcarrier fading amplitudes {h n } is where Q(·) is Gaussian Q-function. The average BER P e can then be obtained by averaging (7) over the joint distribution function (jdf) of {h n }, that is, The multidimensional integration in (8) is not easy to evaluate even by using Monte Carlo integration. This is because the number of subcarriers is usually large (e.g., 64 in IEEE 802.11a wireless LAN systems) and the subcarrier fading gains are usually correlated. (6) results in different dependence on h n in the variance expressions of I, J, and η, thus complicating the analysis of the uplink BER. However, we noticed that its value does not vary much for different values of n, hence it can be approximated as a constant a whose value only depends on N c , that is,

Simplification of BER formula
Later we will show using simulation results that the effect of the approximation made in (9) on the BER is negligible. With this approximation, the term Nc n=1 h 2 n becomes a common factor in the var(I) expression (5), the var(J) expression (6), and the variance expression of η. Thus the conditional uplink BER expression in (7) can be simplified to Denoting the BER can be obtained by averaging (10) over the probability density function (pdf) of the combined subcarrier fading variable β, that is, where f (·) denotes the pdf and ν is given by Comparing (12) and (8), the dimension of integration is reduced from N c to one, provided that the pdf of β can be obtained. However, in general, it is not easy to find the pdf of β for larger number of subcarriers whose fading gains may be correlated, and/or different subcarriers may have different fading characteristics. We will circumvent this problem by transforming the subcarrier-domain integration in (12) into a path-domain integration. This will be elaborated in the next section.

Time-and frequency-domain equivalence of MRC Output
The CIR of a multipath fading channel with N p resolvable paths is typically represented using the tapped delay line model as where g l , ψ l , and τ l are the fading envelope, phase, and delay of the lth path, respectively. Denoting the complex subcarrier fading gains as a vector h with length N c , and the complex path fading gains as a vectorg with length N p , thenh is related tog by discrete Fourier transform (DFT) [14,15], that is, where The term Nc n=1 h 2 n in (10) (12) can now be rewritten as where Compared to (12), (18) is much easier to compute because the pdf of γ is generally easier to analyze than the pdf of β for the following reasons: (i) most practical channels are characterized and represented in the form of CIR or power delay profiles [16,17], which are more directly applicable to (18) than (12) [2,10]). This increases the complexity in determining the pdf of β.
Therefore, we will use (18) and the pdf of the combined path fading variable γ to formulate P e in the next section.

BER FORMULATIONS FOR DIFFERENT FADING CHANNELS
Notice that (18) is equal to the BER expression for a conventional time-domain MRC system, so the problem now is to find the pdf of γ, which is the MRC output of the multiple fading paths. To be most general, we will consider the pdf of γ with arbitrarily correlated paths in Rayleigh, Rician, and Nakagami fading channels. Rayleigh fading is discussed as a special case of Nakagami or Rician fading.

Nakagami fading channels with independent paths
In this subsection, we model the fading path gains {g l } as independent Nakagami-distributed RVs. Nakagami fading distribution, also known as m-distribution, is widely adopted for modelling fading channels because of its good fit to empirical measurements [18,19,20], as well as the tractability it renders to BER evaluation [21]. A variety of fading effects can be modelled as Nakagami fading with different m parameters, including Rayleigh fading as a special case when m equals 1. The Nakagami-m distribution is given by where 2 ]. Γ(·) is the Euler gamma function and E[·] denotes statistical expectation. Since the square of Nakagami RV γ l = g 2 l follows the gamma distribution the MRC signal γ = γ 1 + γ 2 + · · · + γ Np can be viewed as the summation of N p number of gamma variables. If {γ l } are independent with identical m l /Ω l for all values of l, γ follows exactly another gamma distribution with new values of m and Ω given by [5] For independent {γ l } with nonidentical m l /Ω l , the exact pdf of γ becomes much more complicated to derive. In [22], we have shown that the combined output γ in this case can be adequately approximated as a new gamma-distributed RV. For this resultant gamma distribution, its power Ω is given by (23), while its m parameter value can be derived through moment matching to be For equal m l /Ω l , (24) is reduced to (22). Substituting (21) into the BER formula (18) with appropriate m and Ω parameters, we have where 2 F 1 (·) is the hypergeometric function and ν is given in (13).

Nakagami fading channels with correlated paths
Although statistical independence among the diversity branches is desired in MRC systems, there are cases where this assumption is not valid [23,24]. Hence we consider Nakagami fading channels with correlated paths in this subsection. Dual-branch MRC system with correlated Nakagami fading branches is discussed in [5,24]. The study is further generalized to arbitrary number of diversity branches in [23], subject to the conditions of identical branch parameters, that is, m l and Ω l are the same for all values of l. Also, the results of [23] are only applicable to constant or exponential branch correlation models. In [25], arbitrary branch correlation is studied, but the analysis is limited to identical and integervalued m l parameters across the branches. In [21,26], noninteger m l values are considered, but the m l parameters must still be identical across the branches.
In [27], we propose an approach to obtain the fading statistics of correlated {γ l } without any constraints on the fading parameters and correlations of the channel paths. In this approach, Cholesky decomposition is used to transform correlated gamma RVs into linear combinations of independent gamma RVs. Specifically, denote γ = [γ 1 , . . . , γ Np ] T as the set of correlated gamma variables with covariance ma- By Cholesky decomposition, C γ = LL T , where L is a lower triangular matrix with ( j, i)th element denoted as l ji . Let w = [w1 w 2 · · · w Np ] T be a set of independent gamma RVs with identity covariance matrix C w .
Next, let γ = Lw (26) or equivalently then the covariance matrix of Lw is Therefore, the correlated path variables γ have been transformed to weighted sums of independent variables w with weights given by (26) or (27), without affecting the path correlations. By matching the moments of both sides of (27) progressively from top down, the m-parameter m w,i and the power Ω w,i of the elements w i in w can be obtained iteratively using the following equations: Summing up both sides of (27) then gives the resultant combined output Since {w i } are independent, it follows from our earlier analysis in this paper that γ can be approximated as a new gamma distributed RV with Ω given by (23) and m given by It can be shown that the m-parameter expression given in (31) reduces to (24) and (22) for independent paths with nonidentical m l /Ω l and identical m l /Ω l ratios, respectively. Hence (31) is a more general expression.
Finally, it should be clear from the preceding analyses that a closed-form BER formula for MC-CDMA uplink channel with MRC in Nakagami fading channel without any constraint on the fading parameters and correlations of the channel paths is realized in (25), with ν, Ω, and m given by (13), (23), and (31), respectively. Furthermore, with m l = 1 for the fading paths, (25) becomes applicable to channels with Rayleigh fading paths.

Rician fading channels with independent or correlated paths
The Rician distribution is another popular fading model for signal envelops received in channels with direct line-of-sight (LOS) or specular component [28]. When the LOS component is absent, the Rician fading distribution will be reduced to Rayleigh. The BER of the MRC diversity system with Rayleigh diversity branches are available in [28,29]. However, for Rician fading and especially correlated Rician fading branches, the results in [28,30,31] are complicated as they may include hypergeometric functions or sum of integrals of Bessel functions. In this section, we will derive a new one-dimensional integral BER expression based on the characteristic function (CF) [32] of the combined output.
The Rician fading path gains g l follow the pdf expression where Ω l = E[g 2 l ], I 0 (·) is the zeroth-order modified Bessel function of the first kind, and K l is the Rician K factor [28]. When K l = 0, (32) reduces to Rayleigh fading distribution.
It is well known that for Rician fading channel, the complex channel gain can be represented as complex Gaussian RVs, that is, g = [g 1 exp jψ 1 , . . . , g Np exp jψ Np ] T = X c + jX s . (33) Define X = [X c ; X s ], where X c and X s are N p ×1 real Gaussian random vectors, µ = E[X] as the mean vector, and C x as the covariance matrix of X. The CF of γ = Np l=1 g 2 l is given in [32] as follows: where λ k are the eigenvalues of C x , C x = VΛV T , Λ = diag(λ 1 , λ 2 , . . . , λ 2Np ), and k is given by By utilizing an alternative expression for the Gaussian Qfunction given in [33], that is, and with the CF of γ given in (34), the BER expression (18) of MC-CDMA uplink with MRC in correlated Rician fading channel can now be obtained by the new equation as shown: Since only one-dimensional integration is involved and the integration limits are finite, (36) can be easily evaluated numerically. Also, (36) is general enough to cover the case of independent paths with C x being a diagonal matrix, and the Rayleigh fading case with µ = 0.

RESULTS AND DISCUSSIONS
To demonstrate the validity and simplicity of our proposed BER formulation approaches, we compare our results with that in [10], which considers channels with i.i.d. Rayleigh fading paths. Laplace transform and residual method are used in [10] to compute the pdf of Nc n=1 h 2 n and the resultant BER expression is in one-dimensional integration form. In contrast, our BER expression for this case is the closed-form formula in (25). In Figure 1, the analytical BER values com-  puted using (25) in this paper are compared with the BER values taken from [10, Figure 4]. Both sets of BER values are found to match exactly.
In Figures 2 and 3, we use computer simulations of asynchronous MC-CDMA uplink with 128 subcarriers and 10 active users to verify our BER formulas for different fading conditions. There are three types of BER plots in Figures 2 and 3: one simulation plot and two analytical plots (obtained with or without the approximation made in (9)). Figure 2 is for a channel with independent or correlated Rician fading paths with randomly selected K l , Ω l , and correlation values. Figure 3 is for Nakagami channels consisting of independent paths with identical m l /Ω l ratio, or correlated paths with unequal m l /Ω l ratios. Detailed channel specifications are given in the respective figure captions.
The "analytical with approximation (9)" plots in Figures  2 and 3 are obtained by using (36) and (25), respectively, while the "analytical without approximation (9)" plots are obtained by Monte Carlo integration of the conditional BER given in (7). Both Figures 2 and 3 show that the analytical BER values with or without approximation (9) are indistinguishable, hence the effect of the approximation made in (9) on the system BER values is negligible. Although not shown in this paper, the same verification has also been carried out for other values of subcarrier number N c and fading parameters, for example, N c from 16 to 256, Rician K factor from 0 to 20, and Nakagami m parameter from 1 to 20. The same conclusion that the effect of the approximation made in (9) on the system BER is insignificant can be reached.  For Nakagami fading channels, if the fadings in different channel paths are independent and have identical m l /Ω l ratio, the MRC output γ follows an exact gamma distribution. Hence, as seen for the channel condition (a) in Figure 3, both the simulation and analytical BER plots match very well. On the other hand, when the fadings in different channel paths have different m l /Ω l ratios or are correlated, γ is only approximately gamma-distributed. Hence, some mismatch can be observed between the analytical and simulated BER plots for the channel condition (b) in Figure 3. The approximation error associated with the pdf of γ has been discussed in detail in [22,27]. As concluded in these papers, for most cases, the approximation error is very small and acceptable.
As seen in Figures 1, 2, and 3, the MC-CDMA uplink exhibits the familiar error floor at high E b /N 0 level due to the MUI power predominating over the AWGN power. With our simple uplink BER expressions of (36) and (25), the error floor can be easily evaluated by setting E b /N 0 in (13) to infinity. Figure 4 shows the dependence of the error floor values on the number of users in channels with correlated Rician and Nakagami paths. The Rician channel model has the same parameters as the channel condition (b) in Figure 2, while the Nakagami channel model is the same as the channel condition (b) in Figure 3. The Rician channel suffers higher error floor because its channel paths are subject to more severe fading than the Nakagami channel.
Furthermore, with our closed-form or one-dimensional integral BER formulas, the effect of various system or channel parameters on the system performance can also be ob-   tained with ease. As an example, Figure 5 shows how many users the MC-CDMA system can support in order to meet a target BER of 10 −3 in channels with 4 or 8 i.i.d. Rayleigh paths. Such user capacity results can be easily obtained from (25) by simple numerical root finding. Besides, our earlier analysis predicts that the larger the number of channel paths, the more diversity the MC-CDMA system can achieve. This explains why the channel with more paths in Figure 5 can accommodate more users.

CONCLUSIONS
In this paper, we present a way to obtain the analytical BER of MC-CDMA uplink with MRC in channels with correlated Rayleigh, Rician, or Nakagami fading paths. We first achieved a simplified signal model for the MRC output and established its time-frequency equivalence, which states that combining the subcarriers using MRC has exactly the same effect as combining the channel paths using MRC. This principle is exploited to achieve very simplified BER formulas based on (CIR) information of the MC-CDMA system under study. A new closed-form BER formula is derived for channels with correlated Nakagami fading paths using the techniques of Cholesky decomposition and gamma pdf approximation. The BER formula is exact if all the channel paths have identical m l /Ω l ratio; otherwise, it is approximate, but nonetheless adequately accurate. For channels with correlated Rician fading paths, the associated analytical BER formula is derived by appropriate function mapping based on CF and an alternative form of the Gaussian Q-function. The resultant BER formula contains only one-dimensional integration and hence can be easily integrated numerically.