OFDM blind carrier offset estimation: ESPRIT - IEEE Xplore

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 9, SEPTEMBER 2000

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OFDM Blind Carrier Offset Estimation: ESPRIT Ufuk Tureli, Hui Liu, and Michael D. Zoltowski

Abstract—In orthogonal frequency-division multiplex (OFDM) communications, the loss of orthogonality due to the carrier-frequency offset must be compensated before discrete Fourier transform-based demodulation can be performed. This paper proposes a new carrier offset estimation technique for OFDM communications over a frequency-selective fading channel. We exploit the intrinsic structure information of OFDM signals to derive a carrier offset estimator that offers the accuracy of a super resolution subspace method, ESPRIT. Index Terms—Equalizer, estimation, multicarrier modulation, multipath channels, radio receiver.

I. INTRODUCTION

O

RHTOGONAL frequency-division multiplexing (OFDM) has received increasing attention in wireless broadcasting systems for its ability to mitigate the frequency-dependent distortion across the channel bandwidth [1]. While inherently robust against multipath fading, OFDM has been shown to be very sensitive to carrier drifts [2]. A carrier offset at the receiver can cause loss of subcarrier orthogonality, and thus can introduce interchannel interference (ICI) and severely degrade the system performance [2]. Accurate carrier offset estimation and compensation is more critical in OFDM than other modulation schemes. Most existing carrier estimation techniques rely on periodic transmission of reference symbols, which inevitably reduces bandwidth efficiency [3]. Recently, carrier offset estimators that exploit the redundancy of the cyclic prefix (CP) in OFDM [4] were proposed. Although no explicit pilot signal is required, the CP-based algorithms hinge on the availability of an excess CP, i.e., the CP is chosen beyond the length of the fading channel. In this sense, the bandwidth efficiency of the system is still affected since the extra CP acts like pilots. Moreover, such schemes can only estimate the frequency offset to the closest subcarrier because of ambiguity. The other class of blind estimators [5] has two steps for coarse and fine acquisition and restrictions on system design. Another

Paper approved by M. Luise, the Editor for Synchronization of the IEEE Communications Society. Manuscript received March 5, 1998; revised February 11, 1999 and January 28, 2000. This work was supported in part by the National Science Foundation (NSF) CAREER Program under Grant MIP-9703074. This paper was presented in part at the 31st Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, October 1997. U. Tureli was with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195-2500 USA. He is now with the Department of Electrical and Computer Engineering, Stevens Institute of Technology, Hoboken, NJ 07030 USA (e-mail: [email protected]). H. Liu is with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195-2500 USA (e-mail: [email protected]). M. D. Zoltowski is with the School of Electrical Engineering, Purdue University, West Lafayette, IN 47907-1285 USA (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(00)07529-2.

approach was proposed by Luise and Reggianini [6] with dataaided frequency acquisition step followed by a blind decisiondirected tracking algorithm that allows for greater accuracy. This paper proposes a new solution to the carrier offset estimation problem without using reference symbols, pilot carriers, or excess CP. The technique developed here provides a high-accuracy carrier offset estimate by taking advantage of the inherent structure of OFDM signals. Even when the OFDM signal is distorted by an unknown carrier offset, the received signal possesses a certain algebraic structure which will be shown to be sufficient to accomplish blind carrier estimation. The proposed estimator is in analytic form and the acquisition range is not limited to one-half the subcarrier spacing as is the case with some other algorithms [4]. A salient feature of this subspace-based algorithm is that it can offer the performance of a super-resolution subspace algorithm, viz., ESPRIT [7]. II. PROBLEM FORMULATION A. OFDM Principles Conceptually the OFDM modulation is comprised of serial-to-parallel conversion, an inverse discrete Fourier transform (IDFT), and parallel-to-serial conversion. Let , where denotes transpose, be the information bearing sequence at time . OFDM modulation is implemented by applying an -length IDFT operator to the , that is made up of symbols from the data data sequence zeros. because sufficiently stream, padded with wide filter guard bands are needed for reliable communications. and In the IEEE 802.11a standard draft,1 for example, . The unused subcarriers are often referred to as the virtual carriers [8]. Note that it is not necessary to create the virtual carriers at the transmitter, oversampling at the receiver site creates the same effect [8]. For presentational simplicity, as the usable subcarriers and we index subcarriers #0 to # as the virtual carriers. # to # samples of the IDFT output are given by The , where comprises the first columns of inverse DFT matrix . Once is obtained, an the -point CP is added to the IDFT output to cope with the FIR [1]. The removal composite channel of length , of the CP at the receiver end makes the received sequence the circular convolution of the transmitted sequence with the [1]. Within the th channel impulse response block, only the prefix portion of the signal will be affected by leakage from the previous blocks since the channel length . The effect of the channel is a mere scaling on each 1IEEE Standards Website. [Online]. Available: http://standards.ieee.org/ reading/ieee/std/lanman/802.11a.pdf

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subchannel. In particular, the data samples of the th block without the prefix can be expressed as

Similarly the backward vector

is given by

(1) ..

, is the where and diagonal matrix with main diagonal . Clearly, each subchannel can be recovered within a scalar ambiguity by applying the , . DFT to

.

.. .

(5) denotes the complex conjugate. Then, upon defining , the sample covariance matrix can be expressed as

where

B. Carrier Offset , the th received sample In the presence of a carrier offset s is modulated by a residual carrier obtained every . Taking into account the removed prefix, the received -point signal becomes (2)

. Since where , the matrix destroys the orthogonality among the subchannel , the carrier offset carriers and introduces ICI. To recover needs to be estimated and compensated before performing the DFT. The problem addressed in this paper is that of the estima, tion of the carrier offset from the receiver inputs, without the use of any training sequence or known input symbols. A new blind carrier offset estimator with fixed computational cost is presented in this section which resembles the ESPRIT [7] in that it exploits the data structure. The estimator inherits its robustness to frequency-selective channel from ESPRIT. Previously, we proposed to use the data structure using the orthogonality of the carriers in a MUSIC-like algorithm [8]. C. ESPRIT-Like Algorithm The standard ESPRIT algorithm exploits the shift-invariant structure available in the signal subspace of cisoids and estimates the parameters of interest through subspace decomposition and generalized eigenvalue calculation [7] and is not inherently limited in range to a portion of the unit circle [7]. We show in the following that in OFDM, the shift-invariant structure that enables ESPRIT manifests itself directly in the received signal. Given the th block of the received signal (1), one can form blocks of consecutive samples in both the forward and backward directions as follows:

(6) that span the signal subspace The eigenvectors of can be obtained by using spectral value decomposition (SVD) [9, pp. 70–75]. ESPRIT has very good finite sample perforto have rank , we require the mance [7]. However, for should be done considering the estimation acselection of curacy and frequency drifts; if not selected large enough, the sampled covariance matrix will not span the signal space and if selected too large, estimation will be sensitive to model errors as is the carrier offset drifts. The shift-invariant structure of exploited to find the eigenvalues of the diagonal matrix (7) denotes the pseudoinverse. The latter computation where [9, can be performed using QR decomposition because pp. 311–313]. The above gives a similarity transformation of which preserves the eigenvalues of . Since the trace of is , we can estimate from the sum of eigenvalues of , or simply its trace. In particular, can be calculated as (8)

The estimation is in closed form and thus is better than the MUSIC-like searching algorithm [8] in terms of resolution. (3) denotes conjugate transpose. From (2), it can be where easily verified that (4) where

consists of the first

rows of

, and

III. NUMERICAL RESULTS Computer simulations were performed for an OFDM system carriers, used carriers, and the CP length with . The carrier offset is estimated using the algorithm described in Section II-C. The actual carrier frequency offset is , where is the channel spacing. The signal-tonoise ratio (SNR) is defined as (9)


Fig. 1. MSE versus SNR, flat channel.

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Fig. 2. MSE versus SNR, multipath fading.

where is the additive white Gaussian noise vector for data block . The normalized mean square error (MSE) defined below is employed as a performance measure of the estimator (10) where is the estimate of obtained using the ESPRIT-like is the number of Monte Carlo (MC) trials. algorithm and In the first example in Fig. 1, the normalized MSE over 200 MC realizations is shown. The SNR was varied from 0 to 25 dB. In this simulation, the channel is assumed to be flat fading and the only impairment is additive noise. The proposed data blocks and the sample ESPRIT-like estimator uses , smoothed by a factor of 4. vectors are of length For comparison, we also estimated the carrier offset with the CP-based maximum-likelihood (ML) method by Van de Beek et al. [4]. It is seen that the performance of the ESPRIT-like algorithm is superior in performance even for small . The is a noninteger and a rational, which is particular offset not subject to the constraints in [5]. Hence, we show that the algorithm works for arbitrary values of carrier offset. OFDM is proposed for multipath frequency-selective fading channels so it is of interest to compare the methods in a multipath fading environment. Fig. 2 shows the performance of the ESPRIT-like algorithm on this channel with all parameters the same as in the previous setup except for multipath channel whose sampled impulse response is given by . The blind carrier offset estimation algorithm employed by Van de Beek et al. [4] assumes a flat-fading channel and thus suffers from multipath effects in frequency-selective multipath fading channels. The degradation may be significant for channels that have support up to the CP length. In this case, CP-based methods have an error floor which cannot be improved by increasing the SNR [4] as can be seen in Fig. 2. The ESPRIT-like algorithm offers excellent performance as indicated by the large gap between the MSE of ESPRIT and the CP-based method.

IV. CONCLUSION A new technique is presented for blind synchronization of OFDM communications over fading channels encountered in digital broadcasting and cellular communications. ESPRIT-like blind estimation algorithm exploits the structure information of OFDM signals with low fixed complexity. We have compared the acquisition performance of this blind algorithm with the CP-based ML algorithm by Van de Beek et al. [4]. ACKNOWLEDGMENT The authors would like to thank the referees and M. Luise, Editor for Synchronization of the IEEE Communications Society, for their constructive comments and careful reviews. REFERENCES [1] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Commun. Mag., vol. 33, pp. 100–109, Feb. 1995. [2] T. Pollet, M. Van Bladel, and M. Moeneclaey, “BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise,” IEEE Trans. Commun., vol. 43, pp. 191–193, Feb./Mar./Apr. 1995. [3] F. Classen and H. Meyr, “Frequency synchronization algorithms for OFDM systems suitable for communication over frequency-selective fading channels,” in Proc. 1994 IEEE 44th Vehicular Technology Conf., 1994, pp. 1655–1659. [4] J. 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, pp. 1800–1805, July 1997. [5] T. M. Schmidl and D. C. Cox, “Blind synchronization for OFDM,” Electron. Lett., vol. 33, pp. 113–114, Feb. 1997. [6] M. Luise and R. Reggiannini, “Carrier frequency acquisition and tracking for OFDM systems,” IEEE Trans. Commun., vol. 44, pp. 1590–1598, Nov. 1996. [7] R. Roy, A. Paulraj, and T. Kailath, “ESPRIT—A subspace rotation approach to estimation of parameters of cisoids in noise,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 1340–1342, Feb. 1986. [8] H. Liu and U. Tureli, “A high efficiency carrier estimator for OFDM communications,” IEEE Commun. Lett., vol. 2, pp. 104–106, Apr. 1998. [9] G. Golub and C. Van Loan, Matrix Computations: Third Edition. Baltimore, MD: Johns Hopkins Univ. Press, 1996.