Maximum-Likelihood Carrier Frequency Offset Estimation for OFDM ...

Maximum-Likelihood Carrier Frequency Offset Estimation for OFDM Systems in Fading Channels Hao Zhou, Amaresh V. Malipatil and Yih-Fang Huang Department of Electrical Engineering University of Notre Dame, IN 46556 Email: {hzhou,amalipat,huang}@nd.edu Abstract— Carrier frequency offset (CFO) is a major contributor to the inter-carrier interference in OFDM systems. This effect becomes more severe when compounded by the presence of Doppler fading in wireless channels. This paper presents two maximum-likelihood CFO estimation schemes, one in frequency domain and another in time-domain, both under Doppler fading. The properties of the estimators are analyzed and simulation results showing the performance gain of the proposed estimators over other estimation schemes are presented. Apart from improved accuracy, the time-domain maximum likelihood estimator features significantly reduced complexity.

I. I NTRODUCTION Orthogonal frequency-division multiplexing (OFDM) has gained a great deal of popularity lately due to its high spectral efficiency and robustness to multipath. In the last few years, OFDM has been employed in various commercial applications that include wireless local area networks (IEEE 802.11a and HIPERLAN/2), terrestrial digital audio broadcasting (DABT), and terrestrial digital video broadcasting (DVB-T) [1]. It is well known, however, that the performance of OFDM systems is very sensitive to frequency synchronization errors [2], [3]. The local oscillators at the transmitter and receiver may not be synchronized and that results in residual carrierfrequency offset (CFO) at the receiver after down-conversion. Doppler spread, which is present in mobile environments due to changing channel conditions between the transmitter and the receiver, also contributes to the carrier offset. The CFO introduces inter-carrier interference (ICI) which destroys the orthogonality between the subcarriers and attenuates the desired signal, reducing the effective signal-to-noise ratio (SNR) [2]. This results in degraded system performance. Many methods have been proposed in the literature to estimate and compensate for the CFO, see, e.g., [4 – 8]. Most of the conventional methods are based on a frequency-domain (FD) approach, assuming a static channel to estimate the CFO with the phase difference between two successive symbols [4], [5], [6]. The estimates over all the pilot-subcarriers are averaged (or weighted) using the frequency-domain channel estimates to improve the estimation accuracy [6]. In the presence of Doppler fading, the estimation accuracy degrades for these estimators. This paper presents two ML approaches, a frequencydomain approach (FD-MLE) and a time-domain approach (TD-MLE), for fine carrier synchronization under fading channel assumptions. Both of these estimators use the knowledge of

channel autocorrelation and the statistics of the fading process to provide better accuracy. The proposed MLE is shown to be unbiased at high SNRs. Simulation results are provided to show that the FD-MLE and TD-MLE have the same performance and both are superior to other CFO estimators, such as the weighted least-squares (WLS) scheme in [6]. Moreover, since the number of dominant multipaths is normally much smaller than the number of subcarriers, the complexity of TDMLE is considerably smaller than all the frequency-domain based techniques. The system model and the notation used are presented in Section II. In Section III, the FD-MLE is first derived under the fading scenario, afterward the TD-MLE is proposed and their performance and complexity issues are analyzed. In Section IV, simulations comparing the performance of the proposed MLEs and other CFO estimators are presented. The concluding remarks are given in Section V. II. SYSTEM MODEL In this paper, scalars are denoted by lower-case letters, timedomain vectors by underlined lower-case letters, frequencydomain vectors by underlined upper-case letter and matrices by bold upper-case letters. At the transmitter, the ith OFDM symbol is generated from the frequency domain symbols X i = [Xi,0 , Xi,1 , · · · , Xi,N −1 ]T as follows.

1 WH X i (1) N where, N is the FFT length, 1/T is the subcarrier spacing, xi = [xi,0 , xi,1 , · · · , xi,N −1 ]T and W is the N ×N FFT matrix with [W]kn = ej2πkn/N . A cyclic prefix (CP) of length Ng is used to avoid the inter-symbol interference (ISI) and retain the orthogonality of the subcarriers. The received signal is given by xi =

ri,n = hi,n ∗ xi,n + zi,n

(2)

where, hi = [hi,0 , hi,1 , · · · hi,N −1 ]T is wide sense stationary uncorrelated scattering (WSSUS) Rayleigh faded channel. The autocovariance of the multipath channel which also represents the power delay profile is denoted as Rhh = diag{σh2 0 , σh2 1 , · · · σh2 N −1 }. The average channel power is normalized to unity, E[hiH hi ] = trace(Rhh ) = 1. The fading process is assumed to follow Jakes model [9]. The autocorrelation of the fading process is E[hi+1 hH i ] = aRhh , where

the fading coefficient a = J0 (2πFd Ts ), J0 (·) is the zerothorder Bessel function of the first kind and Fd is the maximum Doppler frequency. Ts = (1 + Ng /N )T is the total OFDM symbol duration including the CP. z i is the complex additive white Gaussian noise, i.e., z i ∼ CN (0, σ 2 I). In the presence of a normalized carrier frequency offset # = ∆f T , where ∆f is the actual carrier frequency offset, the received signal is given by yi,n = ri,n · ej

2π" N [(i−1)(N +Ng )+Ng +n]

,

(3)

After down-conversion and sampling, the receiver discards the cyclic prefix and performs the FFT operation. Y i = Wy i = WDi ri ,

2π" N [(i−1)(N +Ng )+Ng +n]

ˆ H i+1

ˆ + X−1 D X(H = ejα# H i+1 i i+1 − H i ) + ! "# $ "

I

"

X−1 Di+1 (Z i+1 − Z i ) ! "# $

, n = 0, 1, · · · , N − 1 (5)

II

jα# ˆ ˆ ˆ Hi E[H i+1 |H i ] = e

CHˆ

i+1

ˆ |H i

" 1 WDi WH Wri = Di [Xi H i + Z i ] !N "# $

(6)

" Di

where, Xi is the diagonal data matrix with diagonal entries [Xi ]k,k = Xi,k , H i = Whi is the CTF and Z i = Wz i is the frequency domain noise. Note that, since WH W = WWH = N I, Z i ∼ CN (0, N σ 2 I). III. M AXIMUM LIKELIHOOD CFO ESTIMATION In OFDM, frequency-selective wideband channel is decomposed into a large number (N ) of narrow-band flatfading channels. Accordingly, a single-tap frequency-domain equalizer on each subcarrier can be used to compensate for the effect of the CTF. Hence, estimation of the CTF is crucial for coherent demodulation in OFDM systems. There are numerous contributions towards pilot-assisted, decision-directed and blind techniques for CTF estimation . Here, a pilot-assisted scheme is assumed; extension to a decision-directed scheme is straight forward. The simplest CTF estimation technique is −1 ˆ Y. the least-squares (LS) estimate H LS = X A. Frequency Domain MLE From (6), the outputs of the FFT for the ith and the i + 1th symbols are as follows. Yi Y i+1

"

"

= Di Xi H i + Di Z i "

(7) "

= Di+1 Xi+1 H i+1 + Di+1 Z i+1

(8)

Assuming that the OFDM symbols are repeated, i.e., Xi+1 = " Xi = X, and noting that Di+1 " = ejα# Di , where α = 2π(N + Ng )/N , Y i+1 can be expressed in terms of Y i as follows. Y i+1

"

= ejα# Y i + ejα# Di X(H i+1 − H i ) + "

Di+1 (Z i+1 − Z i )

(9)

(11)

= V ar[I] + V ar[II] =

Furthermore, Y i = WDi ri =

(10)

To obtain the MLE for the CFO, the conditional probability ˆ ˆ density function f (H i+1 ; # |H i ) is required. It is evident ˆ ˆ from (10) that f (H i+1 ; # |H i ) is Gaussian with mean and covariance given below.

(4)

where Di is the N ×N diagonal matrix with diagonal elements given by [Di ]n,n = ej

−1 ˆ ˆ = X−1 Y and H Y i+1 , Using the LS estimates, H i i i+1 = X we get,

≈

"

"

2(1 − a)X−1 D1 XRHH XH [D1 ]H [X−1 ]H + 2N σ 2 X−1 [X−1 ]H 2β γ(1 − a) ( RHH + I) = CHˆ γ β

(12) (13) "

where (12) follows from observing that Di+1 " = ejα#i D1 ; (13) is obtained under the assumptions that X−1 [X−1 ]H ≈ E{1/|X|2 }I and β = E{|X|2 }E{1/|X|2 } is a constellation dependent constant. It is also assumed here that during closedloop operation, the CFO estimate is fed back for correction and " hence D1 ≈ I. This approximation removes the dependence of CHˆ on # and hence an approximation of MLE can be obtained easily. The index i+1 in the covariance matrix is omitted since it is independent of i. The FD-MLE is obtained by setting the first derivative of the log-likelihood function to zero. ∂ ˆ ˆ ln{f (H (14) i+1 ; # |H i )} = 0 ∂# It is easily seen that the MLE is given by 1 −1 ˆ ˆ ˆ H C−1 H ˆH # #ˆF D−M LE = (# [H i i+1 ] − [H i+1 CH ˆ ˆ H i ]) (15) H 2α Note that CHˆ is time-invariant and hence the inverse can be computed a priori and stored. Also, if the channel is timeinvariant, a = 1 and the FD-MLE simplifies to the scheme proposed by Moose in [5]. B. Time Domain MLE The LS estimator is very simple and computationally less complex. But it is well known that the performance of LS estimator is not satisfactory. Also it does not make use of any available knowledge of the channel statistics. Much better performance can be obtained using improved estimation techniques such as the MMSE estimator and modified LS/MMSE approaches [1], [10], [11], [12]. Here, the optimal low-rank MMSE channel estimation scheme presented in [11] is considered. The rank reduction is achieved using the singular-value decomposition (SVD) of the autocovariance of the CTF. RHH = UΛUH

(16)

−4

10

WLS FD MLE (Moose MLE) TD MLE

Q −5

10

U

0

CFO Estimation Error Variance

FFT

received signal

0

CFO Estimation

Fig. 1.

−6

10

−7

10

OFDM Receiver structure for Channel and CFO estimation −8

10

where U is the unitary matrix consisting of the singular vectors of RHH as the columns. The receiver architecture is shown in Fig. 1. The optimal rank-p frequency-domain CTF estimate is given by H ˆ ˆ H M M SE = U∆U H LS

(17)

%

k = 1, 2, · · · , p k = p + 1, · · · , N

λk λk +β/γ ,

0,

(18)

2

} is the average SNR per subcarrier. Since where γ = E{|X| N σ2 ˆ , yields a U is unitary, the intermediate result in (17), UH H LS time-domain estimate with its components uncorrelated. Apart from providing the MMSE estimate, the diagonal matrix ∆p also truncates the time-domain channel response to retain only the p most dominant multipaths. When CFO is present, each multipath of this time-domain channel estimate undergoes a continuous phase rotation as time progresses. In the following, a maximum-likelihood CFO estimator that uses the phase difference between channel estimates of two successive OFDM symbols is proposed. The length-p time-domain CIR is obtained from the LS estimate as follows. H ˆ ˆ h i+1 = ∆p Up H i+1

(19)

where the p × p matrix ∆p is formed from the first p rows and columns of ∆ and the N × p matrix Up is formed from the first p columns of U. Analogous to (10) we have, ˆ h i+1

ˆ + ∆p UH X−1 D X(H = ejα# h p i+1 i i+1 − H i ) + ! "# $ "

"

(20)

II

Accordingly, it can be ˆ ejα# , C ˆ ), where C ˆ CN (h i h h

ˆ deduced that h i+1 H = ∆p Up CHˆ Up ∆H p

10

15

20

25

30

35

SNR

Fig. 2. Performance comparison of various CFO estimation schemes under static channel conditions

2β 2 γ(1−a) Λp + I). Note γ ∆p ( β while CHˆ is not. Hence, the

that Chˆ is diagonal and real TD-MLE is given by

1 ˆ H −1 ˆ # [h C (21) i ˆ hi+1 ] h α To determine whether the estimator is unbiased consider the following equation. ˆ jα# + w ˆ h i+1 = hi e i+1

(22)

where, wi+1 ∼ CN (0, Chˆ ). Under the assumptions of small CFO and high SNR, the estimation error can be approximated by & ' ˆ jα# H −1 ˆ 1 Im[(hi e ) Chˆ hi+1 ] #ˆ − # ≈ ˆ ejα# )H C−1 h ˆ ] α Re[(h i

≈

1 α

&

ˆe Im[(h i

)

jα# H

i+1 ˆ h ' −1 Chˆ wi+1 ]

ˆ H C−1 h ˆ h i i ˆ h

(23)

It is evident from the above equation that the mean of the ˆ ] = 0. Thus, the proposed estimator estimation error E[ˆ # − #|h i is unbiased. The matrix Chˆ provides weighting based on the knowledge of the channel statistics and the fading process. Also, since Chˆ is diagonal, the inversion is a trivial operation. The complexity of the TD-MLE is O(p) compared to O(N 2 ) of the FD-MLE and O(N ) of other frequencydomain approaches such as those presented in [4], [5], [6]. Moreover, since the dominant paths are chosen using the SVD, the proposed CFO estimator is reliable even under low SNRs. IV. S IMULATION R ESULTS

I

−1 Di+1 (Z i+1 − Z i ) ∆p UH p X ! "# $

5

#ˆT D−M LE =

where, ∆ is a diagonal matrix with diagonal elements [∆]k,k =

0

∼ =

Simulation experiments were performed with the following system parameters: number of subcarriers N = 512, length of cyclic prefix Ng = 52, normalized CFO (frequency deviation / subcarrier spacing) fo = 0.01. The Jakes model was employed to simulate the multipath channel with L = 12 dominant components under static and fading scenarios. The variance of the

−4

−3

10

WLS Moose MLE FD MLE Fad TD MLE Fad

CFO Estimation Error Variance

CFO Estimation Error Variance

10

−5

10

−6

10

−7

10

WLS Moose MLE FD−MLE TD MLE

−4

10

−5

0

5

10

15

20

25

30

35

SNR

10

0

5

10

15

20

25

30

35

SNR

Fig. 3. Performance comparison of various CFO estimation schemes in slow fading channel. Normalized Doppler = 0.005

Fig. 4. Performance comparison of various CFO estimation schemes in fast fading channel. Normalized Doppler = 0.05

CFO estimation error is calculated for 400 OFDM symbols and the results are averaged over 100 monte-carlo simulations for each SNR value. The performance of the proposed estimators is compared with the MLE scheme proposed by Moose [5] (which is a special case of our proposed FD-MLE scheme for static channels) and frequency-domain weighted least squares (WLS) approach presented in [6]. The CFO estimation error variance for different SNRs under time-invariant channel is shown in Fig. 2. In the static channel case, all four schemes performed almost identically, except in the low SNR region, in which the proposed TD-MLE performed the best. Note that, in this case, the proposed FDMLE scheme reduces to the Moose MLE scheme [5], thus they have identical performance. In the case of fading channels, results are shown in Fig. 3 and Fig. 4 with normalized Doppler frequency of 0.005 (slow fading) and 0.05 (fast fading), respectively. Fig. 3 shows that the proposed FD-MLE and the TD-MLE schemes yield comparable results, and they both render better performance compared to the Moose MLE and the WLS approach under all SNRs, and especially in high SNR regions. In the fast fading scenario (Fig. 4), all schemes appear to have a performance floor in moderate and high SNR regions, and the proposed MLEs have lower error floor than the Moose MLE and the WLS scheme. The performance improvement is because the proposed estimators take fading into consideration.

take the fading factor into consideration. In addition to better estimation accuracy, the TD-MLE scheme has the added advantage of reduced complexity since only the most dominant multipaths are used in CFO estimation. Hence, the proposed estimators can be used to estimate the CFO more accurately in fading channels and reduce the ICI in OFDM systems thereby improving the system performance.

V. C ONCLUSION Two maximum likelihood estimators for CFO estimation in fading channel conditions were proposed in this paper. The fading coefficient is considered as a factor in these estimator design. The static channel was discussed as a special case with the fading coefficient a = 1. Simulations showed that the proposed schemes perform comparably with the Moose MLE and the WLS scheme under static channel conditions. Under fading conditions, the proposed estimators are better than the conventional schemes, because the proposed schemes

R EFERENCES [1] L. Hanzo, M. Munster, B. J. Choi and T. Keller, OFDM and MC-CDMA for Broadband Multi-User Communications, WLANs and Broadcasting, IEEE Press, 2003. [2] Jungwon Lee, Hui-Ling Lou, D. Toumpakaris, J. M. Cioffi, “Effect of carrier frequency offset on OFDM systems for multipath fading channels,” in Proc. IEEE GLOBECOMM’04, vol. 6, pp. 3721-3725, Nov 2004. [3] T. Pollet, M.V. Bladel, and M. Moeneclaey, “BER Sensitivity of OFDM Systems to Carrier Frequency Offset and Wiener Phase Noise,”, IEEE Trans. Comm.,, vol. 43, no. 2/3/4, pp. 191-193, Feb/Mar/Apr 1998. [4] S. Kapoor, D. J. Marchok and Y. F. Huang, “ Pilot Assisted Synchronization for Wireless OFDM Systems over Fast Time Varying Fading Channels,” IEEE VTC 98, 48th IEEE, vol. 3, pp. 2077 - 2080, May 1998. [5] Paul H. Moose, “ A Technique for Orthogonal Frequency Division Multiplexing Frequency Offset Correction,”, IEEE Trans. Comm.,, vol. 42, no. 10, pp. 2908-2914, Oct. 1994. [6] P. Y. Tsai, H. Y. Kang and T. D. Chiueh, “Joint Weighted Least-Squares Estimation of Carrier-Frequency Offset and Timing Offset for OFDM Systems Over Multipath Fading Channels,” in IEEE Trans. on Veh. Technol., vol. 54, no. 1, pp. 211-223, Jan 2005. [7] M. Sliskovic, “Carrier and sampling frequency offset estimation and correction in multicarrier systems,” in Global Telecommunications Conference, vol. 1, pp. 285-289, Nov. 2001. [8] A. J. Coulson, “Maximum likelihood synchronization for OFDM using a pilot symbol: algorithms,” in IEEE Journal on Selected Areas in Communications., vol. 19, no. 12, pp. 2486 - 2494, Dec. 2001. [9] Y. R. Zheng and C. Xiao, “Simulation Models with Correct Statistical Properties for Rayleigh Fading Channels,” in IEEE Transactions on Communications., vol. 51, no. 6, pp. 920 - 928, June 2003. [10] J.-J. van de Beek, O. Edfors, M. Sandell, S. K. Wilson and P. O. Borjesson, “On Channel Estimation in OFDM Systems” IEEE VTC 1995, July 1995, pp. 815-819. [11] O. Edfors, M. Sandell, J.-J. van de Beek, S. K. Wilson and P. O. Borjesson, “OFDM Channel Estimation by Singular Value Decomposition” IEEE Trans. Comm.,, vol. 46, no. 7, pp. 931-939, July 1998. [12] D. Schafhuber and G. Matz, “MMSE and Adaptive Prediction of TimeVarying Channels for OFDM Systems,” IEEE Trans. on Wireless Comm., vol. 4, no. 2, pp. 593-602, 2005.