FREQUENCY OFFSET ESTIMATION IN OFDM USING SAMPLE COVARIANCE G. Levin D. Wulich Communications Laboratory Department of Electrical and Computer Engineering Ben-Gurion University of the Negev Beer-Sheva, ISRAEL Tel: ++972-7-646 1537; Fax: ++972-7-6472949; E-Mail: dov@ee.bgu.ac.il ABSTRACT In this paper we consider the performance of the Parametric Weighted Least Square (lW?LS) algorithm for Orthogonal frequency oflset estimation in Frequency-Division Multiplexing (OFDM). No training sequence or redundant information is needed. The correlation between symbols is achieved due to channel spreading, which is assumed to be a Linear Time Invariant (ZTfl jilter with known impulse response. We show that the frequency estimator based on a PWLS algorithm may be used in a tracking mode of the OFDM receivers. INTRODUCTION Orthogonal frequency division multiplexing (OFDM) systems have recently gained increased interest. They are widely used in HDSL, DAB and for digital transmission at HF. One of the main problems in the design of an OFDM receiver is the mismatch of the oscillators in the transmitter and the receiver. The demodulation of a signal with an offset in the carrier frequency can cause a high bit error rate due to degradation in the perilorrnance of the symbol synchronizer and to Inter Channel Interference (ICI) [1]. Many ffequency offset compensation methods have been proposed recently; a training sequence [1,2,3,4] or redundant information [5] is required in many of them. In this paper we consider the Parametric Weighted Least Square (PWLS) algorithm for the frequency offset estimation [6, 7] based on matching, in the weighted least square sense, a sequence of sample correlation to the theoretical values. No training sequence or redundant information is needed. A correlation between symbols is achieved due to channel influence. We suppose that the channel is well described by a Linear Time Invariant (LTI) system and has an a priori known impulse response. The problem of the estimation of the random amplitude complex exponent frequency is very well known in estimation theory and has many solutions [6,8,9]. Here we use, as stated, the Parametric Weighted Least Square (PWLS) algorithm [6]. It is always very interesting to establish the ultimate statistical performance that can be achieved by a given estimation method. The Cramer-Rao Bound (CRB) [10, 11] has proved to be a usefhl tool because it provides a lower bound on the covariance of any unbiased estimate of the parameters vector in question. We use CRB as a criterion of successfid estimation. will THE OFDM SYSTEM BASEBAND MODEL Fig. 1 illustrates the discrete-time baseband equivalent OFDM system. Complex data symbols ~k transformed by the Inverse Discrete Fourier Transform (IDFT) yield a string of symbols bn , which represent the baseband equivalent of an OFDM signal with N-parallel subcarriers. Symbols bn are serially transmitted over a discrete-time channel whose impulse response h(z) is known and assumed to be of length <L. Any mathematical model can be exploited for channel description; here we use the Moving Average (MA) process as a most commonly used model for the LTI channel. The frequency difference between the local oscillators in the transmitter and the receiver is expressed by cq which denotes the normalized frequency as a fraction of the intercarrier frequency spacing l/N. In the following analysis we assume that the channel is a real-valued linear fiker and the transmitted signal is disturbed only by the complex additive white Gaussian noise (AWGN) Wn with power density iV/2. The purpose of the frequency offset estimator is to evaluate u on the basis of receiving symbols cin in order to preset the Carrier Recovery System, which then compensates the residual fi-equency offset and tracks the random phase to make coherent demodulation possible. In particular, as a result of such a compensation and tracking process the ICI is substantially reduced. 0-7803-4902-4/98/$10.00 (c) 1998 IEEE %, h Channel Cn d. h(r) 32’ Fig. 1 Discrete-Time Baseband Equivalent OFDM System PROBLEM DEFINITION We assume that symbols ak are independent and jointly uniformly distributed. Symbols b = [b.,...2 bN_ 1] being a linear combination of a = [a.,... ,aN_l ] have for large IV an approximately Gaussian distribution with independent real and imaginary parts. Moreover, bn constitutes an i.i.d. sequence. Random sequence Cn is a response of the LTI filter to bn and therefore forms a colored Gaussian sequence with independent real and imaginary parts (due to the real-valued coefficients of the filter). As assumed by the suggested model, symbols dn, which act on the input of the frequency offset estimator, are given by: dn = Cnexp{–j2mn} + Wn (1) where: L–1 Cn = ~bkh(n – k) = k=O L–1 L–1 = ~~oRe{bk }h(n - k) +jk~oh{bk)h(n - k) (2) Rdd (k - n)= E{dnd~ } = No = Rcc(k -n)exp{j2m(k -n)) + ~d(k -n) (3) -L+l<(k-n)<L-l where: Rcc(k -n)= ~{CnC; } = -n]) = cr~Rhh(k -n) = G~l_~~>~l)h(l+[k –L+l<(k–n)<L–l (4) cr~=E{bn 2}= NE{ak2}=N& The unbiased sample autocorrelation of dn equals: i& (t)= — 1 M–t M-t Z dnd;+t ~=~ l<t<L–1 (5) where &fdenotes the number of data points. The frequency offset estimation method proposed herein is based on matching sequence fidd (t) to sequence Rdd (t) Model (1) will be called here the Moving Average Exponent (MAEXP) process. Frequency estimation of such processes is described in [6, 7, 8]. In this paper, we propose to explore the PWLS approach. Such a method of estimating the ffequency offset is a direct one and there is no need for decision feedback, which has a potential of error propagation. d. THE ESTIMATION ALGORITHM is stationary and zero-mean (E{ak }= O). Its autocorrelation function is expressed by: with t=(k-n). Parameter a is thus estimated by solving the following least square minimization problem: ~(~) = [~dd - R&j (@]H~&j & = min J(a) a I& where &d = “dd (1),..., fidd (L – 1)~ - R& (a)] (6) is the vector of the sample autocorrelation and Rdd (cx) denotes the vector of autocorrelation lags corresponding to a. W is a positive definite complex weighted matrix. It is shown in [6], that CRB can be attained by appropriate choice of this matrix. 0-7803-4902-4/98/$10.00 (c) 1998 IEEE There is a wide variety of computationally efficient methods for solving (6). Here we use the Newton-Gauss minimization algorithm. This method uses exact first-order derivatives and approximated Hessian to minimize criterion (6) as described in [7]. Fig. 2 shows the block diagram of the frequency offset estimator. The received symbols are stored in a buffer of length L. Then the sample autocorrelation is computed according to (5). This covariance is then updated with any new symbol, which appears at the input. lim A4S(n,k) = M+ca . L-1 = ~=~L+?$ (t)&jd(t lim AZl(n,k) Weighted Matrix derivation: It is shown in [6, 7] that if X{ + L-1 Z &*c(t)Rcc(t t=–L+l + [k – u]) + (11) Z&c(k – n)o~ + o~d(k –n)] where o:- AWGN variance. It can be seen from (5), (8) and(11) that & is an unbiased d%d(~). . , ~ The exact CRB derivation: - Denote: (7) - R&j)(&~ Vm= Re{dm } - R~J*)]-l E{(6 - a)2} = [DWD~]-1 (12) ‘mi-M = ‘{dm] then: (8) According to the definitions of &d and Rdd (equations (3, 4, 5)) it follows that the elements of the matrix S are: S(n, k) = n’Z=o,l,...,l-l A new real process Vz 1= 0,1, ....2ikf– 1 is Gaussian for large N and holds: E(VmVm+M) = O Therefore = E{[&d (n) - Rdd (n)][&d (k) - R&j (k)]* } = the nZ=o,l,..., Al-l Fisher Information (13) Matrix of v~ (1= 0,1, ....2M- 1)may be derived as follows: 1 (9) ‘(M-n)(M-k)x F(a) M-n M-k x ~~1 ~~1 &jd (/ - i?Z)& k,n=l.. x estimator of a with standard deviation cr~ cc 0(— &) ~=~-1= = [E{(Ji& = e-~2zaIk-nl A4+co DERIVATION OF THE PROPOSED METHOD Statistical analysis of the suggested method when dn is described by an auto-regressive process is carried out in [6, 7, 8]. Here we just compute an appropriate weighted matrix W for the MAEXP process and derive the exact Cramer-Rao Bound forgiven frequency offset estimator. da k - n) k,n=l.. L-l By substitution (3) in (10) we receive: STATISTICAL ANALYSES AND cm ~ = + . (/ - m + k -n) L–l In order to check the asymptotic behavior of S, which actually is a covariance matrix of the sample covariance estimator, we assume that A4 + m (and of course L<< M). So the following is valid: = ~trace{r -lzr-l~} da (14) where r is the correlation matrix of real process V1and is defined as: r(n,Z) = E{ VIVn); n,l= 0...2A1-1 (15) The CRB equals therefore: Cl/B(a) =F(a)-1 SIMULATION AND NUMERICAL (16) RESULTS As it has been shown in the previous section, the weighted matrix W depends on the estimated parameter a . Two ways to perform the minimization in order to get the best performance of the estimator are considered. 0-7803-4902-4/98/$10.00 (c) 1998 IEEE r ------ ------- ------ ------- ------ ! Frequency Offset Estimator 1 1 1 : d!nl > Buffer of length L ---.-- ------ . . . . . . ------ Sample Covariance Sample Covariance Calculator ------ . . . . . . .------, 1 [ 1 II I Update Sample Covariance Accumulator ‘ fl~~(t): li 1 1 IzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA I : 1 I CoVariance i : 1 I Vector I I I I I I 1 1 ~ &d(t)zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML I I I I I 1 i Newton-Gauss # I Minimization ; : 1 1 1 1-------------------------------------------- ------------------------------ . Fig. 2 Block diagram of the PWLS based Estimator CONCLUSIONS 1) To compute the first derivative and approximated Hessian of cost fimetion J(cr), when W = W(a) and apply the Newton-Gauss Algorithm for minimization. 2) To set W=l, identity matrix, and solve the problem. To compute weighted matrix W for next step by substitution of the estimated value of & from the previous one. Repeat the steps until convergence. Both ways require computing the weighted matrix at least once for all steps. This task may be very complicated for real time applications and is extremely difficult to implement in the systems with a high bit rate. We propose here to use suboptimal estimation method, when ?7’=1 aJways. The simulations shown later indicate that the estimation accuracy degrades negligibly regarding CRB, while the algorithm becomes much faster. The simulation was performed for the following conditions: N=32, a = 0.2. The symbols a represent QPSK modulation format and a second order Butterworth filter is used to represent the LTI channel. Fig. 3a and Fig, 3b show the estimated mean and the variance of the estimation error as a function of the number of received symbols Cn for SNR=5dB. In Fig. 3b the CRB is also shown for comparison. As can be seen, accuracy degradation because of a suboptirnal estimation is not more than 3dB regarding CRB. Fig. 4a and Fig. 4b show the estimated mean and the variance of the estimated error as a fiumtion of SNR after 300 symbols have been transmitted. The variance of estimation error does not converge to zero and reaches a steady state for large values of SNR The CRB is also shown for comparison. The difference is not more than 3dB. We have presented a PWLS estimator of frequency offset in OFDM. No training or redundant information is needed, The fi-equency offset compensation is obtained by direct estimation and therefore is not affected by error propagation typical of decision feedback estimators. A computation was performed for an assumption that the channel is represented by an LTI filter described well by an MA process with known coefficients and that the distortion is caused by the AWGN only. For instance, such an assumption maybe valid in HDSL. We presented a suboptimal, due to non-optimal choice of the weighed matrix W, frequency offset estimator. The computational complexity of such an estimator is low while its performance is not more than 3dB worst than CRB. Estimated fi-equency offset may be used as frequency preset for conventional carrier recovery systems working in the phase tracking mode. Consequently coherent OFDM may be obtained. If differential encoding is used, such an approach may prevent phase shift accumulation and decrease the bit error rate. ACKNOWLEDGMENT The authors would like to thanks Dr. J. Frances for his suggestions and help. 0-7803-4902-4/98/$10.00 (c) 1998 IEEE ,~.s 02008 -0.2006 --- -- CRB --- Error Variance True Parameter Mean \ ‘: 0.1996J ,& ~m I 300 400 Em 600 7m BOO 9m I om 10.” L — —— zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJI 400 500 600 700 800 900 1 D $00 200 am 0 # Transmitted Symbols Fig. 3b Estimated Error Variance # Transmitted Symbols Fig. 3a Estimated Mean 1o“< , 1 -- CRB --- Error Variance k---- 10-’, “-.L 0.19971 0 10 20 30 40 60 I 60 10“”0 20 30 —— 40 —.— 50 60 Besson and P. Stoics, “Sinusoidal Signals with Random Amplitude: Least-Squares Estimators and Their Statistical Analysis”, IEEE Trans. Signal Processing, vol. 43, pp. 2733-2744, Nov. 1995. [7] O. Besson and P. Stoics, “Estimation the Parameters of a Random Amplitude Sinusoid from its Samples Covariance’s”, ICASSP’96, Atlanta, GA, May 7-10, 1996. [8] O. Besson, “Improved detection of a random amplitude sinusoid by constrain least squares technique”, Signal Processing, vol. 45, no. 3, pp. 347-356, Sept, 1995. [9] S. P. Bruzzone and M. Kaveh, “Information tradeoffs in using the sample autocorrelation function in ARMA parameter estimation”, IEEE Trans., Acoust. Speech Signal Processing, vol. 32, no. 4, pp. 701-714, Aug. 1984. [10] B. Porat, Digital Processing of Random Signals: Theory and Methods, Prentice Hall, 1994. [11] T. Soderstrom and P. Stoics, System Identification, Prentice Hall, London, 1989. [12] P. Stoics, T. Soderstrom, and V. Symonyte, “On estimating the noise power in array processing”, Signal Processing, vol. 26, pp. 205-220, Feb. 1992. [6] O. REFERENCES M. Luise and R. Reggiannini, “Carrier Frequency Acquisition and Tracking for OFDM Systems”, IEEE Trans. Communication, vol. 44, pp. 1590-1598, Nov. 1996. [2] H. Sari, G. Karam, and I. Jeanclaude, “Channel equalization and carrier synchronization in OFDM systems”, in Audio and Video Digital Radio Broadcasting Systems and Techniques, R. De Gaudenzi and M. Luise, Eds. Amsterdam Elsevier, 1994. [3] F. Daflara and O. Adarni, “A new fkequency detector for orthogonal multicarrier transmission techniques”, in Proc IEEE VTC ’95, Chicago, IL, July 1995. [4] M. Luise and R. Reggiannini, “Carrier frequency recovery in all-digital modems for burst-mode transmission”, IEEE Trans. Commun., vol. 43, pp. 1169-1178, Feb./Mar./Apr. 1995. [5] J. Beek, M. Sandell and P. O. Borjesson, “ML Estimation of Time and Frequency Offset in OFDM Systems”, IEEE Trans. Signal Processing, vol. 45, pp. 1800-1805, July 1997. to — SNR [dB] Fig. 4b Estimator Error Variancezyxwvutsrqponmlkj SNR [dB] Fig. 4a Estimated Mean [1] -————.—. 0-7803-4902-4/98/$10.00 (c) 1998 IEEE