We address the problem of carrier frequency offset

(CFO) synchronization in OFDM communications systems in the

context of frequency-selective fading channels. We consider the case where the transmitted symbols have constant modulus, i.e., PSK constellations. A novel blind CFO estimation algorithm is  developed. The new algorithm is shown to greatly outperform a recently published blind technique that exploits the fact that practical OFDM systems are not fully loaded. Further, the proposed algorithm is consistent even when the system is fully

loaded. Finally, the proposed CFO estimator is obtained via a one-dimensional search, the same as with the existing virtual subcarrier-based estimator, but achieves a substantial gain in performance (10-dB SNR or one order of magnitude in CFO MSE)

Ghogho, M.

Swami, A.

English

White Rose Research Online

This is a repository copy of Blind frequency-offset estimator for OFDM systems transmitting constant-modulus symbols.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/693/Article:Ghogho, M. and Swami, A. (2002) Blind frequency-offset estimator for OFDM systems transmitting constant-modulus symbols. IEEE Communications Letters, 6 (8). pp. 343-345.ISSN 1089-7798 https://doi.org/10.1109/LCOMM.2002.802040eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse See Attached Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. IEEE COMMUNICATIONS LETTERS, VOL. 6, NO. 8, AUGUST 2002 343Blind Frequency-Offset Estimator for OFDMSystems Transmitting Constant-Modulus SymbolsMounir Ghogho, Member, IEEE, and Ananthram Swami, Senior Member, IEEEAbstract—We address the problem of carrier frequency offset(CFO) synchronization in OFDM communications systems in thecontext of frequency-selective fading channels. We consider thecase where the transmitted symbols have constant modulus, i.e.,PSK constellations. A novel blind CFO estimation algorithm isdeveloped. The new algorithm is shown to greatly outperforma recently published blind technique that exploits the fact thatpractical OFDM systems are not fully loaded. Further, theproposed algorithm is consistent even when the system is fullyloaded. Finally, the proposed CFO estimator is obtained via aone-dimensional search, the same as with the existing virtualsubcarrier-based estimator, but achieves a substantial gain inperformance (10-dB SNR or one order of magnitude in CFOMSE).Index Terms—Dispersive channels, frequency offset, OFDM.I. INTRODUCTIONTHE presence of carrier frequency offset (CFO) causes lossof orthogonality between the subcarriers in an OFDMscheme and leads to increased bit error rate. Consequently,there has been considerable work done in the area of CFOestimation. A number of pilot-assisted CFO synchronizationtechniques are available in the literature. Blind CFO syn-chronization is attractive because it saves bandwidth, i.e., notraining pilots are required. A blind CFO estimator was recentlyproposed in [1] (see also [2] for more details); this estimator,designed to work with dispersive channels, exploits the fact thatpractical OFDM systems are not fully loaded, i.e., the numberof information-bearing subcarriers is smaller than the size ofthe FFT block.Here, we consider the case where the transmitted sym-bols have constant-modulus (as in various European ACTSprojects, the 6–18-Mbps rates in the IEEE 802.11 5 GHz andHIPERLAN2 standards; [4]). Judicious use of this side infor-mation leads to a simple CFO estimator which significantlyoutperforms the estimators in [1], [2].II. SIGNAL MODELOFDM modulation consists of (usually a power of 2) sub-carriers, equispaced at a separation of , whereis the total system bandwidth. The subcarriers are mutually or-Manuscript received February 20, 2002. The associate editor coordinating thereview of this paper and approving it for publication was Dr. H. Sari. The workof M. Ghogho was supported by the U.K. Royal Academy of Engineering.M. Ghogho is with the School of Electronic and Electrical Engineering, Uni-versity of Leeds, Leeds LS2 9JT, U.K. (e-mail: m.ghogho@ieee.org).A. Swami is with the Army Research Lab, Adelphi, MD 20783 USA (e-mail:a.swami@ieee.org).Publisher Item Identifier 10.1109/LCOMM.2002.802040.thogonal over a time interval of length . Each subcar-rier is modulated independently with symbols having constantmodulus (CM), e.g., PSK constellations. Each OFDM blockis preceded by a cyclic prefix whose duration is longer thanthe delay spread of the overall propagation channel (includingtransmit and receive filters), so that inter-block interference canbe eliminated at the receiver, without affecting the orthogonalityof the sub-carriers. Practical OFDM systems are, in general, notfully loaded in order to avoid aliasing. In this case, some of thesub-carriers at the edges of the OFDM block are not modulated;these subcarriers are referred to as virtual subcarriers (VSCs).Their number is dictated by system design requirements and is,in general, about 10% of . Letdenote the entire set of subcarriers; also letdenote the -element set of active ormodulated subcarriers ( and odd) and letdenote the set of VSCs.1At the receiver, the output of the matched filter is sampledwith period . After discarding the cyclic prefix, thecomplex envelope of the baseband received signal in an OFDMblock can be described as(1)for , where is a CM data sequence,; (a real number, ) is the CFOnormalized to ; is additive noise which is assumed tobe zero-mean, uncorrelated, circularly symmetric and Gaussianwith variance ; is the complex channelresponse at theth subcarrier frequency(2)where are the coefficients of the -tap channel.Since is a CM sequence, in (1) may be rewrittenas where is the angle of , .Note that the unknown parameters, in (1), areparameterized by only ( ) complex parameters—thechannel coefficients . This property is the core of theproposed algorithm.Notation: Let , and denote the real part, theimaginary part and the argument of the complex variable. Letdenote the discrete-time Fourier transform of atfrequency1The assumptions thatN is even andN is odd are not critical.1089-7798/02$17.00 © 2002 IEEE344 IEEE COMMUNICATIONS LETTERS, VOL. 6, NO. 8, AUGUST 2002III. FREQUENCY-OFFSETESTIMATIONWe consider the parameters and as unknowndeterministic parameters. Since the additive noise is white,circularly symmetric and Gaussian, the maximum-likelihoodestimates (MLEs) of , andare obtained by minimizing the norm(3)Setting and assuming that ,2 the MLEsatisfies(4)Substituting into (3), the cost function becomes(5)(6)(7)where we have used Parseval’s theorem and the fact thatfor . The linear relations in (2) should betaken into account in the minimization of the above criterion.Notice that is equivalent to the cost function used inan existing blind CFO estimator [1], which exploits the virtualsub-carriers ( ). The extra term exploits the CM propertyof the transmitted symbols. Let where is aninteger and is a fraction of the sub-carrier spacing.The estimator based solely on cannot determine ; onlycan be estimated. The identifiability conditions of our newestimator are, therefore, the same as those of the VSC-basedestimator (see [3] and references therein).A. Non-Dispersive ChannelsIn this case, and . Using (5), theMLE of is the solution to the -norm problem(8)Note that the VSC-based estimator is equivalently obtained bymaximizing the -norm of the same vector as above. Indeed,using Parseval’s theorem, we have that2If jH j = 0,  becomes nonidentifiable. A channel withL + 1 taps canhave at mostL zeros that coincide with the subcarriers; clearlyN > L ensuresthatH 6 0, 8n 2 N . With N > L, all theH ’s cannot be zero; hencehavingH = 0 for somen 2 N will not affect the final CFO estimator.B. Dispersive ChannelsFirst, we note that is not a function of . Therefore,only needs to be considered in the minimization ofwithrespect to . This should be carried out under the constraintwhich follows from (2). The above constraint states that ,, are quadratic forms of the channel coefficients.We modify as follows(9)The motivation for using this modified criterion is given next.Since our goal is to estimate the CFO by using a 1-D optimiza-tion procedure, we need to eliminate the ’s from the crite-rion in (7). Toward this objective, we use instead ofsince the former can be re-parameterized as follows:whereThe sequence is then described by linear combinationsof the ( ) elements of the parameter vector. Note thatboth and are real-valued vectors. The cost function isno longer the ML cost, but it does lead to a consistent estimateofBy assumption we have that . Settingyields the following closed-form estimator(10)where and denotes the pseudo-inverse.Note that the matrix depends only upon ; hence, itspseudo-inverse can be precomputed. Substitutinginto (5)–(7), we obtain the following cost function:where(11)GHOGHO AND SWAMI: BLIND FREQUENCY-OFFSET ESTIMATOR FOR OFDM SYSTEMS 345In the above criterion, exploits the CM property of thesymbol constellations. Using Parseval’s theorem, the two cri-teria can be merged together; after dropping constant terms, thepseudo-MLE of the CFO is obtained as(12)We refer to the estimator minimizing as the VSC-basedestimator, which is equivalent to the estimator in [1] for a spe-cific choice of . We refer to our estimator in (12) as theVSC&CM-based estimator.We have, therefore, shown that the CM property canbe exploited in blindly estimating without increasingthe dimension of the optimization procedure. Indeed, theVSC&CM-based estimator is obtained by a one-dimensionalsearch, the same as with the existing VSC-based estimator.The computation of the criterion in (12) is obviously moredemanding than that of . However, as we see next, theperformance improvement of the VSC&CM-based estimatorover the VSC-based estimator is more than one order of mag-nitude (i.e., more than 10 dB!) provided (typically), which is usually the case.If the system is fully loaded, then and the VSC-based estimator fails. Our estimator, which in this case shouldbe referred to as the CM-based estimator, continues to performwell. Furthermore, in this case, where is theidentity matrix and hence no matrix inversion is required.However, in this case, the CFO can be unambiguously estimatedonly in the interval 0.5 .IV. PERFORMANCEANALYSISHere, we compare our estimator with the estimator of [1], [2]via Monte Carlo simulations. Only one OFDM block is used. Weconsider a total of sub-carriers where onlysub-carriers are used, i.e., the number of virtual sub-carriers is. The transmitted symbols are drawn from equiprob-able 8-PSK constellations. The channel coefficients are gener-ated using an uncorrelated Rayleigh scattering model with ex-ponential power delay profile, i.e.,, with . The CFO is generated randomly between2and 2. Both the channel coefficients and the CFO are generatedrandomly for each Monte Carlo run. The signal-to-noise ratiois defined as whereis the variance of the noise. Mean-square error (MSE) wasestimated empirically from 200 Monte Carlo runs.A. Performance Versus the SNRFig. 1 displays the MSE versus SNR for a seven-tap channel,i.e., . The performance of the proposed estimator is sig-nificantly better than that of the estimators in [1] and [2]. Fig. 1indicates a 10-dB improvement.B. Unknown Channel OrderHere, we set the actual channel order to and estimatethe MSE when the assumed channel order varies from 0 to 25,Fig. 1. MSE of CFO estimators versus SNR;L = 6.Fig. 2. MSE of CFO estimators vs. assumed channel order; the actual channelorder is 6;SNR = 15 dB.with dB. The results are depicted in Fig. 2. It is seenthat our estimator still performs well if the assumed channelorder is (slightly) larger than the actual channel order. This im-plies that in practice, if is unknown, one only needs to use areasonable upper limit on the possible values of.REFERENCES[1] H. Liu and U. Tureli, “A high-efficiency carrier estimator for OFDMcommunication,”IEEE Commun. Lett., vol. 2, pp. 104–106, Apr. 1998.[2] U. Tureli, H. Liu, and M. D. Zoltowski, “OFDM blind carrier offsetestimation: ESPRIT,”IEEE Trans. Commun., vol. 48, pp. 1459–1461,Sept. 2000.[3] M. Ghogho, A. Swami, and G. Giannakis, “Optimized null-subcarrierselection for CFO estimation in OFDM over frequency-selective fadingchannels,” inProc. Globecom-2001, San Antonio, TX, Nov. 2001, pp.202–206.[4] R. van Nee and R. Prasad,OFDM for Wireless Multimedia Communi-cations. Norwood, MA: Artech House, 2000, pp. 4–6.

Blind frequency-offset estimator for OFDM systems transmitting constant-modulus symbols

M. Ghogho

A. Swami

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Blind frequency-offset estimator for OFDM systems transmitting constant-modulus symbols

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