EURASIP Journal on Applied Signal Processing 2005:5, 730–742 c ○ 2005 Hindawi Publishing Corporation Joint Impact of Frequency Synchronization Errors and Intermodulation Distortion on the Performance

The performance of multicarrier systems is highly impaired by intercarrier interference (ICI) due to frequency synchronization errors at the receiver and by intermodulation distortion (IMD) introduced by a nonlinear ampliﬁer (NLA) at the transmitter. In this paper, we evaluate the bit-error rate (BER) of multicarrier direct-sequence code-division multiple-access (MC-DS-CDMA) downlink systems subject to these impairments in frequency-selective Rayleigh fading channels, assuming quadrature amplitude modulation (QAM). The analytical ﬁndings allow to establish the sensitivity of MC-DS-CDMA systems to carrier frequency o ﬀ set (CFO) and NLA distortions, to identify the maximum CFO that is tolerable at the receiver side in di ﬀ erent scenarios, and to ﬁnd out the optimum value of the NLA output power backo ﬀ for a given CFO. Simulation results show that the approximated analysis is quite accurate in several conditions.


INTRODUCTION
In the last years, several multicarrier code-division multipleaccess (MC-CDMA) schemes [1] have been proposed by combining orthogonal frequency-division multiplexing (OFDM) and direct-sequence code-division multiple-access (DS-CDMA), with the goal of incorporating the advantages of both techniques. Specifically, the low-complexity equalization of cyclic prefixed OFDM systems and the multiple-access interference (MAI) mitigation capabilities offered by DS-CDMA systems make MC-CDMA techniques attractive for future mobile broadband communications [2].
Differently from single-carrier systems, one of the main problems of multicarrier schemes is the high sensitivity to frequency synchronization errors [3]. Indeed, the carrier frequency offset (CFO), which models the frequency mismatch between the transmitter and receiver oscillators, generates intercarrier interference (ICI), thereby destroying the frequency-domain orthogonality of the transmitted data.
The presence of a nonlinear amplifier (NLA) at the transmitter, which introduces both intermodulation distortion (IMD) and out-of-band interference, is another relevant source of impairment in multicarrier systems. Indeed, multicarrier signals are characterized by a high peak-toaverage power ratio (PAPR) [4], and hence they are significantly distorted when, in order to improve the power efficiency, the NLA working point is chosen close to the saturation point.
Obviously, the typical impairments of multicarrier systems add to the sources of degradation that are generally present also in other multiple-access schemes, such as the MAI associated with multiple users in nonorthogonal systems [5], and the intersymbol interference (ISI) introduced by the signal propagation over multipath channels. As a consequence, the degradation introduced by CFO and NLA, as well as MAI and multipath, should be taken into account when evaluating the performance of multicarrier systems.
Specifically, this paper deals with the effects of the mentioned impairments on the bit-error rate (BER) performance. The particular multicarrier scheme herein considered is commonly identified as multicarrier DS-CDMA (MC-DS-CDMA) [1], where multiple users are discriminated by spreading in the time domain. Moreover, in this paper, we focus on the performance at the mobile end (i.e., downlink), assuming that the receiver is equipped with a bank of single-tap equalizers followed by a despreader.
Previous works about MC-DS-CDMA performance considered the presence of either frequency synchronization errors or nonlinear distortions. Indeed, Steendam and Moeneclaey evaluated the signal-to-noise ratio (SNR) degradation produced by the CFO in additive white Gaussian noise (AWGN) for MC-DS-CDMA downlink systems [6]; however, they did not consider the NLA at the transmitter. In addition, Jong and Stark analyzed the effects of multipath and nonlinearities on the BER performance of a multicarrier spread-spectrum system, which basically is an MC-DS-CDMA scheme with one active user [7]. However, they assumed perfect frequency synchronization at the receiver side.
The purpose of our work is to extend the results of [6,7] in order to jointly take into account CFO, nonlinear distortions, and MAI, in multipath fading channels. Our two-step approach to obtain the BER is quite standard. The first step relies on the Gaussian approximation of the interferences, which allows us to evaluate the BER conditioned on a given channel realization. The second step consists in averaging the conditional BER over the channel statistic, which is assumed to be Rayleigh.
The paper is structured as follows. In Section 2, we introduce an MC-DS-CDMA system model where linear signal processing operations (either in the time domain or in the frequency domain) are simply represented by matrix multiplications. Section 3 is dedicated to the BER evaluation in frequency-flat fading channels, assuming quadrature amplitude modulation (QAM). In Section 4, the BER evaluation for flat-fading channels is extended to frequency-selective scenarios characterized by Rayleigh fading. In Section 5, we check the accuracy of the analytical findings by comparing theoretical and simulated results. Finally, in Section 6, some concluding remarks are drawn.

SYSTEM MODEL
Firstly, we introduce some basic notations. We use lower (upper) bold face letters to denote column vectors (matrices), superscripts * , T, and H to represent complex conjugate, transpose, and Hermitian operators, respectively. We reserve E{·} to represent the statistical expectation, and x to denote the greatest integer smaller than or equal to x. The Q-function is defined as Q(x) = (2π) −1/2 +∞ x e −ν 2 /2 dν, 0 M×N is the M × N all-zero matrix, and I N is the N × N identity matrix. We define [A] m,n as the (m, n)th entry of the matrix A, [a] n as the nth entry of the column vector a, (a) mod N as the remainder after division of a by N, and diag(a) as the diagonal matrix with (n, n)th entry equal to [a] n .
We consider the downlink of an MC-DS-CDMA system with N subcarriers, spaced by ∆ f = 1/T, and with K active users. Hence, assuming rectangular pulse shaping, the chip duration is equal to T. The base station transmits N symbols of each user in parallel, one for each subcarrier, by spreading each symbol in the time domain employing a user-dependent spreading code. The spreading code of the kth user, which is the same for all the subcarriers, is denoted by c k = [ck,0 · · · c k,G−1] T , where G is the processing gain. Assuming |c k,g | = G −1/2 , the vector c k has unit norm. As in conventional OFDM systems, the timedomain samples are obtained by N-dimensional IFFT of the data on the N subcarriers, and a cyclic prefix of duration τ CP = LT/N is attached to the IFFT output block in order to avoid channel-induced interference between successive blocks. Consequently, the symbol rate of each user is equal to By defining the N × N unitary FFT matrix F as [F] m,n = N −1/2 exp(− j2π(m−1)(n− 1)/N), and the (N +L)×N cyclic prefix insertion matrix as T CP = [I T CP I T N ] T , where I CP contains the last L rows of the identity matrix I N , the transmitted block, at the input of the NLA, can be expressed by [8] ] n,k the lth symbol of the kth user on the nth subcarrier, and c[g] = [c1,g · · · c K,g ] T is the vector that contains the gth chip of the spreading codes of all the K users. The data symbols {s n,k [l]}, drawn from an M-ary square QAM constellation, are assumed to be independent and identically distributed (i.i.d.) with power σ 2 S . Although in many cases we are dealing with continuous-time signals (i.e., at the NLA input and output, at the receiver input, etc.), throughout the paper we consider the equivalent discretetime signals, to simplify the notation.
In this paper, we model the NLA as a memoryless device characterized by its AM/AM and AM/PM curves. After incorporating both the chip index g and the symbol index l into a single index i = lG + g, by the Bussgang theorem [9], the transmitted block, at the output of the NLA, can be expressed as [10,11] where α represents the average linear amplification gain, andv IMD [i] is the intermodulation distortion (IMD), uncorrelated with the linear part αu IN [i]. The validity of (2), based on a Gaussian distribution of u IN [i] in (1), is justified by the high number of subcarriers usually employed in multicarrier systems (e.g., N > 32) [11]. In this case, the average linear amplification gain can be evaluated as α = R uOUTuIN (0)/R uINuIN (0), where R uOUTuIN (τ) is the continuoustime cross-correlation between the NLA output and input, and R uINuIN (τ) is the continuous-time autocorrelation function of the NLA input. Moreover, the elements of can be predicted in advance by exploiting the knowledge of the continuous-time autocorrelation function R uOUTuOUT (τ) of the NLA output. Such an autocorrelation function can be obtained by means of known closed-form formulas or numerical integration techniques, which require the knowledge of the NLA input statistics, summarized by R uINuIN (τ), of the input power backoff to the NLA, and of the AM/AM and AM/PM curves that characterize the NLA [10].
After the parallel-to-serial conversion, the signal stream , p = 0, . . . , P − 1, is transmitted through a multipath channel, whose discrete-time equivalent impulse response is where Q is the number of paths, ζ q and τ q are the complex amplitude and the propagation delay, respectively, of the qth path, R(τ) is the triangular autocorrelation function of the rectangular pulse-shaping waveform, and T S = T/N is the sampling period. Throughout the paper, we assume that the channel amplitudes {ζ q } are zero-mean complex Gaussian random variables, giving rise to Rayleigh fading, and that the maximum delay spread τ max = max{τ q } is smaller than the cyclic prefix duration τ CP = LT S , that is, h[b] may have nonzero entries only for 0 ≤ b ≤ L. At the receiver side, the samples obtained after matched filtering can be expressed as [12] x where f 0 is the CFO caused by the frequency synchronization error at the receiver side, x AWGN [a] represents the AWGN, with a = iP + p. Throughout the paper, we assume that the timing information is available at the receiver. In multicarrier systems, such an information could be acquired by exploiting timing synchronization algorithms designed to work in the presence of an unknown CFO [13,14,15]. In any case, when timing errors are significant, the BER expressions we will derive can be considered as lower bounds.
The P received samples relative to the gth chip (of the lth symbol) are grouped in the vector x[i] = x[lG + g], thus obtaining [12] x The cyclic prefix elimination, which is common to many multicarrier-based systems, can be represented by the left multiplication of the matrix R CP = [0N×L I N ] with the received vector in (6), which leads to the N × 1 vector [12] y for the AWGN term. By using (1) and (2), the vector y (8) can also be expressed as where n] T are the channel vectors in the time and frequency domain, respectively. Consequently, the recovery of the transmitted data is accomplished by applying the FFT at the receiver, thus obtaining z[lG + g] = Fy[lG + g], which by (9), can be rearranged as from (10), we obtain (12) where The matrix model expressed by (12)  Expression (12) shows the effects of both nonlinear distortions and frequency synchronization errors on the spread data matrix S[l]C. Specifically, the NLA introduces the constant complex gain α and the additive term V IMD [l], which represents the IMD. Moreover, the NLA-distorted signal, after passing through the multipath channel summarized by the diagonal matrix Λ, is affected by the CFO-induced ICI, because the matrix Φ is not diagonal. In addition, the CFO produces two phase-shift effects. The first one is a time-invariant phase shift ϕ = e j2πεL/N e jπε(N−1)/N (the factor e jπε(N−1)/N is contained in the main diagonal of Φ), which adds to the phase shift caused by the NLA. The second effect is a time-varying phase shift, which is represented by the diagonal matrix E[l]. In this paper, we assume that the receiver is able to perfectly compensate for the time-invariant phase-shift term ϕ, which could also be seen as part of the channel-induced distortion. As far as the compensation for the time-variant phase shift is concerned, we consider two alternative hypotheses.

Hypothesis H1
The receiver obtains an accurate phase-shift estimate in each chip interval, as assumed in [6]. As a consequence, by the knowledge of E[l] at the receiver side, the estimated phaseshift matrix can be expressed byÊ

Hypothesis H2
The receiver does not track the time-variant phase shift at the chip level, estimating only the average phase shift within a symbol interval. Hence, by the knowledge of the mean of the elements in E[l], the estimated phase-shift matrixÊ It can be noted that the perfect knowledge of the phase shifts is not completely realistic. However, we consider this hypothesis because it produces a lower bound that is independent of the phase-shift estimation technique.
In this paper, we assume that the receiver performs a single-tap frequency-domain equalization and code despreading. In the absence of NLA-and CFO-induced impairments, the equalizer-despreader detector is able to eliminate both ISI and MAI when orthogonal spreading codes are employed. From the computational complexity point of view, the equalizer-despreader detector is less demanding than multiuser detectors [5], and hence it seems to be more suitable in the downlink situation.
The receiver can perform the despreading operation after the phase-shift compensation and the channel equalization. Thus, by assuming perfect channel state information (CSI) at the receiver side, the equalized data matrix can be con- (14) where under Hypothesis H2. The despreading operation leads to an N-dimensional vector that contains the N decision variables of the kth user, as expressed by The decision over z DESPR,k [l] is successively done according to the proper constellation size M.
It should be pointed out that the K-dimensional vector CΘc * k represents the effect of the CFO-induced timevarying phase shift on the spreading-despreading operation. If the time-varying phase shift can be perfectly compensated for, that is, under Hypothesis H1, this vector becomes equal to Cc * k . Therefore, if the users employ orthogonal spreading codes, the multiuser interference has been perfectly eliminated, because the product S[l]CΘ H1 c * k in (15) is equal to s k [l]. On the contrary, under Hypothesis H2, the multiuser interference cannot be completely eliminated.
which is, in general, different from zero even for orthogonal sequences (characterized by c T k c * . An interesting matter could be the search for spreading sequences that minimize (16), following the lines suggested by [16]. However, this topic is beyond the scope of this paper.
Furthermore, since ρ k,k,H2 = c T k Θ H2 c * k < 1, the timevariant phase shift also introduces an attenuation factor of the useful signal, given by For a given maximum tolerable frequency offset ε, the attenuation factor in (17) limits the maximum value G of the processing gain. Indeed, ρ k,k,H2 is close to 1 only if ε N/(N + L)G ≈ 1/G. On the contrary, under Hypothesis H1, ρ k,k,H1 = c T k Θ H1 c * k = 1, and hence no attenuation is introduced by the spreading-despreading operation. It should be observed that ρ k,k is independent of the user index k in both hypotheses, and hence, in the following, it will be denoted by ρ.

BER OF MC-DS-CDMA SYSTEMS IN FLAT-FADING CHANNELS
In this section, we evaluate the effect of the previously described impairments when the channel exhibits flat-fading behavior. The results herein obtained will be useful for the performance evaluation in frequency-selective fading channels, developed in the next section. In the flat-fading case, all the subcarriers experience the same channel gain. Consequently, since Λ = λI N , (15) can be simplified to For convenience, we consider the scaled version of (18) where z n,AWGN [l] T is the nth row of Z AWGN [l], and m n = [mn,1 · · · m n,N ] T collects the elements on the nth row of M. It can be verified that [17] where m n,n represents the ICI coefficient relative to the n th subcarrier when n = n, and an additional attenuation factor, denoted by m = m n,n , when n = n. Since S[l]CΘc * , by dropping the symbol index l for the sake of simplicity, (19) can be rewritten as z SC,n,k = λ |α|ρms n,k + z ICI,n,k + z MAI,n,k + z MAICI,n,k +z IMD,n,k + z AWGN,n,k , where z ICI,n,k = |α|ρ N n =1 n =n m n,n s n ,k represents the ICI coming from the symbols of the kth user, represents the MAI coming from the symbols of the other users on the nth subcarrier, represents the multiple-access ICI (MAICI) coming from the symbols of the other users on the other subcarriers, represents the IMD, and represents the AWGN. In order to find the BER of the kth user on the nth subcarrier, we will firstly evaluate the conditional bit-error probability P BE,n,k (λ) for a given channel gain λ, and we will successively average the obtained P BE,n,k (λ) over the probability density function (pdf) f λ (λ) of the channel gain.
Therefore, we will determine the BER by making use of the expressions where P BE,n,k (S, λ) is the bit-error probability conditioned on both the channel gain λ and the data symbols in S[l], and f S (S) is the multidimensional pdf of the data symbols in S[l].

Conditional bit-error probability evaluation
From (21), we observe that we have five noise terms, each one with different statistical characterization. Hence, the computation of the conditional probability P BE (λ) seems to be quite complicated. As a consequence, a reasonable way to proceed relies on finding simple (yet accurate) approximations that allow a mathematical treatment. Along this way, we observe that the ICI term z ICI,n,k in (22) is obtained by the linear combination of N − 1 symbols, each one weighted by a different coefficient m n,n . If the frequency offset ε is not too high (e.g., ε < 0.1), the coefficients {m n,n } n =n have comparable amplitudes. Therefore, under the hypothesis of high number N of subcarriers, the ICI term in (22) can be well approximated by a zero-mean Gaussian random variable with power We want also to underline that both the hypotheses of high N and low CFO seem to be realistic. Indeed, multicarrier systems usually employ many active subcarriers, from N = 52 in WLAN environments to N ≈ 6000 in broadcasting applications [18]. Moreover, in the presence of high CFO, the BER will be very high (e.g., BER > 0.1), and therefore in this case the BER analysis seems to be of limited practical interest.
It should be pointed out that the ICI power in (29) does not depend on the user index k as long as σ 2 S remains the same for all the users, as assumed in this paper. However, the results could be extended to take into account different user powers. Moreover, our BER analysis can be easily extended in order to take into account the presence of N VS switched-off subcarriers that are used as guard frequency bands, because a switched-off subcarrier does not contribute to the ICI and to the MAICI. Therefore, the BER analysis remains still valid, provided that the various summations over the subcarrier index n (e.g., (22), (24), etc.) consider only the active subcarriers.
As far as the MAI term is concerned, we similarly observe that z MAI,n,k in (23) is obtained as the sum of the K − 1 elements {s k ,n }, weighted by almost equal elements {ρ k,k }. Therefore, when the number K of users is sufficiently high, we can exploit the Gaussian approximation as we did for the ICI term. The MAI power can be expressed as Since high bit rates are expected in the downlink of fourth generation (4G) systems, also the hypothesis of high K seems to be realistic, because many spreading codes can be assigned to the same user in order to provide higher bit rates (multicode transmission). Therefore, K can be interpreted as the number of spreading codes assigned toK active users, with K < K. Moreover we remark that, under Hypothesis H1, the MAI term can be set to zero by employing orthogonal codes, and hence no accurate MAI characterization is necessary in this case. Because of the previous assumptions about the number K of equivalent users and the number N of subcarriers, the MAICI term can be approximated by a zero-mean Gaussian variable as well, with power equal to We remark that, under Hypothesis H1, also the MAICI term vanishes when the codes are orthogonal. We also observe that the IMD term z IMD,n,k in (25) can equivalently be expressed as z IMD,n,k = G−1 g=0 e − j arg(α) θ g+1 c * k,g m T n v IMD,g , where θ g+1 = [Θ] g+1,g+1 , and v IMD,g is the gth column of V IMD . Since V IMD is evaluated at the output of the receiver FFT, the assumption of high N allows to approximate the pdf of v IMD,g as a multidimensional Gaussian pdf. Moreover, V IMD is transformed in z IMD,n,k by linear operations, and hence z IMD,n,k also can be modeled as a zero-mean Gaussian random variable, with power expressed by which depends on the indexes g and g , can be calculated by exploiting the knowledge of the IMD autocorrelation function, summarized by Rv(g, g ) in (3).
Since the interference terms in (21) are modeled as zeromean Gaussian random variables, also the aggregate noise can be considered as Gaussian. Indeed, the data of different users or different subcarriers are uncorrelated with one another, as well as the ICI, the MAI, and the MAICI terms are uncorrelated with one another and with the useful signal. Moreover, thanks to the Bussgang theorem, the IMD term also is uncorrelated with the other terms. Consequently, the power of the aggregate noise is equal to the sum of the powers of the individual noise terms. Hence, the conditional biterror probability P BE,n,k (λ), for square M-QAM with Gray coding, can be expressed by [19] P BE,n,k (λ)= 1 log 2 with the signal-to-interference-plus-noise ratio (SINR) γ n,k (λ) expressed by γ n,k (λ)= |λ| 2 |αρm| 2 σ 2 S |λ| 2 σ 2 ICI,n,k +σ 2 MAI,n,k +σ 2 MAICI,n,k +σ 2 IMD,n,k +σ 2 AWGN . (35)

Average over the channel statistics
For convenience, we rewrite (34) as We want to highlight that such an approach can be used regardless of the statistical characterization of the channel. However, when the channel is characterized by Rayleigh fading (i.e., f |λ| (λ) = (2|λ|/σ 2 λ )e −|λ| 2 /σ 2 λ , λ > 0), as assumed in this paper, we can circumvent the numerical integration by evaluating a suitable series expansion. As an example, for 4-QAM, (36) becomes P BE,n,k (λ) = Q( γ n,k (λ)), and consequently (27) becomes [20,21] where p F q (·; ·; ·) denotes the generalized hypergeometric function [22], and µ 2 n,k and ν 2 n,k are expressed by As a consequence of (36) and (37), the general M-QAM case leads to a BER that is expressed as the sum of √ M − 1 series expansions. Note that each series expansion in the form of (37) can be conveniently truncated without affecting significantly the BER value (see [21] for a simple and accurate truncation criterion).

BER OF MC-DS-CDMA SYSTEMS IN FREQUENCY-SELECTIVE CHANNELS
For frequency-selective fading channels, we will follow a two-step approach similar to that adopted for frequency-flat channels. By denoting by λ n = [Λ] n,n the channel gain of the nth subcarrier, the frequency-selective counterparts of (27) and (28) can be expressed by (BER) n,k = λn P BE,n,k λ n f λn λ n dλ n , (39) P BE,n,k λ n = S,λn P BE,n,k (S, λ) fλ n |λn λ n |λ n f S (S)dS dλ n , where f λn (λ n ) is the pdf of λ n , fλ n|λn (λ n |λ n ) is the conditional pdf of the channel vector m n,n λ n v IMD,n ,g where v IMD,n,g is the (n, g)th element of V IMD , and z AWGN,n,k = ϕ * e − j arg(α) z T n,AWGNÊ −1 c * k . Equation (43) clearly illustrates the presence of the interference terms (ICI, MAI, MAICI, and IMD). By constructing the conditional random variable t SC,n,k = z SC,n,k |λ n , from (43), we obtain t SC,n,k = λ n |α|ρms n,k + |α|ρ whereλ n ,n = λ n |λ n is the random variable that indicates the channel gain value on the n th subcarrier conditioned on the channel gain on the nth subcarrier. Note that in (44) the IMD relative to the subcarrier of interest and the IMD coming from the other subcarriers are considered separately. As shown in [23, page 532], the (N − 1) × 1 vector λ n = [λ 1,n · · ·λ n−1,nλn+1,n · · ·λ N,n ] T is still a Gaussian random vector, with mean value ηλ n = E{λ n } and covariance Rλ n = E{λ nλ where r λ n λn = E{λ n λ * n } is the statistical correlation between the channels on the n th and the nth subcarrier, rλ nλn = E{λ n λ * n } is the cross-correlation vector between the channel of interest and the other channels, and Rλ n = E{λ nλ H n } is the cross-correlation matrix of the other channels. Consequently, by defining the (N − 1) × 1 zero-mean Gaussian random vector π n =λ n − ηλ n = [π1,n · · · π n−1,n π n+1,n · · · π N,n] T , (44) becomes t SC,n,k = λ n |α|ρms n,k +z ICI,n,k +z MAI,n,k +z MAICI,n,k +z IMD,n,k +z ICI,n,k +z MAICI,n,k +z IMD,n,k + z AWGN,n,k , where z ICI,n,k = |α|ρr −1 λnλn N n =1 n =n m n,n r λ n λn s n ,k , m n,n r λ n λn v IMD,n ,g , z ICI,n,k = |α|ρ N n =1 n =n m n,n π n ,n s n ,k , z MAICI,n,k = |α| N n =1 n =n m n,n π n ,n z IMD,n,k = e − j arg(α) n =n m n,n π n ,n v IMD,n ,g . (53) We want to highlight by (46) that in the frequency-selective case the ICI consists of two different components. The first one, λ n z ICI,n,k , is proportional to the channel amplitude λ n of the useful signal, and therefore it represents the ICI that fades coherently with the useful signal. On the other hand, the second partz ICI,n,k represents the ICI that fades independently of λ n . Indeed,z ICI,n,k in (51) is a zero-mean random variable with power expressed bỹ Also the MAICI and the IMD, similarly to the ICI, consist of two parts. In both cases, the first part is proportional to λ n , while the second one has power independent of λ n . The powers of the second components are expressed, respectively, byσ 2 MAICI,n,k = |α| 2 N n =1 n =n m n,n 2 r λ n λn − r −1 λnλn r λ n λn n =n m n,n m * n,n J T n Rλ n J n n ,n where R v (g, g ) is the same as in (33), and the (N −1)×N matrix J n is obtained from the identity matrix I N by removing the nth row. The conditional SINR can be expressed as γ n,k λ n = λ n 2 |αρm| 2 σ 2 S λ n 2 σ 2 ICI,n,k + σ 2 MAI,n,k + σ 2 MAICI,n,k + σ 2 IMD,n,k +σ 2 ICI,n,k +σ 2 MAICI,n,k +σ 2 IMD,n,k + σ 2 m n n = m n,n r λ n λn , and σ 2 MAI,n,k ,σ 2 ICI,n,k ,σ 2 MAICI,n,k , andσ 2 IMD,n,k are expressed by (30), (54), (55), and (56), respectively. Moreover, as in the previous section, we exploit the hypotheses of a high number of subcarriers and a high number of users in order to approximate as Gaussian the interference terms in (46), by the central limit theorem.

SIMULATION RESULTS
In this section, we present some simulation results in order to validate the approximations introduced in the theoretical analysis. We consider an MC-DS-CDMA system with N = 256 subcarriers, with a subcarrier separation of ∆ f = 1/T = 156.25 kHz and a cyclic prefix of length L = 64. For brevity, we only consider data modulated by 4-QAM, which correspond to a gross bit rate of 4 Mbps per user when the length of the spreading codes is equal to G = 16. We assume that each time-domain tap of the channel suffers independent Rayleigh fading, with an exponentially decaying power delay profile and an rms delay spread equal to 250 nanaoseconds. In the first scenario, we assume that the base station employs Walsh-Hadamard (WH) spreading codes of length G = 16. We also assume Hypothesis H1 to be true. We still remark that the BER performance in this scenario can be considered a lower bound for the case of inaccurate estimate of the CFO-induced time-varying phase shift. Figures 1 and 2 show the BER performance versus the received E b /N 0 when the number of active users is equal to K = G = 16. The received E b /N 0 is defined before the despreading as the ratio between the noiseless signal power and the noise power, while the BER is obtained by averaging the BER over all the subcarriers and all the users. Figure 1 shows the BER versus E b /N 0 for different values of the normalized CFO ε, assuming linear amplification at the transmitter (i.e., without IMD). The good agreement between theoretical analysis and simulated BER for all practical values of ε is evident. This fact clearly testifies that the Gaussian approximation of the ICI leads to accurate results in frequency-selective scenarios. Figure 2 illustrates the impact of the NLA on the BER performance in the absence of CFO. We define the output backoff (OBO) as OBO = P U,MAX /σ 2 U , where P U,MAX and σ 2 U are the maximum power and the mean power, respectively, of the NLA output signal. We assume that the amplifier is perfectly predistorted, that is, it behaves as a clipper of the NLA input envelope. From Figure 2, we deduce that the Gaussian approximation of the IMD is slightly worse than the approximation of the ICI in Figure 1, especially at the saturating BER. Indeed, for a clipping amplifier, the Gaussian approximation of the IMD is very accurate only when the SNR is not too high or the number of subcarriers is very high [24]. However, in this case the BER mismatch seems to be quite small.  In Figures 3a and 3b, we still assume WH spreading codes of length G = 16, Hypothesis H1 as true, and ε = 0, and we focus on the impact of the number K of active users on the BER performance. We also assume that each user employs a fixed spreading code, that is, that the kth user employs the kth row of the G × G Hadamard matrix. Figure 3a shows that at low SNR the BER increases with the number of active users, as expected. However, the BER floor at high SNR does not increase with the number of users.  Such a behavior is due to the fact that the IMD powers in (56) and (60) depend on R v (g, g ), which is highly sensitive to the choice of the spreading codes among the ones provided by the Hadamard matrix. This fact is confirmed by the BER floors of Figure 3b, because R v (g, g ) is the same for K = 4, K = 8, and K = 16. Figure 4 exhibits the joint effect of CFO and NLA on the BER performance when ε = 0.02. The results show that, in the presence of a CFO, it is not convenient to increase the output NLA power backoff at the transmitter beyond a certain value (that obviously depends on ε), because the IMD   reduction is masked by the presence of the ICI induced by the CFO. In order to identify the optimum OBO value, we introduce the total degradation (TD) as [7] TD dB = ∆ Eb/N0 dB + OBO dB , where ∆ Eb/N0 | dB = E b /N 0 (OBO, ε)| dB −E b /N 0 (+∞, ε)| dB is the SNR increase caused by the NLA to achieve a target BER for a fixed CFO ε. If we neglect out-of-band effects, the minimization of the TD can be considered a fair criterion for  the selection of the optimum OBO value, because ∆ Eb/N0 and OBO represent the power penalty on the link budget with respect to the linear scenario, and with respect to the maximum amplifier output power, respectively. Figure 5 shows the TD performance at BER = 10 −3 for different values of the CFO and number of users. It is evident that the optimum OBO mainly depends on the value of the CFO.
In the second simulation scenario, we assume that a rough estimate of the CFO-induced time-varying phase shift is available at the receiver (i.e., Hypothesis H2 is true). In this case, since the MAI cannot be completely eliminated, we assume that the base station employs Gold codes of length G = 31. Figures 6 and 7 display the BER performance for different values of the CFO ε and of the number of users K. It is evident that the BER analysis is quite accurate for E b /N 0 < 30 dB, while a certain mismatch between theoretical and simulated BER exists at high SNR. This fact is due to the nonperfect Gaussian approximation of the MAI terms, as confirmed by the estimation of the kurtosis of the decision variable, which is a standard test for non-Gaussian noise [25]. However, the BER mismatch is smaller when also the NLA-induced distortions are present, as testified by Figures 8  and 9 (OBO = 2.00 dB). Figures 6 and 8 clearly show that, if only a rough phase estimate is available, the maximum tolerable CFO is rather small (e.g., ε < 0.005), because of the additional MAI. Therefore, the phase compensation step plays a crucial role in the BER performance of MC-DS-CDMA systems.

CONCLUSIONS
In this paper, we have evaluated the BER of MC-DS-CDMA downlink systems subject to both CFO and IMD in frequency-selective Rayleigh fading channels. The BER analysis, which is quite accurate in many conditions, allows to identify the optimum OBO value of the NLA, depending on the amount of CFO. The analytical findings have also highlighted the importance of an accurate compensation for the CFO-induced phase shift. Although the analysis has been carried out for M-ary QAM, it can also be extended to M-ary phase-shift keying constellations. Future works may focus on the effects of channel estimation errors and channel coding.