On the maximum likelihood estimation of the ToA under an imperfect path loss exponent
© Valera et al.; licensee Springer. 2013
Received: 16 November 2012
Accepted: 24 May 2013
Published: 11 June 2013
We investigate the estimation of the time of arrival (ToA) of a radio signal transmitted over a flat-fading channel. The path attenuation is assumed to depend only on the transmitter-receiver distance and the path loss exponent (PLE) which, in turn, depends on the physical environment. All previous approaches to the problem either assume that the PLE is perfectly known or rely on estimators of the ToA which do not depend on the PLE. In this paper, we introduce a novel analysis of the performance of the maximum likelihood (ML) estimator of the ToA under an imperfect knowledge of the PLE. Specifically, we carry out a Taylor series expansion that approximates the bias and the root mean square error of the ML estimator in closed form as a function of the PLE error. The analysis is first carried out for a path loss model in which the received signal gain depends only on the PLE and the transmitter-receiver distance. Then, we extend the obtained results to account also for shadow fading scenarios. Our computer simulations show that this approximate analysis is accurate when the signal-to-noise ratio (SNR) of the received signal is medium to high. A simple Monte Carlo method based on the analysis is also proposed. This technique is computationally efficient and yields a better approximation of the ML estimator in the low SNR region. The obtained analytical (and Monte Carlo) approximations can be useful at the design stage of wireless communication and localization systems.
The estimation of a signal time of arrival (ToA), also called time delay, plays an important role in applied signal processing problems, e.g., synchronization , array processing , tracking and positioning of mobile terminals [3–8], or even bioengineering . Let us focus on the estimation of the ToA in wireless radio links. In , two simple (and practically appealing) estimators of the ToA are studied: the maximum correlation (MC) and the maximum likelihood (ML) estimators. The MC estimator of the ToA depends only on the correlation between the transmitted and the received signals. The ML estimator of the ToA, on the other hand, takes also into account the path attenuation, which depends on the distance between the transmitter and the receiver, and the path loss exponent (PLE) of the environment. The performances of both estimators in mobile positioning applications are analyzed in , where it is shown that the ML estimator attains a better accuracy when the PLE is perfectly known a priori. However, the latter assumption is often unrealistic for a practical scenario, because the PLE may change according to variations in the environment and thus may need to be estimated. Tracking the fluctuations of the PLE is specially important in problems that involve the localization of mobile terminals, since the existing positioning techniques based on the ToA estimation are extremely sensitive to errors in the PLE .
The problem of dealing with unknown PLEs has been addressed in several related works. In , a positioning application with several receivers and one transmitter is considered. The PLEs are assumed to be different and random, with either uniform or normal distributions. The availability of different PLEs for each link increases the localization accuracy compared to the identical PLE assumption. In [11, 12], the PLE is estimated from the measurements, whereas in [13–15], several algorithms for the PLE estimation are proposed. The authors of  describe three distributed algorithms for PLE estimation in large wireless networks in the presence of node location uncertainties, m-Nakagami fading and interference. In  and , the algorithms for the estimation of the path loss inside a sensor network are designed using previous path loss measurements among sensors. Specifically, in , the PLEs are estimated for each node applying the ML criterion and using both ToA and received-signal-strength measurements among the sensors. In , a handover algorithm is presented using the least squares estimate of the path loss parameters for each link from a mobile station to a base station. In [17, 18], the sensitivity of the ML estimator of the direction of arrival (DoA) of a received signal under model error, i.e., with a mismatch in the PLE, is investigated. To our best knowledge, the problem of estimating the ToA with both the path attenuation and the PLE unknown has not been tackled yet.
The goal of this paper is to investigate the performance of the ML estimator of the ToA under an imperfect PLE. Specifically, we aim at obtaining analytical or semianalytical approximations of the bias and the mean square error (MSE) of the ML estimator, both given in terms of the PLE error and the signal-to-noise ratio (SNR) of the communication channel. Such approximations are intended to be useful in the design and setup of the communication and localization wireless systems, as they may considerably alleviate the need for a lengthy and computationally expensive simulation of the whole system.
This article is organized as follows. We first describes the signal model and then briefly reviews the expressions of the MC and the ML estimators of the ToA with perfect knowledge of the PLE in Section 2, including formulas for their MSE performances. In Section 3, we carry out an approximate analysis of the bias and the MSE attained by the ML estimator of the ToA under imperfect PLE. We first apply the method in  to study the error between the ML joint estimator of the PLE and the ToA, on one hand, and the ML estimator of the ToA under an imperfect knowledge of the PLE, on the other hand. The difficulty of this study is that the resulting error depends on the ML joint estimates of the ToA and the PLE, which are random variables related to the noise. In order to tackle this limitation and provide an analytical (albeit approximate) expression for the error between the true and the estimated ToA, we analyze the ML estimator of the ToA with mismatched PLE in the second part of Section 3. In particular, we propose a new method to analyze the estimation error based on the Taylor series expansion. In Section 4, we investigate the extension of these results to shadow fading environments . In particular, we first show that the approximate bias is insensitive to the shadow fading, whereas the MSE becomes a random variable (the probability density function (pdf) of which is obtained). Then, we provide a straightforward algorithm to obtain a Monte Carlo estimate of the MSE by drawing only from simple Gaussian distributions (instead of simulating the whole transmission system). In Section 5, we present illustrative computer simulation results based on the transmission of an ultra-wideband (UWB) signal. We have chosen UWB signaling to validate our analysis because it provides an excellent means for wireless positioning due to its high resolution in the time domain . Finally, Section 6 is devoted to the conclusions.
2 Signal model
2.1 Received signal
where r(t) is the received signal, s(t) is the transmitted signal, which is assumed deterministic and possibly complex-valued, and τ≥0 are the path gain and the propagation delay (commonly referred to as ToA) between the transmitter and the receiver, and n(t) is the additive noise at the receiver, assumed to be a circularly-symmetric complex-valued zero-mean white Gaussian process with double-sided power spectral density . In (1), τ is the parameter to be estimated.
2.2 MC and ML estimators of the ToA
In this subsection, we briefly review the MC and ML estimators of the ToA. In the literature, the PLE γ is assumed known, and the effect of the shadow fading is ignored; hence, the path gain is deterministic, known and given by (7) .
respectively, where is the correlation between the transmitted and the received signals, and the function fML(τ)=a2Es−2a ρ(τ) is the log-likelihooda of τ given r(t) . Note that a is a function of the propagation delay, as shown in (7). For conciseness, this dependence is left implicit throughout the paper.
where τ and γ are, respectively, the true values of the propagation delay and the path gain; the path loss exponent () is the effective bandwidth, and is the signal-to-noise ratio (see  for additional details). Note that the expression in Equation (9b) coincides with the Cramér-Rao bound (see, e.g., [23–29]) for the time delay estimation .
3 Performance analysis
where γ0 is the (unknown) true value of the PLE, and δ γ is an (equally unknown) additive error. In this section, we assume that the path gain has the form shown in (7); therefore, the following analysis is valid for environments with no shadow fading.
In former works, the sensitivity analysis of the ML estimator under model error has been investigated for DoA estimation (see, e.g., [18, 17]). Similar ideas can be applied to investigate the mis-modeled estimation problem in (12), as shown in Section 3.1. This approach suffers from several limitations, though. For this reason, we introduce a novel approximate analysis based on a Taylor series expansion in Section 3.2.
3.1 Friedlander’s method
In this subsection, we adapt the methodology proposed in  to the problem of the ToA estimation. Note that, while our objective in the present paper is the characterization of the ToA estimation error under an imperfect PLE, in , the author addresses the DoA estimation for a sensor array under the imperfect knowledge of the channel parameters. Therefore, the log-likelihood function is different for each of the two estimation problems (DoA and ToA).
which is obtained by approximating by its first order Taylor series expansion around and (see , Section III).
Friedlander’s approach provides us with a characterization of the error between two estimators of the ToA, namely, the ‘full’ ML estimator obtained by maximizing the likelihood function −fML(τ,γ) jointly over τ and γ, on one hand, and the conditional ML estimator obtained for a fixed (but imperfect or mismatched) value of the PLE γ, denoted by , on the other. However, the error given in (19) is a function of the random variables , , , , and , which in turn, depend on the realization of the noise process n(t). This dependence makes the derivation of the first and the second moments of analytically intractable.
To tackle this limitation, we propose a different approach that aims at characterizing the error between the estimator for a fixed imperfect PLE, γ, and the true value of the ToA, τ0. Our analysis not only leads to a mathematically tractable approximation but it is also, in our opinion, more meaningful from a practical perspective.
3.2 Performance analysis based on a Taylor series expansion
which vanishes when γ=γ0, i.e., when the PLE is perfectly known.
which is identical to the expression in (9b).
4 Performance in shadow fading environments
In this section, we investigate the impact of the shadow fading on the analysis of Section 3. We also propose a Monte Carlo method for the numerical approximation of the MSE that avoids some of the approximations in the latter analysis.
4.1 Analysis in presence of shadow fading
The analytical approximations of the bias and the MSE of the ML estimator of the ToA are based on the assumption that the path gain can be modeled by Equation (7), which follows from neglecting the shadow fading term in Equation (3), i.e., taking ψdB=0 in (3) or, equivalently, ψ=1 in Equation (6).
If shadow fading is explicitly taken into account, the path gain in Equation (6) becomes a random variable (because the factor ψ is random with log-normal distribution). However, assuming that the channel noise n(t) is independent of the shadow fading factor ψ, the analysis of Section 3.2 can still be carried out conditional on the random variable ψ.
for the MSE of τML. Note that the MSE in (38) is a random variable (while the MSE in Equation (33) is deterministic). Let us note that the approximate bias B(τ0,γ) in (37) is independent of the path gain (and, therefore, of the shadow fading factor ψ) and, hence, deterministic. The approximate MSE ε2(τ0,γ,ψ) of Equation (38), however, depends explicitly on a0(ψ); therefore, it is random. Let us look into its characterization.
and variance . Let us remark that this mean is in agreement with the approximation of Equation (34) in Section 3.2.
Equations (50), (51), and (52) provide a complete statistical characterization of the approximate MSE of the ML estimator of the ToA in the presence of shadow fading.
4.2 Monte Carlo approximation
In the previous sections, we have introduced additional approximations beyond the linearization by Taylor series expansion (see Equations (23) and (32)) in order to attain closed form expressions for the bias and the MSE of the estimation error . One consequence of these approximations is that the formulas in (30) and (37) (for the bias), and (33) and (38) (for the MSE) may not properly represent the effect of the denominator in Equation (22), which can be relevant for the performance of the ML estimator in the low SNR region. In this section, we address this limitation of the analysis by describing a simulation-based (Monte Carlo) method that enables us to obtain accurate numerical estimates of the bias and the MSE of the estimator for a given nominal value γ of the PLE in the presence of shadow fading. The technique only requires the ability to draw from a few Gaussian random variables, avoids some of the approximations in the analysis (namely that of Equation (23)), and has a computational cost well below that of the direct simulation of the transmission system.
Note that is random because it depends on four random variables, ρns,0, , , and a0(ψ). The random path gain, a0(ψ), is related to the shadow fading factor ψ, which is log-normally distributed. Specifically, ψdB=10 log10ψ and .
where the integrals in (58c) and (58f) do no not have a closed form but can be computed numerically.
respectively. According to the Strong Law of Large Numbers, the estimates μ N (τ0,γ) and ε2,N(τ0,γ) converge almost surely toward the true mean and the second order moment of (, Chapter 3). Note that this numerical approximation can be also applied assuming a deterministic path gain, i.e., in the absence of shadow fading effect, by simply setting a0=a0(1) (i.e., ψ=1). In this case, the method requires drawing only from the random variables related to the noise (, , and ).
Let us remark that the approximation procedure described in this section is semianalytical: it essentially relies on the error formula of (55), and the Monte Carlo simulation is only used as a numerical tool to integrate w.r.t. the random variables a0(ψ), ρns,0, and . The simulations required are computationally ‘cheap’ compared to a full simulation of the communication system.
Finally, note also that a similar procedure to estimate the Friedlander’s error derived in Section 3.1 is infeasible due to the fact that the error in Equation (19) depends on the random variables , , , , and , the probability density functions of which are unavailable. They are all related to the ML estimates of the ToA and the PLE ( and ), which, in turn, depend also on the realization of the noise process n(t).
5 Numerical examples
where τp is the pulse-shaping factor.
where ω is the angular frequency, and S(ω) is the Fourier transform of s(t). Note that is used here as an approximation of the central frequency in Equation (4), i.e., we assume . By plugging the parameter values of (65a) and (65b) into Equations (30) and (33), we obtain an analytical characterization of the bias and the MSE, respectively, of the ML ToA estimator (), conditional on the nominal PLE γ for the second-derivative Gaussian pulse. If the shadow fading needs to be considered, (65a) and (65b) can be substituted into Equation (37) for the approximation of the bias and into Equations (50), (51), and (52) for the characterization of the random MSE. Similarly, we can substitute and into Equation (55) in order to carry out a Monte Carlo evaluation of the statistics of the error .
In the remaining of this section, we numerically assess the validity of the approximation formulas that we have derived in Sections 3 and 4. In order to consider realistic scenarios, we select values for the PLE γ0 and the variance of the shadow fading effect, , based on measurements in indoor environments .
5.1 Path loss model
Let us first consider the path loss model in the absence of shadow fading. Assuming a line-of-sight (LOS) wireless communication between a transmitter and a receiver in a residential environment, we use a typical value for the PLE as given in , e.g., γ0=1.7. In order to validate the analytical characterization of the conditional ML estimator () obtained in Section 3, we compare the approximate bias and RMSEs computed by means of the formulas in both sections with the results obtained from the direct simulation of the transmission system. In particular, we consider the following methods to evaluate both the bias and the RMSE of the estimator:
Direct simulation of the transmission system with either perfect knowledge of the PLE (γ=γ0, i.e., δ γ =0) or imperfect PLE (γ≠γ0, i.e., δ γ ≠0). We run NR=6,000 independent simulations and compute the conditional ML estimatorb and the errors and in each case. The errors are then averaged and displayed.
The approximate formula of Section 2 for the MSE of the ML estimator, which can only be applied with a perfect PLE (γ=γ0, i.e., δ γ =0).
With an imperfect PLE (δ γ ≠0), we can also compute the (approximate) expected bias and RMSE via the Monte Carlo approach in Section 4.2, using a population N=6,000 samples. Note that in this case, we consider the deterministic path gain given in (7) (no shadow fading); therefore, it is only necessary to draw from the Gaussian variables , , and .
For a better display, the errors (bias and RMSE) are shown in terms of the transmitter-receiver distance d=c τ, where c is the speed of the light. For an arbitrary estimate , the bias and the RMSE of the corresponding distance estimate are proportional (with constant c) to the bias and the RMSE of .
5.2 Shadow fading environment
An LOS exists between the transmitter and the receiver of the wireless communication system, where the typical values for the PLE and the variance of the shadow fading are, e.g., γ 0=1.7 and , respectively .
There is no LOS (NLOS) between the transmitter and the receiver of the wireless communication system. The typical values for the PLE and the variance of the shadow fading are, e.g., γ 0=3.5 and .
In order to validate the analytical and the numerical characterizations of the conditional ML estimator, , obtained in Section 4, we compare the approximate bias and RMSEs with the results obtained from the direct simulation of the transmission system. Similar to Section 5.1, we consider the following methods to evaluate the bias and the RMSE of the ML estimator :
Direct simulation of the transmission system in the both LOS and NLOS communication, with either perfect (γ=γ0) or imperfect knowledge of imperfect PLE (γ≠γ0). We run NR=6,000 independent simulations and compute the ML estimator and the errors and in each one of them. The errors are then averaged and displayed.
With an imperfect PLE (γ≠γ0), we can approximate the RMSE by using the approximate analysis of the random MSE ε2 obtained in Section 4.2. In particular, we can approximate the RMSE of the ML estimator by the square root of the mean in Equation (50).
With an imperfect PLE (γ≠γ0), we can approximate the expected bias and RMSE via the Monte Carlo approach in Section 4.2, using N=6,000 samples of the random error in Equation (55). Note that in this section, we are considering the shadow fading effect; thus, the path gain is a random variable related to the random shadow fading term ψdB, as shown in Equation (6). Therefore, the Monte Carlo approach requires to draw samples from four Gaussian variables , , , and , i=1,…,N.
5.3 Impact of the approximations
In Sections 3 and 4, we have introduced additional approximations, beyond the linearization by Taylor series expansion, in order to attain analytical expressions for the bias and the MSE of the estimator . The goal of this section is to summarize and point out the impact of such approximations.
In Section 3, we have taken the expectation w.r.t. the noise process n(t) in the denominator of Equation (22). One consequence of this approximation is that the formulas in (30) and (37) (for the bias), and (33) and (38) (for the MSE) do not properly capture the effect of the denominator in Equation (22), which is relevant for the performance of the ML estimator in the low SNR region. Although the impact of this approximation cannot be analyzed theoretically, the numerical results in Figure 4 show that the mismatch between the proposed analytical approximation of the RMSE and the RMSE obtained by direct simulation of the communication system is larger in the low SNR region. This artifact is considerably mitigated when the RMSE is approximated with the Monte Carlo method of Section 4.2. Indeed, it can be seen in Figures 4, 6, and 7 (the latter for a shadow fading scenario) that the Monte Carlo estimates of the RMSE are usable in the whole SNR range (although still better for high values, SNR>20 dB). Note that the Monte Carlo procedure of Section 4.2 requires to draw only from a few simple Gaussian distributions, which is computationally much cheaper than simulating the complete transmission system.
The Taylor series expansion leads to an approximation of the MSE that depends on the sign of the difference δ γ =γ−γ0 between the true and the nominal values of the PLE, as shown in Equation(31). We have followed [, equation (17-9.6), p. 642] in order to remove this dependence, leading to Equation (32). As a consequence, the analytical and the Monte Carlo approximations of the RMSE proposed in the paper are independent of the sign of δ γ . While this approximation is correct for small δ γ , the simulations (see Figures 2, 3, 5, and 6) show that the estimator is more sensitive to negative errors in the PLE.
We have analyzed the performance of the maximum likelihood of a signal time-of-arrival when the path loss exponent of the communication link is not perfectly known. In the first approach, we have modeled the signal received amplitude as a deterministic function of the PLE and the transmitter-to-receiver distance. Within this setup, we have applied a Taylor series expansion, together with other approximations needed for mathematical tractability, in order to obtain closed-form expressions for the bias and the RMSE of the ML ToA estimator. In the second stage, we have extended our analysis to cope with shadow fading effects. In such case, the analytical approximation of the estimator MSE takes the form of a random variable, the mean, variance, and probability density function of which are derived and given in closed form. Additionally, we have introduced a simple Monte Carlo method for the numerical computation of the errors, which removes some of the approximations in the analysis, can be applied both with and without shadow fading, and presents a computational load much smaller than that of the direct simulation of the transmission.
We have carried out extensive computer simulations to assess the validity of the proposed approximation tools. For the evaluation, we have simulated the transmission of a second derivative Gaussian pulse, a waveform commonly used in UWB systems, where the ToA estimation is of great importance for positioning applications. Our simulations show that the analytical approximations, both with and without shadow fading, are very accurate in the medium and high SNR regions. The Monte Carlo technique is equally accurate with mid and high SNRs, while it also yields usable approximations of the bias and RMSE in the lower SNR region. The computer experiments also show that the ML ToA estimator is more sensitive to the underestimation of the PLE than to its overestimation (this result is consistent with the numerical study of ).
a Actually, the negative of the log-likelihood.
b We have solved the optimization problem of Equation (12) numerically in order to compute . In particular, we have generated a regular grid of candidate values of τ (with separation of 10−2 ps between adjacent points of the grid), evaluated the likelihood fML for every candidate, and then selected the best one. It is not the goal of this paper to propose a practical means for the calculation of , but simply to assess its theoretical performance. In a practical receiver, an adaptive algorithm (similar to a timing error detector ) could be used to compute .
Appendix 1: derivatives of the likelihood function f M L ( τ,γ )