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 of Multicarrier DS-CDMA Systems
Luca Rugini Dipartimento di Ingegneria Elettronica e dell’Informazione (DIEI), Universita` degli Studi di Perugia, Via G. Duranti 93, 06125 Perugia, Italy Email: [email protected] Paolo Banelli Dipartimento di Ingegneria Elettronica e dell’Informazione (DIEI), Universita` degli Studi di Perugia, Via G. Duranti 93, 06125 Perugia, Italy Email: [email protected]
Received 5 August 2003; Revised 19 January 2004
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 offset (CFO) and NLA distortions, to identify the maximum CFO that is tolerable at the receiver side in different scenarios, and to find out the optimum value of the NLA output power backoff for a given CFO. Simulation results show that the approximated analysis is quite accurate in several conditions. Keywords and phrases: multicarrier DS-CDMA, carrier frequency offset, intermodulation distortion, Rayleigh fading.
1. INTRODUCTION The presence of a nonlinear amplifier (NLA) at the transmitter, which introduces both intermodulation distor- In the last years, several multicarrier code-division multiple- tion (IMD) and out-of-band interference, is another rele- access (MC-CDMA) schemes [1] have been proposed vant source of impairment in multicarrier systems. Indeed, by combining orthogonal frequency-division multiplexing multicarrier signals are characterized by a high peak-to- (OFDM) and direct-sequence code-division multiple-access average power ratio (PAPR) [4], and hence they are signif- (DS-CDMA), with the goal of incorporating the advan- icantly distorted when, in order to improve the power effi- tages of both techniques. Specifically, the low-complexity ciency, the NLA working point is chosen close to the satura- equalization of cyclic prefixed OFDM systems and the tion point. multiple-access interference (MAI) mitigation capabilities Obviously, the typical impairments of multicarrier sys- offered by DS-CDMA systems make MC-CDMA techniques tems add to the sources of degradation that are generally attractive for future mobile broadband communications present also in other multiple-access schemes, such as the [2]. MAI associated with multiple users in nonorthogonal sys- Differently from single-carrier systems, one of the main tems [5], and the intersymbol interference (ISI) introduced problems of multicarrier schemes is the high sensitivity by the signal propagation over multipath channels. As a con- to frequency synchronization errors [3]. Indeed, the car- sequence, the degradation introduced by CFO and NLA, as rier frequency offset (CFO), which models the frequency well as MAI and multipath, should be taken into account mismatch between the transmitter and receiver oscillators, when evaluating the performance of multicarrier systems. generates intercarrier interference (ICI), thereby destroy- Specifically, this paper deals with the effects of the men- ing the frequency-domain orthogonality of the transmitted tioned impairments on the bit-error rate (BER) perfor- data. mance. The particular multicarrier scheme herein considered MC-DS-CDMA with Frequency Offset and Nonlinear Amplifiers 731 is commonly identified as multicarrier DS-CDMA (MC- We consider the downlink of an MC-DS-CDMA system DS-CDMA) [1], where multiple users are discriminated by with N subcarriers, spaced by ∆ f = 1/T,andwithK ac- spreading in the time domain. Moreover, in this paper, we tive users. Hence, assuming rectangular pulse shaping, the focus on the performance at the mobile end (i.e., down- chip duration is equal to T. The base station transmits N link), assuming that the receiver is equipped with a bank of symbols of each user in parallel, one for each subcarrier, single-tap equalizers followed by a despreader. by spreading each symbol in the time domain employing Previous works about MC-DS-CDMA performance con- a user-dependent spreading code. The spreading code of sidered the presence of either frequency synchronization the kth user, which is the same for all the subcarriers, is = ··· T errors or nonlinear distortions. Indeed, Steendam and denoted by ck [ck,0 ck,G−1] ,whereG is the pro- | |= −1/2 Moeneclaey evaluated the signal-to-noise ratio (SNR) degra- cessing gain. Assuming ck,g G , the vector ck has dation produced by the CFO in additive white Gaussian unit norm. As in conventional OFDM systems, the time- noise (AWGN) for MC-DS-CDMA downlink systems [6]; domain samples are obtained by N-dimensional IFFT of the however, they did not consider the NLA at the transmitter. data on the N subcarriers, and a cyclic prefix of duration = In addition, Jong and Stark analyzed the effects of multi- τCP LT/N is attached to the IFFT output block in or- path and nonlinearities on the BER performance of a mul- der to avoid channel-induced interference between successive ticarrier spread-spectrum system, which basically is an MC- blocks. Consequently, the symbol rate of each user is equal to DS-CDMA scheme with one active user [7]. However, they N/(1 + L/N)GT. × = assumed perfect frequency synchronization at the receiver By defining the N N unitary FFT matrix F as [F]m,n − side. N 1/2 exp(− j2π(m−1)(n−1)/N), and the (N +L)×N cyclic = T T T The purpose of our work is to extend the results of [6, 7] prefix insertion matrix as TCP [ICP IN ] ,whereICP con- in order to jointly take into account CFO, nonlinear distor- tains the last L rows of the identity matrix IN , the transmitted tions, and MAI, in multipath fading channels. Our two-step block, at the input of the NLA, can be expressed by [8] approach to obtain the BER is quite standard. The first step H relies on the Gaussian approximation of the interferences, uIN[lG + g] = TCPF S[l]c[g], (1) 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 where uIN[lG + g] is a vector of dimension P N + L, × to be Rayleigh. S[l] is the N K matrix containing the data symbols, with sn,k[l] = [S[l]]n,k the lth symbol of the kth user on the The paper is structured as follows. In Section 2,we T nth subcarrier, and c[g] = [c1, ··· c , ] is the vector introduce an MC-DS-CDMA system model where lin- g K g that contains the gth chip of the spreading codes of all the ear signal processing operations (either in the time do- K users. The data symbols {sn,k[l]},drawnfromanM-ary main or in the frequency domain) are simply represented square QAM constellation, are assumed to be independent by matrix multiplications. Section 3 is dedicated to the 2 and identically distributed (i.i.d.) with power σS . Although BER evaluation in frequency-flat fading channels, assum- in many cases we are dealing with continuous-time signals ing quadrature amplitude modulation (QAM). In Section 4, (i.e., at the NLA input and output, at the receiver input, etc.), the BER evaluation for flat-fading channels is extended to throughout the paper we consider the equivalent discrete- frequency-selective scenarios characterized by Rayleigh fad- time signals, to simplify the notation. ing. In Section 5, we check the accuracy of the analyti- In this paper, we model the NLA as a memoryless device cal findings by comparing theoretical and simulated re- characterized by its AM/AM and AM/PM curves. After incor- sults. Finally, in Section 6, some concluding remarks are porating both the chip index g and the symbol index l into drawn. a single index i = lG + g, by the Bussgang theorem [9], the transmitted block, at the output of the NLA, can be expressed 2. SYSTEM MODEL as [10, 11] Firstly, we introduce some basic notations. We use lower u [i] = αu [i]+v¯ [i], (2) (upper) bold face letters to denote column vectors (matri- OUT IN IMD ces), superscripts ∗, T,andH to represent complex conju- gate, transpose, and Hermitian operators, respectively. We where α represents the average linear amplification gain, reserve E{·} to represent the statistical expectation, and x and v¯IMD[i] is the intermodulation distortion (IMD), un- to denote the greatest integer smaller than or equal to x. correlated with the linear part αuIN[i]. The validity of (2), = −1/2 +∞ −ν2/2 ν The Q-function is defined as Q(x) (2π) x e d , based on a Gaussian distribution of uIN[i]in(1), is justi- 0M×N is the M × N all-zero matrix, and IN is the N × N fied by the high number of subcarriers usually employed in identity matrix. We define [A]m,n as the (m, n)th entry of multicarrier systems (e.g., N>32) [11]. In this case, the = the matrix A,[a]n as the nth entry of the column vec- average linear amplification gain can be evaluated as α tor a,(a)mod N as the remainder after division of a by N, RuOUTuIN (0)/RuINuIN (0), where RuOUTuIN (τ) is the continuous- and diag(a) as the diagonal matrix with (n, n)th entry equal time cross-correlation between the NLA output and in- to [a]n. put, and RuINuIN (τ) is the continuous-time autocorrelation 732 EURASIP Journal on Applied Signal Processing function of the NLA input. Moreover, the elements of Toeplitz matrices defined by [8] h[0] 0 ··· ··· 0 = H Rv¯ (g, g ) E v¯IMD[lG + g]v¯IMD[lG + g ] (3) . . . . . . . . . . . . = . . . can be predicted in advance by exploiting the knowledge of H0 [ ] .. .. . , h L the continuous-time autocorrelation function R (τ) . uOUTuOUT . .. .. of the NLA output. Such an autocorrelation function can be . . . 0 obtained by means of known closed-form formulas or nu- 0 ··· h[L] ··· h[0] merical integration techniques, which require the knowledge (7) 0 ··· h[L] ··· h[1] of the NLA input statistics, summarized by R (τ), of the uINuIN . . . . ff . . . . input power backo to the NLA, and of the AM/AM and . . . . AM/PM curves that characterize the NLA [10]. = . H1 0 .. [ ] . After the parallel-to-serial conversion, the signal stream h L = = − . . . . uOUT[iP + p] [uOUT[i]]p+1, p 0, ..., P 1, is transmitted ...... through a multipath channel, whose discrete-time equivalent 0 ··· 0 ··· 0 impulse response is The cyclic prefix elimination, which is common to many Q multicarrier-based systems, can be represented by the left h[b] = ζqR bTS − τq ,(4) = 0 × I q=1 multiplication of the matrix RCP [ N L N ] with the re- ceived vector in (6), which leads to the N × 1vector[12] where Q is the number of paths, ζq and τq are the complex j2πε(iP+L)/N amplitude and the propagation delay, respectively, of the qth y[i] = RCPx[i] = e DRCPH0uOUT[i]+yAWGN [i], path, R(τ) is the triangular autocorrelation function of the (8) rectangular pulse-shaping waveform, and TS = T/N is the where D is an N × N diagonal matrix defined by [D]n,n = sampling period. Throughout the paper, we assume that the − = { } exp(j2πε(n 1)/N), and yAWGN [i] RCPxAWGN [i] stands channel amplitudes ζq are zero-mean complex Gaussian for the AWGN term. By using (1)and(2), the vector y[i] = random variables, giving rise to Rayleigh fading, and that y[lG + g]in(8) can also be expressed as the maximum delay spread τmax = max{τq} is smaller than the cyclic prefix duration τ = LT , that is, h[b]mayhave CP S = j2πε((lG+g)P+L)/N H nonzero entries only for 0 ≤ b ≤ L. y[lG + g] e DH αF S[l]c[g]+v˜IMD[lG+g] At the receiver side, the samples obtained after matched + yAWGN [lG + g], filtering can be expressed as [12] (9)
L where H = RCPH0TCP is the equivalent channel matrix ex- j2πf0aT x[a] = e S h[b]uOUT[a − b]+xAWGN [a], (5) pressed by [H]m,n = h[(m − n)mod N ], and v˜IMD[lG + g] = b=0 RCPv¯IMD[lG + g]. Since H is circulant, it can be expressed = H Λ Λ = λ × where f0 is the CFO caused by the frequency synchronization as H F F,where diag( ) is the N√ N diago- λ = error at the receiver side, xAWGN [a] represents the AWGN, nal matrix with elements expressed by NFh,where = ··· − T λ = ··· T with a = iP + p. Throughout the paper, we assume that the h [h[0] h[N 1]] and [λ1 λn] are the timing information is available at the receiver. In multicarrier channel vectors in the time and frequency domain, respec- systems, such an information could be acquired by exploit- tively. Consequently, the recovery of the transmitted data is ing timing synchronization algorithms designed to work in accomplished by applying the FFT at the receiver, thus ob- the presence of an unknown CFO [13, 14, 15]. In any case, taining z[lG + g] = Fy[lG + g], which by (9), can be rear- when timing errors are significant, the BER expressions we ranged as will derive can be considered as lower bounds. = j2πε((lG+g)P+L)/N ΦΛ The P received samples relative to the gth chip (of the z[lG + g] e αS[l]c[g]+vIMD[lG + g] = lth symbol) are grouped in the vector x[i] x[lG + g], thus + zAWGN [lG + g], obtaining [12] (10)
Φ = H × x[i] = e j2πεiP/ND˜ H u [i]+H u [i − 1] + x [i], where FDF is the N N circulant matrix that mod- 0 OUT 1 OUT AWGN = (6) els the ICI due to the CFO, vIMD[lG + g] Fv˜IMD[lG + g] is the IMD after the FFT operation, and zAWGN [lG + g] = = = where [x[i]]p+1 x[iP + p], ε f0T is the normalized FyAWGN [lG + g] represents the AWGN after the FFT opera- CFO, D˜ is a P × P diagonal matrix, defined by [D˜ ]p,p = tion. exp(j2πε(p − 1)/N), that models the block-independent Due to the spreading in the time domain, in order to de- × ··· T phase shift introduced by the CFO, and H0 and H1 are P P code the N data symbols sk[l]=[s1,k[l] sN,k[l]] relative MC-DS-CDMA with Frequency Offset and Nonlinear Amplifiers 733
to the lth interval, the receiver of the kth user has to collect Eˆ[l] = Eˆ[l] H2 is diagonal, with elements { } G consecutive chip vectors z[lG + g] g=0,...,G−1 in (10). By defining the N ×G matrices G−1 ˆ = 1 2πε(lG + g)P E[l] H2 g+1,g+1 exp j G = N = ··· − g 0 Z[l] z[lG] z[lG + G 1] , (13) 2πεP G − 1 ··· − = VIMD[l] = vIMD[lG] vIMD[lG + G 1] , (11) exp j lG + . N 2 Z [ ] = z [lG] ··· z [lG + G − 1] , AWGN l AWGN AWGN It can be noted that the perfect knowledge of the phase shifts is not completely realistic. However, we consider this hypoth- from (10), we obtain esis because it produces a lower bound that is independent of Z[l] = e j2πεL/NΦΛ αS[l]C + V [l] E[l]+Z [l], (12) the phase-shift estimation technique. IMD AWGN In this paper, we assume that the receiver performs T where C = [c[0] ··· c[G − 1]] = [c1 c2 ··· cK ] is the a single-tap frequency-domain equalization and code de- K × G matrix whose rows contain the spreading codes of spreading. In the absence of NLA- and CFO-induced im- the K users, and E[l]isaG × G diagonal matrix defined as pairments, the equalizer-despreader detector is able to elim- j2πε(lG+g)P/N inate both ISI and MAI when orthogonal spreading codes [E[l]]g+1,g+1 = e , g = 0, ..., G − 1. The matrix model expressed by (12) allows a nice repre- are employed. From the computational complexity point of sentation of the MC-DS-CDMA received signal. Indeed, the view, the equalizer-despreader detector is less demanding number of rows of Z[l]in(12) is equal to the number of than multiuser detectors [5], and hence it seems to be more subcarriers, while the number of columns of Z[l]isequalto suitable in the downlink situation. the processing gain. Consequently, we are able to represent The receiver can perform the despreading operation after linear frequency-domain operations (such as the equaliza- the phase-shift compensation and the channel equalization. tion) by matrix multiplications on the left-hand side of Z[l], Thus, by assuming perfect channel state information (CSI) and linear time-domain operations (such as the despreading) at the receiver side, the equalized data matrix can be con- = ∗ −1Λ−1 ˆ −1 by matrix multiplications on the right-hand side of Z[l]. structed as ZEQ[l] ϕ α Z[l]E[l] .From(12), ZEQ[l] Expression (12) shows the effects of both nonlinear dis- becomes equal to tortions and frequency synchronization errors on the spread = Λ−1 Λ Θ data matrix S[l]C. Specifically, the NLA introduces the con- ZEQ[l] M S[l]C + ZEQ,IMD[l]+ZEQ,AWGN[l], (14) stant complex gain α and the additive term VIMD[l], which represents the IMD. Moreover, the NLA-distorted signal, where M = e− jπε(N−1)/N Φ represents the phase-compensated −1 −1 after passing through the multipath channel summarized ICI matrix, ZEQ,IMD[l] = α Λ MΛVIMD[l]Θ represent by the diagonal matrix Λ,isaffected by the CFO-induced the IMD after the channel equalization, ZEQ,AWGN[l] = ∗ −1 −1 −1 −1 ICI, because the matrix Φ is not diagonal. In addition, the ϕ α Λ ZAWGN [l]Eˆ[l] ,andΘ = E[l]Eˆ[l] is a diago- CFO produces two phase-shift effects. The first one is a nal matrix equal to ΘH1 = IG under Hypothesis H1, and j2πεL/N jπε(N−1)/N time-invariant phase shift ϕ = e e (the factor defined as [ΘH2]g+1,g+1 = exp(j(2πεP/N)(g − (G − 1)/2)) e jπε(N−1)/N is contained in the main diagonal of Φ), which under Hypothesis H2. The despreading operation leads to an adds to the phase shift caused by the NLA. The second ef- N-dimensional vector that contains the N decision variables fect is a time-varying phase shift, which is represented by the of the kth user, as expressed by diagonal matrix E[l]. In this paper, we assume that the re- = ∗ ceiver is able to perfectly compensate for the time-invariant zDESPR,k[l] ZEQ[l]ck phase-shift term ϕ, which could also be seen as part of the = Λ−1MΛS[l]CΘc∗ + z [l] (15) channel-induced distortion. As far as the compensation for k DESPR,k,IMD the time-variant phase shift is concerned, we consider two + zDESPR,k,AWGN[l], alternative hypotheses. = ∗ = where zDESPR,k,IMD[l] ZEQ,IMD[l]ck ,andzDESPR,k,AWGN[l] ∗ Hypothesis H1 ZEQ,AWGN[l]ck . The decision over zDESPR,k[l] is successively done according to the proper constellation size M. The receiver obtains an accurate phase-shift estimate in each It should be pointed out that the K-dimensional vec- chip interval, as assumed in [6]. As a consequence, by the ∗ tor CΘc represents the effect of the CFO-induced time- knowledge of E[ ] at the receiver side, the estimated phase- k l varying phase shift on the spreading-despreading operation. shift matrix can be expressed by Eˆ[ ] = Eˆ[ ] = E[ ]. l l H1 l If the time-varying phase shift can be perfectly compen- sated for, that is, under Hypothesis H1, this vector becomes Hypothesis H2 ∗ equal to Cck . Therefore, if the users employ orthogonal The receiver does not track the time-variant phase shift at spreading codes, the multiuser interference has been per- Θ ∗ the chip level, estimating only the average phase shift within fectly eliminated, because the product S[l]C H1ck in (15) a symbol interval. Hence, by the knowledge of the mean is equal to sk[l]. On the contrary, under Hypothesis H2, of the elements in E[l], the estimated phase-shift matrix the multiuser interference cannot be completely eliminated. 734 EURASIP Journal on Applied Signal Processing
T Indeed, when k = k,wehave where zn,AWGN[l] is the nth row of ZAWGN [l], and mn = ··· T [mn,1 mn,N ] collects the elements on the nth row of ∗ = Θ ρk,k ,H2 C H2ck k M. It can be verified that [17] = T Θ ∗ ck H2ck m = [M] (16) n,n n,n − G1 j(2πεP/N)(g−(G−1)/2) ∗ sin π (n − n)mod + ε = e c c , = N k ,g k,g (20) g=0 N sin (π/N) (n − n)mod N + ε − − ff × e jπ((N 1)/N)(n n)mod N , which is, in general, di erent from zero even for orthogo- ∗ G−1 ∗ T = = nal sequences (characterized by ck ck g=0 ck ,g ck,g 0). where mn,n represents the ICI coefficient relative to the n th An interesting matter could be the search for spreading se- subcarrier when n = n, and an additional attenuation fac- quences that minimize (16), following the lines suggested by = = Θ ∗ = tor, denoted by m mn,n, when n n. Since S[l]C ck [16]. However, this topic is beyond the scope of this paper. K = T Θ ∗ ρsk[l]+ k=1 ρk,k sk [l], by dropping the symbol index l for Furthermore, since ρk,k,H2 ck H2ck < 1, the time- k=k variant phase shift also introduces an attenuation factor of the sake of simplicity, (19)canberewrittenas the useful signal, given by zSC,n,k = λ |α|ρmsn,k + zICI,n,k + zMAI,n,k + zMAICI,n,k +zIMD,n,k G−1 + zAWGN,n,k, = 1 j(2πεP/N)(g−(G−1)/2) = sin(πεPG/N) ρk,k,H2 e . (17) (21) G g=0 G sin(πεP/N) where For a given maximum tolerable frequency offset ε, the attenuation factor in (17) limits the maximum value G N =| | of the processing gain. Indeed, ρk,k,H2 is close to 1 only if zICI,n,k α ρ mn,n sn ,k (22) =1 ( + ) ≈ 1 . On the contrary, under Hypothesis n ε N/ N L G /G n =n = T Θ ∗ = H1, ρk,k,H1 ck H1ck 1, and hence no attenuation is introduced by the spreading-despreading operation. It represents the ICI coming from the symbols of the kth user, should be observed that ρk,k is independent of the user index k in both hypotheses, and hence, in the following, it will be denoted by ρ. K zMAI,n,k =|α|m ρk,k sn,k (23) k=1 k=k 3. BER OF MC-DS-CDMA SYSTEMS IN FLAT-FADING CHANNELS represents the MAI coming from the symbols of the other In this section, we evaluate the effect of the previously de- users on the nth subcarrier, scribed impairments when the channel exhibits flat-fading behavior. The results herein obtained will be useful for the K N =| | performance evaluation in frequency-selective fading chan- zMAICI,n,k α ρk,k mn,n sn ,k (24) k=1 n=1 nels, developed in the next section. In the flat-fading case, k=k n=n all the subcarriers experience the same channel gain. Conse- Λ = quently, since λIN ,(15) can be simplified to represents the multiple-access ICI (MAICI) coming from the = Θ ∗ −1 Θ ∗ symbols of the other users on the other subcarriers, zDESPR,k[l] MS[l]C ck + α MVIMD[l] ck (18) ∗ −1 −1 ˆ −1 ∗ = − j arg(α) T Θ ∗ + ϕ α λ ZAWGN [l]E[l] ck . zIMD,n,k e mn VIMD ck (25) For convenience, we consider the scaled version of (18)ob- represents the IMD, and tained by multiplying zDESPR,k[l]withλ|α|. Therefore, we ob- tain zSC, [l] = λ|α|zDESPR, [l], and, by focusing on the nth k k = ∗ − j arg(α) T ˆ −1 ∗ subcarrier, it yields zAWGN,n,k ϕ e zn,AWGNE ck (26) = zSC,n,k[l] zSC,k[l] n represents the AWGN. ∗ In order to find the BER of the kth user on the nth = λ|α|mT S[l]CΘc n k subcarrier, we will firstly evaluate the conditional bit-error − ∗ (19) j arg(α) T Θ probability PBE,n,k(λ) for a given channel gain λ,andwewill + λe mn VIMD[l] ck successively average the obtained PBE,n,k(λ) over the prob- ∗ − j arg(α) T ˆ −1 ∗ + ϕ e zn,AWGN[l] E[l] ck , ability density function (pdf) fλ(λ) of the channel gain. MC-DS-CDMA with Frequency Offset and Nonlinear Amplifiers 735
Therefore, we will determine the BER by making use of the Therefore, when the number K of users is sufficiently high, expressions we can exploit the Gaussian approximation as we did for the ICI term. The MAI power can be expressed as = (BER)n,k PBE,n,k(λ) fλ(λ)dλ, (27) λ K 2 2 =| |2 2 σMAI,n,k αm ρk,k σS . (30) = PBE,n,k(λ) PBE,n,k(S, λ) fS(S)dS, (28) k =1 S k=k where P (S, λ) is the bit-error probability conditioned on BE,n,k Since high bit rates are expected in the downlink of fourth both the channel gain λ and the data symbols in S[l], and generation (4G) systems, also the hypothesis of high K seems f (S) is the multidimensional pdf of the data symbols in S to be realistic, because many spreading codes can be assigned S[l]. to the same user in order to provide higher bit rates (mul- 3.1. Conditional bit-error probability evaluation ticode transmission). Therefore, K can be interpreted as the number of spreading codes assigned to K˜ active users, with From (21), we observe that we have five noise terms, each one K
Since the interference terms in (21)aremodeledaszero- consequently (27)becomes[20, 21] mean Gaussian random variables, also the aggregate noise +∞ ff 2|λ| −| |2 2 can be considered as Gaussian. Indeed, the data of di er- = λ /σλ | | ff (BER)n,k Q γ(λ) 2 e d λ ent users or di erent subcarriers are uncorrelated with one 0 σλ another, as well as the ICI, the MAI, and the MAICI terms 2 +∞ 2 i 1 2µn,k − 2 ν2 1 µn,k are uncorrelated with one another and with the useful signal. = − µn,k /2 n,k (37) e ν2 Moreover, thanks to the Bussgang theorem, the IMD term 2 4 i=0 i! 2 n,k also is uncorrelated with the other terms. Consequently, the power of the aggregate noise is equal to the sum of the pow- × 3 1 −ν2 2F0 i + , ;; n,k , ers of the individual noise terms. Hence, the conditional bit- 2 2 error probability PBE,n,k(λ), for square M-QAM with Gray where pFq(·; ·; ·) denotes the generalized hypergeometric coding, can be expressed by [19] 2 ν2 function [22], and µn,k and n,k are expressed by √ log M | |2 2 2 2 αρm σS σ 1√ µ2 = µ2 = λ , PBE, , (λ)= P , (q, λ), n,k 2 n k n k σAWGN log2 M q=1 σ2 + σ2 + σ2 + σ2 σ2 ν2 = ICI,n,k MAI,n,k MAICI,n,k IMD,n,k λ √1 n,k 2 . Pn,k(q, λ)= σAWGN M (38) √ − −q − (1 2 ) M 1 √ q−1 q−1 2 i 1 × (−1) 2 i/ M 2q −2 √ − As a consequence of (36)and(37), the general M√-QAM case i=0 M 2 leads to a BER that is expressed as the sum of M − 1se- ries expansions. Note that each series expansion in the form 3 of (37) can be conveniently truncated without affecting sig- × Q (2i+1) γ (λ) , M−1 n,k nificantly the BER value (see [21] for a simple and accurate (34) truncation criterion). with the signal-to-interference-plus-noise ratio (SINR) 4. BER OF MC-DS-CDMA SYSTEMS IN γn,k(λ) expressed by FREQUENCY-SELECTIVE CHANNELS For frequency-selective fading channels, we will follow a | |2| |2 2 two-step approach similar to that adopted for frequency-flat λ αρm σS γn,k(λ)= . | |2 2 2 2 2 2 channels. By denoting by λn = [Λ]n,n the channel gain of the λ σICI,n,k +σMAI,n,k +σMAICI,n,k +σIMD,n,k +σAWGN (35) nth subcarrier, the frequency-selective counterparts of (27) and (28) can be expressed by 3.2. Average over the channel statistics = (BER)n,k PBE,n,k λn fλn λn dλn, (39) λn For convenience, we rewrite (34)as = λ λ¯ | λ¯ PBE,n,k λn PBE,n,k(S, ) fλ¯n|λn n λn fS(S)dS d n, √ S,λ¯n M−2 (40) PBE,n,k(λ) = aiQ biγn,k(λ) , (36) i=0 λ¯ | where fλn (λn) is the pdf of λn, fλ¯n|λn ( n λn) is the conditional pdf of the channel vector ffi { } { } where the coe cients ai and bi ,calculatedfrom(34), depend on the modulation size M. Therefore, the resultant ··· ··· T λ¯n = λ1 λn−1 λn+1 λN (41) BER, which is obtained by inserting in (27) the expressions (35), (36), and the specific pdf f (λ) of the channel, can be √ λ of the other subcarriers given the subcarrier of interest λn, expressed as the sum of M − 1 integrals. Each integral can and PBE,n,k(S, λ) is the bit-error probability conditioned on then be evaluated by numerical techniques. both the channel gains in λ and the data symbols in S[l]. We want to highlight that such an approach can be used By multiplying (15)withλn|α|, the scaled decision variable regardless of the statistical characterization of the channel. zSC,n,k[l] can be expressed by However, when the channel is characterized by Rayleigh fad- 2 −|λ|2/σ2 ing (i.e., f| |(λ) = (2|λ|/σ )e λ , λ>0), as assumed T ∗ − j arg(α) T ∗ λ λ zSC,n,k[l] =|α|m ΛS[l]CΘc + e m ΛVIMD[l]Θc in this paper, we can circumvent the numerical integration n k n k by evaluating a suitable series expansion. As an example, ∗ − j arg(α) T ˆ −1 ∗ + ϕ e zn,AWGN[l] E[l] ck . for 4-QAM, (36)becomesPBE,n,k(λ) = Q( γn,k(λ)), and (42) MC-DS-CDMA with Frequency Offset and Nonlinear Amplifiers 737
∗ K ∗ By exploiting S[l]CΘc = ρs [l]+ = ρ , s [l], and drop- where r = E{λ λ } is the statistical correlation be- k k k 1 k k k λn λn n n k =k tween the channels on the nth and the nth subcar- ping the index l,(42)becomes = {λ¯ ∗} rier, rλ¯nλn E nλn is the cross-correlation vector be- N tween the channel of interest and the other channels, and H =| | | | = {λ¯ λ¯ } zSC,n,k α ρmλnsn,k + α ρ mn,n λn sn ,k Rλ¯n E n n is the cross-correlation matrix of the = − × n 1 other channels. Consequently, by defining the (N 1) n =n 1 zero-mean Gaussian random vector πn = λˆn − ηλˆ = K n T [π1,n ··· πn−1,n πn+1,n ··· πN,n] ,(44)becomes + |α|mλn ρk,k sn,k k=1 k=k tSC,n,k = λn |α|ρmsn,k +zICI,n,k +zMAI,n,k +zMAICI,n,k +zIMD,n,k N K (43) + |α| mn,n λn ρk,k sn,k + z˜ICI,n,k + z˜MAICI,n,k + z˜IMD,n,k + zAWGN,n,k, =1 = n k 1 (46) n =n k =k G−1 N − ∗ where j arg(α) + e θg+1ck,g mn,n λn vIMD,n ,g g=0 n=1 N −1 =| | , (47) zICI,n,k α ρrλnλn mn,n rλn λn sn ,k + z , AWGN,n,k n =1 n=n where vIMD,n,g is the (n, g)th element of VIMD,and K = ∗ − j arg(α) T ˆ −1 ∗ zAWGN,n,k ϕ e zn,AWGNE ck .Equation(43)clearly zMAI,n,k =|α|m ρk,k sn,k , (48) illustrates the presence of the interference terms (ICI, MAI, = k 1 MAICI, and IMD). By constructing the conditional random k =k variable tSC,n,k = zSC,n,k|λn,from(43), we obtain N K −1 =| | , (49) zMAICI,n,k α rλnλn mn,n rλn λn ρk,k sn ,k N n=1 k=1 = = ˆ n n k k tSC,n,k = λn|α|ρmsn,k + |α|ρ mn,n λn ,nsn ,k n =1 −1 = G N n n − j arg(α) −1 ∗ zIMD,n,k = e r θg+1c mn,n rλ λ vIMD,n,g , K λnλn k,g n n g=0 n=1 + λ |α|m ρ s n k,k n,k (50) k=1 k=k N N K z˜ICI,n,k =|α|ρ mn,n πn,nsn,k, (51) ˆ + |α| mn,n λn ,n ρk,k sn ,k = n 1 n=1 k=1 (44) n =n n=n k=k N K G−1 =| | − j arg(α) ∗ z˜MAICI,n,k α mn,n πn ,n ρk,k sn ,k , (52) + λ e m θ c v n g+1 k,g IMD,n,g n =1 k =1 g=0 n=n k=k G−1 N G−1 N − j arg(α) ∗ − ∗ ˆ = j arg(α) + e θg+1ck,g mn,n λn ,nvIMD,n ,g z˜IMD,n,k e θg+1ck,g mn,n πn ,nvIMD,n ,g . (53) g=0 = = = n 1 g 0 n 1 n =n n=n
+ zAWGN,n,k, We want to highlight by (46) that in the frequency-selective
ˆ ff where λn ,n = λn |λn is the random variable that indicates case the ICI consists of two di erent components. The first the channel gain value on the n th subcarrier conditioned one, λnzICI,n,k, is proportional to the channel amplitude λn on the channel gain on the nth subcarrier. Note that in of the useful signal, and therefore it represents the ICI that (44) the IMD relative to the subcarrier of interest and the fades coherently with the useful signal. On the other hand, the IMD coming from the other subcarriers are considered sep- second part z˜ICI,n,k represents the ICI that fades independently arately. As shown in [23, page 532], the (N − 1) × 1vector of λn. Indeed, z˜ICI,n,k in (51) is a zero-mean random variable ˆ ˆ ˆ ˆ ˆ T with power expressed by λn = [λ1,n ··· λn−1,n λn+1,n ··· λN,n] is still a Gaussian random vector, with mean value η = {λˆ } and covariance λˆn E n H N = {λˆ λˆ } 2 2 2 −1 2 2 Rλˆ E n expressed by ˜ =| | − , (54) n n σICI,n,k αρ mn,n rλn λn rλnλn rλn λn σS = n 1 −1 n =n η = rλ¯ , λˆn λnrλnλn nλn (45) −1 H Rλˆ = Rλ¯ − r rλ¯ r , which does not depend on the actual channel gain λn. n n λnλn nλn λ¯nλn 738 EURASIP Journal on Applied Signal Processing
G−1 G−1 Also the MAICI and the IMD, similarly to the ICI, consist ∗ ∗ 2 = of two parts. In both cases, the first part is proportional to σ˜IMD,n,k θg+1θg+1ck,g ck,g g=0 g=0 λn, while the second one has power independent of λn.The powers of the second components are expressed, respectively, N N ∗ T (56) × J R J by mn,n mn,n n λˆn n n ,n n=1 n=1 N n=n n=n 2 2 2 −1 2 σ˜ =|α| mn,n rλ λ − r rλ λ MAICI,n,k n n λnλn n n × n=1 Rv(g, g ) n,n , n=n (55) K 2 where Rv(g, g ) is the same as in (33), and the (N −1)×N ma- × 2 ρk,k σS , trix Jn is obtained from the identity matrix IN by removing k =1 k=k the nth row. The conditional SINR can be expressed as
λ 2|αρm|2σ2 γ λ = n S , (57) n,k n 2 2 2 2 2 2 2 2 2 λn σICI,n,k + σMAI,n,k + σMAICI,n,k + σIMD,n,k + σ˜ICI,n,k + σ˜MAICI,n,k + σ˜IMD,n,k + σAWGN
where 5. SIMULATION RESULTS
N In this section, we present some simulation results in order 2 −1 2 2 2 = σICI,n,k αρrλnλn mn,n rλn λn σS , (58) to validate the approximations introduced in the theoretical = = n 1 analysis. We consider an MC-DS-CDMA system with N n =n 256 subcarriers, with a subcarrier separation of ∆ f = 1/T = N K 156.25 kHz and a cyclic prefix of length L = 64. For brevity, 2 −1 2 2 2 2 σ = αr m r ρ σ , MAICI,n,k λnλn n,n λn λn k,k S we only consider data modulated by 4-QAM, which corre- n=1 k=1 n=n k=k spond to a gross bit rate of 4 Mbps per user when the length (59) of the spreading codes is equal to G = 16. We assume that each time-domain tap of the channel suffers independent G−1 G−1 2 −2 ∗ ∗ T ∗ Rayleigh fading, with an exponentially decaying power delay σ = r θg+1θ c ck,g m¯ Rv(g, g )m¯ , IMD,n,k λnλn g +1 k,g n n profile and an rms delay spread equal to 250 nanaoseconds. g=0 g=0 In the first scenario, we assume that the base station = employs Walsh-Hadamard (WH) spreading codes of length m¯ n n mn,n rλn λn , (60) G = 16. We also assume Hypothesis H1 to be true. We still re- mark that the BER performance in this scenario can be con- 2 2 2 2 sidered a lower bound for the case of inaccurate estimate of and σMAI,n,k, σ˜ICI,n,k, σ˜MAICI,n,k,andσ˜IMD,n,k are expressed by (30), (54), (55), and (56), respectively. Moreover, as in the the CFO-induced time-varying phase shift. Figures 1 and 2 previous section, we exploit the hypotheses of a high num- show the BER performance versus the received Eb/N0 when ber of subcarriers and a high number of users in order to the number of active users is equal to K = G = 16. The re- approximate as Gaussian the interference terms in (46), by ceived Eb/N0 is defined before the despreading as the ratio be- the central limit theorem. tween the noiseless signal power and the noise power, while The resultant BER can therefore be obtained by insert- the BER is obtained by averaging the BER over all the subcar- ing (57), (36), and the Rayleigh pdf of |λn| in (39). Also in riers and all the users. Figure 1 shows the BER versus Eb/N0 this case the BER can be calculated by means of series expan- for different values of the normalized CFO ε, assuming lin- sions that involve generalized hypergeometric functions. For ear amplification at the transmitter (i.e., without IMD). The 4-QAM, the BER expression is the same as in (37), with 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 |αρm|2σ2σ2 2 = S λ accurate results in frequency-selective scenarios. µn,k 2 2 2 2 , σ˜ICI,n,k + σ˜MAICI,n,k + σ˜IMD,n,k + σAWGN Figure 2 illustrates the impact of the NLA on the BER 2 2 2 2 2 performance in the absence of CFO. We define the output σICI,n,k + σMAI,n,k + σMAICI,n,k + σIMD,n,k σλ ff = 2 ν2 = . backo (OBO) as OBO PU,MAX/σU,wherePU,MAX and n,k 2 2 2 2 2 σ˜ICI,n,k + σ˜MAICI,n,k + σ˜IMD,n,k + σAWGN σU are the maximum power and the mean power, respec- (61) tively, of the NLA output signal. We assume that the amplifier MC-DS-CDMA with Frequency Offset and Nonlinear Amplifiers 739
100 100
101 101
2 10 ε = 0.1 102 ε = 0.05 103 BER ε = 0.03 BER ε = 0.02 103 104 ε = 0.01 ε = 0.005 104 105 ε = 0 106 105 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60
Eb/N0 (dB) Eb/N0 (dB)
Theoretical K = 4 (theoretical) K = 10 (theoretical) Simulated K = 4 (simulated) K = 10 (simulated) K = 7 (theoretical) K = 13 (theoretical) Figure 1: Impact of the CFO ε on the BER (WH codes, K = G = 16, K = 7 (simulated) K = 13 (simulated) no IMD). (a)
100 100
101
101 102
103 OBO = 0.84 dB 102 BER BER 104 OBO = 1.3dB
105 103 OBO = 2dB 106 Linear 107 104 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60
Eb/N0 (dB) Eb/N0 (dB)
Theoretical K = 4 (theoretical) K = 8 (simulated) Simulated K = 4 (simulated) K = 16 (theoretical) K = 8 (theoretical) K = 16 (simulated)
Figure 2: Impact of the clipping NLA on the BER (WH codes, K = (b) G = 16, no CFO). Figure 3:BERcurvesfordifferentnumberofusersK (WH codes, G = 16, OBO = 0.84 dB, no CFO). is perfectly predistorted, that is, it behaves as a clipper of In Figures 3a and 3b, we still assume WH spreading codes the NLA input envelope. From Figure 2, we deduce that the of length G = 16, Hypothesis H1 as true, and ε = 0, and Gaussian approximation of the IMD is slightly worse than we focus on the impact of the number K of active users on the approximation of the ICI in Figure 1, especially at the the BER performance. We also assume that each user em- saturating BER. Indeed, for a clipping amplifier, the Gaus- ploys a fixed spreading code, that is, that the kth user em- sian approximation of the IMD is very accurate only when ploys the kth row of the G × G Hadamard matrix. Figure 3a the SNR is not too high or the number of subcarriers is very shows that at low SNR the BER increases with the num- high [24]. However, in this case the BER mismatch seems to ber of active users, as expected. However, the BER floor be quite small. at high SNR does not increase with the number of users. 740 EURASIP Journal on Applied Signal Processing
100 100
101 101
102 102 ε = 0.005 BER BER 3 OBO = 0.84 dB 3 10 10 ε = 0.004 OBO = 1.3dB
104 104 ε = 0.003 OBO = 2dB Linear, ε = 0 Linear 105 105 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60
Eb/N0 (dB) Eb/N0 (dB)
Theoretical Theoretical Simulated Simulated
Figure 4: Impact of both CFO and NLA on the BER (WH codes, Figure 6: BER curves for different CFO values (Gold codes, G = 31, K = G = 16, ε = 0.02). K = 25, no IMD).
6 100
101 5 102
4 103 K = 30 BER TD (dB) K = 25 104 K = 20 3 105 K = 15
2 106 0 1 2 3 4 5 6 0 10 20 30 40 50 60
OBO (dB) Eb/N0 (dB)
K = 16, ε = 0.03 K = 16, ε = 0 Theoretical K = 16, ε = 0.01 Linear, ε = 0 Simulated K = 8, ε = 0.01
Figure 5: TD as a function of the OBO (WH codes, G = 16, BER = Figure 7: BER curves for different number of users K (Gold codes, 10−3). G = 31, ε = 0.004, no IMD).
Such a behavior is due to the fact that the IMD powers in reduction is masked by the presence of the ICI induced by (56)and(60)dependonRv(g, g ), which is highly sensitive the CFO. to the choice of the spreading codes among the ones provided In order to identify the optimum OBO value, we intro- by the Hadamard matrix. This fact is confirmed by the BER duce the total degradation (TD) as [7] = floors of Figure 3b,becauseRv(g, g ) is the same for K 4, = 8, and = 16. = ∆ K K TD dB Eb/N0 dB +OBO dB, (62) Figure 4 exhibits the joint effect of CFO and NLA on the = ∆ | = | − ∞ | BER performance when ε 0.02. The results show that, in where Eb/N0 dB Eb/N0(OBO, ε) dB Eb/N0(+ , ε) dB is the the presence of a CFO, it is not convenient to increase the SNR increase caused by the NLA to achieve a target BER for output NLA power backoff at the transmitter beyond a cer- afixedCFOε. If we neglect out-of-band effects, the min- tain value (that obviously depends on ε), because the IMD imization of the TD can be considered a fair criterion for MC-DS-CDMA with Frequency Offset and Nonlinear Amplifiers 741
100 different values of the CFO ε and of the number of users K. It is evident that the BER analysis is quite accurate for Eb/N0 < 30 dB, while a certain mismatch between theoreti- 1 10 cal and simulated BER exists at high SNR. This fact is due to the nonperfect Gaussian approximation of the MAI terms, 102 as confirmed by the estimation of the kurtosis of the deci- sion variable, which is a standard test for non-Gaussian noise
BER [25]. However, the BER mismatch is smaller when also the 103 ε = 0.003 NLA-induced distortions are present, as testified by Figures 8 and 9 (OBO = 2.00 dB). Figures 6 and 8 clearly show that, if ε = 0.002 only a rough phase estimate is available, the maximum toler- 104 able CFO is rather small (e.g., ε<0.005), because of the ad- ε = 0 ditional MAI. Therefore, the phase compensation step plays a crucial role in the BER performance of MC-DS-CDMA sys- 105 0 10 20 30 40 50 60 tems.
Eb/N0 (dB)
Theoretical 6. CONCLUSIONS Simulated In this paper, we have evaluated the BER of MC-DS- Figure 8: BER curves for different CFO values (Gold codes, G = 31, CDMA downlink systems subject to both CFO and IMD in K = 25, OBO = 2.00 dB). frequency-selective Rayleigh fading channels. The BER anal- ysis, which is quite accurate in many conditions, allows to 100 identify the optimum OBO value of the NLA, depending on the amount of CFO. The analytical findings have also high- lighted the importance of an accurate compensation for the 101 CFO-induced phase shift. Although the analysis has been carried out for M-ary QAM, it can also be extended to M-ary
2 phase-shift keying constellations. Future works may focus on 10 the effects of channel estimation errors and channel coding. K = 30 BER 103 K = 25 ACKNOWLEDGMENTS K = 20 The authors would like to thank Prof. Geert Leus for his in- 104 sightful comments and suggestions. This work was partially K = 15 supported by the Italian Ministry of Education, University and Research, PRIN 2002 Project “MC-CDMA: an air inter- 105 0 10 20 30 40 50 60 face for the 4th generation of wireless systems.”
Eb/N0 (dB) REFERENCES Theoretical Simulated [1] S. Hara and R. Prasad, “Overview of multicarrier CDMA,” IEEE Commun. Mag., vol. 35, no. 12, pp. 126–133, 1997. Figure 9: BER curves for different number of users K (Gold codes, [2] K. Fazel and G. P. Fettweis, Eds., Multi-Carrier Spread- G = 31, ε = 0.003, OBO = 2.00 dB). Spectrum, Kluwer Academic Publishers, Dordrecht, Nether- lands, 1997. [3] T. Pollet, M. Van Bladel, and M. Moeneclaey, “BER sensitiv- ∆ ity of OFDM systems to carrier frequency offset and Wiener the selection of the optimum OBO value, because Eb/N0 and OBO represent the power penalty on the link budget with re- phase noise,” IEEE Trans. Commun., vol. 43, no. 2/3/4, pp. spect to the linear scenario, and with respect to the maximum 191–193, 1995. [4] H. Ochiai and H. Imai, “On the distribution of the peak-to- amplifier output power, respectively. Figure 5 shows the TD average power ratio in OFDM signals,” IEEE Trans. Commun., −3 performance at BER = 10 for different values of the CFO vol. 49, no. 2, pp. 282–289, 2001. and number of users. It is evident that the optimum OBO [5] S. Verdu,´ Multiuser Detection, Cambridge University Press, mainly depends on the value of the CFO. Cambridge, UK, 1998. In the second simulation scenario, we assume that a [6] H. Steendam and M. Moeneclaey, “The sensitivity of down- ff rough estimate of the CFO-induced time-varying phase shift link MC-DS-CDMA to carrier frequency o sets,” IEEE Com- mun. Lett., vol. 5, no. 5, pp. 215–217, 2001. is available at the receiver (i.e., Hypothesis H2 is true). In [7] J.-H. Jong and W. E. Stark, “Performance analysis of coded this case, since the MAI cannot be completely eliminated, we multicarrier spread-spectrum systems in the presence of mul- assume that the base station employs Gold codes of length tipath fading and nonlinearities,” IEEE Trans. Commun., vol. G = 31. Figures 6 and 7 display the BER performance for 49, no. 1, pp. 168–179, 2001. 742 EURASIP Journal on Applied Signal Processing
[8] Z. Wang and G. B. Giannakis, “Wireless multicarrier com- Luca Rugini was born in Perugia, Italy, in munications,” IEEE Signal Processing Mag.,vol.17,no.3,pp. 1975. He received the Laurea degree in elec- 29–48, 2000. tronics engineering and the Ph.D. degree in [9] A. Papoulis, Probability, Random Variables and Stochastic Pro- telecommunications from the University of cesses, McGraw-Hill, New York, NY, USA, 3rd edition, 1991. Perugia, Perugia, Italy, in 2000 and 2003, re- [10] P. Banelli and S. Cacopardi, “Theoretical analysis and perfor- spectively. He is currently a postdoctoral re- mance of OFDM signals in nonlinear AWGN channels,” IEEE searcher with the Department of Electronic Trans. Commun., vol. 48, no. 3, pp. 430–441, 2000. and Information Engineering (DIEI), the [11] D. Dardari, V. Tralli, and A. Vaccari, “A theoretical charac- University of Perugia. His research interests terization of nonlinear distortion effects in OFDM systems,” include spread-spectrum and multicarrier IEEE Trans. Commun., vol. 48, no. 10, pp. 1755–1764, 2000. communications. [12]X.Ma,C.Tepedelenlioglu,˘ G. B. Giannakis, and S. Bar- barossa, “Non-data-aided carrier offset estimators for OFDM Paolo Banelli was born in Perugia, Italy, on with null subcarriers: identifiability, algorithms, and perfor- May 19, 1968. He received the Laurea degree mance,” IEEE J. Select. Areas Commun., vol. 19, no. 12, pp. in electronics engineering and the Ph.D. de- 2504–2515, 2001. gree in telecommunications from the Uni- [13] T. M. Schmidl and D. C. Cox, “Robust frequency and timing versity of Perugia, Perugia, Italy, in 1993 and synchronization for OFDM,” IEEE Trans. Commun., vol. 45, 1998, respectively. Since 1998, he has been no. 12, pp. 1613–1621, 1997. an Assistant Professor with the Department [14] H. Minn, M. Zeng, and V. K. Bhargava, “On timing offset of Electronic and Information Engineering estimation for OFDM systems,” IEEE Commun. Lett., vol. 4, (DIEI), University of Perugia. In 2001, he no. 7, pp. 242–244, 2000. joined as a Visiting Researcher the Spin- [15] D. Landstrom,S.K.Wilson,J.-J.vandeBeek,P.¨ Odling,¨ and Comm Group at the Electrical and Computer Engineering (ECE) P. O. B orjesson,¨ “Symbol time offset estimation in coherent Department, University of Minnesota, Minneapolis. His research OFDM systems,” IEEE Trans. Commun.,vol.50,no.4,pp. interests include nonlinear distortions, broadcasting, multiuser de- 545–549, 2002. tection, multicarrier, and block-transmission techniques for wire- [16] K.-W. Yip and T.-S. Ng, “Effects of carrier frequency accuracy less communications. on quasi-synchronous, multicarrier DS-CDMA communica- tions using optimized sequences,” IEEE J. Select. Areas Com- mun., vol. 17, no. 11, pp. 1915–1923, 1999. [17] P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Com- mun., vol. 42, no. 10, pp. 2908–2914, 1994. [18] R. Van Nee and R. Prasad, OFDM for Wireless Multimedia Communications, Artech House, Boston, Mass, USA, 2000. [19] K. Cho and D. Yoon, “On the general BER expression of one- and two-dimensional amplitude modulations,” IEEE Trans. Commun., vol. 50, no. 7, pp. 1074–1080, 2002. [20] L. Rugini, P. Banelli, and S. Cacopardi, “An analytical upper bound on MMSE performance using approximated MMSE multiuser detector in flat Rayleigh fading channels,” in Proc. European Wireless Conference, pp. 952–956, Florence, Italy, February 2002. [21] L. Rugini and P. Banelli, “Symbol error probability of linearly modulated signals affected by Gaussian interfer- ence in Rayleigh fading channels,” Tech. Rep. RT-005- 03, Department of Electronic and Information Engineer- ing, University of Perugia, Perugia, Italy, December 2003, http://www.diei.unipg.it/rt/DIEI RT.htm. [22] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, San Diego, Calif, USA, 5th edition, 1994. [23] S. M. Kay, Fundamentals of Statistical Signal Processing: Es- timation Theory, vol. 1, Prentice Hall, Englewood Cliffs, NJ, USA, 1993. [24] H. Nikopour and S. H. Jamali, “Effects of Cartesian clip- ping noise on the performance of orthogonal frequency di- vision multiplexing system: a theoretical approach,” in Proc. IEEE Global Telecommunications Conference (GLOBECOM ’02), vol. 2, pp. 1254–1258, Taipei, Taiwan, November 2002. [25] S. M. Kay, Fundamentals of Statistical Signal Processing: Detec- tion Theory, vol. 2, Prentice Hall, Englewood Cliffs, NJ, USA, 1998.