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Localization of multiple targets in multi-path environnents

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Localization of multiple targets in multi-path environnents

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Content LOCALIZATION OF MULTIPLE TARGETS IN MULTI-PATH ENVIRONMENTS by Junyang Shen A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) December 2013 Copyright 2013 Junyang Shen Dedication To my family. ii Acknowledgments First of all, I would like to express my gratitude to my advisor Andreas F. Molisch. It has been an honor to be his first PhD student at the University of Southern California. I appreciate all of his time, ideas and funding to make my PhD research productive and stimulating. His deep insight in the field and meticulous scholarship have won my highest respect. IamalsoextremelythankfulforthetimeandinvaluableadviceofProf. Hossein Hashemi, Prof. Bhaskar Krishnamachari and Prof. Qiang Huang, Prof. Robert A. Scholtz, Prof. Mahta Moghaddam, Prof. Jianfeng Zhang and Prof. Moe Win. I would like to thank all of my colleagues including Jussi Salmi, Hao Feng, Seun Sangodoyin, Daphney-Stavroula Zois, Srinivas Yerramalli, Sundar Aditya, Mingyue Ji, Rui Wang, Zheda Li, Joongheon Kim, Vinod Kristem, Pei-Lan Hsu. It is always enjoyable to work with them. I would also like to express my gratitude to the Ming Hsieh Department of Electrical Engineering and Viterbi School of Engineering, especially Diane Demetras, Anita Fung and Gerrielyn Ramos for their generous assistance throughout my studies. During my four years at USC, I have had the pleasure of meeting many cherished friends. I am very grateful to all those who have helped me along the way. My deepest gratitude is owed to my family for all their love and support. If it wasn’t for the inspiration my parents, I would not have begun this journey in iii the first place. Special thanks to my parents-in-law, who put their life on hold in China to help me with the final stage of my thesis in the U.S. Above all, for my dear and loving wife, Xiaoyu, who always believes in me, and always gives me unwavering love and encouragement. Lastly, I must thank my unborn child, who will be with us in two months. You gave me superior courage and spirit to finish my PhD study. I cannot wait to meet you. iv Contents Dedication ii Acknowledgments iii List of Tables ix List of Figures x Abstract xii Chapter 1 Introduction 1 1.1 Active and Passive Localization . . . . . . . . . . . . . . . . . . . . 1 1.2 Localization Techniques . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Localization based on TOAs and Dierence of TOAs . . . . . . . . 3 1.4 Error Sources of TOA based Localization Systems . . . . . . . . . . 4 1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 8 Chapter 2 IndividualTargetLocalization: ATwoStepEstimation (TSE) Method 10 2.1 Overview of Individual Target Localization . . . . . . . . . . . . . . 11 2.2 Architecture of Localization Systems . . . . . . . . . . . . . . . . . 13 2.3 System Model of Individual Localization . . . . . . . . . . . . . . . 16 v 2.4 TSE Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.1 Step 1 of TSE . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.2 Step 2 of TSE . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Comparison of CRLB between TDOA and TOA . . . . . . . . . . . 25 2.6 Performance of TSE . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.8 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.10 Appendix: Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . 42 Chapter 3 Indirect Path Detection with Angle Information: A TDAJ (TOA, DOD, DOA joint) Indirect Path Detec- tion Scheme 44 3.1 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3 IP Detection Scheme based on Dierences of Intersections . . . . . 52 3.4 Distribution of Decision Variables and Evaluations of P f and P d .. 55 3.4.1 Distribution of ⁄ i in DPs and evaluation of p i f ........ 58 3.4.2 Distribution of ⁄ i in IPs and evaluation of p i d ......... 62 3.4.3 Blind Spot of ⁄ i ......................... 63 3.4.4 “Safe region” and outage probability . . . . . . . . . . . . . 65 3.5 TDAJ Scheme with Noisy DOD, DOA and TOA . . . . . . . . . . . 67 3.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.6.1 Simulation Description . . . . . . . . . . . . . . . . . . . . . 69 3.6.2 Comparison between theoretical and simulated PF and PD . 71 3.6.3 Comparison of IP Detection Performance by Dierent ⁄ i .. 73 3.6.4 PF and PD vs. Transmitted Signal Power . . . . . . . . . . 74 vi 3.6.5 Improvement of Localization by IP Detection . . . . . . . . 75 3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.8 Appendix: Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . 78 Chapter 4 Multiple Target Localization based on Time-of-Arrival Measurements: A Sequential Clustering Algorithm in Multi-path Environments 80 4.1 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3 Principle of the Sequential Clustering Algorithm . . . . . . . . . . 88 4.3.1 Step 1: Choose the first three TRPs. . . . . . . . . . . . . . 92 4.3.2 Step 2: Match TOA measurements of the first three TRPs. . 92 4.3.3 Step 3: Match TOA measurements of the next TRP . . . . . 94 4.3.4 Step 4: Choose the final target location estimates based on the threhsold ” and .....................100 4.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.4.1 Comparison with Exhaustive Maximum Likelihood Estima- tion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.4.2 Dierent and ” ........................104 4.4.3 Dierent Number of TRPs . . . . . . . . . . . . . . . . . . . 107 4.4.4 Dierent threshold ......................109 4.4.5 Dierent Path Blockage Probability . . . . . . . . . . . . . . 110 4.4.6 Rich multi-path environments . . . . . . . . . . . . . . . . . 110 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.6 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.6.1 Computation of l k 3 k 1 ,k 2 for three DPs . . . . . . . . . . . . . . 114 4.6.2 Computation of l k i Row(B i≠ 1 ) j 2 ...................115 vii Chapter 5 Future Works and Conclusions 117 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . 119 Reference List 120 viii List of Tables 2.1 Measurement system parameters. . . . . . . . . . . . . . . . . . . . 33 2.2 Comparison between TSE and Quasi-Newton minimization of (2.3) 34 3.1 Blind Spots of Dierent ⁄ i ....................... 65 3.2 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.3 P f and P d of individual rules based on ⁄ 1 , ⁄ 3 , ⁄ 5 for Scenario 1. . . 71 3.4 P f and P d of for Scenario 2 . . . . . . . . . . . . . . . . . . . . . . . 74 3.5 Localization RMSE Comparison . . . . . . . . . . . . . . . . . . . . 76 4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 ix List of Figures 1.1 Passive vs Active Localization . . . . . . . . . . . . . . . . . . . . . 2 1.2 Illustration of Error Sources for TOA based Localization . . . . . . 5 2.1 Illustration of TOA based Location Estimation System Model . . . 17 2.2 Illustration of estimation of ◊◊◊ in step 1 of TSE . . . . . . . . . . . . 22 2.3 CRLB ratio of passive TOA over passive TDOA estimation: (a) contour plot; (b) pcolor plot . . . . . . . . . . . . . . . . . . . . . . 29 2.4 (a): PDF of [ˆ x,ˆ y] by TSE (b): Dierence between the PDF of [ˆ x,ˆ y] by TSE and PDF of Gaussian distribution with mean [¯ x,¯ y] and covariance J ≠ 1 ............................. 33 2.5 Simulation results of TSE for the first configuration. . . . . . . . . . 35 2.6 Simulation results of TSE for the second configuration. . . . . . . . 36 2.7 Simulation Results of combined TSE and TDOA algorithm for the third configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.8 Measurement Setup schematic.. . . . . . . . . . . . . . . . . . . . . 39 2.9 Flow chart of target location estimation using TSE . . . . . . . . . 40 2.10 Distribution of 100 measured signal travel ranges ˆ r 1 , ˆ r 2 and ˆ r 3 at three receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1 Illustration of DPs and IPs . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 Illustration of TOA, DOA and DOD joint location estimation . . . 52 x 3.3 Interpretation of Theorem 1 ...................... 59 3.4 Illustration of Blind Spots of Dierent ⁄ i ............... 66 3.5 P d vs P f in Scenario 2 . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.6 P d vs P f in Scenario 3 . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.7 P d vs P f in Scenario 4 . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.8 P d and P f vs transmitted signal power P dB t (dBm) in Scenario 5 . . 75 4.1 Analysis of Sources of Localization Errors . . . . . . . . . . . . . . 85 4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3 Flow chart of proposed algorithm. . . . . . . . . . . . . . . . . . . . 89 4.4 CDF of RMSE for dierent methods. . . . . . . . . . . . . . . . . . 105 4.5 P m vs. P f for dierent values of and ”.. . . . . . . . . . . . . . . 106 4.6 Comparison between theoretical and true P m .............106 4.7 CDF of RMSE for dierent TRPs with blockage probability p b =0.107 4.8 CDF vs RMSE for dierent number of TRPs with blockage proba- bility p b =0.1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.9 CDF of RMSE for dierent with 3 Tx 4 Rx and p b =0.1.. . . . . 109 4.10 CDF of RMSE for dierent p b , is the optimum choice for each p b .110 4.11 CDF of RMSE for dierent thresholds with p b =0..........112 4.12 CDF of RMSE for dierent thresholds with p b =0.1.........112 4.13 CDF of RMSE for the proposed method and the genie-aided method.113 xi Abstract Passive localization of objects is fast becoming a major aspect of wireless tech- nologies, with applications in logistics, surveillance, and emergency response, etc. The localization can be performed with a variety of localization techniques based on dierent system parameters such as the angle of arrival (AOA), and the signal timeofarrival(TOA)measurements. However,thesemeasuredparametersarecor- rupted by background noise. Further complications in realistic settings arise from the following factors: (i) the direct paths (DPs) between transmitters/receivers and targets might be blocked, (ii) indirect paths (IPs) arising from multi path propagation might be present, (iii) signals reflected by multiple targets cannnot be distinguished by their “signatures” or other unique characteristics. This thesis oers novel solutions to combat theses challenges: • Individual Target Localization with Noisy TOA measurements: Consider a single target localization problem, if there are no IPs or if IPs can be per- fectly identified, the error of target location estimate is due to the noisy TOA measurements. We propose a Two Step Estimation (TSE) method which employs the TOA measurements and the variances of them to per- form approximate maximum likelihood through a two step mechanism. Its complexityislowerthaniterativemethods,andisabletoachievetheCramer Rao Lower Bound (CRLB), which is the lower bound of mean square error xii for any un-biased estimator if TOA measurements suer Gaussian errors. As far as we know, it is the first passive localization method to achieve the CRLB. Another benefit of TSE is that the distribution of the estimated tar- get location is known. Furthermore, simulations and practical experiments are performed to verify the superiority of TSE. • Indirect Path Detection with TOA and Angle Measurements: Becauseofnon- target reflectors, the signals propagate from transmitter to receivers through DP or IPs. In some situations, the DP is blocked and then, the TOA mea- surements based on IPs would usually contain large errors (compared with errors induced by noise). Thus, IP detection algorithm needs to eectively detect the IPs, to avoid the detrimental eect of indirect paths on localiza- tion. We propose a scheme based on channel sounding parameters: TOA, DOD (direction of departure) and DOA (direction of arrival). The principle of the proposed TOA DOD and DOA joint (TDAJ) scheme is as follows: if therewerenomeasurementerrors, anyDPwouldhaveself-consistentparam- etersofmeasurements. Bythiswemeanthatthethreeintersectionsobtained from the pairs of measured parameters are at the same location. In the pres- ence of measurement errors, the decision variables of the Neyman-Pearson (NP) test are the dierences between two of the three intersections in x or y coordinates: only if this dierence lies below a certain threshold, the algo- rithm judges the path to be a DP. The benefit of the TDAJ depends on the measurements of TOA, DOD and DOA instead of multipath channel statistics. Simulation results show that under common localization scenar- ios employing ultrawideband signals, the proposed algorithms can eectively detect IPs and significantly improve localization accuracy. xiii • Multiple Target Localization based on TOA measurements: If there is more than one target to be localized, the system needs to separate the received signals from dierent targets before localizing them. Only the information (such as TOAs) extracted from impulses reflected by the same target should be combined for localization. Thus localization requires to correctly “pair” the TOA measurements of the reflections arriving at the dierent receivers. Further complications in realistic settings arise from the following factors: (i) the number of the targets are unknown, (ii) the DPs between transmitter and target, and/or between target and receiver, might be blocked, (iii) IPs arising from multi path propagation might be present, and (iv) the DP from one target might have a similar time of arrival as the DP or IP from another target,suchthatthosepathscannotbediscriminated(resolved). Wepropose a novel sequential matching algorithm that can accurately localize multiple targets even under these dicult circumstances. This algorithm is based on an iterative approximation to an exhaustive maximum likelihood (EML) estimation of the target locations, yet avoiding the exponential complexity thatanEMLrequires. Itutilizesaclusteringtechniquethatgivesthenumber and locations of targets, and incorporates an IP detection mechanism based on the likelihood of several TOA measurements corresponding to DPs from the same target. Despite its lower complexity, our algorithm achieves almost the same performance as the EML in a variety of simulation settings. We finally study the choice of algorithm parameters and the configurations of the MIMO system nodes to achieve the best localization performance. This thesis aims at providing insights and proposing methods to address the three main challenges of passive localization systems. xiv Chapter 1 Introduction Localization of objects (targets) through the use of wireless signals is of great importance for a variety of applications. It is the canonical problem of radar [51], and thus relevant for both military and civilian applications such as aircraft tracking and anti-collision radars. In recent years, localization has also gained increased importance for a large variety of applications, such as radio frequency identification (RFID)-based tracking [31], robotic wireless sensor networks [39], patient monitoring [59], [64], survivor localization in emergency rescue operations [97], radar sensor network [57], cancer tumor localization [17], and many more. While well-investigated, it still faces many open issues [20, 28, 35, 46, 70]. 1.1 Active and Passive Localization Thetargettobelocalizedcanbeeither“active”,transmitting/receivingsignals, or “passive”, only reflecting signals transmitted by a separate source (and received by separate receivers) as shown in Fig. 1.1. Active target positioning scenarios include cell phone location estimation or wireless sensor positioning in the IEEE 802.15.4a Wireless Sensor Networks (WSNs) [95]. [83] has developed a special concept of active localization to use cooperation between several targets (agents) to improve localization accuracy. The passive target positioning scenario is typical in applications such as sur- vivor rescue, anti-theft surveillance, and automated environment mapping. Radar 1 !"#$%&'()*%+,& -%"%.$%)&/& -%"%.$%)&0& '()*%+& -%"%.$%)&/& 1(,,.$%&'()*%+,& ')(2,3.4%)& -%"%.$%)&5& -%"%.$%)&0& -%"%.$%)&/& '()*%+& Figure 1.1: Passive vs Active Localization is also a “passive” localization technology, which has been studied for many years [79, 56, 30]. In addition, the “passive” localization was recently shown to be a promising technology for cancer detection [38]. For ease of notation, this work will concentrate on passive localization techniques, though many of the results are useful in active localization as well. 1.2 Localization Techniques A variety of localization techniques have been proposed in the literature, which dier by the type of information and system parameters that are used. The three most popular kinds utilize the received signal strength (RSS) [82], angle of arrival (AOA) [96], and time of arrival (TOA) [20], [28], [86], respectively. Note that 2 throughout this thesis, the TOA is used to denote the corresponding signal prop- agation length (product of signal propagation delay and the speed of light), like most of the papers in the literature, e.g., [2, 15, 28, 40, 66]. RSSalgorithmsusethereceivedsignalpowerforobjectpositioning; theiraccu- racies are limited by the fading of wireless signals [82]. AOA algorithms require eitherdirectionalantennasorreceiverantennaarrays 1 . AlgorithmsbasedonTOA measurementsestimatetheobjectlocationusingthetime/distanceittakesthesig- nal to travel from the transmitter to the target and from there to the receivers. They achieve very accurate estimation of object location if combined with high- precision timing measurement techniques [92], such as ultrawideband (UWB) sig- naling, which allows centimeter and even sub-millimeter accuracy, see [94, 63]. Due to such merits, the UWB range determination is an ideal candidate for pre- cise object location systems and also forms the basis for the localization of sensor nodes in the IEEE 802.15.4a standard [95]. 1.3 Localization based on TOAs and Dierence of TOAs The algorithms based on signal propagation length can be further classified into propagation Time-of-Arrival (TOA) and Time Dierence of Arrival (TDOA). TOA algorithms employ the information of the absolute signal travel time from the transmitter to the target and thence to the receivers. The term “TOA” can be used in two dierent cases: 1) there is no synchronization between transmitters 1 Note that AOA does not provide better estimation accuracy than the TOA based methods [75]. 3 and receivers and then clock bias between them exist; 2) there is synchroniza- tion between transmitters and receivers and then clock bias between them does not exist. In this thesis, we consider the second situation with the synchroniza- tion between the transmitter and receivers. Such synchronization can be done by cable connections between the devices, or sophisticated wireless synchronization algorithms [10]. TDOA is employed if there is no synchronization between the transmitter and the receivers. In that case, only the receivers are synchronized. Receivers do not know the signal travel time and therefore employ the dierence of signal travel times between the receivers. It is intuitive that TOA has better performance than the TDOA, since the TDOA loses information about the signal departure time [20]. 1.4 Error Sources of TOA based Localization Systems In TOA based localization systems, the errors of target localization estimates come from the errors of TOA measurements. This section analyzes the sources of the errors in localization. Beforeanalyzingthesourcesoflocalizationerrors, itisnecessarytofirstunder- stand how the target location is estimated. We consider a simple scenario in 2-D dimension for easy understanding. As shown in Fig. 1.2, there are one trans- mitter, two targets and two receivers. The system is trying to localize the two targets based on the TOA measurements. The TOA measurements at receiver 1 and receiver 2 for target 1 are TOA 1,1 = p 1,0 +p 1,1 and TOA 1,2 = p 1,0 +p 1,2 , respectively. The TOA measurements at receiver 1 and receiver 2 for target 2 are 4 !"#$%&'(()" !#"*)(+, !#"*)(+- .)/)'0)"+, .)/)'0)"+- .)12)/(3" 423/5#*) Figure 1.2: Illustration of Error Sources for TOA based Localization TOA 2,1 = p 2,0 +p 2,1 and TOA 2,2 = p 2,0 +p 1,2 , respectively. For each transmitter- receiver pair, the measured TOA graphically presents an ellipse. For every point on the ellipse, the sum of distances to the transmitter and the receiver equals the signal travel distance (product of speed of light and TOA measurement). There- fore, the ellipse denotes the potential location of the target. There would be two ellipses for two transmitter-receiver pairs, and the target location is estimated as the intersection of the two ellipses. The sources of the errors in target localization are [20]: 1. Noise induced errors: The TOA measurement is not the same as the true value, thus, the location of the intersection of TOA ellipses is dierent from the true target location. However, using accurate ranging algorithms (such as UWB signals [15]), the error by noise is very small. 2. The indirect paths induced errors: The Indirect Path (IP) would bring large error to the TOA measurement and the target location estimate. For example, forthepathfromTarget2toReceiver2, thedirectpathisblocked, then the received signal is reflected by non-target reflectors. The measured signaltraveldistancecouldbemuchlargerthanthedirectpathdistance(the 5 dashed line) from Target 2 to Receiver 2 (if the reflector is far away from Target2andReceiver2). Therefore, theintersectionofthetwoTOAellipses might be far away from the true target location. 3. Errors due to wrongly matched impulse response of other targets.: When multiple targets exist, the receivers perceive multiple signals reflected by them. However, multiple targets cannot be distinguished by their “signa- tures” or other unique characteristics. The receivers need to match/pair the TOA measurements from the same targets for localization. The mismatch between the TOA measurements bring errors to the localization. For exam- ple, receiver 1 receives two signals reflected by Target 1 and Target 2. To eectively localize Target 1, we need to match the path p 1,0 +p 1,1 and the path p 1,0 +p 1,1 , since only the two TOA ellipses correspond to these paths would intersect at the location of Target 1. Otherwise, if receiver 1 wrongly assumes the signal through path p 2,0 +p 1,1 to be reflected by target 1, the intersection of this path with the path p 1,0 +p 1,1 might be far away from the true target location. Therefore, a precise localization system needs to have sophisticated algorithms to combat the three kinds of errors in localization. 1.5 Contributions InlightofthethreeerrorsourcesinTOAbasedlocalizationsystems,thisthesis proposes methods to combat the three kinds of errors: 1. A novel, two step estimation (TSE) method is proposed to estimate the tar- get localization to combat the errors induced by noise [72, 74]. We then 6 derive the Cramer-Rao Lower Bound (CRLB) for TOA and show that it is an order of magnitude lower than the CRLB of TDOA in typical setups. The TSE algorithm achieves the CRLB when the TOA measurements are subject to small Gaussian-distributed errors, which is verified by analyti- cal and simulation results. Moreover, practical measurement results show that the estimation error variance of TSE can be 33 dB lower than that of TDOA based algorithms. The detailed information about this algorithm is elaborated in Chapter 2. 2. To combat the errors by IPs, a (TOA, DOD, DOA joint) TDAJ is proposed to perform IP detection based on channel sounding parameters [71, 69, 70]. This work describes how to distinguish between DPs and IPs purely based on measurements of observable propagation parameters, namely the time of arrival(TOA),directionofdeparture(DOD),anddirectionofarrival(DOA). Any combination of two of those parameters allow the computation of a reflection point, and - in the absence of noise - the points computed from the dierent combinations are consistent in the case of a DP. In the presence of noise, the computed points deviate from each other, and we establish rules for how large this deviation might be to consider a path an IP. Simulations showthattheprobabilityofsuccessfulIPdetectionishigherthan90%under common environments. Chapter 3 presents the detailed information of the (TOA, DOD, DOA joint) IP detection scheme. 3. To combat the chanllenges in multi-target localization, we propose a method based on the likelihood that several TOA measurements correspond to direct paths (DPs) reflected by the same targets [68, 73]. Thus, the proposed algorithm is essentially a maximum likelihood estimation method. However, 7 the computation complexity of an exhaustive maximum likelihood (EML) method is very high, as it has to compute the likelihood for all possible combinations of TOA measurements from all TRPs. To circumvent this problem, we employ a sequential matching process: when processing the i- th transmitter-receiver pair (TRP), the proposed algorithm first estimates potential target locations based on the first i≠ 1 TRPs, and then matches the TOA measurements of the i-th TRP with the estimated target locations and the TOA measurements of previous i≠ 1th TRPs. As confirmed by our simulation results, the algorithm possesses almost the same performance as the EML, but has much lower computation complexity. It is verified by simulations that the localization root mean square error (RMSE) can be smaller than 0.1 meter with 95% probability when the standard deviation of TOA measurement is 0.05 meter. Moreover, we studied the transmitter and receiverconfigurationindistributedMIMOradarsystemstoachievethebest localization performance: it is the best to have half of the available nodes as transmitters, and the remainder as receivers. 1.6 Organization of Thesis The rest of the thesis is organized as follows. Chapter 2 concentrates the indi- viduallocalizationproblemandproposesaTwo-stepestimationmethod,Chapter3 considers the problem of IP detection, and introduces a novel IP detection method based on channel characteristics (TOA, DOD and DOA). Chapter 4 studies the multi-target localization problem and presents a sequential matching algorithm 8 for multi-target localization based only on Time-of-arrival measurements in dis- tributed MIMO radar systems. Finally, Chapter 5 provides the conclusions and an outlook on future research directions. 9 Chapter 2 Individual Target Localization: A Two Step Estimation (TSE) Method In this section, the problem of the individual target localization is discussed, under the assumption that the localization error is purely due to noises. The next two chapters will consider the problems of IP detection and multiple target localization. The materials in this chapter have been published in [74]. In individual target localization scenario, one transmitter and multiple, dis- tributed, receiversareusedtoestimatethelocationofapassive(reflecting)object. we furthermore focuse on the situation when the transmitter and receivers can be synchronized, so that TOA (as opposed to time-dierence-of-arrival (TDOA)) information can be used. We propose a novel, Two-Step estimation (TSE) algo- rithm for the localization of the object. We then derive the Cramer-Rao Lower Bound (CRLB) for TOA and show that it is an order of magnitude lower than the CRLB of TDOA in typical setups. The TSE algorithm achieves the CRLB when the TOA measurements are subject to small Gaussian-distributed errors, which is verified by analytical and simulation results. Moreover, practical measurement results show that the estimation error variance of TSE can be 33 dB lower than that of TDOA based algorithms. 10 Theremainderofthissectionisorganizedasfollows. Theoverviewofindividual target localization is presented in Section 2.1. Section 2.2 presents the architec- ture of positioning system. Section 2.3 discusses the system model of individual localization. Section 2.4 derives the TSE, followed by comparison between CRLB of TOA and TDOA algorithms in Section 2.5. Subsection 2.6 analyzes the perfor- mance of TSE. Section 2.7 presents the simulations results. Section 2.8 evaluates the performance of TSE based on UWB measurements. Finally Section 2.9 draws the conclusions. 2.1 Overview of Individual Target Localization The localization algorithms can be divided into two categories: TOA based method or TDOA based method. TOA algorithms employ the information of the absolute signal travel time from the transmitter to the target and thence to the receivers. It requires the synchronization between transmitter and receivers. TDOA is employed if there is no synchronization between the transmitter and the receivers. In that case, only the receivers are synchronized. Receivers do not know the signal travel time and therefore employ the dierence of signal travel times between the receivers. TherearenumerouspapersontheTOA/TDOAlocationestimationfor“active” objects. Regarding TDOA, the two-stage method [8] and the Approximate Max- imum Likelihood Estimation [7] are shown to be able to achieve the Cramer-Rao Lower Bound (CRLB) of “active” TDOA [28]. As we know, the CRLB sets the lowerboundoftheestimationerrorvarianceofanyun-biasedmethod. Twoimpor- tant TOA methods of “active” object positioning are the Least-Square Method [9] and the Approximate Maximum Likelihood Estimation Method [7], both of which 11 achieve the CRLB of “active” TOA. “Active” object estimation methods are used, e.g, for cellular handsets, WLAN, satellite positioning, and active RFID. “Passive” positioning is necessary in many practical situations like crime- preventionsurveillance,assetstracking,andmedicalpatientmonitoring,wherethe target to be localized is neither transmitter nor receiver, but a separate (reflect- ing/scattering) object. The TDOA positioning algorithms for “passive” objects are essentially the same as for “active” objects. For TOA, however, the synchro- nization creates a fundamental dierence between “active” and “passive” cases. Regarding the “passive” object positioning, to the best of our knowledge, no TOA algorithms have been developed. The proposed TSE method aims to fill this gap by proposing a TOA algorithm for passive object location estimation, which furthermore achieves the CRLB of “passive” TOA. The key contributions are: • A novel, two step estimation (TSE) method for the passive TOA based loca- tion estimation. It borrows an idea from the TDOA algorithm of [8]. • CRLB for passive TOA based location estimation. When the TOA measure- ment error is Gaussian and small, we prove that the TSE can achieve the CRLB. Besides, it is also shown that the estimated target locations by TSE are Gaussian random variables whose covariance matrix is the inverse of the Fisher Information Matrix (FIM) related to the CRLB. We also show that in typical situations the CRLB of TOA is much lower than that of TDOA. • Experimental study of the performances of TSE. With one transmitter and three receivers equipped with UWB antennas, we perform 100 experimental measurements with an aluminium pole as the target. After extracting the signal travel time by high-resolution algorithms, the location of the target is 12 evaluated by TSE. We show that the variance of estimated target location by TSE is much (33dB) lower than that by the TDOA method in [8]. Notation: Throughout this chapter, a variable with “hat” ˆ • denotes the mea- sured/estimated values, and the “bar” ¯ • denotes the mean value. Bold letters denote vectors/matrices. E(•) is the expectation operator. If not particularly specified, “TOA” denotes the “TOA” for a passive object. 2.2 Architecture of Localization Systems This section presents the system model of TOA based individual target local- ization systems, and also analyzes the problems of indirect path detection and multi-target separation, which are related to the individual target locations. Foreasyunderstanding,weconsideranintruderlocalizationsystemusingUWB signals. The intruder detection system localizes, and then directs a camera to capture the photo of the targets (intruders). This localization system consists of one transmitter and several receivers. The transmitter transmits signals which are reflected by the targets, then, the receivers localize the targets based on the received signals. Note that the intruder detection can also be performed using other methods such as the Device-free Passive (DfP) approach [90] and Radio Frequency Identifi- cation(RFID)method[4]. However,boththeDfPandRFIDmethodsarebasedon preliminary environmental measurement information like “Radio Map Construc- tion” [90] and “fingerprints” [4]. On the other hand, the TOA based approach considered in our framework does not require the preliminary eorts for obtaining environmental information. 13 Thought, this sections focuses on the Individual Target Localization, it is nec- essary to discuss the relationship between the Individual Target Localization and Multiple Target Separation,Indirect Path Detection tobetterunderstandthearchi- tecture of the system. Multiple Target Separation: If there are more than one intruders, the systemneedstolocalizeeachofthem. Withmultipletargets,eachreceiverreceives impulses from several objects. Only the information (such as TOA) extracted from impulses reflected by the same target should be combined for localization. Thus, the matching of impulses from the same target is very important for target localization and several techniques have been proposed for this purpose. In [33], a pattern recognition scheme is used to perform the Multiple Source Separation. Video imaging and blind source separation techniques are employed for target separation in [58]. We have proposed a multiple-target localization method based on TOA measurements [73, 67]. Indirect Path Detection: The transmitted signals are not only reflected by the intruders, but also by surrounding objects, such as walls and tables. To reduce the adverse impact of non-target objects in the localization of target, the localiza- tion process consists of two steps. In the initial/first stage, the system measures and then stores the channel impulses without the intruders. These impulses are reflected by non-target objects, which is referred to as reflectors here. The radio signal paths existing without the target are called background paths. When the intrudersarepresent, thesystemperformsthesecondmeasurement. Toobtainthe impulses related to the intruders, the system subtracts the second measurement withthefirstone. Theremainingimpulsesafterthesubtractioncanbethroughone 14 ofthefollowingpaths: a)transmitter-intruders-receivers,b)transmitter-reflectors- intruders-receivers, c) transmitter-intruders-reflectors-receivers, d) transmitter- reflectors-intruders-reflectors-receivers 1 . The first kind of paths are called direct paths (DPs) and the rest are called indirect paths (IPs). In most situations, only direct paths can be used for localization. In the literature, there are several meth- ods proposed for indirect path identification [70, 71, 69], [49]. Individual Target Localization: After the Multiple Target Separation and IndirectPathDetection, thepositioningsystemknowsthesignalimpulsesthrough the direct paths for each target. Then, the system extracts the characteristics of directpathssuchasTOAandAOA.Basedonthesecharacteristics, thetargetsare finally localized. Most researches on Individual Target Localization assumes that Multiple Source Separation and Indirect Path Detection are perfectly performed such as [8], [32] and [21]. Note that the three challenges sometimes are jointly addressed,sothatthetargetlocationsareestimatedinonestepsuchasthemethod presented in [73, 67]. Thus, the localization systems need to first perform the Multiple Source Sepa- ration and Indirect Path Detection before the Individual Target Localization. This chapter starts with the simplest problem, Individual Target Localization. In the following of this section, we focus on the Individual Target Localization, underthesameframeworkof[8], [32]and[21], assumingthatMultipleTargetSep- aration and Indirect Path Detection are perfectly performed in prior. In addition, we only use the TOA information for localization, which achieves very high accu- racy with ultra-wideband signals. The method to extract TOA information using background channel cancelation is described in details in [64] and also Section 2.8. 1 Note that here we omit the impulses having two or more interactions with the intruder because of the resulted low signal-to-noise radio (SNR) by multiple reflections. 15 2.3 System Model of Individual Localization For ease of exposition, we consider the passive object (target) location esti- mation problem in a two-dimensional plane as shown in Fig. 2.1. There is a target whose location [x,y] is to be estimated by a system with one transmitter and M receivers. Without loss of generality, let the location of the transmitter be [0,0], and the location of the ith receiver be [a i ,b i ], 1 Æ i Æ M. The trans- mitter transmits an impulse; the receivers subsequently receive the signal copies reflected from the target and other objects. We adopt the assumption also made in [8, 7] that the target reflects the signal into all directions. Using (wired) back- bone connections between the transmitter and receivers, or high-accuracy wireless synchronization algorithms, the transmitter and receivers are synchronized. The errors of cable synchronization are negligible compared with the TOA measure- ment errors. Thus, at the estimation center, signal travel times can be obtained by comparing the departure time at the transmitter and the arrival time at the receivers. Let the TOA from the transmitter via the target to the ith receiver be t i , and r i =c 0 t i , where c 0 is the speed of light, 1Æ iÆ M. Then, r i = Ò x 2 +y 2 + Ò (x≠ a i ) 2 +(y≠ b i ) 2 i=1,...M. (2.1) For future use we define r =[r 1 ,r 2 ,...,r M ]. Assuming each measurement involves an error, we have r i ≠ ˆ r i =e i , 1Æ iÆ M, wherer i isthetruevalue, ˆ r i isthemeasuredvalueande i isthemeasurementerror. In our model, the indirect paths are ignored and we assume e i to be zero mean. 16 [0,0] [, ] xy 11 [, ] ab 22 [, ] ab 33 [, ] ab Transmitter Receiver 1 Receiver 2 Receiver 3 Target Estimation Center Cable for synchronization Figure 2.1: Illustration of TOA based Location Estimation System Model The estimation system tries to find the [ˆ x,ˆ y], that best fits the above equations in the sense of minimizing the error variance = E[(ˆ x≠ x) 2 +(ˆ y≠ y) 2 ]. (2.2) Assuming the e i are Gaussian-distributed variables with zero mean and variances ‡ 2 i , the conditional probability function of the observations ˆ r are formulated as follows: p(ˆ r|z)= N Ÿ i=1 1 Ô 2fi‡ i ·exp 3 ≠ (ˆ r i ≠ ( Ô x 2 +y 2 + Ò (x≠ a i ) 2 +(y≠ b i ) 2 )) 2 2‡ 2 i 4 , (2.3) 17 where z=[x,y]. 2.4 TSE Method In this section, we present the two steps of TSE and summarize them in Algo- rithm 2.4.2. In the first step of TSE, we assume x, y, Ô x 2 +y 2 are independent of each other, and obtain temporary results for the target location based on this assumption. In the second step, we remove the assumption and update the esti- mation results. 2.4.1 Step 1 of TSE In the first step of TSE, we obtain an initial estimate of [x,y, Ô x 2 +y 2 ], which is performed in two stages: Stage A and Stage B. The basic idea here is to utilize the linear approximation [8] [5] to simplify the problem, considering that TOA measurement errors are small with UWB signals. Let v = Ô x 2 +y 2 , taking the squares of both sides of (2.1) leads to 2a i x+2b i y≠ 2r i v =a 2 i +b 2 i ≠ r 2 i . Since r i ≠ ˆ r i =e i , it follows that ≠ a 2 i +b 2 i ≠ ˆ r 2 i 2 +a i x+b i y≠ ˆ r i v =e i (v≠ ˆ r i )≠ e 2 i 2 =e i (v≠ ˆ r i )≠O (e 2 i ). (2.4) 18 where O(•) is the Big O Notation meaning that f(– )= O(g(– )) if and only if there exits a positive real number M and a real number – such that |f(– )|Æ M|g(– )| for all–>– 0 . If e i is small, we can omit the second or higher order terms O(e 2 i ) in Eqn (2.4). In the following of this chapter, we do this, leaving the linear (first order) term. Since there areM such equations, we can express them in a matrix form as follows h≠ S◊◊◊ =Be+O(e 2 )¥ Be, (2.5) where h = S W W W W W W W W W U ≠ a 2 1 +b 2 1 ≠ ˆ r 2 1 2 ≠ a 2 2 +b 2 2 ≠ ˆ r 2 2 2 . . . ≠ a 2 M +b 2 M ≠ ˆ r 2 M 2 T X X X X X X X X X V , S =≠ S W W W W W W W W W U a 1 b 1 ≠ ˆ r 1 a 2 b 2 ≠ ˆ r 2 . . . a M b M ≠ ˆ r M T X X X X X X X X X V , ◊◊◊ =[x,y,v] T , e=[e 1, e 2 ,...,e M ] T , and B =v·I≠ diag([r 1 ,r 2 ,...,r M ]), (2.6) 19 where O(e 2 )=[O(e 2 1 ),O(e 2 2 ),...,O(e 2 M )] T and diag(a) denotes the diagonal matrix with elements of vector a on its diagonal. For notational convenience, we define the error vector ÏÏÏ =h≠ S◊◊◊. (2.7) According to (2.5) and (2.7), the mean of ÏÏÏ is zero, and its covariance matrix is given by =E(ÏÏÏÏ ÏÏ T ) =E(Bee T B T )+E(O(e 2 )e T B T )+E(BeO(e 2 ) T )+E(O(e 2 )O(e 2 ) T ) ¥ ¯ BQ ¯ B T (2.8) where Q = diag [‡ 2 1 ,‡ 2 2 ,...,‡ 2 M ]. Because ¯ B depends on the true values r, which are not obtainable, we use ‚ B (derived from the measurements ˆr) in our calculations. From (2.5) and the definition of ÏÏÏ , it follows that ÏÏÏ is a vector of Gaussian variables; thus, the probability density function (pdf) of ÏÏÏ given ◊◊◊ is p(ÏÏÏ |◊◊◊ )¥ 1 (2fi ) M 2 | | 1 2 exp(≠ 1 2 ÏÏÏ T ≠ 1 ÏÏÏ ) = 1 (2fi ) M 2 | | 1 2 exp(≠ 1 2 (h≠ S◊◊◊ ) T ≠ 1 (h≠ S◊◊◊ )). Then, ln 3 p(ÏÏÏ |◊◊◊ ) 4 ¥≠ 1 2 3 (h≠ S◊◊◊ ) T ≠ 1 (h≠ S◊◊◊ ) +ln| | 4 ≠ M 2 ln2fi (2.9) 20 We assume for the moment that x, y, v are independent of each other (this clearlynon-fulfilledassumptionwillberelaxedinthesecondstepofthealgorithm). Then, according to (2.9), the optimum ◊◊◊ that maximizes p(ÏÏÏ |◊◊◊ ) is equivalent to the one minimizing=( h≠ S◊◊◊ ) T ≠ 1 (h≠ S◊◊◊ )+ln| |.If is a constant, the optimum ◊◊◊ to minimize satisfies d d◊◊◊ =0. Taking the derivative of over ◊◊◊ ,we have d d◊◊◊ =≠ 2S T ≠ 1 h+2S T ≠ 1 S◊◊◊. Thus, the optimum ◊◊◊ satisfies ˆ ◊◊◊ = argmin ◊◊◊ { }=(S T ≠ 1 S) ≠ 1 S T ≠ 1 h, (2.10) which provides [ˆ x,ˆ y]. Note that (2.10) also provides the least squares solution for non-Gaussian errors. However, for our problem, is a function of ◊◊◊ since B depends on the (unknown)values [x,y]. Forthisreason, themaximum-likelihood(ML)estimation method in (2.10) can not be directly used. To find the optimum◊◊◊ , we perform the estimation in two stages: Stage A and Stage B. In Stage A, the missing data ( ) is calculated given the estimate of parameters (◊◊◊ ). Note that◊◊◊ provides the values of [x,y] and thus the value of ‚ B, therefore, can be calculated using ◊◊◊ by (2.8). In the Stage B, the parameters (◊◊◊ ) are updated according to (2.10) to maximize the likelihood function (which is equivalent to minimizing ). These two stages areiterateduntilconvergence. SimulationsinSection2.7showthatcommonlyone iteration is enough for TSE to closely approach the CRLB, which indicates that the global optimum is reached. 21 Choose an initial value Set 0 k = Stage A If , go to Stage B Otherwise, calculate using by (8) 0 k = [] k Ψ [1] k- θ Stage B Perform Maximum Likelihood Estimation of parameter using by (10) [] k θ [] k Ψ Check whether converges θ 1 kk =+ Iteration Over Yes No [0] = Ψ Q Figure 2.2: Illustration of estimation of ◊◊◊ in step 1 of TSE 2.4.2 Step 2 of TSE In the above calculations, ˆ ◊◊◊ contains thsubree components ˆ x, ˆ y and ˆ v. They were previously assumed to be independent; however, ˆ x and ˆ y are clearly not independent of ˆ v. As a matter of fact, we wish to eliminate ˆ v; this will be achieved by treating ˆ x, ˆ y, and ˆ v as random variables, and, knowing the linear mapping of their squared values, the problem can be solved using the LS solution. Let 22 ˆ ◊◊◊ = S W W W W W U ˆ x ˆ y ˆ v T X X X X X V = S W W W W W U x+n 1 y +n 2 v +n 3 T X X X X X V (2.11) where n i (i=1,2,3) are the estimation errors of the first step. Obviously, the estimator (2.10) is an unbiased one, and the mean ofn i is zero. Before proceeding, we need the following Lemma. Lemma 1. By omitting the second or higher order errors, the covariance of ˆ ◊◊◊ can be approximated as cov( ˆ ◊◊◊ )=E(nn T )¥ ( ¯ S T ≠ 1 ¯ S) ≠ 1 . (2.12) where n=[n 1 ,n 2 ,n 3 ] T , and and ¯ S (the mean value of S) use the true/mean values of x, y, and r i . Proof. Please refer to the Appendix. Note that since the true values of x, y, and r i are not obtainable, we use the estimated/measured values in the calculation of cov( ˆ ◊◊◊ ). Let us now construct a vector g as follows g = ˆ ≠ G , (2.13) where ˆ =[ˆx 2 ,ˆ y 2 ,ˆ v 2 ] T ,=[ x 2 ,y 2 ] T and G = S W W W W W U 10 01 11 T X X X X X V . 23 Note that here ˆ is the square of estimation result ˆ ◊◊◊ from the first step containing the estimated values ˆ x, ˆ y and ˆ v. is the vector to be estimated. If ˆ is obtained without error,g=0 and the location of the target is perfectly obtained. However, theerrorinevitablyexistsandweneedtoestimate . Recallingthatv = Ô x 2 +y 2 , substituting (2.11) into (2.13), and omitting the second-order terms n 2 1 ,n 2 2 ,n 2 3 , it follows that, g = S W W W W W U 2xn 1 +O(n 2 1 ) 2yn 2 +O(n 2 2 ) 2vn 3 +O(n 2 3 ) T X X X X X V ¥ S W W W W W U 2xn 1 2yn 2 2vn 3 T X X X X X V . Besides, following similar procedure as that in computing (2.8), we have =E(gg T )¥ 4 ¯ Dcov( ˆ ◊ ˆ ◊ ˆ ◊ ) ¯ D, (2.14) where ¯ D = diag([¯ x,¯ y,¯ v]). Since x, y are not known, ¯ D is calculated as ˆ D using the estimated values ˆ x, ˆ y from the first step. The vector g can be approximated as a vector of Gaussian variables. Thus the maximum likelihood estimation of is the one minimizing ( ˆ ≠ G) T ≠ 1 ( ˆ ≠ G) , expressed by ˆ =( G T ≠ 1 G) ≠ 1 G T ≠ 1 ˆ . (2.15) The value of is calculated according to (2.14) using the values of ˆ x and ˆ y in the first step. Finally, the estimation of target location z is obtained by ˆz=[ˆ x,ˆ y]=[± Ò ˆ 1 ,± Ò ˆ 2 ], (2.16) 24 where ˆ i is the ith item of , i=1,2. To choose the correct one among the four values in (2.16), we can test the square error as follows ‰ = M ÿ i=1 ( Ò ˆ x 2 +ˆ y 2 + Ò (ˆ x≠ a i ) 2 +(ˆ y≠ b i )≠ ˆ r i ) 2 . (2.17) The value of z that minimizes ‰ is considered as the final estimate of the target location. In summary, the procedure of TSE is listed in Algorithm 1: Algorithm 1 TSE Location Estimation Method 1. In the first step, use algorithm as shown in Fig. 2.2 to obtain ˆ ◊◊◊ , 2. In the second step, use the values of ˆ x and ˆ y from ˆ ◊◊◊ , generate ˆ and D, and calculate . Then, calculate the value of ˆ by (2.15), 3. Among the four candidate values of ˆz=[ˆ x, ˆ y] obtained by (2.16), choose the one minimizing (2.17) as the final estimate for target location. Note that one should avoid placing the receivers on a line, since in this case (S T ≠ 1 S) ≠ 1 can become nearly singular, and solving (2.10) is not accurate. 2.5 Comparison of CRLB between TDOA and TOA In this section, we derive the CRLB of TOA based estimation algorithms and show that it is much lower (can be 30 dB lower) than the CRLB of TDOA algo- rithms. TheCRLBof“active”TOAlocalizationhasbeenstudiedin[78]. The“passive” localization has been studied before under the model of multistatic radar [81, 76, 25]. The dierence between our model and the radar model is that in our model 25 the localization error is a function of errors of TOA measurements, while in the radar model the localization error is a function of signal SNR and waveform. The CRLB is related to the 2◊ 2 Fisher Information Matrix (FIM) [34], J, whose components J 11 ,J 12 ,J 21 ,J 22 are defined in (2.18) – (2.20) as follows J 11 =≠ E( ˆ 2 ln(p(ˆr|z)) ˆx 2 ) = M i=1 1 ‡ 2 i ( x≠ a i Ò (x≠ a i ) 2 +(y≠ b i ) 2 + x Ô x 2 +y 2 ) 2 , (2.18) J 12 =J 21 =≠ E( ˆ 2 ln(p(ˆr|z)) ˆxˆy ) = M i=1 1 ‡ 2 i ( x≠ a i Ò (x≠ a i ) 2 +(y≠ b i ) 2 + x Ô x 2 +y 2 ) ◊ ( y≠ b i Ò (x≠ a i ) 2 +(y≠ b i ) 2 + y Ô x 2 +y 2 ), (2.19) J 22 =≠ E( ˆ 2 ln(p(ˆr|z)) ˆy 2 ) = M i=1 1 ‡ 2 i ( y≠ b i Ò (x≠ a i ) 2 +(y≠ b i ) 2 + y Ô x 2 +y 2 ) 2 . (2.20) This can be expressed as J =U T Q ≠ 1 U, (2.21) 26 where Q is defined after Eqn. (2.8), and the entries of U in the first and second column are {U} i,1 = x¯ r i ≠ a i Ô x 2 +y 2 Ò (x≠ a i ) 2 +(y≠ b i ) 2 Ô x 2 +y 2 , (2.22) and {U} i,2 = y¯ r i ≠ b i Ô x 2 +y 2 Ò (x≠ a i ) 2 +(y≠ b i ) 2 Ô x 2 +y 2 , (2.23) with ¯ r i = Ò (x≠ a i ) 2 +(y≠ b i ) 2 + Ô x 2 +y 2 . The CRLB sets the lower bound for the variance of estimation error of TOA algorithms, which can be expressed as [34] E[(ˆ x≠ x) 2 +(ˆ y≠ y) 2 ]Ø Ó J ≠ 1 Ô 1,1 + Ó J ≠ 1 Ô 2,2 =CRLB TOA , (2.24) where ˆ x and ˆ y are the estimated values ofx andy, respectively, and {J ≠ 1 } i,j is the (i,j) th element of the inverse matrix of J in (2.21). For the TDOA estimation, its CRLB has been derived in [8]. The dierence of signal travel time between several receivers are considered: Ò (x≠ a i ) 2 +(y≠ b i ) 2 ≠ Ò (x≠ a 1 ) 2 +(y≠ b 1 ) 2 =r i ≠ r 1 =l i , 2Æ iÆ M. (2.25) Let l=[l 2 ,l 3 ,...,l M ] T , and t be the observations/measurements of l, then, the conditional probability density function of t is p(t|z)= 1 (2fi ) (M≠ 1)/2 |Z| 1 2 ◊ exp(≠ 1 2 (t≠ l) T Z ≠ 1 (t≠ l)), 27 where Z is the correlation matrix of t, Z =E(tt T ). Then, the FIM is expressed as [8] ˇ J = ˇ U T Z ≠ 1 ˇ U (2.26) where ˇ U is a M≠ 1◊ 2 matrix defined as ˇ U i,1 = x≠ a i Ò (x≠ a i ) 2 +(y≠ b i ) 2 ≠ x≠ a 1 Ò (x≠ a 1 ) 2 +(y≠ b 1 ) 2 , ˇ U i,2 = y≠ b i Ò (x≠ a i ) 2 +(y≠ b i ) 2 ≠ y≠ b 1 Ò (x≠ a 1 ) 2 +(y≠ b 1 ) 2 . The CRLB sets the lower bound for the variance of estimation error of TDOA algorithms, which can be expressed as [34]: E[(ˆ x≠ x) 2 +(ˆ y≠ y) 2 ]Ø Ó ˇ J ≠ 1 Ô 1,1 + Ó ˇ J ≠ 1 Ô 2,2 =CRLB TDOA . (2.27) Note that the correlation matrix Q for TOA is dierent from the correlation matrixZforTDOA.AssumethevarianceofTOAmeasurementatith(1Æ iÆ M) receiver is ‡ 2 i , it follows that: Q(i,j)= Y _] _[ ‡ 2 i i =j, 0 i”=j. 28 and Z(i,j)= Y _] _[ ‡ 2 1 +‡ 2 i+1 i =j, ‡ 2 1 i”=j. As an example, we consider a scenario where there is a transmitter at [0, 0], and four receivers at [≠ 6, 2],[6.2, 1.4],[1.5, 4],[2, 2.3]. The range of the target locations is 1 Æ x Æ 10, 1 Æ y Æ 10. The ratio of CRLB of TOA over that of TDOAisplottedinFig. 2.3. Fig. 2.3(a)showsthecontourplotwhileFig. 2.3(b) shows the color-coded plot. It can be observed that the CRLB of TOA is always — in most cases significantly — lower than that of TDOA. 0.01 0.01 0.05 0.05 0.05 0.1 0.1 0.1 0.2 0.2 0.4 0.4 0.4 0.6 0.6 0.6 0.8 0.8 0.8 x y (a) 2 4 6 8 10 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 1 2 3 4 5 6 7 8 9 10 x y (b) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 2.3: CRLB ratio of passive TOA over passive TDOA estimation: (a) contour plot; (b) pcolor plot 2.6 Performance of TSE In this section, we first prove that the TSE can achieve the CRLB of TOA algorithms by showing that the estimation error variance of TSE is the same as 29 the CRLB of TOA algorithms. In addition, we show that, for small TOA error regions,theestimatedtargetlocationisapproximatelyaGaussianrandomvariable whose covariance matrix is the inverse of the Fisher Information Matrix (FIM), which in turn is related to the CRLB. Similar to the reasoning in Lemma 1, we can obtain the variance of error in the estimation of as follows: cov( ˆ ) ¥ (G T ≠ 1 G) ≠ 1 . (2.28) Let ˆ x =x+e x , ˆ y =y +e y , and insert them into , omitting the second order errors, we obtain ˆ 1 ≠ x 2 =2xe x +O(e 2 x )¥ 2xe x ˆ 2 ≠ y 2 =2ye y +O(e 2 y )¥ 2ye y (2.29) Then, the variance of the final estimate of target location ˆ z is cov(ˆ z)=E( S W U e x e y T X V 5 e x e y 6 ) ¥ 1 4 C ≠ 1 E( S W U 1 ≠ x 2 2 ≠ y 2 T X V 5 1 ≠ x 2 2 ≠ y 2 6 )C ≠ 1 = 1 4 C ≠ 1 cov( ˆ ) C ≠ 1 , (2.30) where C = S W U x 0 0 y T X V . 30 Substituting (2.14), (2.28), (2.12) and (2.8) into (2.30), we can rewrite cov(ˆz) as cov(ˆz)¥ (W T Q ≠ 1 W) ≠ 1 (2.31) where W = B ≠ 1 ¯ SD ≠ 1 GC. Since we are computing an error variance, B (2.6), ¯ S (2.5) and D (2.14) are calculated using the true (mean) value of x, y and r i . Using (2.6) and (2.1), we can rewrite B = ≠ diag ([d 1 ,d 2 ,...,d M ]), where d i = Ò (x≠ a i ) 2 +(y≠ b i ) 2 . Then B ≠ 1 ¯ SD ≠ 1 is given by B ≠ 1 ¯ SD ≠ 1 = S W W W W W W W W W W W U a 1 xd 1 b 1 yd 1 ≠ ¯ r 1 Ô x 2 +y 2 d 1 a 2 xd 2 b 2 yd 2 ≠ ¯ r 2 Ô x 2 +y 2 d 2 . . . . . . . . . a M xd M b M yd M ≠ ¯ r M Ô x 2 +y 2 d M T X X X X X X X X X X X V . (2.32) Consequently, we obtain the entries of W as {W} i,1 = x¯ r i ≠ a i Ô x 2 +y 2 Ò (x≠ a i ) 2 +(y≠ b i ) 2 Ô x 2 +y 2 , (2.33) {W} i,2 = y¯ r i ≠ b i Ô x 2 +y 2 Ò (x≠ a i ) 2 +(y≠ b i ) 2 Ô x 2 +y 2 . (2.34) where {W} i,j denotes the entry at the ith row and jth column. From this we can see that W =U. Comparing (2.21) and (2.31), it follows cov(ˆz)¥ J ≠ 1 . (2.35) Then, E[(ˆ x≠ x) 2 +(ˆ y≠ y) 2 ]¥ Ó J ≠ 1 Ô 1,1 + Ó J ≠ 1 Ô 2,2 . 31 Therefore, the variance of the estimation error is the same as the CRLB. Inthefollowing,wefirstemployanexampletoshowthat[ˆ x,ˆ y]obtainedbyTSE areGaussiandistributedwithcovariancematrixJ ≠ 1 ,andthengivetheexplanation for this phenomenon. Let the transmitter be at [0, 0], target at [0.699, 4.874] and four receivers at [-1, 1], [2, 1], [-3 1.1] and [4 0]. The signal travel distance variance at four receivers are [0.1000,0.1300,0.1200,0.0950]◊ 10 ≠ 4 . The two dimensional probability density function (PDF) of [ˆ x,ˆ y] is shown in Fig. 2.4 (a). To verify the Gaussianity of [ˆ x,ˆ y], the dierence between the PDF of [ˆ x,ˆ y] and the PDF of Gaussian distribution with mean [¯ x,¯ y] and covariance J ≠ 1 is plotted in Fig. 2.4 (b). The Gaussianity of [ˆ x,ˆ y] can be explained as follows. Eqn. (2.35) means that the covariance of the final estimation of target location is the FIM related to CRLB. We could further study the distribution of [e x ,e y ]. The basic idea is that by omitting the second or high order and nonlinear errors, [e x ,e y ] can be written as linear function of e: 1. According to (2.29), [e x ,e y ] are approximately linear transformations of ˆ . 2. (2.15) means that ˆ is approximately a linear transformation of ˆ . Here we could omit the nonlinear errors occurred in the estimate/calculation of . 3. According to (2.11), ˆ ¥ ¯ ◊ 2 +2 ¯ ◊ n +n 2 , thus, omitting the second order error, thus, ˆ is approximately a linear transformation of n. 4. (2.10) and (2.39) mean thatn is approximately a linear transformation ofe. Here we could omit the nonlinear errors accrued in the estimate ofS and . Thus, we could approximately write [e x ,e y ] as a linear transformation of e, thus, [e x ,e y ] can be approximated as Gaussian variables. 32 0.69 0.7 0.71 4.87 4.872 4.874 4.876 4.878 −5 0 5 x 10 4 (a) 0.69 0.7 0.71 4.87 4.872 4.874 4.876 4.878 −5 0 5 x 10 4 (b) Figure 2.4: (a): PDF of [ˆ x,ˆ y] by TSE (b): Dierence between the PDF of [ˆ x,ˆ y] by TSE and PDF of Gaussian distribution with mean [¯ x,¯ y] and covariance J ≠ 1 Table 2.1: Measurement system parameters Configuration # [x, y] Receivers Locations [a i , b i ] and TOA Error Variances 1 [3, 8] [-1, 1] (0.1‡ 2 ), [2, 1] (0.13‡ 2 ), [-3, 1.1] (0.12‡ 2 ), [4, 0] (0.095‡ 2 ) 2 [31, 28] [-1, 1] (0.1‡ 2 ), [2, 1] (0.13‡ 2 ), [-3, 1.1] (0.12‡ 2 ), [4, 0] (0.095‡ 2 ) 3 [10, 13] [-1, 1] (1.0‡ 2 ), [1, 1] (1.0‡ 2 ), [1, -1] (1‡ 2 ), [-1, -1] (1‡ 2 ) 4 [10, 13] [-3, 3] (1.0‡ 2 ), [3, 3] (1.0‡ 2 ), [3, -3] (1‡ 2 ), [-3, -3] (1‡ 2 ) 5 [12, 8.5] [-2.1, 3] (0.5‡ 2 ), [1, 3.1] (1.2‡ 2 ), [2.4, 5.1] (1.0‡ 2 ), [-2.8, -1.6] (0.9‡ 2 ) 6 [12, 8.5] [-2.1, 3] (0.5‡ 2 ), [1, 3.1] (1.2‡ 2 ), [2.4, 5.1] (1.0‡ 2 ), [-2.8, -1.6] (0.9‡ 2 ), [-4, -2] (0.7‡ 2 ), [2, 5] (0.8‡ 2 ) 2.7 Simulation Results In this section, we first compare the performance of TSE with that TDOA algorithm proposed in [8] and CRLBs. Then, we show the performance of TSE at high TOA measurement error scenario. For comparison, the performance of a Quasi-Newton iterative method [54] is shown. To verify our theoretical analysis, six dierent system configurations are sim- ulated. The transmitter is at [0, 0] for all six configurations, and the receiver locations and error variances are listed in Table 2.1. Figures 2.5, 2.6 and 2.7 show simulation results comparing the distance to the target (Configuration 1 vs. Con- figuration 2), the receiver separation (Configuration 3 vs. Configuration 4) and 33 Table 2.2: Comparison between TSE and Quasi-Newton minimization of (2.3) Configuration # (Method) ‡ =0.01 ‡ =0.036 ‡ =0.1 ‡ =0.3162 ‡ =0.5623 1(CRLBofTOA) 2.704 ·10 ≠ 5 2.704 ·10 ≠ 4 2.704 ·10 ≠ 3 2.704 ·10 ≠ 2 8.551 ·10 ≠ 2 1 (Quasi-Newton) 2.707 ·10 ≠ 5 (8.4) 2.708 ·10 ≠ 4 (8.4) 2.765 ·10 ≠ 3 (8.4) 2.669 ·10 ≠ 2 (8.4) 8.619 ·10 ≠ 2 (8.4) 1(TSE) 2.707 ·10 ≠ 5 2.709 ·10 ≠ 4 2.775 ·10 ≠ 3 2.776 ·10 ≠ 2 3.045 ·10 ≠ 1 6(CRLBofTOA) 1.137 ·10 ≠ 3 1.137 ·10 ≠ 2 1.137 ·10 ≠ 1 1.137 3.594 6 (Quasi-Newton) 1.137 ·10 ≠ 3 (9.7) 1.139 ·10 ≠ 2 (9.7) 1.158 ·10 ≠ 1 (9.7) 1.1310(10.2) 5.801(11.3) 6(TSE) 1.137 ·10 ≠ 3 1.137 ·10 ≠ 2 1.180 ·10 ≠ 1 1.642 7.871 the number of receivers (Configuration 5 vs. Configuration 6), respectively 2 .In eachfigure, 10000trailsaresimulatedandtheestimationvarianceofTSEestimate is compared with the CRLB of TDOA and TOA based localization schemes. For comparison,thesimulationresultsoferrorvarianceoftheTDOAmethodproposed in [8] are also drawn in each figure. It can be observed that 1. The localization error of TSE can closely approach the CRLB of TOA based positioning algorithms. 2. The CRLB of TDOA based positioning algorithms is much higher (about 30dB for Configuration 3, 5 and 6) than that of TOA based. Moreover, other interesting observations include 1. Distance to Target: Figure2.5showsthatthetargetlocationestimatehas larger error variance when the target is farther from the receivers. 2. Receiver separation: Fig. 2.6 shows that larger size of the receiver cluster leads to smaller location estimate error variance. 3. Number of Receivers: Fig. 2.7 shows that having more receivers achieves lower location estimate error variance. 2 During the simulations, only one iteration is used for the calculation ofB (2.6). 34 Figure 2.5: Simulation results of TSE for the first configuration. Table 2.2 shows the performance of TSE with larger TOA measurement error covariance for Configuration 1 and 6. For comparison, we also show the perfor- mance of the Quasi-Newton method which numerically finds the maximum like- lihood estimate of target location [x,y] for p(ˆ r|z) in (2.3). The Quasi-Newton method is performed by fminunc in MATLAB, using multiple rounds of iterations. The initial guess for the Quasi-Newton method [Â x,Â y] satisfies that Â x≠ ¯ x and Â y≠ ¯ y are both Gaussain random variables with zero mean and variance 1 meter. The average numbers of iterations of Quasi-Newton are shown in the bracket after the values for the mean square estimation errors. Note that on the other hand, there is only one iteration for TSE in all simulations. Noteworthy observations are: 1. For small value of errors‡< 0.5, both TSE and Quasi-Newton approach the CRLB closely. 35 Figure 2.6: Simulation results of TSE for the second configuration. 2. For large value of errors‡> 0.5, the TSE deviates from the CRLB since the second order errors are not negligible in this situation. From the simulation results and previous analysis, we can observe the advan- tages of TSE over iterative methods are: 1. The computational complexity of TSE is much smaller than that of itera- tive methods , while both achieve the CRLB for small TOA measurement error. Thus, TSE is an ideal candidate for ultra-wideband TOA localization systems, which have better than centimeter accuracy [63]. 2. The estimation result of TSE is predictable: the estimated target location is a Gaussian random variable with known covariance matrix. This feature is very help helpful when the estimated target location is further utilized [47]. 36 Figure 2.7: Simulation Results of combined TSE and TDOA algorithm for the third configuration. 2.8 Experimental Results In this section, we apply the TSE algorithm to actual measurement results. In order to obtain reproducible results that show reduced impact of background and indirect paths, we place the target, an aluminum pole, in an anechoic cham- ber. The chamber, part of the UltraLab at the University of Southern California, exhibits very low wall reflectivity in the whole considered frequency range. Dier- ent materials have dierent properties for radio signal reflection. The aluminum pole in this experiment is used to obtain a good reflection of radio signals. There are several previous studies for positioning of objects with dierent materials, e.g., [64] and [63] studied the positioning of human body (chest movement), and [37] focused on the positioning of cancer inside human body. 37 Figure 2.8 (a) shows the measurement setup. Transfer functions are mea- sured by means of an 8720ET Agilent Vector Network Analyzer (VNA), which steps through 801 frequency points from 2 to 8 GHz. The transmit antenna is a UWB horn antenna with strong directionality and the receive antenna is a planar monopole UWB antenna with approximately omnidirectional radiation pattern designed using the Jumping Genes Multiobjective Optimization Scheme [89]. The directional antenna can achieve more accurate localization performance than the omnidirectional antenna, when the target direction is roughly known. For exam- ple, the directional antenna can point to the windows in the intruder detection system. In this experiment, we use the directional antenna for experimental con- venience to enhance signal to interference noise ratio (SINR) and consequently achieve accurate TOA measurements. In the chamber, there is one transmitter, three sequentially measured receive antennas and an aluminum pole as the target. Figure 2.8 (b) shows the transmitter and the third receiver in the chamber. The transmit signals can propagate from the transmitter to the receiver either directly via the target (direct path), through paths that involve both the target and other object (indirect paths), or through paths that do not involve the target at all (background paths). In the first step, the background paths are eliminated through background subtraction [64]. We then subsequently use Maximum Likeli- hood parameter estimation, in particular the RIMAX algorithm [60, 63] to extract the TOAs from the transmitter to three receivers, respectively; we furthermore obtain the strengths, and thus the SNRs, of the direct paths at the receivers. The flow chart of target location estimation using TSE is shown in Figure 2.9. The measurement setup parameters are as follows: • Transmitter Location: [0, 0] meter, • Target Location: [0.699, 4.874] meter, 38 • Receiver 1 Location: [-1.260, -0.501] meter, • Receiver 2 Location: [-1.294, 0.082] meter, • Receiver 3 Location: [1.188, -0.460] meter, • Tx and Rx frequency band: 2-8 Ghz, • Transmitted signal power: -10 dbm. From the TOA estimates, the target is localized using the TSE algorithm. (a) Measurement Setup schematic (b) Pictures of transmitter and the third receiver Figure 2.8: Measurement Setup schematic. There are 100 triples of measured signal travel ranges ˆ r 1 , ˆ r 2 and ˆ r 3 at the three receivers, respectively, where ˆ r i denotes the range measurement at the ith receiver. The mean values of ˆ r 1 , ˆ r 2 and ˆ r 3 are 10.645, 10.114 and 10.280 meters, respectively. Thestandarddeviationsof ˆ r 1 , ˆ r 2 and ˆ r 3 are0.0015,0.0012and0.0016 meters, respectively. The histogram plots of ˆ r 1 , ˆ r 2 and ˆ r 3 are shown in Fig. 2.10. 39 VNA captures channel response RIMAX extracts signal travel ranges of different paths The signal travel range with the highest power is picked out and considered as the distance from transmitter to target to receiver TSE uses the obtained signal travel ranges at three receivers to estimate target location Figure 2.9: Flow chart of target location estimation using TSE Basedonthevarianceof ˆ r 1 , ˆ r 2 and ˆ r 3 andlocationsofthereceivers, theCRLBs ofTOAandTDOAalgorithmsare1.65◊ 10 ≠ 5 m 2 and0.0379m 2 undertheassump- tion of Gaussian distributions of ˆ r 1 , ˆ r 2 and ˆ r 3 . The estimation error variance of the target location obtained with TSE is 1.73◊ 10 ≠ 5 m 2 , while that of the TDOA based algorithm in [8] is 0.0340 m 2 , respectively. Both the error variances of TSE andtheTDOAalgorithmsareclosetotheirCRLBs. However,theCRLBofTDOA is 2.3◊ 10 3 times that of TOA. In other words, the estimation error variance of TSE is expected to be about 33 dB lower than the achievable error variance using TDOA based algorithms. 40 10.641 10.642 10.643 10.644 10.645 10.646 10.647 10.648 10.649 0 10 20 (a): Mean of r 1 is 10.6447 meter, standard deviation of r 1 is 0.0015 meter 10.112 10.113 10.114 10.115 10.116 10.117 10.118 10.119 10.12 0 50 (b): Mean of r 2 is 10.1137 meter, standard deviation of r 2 is 0.0012 meter 10.274 10.276 10.278 10.28 10.282 10.284 10.286 10.288 10.29 10.292 0 50 (c): Mean of r 3 is 10.2802 meter, standard deviation of r 3 is 0.0016 meter Figure 2.10: Distribution of 100 measured signal travel ranges ˆ r 1 , ˆ r 2 and ˆ r 3 at three receivers 2.9 Conclusions This chapter proposes a novel algorithm, called TSE, for positioning of targets. In contrast to previous works, it considers the case where the target is not the transmitter while at the same time transmitter and receivers are synchronized and can exploit the information of time-of-arrival (TOA), instead of the less-accurate time-dierenceofarrival(TDOA).TheTSEproceedsintwosteps, firstcomputing estimates for a parameter set x, y, and Ô x 2 +y 2 , secondly updating the location vector [x,y] from the results of first step. By both theoretical analysis and numerical simulations, we show that for small TOA estimation errors the TSE approaches the CRLB very closely with only one iteration. Further results show the CRLB of passive TOA estimation to be much 41 lower than that of TDOA. This indicates that the synchronization between the transmitter and the receivers can substantially decrease the localization error. ExperimentsalsoshowthattheerrorvarianceofTSEisveryclosetotheCRLB of TOA algorithms under the assumption of Gaussian range measurement errors. In addition, the error variance of TSE is significantly (33 dB in our measurements) lower than that of TDOA algorithms. Ourresultsthusdemonstratethepotentialadvantagesofusingsynchronization between TX and RX in passive object localization, and furthermore provide a practical tool to exploit this potential to its full extent. 2.10 Appendix: Proof of Lemma 1 Let ˆ ◊◊◊ = ¯ ◊◊◊ +n, ˆ S = ¯ S+e S and ˆ h = ¯ h+e h , where ¯ ◊◊◊ , ¯ S and ¯ h are true/mean values and ˆ ◊◊◊ , ˆ S and ˆ h are the measured/estimated values. Obviously, ¯ h≠ ¯ S ¯ ◊◊◊ =0. (2.36) According to (2.5) and (2.7), ÏÏÏ = ¯ h+e h ≠ ( ¯ S+e S ) ¯ ◊◊◊ = e h ≠ e S ¯ ◊◊◊. (2.37) Multiplying both sides of (2.10) by ( ¯ S T +e S T ) ≠ 1 ( ¯ S+e S ), it follows ( ¯ S T +e S T ) ≠ 1 ( ¯ S+e S )( ¯ ◊◊◊ +n) =( ¯ S T +e S T ) ≠ 1 ( ¯ h+e h ), 42 Leaving only the linear perturbation terms by omitting the second order errors, using (2.36), it follows that ¯ S T ≠ 1 ¯ Sn¥ ¯ S ≠ 1 (e h ≠ e S ¯ ◊◊◊ ). Then, we obtain n =( ¯ S T ≠ 1 ¯ S) ≠ 1 ¯ S ≠ 1 (e h ≠ e S ¯ ◊◊◊ ) =( ¯ S T ≠ 1 ¯ S) ≠ 1 ¯ S ≠ 1 ÏÏÏ. (2.38) According to (2.5) and (2.7), substitute ÏÏÏ by Be, it follows that n¥ ( ¯ S T ≠ 1 ¯ S) ≠ 1 ¯ S ≠ 1 Be. (2.39) Then, cov( ˆ ◊◊◊ ) ¥ E(nn T ) =( ¯ S T ≠ 1 ¯ S) ≠ 1 ¯ S T ≠ 1 BE(ee T ) ◊ B T ≠ 1 ¯ S[( ¯ S T ≠ 1 ¯ S) ≠ 1 ] T . Because BE(ee T )B T = , and [( ¯ S T ≠ 1 ¯ S) ≠ 1 ] T =( ¯ S T ¯ ≠ 1 ¯ S) ≠ 1 , it follows that cov( ˆ ◊◊◊ ) ¥ ( ¯ S T ≠ 1 ¯ S) ≠ 1 ¯ S T ≠ 1 ¯ S( ¯ S T ≠ 1 ¯ S) ≠ 1 =( ¯ S T ≠ 1 ¯ S) ≠ 1 , which ends the proof of Lemma 1. 43 Chapter 3 Indirect Path Detection with Angle Information: A TDAJ (TOA, DOD, DOA joint) Indirect Path Detection Scheme This section studies the problem of indirect path (IP) detection. It is an important prerequisite for precise localization, such as in radar systems, to discern “direct path” (DP), i.e., multipath components (MPCs) where the signal propa- gates directly between the target and the localization system nodes (transmitters or receivers), from IPs. Failing to detect the IP paths would bring big errors into localization. The materials in this chapter are available in [70, 71, 69]. In this chapter, we consider a Multi-input-multiple-output (MIMO) systems, whereboththetransmitterandreceiverhavemultipleantennas. Thesignalsprop- agate from transmitters to receivers via dierent DPs or IPs. Note that DPs are pathswheresignalsareonlyreflectedbytruetargets,whileIPsarethepathswhere signals are reflected by additional non-target reflectors. With this system setup, the sophisticated channel sounding algorithms [62, 64] are able to extract chan- nel parameters of each paths, e.g., time of arrival (TOA), direction of departure (DOD), and direction of arrival (DOA). Then, we proposed TDAJ (TOA, DOD, DOA joint) indirect path detection scheme distinguishes whether any triplets of 44 the observable propagation parameters (TOA, DOD and DOA) correspond to DP or IP. Any combination of two of those parameters allow the computation of a reflection point, and - in the absence of noise - the points computed from the dif- ferent combinations are consistent in the case of a DP. In the presence of noise, the computed points deviate from each other, and we establish rules for how large this deviationmightbetoconsiderapathanIP.Severaldierentdecisionrulesaredis- cussed and compared. We derive closed-form equations for these decision criteria and the resulting performance. The situations when the proposed decision rules do not work well are also studied. Simulation results show that under common localization scenarios employing ultrawideband signals, the proposed algorithms can eectively detect IPs and significantly improve localization accuracy. The remainder of this section is organized as follows. Section 3.1 presents the overview of IP detection problems. The system model of IP detection is discussed in Section 3.2. The principle of TDAJ is presented in Section 3.3. In Section 3.4, we study the distribution of the decision variable, evaluations of PF and PD, the “blind spots” of dierent rules and the outage probability. The detailed procedure of TDAJ algorithm is shown in Section 3.5. The simulation results are shown in Section 3.6 and the conclusions are drawn in Section 3.7. In the remainder of this section, we let ¯ • denote the true value, ˆ • denote the estimated or measured value, and • = ˆ •≠ ¯ •. 3.1 Overview Radio positioning technologies employ the measured characteristics (such as time of arrival TOA, direction of departure DOD, and direction of arrival DOA) of multipath components (MPCs) propagating between target and the nodes of 45 the localization system (the nodes can be transmitters or receivers). For the sub- sequent discussion, it is useful to distinguish between the direct path (DP), i.e., a MPC where the signal propagates directly between the target and the localization system nodes and the indirect paths (IPs), where the signal interacts with one or more objects/obstacles in addition to the target and system nodes. For an active localization system, an indirect path is any non-line-of-sight (NLOS) path. Extensive studies [42, 55, 2], show that IPs lead to worse localization accuracy, especially for precise localization. This is because IPs create biases in the DOD, TOAandTOAestimates,resultinginerroneousestimatesoftargetlocation. More importantly, thelocalizationerrorfromIPscannotbemitigatedbyaveragingover multiple measurements - in contrast to noise-induced localization errors. To combat the negative eects of IPs in active localization systems, various techniques have been proposed in the literature. They can be divided into two categories, IP mitigation andIP detection [35]. IPmitigationattemptstoalleviate or counter the bias of channel parameters introduced by IPs [11, 50, 12, 93, 41, 13], while IP detection attempts to dierentiate DPs and IPs so that channel parameters of IPs are not used in the DP based positioning algorithms [29, 48, 28, 91, 3, 49]. Performance evaluation and comparison of existing IP detection schemes were performed in [16]. [13] practically studied the IP detection problem, and proposed a maximum a posteriori probability detection method based on root meansquaredelay,kurtosisandmaximumamplitude. In[29],theIPdetectionwas performedbyanalysingmultipathchannelstatisticssuchasthekurtosis, themean excessdelay,andtheroot-mean-squaredelay. [48]employedtheenergystatisticsof direct-path pulse and certain delay statistics for IP detection. The delay between the first and the strongest impulse was employed in [28]. In [91], the IP detection was based on testing whether angle estimates were matched between transmitter 46 and receiver, or based on whether signal travel range matched power decay. [3] utilized the estimated TOA and Received Signal Strength (RSS) for IP detection. [49] studied characterization of dierences in the channel pulse responses under IP and DP conditions, and proposed non-parametric machine learning techniques to perform IP identification. Thus, there is a wealth of IP detection scheme in the literature; however,theyrelyonfeaturessuchasdelayspreadtodiscernIPs,which in turn rely on the environment, which the system might not know with sucient accuracy. The situation is fundamentally dierent in passive localization systems, where an IP is a path that has interactions with one or more objects/obstacles in addi- tion to the target. In many localization scenarios, especially indoor environments, there are rich IPs during signal propagation. In other words, in passive local- ization systems, the DP implies a single-reflection process (at the target), while IPs undergo multiple reflections (at non-target reflectors). The detection of IP in passive localization requires the discrimination between single-reflections and multiple-reflections; a task that is fundamentally dierent from the detection of line-of-sightpathsencounteredinactivesystems. MostexistingIPdetectionmeth- ods for active targets, as described above, are essentially based on the fact that signal characteristics change after reflection in NLOS paths. Thus, they are not expected to work well for passive target since reflection happens in both IP and NP paths for passive localization. To fill this gap, we focus on IP detection for precise passive target localization. We assume that the DOD, DOA and TOA measurements are available as triplets, from channel sounding algorithms such as SAGE [19] or RiMax [60], [63]. Note that to obtain the DOD and DOA information, the localization system needs antenna arrays or directional antennas. We propose a new TDAJ (TOA, DOD, 47 DOA joint) IP detection scheme, based on such measurements. TDAJ eliminates thedependenceon“tuningparameters”suchas“typical”receivedpoweranddelay spread. The only information needed by TDAJ is the statistical distributions of DOD, DOA and TOA measurement errors, which can be easily obtained from the measurements themselves. For some models of DOD, DOA and TOA errors, such as Gaussian distribution, only a small number of parameters (variance) need to be known. The principle of the algorithm is as follows: if there were no measurement errors, any DP would have self-consistent parameters. By this we mean that the threeintersectionsobtainedfromthepairsofmeasuredparameters(DOAlinewith the TOA ellipse, DOD line with the TOA ellipse, and DOA line with the DOD line) are at the same location. In the presence of measurement errors, the decision variables of the Neyman-Pearson (NP) test are the dierences between two of the three intersections in x or y coordinates: only if this dierence lies below a certain threshold, the algorithm judges the path to be a DP. The TDAJ algorithm is developed under the NP criterion so it is able to achieve desired probability of false alarm (PF) and maximize the probability of detection (PD). Although the TDAJ is designed for passive target localization, its principle can be also used for active target locations. Single-reflection paths can act as “virtual sources” according to the image principle [53] in active systems, and thus make localization in the presence of multipath propagation more accurate than without multipath [75]. To employ the “virtual sources" to improve localization accuracy, we still need to discern the single reflection and the multiple reflector paths. Particularly, we made following contributions in this work: 48 • Propose the idea of IP detection based on the dierences of the three geo- graphical intersections of DOD, DOA and TOA measurements in x or y coordinates. This is a more general form of an idea presented in [69]. • Prove that for small errors of DOD, DOA and TOA measurements, the six decision variables (dierences between the three intersections) are the linear transformations of the measurement errors of DOD, DOA and TOA. Then, the distributions of the decision variables and the thresholds to achieve a given PF are obtained. • Study the relationship between the six decision variables in DPs, and prove that they are linearly dependent as summarized in Theorem 1. • Derive the closed-form expressions of PD given an expected PF. Further definea“saferegion”whichbetterevaluatestheimpactofIPsonlocalization accuracy. • Investigatethesituationswhenthedecisionrulesdonotworkasexpectedand compare the performances of various decision rules via simulations. Simula- tions also show significant localization accuracy improvement by the TDAJ algorithm. 3.2 System Model In this section, we present the system model, assumptions and discuss the importance of IP detection for precise localization systems. For simplicity, we consider a two dimensional scenario, as shown in Fig. 3.1, though extension to three dimensions is straightforward. There is one transmitter at [≠ d 2 ,0], and one receiver at [ d 2 ,0]. We assume that estimates of the tuplets of 49 1 ! 3 ! 2 ! !"#$%&'()"* +),)'-)"* !#".)/* +)0),/1"* 232* 234* !34*566'7%)* 1 p 2 p 3 p 4 p [] ,0 2 d ! [] ,0 2 d [, ] pq [, ] mn ! " Figure 3.1: Illustration of DPs and IPs DOA, DOD, and TOA are available, for all MPCs, from array measurements at the TX and RX. Measurement methods and algorithms for the extraction of these MPC parameters (with automatic pairing) have been discussed, e.g., in [60, 6, 63]. Without restriction of generality, we consider in the following a single MPC, for which we determine whether it is a DP or IP. The target is located at [p,q] and a possible obstacle (which could give rise to additional interactions) is at [m,n]. Then the DP is associated with a tuplet (◊ 1 ;◊ 2 ;p 1 +p 2 ), while an IP corresponds to (◊ 1 ;◊ 3 ;p 1 +p 3 +p 4 ). Note that in the above example, the reflector is on the way from target to receiver. The obstacle might be positioned such that interaction with it occurs “on the way” from TX to target; this case can be treated completely analogously. Furthermore, interactions with multiple reflectors (obstacles) are possible, and could be included in our model; however, for the sake of simplicity, we restrict our attention to the case depicted in Fig. 1. 50 The localization errors happen due to either the measurement noise or the IPs: • Measurement noise leads to errors of DOD, DOA and TOA estimates and finallyerrorsoftargetlocationestimates. Theerrorsfrommeasurementnoise are zero mean, and averaging over multiple runs could mitigate the error of localization. Irrespective of possible averaging, we assume in the following thattheerrorsofDOD,DOAandTOAmeasurementsduetonoisearesmall, and their statistical distributions are known. • IPs should not be used for localization generally 1 , since the association between TOA/DOD/DOA and reflector locations is not bijective. Erro- neously using IPs leads to biases to DOD, DOA and TOA estimates, also resulting in errors of target location estimates. Moreover, the errors of local- ization cannot be decreased by averaging. TheIPdetectionisparticularlyimportantforpreciselocalizationsystemsbased on ultra-wideband signals. This is because the errors of DOD, DOA and TOA measurementsduetonoisearesmallcomparedwiththoseduetoIPs. Forinstance, [63]and[72,74]showthatinananechoicchamber,withultra-wide-bandsignals(2- 8GHz)andsophisticatedchannelsoundingtechniques(RIMAX[63]),thestandard deviation of the signal travel range (proportional to TOA) can be less than 0.5 cm, and the standard deviations of DOD and DOA measurements can be less than 0.5 degree. In[80],with1.5GHzbandwidthsignals,rangingerrorsofabout10cmand angular errors of about 1 degree are achieved. In these two examples, the errors areduetonoise. RegardingerrorsduetoIPs, therangeestimatebias(p 3 +p 4 ≠ p 2 ) and angle estimate bias (◊ 3 ≠ ◊ 2 ) can easily be on the orders of meters, and tens of degrees, respectively. Thus, IPs severely degrade localization accuracy, especially 1 though, see [23], as discussed below. 51 DOD ! DOA ! !"#$%&'()"* +),)'-)"* ./.* ./0* !/0*122'3%)* [] ,0 2 d ! [] ,0 2 d ! " 11 [, ] xy 22 [, ] xy 33 [, ] xy Figure 3.2: Illustration of TOA, DOA and DOD joint location estimation in indoor environments, where a multitude of reflectors such as walls and tables exist. To improve the localization accuracy, it is vital to first identify the IPs. Then, the IPs can either be discarded or combined with the DPs for localization according to the methods described in [13]. 3.3 IPDetection Scheme basedonDierences of Intersections In this section, we present the principle of the proposed IP detection scheme based on dierences of intersections of DOD/DOA, DOD/TOA and DOA/TOA in x or y coordinates. The basic idea is that if DOD, DOA and TOA measurements are consistent, the three intersections from them would be close with each other. Otherwise, the three intersections would be far from each other. 52 AsshowninFig. 3.2,thecoordinatesofreflectionpointsgivingaspecificDOD, DOA, or TOA, respectively, can be written as follows, DOD: y=tan(◊ DOD )(x+ d 2 ), (3.1) DOA: y=tan(◊ DOA )(x≠ d 2 ), (3.2) TOA: x 2 ( l 2 ) 2 + y 2 ( l 2 ) 2 ≠ ( d 2 ) 2 =1. (3.3) Note that l =c 0 ·TOA, where c 0 is the speed of light and TOA is the signal travel time. (3.1) and (3.2) denote two lines and (3.3) is an ellipse. Throughout this chapter, we assume there is no ambiguity in the DOD and DOA estimation: the system can eectively dierentiate ◊ DOD and ◊ DOD ±fi , ◊ DOA and ◊ DOA ± fi . Without loss of generality, we assume that 0 Æ ◊ DOD Æ fi and 0Æ ◊ DOA Æ fi . Practically, the ambiguity can be removed by combining the DOA estimates from several receivers: the correct DOA lines from every receiver should point to the correct target location. Other methods such as using rough prior information of the target location can also remove the DOD and DOA ambiguity. Due to measurement errors or IPs, DOD, DOA and TOA intersect at three points instead of one. Let [ˆ x 1 ,ˆ y 1 ], [ˆ x 2 ,ˆ y 2 ] and [ˆ x 3 ,ˆ y 3 ] denote the locations of 53 intersectionsofDOD/DOA,DOD/TOA,DOA/TOA.Then,therearesixdierence observations of the three intersections as follows: ˆ ⁄⁄⁄ = S W W W W W W W W W W W W W W W W W U ⁄ 1 ⁄ 2 ⁄ 3 ⁄ 4 ⁄ 5 ⁄ 6 T X X X X X X X X X X X X X X X X X V = S W W W W W W W W W W W W W W W W W U ˆ x 1 ≠ ˆ x 2 ˆ y 1 ≠ ˆ y 2 ˆ x 1 ≠ ˆ x 3 ˆ y 1 ≠ ˆ y 3 ˆ x 2 ≠ ˆ x 3 ˆ y 2 ≠ ˆ y 3 T X X X X X X X X X X X X X X X X X V (3.4) Since there are 6 observations, the TDAJ IP detection can be based on indi- vidual rule or fusion rule: • Individual rule: the final decision is based solely on one ⁄ i : if |⁄ i | is larger than a threshold ” i , TDAJ claims an IP, otherwise, claims a DP. • Fusion rule: The final decision is based on a fusion of the six observations: if the number of |⁄ i | larger than a threshold ” i (i=1,2,...,6) is larger than K (K =0,1,...,5), the detection algorithm claims that it is an IP, otherwise, the detection algorithm claims that it is a DP. To evaluate the IP detection performance, the probability of false alarm (PF) and probability of detection (PD) are: P f = Prob (The algorithm claims an IP - - - The path is a DP), P d = Prob (The algorithm claims an IP - - - The path is an IP). (3.5) 54 For each individual measurement ⁄ i , PF and PD can be evaluated as follows: p i f = Prob (|⁄ i |>” i - - - The path is a DP), p i d = Prob (|⁄ i |>” i - - - The path is an IP). (3.6) For fusion rule, the decision strategy is p K f,fr = Prob ((Number of |⁄ i | larger than ” i )> K - - - The path is a DP), p K d,fr = Prob ((Number of |⁄ i | larger than ” i )> K - - - The path is an IP). (3.7) wherethesubscript“fr”denotesthefusionrule,andthesuperscriptK isaparam- eter characterizing the fusion rule. Because a DP means no reflector, the PF is independent of the locations of obstacles/reflectors while the PD is not. In most situations, the locations of the reflectors are unknown, thus, an ideal strategy of TDAJ scheme is based on the Neyman-Pearson rule [34] to choose a threshold ” to achieve a desired PF, and at the same time maximize PD. 3.4 Distribution of Decision Variables and Eval- uations of P f and P d In this section, we study the decision variables ⁄ i and then present the eval- uations of PF and PD. To better evaluate the impact of IPs on localization, we define a “safe region” and outage probability. We also investigate the situations where the TDAJ algorithm does not work well. Let the true/mean values of the locations of the DOD/DOA, DOD/TOA, DOA/TOA intersections be [¯ x 1 ,¯ y 1 ], [¯ x 2 ,¯ y 2 ] and [¯ x 3 ,¯ y 3 ], respectively. The noise 55 induced deviations from the means of the locations of the three intersections are denoted by [ x k , y k ]=[ˆ x k ≠ ¯ x k , ˆ y k ≠ ¯ y k ],k=1,2,3, (3.8) where [ˆ x 1 ,ˆ y 1 ], [ˆ x 2 ,ˆ y 2 ] and [ˆ x 3 ,ˆ y 3 ] are the measured locations of intersections of DOD/DOA, DOD/TOA, DOA/TOA. For notational convenience, we define µ 1 = ◊ DOA , µ 2 = ◊ DOD and µ 3 = l. Taking the derivatives of (3.1) - (3.3) with respect to µ 1 , µ 2 and µ 3 , the deviations of x k , y k and µ k satisfy a k k x k +b k k y k +c k k µ k =0, (3.9) a k k+1,mod3 x k +b k k+1,mod3 y k +c k k+1,mod3 µ k+1,mod3 =0, (3.10) where k=1,2,3, mod3 denotes taking the index modulo 3, and a k 1 =tan¯ µ 1 ,b k 1 =≠ 1,c k 1 = ¯ x k ≠ d 2 cos 2 (¯ µ 1 ) , a k 2 =tan¯ µ 2 ,b k 2 =≠ 1,c k 2 = ¯ x k + d 2 cos 2 (¯ µ 2 ) , (3.11) a k 3 = 2¯ x k ¯ µ 2 3 ,b k 3 = 2¯ y k ¯ µ 2 3 ≠ d 2 ,c k 3 =≠ 2 3 ¯ x 2 k ¯ µ 3 3 ≠ ¯ y 2 k ¯ µ 3 (¯ µ 2 3 ≠ d 2 ) 2 4 . Here a, b and c with superscript k are parameters for the intersection [x k ,y k ].We actually use derivatives (first order Taylor expansion) to approximate the noise induced errors. Note that these above parameters depend on [¯ x k ,¯ y k ] and ¯ µ k , k = 1,2,3, which are actually unknown. We assume that they are known for the purpose of analyzing the distribution of decision variable ⁄ i and the evaluation of PD and PF. In Section 3.5, we will discuss the TDAJ scheme based on estimates of these variables. 56 For notational convenience, we can define the following quantities: – =a 1 1 b 1 2 ≠ a 1 2 b 1 1 , (3.12a) — =a 2 2 b 2 3 ≠ a 2 3 b 2 2 , (3.12b) “ =a 3 3 b 3 1 ≠ a 3 1 b 3 3 . (3.12c) By solving (3.9) and (3.10) for dierent k, we can obtain ÁÁÁ = S W W W W W W W W W W W W W W W W W U x 1 ≠ x 2 y 1 ≠ y 2 x 1 ≠ x 3 y 1 ≠ y 3 x 2 ≠ x 3 y 2 ≠ y 3 T X X X X X X X X X X X X X X X X X V =W µ µ µ. (3.13) where W = S W W W W W W W W W W W W W W W W W U ≠ c 1 1 b 1 2 – c 1 2 b 1 1 – + c 2 2 b 2 3 — ≠ b 2 2 c 2 3 — a 1 2 c 1 1 – ≠ a 1 1 c 1 2 – ≠ c 2 2 a 2 3 — a 2 2 c 2 3 — ≠ b 1 2 c 1 1 – ≠ b 3 3 c 3 1 “ b 1 1 c 1 2 – b 3 1 c 3 3 “ a 1 2 c 1 1 – + c 3 1 a 3 3 “ ≠ a 1 1 c 1 2 – ≠ a 3 1 c 3 3 “ ≠ b 3 3 c 3 1 “ ≠ b 2 3 c 2 2 — b 2 2 c 2 3 — + b 3 1 c 3 3 “ a 3 3 c 3 1 “ a 2 3 c 2 2 — ≠ a 2 2 c 2 3 — ≠ a 3 1 c 3 3 “ T X X X X X X X X X X X X X X X X X V , (3.14) µ µ µ =[ µ 1 , µ 2 , µ 3 ] T . (3.15) According to (3.4) and (3.13), we can evaluate the observations of dierences by ˆ ⁄ ˆ ⁄ˆ ⁄ = ¯ ⁄ ¯ ⁄¯ ⁄ +ÁÁÁ, (3.16) 57 where ˆ ⁄ ˆ ⁄ˆ ⁄ =[ˆ x 1 ≠ ˆ x 2 ,ˆ y 1 ≠ ˆ y 2 ,ˆ x 1 ≠ ˆ x 3 ,ˆ y 1 ≠ ˆ y 3 ,ˆ x 2 ≠ ˆ x 3 ,ˆ y 2 ≠ ˆ y 3 ] T and ¯ ⁄ ¯ ⁄¯ ⁄ =[¯ x 1 ≠ ¯ x 2 ,¯ y 1 ≠ ¯ y 2 ,¯ x 1 ≠ ¯ x 3 ,¯ y 1 ≠ ˆ y 3 ,¯ x 2 ≠ ¯ x 3 ,¯ y 2 ≠ ¯ y 3 ] T are the mean of ˆ ⁄ ˆ ⁄ˆ ⁄ . Discussions: It can be easily proven that ¯ ⁄ i =0 in the case of DPs when ¯ x 1 =¯ x 2 =¯ x 3 and ¯ y 2 =¯ y 2 =¯ y 3 , while ¯ ⁄ i ”=0 in the case of IPs. In the following, the evaluations of p i f and p i d are presented. 3.4.1 Distribution of ⁄ i in DPs and evaluation of p i f In the case of DPs, a k j , b k j , c k j (j =1,2,3) and [¯ x k ,¯ y k ] have the same values for dierent k, and ¯ ⁄ i =0 with i=1,2,...,6. For notational convenience, let ˆ ⁄ DP i denote the value of ⁄ i in the case of DPs and ˆ ⁄ DP ˆ ⁄ DP ˆ ⁄ DP =[ ˆ ⁄ DP 1 , ˆ ⁄ DP 2 ,..., ˆ ⁄ DP 6 ] T . Also, let a j = a k j , b j = b k j and c j = c k j for j=1,2,3. Then, we can show the relations between the 6 dierence observations in the following theorem. Theorem 1. In the case of DP, the six dierence observations ˆ ⁄ i (i=1,2,...,6) are linearly dependent: Ê 1 ˆ ⁄ DP 1 =Ê 2 ˆ ⁄ DP 2 =Ê 3 ˆ ⁄ DP 3 =Ê 4 ˆ ⁄ DP 4 =Ê 5 ˆ ⁄ DP 5 =Ê 6 ˆ ⁄ DP 6 , (3.17) where Ê 1 =1, Ê 2 =≠ b 2 a 2 , Ê 3 =≠ b 2 “ DP b 1 — DP , Ê 4 = b 2 “ DP a 1 — DP , Ê 5 = b 2 “ DP b 3 – DP , Ê 6 =≠ b 2 “ DP a 3 – DP , 58 TOA DOD DOA ሾݔ ଵ ǡݕ ଵ ሿ ሾݔ ଶ ǡݕ ଶ ሿ ሾݔ ଷ ǡݕ ଷ ሿ ߣ ଵ - ɉ ଷ - ɉ ହ ߣ ସ ߣ ଶ െߣ ଺ Figure 3.3: Interpretation of Theorem 1 with – DP =a 1 b 2 ≠ a 2 b 1 , (3.18a) — DP =a 2 b 3 ≠ a 3 b 2 , (3.18b) “ DP =a 3 b 1 ≠ a 1 b 3 . (3.18c) Proof. Please refer to the Appendix. Theorem 1 implies that any one of the six observations contains all the infor- mation to evaluate the level of consistency between DOD, DOA and TOA mea- surements. Thus, we have P f =p i f ,i=1,2,...,6. 59 An interpretation of Theorem 1 is illustrated in Fig. 3.3. The dierence obser- vations ⁄ i (i =1,2,...,6) are also shown in the figure. The TDJA essentially employs linear approximations to set up the relations between the changing of three intersections ([x j ,y j ], j=1,2,3.) andµ 1 , µ 2 andµ 3 as shown in (3.9). Thus, we treat the DOD, DOA and TOA as three lines shown in Fig. 3.3. The changing of the three lines generates a triangle. The shape of the triangle is determined by the absolute DOA, DOD, and TOA, and only the size (scaling) of the triangle dependsontheerror. Thus,thetriangleiscompletelydefined,andallthevaluesof ⁄ i can be calculated, if any one of the ⁄ i is known. This interpretation also allows to show easily that the method in our recent conference paper [69] is actually a special case of the TDAJ method in this chapter. The decision variable in [69] is the distance from DOD/DOA intersection to TOA ellipse. Based on the result of Theorem 1, it is linearly dependent with ⁄ i in case of DPs. Since ¯ ⁄ i =0 in the case of DP, from (3.14) and (3.16), it follows that ⁄ DP i is evaluated by ⁄ DP i =W i,1 µ 1 +W i,2 µ 2 +W i,3 µ 3 , (3.19) where W i,j is the entry of W in (3.14) on the ith row and the jth column. Assume that the probability density functions of DOA, DOD and TOA mea- surementerrorsf µ 1 (x),f µ 2 (x)andf µ3 (x)areknown. Then, let µ i 1 =W i,1 µ 1 , µ i 2 =W i,2 µ 2 and µ i 3 =W i,3 µ 3 , the distributions of µ i 1 , µ i 2 , µ i 3 are g DP µ i 1 (y)= 1 |W i,1 | f µ 1 3 y W i,1 4 , (3.20) g DP µ i 2 (y)= 1 |W i,2 | f µ 2 3 y W i,2 4 , (3.21) g DP µ i 3 (y)= 1 |W i,3 | f µ 3 3 y W i,3 4 . (3.22) 60 Then, the distribution of ⁄ DP i is f DP ⁄ i (z)=(g DP µ i 1 ú g DP µ i 2 ú g DP µ i 3 )(z). (3.23) whereú denotes the convolution operation. Particularly, if µ 1 , µ 2 and µ 3 are Gaussian distributed with variances ‡ 2 1 , ‡ 2 2 and ‡ 2 3 , respectively, the decision variable ⁄ DP i is also a Gaussian random variable with variance ‡ 2 ⁄ DP i =(W i,1 ‡ 1 ) 2 +(W i,2 ‡ 2 ) 2 +(W i,3 ‡ 3 ) 2 . (3.24) For some other special distributions of µ 1 , µ 2 and µ 3 , simple evaluation of f ⁄ i (z) can also be obtained [26]. Based on the definition of PF, given the distribution of f DP ⁄ i (z), we have p i f = Prob(|⁄ i |>” i - - - The channel is a DP) =1≠ ⁄ ” i ≠ ” i f DP ⁄ i (z)dz. (3.25) Whenµ 1 ,µ 2 andµ 3 areGaussiandistributed,wecanobtainasimpleevaluation of p i f by p i f =1≠ ⁄ ” i ≠ ” i 1 2fi‡ ⁄ i exp 3 ≠ z 2 2‡ 2 ⁄ i 4 dz =2Q 3 ” i ‡ ⁄ i 4 . (3.26) where ‡ ⁄ i is given in (3.24) and Q(a)= s Œ a 1 2fi exp 1 ≠ m 2 2 2 dm. 61 Thus, given a desired value of P f , the threshold ” i is computed as ” i =Q ≠ 1 3 P f 2 4 ‡ ⁄ i . (3.27) where Q ≠ 1 (•) is the inverse function of Q(•). 3.4.2 Distribution of ⁄ i in IPs and evaluation of p i d In the case of IP, DOD, DOA and TOA intersect at three intersection points due to both the measurement noise and IP. Particularly, ¯ ⁄ i ”=0 for IPs. Let⁄ IP i denote⁄ i inthecaseofIPs, and ¯ ⁄ IP i bethemeanvalueof⁄ IP i . Accord- ing to (3.16), following similar reasoning of (3.19) - (3.23), the probability density function of ⁄ IP i is f IP ⁄ i (z)=(g IP µ i 1 ú g IP µ i 2 ú g IP µ i 3 )(z), (3.28) with g IP µ i 1 (y)= 1 |W i,1 | f µ 1 3 y≠ ¯ ⁄ IP i W i,1 4 , (3.29) g IP µ i 2 (y)= 1 |W i,2 | f µ 2 3 y≠ ¯ ⁄ IP i W i,2 4 , (3.30) g IP µ i 3 (y)= 1 |W i,3 | f µ 3 3 y≠ ¯ ⁄ IP i W i,3 4 . (3.31) where W i,j is the entry of W in (3.14) on the ith row and the jth column and f µ 1 , f µ 2 and f µ 3 are probability density functions of DOA, DOD and TOA measurement errors, respectively. Based on the definition of PD, given the distribution of f IP ⁄ i (z), 62 p i d = Prob(|⁄ i |>” i - - - The channel is a IP) =1≠ ⁄ ” i ≠ ” i f IP ⁄ i (z)dz. (3.32) When µ 1 , µ 2 and µ 3 are Gaussian distributed with variance ‡ 2 1 , ‡ 2 2 and ‡ 2 3 , respectively, ⁄ IP i is also a Gaussian random variable with non-zero mean ¯ ⁄ IP i and variance ‡ 2 ⁄ IP i =W 2 i,1 ‡ 2 1 +W 2 i,2 ‡ 2 2 +W 2 i,3 ‡ 2 3 . Then, PD is evaluated by p i d =1≠ ⁄ ” i ≠ ” i 1 2fi‡ ⁄ IP i exp 3 ≠ (z≠ ¯ ⁄ IP i ) 2 2‡ 2 ⁄ IP i 4 dz =1≠ Q 3 ≠ ” i ≠ ¯ ⁄ IP i ‡ ⁄ IP i 4 +Q 3 ” i ≠ ¯ ⁄ IP i ‡ ⁄ IP i 4 . (3.33) Inconclusion, wenotethattheaveraged(noise-free)intersectionpointsdepend on the location of the reflector. We have assumed in our derivation a deterministic location. Inpractice,suchalocationcannotbeexpectedtobeknown. Atmost,the distributionofthereflectorlocationscanbeestimated(e.g.,uniformlydistributed), from which further statistics of PF and PD could be derived. 3.4.3 Blind Spot of ⁄ i There are some situations where the indirect paths can not be eectively detected. Here, we call these special situations “blind spots (BSs)”. This section studies the BSs for dierent ⁄ i . There are three dierent kind of situations where the proposed methods do not work well: 63 BS1: The reflectors are on the lines of DOD or DOA. In this situation, the DOD and DOA intersection is on the TOA ellipse when no measurement errors exist. BS2: The locations of reflectors satisfy any of the three conditions: – æ 0, — æ 0 or “ æ 0. This is because under these three conditions, ‡ ⁄ i æŒ according to (3.24). BS3: The locations of reflectors satisfy ⁄ i æ 0. For BS1, the potential locations of reflectors [m,n] satisfy n m≠ d 2 = y m≠ x 2 or n m+ d 2 = y m+ x 2 , (3.34) where [x,y] is the location of the target, [≠ d/2,0] and [d/2,0] are the locations of transmitter and receiver, respectively. Given the locations of target and reflectors, the signal can arrive at them through dierent paths. For simplicity, we consider a reflector location to be BS if the proposed method does not work well for any one of the paths. For example, for transmitter at [≠ 2.5,0], receiver at [2.5,0] and target at [≠ 1,5], the reflector [≠ 2, 5 3 ] is on the line of DOD. The proposed method does not perform well only for the path transmitter-reflector-target-receiver, not the path transmitter-target-reflector-receiver. In this example, we consider the reflector [≠ 2, 5 3 ] to be a BS. For BS2, the reflectors need to satisfy – æ 0, — æ 0 or “ æ 0. Particularly, – æ 0, which is easy to be satisfied, happens when the DOD is parallel to the DOA. Table 3.1 shows the BS positions for dierent ⁄ i . One interesting observation is that BS3 do not exist for ⁄ 5 . Note that though we only consider 0Æ µ 1 Æ fi and 0 Æ µ 2 Æ fi in this chapter, if the full range of µ 1 and µ 2 were considered, 64 Table 3.1: Blind Spots of Dierent ⁄ i Blind Spots BS1 BS2 BS3 ⁄ 1 Reflectors satisfying (3.34) – æ 0 (DOD // DOA), — æ 0 µ 2 = fi 2 ⁄ 2 Reflectors satisfying (3.34) – æ 0 (DOD // DOA), — æ 0 µ 2 =0 or fi ⁄ 3 Reflectors satisfying (3.34) – æ 0 (DOD // DOA), “ æ 0 µ 1 = fi 2 ⁄ 4 Reflectors satisfying (3.34) – æ 0 (DOD // DOA), “ æ 0 µ 1 =0 or fi ⁄ 5 Reflectors satisfying (3.34) — æ 0, “ æ 0 Do not exist ⁄ 6 Reflectors satisfying (3.34) — æ 0, “ æ 0 µ 1 +µ 2 =fi a BS3 would be possible, too. One illustration of the BS is shown in Fig. 3.4, The settings for the illustration are transmitter at [≠ 2.5,0], receiver at [2.5,0] and target at [≠ 1,5]. It is assumed there is only one reflector [m,n] in the range mœ [≠ 8,8] u nœ [0,10]andthesignalpropagationpathcanbeeithertransmitter- reflector-target-receiver, ortransmitter-target-reflector-receiver. Inthefigure, BS1 is shown as blue color dot-dashed lines, BS2 is shown as black color dashed lines and BS3 is shown as red color solid lines. The BS2 of — æ 0 and “ æ 0 does not exist in this setting. The BS2 of – æ 0 is the black lines which are parallel to the blue lines (DOD and DOA lines). One interesting observation is that ⁄ 5 has the smallest BS: only BS1 exist for ⁄ 5 . For the localization of a target, multiple transmitter-receiver pairs are often used. Dierent transmitter-receiver pairs have dierent BSs. Thus, the BSs can be eliminated by combining the IP detection results from multiple transmitter- receiver pairs. 3.4.4 “Safe region” and outage probability In this section, we further study the impact of IPs on target localization, and define the “safe region” and outage probability. In the previous section, we defined the probability of false alarm as “deciding for a DP whenever an IP actually happens”. This has a negative impact on the 65 −5 0 5 0 5 10 λ 1 −5 0 5 0 5 10 λ 2 −5 0 5 0 5 10 λ 3 −5 0 5 0 5 10 λ 4 −5 0 5 0 5 10 λ 5 −5 0 5 0 5 10 λ 6 Figure 3.4: Illustration of Blind Spots of Dierent ⁄ i localization, since IPs add bias to DOD, DOA and TOA measurements, based on which the target is finally localized. However, it can be easily seen that dierent IPs would have dierent impact on the localization accuracy . For example, if the non-targetreflectorisclosetotheDODorDOAlines, andaddslittlebiastoDOD, DOA and TOA measurements, then, this IP would have little negative impact on the localization. There are even situations when the reflector does not change the value of DOD, DOA and TOA, e.g., in the case of BS1, the reflector is essentially on the line connecting the transmitter and the target. Thus, the definition of DP or IP does not fully reflect the impact of non-target reflectors on the localization accuracy. To quantify the IPs that do not lead to significant location bias, one could define a region called “safe region”, where | ¯ ⁄ i |<÷. (3.35) 66 It is easy to understand that larger value of | ¯ ⁄ i | leads to less accurate localiza- tion. The “safe region” means that in this region, the non-target reflectors have lessnegativeimpactonthelocalizationaccuracythanreflectorsoutsidethisregion. Only if the pair of DOD, DOA and TOA is considered to be in “safe region”, they are used for localization. The DP can be considered as a special case of “safe region”, which has the minimum (no) negative impact on localization accuracy. Given locations of reflectors, the value of ¯ ⁄ i can be computed, and we can decide whether the path is in “safe region” or not. Moreover, we can define an outage probability of ⁄ i as follows: P DP outage, i = probability(| ˆ ⁄ i |>” Õ - - -| ¯ ⁄ i |Æ ÷ i ), P IP outage, i = probability(| ˆ ⁄ i |>” Õ - - -| ¯ ⁄ i |>÷ i ). (3.36) where ” Õ together with ÷ are the parameters to control the localization accuracy. There is a tradeo of choosing the values of ” Õ and ÷ : with larger values of either ” Õ or ÷ , IPs are more likely to be considered to be in “safe region”, and subsequently used for localization, however, the localization accuracy is decreased by taking more IPs. 3.5 TDAJ Scheme with Noisy DOD, DOA and TOA In this section, we discuss the computation of threshold ” i given a required PF. Then, the procedures of TDAJ is presented. According to (3.23), if we perfectly know the value of f DP ⁄ i , the value of ” i can be reversely computed to achieve a given PF. However, f DP ⁄ i depends on f µ 1 (•), f µ 2 (•) and f µ 3 (•) which are known and the values of W i,1 , W i,2 and W i,3 which 67 Table 3.2: Simulation Parameters P dB t Target Location Reflector Location Scenario 1 1dBm fixed at [5.2, 10] [4.8, 9.8] Scenario 2 1dBm fixed at [5.2, 10] Uniformly distributed at xœ [≠ 8,8] u yœ [0,10] Scenario 3 1dBm The target and one reflector are uniformly distributed at xœ [≠ 8,8] u yœ [0,10] Scenario 4 25 dBm The target and two reflectors are uniformly distributed at xœ [≠ 8,8] u yœ [0,10] Scenario 5 Changing The target and one reflector are uniformly distributed at xœ [≠ 8,8] u yœ [0,10] are not accurately known. Thus, the values of W i,1 , W i,2 and W i,3 have to be estimated based on the measurements of DOA ˆ µ 1 ,DOD ˆ µ 2 and TOA ˆ µ 3 . The procedure of TDAJ is listed in Algorithm 2: Algorithm 2 DOD, DOA and TOA joint (TDAJ) IP Detection Scheme 1. Compute the three intersections [ˆ x 1 ,ˆ y 1 ], [ˆ x 2 ,ˆ y 2 ] and [ˆ x 3 ,ˆ y 3 ] from DOA, DOD and TOA measurements ˆ µ 1 , ˆ µ 2 and ˆ µ 3 . 2. Compute the matrix W based on (3.11) and (3.14) using the values of [ˆ x 1 ,ˆ y 1 ], [ˆ x 2 ,ˆ y 2 ] and [ˆ x 3 ,ˆ y 3 ] from Step 1. 3. Then, calculate the distribution of f DP ⁄ i (z) according to (3.20) - (3.23). 4. Reversely compute the value of ” i given P f by (3.27). 5. Compare the value of six dierence observations of ⁄ i from (3.4) with threshold ” i ,(i=1,2,...,6) and perform IP detection based on individual rule (3.6) or fusion rule (3.7). 3.6 Simulation Results In this section, extensive simulations are employed to verify the theoretical results, evaluate the performance of TDAJ algorithms and test the improvement of localization accuracy by IP detection. 68 3.6.1 Simulation Description There are 6 scenarios considered for the simulation, and the parameters of them are shown in Table 3.2. In scenario 1, the radio propagation path is designed as transmitter-reflector-target-receiver; in scenario 2 and 3, the sequence of signal arrival at target/reflectors is random: the signal propagation path is either transmitter-target-reflector-receiver or transmitter-reflector-target-receiver with equal probability; in scenario 4 and 5, the signal propagation path is transmitter-object1-object2-object3-receiver, and the target and two receivers are randomly assigned to the three objects. Inallscenarios, thetransmitterandreceiverareat [≠ 2.5,0]and [2.5,0],respec- tively. In this chapter, we assume the locations of all objects (transmitter, target, etc.) areinmeters. Themeasurementerrorsforµ i areGaussian, and thestandard deviation of them (‡ i , i=1,2,3) are given in meters or degrees, respectively. The values of ‡ i , which are dependent of the target and reflector locations, are chosen as follows. First, we employ a signal power loss model commonly used in the radar com- munity [84, 36]. We assume the noise power spectral density is≠ 174 dBm/Hz, the carrier frequency is 3 GHz, and the bandwidth of the signal is 1 GHz. The noise power is P dB n =≠ 174+10log 10 (10 9 )=≠ 84 dBm. Assuming that the wireless sig- nal is reflected by the target and H non-target reflectors, the signal transmission path is divided into H+2 sub-paths 2 . As an example shown in Fig. 3.1, there are two sub-paths p 1 and p 2 for a DP, and three sub-paths p 1 , p 3 and p 4 for an IP. We let p h be the length of the hth (h=1,2,...,H+2) sub-path. The free space path loss through hth sub-path is evaluated by 2 We assume the signal is only reflected once by target and non-target reflectors 69 L dB h =≠ 10log 10 (4fip 2 h ) LetthetransmittedsignalpowerindBbeP dB t , thesignal-to-noiseradio(SNR) in dB may be evaluated as follows [84, 36]: SNR = P dB t ≠ P dB n +G dB TX +10log 10 (A RX )+ H+2 ÿ h=1 L dB h +10log 10 (R t R H r ) = P dB t +88≠ ÿ h 10log 10 (4fip 2 h )≠ 0.22H, (3.37) where G dB TX =15 dB is the transmitter gain in dB, A RX =0.0796 is the receiver antenna aperture, R t =1 m 2 is the radio cross section (RCS) for the target, and R r =0.95 m 2 is the RCS for any non-target reflectors. Second, we model the standard deviations of DOA (‡ 1 ), DOD (‡ 2 ) to be pro- portional to the square root of the Cramer Rao Lower Bounds (CRLBs) [52, 1] : ‡ 1 = 1 2 Û 6 SNRN ar (N 2 ar ≠ 1) 1 sin ¯ µ 1 = 0.05 sin ¯ µ 1 Ô SNR , (3.38) ‡ 2 = 1 2 Û 6 SNRN ar (N 2 ar ≠ 1) 1 sin ¯ µ 2 = 0.05 sin ¯ µ 2 Ô SNR . (3.39) where ¯ µ 1 and ¯ µ 2 are the mean values of DOA and DOD. The standard deviation of the TOA estimate ‡ 3 is assumed to be [15] ‡ 3 = È fi Ô SNR = 0.038 Ô SNR . (3.40) To test whether our model of ‡ i is consistent with practical results [80], we consider a DP with a target at [7.8,14] and the transmitter power P dB t =1 dBm. Then we can compute SNR=18.71 dB, ‡ 1 =0.36 degree, ‡ 2 =0.41 degree and 70 Table 3.3: P f and P d of individual rules based on ⁄ 1 , ⁄ 3 , ⁄ 5 for Scenario 1 Expected P f 0.05 0.1 0.3 0.5 0.9 0.95 ⁄ 1 Simulated P f 0.050 0.100 0.296 0.495 0.898 0.949 P d by (3.33) 0.570 0.689 0.865 0.931 0.990 0.995 Simulated P d 0.576 0.700 0.876 0.938 0.991 0.995 ⁄ 3 Simulated P f 0.049 0.100 0.297 0.495 0.898 0.949 P d by (3.33) 0.542 0.663 0.849 0.921 0.988 0.994 Simulated P d 0.549 0.683 0.872 0.937 0.991 0.995 ⁄ 5 Simulated P f 0.050 0.100 0.296 0.496 0.898 0.949 P d by (3.33) 0.593 0.709 0.877 0.938 0.991 0.995 Simulated P d 0.595 0.711 0.878 0.939 0.991 0.995 ‡ 3 =0.0044 meter. The values of ‡ i are consistent with the results in [72, 74, 80]. Inthesimulationofeveryscenario, 2ú 10 5 samplesarecollected. Foreverysample, the estimates of µ i (i=1,2,3) are Gaussian random variables with ‡ i computed according to (3.38) - (3.40). 3.6.2 Comparison between theoretical and simulated PF and PD The performances TDAJ for Scenario 1 based on ⁄ 1 , ⁄ 3 , ⁄ 5 are listed in Table 3.3. According to the definition in (3.6), PF is computed as the probability of claiming an IP when there are no reflectors and PD is computed as the probability of claiming an IP when there are reflectors. The expected PF, PD computed by (3.33), PF and PD by simulations are shown. From the results, we can observe that: • The TDAJ scheme based on ⁄ i can closely approximate the expected PF, • The closed-form evaluation of PD by (3.33) is consistent with PD obtained from simulations. 71 10 −2 10 −1 10 0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 P f by Simulation P d by Simulation λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 Combined K=0 Combined K=1 Combined K=2 Combined K=3 Combined K=4 Combined K=5 Figure 3.5: P d vs P f in Scenario 2 10 −2 10 −1 10 0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 P f by Simulation P d by Simulation λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 Combined K=0 Combined K=1 Combined K=2 Combined K=3 Combined K=4 Combined K=5 Figure 3.6: P d vs P f in Scenario 3 72 10 −2 10 −1 10 0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 P f by Simulation P d by Simulation λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 Combined K=0 Combined K=1 Combined K=2 Combined K=3 Combined K=4 Combined K=5 Figure 3.7: P d vs P f in Scenario 4 3.6.3 Comparison of IP Detection Performance by Dier- ent ⁄ i The simulated PD vs. PF curves for Scenario 2 - 4 are drawn in Fig. 3.5 to Fig. 3.7, respectively. It can be observed that • In common UWB localization systems (Scenario 2 - 4), the TDAJ can very accurately detect the IP, e.g., P f < 10% and P d > 90%. • The TDAJ scheme based on ⁄ 5 has the best performance among individual rules in the sense that it accurately achieves the expected PF and maximizes thePD.Thisisbecause⁄ 5 hasfewerBSsthanothers, asdiscussedinSection IV. C. InTable3.4, welistthethePFandPDoftheindividualrule⁄ 5 andfusionrule (FR) with K=0 and K=5 for scenario 2, which shows that FR with K=0 and 73 Table 3.4: P f and P d of for Scenario 2 Expected P f 0.03 0.04 0.05 0075 0.1 P f ⁄ 5 0.0294 0.0397 0.0496 0.0691 0.0998 FR (K=0) 0.0334 0.0439 0.0544 0.0743 0.1049 FR (K=5) 0.0260 0.0357 0.0452 0.0640 0.0938 P d ⁄ 5 0.9002 0.9051 0.9095 0.9164 0.9234 FR (K=0) 0.9237 0.9272 0.9302 0.9348 0.9400 FR (K=5) 0.5657 0.5799 0.5933 0.6157 0.6499 K=5 achieve a little higher and lower PF than the expected one, while individual rule based on ⁄ 5 accurately achieves the expected PF. This is because when the model for the ⁄ i is imperfect (the reasons for the imperfect model are analyzed in Section VI. D), the decisions from the dierent ⁄ i are not perfectly correlated. Consequently, we might obtain a false alarm from some ⁄ i but not others. In that case, we can conclude that a “fusion rule” with K =0 has a larger P f than a decision rule based on a single ⁄ i , since any ⁄ i deciding for aP f leads to an overall decision for P f . Conversely, a fusion rule with K =5 will lead to a smaller P f than the individual decision rules, since every ⁄ i must claim a P f . 3.6.4 PF and PD vs. Transmitted Signal Power In Scenario 5, we study the impact of transmitted signal power P dB t . High P dB t leads to smaller standard deviation of DOA, DOD and TOA estimates, and consequently better IP detection performance. The simulated PD and PF vs. P dB t are shown in Fig. 3.8, with expected PF=5%. From Scenario 5, we can observe that an increasing transmitted signal power improves the TDAJ performances: closer to expected PF and higher PD. In terms of PF, TDAJ based on ⁄ 5 is less impacted by the increase of the error level than the Fusion Rule (FL) with K=0 and K=1. Even with P dB t =≠ 5 dBm, TDAJ based on ⁄ 5 can achieve P f =5% and P d > 85%. 74 −30 −25 −20 −15 −10 −5 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Scenario 5 Transmitted Signal Power P t dB (dBm) P f and P d by Simulation P f by λ 1 P f by λ 5 P f Fusion Rule with K=0 P f Fusion Rule with K=1 P d by λ 1 P d by λ 5 P d Fusion Rule with K=0 P d Fusion Rule with K=1 Figure 3.8: P d and P f vs transmitted signal power P dB t (dBm) in Scenario 5 ThemainreasonsforinaccuratePFanddecreasingPDwithhighleveloferrors are: 1. The linear relations between x, y and µ 1 , µ 2 and µ 3 (3.9) are not valid. 2. The threshold ” calculated based on noisy measurements µ 1 , µ 2 and µ 3 is not accurate. 3.6.5 Improvement of Localization by IP Detection In this section, we examine the improvement of localization accuracy by the IP detection. For the localization, the final estimate of the target location is obtained based on the maximum likelihood (ML) estimation. The ML estimation isperformedbyaQuasi-Gaussianiterativemethod[65],withtheinitialvaluebeing the average of the three intersections (DOA/DOD, DOA/TOA and DOD/TOA). 75 Table 3.5: Localization RMSE Comparison Target Location [≠ 1,5] [≠ 2,6] [3,7] [≠ 3,4.6] RMSE without IP Detection 4.2575 5.4581 4.2980 4.6665 RMSE with IP Detection 0.5602 0.8152 0.7863 0.5179 PF 0.0504 0.0498 0.0506 0.0496 PD 0.9832 0.9806 0.9760 0.9797 In the simulation, we consider the problem of localization in an indoor envi- ronment with range xœ [≠ 5,5] u yœ [0,8]. There are totally 2ú 10 5 samples. The errorsofDOA,DODandTOAestimatesaregeneratedbasedonthemodelin(3.37) - (3.40). In 10 5 samples, there is no reflector, and in the remaining 10 5 samples, there is one reflector uniformly distributed in the region x œ [≠ 5,5] u y œ [0,8]. The transmitter and receiver are at [≠ 2.5,0] and [2.5,0], respectively. The target location is shown in Table 3.5. Two localization strategies are simulated and com- pared, the first one is without IP detection but checking whether the average of three intersection [ ˆ x 1 +ˆ x 2 +ˆ x 3 3 , ˆ y 1 +ˆ y 2 +ˆ y 3 3 ] in the in the region xœ [≠ 5,5] u y œ [0,8]. If the average of three intersections is inside the region, for any triplet of DOD, DOA and TOA, the localization is performed. The second one is to first perform IP detection, and only when the triplet is considered to be DP based on the indi- vidual rule with ⁄ 5 , the localization is performed. The root mean square error (RMSE) is utilized to evaluate the localization accuracy, which is defined by RMSE = Û Mean 3 1 ˆ x RMSE ≠ ¯ x 2 2 + 1 ˆ y RMSE ≠ ¯ y 2 2 4 (3.41) where [¯ x,¯ y] is the true target location and [ˆ x RMSE ,ˆ y RMSE ] is the ML estimate of target location using the Quasi-Gaussian iterative method. Table 3.5 shows the values of RMSE without IP detection, RMSE with IP detection, PF and PD for several target locations. It can be observed that the IP detection can increase the localization accuracy by several orders of magnitude. 76 3.7 Conclusion We studied the problem of discriminating IPs and DPs for passive position- ing purposes, from a novel perspective. We utilize the TOA, DOD and DOA information together and test whether the parameter tuplet is consistent with the assumption of a DP. In particular, we test the dierences of the three intersec- tions from DOD, DOA and TOA measurements ⁄ i (i=1,2,...,6) for IP detection. The algorithm is based on Neyman-Pearson criteria to maximize the probability of detection given probability of false alarm. Several individual rules and fusion rules under this framework are discussed. The “blind spots” of individual rules are investigated and it is shown that ⁄ 5 has the smallest “blind spots”. Simula- tions show that the algorithm can eectively detect IPs when the TOA, DOD and DOAmeasurementsareavailablewithsmallerrors. Asverifiedbysimulations, the proposed method can be employed to sort out DPs to increase localization accu- racy. Furthermore, the identified DPs can also be used for other purposes such as generation of environmental maps. 77 3.8 Appendix: Proof of Theorem 1 Since in the case of DP, we have a j = a k j , b j = b k j and c j = c k j for j=1,2,3. With some manipulations, we can rewrite W in (3.14) as W = S W W W W W W W W W W W W W W W W W U ≠ c 1 b 2 – DP ≠ b 2 c 2 “ DP – DP — DP ≠ b 2 c 3 — DP a 2 c 1 – DP a 2 c 2 “ DP – DP — DP a 2 c 3 — DP b 1 c 1 — DP – DP “ DP b 1 c 2 – DP b 1 c 3 “ DP ≠ a 1 c 1 — DP – DP “ DP ≠ a 1 c 2 – DP ≠ a 1 c 3 “ DP ≠ b 3 c 1 “ DP ≠ b 3 c 2 — DP ≠ b 3 c 3 – DP — DP “ DP a 3 c 1 “ DP a 3 c 2 — DP a 3 c 3 – DP — DP “ DP T X X X X X X X X X X X X X X X X X V . (3.42) where – DP =a 1 b 2 ≠ a 2 b 1 , — DP =a 2 b 3 ≠ a 3 b 2 , “ DP =a 3 b 1 ≠ a 1 b 3 . Then, it follows that for i=1,2,...,6, W 1,i = ≠ b 2 a 2 W 2,i =≠ b 2 “ DP b 1 — DP W 3,i = b 2 “ DP a 1 — DP W 4,i (3.44) = b 2 “ DP b 3 – DP W 5,i =≠ b 2 “ DP a 3 – DP W 6,i . According to (3.13), ˆ ⁄ ˆ ⁄ˆ ⁄ DP =ÁÁÁ =W µ µ µ. where µ µ µ is defined in (3.15). Thus, we have 78 ˆ ⁄ DP 1 = ≠ b 2 a 2 ˆ ⁄ DP 2 =≠ b 2 “ DP b 1 — DP ˆ ⁄ DP 3 = b 2 “ DP a 1 — DP ˆ ⁄ DP 4 (3.45) = b 2 “ DP b 3 – DP ˆ ⁄ DP 5 =≠ b 2 “ DP a 3 – DP ˆ ⁄ DP 6 . which finishes the proof. 79 Chapter 4 Multiple Target Localization based on Time-of-Arrival Measurements: A Sequential Clustering Algorithm in Multi-path Environments Inthelasttwochapters,westudiedtheproblemofindividualtargetlocalization and IP detection based on channel sounding characteristics. For the IP detection problem, the proposed TDAJ algorithm is based on angle information. However, the angle information requires a multiple antenna systems, which does not fit low-cost application (such as sensor networks). Motivated by this observation, we considertheproblemofmulti-targetlocalizationinmulti-pathenvironmentsbased on solely on TOA measurements. In particular we address the case that multiple targets cannot be distinguished by their “signatures” or other unique characteristics. Further complications in realisticsettingsarisefromthefollowingfactors: (i)thedirectpathbetweentrans- mitter/receiver and target might be blocked, (ii) indirect paths (IPs) arising from multi path propagation might be present. We propose a novel algorithm that can accurately localize multiple targets even under these dicult circumstances. 80 This algorithm is based on an iterative approximation to an exhaustive maximum likelihood (EML) estimation of the target locations, yet avoiding the exponential complexity that an EML requires. It utilizes a clustering technique that gives the number and locations of targets, and incorporates an IP detection mechanism based on the likelihood of several TOA measurements correspond to direct paths (DPs) from the same target. Despite its lower complexity, our algorithm achieves almost the same performance as the EML in a variety of simulation settings. We finally study the choice of algorithm parameters and the configurations of the MIMO system nodes to achieve the best localization performance. The remainder of this chapter is organized as follows. Section 4.1 overviews the background of the considered problem. Section 4.2 presents the system model. The principle of the proposed method is discussed in details in Section 4.3. The simulation results are shown in Section 4.4, followed by the conclusions in Section 4.5. 4.1 Overview Various signal properties have been proposed as the basis for localization in wireless systems, such as received signal strength (RSS) [18] , Time-of-Arrival (TOA) measurement (TOA) [74], dierence of TOA [8] or angle information [96]. RSSexhibitslowaccuracy,inparticularinenvironmentswithstrongfadingand/or unknown pathless coecient. TOA measurement allows to accurately determine the distance between transceiver and target. TOA measurements can provide very accurate (up to millimeter level accuracy) distance information when signals with ultra-wide bandwidth (UWB) signals [20, 61, 75, 27, 15]. 81 Angular information is, of course, a mainstay of radar systems, and has been used eectively for 70 years [51]. The main diculty in obtaining angular infor- mation lies in the necessity of having a directionally resolving antenna, typically a mechanically or electronically steerable antenna or antenna array. When only simple, low-cost devices are to be used as wireless transceivers, such large and expensive antennas are undesirable. For this reason, this chapter concentrates on systems that only exploit TOA measurements, without having access to angular information. In recent years, the use of multiple-input multiple-output (MIMO) radar for target localization has received intensive attention [43, 44, 30, 21, 22, 88, 45, 85, 87,23,24]. Generally, MIMOradaremploysmultipletransmit/receiveantennasto excite/process the echoes from targets for localization. In contrast to traditional phased-array antennas, where each antenna element is excited by a phase-shifted version of the same signal, MIMO radars use independent signals to excite the transmit antennas. Consequently, MIMO radar can improve the performance of localization due to waveform diversity [43] or spatial diversity [30]. In [43], MIMO radar with co-located antennas was studied. [30] discussed the application and advantage of using spatially distributed antennas in MIMO radar systems. The optimum sensor placement and transmitted energy optimization were studied in [21] and [22], respectively. Ifonlyasingletargetispresent,itcanbelocalizedbasedonTOAsasmeasured at the receivers using maximum likelihood estimation [74]. However, in many practical situations there are multiple targets in the coverage area of the system, and Multi-target localization (MTL) is a very popular and challenging problem [88, 45, 87, 85, 23, 24, 18]. [88, 45] studied the bi-static radar scenario. [87] 82 introducedageneralized-likelihoodratiotest(GLRT)and[85]proposedSubspace- Based Method for MTL. [23, 24] proposed the maximum likelihood estimator for multiple target localization and tracking with colocated MIMO radars. In [18], a compressive sensing based MTL algorithm was proposed. Though MTL has been extensively studied, existing methods still have several critical drawbacks. Firstly, the existing MTL methods in [88, 45, 87, 85, 23, 24] assume that the transmitters or receivers have antenna arrays to obtain target directional information. The directional information makes the MTL problem much easier since multiple targets can be easily separated from their associated angle information. However, as mentioned above, antenna arrays are not desirable in many low-cost applications such as wireless sensor networks [77]. Secondly, the proposedmaximumlikelihoodmethodhasveryhighcomputationalcomplexityfor large number of targets, especially without knowledge of the number of targets. Finally,noneoftheexistingMTLmethods[88,45,87,85,23,24,18]consideredthe adverse impact of indirect-paths (IPs), which are due to surrounding non-target reflectors. The presence of IPs is the main challenge for localization in rich muti- pathenvironmentssuchasindoor[70,49,29]. ThoughtherearemanyIPdetection methods in the literature [93, 48, 91, 49, 47, 70], they suer from a dependence on the statistical knowledge of channel parameters. In this chapter, we consider a very general and practically relevant problem of MTL in distributed MIMO radar systems based on TOA measurements. In other words, we wish to localize passive (reflecting) objects by measuring the signal propagation delay measurement (equivalent to TOA) from multiple transmitters via the object(s) to the multiple receivers. We furthermore assume that the TOA measurements are the only input for the localization algorithm; specifically no angle-of-arrival information is available. In the case of multiple targets, each of 83 them gives rise to a dierent observed TOA measurement at each of the receivers. 1 On the other hand, unlike the active target scenario, the signals reflected by dif- ferent targets do not have unique signatures 2 . Thus, it is important to perform a “pairing” of TOAs that are consistent in the sense that the computed target locations explain the arising TOAs. The pairing task needs to be performed prop- erly by considering nonidealities of the measurements: (i) firstly, noise leads to uncertainties of the TOA. Secondly, due to the non-target reflectors, the signal propagation could take a detour via an additional (non-target) obstacle; this is known as an IP. In other words, an IP is detrimental for localization if the direct path transmitter-target-receiver is blocked, while MPCs involving a target and other reflecting objects can be observed. The IP then introduces a positive bias to the TOA. It is also possible that no signal arrives at the receiver at all, if the blockage of the target from all possible signal paths is suciently strong. Finally, it is possible that the TOAs from two targets coincide. 3 All of these issues make the localization of multiple targets extremely challenging. To solve the problems, we propose an algorithm for MTL problems with IPs based on the following observation (see Fig. 4.1): for a two dimensional sce- nario, every TOA measurement of a transmitter-receiver pair (TRP) geometrically denotes an ellipse. The ellipses corresponding to direct paths (DPs) reflected by the same target will intersect at one point, which is the target location. On the other hand, if an TOA is estimated from an IP or is from other targets, the ellipse corresponding to it will not include the true target location, and a ghost targets 1 strictly speaking, each target gives rise to multiple echoes (multi-path components, MPCs), which can interpreted as the “impulse response" of this particular target. 2 For example, RFID nodes provide a unique identification of targets, and thus do not require the method discussed in this thesis 3 The case in which such coincidence is prohibited by assumption is treated in our conference paper [68]. 84 Receiver 1 Receiver 2 Target 1 Transmitte 1 Ellipse corresponding to IP (of target 1) or DP of target 2 Transmitte 2 Ghost Target Target 2 Figure 4.1: Analysis of Sources of Localization Errors (in comparison to true targets) might be estimated using the IPs (dashed ellipse in Fig. 4.1). Since the TOAs suer errors due to noise, the proposed algorithm performs the IP detection based on the likelihood that several TOAs are DPs reflected by the same targets. Thus, the proposed algorithm is essentially a maximum likelihood estimation method. However, the computation complexity of an exhaustive maxi- mumlikelihood(EML)methodisveryhigh, asithastocomputethelikelihoodfor all possible combinations of TOAs from all TRPs, , which has exponential com- plexity. To circumvent this problem, we employ a sequential matching process: when processing the i-th TRP, the proposed algorithm first estimates potential target locations based on the first i≠ 1 TRPs, and then matches the TOAs of the i-thTRPwiththeestimatedtargetlocationsandtheTOAmeasurementsofprevi- ous i≠ 1th TRPs. As confirmed by our simulation results, the algorithm possesses 85 Receiver 1 Receiver 3 Target 1 Target 2 Transmitter 2 Indirect Path Blockage Transmitter 1 Receiver 2 Reflector Figure 4.2: System Model almost the same performance as the EML, but has much lower computation com- plexity. It is verified by simulations that the localization root mean square error (RMSE) can be smaller than 0.1 meter with 95% probability when the standard deviation of TOA measurement (actually the signal propagation length according to the TOA measurement) is 0.05 meter. Moreover, we studied the transmitter and receiver configuration in MIMO radar systems to achieve the best localization performance: it is the best to have half of the available nodes as transmitters, and the remainder as receivers, consistent with existing MIMO radar systems. 4.2 System Model For simplicity, we consider a two dimensional scenario, though extension to three dimensions is straightforward. Fig. 4.2 shows the model: there areM t trans- mitters and M r receivers, and M tar targets to be localized. The total number of transmitter- receiver pairs (TRPs) is M r M t ; we denote the location of transmitter and receiver for theith TRP are [c i ,d i ] and [a i ,b i ], respectively. Number and loca- tion of TX and RX are known, while number and location of the targets is to be 86 estimated. The location of the wth target is written as [x w ,y w ]. The transmitters are synchronized with the receivers, so that every TRP is able to measure the TOA of each multi-path component (MPC). Assuming orthogonal transmit sig- nals, the receivers can dierentiate signals from dierent transmitters. However, the receivers do not know which MPC is reflected by which target. The signal propagation path from a transmitter to a receiver might be one of the following (see Fig. 4.2): 1. Direct Path (DP): the signal propagates directly from the transmitter via the target to the receiver. 2. Indirect Path (IP): the signal is reflected not only by the target, but also by other objects on its way from the transmitter to the receiver. BoththeDPandIPsmightbeblockedbysomeobjects,sothatthesignalsthrough thesepathsdonotarriveatreceivers. Throughoutthischapterwespeakof“block- age" only if the DP is blocked. We use Fig. 4.1 to analyze the sources of error for MTL. The TOA geograph- ically denotes an ellipse with the transmitter and receiver locations be the foci of the ellipse. The sum of the distances from any point on the ellipse to the two foci is the TOA measurement. For example, in Fig. 4.1, if the TOA corresponding to the dashed ellipse (IP) is used to estimate target 1, the estimated target location would be erroneous. In a practical scenario, the TOAs also contain errors due to noises. Thus, to achieve accurate localization of multiple targets, we need to combat the following three error sources: 1. Distance measurement errors due to system noise. 2. Positive bias of TOA due to IPs. 87 3. Errors due to wrongly matched impulse response of other targets. In summary, the assumptions for our model are: 1. The signal propagation path might be DP, blocked path or IP. One target could create both a DP and IPs at the same time, due to the non-target reflection. The same target might create DP/IP to one TRP, and be blocked with respect to another. 2. TheTOAmeasurementsarecorruptedbyzeromeanadditivewhiteGaussian noises. 3. If the arrival time of consecutive MPCs are close together, the receiver could not resolve them. In our further computations, we will assume that the threshold for resolvability is twice the standard deviation of TOA measure- ments, 4. Paths that do not involve the targets (clutter) are perfectly suppressed through background cancellation techniques [51]. 4.3 Principle of the Sequential Clustering Algo- rithm In this section, we present the proposed sequential clustering algorithm to jointly estimate multiple target locations, with the consideration of IP detection. Before moving forward, we define the transmitter-receiver multi-path compo- nent (TRMPC) be the MPC between a transmitter and a receiver. Note that using orthogonal signals, the receivers are able to dierentiate signals from dier- ent transmitters [44]. However, the signal from each transmitter could be reflected 88 Step 1: Select Three Tx-Rx pairs Are there unprocessed Tx-Rx pairs? Step 2 : Match the RLMs of the first three Tx- Rx pairs , i=4 Step 3a : Match RLMs of the ith TRP with existing potential target locations Yes Step 4: Choose the final estimate of target locations Step 3b : Match RLMs of the ith TRP with RLMs of processed TRPs Yes Yes Figure 4.3: Flow chart of proposed algorithm. by multiple targets, and non-target objects, creating many TRMPCs. As shown in Fig. 4.2, each TRMPC geographically corresponds to an ellipse. Only if the TRMPC corresponds to a DP of a target, the corresponding ellipse could cross the target location. The objective of the proposed method is to estimate the target locations based on these intersections, with high accuracy and low complexity. Given the errors of the TOA measurements, even the ellipses corresponding to DPs of the same target might not intersect at the same location. Therefore, we design a quantity to measure the likelihood that the intersection of several DPs correspond to the same target. Only if the likelihood is above a threshold does the algorithm decide that the DPs correspond to the same target. Assume that there areM r ú M t TRPs andM tar targets. Consider the simplest situation without any IPs or blockage, so that each TRP receives M tar TOA measurements. Then, the EML method would compare the likelihood of all the possible combinations 89 of TOA measurements of all TRPs, which requires M MrMt tar likelihood evaluations. Since the computation complexity is exponential in M r and M t , this approach is not suitable for real time localization applications. ToreducethecomplexityoftheELM,weproposeanovelalgorithmthatworks as follows (see Fig. 4.3; a summary of the notation is given in Table 4.1): it starts by selecting three Tx-Rx pairs (TRP) and pairing the TRMPCs from them, thus obtaining preliminary target locations. Additional TRPs are added one by one, eitherrefiningtheestimatesorgivingrisetoadditionalpossibletargetlocations. In theend,foranytargetlocationestimatedenotedby[x,y],iftheproposedalgorithm considers equal or more than M r M t ≠ +1 TRPs having TOA measurements corresponding to DPs of [x,y], [x,y] is considered to be a valid estimate. Consider, concretely, the situation after processing the first ith TRPs: the algorithm has obtained some potential target location estimates (e.g., after the first 3 TRPs, we have estimates of locations that are consistent with those TRPs. We then check every TOA measurement of the i+1th TRP. If a TOA measure- ment “matches” a possible target location, we note that fact, as it increases the likelihood that the estimateTOA measurementd target location is a “true” one. 4 Clearly,weareveryconfidentthatpreliminarytargetlocationsthatfitwithaTOA measurement of all already-tested TRPs has a high likelihood of being a “true” target location. However, we also allow for the possibility that some of the tested TRPs do not match a TOA measurement even for a true target location. The reason for this to occur can be, e.g., due to blockage. Thus, when we investigate a TOA measurement in the i+1 TRP, we also have to test whether - in conjunction withMRLsofthefirstiTRP,amatchingcanbeachieved; inotherwords, whether 4 Throughout this chapter, “a TOA measurement matches a target location” means that TOA measurement corresponds to the DP of the target. 90 Table 4.1: Notation [c i ,d i ], the location of the transmitter in the ith TRP [a i ,b i ], the location of the receiver in the ith TRP [x w ,y w ], the location of the wth target M r , the number of receivers M tar , the number of targets M t , the number of transmitters N i , the number of TRMPCs from the ith TRP ˆ d i ki , the time-of-arrival measurements for the kth TRMPC at the ith TRP k i , index of the kth TOA measurement for the ith TRP [ˆ x,ˆ y](k 1 ,k 2 ,...,k i ) an estimated target location from [ ˆ d 1 k1 , ˆ d 2 k2 ,..., ˆ d i ki ] l ki k1,k2,...,ki≠ 1 , the distance from [ˆ x,ˆ y](k 1 ,k 2 ,...,k i≠ 1 ) to ˆ d i ki B i , the matching matrix for the k i th TOA measurement after processing the ith TRP two TOA measurements among the first i TRPs, together with the TRP from the i+1thTRP,providesaconsistentestimatesuchthatthreeellipsesintersectwithin one location. Note that when we make a matching decision between a TOA measurement ˆ d i+1 k i+1 for one potential target [ˆ x,ˆ y](k 1 ,k 2 ,...,k i ), we do not need to check the con- sistency with any subset of [ ˆ d 1 k 1 , ˆ d 2 k 2 ,..., ˆ d i k i ], which saves computational eort. A similarargumentholdswhenwedo notmatchaTOAmeasurementtoaparticular target location. Finally, we note that while we allow a potential target to not be matched to TOA measurements from a few TRPs, there is a limit to how many “missed”weallow(thatnumberdependsontheblockageprobabilityandtheprob- ability of miss detection; see below). Consequently, we can eliminate prospective target locations that have too few “matching” TOA measurements even before processing all TRPs. This is the second eect that limits the computational eort of the method. We will now discuss the details of the four steps of the proposed algorithm. 91 4.3.1 Step 1: Choose the first three TRPs. Since our algorithm finds an approximate solution to an NP-hard problem, it might suer from convergence to a local minimum; thus the initialization, i.e., the choice of the initial three TRPs, is very important. Convergence is helped if the firstthreeTRPsprovideasaccurateandcompleteestimatesofthepotentialtarget locations as possible. We argue that choosing TRPs corresponding to ellipses with low eccentricity is useful 5 . This is based on the observation that if the eccentricity of an ellipse is large, it has higher possibility to become larger than 1 (a hyperbola) because of noise-induced errors, and is also more prone to “grazing angle intersections, which in turn create larger errors. Specifically, for the i-th TRP, there are N i TOA measurements 6 based on the TRMPCs. The k i -th (1Æ k i Æ N i ) TOA measurement of the i-th TRP has eccentricity ÷ k i . Then, the largest eccentricity of all TOA measurements for the i-thTRPis÷ i max = max(÷ k i )(1Æ k i Æ N i ). Then, wesort÷ i max inascendingorder, to be [÷ i 1 max ,÷ i 2 max ,...,÷ i MrM t max ]. The first three TRPs chosen in step 1 are those with indices i 1 , i 2 and i 3 . 4.3.2 Step 2: Match TOA measurements of the first three TRPs. The basic idea of this step is that if the TOA measurements of the three TRPs arefromDPsofthesametarget,thenthecorrespondingellipseswillintersectatthe same point. Since the TOA measurements are corrupted by errors, the likelihood 5 The eccentricity is the ratio of the distance between the foci and the major axis of an ellipse. In other words, the flatter the ellipse, the larger the eccentricity. 6 For simplicity, we use k the TOA measurement to denote the TOA measurement for the kth TRMPC. 92 of three TOA measurements is computed and compared with a threshold to decide whether they correspond to the same target (Step 6 of Algorithm 1). Specifically, let l k 3 k 1 ,k 2 be the quantity to evaluate the likelihood that the TOA measurement ˆ d 1 k 1 (from the first TRP), ˆ d 2 k 2 (from the second TRP) and ˆ d 3 k 3 (from the third TRP) are created by DPs from the same target. ‡ l k 3 k 1 ,k 2 is the standard deviation of the l k 3 k 1 ,k 2 . l k 3 k 1 ,k 2 and ‡ l k 3 k 1 ,k 2 are computed in Appendix A. Only if l k 3 k 1 ,k 2 /‡ l k 3 k 1 ,k 2 is smaller than a threshold ” , we consider the TRMPCs corresponding to the three TOA measurements are DPs from the same target. AfteralltheTOAmeasurementsofthethirdTRPhasbeenprocessed,amatch- ingmatrixisconstructed: eachrowofB 3 containsatripleofindices(oneeachfrom TRP 1, TRP 2, and TRP 3) of TOA measurements that are consistent 7 with each other. Based on the TOA measurements with indices in each row ofB 3 , we could estimate the target location based on [ ˆ d 1 k 1 , ˆ d 2 k 2 , ˆ d i k 3 ] using, e.g., the method of [74] or Gauss-Newton iterative method. We will generalize the matching matrix in the subsequent sections to B i , describing the matching after processing of the i-th TRP. We can generally define that the entry on the s i th row, uth column (1 Æ u Æ i)of B i is the index of TOA measurement at the uth TRP, and the s i th row contains the indices of TOA measurements from dierent TRPs corresponding to the same target. We note that after this matching step, not all TOA measurements will be assigned to potential target locations. Firstly, some of the TOA measurements might be IPs, i.e., MPCs that do not correspond to a direct transmitter-target- receiver path. Secondly, even an TOA measurement corresponding to a DP might not be assigned to a target if the DP of this target with respect to one of the other 7 Here, “consistent” means that the corresponding l k3 k1,k2 /‡ l k 3 k 1 ,k 2 is smaller than the threshold ” . 93 two TRPs is blocked - in this case, the (true) target location is not intersected by three ellipses. Thirdly, we have to account for the possibility that the noise is so large that an ellipse does not intersect the ellipses from the other TRPs even though it corresponds to the same direct path. For these reasons, it is necessary to test TOA measurements of the i-th TRPs withi> 3 for possible consistency even with those target locations that are not consistent among the first three TRPs; details will be described in Step 3. We also note that a TOA measurement can be associated with multiple target locations. This is so because MPCs associated with dierent targets might have unresolvablearrivaltimes; arealisticassumptioninparticularwhenthebandwidth of the considered system is low. Note that our previous conference paper [68] dealt with dierent assumptions, namely the absence of MPCs and the full resolvability of TOA measurements. Algorithm 3 Constructing Matching Matrix for Three TRPs 1: for k 3 =1 to N 3 do 2: j=0 3: B 3 be empty 4: for k 1 =1 to N 1 do 5: for k 2 =1 to N 2 do 6: if l k 3 k 1 ,k 2 /‡ l k 3 k 1 ,k 2 <” then 7: j=j+1 8: Row(B 3 ) j =[k 1 ,k 2 ,k 3 ] 9: end if 10: end for 11: end for 12: end for 4.3.3 Step 3: Match TOA measurements of the next TRP This section analyzes the procedure to match the TOA measurements of the ith TRP (i Ø 4). The basic idea is to match the TOA measurements ˆ d i k i (k i = 94 1,2,...,N i ) with the existing potential target locations (associated with matching matrix), and also the TOA measurements of the previous TRPs, as shown in Algorithm 2. The matching process of a TOA measurement of the ith TRP consists of (i) matchingwith the previous matching matrixB i≠ 1 (Step 4- 10 of Algorithm 2) and (ii) matching with (previously unmatched) TOA measurements of previous TRPs (Step 11 - 19 of Algorithm 2): • Match with previous matching matrix B i≠ 1 : for every TOA measurement d i k i of the ith TRP, compute the likelihood that d i k i and TOA measurements with indices in Row(B i≠ 1 ) j 2 are direct paths from the same target. The likelihood is evaluated by l k i Row(B i≠ 1 ) j 2 /‡ l k i Row(B i≠ 1 ) j 2 , where ‡ l k i Row(B i≠ 1 ) j 2 is the standard deviation of l k i Row(B i≠ 1 ) j 2 . l k i Row(B i≠ 1 ) j 2 are ‡ l k i Row(B i≠ 1 ) j 2 are computed in Appendix B. If it is smaller than a threshold, add the set of TOA measure- ments [Row(B i≠ 1 ) j 2 ,k i ]] into the matching matrixB i . • Match with two TOA measurements of two TRPs: for the ithe TRP, choose the u 1 th and u 2 th TRPs (1Æ u 1 ,u 2 Æ i≠ 1 and u 1 ”= u 2 ). For three TOA measurements of these three TRPs, denoted by ˆ d i k i , ˆ d u 1 ku 1 and ˆ d u 2 ku 2 , compute the a matching matrix B i of the ith TRP. This is because of the possibility that the targets are not detected in the previousi≠ 1 TRPs, due to blockage and noises. In Algorithm 2, there is an important parameter , which controls the tradeo between the probability of miss detection, and the overall computational complex- ity. Here means that for M r M t TRPs, only if there are more than M r M t ≠ TRPs having TOA measurements considered to correspond to DPs of a target, this target is considered to be a valid estimate. can also be interpreted as the 95 upper bound of the zero entries of the rows of matching matrixB i : the algorithm finally considers a row (corresponding to a target location estimate) a valid one if the number of zeros in this row is smaller than . This value can be related to statistics of the propagation environment as follows. Let p b be the blockage probability for the DPs of a target [x.y], the probability that there are equal or less than M r M t ≠ TRPs having un-blocked DPs of [x,y] is p ub () = MrMt≠ ÿ i=0 A M t M r i B (1≠ p b ) i p MrMt≠ i b (4.1) Then, the actual probability of miss detection for target [x,y] can be evaluated as P m =p ub ()+ p 1 ≠ p 2 . (4.2) where p 1 is the increase of probability of miss detection due to the noise induced errorsofTOAmeasurements: iftheerrorsarelarge, theproposedalgorithmwould consider the number of TPRs having DPs of [x,y] is equal or less than M r M t ≠ , even though it is actually more than M r M t ≠ . p 2 is the decrease of probability of missed detection due to the IPs: for N Õ dp (N Õ dp Æ M r M t ≠ ) DPs of target [x,y], if there are N Õ ip IPs with the corresponding ellipses happening to be close to [x,y], the proposed algorithm would consider the total number of DPs of [x,y] to be N Õ dp +Nip Õ . Then, if N Õ dp +N Õ ip Ø M r M t ≠ +1 , [x,y] would still be considered to be a valid estimate. Therefore, p ub ()+ p 1 is an upper bound bound of P m . To obtain a closed-form formulation of P m , we consider this upper bound, i.e., we ignore the impact of in IPs (p 2 ¥ 0), let p r um be the probability that a TOA measurement of DP is either blocked or not considered to be a DP of a target. According to Algorithms 1 and 2, for an unblocked DP, the probability that a 96 TOAmeasurementcorrespondingtoitnotbeconsideredtobeDPisapproximately 2ú Q(” ), where Q(” )= 1 Ô 2fi s Œ ” exp 3 ≠ u 2 2 4 du. Then, p r um =p b +(1≠ p b )(2ú Q(” )). Then, the probability of miss detection is P m = MrMt≠ ÿ i=0 A M t M r i B (1≠ p r um ) i (p r um ) MrMt≠ i (4.3) InSectionIV,wecomparethetheoreticalevaluationofP m withthetruemissed detection, and show that Eqn. (4.3) indeed is a good approximate to the true P m . Therefore, is decided by the probability of missed detection given p b and ” . For example, given p b =0.1, ” = Ô 10, M t =3 and M r =4, P m =0.01 requires Ø 5. To reduce the computational complexity given a required P m , when processing the ith TRP, only if iÆ +2 (Step 11 of Algorithm 2), every TOA measurement of the ith TRP needs to be checked with every two TOA measurements of two dierent previous TRPs. For the remaining +3 to M r M t - th TRPs, every TOA measurement is only checked with already estimated target locations,andnotwithwitheverytwoTOAmeasurementsoftwodierentprevious TRPs. This is because even if there is a new estimated target location at the ith TRP when iØ +3 , the target could finally only have maximum M r M t ≠ DPs, and will finally be dropped given the threshold . In summary, the first+2 TRPs are used to find all the possible target locations, and the rest TRPs are used to refine the estimates. Given the definition of , after the matching matrix B i generation is com- pleted, if any row of B i already contains equal to or more than zeros, this row is deleted. This also reduces the computational eort of matching this row with TOA measurements of the remaining TRPs. AftertheithTRPhasbeenprocessed,thei+1thTRPisthenprocessedaccord- ing to the same procedures of Algorithm 2, until every TRP has been processed. If 97 the TOA measurements of several TRPs correspond to DPs from the same target location, their corresponding indices are likely to be included in the rows of the matching matrix. However, if the TOA measurements do not correspond to DPs of the same target, the likelihood of being the DPs of the same target is very likely tobesmallerthanthethreshold. Evenifthelikelihoodislargerthanthethreshold and their indices are in a row of matching matrix, with more TRPs processed, this row will have more zeros, and finally be dropped when the number of zeros of this row is equal or larger than . 98 Algorithm 4 Matching the TOA measurements of the ith TRP 1: j 1 =0 2: R(j 2 )= [0, 0, ..., 0] % Initialize Vector R to be a vector with zeros. 3: for k i =1 to N i do %2.1Match d i k i with Row(B i≠ 1 ) 4: for j 2 =1 to the number of rows of B i≠ 1 do 5: if l k i Row(B i≠ 1 ) j 2 /‡ l k i Row(B i≠ 1 ) j 2 <” then 6: j 1 =j 1 +1 7: Row(B i ) j 1 =[Row(B i≠ 1 ) j 2 ,k i ] 8: R(j 2 )=1 9: end if 10: end for %2.2If iÆ +2 ,match d i k i with two TOA measurements of previous two TRPs 11: if iÆ +2 then 12: for any 1Æ u 1 <u 2 Æ i≠ 1 do 13: if l k i ku 1 ,ku 2 <” then 14: j 1 =j 1 +1; 15: Row(B i ) j 1 =[0,0,...,0]; 16: B i (j 1 ,u 1 )=ku 1 and B i (j 1 ,u 2 )=ku 2 and B i (j 1 ,i)=k i ; 17: end if 18: end for 19: end if 20: end for %2.3Addunmatchedrowsof B i≠ 1 21: for j 2 =1 to the number of rows of B i≠ 1 do 22: if R(j 2 )”=1 then 23: j 1 =j 1 +1 24: Row(B i ) j 1 =[Row(B i≠ 1 ) j 2 ,0]%Addtheunmatchedrowsof B i≠ 1 to B i 25: end if 26: end for %2.4Deletetherowsof B i with equal or more than zeros 27: for j 1 =1 to the number of rows of B i do 28: if number of zeros in Row(B i ) j 1 is equal or larger than then 29: Delete Row(B i ) j 1 from B i 30: end if 31: end for 99 4.3.4 Step 4: Choose the final target location estimates based on the threhsold ” and After the last TRP is processed, there might be several rows of the matching matrix B MtMr , and each row corresponds to a potential target locations. The algorithm makes the final decisions of the target location estimates based on the thresholds ” and . After finishing all the TRP, assume that the number of rows ofB MtMr is M Õ tar , and the qth (1Æ qÆ M Õ tar )rowofB MtMr is [k 1 ,k 2 ,...,k MtMr ]. Then, we could esti- mate the target location [ˆ x,ˆ y](k 1 ,k 2 ,...,k MtMr ) based on the TOA measurements [ ˆ d 1 k 1 , ˆ d 2 k 2 ,..., ˆ d MtMr k M t Mr ] using Gauss-Newton iterative method or [74]. For notational simplicity, we let [ˆ x q ,ˆ y q ] denote the estimated target location from the qth row of B MtMr .Foreach [ˆ x q ,ˆ y q ], if k i ”=0 ( ˆ d i k i contributes to the estimate of [ˆ x q ,ˆ y q ]), we can compute a run length ˜ d i k i = Ò (ˆ x q ≠ c i ) 2 +(ˆ y q ≠ d i ) 2 + Ò (ˆ x q ≠ a i ) 2 +(ˆ y q ≠ b i ) 2 . (4.4) where [c i ,d i ] and [a i ,b i ] are the locations of transmitter and receiver in the ith TRP, respectively. So, for the qth row being [k 1 ,k 2 ,...,k MrMt ], we compute the cumulative square error as follows: e q = MrMt ÿ i=1,k i ”=0 ( ˜ d i k i ≠ ˆ d i k i ) 2 ‡ 2 ˆ d i k i (4.5) Then, the likelihood of the q row of B MtMr is defined as L q =≠ e q . If L q <” , the qth row is deleted from matching matrixB MtMr . 100 Thecomputationalcomplexityoftheproposedmethodisanalyzedasfollows: a TOAmeasurementattheith(iÆ +2 )TRPwillbecheckedwithtwoTOAmea- surements from previous two TRPs. Thus, the number of computed likelihoods is q i≠ 2 u 1 =1 q i≠ 1 u 2 =u 1 +1 N u 1 N u 2 where N i are the number of TRMPCs from the ith TRP. Furthermore, when processing theith (i>+2 ) TRP, each TOA measurement is checked against the existing targets location estimates (corresponding to rows of matching matrix). Then, if the number of target candidates after processing the ith TRP is ˜ M i tar , the number of likelihood computation for all TOA measurements of the ith TRP is N i ˜ M i≠ 1 tar . Thus, the total number of likelihood computation is q +2 i=3 q i≠ 2 u 1 =1 q i≠ 1 u 2 =u 1 +1 N u 1 N u 2 N i + q MrMt i=+3 N i ˜ M i≠ 1 tar . To have a clearer idea about the complexity, we assume that N i =N for all i=1,...,M r M t . Then, the number of likelihood computation is q +2 i=3 (i≠ 1)(i≠ 2) 2 N 3 + q MrMt i=+3 N i ˜ M i≠ 1 tar . Obviously the first item q +2 i=3 (i≠ 1)(i≠ 2) 2 N 3 is polynomial with respect to N and . To let the second item q MrMt i=+3 N i ˜ M i≠ 1 tar also be polynomial with respect to N and M r M t , the number of potential targets ˜ M i≠ 1 tar should still be polynomial instead of growing exponentially with i. For example, the worst situation is when the TOA mea- surements have large variances, and any three or more TOA measurements are considered to be DPs to the same target, then ˜ M MrMt≠ 1 tar = N MrMt≠ 1 , which is exponential with respect to M r M t . Therefore, to maintain a polynomial com- plexity of our algorithm, the TOA measurements need to be accurate so that the number of potential targets is polynomial with respect to N and M r M t . This is a reasonable assumption, since even the ML method could not have good localiza- tion performance if the variances of TOA measurements are large. To compare the computational complexity between the EMK method and the proposed method, we assume that all MPCs are DPs (N i = M tar ) and ˜ M i≠ 1 tar =50 for every i, and 101 consider an example of M tar =6 and M r M t =6, and=2 . The number of likeli- hood computation is 46656 for EML method, while 1464 for the proposed method, which shows a significant dierence of computational complexity between them. 4.4 Simulations This section shows simulation results for the proposed method. We consider the following fundamental setting (unless explicitly stated otherwise): 1. There are M tar targets, and M ntar non-target reflectors randomly distributed in the region xœ [≠ 12.5,12.5]fl yœ [≠ 12.5,12.5] meters. There are multiple transmitters and multiple receivers. The signal propagation path might be blocked. 2. We consider direct paths (transmitter - target - receivers) and “simple” indirect paths (we assume that either Transmitter-Target-Reflector-Receiver or Transmitter-Reflector-Target-Receiver constitute a valid simple IP, but assume that not both occur at the same time for a particular reflector - target combination). Paths involving more reflections are ignored, due to their low power. Therefore, each TRP has at most M tar +M tar M ntar TOA measurements. 3. We set the system bandwidth of Ultra-wideband signals to 3GHz. The stan- dard deviation of the TOA measurement measurement is assumed to be Speed of light 2B = 3ú 10 8 2ú 3ú 10 9 =0.05 meter. While higher accuracy of TOA measure- ment measurements can be obtained from high-resolution algorithms such as CLEAN [14], these algorithms have relatively high computational complex- ity, and are therefore not considered here. 102 4. We consider the possibility that MPCs might be unresolvable if their arrival time is too close. For each TRP, if the dierence between any two run-length measurements at a TRP is smaller than 0.1 meter (2 times the standard deviation of TOA measurement), the TOA measurement with the larger value is ignored. 5. The performance of localization is characterized by the root mean square error (RMSE), the probability of miss detection (P m ) and the false alarm probability (P f ), i.e., the probability that we identify a target location where no physical target is present. This is computed as follows: • The RMSE is simply computed as Ò (x≠ ˆ x) 2 +(y≠ ˆ y 2 ). • The probability of missed detection is the probability that a target is either not detected at all, or detected with the root square error larger than 0.158 meter ( Ô 10 times the standard deviation of TOA measurements). • The false alarm probability P f is N e tar Ntar , where N tar is the number of targets, and N e tar is the number of estimated target clusters, so that every estimated target in the cluster is 0.158 meter farther from any true target 8 . 8 In the simulations, the estimated target location sometimes clustered, and we define a cluster to be a set of estimated target location, where the distance between any two members in this cluster is smaller than 0.158 meter. 103 4.4.1 Comparison with Exhaustive Maximum Likelihood Estimation Method Assume that there are 2 targets, 1 non-target reflectors, 1 transmitter and 3 receivers. First, we compare the performance of the proposed method with (i) the exhaustive maximum likelihood (EML) method and (ii) a genie-aided method. In theEMLmethod,anycombinationoftheTOAmeasurementsatdierentreceivers are considered, including the possibility that any TOA measurement might be an IP. Then, the final decision is chosen as the one combination that maximizes the likelihood as discussed in Section III. D. The genie-aided method knows which run length measurement corresponds to which target, and uses the proposed algorithm to process all TOA measurements to estimate the target location one by one. The genie method does suer from the detrimental eects from the IPs, but not from assigning TOA measurements to wrong targets; clearly it cannot be implemented even with unlimited computational resources and serves as an upper bound on the possible performance. The cumulative distribution function (CDF) of the target location estimation error is shown in Fig. 4.4. We can observe that the three methods have almost the sameCDFcurves,whilethegenie-aidedmethodhassmallerfalsealarmprobability. 4.4.2 Dierent and ” In the proposed algorithm, there are two thresholds and ” to control the system performances (P m and P f ). Larger and larger ” leads to lower P m and higher P f . So, there are dierent pairs of values for and ” leading to the same P m . Consider a scenario with 3 transmitter, 3 receivers, 2 targets and 2 non-target reflectors, all uniformly distributed in xœ [≠ 12.5,12.5]fl yœ [≠ 12.5,12.5] meters. 104 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 10 −1 10 0 Root Mean Square Error CDF EML, Φ=1, Pf = 0.261, Pm = 0.076 Proposed Method, Φ=1, Pf = 0.261, Pm = 0.076 Genieaided Method, Φ=1, Pf = 0.231, Pm = 0.0.076 Figure 4.4: CDF of RMSE for dierent methods. Fig. 4.5 shows P m and P f for dierent values of ” and . The results show that for the same number of TRPs, dierent values of and ” have similar curves of P m vs. P f . Moreover, to achieve the same value ofP m , smaller values of ” requires larger value of . We furthermore compare the values of P m by simulation with the theoretical evaluation in Eqn. (4.3). As shown in Fig. (4.6), the theoretical evaluation in Eqn. (4.3) well approximates the true value of P m . Therefore, Eqn. (4.3) could help specify the parameters ( and ” ), to achieve a desired P m . In the rest of the simulations, the threshold ” is chosen as ” = Ô 10‡ r , where ‡ r =0.05 is the standard deviation of TOA measurements. 105 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Pf Pm δ=√4 δ=√6 δ=√8 δ=√10 Φ=1 Φ=2 Φ=3 Φ=4 Figure 4.5: P m vs. P f for dierent values of and ” . 2 2.2 2.4 2.6 2.8 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 δ P m P m by simulation, Φ=1 Theoretical P m , Φ=1 P m by simulation, Φ=2 Theoretical P m , Φ=2 P m by simulation, Φ=3 Theoretical P m , Φ=3 P m by simulation, Φ=4 Theoretical P m , Φ=4 Figure 4.6: Comparison between theoretical and true P m . 106 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 10 −1 10 0 Root Mean Square Error CDF Tx1Rx6, Φ=2, Pf = 0.610, Pm = 0.007 Tx1Rx10, Φ=2, Pf = 0.227, Pm = 0.004 Tx2Rx5, Φ=2, Pf = 0.075, Pm = 0.003 Tx3Rx4, Φ=2, Pf = 0.034, Pm = 0.003 Figure 4.7: CDF of RMSE for dierent TRPs with blockage probability p b =0. 4.4.3 Dierent Number of TRPs Assume that there are 2 targets, and 2 non-target reflectors, so that each TRP receives at most 6 MPCs. Fig. 4.7 shows the CDF of the RMSE for the proposed method with dierent numbers of TRPs ( p b =0 and=2 ). In the figure, P f is the false alarm probability, and P m is the probability of miss detection. There are several important observations from the simulation results: 1. When the sum of transmitter and receiver numbers M r +M t =7, the con- figuration of M t =3,M r =4 has both smaller RMSE and lower false alarm probability than [M t =2,M r =5] and [M t =1,M r =6]. Generally, for fixed M r +M t , the system configuration with larger M r ú M t leads to smaller RMSE and false alarm probability. This is in line with previous insights for MIMO radar systems. 107 2. A configuration with (M t =2,M r =5) has slightly better performance than the configuration of (M t =1,M r =10) in terms of RMSE; more importantly (M t =2,M r =5) has much smaller false alarm probability. This indicates that TOA measurements with the same Tx are more likely to have ghost targets, compared with the same number of TRPs with dierent pairing of Tx and Rx. Moreover, in Fig. 4.8, the performances of the proposed method for dierent numbers of TRPs are shown. It can be observed that the missed detection proba- bility and false alarm probability are both lowers with more number of TRPs. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 10 −1 10 0 Root Mean Square Error CDF Tx2Rx2, Φ=3, Pf = 0.963, Pm = 0.132 Tx3Rx3, Φ=3, Pf = 0.141, Pm = 0.066 Tx4Rx4, Φ=5, Pf = 0.053, Pm = 0.018 Tx5Rx5, Φ=7, Pf = 0.031, Pm = 0.013 Figure4.8: CDFvsRMSEfordierentnumberofTRPswithblockageprobability p b =0.1. 108 4.4.4 Dierent threshold AsanalyzedinSectionIII,dierent willleadtodierentsystemperformance. Fig. 4.9 shows the CDF of RMSE for dierent threshold with blockage proba- bility p b =0.1. It shows that larger leads to higher false alarm probability and lower missed detection probability. =4 is the optimum choice for p b =0.1 in terms of RMSE and miss detection probability.=4 , the probability of having equal or less thanM r M t ≠ DPs is 0.0256, and the true probability of miss detec- tion 0.013. Since in the simulations, the threshold is ” = Ô 10‡ r where ‡ r =0.05 is the standard deviation of TOA measurements. Therefore, p 1 ¥ 0 and the dif- ference between 0.0256 and 0.013 is the probability that the IPs are considered to be DPs, let to p 2 =0.0126 according to Eqn. (4.2). 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 10 −1 10 0 Root Mean Square Error CDF Tx3Rx4, Φ=1, Pf = 0.013, Pm = 0.353 Tx3Rx4, Φ=2, Pf = 0.015, Pm = 0.162 Tx3Rx4, Φ=3, Pf = 0.032, Pm = 0.054 Tx3Rx4, Φ=4, Pf = 0.069, Pm = 0.013 Figure 4.9: CDF of RMSE for dierent with 3 Tx 4 Rx and p b =0.1. 109 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 10 −1 10 0 Root Mean Square Error CDF Tx3Rx4, p b =0, Φ=2, Pf = 0.034, Pm = 0 Tx3Rx4, p b =0.1, Φ=4, Pf = 0.069, Pm = 0.013 Tx3Rx4, p b =0.2, Φ=5, Pf = 0.098, Pm = 0.032 Figure 4.10: CDF of RMSE for dierent p b , is the optimum choice for each p b . 4.4.5 Dierent Path Blockage Probability Assume that there are 2 targets, and 2 non-target reflectors, and each TRP then receives at most 6 MPCs. The simulation results in Fig. 4.10 show the performances of the proposed method with dierent blockage probability. The results show that higher blockage probability will lead to higher miss detection probability-whichistobeexpected. ThedierencebetweenCDFcurvesofRMSE with dierent blockage probability configurations is small with proper threshold setting. 4.4.6 Rich multi-path environments Simulations are performed to evaluate the performance in rich multi-path envi- ronmentswithalargernumberofMPCs. Inthissimulation,thereare5targetsand 7 non-target reflectors randomly distributed in the region x œ [≠ 12.5,12.5],y œ 110 [≠ 12.5,12.5]. Thus, each target generates a maximum of 8 MPCs (1 DP and 7 IPs), and each TRP pair receives a maximum of 40 MPCs. The simulation results are shown in Fig. 4.11 and Fig. 4.12 for blockage probability p b =0 and p b =0.1, respectively. It can be observed that the dierences between CDF curves with dierent threshold are small forp b =0, while larger forp b =0.1. This is because for small value of , the missed detection probability P m with p b =0 is very low (5% for=1 ), however P m with p b =0.1 is very high (66.6% for=1 ). To further evaluate the performance of the proposed method, we compare its performance with the genie-aided method, which could dierentiate MPCs from dierent targets, however not DPs from IPs. The simulation results are shown in Fig. 4.13, which show that the dierence between the CDF curves of the proposed method and the genie-aided method is negligible.Moreover, the two methods have almost the same missed detection probability, while the false alarm probability of the proposed method is 14% ( 0.240≠ 0.211 0.211 =0.14) percent higher than that of genie-aided method. For larger values of M tar , the computational complexity of EML is prohibitively high for real time implementation (which is why its results are not shown here for those cases), while the proposed method greatly saves the estimation time. In summary, the simulations show that the proposed algorithm has very good performance(95%probabilityofsmallerthan0.1meterRMSE)usingtheproposed matching algorithm. Besides, given the number of M r +M t , it is always best to choose the configuration of maximum M r ú M t , to have smaller RMSE and lower false alarm probability at the same time. important feature of MIMO radar in practical scenario. 111 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 10 −1 10 0 Root Mean Square Error CDF Tx3Rx4, Φ=1, Pf = 0.140, Pm = 0.050 Tx3Rx4, Φ=2, Pf = 0.223, Pm = 0.003 Tx3Rx4, Φ=3, Pf = 0.240, Pm = 0 Tx3Rx4, Φ=4, Pf = 0.244, Pm = 0 Tx3Rx4, Φ=5, Pf = 0.247, Pm = 0 Figure 4.11: CDF of RMSE for dierent thresholds with p b =0. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 10 −1 10 0 Root Mean Square Error CDF Tx3Rx4, Φ=1, Pf = 0.009, Pm = 0.662 Tx3Rx4, Φ=2, Pf = 0.027, Pm = 0.482 Tx3Rx4, Φ=3, Pf = 0.064, Pm = 0.283 Tx3Rx4, Φ=4, Pf = 0.161, Pm = 0.121 Tx3Rx4, Φ=5, Pf = 0.272, Pm = 0.059 Figure 4.12: CDF of RMSE for dierent thresholds with p b =0.1. 112 0 0.02 0.04 0.06 0.08 0.1 10 −1 10 0 Root Mean Square Error CDF Genie−aided Method, Φ=1, Pf = 0.140, Pm = 0.050 Proposed Method, Φ=1, Pf = 0.140, Pm = 0.050 Tx3Rx4, Φ=2, Pf = 0.199, Pm = 0.003 Proposed Method, Φ=2, Pf = 0.223, Pm = 0.003 Genie−aided Method, Φ=3, Pf = 0.211, Pm = 0 Proposed Method, Φ=3, Pf = 0.240, Pm = 0 Figure 4.13: CDF of RMSE for the proposed method and the genie-aided method. 4.5 Conclusion In this chapter, we have proposed a novel algorithm for determining locations of passive targets in a setting with multiple, spatially separated transmitters and receivers. In contrast to previous investigations, we do not assume that any direc- tional information is available. Consequently the system cannot easily determine which TOA measurements are associated with which target. Besides, our formu- lation takes into account a number of practical constraints: TOA measurements suerfromnoise-inducederrors, andcanbeindirectpaths(resultinginbias), com- pletely blocked, or unresolvable from MPCs associated with other targets. Such pairing could theoretically be done by evaluating the likelihoods of all possible pairings of TOA measurements, but such an algorithm would have exponential complexity. Instead the proposed method relies on a step-by-step adding of infor- mation from the dierent TRPs, trimming the set of possible target locations in 113 the process. Simulations show that our algorithm gives performance very similar to that of exhaustive search, while vastly decreasing complexity. Furthermore, we study the problem of assigning transceiver nodes to function as transmitters or receivers. Simulationresultsshowthatgiventhetotalnumberoftransceivernodes in the system, the configuration with the maximum number of TRPs generally has the best performance. 4.6 Appendices 4.6.1 Computation of l k 3 k 1 ,k 2 for three DPs Assume the ith TRP consists of the transmitter [c i ,d i ], and receiver [a i ,b i ]. For each TOA measurement d i k i from the ith TRP, we can construct the following equation for the potential target location [x,y]: Ò (x≠ c i ) 2 +(y≠ d i ) 2 + Ò (x≠ a i ) 2 +(y≠ b i ) 2 =d i k i . (4.6) Let the intersection of the ellipses corresponding to two TOA measurements d i 1 k i 1 andd i 2 k i 2 be[ˆ x,ˆ y](k i 1 ,k i 2 ).Let ˜ d i 3 k i 1 ,k i 2 denotethesumofthedistancesfromthis intersection to the transmitter [c 3 ,d 3 ] and the receiver [a 3 ,b 3 ] of the third TRP. Then, if d i 1 k i 1 , d i 2 k i 2 and d i 3 k i 2 correspond to direct paths of the same target, and the measurement errors are small, we can evaluate l k 3 k 1 ,k 2 as follows: l k 3 k 1 ,k 2 = ˜ d i 3 k i 1 ,k i 2 ≠ ˆ d k 3 i 3 =( — 2 – 3 – 1 — 2 ≠ — 1 – 2 + – 2 — 3 — 1 – 2 ≠ – 1 — 2 ) d i 1 k i 1 +( — 1 – 3 — 1 – 2 ≠ – 1 — 2 + – 1 — 3 – 1 — 2 ≠ — 1 – 2 ) d i 2 k i 2 ≠ d i 3 k i 3 (4.7) 114 where d i 1 k i 1 and d i 2 k i 2 and d i 3 k i 2 are measurement errors ofd i 1 k i 1 andd i 2 k i 2 andd i 3 k i 2 , respectively, and – i = x≠ c i Ò (x≠ c i ) 2 +(y≠ d i ) 2 + x≠ a i Ò (x≠ a i ) 2 +(y≠ b i ) 2 , (4.8) — i = y≠ d i Ò (x≠ c i ) 2 +(y≠ d i ) 2 + y≠ b i Ò (x≠ a i ) 2 +(y≠ b i ) 2 , (4.9) where [x,y] is the true location of target, and [c i ,d i ] and [a i ,b i ] are the locations of transmitter and receiver of the ith (i=1,2,3) TRP, respectively. According to (4.7), the variance of l k 3 k 1 ,k 2 is evaluated by ‡ l k 3 k 1 ,k 2 =( — 2 – 3 – 1 — 2 ≠ — 1 – 2 + – 2 — 3 — 1 – 2 ≠ – 1 — 2 ) 2 Var( d i 1 k i 1 ) +( — 1 – 3 — 1 – 2 ≠ – 1 — 2 + – 1 — 3 – 1 — 2 ≠ — 1 – 2 ) 2 Var( d i 2 k i 2 )+Var( d i 3 k i 3 ) (4.10) where Var(e) denotes the variance of e. 4.6.2 Computation of l k i Row(B i≠ 1 ) j 2 Let [k 1 ,k 2 ,...,k i≠ 1 ]=Row(B i≠ 1 ) j 2 , and [ˆ x TSE ,ˆ y TSE ] be the estimated target location based on [d 1 k 1 ,d 2 k 2 ,...,d i≠ 1 k i≠ 1 ]. Also let ˜ d k i k 1 ,k 2 ,...,k i≠ 1 be the sum of distances from [ˆ x TSE ,ˆ y TSE ] to the transmitter ([c i ,d i ]) and receiver ([a i ,b i ]) the ith TRP. Then, we have l k i Row(B i≠ 1 ) j 2 = l k i k 1 ,k 2 ,...,k i≠ 1 = ˜ d k i k 1 ,k 2 ,...,k i≠ 1 ≠ ˆ d k i i = – i x TSE +— i y TSE ≠ d i k i . (4.11) 115 where – i and — i are defined in (4.8) and (4.9), [ x TSE , y TSE ] is the estimation error of [ˆ x TSE ,ˆ y TSE ], which can be computed [74], and d i k i is the measurement error of d i k i . The variance of l k i Row(B i≠ 1 ) j 2 can also be evaluated as ‡ l k i Row(B i≠ 1 ) j 2 =(– i ) 2 Var( x TSE )+(— i ) 2 Var( y TSE ) +2– i — i Cor( x TSE y TSE )+Var( d i k i ). (4.12) where Cor( x TSE y TSE ) is the cross-correlation between x TSE and y TSE . 116 Chapter 5 Future Works and Conclusions 5.1 Conclusions This thesis investigates the problem of passive target localization in multi-path environments. Firstly, we analyze the three main challenges in practical system employment, e.g., TOA measurement errors due to noises, indirect paths due to multi-paths and unmatched TOA measurements from multiple targets. To build a precise localization, we address the three challenges with novel algorithms: 1. A Two-Step algorithm for individual target localization: we focus on how to accurately estimate target location based on time-of-arrival (TOA) mea- surements with only noise-induced errors. The proposed TSE proceeds in two steps, first computing estimates for a parameter set x, y, and Ô x 2 +y 2 , secondly updating the location vector [x,y] from the results of the first step. BoththeoreticalanalysisandnumericalsimulationsshowthatforsmallTOA estimation errors the TSE approaches the CRLB very closely with only one iteration. Experiments also showed that the error variance of TSE is very close to the CRLB of TOA algorithms under the assumption of Gaussian range measurement errors. In addition, we show the CRLB of passive TOA estimation to be much lower than that of TDOA. This indicates that the synchronization between the transmitter and the receivers can substantially decrease the localization error. 117 2. Indirect path detection based on TOA measurements and angle information: we propose a novel IP detection based on channel sounding results, e.g., time-of-arrival (TOA), direction of arrival (DOA) and direction of depar- ture (DOD). Based on the triplet (TOA, DOD and DOA) of each path, we test whether it is consistent with the assumption of a DP. In particular, we test the dierences of the three intersections from DOD, DOA and TOA measurements ⁄ i (i=1,2,...,6) for IP detection. The algorithm is based on Neyman-Pearson criteria to maximize the probability of detection given probabilityoffalsealarm. Severalindividualrulesandfusionrulesunderthis framework are discussed. The “blind spots” of individual rules are investi- gated for dierent decision variables. Simulations show that the algorithm can eectively detect IPs when the TOA, DOD and DOA measurements are available with small errors. As verified by simulations, the proposed method can be employed to sort out DPs to increase localization accuracy. 3. Multiple target localization based on TOA measurements: for the implemen- tation of low cost localization systems, the angle information might be hard to obtain. For this reason, we introduced a novel algorithm for determin- ing locations of multiple targets based solely on TOA measurements in a distributed MIMO radar system. We consider that TOA measurements suf- fer from noise-induced errors, IPs and MPCs associated with other targets. Such pairing could theoretically be done by evaluating the likelihood of all possible TOA pairings, which has exponential complexity. The proposed method relies on a step-by-step adding of TOA measurements from the dif- ferent TRPs, trimming the set of possible target locations in the process. Simulations show that the performance of our algorithm is very similar to that of exhaustive search, while vastly decreasing complexity. Furthermore, 118 we study the problem of assigning transceiver nodes to function as trans- mitters or receivers. Simulation results show that given the total number of transceivernodesinthesystem,theconfigurationwiththemaximumnumber of TRPs generally has the best performance. 5.2 Future Research Directions Thisthesisaddressesthemainchallengesofprecisepassivelocalizationsystems. In the future, we are interested in investigating the following related aspects: • Improve localization performance with information of surrounding reflectors: the deterministic or statistical information of the surrounding objects can be obtained from floor mapping or an initial background measurement. Then, the locations of reflection points (such as walls) can be identified, and be used to identify IPs and further improve target location estimate based on IPs. 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Abstract (if available)

Abstract Passive localization of objects is fast becoming a major aspect of wireless technologies, with applications in logistics, surveillance, and emergency response, etc. The localization can be performed with a variety of localization techniques based on different system parameters such as the angle of arrival (AOA), and the signal time of arrival (TOA) measurements. However, these measured parameters are corrupted by background noise. Further complications in realistic settings arise from the following factors: (i) the direct paths (DPs) between transmitters/receivers and targets might be blocked, (ii) indirect paths (IPs) arising from multi path propagation might be present, (iii) signals reflected by multiple targets cannnot be distinguished by their ""signatures"" or other unique characteristics. This thesis offers novel solutions to combat theses challenges: Individual Target Localization with Noisy TOA measurements: Consider a single target localization problem, if there are no IPs or if IPs can be perfectly identified, the error of target location estimate is due to the noisy TOA measurements. We propose a Two Step Estimation (TSE) method which employs the TOA measurements and the variances of them to perform approximate maximum likelihood through a two step mechanism. Its complexity is lower than iterative methods, and is able to achieve the Cramer Rao Lower Bound (CRLB), which is the lower bound of mean square error for any un-biased estimator if TOA measurements suffer Gaussian errors. As far as we know, it is the first passive localization method to achieve the CRLB. Another benefit of TSE is that the distribution of the estimated target location is known. Furthermore, simulations and practical experiments are performed to verify the superiority of TSE. ❧ Indirect Path Detection with TOA and Angle Measurements: Because of non-target reflectors, the signals propagate from transmitter to receivers through DP or IPs. In some situations, the DP is blocked and then, the TOA measurements based on IPs would usually contain large errors (compared with errors induced by noise). Thus, IP detection algorithm needs to effectively detect the IPs, to avoid the detrimental effect of indirect paths on localization. We propose a scheme based on channel sounding parameters: TOA, DOD (direction of departure) and DOA (direction of arrival). The principle of the proposed TOA DOD and DOA joint (TDAJ) scheme is as follows: if there were no measurement errors, any DP would have self-consistent parameters of measurements. By this we mean that the three intersections obtained from the pairs of measured parameters are at the same location. In the presence of measurement errors, the decision variables of the Neyman-Pearson (NP) test are the differences between two of the three intersections in x or y coordinates: only if this difference lies below a certain threshold, the algorithm judges the path to be a DP. The benefit of the TDAJ depends on the measurements of TOA, DOD and DOA instead of multipath channel statistics. Simulation results show that under common localization scenarios employing ultrawideband signals, the proposed algorithms can effectively detect IPs and significantly improve localization accuracy. ❧ Multiple Target Localization based on TOA measurements: If there is more than one target to be localized, the system needs to separate the received signals from different targets before localizing them. Only the information (such as TOAs) extracted from impulses reflected by the same target should be combined for localization. Thus localization requires to correctly ""pair"" the TOA measurements of the reflections arriving at the different receivers. Further complications in realistic settings arise from the following factors: (i) the number of the targets are unknown, (ii) the DPs between transmitter and target, and/or between target and receiver, might be blocked, (iii) IPs arising from multi path propagation might be present, and (iv) the DP from one target might have a similar time of arrival as the DP or IP from another target, such that those paths cannot be discriminated (resolved). We propose a novel sequential matching algorithm that can accurately localize multiple targets even under these difficult circumstances. This algorithm is based on an iterative approximation to an exhaustive maximum likelihood (EML) estimation of the target locations, yet avoiding the exponential complexity that an EML requires. It utilizes a clustering technique that gives the number and locations of targets, and incorporates an IP detection mechanism based on the likelihood of several TOA measurements corresponding to DPs from the same target. Despite its lower complexity, our algorithm achieves almost the same performance as the EML in a variety of simulation settings. We finally study the choice of algorithm parameters and the configurations of the MIMO system nodes to achieve the best localization performance. ❧ This thesis aims at providing insights and proposing methods to address the three main challenges of passive localization systems.

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Asset Metadata

Creator Shen, Junyang (author)

Core Title Localization of multiple targets in multi-path environnents

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 12/04/2013

Defense Date 10/24/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Cramer-Rao Lower Bound,DOA,DOD,Indirect Path Detection,indoor localization,Maximum-likelihood Estimation,Mean Square Error,MIMO radar,multi-target localization,OAI-PMH Harvest,TOA,Ultra-wideband signals

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Molisch, Andreas F. (committee chair), Hashemi, Hossein (committee member), Huang, Qiang (committee member), Krishnamachari, Bhaskar (committee member)

Creator Email fly.shenjy@gmail.com,junyangs@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-355576

Unique identifier UC11296538

Identifier etd-ShenJunyan-2202.pdf (filename),usctheses-c3-355576 (legacy record id)

Legacy Identifier etd-ShenJunyan-2202.pdf

Dmrecord 355576

Document Type Dissertation

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Rights Shen, Junyang

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

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