- Research Article
- Open Access

# Joint Symbol Timing and CFO Estimation for OFDM/OQAM Systems in Multipath Channels

- Tilde Fusco
^{1}, - Angelo Petrella
^{2}and - Mario Tanda
^{3}Email author

**2010**:897607

https://doi.org/10.1155/2010/897607

© Tilde Fusco et al. 2010

**Received:**27 May 2009**Accepted:**13 November 2009**Published:**27 December 2009

## Abstract

The problem of data-aided synchronization for orthogonal frequency division multiplexing (OFDM) systems based on offset quadrature amplitude modulation (OQAM) in multipath channels is considered. In particular, the joint maximum-likelihood (ML) estimator for carrier-frequency offset (CFO), amplitudes, phases, and delays, exploiting a short known preamble, is derived. The ML estimators for phases and amplitudes are in closed form. Moreover, under the assumption that the CFO is sufficiently small, a closed form approximate ML (AML) CFO estimator is obtained. By exploiting the obtained closed form solutions a cost function whose peaks provide an estimate of the delays is derived. In particular, the symbol timing (i.e., the delay of the first multipath component) is obtained by considering the smallest estimated delay. The performance of the proposed joint AML estimator is assessed via computer simulations and compared with that achieved by the joint AML estimator designed for AWGN channel and that achieved by a previously derived joint estimator for OFDM systems.

## Keywords

- Root Mean Square Error
- Orthogonal Frequency Division Multiplex
- Orthogonal Frequency Division Multiplex System
- Cyclic Prefix
- Orthogonal Frequency Division Multiplex Signal

## 1. Introduction

In the last years, the interest for filter-bank multicarrier (FBMC) systems is increased, since they provide high spectral containment. Therefore, they have been taken into account for high-data-rate transmissions over both wired and wireless frequency-selective channels. Moreover, they have been considered for the physical layer of cognitive radio systems [1]. One of the most famous multicarrier modulation techniques is orthogonal frequency division multiplexing (OFDM), embedded in several standards such as digital audio and video broadcasting or Wi-Fi wireless LANs IEEE 802.11a/g. Other known types of FBMC systems are Filtered Multitone (FMT) systems, that have been proposed for very high-speed digital subscriber line standards [2] and are under investigation also for broadband wireless applications [3] and, moreover, OFDM based on offset QAM modulation (OQAM), considered by the 3GPP standardization forum for improved down-link UTRAN interfaces [4].

Unlike OFDM, OFDM/OQAM systems do not require the presence of a cyclic prefix (CP) in order to combat the effects of frequency selective channels. The absence of the CP implies on one hand the maximum spectral efficiency and, on the other hand, an increased computational complexity. However, since the subchannel filters are obtained by complex modulation of a single filter, efficient polyphase implementations are possible. Another fundamental difference between OFDM and OFDM/OQAM systems is the adoption in the latter case of pulse shaping filters very well localized in time and frequency [5, 6].

OFDM/OQAM systems are more sensitive to synchronization errors than single-carrier systems. In particular, carrier frequency-offset (CFO) and symbol timing (ST) estimation errors can lead to a performance degradation. For this reason, it is very important to derive efficient synchronization schemes. In the last years several studies have been focused on blind or data-aided synchronization for OFDM/OQAM systems. For example, in [7, 8] blind CFO estimators have been derived. Moreover, in [9] a blind joint CFO and ST estimator is proposed. Furthermore, in [10] a synchronization scheme for data-aided ST and CFO estimation with robust acquisition properties in dispersive channels is developed. Finally, in [11, 12] a full synchronization method utilizing frequency domain scattered pilots in the time domain is proposed. However, all cited estimators are designed for down-link communications.

In this paper we consider the problem of data-aided synchronization for OFDM/OQAM systems in multipath channels. In particular, the joint maximum-likelihood (ML) estimator for CFO, amplitudes, phases, and delays, exploiting a short known preamble, is derived. The ML estimators for phases and amplitudes are in closed form. Moreover, under the assumption that the CFO is sufficiently small, a closed form approximate ML (AML) CFO estimator is obtained. By exploiting the obtained closed form solutions a cost function whose peaks provide an estimate of the delays is derived. In particular, the ST (i.e., the delay of the first multipath component) is obtained by considering the smallest estimated delay. The proposed joint estimator is derived with reference to a down-link scenario; however, by following an approach similar to that considered in [13], it can be easily modified to be exploited for up-link communications. The performance of the proposed joint AML estimator is assessed via computer simulations and compared with that achieved by the joint AML estimator designed for AWGN channel and that achieved by a previously derived joint estimator for OFDM systems. The paper is organized as follows. In Section 2 the OFDM/OQAM system model is described. In Section 3 the proposed data-aided estimator is described. In Section 4 numerical results obtained in AWGN and multipath channel are presented and discussed. Finally, conclusions are drawn in Section 5.

Notation 1.

, superscript denotes the complex conjugation, real part, imaginary part, and absolute value. Moreover, denotes transpose and the argument of a complex number in . Finally, lower case boldface symbols denote column vectors.

## 2. System Model

where is the information-bearing signal, is the number of multipath components, and, , , and denote amplitude, phase, and delay, respectively, of the th path. Moreover, in (1) is a zero-mean complex-valued white Gaussian noise process with independent real and imaginary part, each with two-sided power spectral density . The received signal is filtered with an ideal lowpass filter with a bandwidth of , where denotes the sampling period. The sampled signal is equal to

where is the OFDM/OQAM symbol interval and denotes the number of information-bearing symbols in the burst. Moreover, in (2) is the set of size of used subcarriers, and denote the real and imaginary part of the complex data symbol transmitted on the th subcarrier during the th OFDM/OQAM symbol, while the real-valued and unit-energy pulse-shaping filter is bandlimited within .

## 3. Joint Symbol Timing and CFO Estimator

From (8), it immediately follows that the ML estimator for phase and amplitude of the th path is given by

Moreover, from (16) and (21)–(24) it follows that

presents the highest peaks. Moreover, the lowest among the obtained delays represents an estimate of the ST. If the number of paths is not known in advance, a sufficiently high number of paths should be considered to avoid to lose a strong path and, moreover, to avoid to consider very weak paths. The obtained delays can be substituted in (24) to obtain the CFO estimate, and, finally, phases and amplitudes can be obtained from (13) and (14), respectively. Note that the numerical results reported in the next section show that the considered approximation of (25) leads to a symbol timing estimator with satisfactory performance if the number of subcarriers is sufficiently large.

Moreover, in Appendix is derived an approximate expression for the mean square error (MSE) of the AML CFO estimator in (24) for a single-path channel and in the case of perfect ST synchronization. In particular, in Appendix it is shown that in this case the MSE can be approximated by

where and . Note that for the MSE in (29) is coincident with the MSE of the CFO estimator for OFDM systems proposed by Schmidl and Cox (SC) in [14]. It is worthwhile to emphasize that the MSE in (29) has been derived by neglecting the interference at the output of each matched filter due to adjacent subcarriers. Therefore, the actual performance of the proposed AML CFO estimator presents a floor that is not predicted by (29). However, it is shown in the next section that the approximate expression in (29) can be exploited in the range of moderate values.

## 4. Numerical Results and Comparisons

In this section the performance of the proposed joint AML estimator is assessed via computer simulations. A number of Monte Carlo trials has been performed under the following conditions (unless otherwise stated):

- (1)
- (2)
- (3)
the length of the considered prototype filter (designed with the frequency sampling technique [15]) is , where the overlap parameter is fixed at ;

- (4)
the considered multipath fading channel model is the ITU Vehicular A [16], which has six multipaths with differential delays 0, 0.31, 0.71, 1.09, 1.73, and 2.51 microseconds and relative powers , and ;

- (5)
the channel is fixed in each run but it is independent from one run to another.

In the first set of simulations we have tested the sensitivity of the performance of the derived CFO estimators to the condition (28) and to the interference due to the data burst sent after the training symbol. Specifically, four operating conditions have been considered:

- (1)
in the first case, denoted as OC1, condition (28) is satisfied and, moreover, to reduce the interference due to the data symbols, the useful data in the whole burst is delayed with respect to the preamble of the burst by one OFDM/OQAM symbol interval;

- (2)
in the second case, denoted as OC2, condition (28) is not satisfied and the data burst is not delayed;

- (3)
in the third case, termed OC3, condition (28) is not satisfied and the data burst is delayed;

- (4)
in the fourth case, termed OC4, condition (28) is satisfied and the data burst is not delayed.

## 5. Conclusions

In this paper we have dealt with the problem of data-aided synchronization for OFDM/OQAM systems in multipath channels. In particular, the joint ML estimator for CFO, amplitudes, phases, and delays, exploiting a short known preamble, has been derived. Exploiting the closed form ML estimators for phases and amplitudes and the closed form AML CFO estimator for small CFO values, a cost function that can provide an estimate of the ST, has been obtained. The performance of the joint AML1 estimator for multipath channel has been assessed via computer simulations and compared with that achieved by the joint AML2 estimator designed for AWGN channel. Moreover, a comparison with the performance achieved by the SC estimator for OFDM systems has been made. The results have shown that if it satisfied a condition involving the training symbol and the data burst is delayed by one OFDM/OQAM symbol interval with respect to the training burst, the AML CFO estimators assure a performance similar to that achieved by the SC estimator in multipath channel A, while the AML ST estimators outperform the SC estimator. Moreover, an approximate expression for the MSE of the AML CFO estimators has been derived that can be exploited to predict the actual performance in the range of moderate SNR values. Finally, a comparison between the BER obtained with the adoption of the AML and SC estimators followed by a one-tap equalizer with perfect knowledge of the channel and of the residual synchronization errors has been made. The results have shown that both in AWGN and multipath channel A the AML estimators assure a negligible degradation with respect to the perfectly synchronized system while the adoption of the SC synchronization scheme leads to an error floor due essentially to the inaccuracy of the ST estimates.

## Declarations

### Acknowledgment

This work was supported in part by the European Commission under Project PHYDYAS (FP7-ICT-2007-1-211887).

## Authors’ Affiliations

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