Baghaki and Champagne EURASIP Journal on Advances in Signal EURASIP Journal on Advances Processing (2018) 2018:4 DOI 10.1186/s13634-017-0526-4 in Signal Processing RESEARCH Open Access Joint frequency offset, time offset, and channel estimation for OFDM/OQAM systems Ali Baghaki* and Benoit Champagne Abstract Among the multicarrier modulation techniques considered as an alternative to orthogonal frequency division multiplexing (OFDM) for future wireless networks, a derivative of OFDM based on offset quadrature amplitude modulation (OFDM/OQAM) has received considerable attention. In this paper, we propose an improved joint estimation method for carrier frequency offset, sampling time offset, and channel impulse response, needed for the practical application of OFDM/OQAM. The proposed joint ML estimator instruments a pilot-based maximum-likelihood (ML) estimation of the unknown parameters, as derived under the assumptions of Gaussian noise and independent input symbols. The ML estimator formulation relies on the splitting of each received pilot symbol into contributions from surrounding pilot symbols, non-pilot symbols and additive noise. Within the ML framework, the Cramer-Rao bound on the covariance matrix of unbiased estimators of the joint parameter vector under consideration is derived as a performance benchmark. The proposed method is compared with a highly cited previous work. The improvements in the results point to the superiority of the proposed method, which also performs close to the Cramer-Rao bound. Keywords: OFDM/OQAM, Joint estimation, Filter bank multicarrier, Carrier frequency offset, Sampling time offset, Channel impulse response, Cramer-Rao bound 1 Introduction paramount importance for the successful application of Due to the several benefits of multicarrier modulation OFDM/OQAM in practical systems. (MCM) over single carrier modulation, the former has There exist three main categories of synchronization been considered as the primary choice in the physical and channel estimation methods for OFDM/OQAM sys- layer implementation of telecommunication systems for tems: blind, semi-blind and pilot-based methods. While quite a long time. Among the MCM family, orthogonal blind methods, e.g., [4–7], provide higher spectral effi- frequency division multiplexing (OFDM) has been largely ciency by avoiding the overhead of training sequences, the studied and adopted in many wireless and wireline stan- requirement of a longer observation window for accurate dards [1, 2]. Still, as an alternative and promising form of estimation limits their tracking ability, rendering them MCM for future generations of wireless networks, a vari- less popular in most practical applications. In contrast, ant of OFDM based on offset quadrature amplitude mod- the semi-blind methods only require the transmission of ulation (OFDM/OQAM) has attracted much research a small number of parameters to resolve an estimation interest in recent years, due to its many advantages over ambiguity, e.g., [8] and as such, they offer a useful trade- classical OFDM, including higher spectral efficiency and off between spectral efficiency and estimation accuracy. reduced sensitivity to timing and frequency mismatch However, since in practical scenarios, the training symbol [3]. In spite of these advantages, accurate carrier fre- overhead needed to obtain a better estimation perfor- quency and timing synchronization along with channel mance is usually tolerated, our focus here is on pilot-based estimation (for the purpose of equalization) remain of synchronization and channel estimation. Compared to channel estimation, pilot-based car- rier frequency offset (CFO), and sampling time off- *Correspondence: [email protected] set (STO) estimation has received less attention in the Department of Electrical and Computer Engineering, McGill University, 3480 OFDM/OQAM literature. In [9], a maximum likelihood University Street, H3A 0E9 Montreal, Canada (ML) symbol timing estimator is derived by using two © The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 2 of 19 training symbols per burst transmission. In [10], using the Although the aforementioned synchronization and same preamble structure, the authors extend this work equalization problems have been separately addressed by proposing a joint ML-based estimator of the CFO and throughout the literature on OFDM/OQAM systems, STO. Then, to avoid the computational complexity of a only a few research papers can be found, e.g., [29], that are two dimensional ML search, a feasible joint estimation devoted to STO, CFO and channel estimation at the same method, called approximate maximum-likelihood (AML), time, let alone a joint estimation approach based on a uni- is developed by assuming a small CFO and using only one fied criterion. In fact, to the best of our knowledge, a joint OQAM preamble symbol per burst. The same authors, estimation method for OFDM/OQAM systems, account- in [11], propose a joint least-squares (LS) CFO and STO ing for all the three error sources, i.e., CFO, STO, and CIR, estimation method by using two identical OFDM/OQAM has not yet been developed. Hence, our focus in this paper pilot symbols per burst transmission. Therein, as a time- is to develop and investigate a general estimation method domain method, the estimation is performed before the to fill this need. analysis filter bank (AFB) at the receiver. In [12], by using Specifically, a new formulation of the joint parame- a polyphase network implementation of OFDM/OQAM, ter estimation problem in OFDM/OQAM system is first the preloading technique, and a conjugate-symmetric introduced, which is based on splitting the interference preamble, the CFO and STO are separately estimated. The term on the desired received pilot into adjacent pilot, non- CFO estimation exploits the phase difference between pilot and noise contributions. Then, by assuming Gaus- the adjacent pilots while the frame detection and STO sian noise and independent input symbols, a pilot-based estimation are derived based on the conjugate-symmetry joint ML estimator of CFO, STO, and CIR is derived. property. Moreover, in [13, 14], a joint CFO and STO Such a general approach offers many advantages, includ- estimation method is proposed by using a four-column ing a unified framework for the estimation of multiple preamble per burst transmission, which contains zeros parameters using a common preamble/burst structure in every other subcarrier and every other symbol time and the proper treatment of different types of interfer- index. ence in the estimator derivation. More importantly, a In contrast to the CFO and STO estimation, during significant performance improvement is expected in the the past decade, many pilot-based channel estimation joint estimation of the aforementioned error sources as schemes have been proposed for OFDM/OQAM systems, opposed to their separate estimation. Through numeri- which can be broadly classified into frequency domain cal simulations of wireless OFDM/OQAM transmission and time domain methods. Frequency domain methods, over multipath fading channels, the proposed estimator e.g., [15–23], rely on the assumption that the symbol dura- is evaluated by comparing the accuracy of the result- tion is much longer than the maximum channel delay ing parameter estimates to that obtained with a selected spread. While these methods are generally characterized benchmark approach among a few existing works where by a lower computational complexity, when the above all the three error sources of our focus are estimated1,as condition is not satisfied, they will suffer from a perfor- well as to the CRB. The simulation results show that, the mance degradation. Time domain methods, e.g., [24–26], proposed method is capable of significant improvements attempt to estimate the channel impulse response (CIR) in parameter estimation accuracy, performing close to the by using sequences of pilot tones. In [24], a time domain CRB. In turn, this improved performance leads to a lower CIR estimator is proposed based on the multiple sig- bit error rate (BER) of the compensated (i.e., synchronized nal classification (MUSIC) and LS algorithms. In [25], a and equalized) transceiver system. per-subchannel estimator is proposed in which the CIR This paper is a more developed and improved version on each subcarrier is estimated separately. In [26], the of our previous work [30] addressing the joint estima- authors exploit pilot tone structures in OFDM/OQAM tion problem under a more restrictive set of assumptions. systems to derive two new CIR estimators, namely the lin- Specifically, the new contributions of the current work ear minimum mean square error (LMMSE) and weighted include the following: least-square (WLS) estimators. The former exploits a pri- ori knowledge of the CIR covariance matrix while the • In [30], orthogonality of the OFDM/OQAM latter only requires the knowledge of the channel length; analysis/synthesis filters in the complex domain is both methods are benchmarked against the Cramer-Rao assumed, as opposed to