Joint maximum likelihood timedelay estimation for LTE positioning in multipath channels
 José A del PeralRosado^{1}Email author,
 José A LópezSalcedo^{1},
 Gonzalo SecoGranados^{1},
 Francesca Zanier^{2} and
 Massimo Crisci^{2}
https://doi.org/10.1186/16876180201433
© del PeralRosado et al.; licensee Springer. 2014
Received: 31 May 2013
Accepted: 28 February 2014
Published: 14 March 2014
Abstract
This paper presents a joint timedelay and channel estimator to assess the achievable positioning performance of the Long Term Evolution (LTE) system in multipath channels. LTE is a promising technology for localization in urban and indoor scenarios, but its performance is degraded due to the effect of multipath. In those challenging environments, LTE pilot signals are of special interest because they can be used to estimate the multipath channel and counteract its effect. For this purpose, a channel estimation model based on equispaced taps is combined with the timedelay estimation, leading to a lowcomplexity estimator. This model is enhanced with a novel channel parameterization able to characterize closein multipath, by introducing an arbitrary tap with variable position between the first two equispaced taps. This new hybrid approach is adopted in the joint maximum likelihood (JML) timedelay estimator to improve the ranging performance in the presence of shortdelay multipath. The JML estimator is then compared with the conventional correlationbased estimator in usual LTE conditions. These conditions are characterized by the extended typical urban (ETU) multipath channel model, additive white Gaussian noise (AWGN) and LTE signal bandwidths equal to 1.4, 5 and 10 MHz. The resulting timedelay estimation performance is assessed by computing the cumulative density function (CDF) of the errors in the absence of noise and the rootmeansquare error (RMSE) and bias for signaltonoise ratio (SNR) values between −20 and 30 dB.
Keywords
1 Introduction
Navigation and positioning technologies are every day more important in civil applications, demanding enhancements on accuracy, availability and reliability. Positioning improvements are mainly achieved, thanks to the advances in Global Navigation Satellite Systems (GNSS) and the introduction of new systems, such as Galileo. However, a myriad of possible working conditions are faced in ubiquitous positioning, where the GNSS nominal performance is highly degraded, such as in urban environments or indoors. Thus, the use of complementary terrestrial localization systems is envisaged as a major step towards the realization of anywhere and anytime positioning. A relevant example of these technologies is the Long Term Evolution (LTE), a nextgeneration mobile communications system with promising perspectives on positioning. Indeed, the LTE standard [1, 2] specifies a dedicated downlink reference signal for observed time difference of arrival (OTDoA) localization, i.e. the positioning reference signal (PRS). The LTE OTDoA method is based on the timedelay estimation (TDE) of the signals received from different source transmitters (i.e. cellular base stations). Since the LTE downlink signal is based on the orthogonal frequency division multiplexing (OFDM) modulation, the TDE is typically performed with the LTE pilot subcarriers in the frequency domain. Using this method, recent research studies, such as those in [3–5], have shown the potential of LTE to provide accurate positioning. However, multipath propagation and nonlineofsight conditions are still the main limiting factors in urban environments, once intercell interference is removed. Therefore, countermeasures against multipath are needed in order to achieve the ultimate positioning performance in LTE.
In order to properly understand and mitigate the effect of multipath, it is important to have a good characterization of the propagation conditions and reflect them on the estimation model of the receiver. Many channel models have been proposed to characterize the propagation conditions of the possible LTE working scenarios. Among them, the LTE standard adopted an extension of the tappeddelay line (TDL) channel models specified for the second and thirdgeneration mobile systems, i.e. Global System for Mobile communications (GSM) and Universal Mobile Telecommunications System (UMTS). One of these propagation models is the extended typical urban (ETU) model, described in Annex B of [6] and [7]. The ETU model is of special interest because it defines a powerdelay profile (PDP) with a lineofsight (LoS) signal more attenuated (in average) than the multipath rays. This is a representative example of harsh conditions where LTE is expected to be used. In these conditions, the ranging performance of the correlator or matched filter is relatively poor [8]. The correlationbased TDE can be considered as a conventional timedelay estimation method due to its low complexity, being the maximum likelihood (ML) estimator in additive white Gaussian noise (AWGN) channels. However, in multipath channels, the delayed reflections of the signal induce a notable bias on this conventional estimation. Thus, another estimator is required in order to improve the TDE performance in multipath environments.
The timedelay estimation can be enhanced by modelbased estimators. These estimators use channel estimation models in order to counteract the effect of multipath. The aim of these models is to characterize the response of the channel, instead of identifying the physical multipath components of the specific environment. There are many possibilities for these channel estimation models. On the one hand, the most accurate model corresponds to the estimation of amplitude, phase and delay of every propagation ray. However, it is also the most complex model because of the many unknowns to estimate. Despite its complexity, this estimation model has been widely studied, for instance, with superresolution techniques in [9, 10], with the ML criterion in [11], or in a twostep approach in [12]. On the other hand, channel estimation models can be simplified by defining equispaced or periodic taps relative to the time delay of the first path. Since this model is based on the uniform sampling of the channel, it has been used with the compressed sampling theory for channel estimation, such as in [13, 14], but it can also be found in multipath interference cancellation [15].
Timing synchronization for data transmission does not need to achieve the extreme accuracy required for positioning. This is the reason why, in communication applications, the ML approach is widely applied to channel estimation assuming the time delay to be coarsely corrected in a previous stage, and the residual time delay is considered negligible. There are still some contributions that propose the joint maximum likelihood (JML) estimation of the time delay and channel response in OFDM systems, considering a model based on equispaced or periodic taps, but few of them deal with the specific case of ranging applications. The authors of [16] propose an algorithm based on the JML approach and on the channel length estimation, but providing only coarse timing estimates. In [17], two JML estimators are applied for synchronization of multiple users. The first one is a joint frequency and channel estimator, while the second one is a joint timedelay and channel estimator. A similar JML approach is used for ranging purposes considering an IEEE 802.16 system [18]. An approximation of the JML algorithm is proposed in [19] using early and late estimations in a delay lock loop (DLL). The JML estimation has also been studied for multicarrier ranging considering the optimal placement of pilot subcarriers in [20] and applied without dataaiding after the definition of a unified signal model in [21]. In addition, a very preliminary study of the JML in LTE for a very specific scenario is presented in [22]. Therefore, the joint estimation algorithms found in the literature have mainly focused on communication applications, where a very accurate timedelay estimation is not critical in general. In our ranging application, the representation of the channel has to be improved, especially for those scenarios where multipath highly deteriorates the timedelay estimation. Thus, the channel estimation models have to be adapted to these harsh environments.
The periodictap estimation model is suitable for massmarket receivers, such as mobile phones, because it has a low complexity. However, this model may lead to significant ranging errors with closein multipath. Typically, multipath appears close to the lineonsight ray in urban and indoor environments, producing a critical degradation in ranging applications. This multipath, which is ignored in communications, significantly affects the performance of the periodictap estimation model for lowsampling rates, because shortdelay multipath may not be properly modelled between samples. Therefore, we propose a hybrid estimation model by using equispaced taps together with an arbitrary tap between the first two. This solution improves the characterization of the channel while only adding the complexity of one more estimation parameter. Thus, the introduction of this arbitrary tap with a variable position helps to increase the ranging accuracy in closein multipath environments. This new hybrid JML approach, as well as the periodic JML approach, is studied in this paper and used to assess the achievable positioning performance of LTE, considering a lowcomplexity timedelay estimation that exploits the structure of the LTE OFDM signals. This estimation is analyzed for usual working conditions, represented by typical LTE signal bandwidths with the ETU standard channel model.
The remainder of this paper is organized as follows. Section 2 defines the signal model and describes the main LTE pilot signals. Section 3 defines the propagation channel model. Section 4 reviews the different channel estimation models and derives the joint ML estimator, including the novel channel parameterization. In Section 5, the performance of the joint ML estimators is assessed with numerical results in multipath and noise channels for different signal bandwidths. Finally, conclusions are drawn in Section 6.
2 Signal model
where C is the power of the bandpass signal, N is the total number of subcarriers, b(n) is the complexvalued symbol transmitted at the n th subcarrier, and T is the OFDM symbol period. The symbol b(n) is defined by b(n)=d(n)·p(n), being d(n) the data or pilot symbol assigned with a relative power weight p(n)^{2}, which is constrained by ${\sum}_{n=0}^{N1}p{\phantom{\rule{1pt}{0ex}}\left(n\right)}^{2}=N$. In particular, the downlink physical layer of the LTE specification [1] defines a symbol period T of 66.67 μ s, which corresponds to a subcarrier spacing F_{sc}=1/T of 15 kHz. The transmission grid is defined in time and frequency by resource blocks (RB), which are the minimum resource allocation unit formed by seven OFDM symbols and 12 subcarriers in the normal cyclic prefix configuration. The system bandwidth is scalable from 1.4 to 20 MHz, but the guard bands are left at the edges of the spectrum; thus, only a minimum transmission bandwidth of 6 RB (i.e. 1.08 MHz) and a maximum of 100 RB (i.e. 18 MHz) can be configured [6].
where n is the index of the subcarriers, $H\left(n\right)=\mathcal{F}\left\{{h}_{\mathrm{d}}\left(m\right)\right\}$ is the channel frequency response, being $\mathcal{F}\left\{\xb7\right\}$ the discrete Fourier transform operator, and w(n) are the noise frequency samples, which are statistically uncorrelated with $w\phantom{\rule{1pt}{0ex}}\left(n\right)\sim \mathcal{N}\phantom{\rule{1pt}{0ex}}\left(0,{\sigma}_{\mathrm{w}}^{2}\right)$.
3 Propagation channel model
ETU channel model parameters
Tapk  t_{ k } (ns)  ${\overline{\text{RP}}}_{k}$ (dB) 

1  0  −1.0 
2  50  −1.0 
3  120  −1.0 
4  200  0.0 
5  230  0.0 
6  500  0.0 
7  1,600  −3.0 
8  2,300  −5.0 
9  5,000  −7.0 
4 Timedelay estimation
4.1 Channel estimation models
It is important to note that channel estimation models should be distinguished from propagation channel models. The first ones consider the response of the channel in order to later counteract its effect. The second ones model the physical channel to understand the behaviour of the channel itself. Thus, the propagation channel models are used to simulate the actual channel, while the channel estimation models are used to represent the effect of the channel on the timedelay estimation. Next, we describe those channel estimation models that are typically used, as well as a new model presented in this work.
4.1.1 Singletap model
4.1.2 Arbitrarytap model
where L is the number of taps, h_{ k } is the channel coefficient for the k th tap, τ_{ k } is the relative delay to the first tap (i.e. τ_{0}=0), and τ is the time delay. As it shown in Figure 3, the arbitrarytap model is represented, in that case, by matching the L_{c} propagation taps at delay positions τ_{c,k}. Although the channel response can be accurately reconstructed using this model, the implementation complexity is a major concern. For instance, the number of unknowns substantially increases (without a priori statistics) in dense multipath due to the high number of rays to estimate. Thus, the iterative methods implemented for the TDE, such as superresolution techniques [10], have a high computational burden.
4.1.3 Periodictap model
where L is the number of taps, h_{ k } is the channel coefficient for the k th tap, and τ is the time delay. Ideally, the sampled model would require an infinite number of taps in order to perfectly represent the channel. The solution described in (9) is to truncate the number of taps to L, by assuming that the rest of taps have a negligible contribution. However, this assumption may produce an incorrect characterization of the channel response, leading to the socalled problem of model mismatch. The periodictap model is represented in the example of Figure 3 considering six taps. Since the tap positions are assumed to be equispaced, the closein multipath, i.e. multipath close to the LoS signal, is not properly modelled when it appears between the first two samples at delay 0 and T_{s}. Thus, the multipath energy missed between samples may severely degrade the performance of the timedelay estimation. In the opposite case, if the sampling period T_{s} is small enough, the number of taps L has to expand a similar interval to the multipath dispersion. The design of L is beyond the scope of the paper, but it can be obtained by means of model order selection techniques, such as minimum description length (MDL) or Akaike [16], or by considering the delay spread of the channel, which can be estimated as in [24] and the references therein.
4.1.4 Hybridtap model
where L is the number of taps, h_{ k } is the channel coefficient for the k th periodic tap, h_{L−1} and τ^{′} denote the channel coefficient and delay of the arbitrary tap, respectively, and τ is the time delay. As an example, the hybridtap model is represented in Figure 3, where the arbitrarytap delay values of τ^{′} are fixed within 0 and 1 with respect to τ.
4.2 Onedimensional joint ML estimator
where ${\mathbf{P}}_{\mathbf{A},\tau}^{\perp}=\mathbf{I}{\mathbf{A}}_{\tau}{\left({\mathbf{A}}_{\tau}^{\mathrm{H}}{\mathbf{A}}_{\tau}\right)}^{1}{\mathbf{A}}_{\tau}^{\mathrm{H}}$ is the orthogonal projection matrix onto the subspace orthogonal to that spanned by the columns of A_{ τ }. Thus, the decoupling of $\widehat{\tau}$ and $\widehat{\mathbf{h}}$ leads to the proposed ML timedelay estimator of (26). It is hereafter called onedimensional joint ML (1DJML) timedelay and channel estimator. The 1DJML estimation is computed numerically by minimizing the cost function of $\parallel {\mathbf{P}}_{\mathbf{A},\tau}^{\perp}\mathbf{r}{\parallel}^{2}$ as a function of τ. This optimization is not complex because it is a onedimensional function that is simply evaluated within the range [−1/2,1/2] and then minimized. This range is defined to find the residual time delay after a coarse estimation. The minimum could be obtained by solving the function with a sufficiently fine grid of points. However, the fminbnd function of MATLAB is used for an efficient computation, which finds the minimum in the search interval. Instead of doing an exhaustive evaluation, this function searches the minimum by means of the golden section technique followed by a parabolic interpolation.
Thus, the particular case of the 1DJML estimation for L=1 reduces to the estimation based on the correlation or matched filter output. This confirms the optimality of the matched filter for timedelay estimation in the absence of multipath.
4.3 Twodimensional joint ML estimator
The twodimensional optimization of (29) is computed by an exhaustive search in the τ×τ^{′} region of [−1/2,1/2]×[0,1]. Since the cost function of $\parallel {\mathbf{P}}_{\mathbf{A},\tau ,{\tau}^{\prime}}^{\perp}\mathbf{r}{\parallel}^{2}$ only depends on two parameters, its minimum can be found with the required accuracy by evaluating the function in a sufficiently fine grid of points.
5 Numerical results
The proposed JML timedelay estimator is, in principle, applicable to any multicarrier signal. Multicarrier signals show a flexible allocation of data and pilot resources that facilitates the adoption of the JML estimator. Thus, the JML estimator is used to show the achievable performance with the LTE positioning reference signals for different signal bandwidths. The TDE performance of the 1D and 2DJML estimators for L>1 is compared with that of the 1DJML estimator for L=1, i.e. the conventional correlationbased technique. First, the multipath error envelope (MPEE) is used to characterize the multipath impact on the TDE. Second, the bias and the rootmeansquare error (RMSE) of the JML estimators is statistically assessed over realistic multipath and noise conditions. In order to conduct these analyses, the OFDM signal is assumed to be successfully acquired, being the receiver in signal tracking mode, thus the timedelay estimation range is defined within [−T_{s}/2,T_{s}/2], or [−1/2,1/2] since τ is in T_{s} units.
5.1 Multipath error envelope
where a_{1}, ϕ_{1} and τ_{1} are the amplitude, phase and delay of the multipath ray, respectively. The MPEE is computed considering −1 dB of relative power to the LoS ray within a delay range between 0 and 3 ·T_{s}/2. The multipath ray is added constructively and destructively to the LoS component, i.e. the multipath contribution is inphase (i.e. ϕ_{1}=0) and counterphase (i.e. ϕ_{1}=π), respectively. In this scenario, the LTE PRS is configured for the lowest bandwidth of 6 RB, assuming no data allocation on the transmitted symbol. As it is discussed in Section 2, the 6RB PRS bandwidth is defined by N=12·N_{RB}−4=68 subcarriers, which results in T_{s}=T/N=980.39 ns and a signal bandwidth equal to 1/T_{s}=1.02 MHz.

The effect of increasing the number of taps from L=1 to L=8 in the 1DJML estimator, that is, from using a singletap model to a periodictap model, improves the TDE performance, but there is still a significant bias in both cases.

While the 1DJML estimator for L={1,8} is only unbiased at certain instants (e.g. τ={1.34,1}, respectively), the 2DJML estimator is completely unbiased for values of multipath delay τ_{1} within 0 and 1 due to the matching between the channel estimation model and the propagation channel model.

The effect of decreasing the number of taps from L=8 to L=2 in the 2DJML estimator does not have the same behaviour as in the 1DJML estimator because the hybrid approach is still unbiased for values of multipath delay τ_{1} within 0 and 1.
Maximum TDE errors of the 1DJML in the MPEE
ε_{max,L=1} (m)  ε_{max,L=8} (m)  

Bandwidth (RB)  T_{s}/2 (m)  ϕ_{1}=0  ϕ_{1}=π  ϕ_{1}=0  ϕ_{1}=π 
6  147.06  127.02  −127.07  64.71  −96.11 
25  33.78  29.18  −29.19  14.87  −22.08 
50  16.78  14.49  −14.50  7.38  −10.97 
5.2 Bias and RMSE of the JML estimators over realistic conditions
The multipath error envelope has shown the bias introduced by a particular multipath ray on the timedelay estimation. The results indicate the potential of the new JML approach to improve the TDE performance with respect to the conventional correlatorbased estimator. However, the tworay multipath model does not represent general urban channels. Thus, the ETU channel model is used to assess the performance of the estimators in more realistic conditions.
5.2.1 Low signal bandwidth (i.e. 1.4 MHz)
where N_{ ℓ } is the number of realizations. In Figure 5, the multipath delays of the channel are highlighted with vertical red lines. As it can be seen, this channel model is mainly characterized by the presence of LoS signal and strong multipath in short delays. Thus, most of the multipath energy is concentrated for delays between 0 and T_{s}/2, approximately. In addition, it can be seen (as specified in Table 1) that the delay spread of the ETU model is equal to 5.1·T_{s}. Then, if the periodictap estimation model is used, one can notice as the estimation taps at positions larger than the delay spread have a negligible multipath contribution. Thus, we use this prior information in order to correctly assume the truncation of the number of taps to the delay spread of the ETU model, which in the 1DJML estimator corresponds to L=6 and in the 2DJML estimator corresponds to L=7. Using a higher number of taps, the estimators do not capture more channel energy, thus they may obtain a similar TDE performance (in absence of noise).
5.2.2 Typical signal bandwidths (i.e. 5 and 10 MHz)
The most usual working modes of LTE are based on the 5 and 10 MHz operating bandwidths. These configurations can be identified as typical modes because they are specified for most of the LTE bands, as it is shown in Table 5.6.11 of [6]. The signal bandwidths associated to these modes are 25 RB (i.e. 4.5 MHz) and 50 RB (i.e. 9 MHz), respectively. Thus, the ETU model is applied with these typical bandwidths in order to represent usual LTE positioning conditions.
6 Conclusions
A new technique for joint timedelay and channel estimation is presented in this paper to improve the ranging performance in channels with closein multipath. The proposed algorithm is a joint maximum likelihood (JML) timedelay and channel estimator based on a new hybrid channel estimation model, defined by equispaced or periodic taps and an arbitrary tap between the first two. This novel channel parameterization helps to counteract shortdelay multipath by solving a twodimensional optimization problem with relative low complexity. The technique has been studied for the specific case of positioning in LTE, using the positioning reference signal (PRS). The results have been obtained in usual LTE working conditions, represented by the standardized ETU channel model and typical LTE signal bandwidths of 1.4, 5 and 10 MHz. The hybrid approach significantly improves the ranging performance for the lowest signal bandwidth (i.e. 1.4 MHz), where closein multipath is critical, with respect to the periodic approach (i.e. only equispaced taps) and the conventional correlationbased estimator. For high signal bandwidths, such as 10 MHz, both hybrid and periodic JML estimators still provide smaller timedelay estimation errors than the matched filter, achieving a bias below 5 m.
Declarations
Acknowledgements
This work was supported by the ESA under the PRESTIGE programme ESAP2010TECETN01 and by the Spanish Ministry of Economy and Competitiveness project TEC201128219.
Disclosure
The content of the present article reflects solely the authors view and by no means represents the official ESA view.
Authors’ Affiliations
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