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2784 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 10, OCTOBER 2006 Enhanced Carrier Frequency Offset Estimation for OFDM using Channel Side Information Defeng (David) Huang, Member, IEEE, and Khaled B. Letaief, Fellow, IEEE Abstract— Carrier frequency offset (CFO) in OFDM systems, by the oscillator discrepancy between the transmitter and the which induces the loss of orthogonality among OFDM sub- receiver and/or the Doppler phenomena due to the movement carriers, can result in signiﬁcant performance degradation. As a of the mobile terminals, can destroy the orthogonality among result, it needs to be estimated and compensated for. In this paper, we present a general CFO estimator based on the maximum OFDM sub-carriers and induce ICI (intercarrier interference). likelihood (ML) estimation criterion, with which CFO can be This results in signiﬁcant performance degradation and as a obtained using training OFDM symbols, pilot tones, null sub- result, it is very important that the CFO should be estimated carriers, or a combination of them. Using the proposed CFO and compensated for. A signiﬁcant amount of works [1]–[8] estimator, the performance of CFO estimation can be signiﬁcantly have been devoted to the CFO estimation issue. In [1]–[3], improved by taking advantage of the channel side information. In particular, using the channel statistics information, such CFO estimators were developed using a training sequence with performance improvement can be achieved for low SNR values two or multiple identical components. In [5], null sub-carriers and all SNR values over Rayleigh fading channels and Ricean in OFDM symbols were employed for CFO estimation, and fading channels, respectively. When the complete channel impulse a good way to allocate the null sub-carriers was proposed in response (CIR) information is available, simulation results will [6]. In [7], [8], the cyclic property of OFDM signals was used show that the performance improvement can be more than 6dB. To further demonstrate the capability of the proposed CFO for CFO estimation. estimator, we will consider an OFDM system using the signal All of the above works assumed that the channel side structure of the IEEE WLAN standard 802.11a. Compared with information1 is unavailable to the CFO estimator. However, it previous work using null sub-carriers alone, we will show that is known that the availability of the channel statistics informa- by taking advantage of the pilot tones, null sub-carriers, and tion is beneﬁcial to many channel estimation and space-time channel statistics, the performance of CFO estimation can be improved by about 2dB. coding schemes (See [9]–[11] and references therein). As a result, we can expect that the channel statistics information Index Terms— OFDM, carrier frequency offset, IEEE 802.11a, null sub-carrier. can also be useful for CFO estimation in OFDM. On the other hand, the channel impulse information (CIR) can be available to CFO estimation. As pointed out in [12], channel I. I NTRODUCTION estimation and synchronization can be achieved in an iterative way. In particular, we can expect that the performance of O FDM (Orthogonal Frequency Division Multiplexing) is an enabling technology for future broadband wireless communications due to its high spectral efﬁciency and ca- CFO estimation can be improved using the CIR information estimated in previous iterations. pability in combating multi-path propagations. Many broad- For ﬂat fading channels, some schemes have been proposed band wireless communication standards and proposals such as for CFO estimation with the aid of channel statistics [13]–[15]. 802.11a, DVB-T (Terrestrial Digital Video Broadcasting), and However, only a few works employed channel side informa- 802.15.3a (a proposal that uses the Ultrawideband Communi- tion for CFO estimation over frequency selective channels. cations Spectrum), have adopted OFDM as the key technology. In [16], the second order channel statistics were assumed to To achieve good performance in OFDM systems, the or- be known and the marginal likelihood function of the CFO thogonality among OFDM sub-carriers must be maintained was used for CFO estimation. But, as pointed out in [17], the [1]. However, carrier frequency offset (CFO), which is induced performance of the scheme proposed in [16] is poor due to the bias in the estimation. The use of a training sequence to Manuscript received July 26, 2004; revised February 23, 2006; accepted May 6, 2006. The editor coordinating the review of this paper and approving it jointly estimate the CIR and the CFO based on the maximum for publication is H. Li. This work was supported in part by the Hong Kong likelihood (ML) estimation criterion has also been proposed Telecom Institute of Information Technology. This paper was presented in in [17]. In this paper, we will demonstrate, however, that this part at the 2005 Asia-Paciﬁc Conference on Communications, Perth, Western Australia, October 3-5, 2005. can be only achieved under the assumption that the number of D. Huang was with the Center for Wireless Information Technology, distinct paths in the channel is known. The mismatch between Electrical and Electronic Engineering Department, The Hong Kong University the presumed and the actual number of distinct paths in the of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. He is now with the School of Electrical, Electronic and Computer Engineering, channel can signiﬁcantly impact system performance. The University of Western Australia, Crawley, WA 6009, Australia (e-mail: By approximating the received signal as a Gaussian random huangdf@ee.uwa.edu.au). variable, we propose in this paper a general CFO estimator K. B. Letaief is with the Center for Wireless Information Technology, Electronic and Computer Engineering Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (e-mail: 1 Throughout this paper, the channel side information includes two speciﬁc eekhaled@ee.ust.hk). cases: the channel statistics and the complete CIR (channel impulse response) Digital Object Identiﬁer 10.1109/TWC.2006.04507. information. 1536-1276/06$20.00 c 2006 IEEE HUANG and LETAIEF: ENHANCED CARRIER FREQUENCY OFFSET ESTIMATION FOR OFDM USING CHANNEL SIDE INFORMATION 2785 D Add s(n) it will be demonstrated that the complexity of the global Source IDFT Guard Interval search can be signiﬁcantly reduced by using FFT (Fast Fourier Transform) to simultaneously calculate the measures ~ f1 for all attempted CFO values in the search space. Therefore, complexity is not a critical issue for the proposed scheme especially considering the dramatic development of the VLSI Remove x technology. Guard DFT Sink Interval This paper is organized as follows. In Section II, the system model is described, along with the optimal ML CFO ~ f2 estimation using the Gaussian approximation. The extension of this CFO estimator to other scenarios is presented in Section Fig. 1. Block diagram of the OFDM system. III. Finally, simulation results and concluding remarks are given in Sections IV and V, respectively. based upon the ML criterion, with which the CFO can be II. S YSTEM M ODEL AND ML CFO E STIMATION estimated using training OFDM symbols, pilot tones, and/or We consider an OFDM system with N sub-carriers as null sub-carriers. Using the proposed CFO estimator, the shown in Fig. 1. The transmitted signal is given by performance of CFO estimation can be signiﬁcantly improved N −1 by taking advantage of the channel side information. 1 When the training OFDM symbol is available, simulation s(n) = √ dk ej2πnk/N , n = −Ng , · · · , N − 1 (1) N k=0 results will show that the performance of CFO estimation using the proposed scheme can be much better than that where dk is the data symbol at the kth sub-carrier, and Ng is proposed in [17] owing to the exploitation of the channel the length of the guard interval, which is assumed to be longer side information. For Rayleigh fading channels, simulation than that of the CIR. results will show that the performance of CFO estimation At the receiver, we assume that time synchronization is can be improved signiﬁcantly especially at low SNR values perfectly achieved. After sampling and guard interval removal, by taking advantage of the second order channel statistics. the received signal is given by We note here that the low SNR values are especially of N −1 1 practical interest when MC-CDMA [18] is used. When low x(n) = √ ej2πnφ H(k)dk ej2πnk/N + zn , order modulation, low coding rate, or high performance codes N k=0 such as LDPC codes and turbo codes [12] are used, the n = 0, · · · , N − 1 (2) SNR values of practical interest are also relatively low. For Ricean fading channels, we will show that the performance where φ is the CFO normalized by the sampling interval, H(k) of CFO estimation can be improved for all SNR values with is the channel frequency response at the kth sub-carrier, and the aid of the second order channel statistics. For a channel zn is the additive white Gaussian noise with zero mean and with log-normal distribution, simulation results will show that variance σ 2 . We note here that the CFO can be induced by the performance of CFO estimation can also be improved by the oscillators discrepancy between the transmitter and the using the second order channel statistics. Finally, when the receiver (i.e., f1 and f2 in Fig. 1 are not exactly the same). It complete CIR information is available, further performance can also be induced by the Doppler shift due to the movement improvement can be achieved. of the mobile terminals. Null sub-carriers and pilot tones in OFDM systems were We assume that there are L distinct paths in the CIR. h(l) originally employed to mitigate the adjacent channel inter- is used to denote the CIR at the lth path, and its delay is ference and achieve channel estimation, respectively [5]. By nl samples. The relationship between the channel frequency using null sub-carriers and pilot tones for CFO estimation, response and the CIR is then as follows: transmission efﬁciency can be improved because no extra T H = H(0), H(1), · · · , H(N − 1) training OFDM symbols are required. In this paper, we will show that null sub-carriers and pilot tones can also be used = FN ×L h (3) for CFO estimation due to the ﬂexibility of the proposed CFO where [.]T in the superscript denotes transpose, estimator. Speciﬁcally, we take the signal structure of the IEEE h = [h(0), h(1), · · · , h(L − 1)]T , and FN ×L is an N × L 802.11a as an example. We then demonstrate that by using matrix with [FN ×L ]n,l = √1 e−j2πnnl /N . pilot tones, null sub-carriers, and the second order channel N For convenience, we rewrite (2) into a vector form as statistics, the performance of CFO estimation can be improved follows: by about 2dB compared with the case where only null sub- T carriers are used. x = X(0), X(1), · · · , X(N − 1) In the proposed scheme, a global search is needed to = P(φ)WDH + z (4) estimate the CFO. As a result, the complexity of our scheme 1 √ ej2πnk/N , is higher than the one proposed in [16]. However, the per- where W is the IFFT matrix with W n,k = N formance of our proposed method is much better especially when a large CFO estimation range is required. Furthermore, z = [z0 , z1 , · · · , zN −1 ]T , 2786 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 10, OCTOBER 2006 P(φ) = diag 1, ej2πφ , · · · , ej2π(N −1)φ , Using (11) to ﬁnd φ in general induces great complexity due to the requirement of a global search. To reduce the search D = diag d0 , d1 , · · · , dN −1 , complexity, (11) can be written into the following form:3 and diag(.)is a diagonal matrix with the elements in the main N −1 diagonal given by (.). S(φ) = −ρ(0) + 2Re ρ(m)e−j2πmφ (12) By substituting (3) into (4), we have m=0 x = P(φ)WDFN ×L h + z. (5) where N −1 As pointed out in [19], in a frequency selective fading ρ(m) = A k−m,k x(k)x∗ (k − m) − b∗ (m)x(m), (13) channel, the delay axis is typically divided into bins whose k=m size is comparable to the inverse of the signal bandwidth. For ∗ and (.) in the superscript denotes the conjugate of (.). We relatively narrowband signals, the number of scatters that fall then deﬁne the following variables: in a bin is large. Therefore, we can assume that the entries in h are with a multivariate complex Gaussian distribution.2 As ρ(m), 0 ≤ m ≤ N − 1 ρ (m) = (14) a result, given P(φ) and D, x is a complex Gaussian random 0, N ≤ m ≤ JN − 1 vector, and its average is given by where J is a parameter that characterizes the precision of the E(x) = P(φ)WDFN ×L μh (6) global search. The outputs of the FFT of ρ (m) are then put into (12) for the search of φ with a precision of 1/JN . Since where E(.) is the expectation of (.) and μh is the expectation there are a lot of zeros at the input of the FFT, the complexity of h. After some manipulations, the covariance matrix of x can be signiﬁcantly reduced by using the pruning technique can be shown to be given by [20]. H Cx = E x − E(x) x − E(x) III. CFO E STIMATION IN OTHER S CENARIOS = P(φ)A P(φ)H (7) In the previous section, given the training OFDM symbol where (.) H in the superscript denotes the conjugate transpose and the second-order channel statistics at the receiver, we of (.), obtained a CFO estimator with φ obtained by minimizing (11). In this section, by simply presenting A and b in (11) into a A = WDFN ×L E((h−μh )(h−μh )H )FH ×L DH WH +σ 2 I, N proper form, the CFO estimator can be found in many other scenarios. As a result, our proposed CFO estimator is quite and I denotes the identity matrix. From (7), we have general and can be considered as a generalization of various −1 Cx = P(φ)(A )−1 P(φ)H = P(φ)AP(φ)H (8) schemes that have already been proposed in the literature. where A = (A )−1 and (.)−1 denotes the inverse of (.). A. CFO estimation using null sub-carriers Given φ and D, the likelihood function of x is then given by To improve the transmission efﬁciency of an OFDM system, blind methods or methods using the inherent structure of 1 H −1 p(x; φ, D) = exp − x−E(x) Cx x−E(x) OFDM signals can be employed for CFO estimation. As π N det(Cx ) mentioned in the introduction, null sub-carriers can be used (9) for CFO estimation using the sub-space based method [5]. The where det(.) denotes the determinant of (.). Note that CFO estimation scheme proposed in [2], [3], where the CFO det(Cx ) = det(A ) and is independent of φ. is estimated using an OFDM symbol with several identical Based upon the ML estimation criterion, φ is found by max- components, can also be regarded as a CFO estimation scheme imizing (9), which is equivalent to minimizing the following using null sub-carriers with speciﬁc null sub-carrier allocations measure: H [6]. In [4], it was shown that the sub-space based CFO −1 x − E(x) Cx x − E(x) . (10) estimation is equivalent to the ML CFO estimation. By substituting (6) and (8) into (10), the normalized CFO Assume that the number of null sub-carriers used in an can be obtained as OFDM system is M . The set that contains all the null sub- carrier indices is denoted by a1 , a2 , · · · , aM . Using the null ˆ min φ = φ Λ (x; φ) subcarriers and based on the ML criterion, it can be shown that the CFO estimation can be achieved by minimizing (11) where with A and b given by Λ (x; φ) = xH P(φ)AP(φ)H x − 2Re bH P(φ)H x , (11) A = VVH (15) 1 √ ej2πnai /N , Re(.) denotes the real part of (.), and where V is an N × M matrix with V n,i = N T and b = b(0), b(1), · · · , b(N − 1) = AH WDFN ×L μh . b=0 (16) 2 When the signal bandwidth is very large, this assumption is in general not where 0 is the all-zero vector. accurate. For example, for ultrawideband communications, the log-normal or the Nakagami distribution is more appropriate. 3A similar approach has been used in [17]. HUANG and LETAIEF: ENHANCED CARRIER FREQUENCY OFFSET ESTIMATION FOR OFDM USING CHANNEL SIDE INFORMATION 2787 B. CFO estimation using training OFDM symbols and In a frequency selective channel, the CFO estimation can b = AH WD1 FN ×L μh . (23) also be achieved using a known training sequence with cyclic In an 802.11a system, there are both null sub-carriers and property [17]. Using such a sequence, the CFO and the CIR pilot tones. By taking null sub-carriers as special pilot tones can be jointly estimated. By comparing the sequence used in with zeros transmitted, we can then deﬁne D1 accordingly, [17] with an OFDM symbol, it can be seen that an OFDM and the CFO estimation can also be achieved by minimizing symbol has the same structure as the sequence used in [17]. In (11) with A and b deﬁned by (21)-(23). particular, the guard interval in an OFDM symbol can function as the precursors of the sequence used in [17]. As a result, we D. CFO estimation using training OFDM symbol and com- can use the scheme proposed in [17] for CFO estimation in plete CIR information OFDM with the aid of a training OFDM symbol. Comparing the results in [17] with (11) and after some manipulations, When the training OFDM symbol and the CIR are both it can be seen that the CFO estimation can be achieved by perfectly available at the receiver, the CFO can be obtained minimizing (11) with A and b given by by minimizing (11) with A and b given by (See Appendix II) −1 A=0 (24) A = WDFN ×L FH ×L DH DFN ×L N FH ×L DH WH N (17) and and b = WDFN ×L h. (25) b = 0. (18) When the CIR is obtained with an estimation error,4 we can use the following to represent the estimated CIR Note that for this CFO estimator, FN ×L is required to be known, which implies that the number of distinct paths in ˆ h= h+e (26) the channel should be available. As will be shown by the simulation results in Section IV, the mismatch between the where e is a vector used to denote the CIR estimation error. presumed number and actual number of distinct paths in the We assume that the mean of e is zero and its covariance matrix channel can signiﬁcantly impact system performance. is Λ. By substituting (26) into (5), we have ˆ x = P(φ)WDFN ×L h − P(φ)WDFN ×L e + z. (27) C. CFO estimation using pilot tones and channel statistics Using (27), we can then get the likelihood function of φ as Pilot tones are often used in OFDM systems for channel shown in Appendix II. Through maximizing the likelihood estimation. In this subsection, we will show that pilot tones function, the CFO can then be obtained by minimizing (11) can also be used for CFO estimation. Assume that there with A and b given by are K pilot tones in an OFDM system, which are denoted −1 A= A , (28) by pk , k = 0, 1, · · · , K − 1. Other sub-carriers are data sub- carriers with independent zero mean Gaussian distribution, and A = WDFN ×L ΛFH ×L DH WH + σ 2 I, N (29) the transmitted signal power of each data sub-carrier is Es . and For convenience, we use an N × N diagonal matrix D1 to ˆ b = AH WDFN ×L h. (30) represent the pilot tones as follows: pk , if the nth sub-carrier is the kth pilot tone IV. S IMULATION R ESULTS D1 = n,n 0, otherwise In this section, extensive simulations are conducted to (19) demonstrate the performance of CFO estimation with the aid and an N × N diagonal matrix I2 to represent the data sub- of channel side information. In the simulations, we use the carriers allocation as follows: normalized mean square error (NMSE) as the performance 1, if the nth sub-carrier is a data sub-carrier measure, which is deﬁned by I2 = n,n 0, otherwise. Nt 2 N ˆ (20) N M SE = φt − φ (31) By approximating the received signal x as a Gaussian random Nt t=1 vector, we can obtain the likelihood function. Through max- where Nt is the number of Monte Carlo trials, φ is the actual imizing the likelihood function and as shown in Appendix I, ˆ normalized CFO, and φt is the estimated normalized CFO at we can then obtain the CFO by minimizing (11) with A and the tth trial. b given by When the CIR is perfectly available and as shown in −1 A= A (21) Appendix III, the Cramer Rao bound (CRB) is given by where 1 CRB = 2 H H (32) H 8π h FN ×L DH WH M2 WDFN ×L h A =WD1 FN ×L E h − μh h − μh FH ×L DH WH N 1 4 When an iterative algorithm is used for both CFO estimation and channel + Es WI2 diag FN ×L E hhH FH ×L WH + σ 2 I N estimation, the channel estimation in the ﬁrst several iterations should bear the impact of the residual CFO. This is modeled as the channel estimation (22) error as shown in (26). 2788 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 10, OCTOBER 2006 where M = diag(0, 1, · · · , N − 1). Since we are interested in 1 10 J = 64 the average performance of the CFO estimation for all channel J=4 0 realizations, in the following, we average (32) over all channel 10 HTE realizations and the result is taken as the baseline. −1 When null sub-carriers are used for CFO estimation, some 10 With Channel Statistics sub-channels may not be excited by the transmitted signal. Therefore, if we ﬁx the transmitted signal power, a low 10 −2 NMSE Without Channel Statistics instantaneous received signal power may not mean a poor in- Channel Known stantaneous channel realization. To make the SNR independent 10 −3 of the null sub-carrier allocations, we ﬁx the transmitted signal power as 10 −4 N −1 N= E |dk |2 . (33) 10 −5 k=0 Cramer Rao Bound The SNR is then deﬁned by 10 −6 −10 −8 −6 −4 −2 0 2 4 6 8 10 2 Instantaneous SNR (dB) N −1 k=0 H(k) SN R = . (34) Fig. 2. NMSE performance of the CFO estimation over Rayleigh fading N σ2 channels when the actual normalized CFO is uniformly distributed between −0.25 and 0.25. A. CFO estimation using training OFDM symbols 1 10 We ﬁrst present the CFO estimator proposed in [16] for J = 64 J=4 comparison. This estimator is called the HTE estimator in [17]. 0 10 For convenience, we rewrite the CIR into the following form T With Channel Statistics h = h (0), h (1), · · · , h (N − 1) = Λh (35) 10 −1 where Λ is an N × L matrix with ’1’ in its −2 Without Channel Statistics (nl , l)th (l = 0, 1, · · · , L − 1) entries and ’0’ in other 10 NMSE entries. In the HTE estimator, the CFO is obtained as 10 −3 Channel Known η ˆ 1 φ= arg ϕ(m) (36) (η + 1)π m=1 10 −4 HTE where η is a design parameter and ϕ(m) is given by (37). In −5 ∗ 10 (37), μ(l1 , l2 ) = E h (l1 )h (l2 ) and ak is the mod(k,N )th element of √1 WD1, where 1 is the N × 1 all-one vector, N 10 −6 Cramer Rao Bound −10 −8 −6 −4 −2 0 2 4 6 8 10 and mod(k,N ) is the remainder after k is divided by N . As Instantaneous SNR (dB) shown in [16], [17], along with the increase of η, the estima- tion accuracy is improved. At the same time, the estimation Fig. 3. NMSE performance of the CFO estimation over Rayleigh fading channels when the actual normalized CFO is uniformly distributed between range is decreased, which is given by |φ| ≤ 1/(η + 1). From −0.0025 and 0.0025. (36), it can be seen that the HTE estimator is simple because 1 10 no global search is required. J = 64 In the following, we consider an OFDM system with 64 J=4 Using complete covariance matrix 0 sub-carriers. The length of the guard interval is 12 samples. In 10 Using diagonal elements of covariance matrix the simulations, one training OFDM symbol is used for CFO estimation. The sub-carriers in the training OFDM symbol are 10 −1 BPSK modulated, and their values are arranged based upon an extended m-sequence as proposed in [6] with length 64 as 10 −2 NMSE follows (in hexadecimal):5 −3 10 A4E2F28C20FD59BA. (38) −4 10 Each bit in the extended m-sequence is used to denote the value of the corresponding sub-carrier in the training OFDM 10 −5 symbol. Speciﬁcally, ’1’ is used to denote a sub-carrier with value 1, and ’0’ is used to denote a sub-carrier with value −1. 10 −6 −10 −8 −6 −4 −2 0 2 4 6 8 10 In Figs. 2 and 3, we present the simulation results of the Instantaneous SNR (dB) CFO estimation using the GSM channel model. This model Fig. 4. NMSE performance of the CFO estimation using channel statistics 5 The good performance of using a white noise like sequence for CFO over Rayleigh fading channels when the actual normalized CFO is uniformly estimation is also justiﬁed by [21] using an asymptotic analysis. distributed between −0.25 and 0.25. HUANG and LETAIEF: ENHANCED CARRIER FREQUENCY OFFSET ESTIMATION FOR OFDM USING CHANNEL SIDE INFORMATION 2789 N −1 N −1 N −1 1 ϕ(m) = x(k)x∗ (k − m) μ(l1 , l2 )a∗ 1 ak−m−l2 k−l (37) N −m k=m l1 =0 l2 =0 1 10 ξ2 = −40dB set the non-diagonal elements to be zero. Using the same ξ2 = −30dB channel parameters as those in Figs. 2 and 3, we show in 0 2 10 J=64 ξ = −20dB 2 ξ = −10dB Fig. 4 the performance of the CFO estimator with the diagonal With Channel Statistics 2 ξ = 0dB elements of the covariance matrix set to zero. It can be seen −1 10 that its performance is basically the same as that using perfect covariance matrix information. −2 10 In Fig. 5, we demonstrate the NMSE performance of the NMSE Without Channel Statistics CFO estimation with imperfect channel estimation. The chan- −3 10 nel model and the distribution of the actual normalized CFO values are the same as those used in Fig. 2. In the simulations, 10 −4 we assume that the estimation of the ith path of the CIR is as follows: −5 10 ˆ h(i) = h(i) 1 + f (i) i = 0, 1, · · · , L − 1 (39) Cramer Rao Bound −6 where f (i) is a zero mean Gaussian random variable with 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 Instantaneous SNR (dB) variance σ 2 . We further assume that f (i) is independent for Fig. 5. NMSE performance of the CFO estimation with imperfect channel different values of i. As a result, the covariance matrix of the estimation. estimation error e is given by Λ = ξ 2 diag hhH . (40) is a Rayleigh fading channel model, which is also used in [6], [17]. In Fig. 2, the actual normalized CFO is uniformly From Fig. 5, it can be seen that the performance of the distributed between −0.25 and 0.25. In this case, η is set CFO estimation deteriorates along with the increase of ξ 2 . to 3 for the HTE scheme. In Fig. 3, the actual normalized However, even when ξ 2 is as large as 0dB, the performance CFO is uniformly distributed between −0.0025 and 0.0025. of the CFO estimation is still as good as that of the CFO In this case, η is set to 63 for the HTE scheme. By comparing estimation scheme with the aid of channel statistics. When Fig. 2 with Fig. 3, it can be seen that the performance of the channel estimation is very good (e.g., ξ 2 is -40dB), it can the HTE estimator is much poorer when the actual CFO be seen from Fig. 5 that the performance of CFO estimation variation range is large. This is because there is a bias for can be improved signiﬁcantly compared with the case using the HTE estimator and the bias is proportional to the actual channel statistics. normalized CFO value as can be seen from the simulation We show in Fig. 6 the impact of the presumed number results in [17]. For other schemes, the CFO variation range of distinct paths in the channel on the performance of the basically does not impact system performance. Compared with CFO estimation scheme proposed in [17]. For convenience, the conventional CFO estimation scheme that does not use we use L to denote the presumed number of distinct paths the channel statistics (i.e., the scheme proposed in [17]), a in the channel. In this case, the scheme in [17] is equivalent close observation of Figs. 2 and 3 shows that the performance to minimizing (11) with A and b given by (17) and (18) of CFO estimation can be signiﬁcantly improved using the with the matrix FN ×L in (17) replaced by FN ×L . In the channel side information. For example, when perfect CIR simulations, the channel used is an exponentially decaying information is available and the search precision is in a high Rayleigh fading channel model with the root mean square level (J=64), the performance can be improved by more (rms) delay spread set to 0.5.6 The actual number of distinct than 6 dB for large SNR values. By using the second order paths in the channel is L = 12 and the actual normalized channel statistics, it can be seen from Figs. 2 and 3 that the CFO is set to be uniformly distributed between −0.25 and performance can be improved for low SNR values. When 0.25. From Fig. 6, it can be seen that for high SNR values the search precision is at a low level (J=4), an error ﬂoor (SNR=6dB, 10dB), as long as the presumed number of distinct appears for the proposed scheme. However, when the actual paths is more than 4, a good performance can be achieved. CFO variation range and the instantaneous SNR values are However, when SNR=−2dB, the best performance is achieved both relatively large, it can still be seen from Fig. 2 that the only when L = 2 or L = 6. When SNR=−6dB, the best performance of the proposed scheme is much better than that performance is achieved only when L = 2. Therefore, to use of the HTE estimator. the CFO estimation scheme proposed in [17], we should select In general, the covariance matrix of the CIR is non-diagonal a proper L to achieve a good performance. As a result, it is (e.g., for the GSM channel model used for Figs. 2 and 3). 6 Here we assume that different paths in the CIR are uncorrelated. When As a result, a large number of parameters are needed to be they are correlated, the performance of the channel estimation can be estimated. To reduce the complexity induced by this, we only improved by using the singular value decomposition method [11]. Similarly, estimate the diagonal elements of the covariance matrix and the performance of the CFO estimation might be improved further. 2790 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 10, OCTOBER 2006 1 −3 10 10 J=64 Without Channel Statistics J=4 0 10 With Channel Statistics −1 10 −4 10 −2 10 NMSE NMSE SNR= −6dB 10 −3 L=12 SNR=−2dB SNR=6dB 5 rms=0. −5 SNR=10dB 10 −4 10 J=64 −5 10 −6 −6 10 10 2 4 6 8 10 12 14 16 18 20 22 24 −10 −8 −6 −4 −2 0 2 4 6 8 10 L’ Instantaneous SNR (dB) Fig. 6. NMSE performance of the CFO estimation versus the presumed Fig. 8. NMSE performance of the CFO estimation over log-normal fading number of distinct paths in the channel. channels. 1 10 K=0dB K=20dB by (32) over the Ricean fading channels is basically the same 0 K=40dB 10 for different values of K. In the following, we use the UWB channel model (the −1 10 CM1 model in [22]) to show the performance of the CFO −2 estimation when the CIRs are non-Gaussian distributed. In 10 J=64 the UWB channel model, each distinct path of the CIR is log-normal distributed. We consider 128 sub-carriers in one NMSE −3 10 OFDM symbol and the length of the guard interval is 32 10 −4 samples, which are the same as those in the multi-band OFDM proposal for 802.15.3a [23]. In the simulations, the actual CFO 10 −5 value is set to be uniformly distributed between −1/8 and 1/8 Cramer Rao Bound of the sub-carrier spacing.7 Furthermore, we use one training −6 10 OFDM symbol for the CFO estimation. In the training OFDM −7 symbol, the values of the sub-carriers of the training OFDM 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 symbol are set based on the extended m-sequence [6]. In this Instantaneous SNR (dB) case, similar to (38), we use an extended m-sequence with Fig. 7. NMSE performance of the CFO estimation over Ricean fading length 128 to denote the training OFDM symbol, which is channels. The solid lines denote the scheme in [17]. The dotted lines denote given by the CFO estimation scheme with the aid of channel statistics. CEA7D0E24DADEC697732AFE041851E44. (42) fair to say that the scheme proposed in [17] also partially takes From Fig. 8, it can be seen that the performance of CFO advantage of the channel side information. estimation can still be improved by using the channel statistics over a log-normal fading channel especially for low SNR For Ricean fading channels, we deﬁne K as follows: values. α2 K(dB) = 10log10 (41) B. CFO estimation using null sub-carriers β2 In this subsection, we use the inherent structure (i.e., null where α2 is the signal power of the specular component and sub-carriers and pilot tones) of OFDM signals to achieve the β 2 is the total power of the non-line of sight components. CFO estimation. In the simulations, we use the same signal For the non-line of sight components, we use a two-ray equal structure as that in 802.11a [24], where there are 64 sub- gain Rayleigh fading channel model. The actual normalized carriers in one OFDM symbol. Among the 64 sub-carriers, CFO is assumed to be uniformly distributed between −0.25 48 are data sub-carriers, 4 are pilot sub-carriers, and 12 are and 0.25. The simulation results of the CFO estimation over null sub-carriers. In our simulations, the positions of the 12 Ricean fading channels are shown in Fig. 7. It can be seen null sub-carriers and the 4 pilot tones are also the same as that by using the channel statistics, the performance of the those in 802.11a. The channel used is an 8-ray exponentially CFO estimation can be signiﬁcantly improved along with the 7 In the multi-band OFDM proposal [23], the sub-carrier spacing is about increase of K. On the other hand, when the scheme in [17] is 4 MHz. If we use an oscillator with a precision of 50 ppm and a center employed, the value of K does not have much impact on the frequency of 10 GHz, the actual CFO is then at most about 500 kHz. performance. We note here that the Cramer Rao bound given Therefore, it is between -1/8 and 1/8 of the sub-carrier spacing. HUANG and LETAIEF: ENHANCED CARRIER FREQUENCY OFFSET ESTIMATION FOR OFDM USING CHANNEL SIDE INFORMATION 2791 1 10 With channel statistics and pilot tones The covariance matrix of x can then be shown as (44). Let us 0 With channel statistics Without channel statistics now separate D into the data part and the pilot tones part as 10 follows: D = D1 + D2 (45) −1 10 where D1 is used to denote the pilot tones as deﬁned by (19) −2 10 and D2 is used to denote the data sub-carriers. We assume that E(D2 ) = 0, and that the data at different sub-carriers NMSE −3 10 are independent. By substituting (45) into (44), the covariance matrix of x can be shown as (46). Assume that x is a complex J=64 −4 Gaussian random vector, then the likelihood function can 10 be easily obtained. It can then be seen that maximizing the −5 likelihood function is equivalent to minimizing (11) with A 10 and b given by (21)-(23). −6 10 0 2 4 6 8 10 12 14 16 18 20 A PPENDIX II Instantaneous SNR (dB) CFO E STIMATION WHEN THE CIR IS AVAILABLE Fig. 9. NMSE performance of the CFO estimation using the same signal Given D and h, then from (5) the likelihood function is structure as 802.11a. given by (47). We can then obtain φ by maximizing the above likelihood function, which is equivalent to minimizing decaying quasi-static Rayleigh fading channel with an rms Λ (x; φ) = −2Re hH FH ×L DH WH P(φ)H x . N (48) delay spread of 0.95. We use 10 OFDM symbols for the CFO estimation and the actual normalized CFO is uniformly The above metric is equivalent to (11) with A and b given distributed between −0.25 and 0.25. Fig. 9 shows the CFO by (24) and (25), respectively. estimation performance of such an OFDM system. It can be When there are channel estimation errors, the average of x seen that by using null sub-carriers alone, the performance of is given by (See (27)) the CFO estimation is exactly the same with or without the E(x) = P(φ)WDFN ×L h. ˆ (49) use of the channel statistics. This is not surprising because null sub-carriers do not excite the channel. On the other hand, By assuming that the channel estimation error and the additive by using pilot tones and null sub-carriers, it can be seen from white Gaussian noise are independent, the covariance matrix Fig. 9 that the performance of the CFO estimation can be of x is then given by improved by taking advantage of the channel statistics for Cx = P(φ)WDFN ×L ΛFH ×L DH WH P(φ)H + σ 2 I. (50) N about the entire SNR region. For instance, the performance Given φ and D, the likelihood function of x is then given can be improved by about 2 dB at an instantaneous SNR value by (51). To ﬁnd φ, we can maximize the likelihood function of about 13 dB. given by (51), which is equivalent to minimizing (11) with A V. C ONCLUSION and b given by (28)-(30). In this paper, we proposed a general CFO estimator for A PPENDIX III OFDM systems, with which the performance of CFO estima- C RAMER R AO B OUND WHEN P ERFECT CIR IS AVAILABLE tion can be improved by using more channel side information. When the CIR is known, the derivative of (48) with respect Using the second order channel statistics and for Rayleigh to φ is as follows: fading channels, it was shown that better CFO estimation performance can be achieved for low SNR values. For Ricean dΛ (x; φ) dP(φ)H = −2Re hH FH ×L DH WH N x fading channels, the performance of the CFO estimation can be dφ dφ improved for all SNR values. When the complete CIR infor- mation is available, the performance of CFO estimation can be = 4πjRe hH FH ×L DH WH MP(φ)H x N further improved. Using the proposed CFO estimator, we can (52) take advantage of not only the channel side information, but also the inherent structure of OFDM signals. For example, in where M = diag(0, 1, · · · , N − 1). The second order deriva- an IEEE 802.11a system, the CFO estimation can be achieved tive of (48) with respect to φ is then as follows: by using not only the null sub-carriers, but also the pilot tones d2 Λ (x; φ) = −8π 2 Re hH FH ×L DH WH M2 P(φ)H x . N and channel statistics. dφ2 (53) A PPENDIX I By substituting (5) into (53), we have CFO E STIMATION U SING P ILOT T ONES AND d2 Λ (x; φ) C HANNEL S TATISTICS −E = 8π 2 hH FH ×L DH WH M2 WDFN ×L h. N dφ2 Assume that the transmitted data and the CIR are indepen- (54) dent. From (5), the average of x is given by According to [25], the Cramer Rao bound is then as given by E(x) = P(φ)WE(D)FN ×L μh . (43) (32). 2792 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 10, OCTOBER 2006 H Cx = E x − E(x) x − E(x) H =E P(φ)WDFN ×L h − P(φ)WE(D)FN ×L μh P(φ)WDFN ×L h − P(φ)WE(D)FN ×L μh + σ2 I (44) H Cx = E P(φ)W(D1 + D2 )FN ×L h − P(φ)WD1 FN ×L μh P(φ)W(D1 + D2 )FN ×L h − P(φ)WD1 FN ×L μh +σ 2 I H = P(φ)WD1 FN ×L E h − μh h − μh FH ×L DH WH P(φ)H N 1 +E P(φ)WD2 FN ×L hhH FH ×L DH WH P(φ)H + σ 2 I N 2 H = P(φ)WD1 FN ×L E h − μh h − μh FH ×L DH WH P(φ)H N 1 +Es P(φ)WI2 diag FN ×L E hhH FH ×L WH P(φ)H + σ 2 I N (46) 1 H p x; φ, D, h = exp − x − P(φ)WDFN ×L h x − P(φ)WDFN ×L h (47) π N σ 2N 1 ˆ H ˆ p x; φ, D = exp − x − P(φ)WDFN ×L h x − P(φ)WDFN ×L h (51) π N det Cx R EFERENCES [16] M. G. Hebley and D. P. Taylor, “The effect of diversity on a burst- mode carrier-frequency estimator in the frequency-selective multipath [1] P. H. Moose, “A technique for orthogonal frequency division multiplex- channel,” IEEE Trans. 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Loeliger, “An introduction to factor graphs,” IEEE Signal Processing From 1998, he was an assistant teacher and later Mag., vol. 21, no. 1, pp. 28–41, Jan. 2004. a lecturer with Tsinghua University. Currently, he is [13] W. C. Kuo and M. P. Fitz, “Frequency offset compensation of pilot a lecturer with School of Electrical, Electronic and symbol assisted modulation in frequency ﬂat fading,” IEEE Trans. Computer Engineering at the University of Western Australia. His research Commun., vol. 45, no. 11, pp. 1412–1416, Nov. 1997. interests include broadband wireless communications, OFDM, OFDMA, [14] M. Morelli, U. Mengali, and G. M. Vitetta, “Further results in carrier cross-layer design, multiple access protocol, and digital implementation of frequency estimation for transmissions over ﬂat fading channels,” IEEE communication systems. Commun. Lett., vol. 2, no. 12, pp. 327–330, Dec. 1998. Dr. Huang serves as an Editor for the IEEE Transactions on Wireless Com- munications. He received the Hong Kong Telecom Institute of Information [15] O. Besson and P. Stoica, “On frequency offset estimation for ﬂat-fading Technology Postgraduate Excellence Scholarships in 2004. channels,” IEEE Commun. Lett., vol. 5, no. 10, pp. 402–404, Oct. 2001. HUANG and LETAIEF: ENHANCED CARRIER FREQUENCY OFFSET ESTIMATION FOR OFDM USING CHANNEL SIDE INFORMATION 2793 Khaled B. Letaief (S’85-M’86-SM’97-F’03) re- Series (as Editor-in-Chief) and the IEEE Transactions on Communications. He ceived the BS degree with distinction in Electri- has been involved in organizing a number of major international conferences cal Engineering from Purdue University at West and events. These include serving as the Technical Program Chair of the Lafayette, Indiana, USA, in December 1984. He 1998 IEEE Globecom Mini-Conference on Communications Theory, held in received the MS and Ph.D. Degrees in Electri- Sydney, Australia as well as the Co-Chair of the 2001 IEEE ICC Communi- cal Engineering from Purdue University, in August cations Theory Symposium, held in Helsinki, Finland. In 2004, he served as 1986, and May 1990, respectively. From January the Co-Chair of the IEEE Wireless Communications, Networks and Systems 1985 and as a Graduate Instructor in the School of Symposium, held in Dallas, USA as well as the Co-Technical Program Chair Electrical Engineering at Purdue University, he has of the 2004 IEEE International Conference on Communications, Circuits taught courses in communications and electronics. and Systems, held in Chengdu, China. He is the Co-Chair of the 2006 From 1990 to 1993, he was a faculty member at IEEE Wireless Ad Hoc and Sensor Networks Symposium, held in Istanbul, the University of Melbourne, Australia. Since 1993, he has been with the Turkey. In addition to his active research and professional activities, Professor Hong Kong University of Science and Technology where he is currently Letaief has been a dedicated teacher committed to excellence in teaching Chair Professor and Head of the Electronic and Computer Engineering and scholarship. He received the Mangoon Teaching Award from Purdue Department. He is also the Director of the Hong Kong Telecom Institute University in 1990; the Teaching Excellence Appreciation Award by the of Information Technology as well as the Director of the Center for Wireless School of Engineering at HKUST (four times); and the Michael G. Gale Information Technology. His current research interests include wireless and Medal for Distinguished Teaching Highest university-wide teaching award mobile networks, Broadband wireless access, OFDM, CDMA, and Beyond and only one recipient/year is honored for his/her contributions). 3G systems. In these areas, he has published over 280 journal and conference He is a Fellow of IEEE, an elected member of the IEEE Communications papers and given invited talks as well as courses all over the world. Society Board of Governors, and an IEEE Distinguished lecturer of the Dr. Letaief served as a consultant for different organizations and is currently IEEE Communications Society. He also served as the Chair of the IEEE the founding Editor-in-Chief of the IEEE Transactions on Wireless Commu- Communications Society Technical Committee on Personal Communications nications. He has served on the editorial board of other prestigious journals, as well as a member of the IEEE ComSoc Technical Activity Council. including the IEEE Journal on Selected Areas in Communications — Wireless