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Time-of-Arrival Estimation Based on Information Theoretic Criteria Andrea Giorgetti, Member, IEEE, and Marco Chiani, Fellow, IEEE
Abstract—The possibility to accurately localize tags by using wireless techniques is of great importance for several emerging applications in the Internet of Things. Precise ranging can be obtained with ultra wideband (UWB) impulse radio (IR) systems, where short impulses are transmitted, and their time-of-arrival (ToA) is estimated at the receiver. Due to the presence of noise and multipath, the estimator has the difficult task of discriminating the time intervals where the received waveform is due to noise only, by those where there are also signal components. Common low-complexity methods use an energy detector (ED), whose output is compared with a threshold, to discriminate the time intervals containing noise only from those containing signal plus noise. Optimal threshold design for these methods requires knowledge of the channel impulse response and of the receiver noise power. We propose a different approach, where ToA estimation is based on model selection by information theoretic criteria (ITC). The resulting ToA algorithms do not use thresholds, and do not require any information about the channel or the noise power level. These blind, universal ToA estimators show, for completely unknown multipath channels and in the presence of noise with unknown power, excellent performance when compared with ideal genie-aided schemes. Index Terms—Information theoretic criterion, model selection, radar, ranging, RF-ID, time-of-arrival, ultrawide band systems.
I. INTRODUCTION
R
ELIABLE localization with sub-meter accuracy in harsh propagation environments, as inside buildings, is of great importance in large set of emerging wireless sensor network (WSN) applications: commercial, public safety, automotive safety, search and rescue operations, and military systems [1], [2]. Given a set of measurements between nodes in a network, localization algorithms are capable to find the unknown positions
Manuscript received February 22, 2012; revised August 04, 2012 and November 29, 2012; accepted December 05, 2012. Date of publication January 11, 2013; date of current version March 20, 2013. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Philippe Ciblat. This research was supported, in part, by the European Commission under the FP7 ICT project Content and cOntext aware delivery for iNteraCtivemicro-multimEdia healthcaRe applicaTiOns (CONCERTO), Grant no. 288502; and by the Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR) under the PRIN 2009 project no. 2009A85F98_003. This paper was presented, in part, at the IEEE International Conference on Ultrawide-band (ICUWB), Bologna, Italy, September 2011. The authors are with the Department of Electrical, Electronic, and Information Engineering “Guglielmo Marconi” (DEI), CNIT, University of Bologna, 47521 Cesena (FC), Italy (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2013.2239643
of nodes. Measurements between nodes can be broadly classified in range-based, angle-based, and proximity-based. Among them, range-based systems (i.e., based on distance estimates) have gained an increasing attention for high-definition localization accuracy when low complexity devices are available [3], [4]. The UWB technology offers, even in critical environments, the potential of achieving high ranging accuracy through signal ToA measurements, due to its ability to resolve multipath and penetrate obstacles [5]–[8]. Ranging techniques based on ToA estimation of the first arriving signal path have to face the effects of noise, multipath propagation, obstacles, interference and clock drift [4], [9]–[11]. Moreover, in dense multipath channels, the first path is often not the strongest, making estimation of the ToA a challenging task [12]–[14]. Maximum likelihood (ML) ToA estimators are known to be asymptotically efficient, that is, they achieve the Cramér Rao Bound for increasing signal-to-noise ratio (SNR). However, they require sampling at Nyquist rates or higher, which can be impractical due to the large bandwidth of UWB signals. For this reason, ToA estimation schemes based on ED are gaining interest due to their low complexity implementation with sub-Nyquist sampling rates [15]–[17]. These schemes rely on the energy collected at sub-Nyquist rates over several time slots. A similar approach can be obtained through low-rate sampling of correlation outputs if a-priori knowledge of a signal template is available [18]. After the ED, a simple technique to detect the first arriving path is threshold crossing (TC), where the output of the ED is compared with a threshold and the first crossing gives the ToA [19], [20]. The threshold value strongly influences the performance of TC ToA estimation, and needs to be carefully designed according to the operating conditions. When the threshold is small, we expect a high rate of early detection events (i.e., false alarm, due to noise and interference prior to the first path). On the other hand, especially in non-line-of-sight (NLOS) propagation where the first path is often weak, increasing the threshold will increase the missed detection rate. Optimal threshold design, to minimize the root mean-square error (RMSE), requires knowledge of the channel impulse response (CIR) and of the receiver noise power. Otherwise, if only channel statistics and receiver noise power are available, it is possible to design an optimum threshold (OT) which minimizes the average RMSE, where the average is taken over the channel statistics. A simple criterion to determine a sub-optimal threshold, based on the evaluation of the probability of early detection and noise power knowledge, is proposed in [19]. Also, several other sub-optimal techniques have been proposed to reduce early
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detections in TC, such as jump back and search forward (JBSF) criterion (based on the detection of the strongest sample and a forward search procedure over a predefined time window), and serial backward search (SBS) criterion (based on the detection of the strongest sample and a search-back procedure over a predefined time window) [20], [21]. Despite their low complexity, all these approaches need to set a threshold plus other parameters like, e.g., the window length in JBSF and SBS, for which the noise power and some channel statistics must be known [10]. Hence, channel adaptive strategies are needed, since the threshold value and the other estimator parameters, giving good results for a given environment, would not be the best choice for a different one. Unfortunately, in many practical operating conditions and owing to the need to have low complexity receivers, we cannot assume to have reliable channel statistics at the receiver. Moreover, noise power is only known with some degrees of uncertainty because of temperature variations, calibration errors, and presence of interferers [22]. In this paper, we propose practical blind ED based ToA estimation schemes for UWB IRs extremely robust to channel statistics and noise power uncertainties. Instead of using a threshold to discriminate noise-only bins from signal-plus-noise bins, we propose to estimate the number of the noise-only bins by using model order selection methods. In other words, model selection is used as a non-linear excision filter, to locate and remove the noise-only bins. Once the noise-only bins have been identified, the ToA is that of the earlier signal-plus-noise bin. The resulting estimators are completely blind, since they do not require any information about the channel and the noise power, and they eliminate the need to set a threshold. The contributions of this paper can be summarized as follows: • We introduce the concept of ToA estimation based on model order selection by ITC. • We derive the exact expression and approximations of the log-likelihood functions needed for model selection. • We investigate several penalty functions for model selection, leading to low-complexity algorithms. • We numerically compare the performance of the proposed algorithms with TC based ToA estimators. We show that the novel blind ToA estimators achieve, in realistic multipath channels, a RMSE very close to that obtained with genie-aided TC estimation where complete channel model and noise power knowledge is assumed. The proposed algorithms can be applied in all contexts where robust ToA estimation needs to be performed in the presence of noise uncertainty and where information about the channel model are not available, such as, e.g., in radar systems. It can be also extended to systems where matched filtering is used instead of the ED. The paper is organized as follows. Section II presents the system model, derives the statistical distribution of the ED output and its approximations. Section III presents two ITC-based ToA estimation techniques. Section IV presents the exact as well as approximate expressions for the log-likelihood function (LLF) required by ITC. In Section V we study the penalty terms required by ITC. Section VI presents numerical results for the proposed blind ToA estimation techniques,
compared with TC, in realistic multipath channels. Section VII concludes the paper and summarizes important findings. In the rest of the paper we use the following notation: stands for the real part of a complex number; stands for expectation; denotes a real Gaussian distribution with mean and variance ; denotes a circularly-symmetric complex Gaussian (CSCG) distribution, where the real and imaginary parts are independent, identically distributed (i.i.d.) ; denotes a Chi-square distribution with degrees of freedom and non-centrality parameter . II. SYSTEM MODEL A. System Description We consider a typical signal structure in which the transmitter sends a packet with a preamble, known at the receiver, used for acquisition, synchronization, and ranging. The preamble is divided into frames each with duration . Each frame is further decomposed into smaller intervals, called chips, each with duration . A single unitary energy pulse , with duration , is transmitted in each frame in a position specified by a user-specific pseudo-random time-hopping (TH)1 sequence having period [5], [10]. A generic ED-based ToA estimator scheme is shown in Fig. 1. After bandpass filtering with bandwidth and center frequency to eliminate the out-of-band noise, the received signal due to one preamble is [10] (1) where is the received pulse given by the continuous convolution ,2 is the CIR, and represents the bandlimited thermal noise with one-sided power spectral density (PSD) . We consider frequency-selective fading channels, represented as [24] (2) is the number of received multipath components, and denote the amplitudes and delays of the paths, respectively, and is the Dirac-delta function. Assuming that the receiver knows the TH sequence of the desired user, the goal is to estimate the ToA of the first path by observing the received signal (1). A common approach to precise ToA estimation is to adopt a two-step procedure which first leads to a coarse estimate through frame acquisition and then a finer estimate. Acquisition of a UWB signal is commonly achieved by peak detection of the received pulse and confine the ToA into an uncertainty region of one frame interval [13], [15], [18]. After acquisition, the ToA where
1Without loss of generality we include only TH in the signal model because direct sequence (DS) signaling, if present, does not play any role in non-coherent standard adopts ToA estimation. As a special case of DS, the IEEE ternary sequences [23]. 2
includes the effects of bandpass filtering.
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Fig. 1. The ToA estimation scheme. After bandpass filtering, the received signal, , composed of pulses is decomposed into time intervals then processed energy profiles (each one composed of bins). Based on these observations, the ToA is estimated by ITC. by an ED to extract
estimator utilizes a portion of consisting of sub-intervals , each of duration , with . Note that is chosen to guarantee that only one received pulse stays within each sub-interval to avoid ambiguities in ToA estimation [17].3 The observed signal forms the input to the ED, which is composed of a square-law device followed by an integrate-and-dump (ID) device whose integration time is seconds. The ED output is then sampled every seconds, thus samples are collected in each sub-interval with indexes corresponding to time slots or bins. The true ToA, , is contained in the time slot . In the absence of other information the system can assume that is uniformly distributed in the uncertainty region , with ; as a consequence the discrete random variable (r.v.) is uniformly distributed on the integers [10], [18]. Note that the first samples contain noise only (the noise region), followed by the sample containing the first path, and the remaining samples that contain possibly the echoes of the useful signal (the multipath region), in addition to noise. The ED integration time, , determines the time resolution and thus the minimum achievable RMSE on ToA estimation is given by [10], [17], [19]. For notational convenience, the collected samples ( intervals with samples each) at the output of the ED are arranged in the energy matrix , with elements
channel) during the transmission of pulses, the signals , are all equal and thus can be seen as replicas of the same received pulse. In the following we consider the channel to be slowly-varying so that we can consider the CIR to be constant during the transmission of frames. In conventional threshold based ToA estimators, a simple and common way to combine these replicas consists in building a vector through simple column averaging (averaging filter), that is [10] (4) A widely adopted ToA estimation technique is threshold crossing (TC) which consists in deciding for the leading edge as the smallest index in (4) for which is greater than a suitable threshold [10]. In contrast, our ITC-based ToA estimator does not need thresholds, and uses as statistic the energy matrix , as explained in Section III. B. Statistical Analysis of the Energy Matrix Since our estimator is based on the observed energy matrix , we need first to analyze the statistical distribution of the r.v.s (3). Considering the equivalent low-pass (ELP) representation , where is the center frequency of the bandpass signal, we can rewrite (3) as (5)
(3)
where is a complex bandlimited signal with frequency band . Therefore, can be expanded using the sampling theorem, giving [25], [26]4
, is the received signal where after the de-hopping process in the sub-interval . Note that in the absence of noise and assuming time-invariant CIR (static
(6)
3The choice of and the acquisition phase ensure that neither intersymbol interference nor interframe interference occur.
4Strictly speaking this is an approximation. The discussion about its accuracy can be found in [25], [26].
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where we assume Then, we write
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and
.
where and are the ELP representations of the received pulse and noise, respectively, after the bandpass filter. In particular, is a bandlimited CSCG process with two-sided PSD and therefore the r.v.s are i.i.d. in , and . The terms for the bins containing both the signal and the noise can be rewritten as
(7) while for the noise-only bins the are as in (7) but without the contribution from . Note that the r.v.s are i.i.d. in , and statistically independent in . 1) Exact Distribution: Since the are Gaussian and are i.i.d. in , we have , i.e., the energies in (7) are non-central Chi-square distributed r.v.s whose probability distribution function (p.d.f.) is [26]
Two approximations are used in this paper, one based on the central Chi-square distribution, the other on the Gaussian distribution. 2) Central Chi-Square Approximation: A possible useful approach consists in approximating the non-central Chi-square distribution with a central one.5 By matching the second-order moment (11) of the non-central Chi-square distribution (8), with that of the central Chi-square distribution (10), obtained from (11) with , we then approximate with , where (12) we get . Note that for 3) Gaussian Approximation: Another choice to derive simplified ITC-based ToA estimators is based on the Gaussian approximation. To this aim we recall that, for the Central Limit Theorem, for large the distribution of can be approximated by a Gaussian distribution with the same mean and variance (given by (11)). III. INFORMATION THEORETIC CRITERIA FOR TOA ESTIMATION
(8) for , with degrees of freedom, non-centrality parameter , , and where is the -order modified Bessel function of the first kind [27, ch. 9, p. 374]. It is important to note that, going back to continuous-time signals, the non-centrality parameter is equal to the energy of the received pulse in the bin, i.e.,
(9) so that Clearly, the bins containing only noise have the energy samples are central Chi-square r.v.s with p.d.f. [26]
(10) where is the Gamma function [27, ch. 6, p. 255]. The mean and second-order moment of (8) are respectively [26] (11) The same equations can be used for the distribution (10) by . setting To derive simplified ITC-based ToA estimators, it will be useful to find approximations for the exact p.d.f. that do not involve the -order modified Bessel function of the first kind.
We propose a blind ToA estimator using the energy matrix and based on model order selection theory [28]–[38]. Let us start by defining with the terms signal-plus-noise bins and noise-only bins, two disjoint sets of bins, corresponding to ED samples containing energy from the received pulse plus noise, and containing only noise, respectively. The bin containing the leading edge is determined once the set of signal-plus-noise bins is known, simply as the earliest among the signal-plus-noise bins. So, the problem is essentially a classification problem: we have to partition the set of bins into two sets of signal-plus-noise bins and noise-only bins. To this aim we adopt model order selection to estimate the number of bins, among the weakest in the observation window, due to noise only. This can be also interpreted as a non-linear excision filter which allows the deletion of the noise-only bins. More precisely, we define a family of models to fit the observed data , in which the th model describes the case where, after sorting the bins according to some criteria (described later), the first are noise-only bins and the remaining are signal-plusnoise bins, with a number of free adjusted parameters univocally related to .6 In the following, we propose two different algorithms, related to two different sorting criteria, that we name ToA Estimation based on Model Order Selection from All bins (TEMOS-A) and ToA Estimation based on Model Order Selection from Early bins (TEMOS-E). Accordingly, denoting by the number of 5Although there are several methods to approximate a non-central Chi-square distribution, most of these aim at finding a distribution with good tail fit. On the contrary, since we are interested in evaluating the log-likelihood function (see Section IV), we look for a good approximation that works reasonably well on the whole support of (8), to guarantee a global goodness of fit. In this regard, we found that our choice to approximate (8) with a central Chi-square distribution gives very good results, as confirmed in Section VI. 6The parameter is a positive integer, as it represents the number of free adjusted parameters in model order selection. Since we assume that the noise variance is not known, one of the free adjusted parameters is the noise variance.
GIORGETTI AND CHIANI: TIME-OF-ARRIVAL ESTIMATION BASED ON INFORMATION THEORETIC CRITERIA
The ML estimate of the noise variance sumed noise-only bins, is simply
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, based on the pre-
(16)
Fig. 2. Illustration of the ordered vector model order hypothesis for TEMOS-A. erage bin energies, in the
of av-
bins used for model selection, will indicate the matrix of collected energies and the corresponding vector of elements obtained by column averaging through (4). A. TEMOS-A: ToA Estimation From all Bins For TEMOS-A, sorting is based on energy and all available . More precisely, we first sort the bins are used, i.e., vector in ascending order, , and define as the vector containing the permutation indexes, i.e., . In the th model order hypothesis there are noise-only bins with unknown noise variance , and signal-plus-noise bins with unknown signal energies (corresponding to ). Therefore, the noise-only bins and the signal-plus-noise bins have indexes and , respectively (see Fig. 2). We then define the unknown parameters vector for the th model as (13)
Now the problem is to estimate the number of noise-only bins, , given observations (the rows in ). According to the theory of model order selection based on ITC, we estimate as (14) where is the log-likelihood of the observed data , which depends on the parameter vector , and is a penalty function associated with the th model. These two functions will be studied in Sections IV and V. The vector in (14) is the estimate of the model7 unknown parameters for the (15) where the estimate is obtained through the ML principle. 7With
the
a slight abuse of notation model order hypothesis.
stands for the noise variance estimate in
For the signal-plus-noise bins we should use ML estimation of the non-centrality parameters corresponding to the signal energy. However, the ML estimator of the non-centrality parameter of a Chi-square r.v. cannot be expressed in closed-form, since the maximization of the likelihood function leads to a numerical solution of an equation involving modified Bessel functions [39], [40]. Thus, instead of the ML, we use the simpler and yet accurate estimator8 (17) Once all functions and parameters have been obtained, the model order selection method (14) provides as output the estimated number of noise-only bins. Then, the bin indicating the ToA corresponds to the earliest bin among the signalplus-noise bins (18) giving the ToA estimate
.
B. TEMOS-E: ToA Estimation From Early Bins To simplify the previous algorithm we observe that the leading edge will be always chosen between the first bin and the bin containing the maximum received energy. Thus, in TEMOS-E we restrict the observation window. To further reduce complexity, sorting is just based on the time indexes. More precisely, we define the set of early bins as those with indexes where and . Accordingly, since no ordering based on energy is required, for this algorithm the permutation vector reduces to . Then, similar to TEMOS-A, in the model order hypothesis there are noise-only bins with unknown noise variance , followed by signal-plus-noise bins with unknown signal energies (corresponding to ). The noise-only bins and the signal-plus-noise bins have indexes and , respectively (see Fig. 3). With these sets, we estimate the number of noise-only bins, , by applying (13)–(17). For TEMOS-E the bin containing the leading edge corresponds to the first signal-plus-noise bin (19) and the ToA estimate is . While TEMOS-A is general and can be applied without any assumption on the channel model, TEMOS-E is based on the assumption that there are no noise-only bins between the bin 8The operator stands for the positive part of . It , then the ML estimate of is dominated has been shown that if by the simplified estimator (17) with squared error as the loss function [40]. As will be shown in Section VI such simple estimator gives very good results.
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Expressions (20) and (21) involve the -order modified Bessel function of the first kind. In the following we provide two approximations of the LLF that require less computational effort with negligible performance loss. B. LLF by Central Chi-Square Approximation (CCSLL) By using the central Chi-square approximation introduced in Section II-B2, the LLF can be simplified and rewritten as9 Fig. 3. Illustration of the vector of average energies of model order hypothesis for TEMOS-E. early bins, in the
containing the leading edge and the bin with maximum energy. Hence, TEMOS-E may suffer a degradation if the received signal has a very large bandwidth and the integration time is smaller than the path interarrival times.
(22) with
IV. THE LOG-LIKELIHOOD FUNCTION The LLF of the matrix , , can be evaluated from the analysis in Section II-B. We derive the exact expression , named exact log-likelihood (ELL), as well as of simpler expressions obtained by approximating the non-central Chi-square distribution with central Chi-square and Gaussian distributions, called central Chi-square log-likelihood (CCSLL) and Gaussian log-likelihood (GLL), respectively.
(23) C. LLF by Gaussian Approximation (GLL) By using the Gaussian approximation discussed in Section II-B3, the LLF can be simplified as
A. LLF From the Exact Statistical Distribution (ELL) From the p.d.f. derived in Section II-B we have
(20) (24) is given by (8) for and by (10) for where . In TEMOS-A, considering that for is evaluated through (17), and that (16) is the average noise energy evaluated from the smallest bins (due to ordering), it is always true that for . Hence, for TEMOS-A the LLF can be also written more explicitly as
(21)
Note that (22) and (24) are simpler to implement than (20) and (21). V. THE PENALTY FUNCTION Model selection has been first addressed in [28], [36] based on ITC, and in [37] based on a Bayesian approach. Recent developments can be found in [30], [33], [34], [38]. The performance of model order selection strategies is in general measured by their tendency to under-parameterization or over-parameterization, i.e., their aptitude to identify a model order which is smaller or larger than the true one [30]. Therefore, for our ToA estimation problem (14) we investigated several penalty terms, with different complexities and performance. In the following the most interesting, which we found to be the minimum description length (MDL), the efficient detection criteria (EDC), the Consistent AIC With Fisher Information (CAICF), and a new simplified version of CAICF suitable for ToA estimation, will be presented. 9In this expression and in the following we omitted all terms that do not depend on .
GIORGETTI AND CHIANI: TIME-OF-ARRIVAL ESTIMATION BASED ON INFORMATION THEORETIC CRITERIA
A. Minimum Description Length (MDL) Note that the MDL criterion in [36] is in general different from the Bayesian information criterion (BIC) [37], although for large samples the two will select the same model [38]. For the sake of simplicity we will here refer to the simplified version of the MDL which is equivalent to the BIC, whose penalty function is [36] (25)
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The calculation of (28) with the exact distribution, i.e., starting from (20), is quite cumbersome, and leads to complicate expressions. Therefore, we proceed considering the Gaussian approximation (24) which leads to an easy-to-handle closed-form expressions for (28). We name this approach penalty under Gaussian approximation (PGA). In the Appendix it is shown that under this approximation the FIM (28) for both TEMOS-A and TEMOS-E assumes the form (29) with elements (30) at the bottom of the page. D. Simplified CAICF
B. Efficient Detection Criteria (EDC) EDC is a family of criteria where the penalty functions satisfy conditions for consistency [41]. In this case a possible penalty function is (26) which is double the penalty function of MDL.
Based on the derivation of the FIM we propose a simplified penalty function which leads to a negligible performance loss with a significant complexity reduction. In fact, in the specific ToA application, the number of noise-only bins is in general quite large. Therefore, all terms in the FIM that involves can be neglected. If we consider only the FIM involving the , the penalty function (29) can be parameters simplified as
C. Consistent AIC With Fisher Information (CAICF) The CAICF is an extension of the Akaike Information Criterion (AIC) which makes AIC asymptotically consistent and penalizes over-parameterization [33], [34]. In CAICF the penalty term is [33]10 (27) is the Fisher information matrix (FIM) for the where model with elements
(28) 10Note
that the first term is the penalty function in AIC.
..
.
(31)
We name (31) simplified penalty under Gaussian approximation (SPGA). Note that (31) has a complexity comparable to the LLF, i.e., the first term in (14). VI. RESULTS In this section we provide some numerical results to show the effectiveness of the proposed blind ITC-based ToA estimation techniques in realistic multipath channels. The performance are derived considering a UWB IR signal with root raised cosine (RRC) bandpass pulse, compliant with IEEE specifications, with , roll-off factor and a duration parameter . We also consider ,
.. .
(29)
(30)
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, and an ideal bandpass filter with bandwidth .11 The investigated number of frames are , and , corresponding to preamble consisting of 1, 16, and 64 ternary sequences, respectively, for the IEEE .12 All results are obtained through simulations considering the IEEE channel models CM1 (LOS residential) and CM4 (NLOS office) with 1000 channel realizations for each curve. In the following, we define the signal-to-noise ratio as where is the overall received pulse energy. As a benchmark for performance analysis we choose TC estimation which has been widely analyzed in the literature. TC methods require accurate setup of the threshold, , which in turn require channel and noise power knowledge. Assuming all these parameters are perfectly known, it is possible to find the threshold-to-noise ratio (TNR), defined as , that guarantees optimal performance (i.e., minimum RMSE). In this paper, such optimal TNR, , has been found through exhaustive a-posteriori search, as the value that minimizes the average RMSE, averaged over all CIR realizations, for each SNR.13 Once and the noise variance are known, the threshold is readily determined. Since the noise power is in general not perfectly known, but estimated with some accuracy, the corresponding threshold can be computed as , where is the estimate of . Two possible estimates of are: 1) estimated noise power from the first bin (ENPF) in , i.e., ; 2) estimated noise power from the smallest bin (ENPS) in , i.e., . In the following we use also the term genie-aided OT algorithm, to indicate the TC ToA estimation with the OT obtained assuming perfect channel model and noise power knowledge (provided by a genie).
Fig. 4. Average RMSE for OT techniques and the proposed blind TEMOS-A (blue curves), 256 (green algorithm (ELL and CAICF with PGA) with channel model CM1. curves), and 1024 (red curves), in the IEEE
A. Performance of TEMOS-A In Fig. 4 the average RMSE for TC techniques and the proposed blind TEMOS-A algorithm are compared in the IEEE channel CM1. In particular, the blind ToA estimation algorithm considered is based on the exact LLF (21) and CAICF with PGA (27)–(30). As can be seen, despite the choice of an optimal TNR, both OT with ENPF and ENPS gives very poor performance because of the insufficient accuracy of noise power estimation, which results in choosing a wrong threshold. On the contrary, despite the proposed algorithm is completely blind, its performance is remarkably close to the genie-aided OT even for few received pulses, i.e., . In Fig. 5 the performance of TEMOS-A with various forms of the LLF, as discussed in Section IV, are compared in the IEEE channel CM4. It can be noted that the central Chi-square approximation and the SPGA have very good 11The corresponding Nyquist rate is pling rate at the output of the ID device is i.e., the sampling rate is
, while the sam. Hence, the Nyquist rate.
12The
IEEE 802.15.4a standard proposes a preamble consisting only of am) using a length-31 ternary sequence with plitude modulated pulses (i.e., an ideal periodic autocorrelation function [42]. This preamble enables robust sequence acquisition and allows the use of both coherent and non-coherent ToA estimation. 13An approximation of the optimal TNR which is mildly dependent on channel model has been found analytically in [19].
Fig. 5. Average RMSE for TEMOS-A, considering different combinations of approximated LLFs and penalty terms (CAICF with PGA and SPGA), with (violet curves), 256 (orange curves), and 1024 (brown curves), in channel model CM4. TEMOS-A is also compared with the IEEE the genie-aided OT technique.
performance with respect to the exact LLF and PGA, while the main degradation is introduced by the Gaussian approximation of the LLF. Performance of MDL and EDC for TEMOS-A are not provided since they perform much worse than CAICF, and in any case do not give significant complexity reduction. B. Performance of TEMOS-E In Figs. 6 and 7, the performance of TEMOS-E estimation algorithms based on combinations of two forms of the LLF and all the penalty terms, are compared in both channel models CM1 and CM4. It can be noted that even for this algorithm CCSLL is an excellent approximation. As far as the penalty function is concerned, SPGA performs very well, irrespective of the number of pulses and channel models, while when the number of samples is small (i.e., in Fig. 6), EDC gives the best performance in the entire SNR range. On the contrary, the MDL exhibits good performance at low SNR (below
GIORGETTI AND CHIANI: TIME-OF-ARRIVAL ESTIMATION BASED ON INFORMATION THEORETIC CRITERIA
Fig. 6. Average RMSE for the genie-aided OT technique and the proposed , in the IEEE channel blind TEMOS-E algorithm, with model CM1 (blue curves) and CM4 (violet curves). Different combinations of LLF (ELL and CCSLL) and penalty term (CAICF with SPGA, MDL and EDC) are considered.
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It is also important to note that although TEMOS-E with MDL is completely blind, it slightly outperforms the genieaided OT for some values of the SNR. In fact, looking at Fig. 7, we can see that in the SNR range between to for both CM1 and CM4, TEMOS-E with MDL has a SNR gain of about with respect to genie-aided OT. This slight performance improvement is due to the fact that the OT is determined a-posteriori as a value which minimizes the average RMSE over the channel ensemble, while TEMOS-E estimates the bin containing the leading edge for each channel realization. Finally, note that since TEMOS-A applies the ITC to ordered bins, even a small error in may cause a large error in ToA estimation (18). On the contrary, TEMOS-E applies ITC directly on unordered bins and a small error in corresponds to a small error in ToA estimation (19). Hence, TEMOS-E is less sensitive to model order selection performance than TEMOS-A. For this reason, while TEMOS-A requires sCAICF, TEMOS-E have quite good performance with all the penalty terms considered, especially when is high. VII. CONCLUSION
Fig. 7. Average RMSE for the genie-aided OT technique and the proposed , in the IEEE channel blind TEMOS-E algorithm, with model CM1 (red curves) and CM4 (brown curves). Different combinations of LLF (ELL and CCSLL) and penalty term (CAICF with SPGA, MDL and EDC) are considered.
for CM4 and below for CM1) but poor performance at high pSNR, where the RMSE floor is higher than that of genie-aided OT. When the number of pulses is relatively high (i.e., in Fig. 7), MDL in general outperforms the other techniques. C. Comparison While TEMOS-A is practically equivalent to the genie-aided, the RMSE floor introduced by TEMOS-E is in general slightly higher than that of genie-aided OT: for instance the genie-aided OT floor translates to a RMSE of , while that of TEMOS-E is .14 14The RMSE floor can be controlled by the integration time and can be reduced by increasing the sampling rate at the cost of a higher complexity [9], [10], [15], [17], [19]–[21]. The analog-to-digital conversion (ADC) resolution may have also an impact on complexity and ToA estimation performance [17].
This paper proposed blind ToA estimation techniques based on model selection for UWB IR systems. The new ToA algorithm, based on ITC, eliminates the need to set a predefined threshold, typical of TC ToA estimation algorithms. In particular, the algorithm is capable to detect the noise-only bins and signal-plus-noise bins using model order selection methods, from which the ToA is immediately found. The performance of the proposed ToA estimators have been compared with TC algorithms in realistic multipath channels. Model selection based algorithms exhibit excellent performance, comparable with genie-aided TC ToA estimation with perfect channel and noise power knowledge. Requiring only the collection of received energy as input, the proposed ToA estimators do not need any information about the channel model or the noise power level, and are suitable for low-complexity implementation. APPENDIX The FIM (28) can be evaluated for both TEMOS-A and TEMOS-E. We define the parameter vector of elements where the first elements refers to the estimated signal energy of presumed signal-plus-noise bins. Therefore, the FIM can be conveniently partitioned into four submatrices ..
.
.. .
that can be calculated separately. The elements of the FIM can be derived starting from the exact log-likelihood (that leading to (20)). However, the resulting expressions are quite complicated. Fortunately, we found that a negligible performance loss is paid when, for the FIM, we use the Gaussian
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approximation of the log-likelihood of observed data , as in (24)
(32) leading to more manageable expressions. Note that in (32) for . TEMOS-E we have The first submatrix contains Fisher information elements that refer to signal energy estimates. From (32) the diagonal terms of the first submatrix are
(33) where the expectation is evaluated considering (11). Regarding the off-diagonal terms it is straightforward to show that
(34) refers to the Fisher informaThe element tion of the noise variance estimate and can be written as
(35) Finally, the two submatrices with the Fisher information between signal energy estimate and noise variance estimate can be written as
(36)
ACKNOWLEDGMENT The authors wish to thank Davide Dardari and Moe Z. Win for helpful discussions, and the Editor and anonymous reviewers for their valuable comments and suggestions. REFERENCES [1] D. Dardari, E. Falletti, and M. Luise, Satellite and Terrestrial Radio Positioning Techniques—a Signal Processing Perspective. London, U.K.: Elsevier, 2011. [2] M. Z. Win, A. Conti, S. Mazuelas, Y. Shen, W. Gifford, D. Dardari, and M. Chiani, “Network localization and navigation via cooperation,” IEEE Commun. Mag., vol. 49, no. 5, pp. 56–62, May 2011. [3] H. Soganci, S. Gezici, and H. V. Poor, “Accurate positioning in ultrawideband systems,” IEEE Wireless Commun., vol. 18, no. 2, pp. 19–27, Apr. 2011. [4] S. Gezici, Z. Sahinoglu, H. Kobayashi, and H. V. Poor, “Ultra wideband geolocation,” in Ultrawideband Wireless Communications. New York, NY: Wiley, 2006. [5] M. Z. Win and R. A. Scholtz, “Ultra-wide bandwidth time-hopping spread-spectrum impulse radio for wireless multiple-access communications,” IEEE Trans. Commun., vol. 48, no. 4, pp. 679–691, Apr. 2000. [6] D. B. Jourdan, D. Dardari, and M. Z. Win, “Position error bound for UWB localization in dense cluttered environments,” IEEE Trans. Aerosp. Electron. Syst., vol. 44, no. 2, pp. 613–628, Apr. 2008. [7] R. J. Fontana and S. J. Gunderson, “Ultra-wideband precision asset location system,” in Proc. IEEE Conf. Ultra Wideband Syst. Technol. (UWBST), Baltimore, MD, USA, May 2002, vol. 21, pp. 147–150. [8] W. C. Chung and D. Ha, “An accurate ultra wideband (UWB) ranging for precision asset location,” in Proc. IEEE Conf. Ultra Wideband Syst. Technol. (UWBST), Reston, VA, USA, Nov. 2003, pp. 389–393. [9] D. Dardari, A. Giorgetti, and M. Z. Win, “Time-of-arrival estimation of UWB signals in the presence of narrowband and wideband interference,” in Proc.IEEE Int. Conf. Ultra-Wideband (ICUWB), Singapore, Sep. 2007, pp. 71–76. [10] D. Dardari, A. Conti, U. Ferner, A. Giorgetti, and M. Z. Win, “Ranging with ultrawide bandwidth signals in multipath environments,” Proc. IEEE, vol. 97, no. 2, pp. 404–426, Feb. 2009. [11] S. Gezici, Z. Tian, G. B. Giannakis, H. Kobayashi, A. F. Molisch, H. V. Poor, and Z. Sahinoglu, “Localization via ultra-wideband radios: A look at positioning aspects for future sensor networks,” IEEE Signal Process. Mag., vol. 22, pp. 70–84, Jul. 2005. [12] N. Decarli, D. Dardari, S. Gezici, and A. A. D’Amico, “LOS/NLOS detection for UWB signals: A comparative study using experimental data,” in Proc. IEEE Int. Symp. Wireless Pervasive Comput. (ISWPC), Modena, Italy, May 2010, pp. 169–173. [13] K. Yu and I. Oppermann, “Performance of UWB position estimation based on time-of-arrival measurements,” in Proc. Int. Workshop Ultra Wideband Syst. (IWUWBS), Kyoto, Japan, May 2004, pp. 400–404. [14] D. Dardari, A. Conti, J. Lien, and M. Z. Win, “The effect of cooperation on UWB based positioning systems using experimental data,” EURASIP J. Appl. Signal Process. (Special Issue on Wireless Cooperative Networks), vol. 2008, 2008. [15] I. Guvenc, Z. Sahinoglu, and P. V. Orlik, “TOA estimation for IR-UWB systems with different transceiver types,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1876–1886, Jun. 2006. [16] A. Rabbachin, I. Oppermann, and B. Denis, “ML time-of-arrival estimation based on low complexity UWB energy detection,” in Proc. IEEE Int. Conf. Ultra-Wideband (ICUWB), Waltham, MA, USA, Sep. 2006, pp. 599–604. [17] A. A. D’Amico, U. Mengali, and L. Taponecco, “Energy-based TOA estimation,” IEEE Trans. Wireless Commun., vol. 7, no. 3, pp. 838–847, Mar. 2008. [18] S. Gezici, Z. Sahinoglu, A. F. Molisch, H. Kobayashi, and H. V. Poor, “Two-step time of arrival estimation for pulse-based ultra-wideband systems,” EURASIP J. Adv. Signal Process., vol. 2008, pp. 1–11, 2008, Article ID 529134. [19] D. Dardari, C.-C. Chong, and M. Z. Win, “Threshold-based time-ofarrival estimators in UWB dense multipath channels,” IEEE Trans. Commun., vol. 56, no. 8, pp. 1366–1378, Aug. 2008. [20] I. Guvenc and Z. Sahinoglu, “Threshold-based TOA estimation for impulse radio UWB systems,” in Proc. IEEE Int. Conf. Ultra-Wideband (ICUWB), Zurich, Switzerland, Sep. 2005, pp. 420–425.
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Andrea Giorgetti (S’98–M’04) received the Dr.Ing. degree (summa cum laude) in electronic engineering and the Ph.D. degree in electronic engineering and computer science, both from the University of Bologna, Italy, in 1999 and 2003, respectively. From 2003 to 2005, he was a Researcher with the National Research Council, and since 2006, he has been an Assistant Professor at the University of Bologna. During spring 2006, he was with the Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology (MIT), Cambridge, MA, USA. Since then he is a Research Affiliate of LIDS, MIT. His research interests include ultra-wide bandwidth communications systems, active and passive localization, wireless sensor networks, and cognitive radio. Dr. Giorgetti is Technical Program Co-Chair of the Cognitive Radio and Networks Symposium at the IEEE International Conference on Communications (ICC), Budapest, Hungary, June 2013, and the Cognitive Radio and Networks Symposium at the IEEE Global Communications Conference (Globecom), Atlanta, GA, USA, December 2013. He was Technical Program Co-Chair of the Wireless Networking Symposium at the IEEE International Conference on Communications (ICC), Beijing, China, May 2008, and the MAC track at the IEEE Wireless Communications and Networking Conference (WCNC), Budapest, Hungary, April 2009. He is an Editor for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and for the IEEE COMMUNICATIONS LETTERS.
Marco Chiani (M’94–SM’02–F’11) received the Dr.Ing. degree (summa cum laude) in electronic engineering and the Ph.D. degree in electronic and computer engineering from the University of Bologna, Italy, in 1989 and 1993, respectively. He is a Full Professor in Telecommunications and the current Director of the Industrial Research Center on ICT at the University of Bologna. During summer 2001, he was a Visiting Scientist at AT&T Research Laboratories, Middletown, NJ, USA. Since 2003, he has been a frequent visitor at the Massachusetts Institute of Technology (MIT), Cambridge, MA, USA, where he currently holds a Research Affiliate appointment. He is leading the research unit of the University of Bologna on cognitive radio and UWB (European project EUWB), on Joint Source and Channel Coding for wireless video (European projects Phoenix-FP6, Optimix-FP7, Concerto-FP7) and is a consultant to the European Space Agency (ESA-ESOC) for the design and evaluation of error correcting codes based on LDPCC for space CCSDS applications. His research interests include wireless communication systems, MIMO systems, wireless multimedia, error correcting codes, cognitive radio and UWB. Dr. Chiani recently received the 2011 IEEE Communications Society Leonard G. Abraham Prize in the Field of Communications Systems, the 2012 IEEE Communications Society Fred W. Ellersick Prize, and the 2012 IEEE Communications Society Stephen O. Rice Prize in the Field of Communications Theory. He is the past chair (2002–2004) of the Radio Communications Committee of the IEEE Communication Society and past Editor of Wireless Communication (2000–2007) for IEEE TRANSACTIONS ON COMMUNICATIONS. In 2011, he was named Fellow of the IEEE for “contributions to wireless communication systems.” He is a Distinguished Lecturer for the IEEE ComSoc (2011 and 2012). In 2012, he was appointed Distinguished Visiting Fellow of the Royal Academy of Engineering, U.K.