Performance comparison of time-of-arrival estimation techniques for LTE signals in realistic multipath propagation channels

2.1 Theoretical bound analysis

The theoretical Cramer-Rao lower bound (CRLB) is commonly used to indicate the maximum achievable accuracy of an unbiased estimator under moderate to high SNR conditions. In Wang and Fattouche (2010), the CRLB for the TOA estimation with generic orthogonal frequency division multiplexing (OFDM) signals in an additive white Gaussian noise (AWGN) environment was derived based on an approximation of the mean square bandwidth of the OFDM signal and was compared to that of a pseudo-random noise (PRN) signal. It was proven that the required SNR of the OFDM signal is 4.8 dB lower than the PRN signal to achieve the same ranging accuracy in the asymptotic region. In the presence of AWGN plus a zero-mean complex Gaussian process modeled interference, a closed-form CRLB expression of the TOA estimation was studied for OFDM systems (Karisan et al., 2011). The CRLB and Ziv-Zakai bound for characterizing the TOA estimation performance of realistically modeled OFDM signals, e.g., those in the third-generation partnership project (3GPP) LTE specification, in the AWGN channel were derived in Driusso et al. (2014). These bounds were then used to analyze the TOA estimation behavior in both the asymptotic and threshold regions considering two sets of subcarrier power distributions. The analysis revealed the specific strategy for subcarrier power distribution to achieve a satisfactory asymptotic or threshold performance. In del Peral-Rosado et al. (2012b) and Gentner, Sand, et al. (2012), the CRLB expression in Wang and Fattouche (2010) was extended for analyzing the theoretical limits of the TOA estimation accuracy with LTE pilot signals. The impact of inter-cell interference in the LTE network and the effectiveness of inter-cell interference coordination techniques were also analyzed in del Peral-Rosado et al. (2012b) by computing the equivalent signal-to-interference plus noise ratio in the AWGN channel. A more rigorous CRLB for the TOA estimation with LTE signals in the AWGN channel was provided in Xu et al. (2016) without exploiting the approximation of the mean square bandwidth. A comparison of the CRLBs derived from several LTE pilots or a combination of these pilots was given.

For multipath channels, the closed-form expression of the Fisher information matrix for all the channel parameter estimations in OFDM systems was derived (Wang et al., 2013). Based on this, the effects of signal and channel parameters on the ranging accuracy were then quantified in multipath channels. Researchers in del Peral-Rosado et al. (2018) calculated the CRLBs for several channel estimation models, i.e., the periodic-tap model, hybrid-tap model, and arbitrary-tap model. These theoretical expressions can be used to deduce the achievable accuracy of several joint maximum likelihood (ML) time-delay and channel estimators developed from these channel estimation models in static multipath channels. The CRLBs for the TOA estimation using the PRS with 5G radio access were derived considering a multipath channel with AWGN, carrier frequency offset, and Wiener phase noise (Luan, 2017). The effects of flexible bandwidth, subcarrier spacing, and power allocation on the TOA estimation accuracy were investigated to guide the PRS redesign for 5G-based positioning. However, it should be noted that no theoretical CRLB exists for varying multipath channels (Xu et al., 2016).

2.2 Error modeling

In general, the TOA measurements may be corrupted by many error sources, such as thermal noise, non-line-of-sight (NLOS) propagation, multipath effects, clock drift, and receiver motion. Empirical or theoretical error models for the TOA or TDOA measurements in LTE systems have been established for a variety of scenarios. A measurement campaign was performed in an indoor scenario (Gentner, Muñoz, et al., 2012) during which the receiver generated the TOA estimation by detecting the first correlation peak above a heuristic threshold using the synchronization signals with scattered pilots. An approximate Gaussian distributed TDOA error model was then built through statistical analysis. In Kong and Kim (2016), the analytical expressions for the probability density function (PDF) of the TOA errors caused by the scattering environment, multipath effects, and thermal noise were derived, respectively. Note that the TOA estimation was related to the resolved first arrival path measured from the PRS. A Gaussian scatter distribution model was assumed and both the Rician and Rayleigh channels were considered for representing outdoor multipath environments. The overall TOA and TDOA error distributions were then obtained by integrating all the error PDFs due to a variety of factors. Specifically, for a low LTE signal bandwidth, the TOA errors introduced by multipath channels were modeled as a skew-t distribution (Müller et al., 2016). The parameters of the skew-t distribution were obtained by fitting training data of the TOA errors obtained from several types of TDL channel models. It was demonstrated that the application of these error models in the positioning algorithm yielded an improved positioning accuracy.

2.3 Computer simulations

Various channel models have been specified by the 3GPP to characterize the multipath fading propagation conditions present in typical LTE channels. Through a large number of channel realizations, computer simulations can be implemented to statistically test the LTE positioning capabilities in complex propagation channels.

It was not until the end of 2009 that the dedicated PRS devoted to the OTDOA technique had been introduced by the 3GPP in Release 9 (3GPP, 2010; Sven, 2014). Before that, some pioneering work was done using synchronization signals and scattered pilot signals in the 3GPP LTE. In Mensing et al. (2007), two symbol timing synchronization approaches for OFDM systems, i.e., the Schmidl-Cox algorithm and the Minn algorithm, were examined with respect to their positioning capabilities using the scattered pilot signals in the WINNER C2 wide area channel model. To combat the inter-cell interference, an interference cancellation scheme combined with the conventional correlation-based timing estimation algorithm was developed in Mensing et al. (2009b). Mensing et al. (2009a) further integrated a data feedback approach using already decided data symbols into the method in Mensing et al. (2009b) to overcome the limited number of pilot symbols. These two algorithms were tested with the second synchronization signal (SSS) considering both an AWGN channel and a typical urban multipath channel. The authors in Benedetto et al. (2011) applied a parabolic interpolation around the maximum of the correlation function obtained with the primary synchronization signal (PSS) to refine the TOA estimation with performance evaluation in an AWGN channel. However, this algorithm has been shown to be vulnerable to harsh multipath conditions where the first arriving path is severely attenuated and a few close-in multipaths exist (Gadka et al., 2019).

The correlation-based peak or FPD algorithms are currently widely used for LTE-based positioning. In del Peral-Rosado et al. (2012c), the accuracy of the conventional ML estimator based on the time-domain correlation results using the PRS was evaluated for severe multipath scenarios represented by the TDL channel models and geometric-based stochastic channel models. The multipath resolution capabilities for a range of LTE PRS bandwidths were also characterized. The effect of multipath on the first peak estimator using the PRS was then assessed in del Peral-Rosado et al. (2012a) for the TDL channel models in a noisy environment. In addition, the joint impact of inter-cell interference and multipath was also analyzed. Another first arriving path detection method with two threshold settings on the basis of the time-domain correlation results was proposed in Huang and Xu (2013) and Xu et al. (2016). Their estimation performances with the PRS or CRS were analyzed using simulations for the TDL channel models. In Rydén et al. (2015), the horizontal and vertical indoor positioning performances of the LTE OTDOA technique using the first peak-based TOA estimator were evaluated with 3D multiple-input, multiple-output (MIMO) channel models. Two simulation scenarios of outdoor-only and outdoor-indoor network deployments were under consideration. The results indicated that the indoor LTE positioning accuracy could be enhanced by deploying more small cells indoors and outdoors.

Some other TOA estimation algorithms have also emerged for coping with LTE signals. In Panchetti et al. (2013), the phase information of the cross-correlation product between the channel frequency response (CFR) estimations obtained at two adjacent OFDM symbols containing the PRS was extracted for the TOA estimation. Performance analysis of this approach in both the AWGN and TDL multipath channels was carried out by simulations. According to a hybrid-tap model, which includes equally spaced taps plus an arbitrary tap within the first two taps, a joint ML time-delay and channel estimation technique was proposed by solving the corresponding two-dimensional optimization problem (del Peral-Rosado, Parro-Jiménez, et al., 2014). The performance improvement of this two-dimensional estimator over the one-dimensional one based on a periodic-tap model using the PRS was demonstrated by simulations for a combined multipath propagation and AWGN channel. An iterative TOA estimation algorithm, which excluded the contribution of already detected paths in the CIR at each iteration, was developed in Rydén et al. (2016) to overcome the poor accuracy and complex parameter tuning in the threshold-based nonit-erative method. The performance was evaluated using an International Telecommunication Union (ITU) Macro 3D channel model and outdoor-indoor network deployment scenario. Chen et al. proposed a novel OTDOA technique for three-dimensional positioning in LTE systems (Chen & Wu, 2016). A compressive sensing channel estimation method exploiting the inherent sparsity of the CIR was used for the TOA estimation. The positioning performance was assessed in a simulation scenario considering the EPA and ETU channel models. Simulation results showed that 90% of the position errors provided by the proposed scheme were less than 7 m. The estimation of signal parameters by the rotational invariance technique (ESPRIT) was used in Dan et al. (2018) for joint TOA and direction of arrival (DOA) estimation with the CRS considering a UE equipped with two antennas. The positioning performance was only evaluated for static multipath channels.

2.4 Experiment testing

Another straightforward way is to conduct experiments for evaluating the LTE positioning capabilities. The common FPD approach was employed in Medbo et al. (2009) for producing the ranging measurements, which detected the first arriving signal by the first occurrence of a signal level above a −30 dB detection threshold relative to the strongest peak in the power delay profile (PDP). Provided with the real channel data measured by a channel sounder using the PRS, the received signal strengths and positioning errors were analyzed. For the LTE-based indoor positioning system developed in Gentner, Munoz, et al. (2012), the TOA was estimated by detecting a correlation peak before the maximum peak with a threshold of 8 dB below the maximum peak. The ranging and positioning accuracy obtained with this system was evaluated through a measurement campaign for an outdoor to indoor scenario. Shamaei et al. developed an SDR LTE positioning receiver using four main stages for timing estimation: cell acquisition, system information extraction, signal tracking, and timing information extraction (Shamaei et al., 2018b). At the last stage, the TOA was measured from the CRS by a first peak estimator, which computes an adaptive threshold according to the varying noise level and a constant false alarm rate. The positioning performance obtained with this SDR receiver was examined by field tests conducted with an unmanned aerial vehicle (UAV) and a ground vehicle. The authors in Gadka et al. (2019) proposed two peak detection strategies for low and high bandwidths of LTE downlink signals. Simulated and experimental results were used for obtaining an adaptive threshold criterion based on SNR values. The estimation accuracy was assessed through a measurement campaign carried out based on a real LTE network in an urban scenario. The LTE-based navigation performance was tested using an LTE SDR made up of a universal software radio peripheral (USRP) B200mini radio frontend and an open source software implementation of a UE (Ikhtiari, 2019). Only the cell acquisition approach with the PSS and SSS was used to estimate the TOA. Through the experiments conducted in both mild and severe multipath conditions, the authors claimed that the biggest contributor of positioning error was the UE clock drift. A few solutions for combating the clock drift issue were discussed.

Another category of LTE signal TOA estimator makes use of the SRA, e.g., ESPRIT, based on the channel frequency measurements (Li & Pahlavan, 2004). In Driusso et al. (2017, 2016), a TOA estimation algorithm, named the ESPRIT and Kalman filter for time-of-Arrival Tracking (EKAT), using the CRS, was proposed to increase the robustness to multipath effects. Signal combination in multiple domains, i.e., time, frequency, spatial, and cell ID domains, was adopted for performance enhancement. Live data of commercial LTE signals were gathered in different propagation environments to demonstrate the advantage of the ranging accuracy provided by the proposed algorithm over other existing methods in Benedetto et al. (2011), Dardari et al. (2008), Giorgetti and Chiani (2013), and Knutti et al. (2015). The EKAT algorithm was also used in Pittino et al. (2017) to evaluate the positioning performance with real collected LTE data in both outdoor and indoor testing environments.

One disadvantage of the SRA is the high computational cost, which motivates the application of the computationally efficient closed-loop approaches. The SDR LTE positioning receivers presented in del Peral-Rosado, Parro-Jiménez, et al. (2014) and del Peral-Rosado et al. (2013) implemented a tracking architecture based on a DLL and a phase-lock loop (PLL) using the CRS for time and frequency synchronization. They employed a Fitz estimator and a ML estimator for TOA estimation in the DLL, respectively. An experimental testbed based on an emulated LTE network with a static UE was built. The real LTE signals generated in the testbed were captured by a USRP platform for performance validation of the SDR receiver in del Peral-Rosado et al. (2013). The positioning results obtained from the SDR receiver in del Peral-Rosado, Parro-Jiménez, et al. (2014) with two low-cost hardware platforms, i.e., a USRP platform and a digital video broadcasting-terrestrial (DVB-T) dongle, were compared. In Shamaei et al. (2018a), the authors assessed the ranging accuracy for only using the signal tracking stage based on the SSS in Shamaei et al. (2018b) by experimental results of a ground vehicle. Field tests were conducted in Shamaei et al. (2017) to compare the LTE positioning accuracy obtained with the CRS-based receiver (Shamaei et al., 2018b) and the SSS-based receiver (Shamaei et al., 2018a). In Shamaei and Kassas (2018), another SDR LTE positioning receiver architecture was presented with two main stages, i.e., acquisition and tracking. Provided with the initial TOA estimation from the ESPRIT algorithm, a PLL-aided DLL was developed to track the LTE CRS. The effectiveness of this SDR receiver was verified by experiments on a ground vehicle in an urban environment. The pseudorange accuracy of the two SDR receivers proposed by Shamaei and Kassas (2018) and Shamaei et al. (2018b) was also compared with several state-of-the-art algorithms.

In addition, an iterative TOA estimation algorithm with multi-access interference mitigation using the PSS was developed and examined by a generic OFDM-based positioning testbed (Staudinger & Gentner, 2011; Staudinger et al., 2011). With the same collected data as in Driusso et al. (2017), the performance of another TOA estimator aiming at rejecting NLOS measurements was evaluated in Knutti et al. (2015). Several peaks exceeding a predefined threshold were detected in the CIR estimation obtained from the CRS. The TOA measurement would be discarded if the first peak is not the highest among all detected peaks. Laboratory experiments were carried out in del Peral-Rosado et al. (2018) to evaluate the ranging performances of several joint ML time-delay and channel estimators. A controlled scenario only with the effects of multipath and noise was considered by correcting the tracked receiver clock offset.

2.5 Other related work

Different from mitigating the multipath effects on the TOA estimation, another strategy is to exploit multipath signals for assisting positioning. Such an algorithm for LTE signals was provided in Ulmschneider and Gentner (2016) through estimating and tracking the parameters of all the arriving multipath components. An experimental site with a special scattering object was carefully chosen to demonstrate its effectiveness.

There are a number of TOA estimation algorithms for generic OFDM signals (Abrudan et al., 2013; Babich et al., 2017; Noschese et al., 2017, 2018; Wang et al., 2015; Yang et al., 2012) or other OFDM-based signals, e.g., WiFi signal (Lee et al., 2018), ultra-wideband (UWB) signal (Saberinia & Tewfik, 2008; Xu et al., 2008), and narrowband Internet-of-Things (NB-IoT) (Hu et al., 2019; Radnosrati et al., 2017). The channel estimation approaches for OFDM systems (Liu et al., 2014) also incorporate the TOA estimation required by the position determination. It is worth noting that these algorithms may also be applicable to LTE signal TOA estimation for positioning.

The above survey shows that a number of advanced TOA estimation algorithms for LTE signals aiming at counteracting multipath effects have been developed and evaluated by experiments with relatively mild multipath propagation conditions. Nevertheless, their performance assessment and comparison under complex multipath propagation conditions are not available in the literature. Hence, this paper aims to fill this gap.

5.1 First peak detection technique

A simple, yet efficient method, which compares the PDP to a predefined threshold, is referred to as the FPD technique. The index of the first threshold-exceeding PDP sample is chosen as . The TOA estimation can then be computed as . This technique is susceptible to the threshold settings. Thus, two specific threshold criteria are considered here for evaluating the TOA estimation performance.

5.1.1 Threshold Criterion 1

Threshold criterion 1 is based on a normalized adaptive threshold as (Guvenc & Sahinoglu, 2005)

8

where T_norm is the normalized threshold. max{⋅} and min{⋅} denote the maximum and minimum operators, respectively. Due to the involvement of the maximum value of , the threshold T₁ is tightly related to the channel properties.

Figure 3 shows the cumulative distribution function (CDF) of the TOA estimation errors provided by the FPD technique based on threshold criterion 1 for different channel models and SNR values when N_RB = 100. The errors with absolute values higher than 200 m are truncated to 200 m for a clearer display. There, T_norm = 0.4 is assumed. Table 2 lists the root mean square (RMS) values of the TOA estimation errors for all the simulation conditions. Under each condition, we further separated LOS and NLOS scenarios. The RMS values for the two scenarios are listed next to the combined RMS values in the bracket (⋅⋅). This same format of presenting the RMS values for overall and separate LOS and NLOS scenarios will be used in the following tables and figures.

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FIGURE 3

CDF of the TOA estimation errors provided by the FPD technique based on threshold criterion 1 for different channel models and SNR values when N_RB = 100

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TABLE 2

RMS of the TOA estimation errors (m) provided by the FPD technique based on threshold criterion 1 for different channel models and SNR values when N_RB = 100. The numbers in the bracket (··) indicate the RMS for LOS and NLOS scenarios

As can be observed from Figure 3 and Table 2, the estimation accuracy degrades obviously for SNR =−15 dB. This is due to a high probability of early estimations in the presence of high noise. For SNR values higher than −10 dB, the technique yields a nearly constant accuracy level. The high positive TOA estimation errors are due to the channel NLOS conditions, i.e., the first arriving signal experiences a much deeper fading compared to other later arriving signals. Under each condition, the estimation accuracy in NLOS scenarios is degraded compared to LOS scenarios. The technique can be improved to better mitigate the noise effects by using a larger normalized threshold. It should be pointed out that if the normalized threshold is too large, it would cause late estimations for the NLOS cases. The step-shape cumulative probability shown in Figure 3 is caused by the discrete TOA estimations.

Figure 4 displays the effect of N_RB (equivalent to signal bandwidth) on estimation performance of the FPD technique based on threshold criterion 1, considering three channel models and a fixed SNR of 0 dB. The RMS value of the overall TOA estimation errors and those for LOS and NLOS scenarios are also given in the legend of Figure 4. Understandably, a larger N_RB implies a smaller sampling period and an improved multipath resolution capability. Therefore, for most scenarios, a higher signal bandwidth is beneficial for improving the estimation accuracy. One exceptional case is that for the ETU model, where the RMS is slightly larger when increasing N_RB from 50 to 100. This is because the strong later arriving paths in the ETU model become distinguishable for N_RB = 100. One of the later arriving paths may be mistaken as the first arriving path when the real first arriving path experiences a deep fading, leading to large estimation errors.

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FIGURE 4

CDF and RMS of the TOA estimation errors provided by the FPD technique based on threshold criterion 1 for different channel models and numbers of RBs when SNR = 0 dB

5.1.2 Threshold Criterion 2

Another criterion to determine a threshold is based on a fixed early detection probability. As the noise-only bin in PDP follows a central 𝜒² distribution with ν = 2N_slot degrees-of-freedom (DOF), the early detection probability can be modeled as (Dardari et al., 2008)

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where N_TOA is the number of possible TOA values, which should be set empirically for practical implementation. q_noise is the probability of a noise-only PDP sample above a threshold T₂ and can be computed as (Giorgetti & Chiani, 2013; Torrieri, 2005)

10

where σ² is the noise variance for the real or imaginary part of the noise component w[n] in the CIR estimation for each slot. Therefore, the threshold T₂ can be determined in terms of a preset P_ed according to Equations (9) and (10).

The CDF of the TOA estimation errors obtained with the FPD technique based on threshold criterion 2 for different channel models and SNR values when N_RB = 100 is illustrated in Figure 5. The number of the TOA estimations and RMS values of the TOA estimation errors for all the simulation conditions are summarized in Table 3. P_ed is set to 10⁻⁶, and N_TOA is computed based on a possible TOA range of 0 to 300 m. The noise variance in the PDP is also assumed to be available.

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FIGURE 5

CDF of the TOA estimation errors provided by the FPD technique based on threshold criterion 2 for different channel models and SNR values when N_RB = 100

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TABLE 3

Number of the TOA estimations and RMS of the TOA estimation errors (m) provided by the FPD technique based on threshold criterion 2 for different channel models and SNR values when N_RB = 100. The numbers in the bracket (··) indicate the RMS for LOS and NLOS scenarios

This threshold criterion does not involve the signal strength information and thus is not related to the channel properties. In case of low SNR conditions, no threshold-crossing may occur. This enhances the capability of noise resistance, but yields less measurements. As can be seen from Table 3, the RMS of the TOA estimation errors first decreases when increasing the SNR as the number of high positive estimation errors is reduced. However, if the SNR further increases, the RMS increases instead. This can be explained by the fact that the TOAs of the received signals may not correspond to discrete time indices in the CIR according to {n_l} = {τ_l} N_r/N_c. The spectral leakage effect occurs in the IDFT operation for estimating the CIR, but is not taken into account in the threshold criterion 2. The leaked signal energy in the noise-only bins becomes stronger and thus causes more early estimation results for higher SNR signals as shown in Figure 5. Similarly, for the same SNR value (≥ −5 dB), the EPA model with stronger power distributed at the short-delay paths yields a higher probability of early estimation results and a larger RMS value in comparison with the other two models. From this point of view, a smaller early detection probability should be set to alleviate early estimations. Meanwhile, this would decrease the number of measurements in low SNR conditions or yield wrong estimation results for channel realizations with strong, later arriving signals.

Figure 6 shows the CDF and RMS of the TOA estimation errors provided by the FPD technique based on threshold criterion 2 for different channel models and numbers of RBs when SNR = 0 dB. It can be clearly seen that a larger number of RBs enhances the estimation accuracy due to the higher timing resolution. Moreover, as the technique is susceptible to the spectral leakage effect, the LOS cases with a stronger first arriving path lead to a higher RMS value relative to the NLOS cases.

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FIGURE 6

CDF and RMS of the TOA estimation errors provided by the FPD technique based on threshold criterion 2 for different channel models and numbers of RBs when SNR = 0 dB

5.2 Information theoretic criteria technique

The ITC technique was initially intended to provide a TOA estimation for UWB signals based on the statistical properties of the energy detector outputs (Giorgetti & Chiani, 2013). It is also applicable to LTE signals by using the underlying statistical characteristics of the PDP.

Unlike the integration operation in Equation (7) for other techniques, the PDP estimations over N_slot slots are arranged in a N_slot-by-2M matrix 𝐏 for the ITC method. Theoretically, the noise-only bin in 𝐏 follows a central 𝜒² distribution, whereas the signal-plus-noise bin follows a non-central 𝜒² distribution with a non-centrality parameter depending on the path gain. There are two implementation schemes for this technique in Giorgetti and Chiani (2013). Here, we only elaborate the ITC technique based on early bins for the reader’s convenience. The three main steps of the ITC technique are illustrated as follows:

1. Build the averaged PDP vector by computing the mean of each column in 𝐏 and find the index m_max of the maximum in . The first m_max + 1 columns in 𝐏 and are regarded as 𝐏_E and , respectively.

2. The cost function values for all the possible model orders k = 1,…,m_max are computed. For the kth model, the first k columns in 𝐏_E correspond to noise-only bins, while the remaining columns correspond to signal-plus-noise bins. contains the information to calculate the distribution parameters for each bin in 𝐏_E. The log-likelihood of each element in 𝐏_E is then evaluated. The cost function is computed using all the loglikelihood values and the penalty function associated with the kth model.

3. The TOA estimation is , where is the model order which minimizes the cost function.

Note that the detailed equations used in the above steps can be found in Giorgetti and Chiani (2013). As suggested in Driusso et al. (2017), a windowing operation is applied to the PDP to reduce the biasing effect of sidelobes. The central chi-square approximation is used in computing the log-likelihood, and the efficient detection criteria is used for the penalty function. The CDF and statistical results of the TOA estimation errors obtained with the ITC algorithm for different channel models and SNR values when N_RB = 100 are given in Figure 7 and Table 4, respectively.

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FIGURE 7

CDF of the TOA estimation errors provided by the ITC algorithm for different channel models and SNR values when N_RB = 100

Figure 7 indicates that the technique generates early estimation results in most of the channel realizations for SNR higher than −10 dB. This is caused by the high sensitivity to the spectral leakage effect occurring in the CIR. Because of this, the RMS value for LOS cases in the EVA/ETU channel is even higher than that for NLOS cases as shown in Table 4. Nevertheless, when SNR = −15 dB, the technique provides poor estimation accuracy owing to the large positive estimation errors. This is because the detected PDP bin with maximum energy is far from that of the first arriving signal in the presence of high noise. Some noise-only bins exist between the bins containing the first arriving signal and with maximum energy, failing to comply with the PDP bin classification in Step 2 of the ITC algorithm. Thus, as can be observed from Table 4, the estimation accuracy first becomes better when increasing the SNR, but then becomes worse with further increase of the SNR. Overall, this technique is robust to the harsh multipath conditions for medium SNR values.

Figure 8 shows the estimation performance provided by the ITC method for different N_RB values and three channel models when SNR = 0 dB. Since the ITC approach generates discrete TOA estimations like the FPD technique does, it provides an improved estimation accuracy if a higher N_RB is used in the LTE system.

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FIGURE 8

CDF and RMS of the TOA estimation errors provided by the ITC method for different channel models and numbers of RBs when SNR = 0 dB

5.3 Super-resolution algorithm

Because the CFR for a multipath propagation channel in Equation (2) is a harmonic signal model, the TOA estimation problem is equivalent to the spectral estimation problem by exchanging the time and frequency variables. Based on this, the application of an SRA to frequency-domain channel measurements was investigated for TOA estimation in Li and Pahlavan (2004).

Here, we adopt the ESPRIT as a representative of SRAs for illustration. This algorithm has been well documented in Driusso et al. (2017) and Shamaei and Kassas (2018). We only summarize the key steps as follows:

1. Construct a P-by-(2M − P + 1) frequency-domain data matrix using the CFR and perform singular value decomposition (SVD) of the data matrix.

2. Estimate the channel length by the minimum description length (MDL) criterion using the singular values obtained in Step 1.

3. Construct the ESPRIT rotational matrix and compute its eigenvalues.

4. Compute the delay estimations for all the detected paths using the phase information in the eigenvalues obtained in Step 3. The smallest delay estimation is considered as .

According to Yang et al. (2000), we can reasonably adjust the true TOA to be a positive value after coarse symbol synchronization during the TOA estimation. Owing to this intrinsic constraint, the results with negative are discarded. Figure 9 illustrates the CDF of the TOA estimation errors using the SRA for different channel models and SNR values when N_RB = 100. The number of the TOA estimations and RMS values of the TOA estimation errors for all the simulation conditions are given in Table 5. The design parameter P is assumed to be M (Driusso et al., 2017; Shamaei & Kassas, 2018).

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FIGURE 9

CDF of the TOA estimation errors provided by the SRA for different channel models and SNR values when N_RB = 100

The algorithm’s performance is tightly related to the channel length estimation, which is in turn dependent on the SNR condition. As mentioned in Driusso (2016), Liavas et al. (1999), and Via et al. (2006), our simulations also indicate that the MDL criterion tends to underestimate or overestimate the channel length for relatively low or high SNR values, resulting in a large number of invalid negative TOA estimations. Accordingly, the number of estimations is sharply reduced, especially for the ETU channel with SNR =−15 or 5 dB. Moreover, the RMS of the TOA estimation errors increases with the decreasing SNR. Contrary to other techniques, the TOA estimations obtained with the SRA are not confined to discrete values. Overall, the SRA performs well in a wide range of SNR conditions and several channel types due to its promising time-domain resolution of the channel response. This algorithm, however, also has a high computational cost, making it impractical for real-time implementation.

Figure 10 shows the statistical results of the TOA estimation errors using the SRA in case of three N_RB values for three channel models and a fixed SNR of 0 dB. The CDF curves and RMS values demonstrate the improvement in the estimation accuracy with a larger N_RB value and the better accuracy in LOS scenarios.

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FIGURE 10

CDF and RMS of the TOA estimation errors provided by the SRA for different channel models and numbers of RBs when SNR = 0 dB

5.4 Delay-lock loop

The DLL with a block diagram shown in Figure 11 takes advantage of a closed-loop architecture to estimate the TOA. With consideration of the OFDM modulation in the LTE signal, the correlation process is performed in the frequency domain. At the beginning of each loop updating period, the integer and fractional parts of the TOA estimation are used to control the DFT window position and perform phase rotation, respectively. Next, the correlation results in early and late branches are obtained by shifting the CRS sequence with ξ and −ξ samples, respectively. Then, a discriminator exploits the early and late correlation results to calculate the residual in the TOA estimation. Finally, the filtered residual is used to adjust the TOA estimation for the next updating period. The TOA can be tracked in a locked status when the energies in early and late branches are close to each other.

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FIGURE 11

Block diagram of the DLL for tracking the LTE signal TOA

The main weakness of the DLL for tracking LTE signals is its narrow pull-in range in the delay domain (Shamaei & Kassas, 2018; Yang et al., 2000). This implies that an accurate TOA estimation must be provided by other techniques for initializing the DLL. However, as shown by the previous simulation results, there is still a high probability of large TOA estimation errors in complex multipath propagation channels. If an inaccurate initial estimation is used, the DLL may track one of the later arriving signals instead and yield an unacceptable estimation error.

In the multipath propagation channel, the multipath effect is the dominant factor of the tracking error in the DLL (Yang et al., 2000). Following the common metric for analyzing the multipath mitigation performance of a closed-loop architecture (Townsend et al., 1995), Figure 12 presents the multipath error envelope provided by the DLL for different N_RB values. A two-path model with a fixed multipath to direct signal power ratio (MDR) of −1 dB is used, and a noise-free environment is assumed in the simulations. The correlator spacing ξ is set to 0.5 samples (Shamaei & Kassas, 2018; Yang et al., 2000). The two curves for each N_RB value correspond to in-phase and out-of-phase multipath cases. As can be seen, the DLL suffers from a TOA tracking bias, depending on the multipath parameters and signal bandwidth. A higher signal bandwidth brings a considerable reduction in the TOA tracking bias.

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FIGURE 12

Multipath error envelope provided by the DLL for different numbers of RBs and a fixed MDR of −1 dB

6.1 Accuracy comparison

To allow an intuitive comparison, Figure 13 compares the RMS values of the TOA estimation errors as a function of SNR for different channel models obtained with the aforementioned advanced algorithms except for the DLL when N_RB = 100. It demonstrates that in cases of medium SNR values, all the advanced algorithms provide comparable TOA estimation accuracy in these multipath propagation channels.

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FIGURE 13

RMS of the TOA estimation errors versus SNR for different channel models obtained with several advanced TOA estimators when N_RB = 100

Nevertheless, in cases of relatively low and high SNR values, the performance differences among these methods are notable. If only the statistical accuracy is taken into account, Table 6 summarizes the preferred algorithms under different conditions. The FPD technique with threshold criterion 1 can be used for the EPA channel in a high SNR condition. The FPD technique based on threshold criterion 2 is preferred for low SNR values. The SRA is a promising method except for the ETU channel with a low SNR value.

6.2 Complexity comparison

In this subsection, we analyze and compare the computational complexity associated to the number of multiplications involved in the aforementioned techniques. Since all techniques require the CFR estimation, we only examine the signal processing procedures after the CFR estimation for generating one TOA estimation in the following analysis.

The FPD and ITC techniques both exploit the PDP estimations over N_slot slots. For each slot, the IDFT operation for the CIR estimation requires O(2Mlog(2M)) multiplications. The PDP estimation is then obtained by a modulus square operation with O(2M) multiplications. The remaining procedures in the FPD technique incorporate a few times of peak finding in the integrated PDP. For the ITC, the PDF of central chi-square distributions for different distribution parameters can be stored in a look-up table. The evaluation of the log-likelihood for calculating the cost functions can thus be achieved by table look-up operations, leading to a significantly decreased complexity. The final step is to find the model order, which minimizes the cost function. The computational cost of these operations is negligible compared to the calculation of the PDP estimations. Therefore, the total numbers of multiplications required by the FPD and ITC approaches can both be approximated as O([2Mlog(2M) + 2M]N_slot).

For the SRA, the most demanding procedures are those matrix operations. The SVD of the frequency-domain data matrix requires O(P²K) multiplications. Computing the MDL criterion requires many fewer multiplications than the SVD operation. The number of multiplications to calculate the ESPRIT rotational matrix is . The eigenvalue decomposition of the ESPRIT rotational matrix requires multiplications. Therefore, the total number of multiplications used by the SRA can be expressed by . In our simulations, we have P = M, K = M + 1, and . Thus, the number of multiplications can be approximated as .

The main computational complexity of the DLL comes from the correlation process. To compute an early and a late correlation result, the DLL consumes O(4M) multiplications.

Therefore, the complexity of SRA is much higher than those of other techniques. The DLL is the most computationally efficient one. The complexities of the FPD and ITC algorithms are in the same order of magnitude if we ignore the large number of table loop-up operations required by the ITC algorithm.

6.3 Discussion for each algorithm

The FPD method largely depends on the predefined threshold to achieve a desirable performance. Two threshold criteria are examined to assess its performance. Criterion 1 computes an adaptive threshold using the maximum and minimum values of the channel estimation result and a preset normalized threshold. This criterion leads to a number of early estimations and thus a poor TOA estimation accuracy for low SNR values. A high, normalized threshold would be beneficial to enhance noise resistance. However, late estimations may occur when using a high, normalized threshold for the propagation condition if the first arriving path is much more attenuated than other later arriving paths. Hence, the normalized threshold in this criterion should be chosen carefully based on the propagation channel and SNR condition.

Criterion 2 used for the FPD method determines the threshold according to a fixed early detection probability. The threshold is thus irrelevant to the channel properties. It should be noted that the spectral leakage effect in the CIR estimation introduces an increased number of early estimations in this technique for a higher SNR condition. Moreover, this technique eliminates the early estimations, but also reduces the number of measurements for a low SNR value. Therefore, both the measurement availability and estimation accuracy need to be considered for choosing a reasonable early detection probability.

The ITC technique is also highly sensitive to the spectral leakage occurring in the CIR estimation. The TOA estimation performance obtained with the ITC algorithm adversely deteriorates for increasing SNR values except for very low SNR values. The technique is robust to the harsh multipath condition for medium SNR values and does not require a parameter tuning for different conditions. Nevertheless, the performance would experience a drastic degradation if the SNR value is too low or too short an integration time is used.

The SRA has a superior multipath resolution capability at the cost of a high computational complexity. It incorporates a channel length estimation procedure based on the MDL criterion. Either an underestimated channel length in a low SNR condition or an overestimated channel length in a high SNR condition would reduce the number of estimations. Additionally, this algorithm provides an improved TOA estimation accuracy for a higher SNR value.

The DLL method is different from the others owing to its closed-loop architecture. It requires an accurate TOA estimation provided by other techniques for initialization. An improper initial value would lead to the DLL’s failure to track the LTE signal TOA. In addition, it still suffers from a notable tracking bias due to the multipath effects.