On the Impact of Carrier Frequency Offsets in OFDM/SDMA Systems

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1 On the Impact o Carrier Frequency Osets in OFDM/SDMA Systems Sheng Zhou, Kai Zhang and Zhisheng Niu State Key Lab on Microwave and Digital Communications Dept. o Electronic Engineering, Tsinghua University, Beijing , China zhouc@mails.tsinghua.edu.cn, zhangkai98@mails.tsinghua.edu.cn, niuzhs@tsinghua.edu.cn Abstract The combined OFDM/SDMA approach has raised lots o research interests recently as it appears to be quite suitable or uture broadband wireless transmission. In order to build practical OFDM/SDMA systems, we need the perormance evaluation o such systems under real-world conditions. Given that plain single-user OFDM is highly sensitive to carrier requency oset (CFO), the perormance degradation o an OFDM/SDMA system due to CFO can be predicted. However, the property o the degradation is not clear. In this paper, the impact o CFO on the perormance o an uplink OFDM/SDMA system is analyzed. Dierent rom the existing analysis, we jointly consider the impact o CFO on channel estimation and multi-user signal detection. Analytical expressions are derived or the variance o channel estimation error and multi-user signal detection error in the presence o CFO, whose accuracies are validated by simulation results. Observations related to channel estimation accuracy and the interaction o the CFO eects rom dierent users are also given. I. INTRODUCTION The growing demand o high speed wireless Internet access has led to extensive research on systems with ine ability to combat channel multi-path and to provide high spectral eiciency. First, orthogonal requency division multiplexing (OFDM) has become a popular technique in broadband wireless communications due to its advantage in mitigating channel selectivity while providing a high spectrum eiciency [1]. Second, with the development o the multiple-input multipleoutput (MIMO) technique, space division multiple access (SDMA) can urther optimize the spectrum eiciency by reusing the bandwidth among dierent users while utilizing their spatial signatures []. Moreover, the inherent parallelism o the OFDM systems allows per-subcarrier SDMA processing, resulting in a considerably lower implementation cost than the single-carrier SDMA systems. Hence, the OFDM/SDMA systems are considered quite suitable or uture wireless broadband networks, or the combined OFDM/SDMA approach can beneit rom the advantages o both OFDM and SDMA [6]. However, while making use o the beneits, the combination o OFDM and SDMA also inherits the disadvantages rom each technique. Carrier requency osets (CFO) exist between user terminals and the base station because o the precision limitation o oscillators and the Doppler eect. It has been shown that OFDM systems are very sensitive to CFO, which leads to perormance degradation by introducing inter-carrierintererence (ICI) [] [4]. Analysis o the impact o CFO in single-input single-output (SISO) OFDM systems can be ound in [] [5]. For uplink OFDM/SDMA systems, dierent CFO rom multiple users should be considered, and analysis can be ound in [9], in which the author has shown that the ICI induced by multiple CFO is a mixture o signals rom dierent users on dierent subcarriers, however, the quantitative expression or the degradation is not derived. Additionally, in an uplink OFDM/SDMA system, co-channel signals rom dierent users should be separated by multi-user detection (MUD [8]), and the perormance o MUD relies on the channel estimation (CE) accuracy [10], which can also be aected by CFO. Thereore, when jointly consider CE and MUD, the perormance degradation o MUD in the presence o imperect channel estimation due to CFO may be signiicant. Although various papers treat CFO estimation schemes (e.g., [9], [16], [17]) and channel estimation algorithms in the presence o CFO (e.g., [14]), we still need theoretical results o the impact o CFO on the perormance when implementing CE and MUD into a practical OFDM/SDMA system, which is important to the system design. In [6], a ull OFDM/SDMA solution under practical conditions is proposed, and the eect o CFO in an MMSE-OFDM-SDMA system is shown. However, the results are mainly based on simulation. In this paper, we theoretically analyze the imperect channel estimation due to CFO and its impact on MUD. In Section II, the system model is set up. In order to use per-subcarrier analysis, the model is urther simpliied in Section III by considering multiple CFO eects as an additional noise at the transmitter side o each user. Then the variance o channel estimation error in the presence o CFO is calculated in Section IV, based on which the expression or the variance o signal detection error ater MUD is derived in the case o three dierent situations in Section V. Numerical examples and discussions are given in Section VI, where the accuracy o our analytical expression is validated. Finally, Section VII contains our conclusions. II. SYSTEM MODEL The system under consideration is schematized in Fig. 1, which illustrates an uplink OFDM/SDMA system with U simultaneous user terminals and one base station (BS). Each user terminal has a single antenna or low cost, while the BS has an array o A antennas. We assume that A > U. U users simultaneously transmit OFDM modulated (with N subcarriers) symbols to the BS. Then the BS uses the

2 TABLE I VALUES OF I n FOR N = 18 n I n 1 I π n Fig. 1. System model or OFDM/SDMA in the uplink Zero-Forcing (ZF) scheme to separate the signals o the U users, based on the channel knowledge obtained by requency domain Least Square (LS) channel estimator. Without CFO, the ideal requency domain signal model is given by Y = HX + N n, (1) where H denotes an NA NU block diagonal matrix. The kth A U block diagonal element is H k, the MIMO channel matrix or the kth subcarrier, and the entries are i.i.d. complex Gaussian variables with zero mean and unitary variance. The NU 1 transmitted MIMO OFDM vector is given by X = (x[1] T, x[] T,..., x[n] T ) T,where x[k] denotes the U 1 requency domain MIMO transmit signal vector or the kth subcarrier. We assume all the elements o X are i.i.d and with unitary variance. The NA 1 received MIMO OFDM vector is given by Y = (y[1] T, y[] T,..., y[n] T ) T,where y[k] denotes the A 1 requency domain MIMO receive signal vector or the kth subcarrier. The NA 1 vector N n represents the receiver noise, with i.i.d zero-mean, complex Gaussian elements o variance σn. Even ater carrier requency synchronization, we assume that each user still has a normalized residual CFO jitter u, which is deined relative to the subcarrier spacing. Hence, with R the sample rate, it corresponds to an absolute requency oset o R u /N Hz. All the CFO jitters are assumed to be i.i.d Gaussian with zero mean and a variance o σ or each user 1. In the presence o CFO, the contribution o user u to the data symbol ya[k] u that the BS receives on the kth subcarrier and at the ath antenna during an OFDM symbol is given by [] [4] ya[k] u = h u a[k]x u [k]i 0 + h u a[i]x u [i]i i k, () where the noise actor is omitted or clarity, and I n = sin(π(n u )) π(n u )() N sin( π N (n u )) e j N. () There are two eects existing due to CFO. One is the rotation and attenuation o the useul signal given by the irst term in (). The other is the leakage rom other subcarriers into the subcarrier o interest given by the second term in (), 1 For simplicity, we assume the CFO jitter variances o dierent users are the same. which is commonly considered ICI. Then we can rewrite the signal model as Y = H eq X + N n, (4) where H eq is no longer block diagonal because o the ICI eect. III. SIMPLIFIED SIGNAL MODEL WITH CFO JITTERS In this section we simpliy the signal model in (4) in order to use the original block diagonal channel matrix H or urther analysis. Here we consider all the eects caused by the CFO jitters as an additional noise to the useul signal. Assuming that the channel coeicients h u a[i] are highly correlated around the kth subcarrier, and since I n decreases approximately as 1/n with increasing n when n < N (see Table I), we can approximate () as ya[k] u h u a[k] x u [k]i 0 + x u [i]i i k. (5) We deine the equivalent average noise power conditioned on u as P d = E x u [k] x u [k]i 0 x u [i]i i k, (6) where E{ } represents the expectation operator. Since transmit signals x u [k] are independent and with unitary variance, we can urther write P d as P d = 1 I 0 + I i k. (7) For suiciently large N, as in [4], we have some approximations I i k i= I 0 1 (π u ), 1 I 0 (π u ), I i k I 0 = 1 I 0, where we have assumed a small residual requency oset u, which leads to sin( π N u ) π N u, and we have also used Taylor series o sin(π u ). (8)

3 Subsequently, we get P d 4(π u ). (9) Further more, since u is random, we can average P d over u, and deine σd = E{P d } = 4(πσ ). (10) As a result, we can rewrite (5) as y u a[k] h u a[k] (x u [k] + n d ), (11) where n d denotes the equivalent additional noise with zero mean and variance σd. Now the CFO eects equivalently result in the orm o an additional noise n d at the transmitter side. Hence, the equivalent signal model can be simpliied to Y = HX + N n = HX + HN d + N n, (1) where we denote X = X + N d, and the equivalent NU 1 noise vector N d = (n d [1] T,..., n d [N] T ) T, o which the element n d [k] is a U 1 vector with elements o variance σ d. Then we can use the original block diagonal channel matrix H in order to analyze the problem or each subcarrier k y[k] = H k x [k] + n n = H k x[k] + H k n d + n n, (1) where the A 1 vector n n has i.i.d complex Gaussian elements o variance σ n. Hence, we can ocus our analysis on the kth subcarrier in the ollowing sections. IV. IMPACT OF CFO ON CHANNEL ESTIMATION Frequency domain LS channel estimator [7] is used in this section. For realistic systems, channel estimation algorithms are more complicated and time domain approaches are oten used [11] [1]. However, the time domain LS CE is equivalent to the requency domain one in nature [1]. In other words, one can always transorm a time domain LS estimator into a requency domain one by DFT operation. So our analysis on requency domain estimator will be o universality. We will show how the CFO jitters aect the perormance o the channel estimator. The U U training symbol matrix is designed to be diagonal T = t 1... t U, (14) where the U columns represent training symbols sent on adjacent subcarriers by dierent users making use o the high correlation o these channels, which is called a requency division scheme. The positions o the training symbols are shown in Fig.. Each training symbol has a unit magnitude, and the phase is randomly chosen, and those on dierent subcarriers or o dierent users are assumed to be independent. The result is dierent rom [4], because in order to get the raw signal model we don t have the assumption in [4] that the receiver compensates the rotation and attenuation o the useul signal caused by I 0. Fig.. Positions o the training symbols Here the training symbols t u are o the target estimation set, while t u and t # u are o adjacent sets. Although or practical MIMO-OFDM systems, the estimation symbols may not be periodically spaced, and can be more sparsely distributed over the whole requency band, it is suicient to analyze the basic training symbol distribution in Fig. to relect the impact o CFO in requency division estimation schemes. The LS channel estimation can be obtained as Ĥ k = ỹ[k]t 1, (15) where Ĥk represents the estimation o H k, and rom (1), ỹ[k] = H k T + n n, we have Ĥ k = H k + H k Ñ d T ÑnT, (16) }{{} E k where E k denotes the estimation error, and Ñn is an A U receiver noise matrix, and Ñd = T T. Because o the rotation, attenuation and ICI eects due to CFO, T is given by (We give a user example or illustration ) T I 0 t 1 I 1 t 1 + I t # 1 I t 1 + I 1 t # 1 I t + I 1 t I 0 t I 1 t + I t # I 1 t + I t I t + I 1 t I 0 t, (17) where we have eliminated the small items (e.g., I t # 1 ) or clarity, and during the calculation, we only consider the adjacent pilot subsets. We can see that because o the ICI eect, the equivalent transmit training symbol is no longer diagonal. Notice that without CFO, the estimation error is only Ñ n T 1. Then we can get the variance o each element in E k, which is the average estimation error variance o each element in H k σ e ( + 6 U 1 1 π i )(πσ ) + σ n, (18) where we have used the i.i.d eature o the elements in H k, the approximates in (8) and In 1 I 0 π n shown in Table I. The average error variance slightly increases with U, so actually the estimation error is not sensitive to the MIMO coniguration. For large number o U (larger than 5 is enough, see Table I), we have σe 4(πσ ) + σn.

4 V. IMPACT OF CFO ON MULTIUSER DETECTION Linear ZF multiuser signal detector [8] calculates an estimate ˆx[k] o the transmitted signal x[k] by ˆx[k] = (ĤH k Ĥk) 1 Ĥ H k y[k], (19) where Ĥk is the estimated channel matrix on the kth subcarrier, and H denotes the conjugate transpose. Since in the case o a requency-lat channel with a suicient low noise level, the MMSE detection can be approximates by the ZF detection, our analysis is appropriate or MMSE detection and related successive cancelation approaches (e.g., SIC-successive intererence cancellation [6]) under high SNR. From (16) and (1), we have ˆx[k] =((H k + E k ) H (H k + E k )) 1 (H k + E k ) H y[k] = ((H k + E k ) H (H k + E k )) 1 (H k + E k ) H E k x [k] + ((H k + E k ) H (H k + E k )) 1 (H k + E k ) H n n + x [k]. (0) Since elements in E k are with small variance, so (0) can be approximated by a Taylor series [10] ˆx[k] (H H k H k ) 1 H H k E k x [k] + (H H k H k ) 1 (H H k + E H k )n n (H H k H k ) 1 (H H k E k + E H k H k ) (H H k H k ) 1 H H k n n + x [k]. (1) To get the signal estimation error e k = ˆx[k] x[k] and its variance, calculation interacting with results in the last section is needed. For comparison, we consider three dierent situations. A. CFO compensated Channel Estimation Here we assume that the CFO eect is compensated in CE, which can be accomplished by using some joint algorithms (e.g., [14]), thus the channel estimation error is only caused by channel noise. Now we have E k = ÑnT 1. With (1), and the reasonable assumption that x [k], E k and n n are statistically independent, and also tr [ E { x [k]x H [k] }] = tr [ E { x[k]x H [k] }] = U, the covariance matrix o e k is R ek σ n(h H k H k ) 1 (1 + U) + σ di U U, () where we urther leave out the high order terms. The variance o this estimation error or each user can be obtained by averaging the total error variance over U users P e (u) = tr [E {R e k }] = σ n(1 + U) U A U + 4(πσ ), () where the result in [15] is used that tr [ E { (H H k H k) 1}] = U/(A U) in the case o A > U. In (), the irst term represents the impact o the channel noise, including the channel estimation error due to the channel noise, and the second term represents the equivalent transmitter noise due to CFO described in (10) and (11). Because ZF detection is unbias and CFO eects are considered equivalent zero-mean noise, elements in e k are zero-mean. B. Independent CFO Independent CFO means that the CFO in channel estimation is independent with the one in data transmission, which corresponds to the situation when CFO is rapidly changing with time. There is no CFO compensation in CE. Using the result in (18), and ollowing the similar calculations in V-A, we get the covariance matrix o e k R ek σn(h H k H k ) 1 (1 + U) + ( U 1 1 π i )σ di U U. (4) Then the variance o this estimation error or each user can be obtained by averaging the total error variance over U users P e (u) = σ n(1 + U) A U + (7 + 6 U 1 1 π i )(πσ ), (5) where the irst term has the same meaning with the one in V-A. However, or the second term, besides the equivalent transmitter noise part, the dierence between (5) and () is the additional channel estimation error due to the CFO eects. For large number o U, we have P (u) e C. Correlated CFO σ n (1+U) A U + 8(πσ ). In this situation, we assume that the CFO in channel estimation is the same with the one in data transmission, which corresponds to the slow changing CFO and holds or most o the real cases. Now the calculation should be careully done, or elements in x [k] and E k are no longer independent. Actually, in the derivation o the error variance, ater eliminating the high order terms, σ in () and (4) only comes rom the irst and the last terms in (1). So we write the uth row o (H H k H k) 1 H H k ÑdT 1 x [k] + x [k], which is the part o the estimated signal without the contribution o channel noise, denoted by ˆx u [k] d ˆx u [k] d =( I 0 ) x u [k]i 0 + u j=1 U j=u+1 ( t u t j I u j U + t u t j I u j x u [i]i i k ) x j [k] ( t # u I u j+u + t ) u I u j x j [k]. t j t j (6) Then we can write the variance o the estimation error or the uth user P e (u) = σ n(1 + U) A U + E { (ˆx u [k] d x u [k]) } (1 + 6 U 1 1 σ n(1 + U) π i )(πσ ) (7) A U +, where we have used the approximates in (8), and the independency among the transmit signals and the training symbols. For large number o U, we have P e (u) σ n (1+U) A U + (πσ ).

5 16 x independent CFO, sim independent CFO, analy correlated CFO, sim correlated CFO, analy CFO compensated CE, sim CFO compensated CE, analy 10 P e (u) BER σ = σ = σ = σ = σ SNR(dB) Fig.. Variance o signal estimation error, or σ n = 10,U=,A=4. Fig. 5. Uncoded BER perormance, or correlated CFO,U = 4, A = independent CFO correlated CFO CFO compensated CE BER 10 BER σ =1 10,U=4 5 σ =5 10,U=4 σ =1 10 4,U= σ =4 10 4,U= σ =1 10 6,U=1 5 σ =5 10,U= SNR(dB) 10 8 σ =1 10 4,U=1 σ =4 10 4,U= SNR(dB) Fig. 4. Uncoded BER perormance, or σ = ,U =, A = 4. Fig. 6. Uncoded BER perormance, or independent CFO,A = 5. VI. NUMERICAL RESULTS AND DISCUSSION In this section, we present some simulation results in order to validate and discuss our theoretical analysis. In these Monte- Carlo simulations, the number o OFDM subcarriers N = 18, and 16-QAM modulation is applied. Channel matrix with i.i.d. complex Gaussian elements and CFO o every user are randomly generated or each run. At the beginning o each run, channel estimation takes place, ater which is the data transmission. We divide the discussion into three subsections. A. Validation o the Theoretical Results Figure. shows the signal estimation error variance P e (u) as a unction o CFO variance σ o user u under three situations mentioned in the last section. There is a good agreement between our analytical results and the simulation results. When σ is small, channel noise dominates P e (u), while the CFO is the main actor o P e (u) when σ is relatively large. Thus, or the systems with high SNR, the CFO can be considered one o the main actor o the perormance degradation. Although the CFO estimation algorithms (e.g., [16] [17]) can reduce the residual CFO with increasing SNR, the relationship is only linearity in terms o log, and the absolute value o the linearity coeicient is much smaller than 1, so the ratio σ σ n will still increase with decreasing σ n even though σ decreases too. B. Does Accurate CE Lead to Better Perormance? In Figure., we can also see that the perormance o correlated CFO is better than that o the CFO compensated CE, which is already predicted by our analytical results. The conclusion can be urther conirmed by the uncoded BER perormance comparison shown in Fig. 4. In CFO compensated CE, the estimated channel matrix is more accurate (the estimation error variance is σ n) than that o correlated CFO (the estimation error variance is given in (18)), so the results sounds strange at irst glance. However, back to (4), it is clear

6 that i we can estimate equivalent channel matrix H eq instead o H, the perormance o ZF detector in the presence o CFO can be better. In the situation o correlated CFO, we assume no special compensation or CFO (e.g., by special pilot design) in CE to get accurate channel inormation, but making use o the CFO correlation between CE and data transmission, we actually estimate the equivalent channel matrix to some extent. That s why we have better perormance in correlated CFO. This act implies that in OFDM/SDMA systems, channel estimation and CFO compensation can be separately designed, and increasing channel estimation accuracy (in terms o combating the impact o CFO) may not optimize system perormance. Hence, additional CFO compensation in CE will not be recommended in OFDM/SDMA systems. C. Impact o CFO Among Dierent Users Looking into (6), we can see that ˆx u [k] d is only aected by the CFO o the uth user, or all the I n are the unction o the uth user s. This implies that using ZF detector, the CFO on a certain user only aects the signal estimation o that user, and so are the ormer two situations. The reason is that ZF detector can ully suppress multi-user intererence. Although, the CFO on one user will aect all the elements in the estimated channel matrix Ĥk, which leads to perormance degradation o other users due to the imperect channel inormation in ZF detection, the degradation is barely noticeable, or the CFO is generally very small. So the variance o signal estimation error or those who have negligible CFO is approximated by P (u) e σ n (1+U) A U. Simulation results in Fig. 5 validate the analysis above. In the simulation, only user 1 has CFO jitter, while the other users have negligible CFO. The uncoded BER perormance o user 1 (solid lines) degrades more rapidly than the one o the other users (dashed lines). The perormance degradation o other users is barely noticeable. This result implies that the OFDM/SDMA systems with ZF detector have advantage over the orthogonal requency division multiple access (OFDMA) systems in terms o the interaction o CFO impacts among users, because in the uplink OFDMA systems, each o all users occupies dierent subcarriers, and ICI directly leads to multi-user signal intererence. Because the CFO o each user mainly aects the perormance o its own, the degradation due to CFO is not sensitive to the number o users. The second term in (), (5) and (7) is not only insensitive to U, but also irrelative to the number o receive antennas A. So the additional signal estimation error variance due to CFO jitters is not sensitive to MIMO conigurations. The result is urther validated in Fig. 6, where the uncoded BER perormances o dierent MIMO conigurations tend to be the same when SNR is high (i.e., when the impact o CFO dominates the signal estimation error). VII. CONCLUSION We have investigated the perormance degradation o uplink OFDM/SDMA systems due to CFO, and derived the analytical expressions or the variance o channel estimation error and multi-user signal detection error. Ater jointly analyzed CE and MUD, we have observed that increasing channel estimation accuracy in terms o combating CFO may not optimize system perormance. The result suggests independently design CE and CFO, additional CFO compensation in CE may not be recommended in OFDM/SDMA systems. We have also ound that CFO o dierent users have slight interaction, and the impact o CFO on each user is not sensitive to the MIMO conigurations, which gives conidence to implementing MUD in OFDM/SDMA systems. ACKNOWLEDGMENT The authors would like to express their sincere thanks to Hitachi R&D Headquarter or the continuous supports. REFERENCES [1] J. Bingham, Multicarrier modulation or data transmission: an idea or whose time has come, IEEE Commun. Mag., vol. 8, pp.5-14, May [] B. 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