162 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vol.106 (3) September 2015 ANALYSIS AND OPTIMIZATION OF AUTO-CORRELATION BASED FREQUENCY OFFSET ESTIMATION I.M. Ngebani∗ , J.M. Chuma† and S. Masupe‡ ∗ Dept. of Information Science and Electronics Engineering, 38 Zheda Road, Zhejiang University, Hangzhou 310027, China E-mail: iboz55@gmail.com † College of Engineering and Technology, Botswana International University of Science and Technology, Private Bag 14, Palapye, Botswana E-mail: chumaj@biust.ac.bw ‡ College of Engineering and Technology, Botswana International University of Science and Technology, Private Bag 14, Palapye, Botswana E-mail: masupes@biust.ac.bw Abstract: In this letter, a general auto-correlation based frequency offset estimation (FOE) algorithm is analyzed. An approximate closed-form expression for the Mean Square Error (MSE) of the FOE is obtained, and it is proved that, given training symbols of fixed length N, choosing the number of summations in the auto-correlation to be � N3 � and the correlation distance to be � 2N 3 � is optimal in that it minimizes the MSE. Simulation results are provided to validate the analysis and optimization. Key words: Auto-correlation, frequency offset estimation, optimization, performance analysis, un-biased estimator. 1. INTRODUCTION Carrier Frequency Offset (CFO), caused by frequency deviation between a transmitter and a receiver exists in most communication systems and may result in severe performance degradation or even system failure. Therefore, estimation and compensation of frequency offset in communication systems is important in order to allow coherent demodulation of the transmitted signals. Compared to single-carrier modulation, Orthogonal Frequency Division Multiplexing (OFDM) is more sensitive to frequency offset because it introduces Inter-Carrier Interference (ICI) and destroys the orthogonality among sub-carriers [1]. To mitigate the negative impact of frequency offset, continuous efforts have been made to develop efficient Frequency Offset Estimation (FOE) algorithms. FOE can be done in the time or frequency domain. In OFDM systems, time-domain algorithms are typically used to estimate the initial frequency offset and frequency-domain algorithms are used to track the residual frequency offset. Time-domain FOE algorithms generally rely on the auto-correlations between two specially designed training signal segments [2–5]. Further enhancements of utilizing training signals composed of multiple identical segments have been proposed in [7, 8]. [9] gives a comparative study of the Schmidl-Cox (SC) [5] and Morelli-Mengali (MM) [6] algorithms for frequency offset estimation in OFDM, along with a new least squares (LS) and a new modified SC algorithm. In [10], the author proposes a novel maximum likelihood (ML) based algorithm for estimating the timing offset and carrier frequency offset in OFDM systems under dispersive fading channels. Although auto-correlation based FOE algorithms have been used in many practical systems, the performance Figure 1: Autocorrelation based FOE analysis and optimization of the algorithms has not yet been thoroughly investigated. In this letter, a general auto-correlation based FOE algorithm is analyzed, a closed-form expression for the Mean Square Error (MSE) is derived, and it is proved that if the training symbol length is fixed to be N, to minimize the MSE, the optimal number of summations in the auto-correlation should be � N3 � and the optimal auto-correlation distance equals � 2N 3 �. This letter is organized as follows: Section 2 introduces a general auto-correlation based frequency offset algorithm. The main result is presented in Section 3. Section 4 presents simulation results and some discussions. Finally, conclusions are drawn in Section 5. 2. AUTO-CORRELATION BASED FREQUENCY OFFSET ESTIMATION A quasi-static dispersive channel that contains L resolvable multi-paths can be denoted by {hl }L−1 l=0 . Let sn be the n-th transmitted training symbol with unit energy, then the n-th received symbol can be expressed as yn = e jθn L−1 ∑ hl sn−l + vn , (1) l=0 where vn is the AWGN with zero mean and variance σ2 and θn is the rotation angle at the n-th symbol caused by Vol.106 (3) September 2015 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 163 the frequency offset. In (1), it is assumed that the rotation angles for L consecutive symbols are approximately the same, this is valid if the frequency offset is not absurdly large. Let Δ fs be the true frequency offset and Ts be the symbol interval, then θn can be expressed as θn = nΔθ, where Δθ is the rotation angle per symbol, and is defined as Δθ � 2πTs Δ fs . (2) Auto-correlation based FOE relies on training symbols of length N that are composed of multiple identical segments, each segment has Ms symbols. A sensible design should have Ms � L. frequency offset equals The auto-correlation metric between yn and yn+D1 is 1 M1 † Q(M1 ) = ∑ (yn )(yn+D1 ), M1 n=1 Figure 2: Illustration of angle approximation induced by ṽ(M1 ) Δ fˆs = (3) where ()† denotes complex conjugation, D1 is called the“auto-correlation distance”, M1 is the number of summations in the auto-correlation and is called the “complementary auto-correlation distance”. Fig.1 illustrates the autocorrelation based FOE, from Fig.1 it is clear that D1 = N − M1 . ∠Q(M1 ) + 2πdˆ . 2πD1 Ts (8) In the autocorrelation based FOE algorithm introduced above, the FOE precision is mainly determined by M1 and the range of resolved frequency offset is determined by M2 . In the following, we analyze the performance of the auto-correlation based FOE algorithm, and show how to optimize the algorithm. Having obtained Q(M1 ), the frequency offset can be estimated as [2, 3] ∠Q(M1 ) . (4) 2πD1 Ts   If Δ fs is in the range − 2D11 Ts , 2D11 Ts , equation (4) can provide correct estimation, otherwise there exists a 2π or multiples of 2π phase ambiguity. In this case, the correct rotated angle should be ∠Q(M1 ) + 2πd instead of ∠Q(M1 ), where d is an integer. To resolve the phase ambiguity, another auto-correlation metric with a shorter auto-correlation distance D2 � (N − M2 ) can be used, i.e., calculating Δ fˆs Q(M2 ) = = 1 M2 † ∑ (yn )(yn+D2 ), M2 n=1 (5) 3. PERFORMANCE ANALYSIS AND PARAMETER OPTIMIZATION For the auto-correlation based FOE algorithm, clearly, the larger the auto-correlation distance (i.e., D1 or D2 ) is, the finer the estimated frequency offset, and the better the performance. However, given a fixed training symbol length N, large auto-correlation distances mean smaller complementary auto-correlation distances (i.e. M1 or M2 ). The smaller the complementary auto-correlation, the lesser the number of samples used to calculate the auto-correlation metric and thus leading to poor performance. Therefore, given N, there is an optimal auto-correlation distance where the MSE is minimized. where M2 is the corresponding complementary auto-correlation distance. Clearly, the two auto-correlation metrics have the relation Since M2 is only used to resolve the ambiguity, it is sufficient to choose M2 to satisfy the following inequality D1 ∠Q(M2 ) ≈ ∠Q(M1 ) + 2πd, D2 (6) −π < 2π(N − M2 )Δ fs Ts < π. (7) In the following, we only focus on how to optimize the parameter M1 . We first derive the MSE of the estimated frequency offset with complementary auto-correlation distance M1 . and the 2πd phase ambiguity can be estimated as D  1 D2 ∠Q(M2 ) − ∠Q(M1 ) ˆ d= , 2π where �·� is the rounding operation. Then, the estimated (9) Because of the repeated segments, D1 is a multiple of Ms , 164 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS To get optimal FOE performance, M1 should be chosen to satisfy opt M1 = arg min {R} . (20) and yn+D1 equals yn+D1 e j(n+D1 )Δθ = L−1 ∑ hl sn+D1 −l + vn+D1 l=0 L−1 Vol.106 (3) September 2015 M1 (10) The following theorem summarizes the main result of this opt letter, which gives M1 , and the minimum MSE. Let us define zn � ∑L−1 l=0 hl sn−l . Assuming independent   and unit energy training symbols sn , we have E �zn �2 = 2 �h�2 � ∑L−1 l=0 �hl � , and as M1 gets large, we have the following approximation Theorem 1: For a system with N training symbols for FOE, the optimal complementary auto-correlation distance that minimizes the MSE of the estimated frequency offset is   N opt M1 = 3 e = j(n+D1 )Δθ ∑ hl sn−l + vn+D1 . l=0 1 M1 ∑ �zn �2 ≈ �h�2 . M1 n=1 (11) Substituting (10) into (5) and using the above approximation, Q(M1 ) can be expressed as Q(M1 ) = 1 M1 ∑ �zn �2 e jD1 Δθ + ṽ(M1 ) M1 n=1 ≈ �h�2 e jD1 Δθ + ṽ(M1 ), (12) where ṽ(M1 ) is called the “noise term” for FOE and is given by ṽ(M1 ) = A + B +C, (13) where A, B and C are defined as: A � B � C � 1 M1  † j(n+D1 )Δθ  , ∑ vn zn e M1 n=1  1 M1  vn+D1 z†n e− jnΔθ , ∑ M1 n=1  1 M1  † vn vn+D1 . ∑ M1 n=1 (14) (15) (16) (17) where α is the angle induced by noise term ṽ(M1 ). Note that α �= ∠ṽ(M1 ), instead, it is the angle between Q(M1 ) and e jD1 Δθ (See Fig.2). The estimation of Δ fs in equation (8) can be derived as α . Δ fˆs = Δ fs + 2πD1 Ts where SNR � (18) Δ fˆs is later shown to be an unbiased estimator, and the MSE of the estimated frequency offset is given by   E �α�2 R � 2 2 2. (19) 4π D1 Ts �h�2 . σ2 Proof: Expanding the expectation of �ṽ(M1 )�2 in (13), we have       E �ṽ(M1 )�2 = E �A + B�2 + E (A + B)C†     +E C(A + B)† + E �C�2 . Since vn is a complex with zero  Gaussian  random variable  † = 0 and E C(A + B)† = 0. mean, we have E (A + B)C   Therefore, E �ṽ(M1 )�2 can be simplified to     σ4 E �ṽ(M1 )�2 = E �A + B�2 + . M1 Case 1: M1 ≤ Using equation (12) and resolving the 2πd ambiguity, we obtain ∠Q(M1 ) + 2πd = D1 Δθ + α, and the corresponding minimum MSE is approximately   1 1 2 min + , R ≈     N 2 SNR SNR2 8π2 Ts2 N − N3 3 (21)  N−1  2 In this case, there is no overlap between vn and vn+D1 for n = 1, 2, · · · , M1 , so A and B are independent zero mean circular complex Gaussian random variables. Since ṽ(M1 ) does not favor any specific direction, we have E [α] = 0. This makes Δ fˆs given in equation (18) an unbiased estimator.∗ Based on the illustration in Fig.2, assuming M1 is large, in high SNR scenarios, the angle α can be approximated as α≈ �ṽ(M1 )� sin ϕ , �h�2 (22) where ϕ is the angle between ṽ(M1 ) and e jD1 Δθ . In this ∗ It is important to note that the distribution of α in equation(18) is unknown even though the first and second moments are known. Since the distribution is unknown, the CRLB cannot be derived for this dedicated case. Vol.106 (3) September 2015 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 165 because Ak+D1 + Bk can be written as   2ℜ v†k+D1 zk+D1 e j(k+D1 )Δθ e jD1 Δθ Ak+D1 + Bk = . M1 case, R can be approximated as     E �ṽ(M1 )�2 − E cos 2ϕ �ṽ(M1 )�2 R≈ . 8π2 D21 Ts2 �h�4 We also have   E cos 2ϕ�ṽ(M1 )�2 = 0, (23) where we have applied the property that ϕ is uniformly distributed and independent to the length of �ṽ(M1 )�.  The expectation E �ṽ(M1 )�2  A+B = D1 M1 n=1 k=D1 +1 ∑ (An + Bn+M1 −D1 ) + ∑ ≈ = 1 8π2 Ts2 M1 (N − M1 )2  1 2 + SNR SNR2  The optimization problem (20) is now equivalent to   opt M1 = arg max M1 (N − M1 )2 . M1 . (25) (26) It is not difficult to show that opt M1 1 + SNR 2 Rmin =   N   N 2 . 2 2 8π Ts N − 3 3  N−1  Case 2: M1 > 2 (28) (31)  =u(M1 ) Using similar arguments as in Case 1, we have E [α] = 0, which leads to an unbiased estimation of Δ fˆs given by equation (18). Based on the illustration in Fig.2, we can approximate the angle α as �u(M1 )� sin ϕ , (32) α≈ �h�2  2(N − M )�h�2 σ2 σ4 1 + , E �u(M1 )�2 ≈ M1 M12 (33) and the corresponding MSE equals R2 ≈ In this case, A and B are NOT independent anymore because the (k + D1 )-th term in A, which is , (29) vk+D1 z†k+D1 e− jkΔθ vk+D1 z†k e− jkΔθ = , M1 M1 (30) M1 � +w(M1 ). Following the same procedure as in Case 1, we have 2 SNR and the k-th term in B, which is Bk =  n=1 (27) and the corresponding minimum MSE is Ak+D1 = D1 where ϕ is the angle between u(M1 ) and e jD1 Δθ .   N = , 3 v†k+D1 zk+D1 e j(k+D1 )Δθ e jD1 Δθ  w(M1 ) ∑ (An + Bn+M1 −D1 ) +C ṽ(M1 ) = 2�h�2 σ2 + σ4 8π2 Ts2 �h�4 M1 (N − M1 )2 Ak + Bk−D1 , where w(M1 ) is the summation of correlated terms and is along the direction of e jD1 Δθ , so it has no contribution to the angle α. Then, ṽ(M1 ) can be re-written as Using the relation D1 = N − M1 and combining equations (23) and (24), R becomes R1  equals     σ4 E �A�2 + E �B�2 + M1 2 2 4 2�h� σ + σ ≈ . (24) M1   E �ṽ(M1 )�2 = Regrouping the terms in A + B, we obtain are correlated, and the terms Ak+D1 + Bk for k = 1, 2, · · · , (M1 − D1 ) are along the same direction as e jD1 Δθ , 2 SNR 8π2 Ts2 M12 (N − M1 ) Define L(M1 )  opt 1 + 1 SNR2 8π2 Ts2 M1 (N − M1 )2 8π2 Ts2 M1 (N−M1 )2 SNR . (34) opt and D  L(M1 ), where M1 is given by equation (27). R1 and R2 given by equations (25) and (34), respectively can then be re-written as   1 R1 (M1 ) = L(M1 ) 2 + SNR   2N 1 R2 (M1 ) = L(M1 ) −2+ M1 SNR We know that L(M1 ) ≥ D, and the minimum value of R1 1 is Rmin 1 = D(2 + SNR ). To complete the proof we show that 166 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS respectively. 5 Mean Squre Error of Frequency Offset Estimate 10 Simu:SNR=0dB Simu:SNR=5dB Simu: SNR=10dB Theoretic:SNR=0dB Theoretic:SNR=5dB Theoretic: SNR=10dB 4 10 As a last comment, from the closed-form MSE formulas, we can see that, when N is fixed, the MSE of FOE is just a function of M1 and SNR, and is independent of Δ fs . 5. 3 10 2 10 0 100 200 300 Value of M1 (Samples) 400 500 Figure 3: Validation of approximated analysis and parameter optimization.   opt 1 R2 (M1 ) > R1 (M1 ) = D 2 + SNR . L(M1 ) ≥ R2 (M1 ) ≥ D  2N 1 −2+ D M1 SNR  (35) (36) (37) R2 is bounded by  2N 1 −2+ (N/2) SNR R2 (M1 ) > D R2 (M1 ) > D(2 + A) = Rmin 1  (38) (39) Therefore, the minimum MSE in Case 1 is a global minimum. 4. Vol.106 (3) September 2015 SIMULATION VALIDATIONS AND DISCUSSION To validate the analysis and optimization, we consider a communication system that has N = 500 symbols, Δ fs = 10kHz, and 1/Ts = 1MHz. To satisfy (9), we choose M2 = 480 symbols. The simulated and theoretical results of MSE vs. M1 are shown in Fig.3. It can be seen that the MSE calculated from our analysis matches the simulated MSE very well, and  500 the minimum MSE is achieved when M1 = 167 = 3 , as predicted by Theorem 1. From Fig.3, it can be observed that the curve for SNR = 10dB is more symmetric than the curve for SNR = 0dB and the local minimum in the curve of SNR = 10dB is closer to the global minimum. This is because at high 1 SNRs, the SNR 2 term in (25) and (34) can be ignored 2 and R ≈ and the MSE becomes R ≈ 8π2 T 2 M (N−M )2 (SNR) 2 s 1 1 , for Case 1 and Case 2, respectively.   They are symmetric to the center M1 = N−1 and reach 2 N    the same minimum when M1 = 3 and M1 = N − N3 , 8π2 Ts2 M12 (N−M1 )(SNR) CONCLUSION In this letter, a general auto-correlation based FOE algorithm was analyzed, closed-form expressions of the MSE were derived, and it was proved that the optimal   complementary auto-correlation distance equals N3 , where N is the total number of training symbols. The results obtained in the letter can be of practical usage when designing training symbols in the implementation of auto-correlation based FOE algorithms. REFERENCES [1] T. Pollet, M. V. Bladel, and M. Moeneclaey “BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise ,” IEEE Trans. Commun., vol. 43, no. 234, pp. 191-193, Feb./Mar./Apr. 1995. [2] J. Van De Beek, M. Sandell, and P. O. Borjesson, “ML estimation of time and frequency offset in OFDM systems,” IEEE Trans. Signal Processing, vol. 45, no. 7, pp. 1800-1805, July 1997. [3] M. Morelli, A. Andrea, and U. Mengali, ”Feedback frequency synchronization for OFDM applications,” IEEE Commun. Lett., vol. 5, no. 1, pp. 28-30, Jan. 2001. [4] P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun., vol. 42, pp. 2908-2914, Oct. 1994. [5] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613-1621, Dec. 1997. [6] M. Morelli and V. Mengali, “An improved frequency offset estimator for OFDM applications,” IEEE Commun. Lett., vol. 3, no. 3, pp. 75-77, Mar. 1999. [7] Y. H. Kim, I. Song, S. Yoon and S. R. Park, ”An efficient frequency offset estimator for OFDM systems and its performance characteristics,” IEEE Trans. Veh. Technol., vol.50, pp. 1307-1312, Sep. 2001. [8] Z. Zhang, K. Long, and Y. Liu,, “Complex efficient carrier frequency offset estimation algorithm in OFDM systems,” IEEE Trans. Broadcast, vol. 50, no. 2, pp. 159-164, June 2004. [9] Z. Cvetkovic, V. Tarokh, and S. Yoon, “On frequency offset estimation for OFDM,” IEEE Trans. Wireless Commun, vol.12, no.3, pp.1062,1072, Mar 2013. Vol.106 (3) September 2015 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS [10] C. Wen-Long “ML Estimation of Timing and Frequency Offsets Using Distinctive Correlation Characteristics of OFDM Signals Over Dispersive Fading Channels,” IEEE Trans. Vehicular Technology, vol.60, no.2, pp.444,456, Feb. 2011 167 168 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS NOTES Vol.106 (3) September 2015