Maximum-Likelihood Carrier Frequency Offset Estimation for OFDM Systems Over Frequency-Selective Fading Channels

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1 Maximum-Likelihood Carrier Frequency Offset Estimation for OFM Systems Over Frequency-Selective Fading Channels Tao Cui and Chintha Tellambura epartment of Electrical and Computer Engineering niversity of Alberta Edmonton, AB, Canada T6G 2V4 taocui, Abstract This paper considers carrier frequency offset CFO) estimation for OFM systems over frequency-selective fading channels We derive three new maximum-likelihood ML) CFO estimators The first estimator takes the pre-ft signal at the receiver as Gaussian and averages the likelihood function over the Gaussian variable The other two estimators first remove the channel impulse response CIR) and average the resulting likelihood function over time-domain transmitted symbols We demonstrate that the presence of virtual carriers is critically important for estimation All our proposed estimators can achieve the asymptotic Cramér Rao Bound ACRB) and can also be enhanced using embedded pilots I INTROCTION The considerable interest in Orthogonal Frequency-ivision Multiplexing OFM), a bandwidth efficient signalling scheme for wireless communication, is mainly due to its remarkable resistance to frequency-selective fading and impulsive noise [] In OFM, the available bandwidth is sliced into several parallel narrow subchannels so that each subchannel is only subject to flat fading, which simplifies equalization to mitigate frequency-selective fading These features have made OFM a standard for high-speed mobile communications OFM has been used in Europe for igital Audio Broadcasting, ETSI-BRAN igh Performance Local Area Networks iperlan/2) and IEEE 802a Additionally, OFM forms the basis for the fourth-generation broad-band systems that will transmit multimedia to mobiles and portable personal communications While OFM systems are robust to frequency selective fading, they are more sensitive to synchronization errors than single-carrier systems [2] In particular, the presence of carrier frequency offset CFO) introduces inter-carrier-interference ICI) that would significantly degrade the system performance [2] To mitigate this effect, various CFO estimation methods have been proposed In [3], repeated symbols are used for frequency synchronization The signal redundancy in the cyclic prefix CP) is exploited for CFO estimation in [4] In [5], a blind estimation method is developed but some restrictions on CFO values and carrier spacing are imposed In [6], Liu et al exploit the presence of virtual carriers in OFM and propose blind estimation methods reminiscent of spectral analysis techniques in array processing, ie, MSIC and ESPRIT ML estimators are derived in [7] [9] In [8], [9], the CIR is first removed from the likelihood function and the frequency domain transmitted signals are removed by summing over all the possible signals In this paper, we derive three new ML estimators MLE) for CFO estimation The CIR and the transmitted symbols together are modelled as a Gaussian process for the first MLE, and they are removed by averaging the likelihood function over the resulting Gaussian variable In MLE2, we first average the likelihood function over the CIR Since the transmitted signal can be treated as Gaussian, we next average the likelihood function over the Gaussian variable, resulting in MLE2 In MLE3, the CIR is obtained by maximizing the likelihood function for a given CFO The CIR is removed by substituting the CIR estimate back into the likelihood function The MLE3 is then derived by averaging over the time domain signal We show that the estimators in [6], [7] and all our proposed estimators are equivalent in high SNR, when only virtual carriers exist Moreover, all the three estimators can use embedded pilots to improve their accuracy and to perform joint estimation of CIR and CFO Notation: boldface letters will be used for matrices and column vectors ) denotes ermitian conjugate transpose) I N denotes the N N identity matrix diagx stands for the diagonal matrix with the column vector x on its diagonal tra) = N i= a ii is the trace of matrix A The Column- Circulant down matrix of a vector a = [a,a 2,,a n ] T a k can be scalar, vector or matrix) with ξ block columns is defined as a a n a n ξ+2 C ξ) [a] = a 2 a a n ξ+3 ) a n a n a n ξ+ The Column-Circulant up matrix with ξ block columns is /05/$2000 C) 2005 IEEE 2506 Authorized licensed use limited to: NIVERSITY OF ALBERTA ownloaded on ecember 2, 2009 at 9:56 from IEEE Xplore Restrictions apply

2 defined as a a 2 a ξ C ξ) [a] = a 2 a 3 a ξ+ 2) a n a a ξ II SYSTEM MOEL In an OFM system, source data are grouped and mapped into d k, which is selected from a complex signal constellation Q with unitary energy Complex data are modulated by inverse discrete Fourier transform IFT) on N parallel subcarriers The symbol interval and block interval are denoted by T s and NT s The resulting OFM symbol during the mth block interval for brevity we omit m) that comprises N samples is given by xn) = N N k=0 Xk)e j2πkn/n), n =0,, 2,,N Xk) = d k p k 3) k I d k I p 4) 0 k I v and I d is the index set of data subcarriers, I p is the index set of subcarriers reserved for pilot symbols with N p elements, I v is the index set of subcarriers reserved for VCs with N v elements and the number of data subcarriers is N d Wehave N d + N v + N p = N The guard interval, inserted to prevent inter-blockinterference, includes a cyclic prefix which replicates the end of the IFFT output samples The number of samples in the guard interval N g is assumed to be larger than the delay spread of the channel The signal is transmitted over a frequency selective fading channel modelled as L ht) = α l δt τ l ) 5) l=0 α l CN0,σ 2 l ) and τ l is the delay of the lth tap When a CFO exists, the received signal after sampling is given by L yn) =e j2π n N hl)xn l)+wn) 6) l=0 wn) is an Additive White Gaussian Noise AWGN) sample, wn) CN0,) 2 Channel taps h l l =0,,L ) represent the sampled overall channel impulse response which comprises the transmit/receive filters and the physical channel ht)) L is the total number of propagation paths The CFO normalized by the block interval NT s ) is denoted by As usual, we assume the channel stays constant within each OFM symbol For convenience, 6) can be written in vector form as y = Γ)C L) [x]h + w 7) or equivalently y = Γ)F X F L h + w 8) x = [x0),x),,xn )] T, y = [y0),y),,yn )] T, h = [h 0,h,,h L ] and w = [w0), w),, wn )] denote transmitted vector, received vector, channel vector and additive noise, respectively The N N discrete Fourier transform FT) matrix F has the i, j)-th entry [F] i,j = e j 2π N i )j ) 9) N and F L =[f Ih 0), f Ih ),,f Ih L )] is the relevant N L submatrix of F, I h is the index set of channel taps and f i is the ith column of F The diagonal matrices Γ) ξ =expj2π/n) and Γ) =diag[,ξ,,ξ N ] 0) X =diag[x0),x),,xn )] ) Let X =[X0),X),,XN )] T x = F X/ N and C L) [x] =F X F L III MAXIMM LIKELIOO ESTIMATION OF CFO The received symbol vector y is Gaussian with mean Γ)C L) [x]h and covariance matrix σ2 ni N The likelihood function for the unknown parameters x, h and is given by Λy x, h,) = exp 2 y Γ)C L) [x]h 2 2) We next consider ML CFO estimation using the likelihood function A MLE When the number of subcarriers N and the channel length L is large, the signal g = C L) [x]h can be modelled as a zeromean complex Gaussian process via the central limit theorem The autocorrelation matrix of g can be evaluated as ) R g =Egg = E C L) [x]hh C L) [x] =E F X F L hh F L X F 3) =F rλ d + Λ p F L R h F L Λ ) p F r = L l=0 σ2 l, R h = Ehh is the covariance matrix of h and k Id Λ d =diag[λ 0,λ,,λ N ],λ k = 0 otherwise 4) k Ip Λ p =diag[λ 0,λ,,λ N ],λ k = 0 otherwise 5) 2507 Authorized licensed use limited to: NIVERSITY OF ALBERTA ownloaded on ecember 2, 2009 at 9:56 from IEEE Xplore Restrictions apply

3 The average of Λy x, h, ) over g gives the marginal likelihood function Λy ), which removes the likelihood function dependence on h and x The marginal likelihood function can be expressed as Λy ) =E g exp 2 y Γ)g 2 6) E X denotes the statistical average with respect to the random vector X Eq 6) can be interpreted as the characteristic function CF) λ Q s) = EexpsQ) of the quadratic form Q = y Γ)g 2 7) of the zero-mean Gaussian vector g evaluated for s = /)IfR 2 g is non-singular eg, no VCs), the CF is given by [0, p 595, eq B-3-20)] When VC exists, R g becomes singular, which means some of the random variables in g are linearly dependent To make the CF approach still applicable in this case, we assume ζ k Iv Λ v ζ) =diag[λ 0,λ,,λ N ],λ k = otherwise sing [0, p 595, eq B-3-20)], we can obtain Λy, ζ) = exp 2 y Γ)Gζ)Γ) y Let ζ 0, wehave Λ y ) = lim Λy, ζ) = exp ζ 0 2 y Γ)GΓ) y 8) 9) 20) G = R g R g + I N ) Eq 20) follows from that the limitation and integral operations are interchangeable The ML estimate of can be obtained by maximizing ˆ = arg max y Γ)GΓ) y 2) B MLE2 In MLE2 and MLE3, we assume that no pilot exists The existence of pilots will be discussed in the full journal version As [9], we first evaluate the marginal likelihood function Λy x,), which is the average of Λy h, x,) with respect to h and can be expressed as Λy x,)=e h Λy h, x,) 22) sing [0, p 595, eq B-3-20)] and dropping irrelevant factors, 22) becomes Λy x,) = exp 2 y y y Γ)C L) [x] ) ) 23) G [x] Γ) y C L) G = R h R h + I 2 N ) We define N L matrix Ξ i = Ih j),j =0,,,L Ξ i,j = 24) 0 otherwise and i = I h k), Ω = diag[ω 0,ω,,ω N ],ω i = k =0,,,L 0 otherwise 25) It can be readily verified that C L) [x] =CN) [x]ξ, ΞΞ = Ω and ) y y =y Γ)C N) [x] C N) [x] Γ )y ) + y Γ)C N) [x v] C N) [x] Γ )y 26) + y Γ)C N) [x] C N) v]) [x Γ )y + y Γ)C N) [x v] C N) v]) [x Γ )y x v = F Λ v / N, C N) [x v]=f Λ v F and k Iv Λ v =diag[λ 0,λ,,λ N ],λ k = 0 otherwise 27) ence, 22) becomes Λy x,) = exp β =y Γ)C N) β 2 =y Γ)C N) [x v] 2 β + β 2 + β 3 + β 4 ) ) [x]i N ΞGΞ) C N) [x] Γ )y ) [x] Γ )y C N) β 3 =y Γ)C N) [x] C N) β 4 =y Γ)C N) [x v] v]) [x Γ )y C N) v]) [x Γ )y 28) 29) Let z = Γ )y and note that for two vectors a and b of length N the following relationship holds a C N) [b] =bt C N) [a ] 30) Therefore β β 4 can be rewritten as β =x C N) [z]i N ΞGΞ) ) β 2 =x C N) [z] C N) [z] xv ) β 3 =x v C N) [z] C N) [z] x ) β 4 =x v C N) [z] C N) [z] xv C N) [z] ) x 3) Without pilots, the time domain transmitted signal x can be modelled as a zero-mean complex Gaussian process by using the central limit theorem when the number of subcarriers N is large The covariance matrix of x can be obtained as R x = Exx = N F Λ d F 32) 2508 Authorized licensed use limited to: NIVERSITY OF ALBERTA ownloaded on ecember 2, 2009 at 9:56 from IEEE Xplore Restrictions apply

4 Λ d is given in 4) The average of Λy x,) with respect to x generates the marginal likelihood function, which is given by Λy ) =E x Λy x,) 33) sing [0, p 595, eq B-3-20)] and following the same approach as 8)-20) for the singular case, 33) yields Λ 2 y ) = det I N + R ) exp xa σ 2 x v B + C)x v n 2 34) A =C N) [z]i N ΞGΞ) ) B =C N) [z] C N) [z] C =B I N + R xa Therefore, MLE2 is given by C MLE3 C N) ) R x B [z] ) 35) ˆ = arg max Λ 2 y ) 36) Two possible approaches may be taken to estimate according to the ML criterion One is the Bayesian approach adopted in MLE2 We use the alternative approach to derive MLE3 If we keep and x fixed and let h vary in 2), Λy x, h,) achieves a maximum for [ ) ] ) hx,)= C L) [x] L) C [x] C L) [x] Γ )y = F ) ) L Λ d F L [x] Γ )y C L) Substituting 37) into 2), we can obtain Λy x,) =exp =exp C L) C L) y Γ)C L) ) [x] Γ )y 2 [x] F L Λ d F L ) y y y Γ)C L) [x] F ) L Λ d F L ) ) [x] Γ )y 37) 38) Following the same procedure as 24)-35), the marginal likelihood function can be derived as Λ 3 y ) = det I N + R ) exp xa σ 2 x v B + C)x v n 2 39) A =C N) [z] I N Ξ F ) ) L Λ d F L Ξ ) B =C N) [z] C N) [z] C =B I N + R ) xa 2 R x B Therefore, MLE3 is given by C N) [z] ) 40) ˆ = arg max Λ 3 y ) 4) IV PERFORMANCE OF ESTIMATORS Since the MLE is equivalent to the one in [6] without pilots, we expect that MLE is unbiased ence MLE2 and MLE3 are asymptotically unbiased If pilots exist, we assess the performance of our proposed CFO estimator using the approach in [], which show that the expectation of the ML estimator in high SNR is given by Eˆ Eġ) E g) 42) g) denotes the cost function of the estimator given by 2) in this case The first and second derivatives of g) are given by ġ) and g) It can be proved that Eˆ = 43) MLE is thus unbiased From 2), 34) and 39), g) is periodic of period N The range of MLE is [0,N) if L<N v The integer CFO and fractional CFO do not need to perform separately To assure that MLE only has a unique minimum, the VCs are placed distinctly and N v = L +2 in [2] MLE2 and MLE3 asymptotically follow the same properties If neither pilot nor VC exists, it can be readily verified that G in 20) becomes a diagonal matrix In this case, Eq 20) becomes a constant, which means that MLE cannot estimate CFO without VC Likewise, MLE2, MLE3 and the estimators in [6], [7] cannot work without VC, which means all the ML CFO estimators are VC driven In MLE-MLE3, pilots embedded for channel estimation can be used to assist CFO estimation This suggests the possibility of joint estimation of CIR and CFO V NMERICAL RESLTS We now present numerical results to illustrate the effectiveness of the proposed ML estimators for a practical OFM system We assume the following system specifications: Both the data and pilot symbols are chosen from the QPSK alphabet, denoted by Q The carrier frequency of the OFM system is 5Gz and the data rate is 2Mz 2509 Authorized licensed use limited to: NIVERSITY OF ALBERTA ownloaded on ecember 2, 2009 at 9:56 from IEEE Xplore Restrictions apply

5 0 0 2 MLE MLE2 MLE3 CVE MME ACRB N p =0 N p =4 N =6 p 0 3 MSE of CFO 0 3 MSE of CFO SNR db) SNR db) Fig MSE of CFO versus SNR in a QPSK OFM system with =025, N =64, N p =0and N v =2 The 6-ray COST 207 T model with the PP [089, 0379, 0239, 0095, 006, 0037] and delay profile [00, 02, 05, 6, 23, 50]µs is considered The channel remains constant during each OFM data block but varies from one block to another All the paths are rounded to integer and we do not consider leakage The PP is assumed perfectly known at the receiver A normalized CFO of 025 is considered The number of subcarriers N =64, the number of VCs N v =2and VCs are distinctly placed [2] to preserve the uniqueness of the estimator In Fig, the MSE Eˆ ) 2 ) of proposed estimators with no pilot are compared with those of the Morelli and Mengali Estimator MME) [8], the Chiavaccini and Vitetta Estimator CVE) [9] and the asymptotic Cramer-Rao bound ACRB) given by [9] as [ π ACRB = 2 2 N v N 2 ) 3N 2 2π2 N 2 exp[j2πi j)/n ] 2 i I v j I v,i / I v 44) In low SNR, all the estimators perform similarly and CVE performs slightly better than the other estimators Both MME and CVE show error floors in high SNR The error floors of MME and CVE are 0005 and 00007, respectively MLE2 and MLE3 perform almost identically for any SNR and both performances approach that of MLE Our proposed estimators asymptotically approach the ACRB Fig 2 shows the performance of MLE with different number of pilots The optimal pilots for CFO estimator can be obtained by minimizing the CRB for MLE ere, the pilots are equispaced and are randomly chosen from Q They vary each frame MLE with N p = 4 gains 2 db over that of N p =0 The gain can be as large as 4 db when N p =6 These verify the performance enhancement using pilots The CIR and CFO can thus be jointly estimated using these pilots Fig 2 MSE of CFO versus SNR with N p =0, 4, 6 in a QPSK OFM system with =025, N =64and N v =2 VI CONCLSION Three new ML CFO estimators have been derived in this paper MLE2 and MLE3 perform close to MLE in high SNR and all of them achieve the ACRB We show that the estimators in [6], [7] and MLE are equivalent and are VC driven estimators All our proposed estimators can utilize the pilots to enhance CFO estimation, enabling joint channel and CFO estimation The proposed estimators can be implemented by an efficient FFT algorithm REFERENCES [] B LeFloch, M Alard, and C Berrou, Coded orthogonal frequency division multiplex, Proc IEEE, vol 83, pp , June 995 [2] T Pollet and M Moeneclaey, The effect of carrier frequency offset on the performance of band limited single carrier and ofdm signals, in Proceeding of IEEE GLOBECOM96, London, K, 996, pp [3] P Moose, A technique for orthogonal frequency division multiplexing frequency offset correction, IEEE Trans Commun [4] J J van de Beek, M Sandell, and P O Borjesson, ML estimation of time and frequency offset in OFM systems, IEEE Trans Signal Processing, vol 45, no 7, pp , Jul 997 [5] M Schmidl and C Cox, Blind synchronization for OFM, Electron Lett, vol 33, pp 3 4, Feb 997 [6] Liu and Tureli, A high-efficiency carrier estimator for OFM communications, IEEE Commun Lett, vol 2, no 4, pp 04 06, April 998 [7] F affara and A Chouly, Maximum likelihood frequency detectors for orthogonal multicarrier systems, in Proceeding of IEEE International Conference on Communications, Geneva, May 993, pp [8] M Morelli and Mengali, Carrier-frequency estimation for transmissions over frequency selective channels, IEEE Trans Commun, vol 48, no 9, pp , Sep 2000 [9] E Chiavaccini and G M Vitetta, Maximum-likelihood frequency recovery for OFM signals transmitted over multipath fading channels, IEEE Trans Commun, vol 52, no 2, pp , Feb 2004 [0] M Schwartz, W R Bennett, and S Stein, Communication Systems and Techniques New York: McGraw-ill, 966 [] M Meyrs and L Franks, Joint carrier phase and symbol timing recovery for PAM systems, IEEE Trans Commun, vol 28, no 8, pp 2 29, Aug 980 [2] X Ma, C Tepedelenlioglu, G B Giannakis, and S Barbarossa, Nondata-aided carrier offset estimators for OFM with null subcarriers: identifiability, algorithms, and performance, IEEE J Select Areas Commun, vol 9, no 2, pp , ec Authorized licensed use limited to: NIVERSITY OF ALBERTA ownloaded on ecember 2, 2009 at 9:56 from IEEE Xplore Restrictions apply