residual carrier frequency offset estimation and correction in ofdm ...

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

RESIDUAL CARRIER FREQUENCY OFFSET ESTIMATION AND CORRECTION IN OFDM MIMO SYSTEMS Antonio Pascual-Iserte Universitat Politècnica de Catalunya Centre Tecnològic de Telecomunicacions de Catalunya Barcelona, Spain email: [email protected]

Llu´ıs M. Ventura Centre Tecnològic de Telecomunicacions de Catalunya Barcelona, Spain email: [email protected]

A BSTRACT This paper addresses the problem of estimating and correcting the residual carrier frequency offset in a multi-input-multioutput (MIMO) system employing orthogonal frequency division multiplexing (OFDM). The presented technique is based on the maximum likelihood criterion and exploits the pilot carriers of the OFDM modulation by assuming that no inter-carrier interference is present. Some implementation aspects of the estimator are also discussed, presenting two alternatives: the grid search and the truncated discrete Fourier transform (DFT) approaches. The evaluation of the estimator is performed by deriving and comparing the corresponding Cramer-Rao bound with the mean square error. Additional simulations are provided in terms of symbol error rate for the case of applying orthogonal space-time block coding to the data carriers. I.

I NTRODUCTION

Orthogonal frequency division multiplexing (OFDM) is a technique for efficiently transmitting information over a fading channel, which is based on the division of the transmission bandwidth into a set of frequency subchannels, each of them transmitting a different stream of symbols. The modulation and demodulation are based on the application of the efficient inverse fast Fourier transform (IFFT) and FFT, respectively. On the other hand, the quality and the rate of the communication can be increased by combining OFDM with spatial diversity techniques, i.e., exploiting multiple antennas at the transmitter and/or the receiver. The most general case corresponds to a multi-input-multi-output (MIMO) channel, with multiple antennas at both sides. Although these techniques are very promising, their practical deployment may encounter some problems [1]. Particularly, OFDM is very sensitive to mismatches in the sampling clocks and the carrier frequency offset (CFO) of the local oscillators at the transmitter and the receiver. More concisely, the CFO can generate inter-carrier interference (ICI) between the frequency subchannels, increasing the symbol error rate (SER). The CFO can be compensated in two stages. First, a CFO correction is performed in an acquisition stage, which frequently is not enough, resulting in a residual CFO that has to be compensated using a second tracking stage. Usually, these 0 This work was partially supported by the Catalan Government under grants 2005SGR-00996 and 2005SGR-00690; and by the Spanish Government under projects TEC2005-08122-C03 (ULTRA-PROCESS) and 2A103 (MIMOWA) from MEDEA+ program.

c 1-4244-1144-0/07/$25.002007 IEEE

Xavier Nieto Gigle Semiconductor Barcelona, Spain email: [email protected]

stages can be carried out using the preamble and the pilot carriers of the OFDM signal. The preamble is composed of several known OFDM symbols at the beginning of the frame that are used for the first CFO correction and the channel estimation. On the other hand, the pilot carriers are known symbols transmitted in all the OFDM symbols multiplexed with the data subcarriers that can be used for the residual CFO correction. Many papers have considered the single-antenna case. For example, in [2], the preamble (but note the pilot carriers) is exploited in the acquisition mode. The use of the cyclic prefix (CP) in the tracking mode is considered in [3]. [4] deals with the blind estimation of the channel and the CFO assuming constant-modulus symbols. Finally, [5] describes a technique for the CFO estimation in a multi-user MC-CDMA system. The number of methods for systems with multiple antennas is low. Some examples are the use of blind techniques, as in [6], where neither the preamble nor the pilot subcarriers are used. In [7], a technique for joint estimating the channel and the CFO in a MIMO channel using the preamble but not the pilot subcarriers is presented. See [8], and references therein, for a more complete description of the state of the art. In this paper, we present a technique for estimating and correcting the residual CFO exploiting the pilot carriers in an OFDM frame. The estimation follows the maximum likelihood (ML) criterion and can be applied to the general case of a MIMO channel. Some implementation advices are also provided and the asymptotic efficiency of the estimator is evaluated by its comparison with the Cramer-Rao bound (CRB) [9]. Finally, some simulations results in terms of SER are given assuming that orthogonal space-time block coding (OSTBC) [10] is applied to the data symbols on a per-carrier basis. The performance of our proposal is compared with the technique presented recently in [11], where a multi-user OFDM system is considered leading to a similar notation to ours. The paper is organized as follows. The system and signal models are presented in Section II., whereas Section III. addresses the CFO correction problem. The CRB is deduced in Section IV., and some simulation results and conclusions are drawn in Section V. II. S YSTEM AND S IGNAL M ODELS Consider the transmission through a frequency-selective MIMO channel with nT and nR transmit and receive antennas, respectively, where h(q,p) (n) is the impulse response between the pth transmit and the qth receive antennas. Let us assume a N -carriers OFDM modulation with carrier index k =


cordingly, the received signal sample in the frequency domain corresponding to the qth receive antenna, lth OFDM symbol, and kth subcarrier can be approximated as n T (q) (q,p) (p) H [k]S [k, l] ej(αϕq +lϕq ) +W (q) [k, l], XR [k, l]

phase φq n

αϕ q + lϕ q

ϕ q = ( N + L)φq

p=1

n ( l = 0)

N+L ( l = 1)

( l = 2)

( l = 3)

Figure 1: Approximation of the phase shift for residual CFO. 0, . . . , N −1 and that the data frame is composed of nL OFDM symbols, where l is the symbol index (l = 0, . . . , nL − 1). The complex symbol modulating the kth subcarrier at the pth transmit antenna during the lth OFDM symbol is denoted by S (p) [k, l]. These symbols can be data or pilots. The data symbols can be obtained, for example, from the application of an OSTBC on a per-carrier basis to the information symbols streams, whereas the pilot carriers are defined in the standard. We will assume that nP pilots are available at the carrier positions k1 , . . . , knP at each OFDM symbol. The complete data frame in the time domain for the pth transnL −1 (p) (p) xT (n − mit antenna can then be written as xT (n) = l=0 (p) l(N + L), l), where xT (n, l) (n = −L, . . . , N − 1) contains the L samples of the CP and the N samples of the payload in the lth OFDM symbol for the pth transmit antenna. Accordingly, the signal at the qth receive antenna is expressed as n T (q) (p) (q,p) xT (n) ∗ h (n) ejφq n + w(q) (n), (1) xR (n) = p=1

where w(q) (n) represents the additive white Gaussian noise (AWGN) and φq is the incremental phase between consecutive samples due to the CFO. This model assumes that the CFO can be different at each receive antenna due to, for example, having a different local oscillator at each RF chain; however, the model can be directly extended to the case of having a common CFO, as will be explained later. Usually, the channel estimation is performed using the preamble, which also allows for a first estimation and correction of the CFO. Although this correction can be good enough so as to assume that no ICI is present in the resulting signals, a residual CFO always remains producing a non-negligible phase shift between consecutive OFDM symbols that can deteriorate the signal detection very importantly, as will be shown later. In the following, we address the problem of correcting this effect using the pilot carriers and assuming that the channel is known (note that the study of the impact of using imperfect channel estimates is out of the scope of this paper). Since the incremental phase due to the residual CFO is expected to be low, the temporal samples corresponding to the same OFDM symbol can be approximated to share a common phase shift, as shown in Fig. 1. There, ϕq = (N + L)φq is the incremental phase between consecutive OFDM symbols. Ac-

(2) where H (q,p) [k] is the channel response at the kth carrier, α = N/(2(N + L)) accounts for the phase shift at the first OFDM symbol (see l = 0 in Fig. 1), and W (q) [k, l] is the AWGN. III. P ROBLEM S TATEMENT AND S OLUTION In this section, the estimation and correction of the residual CFO is addressed exploiting the received samples at the pilot carriers for the nL OFDM symbols in the frame. First, the ML estimation is formulated in terms of the phase shift ϕq and, then, some comments on efficient implementations are given. A.

Maximum Likelihood Estimation and CFO Correction

Before deriving the expression of the ML estimator, a conve(q) nient notation is introduced. Let us define xR ∈ CnL nP ×1 as the column vector containing all the frequency samples received at all the pilot subcarriers and the qth antenna for the nL OFDM symbols in the data frame, i.e., T (q) (q) (q) (q) xR = xR [0]T · · · xR [nL − 1]T , where xR [l] = (q) T (q) XR [k1 , l] · · · XR [knP , l] . Based on this definition, the following notation is obtained from (2): (q)

xR exp(jαϕq )T(q) ϕ(q) + w(q) ∈ CnL nP ×1 , (3) where the noise vector is w(q) = w(q) [0]T · · · w(q) [nL − T T 1]T and w(q) [l] = W (q) [k1 , l] · · · W (q) [knP , l] . T(q) ∈ CnL nP ×nL is a block-diagonal matrix, where its nL blocks are nL −1 with the noise-free received the column vectors {t(q) [l]}l=0 pilot carriers at each OFDM symbol assuming no CFO; i.e., t(q) [l] is vector with nP components such that t(q) [l] m = nT (q,p) [km ]S (p) [km , l], m = 1, . . . , nP . Finally, the p=1 H (q) vector ϕ ∈ CnL ×1 contains the phase shifts at the nL received OFDM symbols at the qth antenna, i.e., ϕ(q) = T 1 exp(jϕq ) · · · exp(j(nL − 1)ϕq ) . The ML criterion leads to a non-linear least squares estimation problem, which is formulated as 2 (q) (4) ϕ q = arg min xR − exp(jαϕq )T(q) ϕ(q) ϕq (q)H (q) (q)H = arg min xR xR − exp(jαϕq )xR T(q) ϕ(q) (5) ϕq

(q) −e−jαϕq ϕ(q)H T(q)H xR + ϕ(q)H T(q)H T(q) ϕ(q)

(q) (6) = arg max Re exp(−jαϕq )ϕ(q)H T(q)H xR , ϕq

where the last equality follows from the fact that (q)H (q) xR xR does not depend on ϕq and that T(q)H T(q) is a diagonal positive semidefinite matrix (consequently, nL −1 (q) t [l]2 and does not ϕ(q)H T(q)H T(q) ϕ(q) = l=0 depend on ϕq either).


As explained, this signal model corresponds to having a different CFO at each antenna. In case that a joint first CFO correction is performed among all the receive antennas, then the resulting residual CFO would be the same (in the following, this common CFO will be represented by ϕ). In this situation, the previous estimator can be applied directly by using the following notation resulting from the columnwise stacking of all the samples received at all the antennas: xR exp(jαϕ)Tϕ + w ∈ CnR nL nP ×1 ,

(7)

where ϕ = [ 1 exp(jϕ) · · · exp(j(nL − 1)ϕ) ]T , T (1)T (n )T T xR = xR · · · xR R , w = w(1)T · · · w(nR )T , T and, finally, T = T(1)T · · · T(nR )T . Note that in this signal model, it is still true that T fulfills that TH T = nR (q)H (q) T is diagonal and positive semidefinite; thus, q=1 T the estimation problem can be written as (8) ϕ = arg max Re exp(−jαϕ)ϕH TH xR . ϕ

Once the phase shift has been estimated, it can be corrected by counteracting the phase variation in the received signals in the time or frequency domains. The time domain cor(q) (q) rection is given by x R (n) = xR (n) exp(−j φq n), where q /(N + L); whereas the frequency domain correction φq = ϕ (q) [k, l] = X (q) [k, l] exp(−j(αϕ is described by X q + lϕ q )). R R Note that after the CFO correction in the time domain, the OFDM demodulation based on the FFT has to be applied to (q) x R (n) followed by the convenient detection stage over the frequency samples (for example, if OSTBC has been employed at the transmitter, the corresponding decoding has to be applied after the FFT on a per carrier basis). On the other hand, the CFO correction in the frequency domain does not require new recomputations of the FFT, thus, reducing the complexity, which may be a critical point in real-time applications. Note, however, that the time domain correction is expected to perform better since it can also correct the ICI effects when the residual CFO produces a non-negligible phase shift between the samples within a single OFDM symbol (see Fig. 1). In [11], the residual CFO correction in a multi-user OFDM system is considered, leading to the same notation as ours since the same assumptions concerning the channel knowledge and the presence of no ICI are taken. However, there, an additional approximation is applied by considering that both the phase ϕq and the number of OFDM symbols in the frame nL are low, implying that exp(j(α + l)ϕq ) 1 + j(α + l)ϕq , l = 0, . . . , nL − 1. Thanks to this, a closed-form linear estimator for ϕq is obtained. In the simulations section, a comparison between our proposal and that shown in [11] will be shown. B.

1)

Grid Search

The solution to the estimation problem (6) can be found by an exhaustive search over the margin of possibly values of phase shifts [−ϕmax , ϕmax ]. If M + 1 points are to be calculated in this margin, then, the value of the function (q) exp(−jαϕq )ϕ(q)H T(q)H xR has to be calculated at ϕq = max , n = 0, . . . , M . −ϕmax + n 2ϕM 2)

Truncated Discrete Fourier Transform (DFT)

The previous strategy can be simplified in terms of computational load by taking advantage of the particular structure of the expression to be maximized, i.e., of (q) exp(−jαϕq )ϕ(q)H T(q)H xR . Let us define the vector r(q) = (q) (q) (q) T(q)H xR = [r0 · · · rnL −1 ]T ∈ CnL ×1 . Using this definition, the function to be maximized can be written as n L −1

(q) (q) Re e−jαϕq ϕ(q)H T(q)H xR = Re e−jαϕq rl e−jlϕq . l=0

Note that the sum in the previous expression has the same structure as a Fourier transform; thus, the existing efficient DFT algorithms can be applied. Let us assume that we apply a Kpoints DFT, then, the expression of the sum is given by n n L −1 L −1 (q) −jlk 2π (q) −jlϕ q K = rl e rl e , (9) 2π l=0

l=0

ϕq =k

K

which corresponds to the evaluation of the sum at the phase shift k 2π K . If the search margin for the phase is [−ϕmax , ϕmax ], then the values of the index k in the DFT should be {0, 1, . . . , kmax , K − 1, K − 2, . . . , K − kmax }, where ϕmax = kmax 2π/K. It should be remarked that in this DFT, only nL inputs out of K are active and 2kmax + 1 outputs out of K have to be calculated. That means that if an efficient implementation of the DFT, by means of the butterfly architecture commonly used for FFT, is exploited, many of the sub-FFT blocks will be inactive and, therefore, the computational load can be decreased importantly. Besides, the sub-FFT blocks that have to be used depend only on K, nL , and kmax , and, therefore, the optimum architecture could be found once off-line. Fig. 2 shows the computational cost of the ML estimator in terms of number of complex multiplications as a function of nL assuming a maximum phase shift ϕmax = 28.8o and a estimation resolution of ∆ϕ = 2π/K, K = 214 . In addition to the grid search and the truncated FFT approaches presented previously, we add the full FFT technique, in which all the sub-FFT blocks are active, although many of them are not necessary. The main conclusion is that the technique requiring the lowest computational load is the truncated FFT approach.

Implementation Aspects

One of the most critical points when implementing this kind of estimation strategies in a realistic deployment is the computational complexity. In this subsection, two implementation proposals are detailed to solve the estimation problem described by the maximization in (6) (note that the case of having a common CFO at all the receive antennas is equivalent).

IV. C RAMER -R AO B OUND The performance evaluation of the ML estimator can be carried out in terms of the mean square error (MSE). In this section, we provide some simulations results showing the mean MSE and its comparison with the CRB [9], which allows to get an insight into the theoretical absolute performance of the estimator.


Computational Cost for the ML Estimator

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Figure 2: Computational cost of the grid search, truncated FFT, and full FFT implementation approaches (maximum phase: ϕmax =28.8o , estimation resolution: ∆ϕ = 2π/214 ).

Figure 3: Comparison between the actual MSE of the ML estimator and the CRB (CFO=2 kHz, nT = 4). OSTBC decoding without CFO correction

OSTBC decoding with CFO correction

1.5

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(q) xR

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The CRB, i.e., the lower bound on the achievable MSE for any unbiased estimator, can be derived by calculating (q) d2 Λ(ϕq ; xR )/dϕ2q and computing its expectation along the re(q)

ceived samples xR . Using similar steps to those used to obtain (6), the second derivative can be shown to be: (q) d2 Λ(ϕq ; xR ) 2 (q) H jαϕq (q) (q) = − Re x e T Aϕ , (10) R dϕ2q σ2 where A = diag{α2 , (α+1)2 , . . . , (α+nL −1)2 } ∈ RnL ×nL . (q) The expectation with respect to xR is then calculated as: (q) d2 Λ(ϕq ; xR ) 2 (q) H jαϕq (q) (q) E Re E x T Aϕ = − e R dϕ2q σ2

H H 2 = − 2 Re ϕ(q) T(q) T(q) Aϕ(q) σ nL −1 2 (α + l)2 t(q) [l]2 , (11) =− 2 σ l=0

obtaining the following closed form expression for the CRB: 1 σ2 = n −1 CRB = − . (q) L 2 t(q) [l]2 d2 Λ(ϕq ;xR ) 2 (α + l) l=0 E dϕ2 q

From the CRB expression it can be noted that the minimum achievable variance does not depend on the actual value of the parameter ϕq . Note that this is true whenever the phase shift due to the CFO can be assumed to be constant during the time duration of a single OFDM symbol, as shown in Fig. 1.

1

Quadrature component (Q)

Quadrature component (Q)

1

(q)

The log-likelihood function is defined as Λ(ϕq ; xR ) =

(q) (q) log p(xR ; ϕq ) , where p(xR ; ϕq ) is the probability density function corresponding to the signal model (3) assuming AWGN with variance σ 2 : Λ(ϕq ; xR ) = −nP nL log(πσ 2 ) −

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Figure 4: Signal constellation at the output of the Alamouti’s OSTBC decoder without and with residual CFO correction (CFO=2 kHz, nT = 2, nR = 1). Fig. 3 shows the mean MSE for the ML estimator and the CRB, concluding that the ML performs efficiently, even for low values of the SNR. The simulation parameters correspond to IEEE 802.11n [12] (N = 64, L = 16, nP = 4, and a sampling frequency of 20 MHz). The channel is a single realization of a Rayleigh channel with 16 taps and 4 transmit antennas, considering an independent CFO at each receive antenna. Different sizes for the data frame have been taken, showing that, as nL increases, the quality of the estimator improves, although this improvement tends to saturate. The CFO in this case corresponds to 2 kHz, which is equivalent to a phase shift of ϕq = 2.88o between consecutive OFDM symbols. As commented previously, the quality of the estimator does not depend on the actual value of ϕq , whenever this phase shift is low enough so that the approximation shown in Fig. 1 is accurate. This effect will be shown in the next section based on simulations for the evaluation of the SER for different values of the residual CFO. V. S IMULATION R ESULTS AND C ONCLUSIONS In this section, we provide some simulation results to evaluate the benefits of using the residual CFO correction in terms of SER when the data carriers are encoded using OSTBC [10] and taking the same system parameters as before. Fig. 4 shows the constellation for QPSK symbols when Alamouti’s OSTBC (nT = 2, nR = 1) is applied to the data carriers for the simple channel h(q,p) (n) = δ(n). This constellation is observed after the OSTBC decoder and before applying hard decisions. The figure on the left corresponds to a CFO


SER vs SNR (Alamouti−16QAM with Rayleigh Channel)

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Figure 5: SER using Alamouti OSTBC (nT =2, nR =1). OFDM frames of 24 symbols (nL = 8 of them for CFO estimation). Performance of the Estimators

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CFO=.5KHz ML estimator CFO=.5KHz estimator [9] CFO=1KHz ML estimator CFO=1KHz estimator [9] CFO=1KHz ML estimator CFO=1KHz estimator [9] CFO=2KHz ML estimator CFO=2KHz estimator [9]

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Figure 6: Mean MSE for the estimator proposed in this paper and the one described in [11]. of 2 kHZ, i.e., when the phase shift is ϕq = 2.88o . Note that even in this case, when the phase shift is very low, the residual CFO dramatically destroys the constellation producing a high SER. The constellation can be recovered by using the presented technique, resulting in the corrected constellation on the right. Fig. 5 shows the average SER vs. SNR for Alamouti’s OSTBC, 16QAM, a Rayleigh distributed channel with 16 taps, and different values of CFO. The SER has been evaluated for the case of performing no CFO correction and when carrying out such correction both in the time and frequency domains. The data frames are composed of 24 OFDM symbols, although only nL = 8 of them are used for the CFO estimation. As can be concluded from the figure, if no correction is performed, the SER rapidly deteriorates, even for very low values of the residual CFO. The correction performs very well, showing that for a broad range of CFO values, the SER performances are close to the case of having no CFO. This effect was predicted by the CRB, which stated that the estimation MSE was independent of the actual value of the CFO. Even for the concrete case of a CFO of 20 kHz, which is equivalent to non-negligible ICI and a phase shift of ϕq = 28.8o , the time-domain correction provides a SER very close to the case of having no CFO. However, the

frequency-domain correction does not produce so good results since the ICI effects are not counteracted in the correction. Finally, in Fig. 6, the MSE of our proposed estimator is compared with [11] (the technique proposed in [11] is summarized at the end of subsection III.A. in this paper). From the simulations it is deduced that as the number of OFDM symbols nL increases, the performance of our technique always improves, whereas the MSE corresponding to [11] begins to increase since the approximation exp(j(α + l)ϕq ) 1 + j(α + l)ϕq , l = 0, . . . , nL − 1 is less accurate. As a conclusion, a technique for correcting the residual CFO in a MIMO-OFDM system has been presented using the pilot carriers. Some efficient implementations have been proposed and the asymptotic efficiency of the estimator has been evaluated using the CRB. Finally, some simulations have shown the important performance gains even for high values of the CFO. R EFERENCES [1] R. Narasimhan, “Performance of Diversity Schemes for OFDM Systems with Frequency Offset, Phase Noise, and Channel Estimation Errors,” IEEE Trans. Commun., vol. 50, no. 10, pp. 1561–1565, October 2002. [2] J. Lei and T.-S. Ng, “A Consistent OFDM Carrier Frequency Offset Estimator Based on Distinctively Spaced Pilot Tones,” IEEE Trans. Wireless Commun., vol. 3, no. 2, pp. 588–599, March 2004. [3] J.-J. van de Beek, M. Sandell, and P. O. Börjesson, “ML Estimation of Time and Frequency Offset in OFDM Systems,” IEEE Trans. Signal Process., vol. 45, no. 7, pp. 1800–1805, July 1997. [4] T. Roman, A. Richter, and V. Koivunen, “Blind CFO Estimation for OFDM with Constant Modulus Constellations: Performance Bounds and Algorithms,” in Proc. European Signal Processing Conference (EUSIPCO’06), September 2006. [5] M. Guainazzo, M. Gandetto, C. Sacchi, and C. Regazzoni, “Maximum Likelihood Estimation of Carrier Offset in a Variable Bit Rate Orthogonal Multicarrier CDMA,” in Proc. Int. Symposium on Image and Signal Processing Analysis (ISPA’03), vol. 2, September 2003, pp. 1181–1185. [6] R. Ambati and U. Tureli, “Experimental Studies on an OFDM Carrier Frequency Offset Estimator,” in Proc. IEEE International Conference on Communications (ICC’03), vol. 3, May 2003, pp. 2056–2060. [7] J. Li, G. Liao, and Q. Guo, “MIMO-OFDM Channel Estimation in the Presence of Carrier Frequency Offset,” EURASIP Journal on Applied Signal Proc., vol. 2005, no. 4, pp. 525–531, 2005. [8] T. Roman, “Advanced Receiver Structures for Mobile MIMO Multicarrier Communication Systems,” Ph.D. dissertation, Helsinki University of Technology, April 2006. [9] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice Hall International, 1993. [10] V. Tarokh, H. Jafharkani, and A. R. Calderbank, “Space-Time Block Codes from Orthogonal Designs,” IEEE Trans. Trans. Inf. Theory, vol. 45, no. 5, pp. 1456–1467, July 1999. [11] L. Häring, S. Bieder, and A. Czylwik, “Residual Carrier and Sampling Frequency Synchronization in Multiuser OFDM Systems,” in Proc. IEEE Vehicular Technology Conference (VTC’06), vol. 4, May 2006, pp. 1937–1941. [12] E. W. C. P. (EWC), “Interoperability PHY specification v1.27,” December 2005.