Multi-TOA Based Position Estimation for IR-UWB

Transcription

1 Multi-TOA Based Position Estimation for IR-UWB Genís Floriach, Montse Nájar and Monica Navarro Deartment of Signal Theory and Communications Universitat Politècnica de Catalunya (UPC), Barcelona, Sain Centre Tecnològic de Telecomunicacions de Catalunya (CTTC) Castelldefels, Barcelona, Sain s: Abstract The roblem of localizing an IR-UWB transmitter from the signals received at several anchors is considered. The ositioning roblem is tyically solved in a two-ste aroach where in the first ste the Time of Arrival (TOA) is estimated indeendently at each anchor, and the osition estimate is found in a second ste. However, this aroach can be imroved, esecially in challenging scenarios, if the ositioning roblem is treat as a whole, that is, the target osition is estimated directly from the signals received on each anchor (Direct Position estimation DPE). In this aer, we resent a different aroach that sits halfway between these two aroaches. The algorithm is based on a soft two-stes aroach, where several ossible TOA estimators are selected in the first ste, and then the best estimators are used to find the osition. The erformance of the method is assessed under the framewor of the IEEE a channel models. I. INTRODUCTION Recently, UWB (Ultra Wide Band) technology [1], [2] has drawn attention due to its recise localization caabilities. Secially in indoor wireless environment, where the dense multiath maes osition estimation a very challenging tas. The use of extremely short time domain ulses with several GHz of bandwidth offers a high time resolution which allows resolving the multiath comonents as well as enetrating obstacles. To rovide accurate osition information of nodes, signals are exchanged between the node and a number of reference nodes ( Anchor nodes) whose ositions are nown. The osition of a target node can be estimated by the target node itself (self-ositioning), or it can be estimated by a central unit that gathers osition information from the reference nodes (remote-ositioning). There are several algorithmic aroaches to wireless location with various degrees of recision and accuracy. In general, triangulation algorithms that combine the distances estimated between different reference nodes and the target to be located are alied. This rocedure is the so called two-stes aroach. Deending on the algorithm, distance measurements are based on different signal arameters being the delay or Time of Arrival (TOA) the most suitable for UWB systems as it taes advantage of the high time resolution of UWB signals [1], This wor has been suorted by the Sanish Ministry of Economy and Cometitiveness (Ministerio de Economía y Cometitividad) through roject TEC C2-1-R (RUNNER), and from the Catalan Government (AGAUR) through grant 214 SGR 6. [3], [4]. However, as roved in [5], the two-ste rocedures are subotimal because in the first stage, the measurements at distinct anchors are indeendent and ignore the constraint that all measurements must be consistent with a single emitter. Therefore, we can achieve better erformance, esecially in challenging scenarios, if we merge the two ositioning stes into a single stage and then estimate the target osition directly from the signals received on each anchor node (Direct osition estimation DPE). Nonetheless, the imroved accuracy of DPE comes at cost of higher comlexity and comutational load as well as the requirement of transmitting the whole signal to a central rocessing node. The Cramer-Rao lower bounds of DPE and two-ste for the single-ath model have been develoed in [5]. The analytical lower bounds rove that DPE attains lower variance than twoste. However, the single-ath model is not aroriate for UWB channels as it resents many multiath comonents. The Maximum-lielihood (ML) for dense multiath channels is nown [4], however, it is too comlex for ractical imlementation. That is why many roosals aim for a simlified multiath model. The authors in [6] roose the use of signal classification (MUSIC) method and focusing matrices. However, it assumes that the number of multiath reflectors is nown and that they are smaller that the number of receivers. In [7] the authors develoed and aroximation of the ML for dense multiath channels by treating the received signal as Gaussian. Recent wors [8] suggest the use of the Exectation- Maximization algorithm to find the Maximum Lielihood, however, it has not been tested in dense multiath scenarios lie the ones we find in UWB. In this aer, we revisit the DPE aroach resented by the authors in [9] and roose a novel technique to overcome the drawbacs resent in this aroach. The roosed algorithm, unlie the DPE aroach, does not wor with the comlete received signal but instead it wors with a set of measurements or ossible TOA estimators, all these measurements are later used to determine the node osition. Consequently, it reduces the ambiguity in the selection of the roer ea resent in all Threshold-based TOA estimators. Moreover, the amount of data required to be transmitted to the central node is much lower and the comutational comlexity is reduced as well. The rest of the aer is organized as follows: Section II introduces the IR-UWB signal model, Section III and ISBN EURASIP

2 IV reviews the two-ste ositioning scheme based on TOA estimates and DPE scheme roosed by the authors in [3] and [9] resectively. The roosed Multi-TOA aroach is introduced in Section V. Performance evaluation is given in Section VI and conclusions are drawn in VII. II. IR-UWB SYSTEM MODEL The IR-UWB signal model is considered to be a train of unit energy gaussian ulses (t) of very short duration that roagates through an M-ath ta delay channel. The channel imulse resonse is M 1 m= h mδ(t τ m ), with τ < τ 1 <... < τ M 1, being τ the TOA that has to be estimated. The received signal is then exressed as: y(t) = M 1 m= = h m (t T f τ m ) + w(t) (1) where T f is the ulse reetition eriod also referred to as frame eriod and w(t) is zero-mean white noise with variance σ 2. The signal associated to the -th transmitted ulse, in the frequency domain is given by: Y (ω) = M 1 m= h m S (ω)e jwτm + V (ω) (2) The frequency comonent associated to the shifted ulse is given by S (ω) = P (ω)e jωt f, with P (ω) being the Fourier Transform of the ulse (t) and V (ω) is the noise associated to the -th symbol. Samling (2) at ω n = ω n for n =, 1,..., N 1 where ω = 2π N, rearranging the frequency domain samles Y [n] into the vector Y C N 1 and exlicitly searating the LOS term yields Y = h S e τ + Ṽ (3) where Ṽ = S E τ h + V and S C N N is a diagonal matrix whose comonents are the frequency samles of S (ω). The matrix E τ C N M 1 contains the delay signature vectors associated to each arriving delayed signal of the multiath, E τ = [e τ1... e τm... e τm 1 ] (4) with e τm = [1 e jωτm... e jω(n 1)τm ] T. The channel fading coefficients, excet for h are arranged in the vector h = [h... h M 1 ] T R M 1 1, and the noise samles in vector V C N 1. III. TOA-BASED POSITION ESTIMATION FOR IR-UWB Next, we review the TOA-based ositioning scheme for IR- UWB systems roosed by the authors in [3]. The estimator resented in this section is the basis for the more sohisticated estimators that will be resented in Section IV and V. In this case, the osition estimate is obtained by first measuring the distance to the anchor nodes at nown locations and then erforming a trilateration. To measure the distance, we consider the TOA estimation algorithm roosed by the authors in [3]. This estimator is based on calculating the seudo-eriodogram or ower delay rofile, defined as the signal energy distribution with resect to roagation delays. Namely, P l (τ) = e H τ R l e τ (5) where l = 1,..., N A denotes the anchor index, the suerscrit (.) H denotes the transose comlex conjugate, and R l is the samle covariance matrix defined as, R l = 1 N f N f =1 ( Y (l) Y (l) ) H (6) where N f is the number of observed frames. The TOA estimate is the first ea in the eriodogram P l that exceeds a certain threshold γ. The selection of the threshold is always a critical factor in all TOA estimation techniques based on threshold decisions. There are some roosals for the estimation of threshold values for secific channels [1], [11]. However, most of the roosals require some calibration and heuristic adjustment of some of the arameters that define the threshold level. IV. DIRECT POSITION ESTIMATION FOR IR-UWB Here, we briefly review the DPE aroach for IR-UWB [9] considered for baseline erformance. In this case, the ositioning roblem is solved on a single ste erforming a search directly over the satial coordinates. Defining Y = (Y (1),..., Y (l),..., Y (N A) ) T, with Y (l) = h,l Se τ,l +Ṽ(l) according to (3) (symbols index are droed in order to ease notation), the system model can be written in terms of LOS contribution as, Y = S e + Ṽ (7) where S = diag(h,1 S,..., h,na S) is a bloc diagonal matrix with ulse sectral comonents weighted by the LOS channel fading coefficients, e = (e f1(),..., e fna ()) T is the delay signature vector as a function of the target satial coordinates = [x, y] T, and Ṽ = (Ṽ(1),..., Ṽ(NA) ) T. The delay signature vectors e fl () are defined as in (4) but in this case the delay is related to the osition vector by the geometrical relation f l () = l /c, with l being the two-dimensional coordinates of the l-th anchor. The DPE osition estimate is the osition vector that maximizes the following function, ˆ = argmax e H Re (8) where R = YY H. Intuitively what this algorithm does is reresent the eriodograms in terms of the satial coordinates and then add them u. We refer the reader to [9] for further information about this aroach. Although DPE techniques are exected to rovide better results than the ure two-ste aroach, the roosed DPE in this section is still far from the otimum. The main sources of errors in the DPE come from the fact that UWB channels are not classical LOS scenarios where the LOS is resent with stronger amlitude than other multiath comonents. In fact, ISBN EURASIP

3 there are channel realizations where delayed multiaths are considerably larger than the LOS. As the eriodograms estimated in each anchor are not functions to be maximized but instead we are interested in the first ea, it is not otimal to combine the signals adding them because then we lose the idea that first eas are more imortant than strong eas. To solve this roblem we roosed a new aroach based in Multi-TOA estimation. V. MULTI-TOA BASED POSITION ESTIMATION FOR IR-UWB Selection of the threshold is always a critical factor that may dro the erformance drastically if it is not chosen roerly. Then, the roosed solution is the following: instead of setting a threshold and eeing the first ea above it, all rominent eas are selected and then all otential TOAs are used to find the osition. All the otential TOAs are gathered in the central rocessing unit and the combination of TOAs that minimize the trilateration cost function are the ones selected. In other words, the osition estimate will be the one that minimizes the following cost function: ˆ = argmin [ min (τ l [ l ]c l ) 2] (9) where = [ 1... l... NA ] l {1,..., m l }, m l is the number of eas detected in the l-th anchor node and τ l [ l ] is the otential l -th TOA detected in the l-th anchor node. However, not all selected eas are equally liely to be the true TOA. For examle in the Fig. 1, the last ea to be the true TOA would mean that all the revious eas were noise or ulse sidelobes of that ea and such thing is highly unliely. Somehow, we must add information of the energy of the ea and the relative osition of the eas in the eriodogram. Therefore, it is convenient to weight each ea by the robability of that ea being the true TOA. By weighting the eas we imrove the overall erformance of the osition estimation technique because the number of eas gets reduced (last eas will have robability so we remove them) so it will be easier to minimize the cost function, and moreover we reduce the robability of choosing aths that do not reresent the osition of the target. P(m) m Fig. 1: Periodogram realization However, as it haened with threshold selection (in the ure TOA eriodogram aroach), finding these robabilities is not an easy tas. Then, the roosed weighting criterion is a heuristic aroach based on the distribution of the energy of the TOA, i 1 W eight eai = (1 W eight ea ) P (E T OA < E eai ) = (1) where E T OA is the random variable that reresents the energy of the TOA (its distribution has been found emirically), W eight ea = and E eai is the Energy of a ith detected ea. Finally, the osition estimate is given by, ˆ = argmin [ min (τ l [ l ]c l ) 2 + ] λ( W eight l [ l ]) (11) where W eight l [ l ] is the weight of the l ea of the lth beacon and the arameter λ is a trade-off between consistency and robability of the delays. By using this aroach, the roblematic of strong multiath comonents resent in the DPE is mitigated because now we tae into account the fact that earlier eas are more imortant than stronger eas and the fact that eas should be consistent. Moreover, it solves the ambiguity in the roer selection of the ea (resent in the two stes aroach) because several eas are selected. In addition, whereas the DPE aroach requires transmitting the whole signal to the central rocessing node, the Multi-TOA solution just requires transmitting the otential TOAs, which involves far less data. A. Imlementation When it comes to solving the cost function in (11) we must deal with a function that deends on discrete variables (the variable ). Addressing this discrete otimization roblem by brute force, which is enumerating all the ossible combinations, is not feasible as the number of combinations grows exonentially with the number of anchor nodes. Therefore, we roose the use of Branch and Bound [12] algorithm to try to reduce the search sace. Branch and Bound is a tree-search based algorithm where each node of the tree is a subroblem, then after solving a subroblem we can set a bound about how good the final solution (leaves of the tree) could otentially be. If that bound is better than an already found solution we have to exand this node and exlore its children. Otherwise, we can rune that node, thus reducing the search sace. From this oint onwards, we will refer to the anchor nodes as beacons and the nodes of a tree as nodes. In our articular roblem, the elements of the tree would be: Nodes: A articular trilateration using fewer beacons than the available. The deth of the tree is equal to the number of beacons ISBN EURASIP

4 Lower Bound: The cost of that articular trilateration. Children: The children of a given arent node are the trilaterations using the same arameters of the arent node but adding the TOA information of the new beacon (the number of children is equal to the number of otential TOAs estimated by the new beacon). Leaves: A ossible solution of the roblem. There are as many leaves as number of combinations. The way to exlore the tree is a ey factor to find the solution quicly. The roosed technique is the best-first search strategy which exlores the tree by exanding the most romising node. It is imortant to note that although the worst case comlexity of Branch and Bound is exonential, in ractice, for our case of study it wors well. Algorithm 1 is a seucode imlementation of the roosed algorithm, where the Priority queue is a queue where each element has an associated riority (in our case the lower the bound the higher the riority), then an element with high riority is served before an element with low riority, and TreeNode is an structure that stores the beacons used, the otential TOA used for each beacon, the value of the bound and the estimated osition if only these beacons were available. Algorithm 1 Comutation of osition Define: Priority queue PQ, struct TreeNode 1: Select a beacon arbitrarly, call it b 1 2: for all TOAs in b 1 do First Tree layer 3: Create a TreeNode, call it node 4: Set node.bound = 5: Set rest of node arameters 6: Put node into PQ 7: end for 8: Solution = null 9: J = 1: while PQ is not emty do 11: Remove most riority node of PQ, call it tonode 12: if tonode.bound < J then 13: Exlore children of tonode 14: Put children into PQ 15: for all children do 16: if child is leaf & child.bound < J then 17: udate J 18: Solution = child 19: end if 2: end for 21: end if 22: end while VI. NUMERICAL RESULTS In this section, the erformance of the roosed method is evaluated by means of numerical simulations with the IEEE a UWB channel models [13]. In articular, we focus on the CM3 office LOS and CM4 Office NLOS models which are indoor office environments characterized to exhibit dense multiath. The simulation setu is configured as follows: A single target is laced within a square room of 6 6 m 2 and is continuosly transmitting gaussian monocycle ulses of duration T = 1 ns. The transmitted ulses are received by N A = 4 beacons laced at the corners of the room. The received signal is low ass filtered to avoid aliasing and samled at 2GHz. In all scenarios, the signal to noise ratio (SNR) at each beacon is set to 4dB. It is imortant to mention, however, that similar results were obtained for SNRs down to -2dB. Fig. 2a and Fig. 2b deicts the cost function (8) and (11) obtained for the DPE and Multi-TOA resectively in a articular realization for the CM3 Office LOS scenario. As mentioned before, in UWB channels the LOS can be attenuated with resect to multiath comonents. This fact can lead the DPE aroach to estimate the osition erroneously as we can see in Fig. 2a. However, the Multi-TOA aroach does not suffer from such channel roagation feature, and manages to estimate the osition roerly. For the evaluation of the osition estimation accuracy, we considered 5 channel realizations for each scenario. The osition error of DPE, Multi-TOA and TOA-based two-ste aroach is deicted in Fig. 3 in terms of the cumulative distribution function (CDF) for Office LOS (CM3) and Office NLOS (CM4) scenarios. The arameter λ in (11) for the Multi-TOA aroach is obtained emirically. In Fig. 3a there is also deicted the CDF for the DPE when using strictly LOS channels, that is, channels where the LOS is always the one with higher energy. In this case we can see that the erformance is similar to the other aroches. Such disarity between the two DPE curves shows that it is not robust to the strong attenuation of the LOS, this fact is emhasized in the Fig. 3b where DPE erforms extremely oorly. Fig.3a also shows the CDF for the two-stes when the threshold is deviated a 3% and a 4% from the otimum value. These curves show the sensibility of the two-ste to the threshold value. From Fig. 3 it can be concluded that the roosed Multi- TOA outerforms the DPE aroach. Comaring Fig. 3a and Fig. 3b it can also be concluded that the roosed algorithm are robust to the NLOS roagation imairments. VII. CONCLUSIONS A novel osition estimator for IR-UWB localization has been introduced and assessed under realistic channel models develoed by the IEEE a standardization grou. The roosed algorithm reduces the comlexity of DPE aroaches, since it does not require the samled received signal but only the set of measurements (TOA estimates). Such comlexity reduction does not incur in information loss for the ositioning roblem. Moreover, we show by means of numerical results that the roosed scheme erforms better in terms of osition estimate accuracy and is robust to NLOS roagation. Further wor will focus on an analytical analysis of the roer weighting as well as an accurate analysis of the comlexity of the roosed algorithm. ISBN EURASIP

5 Emirical CDF Multi-TOA DPE 2-Stes-TOA Th ot 2-Stes-TOA Th 3% (a) Joint Periodogram realisation for DPE.1 2-Stes-TOA Th 4% DPE ure LOS error[m] (a) Office LOS (CM3) Multi-TOA DPE 2-Stes-TOA Th ot.7 Emirical CDF (b) Cost function for Multi-TOA Fig. 2: Target localization results with 4 beacons. REFERENCES [1] S. Gezici, Z. Tian, G. B. Giannais, H. Kobayashi, A. F. Molisch, H. V. Poor, and Z. Sahinoglu, Localization via ultra-wideband radios: a loo at ositioning asects for future sensor networs, IEEE signal rocessing magazine, vol. 22, no. 4,. 7 84, 25. [2] Z. Xiao, Y. Hei, Q. Yu, and K. Yi, A survey on imulse-radio UWB localization, Science China Information Sciences, vol. 53, no. 7, , 21. [3] M. Navarro and M. Najar, Frequency domain joint TOA and DOA estimation in IR-UWB, IEEE transactions on wireless communications, vol. 1, no. 1,. 1 11, 211. [4] O. Bialer, D. Rahaeli, and A. J. Weiss, Efficient time of arrival estimation algorithm achieving maximum lielihood erformance in dense multiath, IEEE Transactions on Signal Processing, vol. 6, no. 3, , 212. [5] A. G. Amigó, P. Closas, A. Mallat, and L. Vandendore, Cramér-Rao bound analysis of UWB based localization aroaches, in 214 IEEE International Conference on Ultra-WideBand (ICUWB). IEEE, 214, [6] E. Miljo and D. Vucic, Direct osition estimation of uwb transmitters in multiath conditions, in Ultra-Wideband, 28. ICUWB 28. IEEE International Conference on, vol. 1. IEEE, 28, [7] O. Bialer, D. Rahaeli, and A. J. Weiss, Maximum-lielihood direct osition estimation in dense multiath, IEEE Transactions on Vehicular Technology, vol. 62, no. 5, , 213. [8] E. Tzoreff and A. J. Weiss, Exectation-maximization algorithm for direct osition determination, Signal Processing, vol. 133, , error[m] (b) Office NLOS (CM4) Fig. 3: Emirical CDF of the osition error using 4 beacons for Multi-TOA, DPE and 2-Stes-TOA. [9] M. Navarro, P. Closas, and M. Nájar, Assessment of Direct Positioning for IR-UWB in IEEE a channels, in 213 IEEE International Conference on Ultra-Wideband (ICUWB). IEEE, 213, [1] I. Guvenc and Z. Sahinoglu, Threshold selection for UWB TOA estimation based on urtosis analysis, IEEE Communications Letters, vol. 9, no. 12, , 25. [11] D. Dardari, C.-C. Chong, and M. Win, Threshold-based time-of-arrival estimators in UWB dense multiath channels, IEEE Transactions on Communications, vol. 56, no. 8, , 28. [12] J. Clausen, Branch and bound algorithms-rinciles and examles, Deartment of Comuter Science, University of Coenhagen,. 1 3, [13] A. F. Molisch, K. Balarishnan, C.-C. Chong, S. Emami, A. Fort, J. Karedal, J. Kunisch, H. Schantz, U. Schuster, and K. Siwia, IEEE a channel model-final reort, IEEE P82, vol. 15, no. 4,. 662, 24. ISBN EURASIP