Research Article  Open Access
Simple Algorithms for Estimating the Symbol Timing Offset in DCTBased Multicarrier Systems
Abstract
Multicarrier modulation based on discrete cosine transform (DCTMCM) is a candidate technique for future wireless communication. DCTMCM provides a better usage of the spectrum, allowing for much lower outofband radiation than conventional OFDM. In this paper, we address the problem of correcting the symbol timing offset in DCTMCM. Two sliding windowbased correlation algorithms are proposed. The new approaches consider a prefix and a suffix that are inserted into each data symbol, which are the replica of part of the useful data. The performance of the proposed approaches is verified and tested, and their usefulness and effectiveness are illustrated by computer simulations.
1. Introduction
The next generation of communication systems (5G and beyond) will employ higher bandwidths and data rates that are able to provide enhanced data services to users, also improving the efficiency of spectrum usage. Consequently, channel partitioning methods will be the dominant medium access techniques due to their appealing features [1], such as higher system performance and spectrum efficiency, typically measured in (bit/s)/Hz per unit area or cell.
Discrete multitone (DMT) modulation and orthogonal frequency division multiplexing (OFDM) are examples of channel partitioning techniques that employ the strategy of replicating the data into each transmitted symbol. This replica of part of the data, usually referred to as redundant samples, leads to a channel partitioning that provides a set of ideally independent subchannels in which channel equalization can be easily performed [1]. DMT and OFDM are efficiently implemented using discrete Fourier transforms (DFTs). These techniques, however, have drawbacks, such as a sensitivity to offsets in time and carrier frequency. Moreover, the temporal pulse shape of DFT is a rectangular window, thus DFTbased multicarrier modulation (DFTMCM) suffers from high sidelobe radiation.
As alternatives to DFTbased solutions, several authors have proposed the use of discrete cosine transforms (DCTs) [2–6]. DCTbased multicarrier modulation (DCTMCM) offers benefits such as excellent spectral compaction and energy concentration and less intercarrier interference leakage to adjacent subcarriers or that DCT uses only real arithmetic [2, 3]. Furthermore, DCTMCM provides a better usage of the spectrum allowing for much lower outofband radiation. As has been widely reported (e.g., [3]), the bandwidth of a DCTMCM system can be only half of the bandwidth required by DFTMCM with the same number of subcarriers.
In any multicarrier modulation (MCM) systems, symbol timing estimators play an important role in the receiver to find the start of the symbol of the received signal. Numerous methods for DFTMCM have been proposed since the publication of the maximum likelihood (ML) estimation algorithm in an additive white Gaussian noise (AWGN) channel and the method of Schmidl and Cox [7–9]. We refer the reader to the studies in [10–12] for a detailed list of recent synchronization approaches and their implementations in some hardware/software environments for different realworld applications. Unfortunately, this field has received less attention for the DCTMCM, even though synchronization is also crucial and critical. In [13], two joint maximum likelihood frequency offset and phase offset estimators, considering only AWGN channels, were presented. More recently, a new symbol synchronization method for optical fast OFDM is reported in [14]. As a result, the problem of a feasible technique for DCTMCM still remains open.
The main novelty of this paper is the proposal of simple algorithms to carry out precise symbol timing estimation for DCTMCM. The new approaches exploit the redundancy inserted as both left and right extensions in DCTbased systems. It is shown by computer simulations that the proposed approaches yield efficient and feasible time estimation solutions, providing satisfactory results in several communication scenarios.
The rest of this paper is organized as follows. In Section 2, the system model for the considered multicarrier transceivers is described. Section 3 presents the proposed algorithms for DCT2e and DCT4ebased systems. Section 4 provides a performance evaluation of the proposed algorithms and its comparison to DFTbased systems, and, finally, concluding remarks are given in Section 5.
2. System Model
Figure 1 is a general block diagram to implement MCM. At the transmitter, the data are processed by an point inverse transform , with being the number of subchannels or subcarriers. At the receiver, a discrete transform is performed.
In DFTMCM (e.g., OFDM/DMT systems), is an IDFT, the redundancy of length , e.g., a cyclic prefix (CP) (see Figure 2(a)), is introduced at the beginning of each length data symbol to be transmitted, is a timedomain equalizer that shortens the effective channel to an appropriate length, is a DFT, and, finally, the frequencydomain equalizer (FEQ) block corrects the dispersive effect of the transmission channel.
(a)
(b)
(c)
(d)
In DCTMCM, can be any kind of inverse DCT [16], with two different types of redundancy: symmetric extension (SE) or zeropadding (ZP). This work focuses on TypeII even discrete cosine transform (DCT2e) and TypeIV even discrete cosine transform (DCT4e), assuming SE [2, 5]. Unlike DFTbased systems, two symmetrical extensions of samples are appended: a prefix or left extension (LE), and a suffix or right extension (RE). Figures 2(b) and 2(c) show examples of the symmetric extensions for DCTMCM.
In Figure 1, the received sequence in presence of a temporal delay and carrier frequency offset can be expressed as where and is a term related to the noise. An additional constraint of DCTMCM is that the channel impulse response must be symmetric, as shown in Figure 2(d). It is important to highlight that some channels satisfy this condition, such as chromatic dispersion in singlemode fibers [4]. If not, the prefilter of Figure 1 is located at the receiver to enforce the symmetry in . There are some appropriate techniques to design this prefilter [2, 6]. For the sake of simplicity, we adopt a model of symmetric discretetime channel for DCTMCM. If this condition is not satisfied, then the training symbol used in [6] can be employed to perform an initial timing symbol synchronization using the algorithms proposed in this paper, and then the channel estimation of can be carried out.
Finally, the channel length of is assumed to be less than the length of the redundancy (SE or CP); i.e., . For a more thorough description of DCTMCM, we refer the reader to [2, 5, 16].
3. Proposed Metrics
An analysis using the correlation between two signals provides a quantitative measure of the similarity between them. In STO estimation, the idea behind [7–9] and other approaches in [10, 11] is to use correlation functions to find the similarities that are shared by the data part of the symbol and the redundant samples in the prefix/suffix.
In DCTMCM, the maximum value or peak is also reached (in the absence of noise) when there exists a set of samples that are pairwise correlated. The time position of this maximum value is useful in finding the symbol timing and the phase of the correlation could yield the frequency estimate. Given that in DCTMCM the redundancy follows a mirror (anti)symmetry and the channel impulse response is a wholesample symmetry (WS) sequence, there are two blocks of samples that are pairwise correlated. For example, let us consider the absence of noise, the fact that is an integer number and , and the time instants represented in Figure 3. At the start of the symbol, specifically at instants and , we have In addition, at the end of the symbol, the receiving signal presents identical (for the HS extension) or opposite (for the HA extension) values in two consecutive samples, and . That is, we have where for the HS extension or for the HA extension (see Figure 2 and [2, 16]). Notice that there are other sets of identical samples at the start of the received symbol: and identical (for the HS extension) or opposite (for the HA extension) samples at the end of the received symbol: Considering the above, we propose new blind algorithms to perform tight timing offset and coarse frequency synchronization for DCT2eMCM and DCT4eMCM. The first proposed approach estimates the timing offset using the maximum of the metric:where The normalization term considers the energy of the windowed signals: where
(a)
(b)
The second proposed method is based on Schmidl and Cox’s approach. We adapt the metric of [8] to the scheme of prefix and suffix used in DCTMCM, and, as a result, the timing offset can be estimated using the correlation peak given bywhere Finally, assuming perfect symbol synchronization at and absence of noise, we have that for those pairwise correlated samples. Considering the above condition, a coarse fractional CFO estimator () for DCTMCM can be carried out by means of the expression
4. Experimental Study
In our experiments, we have assumed systems with 512 subcarriers and 4QAM modulation. Each experiment consists of an average of 10,000 simulation runs with STO independently generated and uniformly distributed over . For the first experiment, the length of the redundant samples is modified to study its effects on different parameters. For the rest, we have considered CPs of 32 samples for the DFTMCM and the same number of redundant samples for each prefix (LE) and suffix (RE) in DCTMCM. The setup used in our experiments is summarized in Table 1.

Example 1. The influence of the number of redundant samples () is studied. The performance of the proposed estimators (6)–(10), for both DCT2e and DCT4e, is assessed and compared with that derived in [10] based on [7], for SNR values of 10 and 16 dB and over discrete memoryless additive white Gaussian noise (AWGN) channels, with no intersymbol interference. The SNR is defined by the ratio of the power of the received signal and the variance of the AWGN. Figure 4 shows the resulting meansquare error (MSE) of the STO estimates evaluating the trial and the estimate values of the timing offset. As was documented in [7], it can be seen in Figure 4 that the performance of the DFTMCM estimator is asymptotically independent of the number of redundant samples, provided that the amount of data for the CP is longer than a certain threshold value, which decreases with the SNR (see Figure 4 and [7, 10]). As the length increases beyond the thresholds, the DFTMCM time estimator does not improve. On the contrary, the behaviour of the proposed schemes for DCTMCM is no longer flat. MSE decays monotonically and the improvement is achieved by increasing the SNR and/or the number of redundant samples. Note that MSEs for the DFTMCM saturate at and , and that the performance of the proposed schemes is much better, with MSE, when SNR and the number of redundant data is greater than 22 (see (6)) or 17 (see (10)).
(a)
(b)
We also use our analysis to show how BER changes with the length of the redundant samples. Figure 5 depicts the uncoded BER curves, and, given that the results have been practically indistinguishable for DCT2e and DCT4e, only one curve to represent DCTMCM is included. Notice that the BER curves become practically saturated for DFTMCM, and increasing the number of redundant samples does not improve BER. An immediate conclusion that can be drawn from these plots is that the proposed metrics exhibit good performance in the estimation of the STO when the length of the redundant samples is at least onehalf of the maximum considered offset. For DCTMCM, the performance of the proposed approaches for SNR = 10 dB does not improve as the length of the redundant sample increases from 20 to 32, whereas it is possible to obtain BER values lower than for SNR = 16 dB.
Example 2. In our second set of experiments, all of the results are obtained over three different channels: AWGN and two sets of 100 wireless fading channels each, according to the ITU Pedestrian A and Vehicular A channels [15]. The multipath channels were generated with Matlab’s stdchan using the channel models itur3GPAx and itur3GVAx with a carrier frequency GHz and two different sets of parameters: (a) 4 km per hour as pedestrian velocity, ns and length ; and (b) 100 km per hour as mobile speed (the moving speed has been chosen considering that the average vehicle speed on European highways lies in the range of 90100 km per hour under freeflow conditions), ns and length . These channels are referred to as PED200 and VEH200, respectively.
Figure 6(a) shows the resulting meansquare error (MSE) of the STO estimates. As can be seen, the performance of is much better than the others, with the exception of the VEH200 channel, for which [7] provides the best values of MSE.
(a)
(b)
The estimated temporal offsets have also been obtained. Based on the fact that an accuracy of less than three samples at low SNR makes the approach robust in fading channels [7, 10], we have obtained the probability that the estimated error in the time offset is greater than two samples. The numerical results for the three different transceivers over the channels are displayed in Figure 6(b). These curves show that DCTbased methods significantly improve the performance compared to the DFTMCM over the entire range of SNR values considered in our experiments.
Next, we compare the BER performance for the three different channels. Figure 7 depicts the obtained results versus the received SNR. In this figure, perfect synchronization (PS) represents the case where STO is perfectly known at the receiver. Interestingly, for a particular channel, the BER performance of the proposed metrics and is practically indistinguishable, and for this reason we only represent one curve. In addition, the proposed methods have the best performance. They are close to the perfect synchronization for the AWGN and PED200 channels, and a performance degradation occurs for the VEH200 channel. On the other hand, the DFTMCM system is severely affected by a timing offset, especially for the PED200 and VEH200 channels.
Example 3. Lastly, we consider a more complex scenario, where a normalized CFO is added following a uniform distribution in the interval , which is estimated by the approach in [7] for DFTMCM, whereas (13) is employed for DCTMCM. Figures 8(a) and 8(b) characterize the MSE and of the STO estimates, respectively, whereas Figure 9 plots the resulting BER. Comparing the performance of the proposed algorithms with DFTMCM, it is proved that the new approaches work properly in AWGN and PED200 channels, significantly improving the performance and reducing the BER. The proposed schemes also perform well for the VEH200 channel, although in this case the CFO has an effect on our schemes. On the other hand, the BER is practically flat for DFTMCM in the more challenging realistic scenarios of PED200 and VEH200.
(a)
(b)
5. Conclusions
DCTMCM offers benefits such as excellent spectral compaction and energy concentration and allows a better usage of the spectrum. In this work, new techniques of symbol timing estimation for DCTMCM systems are proposed. The new metrics are based on the fact that the correlation between two signals provides a quantitative measure of their similarity, and they take into account the fact that the redundant samples in DCTMCM are inserted as left (prefix) and right (suffix) extensions satisfying (anti)symmetry properties. A wide set of simulations have been carried out to test the effectiveness of the proposed approaches. It can be established from the results of our experiments that the proposed techniques perform well, especially in Gaussian and low dispersive channels. It is further shown that considering the BER and the probability of the estimate error in the time offset, a performance gain over the conventional DFTbased technique is obtained when the length of the redundancy increases.
Data Availability
The data used to support the findings of this study are not needed actually. The performance study was based on the data randomly generated by the simulation code.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This work was partially supported by the Spanish Ministry of Economy and Competitiveness through Project TEC201564835C31R.
References
 J. Cioffi, “Multichannel modulation,” in Digital Communications, https://web.stanford.edu/group/cioffi/doc/book/chap4.pdf. View at: Google Scholar
 N. AlDhahir, H. Minn, and S. Satish, “Optimum DCTbased multicarrier transceivers for frequencyselective channels,” IEEE Transactions on Communications, vol. 54, no. 5, pp. 911–921, 2006. View at: Publisher Site  Google Scholar
 P. Tan and N. C. Beaulieu, “A comparison of DCTbased OFDM and DFTbased OFDM in frequency offset and fading channels,” IEEE Transactions on Communications, vol. 54, no. 11, pp. 2113–2125, 2006. View at: Publisher Site  Google Scholar
 X. Ouyang and J. Zhao, “Singletap equalization for fast OFDM signals under generic linear channels,” IEEE Communications Letters, vol. 18, no. 8, pp. 1319–1322, 2014. View at: Publisher Site  Google Scholar
 F. CruzRoldán, M. E. DomínguezJiménez, G. SansigreVidal, J. PineiroAve, and M. BlancoVelasco, “Singlecarrier and multicarrier transceivers based on discrete cosine transform typeIV,” IEEE Transactions on Wireless Communications, vol. 12, no. 12, pp. 6454–6463, 2013. View at: Publisher Site  Google Scholar
 F. CruzRoldán, M. E. DomínguezJiménez, G. SansigreVidal, D. Luengo, and M. Moonen, “DCTbased channel estimation for single and multicarrier communications,” Signal Processing, vol. 128, pp. 332–339, 2016. View at: Publisher Site  Google Scholar
 J.J. Van De Beek, M. Sandell, and P. O. Börjesson, “ML estimation of time and frequency offset in OFDM systems,” IEEE Transactions on Signal Processing, vol. 45, no. 7, pp. 1800–1805, 1997. View at: Publisher Site  Google Scholar
 T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Transactions on Communications, vol. 45, no. 12, pp. 1613–1621, 1997. View at: Publisher Site  Google Scholar
 T. Kung and K. K. Parhi, “Optimized joint timing synchronization and channel estimation for OFDM systems,” IEEE Wireless Communications Letters, vol. 1, no. 3, pp. 149–152, 2012. View at: Publisher Site  Google Scholar
 Y. S. Cho, J. Kim, W. Y. Yang, and C. G. Kang, MIMOOFDM Wireless Communications with Matlab, IEEE Press, John Wiley & Sons, 2010.
 A. A. Nasir, S. Durrani, H. Mehrpouyan, S. D. Blostein, and R. A. Kennedy, “Timing and carrier synchronization in wireless communication systems: a survey and classification of research in the last 5 years,” EURASIP Journal on Wireless Communications and Networking, vol. 2016, no. 1, article 180, 2016. View at: Publisher Site  Google Scholar
 T. H. Pham, V. A. Prasad, and A. S. Madhukumar, “A HardwareEfficient Synchronization in LDACS1 for Aeronautical Communications,” IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 26, no. 5, pp. 924–932, 2018. View at: Publisher Site  Google Scholar
 F. Gao, T. Cui, A. Nallanathan, and C. Tellambura, “Maximum likelihood based estimation of frequency and phase offset in DCT OFDM systems under noncircular transmissions: algorithms, analysis and comparisons,” IEEE Transactions on Communications, vol. 56, no. 9, pp. 1425–1429, 2008. View at: Publisher Site  Google Scholar
 J. Zhao, S. K. Ibrahim, D. Rafique, P. Gunning, and A. D. Ellis, “Symbol synchronization exploiting the symmetric property in optical fast OFDM,” IEEE Photonics Technology Letters, vol. 23, no. 9, pp. 594–596, 2011. View at: Publisher Site  Google Scholar
 3GPP, “Technical specification group radio radio access network. user equipment (UE) radio transmission and reception (FDD) (release 7),” 3GPP TS 3GPP TS 25.101, 3rd Generation Partnership Project, 2007. View at: Google Scholar
 F. CruzRoldán, M. E. DomínguezJiménez, G. SansigreVidal, P. AmoLópez, M. BlancoVelasco, and A. BravoSantos, “On the use of discrete cosine transforms for multicarrier communications,” IEEE Transactions on Signal Processing, vol. 60, no. 11, pp. 6085–6090, 2012. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2018 Fernando CruzRoldán et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.