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Towards an understanding of the impact of dependent blocking on localization performance
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Towards an understanding of the impact of dependent blocking on localization performance
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Towards an Understanding of the Impact of Dependent Blocking on Localization Performance Sundar Aditya A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL, UNIVERSITY OF SOUTHERN CALIFORNIA, in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) Advisor: Prof. Andreas F . Molisch August 2018 Copyright 2018 Sundar Aditya 1 Dedication To Anjali, without whose seemingly limitless reserves of strength and courage, this thesis would not have materialized. To Amma, my academic inspiration, and to whom I owe everything. To Rugmani and Shivakumar Shankar, for their unconditional support during trying times. To Pati, who simply put, is a treasure. Acknowledgements It takes a village to produce a PhD graduate and there are a number people to whom I owe a deep debt of gratitude for supporting me on this journey. Firstly, my heartfelt thanks to my advisor, Prof. Andreas F. Molisch. On numerous occasions, he has helped me stay on course and not abandon lines of investigation that I had immaturely considered to be dead-ends; without his persistence and patience, this dissertation would certainly not have seen the light of day. Apart from being a phenomenal scientist and advisor, his understanding of the challenges involved in managing what academia euphemistically calls the two-body problem means a lot to me personally. I have also had the great pleasure of collaborating with Prof. Harpreet S. Dhillon from Virginia Tech in producing a sizable chunk of the research contained in these pages. His able guidance, coupled with his immense technical acumen, were the driving forces behind the success of my stint as a visiting researcher at Virginia Tech and I consider myself fortunate to have him as a mentor. The labor of love that is a PhD requires copious amount of professional as well as personal support and I am eternally grateful to my family for the latter. With both parents working, I was largely raised by my grandmother, whom I address as Pati. During my childhood, she used to get me ready for school, pack my lunch and feed me once I got back home. Her role in my growth and development cannot be understated; my very first words were spoken in her presence. Even today, when I meet a friend or a teacher from my school days, the first question that they invariably ask is “How is your grandmother?" I doubt there can be any greater endorsement of her impact in my life. Words are insufficient to fully express the depth of my gratitude and thanks to my mother, Dr. Viha Sudha. It is a great testament to Amma’s parenting skills that her considerable academic achievements have always been a source of inspiration, rather than pressure, for me. Her optimism, resolve and courage in the face of tremendous adversity over the course of her storied life have been a massive source of strength for me and served as a reminder that the challenges encountered in research were pale in comparison. My greatest supporter throughout this endeavor has been my wife, Gitanjali Shankar. We have been grappling with the aforementioned two-body problem since August 2012 - almost the entire duration of my studies. Many others who have found themselves on the same boat can testify that it is never easy; under these trying circumstances, her wellspring of quiet strength and dignity has been nothing short of inspirational. Anjali is the funnier one among us and the fact that her sense of humour (I would’ve been mercilessly mocked if I had used ‘humor’ instead!) has not been diminished by the strains of a trans-oceanic relationship is a testament to her courage. I have lost count of the number of weeks that I have picked myself up to fight on, motivated only by a few hours of cheery banter with her on Sundays, often over a dodgy internet connection. She has been by my side through thick and thin and this milestone is as much hers as it is mine. While the joy of this occasion is immense, it still pales in comparison to the pride I feel in being her husband. Along with Amma and Anjali, my wonderful in-laws, Rugmani and Shivakumar Shankar, also deserve a lot of credit for being my cheerleaders throughout. They are among the most positive people that I have encountered and I deeply cherish their unconditional support. I would also like to extend my sincere thanks to all the professors at USC who have been instrumental in my professional development; in particular, it has been the privilege of my academic life to have been taught by Professors Solomon W. Golomb and Robert A. Scholtz. I am especially thankful to Professors Mahta Moghaddam, Bhaskar Krishnamachari and David Kempe for their critique of my research at various stages that have helped me immensely in producing this dissertation. I am also hugely indebted to my professors at IIT Madras, my alma mater, for laying the solid technical foundation that contributed to my success in graduate school. Through the grind of graduate school, many bonds of friendship are formed with peers over mutually shared experiences and I am grateful to the following groups of people for their wonderful company over the years: Nachikethas A. J., a dear friend and fellow alumnus of IIT Madras who joined USC at the same time as me which created an instant connection that has only been strengthened by many shared interests like tennis, mathematics etc. Roommates: Seven years in LA have inevitably produced their fair share of roommates, many of whom have gone on to become very good friends. In no particular order, I would like to thank Kiran Matam, Sanmukh Kuppannagari, 4 Aravind Bhimarasetty and Gopi Neela not only for their lively camaraderie, but also for their impressive culinary skills, which played no small part in alleviating my homesickness. WiDeS group members: I would also like to thank the following friends and colleagues, in no particular order, for the valuable technical discussions over the years: Celalettin Umit Bas, Vishnu Ratnam, Rui Wang, Hao Feng, Vinod Kristem, Oluwaseun Sangodoyin, Zheda Li and Ranjit S. Patil. In particular, I would like to acknowledge the efforts of Ranjit, Umit, Rui and Seun in helping me with some of my measurements. Virginia Tech friends: In addition to Prof. Harpeet Dhillon’s efforts, the success of my stint at Virginia Tech owes a great deal to my wonderful set of friends there who warmly accepted me as one of their own and helped me settle down comfortably at a new place. Once again, in no particular order, I would like to thank Priyabrata Parida, Ishaan Goswami, Suvankar Biswas, Bidisha Barath, Jaya Kartheek Devineni and Mehrnaz Afzhang for all the fun times in the charming town of Blacksburg, Virginia. I would also like to acknowledge Dr. Kristin Conover and Dr. Sunil Obediah from the USC Student Counselling Services for their assistance at a crucial stage during my PhD. Finally, I wish to thank King Abdulaziz City for Science and Technology (KACST) for their generous support of my research. 5 Contents List of Figures 7 List of Acronyms 9 Abstract 11 1 Introduction 15 1.1 Bayesian Multi-Target Localization . . . . . . . . . . . . . . . . . 22 1.2 Blind spot Probability Analysis using Stochastic Geometry . . . . . 24 1.3 Mean Square Error Analysis . . . . . . . . . . . . . . . . . . . . . 26 1.4 Common Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 Bayesian Multi-Target Localization (MTL) 29 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.4 Additional Notation . . . . . . . . . . . . . . . . . . . . . 31 2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3 Bayesian MTL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4 MTL Algorithm using Blocking Statistics . . . . . . . . . . . . . . 42 2.4.1 Complexity of Bayesian MTL Algorithm . . . . . . . . . . 44 6 2.4.2 Limitations of Bayesian MTL Algorithm . . . . . . . . . . 45 2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.5.1 Comparison to Prior Art . . . . . . . . . . . . . . . . . . . 46 2.5.2 Effect of Correlated Blocking . . . . . . . . . . . . . . . . 47 2.5.3 Comparison with genie-aided method . . . . . . . . . . . . 52 2.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.6.1 Measurement setup . . . . . . . . . . . . . . . . . . . . . . 54 2.6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . 59 3 Blind spot Probability Analysis under Dependent Blocking 63 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.1.4 Additional Notation . . . . . . . . . . . . . . . . . . . . . 67 3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3 Analysis of blind spot Probability . . . . . . . . . . . . . . . . . . 71 3.4 Characterizing Unshadowed Area . . . . . . . . . . . . . . . . . . 75 3.5 A Tractable Approximation forb(;z) . . . . . . . . . . . . . . . . 81 3.6 Asymptotic blind spot Probability . . . . . . . . . . . . . . . . . . 86 3.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4 Localization Outage Analysis 97 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.1.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.1.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.1.3 Additional Notation . . . . . . . . . . . . . . . . . . . . . 100 4.1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.3 Characterizing SPEB distribution . . . . . . . . . . . . . . . . . . . 104 4.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5 Future Work 121 A Blocking model from Chapter 2 125 B Proofs from Chapter 3 127 B.1 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 B.2 Proof of Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 B.3 Proof of Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 B.4 Proof of Lemma 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 C Proofs from Chapter 4 133 C.1 Proof of Lemma 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 C.2 Proof of Lemma 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 C.3 Proof of Lemma 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Bibliography 137 8 ListofFigures 1.1 A typical localization use-case scenario . . . . . . . . . . . . . . . . . 15 1.2 Basics of localization. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3 Example of MPC overlap . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4 Example of a germ-grain model. . . . . . . . . . . . . . . . . . . . . . 24 2.1 Correlated blocking: An example . . . . . . . . . . . . . . . . . . . . . 30 2.2 Example network where dependent blocking can be significant . . . . . 48 2.3 Comparing Bayesian MTL algorithm with prior art . . . . . . . . . . . 49 2.4 A sample simulation instance of the Bayesian MTL Algorithm . . . . . 50 2.5 Performance of Bayesian MTL algorithm under dependent blocking . . 51 2.6 Comparison of Bayesian MTL algorithm with a genie-aided method . . 52 2.7 Hardware used for localization experiments . . . . . . . . . . . . . . . 53 2.8 The measurement environment. . . . . . . . . . . . . . . . . . . . . . . 55 2.9 Layout of target and anchor locations used in the measurements . . . . 56 2.10 Results from a sample measurement . . . . . . . . . . . . . . . . . . . 58 2.11 ROC curve from measurements . . . . . . . . . . . . . . . . . . . . . . 59 3.1 Example of a localization scenario consisting of anchors, targets and obstacles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2 Visible region around a typical target, for the line segment obstacle model where all the obstacles have lengthL and face the target (! i = i +=2). 68 3.3 Illustration of the quasi worst-case obstacle orientation . . . . . . . . . 70 9 3.4 Graphical illustration of the proof of Theorem 1 . . . . . . . . . . . . . 74 3.5 Shadowed area (shaded gray) due to a single obstacle. . . . . . . . . . . 76 3.6 Structure of theA v (z) when there are at least two obstacles . . . . . . . 77 3.7 Average fraction of the shadowed area induced by far-off obstacles . . . 79 3.8 LoS probability at a distanceR for line process obstacle model . . . . . 87 3.9 Unshadowed region induced by line process obstacle model . . . . . . . 88 3.10 Poisson-V oronoi cell as unshadowed region . . . . . . . . . . . . . . . 90 3.11 By capturing most of the blocking correlation,b (2+) (;z) yields an ac- curate approximation ofb(;z). In contrast, by ignoring the blocking correlation,b ind (;z) significantly underestimatesb(;z). . . . . . . . 91 3.12 The accuracy of b (2+) (;z) implies that it can be used to determine the anchor intensity that satisfiesb (2+) (;z) b(;z) , for some threshold,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.13 Plot ofb as (;z as ) as a function of . . . . . . . . . . . . . . . . . . . 94 3.14 Validity of the asymptotic regime . . . . . . . . . . . . . . . . . . . . . 95 4.1 Feasibility of Eqn. (4.44) . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2 Comparison ofF S;app (:) with respect toF S (:) . . . . . . . . . . . . . 114 4.2 Fig. 4.2 contd. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2 Fig. 4.2 contd. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.3 Impact of SNR heterogeneity onF S;app () . . . . . . . . . . . . . . . . 117 4.4 Impact ofd max d min on SNR heterogeneity . . . . . . . . . . . . . . 118 5.1 Illustration of Johnson-Mehl and Poisson-V oronoi cells . . . . . . . . . 122 5.2 Illustration of a Manhattan line process based model for obstacles . . . 123 A.1 LoS blocking model from Chapter 2 . . . . . . . . . . . . . . . . . . . 125 A.2 EvaluatingP(k t ): An example . . . . . . . . . . . . . . . . . . . . . . 126 B.1 EvaluatingP(V (p;z) = 1) . . . . . . . . . . . . . . . . . . . . . . . . 130 B.2 Overlap situations betweenA sh (p 1 ;z) andA sh (p 2 ;z) . . . . . . . . . 131 10 ListofAcronyms ACK Acknowledgment AoA Angle of Arrival BPP Binomial Point Process ccdf Complementary cumulative distribution function cdf Cumulative distribution function CRLB Cramer-Rao Lower Bound DP Direct Path GDOP Geometric Dilution of Precision iid Independent and identically distributed IP Indirect Path LoS Line of Sight MIMO Multiple Input Multiple Output ML Maximum Likelihood MPC Multipath Component MSE Mean Square Error MTL Multi-Target Localization NLoS Non Line of Sight PPP Poisson Point Process pdf Probability density function RF Radio Frequency RFID Radio Frequency Identification 11 RII Ranging Information Intensity ROC Region of Convergence RSSI Received Signal Strength Indicator RX Receiver SINR Signal to Interference plus Noise Ratio SNR Signal to Noise Ratio SPEB Squared Position Error Bound SS Signal Strength TDoA Time Difference of Arrival ToA Time of Arrival TX Transmitter 12 Abstract The ability to accurately determine the location of (i.e., localize) one or more targets remotely is an essential requirement for many applications, such as navigation, search-and- rescue operations etc. The canonical localization problem involves estimating the target locations in an environment by analyzing wireless signals emanating from them at a collection of known anchor locations 1 . If the targets transmit their own signals, it is referred to as active localization; on the other hand, if the targets only reflect the incoming signals from the anchors (e.g., tumors in medical imaging), it is referred to as passive localization. In either case, localization typically involves the following two steps: (a) Ranging: In this step, each anchor obtains an estimate of its round-trip distance (range) or bearing to each target using a location-dependent feature of the received signal, such as time-of-arrival, signal strength or angle-of-arrival. (b) Multilateration or Triangulation: Once all the range and/or bearing estimates are available at the anchor locations, they are combined to obtain position estimates for all the targets in this step. Throughout this dissertation, we assume time-of-arrival based ranging and subsequent local- ization. To obtain accurate range estimates, it is desirable for the direct paths (DPs) from the anchors to the targets (i.e., going from a transmitter and reflecting off a target to a re- ceiver) to be unblocked. However, in many environments (especially indoors), obstacles in the environment (such as walls) can cause the DPs to be blocked. If the obstacle locations are unknown, then the blocking of the DPs can be modeled as a stochastic phenomenon which influences both the design of localization networks - in terms of determining the number of anchors required, as well as their positions - and the performance of any localization algorithm in those networks. As a result, the accurate characterization of blocking is important for a network and algorithm design perspective. A common assumption is to consider the DP blocking events to be mutually independent. However, the distributed (i.e., non-point) nature of the obstacles induces a statistical dependence between the blocking of different DPs in general, with the extent of dependence being a function of the size of the obstacles, as well as the geometry of the links. This phenomenon is commonly referred to as correlated blocking or dependent blocking and we use both these terms interchangeably from here on. This dissertation, comprising five chapters, primarily focuses on modeling and understanding the dependent blocking phenomenon and presents some insights into its impact on localization performance. 1 An anchor consists of a transmitter and a receiver. An anchor is said to be in monostatic configura- tion if the transmitter and receiver are co-located (i.e., a transceiver). Otherwise, an anchor is said to be in multistatic configuration. 13 In Chapter 2, the impact of dependent blocking on the performance of localization algorithms is investigated. In particular, the problem of localizing an unknown number of passive (i.e., reflecting) targets by single antenna anchors that cannot determine directions of departure and arrival is addressed. In addition to dependent blocking, we also consider the presence of multipath propagation and assume that all the targets have the same radar signature. In its most general form, this problem can be cast as a Bayesian estimation problem, where the propagation path of every multipath component is also estimated, along with the target locations. However, when the environment map is unknown, this problem is ill-posed and hence, a tractable approximation is derived where only the DPs are identified. In this framework, the dependent blocking caused by distributed obstacles in the environment is modeled as a Bayesian prior. A sub-optimal polynomial-time algorithm to solve the Bayesian multi-target localization problem with dependent blocking is proposed and its performance evaluated using simulations and measurements. In particular, when the DP blocking events are highly dependent, assuming them to be independent and having a constant probability (as was done in previous papers) resulted in poor detection performance, with false alarms more likely to occur than detections. The impact of dependent blocking on localization network design is studied in Chapters 3 and 4. The blocking of DPs between anchors and targets by distributed obstacles in an environment can create blind spots in a localization network if there is an insufficient number of unblocked anchors. In Chapter 3, we characterize the blind spot probability of a network, using tools from stochastic geometry to model the spatial randomness of the obstacle and anchor locations. In particular, a homogeneous Poisson point process is used to model the anchor locations, while a germ-grain model is employed to represent the obstacle locations and shapes. Using this model, we first characterize the regime over which the independent blocking assumption underestimates the blind spot probability of the typical target, which in turn, is characterized as a function of the distribution of the unshadowed area, as seen from the target location. Since this distribution is difficult to characterize exactly, we formulate the nearest two- obstacle approximation, which is equivalent to considering the effects of dependent blocking for only the nearest two obstacles from the target, while assuming independent blocking due to the remaining obstacles. As nearby obstacles induce greater blocking correlation than farther ones, this approximation achieves a reasonable trade-off between accuracy and tractability. Based on this quasi-independent blocking assumption, we derive a closed form (approximate) expression for the blind spot probability, which helps determine the intensity with which anchors need to be deployed so that the blind spot probability at a typical target location is bounded by a threshold,. Even if a target is not situated in a blind spot, the localization mean square error (MSE) depends on the relative geometry of the anchors, i.e., the point process formed by the unblocked 14 anchors, as seen from the target location. An outage occurs if the positioning error exceeds a pre-defined threshold, th . From a design perspective, characterizing the distribution of the positioning error over an ensemble of target and anchor locations is essential for providing probabilistic performance guarantees against outage. However, even for the simplest case where the anchors are deployed according to a Binomial point process and do not experience any blocking, this problem has remained unsolved. Hence, in Chapter 4, we tackle this open problem and derive a tractable approximation for the MSE distribution. Finally, some promising ideas for future work are proposed in Chapter 5 to conclude this dissertation. 15 Chapter 1 Introduction Accurate localization is a necessary and vital component of numerous applications such as navigation, search-and-rescue operations, surveillance, medical imaging etc., including an ever-growing list of emerging applications such as location-based advertising [1] and social networks [2], crowd-sensing [3] and crowd-sourcing [4], inventory tracking [5], assisted living [6], remote RFID [5, 7, 8] and so on. Generally speaking, a typical use-case scenario consists of a network of nodes with known coordinates, called anchors, deployed over a region of interest that contains one or more (possibly moving) targets that need to be localized, as shown in Fig. 1.1. Under Target Anchor Obstacle IP DP Blocked DP Figure 1.1: A typical localization use-case scenario consisting of anchors, targets and possibly obstacles, which may give rise to multipath. The location of the obstacles (and therefore, the propagation paths of the multipath components contributed by them) may or may not be known. 17 Anchor 1 Anchor 2 Anchor 3 Target r 1 r 2 (a) Each distance (range) value constrains the target to lie on a circle of radius equal to the range, with the corresponding anchor at the center, and the intersection of three or more such cir- cles provides an unambigu- ous solution for the target lo- cation. Anchor 4 Anchor 1 Anchor 3 Anchor 2 Target (b) The difference of a pair of range values from two different anchors constrains the target to lie on a hy- perbola. In this figure, the range to Anchor 1 is sub- tracted from all the other range values. A minimum of four anchors is required for unambiguous 2D local- ization in this case. Anchor Array 2 Anchor Array 1 θ 1 θ 2 Target (c) The angles formed by the target (with respect to the horizontal) at two or more anchor locations can be used to obtain an unambiguous solution for the target loca- tion. Figure 1.2: Basics of localization. this setting, target localization reduces to a relatively simple application of geometric and trigonometric principles, as shown in Fig. 1.2. In Fig. 1.2a, the unknown (2D) coordinates of the target can be determined unambiguously if its distance (also known as range) to at least three anchors is known. Each range value constrains the target to lie on a circle of radius equal to the range, with the corresponding anchor at the center, and the target location can be unambiguously determined by solving for the intersection of three or more such circles 1 . In a similar manner, the difference between pairs of ranges can also be used for localization, as illustrated in Fig. 1.2b. In this case, each range difference constrains the target to lie on a hyperbola and the intersection of three or more such curves unambiguously determines the target location. These 1 For 3D localization, the ranges to at least four anchors are required for unambiguous localization. 18 range-based localization techniques are referred to as multilateration 2 . Alternately, in Fig. 1.2c, the target coordinates can be determined if the angles formed with respect to the horizontal at two or more anchor locations are known. This technique is called triangulation. The distance between two points can be accurately determined by transmitting a short pulse [9], known as a ranging signal, from one of the locations and measuring its time-of-arrival (ToA) at the other point using a common reference clock. Thus, if the anchors in Fig. 1.1 are a network of wireless transceivers, then localization using ranges obtained from the ToA of the ranging signal (Fig. 1.2a) is commonly referred to as ToA-based localization. Similarly, the technique in Fig. 1.2b using range differences is also known as time-difference-of-arrival (TDoA)-based localization. Alternately, the range between a target and an anchor can also be obtained from the ratio of the transmitted and received powers of the ranging signal, as the signal strength decreases monotonically with distance. Thus, localization using ranges obtained in this manner is known as signal strength (SS) or received signal strength indicator (RSSI) based localization. The ranging signal can also be used to measure the angles required for triangulation if the anchors contain an antenna array, as shown in Fig. 1.2c. In particular, the angle-of-arrival (AoA) of the ranging signal at an anchor can be estimated from the variation of the signal phase across the array elements [10]. As a result, the localization technique illustrated in Fig. 1.2c is also known as AoA-based localization. In the above discussion, there is an implicit assumption that the targets in Fig. 1.1 have radio-frequency (RF) circuitry that enables them to either transmit or receive ranging signals. This holds true for a number of applications (e.g., location-based advertising, where the targets are smartphones), which are collectively categorized as active localization scenarios. On the other hand, there are many applications where the targets do not have any RF circuitry and only reflect or scatter the incoming signals from the anchors (e.g., tumors in medical imaging); these are collectively categorized as passive localization scenarios. Furthermore, for passive localization, the transceivers can also be replaced by disjoint transmitter (TX) and receiver (RX) 2 In some texts, the technique in Fig. 1.2a using absolute range values is referred to as trilateration, regardless of the number of anchors, whereas multilateration is used to exclusively refer to the technique in Fig. 1.2b involving range differences. 19 nodes (i.e., the multistatic anchor configuration). In this case, an anchor is functionally equivalent to a TX-RX pair. Such an anchor architecture is especially popular in the radar community, where it is known as distributed multiple input multiple output (MIMO) radar [11]. Finally, the classification of localization techniques, based on the properties of the ranging signal (e.g., ToA, AoA etc.), can be extended to passive localization as well, with the understanding that the propagation path of the ranging signal involves two hops (i.e., anchor!target!anchor for the monostatic transceiver anchor model and TX!target!RX for the multistatic distributed MIMO radar anchor model). For instance, for passive ToA-based localization under the distributed MIMO radar anchor model, each range value associated with an anchor-target pair constrains the target to lie on an ellipse instead of a circle, with the corresponding TX and the RX lying at its foci, and the target location can be obtained from the intersection of three or more ellipses, similar to Fig. 1.2a. The choice of a suitable localization technique for a given application depends on a number of factors, such as cost, spatial constraints, accuracy of the range measurements, desired localization accuracy, algorithmic complexity etc. For instance, angle-based localization may not be suitable for low-cost sensor network-based applications, as antenna arrays are typically larger and more expensive to realize than single-antenna solutions. On the other hand, while RSSI-based localization is simple and relatively inexpensive to implement, the accuracy of the range measurements are especially sensitive to channel propagation phenomena, such as log-normal shadowing, where the received signal power varies spatially in a random manner 3 as a result of propagation through obstacles in the environment [12]. For high accuracy (e.g., centimeter-level) applications, ToA-based localization using a high bandwidth (of the order of GHz) ranging signal is especially attractive, since a large bandwidth provides fine time resolution (because time and frequency domains form a Fourier-transform pair), which improves the accuracy of the range measurements [13]. As a result, wideband ToA-based localization has attracted a lot of research interest [14], and is therefore the main focus of this dissertation 4 . However, in order to consistently 3 This variation is typically modeled as a log-normal random variable, which explains the nomencla- ture. 4 Narrowband ToA-based localization using cellular base stations as anchors has also been extensively studied, although the accuracy is typically on the order of tens of meters [15]. 20 achieve high localization accuracy under a variety of environmental conditions, a number of challenges need to be addressed, which are briefly described below: a) Blocking: For accurate range estimation, the ToA of the ranging signal along the direct path (DP), corresponding to the line-of-sight (LoS) link between a target and an anchor, needs to be estimated accurately. However, the DPs between targets and anchors can be blocked by non-target obstacles in the environment. If the obstacles locations are known, then it may be possible to deploy anchors in such a way that LoS is never blocked. However, in many situations, the obstacle locations may not be known apriori; for example, 1. In tracking wildlife in a forest environment, the trees may act as obstacles. In this case, it is unreasonable to assume that all the obstacle locations are known in this case. 2. On a road or in a shopping mall, vehicles and humans may respectively act as obstacles intermittently. In such scenarios, LoS blocking can be modeled as a stochastic phenomenon. A common assumption in the literature is to consider the blocking across different links to be statistically independent [16]. However, due to the distributed (i.e., non-point) nature of most obstacles, the blocking of two or more links exhibits statistical dependence in general (e.g., when multiple anchors are blocked to a target by a common obstacle). This is referred to as dependent blocking which gives rise to the following phenomena: (i) Blind spots: A target is said to be in a blind spot if it does not have LoS to at least three anchors, since it cannot be localized in that case (Fig. 1.2a). Ignoring the dependence in DP blocking events can result in the underestimation of the blind spot probability at a typical target location. For instance, if two anchors, situated close to one another, are each blocked from a target with probability p, then the joint blocking probability for the anchors is also approximatelyp, which is greater than p 2 , the result obtained by assuming independent blocking. (ii) Unfavorable geometries: When there are more than three unblocked anchors to a target, it is desirable for the target to be situated within their 21 convex hull, since the achievable accuracy in such configurations is high, as reflected in metrics like the Cramer-Rao lower bound (CRLB) [17]. However, an obstacle that is really close to a target causes all unblocked anchors to be on only one side of the target, thereby giving rise to an unfavorable geometry. b) Multipath: The same obstacles that can block DPs may also cause the ranging signal to be reflected off them, which results in multiple copies of the rang- ing signal arriving at different times with varying strengths (Fig. 1.1). This phenomenon is referred to as multipath and each such copy is referred to as a multipath component (MPC). In particular, the DP component is also an MPC and the other MPCs, arising due to reflections off obstacles, are known as indirect paths (IPs, Fig. 1.1). It is easy to see that the ToA of the DP component is smaller than that of the IPs. However, multipath, when coupled with blocking, poses the following challenges to accurate ranging, which in turn affect the localization accuracy: (i) The strongest MPC [i.e., having the the highest signal-to-noise ratio (SNR)] need not correspond to the DP signal from a target, since a blocked LoS link may result in the DP component being either completely absent or severely attenuated with respect to the other IPs. As a result, ranging using the ToA of the strongest MPC can cause large errors. (ii) Due to DP blockage (Fig. 1.1), the first arriving MPC need not correspond to the DP either. As a result, ranging using the ToA of the earliest arriving MPC can also cause large errors. (iii) In spite of the large bandwidth, the time resolution is not infinite and therefore, the ranging accuracy is hampered when the DP signal overlaps with the other MPCs and is not resolvable (Fig. 1.3). Resource Allocation: In addition to blocking and multipath, which may be viewed as challenges imposed by the environment, other system-level issues, such as the efficient use of power and bandwidth resources [18, 19, 20, 21], interference management [22] and scheduling [23], also play an important role in the performance of localization systems. 22 t Received signal False ToA True ToA Figure 1.3: The two MPCs in blue appear as a single MPC (orange), since their ToAs are too close to be resolved. This leads to a ranging error since the measured ToA is not equal to the true ToA. The time resolution is inversely proportional to the bandwidth of the ranging signal. Some of the key metrics used to analyze the performance of localization systems are: (i) the target detection probability, (ii) the localization mean square error (MSE) which captures the positioning accu- racy of detected targets, (iii) the blind spot probability, which is a lower bound to the probability that a target is not detected, and (iv) the false alarm probability. My research efforts, culminating in this dissertation, have largely focused on the impact of dependent blocking on some of these metrics for ToA-based localization systems. Since the source of dependent blocking is (a) the unknown locations of non-point obstacles, and (b) the spatial distribution of anchors around a particular target, a key theme in my work is the use of stochastic geometry to model these phenomena. One strand of my research has been devoted to the development of a blocking aware localization algorithm that exploits dependent blocking to improve the detection probability, while another has looked at the design of localization networks that can provide probabilistic performance guarantees in terms of the localization error 23 as well as the occurrence of blind spots (e.g., a localization error of at most 1m more than 90% of the time, with a blind spot probability of at most 1%, say). Together, they have resulted in three major works, which are summarized below and which form the content of the next three chapters. While stochastic geometry has been used extensively to analyze wireless commu- nication systems, its use in studying localization/radar systems has been relatively minimal. This dissertation represents one of the first attempts to use stochastic geome- try to model and analyze the impact of random blockages on localization performance and the final chapter (Chap. 5) presents some ideas for future work based on the insights obtained in the first four chapters. Finally, although 2D localization has been assumed throughout, all the results presented here can be readily extended to the 3D case as well. 1.1 BayesianMulti-TargetLocalization The problem of localizing an unknown number of passive targets, all having the same radar signature (i.e., having the same reflection properties), by a distributed MIMO radar consisting of single antenna TXs and RXs is addressed in Chapter 2. Since all the targets ‘look the same’ from a signal perspective, it is not clear which received signal corresponds to which target. This data association is important, as an incorrect association could result in ghost targets being detected [16]. The problem is further compounded by dependent blocking and multipath. In its most general form, multi-target localization can be cast as a Bayesian estimation problem, where the propagation path of every MPC is estimated. However, when the environment map is unknown, this problem is ill-posed and hence, a tractable approximation is derived, where only the DPs corresponding to each target are identified. Under this framework, dependent blocking by obstacles in the environment appears as a Bayesian prior and the resulting localization algorithm works as follows: when three or more ellipses intersect at a point (see Fig. 1.2a), we first assume that they are DPs. We then compute the joint probability that LoS exists to the TXs and RXs in question at the point of intersection 5 . If this probability is sufficiently high, then we conclude that a target is 5 This joint LoS probability captures dependent blocking in its entirety. 24 present at the ellipse intersection point. The main contributions of this work are as follows: The general problem of localizing all targets and obstacles in an unknown environment is cast as a Bayesian estimation problem. This problem is shown to be ill-posed, following which we derive a tractable approximation called the Bayesian Multi-Target Localization (MTL) problem, where the objective is to localize only the targets. This is also a Bayesian estimation problem, where the joint DP blocking distribution plays the role of a prior. We propose a sub-optimal polynomial-time algorithm to solve the Bayesian MTL problem, which can be used even when only empirical blocking statistics, obtained via measurements or simulations, are available. This work has led to the following publications: R. S. Patil, S. Aditya, A. F. Molisch and H. Behairy, “An Experimental Investiga- tion of the Bayesian Passive Multi-Target Localization Algorithm",Workshop onAdvancesinNetworkLocalizationandNavigation(ANLN),Intl. Conf. on Communications(ICC)2018,KansasCity S. Aditya, A. F. Molisch, N. Rabeah and H. Behairy, “Localization of Multiple Targets with Identical Radar Signatures in Multipath Environments with Corre- lated Blocking",IEEETrans. WirelessCommun., vol. 17, no. 1, Jan. 2018, pp. 606-618. S. Aditya, A. F. Molisch and H. Behairy, “Bayesian Multi-Target Localization using Blocking Statistics in Multipath Environments",WorkshoponAdvances inNetworkLocalizationandNavigation(ANLN),Intl. Conf. onCommunica- tions(ICC)2015,London 25 (a) A realization of the germ point process. r (b) The grains are circles of radiusr. Figure 1.4: Example of a germ-grain model. 1.2 BlindspotProbabilityAnalysisusingStochastic Geometry The impact of dependent blocking on the blind spot probability at a typical target location is investigated in Chapter 3, using tools from stochastic geometry to model the randomness in the anchor locations, as well as the obstacle locations and shapes. This kind of analysis characterizes the performance over an ensemble of environment realizations, instead of a particular snapshot. Over an ensemble of realizations, the anchor locations can be viewed as a spatial random process, or a point process. A popular model for the anchor locations in localization networks is the homogeneous Poisson point process (PPP), which is characterized by an intensity parameter,, where the number of points over a 2D region of area A is a Poisson random variable with mean A. Similarly, if the map of the environment is unknown, then the obstacles can be viewed as a random congregation of shapes, which can be modeled using random shape theory. As an illustrative example, consider Fig. 1.4, which is obtained by starting with a realization of a point process (Fig. 1.4a) and drawing a circle of radius r with each point as the center (Fig. 1.4b). This results in a random collection of circles, which could be used to model trees in a forest, say 6 . Apart from the location of the shapes, which 6 To avoid the circles (trees) from overlapping with one another, the distance between any two points should be at least2r. Point processes exhibiting these distance constraints are called Matern hard-core point processes [24, Chap. 5]. 26 is governed by the underlying point process, randomness can also be introduced in the shapes themselves; for instance, by makingr a realization of a random variable, R, with a probability distribution function (pdf),f R (). In general, such shape-based random processes are called germ-grain models, where the points from the underlying point process are referred to as germs, and the shapes as grains. A special case of a germ-grain model is the Boolean model [24, Chap. 3], which satisfies the following conditions: The germs are distributed according to a homogeneous PPP. The grains are drawn independently from a grain distribution that is independent of the germ PPP. For the blind spot probability analysis, we assume a Boolean model for the obsta- cles. Using the grain size as a parameter, this model captures the dependent blocking of anchors induced by obstacles, unlike previous works that assume independent anchor blocking. The main contributions of this work are as follows: Using a homogeneous PPP to represent the anchor locations and a Boolean model for the obstacle locations and shapes, we express the blind spot probabil- ity at a given target location as a function of the probability distribution of the unshadowed area surrounding the target. We then show that the blind spot probability under the independent anchor blocking assumption depends only on the mean unshadowed area, instead of the entire probability distribution. In addition, we derive the conditions under which the independent blocking assumption underestimates the true blind spot probability. We then demonstrate that the probability distribution of the unshadowed area is difficult to characterize in closed form. As a result, we propose an approximate solution for characterizing the unshadowed area whereby in each environment realization, the unshadowed area is evaluated exactly up to the location of the second nearest obstacle and the remaining value beyond that is approximated by its mean. We refer to this as the nearest two-obstacle approximation and we show that it is equivalent to considering dependent blocking up to the 27 location of the second nearest obstacle and assuming independent blocking for farther obstacles, where the impact of blocking correlation is relatively minimal. In other words, the nearest two-obstacle approximation engenders a quasi-independent blocking assumption. Using the nearest two-obstacle approximation, we derive a closed form approx- imation for the blind spot probability as well as the conditions under which it yields a tighter bound on the true blind spot probability, relative to the indepen- dent blocking assumption. As a result, our work provides useful design insights, such as the intensity with which anchors need to be deployed so that the blind spot probability over the entire region is less than a threshold,. The following publications have resulted from this work: S. Aditya, H. S. Dhillon, A. F. Molisch and H. Behairy, “A Tractable Analysis of the blind spot Probability of Localization Networks under Correlated Blocking", submitted toIEEETrans. WirelessCommun., Dec 2017. S. Aditya, H. S. Dhillon, A. F. Molisch and H. Behairy, “Asymptotic blind spot Analysis of Localization Networks under Correlated Blocking using a Poisson Line Process", IEEE Wireless Commun. Lett., vol. 6, no. 5, Oct. 2017, pp. 654-657. S. Aditya, A. F. Molisch, H. S. Dhillon, H. Behairy and N. Rabeah, “blind spot Analysis of Localization Networks using Second-order Blocking Statistics", Intl. Conf. onUbiquitousWirelessBroadband(ICUWB)2016,Nanjing 1.3 MeanSquareErrorAnalysis Even if a target is not situated in a blind spot, the localization accuracy depends on the relative geometry of the anchors, with respect to the target. Specifically, the positioning MSE is a function of (a) the distance-dependent SNRs of the unblocked anchor-target links, and (b) the pairwise angles subtended by the anchors at the target. In particular, an outage occurs if the positioning error exceeds a pre-defined threshold, th . From a design perspective, characterizing the distribution of the positioning error 28 over an ensemble of target and anchor locations is essential for providing probabilistic performance guarantees against outage. The same obstacle and anchor model used for the blind spot probability analysis is suitable for capturing the geometry of the unblocked anchors and their effect on the localization accuracy. However, even for the relatively simple no-blocking case, withN anchors uniformly and independently distributed around a target [known as the Binomial point process (BPP)], this problem is difficult and has remained unsolved. This is largely because the MSE expression is a tightly coupled function of the anchor distances and the angular positions, which renders a stochastic geometry-based analysis intractable. As a result, the focus of this work is on the no-blocking case and the contributions are as follows: For a given target, we assume that the anchors that are within its communication range are distributed according to a BPP over an annular region centered at the target. For this setup, we model the SNR of the anchor-target links using a distance-based pathloss model. As a result, the MSE is a function of the anchor distances and angular positions, relative to the target. We formulate an approximate expression for the MSE, where the coupling between the anchor distances and angular positions is removed. In particular, the MSE is approximated in terms of the product of two independent random variables. The complementary cumulative distribution function (ccdf) of this approxi- mation is derived in closed form and shown to be accurate at estimating the true ccdf of the MSE. Thus, from a design perspective, this result is useful in determining the number of anchors so that the outage probability is at most, for any2 (0; 1). The publications resulting from this work are as follows: S. Aditya, H. S. Dhillon, A. F. Molisch and H. Behairy, “Characterizing the Impact of SNR Heterogeneity on Time-of-Arrival based Localization Outage Probability", submitted toIEEETrans. WirelessCommun., Apr. 2018. 29 1.4 CommonNotation Throughout this dissertation, bold lowercase Latin (e.g.,a) or Greek letters (e.g.,) are used to represent vectors, unless otherwise mentioned. In particular,1 denotes the all-one vector. R denotes the set of real numbers. P() denotes the probability measure and the expectation operator is denoted either byE X [:], to explicitly indicate expectation with respect to a random variable, X; or byE[:], when the context is clear. 30 Chapter 2 BayesianMulti-Target Localization(MTL) 2.1 Introduction In this chapter, we address the problem of localizing an unknown number of point targets, all having the same radar signature, distributed in an unknown environment, using a distributed MIMO radar consisting of single antenna TXs and RXs. 2.1.1 RelatedWork There has been a considerable amount of literature on MIMO radar over the last decade. The fundamental limits of localization in MIMO radar networks were studied in [25]. A number of works have dealt with MTL using co-located antenna arrays at the TX and RX [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. The single-target localization problem using widely-spaced antenna arrays was investigated in [39, 40] and the multi-target case in [41]. However, none of these works address the issues of blocking and multipath common in an indoor environment. The works closest to ours are [16] and [42], where MTL in a distributed MIMO radar setting is addressed. The experiments and the system model in [42] do not consider the effect of blocking and a brute-force method is used for data association, which is computationally infeasible for a large number of targets, as shown in Section 2.4. On the other hand, [16] considers 31 Figure 2.1: Correlated blocking: An example the effect of blocking, but relies on the assumption of a constant and independent blocking probability for all DPs. In reality however, the DP blocking events in any environment are not mutually independent. As shown in Fig. 2.1, the location of the two TXs is such that if one of them has LoS to the target, it is highly likely that the other does as well. Similarly, if one of them is blocked with respect to the target, it is highly likely that the other is as well. In other words, the DP blocking events are, in general, correlated and the extent of correlation depends on the network geometry. In this work, we investigate how correlated blocking can be exploited to obtain better location estimates for the targets. 2.1.2 Contributions The main contributions of this work are as follows: The general problem of localizing all targets and obstacles in an unknown envi- ronment is cast as a Bayesian estimation problem which serves to reconstruct the propagation path of each extracted multipath component, akin to ray tracing. We show this problem to be ill-posed and proceed to derive a tractable approxi- mation called the Bayesian MTL problem, where the objective is to localize only the targets, but not the obstacles. This is also a Bayesian estimation problem where the joint DP blocking distribution plays the role of a prior. We propose a sub-optimal, polynomial time algorithm to solve the Bayesian MTL problem, which can be used even when only empirical blocking statistics, obtained via measurements or simulations, are available. 32 2.1.3 Organization This chapter consists of six sections. In the system model in Section 2.2, we define decision variables to decide if an MPC is a DP, IP or a noise peak. The generalized problem of localizing all targets and obstacles is formulated as a Bayesian estimation problem in Section 2.3, where along with the target and obstacle locations, the aforementioned decision variables are the estimation parameters. Furthermore, this problem is shown to be ill-posed and a more tractable approximation called the Bayesian MTL problem is derived, where the objective is to localize only the targets. In Section 2.4, the brute force solution to the Bayesian MTL problem is shown to have exponential complexity in the number of targets and TX-RX pairs (TRPs). As a result, a sub-optimal polynomial time algorithm taking correlated blocking into account is proposed instead. Simulation results for the proposed algorithm are presented in Section 2.5, while experimental results from a measurement campaign are discussed in Section 2.6. Finally, Section 2.7 concludes this chapter. 2.1.4 AdditionalNotation The following notation is specific to this chapter: For a collection of scalarsfa ij :i2 J 1 ;j2J 2 g, whereJ 1 andJ 2 are discrete index sets,vec(a ij ) denotes the column vector containing alla ij , ordered first according to indexi, followed byj and so on. For continuous random variables X and Y,f(X; Y) denotes their joint pdf,f(X) the marginal pdf of X, andf(XjY) the conditional pdf of X, given Y. 2.2 SystemModel Consider a network consisting of M TX TXs and M RX RXs, each equipped with a single omni-directional antenna and deployed in an unknown environment. An unknown number of stationary point targets are present and the objective is to localize all of them. We assume that the environment has non-target obstacles, which can either block some target(s) to some TX(s) and/or RX(s), and/or give rise to IPs. All TX and RX locations are assumed to be known. The number of TRPs, denoted byI, equalsM TX M RX and the TX and RX forming thei-th TRP (i = 1; ;I) is given by a lookup table, in general (e.g., Table 2.1). 33 TRP No. 1 2 3 4 5 6 7 8 9 TX No. 1 2 3 1 2 3 1 2 3 RX No. 1 1 1 2 2 2 3 3 3 Table 2.1: TRP indexing notation for a 3 TX, 3 RX case Let the TX and the RX for the i-th TRP be located at (c i ;d i ) and (a i ;b i ), re- spectively. We assume that the TXs use orthogonal signals so that the RXs can distinguish between signals sent from different TXs. For each TRP, the RX extracts the channel impulse response from the received signal; an MPC is assumed to exist at a particular delay (within the resolution limit of the RX) when the amplitude of the impulse response at that delay bin exceeds a threshold; alternately, a maximum likelihood (ML) estimator or other high-resolution algorithms can be used to extract the amplitudes and delays of all the MPCs [43, 44]. All MPCs that do not involve a reflection off a target (e.g., TX!obstacle!RX) are assumed to be removed by a background cancellation technique. For stationary or even slow-moving targets, a simple way to achieve this is to measure the impulse responses for all the TRPs when no targets are present. This set of template signals can then be subtracted from the signals obtained when the target(s) are introduced, which would remove MPCs of the form TX!obstacle!RX since they appear twice [45]. Other background subtraction techniques for localization and tracking applications are described in [46, 47]. An MPC involving more than two reflections is assumed to be too weak to be detected. Finally, two or more MPCs could have their delays so close to one another that they can be unresolvable due to finite bandwidth. Under this model, each extracted MPC could be one or more of the following: 1. A DP to one or more targets, which occurs when a target has LoS to both the TX and RX in question. 2. An IP of the first kind, which is of the form TX!target!obstacle!RX. 3. An IP of the second kind, having the form TX!obstacle!target!RX. 4. A noise peak. Each MPC gives rise to a ToA estimate which, in turn, corresponds to a range estimate. If only additive white Gaussian noise (AWGN) is present at the RXs, then 34 each ToA estimate is approximately perturbed by zero-mean Gaussian errors whose variance depends on the SNR via the CRLB and the choice of estimator [17]. For simplicity, it is assumed that all ToA estimation errors have the same variance ^ 2 . The extension to the general case where the variance is different for each MPC is straightforward. Thus, for a DP, the true range of the target from its TRP is corrupted by AWGN of variance 2 =c 2 ^ 2 , wherec is the speed of light in the environment. Suppose thei-th TRP hasN i MPCs extracted from its received signal. Letr ij de- note the range of thej-th extracted MPC at thei-th TRP and letr i = [r i1 r iN i ] T 2 R N i 1 denote the vector of range estimates from thei-th TRP. Similarly, let r = [r 1 r I ] T 2R N 1 N 2 :::N I 1 denote the stacked vector of range estimates from all TRPs. Ifr ij is a DP corresponding to a target at (x t ;y t ), then the conditional pdf of r ij , given (x t ;y t ), is Gaussian and denoted byf DP (r ij jx t ;y t ) and has the following expression: f DP (r ij jx t ;y t ) = 1 p 2 exp (r ij r i (x t ;y t )) 2 2 2 (2.1) wherer i (x t ;y t ) = p (x t a i ) 2 + (y t b i ) 2 + p (x t c i ) 2 + (y t d i ) 2 r i (x t ;y t ) denotes the range of a target at (x t ;y t ) from thei-th TRP. Similarly, let f IP;1 (r ij jx t ;y t ;u m ;v m ) andf IP;2 (r ij jx t ;y t ;u m ;v m ) denote the conditional IP pdfs of the first and second kind, respectively, given a target at (x t ;y t ) and an obstacle at (u m ;v m ). These pdfs are also Gaussian, f IP;1 (r ij jx t ;y t ;u m ;v m ) = 1 p 2 exp (r ij l i (x t ;y t ;u m ;v m )) 2 2 2 (2.2) f IP;2 (r ij jx t ;y t ;u m ;v m ) = 1 p 2 exp (r ij m i (x t ;y t ;u m ;v m )) 2 2 2 (2.3) wherel i (x t ;y t ;u m ;v m ) = p (c i x t ) 2 + (d i y t ) 2 + p (x t u m ) 2 + (y t v m ) 2 + p (u m a i ) 2 + (v m b i ) 2 m i (x t ;y t ;u m ;v m ) = p (c i u m ) 2 + (d i v m ) 2 + p (u m x t ) 2 + (v m y t ) 2 + p (x t a i ) 2 + (y t b i ) 2 l i (x t ;y t ;u m ;v m ) andm i (x t ;y t ;u m ;v m ) respectively denote the path length between thei-th TRP, a target at (x t ;y t ) and an obstacle at (u m ;v m ) for an IP of the first and second kind. Finally, the range of a noise peak is modelled as a uniform random 35 variable in the interval [0;R obs ], whereR obs denotes the maximum observable range in the region of interest. Let the number of targets and obstacles be denoted byT andM, respectively. To determine all the unknowns, every MPC needs to be accounted for. Hence, we define the following variables, k it = 8 < : 1; ift-th target is NOT blocked to thei-th TRP 0; else (2.4) g imt = 8 > > > < > > > : 1; if9 an IP of the first kind between thei-th TRP,m-th obstacle andt-th target 0; else (2.5) h imt = 8 > > > < > > > : 1; if9 an IP of the second kind between thei-th TRP,m-th obstacle andt-th target 0; else (2.6) The values ofk it ,g imt andh imt (i2f1; ;Ig,m2f1; ;Mg,t2f1; ;Tg) capture the ground truth regarding the existence of DPs and IPs and depend on the map of the environment, which is unknown. Therefore, these quantities need to be estimated fromr. To do this, we define the following decision variables to determine if an MPCr ij is a DP, IP or noise peak, ~ k ijt = 8 < : 1; ifr ij is a DP to thet-th target 0; else (2.7) ~ g ijmt = 8 > > > < > > > : 1; ifr ij is an IP of the first kind between the m-th obstacle andt-th target 0; else (2.8) ~ h ijmt = 8 > > > < > > > : 1; ifr ij is an IP of the second kind between the m-th obstacle andt-th target 0; else (2.9) 36 ~ z ij = 8 < : 1; ifr ij is a noise peak 0; else (2.10) Since two or more resolvable MPCs cannot be DPs to the same target or IPs of a particular kind between a given target-obstacle pair, it follows that the estimates of k it ,g imt andh imt , denoted by ^ k it , ^ g imt and ^ h imt , respectively, are given by: ^ k it = N i X j=1 ~ k ijt (2.11) ^ g imt = N i X j=1 ~ g ijmt (2.12) ^ h imt = N i X j=1 ~ h ijmt (2.13) Before concluding this section, we define the following vectors which shall be useful when the Bayesian MTL problem is defined in the next section Ground truth:k =vec(k it ); g =vec(g imt ); h =vec(h imt ) (2.14) DP/IP/noise peak decisions: ~ k =vec( ~ k ijt ); ~ g =vec(~ g ijmt ); ~ h =vec( ~ h ijmt ); ~ z =vec(~ z ij ) (2.15) Estimates of ground truth: ^ k =vec( ^ k it ); ^ g =vec(^ g imt ); ^ h =vec( ^ h imt ) (2.16) 2.3 BayesianMTL Using the notation from the previous section, the MTL problem in multipath environ- ments with correlated blocking is formulated as a Bayesian estimation problem in this section. We first show that the obstacle locations cannot be determined uniquely, in general, as they are not point objects. Then, we show that that the distribution ofk in (2.14) captures correlated blocking in its entirety and acts as a prior. We also assume a single error at most between the entries of ^ k t andk t in order to obtain a tractable algorithm for the MTL problem in Section 2.4. Let tar =f(x t ;y t ) :t = 1; ;Tg and sc =f(u m ;v m ) : 1; ;Mg denote the collection of target and obstacle locations, respectively, and let ~ p dec = [ ~ k; ~ g; ~ h; ~ z] 37 denote the vector of decision variables. Using the terminology defined in Section 2.2, determining the location of all targets and obstacles can be formulated as a Bayesian estimation problem in the following manner, maximize T;M;tar;sc; ~ p dec ;k;g;h f(rj~ p dec ; tar ; sc ;k;g;h)f( tar ; sc ) P( ^ k; ^ g; ^ hj tar ; sc ;k;g;h)P(k;g;hj tar ; sc ) (2.17) subject to (2:11); (2:12); (2:13) X j;t ~ k ijt + X j;m;t (~ g ijmt + ~ h ijmt ) + X j ~ z ij N i ;8i (2.18) ~ k ijt ; ~ g ijmt ; ~ h ijmt ; ^ k it ; ^ g imt ; ^ h imt 2f0; 1g;8i;j;t;m (2.19) where the first term in the objective (2.17) denotes the likelihood function and the remaining three terms denote the prior. A detailed explanation of all the terms and constraints in (2.17)-(2.19) is provided below: (a) The term f( tar ; sc ) denotes the prior joint distribution of the target and obstacle locations. It is reasonable to assume that the target and obstacle locations are independent of each other. Hence,f( tar ; sc ) =f( tar )f( sc ). In addition,f( tar ) andf( sc ) are both assumed to be uniform pdfs over the region of interest. (b) The discrete distributionP(k;g;hj tar ; sc ) represents the geometry of the environment, such as the blocked DPs for each TRP, the IPs (if any) be- tween a target-obstacle pair etc. Let TX =f(c i ;d i ) : i = 1; ;Ig and RX =f(a i ;b i ) :i = 1; ;Ig denote the collection of TX and RX locations, respectively. TX and RX are known quantities and for a given set of values for tar and sc , the set env =f TX ; RX ; tar ; sc g completely describes all the propagation paths in the environment and the values ofk,g andh are deterministic functions of env , denoted by k (det) ( env ), g (det) ( env ) and h (det) ( env ), respectively 1 . Hence, P(k;g;hj tar ; sc ) = 1 k (det) (env) (k) 1 g (det) (env) (g) 1 h (det) (env) (h) (2.20) 1 This is akin to ray-tracing 38 where 1 y (x) equals 1 ifx =y and 0, otherwise. (c) The estimates ^ k it , ^ g imt and ^ h imt may differ from their respective ground truths, k it , g imt andh imt due to noise or IPs. Assuming that ^ k it (or ^ g imt , ^ h imt ) is conditionally independent of other estimates, givenk it (org imt ,h imt ), we get P( ^ k; ^ g; ^ hj tar ; sc ;k;g;h) =P( ^ k; ^ g; ^ hjk (det) ( env );g (det) ( env );h (det) ( env )) (2.21) = Y i;t;m P( ^ k it jk (det) it ( env ))P(^ g imt jg (det) imt ( env ))P( ^ h imt jh (det) imt ( env )) (2.22) where (2.21) follows from (2.20). (d) ~ p dec is a sufficient statistic for estimatingk,g andh. Hence, the likelihood function, f(rj~ p dec ; tar ; sc ;k;g;h), equals f(rj~ p dec ; tar ; sc ). Further, f(rj~ p dec ; tar ; sc ) decomposes into product form as the noise terms on each r ij are mutually independent. Thus, f(rj~ p dec ; tar ; sc ;k;g;h) =f(rj~ p dec ; tar ; sc ) = Y i;j f(r ij j~ p dec ; tar ; sc ) wheref(r ij j~ p dec ; tar ; sc ) = Y t;m (f DP (r ij jx t ;y t )) ~ k ijt (f IP;1 (r ij jx t ;y t ;u m ;v m )) ~ g ijmt (f IP;2 (r ij jx t ;y t ;u m ;v m )) ~ h ijmt 1 R obs ~ z ij (2.23) (e) Finally, constraint (2.18) ensures that the number of DPs, IPs and noise peaks received at thei-th TRP is at leastN i , the number of resolvable MPCs extracted at thei-th TRP. 39 After taking natural logarithms, (2.17) can be re-written as follows to obtain problemP 1, wherek,g andh are no longer unknowns due to (2.20): P 1 : minimize T;M;~ p dec ;tar;sc 1 2 2 4 X i;j;t ~ k ijt (r ij r i (x t ;y t )) 2 3 5 + 1 2 2 4 X i;j;t;m ~ g ijmt (r ij l i (x t ;y t ;u m ;v m )) 2 3 5 + 1 2 2 4 X i;j;t;m ~ h ijmt (r ij m i (x t ;y t ;u m ;v m ) 2 3 5 + 2 4 X i;j;t ~ k ijt + X i;j;m;t (~ g ijmt + ~ h ijmt ) 3 5 log p 2 + 0 @ X i;j ~ z ij 1 A logR obs X i;t logP( ^ k it jk (det) it ( env )) X i;m;t logP(^ g imt jg (det) imt ( env )) X i;m;t logP( ^ h imt jh (det) imt ( env )) (2.24) subject to (2:11); (2:12); (2:13); (2:18); (2:19) Typically, sc represents a finite collection of points belonging to distributed non-point objects (e.g., a wall), where reflection takes place. A minimum of three reflections are needed at each (u m ;v m ) for uniquely determining sc , which need not be satisfied in all circumstances. Hence,P 1 is ill-posed if the map of the environment is unknown 2 . To makeP 1 tractable, we restrict ourselves to localizing only the targets by retaining those terms and constraints involving just the DPs in (2.17)-(2.19). This gives rise to the following approximation,P 2, which is also a Bayesian estimation problem that accounts for all the DPs P 2 : minimize T; ~ k;tar 0 @ X i;j;t ~ k ijt 1 A log p 2 logP(kj tar ) 2 If the map of the environment is known, thenP1 is not ill-posed and the IPs can be re-cast as virtual DPs, obtained from virtual TXs and RXs [48]. 40 + 1 2 2 4 X i;j;t ~ k ijt (r ij r i (x t ;y t )) 2 3 5 X i;t logP( ^ k it jk it ) (2.25) subject to ~ k ijt ; ^ k it 2f0; 1g;8i;j;t;m (2.26) X j ~ k ijt = ^ k it (2.27) The joint DP blocking distributionP(kj tar ) inP 2 is no longer a discrete-delta function, like (2.20). Instead, it depends on the distribution of obstacle locations in the environment. From (2.4),k it = 0 if either the TX or the RX of thei-th TRP does not have LoS to (x t ;y t ); hence,k it can be expressed as a product of two terms in the following manner: k it =v i T ;t w i R ;t ; (2.28) wherev i T ;t = 8 < : 1; if thei T -th TX has LoS to (x t ;y t ) 0; else w i R ;t = 8 < : 1; if thei R -th RX has LoS to (x t ;y t ) 0; else k it can be interpreted as a Bernoulli random variable when considering an ensemble of settings in which the obstacles are placed at random. For vectorsk t = [k 1t ; ; k It ], v t = [v 1;t ; ; v M TX ;t ] andw t = [w 1;t ; ; w M RX ;t ], it can be seen thatk t = w t N v t , where N denotes the Kronecker product. k t is a vector of dependent Bernoulli random variables (Fig. 2.1) and shall henceforth be referred to as the blocking vector at (x t ;y t ). Note thatk = vec(k it ) = [k 1 ; ; k T ] and therefore, P(kj tar ) = P(k 1 ; ;k T ). In general, two or more blocking vectors may also be dependent as nearby targets can experience similar blocking. Thus, the joint dis- tributionP(k 1 ; ;k T ) captures correlated blocking in its entirety. Consequently, target-by-target localization is not optimal, in general. However, for ease of com- putation, we resort to such an approach in this chapter, thereby implicitly assuming independent blocking vectors at distinct locations, i.e.,P(kj tar ) Y t P(k t ). Among the 2 I possible values,k t can only take on (2 M TX 1)(2 M RX 1) + 1 physically realizable values, which can be expressed in the form w t N v t (e.g., 41 k t = [1 1 0 1 1 0 1 1 0] = [1 1 1] N [1 1 0] for the TRP indexing notation in Table 2.1). These are referred to as consistent blocking vectors while the remaining values are inconsistent (e.g., k t = [1 1 0 1 1 1 1 0 0]). Ifk t is inconsistent, then P(k t ) = 0. To characterizeP( ^ k it jk it ), a distinction between two kinds of estimation errors needs to be made: a) The DP corresponding to thet-th target at thei-th TRP may not be detected if the noise pushes the range estimate far away from the true value. As a result, ^ k it = 0 whenk it = 1. If the noise is independent and identically distributed (iid) for all TRPs, we may assume thatP( ^ k it = 0jk it = 1) = 01 (8 i;t), where 01 is determined by the SNR and the ToA estimator. b) If the DP for thet-th target at thei-th TRP is blocked, but a noise peak or IP is mistaken for a DP because it has the right range, then ^ k it = 1 andk it = 0. P( ^ k it = 1jk it = 0) depends on the obstacle distribution and varies according to TX, RX and target locations. However, in the absence of IP statistics, we make the simplifying assumption thatP( ^ k it = 1jk it = 0) = 10 , for alli;t. The availability of empirical IP statistics would obviously improve localization performance. Let ^ k t = [ ^ k 1t ; ; ^ k It ] denote the estimated blocking vector at (x t ;y t ). While ^ k t can, in principle, take on all 2 I values, a false alarm is less likely if ^ k t is a short Hamming distance away from a consistent vector having high probability. LetK denote the set of consistent blocking vectors. We restrict ^ k t to be at most a unit Hamming distance from some element inK. This assumption is reasonable when the number of obstacles is small and the SNR at all RXs is sufficiently high. Given ^ k t , let K t K denote the set of consistent vectors that are at most a unit Hamming distance away from ^ k t . Then, P( ^ k t ) = X kt2Kt P( ^ k t jk t )P(k t ) = X kt2Kt Y i P( ^ k it jk it ) ! P(k t ) (2.29) 42 8 > < > : X kt2Kt 01 01 (1 01 ) 11 10 10 (1 10 ) 00 P(k t ); ifK t 6=; 0; otherwise (2.30) where 01 =jfi : ^ k it = 0;k it = 1gj 11 =jfi : ^ k it = 1;k it = 1gj 10 =jfi : ^ k it = 1;k it = 0gj 00 =jfi : ^ k it = 0;k it = 0gj Using (2.29) and assuming independent blocking vectors at distinct points (i.e., target-by-target detection),P 2 can be reduced to the Bayesian MTL problemP 3, given below: P 3 : minimize T; ~ k;tar 2 4 1 2 0 @ X i;j;t ~ k ijt (r ij r i (x t ;y t )) 2 1 A 0 @ X i;j;t ~ k ijt 1 A log p 2 3 5 X t logP( ^ k t ) (2.31) subject to (2:26); (2:27) A matchingq t (r) =fr ij 2 rj ~ k ijt = 1g is the set of DPs corresponding to thet-th target. Given q t (r) and a point (x t ;y t ), the term in square parentheses in (2.31) determines if the ellipses corresponding to the MPCs inq t (r) pass through (x t ;y t ) or not. The other term in (2.31) plays the role of a prior by determining the probability of the blocking vector, ^ k t , obtained fromq t (r), at (x t ;y t ). The objective in (2.31) is minimized only when both these quantities are small. To solveP 3, a mechanism for obtaining the matchings for all targets is required, for which it is necessary to distinguish between DPs and IPs. Since the IP distribution is unknown, none of the conventional tools such as Bayesian, minimax or Neyman-Pearson hypothesis testing can be used for this purpose. In the next section, we describe our DP detection technique and propose a polynomial-time algorithm to solveP 3. 43 2.4 MTLAlgorithmusingBlockingStatistics The number of matchings possible forT targets,M obstacles andI TRPs is ( I 3 N 3 + I 4 N 4 + I I N I ) T , whereN = (2M + 1)T is an upper bound on the number of MPCs extracted at each TRP, ignoring noise peaks. The computational complexity of a brute-force search over all possible matchings for solvingP 3 isO(N IT ), which is intractable for a large number of TRPs and/or targets. However, for the optimal matching, both the terms in the square parenthesis in (2.31) are expected to be small. We use this to define a likelihood function that enables to obtain an accurate solution forP 3. To obtain accurate matchings in a tractable manner, we employ an iterative approach. Consider, without loss of generality, a matchingq (i1) t (r) for thet-th target consisting of MPCs from the firsti 1 TRPs (3iI). The size ofq (i1) t (r) is at mosti1. Let (^ x (i1) t ; ^ y (i1) t ) denote the estimate of the target location obtained from q (i1) t (r) (e.g., using the two-step estimation method [17]). For an MPCr ij i from thei-th TRP, letq (i) t;temp (r) =q (i1) t (r)[r ij i and let ^ k (i) t denote thei-length partial blocking vector at (^ x (i1) t ; ^ y (i1) t ), obtained fromq (i) t;temp (r). Ifq (i) t;temp (r) consists entirely of DPs from (x t ;y t ), then (i) the ellipses corresponding to its constituent MPCs should pass close to (x t ;y t ), and (ii) the blocking vector ^ k (i) t should have high probability. This motivates the definition of a blocking-aware vector likelihood function,l B (q (i) t;temp (r)), defined as follows: l B (q (i) t;temp (r)) = L E (q (i) t;temp (r)) (q (i) t;temp (r)) ; logP( ^ k (i) t ) ! (2.32) whereL E (q (i) t;temp (r)) =r ij i r i (^ x (i1) t ; ^ y (i1) t ) (2.33) and(q (i) t;temp (r)) is the standard deviation ofL E (q (i) t;temp (r)). If the ellipses corresponding to the MPCs inq (i) t;temp (r) pass through the vicinity of (x t ;y t ), thenL E (q (i) t;temp (r)) should be very small in magnitude. Under this condition, it can be shown by a first order Taylor series expansion thatL E (q (i) t;temp (r)) is approxi- mately a zero-mean Gaussian random variable [16]. Hence, ifjL E (q (i) t;temp (r))=(q (i) t;temp (r))j , where is an ellipse intersection threshold, then we conclude that r ij i passes through (^ x (i1) t ; ^ y (i1) t ). 44 If the above ellipse intersection condition is satisfied, then the term logP( ^ k (i) t ), which denotes the blocking likelihood ofq (i) t;temp (r) at (^ x (i1) t ; ^ y (i1) t ), needs to be small as well. Hence, if logP( ^ k (i) t ) , where (> 0) is a blocking thresh- old, then we defineq (i) t (r) = q (i) t;temp (r) and compute a refined target location es- timate (^ x (i) t ; ^ y (i) t ) from q (i) t (r). Otherwise, q (i) t (r) = q (i1) t (r) and (^ x (i) t ; ^ y (i) t ) = (^ x (i1) t ; ^ y (i1) t ). This motivates an algorithmic approach that is divided into stages, indexed byi. In general, let (z 1 ;z 2 ;:::;z I ), a permutation of (1; 2; ;I), be the order in which TRPs are processed. At the beginning of thei-th stage (3iI), eachq (i1) t (r) has at mosti1 entries. During thei-th stage, all the DPs among the MPCs of thez i -th TRP are identified to obtain a set of matchingsfq (i) t (r)g for each targett. Since the blocking likelihood logP( ^ k (i) t ) is non-decreasing ini, a matching and its corresponding target location can be removed from consideration if at any stage its blocking likelihood exceeds the blocking threshold,. To initialize the algorithm, letP (a;b;j a ;j b ) denote the points of intersection of the ellipses corresponding to thej a -th MPC of thea-th TRP and thej b -th MPC of theb-th TRP. For the initial set of matchings (i.e.,i = 3), P (z 1 ;z 2 ;j z 1 ;j z 2 ) is computed for allj z 1 ;j z 2 (1j z 1 N z 1 ; 1j z 2 N z 2 ). There can be at most four points in anyP (z 1 ;z 2 ;j z 1 ;j z 2 ) and each such point is an ML estimate of the target location for the matchingq (2) t (r) =fr z 1 ;jz 1 ;r z 2 ;jz 2 g. Hence, the target location estimate (^ x (2) t ; ^ y (2) t ) need not be unique. Furthermore, in thei-th stage (i 4) we also computeP (z u ;z i ;j zu ;j z i ) for allj zu ;j z i (8u<i) to identify previously blocked targets. Thus, any intersection of two ellipses is a potential target location to begin with. At each such location, the likelihood of a target being present is updated depending on the number of other ellipses passing around its vicinity. Unlikely target locations, corresponding to matchings whose likelihood [given by l B (:)] does not satisfy the thresholds and , are eliminated at each stage. The number of targets that remain at the end is the estimate ofT . The thresholds and can be chosen to trade off between the probabilities of detection and false alarm (see Section 2.5). In updating matchings at each stage, a matching is consistent (inconsistent) if the corresponding blocking vector is consistent (inconsistent). If ^ k (i) t is inconsistent, then letK (i) t denote the set of consistenti-length partial blocking vectors that are at most a unit Hamming distance away from ^ k (i) t . The following cases are of interest then: 45 (a) IfK (i) t is empty, thenP( ^ k (i) t ) = 0 (from (2.29) and (2.30), which holds for partial blocking vectors as well) and logP( ^ k (i) t ) =1. Hence, we conclude that a target is not present at the estimated location. (b) IfK (i) t is not empty, then each element ofK (i) t is a feasible ground truth. In particular, an element inK (i) t whose Hamming weight is lower than that of ^ k (i) t represents a ground truth where exactly one MPC in q (i) t;temp (r) is not a DP. For each such element, a new matching can be derived by removing the corresponding non-DP from q (i) t;temp (r) and evaluating a new blocking likelihood. On the other hand, an element ofK (i) t whose Hamming weight exceeds that of ^ k (i) t represents a ground truth where one DP is absent from q (i) t;temp (r) due to noise. Unlike the previous case, no modification of the matching is possible and the blocking likelihood of ^ k (i) t is computed according to (2.29)-(2.30). In this manner, it is possible that multiple matchings may exist for a single potential target location, each corresponding to a different ground truth. All the matchings whose blocking likelihood satisfies the threshold are retained, since it is premature to determine the most likely ground truth until all TRPs are considered. After the I-th TRP has been processed, if multiple matchings still exist for thet-th target, then the one that minimizes the objective function in (2.31) is declared the true matching and the corresponding (^ x (I) t ; ^ y (I) t ) is the location estimate for thet-th target. Thus, by construction, the only inconsistent matchings at each stage are due to missing DPs, as specified in (b) above. Moreover, a finite value of ensures that an inconsistent ^ k (i) t is always a unit Hamming distance away from consistency, due to (2.30). Algorithm 1 lists the pseudocode of the Bayesian MTL algorithm. 2.4.1 ComplexityofBayesianMTLAlgorithm Let ^ T (i) denote the number of targets identified at the end of stagei. The following relation holds, ^ T (i) ^ T (i 1) + i 1 2 N 3 ; (i = 4; ;I) (2.34) and, ^ T (3)N 3 (2.35) 46 At the end ofi 1-th stage, each target can have at most (i 1) matchings. Hence, O(i ^ T (i1)N) likelihood computations are carried out in thei-th stage due to existing targets. The second term in (2.34) is an upper bound on the number of new targets that can be identified in theith stage and the number of likelihood computations due to these isO(i 2 N 3 ). At each stage, the number of potential targets increases at most polynomially inN andI (2.34). Hence, the number of likelihood computations is also polynomial inN andI. The reduction in complexity occurs because target locations are determined by ‘grouping’ pair-wise ellipse intersections that are close together. Since there are onlyO(I 2 N 2 ) ellipse intersections to begin with, it is intuitive that the proposed algorithm terminates in polynomial-time. 2.4.2 LimitationsofBayesianMTLAlgorithm The Bayesian MTL algorithm assumes complete knowledge of the distribution of k t at all locations (x t ;y t ). This would have to be obtained either from very detailed theoretical models or exhaustive measurements, neither of which might be feasible in practice. A sub-optimal, but more practical, alternative could involve the use of second-order statistics of k t . In particular, the Mahalanobis distance, defined as q ( ^ k t m t ) T C 1 t ( ^ k t m t ), wherem t andC t respectively denote the mean vector and covariance matrix ofk t and (:) T and (:) 1 denote the matrix transpose and inverse operations, respectively, can be compared to a threshold 2 as the basis for a blocking likelihood decision. Even in this simplified case, one still needs the mean blocking vector and the covariance matrix at each point. In practice, these can be measured at only at a fixed set of grid points. Hence, the accuracy of the algorithm depends on the grid resolution of the measured data. 2.5 SimulationResults In this section, we simulate the performance of the Bayesian MTL algorithm intro- duced in Section 2.4. In Section 2.5.1, the algorithm is validated by reproducing the results described in prior art for independent blocking, which is a special instance ofP 3. The importance of considering correlated blocking and the accuracy of the 47 matchings obtained by the Bayesian MTL algorithm are discussed in Sections 2.5.2 and 2.5.3, respectively. Unless otherwise mentioned, we use the following settings for our simulation results: G = [10m; 10m] [10m; 10m] is the region of interest. obstacles are modelled as balls of diameterL; obviously, the blocking correlation increases with L. The standard deviation of the ranging error, , is assumed to be 0:01m. Two or more MPCs that are within a distance of 2 are considered to be unresolvable; in that case, the earliest arriving peak is retained and the other peaks are discarded. For a given, 01 = 10 = 2Q() was assumed, whereQ(x) = 1 Z x e x 2 =2 = p 2dx. A target is considered to be missed if there is no location estimate lying within a radius of 3 from the actual coordinates. Similarly, a false alarm is declared whenever there is no target within a radius of 3 from an estimated target location. For a given network realization, let ^ T D and ^ T F denote the number of detections and false alarms, respectively. Then, the detection and false alarm probabilities, denoted byP D and P F , respectively, are calculated as follows, P D =E[ ^ T D =T ] (2.36) P F =E[ ^ T F =( ^ T D + ^ T F )] (2.37) where the expectation is over the ensemble of network realizations 2.5.1 ComparisontoPriorArt In [16], the probability of any DP being blocked was assumed to be constant through- outG and independent of other blocking probabilities. Target detection was achieved if there existed a matching of size at leastI , where denotes the maximum number of undetected DPs permitted, regardless of consistency. We now proceed to demonstrate how this criterion is a special case of the Bayesian MTL Algorithm, obtained by assuming independent blocking with constant blocking probabilities (henceforth referred to as the i.c.b assumption). Letp los denote the probability that LoS exists between any two points inG. The probability that a DP is blocked is then given byp b = 1p 2 los . Taking into account both blockage and missed detection by 48 noise, the probability that a DP is undetected (denoted byp dp ) is given by p dp = 2(1p b )Q() +p b (2.38) The blocking likelihood of a matching with undetected DPs equals log((1 p dp ) I p dp ). Ifp dp < 1=2, then the blocking likelihood monotonically increases with . Hence, for a given , the corresponding blocking threshold,(), can be set as follows () = log((1p dp ) I p dp ) (2.39) which ensures that the detected targets have matchings of size at leastI . To validate the Bayesian MTL algorithm, we compared it with the prior art proposed in [16], under the i.c.b assumption. The comparison was done on the network shown in Fig. 2.2. To model the i.c.b condition, the values forL andp los were chosen to be 0:001m and 0.9, respectively. With probability 1p los , an obstacle was placed independently and uniformly along each line segment between a node (TX/RX) and a target. The two algorithms were evaluated over 100 realizations for three values of(= 1; 2 and 3) and (= 1; 3 and 6). For each value of , the threshold() for the Bayesian MTL algorithm was chosen according to (2.39). The region of convergence (ROC) curves, plottingP D versusP F , for both algorithms are shown in Fig. 2.3. As expected, they yield identical missed-detection and false alarm rates. Increasing loosens the compactness constraint on the ellipse intersections around a potential target location, while increasing relaxes the constraint on the probability of a blocking vector/matching. Hence, bothP F andP D are non-decreasing in and , as seen in Fig. 2.3. In the special case where only three ellipse intersections are sufficient to declare the presence of a target ( = 6), the false alarm rates are very high. This is in agreement with the results reported in [49]. 2.5.2 EffectofCorrelatedBlocking To highlight the effect of correlated blocking, the value ofL was increased to 5m and the obstacle centers were distributed according to a homogeneous PPP of intensity = 0:0075m 2 , which amounts to three obstacles inG, on average, per realization. 49 -10 -8 -6 -4 -2 0 2 4 6 8 10 x (m) -10 -8 -6 -4 -2 0 2 4 6 8 10 y (m) TXs RXs Targets Figure 2.2: A network consisting of 3 TXs at (8m; 7m), (7m; 8m) and (7m; 7m), 3 RXs at (7m; 7m), (8m; 7m) and (7m; 8m) (i.e.,I = 9 TRPs) and 2 targets at (0m; 0m) and (0m; 5m). The TX and RX locations are such that the LoS block- ing probabilities are independent only if L is very small. For L = 0:001m, the independent blocking assumption holds. The blocking distribution for the PPP obstacle model is derived in Appendix A. A total of 100 network realizations were considered, with M TX = M RX = 3 and T = 2, which corresponds toN = T (2M + 1) = 2(2 3 + 1) = 14 MPCs per TRP, on average. LetS sc G denote the region occupied by obstacles in a given realization. The TX, RX and target locations were uniformly and independently distributed over the region GnS sc , where ‘n’ denotes the set difference operator. Under the i.c.b assumption for the above settings, p los = exp(Ld avg ), where d avg = 10:1133m is the average distance between a target and a node. Hence, from (2.38),p dp = 0:5329 > 1=2 for = 3. The distribution of the average number of DPs at a point is tabulated in Table 2.2 for both the true blocking distribution and the 50 Figure 2.3: ROC curves plottingP D versusP F for the i.c.b condition. The prior art in [16] is a special case of the Bayesian MTL algorithm. Since the blocking distribution under the i.c.b condition does not vary spatially, the achievable set of (P F ;P D ) points is discrete. i.c.b assumption. As per the true blocking distribution, a target has LoS to all TXs and RXs (i.e., 9 DPs) over 66% of the time and the probability that a target has only 3 DPs is a little over 1%. As a result, a matching of size 3 is more likely to be a false alarm. However, sincep dp > 1=2, a matching of size 3 is more probable than a matching of size 9 (which occurs with less than 1% probability) under the i.c.b assumption. As a result, false alarms are identified first, followed by detections, as the value of increases (Fig. 2.4). This is reflected in the ROC curves plotted in Fig. 2.5, where the i.c.b assumption gives rise to very high false alarm rates. 51 -10 -8 -6 -4 -2 0 2 4 6 8 10 x (m) -10 -8 -6 -4 -2 0 2 4 6 8 10 y (m) μ = 3 TXs RXs True target locations Estimated target locations Scatterer midpoints (a) True blocking distribution: Both tar- gets detected and no false alarms -10 -8 -6 -4 -2 0 2 4 6 8 10 x (m) -10 -8 -6 -4 -2 0 2 4 6 8 10 y (m) μ = 6 TXs RXs True target locations Estimated target locations Scatterer midpoints (b) True blocking distribution: Both tar- gets detected plus lots of false alarms -10 -8 -6 -4 -2 0 2 4 6 8 10 x (m) -10 -8 -6 -4 -2 0 2 4 6 8 10 y (m) μ = 6.5 TXs RXs True target locations Estimated target locations Scatterer midpoints (c) i.c.b assumption: Only false alarms, both targets undetected -10 -8 -6 -4 -2 0 2 4 6 8 10 x (m) -10 -8 -6 -4 -2 0 2 4 6 8 10 y (m) μ = 8 TXs RXs True target locations Estimated target locations Scatterer midpoints (d) i.c.b assumption: Both targets detected plus lots of false alarms Figure 2.4: A sample simulation instance whenL = 5m. Since both target are not blocked to any TX/RX, each has a maximal matching of size 9 while false alarms have a matching of size less than 9. For the true blocking distribution, maximal matchings have a lower blocking likelihood on average than smaller matchings (see Table 2.2). In contrast, smaller matchings have a lower blocking likelihood under the i.c.b assumption. Hence, as increases, the true targets are detected first followed by an increase in false alarms [(a) and (b)] whereas under the i.c.b condition, false alarms appear first, followed by the true targets [(c) and (d)]. 52 Avg. no. of DPs < 3 3 4 5 6 7 8 9 True 0.0700 0.0150 0.0750 0 0.1750 0 0 0.6650 i.c.b. 0.3367 0.1961 0.2578 0.2259 0.1320 0.0496 0.0109 0.0011 Table 2.2: Probability distribution of the average number of DPs at a point forL = 5m and = 0:0075m 2 . Figure 2.5: Ignoring correlated blocking can result in false alarms being more likely to occur than detections. 53 Figure 2.6: Comparison with genie-aided method. 2.5.3 Comparisonwithgenie-aidedmethod In many radar applications, a missed-detection is more costly than a false alarm. As a benchmark, the missed-detection probability of the Bayesian MTL algorithm is compared with that of a genie-aided method, which involves running the Bayesian MTL Algorithm on the true target matchings, in Fig. 2.6. It can be seen that the proposed algorithm performs as well as the genie-aided method 3 . 3 The false alarm probability increases with. Hence, for 6, the slightly better detection performance of the Bayesian MTL algorithm can be attributed to false alarms around the vicinity of targets which the genie-aided method misses due to insufficient DPs. 54 (a) Pozyx Anchor (MTL target) (b) Pozyx Tag (MTL anchor) Figure 2.7: Components of the Pozyx test bed. 2.6 ExperimentalResults In this section, we describe the performance of the Bayesian MTL algorithm on range measurements conducted in an indoor environment. Our main focus is to determine if dependent blocking manifests itself in the measured data and verify if the Bayesian MTL algorithm takes advantage of it to provide better detection performance. Our experimental approach involved ‘simulating’ passive localization with the help of a commercially available active localization testbed, instead of using a more conventional distributed MIMO channel sounder. Specifically, we used the Pozyx testbed [50], consisting of anchors (transceivers) and tags that communicate using ultrawideband signals (see Section 2.6.1 for more details). The advantages of this approach are as follows: A commercial localization testbed simplifies the signal processing considerably by addressing a number of system level issues (e.g., synchronization, ranging protocol etc.) which, while important, are secondary to the main goal of analyzing the performance of the Bayesian MTL algorithm in the presence of dependent blocking. In addition, the use of active localization for ranging circumvents the need to perform background subtraction, which further simplifies the signal processing. In essence, our experimental setup enabled us to obtain ranges in a much simpler manner than using a wideband distributed MIMO channel sounder. 55 2.6.1 Measurementsetup The Pozyx localization test bed [50] consists of proprietary hardware for anchors and tags, shown in Fig. 2.7. The typical use-case scenario for the test bed is a distributed active localization application, where each tag obtains an estimate of its distance (range) to each of the anchors and then estimates its own position using them. Ranging between a tag and an anchor takes place using a two-way protocol, where (i) the tag first transmits a ranging signal (of bandwidth 500 MHz), and (ii) the anchor, upon receiving the ranging signal, transmits an acknowledgment (ACK) signal back to the tag. The time elapsed between the transmission of the ranging signal and the arrival of the ACK signal is equal to the signal propagation delay corresponding to twice the range, from which the range is estimated. In particular, the range is determined at a tag using an in-built routine operating on the response of the propagation channel to the ACK signal. It needs to be noted that the signal on which the ranging routine acts is unavailable to us for analysis. Instead, the ranging routine provides at most a single range value for each anchor-tag pair 4 . Unlike the scenario with distinct TX and RX nodes discussed in Section 2.2, each range estimate in this case corresponds to a circular ambiguity region for the tag location, as illustrated in Fig. 1.2a. Thus, the Pozyx test bed is ideally suited for the self-localization of tags in a decentralized manner, independent of other tags present in the environment. Thus, ifM tag > 1 tags are present, then multi-tag localization reduces toM tag decoupled instances of the single-tag localization case. On the other hand, it is evident from the problem formulationsP 1 throughP 3 that for passive MTL, the (joint) localization ofM tag targets is coupled when the targets have the same radar signature, since it is not known beforehand which range value corresponds to which target. Moreover, due to the passive nature of the targets, ranging needs to be carried out by the TXs and RXs. Thus, in order to simulate passive localization using the Pozyx testbed, we swap the roles of tag and anchor in our setup. Thus, a Pozyx tag is an MTL anchor and a Pozyx anchor is an MTL target in our experiments. For the remainder of this section, the terms anchor and target shall refer to an MTL anchor and an MTL target, respectively. 4 If LoS exists between the tag and the anchor, the range is an estimate of the anchor-tag distance. Otherwise, in case of blocked LoS, the range corresponds to the path length of the first arriving IP that has an adequate SNR to be detected. However, if none of the MPCs are strong enough to be detected, then the routine returns a ‘Ranging error’ message. 56 Figure 2.8: The measurement environment. Thus, each anchor measures its distance to all of the targets that it can detect using the in-built ranging routine. In order to simulate disjoint TX and RX nodes as described in Section 2.2, the distance estimates from pairs of anchors are added for each target, to create an elliptical ambiguity region for the target locations. Thus, for M a anchors,M a (M a 1)=2 distinct anchor pairings are possible, which results in M a (M a 1)=2 TRPs, where one of the anchors in each pair acts as a TX node and the other as a RX node (see Table 2.4). The choice of TX and RX node for a particular pair of anchors is arbitrary. The range estimates at each TRP are then arranged in ascending order to simulate the order in which the reflected MPCs arrive in the case of passive targets 5 . The collection of ordered range estimates are then used as an input to run the Bayesian MTL algorithm. In the following section, we describe our measurement results, obtained using this setup. 2.6.2 Results The measurements were conducted in a seminar room (Fig. 2.8), consisting of obsta- cles in the form of chairs and walls which can give rise to IPs if the DP is blocked. Four anchors and two targets were used, each stationary and mounted on a tripod at a height of 0:9m above the ground. The anchors were connected to a laptop computer to form a centrally-controlled network for initiating distance measurements. Each distance estimate between an anchor-target pair was obtained by averaging over 50 5 While the Pozyx system provides information on which ranges belong to a specific target by means of a device ID, we deliberately discard this information to faithfully emulate the passive ranging scenario. 57 Anchor No. 1 2 3 4 Coordinates (m; m) (0; 0) (0; 3:05) (5:49; 3:05) (5:49; 0) Table 2.3: Anchor locations. 0 1 2 3 4 5 6 x(m) -0.5 0 0.5 1 1.5 2 2.5 3 3.5 y(m) Anchors Target locations Figure 2.9: Layout of the target and anchor locations. The square tile pattern (of side length 2 ft) in the carpet (see Fig. 2.8 provides a convenient Cartesian coordinate system. Thus, with Anchor 1 as the origin, Anchors 2, 3 and 4 were at points (0; 10ft), (18ft; 10ft) and (18ft; 0ft), respectively. measurements. The anchor locations are listed in Table 2.3 and also plotted in Fig. 2.9, along with the target locations considered. From the 496 feasible measurement re- alizations for the targets in Fig. 2.9 (i.e., 32 2 ), 100 were arbitrarily chosen for our measurements. For each measurement realization, we consider the scenario where LoS from Anchor 2 to each target is blocked independently with probability 1=3. As a result, the only blocking vectors that have non-zero probabilities (based on (2.4) and Table 2.4) are1, the all-one vector (probability equal to 2=3) and [0 1 1 0 0 1] (probability equal 58 TRP No. 1 2 3 4 5 6 TX node A1 A1 A1 A2 A2 A3 RX node A2 A3 A4 A3 A4 A4 Table 2.4: TRP labelling scheme, where A1 denotes Anchor 1, and so on. Blocking scenario Probability Anchor 2 measurements Both targets unblocked 4=9 [d (u) 21 d (u) 22 ] Only target 1 blocked 2=9 [d (b) 21 d (u) 22 ] Only target 2 blocked 2=9 [d (u) 21 d (b) 22 ] Both targets blocked 1=9 [d (b) 21 d (b) 22 ] Table 2.5: Generating range estimates corresponding to TRP 2 when Anchor 2 is blocked to each target, independently, with probability 1=3 to 1=3). On the other hand, if we ignore the blocking dependence and consider the i.c.b assumption with a blocking probability of 1=3, then all 2 6 = 64 binary vectors have non-zero probabilities, given by: FP( ^ k t )j ind = (2=3) k ^ ktk 1 (1=3) 6k ^ ktk 1 (2.40) wherekk 1 denotes theL 1 norm. To block the LoS component from Anchor 2 to a given target, we used a large piece of cardboard wrapped in aluminum foil and placed it close to Anchor 2, as shown in Fig. 2.8. This was done to suppress the MPC due to diffraction and thereby, ensure that the first detected IP had a much larger path length than the (blocked) DP component 6 . However, the size and the location of the blocking obstacle resulted in both targets being blocked simultaneously. Thus, in order to model the true blocking distribution, two sets of distance measurements were carried out for Anchor 2: one with the blocking obstacle (i.e., the blocked case) and another without (the unblocked case), with the corresponding range to thet-th target (t = 1; 2) denoted byd (b) 2t and d (u) 2t , respectively. The distance measurements corresponding to Anchor 2 were then obtained by sampling from the probability distribution given in Table 2.5. 6 We had initially used smaller obstacles and noticed that (a) the first detected IP was the diffracted path around the obstacle, and (b) that its range was sufficiently close to that of the DP component for the Bayesian MTL algorithm to detect it as a DP. 59 0 1 2 3 4 5 6 x (m) 0 1 2 3 4 5 6 y (m) δ = 3, μ = 8, σ = 0.15 m Anchors True target locations Estimated locations (no blocking assumption) Estimated locations (Anchor 2 blocked with prob.1/3) Estimated locations (independent blocking) Target missed by all 3 cases Figure 2.10: For each target, the detection circle of radius 3 is shown. One of the targets is missed by all the cases while the other target is detected even under the i.c.b assumption. However, the i.c.b assumption gives rise to more false alarms. The value of was chosen to be 0:15m. The results from a sample realization are shown in Fig. 2.10. Anchor 2 was blocked for this realization and hence, no target is detected under the no blocking assumption. However, under the true blocking distribution and the i.c.b assumption, only one of the targets is detected, but the i.c.b case gives rise to more false alarms.The ROC curve is plotted in Fig. 2.11, where, in addition to the true blocking distribution and the i.c.b assumption, the achieved (P F ;P D ) points corresponding to the no-blocking assumption (i.e.,P(1) = 1 and P(k t ) = 0 for allk t 6= 1) is plotted as well. It is easily seen that the ROC curve corresponding to the true blocking distribution lies above the curves corresponding to the incorrect blocking distributions. Thus, for a fixedP F , the highest value of P D is obtained by taking dependent blocking into account, thereby highlighting its 60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 P F 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P D δ = 3 Anchor 2 blocked with prob 1/3 No blocking assumption Indep. blocking assumption with prob 1/3 Figure 2.11: ROC curve obtained by fixing = 3 and varying. importance in improving the reliability of localization. 2.7 SummaryandConclusions In this chapter, we considered the impact of environment-induced correlated blocking on localization performance. We first provided a theoretical framework for MTL by formulating the general problem of localizing all the targets and obstacles in an unknown environment as a Bayesian estimation problem. We then proceeded to derive a more tractable approximation, known as the Bayesian MTL problem, where the objective was to localize only the targets, but not the obstacles. We then proposed a sub-optimal polynomial-time algorithm - at the heart of which was a blocking-aware vector likelihood function that took correlated blocking into account - to solve this problem. The algorithm relies on two thresholds, and , to detect targets and 61 works with either theoretical or empirical blocking statistics that may be obtained via measurements or simulations. Our simulations showed that ignoring correlated blocking can lead to poor detection performance, with false alarms being more likely to occur than detections, which was further validated by our experimental results. 62 Algorithm 1 Bayesian MTL algorithm Obtain the TRP processing order (z 1 ;z 2 ; ;z n ) [16] t = 0 . (Initial set of matchings) for eachj z 1 ;j z 2 do for each ellipse intersection (x;y) corresponding tor z 1 ;jz 1 andr z 2 ;jz 2 do ifl B (fr z 1 ;jz 1 ;r z 2 ;jz 2 g) (;) then t =t + 1 q (2) t (r) =fr z 1 ;jz 1 ;r z 2 ;jz 2 g (^ x (2) t ; ^ y (2) t ) = (x;y) end if end for end for ^ T (2) =t . ^ T (i) denotes the number of estimates at the end of thei-th stage fori = 3 toI do fort = 1 to ^ T (i 1) do . (Updating existing matchings) if9 anyr z i ;jz i such thatL E (q (i1) t (r)[r z i ;jz i ) then q (i) t;temp (r) =q (i1) t (r)[r z i ;jz i else q (i) t;temp (r) =q (i1) t (r) end if DeriveK (i) t fromq (i) t;temp (r) for each ^ k (i) t 2K (i) t do if log ^ k (i) t then Deriveq (i) t (r) fromq (i) t;temp (r) according to ^ k (i) t end if end for end for Update ^ T (i) and sett = ^ T (i) for eachj z i ;j zu (u = 1; ;i 1) do. (New targets, previously unidentified due to blocking) for each ellipse intersection (x;y) corresponding tor z i ;jz i andr zu;jzu do ifl B (fr z i ;jz i ;r zu;jzu g) (;) then t =t + 1 q (i) t (r) =fr z i ;jz i ;r zu;jzu g (^ x (i) t ; ^ y (i) t ) = (x;y) end if end for end for ^ T (i) =t end for 63 Chapter 3 BlindspotProbabilityAnalysis underDependentBlocking 3.1 Introduction In the previous chapter, we saw how dependent blocking can be leveraged to improve the detection performance of localization algorithms. However, this is of little advan- tage if a target is in a blind spot anc cannot be localized because it is has LoS to fewer than three anchors. If the map of the environment is known, then a deterministic, blind spot eliminating anchor placement can be obtained by solving a variant of the art-gallery problem [51, 52]. But, in many applications, the map of the environment may not be known beforehand; for instance, In a forest environment where the target(s) are wildlife, the trees could act as obstacles. In this case, it is unreasonable to assume that all obstacle locations are known. On a road or in a shopping mall, vehicles and humans may respectively act as obstacles intermittently. The above examples encompass a diverse range of indoor and outdoor situations, where the obstacles can either be static or dynamic. To the best of our knowledge, the existing literature on the art gallery problem does not address the question of 65 eliminating or minimizing the occurrence of blind spots when the environment map is unknown. To address this gap, we consider a stochastic geometry based framework in this chapter where we use random shape theory to model the obstacle locations and shapes and a homogeneous PPP to model the anchor locations 1 . Due to the probabilistic nature of the anchor and obstacle locations, it is not possible to completely eliminate blind spots. This motivates the analysis of the blind spot probability of a typical target over a localization network, which is a performance measure over an ensemble of environment realizations instead of a particular snapshot. Apart from capturing the uncertainty in the obstacle locations, random shape theory also enables us to model dependent blocking phenomenon caused by obstacles of varying sizes and shapes. This is important since ignoring the correlation in LoS blocking events and assuming independent blocking across links instead can result in the underestimation of the blind spot probability at a given (target) location. For instance, if two anchors, situated close to one another, are each blocked to a target with probabilityp, then the joint blocking probability of the two anchor-target links is also approximatelyp, which exceedsp 2 , the result obtained by assuming independent blocking. In this chapter, we analyze the relationship between the blind spot probability and the statistics of the obstacle sizes and locations and the anchor point process. In doing so, we wish to determine the intensity with which anchors need to be deployed so that the blind spot probability over the entire region is less than a threshold. For the ease of exposition, we use vision as an analogy to describe the presence/absence of LoS for the remainder of this chapter; specifically, an anchors is said to be invisible (visible) to a target if the LoS between the anchor and the target is blocked (unblocked). 3.1.1 RelatedWork The PPP was used to model base station locations while investigating the hearability problem for localization in cellular networks [53, 54], where similar to the visibility 1 For a number of commercial applications, the anchors would typically be cellular base stations that also provide other wireless communication services. The PPP is a standard model for base station deployment in wireless communication. Furthermore, for some other applications (e.g., dropping anchors from the air to provide wildlife tracking capability in a forest), a deterministic placement of anchor nodes is inherently impossible and a point process model for the anchor locations is appropriate. 66 analogy, the hearability metric was defined as the number of base stations whose SINR (signal-to-interference-plus-noise ratio) at a target mobile station crossed a particular threshold. However, independent log-normal shadowing was assumed for all links and the blocked LoS senario was not specifically addressed. The Boolean model has been used to analyze the impact of blocking on the performance of urban cellular networks [55], and mm-wave systems [56, 57, 58, 59]. In [55, 57, 58], independent blocking was assumed across different links, while in [59], the spatio-temporal correlation between the LoS/NLoS states of two links was investigated. The effect of correlated shadowing on the interference distribution of wireless networks in urban areas was studied in [60], using a Manhattan line process to model building locations. In [61], we partially considered the impact of correlated blocking by estimating the blind spot probability at a given (target) location using approximate second-order blocking statistics and in [62], the worst-case impact of correlated blocking on the blind spot probability was investigated by considering infinitely large obstacles modeled by a line process. In general though, to the best of our knowledge, stochastic geometry models for correlated shadowing or blocking in wireless networks is an emerging field. 3.1.2 Contributions The main contributions of this chapter are as follows: We model the anchor locations using a homogeneous PPP and the obstacle locations and shapes using random shape theory (specifically, a Boolean model). From the perspective of a typical target, the anchors that are within communi- cation range are constrained to lie in a circular region, centered at the target. The obstacles lying within this circle partition it into visible and shadowed regions, where the anchors lying in the shadowed region are invisible to the target. Under these conditions, we express the blind spot probability at a typical target location in terms of the probability distribution of the visible area (i.e., the area of the visible region surrounding a typical target). We then show that the blind spot probability under the independent anchor blocking assumption depends only on the mean visible area, instead of the 67 entire probability distribution. In addition, we derive the conditions under which the independent blocking assumption underestimates the true blind spot probability. We then demonstrate that the visible area distribution is difficult to characterize in closed form. As a result, we propose an approximate solution for characteriz- ing the visible area whereby in each environment realization, the visible area is evaluated exactly up to the location of the second nearest obstacle and the remaining value beyond that is approximated by its mean. We refer to this as the nearest two-obstacle approximation and we show that it is equivalent to consid- ering correlated blocking up to the location of the second nearest obstacle and assuming independent blocking for farther obstacles, where the impact of block- ing correlation is relatively minimal. In other words, the nearest two-obstacle approximation engenders a quasi-independent blocking assumption. Using the nearest two-obstacle approximation, we derive a closed-form approx- imation for the blind spot probability as well as the conditions under which it yields a tighter bound on the true blind spot probability, relative to the indepen- dent blocking assumption. As a result, our work provides useful design insights, such as the intensity with which anchors need to be deployed so that the blind spot probability over the entire region is less than a threshold,. At the other end of the spectrum, when there is a sufficiently large number of obstacles and their lengths tend to infinity, the Boolean model approaches a random collection of lines, known as a line process, which induces a random polygon tessellation ofR 2 . We show that the visible region is a Poisson-V oronoi cell, whose area distribution is known in closed form. This enables us to derive a closed-form expression for the asymptotic blind spot probability, which captures the worst case impact of dependent blocking and provides an upper bound for the scenario when the obstacles have finite dimensions. Taken together, these contributions provide useful design insights, such as the intensity with which anchors need to be deployed so that the blind spot probability over the entire region is less than a threshold,. 68 3.1.3 Organization This chapter is divided into eight sections. The system model is described in Sec- tion 3.2, where the anchor locations are modeled using a homogeneous PPP and the obstacles are represented using line-segments of random lengths and orientations. In Section 3.3, the blind spot probability at a typical target location is characterized as function of the distribution of the surrounding visible area. Additionally, the blind spot probability under the independent anchor blocking assumption is also character- ized and the conditions under which it underestimates the true blind spot probability are derived. The nearest two-obstacle approximation is introduced in Section 3.4 to characterize the visible area in a tractable manner, which is then used to derive an approximate expression for the blind spot probability in Section 3.5, that takes into account the impact of correlated blocking up to the second nearest obstacle. The limiting case when the obstacle lengths tend to infinity is considered in Section 3.6, where the notion of the asymptotic blind spot probability is defined and a closed form expression for it is derived. Numerical results to validate our approximations are presented in Section 3.7. Finally, Section 3.8 concludes this chapter. 3.1.4 AdditionalNotation In this chapter, 2 (:) denotes the Lebesgue measure inR 2 (i.e., for a setS R 2 , 2 (S) denotes the area ofS). S and T denote set union and intersection, respectively, and? denotes the empty set. A real functionh, with argumentt and parameters given by a vector, a, is denoted by h(t;a). Finally, for a function h : R! R, graph(h) ,f(x;y)2 R 2 : y = h(x)g and epi(h) ,f(x;y)2 R 2 : y h(x)g denote its graph and epigraph, respectively [63]. 3.2 SystemModel Consider an environment inR 2 consisting of point targets and distributed obstacles. Intuitively, the i-th obstacle can be parametrized by the tuple (p i ;S i ;! i ), where S i R 2 denotes the ‘shape’ of the obstacle (e.g., a rectangle),p i = (r i ; i )2R 2 its ‘location’ in polar coordinates ( i 2 [0; 2)) (e.g., the geometric center of the rectangle), and! i 2 [0; 2) its ‘orientation’ with respect to the positivex-axis, as 69 Target Anchor Obstacle Blocked LoS Figure 3.1: Example of a localization scenario consisting of anchors, targets and obstacles. Figure 3.2: Visible region around a typical target, for the line segment obstacle model where all the obstacles have lengthL and face the target (! i = i +=2). 70 shown in Fig. 3.1. The collection of obstacles, S i (p i ;S i ;! i ), forms a germ-grain model if the following conditions are satisfied [64]: (i) The set of pointsfp i g, known as germs, form a point process inR 2 . (ii) The setf(S i ;! i )g, known as grains, is drawn from a family of closed sets S . The obstacles are assumed to be opaque to radio waves; therefore, the obstacle thickness does not influence the existence of LoS and hence, it is sufficient to letS be the set of line-segments of length at mostL, whereL is the maximum obstacle length (i.e.,S, [0;L]). Without loss of generality, the germs can be chosen to be the mid- points of the line-segments 2 . Thus, , [0;) is sufficient to encompass all obstacle orientations. We assume the germs to be distributed according to a homogeneous PPP with intensity 0 . The obstacle lengths and orientations are modeled as samples drawn from a joint distribution, supported onS , whose pdf is denoted byf L;W (;), where L and W denote the random variables representing the obstacle length and orientation, respectively. A localization network consisting of single-antenna anchors is deployed overR 2 and we assume the anchor locations to also form a homogeneous PPP, with intensity, independent of the obstacle germ process. Due to the stationarity of the PPP, it can be assumed without loss of generality that a target is situated at the origin,o, which we refer to as the typical target. A transmit power constraint further restricts our attention to a disc of radiusR, centered aroundo and denoted byD o (R), in which anchors must lie for the typical target to be localized. From the target’s perspective, each obstacle induces a shadow region, which is the set of points that it renders invisible to the target, as illustrated in Fig. 3.2. Consequently, the anchors that lie in a shadow region are invisible to the target. Intuitively, the blocking probability of an anchor (and consequently, the blind spot probability) increases with the area of the shadow regions, which in turn depends on obstacle mid-point PPP andf L;W (;). Sincef L;W (;) is usually unknown, we assume all obstacle lengths are equal toL and! i = i +=2 (see Fig. 3.2). Ifr i R (e.g., obstacle with mid-pointp 1 in Fig. 3.3), then such a rotation of thei-th obstacle to face the (typical) target maximizes the shadow region 2 In general, the germs need not be the geometric centers of their corresponding grains. 71 L L Figure 3.3: Illustration of the quasi worst-case obstacle orientation, where all obsta- cles are assumed to face the typical target (illustrated using dotted lines) and have maximum length, L. This maximizes the shadowed area due to obstacles whose mid-points are withinD o (R) (e.g.,p 1 above. The shadow regions due to the original and ‘rotated’ orientations are represented using the plain and striped grey regions, respectively.), while neglecting the shadowed region induced by obstacles whose mid-points lie outsideD o (R) (e.g.,p 2 above). induced by it; on the other hand, if r i > R (e.g., obstacle with mid-point p 2 in Fig. 3.3), this rotation eliminates any shadow region otherwise induced by thei-th obstacle, thereby ignoring the blocking caused by it. As a result, this assumption corresponds to a quasi worst-case orientation for the obstacles that emphasizes the (greater) influence of nearer obstacles on correlated anchor blocking and subsequently, the blind spot probability. Thus, the obstacles whose mid-points lie withinD o (R) partition it into shadowed and visible regions, where for ToA-based localization, the target can be localized if it there are at least three anchors in the visible region. Consequently, the typical target is said to be in a blind spot if this condition is not satisfied. As blind spots are undesirable, the blind spot probability of the typical target is an important metric from a network design perspective. In the following section, we develop the relationship between the blind spot probability at a typical target location and its surrounding visible area distribution, which is a function of the obstacle intensity ( 0 ) and size (L). 72 Remark 1. The stationarity of the anchor and obstacle germ PPPs ensure that the statistics of the visible region surrounding any target location is the same. Hence, even if multiple targets are present (e.g., Fig. 3.1), it is sufficient to analyze the single target case in order to bound the blind spot probability at all target locations. This helps define the notion of a typical target at the origin. 3.3 AnalysisofblindspotProbability For the parameter vector z = [ 0 L R], we define the visibility random variable, denoted byV (p;z) forp = (r;)2D o (R), in the following manner: V (p;z) = 8 < : 1; ifp is visible too 0; else. (3.1) LetV(z) =fp2D o (R) :V (p;z) = 1g denote the visible region around the target and letA v (z) = 2 (V(z)) denote its area, which we refer to as the visible area. The typical target is in a blind spot if and only if there are fewer than three anchors in V(z). Thus, the blind spot probability, conditioned on the random variableA v (z) and denoted byg(A v (z);), with parameter, has the following expression: g(A v (z);),P(blind spotjA v (z)) = 2 X k=0 P(k anchors present in the visible region of areaA v (z)) (3.2) =e Av (z) 1 +A v (z) + (A v (z)) 2 2 : (3.3) Remark 2. The definition of a blind spot can be generalized to the absence of at least k v visible anchors inV(z), due to which the summation limits in (3.2) would run from 0 tok v 1. This is useful to analyze the blind spot probability for other localization techniques such as time-difference of arrival (TDoA) based localization for which k v = 4. The unconditional blind spot probability,b(;z), is then obtained by averaging 73 over the distribution ofA v (z) as given below, b(;z) = R 2 Z 0 g(t;)f Av (z) (t)dt; (3.4) wheref Av (z) (:) is the pdf ofA v (z), which fully captures the statistics of correlated anchor blocking due to obstacle sizeL and intensity 0 . The visible anchors can be interpreted as a point process derived by sampling from the underlying anchor PPP, where an anchor at pointp is selected with a probability equal toP(V (p;z) = 1). Furthermore, the sampling process is also correlated across anchor locations due to correlated blocking (i.e., the probability that an anchor atp is selected also depends on the selection of other anchors inD o (R)). However, if we ignore this correlation and assume that each anchor is sampled independently of the other anchors, we obtain the well-known independent blocking assumption, for which the resulting blind spot probability is given by the following lemma: Lemma 1. The blind spot probability under the independent anchor blocking assump- tion, denoted byb ind (;z), is given by: b ind (;z) =e E[Av (z)] 1 +E[A v (z)] + (E[A v (z)]) 2 2 =g(E[A v (z)];): (3.5) Proof. See Appendix B.1. From Lemma 1, it can be seen that the mean visible area,E[A v (z)], completely characterizes the blind spot probability if independent anchor blocking is assumed. For the system model from Section 2.2,E[A v (z)] is given by the following lemma: Lemma 2. For a parameter vectorz, the average visible area,E[A v (z)], overD o (R) is given by: E[A v (z)] = 2 R Z 0 exp( 0 2 (S V (p;z)))rdrd; (3.6) whereS V (p;z) =f(;)2R 2 : 0 tanjjL=2; 0 secjjrg (3.7) 74 and 2 (S V (p;z)) = 2 r Z 0 min arctan L 2 ; arccos r d: (3.8) Proof. See Appendix B.2. The relationship betweenb(;z) andb ind (;z) is given by the following theorem: Theorem 1. b ind (;z)b(;z) overf(;z) :E[A v (z)] 3:3836g. Proof. As a twice-differentiable function oft, we have d dt g(t;) =( 3 =2)t 2 e t (3.9) d 2 d 2 t g(t;) = ( 3 =2)te t (t 2): (3.10) From (3.10), the second derivative ofg(t;) is non-negative whent 2=. Hence, g(t;) is a convex function int over this regime [63]. Lett 0 denote the solution to the following equation: 1 =g(0;) =g(t 0 ;)t 0 d dt g(t;) t=t 0 (3.11) =) 1 =e t 0 (t 0 ) 3 2 + (t 0 ) 2 2 +t 0 + 1 : (3.12) Eqn. (3.12) is a mixed polynomial-exponential equation int 0 and solving fort 0 numerically, we obtain (up to four digits of precision), t 0 = 3:3836 : (3.13) Geometrically,t 0 determines thex-coordinate of the point at which the liney(t;) R 2 , passing through (0;g(0;)), is tangential to epi(g(:;)), as shown in Fig. 3.4. The equation ofy(t;) is as follows: y(t;) =g(t 0 ;) + (tt 0 ) d dt g(t;) t=t 0 ; t 0: (3.14) Let g con (t;) = 8 < : g(t;); t>t 0 y(t;); tt 0 : (3.15) 75 0 1 2 3 4 5 6 7 8 9 10 λA v 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 g(A v ;λ) y(A v ;λ) λt 0 Figure 3.4: ForE[A v (z)]t 0 = 3:3836, the set of pointsf(E[A v (z)];b(;z))g (i.e., the grey shaded region) lies above the setf(E[A v (z)];b ind (;z))g, shown by the blue curve. For 0 t t 0 , the supporting hyperplane at each point, (t;g con (t;)), on the boundary of epi(g con (:;)) isy(:;). Similarly, there also exists a supporting hy- perplane at each boundary point, (t;g con (t;)), of epi(g con (:;)) fort > t 0 , since g con (;) g(;), a convex function in its argument over this interval [63]. Thus, g con (t;) is a convex function int fort 0. Consequently, ifE[A v (z)]>t 0 , then b ind (;z) =g(E[A v (z)];) (3.16) a =g con (E[A v (z)];) (3.17) b E[g con (A v (z);)] (3.18) c E[g(A v (z);)] (3.19) 76 d =b(;z); (3.20) where (a) follows from (3.15), (b) from Jensen’s inequality, (c) from the fact that g con (t;)g(t;) for allt 0, and (d) from the definition ofb(;z) in (3.4). Remark 3. A geometric interpretation of Theorem 1 is seen in Fig. 3.4, where, as a result of (d) in 3.16, the feasible values for the ordered pair (E[A v (z)];b(;z)) is given by the convex hull of graph(g(:;)). On the other hand, the feasible values for (E[A v (z)];b ind (;z)) is graph(g(:;)), which forms the lower boundary of its convex hull whenE[A v (z)]t 0 = 3:3836. It is important to note that Theorem 1 represents a sufficient, but not necessary, condition as the proof is a consequence of the convexity properties ofg(;) that do not depend onf Av (z) (:). Thus, the inequality b ind (;z) b(;z) may still hold over a setZf(;z) : E[A v (z)] 3:3836g for some choice(s) off Av (z) (:). From a design perspective, it is desirable to have at least three unblocked anchors, on average (i.e., E[A v (z)] 3). Hence, from Theorem 1, it is clear that the independent blocking assumption underestimates the true blind spot probability for most practical scenarios and that correlated blocking should be taken into account while designing a localization network that meets a desired blind spot probability threshold. From (3.2)-(3.4), it is evident that the distribution of the visible area plays a critical role in determining the blind spot probability of the typical target, for a given anchor intensity. In the next section, we attempt to characterize this distribution. 3.4 CharacterizingUnshadowedArea The visible area around the typical target depends on the number of obstacles as well as their locations. To capture this dependence, we define the following: Definition 1. LetV(p (k) ;z) denote a realization ofV(z) whenk(> 0) obstacle(s) are present, with the obstacle locations determined by p (k) = [r (k) (k) ], where r (k) = [r 1 r k ] (r i r j ;i<j), (k) = [ 1 k ], and thei-th nearest obstacle mid-point is located at (r i ; i ), (i = 1; ;k). The special case when k = 0 is denoted byV(?;z) and is equal toD o (R). 77 (a) Entire obstacle causes blocking (b) Only a part of the obstacle causes blocking Figure 3.5: Shadowed area (shaded gray) due to a single obstacle. Definition 2. LetA (k) v (p (k) ;z) denote the visible area corresponding toV(p (k) ;z) (i.e., A (k) v (p (k) ;z) , 2 (V(p (k) ;z))). In particular, A (k) v (p (k) ;z) is a realization of the random variable A v (z), conditioned on the presence of k obstacles whose locations are given byp (k) . Fork< 2,A (k) v (p (k) ;z) is easy to characterize, A (0) v (?;z) =R 2 (3.21) A (1) v (p 1 ;z) =R 2 (p 1 ;z) 2 R 2 1 2 r 1 x(p 1 ;z) | {z } Shadowed area ; (3.22) where(p 1 ;z) = 8 < : 2 arctan L 2r 1 ; 0r 1 p R 2 (L=2) 2 2 arccos r 1 R ; p R 2 (L=2) 2 r 1 R (3.23) x(p 1 ;z) = 8 < : L; 0r 1 p R 2 (L=2) 2 2 p R 2 r 2 1 ; p R 2 (L=2) 2 r 1 R: (3.24) In particular, the term in parenthesis in (3.22) denotes the shadowed area (Fig. 3.5). Fork 2, overlaps may occur between the shadow regions corresponding to different obstacles (see Fig. 3.2). In order to accurately determineA (k) v (p (k) ;z), the 78 L Figure 3.6: The shaded region denotes the area shadowed by the nearest two obstacles. The additional shadow region induced by the third nearest obstacle onwards must intersect the part-annular checkered region. areas of all overlapping shadowed regions should be counted exactly once. We first attempt to characterize the shadow region overlap corresponding to the nearest two obstacles, for which we define the following: Definition 3. LetA sh (p;z)D o (R) denote the shadow region induced by an ob- stacle whose mid-point is atp (e.g., Fig. 3.5). The azimuthal end-points ofA sh (p;z), denoted byl(p;z) andu(p;z), are given by the following expressions: l(p;z) = (p;z) 2 mod 2 (3.25) u(p;z) = + (p;z) 2 mod 2; (3.26) where(p;z) is given by (3.23). Thus, the azimuthal span ofA sh (p;z), denoted by the intervalI(p;z) [0; 2), has the following expression: I(p;z) = [min(l(p;z);u(p;z)); max(l(p;z);u(p;z))]: (3.27) A typical overlap between a pair of shadow regionsA sh (p 1 ;z) andA sh (p 2 ;z) is illustrated in Fig. 3.6 and the extent of overlap can be characterized by the following lemma. 79 Lemma 3. Let(p (2) ;z)2 [0; 1] denote the fraction ofA sh (p 2 ;z) that overlaps withA sh (p 1 ;z) in the azimuth. Then, (p (2) ;z) = max 0; (p (2) ;z) (p 2 ;z) ! (3.28) where (p (2) ;z) = 8 > > > > > > > > > < > > > > > > > > > : min(u(p 1 ;z);u(p 2 ;z)) max(l(p 1 ;z);l(p 2 ;z)); if l(p 1 ;z)u(p 1 ;z);l(p 2 ;z)u(p 2 ;z) 2 (max(l(p 1 ;z);l(p 2 ;z)) min(u(p 1 ;z);u(p 2 ;z))); if l(p 1 ;z)>u(p 1 ;z);l(p 2 ;z)>u(p 2 ;z) max(u(p 2 ;z)l(p 1 ;z);u(p 1 ;z)l(p 2 ;z)); else. (3.29) Proof. See Appendix B.3 The visible region beyond a radiusr 2 can be decomposed into the union of two sets,V in (p (k) ;z) andV out (p (k) ;z), which are defined as follows: V in (p (k) ;z) =fp2V(z) :r>r 2 ;2I(p 1 ;z)[I(p 2 ;z)g (3.30) V out (p (k) ;z) =fp2V(z) :r>r 2 ; = 2I(p 1 ;z)[I(p 2 ;z)g: (3.31) V in (p (k) ;z) is the (vertically) striped region in Fig. 3.6 andV out (p (k) ;z) is a subset of the annular region fromr 2 toR, excluding the azimuthal end points ofA sh (p 1 ;z)[ A sh (p 2 ;z), i.e., the checkered region in Fig. 3.6. Using the terminology defined so far,A (k) v (p (k) ;z) can be expressed as follows: A (k) v (p (k) ;z) =A n2 (p (2) ;z) +A f (p (k) ;z); (3.32) whereA n2 (p (2) ;z),r 2 2 (p 1 ; ~ z) 2 r 2 2 1 2 x(p 1 ; ~ z)r 1 (3.33) ~ z = [ 0 Lr 2 ] (3.34) A f (p (k) ;z), 2 (V in (p (k) ;z)) + 2 (V out (p (k) ;z)): (3.35) In (3.32)-(3.35),A n2 (p (2) ;z) denotes the visible area up to the location of the second nearest obstacle (i.e., the area of the white region in Fig. 3.6) andA f (p (k) ;z) denotes the remaining visible area, beyond the second nearest obstacle. 80 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 L/R 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Average fraction Avg. fraction of shadowed area induced by far-off obstacles Avg. no. of obstacles, (i.e., λ 0 π R 2 ) = 4, 6, 8 Figure 3.7: They-axis plots the average fraction of the shadowed area induced by far-off obstacles (i.e., beyond the second nearest obstacle). The curves have been generated by averaging over 10 6 Monte-Carlo simulations. Fork> 2, evaluating the pairwise shadow region overlaps is, in general, insuffi- cient, as more than two obstacles may contribute to a common overlapping region. Since it is not straightforward to ensure that the areas of all overlapping shadowed regions are counted exactly once,A f (p (k) ;z) is difficult to compute exactly. Conse- quently,f Av (z) (:) is hard to characterize in closed form, as well. Hence, we focus on approximatingA f (p (k) ;z) in the remainder of this section, which shall then be used to derive a tractable approximation forb(;z) in the following section. Since nearer obstacles induce larger shadow regions, it is intuitive that the nearest two obstacles should be responsible for a large fraction of the total shadowed area. To quantify this notion, let (z) =E[A f (p (k) ;z)=A (k) v (p (k) ;z)] denote the average fraction of the shadowed area contributed by the far-off obstacles, where the expecta- 81 tion is over both the number as well as the locations of the obstacles. (z) is plotted in Fig. 3.7 as a function of the normalized obstacle length,L=R, by averaging over 10 6 Monte-Carlo simulations. Unsurprisingly, (z) increases with the number of obstacles as there is a greater possibility of a non-overlapping far-off shadow region. However, the likelihood of such an outcome reduces with increasing obstacle size and therefore, (z) is monotonically decreasing inL=R. Hence, when there are a small number of obstacles on average, the nearest two account for most of the shadowed area (in excess of 60%, on average, when the average number of obstacles is at most eight, as seen in Fig. 3.7). Thus, conditioned onp (2) , it is reasonable to approximate the shadowed area due to the remaining obstacles by its mean value. In other words, A f (p (k) ;z) can be approximated by its mean value, conditioned onp (2) . We refer to this as the nearest two-obstacle approximation, which is formally expressed below: Approximation 1 (Nearest two-obstacle approximation). For k 2 and a small number of obstacles on average 3 , A (k) v (p (k) ;z)A (2+) v (p (2) ;z) ,A n2 (p (2) ;z) +E[A f (p (k) ;z)jp (2) ] (3.36) A n2 (p (2) ;z) +E[ 2 (V out (p (k) ;z))jp (2) ]: (3.37) In evaluating the conditional mean ofA f (p (k) ;z) in (3.36), givenp (2) , we average over both the number and the locations of the far-off obstacles, i.e., over both k and p (3:k) , respectively. The approximation in (3.37) is obtained from (3.35) by ignoring the termE[ 2 (V in (p (k) );z)jp (2) ] (i.e., the average area of the striped region in Fig. 3.6) for the sake of tractability. However, it is easy to observe from Fig. 3.6 that the area of the striped region increases with increasing obstacle size. As a result, the approximation in (3.37) may not be reasonable beyond a certain value ofL. In the following lemma, we derive an expression forE[ 2 (V out (p (k) ;z))jp (2) ]. Lemma 4. Conditioned on the nearest two obstacles, the average visible area over V out is given by E[ 2 (V out (p (k) ;z))jp (2) ] = 2(p 1 ;z) (1(p (2) ;z))(p 2 ;z) 3 Based on Fig. 3.7, at most eight obstacles on average is a reasonable heuristic. 82 R Z r 2 exp 0 @ 2 0 r Z r 2 min arctan L 2 ; arccos r d 1 A dr: (3.38) Proof. See Appendix B.4. Remark 4. The nearest two-obstacle approximation characterizes the visible area beyond the second nearest obstacle only by its mean. However, from Lemma 1, this is equivalent to assuming independent blocking beyond the second nearest obstacle. Hence, the nearest two-obstacle approximation can also be interpreted as a ‘quasi- independent blocking assumption’. In the next section, we derive a tractable approximation for b(;z) using the nearest two-obstacle approximation. 3.5 ATractableApproximationforb(;z) Let b k (;z) denote the blind spot probability, conditioned on k obstacles being present, for anchor intensity and parameter vectorz. By first conditioning and then averaging over the obstacle locations,b k (;z) can be expressed as follows: b k (;z) = 2 Z 0 R Z 0 f (k) (p 1 )dp 1 2 Z 0 R Z r k1 g(A (k) v (p (k) ;z);)f (k) (p k jp (k1) )dp k (3.39) = 2 Z 0 R Z 0 f (k) (p 1 )dp 1 2 Z 0 R Z r k1 g(A (k) v (p (k) ;z);)f (k) (p k jp k1 )dp k ; (3.40) where dp i = r i dr i d i (i = 1; ;k) and f (k) (p i jp (i1) ) in (3.39) denotes the conditional pdf of the location of thei-th (2 i k) nearest obstacle given the location(s) of the other obstacles that are closer to the target than it, when a total ofk obstacles are present. Similarly,f (k) (p 1 ) denotes the pdf of the location of the nearest obstacle. The simplification in (3.40) is a result of the Markov property, sincer i lies 83 in the interval [r i1 ;R] and is therefore independent ofr j forj2f1; ;i 2g, givenr i1 . Fork< 2,b k (;z) is expressed as follows: b 0 (;z) =g(A (0) v (?;z);) (3.41) b 1 (;z) = 2 Z 0 R Z 0 g(A (1) v (p 1 ;z);) 1 R 2 dp 1 : (3.42) Fork 2, the nearest two-obstacle approximation is used to simplifyb k (;z), as given below, b k (;z)b (2+) k (;z) (3.43) , 2 Z 0 R Z 0 f (k) (p 1 )dp 1 2 Z 0 R Z r 1 f (k) (p 2 jp 1 )dp 2 2 Z 0 R Z r k1 g(A (2+) v (p (2) ;z);)f (k) (p k jp k1 )dp k (3.44) = 2 Z 0 R Z 0 f (k) (p 1 )dp 1 2 Z 0 R Z r 1 g(A (2+) v (p (2) ;z);)f (k) (p 2 jp 1 )dp 2 ; (3.45) whereA (2+) v (p (2) ;z) is given by (3.37). The expressions forf (k) (p 1 ) andf (k) (p 2 jp 1 ) are as follows: f (k) (p 1 ) = k R 2 R 2 r 2 1 R 2 k1 (3.46) f (k) (p 2 jp 1 ) = k 1 (R 2 r 2 1 ) R 2 r 2 2 R 2 r 2 1 k2 (3.47) with (3.46) and (3.47) following as a result of thek obstacle mid-points being inde- pendently and uniformly distributed overD o (R). Using (3.41)-(3.47), an approximate expression forb(;z) can be derived by first conditioning and then averaging over the number of obstacles,k, in the following manner: b(;z) = 1 X k=0 b k (;z)e 0 R 2 ( 0 R 2 ) k k! (3.48) 84 e 0 R 2 b 0 (;z) +b 1 (;z) 0 R 2 + 1 X k=2 b (2+) k (;z) ( 0 R 2 ) k k! ! (3.49) =e 0 R 2 2 4 g(A (0) v (?;z);) + 0 @ 2 Z 0 R Z 0 g(A (1) v (p 1 ;z);) 1 R 2 r 1 dr 1 d 1 1 A 0 R 2 + 2 Z 0 R Z 0 dp 1 2 Z 0 R Z r 1 g(A (2+) v (p (2) ;z);) 1 X k=2 f (k) (p 1 )f (k) (p 2 jp 1 ) ( 0 R 2 ) k k! ! dp 2 3 5 (3.50) =e 0 R 2 2 4 g(A (0) v (?;z);) + 0 @ 2 Z 0 R Z 0 g(A (1) v (p 1 ;z);) 1 R 2 r 1 dr 1 d 1 1 A 0 R 2 3 5 + 2 Z 0 R Z 0 r 1 dr 1 d 1 2 Z 0 R Z r 1 g(A (2+) v (p (2) ;z);) 2 0 e 0 r 2 2 r 2 dr 2 d 2 (3.51) ,b (2+) (;z): (3.52) For all practical purposes, the average number of obstacles is rarely less than two. Hence, the third in the summation in (3.51) is the most significant. We now proceed to determine the conditions under whichb (2+) (;z) is a good approximation forb(;z). Theorem 2. Given z, b (2+) (;z) b ind (;z) overf(;z) : E[A v (z)jK 2 ] 3:3836g, where K 2 denotes the event that there are at least two obstacles present in D o (R). Proof. By conditioning on K 2 and K c 2 ,E[A v (z)] can be expressed as follows: E[A v (z)] =E[A v (z)jK c 2 ]P(K c 2 ) +E[A v (z)jK 2 ]P(K 2 ): (3.53) Clearly,E[A v (z)jK 2 ]E[A v (z)jK c 2 ] as the average visible area can only decrease as the number of obstacles increases. Sinceg(x;) is convex if and only ifx 2, the following holds, from Jensen’s inequality, forE[A v (z)jK c 2 ]E[A v (z)jK 2 ] 2, b ind (;z) =g(E[A v (z)];) (3.54) =g(E[A v (z)jK c 2 ]P(K c 2 ) +E[A v (z)jK 2 ]P(K 2 );) (3.55) g(E[A v (z)jK c 2 ];)P(K c 2 ) +g(E[A v (z)jK 2 ];)P(K 2 ): (3.56) 85 Furthermore, from Theorem 1, we have the following inequality forE[A v (z)jK c 2 ] 3:3836, g(E[A v (z)jK c 2 ];)E[g(A v (z);)jK c 2 ]: (3.57) By further conditioning K 2 on the obstacle locations,E[A v (z)jK 2 ] can be expressed as follows: E[A v (z)jK 2 ] =E p (2)[A n2 (p (2) ;z) +E[A f (p (k) ;z)jp (2) ]]: (3.58) Remark 5. E[A v (z)jK 2 ] can also be obtained by averaging the expression in (3.32) overk. The expression in (3.58) is an equivalent representation of the same quantity. Again, from Theorem 1, the following inequality holds for E[A v (z)jK 2 ] 3:3836 g(E[A v (z)jK 2 ];) =g(E p (2)[A n2 (p (2) ;z) +E[A f (p (k) ;z)jp (2) ]];) (3.59) E p (2)[g(A n2 (p (2) ;z) +E[A f (p (k) ;z)jp (2) ];)]: (3.60) Thus, from (3.53)-(3.60), forE[A v (z)jK c 2 ]E[A v (z)jK 2 ] 3:3836, we have b ind (;z) =g(E[A v (z)];) E p (2)[g(A n2 (p (2) ;z) +E[A f (p (k) ;z)jp (2) ];)]P(K 2 ) +E[g(A v (z);)jK c 2 ]P(K c 2 ) =b (2+) (;z): (3.61) Remark 6. Similar to Theorem 1, Theorem 2 represents a sufficient, but not necessary, condition. Theorem 3. Givenz and,b (2+) (;z)b(;z)c(;z), wherec(;z)2 (0; 1) is a decreasing function in. Proof. Conditioning on K 2 and K c 2 ,b(;z) andb (2+) (;z) can be expressed as fol- lows: b(;z) =E[g(A v (z);)jK c 2 ]P(K c 2 ) +E[g(A v (z);)jK 2 ]P(K 2 ) 86 =E p (2)[E[g(A n2 (p (2) ;z) +A f (p (k) ;z);)jp (2) ]]P(K 2 ) +E[g(A v (z);)jK c 2 ]P(K c 2 ) (3.62) b (2+) (;z) =E p (2)[g(A n2 (p (2) ;z) +E[A f (p (k) ;z)jp (2) ];)]P(K 2 ) +E[g(A v (z);)jK c 2 ]P(K c 2 ): (3.63) Similar to (3.36), the conditional expectation in (3.62), givenp (2) , is over bothk and p (3:k) . Let g 1 (p (2) ;;z) =E[g(A n2 (p (2) ;z) +A f (p (k) ;z);)jp (2) ] (3.64) g 2 (p (2) ;;z) =g(A n2 (p (2) ;z) +E[A f (p (k) ;z)jp (2) ];) (3.65) R 1 (;z) :=fp (2) 2D o (R)D o (R) :g 1 (p (2) ;;z)g 2 (p (2) ;;z); r 1 r 2 g (3.66) R 2 (;z) :=fp (2) 2D o (R)D o (R) :g 1 (p (2) ;;z)<g 2 (p (2) ;;z); r 1 r 2 g (3.67) F 1 (;z) :=fp (2) 2D o (R)D o (R) :A n2 (p (2) ;z) 2=; r 1 r 2 g (3.68) F 2 (;z) :=fp (2) 2D o (R)D o (R) :A n2 (p (2) ;z)< 2=; r 1 r 2 g (3.69) Sinceg(x;) is convex wheneverx 2, it follows thatg(;) is convex over the set ofA n2 (p (2) ;z) resulting fromF 1 . Hence, from Jensen’s inequality,F 1 R 1 . As a result,F 2 R 2 , sinceR 1 [R 2 =F 1 [F 2 . Hence, from (3.62)-(3.69), b (2+) (;z)b(;z) =P(K 2 ) 0 B @ Z R 1 (;z) (g 2 (p (2) ;;z)g 1 (p (2) ;;z))f(p (2) )dp (2) + Z R 2 (;z) (g 2 (p (2) ;;z)g 1 (p (2) ;;z))f(p (2) )dp (2) 1 C A; (3.70) (3.71) wheref(p (2) ) denotes the pdf ofp (2) . Since the integral overR 1 (;z) is non-positive, 87 we have b (2+) (;z)b(;z)P(K 2 ) Z R 2 (;z) (g 2 (p (2) ;;z)g 1 (p (2) ;;z))f(p (2) )dp (2) (3.72) P(K 2 ) Z F 2 (;z) (g 2 (p (2) ;;z)g 1 (p (2) ;;z))f(p (2) )dp (2) (3.73) P(K 2 ) 1 min u2F 2 (;z) g 1 (u;) Z F 2 (;z) f(p (2) )dp (2) (3.74) :=c(;z); (3.75) wherec(;z) :=P(K 2 ) 1 min u2F 2 (;z) g 1 (u;) P(p (2) 2F 2 (;z)) is non-negative and decreasing in and is bounded above by one. Remark 7. From Theorems 2 and 3,b ind (;z) b (2+) (;z) b(;z) +c(;z), for sufficiently large. It is worth pointing out that this inequality relation makes no assumption on the number of obstacles. This implies thatb (2+) (;z) may be a relatively more accurate approximation ofb(;z) thanb ind (;z) as increases, but its accuracy in absolute terms is restricted to when the number of obstacles is small, according to Approximation 1. To summarize, it is intuitive that obstacles which are closer to the typical target induce greater blocking correlation, with the extent of correlation decreasing with distance. Hence, by taking into account the impact of correlated blocking due to the nearest two obstacles,b (2+) (;z) achieves a reasonable trade-off between accuracy and tractability. 3.6 AsymptoticblindspotProbability For the special case whenL!1, the obstacles become a random collection of lines, referred to as a line process. This captures the worst-case impact of correlated blocking 88 Figure 3.8: If the projection,p, ofo onto a linel p lies inside the above disk, thenl p intersects the diameter fromo to the point (R; 0 ). and the resulting asymptotic blind spot probability at a typical target location is an upper bound for the more realistic scenario when the obstacles have finite dimensions. In Cartesian coordinates, a line inR 2 can be expressed as follows: x cos +y sin =r; (3.76) where the pointp = (r;) (r2 [0;1);2 [0; 2)) in polar coordinates denotes the projection ofo onto the line (e.g., Fig. 3.8) and uniquely identifies a line. Hence, the obstacle line process, denoted byX lp , is completely defined by the point process, X p , formed by the collection of projection points, which also coincides with the line- segment mid-points from the obstacle model considered in proposed in Section 3.2. Thus,X p is a homogeneous PPP of intensity 0 . X lp splitsR 2 into a collection of non-overlapping convex polygons, denoted by C(X lp ), that form a tessellation ofR 2 . LetC o (z as )2C(X lp ) denote the polygon containing the typical target ato, wherez as = [ 0 R]. The anchors having LoS too are constrained to lie inC o (z as )\D o (R) (Fig. 3.9) and hence, the typical target is in a blind spot if and only if there are fewer than three anchors present inC o (z as )\D o (R). The special case whenC o (z as )D o (R) is of particular interest as the blind spot probability of the typical target depends only on the area distribution ofC o (z as ) (instead ofC o (z as )\D o (R)), for which accurate closed form approximations exist. IfC o (z as )*D o (R), then there exists at least one direction 0 2 [0; 2) such that V ((R; 0 );z as ) = 1, whereV (:;:) is given by (3.1), withz as being the asymptotic 89 Figure 3.9: If the obstacles are lines, then the unshadowed region is a convex region. In particular, it is a convex polygon if no point at a distanceR from the target has LoS to it. parameter vector asL!1. For this condition to be satisfied, no pointp2X p should lie in a disk of diameterR, centered at (R=2; 0 ) (see Fig. 3.8). Therefore, P(V ((R; 0 );z as ) = 1) = exp( 0 R 2 =4); (3.77) for any 0 2 [0; 2). Hence, for a sufficiently small2 (0; 1) such thate 0 R 2 =4 < ,C o (z as )D o (R) with probability greater than 1. Therefore, we develop our analysis in this section under the assumption thatC o (z as )D o (R) by considering a sufficiently large number of obstacles determined by the parameter (i.e., 0 R 2 > 4 log(1=). Remark 8. Similar to , the notion of a small number of obstacles can also be formalized by requiringP(V ((R; 0 );z as ) = 1)>, for a suitably large2 (0; 1). For example, in Fig. 3.7, eight obstacles on average corresponds to = exp(2). Similar to (3.4), the asymptotic blind spot probability of the typical target, denoted byb as (;z), is defined as follows: b as (;z), 1 Z 0 g(t;)f Av (zas (t)dt; (3.78) 90 whereA v (z as ) denotes the area ofC o (z as ) andf Av (zas) (:) its pdf. To characterize f Av (zas) (:), we define the following terms Definition 4 (V oronoi cell). For a countable set of pointsAR 2 , the Voronoi cell of x2A, denoted byV x (A), is defined as follows: V x (A) =fy2R 2 :kyxk 2 inf u2Anx kyuk 2 g (3.79) wherek:k 2 denotes theL 2 -norm. In other words,V x (A) contains all the points inR 2 that are closer tox than any other point inA. Definition 5 (Poisson-V oronoi cell). IfA is a realization of a homogeneous PPP of intensity, thenV x (A) is referred to as a Poisson-Voronoi cell with parameter, for x2A. In particular, the expected area ofV x (A) equals 1=. From Definitions 4 and 5, we obtain the following result, reproduced from [65]). Theorem 4. The area distribution ofC o (z as ) coincides with that of a typical Poisson- Voronoi cell with parameter 0 =4. Proof. The point process 2X p is a homogeneous PPP of intensity 0 =4. By the Slivnyak-Mecke Theorem [24],A = 2X p [fog has the same distribution as 2X p . Hence, from Definition 5,V o (A) is a Poisson-V oronoi cell with parameter 0 =4. By construction,V o (A) coincides withC o (z as ), as illustrated in Fig. 3.10. Therefore, C o (z as ) has the same area distribution asV o (A). For a typical Poisson-V oronoi cell with parameter 0 =4, the pdf of its area is well-approximated by a three-parameter Gamma distribution [66, 67], given by: f Av (zas) (t) 8 < : a 0 b (c 0 =a 0 ) 0 (c 0 =a 0 ) 0 4 c 0 t c 0 1 e b 0 ( 0 t=4) a 0 ; t 0 0; else (3.80) wherea 0 = 1:07950,b 0 = 3:03226,c 0 = 3:31122 and (z) = R 1 0 x z1 e x dx for z> 0. Hence,b as (;z) can be obtained by substituting (3.80) in (3.78) and evaluating the integral. 91 Figure 3.10: The interior of the polygon,C o (z as ), surrounded by the red lines is the V oronoi cell,V o (A), whereA containso and the points shown in black circles. 3.7 NumericalResults For each (;z), the following cases were evaluated: (i)b(;z), obtained by averag- ing over 50000 Monte-Carlo simulations, (ii)b (2+) (;z), given by (3.51), and (iii) b ind (;z). In accordance with Approx 1, we consider an average of eight obstacles (i.e., 0 R 2 = 8) to represent the regime where the number of obstacles is small. For a fixed average number of anchors, the impact of correlated blocking, which is a function of the normalized obstacle length,L=R, on the blind spot probability is shown in Fig. 3.11. For small values ofL=R (low blocking correlation), the difference between the three cases is minimal, which is intuitive. However, even for moderate blocking correlation (L=R = 0:5), b ind (;z) significantly underestimates b(;z). In contrast, b (2+) (;z) accurately estimates b(;z) across all levels of blocking correlation. For three different cases ofL=R, which capture low, moderate and high blocking correlation, the blind spot probability is plotted as a function of the average number of anchors,R 2 , in Fig. 3.12. For all the cases,b ind (;z) decreases faster thanb(;z) 92 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 L/R 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 P(blind-spot) Avg. no. of obstacles, λ 0 πR 2 = 8, Avg. no. of anchors, λπR 2 = 6 b(λ,z) b (2+) (λ,z) b ind (λ,z) Figure 3.11: By capturing most of the blocking correlation, b (2+) (;z) yields an accurate approximation ofb(;z). In contrast, by ignoring the blocking correlation, b ind (;z) significantly underestimatesb(;z). with increasing , with the rate of divergence being proportional to L=R. Since the nearest two-obstacle approximation captures most of the blocking correlation, the rate of decrease of b (2+) (;z) with respect to is almost identical to that of b(;z), leading to a more accurate approximation. Hence, from a design perspective, b (2+) (;z) can be used to determine such thatb(;z) b (2+) (;z) . It is worth pointing out thatb (2+) (;z) b(;z) for high blocking correlation (3.12c). Although this is consistent with the statement of Theorem 3, we believe that the effect of ignoring the termE[ 2 (V in (p (k) );z)jp (2) ], which is the average area of the striped region in Fig. 3.6 may also be a contributing factor. As pointed out in Approximation 1, E[ 2 (V in (p (k) );z)jp (2) ] increases withL. Hence, by neglecting its contribution to 93 A (2+) v (p (2) ;z) in (3.33), the unshadowed area beyond the second nearest obstacle is systematically underestimated, which may contribute tob (2+) (;z) being greater than b(;z). The approximation error can be further reduced by considering the blocking correlation beyond second nearest obstacle; however, this comes at the expense of tractability as we would have to deal with more complex overlaps involving the shadowed regions. At the other end of the spectrum, for a large average number of obstacles cor- responding to = 10 2 ,b as (;z as ) is plotted as a function of in Fig. 3.13. The closed form expression obtained in (3.78) mirrors the empirically observed results, thereby justifying the approximation thatC o (z as )D o (R) for the chosen value of. Although the asymptotic blind spot probability analysis was formulated forL! 1, it is easy to see that ifC o (z as )D o (R), then clearly,b(;z) =b as (;z as ) for L 2R, since each obstacle forms a secant. However, we observe from Fig. 3.14 that the asymptotic regime begins at much smaller obstacle, starting from approximately R=2. 3.8 Summary In this chapter, we formulated a probabilistic version of the well known art gallery problem when the map of the environment is unknown. We defined the blind spot probability metric and set out to analyze the impact of correlated blocking on it at a typical target location. To model the uncertainty in the obstacle locations, we consid- ered a novel stochastic geometry based approach where the obstacles were modeled as random line-segments using a germ-grain model and the anchor locations were modeled using a homogeneous PPP Under these conditions, we characterized the blind spot probability as a function of the pdf of the unshadowed area. Furthermore, we showed that the blind spot probability under the independent anchor blocking as- sumption depends only on the mean unshadowed area, instead of the entire probability distribution, and derived the conditions under which the independent blocking assump- tion underestimates the true blind spot probability. Since the pdf of the unshadowed area is difficult to characterize in closed form, we derived an approximate expression for the blind spot probability when there are a small number of obstacles present by formulating the nearest two-obstacle approximation, which captures the blocking 94 4 5 6 7 8 9 10 Avg. no. of anchors, λπR 2 10 -3 10 -2 10 -1 10 0 P(blind-spot) L/R = 0.1 b(λ,z) b (2+) (λ,z) b ind (λ,z) (a) Low blocking correlation:L=R = 0:1 4 5 6 7 8 9 10 Avg. no. of anchors, λπR 2 10 -2 10 -1 10 0 P(blind-spot) L/R = 0.5 b(λ,z) b (2+) (λ,z) (closed-form) b ind (λ,z) (b) Moderate blocking correlation:L=R = 0:5 4 5 6 7 8 9 10 Avg. no. of anchors, λπR 2 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 P(blind-spot) L/R = 1 b(λ,z) b (2+) (λ,z) b ind (λ,z) (c) High blocking correlation:L=R = 1 Figure 3.12: The accuracy ofb (2+) (;z) implies that it can be used to determine the anchor intensity that satisfiesb (2+) (;z)b(;z), for some threshold,. 95 3 4 5 6 7 8 9 10 Avg. no. of anchors in C o (z as ) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 P(blind-spot) δ = 0.01 b(λ,z as ) (Simulation) b(λ,z as ) (Closed-form) b ind (λ,z as ) Figure 3.13: b as (;z as ) accurately characterizes the blind spot probability if there exists a sufficiently high intensity of ‘large’ obstacles. correlation up to the second nearest obstacle and assumes independent blocking due to farther obstacles. This yields a reasonable trade-off between accuracy and tractability, wherein our approximation is more accurate than the independent blocking assumption in estimating the true blind spot probability, as the anchor intensity increases. On the other hand, for the limiting case of of infinitely long obstacles, correspond- ing to the worst-case impact of dependent blocking on the blind spot probability, the unshadowed region was shown to be the Poisson-V oronoi polygon, for a sufficiently large number of obstacles. Using its known area distribution, we derive a closed form expression for the asymptotic blind spot probability, which yields an upper bound for the finite obstacle size scenario. Our analysis in this chapter illustrates a solution approach for more general 96 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 L/R 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 P(blind-spot) δ = 0.01 Correlated blocking Avg. no. of anchors in C o (z as )=6, 5, 4 Independent blocking Figure 3.14: The dashed blue, solid red and solid black curves plotb(;z),b as (;z as ) andb ind (;z), respectively. ForLR=2, the line assumption for obstacles holds, given a sufficiently high intensity of obstacles. obstacle models that can be described as follows: (a) Model the randomness in the obstacle locations and shapes. (b) Characterize the visible area distribution. (c) Obtain an (approximate, if necessary) expression/upper bound for the blind spot probability based on the visible area distribution. (d) Using (c), determine the intensity with which anchors need to be deployed so that the blind spot probability over a region of interest is less than a threshold, . 97 Chapter 4 LocalizationOutageAnalysis 4.1 Introduction In order to provide a reliable quality of service in terms of accuracy (e.g., an error of at most 1m at least 90% of the time), it is important to characterize the probability distribution of the positioning error over an ensemble of operating conditions, espe- cially for safety critical applications like autonomous vehicles or E911 emergency services. A commonly used metric for this purpose is the localization mean square error (MSE), which is a function of the anchor locations, the transmit powers, the propagation environment, as well as the choice of ranging and localization algorithms. In this chapter, we focus on the impact of the anchor locations, relative to a target, on the MSE. Specifically, we consider a lower bound for the MSE, known as the squared position error bound (SPEB) [68, 69], which is satisfied by any positioning algorithm that provides an unbiased 1 estimate of the target location. The SPEB is a function of (a) the distance-dependent SNRs of the anchor-target links, and (b) the pairwise angles subtended by the anchors at the target location; crucially, it does not depend on a specific localization algorithm. As a result, it is well-suited as a metric to analyze the impact of the anchor geometry on the positioning error. If the SPEB exceeds a pre-defined threshold, th , then the target is said to be in outage. Over an ensemble of target and/or anchor locations, the SPEB (and the MSE, as well) 1 An estimate ^ p of a target locationp is said to be unbiased ifE[^ p]=p. 99 is a random variable and characterizing its complementary cumulative distribution function (ccdf) in closed form (i.e.,P(SPEB>u), as a function ofu) is important from a design perspective, as it can be used to determine a deployment of anchors that can guarantee an outage probability of at mostp out 2 . GivenN anchors in a region, a natural model for capturing the randomness in the anchor locations is the well-known binomial point process (BPP) [24, Chap. 2], in which the anchors are distributed independently and uniformly over the region. In this chapter, we attempt to derive a closed form expressions for the SPEB ccdf for such an anchor model 3 . To solve this difficult problem, previous works have assumed all links to have the same SNR (i.e., SNR homogeneity), which is equivalent to assuming all anchors to be equi-distant from the target, thereby neglecting the impact of distance variation among the anchors on the positioning error. In this work, we take SNR heterogeneity into account and derive an accurate approximation for the SPEB ccdf. Due to the difficulty of the analysis, we do not assume any blockages due to obstacles, unlike the previous chapter. Our approach is summarized below. 4.1.1 Methodology For a given target, we assume that the anchors that are within its communication range are distributed according to a BPP over an annular region centered at the target. For this setup, we model the SNR heterogeneity across different anchor-target links using a pathloss model. As a result, the SPEB metric is a function of the anchor distances and angular positions, relative to the target. GivenN anchors, we rearrange the SPEB expression and reduce it in terms of the product of two random variables, X N and Y N . While X N depends only on the anchor distances, Y N depends on both the distances and angular positions of the anchors. In particular, Y N and X N are statistically dependent. 2 For a given error threshold, th , an outage probability of at mostpout can be guaranteed if and only if the conditionP(SPEB> th )pout is satisfied, which poses a constraint on the shape of the SPEB ccdf. 3 Typically, the number of anchors,N, is also a random variable, often modeled as having a Poisson distribution. Together with the randomness in the anchor locations, this corresponds to the well-known homogeneous PPP, which has been used to analyze the localization performance of a variety of wireless networks [53, 70, 54, 62, 71]. Hence, the results presented in this chapter for the BPP anchor model can be readily extended for the PPP case by averaging over the distribution ofN. 100 We then proceed to demonstrate that a closed form characterization of the condi- tional distribution of Y N , given X N , is intractable. Hence, through constrained moment matching, we derive an approximation for Y N , denoted by V N , which depends only on the angular positions of the anchors and has the same mean as Y N . In particular, X N and V N are statistically independent. Consequently, the SPEB can be approximated in terms of the product of inde- pendent random variables, X N and V N , and we derive a closed form expression for the ccdf of this approximation (see Theorem 5 in Section 4.3), which is the key result of this chapter. Through simulations, we verify that the derived SPEB ccdf accurately estimates the true ccdf. Thus, from a design perspective, our contribution is useful in determining the number of anchors required in order to satisfyp out , for any 2 (0; 1). We also show that the accuracy of our approach is superior to that of other approaches that ignore SNR heterogeneity, which serves to highlight the impact of SNR heterogeneity on the SPEB (and consequently, the MSE) distribution. 4.1.2 RelatedWork There have been a number of recent works that have focused on the impact of anchor geometry on the localization error performance; specifically, for the SPEB metric, the impact of the target being situated within the convex hull of the anchors was investigated in [72], while scaling laws with respect to the number of anchors within communication range were derived in [73]. A related but simpler metric, known as the geometric dilution of precision (GDOP) has been studied extensively for the BPP anchor model. The GDOP corresponds to a special case of the SPEB when all the anchor-target links have the same SNR. The asymptotic distribution of the GDOP, as the number of anchors approaches infinity, was derived using U-statistics in [74]. For the more realistic case of a finite number of anchors, the max-angle metric was proposed and analyzed in [75] and shown to be correlated to the GDOP. An approximate GDOP distribution was presented in [76], using the order statistics of the inter-node angles, while the exact GDOP distribution was characterized in [77]. 101 To the best of our knowledge, this is the first work that considers the more realistic scenario where the anchor-target links may have different SNRs (due to the anchors being situated at different distances from the target), which increases the difficulty of the problem considerably, as highlighted in Section 4.2. 4.1.3 AdditionalNotation In this chapter, uppercase letters in sans serif font are used for scalar random variables (e.g., X), while random vectors are underlined and similarly represented (e.g., X). Matrices are represented using bold uppercase letters (e.g.,A). For square matrices, the trace and inverse operators are respectively denoted by tr() and () 1 .C denotes the complex numbers,i2 C the imaginary unit, and Im(z) the imaginary part of z2C. For random variables X and Y,f X (),F X () and' X () denote the marginal probability density function (pdf), the marginal ccdf and the characteristic function of X, respectively, whileF XjY (:jy) denotes the conditional ccdf of X, given Y =y. For x2R, the sine and cosine integrals, denoted by Si(x) and Ci(x), respectively, are as follows: Si(x) = x Z 0 sint t dt; (4.1) Ci(x) = 1 Z x cost t dt; (4.2) and the function H :R!C is defined as follows: H(x) := Si(x)iCi(x); x2R: (4.3) 4.1.4 Organization This chapter is divided into five sections. The system model is described in Section 4.2, where the anchors are modeled by a BPP over an annular region surrounding the target, and a distance-dependent pathloss model is assumed for the SNRs of the anchor-target links. Under these conditions, we illustrate the difficulty of characterizing the SPEB distribution in Section 4.3, which motivates the derivation of a tractable approximation for the SPEB ccdf later on in the same section. In Section 4.4, we compare the accuracy 102 of our approach with other bounds and approximations that do not consider SNR heterogeneity. Finally, Section 4.5 concludes this chapter. 4.2 SystemModel Consider a target situated inR 2 that needs to be localized. Since we are interested in the anchor geometry relative to the target, we can assume, without loss of generality, that the target is situated at the origin,o. Centered at the target, considerN 3 an- chors deployed according to a BPP over an annular region fromd min tod max (d max > d min > 0) 4 , denoted byA o (d min ;d max ), and let (R k ; k ) denote the location of the k-th anchor in polar coordinates (R k 2 [d min ;d max ]; k 2 [0; 2); k = 1; ;N). Lets(t), having Fourier transformS(f), denote the ranging signal transmitted by the anchors 5 and lety(t; R k ; k ) denote the signal received from thek-th anchor, which can be modeled as a superposition of a number of MPCs in the following manner: y(t; R k ; k ) = L(R k ; k ) X l=1 l (R k ; k )s(t l (R k ; k )) + k (t); k2f1; ;Ng; (4.4) where the location-dependent quantitiesL(R k ; k ), l (R k ; k )2C and l (R k ; k )2 R respectively denote the number of observed MPCs, the complex amplitude of the l-th MPC and its ToA. k (t) is the measurement noise, which is modeled as a zero- mean complex Gaussian random process, having a power spectral density ofN 0 . We assume that LoS exists from the target to all the anchors. Hence, the first arriving MPC from each anchor corresponds to the DP and depends on the anchor position as follows: 1 (R k ; k ) = 1 (R k ) = R k c ; k2f1; ;Ng; (4.5) wherec denotes the speed of light in free space. Concerning IPs, we assume no prior knowledge of their statistics. Under these conditions, the MSE of an unbiased estimate 4 dmax can be interpreted as the distance beyond whichs(t) is too weak to be detected by the target. 5 We assume that the anchors coordinate their transmissions to avoid interference at the target. As a result, ToA/range estimation is noise-limited. 103 of the target location can be bounded using the CRLB [68, 69], as follows: MSE tr 0 @ N X k=1 (R k ; k )u( k )u( k ) T ! 1 1 A (4.6) := S(R (N) ; (N) ); (4.7) where R (N) = [R 1 ; ; R N ] T ; (4.8) (N) = [ 1 ; ; N ] T ; (4.9) (R k ; k ) = 8 2 2 (1(R k ; k )) (R k ; k ) c 2 ; (4.10) (R k ; k ) = j 1 (R k ; k )j 2 N 0 1 Z 1 js(t)j 2 dt; (4.11) = 2 4 0 @ 1 Z 1 f 2 jS(f)j 2 df 1 A , 0 @ 1 Z 1 jS(f)j 2 df 1 A 3 5 1=2 ; (4.12) andu( k ) = [cos( k ) sin( k )] T : (4.13) S(R (N) ; (N) ) is commonly known as the squared-position error bound (SPEB) [68, 69] in localization terminology. The term(R k ; k ) is referred to as the ranging information intensity (RII) from the k-th anchor and is a measure of the ranging accuracy associated with thek-th anchor 6 . It is a function of the DP SNR, (R k ; k ), the effective bandwith, , and the path overlap factor, (R k ; k )2 [0; 1], which determines the extent of overlap between the DP and subsequent MPCs, due to finite bandwidth 7 . For simplicity, we assume(R k ; k ) = 0 for allk, which corresponds to the case where the DP does not overlap with any other MPC, thereby resulting in the most accurate estimate of R k . Furthermore, (R k ; k ) is a function of the DP attenuation,j 1 (R k ; k )j 2 , for which the following pathloss model is assumed: j 1 (R k ; k )j 2 =j 1 (R k )j 2 = (d min =R k ) 2 : (4.14) Remark 9. For anchors having LoS to the target, the inverse-square law pathloss model in (4.14) is a reasonable assumption for the DP component if there is zero path 6 (R k ; k ) is the reciprocal of the CRLB for an unbiased estimate of R k [14]. 7 The expression for(R k ; k ) can be found in [69]. 104 overlap, which, in turn, can be assumed whend max <d break , whered break denotes the breakpoint distance associated with the ground reflection [78], since zero overlap between the DP and the ground-reflected path can be rarely achieved. Apart from the RIIs, which depend primarily on the ranges, S(R (N) ; (N) ) also depends on the angular geometry of the anchors, which is captured in (4.6) by the outer productu( k )u( k ) T , whereu( k ) is the unit vector in the direction of the k-th anchor. In summary, the k-th term in the summation in (4.6) represents the contribution of thek-th anchor to S(R (N) ; (N) ). From (4.6)-(4.14), S(R (N) ; (N) ) can be expressed as follows: S(R (N) ; (N) ) = N X k=1 R 2 k T s N1 X j=1 N X k=j+1 R 2 j R 2 k sin 2 ( j k ) ; (4.15) whereT s = 8 2 2 d 2 min N 0 c 2 1 Z 1 js(t)j 2 dt: (4.16) Since S(R (N) ; (N) ) does not depend on any particular positioning algorithm, it is well-suited as a metric to analyze the impact of anchor geometry on the MSE. Moreover, many positioning algorithms have been proposed in recent years that have been shown to satisfy (4.6) with equality [17, 79, 80, 81]. Hence, for the remainder of this chapter, we assume that the MSE is identical to S(R (N) ; (N) ). For the special case when all the anchors are at the same distance R from the target (i.e., all links have the same SNR), S(R (N) ; (N) ) reduces to another well-known metric called the Geometric Dilution of Precision (GDOP), which is denoted by G(R; (N) ) and has the following expression 8 : G(R; (N) ) = NR 2 T s N1 X j=1 N X k=j+1 sin 2 ( j k ) = 1 T s G 1 (R)G 2 ( (N) ); (4.17) 8 Technically, the GDOP is defined as the square root of G(R; (N) ) [82] and thus, has the units of distance. However, in order to have a fair comparison with S(R (N) ; (N) ) (which has units of distance-squared), we slightly abuse the notation and refer to G(R; (N) ) as the GDOP here. 105 where G 1 (R) = R 2 ; (4.18) and G 2 ( (N) ) = N N1 X j=1 N X k=j+1 sin 2 ( j k ) : (4.19) Compared to S(R (N) ; (N) ), G(R; (N) ) is more tractable for a statistical charac- terization, since it can be decomposed into a product of two independent random variables, G 1 (R) and G 2 ( (N) ), as shown in (4.17). However, since the sin 2 () terms are weighted differently in the denominator of (4.15), it is, in general, not possi- ble to express S(R (N) ; (N) ) as S 1 (R (N) )S 2 ( (N) ), for some S 1 () and S 2 (), in much the same way as it is generally not possible to represent an expression like a 1 x 1 + +a M x M ash 1 (a 1 ; ;a M )h 2 (x 1 ; ;x M ), for some scalar-valued real functions, h 1 () and h 2 () and arbitrary real values of a i and x i (i = 1; ;M). Hence, for the sake of tractability, we formulate an approximation that allows a decomposition of S(R (N) ; (N) ), along the lines of (4.17), in the following section. 4.3 CharacterizingSPEBdistribution Although S(R (N) ; (N) ) cannot, in general, be decomposed as a product of indepen- dent random variables, a partial decomposition can be obtained as shown in the lemma below: Lemma 5. The expression for S(R (N) ; (N) ) in (4.15) can be re-written as follows: S(R (N) ; (N) ) = 4 T s X N Y N ; (4.20) where X N = N X k=1 A k ; (4.21) A k = R 2 k ; (4.22) Y N = 1 N X k=1 B k;N cos 2 k ! 2 N X k=1 B k;N sin 2 k ! 2 ; (4.23) and B k;N = A k X N ; k = 1; ;N: (4.24) Proof. See Appendix C.1. 106 While X N depends only on R (N) , Y N is a function of both R (N) and (N) . Moreover, since Y N is a function of X N , the two random variables are statistically dependent. LetF S () denote the ccdf of S(R (N) ; (N) ), which can be expressed as follows: F S (u) =P(S(R (N) ; (N) )>u) =P X N Y N 4 uT s = 1E X N F Y N jX N 4 uxT s x : (4.25) Before proceeding to derive an expression for F S (), we consider the GDOP special case, for which the evaluation of (4.25) is relatively simpler. Corollary 1. For the special case when R (N) = R1 in (4.20)-(4.24), S(R (N) ; (N) ) reduces to G(R; (N) ), which can be re-written as follows: S(R1; (N) ) = G(R; (N) ) = 4R 2 T s NW N ; (4.26) where W N = 1 1 N 2 2 4 N X k=1 cos 2 k ! 2 + N X k=1 sin 2 k ! 2 3 5 : (4.27) From (4.26), the ccdf of G(R; (N) ), denoted by F G (), can be obtained as follows: F G (u) =P(G(R; (N) )>u) = 1E R F W N jR 4r 2 T s Nu r = 1E R F W N 4r 2 T s Nu ; (4.28) where (4.28) follows from the independence of R and W N . As a result, the marginal distributions of R and W N completely characterizeF G (). In particular, the ccdf of 107 W N has the following expression [77], F W N (u) = 8 > > > > > > < > > > > > > : 1; u< 0 N p 1u 1 Z 0 J 1 N p 1uy (J 0 (y)) N dy; u2 [0; 1]; 0; u> 0 (4.29) whereJ 0 (:) andJ 1 (:) denote the zeroth and first order Bessel functions, respectively, while the pdf of R is given by f R (r) = 2r d 2 max d 2 min 1(r2 [d min ;d max ]): (4.30) We now focus our attention back to the general case of deriving a closed form expression forF S () from (4.25), by characterizing the marginal distribution of X N and the conditional distribution Y N , given X N . Lemma 6. The characteristic function of X N is given by: ' X N (t) = (' A 1 (t)) N ; (4.31) where' A 1 (t) = 1 d 2 max d 2 min d 2 max exp i t d 2 max d 2 min exp i t d 2 min +tH t d 2 max tH t d 2 min ; (4.32) and H() is given by (4.3). Proof. See Appendix C.2. From' X N (t), the ccdf of X N can be evaluated as follows [83]: F X N (x) = 1 2 + 1 1 Z 0 Imfexp(itx)' X N (t)g t dt: (4.33) Remark 10. We have chosen to characterize X N by its ccdf instead of its pdf, since the ccdf can be obtained from the characteristic function by evaluating a single integral, which is computationally less intensive than the double integral required to obtain the pdf. Since X N is non-negative, the expected value ofh(X N ), for a differentiable 108 real functionh(), can be expressed in terms ofF X N (), by considering the following relation: h(X N ) =h(0) + X N Z 0 h 0 (u)du =h(0) + 1 Z 0 h 0 (u)1(X N >u)du; (4.34) whereh 0 () denotes the derivative ofh(). Thus, by applying the expectation operator on both sides of (4.34), we obtain E[h(X N )] =h(0) + 1 Z 0 h 0 (u)F X N (u)du: (4.35) While the marginal distribution of X N is fairly tractable, as it is the sum ofN iid random variables, the same cannot be said ofF Y N jX N (jx). To illustrate this, consider the expression for Y N , given X N =x: Y N = 1 1 x 2 N X k=1 A 2 k 2 x 2 N1 X j=1 N X k=j+1 A j A k (cos 2 j cos 2 k + sin 2 j sin 2 k ): (4.36) Let A (N) = [A 1 ; ; A N ] T . Given X N =x, it is easily seen from (4.21) and (4.36) that A (N) is a vector of identically distributed, but not independent random variables. Hence, in order to characterizeF Y N jX N (jx), the conditional joint distribution of A (N) , given X N =x, is required, which is not easy to express in closed form. From (4.23), it is clear that the dependence between X N and Y N is induced by the collection of random variables,fB k;N :k = 1; ;Ng. For the sake of tractability, it is desirable to remove this dependence while still preserving some of the statistical properties of Y N such as its mean and variance. This can be done by assuming B k;N m (m 0) and scaling the resulting approximation by a factorv to obtain the following approximation for Y N : Approximation 2. Y N V N , where V N :=v 0 @ 1m 2 2 4 N X k=1 cos 2 k ! 2 + N X k=1 sin 2 k ! 2 3 5 1 A ; (4.37) 109 where the values ofv andm are obtained by moment matching with Y N . Remark 11. For the special case when R (N) = R1, the approximation in (4.37) reduces to an equality (i.e., Y N = V N = W N ), withv = 1 andm = 1=N. From (4.37), the mean and variance of V N are given by: E[V N ] =v(1m 2 N): (4.38) 2 V N =E[(V N E[V N ]) 2 ] = 2 N 2 v 2 m 4 ; (4.39) where (4.38) follows as a result of (N) being an iid uniform random vector over [0; 2). EquatingE[V N ] and 2 V N with the corresponding quantities for Y N , i.e., E[Y N ] and 2 Y N , we obtain the following expressions form andv: m = 1 p v 2 Y N N(N 1) ! 1=4 ; (4.40) v =E[Y N ] +N s 2 Y N N(N 1) : (4.41) However, since Y N is non-negative, a similar requirement on V N imposes the follow- ing constraint onm: V N 0; =) m 2 max (N) 2 4 N X k=1 cos 2 k ! 2 + N X k=1 sin 2 k ! 2 3 5 | {z } Squared-distance ofN-step random walk 1: (4.42) The term in square parentheses in (4.42) can be interpreted as the squared-distance of anN-step two-dimensional random walk with unit step size; thus, it has a maximum value ofN 2 , obtained when all the steps are in the same direction (i.e., k = , for allk). Therefore, 0m 1=N: (4.43) 110 0 5 10 15 20 25 30 N -800 -700 -600 -500 -400 -300 -200 -100 0 (E[Y N ]) 2 σ 2 Y N − (N 2 −N) Verifying the inequality given by (49) Figure 4.1: Since (4.44) is not satisfied, it follows that (4.40), (4.41) and (4.43) cannot be satisfied simultaneously. The values ofE[Y N ] and 2 Y N were obtained empirically from 10 6 samples. For a closed form characterization ofE[Y N ], see Lemma 7. From (4.40) and (4.41), the upper bound on m, given by (4.43), reduces to the following equivalent constraint on the second-order statistics of Y N : (E[Y N ]) 2 2 Y N (N 2 N) 0: (4.44) However, from Fig. 4.1, it can be seen that (4.44) is not satisfied for anyN 3; in fact, the expression on the left-hand side of (4.44) becomes increasingly negative as N increases. Thus, it follows that (4.40), (4.41) and (4.43) are not satisfied simultaneously. In particular, the expression form in (4.40) is greater than 1=N. As a 111 result, we optimize for the values ofm andv in the following manner: min m;v j 2 Y N 2 V N j (4.45) subject to (4:43); E[Y N ] =E[V N ]; (4.46) where the above optimization problem can be viewed as constrained moment matching, due to the non-negativity constraint on V N imposed by (4.43). From (4.46) and (4.38), the objective function in (4.45) can be represented in terms of a single parameter,m, as follows: Y 2 N N(N 1)(E[Y N ]) 2 m 4 (1m 2 N) 2 (4.47) Form> 0, the expression in (4.47) is initially a monotonically decreasing function of m and attains a minimum value of zero, for m given by (4.40). However, as observed previously, this value ofm does not lie in the feasible region, 0m 1=N. Consequently, the minimum value of (4.47) over the interval [0; 1=N] is attained atm = 1=N. Thus, the optimal solutions form andv, denoted bym opt andv opt , respectively, are given by: m opt = 1=N; (4.48) v opt := E[Y N ] 1m 2 opt N ; (4.49) where the expression forE[Y N ] is given by the following lemma: Lemma 7. The mean of Y N is given by E[Y N ] = 1N 0 @ 2 max 2 min 2 + 2 max Z min 1 Z 0 u Imf' T (N) (u) (t)g t dt du 1 A ; (4.50) where min = d 2 max d 2 max + (N 1)d 2 min ; (4.51) max = d 2 min d 2 min + (N 1)d 2 max ; (4.52) and' T (N) (u) (t) =' A 1 ((1u)t)(' A 1 (ut)) N1 : (4.53) 112 Proof. See Appendix C.3. Remark 12. Incidentally, note thatE[B k;N ] = m opt = 1=N. To see this,E[B k;N ] can be expressed as follows: E[B k;N ] =E X N E[A k jX N =x] x ; k = 1; ;N: (4.54) Since A (N) is a vector of identically distributed random variables, given X N =x, we have N X k=1 E[A k jX N =x] =E[X N jX N =x] =x =NE[A k jX N =x]; for any k2f1; ;Ng (4.55) =) E[A k jX N =x] =x=N: (4.56) Substituting (4.56) in (4.54), we get E[B k;N ] = 1=N =m opt : (4.57) While, in retrospect, approximating B k;N by its mean may seem like an obvious choice, the optimality of this approach from a constrained moment matching perspective is not self-evident. Substitutingm opt andv opt in (4.37), we obtain V N =v opt W N : (4.58) Using Approximation 2, S(R (N) ; (N) ) can be approximated as follows: S(R (N) ; (N) ) 4 T s X N V N ; (4.59) where V N is given by (4.58). We now proceed to derive an approximate expression for the ccdf of S(R (N) ; (N) ) using (4.59). Theorem 5. The SPEB ccdf,F S (), can be approximated as follows: F S (u) 1E X N F W N 4 T s uX N v opt :=F S;app (u); (4.60) 113 whereF W N () is given by (4.29),v opt by (4.49) and Lemma 7, and the distribution of X N by (4.33). Proof. From (4.59), we get F S (u) 1E X N F V N jX N 4 T s ux x (4.61) (a) = 1E X N F W N jX N 4 T s uxv opt x (4.62) (b) = 1E X N F W N 4 T s uX N v opt (4.63) :=F S;app (u); (4.64) where (a) follows from (4.58), and (b) from the independence of X N and W N . In the next section, we present numerical results pertaining to Theorem 5. 4.4 NumericalResults For our simulations, we chosed min = 1m andd max = 10m. For comparison, we considerF S (:), obtained from 10 6 realizations of (4.20), F S;app (:) obtained from Theorem 5,F G () from (4.28) and the ccdfs corresponding to the following GDOP- based bounds: Upper and lower bounds to S(R (N) ; (N) ), based on G(R; (N) ): Let R (1) and R (N) denote the distance of the nearest and farthest anchors, respectively. S(R (N) ; (N) ) can then be bounded as follows: G(R (1) ; (N) ) S(R (N) ; (N) ) G(R (N) ; (N) ): (4.65) As a result,F S () can be bounded using (4.28), in the following manner: 1E R (1) " F W N 4R 2 (1) T s Nu !# F S (u) 1E R (N) " F W N 4R 2 (N) T s Nu !# ; (4.66) 114 where the ccdfs of R (1) and R (N) are given by: F R (1) (r) = d 2 max r 2 d 2 max d 2 min N 1(r2 [d min ;d max ]): (4.67) F R (N) (r) = " 1 r 2 d 2 min d 2 max d 2 min N # 1(r2 [d min ;d max ]): (4.68) The ccdf curves are plotted as a function of the SPEB, scaled by the term T s , in Fig. 4.2 forN2f3; ; 8g. For all the values ofN considered, it can be seen that F S;app is accurate at estimatingF S (). From a design perspective, the accuracy of F S;app () at estimating the tail ofF S () is especially useful, as it captures the outage regime. Specifically, for outage probabilities below 1%, both curves coincide, whereas for a 10% outage probability, the MSE threshold, th , is slightly larger forF S;app () thanF S (). Consequently, for a given value of th ,F S;app () can be used to determine the value ofN such that the outage probability is at most 10%. In contrast toF S;app (), we observe that the GDOP-based bounds given by (4.66) become progressively loose, whileF G () becomes increasingly inaccurate, as the value ofN increases. To quantify this, we use the Kolmogorov-Smirnov (KS) statistic as an error metric, which is defined as follows between a ccdf,F (), andF S (): D KS (F ) = sup x jF (x)F S (x)j: (4.69) In Fig. 4.3,D KS () is plotted as a function ofN, for all the ccdfs considered. Consis- tent with the insight obtained from Fig. 4.2, we observe that the error increases withN for the GDOP-based ccdfs, whileD KS (F S;app ) is nearly constant for all values ofN. This highlights the importance of considering distance-dependent SNR heterogeneity, especially when the gap betweend min andd max is large. To quantify the impact of the difference betweend min andd max , Fig. 4.4 plotsD KS () as a function of thed max , forN = 5. While the accuracy ofF S;app () slightly deteriorates with increasing d max , as a consequence of B k;N E[B k;N ] = 1=N becoming less accurate due to the larger variance of B k;N , the resulting error is still smaller than the ones obtained for the GDOP-based ccdfs. Hence,F S;app () still provides the most accurate estimate ofF S (), among previously known approaches. 115 0 100 200 300 400 500 T s u(m 2 ) 10 -2 10 -1 10 0 CCDF N=3 F S (·) F S,app (·) F G (·) Upper bound (eqn. 4.68) Lower bound (eqn. 4.67) (a) 0 50 100 150 200 T s u (m 2 ) 10 -2 10 -1 10 0 CCDF N=4 F S (·) F S,app (·) F G (·) Upper bound (eqn. 4.68) Lower bound (eqn. 4.67) (b) Figure 4.2: F S;app (:) accurately estimates F S (), which is useful from a design perspective for providing probabilistic guarantees against outage. 116 0 50 100 150 200 T s u (m 2 ) 10 -3 10 -2 10 -1 10 0 CCDF N=5 F S (·) F S,app (·) F G (·) Upper bound (eqn. 4.68) Lower bound (eqn. 4.67) (c) 0 50 100 150 200 M 2 u(m 2 ) 10 -4 10 -3 10 -2 10 -1 10 0 CCDF N=6 F S (·) F S,app (·) F G (·) Upper bound (eqn. 4.68) Lower bound (eqn. 4.67) (d) Figure 4.2: continued from the previous page. 117 0 50 100 150 200 T s u (m 2 ) 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 CCDF N=7 F S (·) F S,app (·) F G (·) Upper bound (eqn. 4.68) Lower bound (eqn. 4.67) (e) 0 50 100 150 200 T s u (m 2 ) 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 CCDF N=8 F S (·) F S,app (·) F G (·) Upper bound (eqn. 4.68) Lower bound (eqn. 4.67) (f) Figure 4.2: continued from the previous page. 118 3 4 5 6 7 8 N 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D KS (·) F S,app (·) F G (·) Lower bound (eqn. (66)) Upper bound (eqn. (66)) Figure 4.3: By taking SNR heterogeneity into account,F S;app () is more accurate at estimatingF S () than the GDOP-based ccdfs. 4.5 Summary In this chapter, we set out to characterize the impact of distance-based SNR hetero- geneity on the error performance of ToA-based localization, using the SPEB metric, S(R (N) ; (N) ). We considered anchors deployed according to a BPP over an annular region centered around a given target and assumed a distance-dependent inverse-square law pathloss model to capture the SNR heterogeneity. For this setup, S(R (N) ; (N) ) was shown to be a tightly coupled function of the anchor distances (R (N) ) and angular positions ( (N) ) and as a result, its ccdf,F S () was difficult to characterize in closed form. Hence, we formulated an approximation for S(R (N) ; (N) ), where the coupling 119 0 5 10 15 20 d max 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 D KS (·) N=5 F S,app (·) F G (·) Lower bound (eqn. (66)) Upper bound (eqn. (66)) Figure 4.4: Although B k;N E[B k;N ] = 1=N becomes less accurate as the difference betweend max andd min increases,F S;app () is still more accurate than the GDOP- based ccdfs, as it takes SNR heterogeneity into account. 120 between R (N) and (N) was removed by constrained moment matching, which en- abled us to derive an approximate ccdf of S(R (N) ; (N) ) (i.e.,F S;app ()) in closed form. Through simulations, we observed thatF S;app () was accurate at estimating F S (), especially at the tail, which corresponds to the outage regime. In particular, from a design perspective, it was observed thatF S;app () can be used to determine the number of anchors needed to guarantee an outage probability of at most 10%. Finally, by comparing the accuracy ofF S;app () with GDOP-based ccdfs (obtained from assuming SNR homogeneity) using the KS statistic,D KS (), we demonstrated that SNR heterogeneity has a considerable impact onF S (). 121 Chapter 5 FutureWork In this chapter, we present some ideas for future work based on the material contained in the previous chapters. With respect to multi-target localization, it was already seen in Chapter 2 that target-by-target localization given by P 3 in (2.31) is sub- optimal since it only captured the blocking correlation among anchors at a given target location but not across locations. It is intuitive that targets which are closely situated are more likely to have similar blocking vectors; hence, joint multi-target localization can reduce false alarms by leveraging this information. While P 2 in (2.25) may completely capture the blocking correlation even across target locations, it may not be tractable as it involves the evaluation of a joint distribution,P(k), whose support grows exponentially with the number of targets. A deeper investigation of this trade- off between tractability versus false alarm probability, resulting in a simplification of P 2 where blocking correlation across target locations is partially captured by would be of significant interest to the localization community. The use of stochastic geometry as a modeling and analysis tool for localization and radar networks is still in an early stage with many unexplored questions. For the line segment obstacle model, the notion of the asymptotic blind spot probability could be generalized to any obstacle intensity where for a low obstacle intensity, the unshadowed region is a Johnson-Mehl cell, instead of a Poisson-V oronoi cell, as illustrated in Fig. 5.1. The area pdf of a Johnson-Mehl cell was approximated by a beta distribution in [84]. Apart from providing an upper bound for the blind spot probability, it can also be the basis for assuming a parametric form for the pdf of 123 Shadow region Target Anchor Obstacle (a) Johnson-Mehl (JM) cell Shadow region Target Anchor Obstacle (b) Poisson-V oronoi (PV) cell Figure 5.1: For L 2R and any obstacle intensity, the unshadowed region is a Johnson-Mehl cell, which is the intersection of a Poisson-V oronoi cell and a disc. As the obstacle intensity increases, the unshadowed region tends to a Poisson-V oronoi cell. the unshadowed area,A v (z), for any finite obstacle lengthL, with the parameters estimated by moment matching 1 . This may provide a better understanding of the impact of the number of obstacles ( 0 ), their size (L) and the communication range (R) on the blind spot probability than the integral expressions in (3.51). The connections between the art gallery problem and the blind spot probability present a promising direction for future research. For a given realization of the obstacle locations, the solution to the art gallery problem determines the minimum number of anchors needed for ensuring that there are no blind spots 2 . Hence, over an ensemble of obstacle realizations, the collection of such anchor locations can be interpreted as an optimized point process. For a given blind spot probability, the difference between the properties of the optimized point process and the PPP is a measure of the penalty incurred by resorting to the sub optimal PPP anchor model, 1 The beta and the three parameter gamma distributions for the area pdfs of the Johnson-Mehl and Poisson-V oronoi cells, respectively, were similarly estimated by curve fitting using moment matching. This is a standard technique in the stochastic geometry literature for approximating distributions whose closed form characterization may be intractable. 2 Since this problem is NP hard, the word ‘solution’ in this context refers to the output of any algorithm that provides a constant factor approximation [85]. 124 Target Fixed Obstacles Fleeting Obstacles (a) Target (b) Figure 5.2: Consider independent realizations of a 1D PPP on thex andy axes of a Cartesian coordinate system. At each point on thex/y-axis, a vertical/horizontal line. The resulting collection of lines forming a random grid is referred to as a Manhattan Line Process (MLP) which is a suitable model for obstacles such as buildings which have a grid like structure. The Boolean model may still be suitable for fleeting obstacles like vehicles. which would be of interest to a network planner. It is easy to see that the penalty incurred by the PPP anchor model would depend on statistical characteristics of the obstacle model. While the Boolean model was chosen for its simplicity in Chapter 3, it may not adequately capture all environments; for instance, in an outdoor urban environment with buildings as obstacles, the unshadowed region surrounding a target is likely to resemble the shapes shown in Fig. 5.2. Such obstacles can be modeled with the Manhattan line process (MLP) [24] as a starting point. Since the network planner would typically need to satisfy constraints on both the blind spot probability and positioning accuracy simultaneously (e.g., 1% blind spot probability and a median positioning error of 0:5m), the characterization of the SPEB distribution for the case when anchors experience dependent blocking is of considerable interest from a design perspective. 125 Appendix A BlockingmodelfromChapter2 Let the obstacles be represented by balls of diameterL, whose centers are distributed according to a homogeneous PPP with intensity. For LoS to exist between two points separated by a distanced, no obstacle center should lie within a rectangle of sidesL andd (Fig. A.1). Therefore, the LoS probability is exp(Ld). Consider a consistent blocking vectork t =w t N v t at (x t ;y t ). The set of nodes that are blocked/unblocked at (x t ;y t ) is determined byv t andw t . For each unblocked node, there exists a rectangle which cannot contain any obstacle center. The LoS Figure A.1: LoS is obstructed if there exists at least one obstacle center within a distance ofL=2 from the LoS path 127 Figure A.2:k t = [0; 0; 0; 1; 0; 0; 1; 0; 0] = [0; 1; 1] N [1; 0; 0] polygon,S los , is the union of such rectangles (shaded grey in Fig. A.2). In contrast, for each blocked noden, there exists an NLoS polygonS n - the portion of its rectangle not contained inS los - which must contain at least one obstacle center. LetN bl denote the number of blocked nodes. Then, P(k t ) exp(Ar(S los )) N bl Y n=1 (1 exp(Ar(S n ))) (A.1) where Ar(:) denotes the area operator, acting on sets inR 2 . The expression in (A.1) is a lower bound since it ignores overlapping NLoS polygons which may share obstacle centers (e.g., TX 2 and TX 3 in Fig. A.2). The bound is met with equality when none of the NLoS polygons overlap. 128 Appendix B ProofsfromChapter3 B.1 ProofofLemma1 The unshadowed area,A v (z), can be expressed as follows: A v (z) = 2 Z 0 R Z 0 V (p;z)rdrd (B.1) ) E[A v (z)] = 2 Z 0 R Z 0 E[V (p;z)]rdrd = 2 Z 0 R Z 0 P(V (p;z) = 1)rdrd = 2 R Z 0 P(V (p;z) = 1)rdr; (B.2) where (B.2) follows from the radial symmetry of the system model considered in Section 3.2. For independent anchor blocking, the unblocked anchors can be viewed as a point process obtained by independently sampling the anchor PPP, where the sampling probability of an anchor at p2D o (R), with respect to the typical target, equals P(V (p;z) = 1). As a result, the unblocked anchors to the typical target form a 129 non-homogeneous PPP whose intensity, ind (p;z), is given by ind (p;z) =P(V (p;z) = 1): (B.3) For a non-homogeneous anchor PPP with intensity ind (p;z), the number of anchors over a circle of radiusR has a Poisson distribution with mean (;z), given by (;z) = 2 Z 0 R Z 0 ind (p;z)rdrd =E[A v (z)]; (B.4) where (B.4) is obtained from (B.2). Hence, the blind spot probability due to indepen- dent anchor blocking is given by b ind (;z) =e (;z) 1 + (;z) + ((;z)) 2 2 =e E[Av (z)] 1 +E[A v (z)] + (E[A v (z)]) 2 2 =g(E[A v (z)];): (B.5) B.2 ProofofLemma2 An obstacle with mid-point at (;) (in polar coordinates) can block the LoS path between the typical target and an anchor atp2D o (R) if and only if the following conditions are satisfied (see Fig. B.1) 0 tanjjL=2 (B.6) 0 secjjr: (B.7) Hence,V (p;z) = 1 is unblocked if and only if there are no obstacle mid-points in the setS V (p;z) =S V 1 (p;z) T S V 2 (p;z) (Fig. B.1), where S V 1 (p;z) =f(;)2R 2 : 0 tanjjL=2g (B.8) S V 2 (p;z) =f(;)2R 2 : 0 secjjrg: (B.9) 130 From (B.8) and (B.9), the azimuthal end-points of S V (p;z) at a radial distance 2 [0;r] are given by min arctan L 2 ; arccos r . Therefore, P(V (p;z) = 1) =P(no obstacle mid-point inS V (p;z)) =e 0 2 (S V (p;z)) (B.10) where 2 (S V (p;z)) = r Z 0 +min(arctan(L=(2));arccos(=r)) Z min(arctan(L=(2));arccos(=r)) dd = 2 r Z 0 min(arctan(L=(2)); arccos(=r))d: (B.11) Substituting (B.10) and (B.11) in (B.2) completes the proof. B.3 ProofofLemma3 Suppose there exists an overlap betweenA sh (p 2 ;z) andA sh (p 1 ;z). Then, let (p (2) ;z) denote the azimuthal width ofA sh (p 2 ;z)\A sh (p 1 ;z). The feasible sce- narios for the end-points of the intervalsI(p 1 ;z) andI(p 2 ;z) are as follows: Case 1:l(p 1 ;z)u(p 1 ;z) andl(p 2 ;z)u(p 2 ;z) This corresponds to whenf(r; 0) : 0 r Rg = 2A sh (p 1 ;z)[A sh (p 2 ;z) (Fig. B.2a). For an overlap to occur betweenA sh (p 2 ;z) andA sh (p 1 ;z), one of the following conditions must be satisfied: a) l(p 1 ;z)l(p 2 ;z)<u(p 1 ;z) (top, Fig. B.2a). b) l(p 2 ;z)l(p 1 ;z)<u(p 2 ;z) (bottom, Fig. B.2a). Hence, (p (2) ;z) = min(u(p 1 ;z);u(p 2 ;z)) max(l(p 1 ;z);l(p 2 ;z)): (B.12) Case 2:l(p 1 ;z)u(p 1 ;z) andl(p 2 ;z)>u(p 2 ;z) This corresponds to when f(r; 0) : 0 r Rg = 2A sh (p 1 ;z) andf(r; 0) : 0 r Rg2A sh (p 2 ;z) (Fig. B.2b). For an overlap to occur, exactly one of the following conditions must be satisfied: 131 S v (p;z) for p=(5,0) and L = 5m 1 2 3 4 5 30 210 60 240 90 270 120 300 150 330 180 0 L/2 ρ tanβ r (ρ,β) )β ρ sec β Figure B.1: Forp = (5; 0), the region enclosed by the blue curve corresponds to S V 1 (p;z) =f(;)2R 2 : 0 tanjjL=2g. Similarly, the region enclosed by the black curve corresponds toS V 2 (p;z) =f(;)2R 2 : 0 secjjrg. Hence, the LoS path too is unblocked if and only if there is no obstacle mid-point in the shaded region, which corresponds toS V (p;z) =S V 1 (p;z)\S V 2 (p;z). a) u(p 2 ;z)<l(p 1 ;z)<l(p 2 ;z)u(p 1 ;z) (top, Fig.B.2b). b) u(p 2 ;z)>l(p 1 ;z) (bottom, Fig.B.2b). Hence, (p (2) ;z) = max(u(p 2 ;z)l(p 1 ;z);u(p 1 ;z)l(p 2 ;z)): (B.13) Case 3:l(p 1 ;z)>u(p 1 ;z) andl(p 2 ;z)u(p 2 ;z) This corresponds to when f(r; 0) : 0 r Rg2A sh (p 1 ;z) andf(r; 0) : 0 r Rg = 2A sh (p 2 ;z) (Fig. B.2c). For an overlap to occur, exactly one of the following conditions must be satisfied: a) l(p 2 ;z)<u(p 1 ;z) (top, Fig. B.2c). 132 (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4 Figure B.2: Feasible overlap situations betweenA sh (p 1 ;z) andA sh (p 2 ;z) in the azimuth coordinate. The size of the interval shaded grey denotes(p (2) ;z). b) u(p 2 ;z)>l(p 1 ;z) (bottom, Fig. B.2c). Hence, (p (2) ;z) = max(u(p 2 ;z)l(p 1 ;z);u(p 1 ;z)l(p 2 ;z)): (B.14) Case 4:l(p 1 ;z)>u(p 1 ;z) andl(p 2 ;z)>u(p 2 ;z) This corresponds to when f(r; 0) : 0rRg2A sh (p 1 ;z)\A sh (p 2 ;z) (Fig. B.2d). For an overlap to occur, one of the following conditions must be satisfied: a) l(p 1 ;z)<l(p 2 ;z) (top, Fig. B.2d). b) u(p 1 ;z)>u(p 2 ;z) (bottom, Fig. B.2d) Hence, (p (2) ;z) = 2 (max(l(p 1 ;z);l(p 2 ;z)) min(u(p 1 ;z);u(p 2 ;z))): (B.15) From the expressions in (B.12)-(B.15), it is easily seen that(p (2) ;z) is negative if A sh (p 1 ;z)\A sh (p 1 ;z) =?. Hence, from the above cases, the fraction ofA sh (p 2 ;z) that overlaps withA sh (p 1 ;z) is given by, (p (2) ;z) = max 0; (p (2) ;z) (p 2 ;z) ! ; (B.16) 133 where(p (2) ;z) = 8 > > > > > > > > > < > > > > > > > > > : min(u(p 1 ;z);u(p 2 ;z)) max(l(p 1 ;z);l(p 2 ;z)); if l(p 1 ;z)u(p 1 ;z);l(p 2 ;z)u(p 2 ;z) 2 (max(l(p 1 ;z);l(p 2 ;z)) min(u(p 1 ;z);u(p 2 ;z))); if l(p 1 ;z)>u(p 1 ;z);l(p 2 ;z)>u(p 2 ;z) max(u(p 2 ;z)l(p 1 ;z);u(p 1 ;z)l(p 2 ;z)); else. (B.17) B.4 ProofofLemma4 LetA out (p (2) ;z) = f(r;) 2 D o (R) : r > r 2 ; = 2 I(p 1 ;z)[I(p 2 ;z)g V out (p (k) ;z). Similar to (B.2), we have E[ 2 (V out (p (k) ;z))jp (2) ] = Z p2Aout(p (2) ;z) P(V (p;z) = 1)rdrd: (B.18) Due to rotational symmetry, the integral in (B.18) does not depend on azimuthal coordinate,. Hence, E[ 2 (V out (p (k) ;z))jp (2) ] =' span (A out (p (2) ;z)) R Z r 2 P(V (p;z) = 1)rdr; (B.19) where' span (A out (p (2) ;z)) = 2(p 1 ;z) (1(p (2) ;z))(p 2 ;z): (B.20) In (B.20),' span (A out (p (2) ;z)) denotes the azimuthal width ofA out (p (2) ;z). Substi- tuting (B.10) and (B.11) in (B.18), we get the desired result. 134 Appendix C ProofsfromChapter4 C.1 ProofofLemma5 From (4.15), we have S(R (N) ; (N) ) = N X k=1 R 2 k T s N1 X j=1 N X k=j+1 R 2 j R 2 k sin 2 ( j k ) (C.1) = 2 N X k=1 R 2 k T s N1 X j=1 N X k=j+1 R 2 j R 2 k (1 cos(2 j 2 k )) (C.2) = 2 N X k=1 R 2 k T s N1 X j=1 N X k=j+1 R 2 j R 2 k (1 cos 2 j cos 2 k sin 2 j sin 2 k ) (C.3) 135 = 4 N X k=1 R 2 k T s 2 4 N X k=1 R 2 k ! 2 N X k=1 R 2 k cos 2 k ! 2 N X k=1 R 2 k sin 2 k ! 2 3 5 ; (C.4) where (C.4) is obtained from the following identity N1 X j=1 N X k=j+1 a j a k = 1 2 N X k=1 a k ! 2 1 2 N X k=1 a 2 k ; a k 2R8k: (C.5) Let A k = R 2 k ; k2f1; ;Ng; (C.6) X N = N X k=1 A k ; (C.7) and B k;N = A k X N : (C.8) Using (C.6)-(C.8), (C.4) can be expressed as follows: S(R (N) ; (N) ) = 4 T s X N Y N ; (C.9) where Y N = 1 N X k=1 B k;N cos 2 k ! 2 N X k=1 B k;N sin 2 k ! 2 : (C.10) C.2 ProofofLemma6 Since A (N) is an iid random vector, the characteristic function of X N = A 1 ++A N is given by ' X N (t) = (' A 1 (t)) N (C.11) From (4.22) and (4.30), the pdf of A 1 can be expressed as follows: f A 1 (a) = (1=2)a 3=2 f R (a 1=2 ) (C.12) 136 ) ' A 1 (t) =E[exp(itA 1 )] = 1 Z 1 cos(ta)f A 1 (a) da +i 1 Z 1 sin(ta)f A 1 (a) da =I c (t;d min ;d max ) +iI s (t;d min ;d max ); (C.13) whereI c (t;d min ;d max ) := 1 Z 1 cos(ta)f A 1 (a) da; (C.14) andI s (t;d min ;d max ) := 1 Z 1 sin(ta)f A 1 (a) da: (C.15) Integrating (C.14) and (C.15) by parts, we get I c (t;d min ;d max ) = 1 d 2 max d 2 min d 2 max cos t d 2 max d 2 min cos t d 2 min tSi t d 2 min +tSi t d 2 max ; (C.16) I s (t;d min ;d max ) = 1 d 2 max d 2 min d 2 max sin t d 2 max d 2 min sin t d 2 min tCi t d 2 max +tCi t d 2 min ; (C.17) where Si() and Ci() are given by (4.1) and (4.2), respectively. Combining (C.13), (C.16), (C.17) and (4.3), we get ' A 1 (t) = 1 d 2 max d 2 min d 2 max exp i t d max 2 d 2 min exp i t d min 2 +tH t d 2 max tH t d 2 min : (C.18) C.3 ProofofLemma7 B 1;N ; ; B N;N form a collection of identically distributed, but not independent, random variables. In addition, j and B k;N are also independent random variables, 137 for anyj;k. Hence, from (4.23), we obtain E[Y N ] = 1NE[B 2 1;N ]; (C.19) For a fixedb2 R, the functionx=(x +b) is increasing inx. Hence, from (4.24), it follows that B 1;N is increasing in A 1 , for fixed A 2 ; ; A N . As a result, B 1;N 2 [ min ; max ], where min = d 2 max d 2 max + (N 1)d 2 min ; (C.20) max = d 2 min d 2 min + (N 1)d 2 max : (C.21) As B k;N is non-negative for allk,E[B 2 1;N ] can be expressed as follows: E[B 2 1;N ] = 2 max Z min uF B 1;N (u)du: (C.22) From (4.24),F B 1;N (u) can be expressed as follows: F B 1;N (u) =F T (N) (u) (0); (C.23) where T (N) (u) = A 1 (1u)u N X j=2 A j (C.24) Since A (N) is an iid random vector, the characteristic function of T (N) (u) has the following expression: ' T (N) (u) (t) =' A 1 ((1u)t)(' A 1 (ut)) N1 : (C.25) Similar to (4.33),F T (N) (u) (0) can be evaluated from' T (N) 1 (u) (t), as follows: F T (N) (u) (0) = 1 2 + 1 1 Z 0 Imf' T (N) (u) (t)g t dt: (C.26) Combining (C.19)-(C.26), we get E[Y N ] = 1N 0 @ 2 max 2 min 2 + 2 max Z min 1 Z 0 u Imf' T (N) (u) (t)g t dt du 1 A : (C.27) 138 Bibliography [1] S. Steineger, M. Neun, A. Edwardes, and B. Lenz, “Foundations of location based services,” 2006. [Online]. 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Abstract (if available)
Abstract
The ability to accurately determine the location of (i.e., localize) one or more targets remotely is an essential requirement for many applications, such as navigation, search-and-rescue operations etc. The canonical localization problem involves estimating the target locations in an environment by analyzing wireless signals emanating from them at a collection of known anchor locations. If the targets transmit their own signals, it is referred to as active localization
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Asset Metadata
Creator
Aditya, Sundar
(author)
Core Title
Towards an understanding of the impact of dependent blocking on localization performance
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
07/29/2018
Defense Date
05/02/2018
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
Bayesian estimation,blind spot probability,dependent blocking,multi-target localization,OAI-PMH Harvest,Poisson line process,Poisson point process,positioning algorithms,random geometry,squared position error bound
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application/pdf
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Language
English
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(provenance)
Advisor
Molisch, Andreas (
committee chair
), Dhillon, Harpreet (
committee member
), Kempe, David (
committee member
), Krishnamachari, Bhaskar (
committee member
)
Creator Email
sundarad@usc.edu,sundaraditya89@gmail.com
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https://doi.org/10.25549/usctheses-c89-38689
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UC11669154
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etd-AdityaSund-6548.pdf (filename),usctheses-c89-38689 (legacy record id)
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etd-AdityaSund-6548.pdf
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Aditya, Sundar
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University of Southern California Dissertations and Theses
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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...
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Tags
Bayesian estimation
blind spot probability
dependent blocking
multi-target localization
Poisson line process
Poisson point process
positioning algorithms
random geometry
squared position error bound