IEEE COMMUNICATIONS LETTERS, VOL. 8, NO. 3, MARCH 2004 171

Frequency Offset Estimation WithFast Acquisition in OFDM System

Zhongshan Zhang, Ming Zhao, Haiyan Zhou, Yuanan Liu, Member, IEEE, and Jinchun Gao

Abstract—A new carrier frequency offset estimation schemein orthogonal frequency-division multiplexing (OFDM) system isproposed. The carrier frequency offset estimation includes acqui-sition and tracking, and the acquisition range is as large as onehalf of overall signal bandwidth. The proposed tracking estimatoris a maximum-likelihood estimator, and in AWGN channel, theCramer-Rao lower bound is met at high signal-to-noise ratio(SNR); in multipath channel, the tracking algorithm works wellat moderate SNR. Timing synchronization can be also performedduring the course of acquisition.

Index Terms—Frequency offset estimation, multipath channel,orthogonal frequency-division multiplexing (OFDM).

I. INTRODUCTION

ORTHOGONAL frequency-division multiplexing(OFDM) systems are very sensitive to carrier fre-quency offset, and frequency offset with even a small

fraction of subcarrier spacing will degrade the performanceof OFDM receiver greatly [1]. Many methods have beenpresented for frequency offset estimation [1]–[4]. All thesealgorithms transmit identical patterns at the transmitter; basedon the phase rotations between these patterns at the receiver,carrier frequency offset can be estimated. In this letter, a newpattern is proposed for carrier frequency offset estimation.Based on the special structure of a training-symbol-block, theacquisition range for the proposed scheme is as large as onehalf of overall signal bandwidth, and high performance trackingis also obtained. The same training-symbol-block can be alsoused for accurate timing synchronization. Section II describesthe frequency offset acquisition algorithm and Section IIIpresents the tracking algorithm. Simulation results are given inSection IV, and conclusions are drawn in Section V.

II. CARRIER FREQUENCY OFFSET ACQUISITION

A new carrier frequency offset estimation scheme is proposedin this letter. A special training-symbol-block is used for bothcarrier frequency offset estimation and timing synchronization.At the transmitter, the training-symbol-block contains twoequal-length training symbols in time domain, and the secondtraining symbol is the inverse repeat of the first one. Thetraining-symbol-block has the following form:

(1)

Manuscript received June 2, 2003. The associate editor coordinating thereview of this letter and approving it for publication was Prof. P. Loubaton.This work was supported in part by 863 project “the Research of WirelessTransportation Techniques in the Next Generation Mobile Communicationunder 2001AA121031”, by the Key Project of Ministry of Education of Chinaand Doctorial Foundation, and by the Panasonic Joint Project.

The authors are with the Center for Wireless Communication, BeijingUniversity of Posts and Telecommunications (BUPT), Beijing 100876, China(e-mail: adhoc@bupt.edu.cn; manyzhao@163.com; mumuzhou@263.net;yuliu@bupt.edu.cn; gjc@bupt.edu.cn).

Digital Object Identifier 10.1109/LCOMM.2004.823423

Fig. 1. Expectations of M(d) for correct synchronization (d = 0) andincorrect synchronization (d <> 0).

where denotes the DFT length of training symbols.At the receiver, without considering of the channel atten-

uation and additive white Guassian noise (AWGN), the rela-tionship between corresponding samples in a received training-symbol-block is:

(2)

where denotes the th sample of and denotes the carrierfrequency offset normalized to a subcarrier spacing of trainingsymbols.

A timing metric is defined:

(3)

where is a time index corresponding to the first sample in awindow of samples.

Fig. 1 shows the expectations of when dBand . Without loss of generality, the start position indexof a training-symbol-block is assumed to be zero. It is shownin Fig. 1 that the expectation of is a function of withperiod of , and within each period, a main lobe appears;the expectation of is a constant independent of .Within one period (without loss of generality,is assumed), is satisfied only in a smallvicinity which stands in the center of the main lobe; in the mainlobe, increases as approaches to the main lobe center.This property is the basis for the proposed carrier frequencyoffset acquisition algorithm.

The detailed illustration of acquisition is shown in Fig. 2.Since the expectation of is a function of with period of

1089-7798/04$20.00 © 2004 IEEE

172 IEEE COMMUNICATIONS LETTERS, VOL. 8, NO. 3, MARCH 2004

Fig. 2. Frequency offset acquisition.

, the acquisition range of the proposed algorithm is limitedwithin subcarrier spacings, which equals to one half ofoverall signal bandwidth. At the transmitter, 2 identical training-symbol-blocks are transmitted. At the receiver, samples ofthe training sequence are buffered. The length of guaranteesone integral training-symbol-block being buffered.

Because the carrier frequency offset is fullyunknown at the start of acquisition, a methodof lookup is used: presupposed values

where and )are used to compensate for the carrier frequency offsetof the buffered samples (multiplying the bufferedsamples by where and

). denotes the lookup interval (in thefollowing section we will know that the tracking range for theproposed scheme is subcarrier spacing,and guarantees the remainingcarrier frequency offset being not beyond the tracking range).

For th compensated training sequence, themaximum ( ) is represented as whichimplies the appearance of an integral training-symbol-block; if

for each , then denotes thecarrier frequency offset estimated by acquisition algorithm. The

estimated start of that integral training-symbol-block is alsothe output of acquisition.

After acquisition, if the remaining carrier frequency offset ex-ceeds the tracking range, tracking algorithm will not work cor-rectly, and this is called Missing Lock. For usable SNR, ifis large enough, the probability of Missing Lock ( ) is neg-ligible. For example, when , dB and thecarrier frequency is assumed to be 2.5 subcarrier spacings, ifis set to 0.2, is far less than .

III. CARRIER FREQUENCY OFFSET FINE ADJUSTMENT

After acquisition, the remaining frequency offset needs to befurther corrected. For a received training-symbol-block under

certain SNR condition, the log-likelihood function for the car-rier frequency offset , is the logarithm of the probabilitydensity function . Using the correlation properties ofsamples in , the log-likelihood function can be written as

(4)

Like in [2], the 2-D complex-valued Guassian distributionis

(5)

where

(6)

(7)

The carrier frequency offset can be estimated by maxi-mizing the log-likelihood function. By taking the partial deriva-tive of with respect to and setting the result to zero, themaximum-likelihood estimate of is given by (8) at the bottomof the following page.

ZHANG et al.: FREQUENCY OFFSET ESTIMATION WITH FAST ACQUISITION IN OFDM SYSTEM 173

In order to make the proposed tracking algorithm workcorrectly, for each

should be satisfied, i.e., the tracking range issubcarrier spacing. By using the method

in [5], its Cramer–Rao lower bound is

(9)

IV. SIMULATION RESULTS

In this letter, it is assumed that the channel character doesnot change during one training-symbol-block period. A wirelesssystem operating at 5 GHz and with bandwidth of 10 MHz isassumed. An outdoor dispersive, fading environment is chosen:the channel is modeled to consist of 4 independent Rayleigh-fading taps with path delay of 0, 5, 9, and 12 samples andpath gains are:

(10)

Since the proposed acquisition algorithm reliably guaranteesthe remaining carrier frequency offset being not beyond thetracking range, only tracking performance is evaluated in thissection. The performance of the proposed tracking algorithmwith (not including cyclic-prefix) is compared tothat of Schmidl’s algorithm (fractional part carrier frequencyoffset estimator) [3] with training symbol length of 256 anda cyclic-prefix length of 16. In simulation, QPSK modulationis assumed. Estimation variance error is used for evaluatingthe algorithms’ performances. Variance error is defined as

where denotes the true carrier frequency

offset. Carrier frequency offset is assumed to be 7.813 kHz,which is not beyond the tracking ranges of both algorithms.Simulation results are shown in Fig. 3. In the AWGN channel,for high SNR, the performance of proposed algorithm is about7.27 dB better than that of Schmidl’s algorithm. In multipathchannel, the performance of Schmidl’s algorithm will notdegrade too much comparing to that in AWGN channel, pro-vided that the cyclic-prefix is longer than the channel impulseresponse; while for the proposed algorithm, is estimatedbased on the maximum power tap signal, and other taps becomeinterference noise, which reduces the effective SINR. For large

, the Cramer-Rao lower bound of the proposed tracking algo-rithm in multipath channel nearly equals to .That for Schmidl’s algorithm is ( denotes onehalf of the training symbol length, and in this letter, ).For the proposed tracking algorithm, ifis satisfied, its tracking error will be smaller than that ofSchmidl’s algorithm.

Fig. 3. Performance of Schmidl’s tracking algorithm (fractional part carrierfrequency offset estimator) and that of the proposed tracking algorithm.

V. CONCLUSIONS

In this letter, a new carrier frequency offset estimation schemeis proposed. The acquisition range for the proposed scheme isas large as one half of the total signal bandwidth. The proposedtracking estimator is an ML estimator, and in AWGN channel,its performance is about 7.27 dB better than that of Schmidl’sestimator at high SNR. In multipath channel, at moderate SNR,comparison of the proposed estimator with Schmidl’s estimatoralso illustrates the superior performance of the proposed schemewith regard to estimation accuracy; at high SNR, as SNR in-creases, a performance floor appears in the proposed estimator.Accurate timing synchronization can be also performed duringthe course of acquisition.

REFERENCES

[1] P. H. Moose, “A technique for orthogonal frequency division multi-plexing frequency offset correction,” IEEE Trans. Commun., vol. 42, pp.2908–2914, Oct. 1994.

[2] J.-J. van de Beek and M. Sandell, “ML estimation of time and frequencyoffset in OFDM systems,” IEEE Trans. Signal Processing, vol. 45, pp.1800–1805, July 1997.

[3] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchro-nization for OFDM,” IEEE Trans. Commun., vol. 45, pp. 1613–1621,Dec. 1997.

[4] M. Louise and R. Reggiannini, “Carrier frequency acquisition andtracking for OFDM systems,” IEEE Trans. Commun., vol. 44, pp.1590–1598, Nov. 1996.

[5] D. Rife and R. Boorstyn, “Single-tone parameter estimation from dis-crete-time observations,” IEEE Trans. Inform. Theory, vol. IT-20, pp.591–598, Sept. 1974.

(8)