University of Southern California Dissertations and Theses

Improvement in hyperbolic position location systems

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Improvement in hyperbolic position location systems

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Content IMPROVEMENT IN HYPERBOLIC POSITION LOCATION SYSTEMS by Ziba Ebrahimian A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) December 2006 Copyright 2006 Ziba Ebrahimian Acknowledgements I would first like to give my sincere thanks to my supervisor, Professor Robert A. Scholtz, who gave me so much encouragement and helpful advice that always guided me in the right direction. If it was not for his dedication to research and science, I would not have started and finished my dissertation successfully. I would like to give my special thanks to Professor Francis Bonahon, who offered me much-appreciated advice, support and thought-provoking ideas throughout my research. I also appreciate the helpful comments, encouragement and support given to me by Prof. William C. Lindsey. I am also profoundly indebted to my fellow student colleagues (Robert Weaver, Majid Nematian and Sanghyun Chang) who provided me with invaluable support, assis- tance and offered their time as I collected necessary data for the experiment. Finally, I would like to express my sincere appreciation to Jordan Melzer and Terry Lewis for their editorial support. Additionally, I would like to thank Milly Montenegro, Gerrielyn Ramos, and Mayumi Thrasher, for their invaluable assistance throughout my graduate studies. This work was supported in part by the Army Research Office under MURI Contract DAAD19-01-1-0477. It is not sufficient to express my gratitude with only a few words to my parents (espe- cially my mom) for their love and support; and to my friends for their encouragement. ii This dissertation is dedicated to them, and to those who dedicated their lives to a better life. To all of you, thank you. iii Table of Contents Acknowledgements v List of Figures vi 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Overview of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . 3 2 New Methods for NLOS Omission in Position Location Systems 5 2.1 Consistent Groups of Signals From an Array of Omni-Directional Anten- nas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Grouping Algorithm for Omni Directional Receivers . . . . . . 9 2.2 Determining the Groups of Signals for an Array of Directive Antennas in 3-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.1 Grouping Algorithm for Directional Receivers . . . . . . . . . 15 2.3 Algorithm to Distinguish Between Transmitter and Reflector Sources . 18 2.4 Using Four Reflectors to Locate the Transmitter Without Having LOS 24 2.4.1 Algorithm to Locate the Transmitter Using Located Reflectors . 25 2.5 Virtual Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5.1 Ambiguity in the Algorithms for Outdoor Systems . . . . . . . 31 2.6 Threshold Setting for the Algorithms . . . . . . . . . . . . . . . . . . . 33 2.6.1 Estimator Accuracy . . . . . . . . . . . . . . . . . . . . . . . . 34 2.6.2 Hyperbolic Location Systems . . . . . . . . . . . . . . . . . . 40 2.6.3 Threshold Setting . . . . . . . . . . . . . . . . . . . . . . . . . 44 3 Receiver Sites for Accurate Indoor Position Location Systems 48 3.1 Hyperbolas of Uncertainty Area and Their Features in 2-D and 3-D . . 48 3.1.1 Uncertainty Region in 3-D . . . . . . . . . . . . . . . . . . . . 52 3.2 Best Locations of the Receivers for 2-D . . . . . . . . . . . . . . . . . 54 3.3 Best Locations of the Receivers for 3-D . . . . . . . . . . . . . . . . . 59 3.4 Number of the Receivers to Cover an Area in 2-D and 3-D . . . . . . . 66 3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 iv 4 Experimental Results 69 4.1 Measurement Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3 Precision Analysis with Simulated AWGN Noise . . . . . . . . . . . . 77 5 Conclusions and Future Works 84 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . 85 Reference List 86 v List of Figures 2.1 Example 1: source localization with arrays of omni-directional receivers 6 2.2 Timing diagram of example 1 . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Flow chart of the Grouping algorithm for omni directional receivers . . 12 2.4 Example2: source localization with arrays of directive receivers . . . . 14 2.5 Timing diagram of example2 . . . . . . . . . . . . . . . . . . . . . . . 15 2.6 Source localization by the triangulation method for directive receivers . 16 2.7 Example3: source localization when there are insufficient LOS signals but at least four reflectors are detected . . . . . . . . . . . . . . . . . . 25 2.8 Timing diagram of example3 . . . . . . . . . . . . . . . . . . . . . . . 26 2.9 Virtual source in 2-D . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.10 Ambiguity ellipse for 2-D . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.11 Three Ambiguity ellipses 2-D . . . . . . . . . . . . . . . . . . . . . . 33 2.12 Ambiguity ellipsoid for 3-D . . . . . . . . . . . . . . . . . . . . . . . 33 2.13 Three Ambiguous ellipsoids for 3-D . . . . . . . . . . . . . . . . . . . 34 2.14 Uncertainty ellipse [14] . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.15 Geometry of transmitter position, mean location estimate, CEP, estima- tor bias vector, and particular location estimate [14] . . . . . . . . . . . 38 2.16 Geometry of transmitter and N station [14] . . . . . . . . . . . . . . . . 41 2.17 Position error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1 hyperbola of LOP line for two receiving station atA andB and nominal source atx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 vi 3.2 Geometry of a hyperbolic location system with nominal emittert posi- tionx and receiving stations at A, B, and C. The uncertainty area approx- imated with a parallelogram . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Hyperboloid of LOPs for three sensors in a rectangular shape area . . . 51 3.4 Parallelograms of uncertainty of two pairs sensors meet each other at the corners. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.5 Shape of uncertainty for every triplet of sensors for 3-D. . . . . . . . . 52 3.6 Shape of uncertainty for pairs sensors for 3-D with Z configuration. . . 53 3.7 Shape of uncertainty for pairs sensors for 3-D with Y configuration. . . 55 3.8 Sensors located on the the vertices of the triangular area . . . . . . . . 56 3.9 FunctionS for0<®;¯ <¼ . . . . . . . . . . . . . . . . . . . . . . . 56 3.10 The domain of functionS for the points inside the triangle of the three sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.11 Function S for the domain in 3.18(Á 1 =¼=6;Á 2 =¼=4) . . . . . . . . 58 3.12 Sensors located on the points other than the vertices of the triangular area 58 3.13 Tetrahedral volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.14 Geometry of bad edges: case 1 . . . . . . . . . . . . . . . . . . . . . . 60 3.15 Geometry of the sensors for an arbitrary point inside the tetrahedron for case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.16 Geometry of bad edges: case 2 . . . . . . . . . . . . . . . . . . . . . . 61 3.17 Geometry of bad edges: case 3 . . . . . . . . . . . . . . . . . . . . . . 62 3.18 The geometry of bad edges: case 4 . . . . . . . . . . . . . . . . . . . . 62 3.19 Geometry of bad edges: case 5 . . . . . . . . . . . . . . . . . . . . . . 63 3.20 Geometry of the sensors for an arbitrary point inside the tetrahedron for case 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.21 Sensors located on the points other than the vertices of the volume . . . 64 3.22 Regular tetrahedron placed in a cube . . . . . . . . . . . . . . . . . . . 65 vii 3.23 n-polygon as an arbitrary area . . . . . . . . . . . . . . . . . . . . . . 67 3.24 The sensors are located on the vertices of the area . . . . . . . . . . . . 67 3.25 n-polygon with h-holes as an arbitrary area. The sensors are located on the vertices of the area and of its holes . . . . . . . . . . . . . . . . . . 68 4.1 Template signal used in the experiments . . . . . . . . . . . . . . . . . 69 4.2 The experiment plan for Tx1: 3D image . . . . . . . . . . . . . . . . . 70 4.3 The experiment plan for Tx1: top view. . . . . . . . . . . . . . . . . . 71 4.4 The experiment plan for Tx2: side view . . . . . . . . . . . . . . . . . 71 4.5 The experiment plan for Tx2: top view . . . . . . . . . . . . . . . . . . 72 4.6 The experiment plan for Tx3: side view . . . . . . . . . . . . . . . . . 72 4.7 The experiment plan for Tx3: top view . . . . . . . . . . . . . . . . . . 73 4.8 The experiment plan for Tx2 and Tx3: 3D image . . . . . . . . . . . . 73 4.9 Table 1: The difference in timing error for different guesses . . . . . . 79 4.10 Indexing of the receivers for Tx1 . . . . . . . . . . . . . . . . . . . . . 80 4.11 Results for Tx1 for some of the groups . . . . . . . . . . . . . . . . . 80 4.12 Block diagram of the receiver’s front end . . . . . . . . . . . . . . . . 81 4.13 Spectrum of the template signal and filter response of the front end . . . 81 4.14 Error curves for Tx1 by MMSE method . . . . . . . . . . . . . . . . . 82 4.15 Error curve for Tx2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.16 Error curve for Tx3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.17 Table 2: error(inch) versus SNR(dB) for Tx2, with closed form method 83 viii Abstract Non-Line of Sight (NLOS) error is usually the largest cause of degradation in local- ization accuracy and is quite common in all environments. In the first part of this work, four algorithms are given to distinguish between line-of-sight (LOS) and reflected sig- nals. These algorithms exploit timing equations in an RF emitter location system. The algorithms are designed to be applied to both directional and omnidirectional receivers. It is known that hyperbolic radio-location systems possess intrinsic uncertainty that depends on the sensors’ geometry and the modulation’s time resolution. In the second part of this work, the best locations for the receivers to minimize the uncertainty in 2-dimensional (2-D) and 3-dimensional (3-D) models for indoor hyperbolic position location systems are found. Next, we derived the number of sensors that is sufficient to cover any area in 2-D and 3-D cases. Three different measurements were done with Ultra-Wide-Band signals. The trans- mitter was located with good precision, and the accuracy of the first and third algorithms were verified for omnidirectional receivers by the experiments. The precision of the first and third algorithms was analyzed by simulated Additive White Gaussian Noise (AWGN). ix Chapter 1 Introduction 1.1 Motivation One of the advantages of Ultra-Wide-Band (UWB) technology is its very short pulse widths, which allow multipath be resolved. The fine time resolution of UWB radio also makes it attractive for precision localization. Ranging and positioning are expected to play a leading role in the advanced design of wireless communication networks in the coming years [6]. Source localization has been of considerable interest in radar, sonar, navigation, geophysics and tracking. In most of these applications the source is located using the time-of-arrival (TOA), time- difference-of-arrival (TDOA) or angle-of-arrival (AOA). Measurement error and non-line-of-sight (NLOS) error are two major sources of localization error in location techniques. Among these two sources of error, it has been remarked ([5, 9, 15]) that NLOS error usually causes more degradation in localization accuracy and is quite common in all environments [5]. One other source of error in radio navigation systems is emitter position-fix uncer- tainty, which is a function of hyperbolic line-of-position (LOP) crossing angles and therefore depends on the geometry of the sensors. Another source of error in such sys- tems is a phenomenon that is called fix ambiguity [1]. Based on [1], fix ambiguity results when one Loran LOP crosses another LOP in two separate places. Near the base line extension, the “ends” of a hyperbola can wrap around so that they cross another LOP 1 twice, once along the baseline, and again along the base line extension. A third LOP would resolve the ambiguity [1]. For spherical positioning, which is usually based on time-of-arrival measurements, techniques have been prepared to mitigate NLOS error, as well as measurement error. NLOS identification techniques, proposed in some literature (e.g. [15]), try to find some distinct properties of NLOS range measurement, such as standard deviation to distin- guish between LOS and NLOS. Another approach in positioning with radiolocation is based on angle-of-arrival mea- surements. This technique usually uses directive sensors to obtain AOA, then, the inde- pendent measurements are used to solve a triangulation problem. Again, the main draw- back of this technique for terrestrial systems is the possibility of error in estimating the directions caused by multipath reflections ([10]). The previous work on hyperbolic positioning (which is based on TDOA measure- ment), have largely focused on improvement of processing accuracy, and development of efficient algorithms by different techniques, including linear approximation/direct numerical optimization, likelihood/least-squares and iterative versus closed-form algo- rithms [13, 4, 12, 8, 3]. Having LOS signals was implied in all of these literatures. Vir- tually none of them has focused on omitting or mitigating NLOS error in the localization process, or on decreasing the intrinsic uncertainty in hyperbolic systems by finding an appropriate geometry for the receiving stations. The work presented here address both problems in TDOA systems, and also NLOS mitigation in AOA systems. To find the source position in hyperbolic systems, closed form computations (e.g., the Fang’s estimator given in [7]) or iterative algorithms (based on maximum likelihood estimation (MLE)) have been developed. The algorithms presented inx2.1.1 andx2.4.1 are applicable to the iterative methods which give the MLE solution. 2 1.2 Overview of the Dissertation In this dissertation, we address distinguishing between LOS and NLOS signals, for both techniques (TDOA and AOA), exploiting the timing relations between the arrivals and the estimated location. Therefore, the source can be located without NLOS errors and the localization precision is improved. For TDOA, one transmitter andn omni-directional sensors are considered receivers. For AOA, one transmitter and n arrays with directive sensors are considered. The receivers are considered to be connected in order to have a common clock available. Only NLOS error is considered and all other kinds of error are ignored. Also, we find the locations for the receivers in 2-D and 3-D that minimize location uncertainty for TDOA indoor systems. For this purpose, at first a triangular area and a tetrahedron volume are considered as the building block in 2-D and 3-D respectively. For each case, we use the minimum number of sensors necessary for receiving clear LOS signals, that is, three and four for the 2-D and 3-D cases respectively. Then we use the results to position the receivers in more complex areas and volumes. This method is similar to that used for determining the minimum number of guards sufficient to cover the interior of ann-wall art gallery room in the “Art Gallery Theorem” [11]. The first part of the second chapter addresses NLOS omission in hyperbolic location systems which applies minimum Least Square (LS) method to locate transmitter. In this chapter, first an algorithm is given to extract the group of received signals coming from the same source from all the first-arriving signals at a collection of omni-directional sensors. Next, an algorithm is given to group the signals received by the directive sensors that come from the sources. Another algorithm is given to distinguish between the group of LOS signals and the reflections. By applying this algorithm, the transmitter can be 3 located with a higher precision because reflections do not contribute in the localiza- tion process. Finally an algorithm is given to locate the transmitter when there are not enough LOS signals but at least four reflectors (with single bounced reflections) have been located in the first or second algorithm. Then, because of measurement noise, a threshold setting procedure is given to apply the algorithms to real data. Receiver sites that minimize the uncertainty in the source location are found in chap- ter 3. Experimental results for localizing a transmitter are given in chapter 4. Also in this chapter, localization accuracy with different signal-to-noise-ratios (with simulated AWGN noise) is analyzed. 4 Chapter 2 New Methods for NLOS Omission in Position Location Systems 2.1 Consistent Groups of Signals From an Array of Omni-Directional Antennas It is well known that to locate a transmitting source in 3-D, the source’s signal must be received through direct path by at least four receivers that are not coplanar. Here we assume that ² There is a transmitter in the area and there are at least four omni directional receivers around the area (Fig. (1)). ² All the first incoming signals to the receivers are LOS or the one-bounce signals. ² All the sensors in the receiving arrays are connected in order to have the same clock. ² At least four non-coplanar sensors receive LOS signals. Fig. 2.1 shows an example of source localization with the above assumptions. This figure includes ² Single transmitter shown as Tx. ² Eight receivers, indicated with A1-A8. 5 Tx S1 S2 S3 S4 S5 S6 S7 S9 S8 A1 A2 A3 A4 A8 Ref2 Ref3 Ref1 A5 A6 A7 o S10 S11 Figure 2.1: Example 1: source localization with arrays of omni-directional receivers ² Three reflectors denoted by Ref1, Ref2 and Ref3. ² S1-S4 which represent LOS signal paths. ² S6-S9, and S11 as the reflections. In Fig. 2.1 signalsS1;S2;S3 andS4 from the transmitter are received by four receivers (non-coplanar), and signalS6;S7;S8;S9 andS11 are received from reflectors. Fig. 2.2 shows the timing diagram of the signals in Fig. 2.1. It is assumed that the transmitter sends its signal at t = 0. ¢t is the difference between the transmitter’s clock and receivers’ clock, and t r i is the arrival time of the earliest signal at receiver i, which is measured relative to the receiver clock. Assuming that direct-path signals from a source are detected by (at least) four receivers, the source can be localized by the hyperbolic method. Once the source position has been found, the propagation timet p i between the source, i.e., the transmitter or the reflector, and receiveri can be calculated using 6 Tx clk A1 clk A2 clk A3 clk S1 S3 S2 S4 t Δ 1 r t A4 clk A5 clk S8 S6 A6 clk A7 clk A8 clk S7 S9 S11 t 0 Figure 2.2: Timing diagram of example 1 t p i =R i =c; (2.1) in whichR i is the range of the source of the signal from sensori obtained by the hyper- bolic method, andc is the propagation velocity. If the located source is transmitter,¢t can be easily obtained as t p i ¡t r i =¢t i=1;2;:::;m; (2.2) in whicht p i is the propagation time of the signal to theith receiver obtained by (2.1). If the located source is a reflector (e.g. Ref1 in Fig. 2.1), the timing equation for signal S6;S7;S8 andS9 is 7 t p i +t mid =t r i +¢t i=6;7;8;9; (2.3) where t p i is obtained from (2.1), t r i is available in the receiver and ¢t and t mid (the propagation time between the transmitter and the reflector) are unknown. Equation (2.3) can be rewritten as t p i ¡t r i =¢t¡t mid i=6;7;8;9: (2.4) As is evident from (2.2) and (2.4), for all the the signals in a source-consistent group, i.e. coming from the same source, the difference betweent p i andt r i is constant, t p i ¡t r i =K; (2.5) where K depends on the source location and not on i (the index of the signals in a group). In addition to the transmitter behaving like a point source, it is implicit in the above timing equations that the reflector also behaves like a point reflector. If by mistake the signals being considered in a group are from the different sources (e.g. some LOS signals and some reflections), then localization by the hyperbolic method provides no answer or a wrong location for the source. If no answer is obtained in the localization process, it implies that the signals considered in the group are not from the same source. If a wrong location is obtained, forming the timing equations (2.4) does not provide a constant for all the signals in the group. This property can be exploited to determine if a group of signals comes from the same source, i.e., if the group of signals is source-consistent. For this purpose, any group of the received sig- nals can be conjectured to be source-consistent, and the conjecture can be checked by forming the timing equations and check if (2.5) holds or not. 8 With a large number of receivers, both a source and several reflectors may be located. Section 2.1.1 addresses this problem and gives an algorithm to locate the source and reflectors. 2.1.1 Grouping Algorithm for Omni Directional Receivers This section presents an algorithm to find source-consistent groups of signals. The algo- rithm is described through the following steps. Step 1: Guessing the signals in a source-consistent group It is guessed that the first arrivals to the four receivers, are in a group . Step 2: Localization The minimum Least square (LS) method is applied to the TDOA of group of signals in the above conjecture and the source location is determined. Step 3: Checking the validity of Localization I The validity of the source location (from the previous step) in the accessible area is checked (accessible area is the area where the source location is valid; for example, in indoor systems, the accessible area is the area restricted to the walls and ceiling and floor). If the guess is correct, the location is valid. However, if the conjecture is not correct, the source location may or may not be valid and the guess could be passed or rejected in this step. Step 4: Determining the propagation times and K Thet p i s are obtained by (2.1). Then, the constantK i s are obtained via (2.5) 9 K 1 =t p 1 ¡t r 1 K 2 =t p 2 ¡t r 2 K 3 =t p 3 ¡t r 3 K 4 =t p 4 ¡t r 4 (2.6) The group is consistent if and only if K 1 =K 2 =K 3 =K 4 =K: (2.7) Step 5: Checking the validity of Localization II In this step, verification of the grouping hypothesis is performed by determination if the following equation is consistent 2 6 6 6 6 6 6 6 4 K 1 K 2 K 3 K 4 3 7 7 7 7 7 7 7 5 ¡ 2 6 6 6 6 6 6 6 4 ^ K ^ K ^ K ^ K 3 7 7 7 7 7 7 7 5 ¼0: (2.8) ^ K, which is the estimation of K, can be obtained from one of the signals in the group, for example ^ K =K 1 ; (2.9) or it can be estimated as ^ K = 1 4 (K 1 +K 2 +K 3 +K 4 ): (2.10) Step 6: 10 If all the possible groups have been checked for consistency, stop, otherwise, go to the Step 1. Fig. 2.3 shows the flowchart of the algorithm. Note that there are two types of grouping ² Consistent groups which are groups of signals for which equation 2.8 is satisfied. ² Inconsistent groups which are groups of signals for which equation 2.8 is not satisfied. As an example of consistent group, assume that in Fig. 2.1 we guess thatS1;S2;S3 and S4, (which are the earliest signals to receivers A1-A4) are one group (this is a correct guess). With the grouping guess, (2.8) is satisfied. Therefore, the signals in the group are from the same source (here the transmitter) and a correct location for the source has been obtained in Step 2. Now assume that a wrong guess is made in Step 1, with signals S1;S2;S3 and S6 being considered to be in a group. Then the hyperbolic localization method is applied in Step 2 to this group. If grouping is incorrect, equation 2.8 is not satisfied which implies a wrong location for the source. Note that even if this location is valid through Step 3, thet p i s andK i s are determined in Step 4. Since a wrong location has been determined for the source, the K i s are not the same for all the signals in the group. In the same way, all the other wrong conjectures are rejected in Step 3 or Step 5, and the only real groups that remain areS1;S2;S3;S4 andS6;S7;S8;S9. Due to the measurement noise in the real world, the right-hand side (RHS) of (2.8) for a correct guess is not exactly a zero vector, but a vector with small elements close to zero (their statistical characteristics depend on the the noise). When a wrong guess 11 Make J group of signals Use the signals in the jth group to locate the transmitter Is the determined location valid? Is the determined location valid? Yes no no Yes 1 2 3 4 5 The group is consistent j J ? = j=0 j=j+1 no The group is inconsistent } { , i p t K Find Figure 2.3: Flow chart of the Grouping algorithm for omni directional receivers passes through Step 5, since a wrong location has been determined for the source, right- hand side of (2.8) is a vector with non-zero elements that are far from each other. For omni-directional receivers, a group can be formed from every set of earliest signals from four non-coplanar receivers. However, a group can have a larger number of the receivers. Note that the size of the vectors in 2.8 is always m x 1 where m is the number of the signals in a group. Withn receivers, the number of groups that can be formed isJ = ¡ n m ¢ . The signals that do not lie in any consistent group are reflections and ignored in the localization process. 12 If among the consistent groups, more than one group can locate the same source, this implies that more than m signals are available. Thus, the signals in those similar groups can be joined together to give a more precise location for the source. If there were 4 · n < 8 receivers in the area, with the assumptions given in the section 2.1, only one source would be located, which is the transmitter. When there aren¸ 8 receivers in the area (Fig.2.1), multiple consistent group sig- nals may be distinguished by the algorithm. Therefore, multiple sources may be located in which one of them is the transmitter and the others are reflectors. In this situation, the transmitter must be distinguished from the reflectors. Later, another algorithm is introduced to distinguish between the transmitter and the reflectors. Note that in the example shown in Fig. 2.2, the earliest arriving signals S1¡S4;S6¡S9 (S5 is excluded because it is completely blocked) were used in the algorithm. As observed, for every area withn omni directional receivers, the algorithm provided in this section can handlen¡4 blockages of LOS signals, provided that LOS signals are received by four non-coplanar receivers. However, when directive receivers (x2.2) are used, the number of blockages that can be handled isn¡2, wheren is the number of the arrays. A method for separating LOS signals from reflections ,for directive receivers, is introduced in the next section. 2.2 Determining the Groups of Signals for an Array of Directive Antennas in 3-D To locate every source by directive antenna-array receivers, there must be at least two LOS signals received by two receivers. Thus to be able to locate the transmitter in the present work, the assumptions are 13 S1 TX A1 A2 S5 S6 S7 S11 S10 S2 S8 S9 Ref1 Ref2 Ref3 S12 S13 S14 S15 S16 S17 A3 A4 S3 S4 Ref4 Figure 2.4: Example2: source localization with arrays of directive receivers ² There is a single transmitter in the area. ² The receivers’ antennas are pencil beam arrays where the beams are assumed to be stacked as indicated in Fig. 2.4). ² All the earliest arriving signals to the receivers are LOS or the one bounce reflec- tions. ² All the receivers in the arrays are connected in order to have the same clock. ² At least two receivers for different arrays receive LOS signals. With this configuration, the directions of the received signals are known approximately. Therefore, if two known signals from a source are received by the two receivers in differ- ent arrays, the transmitter can be located by triangulation (Fig. 2.6), and the propagation time between the signal source and the arrays can be obtained by equation (2.1). Here- after, the additional steps that must be taken to locate a transmitter, are similar to those 14 Tx clk A1 clk A2 clk A3 clk S1 S11 S5 S6 S3 S14 S16 S4 A4 clk ∆t 1 r t S9 t 0 Figure 2.5: Timing diagram of example2 described in section 2.1 except that here every two signals can form a group. Note that applying the triangulation method to a pair of signals requires that the direction of the signals are consistent. More details are given by an example in the next section. 2.2.1 Grouping Algorithm for Directional Receivers In this section the algorithm is described with the example shown in Fig. 2.4. Step 1: Forming a hypothesis of a signal group It is hypothesized that the first arrivals to the two receivers, are in a consistent group. Step 2: Verifying the compatibility of the signals directions In this step, it is determined if the two selected signals in the group can be used in triangulation. If the signals directions are not compatible, i.e., their pencil beams do not intersect, the group is not valid. Step 3: Localization Because one side of the triangle shown in Fig. 2.6 and the angle of the arrivals (µ 1 and µ 2 ) are available, the triangulation method can be applied to the group signals and the source location is determined. 15 1 θ 2 θ 1 R 2 R Rx1 Rx2 Tx Figure 2.6: Source localization by the triangulation method for directive receivers Step 4: Checking the validity of Localization I The validity of the source location (from the previous step) in the accessible area is verified. (If the conjecture is not correct, the source location may or may not be valid and the hypothesis could be rejected or passed in this step). Step 5: Determining the propagation time The set of the propagation times t p i to receiver i and thefK i g are determined by (2.1) and (2.5) for the signals in the group. Step 6: Checking the validity of Localization II Similar to Step 5 in section 2.1.1, in this step, verification of the grouping hypothesis is performed by determination if the following equation is consistent 2 4 K 1 K 2 3 5 ¡ 2 4 ^ K ^ K 3 5 ¼0: (2.11) where ^ K is the estimation ofK. 16 Step 7: If all the possible groups have been checked for consistency, cease, otherwise, go to the Step 1. As an example, assume that in Fig. 2.4, we guess thatS1 andS3, (which are the earliest signals to receivers A1, A3) are one consistent group (this is a correct guess). With this hypothesis, (2.8) is satisfied. Therefore, the signals in the group are from the same source (here the transmitter) and a correct location for the source has been obtained in Step 2. With the above hypothesis, (2.11) is consistent. Therefore the signals in the group (S1;S3) are coming from the same source and a correct location for the source has been obtained in Step 3. Now assume that a wrong hypothesis is made in Step 1; signals S1 and S6 are considered to be in a group. If this group passes Step 2, then the source is located by triangulation with this group in Step 3 and an incorrect location is determined for the source. Next, if the location is validated in Step 4, t 0 p i (the propagation times) and the constant K i are determined in Step 5. Since a wrong location has been determined for the source, theK i s are not the same for all the signals in the group. In Step 6, (2.11) is checked and it is not consistent. In the same way, all other wrong conjectures will be rejected in Step 2, Step 4 or Step 6. Thus the only detected sources are transmitter,Ref1 andRef4 which have at least two LOS signals. As observed, for the directive receivers, an every set of two signals in the near-field (received by receivers in different arrays) can form a group. However a group can be constructed by selecting a larger set of signals which must be consistent. As before, the size of matrixA ism x3 wherem is the number of the signals in a group. 17 Similar to omni-directional case, withn receivers, totally ¡ n m ¢ groups can be formed. The signals that do not lie in any consistent group passed through the algorithm, are ignored in the localization process. If among the groups that pass the algorithm, more than one group locate the same source, it implies that more than m signals are available from the same sources. Thus, the signals in those similar groups can be joined together to give a more precise location for the source. As mentioned inx2.1, with the measurement noise, the right-hand side of (2.11) for a correct hypothesis is a vector with small elements close to zero. But with a wrong hypothesis, it is a vector with non-zero elements that are far from each other. If multiple groups are distinguished, multiple sources will be located (one of them is the transmitter). In this situation, the transmitter must be distinguished from the reflec- tors by applying the algorithm introduced in section 2.3. 2.3 Algorithm to Distinguish Between Transmitter and Reflector Sources We use the time equations of (2.2) and (2.3) to distinguish between the transmitter and the located reflectors when there areL> 1 sources located by the previous algorithms. Note that for the known and located transmitter and reflectors, the distance between the transmitter and thejth reflector ¡ ! R mid j can be found as ¡ ! R mid j = ¡ ! R x ¡ ¡ ! R ref j (2.12) 18 in which ¡ ! R x is the position of the transmitter, and ¡ ! R ref j is the position of thejth reflec- tor in a global coordinate system. Therefore the propagation time ~ t mid j between the transmitter andjth reflectors is t mid j =j ¡ ! R mid j j=c j =1;2;:::;L¡1; (2.13) wherec is the propagation velocity. The time equation for the reflected signals fromjth reflector (the signals in the source-consistent group of the reflector) is t p i +t mid j ¡t r i ¡¢t=0 i=1;2;:::;m (2.14) where indices i and j denote the ith signal in the group of the jth reflector. The time equation for the transmitter was described in (2.2). While the transmitter is unknown, it is the only source (among all the located sources) that gives a correct ¢t to be applied to the time equations of the reflectors (2.14). Therefore a guess of transmitter’s group can be checked by verifying the time equation 2.14. In the next section, we illustrate this algorithm with the example. shown in Fig. 2.1. Example After applying the algorithm described inx2.1.1, two sourcesSource1 andSource2, the transmitter and Ref1, are located. The groups related to these sources are S1;S2;S3;S4 and S6;S7;S8;S9, and their related timing equations are (2.2) and (2.14) (with m = 4, L = 2 ) respectively. The algorithm is described in the follow- ing. Step 1: Guessing the transmitter Assume that among two sources located in Fig. 2.1, it is guessed thatSource1 is the transmitter. ThereforeSource2 is the reflector (that is a correct hypothesis). Step 2: Obtaining¢t 19 By (2.2),¢t is obtained from one of the signals (e.g. S1) of the group signals of the assumed transmitter (that are signalsS1;S2;S3 andS4). Thus¢t is obtained as ¢t=t p 1 ¡t r 1 : (2.15) Step 3: Determining the propagation time between the assumed transmitter and other sources Assuming pointo (Fig. 2.1) as the center of a global coordinate system, the distance between the assumed transmitter andRef1 is found by ¡ ! R mid 1 = ¡ ! R x ¡ ¡ ! R ref 1 (2.16) in whichR x is the range of the assumed transmitter. Therefore the propagation time between the assumed transmitter andRef1 is t mid 1 =j ¡ ! R mid 1 j=c: (2.17) Step 4: Constructing the timing matrix Using the time equations (2.2) and (2.14), matrixB is constructed as below B= 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 t p 1 0 t r 1 ¢t t p 2 0 t r 2 ¢t t p 3 0 t r 3 ¢t t p 4 0 t r 4 ¢t t p 6 t mid 1 t r 6 ¢t t p 7 t mid 1 t r 7 ¢t t p 8 t mid 1 t r 8 ¢t t p 9 t mid 1 t r 9 ¢t 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : (2.18) 20 As seen, every row of this matrix is related to a signal in the group of the located sources. The size of matrix B is (mL)x4. The first four rows are related to the first group (S1;S2;S3;S4) and the last four rows are related to the second group (S6;S7;S8;S9). The first column of the matrix is containedt p (obtained in the third step of the previous algorithm), the second column is propagation time between the assumed transmitter and the reflector(s), the third column is received timet r , and the third column is¢t obtained in Step 2. Step 5: Check the hypothesis Now it is checked if the following equation is satisfied or not B 2 6 6 6 6 6 6 6 4 1 1 ¡1 ¡1 3 7 7 7 7 7 7 7 5 ¼0: (2.19) With the conjecture inStep1, (2.19) is held, therefore it is a correct hypothesis. Now it is proved that a wrong guess can not pass through Step 5. Assume that a wrong hypothesis would be made in Step 1 such thatSource2 (actuallyRef1) would be assumed the transmitter, thenSource1 (transmitter) would be assigned to be a reflector. Next, ¢t is determined in Step 2 from one of the signals (e.g. S6) in the group of the assumed transmitter (Ref1) as ¢t 0 =t p 6 ¡t r 6 : (2.20) After computing the propagation time between the assumed transmitter and reflector, that ist mid 1 , in Step 3 matrixB 0 is constructed as 21 B 0 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 t p 1 t mid 1 t r 1 ¢t 0 t p 2 t mid 1 t r 2 ¢t 0 t p 3 t mid 1 t r 3 ¢t 0 t p 4 t mid 1 t r 4 ¢t 0 t p 6 0 t r 6 ¢t 0 t p 7 0 t r 7 ¢t 0 t p 8 0 t r 8 ¢t 0 t p 9 0 t r 9 ¢t 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; (2.21) and the following equation is checked B 0 2 6 6 6 6 6 6 6 4 1 1 ¡1 ¡1 3 7 7 7 7 7 7 7 5 ¼0: (2.22) In the following it is shown by contradiction that (2.22) does not hold. Assume that (2.22) is held, since (2.19) is also true, rewriting (2.19) and (2.22) in the form of the linear combination of the columns of matricesB andB 0 gives ~ b 1 ¤1+ ~ b 2 ¤1+ ~ b 3 ¤(¡1)+ ~ b 4 ¤(¡1)=0; (2.23) ~ b 0 1 ¤1+ ~ b 0 2 ¤1+ ~ b 0 3 ¤(¡1)+ ~ b 0 4 ¤(¡1)=0: (2.24) whereb i s andb 0 i s,i=1;2;3;4 are the column vectors of matrixesB andB 0 respectively. Subtracting (2.23) from (2.24) yields ( ~ b 1 ¡ ~ b 0 1 )¤1+( ~ b 2 ¡ ~ b 0 2 )¤1+( ~ b 3 ¡ ~ b 0 3 )¤(¡1)+( ~ b 4 ¡ ~ b 0 4 )¤(¡1)=0: (2.25) 22 But ~ b 1 = ~ b 0 1 ~ b 3 = ~ b 0 3 ~ b 4 ¡ ~ b 0 4 =k¤I ; (2.26) in whichI is unitary vector and k =(¢t¡¢t 0 )=(¢t¡(¢t+t mid 1 ))=¡t mid 1 : (2.27) Applying (2.26) to (2.25) yields ( ~ b 2 ¡ ~ b 0 2 )¤1=kI: (2.28) Replacing ~ b 2 and ~ b 0 2 with their values from (2.18) and (2.21) gives ( ~ b 2 ¡ ~ b 0 2 )= 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 ¡t mid 1 ¡t mid 1 ¡t mid 1 ¡t mid 1 t mid 1 t mid 1 t mid 1 t mid 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 =kI: (2.29) As seen, the vector in the left side of (2.29) contains both positive and negative elements while the right side shows that all the components must have the same sign as k, that is a contradiction. Therefore (2.22) can not hold¤. 23 As seen, by applying the above algorithms, all NLOS signals are excluded from the localization process and the transmitter is located by only LOS signals. As mentioned before, the right-hand side of (2.19) may be affected by the measure- ment noise. If the last assumption inx2.1 andx2.2 is not satisfied, some groups signals may be distinguished in the algorithms 2.1.1 and 2.2.1, but algorithm 2.3 can not determine a transmitter since all the group signals are related to the reflectors. However, if at least four reflectors are located (by the algorithms 2.1.1 or 2.2.1), the transmitter can still be localized. In the next section another algorithm is introduced to locate the transmitter in the case that there are insufficient LOS signals from the transmitter but at least four reflectors are located. 2.4 Using Four Reflectors to Locate the Transmitter Without Having LOS Assuming that there are at least four one-bounce reflectors detected by the algorithms described inx2.1.1 orx2.2.1, the timing equations introduced in (2.3) can be used to locate the transmitter even when there are not enough LOS signals from the transmit- ter (i.e., four and two LOS signals for omni directional and directive receivers respec- tively). When the algorithm ofx2.3 can not determine the transmitter among the sources detected in sections 2.1.1 or 2.2.1, it implies that all the detected sources are reflectors. Assuming that the located sources are one-bounce reflectors, they can be considered as the known secondary sensors receiving LOS signals from the transmitter. Therefore, subtracting the propagation time t p from the receiving time t r for each reflector yields the difference of t mid , that is the propagation time between the transmitter and these secondary sensors, and¢t. 24 S1 TX S5 S6 S11 S10 S2 S12 Ref3 Ref1 Ref4 S7 S9 S8 S3 S4 Ref2 1 A 2 A 3 A 4 A Figure 2.7: Example3: source localization when there are insufficient LOS signals but at least four reflectors are detected t r i ¡t p i =t mid ¡¢t i=1;2;:::: (2.30) This equation implies that the relative time difference between these secondary sen- sors and transmitter is available. Thus, the transmitter can be localized by the hyperbolic method regardless of the type of the main sensors (i.e. directive or omni directional). The procedure is described by the example shown in Fig. 2.7 in the following algorithm. 2.4.1 Algorithm to Locate the Transmitter Using Located Reflec- tors Assume that in Fig. 2.7, by the algorithm introduced in section 2.2.1, four reflec- tors Ref1;Ref2;Ref3 and Ref4 are detected with group signals (S5;S6), (S7;S8), (S9;S10) and (S11;S12) respectively. The transmitter is located by following steps Step 1: Forming the timing matrix 25 Tx clk S10 S12 S7 S9 S11 ∆t S5 S8 1 1- A clk 2 1- A clk 1 2- A 2 2- A 1 3- A 2 3- A 1 4- A 2 4- A clk clk clk clk clk clk S6 Figure 2.8: Timing diagram of example3 For the group signals of the detected reflectors, matrixF is constructed as below F = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 t r 5 t p 5 t r 6 t p 6 t r 7 t p 7 t r 8 t p 8 t r 9 t p 9 t r 10 t p 10 t r 11 t p 11 t r 12 t p 12 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : (2.31) As seen, the first two rows are related to group signals of Ref1, the second two rows are associated toRef2 and so on. The first column is available in the receivers and the second column is determined in Step 3 or Step 4 of the algorithms introduced in sections 2.1.1 or 2.2.1 respectively. Step 2: Forming the timing equations 26 Applying (2.30) to the group signals, the difference between t mid and ¢t is deter- mined by the following equation F 2 4 1 ¡1 3 5 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 a 5 a 6 a 7 a 8 a 9 a 10 a 11 a 12 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 =~ a; (2.32) where a i =t mid i ¡¢t i=5;6;:::;12: (2.33) Note that a 5 =a 6 a 7 =a 8 a 9 =a 10 a 11 =a 12 (2.34) since every two signals belong to a reflector. Therefore, by selecting one signal from each group, vector~ a can be reduced to 27 ~ a 0 = 2 6 6 6 6 6 6 6 4 a 5 a 7 a 9 a 11 3 7 7 7 7 7 7 7 5 ; (2.35) since the signals from the same group do not give extra information regardingt mid and ¢t. Step 3: Finding the time difference of the arrival to the reflectors Considering one of the reflectors as the origin (e.g. Ref1), the relative difference of the arrival time of LOS signals to the secondary sensors (reflectors) is calculated by subtracting the elements of vector ~ a 0 from the element related to the reflector considered as the origin (a 5 ) ~ a 0 ¡a 5 ¤I = 2 6 6 6 6 6 6 6 4 0 t mid 2 ¡t mid 1 t mid 3 ¡t mid 1 t mid 4 ¡t mid 1 3 7 7 7 7 7 7 7 5 = ~ d; (2.36) whereI is unit vector. Step 4: Localization Considering the locations of the reflectorsRef2,Ref3 andRef4 as the (secondary) receivers sites, the transmitter is located by hyperbolic method with the time information of vector ~ d. 28 2.5 Virtual Sources Here we show that algorithm 2.1.1 may recognize consistent group(s) in which the sig- nals are not from the same source. In other words, virtual source(s) which do not exist may be determined by the algorithm. However, these virtual sources can be separated from the transmitter ( with the main group(s), i.e., group(s) which consist of LOS sig- nals) by the fact that the time offset (¢t) found by the main group(s) is always larger than the one determined by these virtual consistent groups. We prove this fact for 2-D by the example shown in Fig. 2.9. In Fig. 2.9, LOS signalsS1¡S3 from transmitterTx is received by receiversA;B andC; but sensorD receives reflectionS4. Algorithm 2.1.1 recognizes groupS1¡S3 as a consistent group, with the following parametric equations t p 1 =t r 1 +¢t t p 2 =t r 2 +¢t t p 3 =t r 3 +¢t: (2.37) IfS1;S2;S4 also forms a consistent group with the following equations t 0 p 1 =t r 1 +¢t 0 t 0 p 2 =t r 2 +¢t 0 t 0 p 4 =t r 4 +¢t 0 ; (2.38) it implies that a source is located in positionTx 0 . Equations (2.37) and (2.38) yield t p 1 ¡t 0 p 1 =¢t¡¢t 0 t p 2 ¡t 0 p 2 =¢t¡¢t 0 : (2.39) But note thatS4 is a reflection, sot r 4 is longer than a normal LOS (if received byS4), and then it pushes the transmitter position upward so that Tx 0 be in a higher distance 29 from sensor D. However, signals S1;S2 are still LOS, so position Tx 0 is on the same hyperbola (corresponds to sensorsA;B) thatTx is located. This fact can be seen by t p 1 ¡t p 2 =t 0 p 1 ¡t 0 p 2 =t r 1 ¡t r 2 : (2.40) Therefore, t p 1 >t 0 p 1 : (2.41) Considering 2.39 and 2.41, we have ¢t>¢t 0 (2.42) Thus, among all consistent groups, the main group (which consist of only LOS signals), gives maximum¢t. It can be seen that as long as the hyperbola corresponding to sensorsB;D and sig- nalsS2;S4 (h BD ) crosses the hyperbola related to sensorsA;B (h AB ), the virtual source can be recognized by the algorithm. It implies that the virtual source will be detected by the algorithm if d(D;Tx 0 )¡d(B;Tx 0 )<d(D;P)¡d(B;P); (2.43) whered(:;:) stands for the distance between two points andP is the intersection ofh AB and the connecting line ofA andB. Same technique can be applied to 3-D to show that the main group always gives¢t max . The above discussion is valid as long as the transmitter is inside the convex hull of the receivers. In the next section, we show that if transmitter is outside of the convex hull, the virtual group(s) can cause the same¢t as the main group. 30 A B D C Tx x Tx’ x S1 S2 S3 S4 P hBD hAD hAB r1 r2 Figure 2.9: Virtual source in 2-D 2.5.1 Ambiguity in the Algorithms for Outdoor Systems If the possible location of the source is outside the region surrounded by the receivers, then there may be ambiguity in the signal grouping ofx2.1.1 andx2.2.1, and the source- consistent grouping algorithms may include the signals from different sources in one group. We explain the problem with two examples and in both assume that¢t is known. Fig. 2.10, shows the situation for 2-D, in which one LOS signal has been blocked when m = 3. Since¢t is known, the possible location of the source is on the intersection of three circles with centers A, B, C; and radiuses determined fromt r 1 +¢t andt r 2 +¢t andt r 3 +¢t respectively. Now assume that signal S3 is blocked, and sensor C receives reflection S4 instead. If the location of the reflector is such that d(Tx;Ref)+d(Ref;C)=d(C;M); (2.44) 31 Tx A B C S1 S2 x M S4 Figure 2.10: Ambiguity ellipse for 2-D where (d(:;:) stands for the distance between the two points). Point M will be recog- nized as the source location. Since the algorithms works based on the comparison of ¢ts, it would identify signals S1; S2 and S4 in the same group. Because point M is outside of the region surrounded by the receivers, for indoor applications, the ambiguity caused by the two LOS and one reflection will not occur, or will be excluded in Step 3 of the algorithm inx2.1.1 . Equation (2.44) implies that the reflectors located on the ellipse identified with two foci atC and Tx, and with radiusd(C;M) can cause the ambiguity (corresponds to two LOS S1;S2 and one reflection). When B and C receive LOS signal but A receives reflection, the ambiguity ellipse has two foci at A and Tx and radius d(A;N) as indi- cated in Fig. 2.11. In this figure, the ambiguity ellipse corresponding to the reflection in sensorB was also shown. A similar example for 3-D is shown in Fig. 2.12. As indicated, when¢t is known, Tx is on the intersection of four spheres with centers A; B; C; D; and radiuses t r i + ¢t; i=1;2;3;4. When the LOS to receiverC is blocked, and reflectionS5 is received byC instead, with the condition in (2.44), the algorithm identifies signalsS1; S2; S4; S5 in the same 32 Tx A B C M N Q Figure 2.11: Three Ambiguity ellipses 2-D Tx A B C D M S1 S2 S4 Ref S5 Figure 2.12: Ambiguity ellipsoid for 3-D group, and recognizes point M as the true position of the source. Note that for 3-D, equation (2.44) identifies an ellipsoid with two foci atTx andC, and radiusd(C;M;). Fig. 2.13 shows four different ambiguity ellipsoids correspond to the reflections in the different receivers. 2.6 Threshold Setting for the Algorithms When the measurement noise is counted, the timing equations for the signals (equations (2.2) and (2.3)) change to 33 Tx A B C D M N O P Figure 2.13: Three Ambiguous ellipsoids for 3-D t p i ¡t r i =¢t+" i ; i=1;2;:::;m t p i ¡t r i =¢t¡t mid +" i ; i=1;2;:::;m (2.45) where " i accounts for noise, including receiver noise, propagation noise, and errors in time of arrival measurements and assumed station positions. Depending on the location accuracy, a threshold can be set to distinguish between wrong and correct hypothesis. To set the threshold, we use the estimator accuracy equations given in [14]. 2.6.1 Estimator Accuracy Here we summarize the position accuracy estimation found in [14]. Assume that a set ofN measurementsr i ;i=1;2;:::;N, is the sum of a known function f i (x) and additive error noise r i =f i (x)+n i i=1;2;:::;N; (2.46) 34 wherex is then-dimensional vector of position coordinates in 2-D or 3-D, whose com- ponents are to be estimated . Given in [14], theseN equations can be written in vector form r=f(x)+n: (2.47) Next f(x) is linearized by expanding it with Taylor series about a reference point x 0 . Keeping the first two terms yields f(x)'f(x 0 )+G¤(x¡x 0 ): (2.48) Under the assumptions that the measurement error n is a multivariate random vector with zero-mean, positive definite covariance matrixN and a Gaussian distribution, it is shown in [14] that the least square estimator^ x ofx is ^ x=x 0 +(G T N ¡1 G) ¡1 G T N ¡1 (r¡f(x 0 )) (2.49) The bias of the estimator^ x is b=E[^ x]¡x =(G T N ¡1 G) ¡1 G T N ¡1 ff(x)¡f(x 0 )¡G(x¡x 0 )+E[n]g (2.50) The covariance matrix of^ x is P =E[(^ x¡E[^ x])(^ x¡E[^ x] T )]=(G T N ¡1 G) ¡1 : (2.51) Whenr is a Gaussian random vector, (2.49) shows that the probability density function of^ x is 35 f ^ x (®)= 1 (2¼) n=2 jPj 1=2 (exp[¡(1=2)(®¡m) T P ¡1 (®¡m)]) (2.52) in whichm = E[^ x] and P is the covariance matrix of^ x. The loci of constant density function values are described by equations (®¡m) T P ¡1 (®¡m)=· (2.53) where· is a constant by which the size of then-dimensional region surrounded by the surface is determined. For 2-D and 3-D, this region is ellipse and ellipsoid respectively. In the general case ofn dimension, it can be considered a hyperellipsoid with the prin- ciple axes of lengths2 p ·¸ i ;i = 1;2;:::;n, where¸ i ;i = 1;2;:::;n are the eigenvalues ofP [14]. The probability that^ x lies inside the region of (2.53) is p e (·)= Z Z R ¢¢¢ Z f ^ x (®)d® 1 d® 2 ¢¢¢d® n (2.54) in which the regionR of integration is R =f® :(®¡m) T P ¡1 (®¡m)··g: (2.55) Note that by definition, P is symmetric and positive semidefinite; and (2.51) implies thatP ¡1 exists. SoP can not have zero eigenvalues, and therefore bothP andP ¡1 are positive definite [14]. Thus, an orthogonal matrixA exists that diagonalizesP ¡1 . After changing variables as ° =®¡m ³ =A T °; (2.56) 36 the region of integration becomes R 2 =f³ : n X i=1 ³ 2 i ¸ i ··g; (2.57) that is the interior of a hyperellipsoid with principle axes of lengths2 p ·¸ i [14]. The³ i are the components of³. In [14] it is shown that p e (·)= n 2 n=2 ¡(n=2+1) Z p · 0 ½ n¡1 exp(¡ ½ 2 2 )d½; (2.58) where ½=( P n i=1 ´ 2 i ) 1=2 ´ i =³ i = p ¸ i i=1;2;:::;n: (2.59) Forn=1; 2 and3 the above integral can be simplified [14] p e (·)= erf( p ·=2); n=1 p e (·)=1¡exp(¡·=2); n=2 p e (·)= erf( p ·=2)¡( p 2·=¼)exp(¡·=2) ;n=3: (2.60) Whenp e is specified,· is obtained from (2.60) by numerical method. Then, the size of the region of (2.55) is obtained. For 2-D, the covariance matrix can be described as P = 2 4 ¾ 2 1 ¾ 12 ¾ 21 ¾ 2 2 3 5 (2.61) 37 1 2 κλ 2 2 κλ 1 γ 2 γ 1 ζ 2 ζ θ Figure 2.14: Uncertainty ellipse [14] Mean estimate Particular estimate Transmitter Bias vector CEP Figure 2.15: Geometry of transmitter position, mean location estimate, CEP, estimator bias vector, and particular location estimate [14] The eigenvalues ofP can be obtained by simple calculation ¸ 1 = 1 2 h ¾ 2 1 +¾ 2 2 + p (¾ 2 1 ¡¾ 2 2 ) 2 +4¾ 2 12 i ¸ 2 = 1 2 h ¾ 2 1 +¾ 2 2 ¡ p (¾ 2 1 ¡¾ 2 2 ) 2 +4¾ 2 12 i : (2.62) The shape of the concentration ellipse is shown in Fig. 2.14. Another measure of uncertainty in the location estimator, is circular error probability (CEP), which is the radius of the circle that has its center at the mean, and contains half of the realization of the random vector [14]. Fig. 2.15 shows the geometric relation between the bias vector, mean estimate and CEP. 38 Definition of CEP implies that it can be determined by solving the following equa- tion [14] 1=2= Z Z R f ^ x (®)d® 1 d® 2 (2.63) in which R =f® :j®¡mj·CEPg: (2.64) Here we extend the CEP definition to any percentage of the realizations of the ran- dom vector: Definition: CEP d is the radius of the circle that has its center at the mean and contains d percent of the realizations of the random vector. Therefore,CEP d can be obtained by solving this equation d 100 = Z Z R f ^ x (®)d® 1 d® 2 (2.65) in which R =f® :j®¡mj·CEP d g: (2.66) From the equation given in [14], it can be concluded that CEP d may be computed by d 200° 2 (1+° 2 )= Z [(CEP d ) 2 =4¸ 2 ](1+° 2 ) 0 exp(¡x)I 0 µ 1¡° 2 1+° 2 x ¶ dx (2.67) in which 39 ° 2 = ¸ 2 ¸ 1 (2.68) The similar definition to CEP for 3-D is spherical error probability (SEP) , which is defined as the 50th percentile value of the three-dimensional position error statistics. Thus, half of the results are within a sphere with radius SEP. Again we extend this definition to any percentage of the realization, defined as SEP d (for d percent). Thus, SEP can be computed by d 100 = Z Z R f ^ x (®)d® 1 d® 2 d® 3 (2.69) where R =f® :j®¡mj·SEP d g: (2.70) 2.6.2 Hyperbolic Location Systems We may summarize the technique as described in [14] as follow. Suppose that the arrival times t 1 ;t 2 ;:::;t N of a signal transmitted at time t 0 are measured at N stations located at the points with position vectors s 1 ;s 2 ;:::;s N (Fig. 2.16). If the propagation path between the transmitter and stationi is shown withD i , then [14] t i =t 0 +D i =c+" i i=1;2;:::;N; (2.71) wherec is the signal velocity. Rewriting (2.71) in matrix form, yields t=t 0 1+D=c+" (2.72) 40 Transmitter Station 1 Station 2 Station N D1 D2 DN . . . Figure 2.16: Geometry of transmitter and N station [14] where1 is a column vector of ones, andt,D, and" areN¡dimensional column vectors corresponding tot i ,D i , and" i respectively. In hyperbolic systems, t 0 is not estimated. Instead, it is eliminated by measuring the relative arrival times t i ¡t i+1 =(D i ¡D i+1 )=c+n i i=1;2;:::;N: (2.73) Note that t p =t¡t 0 =t r +¢t; (2.74) so t=t 0 +t r +¢t; (2.75) and then t i ¡t i+1 =t r i ¡t r i+1 : (2.76) Therefore, (2.71) can be rewritten as 41 t r i ¡t r i+1 =(D i ¡D i+1 )=c+n i i=1;2;:::;N: (2.77) Assuming that the relative arrival times are determined by subtracting measured arrival times,n i , the measurement error, is n i =" i ¡" i+1 i=1;2;:::;N: (2.78) If successive" i have equal means, thenn i have zero means. We assume that the time differencet r i ¡t r i+1 does not exceed the time between successive pulse transmissions, otherwise a potential ambiguity may have occurred [14]. Rewriting (2.77) in matrix form yields Ht r =HD=c+n (2.79) whereH is H = 2 6 6 6 6 6 6 6 4 1 ¡1 0 ¢¢¢ 0 0 0 1 ¡1 ¢¢¢ 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 ¢¢¢ 1 ¡1 3 7 7 7 7 7 7 7 5 (2.80) Herer=Ht;f(x)=HD=c andx=R =[x;y;z]. G can be calculated as G=HF=c: (2.81) From (2.78) n=H" (2.82) 42 Then the covariance matrix of the measurement errors is N=HN " H T (2.83) whereN " is the covariance matrix of the arrival time errors. Assuming " i as the sum of a constant bias plus zero mean Gaussian noise, for the hyperbolic location system the least square estimator is ^ R=R 0 +c(F T H T N ¡1 HF) ¡1 F T H T N ¡1 (Ht¡HD 0 =c) (2.84) in which R 0 = x 0 = [x 0 ;y 0 ;z 0 ] T , and D 0 is a vector with components D 0i ; i = 1;2;:::;N:, andD 0i =kR 0 ¡s i k. Also F = 2 6 6 6 4 (R 0 ¡s 1 ) T =D 01 . . . (R 0 ¡s N ) T =D 0N 3 7 7 7 5 (2.85) Every row of vector F is the unit vector pointing from one of the stations s i to the reference pointR 0 . Replacing (2.81) in (2.51) yields the covariance matrix of ^ R P=c 2 (F T H T N ¡1 HF) ¡1 : (2.86) Note that with the point reflector assumption and Gaussian noise, the above discussion is still valid for the reflector localization with hyperbolic method, when there are enough reflected signals from the same reflector. 43 2.6.3 Threshold Setting To set the threshold, whenP is determined, the size of the uncertainty region is specified. Then, the variation in the source placement, defined as ¡! ±R i , shown in Fig. 2.17 (for 2- D), can be obtained . 1 γ 2 γ A B C A i R + δ A i R - δ Figure 2.17: Position error In this figure, ~ ±R 1+ and ~ ±R 1¡ respectively stand for the positive or negative error of the estimator relative to the true position (x), along the axis throughA andx. We define the (allowable) positive and negative timing errors as ½ i+ = ~ ±R i+ =c ½ i¡ = ~ ±R i¡ =c; i=1;2;:::;m (2.87) Because of the symmetry 44 ½ i+ =½ i¡ i=1;2;:::;m: (2.88) Thus, the total (allowable) timing error is defined as ½ i =2½ i+ i=1;2;:::;m; (2.89) which describes the variation in the location of realizations (from the estimated source location). For LOS signals in a source-consistent group, we have following equation t p i ¡t r i =" 0 i =¢t+" i ; i=1;2;:::;m: (2.90) Note that ¾ " 0 i =¾ " i : (2.91) Because" is zero mean,¢t can be estimated as c ¢t= ¹ " 0 = 1 m X i " 0 i : (2.92) Subtracting ^ ¢t from 2.90 results in " i =" 0 i ¡ c ¢t; i=1;2;:::;m: (2.93) 45 The variation in" 0 i ; i=1;2;:::;m (which is equal to that in" i ) is checked to see if it is in the appropriate range. For this purpose, the variance of½ i ’s is compared with that of" i ; as ¾ 2 ½ i ¸¾ 2 " 0 i ;8i (2.94) or equivalently j½ i j¸j" 0 i ¡ ^ " 0 i j;8i (2.95) When this test is satisfied, the hypothesis is correct. This check implies that for a correct hypothesis, the timing error in theith signal must have the following constraint ½ i¡ ·" i ·½ i+ : (2.96) From the discussion above, the threshold setting process can be summarized in the following steps (1) Choosep e (·) (2) Calculate· from (2.60) (3) Obtainx 0 (4) Form matrixP by equation (2.86) (5) Obtain¸ 0 i s (6) Obtain the size of the uncertainty ellipse, and then±R i (7) Compute the domain of error for signalsS i ; i=1; 2; :::; m by (2.92-2.96). Now the threshold vectors are available and the checking steps of the algorithms (related to the omni-directional receivers) can be modified accordingly. 46 It must be mentioned that other estimates for ¢t, or other threshold criteria than (2.96), can be used to apply different methods for threshold setting. 47 Chapter 3 Receiver Sites for Accurate Indoor Position Location Systems 3.1 Hyperbolas of Uncertainty Area and Their Features in 2-D and 3-D The tangent line to hyperbola at a point is the bisector of the angle between lines from the point to two foci. This fact at pointx for two sensorsA andB is shown in Fig. 3.1. A parametric equation for pointx on the hyperbola is d(x;A)¡d(x;B)=t AB ; (3.1) in which t AB is the difference between propagation times of the received signals at sensors A and B. Here distances are normalized by propagation velocity and hence are measured in time units. If the propagation time difference t AB increases by the time resolution4t, typically of the order of the reciprocal of the signal bandwidth, the emitter locationx is now on pointx 0 on the next hyperbola with equation d(x¶;A)¡d(x¶;B)=t AB +4t: (3.2) For the rest of the paper we assume that4t is very small. With this assumption, tangent lines at pointx andx 0 are parallel and the orthogonal distance between them can be obtained by subtracting (3.2) from (3.1) and applying approximation as 48 x u x’ u’ α α ε A B Figure 3.1: hyperbola of LOP line for two receiving station at A and B and nominal source atx ²=j4t=2sin®j; (3.3) in which without loss of generality, we can ignore the absolute value. As indicated in Fig. 3.2, with two pairs of sensors AB and AC, the uncertainty area at point x is a curved parallelogram that its edges can be approximated with tangent lines at the point as below l i =² i =cos(90¡(¯+®))=² i =sin(¯+®); (3.4) in which² 1 and² 2 correspond to the hyperbolas of pairs of sensorsAB andAC respec- tively. The area of the parallelogram is S =l 1 l 2 sin(¯+®): (3.5) Replacing (3.3) and (3.4) in (3.5) yields S = 4t 1 4t 2 4sin®sin¯sin(¯+®) : (3.6) 49 t Δ α α x ' x β β C A B 1 l 2 l 1 ε 2 ε d Figure 3.2: Geometry of a hyperbolic location system with nominal emittert positionx and receiving stations at A, B, and C. The uncertainty area approximated with a paral- lelogram The largest diagonal of the parallelogram is d= q l 2 1 +l 2 2 §2l 1 l 2 cos(®+¯): (3.7) Replacing (3.3) and (3.4) in (3.7) and manipulating gives d= 4t 2sin(®+¯) q 1 sin 2 ® + 1 sin 2 ¯ § 2cos(®+¯) sin®sin¯ ; (3.8) in which¡ sign correspond to the points outside the convex hull of the three sensors. The hyperbolas of line of positions for three sensors located on three corners of a rectangular area, with equal4t for all three pairs of sensors, are shown in Fig. 3.3. As seen, when 4t 1 =4t 2 =4t 3 =4t; (3.9) three set of hyperbolas always cross at the same point. In the other words, at any point, the hyperbolas of the third pair of sensors is the diagonal of the parallelogram of the other two set of hyperbolas crossing at the point. This fact can be proved by noting that 50 Figure 3.3: Hyperboloid of LOPs for three sensors in a rectangular shape area the parametric equations of hyperbolas of sensorsAB andBC at pointx in Fig. 3.4 can be written as d(x;A)¡d(x;B)=t AB d(x;B)¡d(x;C)=t BC d(x;A)¡d(x;C)=t AC : (3.10) Shifting the propagation time differences of (3.10) by4t, gives the new locationx 0 for the emitter with the parametric equations d(x¶;A)¡d(x¶;B)=t AB +4t d(x¶;B)¡d(x¶;C)=t BC ¡4t d(x¶;A)¡d(x¶;C)=t AC : (3.11) As seen, in the absence of measurement noise, the same hyperbola of the pair of sensorsAC crosses at pointsx andx 0 . For the rest of the paper it is assumed that (3.9) is valid. 51 x x’ a b c Figure 3.4: Parallelograms of uncertainty of two pairs sensors meet each other at the corners. 3.1.1 Uncertainty Region in 3-D For the 3-D case we need at least four sensors (that are not coplanar) to locate a source. For a differential delay4t, the location of the transmitter is somewhere on a hyperboloid of two sheets for every pair of sensors [2]. Similar to the 2-D case, it can be proved that the tangent plane at every point on these hyperboloid sheets is the bisector plane of the sensor pair. Assuming four sensors, there are ¡ 4 2 ¢ = 6 hyperboloid sets, however three of them are enough to locate the transmitter. Every three sets of hyperboloids at a point make a curved parallelepiped, that can be approximated, when4t is small, with a parallelepiped with tangent planes of the hyperboloid at that point as its faces, as indicated in Fig. 3.5. As in the 2-D case, the orthogonal distances of the faces of 1 ε 1 ε 2 ε 3 ε 1 ε 1 ε 2 ε 3 ε x Figure 3.5: Shape of uncertainty for every triplet of sensors for 3-D. 52 parallelepiped, i.e., orthogonal distances of the tangent planes at pointx can be written as ² 1 =j4t=2sin®j ² 2 =j4t=2sin¯j ² 3 =j4t=2sin°j; (3.12) where®,¯ and° are the angles of the sensor triplets at pointx. Adding other hyperboloids decreases the parallelepiped of uncertainty. The form of uncertainty depends on the manner that sensor pairs are selected. Figure 3.6 represents the parallelepiped of uncertainty for Z configuration (e.g., BA, AC and CD), where the top and bottom faces represent the tangent planes corresponding to sensor pairs AC, the left and right sides are related to CD, and the back and front faces correspond to BA. For this configuration, the uncertainties are the space between the faces of the parallelepiped and plainsabgh,adgf, andafh. a b c d e f g h AC CD BA x ' x t Δ + t Δ + t Δ + Figure 3.6: Shape of uncertainty for pairs sensors for 3-D with Z configuration. For proof, note that for Z configuration, the parametric equations for the hyperboloid of two sheets of three pairs of sensors at pointx are: 53 8 > > > < > > > : d(x;B)¡d(x;A)=t BA d(x;A)¡d(x;C)=t AC d(x;C)¡d(x;D)=t CD (3.13) The parametric equation for the hyperboloid of sensor pairs AD crossing at pointx can be obtained as: d(x;C)¡d(x;D)=t CD d(x;A)¡d(x;C)=t AC d(x;A)¡d(x;D)=t AD (3.14) The same hyperboloid crosses at pointx¶ , which is obtained with a small changes to the hyperboloids of sensors pairs CD and AC, because d(x;C)¡d(x;D)=t CD +4t d(x;A)¡d(x;C)=t AC ¡4t d(x;A)¡d(x;D)=t AD (3.15) Therefore, plainabgh corresponds to the hyperboloid of sensor pairs AD. With the similar method, it can be shown that plains adgf and afh correspond to the sensors pairs BC and BD respectively. For the Y configuration (e.g.,AB,AC andAD) of selecting sensor pairs, the uncer- tainty is shown if Fig. 3.7. As seen the uncertainties reduce from parallelepiped to the space between its faces and planesabgh,adgf andbdhf. The proof is left to the reader. 3.2 Best Locations of the Receivers for 2-D For the 2-D case, we consider two different criterions, the area of uncertainty region and its largest diagonal, to be minimized. 54 a b c d e f g h AB AC AD x ' x t Δ + t Δ + t Δ + Figure 3.7: Shape of uncertainty for pairs sensors for 3-D with Y configuration. Considering the largest area of uncertainty as the criterion, we must choose the loca- tions of the sensors in the triangular shape areaV 1 V 2 V 3 (Fig. 3.8) in such a way that the area of the triangle of uncertainty (shown in Fig. 3.4) for the worst point (which is the point with the highest area of uncertainty) is minimized. Similar to 3-D case, it can be proved that for the points inside the triangle of three sensors, there are always at least two obtuse angles correspond to independent hyperbolas. note that in Fig. 3.4 parallelogramsxax 0 b andxcbx 0 have the same area. Therefore, we can choose any two angles to compute the area of uncertainty. SupposeÁ 1 , Á 2 andÁ 3 are the angles of the triangle connecting three sensorsA, B andC as indicated in Fig. 3.8, and assume Á 1 ·Á 2 ·Á 3 : (3.16) For the points inside the triangle of three sensors, as shown in Fig. 3.8 we have 2®+2¯+2° =2¼ Á 1 ·2®·¼ Á 2 ·2¯·¼ Á 3 ·2°·¼; (3.17) 55 in which 2®, 2¯ and 2° are the sight angles of three pairs of sensors BC, AC, and AB of the point x respectively. Therefore, the domain of ® and ¯ for the points inside the triangle of the three sensors is Á 1 =2< ® <¼=2 Á 2 =2< ¯ <¼=2 ¼=2< ®+¯ < (¼¡Á 3 =2): (3.18) This domain is shown in Fig. 3.10. 2α 2β 2γ 1 ϕ 2 ϕ 3 ϕ A B C x 1 V 2 V 3 V Figure 3.8: Sensors located on the the vertices of the triangular area FunctionS for0 < ®;¯ < ¼=2 is shown in Fig. 3.9. In the region(®+¯) < ¼=2, functionS is monotone decreasing. 0 10 20 30 40 50 60 70 80 90 0 50 100 0 0.5 1 1.5 2 2.5 x 10 4 alpha beta Figure 3.9: FunctionS for0<®;¯ <¼ 56 This function in the domain of (3.18),with a larger scale, is shown in Fig. 3.11. In this figure, functionS has its maximum value at the point® =Á 1 =2;¯ =¼=2, which is S = 4t 2 1 2sinÁ 1 : (3.19) It can be seen that there is an absolute minimum at point (¼=3;¼=3). 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 alpha beta Figure 3.10: The domain of function S for the points inside the triangle of the three sensors So, this is the maximum uncertainty if we locate the sensors on the vertices of the area, as shown in Fig. 3.8. Now assume that we put the sensors on the points other than the vertices, as indicated in Fig. 3.12. In this case, the sharpest corner, (i.e.,Á 1 ) have the highest uncertainty area; and for this corner, the domain of functionS is 2®+2¯ =Á 1 : (3.20) It is obvious from Fig. 3.9 that functionS for all the points of (3.20) are greater than (3.19). Therefore, the corners of the area are the best place for the sensors to be located. 57 0 10 20 30 40 50 60 70 80 90 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 1.2 1.4 alpha beta Figure 3.11: Function S for the domain in 3.18(Á 1 =¼=6;Á 2 =¼=4) V1 V3 V2 B C A 1 ϕ 2 ϕ 3 ϕ Figure 3.12: Sensors located on the points other than the vertices of the triangular area Considering (3.19) and (3.20), we can conclude that the maximum value of function S for the points inside the area (i.e., the value of functionS for the worst point) depends on Á 1 , the minimum angle of the triangle. In the other words, for an area with larger minimum angle, we get better (smaller) uncertainty for the worst point. This fact was verified with simulation, by computing maximum value of functionS inside the triangle for the different values ofÁ 1 ,Á 2 andÁ 3 . Suppose we choose the largest diagonal of uncertainty as the criterion. It can be shown that when the sensors are on the vertices of the area, functiond has its maximum value on the point O (Fig. 3.10) with coordinates ® = Á1=2;¯ = ¼=2, which is 58 d(Á 1 =2;¼=2)= 4t 2cos(Á 1 =2) p (cotÁ 1 =2) 2 +4: (3.21) Similar to functionS, it can be proved that to minimize the diagonal of uncertainty for the worst point, the sensors must be located on the corners of the area. Note that the other choices of sensors locations (i.e., locating them in such a way that some points of the area are on the baseline extension of two sensors), cause location fix ambiguity described in [1]. 3.3 Best Locations of the Receivers for 3-D For 3-D case, since determining the volume or the edges of the parallelepiped of uncer- tainty is difficult, the multiplication of LOP errors (² i ’s), is considered as the criterion to be minimized . So we locate four sensors in a tetrahedral shape areaV 1 V 2 V 3 V 4 (Fig. 3.13) in such a way that the following equation is minimized S =² 1 ² 2 ² 3 = 4t 3 8sin®sin¯sin° : (3.22) Suppose we locate the source with three pairs of sensors at each point, and assume that we place four sensors A, B, C, and D on the vertices of the volume, as indicated in Fig. 3.13. Consider the angles between the lines from the point to the sensors (i.e. \AxB;\AxC;:::), we use the following theorem Theorem: For the points inside the tetrahedron of four sensors, we can always find at least three obtuse angles that correspond to independent hyperbolas. Proof : For proof note that every angle corresponds to two vertices of the tetrahedron and so to an edge. We call the edges corresponding to the acute angles as the “bad edges”. We assume five cases for bad edges in the tetrahedron: 59 A B C D 1 V 2 V 3 V 4 V Figure 3.13: Tetrahedral volume Case 1: Assume that there are three bad edges converging at one vertex (by symme- try, we can assume every vertex of the tetrahedron as the meet point). This case is shown in Fig. 3.14 in which bad edges are shown with thick lines. We prove by contradiction that this case is impossible. D B A C X ' α ' β ' γ Figure 3.14: Geometry of bad edges: case 1 60 Consider the vector from the arbitrary point x inside the tetrahedron to antenna A, we draw the plane perpendicular to this vector (Fig. 3.15). If all three angles are acute, then pointx will not be in the convex hull of the set of four sensors (since the receivers are in the same side of the plane), that is a contradiction since we assumed that pointx is inside the tetrahedron. A D C B X Figure 3.15: Geometry of the sensors for an arbitrary point inside the tetrahedron for case 1 Case 2: There are three bad edges which are the edges of a face of the tetrahedron as indicated in Fig. 3.16. Then the other three angles that correspond to independent hyperbolas are obtuse and the theorem still stands. D B A C X ' α ' β ' γ Figure 3.16: Geometry of bad edges: case 2 61 Case 3: Three bad edges with Z configuration as indicated in Fig. 3.17. Again as the previous case, three independent obtuse angles are left. D B A C X ' α ' β ' γ Figure 3.17: Geometry of bad edges: case 3 Case 4: Four bad edges with the configuration shown in Fig. 3.18. It can be seen that this case is similar to case 1 and therefore is impossible. D B A C X ' α ' β ' γ Figure 3.18: The geometry of bad edges: case 4 Case 5: Four bad edges with the configuration shown in Fig. 3.19. It can be proven by contradiction that this case is also impossible. Assume that we draw the vectors from 62 point x inside the tetrahedron to the sensors A and D (Fig. 3.20), and also draw the planes perpendicular to these vectors. The angle between vector ¡ ! xA and ¡ ! xD is a “good angle” (obtuse) and therefore pointD is located under the plane perpendicular to vector ¡ ! xA. From Fig. 3.19, we must have \AxC <¼=2 \CxD <¼=2 \AxB <¼=2 \BxD <¼=2; (3.23) so pointsB andC must locate above the plane perpendicular to ¡ ! xA and on the left side of the plane perpendicular toD as indicated in Fig. 3.20. D B A C X ' α ' β ' γ Figure 3.19: Geometry of bad edges: case 5 Now consider an arbitrary plane crossingx, shown by the dashed line in Fig. 3.20. All four sensors are located on one side of this plane, therefore point x is not in the convex hull of the set of the sensors, so this case can not exist.¤ Therefore, if we locate the sensors on the corners of the tetrahedron of volume, for all the points inside the tetrahedron, we always can find (at least) three “good (obtuse) 63 angles”. We can use the corresponding pair of sensors to locate the source and have small uncertainty (based on functionS). A D Plane perpendicular to xA Plane perpendicular to xD C B x Arbitrary plane Figure 3.20: Geometry of the sensors for an arbitrary point inside the tetrahedron for case 5 Otherwise, suppose we locate the sensors on the points other than the vertices, as indicated in Fig. 3.21. We show that for the worst point outside the tetrahedron of the receivers(which is the sharpest vertex of the tetrahedron V1V2V3V4 ), all the angles are acute, resulting in larger uncertainty (higherS) A B C D X α 1 V 2 V 3 V 4 V Figure 3.21: Sensors located on the points other than the vertices of the volume First we give the definition for “Deficiency”, and also a theorem about tetrahedrons. 64 Definition: Given a solid angle with plane anglesA 1 ;A 2 :::A n , the deficiency of the solid angle is the ”gap” left over if we were to unfold the solid angle and lay it flat. In other words, deficiency of the solid angle is Deficiency=2¼¡(A 1 +A 2 +:::+A n ): (3.24) ’Deficiency’ measurement is another way other than “solid angle” to determine the “pointyness” of a corner. Descartes’ Theorem: The sum of the deficiencies of the solid angles in a polyhedron is always4¼. It is known that a regular tetrahedron can be embedded in a cube, in which four vertices of the tetrahedron are placed on the vertices of the cube, as shown in Fig. 3.22. Figure 3.22: Regular tetrahedron placed in a cube Since every plane angle of the regular tetrahedron is¼=3, the deficiency for every vertex is¼. Based on Descartes’ theorem for any tetrahedron X µ i =4¼; (3.25) in which µ i ’s are the deficiencies of the vertices. In a tetrahedron, the sharpest corner must have the highest deficiency; and from (3.25) it is obvious that at least one of the 65 deficiencies must be greater than ¼, that is, greater than the deficiency of a corner of a regular tetrahedron. While the corner of a regular tetrahedron can be fitted in the corner of a cube, so will a sharper corner. Thus the sharpest corner of a tetrahedron, which is sharper than the corner of a regular tetrahedron, can be placed inside the corner of a cube. Now consider Fig. 3.21 in which four antennas are placed in a tetrahedral volume, on the points other than vertices. The sharpest corner of the volume is the worst point where functionS has its maximum value; and we know that at least one of the corners of tetrahedron (the sharpest corner) can be placed inside the corner of a cube, so all of the angles between every pair of antennas at that corner must be less than¼=2, thus the angles used in (3.22) are less than or equal to¼=4. Therefore, for the worst point inside the tetrahedral volume for the case shown in Fig. 3.13, value of functionS is always smaller than that of the worst point for the case shown in Fig. 3.21. Thus, to have the best result, receivers must be placed on the corners of the tetrahedral volume so that all of the points of the volume are located inside the tetrahedron of the receivers. 3.4 Number of the Receivers to Cover an Area in 2-D and 3-D Now the results obtained in the previous sections can be extended to an arbitrary area. For 2-D case, suppose we have a n¡ polygon area as shown in Fig. 3.23. We triangulate the area as indicated and place the sensors on the corners. Since the sensors are assumed to be omnidirectional, we can keep one sensor on every corner and remove duplicates. Therefore, no matter how the area is triangulated, the sensors are located on the vertices as indicated in Fig. 3.24. However, for locating a source at any point, to 66 Figure 3.23: n-polygon as an arbitrary area have the least uncertainty, we can use the triplet of sensors that see the point and make a triangle with maximum value of minimum angle (i.e., a triangle that is closest to an equilateral triangle). Figure 3.24: The sensors are located on the vertices of the area Therefore the number of the sensors to cover a n¡ polygon is n. This result is compatible with ’Chvatal’s Art Gallery theorem’ [11] which says thatbn=3c guards are occasionally necessary and always sufficient to see the entire interior of a polygon ofn edges. Since in an art gallery we need one guard for each triangle; and here we need three guards, the number of guards here is almost3¤bn=3c. We can extend this conclusion to a polygon with h holes and n vertices in total as indicated in Fig. 3.25. After triangulation and locating the sensors on the corners and removing the duplicates of the sensors, we are left with one sensor on every vertex of the area and of its holes as indicated; and thereforen sensors are sufficient to cover the area. In an art gallery the number of guards that suffices to dominate any triangulation is(n+2h)=3 [11]. 67 Figure 3.25: n-polygon with h-holes as an arbitrary area. The sensors are located on the vertices of the area and of its holes For the 3-D case, consider that everyn¡polyhedron volume can be partitioned into tetrahedrons. Similar to the 2-D case, we divide the volume into tetrahedrons, locate the sensors on the corners (as concluded in the previous section), remove the duplicate sensors and leave one sensor on each vertex. Thereforen sensors are adequate to cover an¡polyhedron. With a similar method, we can conclude that for apolyhedron with h holes andn vertices in total,n sensors (located on the vertices) are sufficient to cover the volume. So we have the following theorem: Theorem: n sensors are always sufficient to localize an emitter in an entire interior of an¡polygon in 2-D (n¡polyhedron in 3-D) with LOS measurements. This result can be generalized to a polygon (polyhedron) withh holes andn vertices in total. 3.5 Simulation Results Simulation was done by MATLAB for the 2-D case for triangular-shaped areas with three sensors, with functionS defined in (3.6) as the criterion. The result was compatible with the theory and confirmed that the best place for the sensors is on the corners of the triangle. 68 Chapter 4 Experimental Results 4.1 Measurement Plans In this chapter, all the position and error units are inch, and SNR is described in dB. Three different experiments were done in UltRa Lab with UWB pulses of 0.5nsec width (The pulse-width definition is full width of half maximum). The UWB template signal is shown in Fig. 4.1. The experiments were done by time domain antennas. The data of the receivers was gathered by digital oscilloscope and averaged on 100 consecutive pulse. The experiments were accomplished by three different plans and locations for the source as described here. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10 −5 −3 −2 −1 0 1 2 3 x 10 −3 t nsec mV Figure 4.1: Template signal used in the experiments 69 Tx1 l d4 a4 a1 d1 a2 d2 b2 c2 b1 c1 a3 b3 c3 d3 b4 c4 h z x y = Rx = Tx (0,0,0) Figure 4.2: The experiment plan for Tx1: 3D image In the first experiment, one transmitter (denoted as Tx1) and sixteen receivers (a1, b1, ..., d4) were involved. The receivers were covering a cubic area, below and above the platform in the lab as shown in Fig. 4.2. The floor of the platform is a metal grate with ”x” holes between ribs Fig. 4.3 is the image of the area and antennas from the top. As seen, the middle of South-West column of the platform was assumed as the origin. In the second experiment, eight receivers were used to cover a cubic area in UltRa Lab as shown in figures 4.4 and 4.5. The transmitter was located between the wall of the chamber, which is a reflector, and another reflector as shown. The third experiment was the same as the second one, except that the reflector was between the transmitter and the wall, as shown in figures 4.6 and 4.7. The distance between the reflector and transmitter in both experiments was 24 inches. 70 " 80 o Tx1 z x y = Rx = Tx (0,0,0) N Figure 4.3: The experiment plan for Tx1: top view. Tx2 Figure 4.4: The experiment plan for Tx2: side view Transmitters for these three experiments were located at Tx1=[¡40:00 ¡63:7556:25] Tx2=[152:00 ¡135:7543:25] Tx3=[32:00 ¡135:7543:25] (4.1) 71 " 80 Tx2 (0,0,0) z x y = Rx = Tx Figure 4.5: The experiment plan for Tx2: top view Tx3 Figure 4.6: The experiment plan for Tx3: side view Two different approaches, MMSE method and closed form method (introduced in [7]), were used for source localization. Assumingm=4 (m is the number of the signal in a group), there are ¡ 16 4 ¢ = 1820 combinations (groups) of the receivers for the first experiment, from which the combinations of the coplanar sensors must be excluded for localization. For the second and third experiments, the number of combinations are ¡ 8 4 ¢ =70. In the next section, the results of different approaches are given. 72 " 80 Tx3 (0,0,0) z x y = Rx = Tx Figure 4.7: The experiment plan for Tx3: top view Tx2 Tx3 l e1 f1 e4 f4 b1 c1 b4 c4 h (h+l) = Rx = Tx z x y Figure 4.8: The experiment plan for Tx2 and Tx3: 3D image It must be mentioned that the nature of the closed form computations (first method) prevents applying the algorithms introduced in chapter 2, in the localization process. 73 4.2 Measurement Results In indoor systems, the results of localization process can be restricted to a given volume. Here, two different strategies were used for restricting area, which are described in each localization method. In the first restricting scheme, the results were restricted to the volumes under test. For the first measurement (Tx1) this area consists of the spaces under and above the platform. When the receivers involved in the localization process are under the platform, only the answers beneath the platform are considered, while for the sensors above the platform, the upper cube of the platform is the restricting area. With this restriction, the number of the combinations for the receivers is reduced to2 ¡ 8 4 ¢ =140. For the second and third measurements, the area under test was the cubic volume restricted to the wall of the chamber, and the plane of four receivers closer to the plat- form. Tx2 was located a little outside of the cubic area restricted by the receivers, while Tx3 was inside of the cube. In the second limiting plan, the results were restrained to be in the convex hull of the receivers. Minimum Mean Squared (MMSE) method From equation (2.47), maximum likelihood estimation (MLE) of the transmitter location is c Tx=arg max x f(rjx): (4.2) Considering equation 2.79, with Gaussian noise assumption, MLE becomes MMSE esti- mation 74 c Tx=arg min D (kHt r ¡HD=ck): (4.3) In this method, all areas were searched to find a point whose time difference of arrivals are close to those that were measured. Threshold was set with the technique described in section 2.6.3. Error was defined as e=k c Tx¡Txk (4.4) The results for the three positions of the transmitter, for p e = 0:1 (p e was given in 2.54), and the related errors are given below. 1. Restricted to the cube d Tx1=[¡39:1429 ¡64:963358:9306] e=3:0647 d Tx2=[146:6952 ¡129:992151:6667] e=11:4950 d Tx3=[33:7060 ¡133:237346:1970] e=4:2319 (4.5) 2. Restricted to the convex hull of the receivers d Tx1=[¡41:1294 ¡63:405957:1412] e=1:4793 d Tx2=[140:0500 ¡136:050045:9000] e=12:2440 d Tx3=[33:7545 ¡135:254546:6727] e=3:8780: (4.6) The table in Fig. 4.9 shows how " i s are variant for different guesses. The first column of this table contains the indices of the receivers, which are shown in Fig. 4.10. 75 The third column of the table (shown asv:d:t (variation in¢t)) represents v:d:t =" imax ¡" i min =" 0 imax ¡" 0 i min : (4.7) The fifth column of the table, indicated by State 1, shows if the transmitter is in the convex hull of the receivers in the group (represented by ’1’) or not (denoted by ’0’). The last column of the table is the same as the fifth column except that it indicates whether the estimated location is in the convex hull of the receivers or not. As seen, for bad choices of the receivers’ indices v:d:t is higher. Fig. 4.11 shows how the v:d:t increases with error. Closed form method In this technique, the closed form solution given in [7] was used to obtain the position of the transmitter. As mentioned before, the algorithms can’t been applied to this method. The estimated positions for the transmitters attained by this solution with the related errors are given in following. The result from the sets of four signals were averaged to obtain the final transmitter’s position. 1. Restricted to the cube d Tx1=[¡39:3363 ¡61:928658:7224] e=3:1418 d Tx2=[145:4228 ¡123:054551:0155] e=16:2708 d Tx3=[31:5418 ¡136:399645:5983] e=2:4792 (4.8) 2. Restricted to the convex hull of the receivers d Tx1=[¡41:0680 ¡63:096456:1605] e=1:2554 d Tx2=[142:5757 ¡129:446744:5698] e=11:4145 d Tx3=[33:5865 ¡135:851347:0004] e=4:0734: (4.9) 76 4.3 Precision Analysis with Simulated A WGN Noise To analyze noise effect on localization precision, receiver front end was simulated with a low pass fourth order Butter Worth filter, as shown in Fig. 4.12. The bandwidth of the filter was selected as 95 percent of the captured energy band- width of the template signal. The spectrum of the template signal and the response of the filter are shown in Fig. 4.13. Signal-to-noise ratio (SNR) was defined as the signal power to the noise power at the output of the receiver’s front end filter SNR = P x o P n o = 1 N P n x 2 o [n] P n o (4.10) wherex o [n] is thenth sample of the received signal at the output of the front end filter (when sampled), and N is the number of the samples of this discrete signal. P xo and P no are the power of the discrete signal and discrete noise at the output of the front end filter respectively. The sampling frequency wasf s =12.5psec. Since in the experiments the signal were averaged on 100 scans, we assume that the captured signal is almost noise-free. Since signal power was different at various receivers, for the first measurement, SNR was defined at receiver a1 (Fig. 4.2); and for the second and third measurements, at receiver b1 (Fig. 4.8). Power gain of the filter was defined as G= 1 N X n jH[n] 2 j (4.11) in whichH is the discrete frequency response of the front end filter. When the input noise of the filter is white, the power of the output noise can be computed by 77 P n i = P n o G (4.12) For a given SNR, the power of the input noise was computed by 4.10-4.12. In the other words, the same noise variance was added to all the signals. The error curve for Tx1 when the algorithms in 2.1.1 and 2.3 are applied is shown in Fig. 4.14 with solid line. The other curve in this figure shows the error for Tx1 with MMSE method, when the algorithms are not applied. For this curve, all the valid locations of the transmitter obtained by different groups of signals, are averaged to find the final transmitter location. In both cases, the results are averaged onn = 10 run. Comparing these two curves shows the improvement in source localization by the algorithms. Figures 4.15 and 4.16 show error curves for the second and third measurements, with MMSE method, averaged onn=10 run. As observed by these figures, and also by results given in 4.5- 4.9, when the trans- mitter is close to a reflector, error increased. Precision gets worse when the transmitter is outside the cube of the receivers. Similar curves for closed form method could not be generated, because in a noisy environment, the method occasionally would not give any result. This fact can be seen by the table given below, which shows the errors for different SNR onn = 10 runs. As seen, even for the runs for which the method provides a result for transmitter location, the average of the error on those runs is higher than that of MMSE method with the applied algorithm (Fig. 4.15). 78 Receiver's index Estimated Tx1(inch) v.d.t (nsec) Error(inch) State 1(Tx1)SState 2(point) 1 3 6 7 -34.6000 -60.6000 62.8000 0.0053 1.0607 1 1 2 3 5 7 -35.9000 -62.2000 51.5000 0.0052 2.6954 1 1 1 3 6 8 -35.2000 -61.5000 56.6000 0.0054 3.6352 1 1 3 5 7 8 -34.1000 -59.9000 67.1000 0.0032 4.0255 0 0 3 6 7 8 -34.6000 -60.2000 62.9000 0.004 4.0404 0 0 3 5 6 7 -35.9000 -61.7000 51.5000 0.0054 4.4368 0 0 1 6 7 8 -35.5000 -64.2000 51.4000 0.0232 5.5484 0 0 1 3 5 8 -34.8000 -68.1000 51.2000 0.0038 5.6254 0 0 2 5 7 8 -39.7000 -69.5000 46.0000 0.006 5.6644 1 1 1 2 4 7 -33.7000 -60.1000 63.1000 0.0044 5.6943 0 0 2 3 4 7 -34.6000 -61.6000 50.1000 0.0015 6.102 0 0 1 5 7 8 -34.2000 -60.9000 56.6000 0.0035 6.1494 0 0 2 6 7 8 -33.3000 -59.4000 67.9000 0.0047 6.1664 1 0 1 4 5 7 -44.1000 -66.1000 60.6000 0.008 6.2797 0 0 1 5 6 8 -62.0000 -75.5000 56.6000 0.0045 6.2917 0 0 2 4 5 7 -39.2000 -63.3000 61.8000 0.0082 6.2964 1 0 1 5 6 7 -40.6000 -64.5000 56.7000 0.0056 6.3028 0 0 1 3 5 6 -42.3000 -65.3000 58.6000 0.0058 6.423 1 1 3 4 5 7 -44.8000 -61.3000 51.4000 0.0041 9.8972 0 0 1 3 4 5 -40.2000 -61.1000 56.7000 0.0042 9.9967 0 0 1 2 5 7 0 -53.5000 100.0000 0.0022 10.2954 1 0 2 4 5 6 1.0000 -64.1000 51.9000 0.009 10.3143 1 1 2 3 4 5 -48.7000 -60.8000 51.4000 0.0029 10.3564 0 0 2 3 4 8 -34.0000 -62.7000 67.5000 0.0021 13.8768 0 0 1 3 4 8 -34.8000 -64.9000 56.5000 0.0023 14.1257 0 0 1 4 6 8 -34.8000 -70.0000 66.9000 0.0031 14.438 0 1 2 4 7 8 -34.9000 1.0000 67.1000 0.0042 14.8565 0 0 2 4 5 8 -44.4000 -64.1000 56.7000 0.0076 15.764 0 0 1 3 5 7 -44.5000 -63.8000 56.2000 0.0201 24.9436 0 1 2 3 6 7 -44.0000 -63.8000 56.7000 0.2538 41.2316 0 0 1 4 5 8 -44.0000 -64.1000 56.7000 0.0264 41.6512 0 0 2 3 5 8 -44.4000 -65.1000 66.7000 0.0692 60.1592 0 0 3 4 7 8 -44.5000 -65.0000 66.5000 0.357 65.8506 0 0 2 4 6 8 -44.2000 -65.0000 66.6000 0.5708 84.4968 1 0 11 12 14 16 -44.3000 -65.1000 66.6000 0.1331 92.9193 0 0 9 12 15 16 -125.0000 -65.5000 108.0000 3.9463 99.5295 0 0 9 13 14 15 -125.0000 -50.6000 108.0000 3.8113 100.3792 0 0 9 13 15 16 -125.0000 -68.7000 109.8000 3.6506 100.5838 0 0 9 14 15 16 -125.0000 -78.6000 108.0000 4.9426 100.616 0 0 12 13 14 16 -49.8000 -140.3000 108.0000 3.2872 102.4262 0 0 12 13 15 16 -125.0000 -88.0000 108.0000 3.6452 103.5297 0 0 12 14 15 16 -125.0000 -79.2000 113.3000 2.6906 112.1576 0 0 Figure 4.9: Table 1: The difference in timing error for different guesses 79 Tx1 l 16 13 1 4 5 8 6 7 2 3 9 10 11 12 14 15 h (0,0,0) Figure 4.10: Indexing of the receivers for Tx1 0 20 40 60 80 100 120 140 160 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 error(inch) v.d.t Figure 4.11: Results for Tx1 for some of the groups 80 Butterworth 4th order H(w) ) (t x i ) (t n i ) (t x o ) (t n o Figure 4.12: Block diagram of the receiver’s front end −1.5 −1 −0.5 0 0.5 1 1.5 x 10 10 −300 −250 −200 −150 −100 −50 0 Frequency (Hz) |H| 2 (dB) Butterworth, 4th order Figure 4.13: Spectrum of the template signal and filter response of the front end 81 0 5 10 15 20 25 0 10 20 30 40 50 60 70 SNR (dB) Position Error (inch) Tx1, MMSE method, strategy II without algorithm with algorithm, Pe=0.1 Figure 4.14: Error curves for Tx1 by MMSE method 0 5 10 15 20 25 0 20 40 60 80 100 120 SNR (dB) Error (inch) Tx2, p e =0.1, strategy II Figure 4.15: Error curve for Tx2 82 0 5 10 15 20 25 0 10 20 30 40 50 60 70 80 SNR (dB) Error (inch) Tx3, p e =0.1, strategy II Figure 4.16: Error curve for Tx3 SNR=0 SNR=5 SNR=10 SNR=15 SNR=20 SNR=25 120.668 121.5784 123.5271 22.3936 16.4702 16.212 130.5005 115.8769 32.6419 22.4855 17.042 16.171 125.5451 121.0124 NaN 15.3491 17.1434 17.2064 128.7543 98.3678 NaN 17.1494 18.4163 17.2508 129.8469 116.3961 76.7641 16.5952 16.2708 16.237 115.3841 118.3982 30.415 25.9651 17.2233 18.3336 119.179 119.777 101.3934 17.2616 16.4251 16.2868 119.4105 82.2644 88.2581 17.2571 18.5097 16.3082 128.4324 110.1364 68.249 18.1309 17.4491 17.207 107.128 106.8767 83.3164 13.5612 17.1889 16.1625 Figure 4.17: Table 2: error(inch) versus SNR(dB) for Tx2, with closed form method 83 Chapter 5 Conclusions and Future Works 5.1 Conclusions Separating LOS signals from the reflections was achieved to localize the transmitter with clear LOS. In chapter 2, first the algorithm was proposed to extract the group of the signals coming from the same source from all the first arrivals to the omni-directional receivers. No analogous method was seen in the literature. Next, an algorithm was proposed to group the signals of the same sources received by the directive sensors. The last given algorithms helps locate the transmitter without any or enough LOS, by four located sources. By mathematically applying these algorithms, the source is located with a higher precision since reflections do not interfere in the localization process. The accuracy of the first and third algorithms were verified for omnidirectional receivers by experiment. Three different experiments were done, and the transmitter was located with good accuracy. For the experiments’ plans, the results from chapter 3 were also applied to locate the receivers with better precision. The error curves based on different Signal to Noise Ratio (SNR) were generated. The results showed that applying algorithms results in better precision for source localization. This improvement is more significant in a noisy environment. Due to expense constraints, no experiment was performed for directive antennas. The study showed that the timing equation is a strong tool to distinguish between LOS and reflections. In transmitter localization applications, the timing equation can 84 also be used to locate the transmitter even without LOS signals assuming that at least four reflectors can be located. This is a novel and innovative method whose utility has yet to be determined. As said, the algorithms were extended based on a measurement noise-free assump- tion. However, the contributing measurement noise changes the last steps of the algo- rithms (to check the validity of the answer). To apply the algorithms in a real noisy environment, a threshold setting method was introduced in chapter 2. 5.2 Future Research Directions Other threshold setting processes than the one introduced in this work can be extended, and their quality may be compared. Also, the probability of correct localization may be computed for the different methods. The appropriate sites for receivers can be obtained with other criteria, such as mini- mizing probability of LOS blockage. In addition, the precision degradation or improve- ments, and the best sites of receivers can be studied for a fewer or larger number of them than given in chapter 3. The possible ambiguous locations for the reflectors can be developed for the algo- rithm of the directive receivers given inx2.2.1 (such as found inx2.5 for the omnidirec- tional receivers). In this work, NLOS omission for single transmitter localization was addressed. However, exploiting the time equations for multiple transmitter problems can also be studied. 85 Reference List [1] Nathaniel Bowditch and the National Imagery and Mapping Agency Staff. The American Practical Navigator: “Bowditch”- 2002 Bicentennial Edition. the U.S. Department of Defense, 2002. [2] Michael Brandstein and Darren Ward. “Microphone arrays : signal processing techniques and applications”. Springer, New York, 2001. [3] Y .T. Chan and K.C. Ho. “A simple and efficient estimator for hyperbolic location”. IEEE Transactions on Acoustics, Speech, and Signal Processing, 42(8):1905 – 1915, Aug 1994. [4] J. C. Chen, R. E. Hudson, and Kung Yao. “Maximum-likelihood source local- ization and unknown sensor location estimation for wideband signals in the near-field”. IEEE Transactions on Acoustics, Speech, and Signal Processing, 50(8):1843 – 1854, Aug 2002. [5] Pi-Chun Chen. “A non-line-of-sight error mitigation algorithm in location esti- mation”. Wireless Communications and Networking Conference, WCNC. IEEE, 1:316 – 320, Sept 1999. [6] Maria-Gabriella Di Benedetto and Guerino Giancola. Understanding Ultra Wide Band Radio Fundamentals. Prentice Hall, June 2004. [7] B.T. Fang. “Simple solutions for hyperbolic and related position fixes”. IEEE Transactions on Aerospace and Electronic Systems, 26(5):748 – 753, Sept 1990. [8] Yiteng Huang, J. Benesty, G.W. Elko, and R.M. Mersereati. “Real-time passive source localization: a practical linear-correction least-squares approach”. IEEE Transactions on Speech and Audio Processing, 9(8):943 – 956, Nov 2001. [9] Wuk Kim, Jang Gyu Lee, Gyu-In Jee, and ByungSoo Kim. “Direct estima- tion of nlos propagation delay for mobile station location”. Electronics Letters, 38(18):1056 – 1057, Aug 2002. 86 [10] Asis Nasipuri and Kai Li. “A directionality based location discovery scheme for wireless sensor networks”. International Conference on Mobile Computing and Networking, Proceedings of the 1st ACM international workshop on Wireless sen- sor networks and applications, pages 105 – 111, Sept 2002. [11] Joseph O’Rourke. Art Gallery Theorems and Algorithms. Oxford University Press, 1987. [12] H. Schau and A. Robinson. “Passive source localization employing intersect- ing spherical surfaces from time-of-arrival differences”. IEEE Transactions on Acoustics, Speech, and Signal Processing, 35(8):1223 – 1225, Aug 1987. [13] J. Smith and J. Abel. “The spherical interpolation method of source localization”. IEEE Journal of Oceanic Engineering,, 12(1):246 – 252, Jan 1987. [14] Don J. Torrieri. “Statistical theory of passive location systems”. aes, 20(2):183– 198, March 1984. [15] M.P. Wylie and J. Holtzman. “The non-line of sight problem in mobile location estimation”. 5th IEEE International Conference on Universal Personal Communi- cations, 2(18):827 – 831, Sept 1996. 87

Abstract (if available)

Abstract Non-Line of Sight (NLOS) error is usually the largest cause of degradation in localization accuracy and is quite common in all environments. In the first part of this work, four algorithms are given to distinguish between line-of-sight (LOS) and reflected signals. These algorithms exploit timing equations in an RF emitter location system. The algorithms are designed to be applied to both directional and omnidirectional receivers.

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Creator Ebrahimian, Ziba (author)

Core Title Improvement in hyperbolic position location systems

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 12/14/2008

Defense Date 06/20/2006

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

Tag arrays,hyperbolic positioning systems,localization,OAI-PMH Harvest,positioning systemsp,UWB

Language English

Advisor Scholtz, Robert A. (committee chair), Bonahon, Francis (committee member), Lindsey, William C. (committee member)

Creator Email zibae@yahoo.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m221

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