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Position Location of AD HOC Networks Based on a Dead Reckoning Scheme Edición Única

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(2) INSTITUTO TECNOLÓGICO Y DE ESTUDIOS SUPERIORES DE MONTERREY CAMPUS MONTERREY ELECTRÓNICA, COMPUTACIÓN, INFORMACIÓN Y COMUNICACIONES PROGRAMA DE GRADUADOS EN ELECTRÓNICA, COMPUTACIÓN, INFORMACIÓN Y COMUNICACIONES. POSITION LOCATION OF AD HOC NETWORKS, BASED ON A DEAD RECKONING SCHEME THESIS JORGE LARA MORALES. FEBRUARY, 2004.

(3) © Jorge Lara Morales. 2004.

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(5) Instituto Tecnologico y de Estudios Superiores de Monterrey Campus Monterrey Electrónica, Computación, Información, y Comunicaciones Programas de Graduados en Electrónica, Computación, Información y Comunicaciones The members of the thesis committee hereby approve of the thesis of Jorge Lara Morales in partial fulfillment of the requirements for the degree of Master of Science in. Electronic Engineering Major in Telecommunications Thesis Committee. David Muñoz Rodríguez. Ph.D. Advisor. Jose Ramon Rodriguez Cruz, Ph.D. Synodal. Cesar Vargas Rosales, Ph.D. Synodal. David Garza Salazar P h. D. Director of the Graduate Program. February 2004.

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(7) Acknowledgments. When I arrived to the Instituto Tecnológico y de Estudios Superiores de Monterrey campus Monterrey to obtain the the degree in Master of Sciences, people as David Muñoz Rodriguez, Ph.D., Cesar Vargas Rosales. Ph.D., Jose Ramon Rodriguez Cruz, Ph.D. and other teachers showed me not only what i needed to obtain the degree, but also to admire Lord's creations in a different way that the one I used to know and with that, i could expand not only my mind but all my being, becoming a better person for society and getting closer to what God wants me to be. To all of them, to my classmates and friends, i am totally gratefull because they made my time in Monterrey one of the best pages in the book of my life. To all my family, especially to my father and mother for the love and principles they gave me.. V.

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(9) Thanks, Lord For that especial piece of heaven born on July 26, 1979.. vii.

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(11) Abstract. Wireless technology is facing the challenges of the new information sharing era with great advancements in every single area, using solutions like Ad hoc networks, where mobile nodes communicate via multihop wireless links, facilitating network connectivity without the aid of any preexisting networking infrastructure. These networks need different types of routing, security, clustering and position location schemes. In the position location field we propose a technique based on a dead reckoning scheme, we analyze the error created by the environmental conditions, and therefore we estimate the probability of the distance error as a function of these conditions and the number of nodes involved in the position estimation. A dead reckoning scheme is based on the sum of consecutive distance vectors that define the final position of the mobile. These distance vectors are modify by errors created by environmental conditions and the systems involved in the location estimation, the error and the dead reckoning estimated position define an area where the mobile can be located. Therefore the study of the behavior of this error has especial relevance. This thesis shows the advantages and boundaries of this scheme, supported by a simulation of different environmental conditions which shows that under certain conditions a probability density function can be used to represent this position error.. ix.

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(13) Abstract. Resumen La tecnología inalámbrica está enfrentando los cambios de la nueva era del intercambio de information con grandes avances en todas las areas, usando soluciones como las redes auto configurables, en donde los nodos se comunican usando multiples enlaces inalámbricos, lo que facilita la conexión sin la necesidad de infraestructura preexistente. Estas redes tienen diferentes necesidades de election de ruta, seguridad, agrupación jerárquica y esquemas de posición y localization. En el campo de la posición y localization es en el que proponemos una técnica basada en un esquema de reconocimiento muerto. Analizamos el error creado por las condiciones ambientales y por lo tanto estimamos la probabilidad de la distancia del error generado en función de las condiciones ambientales y de los nodos involucrados en las posición estimada. El esquema de reconocimiento muerto esta basado en la suma consecutiva de vectores los cuales definen la posición final del móvil. Estos vectores dc distancia son modificados por errores creados por las condiciones ambientales y los sistemas involucrados en la estimation de la posición. Este error y las posición estimada usando reconocimiento muerto definen una area en donde el movil puede ser localizado. Es por esto que el estudio del comportamiento del error cobra especial relevancia. Esta tesis muestra las ventajas y limitaciones del esquema, apoyado por una simulation que recrea diferentes condiciones ambientales se muestra que bajo ciertas condiciones una función de probabilidad puede ser usada para representar el error de la posición.. xi.

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(15) Contents Acknowledgments. v. Dedicatory. vii. Abstract. ix. Abstract. xi. List of Figures. xv. Chapter 1 Introduction 1.1 Problem Description 1.2 Objective 1.3 Justification 1.4 Contribution 1.5 Thesis Organization. 1 1 2 3 3 4. Chapter 2 Basic Concepts 2.1 Ad Hoc Networks 2.2 Position Location Systems 2.2.1 Angle Of Arrival (AOA) 2.2.2 Direction Of Arrival (DOA) 2.2.3 Time Of Arrival (TOA) 2.2.4 Time Sum Of Arrival (TSOA) 2.2.5 Time Difference Of Arrival (TDOA) 2.2.6 Multilateration or Trilateration Systems 2.3 Delay Spread 2.4 Smart Antennas 2.5 Gyroscope and Digital Compasses 2.6 Error Behavior xiii. 5 5 6 6 7 7 8 9 9 10 11 14 15.

(16) xiv. CONTENTS. Chapter 3 Dead Reckoning 3.1 Dead Reckoning and the Position Location Problem 3.2 The Error Function (r, 0) 3.3 Error with Uniform Angle Distribution 3.4 Linear Behavior in the Sine Case (fy(y)) 3.5 Error with Gaussian Angle Distribution. 17 17 18 20 21 22. Chapter 4 Numerical Results 4.1 The Density and Distribution Functions 4.2 Gaussian, Weibull and Gamma Approximations 4.2.1 Probability to Probability plot (PPplot) 4.3 The Gamma Approximation Parameters 4.3.1 Fixed Exponential Distance 4.3.2 Fixed Gaussian or Uniform Angle 4.3.3 Gamma Approximation Parameters (surfaces) 4.4 Numerical Conclusions. 27 27 30 32 35 36 40 44 47. Chapter 5 Conclusions and Further Research 5.1 Conclusions 5.2 Further Research. 51 51 52. Appendix A Exponential Sum with Different Parameters. 53. Appendix B Exponential Sum with Same Parameters. 67. Appendix C Some Random Variables. 69. Appendix D Sinusoidal Distribution, Gaussian Distributed Angle. 71. Appendix E Sinusoidal Distribution, Uniform Distributed Angle. 75. Appendix F Gamma Parameters for Angles Above 35°.. 79. Bibliography. 81. Vita. 91.

(17) List of Figures 1.1 Error in the new estimated position. 3. 2.1 2.2 2.3 2.4 2.5 2.6 2.7. The Angle Of Arrival (AOA) The Time Of Arrival (TOA) Direction finding Position Location Solution Smart antenna systems with a different beam for each subscriber The Direction of Arrival of a plane wave incident Linear equally spaced array oriented along the X axis Difference between actual position and estimated position. 7 8 9 11 12 13 15. 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10. Difference between actual position and estimated position, as (r, 0) function. Error as a function of distances and angles Angle distribution function. Gaussian or Uniform Horizontal f x ( x ) component of the sinusoidal distribution (Uniform angle). Vertical fy(y) component of the sinusoidal distribution (Uniform angle). . The sinusoidal behavior and its linearity Horizontal f x ( x ) component of the sinusoidal distribution (Gaussian angle). Vertical fy(y) component of the sinusoidal distribution (Gaussian angle). . The final error is the sum of all the errors (vectors) The sum of exponential random variables will result in a m-Erlang. 18 18 19 20 21 21 23 23 24 24. 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10. Histogram and Empirical Distribution Function MD[5] 0 ~ U[-10,10]. . Histogram and Empirical Distribution Function MD[100] 0 ~ G[35]. ... Histogram and Empirical Distribution Function MD[250] 0 ~ U[-60, 60]. Histogram and Empirical Distribution Function MD [250] 0 ~ G[60]. ... Simulation vs approximation MD[5] 0 ~ U[-10,10] Simulation vs approximation MD[100] 0 ~ G[20] Ppplot, MD[5] 0~ U[-10,10] Ppplot, MD[50] 0 ~ G[30] Ppplot, MD[200] 0~U[-45,45] and 0 ~ G[45] Gamma parameter a when MD is fixed, 9 ~ U[—60, 60]. 28 28 29 29 30 31 32 33 34 36. xv.

(18) xvi. LIST OF FIGURES. 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25. Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma MD[5] MD[50] MD[5] MD[50]. parameter A when MD is fixed, 0 ~ U[-60, 60] parameter a when MD is fixed, 0 ~ G[60] parameter A when MD is fixed, 0 ~ G[60] parameter a when 9 ~ U is fixed parameter A when 0 ~ U is fixed parameter a when 0 ~ G is fixed parameter A when 0 ~ G is fixed parameter a 3D surface representation 0 ~ parameter A 3D surface representation 0 ~ U parameter a 3D surface representation 0 ~ G parameter A 3D surface representation 0 ~ G 0~U[-2.5,2.5] 0 ~ U[-2.5,2.5] 0 - G[5] 0 ~ G[5]. U. 37 38 39 40 41 42 43 44 44 45 46 47 48 48 49.

(19) Chapter 1 Introduction Wireless technology is facing the challenges of the new information sharing era with great advance in every single area, using solutions like Ad hoc networks [26, 54, 55, 61, 72, 86, 87], mobile nodes which communicate via multihop wireless links [55]; facilitating network connectivity without the aid of any preexisting networking infrastructure. These networks need different types of routing, some of them described by Avudainayagam, A, Samar, P. and others in [9, 80, 83, 88, 94]; security schemes like Bhargava, S., Candolin, C. and others shown in [12, 18, 41]; clustering methods. Basagni, S., Gupta, G. [10, 42, 88] and position location schemes Adusei, I.K.. Chakrabarti, S., Enge, P.K. [2. 25. 32, 51]. If we focus on the position location field [2, 7, 8, 17, 19, 20, 21, 23, 25, 44, 57, 58, 69, 74, 78, 89], we will find that one problem in wireless ad hoc networks is the environment which adds errors to the measure (angle, time or distance), and these errors produce mistakes in the new estimated position, especially when we do not have GPS equipment [28, 32, 60]. For wireless and cellular providers, the Federal Communication Commission (FCC) requirement for mobile terminal location during emergency (E911) calls [28] specifies an accuracy, within 50 mts over 67% of the time for handset-based solutions, and 100 mts over 67% of the time for network-based solutions. It is in the ad hoc wireless network configuration, specifically in the position location area, where we found a possible solution using a dead-reckoning scheme [3, 27, 56, 65. 73. 79].. 1.1. Problem Description. According to the FCC requirement for mobile terminal location, there is a need to know the position of the mobile in an Ad hoc wireless network configuration in order to solve the problem of the location estimation during emergency calls (E-911) [28] for short term, 1.

(20) CHAPTER 1.. 2. INTRODUCTION. medium term, and long term location estimation technology, and also for new services based on the actual position of the node (intelligent traffic control, vehicular fleet management, position based services, multimedia communication [46], etc). Solutions have been proposed for cellular networks by Adusei, I.K., Caffery, J.J. [2, 19, 20], and for PCS networks. A random walk model was suggest by Guoliang Xue [40]. Some other authors consider diverse techniques from graphs [51] to GPS [28, 63, 84] even solutions for highways [74, 82] and indoor environments [69]. But, none of them has been revealed as the main position location solution, and maybe this is because the specific solution depends on a specific problem. In our case the ad hoc network configuration increases the problem due to the mobility of every single node, but even in this field some solutions have been proposed by Amouris, K.N., Stojmenovic, I. [7, 87]. Other solutions involving Dead Reckoning were given by Agarwal, A., Krantz, D., Randell, C. [3, 56, 73] even for walking robots [79]. Random walk can be represented as a man who starts from a point (0, 0) and walks I yards in a straight line: he then turns through any angle whatever and walks another I yards in a straight line. He repeats this process n times. If we consider the stared position (0, 0) and the n times I as the connections between nodes, we could not represent our model as a random walk, because the characterization of the random walk requires that the steps occur at regular spaced time points, the walk is isotropic, and the jump distance has a distribution. The way we face the position location problem falls between two classifications. One includes random walk [39, 24], Brownian motion [50]. and Rayleigh-Pearson walk solutions and on the other hand, we find the worst case where there is no angle, modeled by the m-Erlang distribution (see Appendix B). We arc placed just in the middle, because we consider different kinds of angle distribution functions for an specific range of angles (up to 60°). Therefore, our case does not fall under the random walk classification nor the m-Erlang case.. 1.2. Objective. It would be great to know the exact position of any mobile in the Ad hoc wireless network, but because of the mobility, the environment conditions [13. 53], and lack of fully trusted position nodes (with global position devices [28, 32, 60, 63, 67, 68, 84]) in the network we have an error in the measurement, so we propose an alternative solution to.

(21) 13.. 3. JUSTIFICATION. the position location of Ad hoc networks, based on a dead reckoning scheme. We analyze the error as the sum of random vectors [1, 6. 40, 91] under several environmental conditions. When we see the problem as the difference between the estimated position of the node and the actual position of the node as shown in Figure 1.1, we can analyze the problem as a sum of random vectors under specific angle distribution and range.. Estimated position of the node. *••''' Error. Actual position of the node. Figure 1.1: Error in the new estimated position. We need to find out how many nodes can be involved in the position location estimation for a specific node using a Dead Reckoning scheme, to fullfill the requirements for position based services or even FCC requirements, for different environmental conditions.. 1.3. Justification. It is important to know how accurate a position location system can be using dead reckoning on ad hoc networks. Therefore, we propose a position location technique based on a dead reckoning scheme to define the estimated position according to the environment conditions.. 1.4. Contribution. Taking into consideration the new wireless communication area where mobile devices share information through Ad Hoc networks, the needs of position location information leads us through an amazing research trying to find out if there is a place for schemes like Dead Reckoning in the ad hoc position location field..

(22) CHAPTER 1.. 4. INTRODUCTION. More specifically, it shows how many nodes can be involved in the measurements in order to keep the location error within an acceptable range. Therefore, the information obtained can be used for routing, marketing, E-911 services, etc. And this research will show us how accurate Dead Reckoning can be on solving the position location problem under ad hoc networks.. 1.5. Thesis Organization. This document is organized in five chapters and several appendices. Chapter one contains the introduction, problem description, justification, contribution and thesis organization. Chapter two has basic concepts of Ad Hoc wireless networks, position location techniques and their impairments, smart antennas and heading instruments. Chapter three defines our model based on a dead reckoning scheme and shows the scope,cases and limitations for this technique. Chapter four containe the simulation for different environmental conditions, considering Exponential distributed distance error with Gaussian or Uniform distributed angles. Finally, chapter five presents the conclusions and ongoing research..

(23) Chapter 2 Basic Concepts This ehaptcr describes the basic concepts that will help us to know how the position location systems work, which variables are involved in the position estimation, what smart antenna means and what kind of devices we will need to estimate a North—South, East—West reference.. 2.1. Ad Hoc Networks. Ad hoc networks are dynamically formed, wireless multihop networks, where mobile nodes communicate via multihop wireless infrastructures [26]. They represent a new model shift in the way networks are designed and operated. These nodes discover several routes to each other by exchanging topology information [102]in the form of control messages. The network is a collection of nodes that do not need to rely on a predefined infrastructure to keep the network connected. Most or all nodes participate in network functions, such as routing [9, 80, 94]and network management, depending on their own capacity. The current trend in military ad hoc networking [26] is to deploy technologies based on open standards that are widely used in the civilian environment, for example, IP technology [11, 33, 59, 64, 77]. The deployment of such networks will be of great importance in applications such as disaster relief and battlefield communication [61], where network connectivity needs to be provided without the aid of any pre-existing communication infrastructure. Ad hoc networks arc autonomously formed with a large number of heterogeneous nodes (ranging in complexity from sensors to palmtops and fully functional laptops and routers) without the aid of any pre-existing communication infrastructure. Therefore, tasks are distributed over and carried out by groups of collaborating nodes. In addition, since they usually run on battery, they will have to be power-conscious to conserve energy [15, 76, 93, 95]; hence the functions offered by a node will depend on its available power level and capability..

(24) CHAPTER 2. BASIC CONCEPTS. Furthermore, their topology can be highly dynamic due to autonomous mobility of nodes and physical characteristics of wireless links [88]. Finally, energy limitation, constrained wireless communication bandwidth and quality, and node mobility result in networks becoming partitioned more frequently. These make managing ad hoc networks significantly more challenging than managing stationary and wireline networks [78].. 2.2. Position Location Systems. Radio frequency position location systems can be classified into two broad categories: direction finding and range based systems. [74]. These direction finding and range based position location systems and techniques may be used individually or in combination, in a number of different configurations, to produce a position location solution . Direction finding systems estimate the position location of a mobile source by measuring the Direction Of Arrival (DOA), or Angle Of Arrival (AOA).. 2.2.1. Angle Of Arrival (AOA). AOA location methods [49], estimate the mobile location by first measuring the arrival angles of a signal from a mobile at several base stations through the use of directive antennas or antenna arrays [31, 34, 35, 71, 74, 102]. Simple geometric relationships are then used to determine the location. The approach for the AOA location method is illustrated in Figure 3.6 [23]. As the Figure 2.1 indicates, the AOA method can provide a location estimate with only two base stations since straight lines (nodes in our case), defining the lines of position, are used for positioning. The performance of an AOA location system is limited by the accuracy to which AOA estimates can be made. This is a characteristic of the hardware and estimation algorithm used. When such a location system is employed in a wireless network, performance becomes highly dependent upon the propagation environment since scattering [4] near and around the mobile and base station will affect the measured AOA. The scatterers cause multiple signals to appear at the base station from the mobile (and vice versa) which introduces error into the AOA estimates. In the absence of a Line Of Sight (LOS) signal component, the antenna array will lock on to a reflected signal that may not be coming from the direction of the mobile [4]. Even if a LOS component is present, multipath will still interfere with the angle measurement. The.

(25) 2.2. POSITION LOCATION. SYSTEMS. accuracy of the AOA method diminishes with increasing distance between the mobile and base station due to the scattering environment and fundamental limitations of the devices used to measure the arrival angles.. 'Lines\of position X,. Figure 2.1: The Angle Of Arrival (AOA).. 2.2.2. Direction Of Arrival (DOA). The DOA measurement restricts the location of the source along a line in the estimated DOA [29]. When multiple DOA measurements are made simultaneously by multiple mobile stations, a triangulation method may be used to form a location estimate of the source at the intersection of these Lines Of Bearing (LOB) [74]. In theory, direction finding systems require only two receiving sensors to locate a mobile user, but in practice, finite angular resolution, multipath, and noise often dictate the need for more than two sensors. Range based position location systems may be categorized as ranging, range sum, or range difference systems. The type of measurement used in each of these systems identifies geometry for the position location solution [51]. Several base station receivers are used to simultaneously record propagation delay, or range, measurements between a mobile transmitter and the fixed receivers.. 2.2.3. Time Of Arrival (TOA). Ranging position location systems locate a mobile source by measuring the absolute or differential distance between a source and a receiver [4, 5, 22]. The distance, d, traveled.

(26) CHAPTER 2. BASIC CONCEPTS. by a propagating wave is expressed by d — cr, in which r is the propagation delay and c is the speed of light, 3 x 108 m/s. Thus, range estimates arc made by measuring the Time Of Arrival (TOA) of the signal propagating between the mobile user and each base station receiver. The TOA estimate defines a sphere of constant, range around the receiver. The intersection of multiple spheres produced by multiple range measurements from multiple base station receivers provides the position location estimate of the mobile user [74].. X,. Figure 2.2: The Time Of Arrival (TOA).. Ranging systems are also known as TOA or spherical position location systems. Realistic ranging systems are unable to measure the exact range between the user and a base station directly, and as a result, range measurements contain a bias term. This bias term can be calculated using an additional range measurement by an additional base station. Ranging systems of this type are often called pseudo range systems.. 2.2.4 Time Sum Of Arrival (TSOA) Range sum position location systems measure the relative sum of ranges between the source and the fixed receivers. These systems measure the Time Sum Of Arrival (TSOA) of the propagating signal from a mobile user to two base station receivers to produce a range sum measurement. The range sum estimate defines an ellipsoid with foci at two receivers, and when multiple range sum measurements are obtained, the position location estimate of the user occurs at the intersection of the ellipsoids. Consequently, range sum position location systems are also known as TSOA or elliptical position location systems..

(27) 2.2. POSITION LOCATION SYSTEMS. 2.2.5. Time Difference Of Arrival (TDOA). Range difference position location systems measure the relative difference in ranges between a source and receiver. These systems measure the Time Difference Of Arrival (TDOA) of the propagating signal from a mobile to two base station receivers to produce a range difference measurement. The range difference measurement defines a hyperboloid of constant range difference with the base stations at the foci. When multiple range difference measurements are obtained, producing multiple hyperboloids, the position location estimate of the user occurs at the intersection of the hyperboloids. Consequently, range difference position location systems are also known as TDOA or hyperbolic position location systems [100].. 2.2.6. Multilateration or Trilateration Systems. Range based systems can also be classified as either multilateration or trilateration systems [74]. Multilateration position location systems are those systems which utilize measurements from four or more base station receivers to estimate the three dimensional location of the mobile user. In a multilateration hyperbolic system, four or more base stations produce three or more range difference measurements.. Figure 2.3: Direction finding Position Location Solution..

(28) 10. CHAPTER 2. BASIC CONCEPTS. Trilateration position location systems are those which utilize measurements from three base station receivers to estimate the two dimensional location of the mobile transmitter. In a trilateration ranging position location system, three range measurements are produced from the three base stations, whereas in a trilateration hyperbolic position location system, two range difference measurements are produced from three base stations. The additional measurement by ranging systems is required to reduce the ambiguities due to multipath, signal degradation, and noise [37, 30. 48, 23]. Position location systems which are able to measure the change in frequency of a transmitted signal are called Dopplcr systems. When either the source or receiver is moving, the received signal is subjected to a Doppler shift. This change in frequency is proportional to the relative direction and velocity of movement. By measuring the change in frequency, the rate of change between a mobile and base station can be determined. If the trajectory of a moving base station, such as a satellite based PCS base station, is known, then the user's location can be uniquely determined from the Doppler frequency changes. Doppler systems generally require that the velocity of either the receiver station or user be high enough to generate an easily resolvable Doppler shift [74].. 2.3. Delay Spread. In digital wireless communication systems, transmitted information reaches the receiver after passing through a radio channel, which can be represented as an unknown, time varying filter. Transmitted signals are typically reflected and scattered, arriving at the receiver through multiple paths. When the relative path delays arc on the order of a symbol period or more, images of different symbols arrive at the same time, causing intersymbol interference (ISI) [37, 30, 48, 62]. The information about the frequency selectivity of the channel and the corresponding time domain root mean squared (rms) delay spread can be very useful for improving the performance of the wireless radio receivers through transmitter and receiver adaptation[30]. In channel estimation using channel interpolators, instead of fixing the interpolation parameters for the worst expected channel dispersion as commonly done in practice, the parameters can be changed adaptivcly depending on the dispersion information. Instantaneous rms delay spread provides information about local (small-scale) channel dispersion, is obtained by estimating the channel impulse response (CIR) in time domain. The real and imaginary parts of the taps in CIR, can be written as functions of a Exponential distributed independent random variables..

(29) 2.4. SMART. 2.4. 11. ANTENNAS. Smart Antennas. Smart antennas offer a broad range of ways to improve wireless system performance. In general, smart antennas, as shown in Figure 2.4. have the potential to offer enhanced range and reduced infrastructure costs in early deployments, enhanced link performance as the system is built out, and increased long term system capacity. In general smart antenna can be applied to a broad range of wireless technologies, including FDMA, TDMA, and CDMA systems [31, 34, 35, 71, 74, 82, 85, 102].. b Figure 2.4: Smart antenna systems with a different beam for each subscriber.. Smart antennas provide enhanced coverage through range extension, hole filling, and better building penetration. Given the same transmitter power output at the base station and subscriber unit, smart antennas can increase range by increasing the gain of the base station antenna. The uplink power received from a mobile unit at a base station is expressed by Pr = Pt + Gs + Gb-PL,. (2.1). in which Pr is the power received at the base station, Pt is the power transmitted by the subscriber, Gs s is the gain of the subscriber unit antenna, and Gb is the gain of the base station antenna. On the uplink, if a certain received power, Pr.min, rnin ' is required at the base station, by increasing the gain of the base station. Gb the link can tolerate greater path loss, PL .We write GdBi = lQlog(-£-) = lOlog(G),. (2.2).

(30) CHAPTER 2. BASIC CONCEPTS. 12. in which GdBi is the gain of an antenna in dB, relative to an isotropic antenna, if the antenna is lossless, then the gain of an isotropic antenna is G,;so = 1, so we write PL(d} = PL(do). Wnlog(— ) do. Xa. (2.3). Therefore, by increasing the tolerable path loss, we can increase the reception range, d of the base station. Since smart antennas can allow higher gain compared to conventional antennas smart antennas systems can provide range extension [74]. In order to improve the range on the downlink, we can use smart antennas at the subscriber receiver or at the base station transmitter. Since smart antennas are not usually feasible at mobile and portable subscriber terminals, we may consider downlink beamforming at the station to increase range in balanced systems. Smart antennas may play a role in subscriber equipment for fixed wireless applications. Through range extension, initial deployment costs to install a wireless system can be reduced. When initially deploying cellular wireless networks, systems are often designed to meet coverage requirements. Even with only a few customers in a system, a sufficient number of base stations must be deployed to provide coverage to critical areas. As more customers arc added to a cellular network, system capacity can be increased by decreasing the coverage range of base stations and adding additional cell sites.. Figure 2.5: The Direction of Arrival of a plane wave incident.. Smart antennas provide protection against system perturbations and reduced sensitivity to nonideal behavior. Smart antennas can be used to allow the subscriber and base.

(31) 13. 2.4. SMART ANTENNAS. station to operate at the same range as a conventional system, but at lower power. This may allow FDMA and TDMA systems to be rechannelized to reuse frequency channels more often than systems using conventional fixed antennas, since the carrier to interference ratio is much greater when smart antennas are used. In CDMA systems, if smart antennas are used to allow subscribers to transmit less power for each link, then the Multiple Access Interference is reduced, which increases the number of simultaneous subscribers that can be supported in each cell. Smart antennas can also be used to spatially separate signals, allowing different subscribers to share the same spectral resources, provided that they are spatially separable at the base station. This Space Division Multiple Access (SDMA) allows multiple users to operate in the same cell, on the same frequency/time slot provided, using the smart antenna to separate the signals. Smart antennas use an array of low gain antenna elements which are connected by a combining network. An arbitrary array of elements is shown in Figure 2.6. Here, <p is the azimuthal angle and 6 is the elevation angle of a plane wave incident on the array. The horizon is represented by 9 = ir/2. In general, the array may consist of a number of antenna elements distributed in any desired pattern; however, the array is frequently implemented as a linear equally spaced (LES), uniform circular, or uniformly spaced planar array of similar, co-polarized, low gain elements which are oriented in the same direction.. zi. Figure 2.6: Linear equally spaced array oriented along the X axis..

(32) 14. 2.5. CHAPTER 2. BASIC CONCEPTS. Gyroscope and Digital Compasses. There are a substantial number of gyro applications for which a drift rate of around one degree/second is not unacceptable, but low cost, low weight and volume, high maximum rate, high reliability and good dormancy are of paramount importance. Existing mechanical instruments of this class usually fail to meet all these requirements, with maximum rate being a particular problem. Optical gyroscopes are likely to be to large and too expensive for such applications [16, 43, 70, 97, 99]. Magnetic heading systems, also called magnetic compasses, composed of fluxgate magnetometers are widely used to measure magnetic heading directions (azimuth). They are used for vehicle navigation, guidance and control, and are especially useful in integrated navigation systems. Compared with other types of heading instruments such as gyroscopes, radio compasses, etc. [38, 45, 52, 66, 90. 16, 43, 70, 97, 99], the magnetic heading systems have the merits of simpler construction, smaller size, lighter weight, quicker reaction, higher reliability, freedom of drift and low cost. The conventional magnetic heading systems only have medium accuracy of measurement, say, 1 to 1.5 degrees. The aim is to apply these heading systems to a high precision measurement requirement of the earth's magnetic field in an environment where the compass deviation error of the vehicle is very large, e.g. 20 to 30 degrees..

(33) 15. 2.6. ERROR BEHAVIOR. 2.6. Error Behavior. Error is basically a position difference between the actual position of the node and the position estimated. This depends on the environmental conditions and the accuracy of the equipment. But which errors exist in the measurement? And how do the errors affect the measurement? This mostly depends on what kind of technology is used, but basically they are angle, distance and time error and a combination of them.. I Actual position of the node ) Estimated position of the node. Figure 2.7: Difference between actual position and estimated position.. The difference between the actual position and the estimated position as we can see in Figure 2.7 define the error behavior that we will analyze in chapter 3, to obtain the horizontal (fx(x)), vertical (fY(y)), the result error r and its angle 8..

(34) 16. CHAPTER 2. BASIC CONCEPTS.

(35) Chapter 3 Dead Reckoning Navigation using inertial sensing has been important throughout history. One of the earliest techniques, used in particular by sailors, was dead (or 'deduced') reckoning [73]. This technique involved combining compass heading with knowledge of the sea currents and the speed of the vessel measured by the time taken by an object thrown overboard to travel a fixed length along the side of the ship. This was the technique used by Columbus in his voyages to discover the New World. While full inertial sensing provides heading, velocity and acceleration in three dimensions, dead reckoning uses measures of heading and velocity to provide a two dimensional positioning solution based on a known started point. It is well suited for already instrumented vehicles such as aircraft, ships and automobiles. On a smaller scale, dead reckoning has also been used for robot navigation and, of special interest to wearable computer users; it has been used to aid navigation for walking robots [3, 79]. The use of motion sensors for virtual and augmented reality head trackers has become commonplace [16], demonstrating that these sensors can be used satisfactorily for position prediction in small areas with a high degree of accuracy.. 3.1. Dead Reckoning and the Position Location Problem. If we want to know the position of a node in an Ad hoc wireless network, we need a trusted position node (fixed position node, node with GPS equipment, or low mobility node), and then to calculate the distance between the node that we want to locate and the position trusted node, in our case using dead reckoning. We also consider the error generated by the environment (time of arrival (TOA), angle of arrival (AOA), delay spread, etc), and we add this estimated error to our obtained position.. 17.

(36) CHAPTER 3. DEAD RECKONING. 18. 3.2. The Error Function (r, 0}. We want to characterize the error, so let us first separate the error distance (r) into its ( x , y ) components.. I Actual position of the node I Estimated position of the node. r sin9. r cos 6. Figure 3.1: Difference between actual position and estimated position, as (r,6) function.. These components ( x , y) arc a function of the angle distribution involved (9} in every error caused by the estimation of the actual node position. The straight line between the first node and the last one sets an angle reference. In Figure 3.2 we can sec that every error has and angle.. Estimated position of the node. Q Actual position of the node. Figure 3.2: Error as a function of distances and angles..

(37) 19. 3.2. THE ERROR FUNCTION (R, 0). The error angle distribution function depends on the delay spread, and the analysis of this error is pretty similar to that of the scattering (if we see it as vectors). In some cases the scatter analysis is based on a hyperbolic or elliptical area, in our case the area is defined by the angle, and the A parameter of the exponential error (we do not consider many errors caused by the multiple nodes involved). For our case we consider a Uniform and a Gaussian distributed angle (If the variance in the Gaussian function is short enough the angle error distribution function has a Laplacian performance). In Figure 3.3 we can see how the angle is distributed. The horizontal line is the reference, the line between the fist node (trusted position node) and the one that we want to locate.. Gaussian r. /. Uniform. Figure 3.3: Angle distribution function, Gaussian or Uniform..

(38) CHAPTER 3. DEAD RECKONING. 20. 3.3. Error with Uniform Angle Distribution. If we consider a Uniform angle distribution where — TT < 9 < TT, the horizontal and vertical components are 0 < x < r and —r<y<r where r will be the distance or amplitude of the exponential error, therefore this will lead us to the following equations (see appendix D and E), and their graphical representations: —r < x < r, /y(y. =. < y < r.. (3.1) (3.1. Sinusoidal distribution with uniform angle distribution. 0. 0.1. 0.2. 0.3. 04. 0.5. 0.6. 0.7. 0.8. 0.9. 1. Figure 3.4: Horizontal fx(%) component of the sinusoidal distribution (Uniform angle)..

(39) 21. 3.4. LINEAR BEHAVIOR IN THE SINE CASE (FY(Y)}. Sinuso'dal distribirtion with uniform angte dstribution. -04. -0.2. 0 X. 0.2. 0<. 06. Figure 3.5: Vertical fy(y) component of the sinusoidal distribution (Uniform angle).. 3.4. Linear Behavior in the Sine Case. (fy(y)). In the sine case (fv(y}} we can see a linear behavior for angles close to 0 degrees (—45° < 9 < 45°) for any angle distribution function.. Figure 3.6: The sinusoidal behavior and its linearity.. The linearity makes the angle distribution function dominate over the sine on the /y(y), then for angles between —45° and 45°, the sinusoidal vertical function will have.

(40) 22. CHAPTERS.. DEAD RECKONING. the same performance as the angle distribution function. In other words, if we have a Gaussian angle distribution function the fy(y) will be also Gaussian, and if we have a Uniform distribution function the fy(y) results in a Uniform function (within the specified range only). But this happens only on the vertical component, for the cosine (fx(x)) has a different performance depending on the angle distribution function.. 3.5. Error with Gaussian Angle Distribution. We can see for the Gaussian angle distribution function complex equations (see appendix D). These solutions vary in function of n and a in the Gaussian function, but for our case we can consider that the Gaussian angle is centered at 0°; therefore // will be 0 leaving the Gaussian distribution depending only on a. The variance delimit the vector's decomposition. Therefore angles closed to 0° will result in errors mostly dominated for the (fx(x)). We try to represent (fx(x)) as a Cauchy distribution function for a positive or a negative part depending on the a value. Considering that the Cauchy approximation will be valid only for special and specific angles, and this leaves the fy(y] representation out, the research on the Cauchy (or Levy) approximations was abandoned. Let us consider that this approximation is valid only for one node, one error, but the sum of Cauchy distribution functions is also another Cauchy (see Appendix C). -(cos-'(f )) 2 /2<r 2 + e -(2^-cos- 1 (S))V2<T 2. fx(x] = -==cry 27r|r sm(cos-1 (^)J | J. \. '. fc\. I. g - f s i n - a r T. •/. 1 /T \ \ I. -. ; -r < x < r -. ^. '. _( 2 7r-sin. fy(v} = 7^-, • -i ,,<Tv27r|rcos(sm (*))| m. ;. o<y<r. (3.4). (3.3).

(41) 3.5. ERROR WITH GAUSSIAN ANGLE. 23. DISTRIBUTION. 0.5. 0.45. 0.4. 0.35. 0.3. 0.25. 02. 0.15 0.1 0.05. -1. -0.8. -06. -04. -02. 0. 0.2. 0.4. 0.6. 08. 1. Figure 3.7: Horizontal f x ( x ] component of the sinusoidal distribution (Gaussian angle).. -1. -0,8. -0.6. -0.4. -02. 0. 0.2. 0.4. 0.6. 0.8. 1. Figure 3.8: Vertical fy(y] component of the sinusoidal distribution (Gaussian angle).. Most of the times the error will depend on the nodes (or jumps) involved in the estimation, so the error in the worst case will be cumulative (the sum of the errors in every node). Therefore, the error becomes a function depending on the distance (basically on the A parameter of the exponential error), the angle (9) and the number of the nodes involved (AT)..

(42) CHAPTER 3. DEAD RECKONING. 24. Final error. Figure 3.9: The final error is the sum of all the errors (vectors). If the angle is 0°, this leaves us with a sum of exponential random variables having the same parameter, and therefore, the resulting final error will be an m-Erlang distribution function (see appendix B). Same exponential parameter, 0 degrees angle X. 0=0°. Final error m-Erlang. Figure 3.10: The sum of exponential random variables will result in a m-Erlang.. When the sum of exponential random variables involves different parameters, the final error will not be an m-Erlang distribution function; it will result in the equation 3.5 (see appendix A). (3-5). 1=1. j =. But for different angles the sum of Gaussian or Uniformly distributed errors is not easy to analyze mathematically. This is because we do not have the characteristic function for f x ( z ) and fy(y) the equation (3.3) and (3.4) or (3.1) and (3.2) for the Uniform, and the convolution leads us to a cosine transform that does not have a closed form expression. So we have to do it using a simulation that we will see in the next chapter..

(43) 3.5. ERROR WITH GAUSSIAN ANGLE DISTRIBUTION. 25. Convolution for fx(x). (3.6). Convolution for fy(y). ^ '.

(44) 26. CHAPTER 3. DEAD RECKONING.

(45) Chapter 4 Numerical Results In the simulation we recreate the final error (fE(0.\.N}) as a function of the angle 6, the mean error distance A (depending on the environment conditions) and the number of nodes involved N. This simulation has two density functions for the angle (Gaussian and Uniform) and an Exponential distribution for the error distance. Because of the different environmental conditions, the angles have mean values between 0° and ±60°, and distance mean values between 5 to 250 mts. For every round in the simulation we create one distance matrix having an exponential distribution with mean value A (5,10,15,... 25Qmts), and for these errors another matrix with 0 (0°, ±5°, ±10°, • • • ± 60°) Gaussian or Uniformly distributed. The exponential error is divided into its cartesian (x, y] components having the final result z = \fx2 + y2 which generates a Data Base of values. We compared the behavior of these values and obtain parameters to approach the result with a specific random variable, either Gaussian, Weibull or Gamma. With the Data Base, we build a normalized histogram, having values for every environmental condition (for example 9 = 5°. and A = Wmts). For representation we will use the following format e.g. MD[50] 6 ~ U[a, b] means 50 mts mean distance value with Uniform distributed angle over a° to b°. and MD[150] 0 ~ G[a] means 150 mts mean distance value with Gaussian distributed angle with a° mean value, angles arc in degrees and distances in meters.. 4.1. The Density and Distribution Functions. Once we have the Data Base of values, a histogram is made. This histogram is normalized and a distribution function is obtained. This /£((9,A, JV)) has the same behavior, and is only different for the first node that will always be an exponential distribution function.. 27.

(46) CHAPTER 4. NUMERICAL. 28. RESULTS. The performance is pretty similar for all distances (A) and angles (9), changing only the scale, but the error is mostly a function of the nodes because it increases with the number of nodes involved. The CDF tell us how far the mobile stays in range for accurate position estimation according to the FCC requirement for position based services (that does not need to fill all those requirements).. Histograms-U[-10,10]. Empncal Disintxitwi Function fl - U [-10,10]. '. /'. !. —. j. 2nodes 3nodes 4 nodes Snodes 6 nodes 7 nodes — anodes 9 nodes 10 nodes 11 nodes. 1. J ' / '' ' ' ' ' (, '. '/ ' ('. 11 nodes 12 nodes — I3nodes — 14 nodes ISnodes. 16 rates. ,. , (. 12 nodes — iSnorJes. ''. — Unodes 15 nodes 16 nodes 17 nodes 18 nodes 19 nodes 20 nodes j 21 nodes. '. 17 nodes 16 nodes. ; '. 19 nodes 20 nodes. '. 21 nodes. Figure 4.1: Histogram and Empirical Distribution Function MD[5]. Histogram*-G [35]. mode. — — — —. I , '. 0 ~ £/[—10,10].. Empirical OsiribulkKi Function fl - G (35]. Figure 4.2: Histogram and Empirical Distribution Function MD[10Q]. 6 ~ G[35]..

(47) 4.1. THE DENSITY AND DISTRIB UTION. Histogram 9 - U [-60,60]. FUNCTIONS. 29. Empirical Dishibulion Functor 0 - U [-€. 11 nodes — t2nodes 13 nodes U nodes — 15 nodes 16 nodes !7nodet 18 nodes 19 nodes 20 nodes 21 nodes. Figure 4.3: Histogram and Empirical Distribution Function M£>[250]. 9 ~ £/[-60,60].. Empncal DstnbuSon Function 6 - G (60]. I node 2 nodes Snodes 4nodes Snodes 6 nodes 7nottes Bnodes • • • • Snodes 10 nodes. — — — ---—. II nodes — 12 nodes — I3nodes 14 nodes 15 nodes 16 nodes !7nodes 18 nodes 19 nodes 20nodes 21 nodes. Figure 4.4: Histogram and Empirical Distribution Function MD[250]. — inode — 2nodes — 3nodes — 4nodes Snodes ••--- 6 nodes — 7nodes 6nod« 9 nodes lOnodat 11 nodes — I2nodei •— I3nodes •— 14 nodes 15 nodes <6nodes • ••• 17 nodes 18 nodes 19 nodes 20 nodes 21 nodes. 6 ~ G[60]..

(48) CHAPTER 4. NUMERICAL RESULTS. 30. 4.2. Gaussian, Weibull and Gamma Approximations. We need to identify if these histograms resemble a known probability density function, so we compare it with some probability functions that seems to match the histogram; Gaussian, Weibull and Gamma distribution functions, using parameters obtained (via software, using the mean value and variance) from the Data Base.. MO5 8~U[-10.10] forl nodes. MDS B-U|-!0.10] forionodes. Figure 4.5: Simulation vs approximation MD[5]. MDS e-UI-10,10] lor5nodei. MDS e-U|-10.lO] Iw 21 nodes. 6 ~ [/[—10,10]..

(49) 4.2. GAUSSIAN, WEIBULL AND GAMMA. 31. APPROXIMATIONS. The graphics show that the Weibull is a good approximation and that for several nodes the Gaussian approximation is close to the simulation, but the Gamma approximation is considerably better than the others for all cases.. M0100. fl-G|20]. lorSnodes. MD 100. fi-G[20]. tor7notes. 1500. 2000. 2500. 3000. MD100 «-G[20] Ior20no<!es. 500. 1000. 1500. 2000. 2500. 3000. 3500. 0. 500. 1000. 1500. Figure 4.6: Simulation vs approximation MD[100]. 2000. 2500. 3000. 3500. 9 ~ (7[20].. 4000. 4500.

(50) CHAPTER 4. NUMERICAL RESULTS. 32. 4.2.1. Probability to Probability plot (PPplot). If we want to know how accurate the approximation is, there is a technique called probability to probability plot (ppplot), that in our case displays the approximation against the simulation. If the line is closer to the reference it means that the approximation is more accurate. PPPlot, tan robe, MD [5] 0-U|-10,10]. PPPtot, tor 3 nodes, MO [5] B-U|-10,1G1. — Reference — Smulation vs Gamma Approximation Simulation vs Gaussian Approximation Simulation vs Weibull Approximation. 0.1. 02. 0.3. 04. 0.5. 0.6. 0.7. Simulation. PPPtot, tor 12 nodes, MD|5]. fl-U[-10,1Q]. PPPtot.faM9 nodes. MO |5) 6~U|-10,10]. - Reference. — Reference — Simulation vs Gamma Approximation Simulation vs Gaussian Approximation Simulation vs Weibull Approximation. - Simulation vs Gamma Approximatwr Simulation vs Gaussian Approximation Simulation vs Weibull Approximation. I 0.5. 0.1. 0.2. 0.3. 0.4. 0.5 Simulation. 0.6. 0.7. 0.8. 0.1. 0.2. 0.3. 0.4. 0.5. Figure 4.7: Ppplot, MD[5] 0 ~ f/[-10,10].. 0.6. 0.7. 0.8. 0.9. 1.

(51) 4.2. GAUSSIAN, WEIBULL AND GAMMA. 33. APPROXIMATIONS. This ppplot illustrates that all three approximations are close to the reference (the straight line at 45°), when we see the performance of the approximation across the nodes (more than 3 nodes involved in the estimation).. PPPtot, for 6 nodes. MD 50 e-G|30]. PPPkt lor 2 nodes, MD 50 1-G|30]. — Reference Simulation vs Gamma Approximation Simulation vs Gaussian Approximation Simulation vs Weibull Approximation. 1,5. 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. 0.7. 0.1. 0.2. 0.3. 0.4. Simulation. PPPIot, lor H nodes, MD 50 9-0(301. 0.3. 0.4. 0.7. — Reference — Simulation vs Gamma Approximation Simulation vs Gaussian Approximation Simulation vs Weibuj[Approximation. - Simulation vs Gamma Approximation Simulation vs Gaussian Approximation Simulation vs Weibuil Approximation. 0.2. 0.6. PPPtot.for21 nodes, MD 50 B~G|30]. - Reference. 0.1. 0.5 Simulation. 0.5. 0.6. 0.7. 0.8. 0.9. 0.1. Figure 4.8: Ppplot, MD[50]. 0.2. 0.3. 0.4. 0.5 Simulation. 8 ~ G[30].. 0.6. 0.7.

(52) CHAPTER 4. NUMERICAL. 34. RESULTS. It is clear how all approximations approach the reference. However the Gaussian approximation will not create good approximation for the one node error (because the error is entirely exponential), the Weibull seems to be better than the Gaussian approximation for low nodes implicated, but it is never better than Gamma. PPPIol, for 3 nodes. MD 200 6 - G |45]. PPPkrt,for3nodes.MD|200] e-Ul-45,45]. — Reference — Simulation vs Gamma Approximation Simulation vs Gaussian Approximation Simulation vs Weibull Approximation. 07. I" 1 0.5. <. 0.4. 0. 0.1. 02. 0.3. 0.4. 0.5. 0.6. 07. 08. 0.9. 0. 0.1. 02. Simulation. Figure 4.9: Ppplot, ML>[200]. 0~C7[-45,45]. 0.3. 04. 0.5 Simulation. and. 0.6. 0.7. 0.8. 6 ~ G[45].. This behavior does not matter if the angle distribution function is Gaussian or Uniform; the Gamma approximation is the best for both cases. The ppplot illustrates that the Gamma approximation is better than the others; therefore, we will use the Gamma approximation to represent the simulation..

(53) 4.3. THE GAMMA APPROXIMATION. 4.3. PARAMETERS. 35. The Gamma Approximation Parameters. Now that we have the Gamma approximation as the more accurate, we have to evaluate its parameters. If we keep in mind the Gamma random variable (see Appendix C) it is a function of two variables (a, A). We construct this simulation using three parameters /£•($. A, A/"), where 0 is the angle of the error (Gaussian or Uniform distributed), A the mean value of the exponential distributed distance and N the number of nodes involved in the estimation..

(54) CHAPTER 4. NUMERICAL. 36. 4.3.1. RESULTS. Fixed Exponential Distance. If we fix the exponential distributed mean value (A), we can graphically show the performance of the Gamma parameter a as function of the nodes. For the Uniform distributed angle the behavior is lineal, and seems to follow the nodes, this means that the Gamma parameter a can represent the number of nodes (TV) for a Uniform distributed angle.. M0|5| 9-U[-30,30|. MD|50] !-. MD(100]. MD[200] 9~U[-30,30l. fl-U[-30.30]. 5. 10. 15. Nodes. Figure 4.10: Gamma parameter a when MD is fixed, 0 ~ U[—60,60]..

(55) 4.3. THE GAMMA APPROXIMATION. PARAMETERS. 37. For the Gamma parameter A the graph does not seem too linear, but it is because of the scale, and its behavior is similar to than of the mean angle distribution function, so the Gamma parameter A can represent the mean angle distribution function for the Uniform distribution angle.. MD|5) e-UI-30,30]. MD|50| t - U [-30,30]. M0|100| 6-U|-30,30]. MO [200] 8 - U 1-30,30]. 101. 100.5 100. 99.5. 97.5. 97. 96.5. 96. 5. 10. 15. Nodes. Figure 4.11: Gamma parameter A when MD is fixed, 9 ~ [/[—GO, GO]..

(56) CHAPTER 4. NUMERICAL. 38. RESULTS. In the Gaussian ease the Gamma parameter a varies in a similar way for angles below 35°, but for higher mean values of the angles (9), the behavior decreases gradually. This characteristic is a function of the mean value of the angle (9) over the nodes (N). MD[10| 8-G[60|. MD[7S] e-G|60]. MD[150] 8-G|60]. MD[250] e-G|60]. Figure 4.12: Gamma parameter a when MD is fixed, 9 ~ G[60]..

(57) 4.3. THE GAMMA APPROXIMATION. 39. PARAMETERS. The Gamma parameter A displaysn similar behavior. For angles below 35° the behavior is like the mean angle distribution function for Uniform distribution angle, and is also a function of the angle ($)more than (N).. MD[10] 8-G|60]. MD|75] 9-GI60) — e.ooo — e.005. 13. — 9.010 9 = 015 — .020 . 025 • 030 .035 . 040 — .045 . 050. 125. -' 12 ' ^ § E 115. 9-055 9 = 060. a " £ 3. 90 & a E ffl £. 85 x-'. 3. ~~-~". 80. 10.5. 10. 9.5 -. 95. E E. „ ". ^ l , - —-. — 9.000 — 9.005 — 9.010 9 = 015 — 9-020 9.025 9-030 9-035 9-040 — 9-045 9-050 9.055 9.060. ".. •'///• "^" i^fc^-js^= F • - -----^^i5-^ 5 '^^. =OT. MD[250]. M0|150] 9-G|60]. fl-G[60] - 9.000 - 9-005 -9-010 9-015 9-020 9.025 9.030 9-035 -9.040 9-045 -8.060 8.055 8 . 060. -000 .005 .010 .015. -020 = 025 9.030 9.035 9 = 040 9.045 9 = 050 9.055 8.060. 5. 10. 15. Nodes. ..._. 10. 15. Nodes. Figure 4.13: Gamma parameter A when MD is fixed, 6 ~ G[60]..

(58) CHAPTER 4. NUMERICAL. 40. 4.3.2. RESULTS. Fixed Gaussian or Uniform Angle. If we fix the mean distributed angle at 9 value, we can graphically show the performance of the Gamma parameter Q as a function of the nodes. For the Uniform distributed angle the behavior is lineal, and seems to confirm that the Gamma parameter a can be represented by the Uniform distributed angle (0). MO [250] 9-UI-5, 51. MD|250] 0 - U [-2.5, 2.5). I - 005 rats. X.005 mts. X.OIOmts. X-OIOmts. X-015mts. X-Ot5mts. I.020 rats. X * 020 mts. X.025 mis. X . 025 mts. X . 050 rats. X-050mts. X . 075 mts. X - 075 mts. X= 100 mts. X = 100 mts. X . 1 2 5 mts. X - 1 2 5 mts. X= 150 mts. X . 150 mts. X . 175 mts. X.175 mts. X.200 mts. X-200mts. X = 225 mts. X-225mts. X - 250 mts. X.250 mts. 10. 15. Nodes. MD[250] 6-U[-17.5.17.5]. MD|250] B-U[-275,27.5]. X . 005 mts. X . 005 mts. X-OIOmts. X-OtOmts. X.OlSmts. X-015mt8. X.020 mts. X-020 mts. X-025mts. X . 025 mts X . 050 mts. X-OSOmts. X . 075 mts. X-075 mts. X-IOOmts. X.100 mts. X - 1 2 5 mts. X-125mts. 150 mts. X- 150 mts. X - 175 mts. X = 175 mts. 200mts. X = 200 mts X - 2 2 5 mts. X . 225 mts X = 250 mts. X.250 mis. 10. 15. Nodes. Figure 4.14: Gamma parameter Q when 9 ~ U is fixed..

(59) 4.3.. THE GAMMA APPROXIMATION. 41. PARAMETERS. The Gamma parameter A can represent the exponential distributed mean value (A) when the angle is Uniform distributed. MO |2501 8 - U 1-2.5, 2.51. MO 12501 9-UI-5. 5]. 250. 250. 200. 200. 150. I 150 — 1 = 005 Ms — 1.010MS. t. — l = 005Ms — 1.010MS — 1-015MS. £. — 1-01 5 MS 1 . 020 Ms § 100 — 1.025 MS. I. too. 1-020 MS — 1-025MS 1-OSOMS. 1-050MS 1.075MS. 1-100 Ms 1.125 MS — 1.150 MS 1.1 75 MS 1.200 MS 1.225 MS 1.250 MS. 50. 5. 10. 15. 20. 50 -. -. -. 5. 10. 15. Nodes. Nodes. MO |2501 8 - U 1-175, 17.51. MO |250] e - U 1-27 5. 2751. - 1 = 005 mis. - 1-010 mis - 1 = 015MS. - 1-020 MS - 1.025MS. 1-050 MS 1.075 MS 1-100 Ms •• 1-125MS - 1.150MS. •• 1-175 MS 1-200 Ms 1.225 MS 1.250 MS 20. Figure 4.15: Gamma parameter A when 6 ~ U is fixed.. 1.075 Ms 1-100 Ms 1-125 Ms — 1-150 MS •— • 1.1 75 MS 1.200 Ms 1-225 MS 1-250 MS 20. - 1.005MS - 1-010MS -1.015MS 1.020mt8. - 1-025 mis 1-050 mis 1.075 mis 1-100 Ms 1-125 MS - 1-150MS 1.175MS. 1.200 MS 1-225 Ms 1.250 MS 20.

(60) CHAPTER 4. NUMERICAL RESULTS. 42. In the Gaussian case the Gamma parameter a cannot accurate by represent the nodes (N) for angles above 35°. We can see that lincality for lower angles does not have the same tendency for higher angles, and for 50° to 60° this is definitely not as precise as we expected (if we follow the Uniform distributed angle case).. MO[250] B - G H. MO[250| e - G [ 1 5 ]. MO [250] 8 - G H O ]. MO[250] 6-G[60]. X.020MS X - O S S mis X-050 mis X , 075 Ms X , 100 mis X . 125 mis X.150ms = 175 mis X.200 mis X,225 mis 2SOmts. Figure 4.16: Gamma parameter a when 9 ~ G is fixed.. X.005 nils X - 010 mis X-015 mis X,020 mis X-025 Ms k-050 ml! X « 075 nils X- 100 mis X . 125 MS X-150m1s X = 175 mis X.200 MS I-225 ml! X-250MS.

(61) 4.3. THE GAMMA APPROXIMATION. PARAMETERS. 43. The Gamma parameter A cannot accurately represent the exponential distributed mean value (A) for angles above 35°. We can sec how something similar happens in the Gamma parameter a case. We cannot have an accurate representation for mean Gaussian distributed angles above 35°.. MD[250| 8-615]. MD |250i e - G | 1 5 ). 250. 250. 200. 200. x & 150. ^ S 150. I ". — 1 = 005 Ms X. 01 5 mis — X.020MS X . 025 mis X = 050 mis X . 075 mis X=100mts — X . 125 mis — X,150m1s X . I 75 mis X. 200 mis. § 100. -. B " cd E. — X = 005Ms. — X. 015 Ms — X,020mts - X-025mts X. 050ms X . 075 mts X.lOOmts — X. 125 mis — X. 150 mts X. 175 Ms X-200MS X. 225 mts X = 250mts. S '00. 50. X = 250 mis 5. 10. 15. 20. 0 5. Nodes. 10. 15. 20. Nodes. MD|250| 8 - G. MD|250] 9-GI60]. — X = 005mts. ^. .. ,. X-015mts — X. 020 mts X. 025 mts X-050mts X. 075 mts X-IOOmts — X « 125 mts — X*150mts X = 175 mts X. 200 mis X = 225 mts X-250mts. I ISO. - X.005MS - X.010mts X-015MS - X = 020mts X.025MS X . 050 mts X-075mE X.lOOmts - X.125mts - X.150ml! •• X = 175 mts X.200 mts X,225mts X.250MS. 20. Figure 4.17: Gamma parameter A when 9 ~ G is fixed..

(62) CHAPTER 4. NUMERICAL. 44. 4.3.3. RESULTS. Gamma Approximation Parameters (surfaces). A 3D representation of the parameters can be more explicit sometimes. We show the graphics from different angles to show more precisely the behavior of the Gamma parameter Q and A. The result is a 3D representation of the above 2D cases.. 1node 2 nodes 3 nodes I nodes 5 nodes Snodes 7 nodes "nodes 9 nodes C3 to nodes II nodes CD 12 13 nodes QUnodes 15 nodes 16 nodes I7nodes 18 nodes 19 nodes 20 nodes 21 nodes. node 2 nodes 3 nodes 4 nodes 5 nodes (nodes 7 nodes ! nodes C3 Snodes l~l 10 nodes [~] 11 nodes [~l 12 nodes I 113 nodes CD 14 nodes 15 nodes 16 nodes 17 nodes IB nodes 19 nodes 20 nodes 250. Angte 11 degrees (!) Error in meters (1). 0. Angte in degrees («}. Error in meters (1). Figure 4.18: Gamma parameter a 3D surface representation 9. U.. trade 2 nodes 3 nodes trades 5 nodes Crudes 7 nodes 6 nodes 9nodes tOnodes 11 nodes 12nodes 13mdes 14 nodes ISnodes lenodes 17nodes ISnodes 19 roles 20 nodes 21 nodes. 300-^ ^£ g 200-. I. £. <g E 100E. 0. 00. Angle in degree Error in meters (X) Error In meters (X). Angle in degrees (6). Figure 4.19: Gamma parameter A 3D surface representation 0. U..

(63) 4.3.. THE GAMMA APPROXIMATION. PARAMETERS. 45. For Uniform distributed angles, the behavior is pretty similar for all cases. It is more linear for both parameters, and can accurately represent the mean Uniform distributed angle value (8) and the exponential distributed mean value (A).. 1 node 2 nodes 3 nodes 4 nodes 5 nodes 6 nodes 7 nodes 6 nodes CD 9 nodes F~1 10 nodes. Error in meters (X). Angle in degrees (8). Figure 4.20: Gamma parameter a 3D surface representation 6. G.. For Gaussian distributed angles, the behavior is pretty similar for some cases. Angles below 35° can accurately represent for those cases the mean Gaussian distributed angle value (0) and the exponential distributed mean value (A)..

(64) CHAPTER 4. NUMERICAL RESULTS. 46. For mean Gaussian distributed angles above 35° we cannot precisely represent those parameters as an exponential distributed mean value (A) and the number of the nodes (JV), so for angles above 35° the Gamma representation using exponential distributed mean value (A) and the number of nodes (AT) is no longer valid.. Error in meters (X). Angle in degrees (fl). Figure 4.21: Gamma parameter A 3D surface representation 6. G..

(65) 4.4. NUMERICAL. 4.4. 47. CONCLUSIONS. Numerical Conclusions. For Uniform distributed angles the error seems to be similar to the Gamma approximation, even for 60° angles. This is because this uniformity redeems the error. The Gamma Random Variable can be used. It is very accurate and will be the function that defines the error for Uniform distributed angles. In Figure 4.22 we can see an environment where the mean error distance value is 5 meters and the Uniform angle distribution is 5°. If we consider the FCC requirements, only 10 nodes can be involved in the position estimation. If there are more they will not be under these requirements, but maybe the position information can be used for other position based services.. — 1 node — 2 nodes — anodes — Anodes — 5n. 6 nodes 7 nodes — 8 nodes 9nodes 10 nodes 11n — 12 nodes 13 n 14 n — 15 n 16 nodes 17 nodes. Figure 4.22: MD[5]. 6 ~ f/[-2.5, 2.5].. The simulations show us that the Gamma approximation function can be used under certain angle conditions (for angles below 30°). Depending on the distribution of the nodes, the network nodes density, and which nodes arc used to estimate the position of the node, the angle distribution function will be Gaussian or Uniform; this defines the behavior of the Gamma approximation..

(66) CHAPTER 4. NUMERICAL RESULTS. 48. ,/ / '. — — — — —. '. ( /. / '/ / '. /"'. , '. '. trade 2nooes 3 nodes 4 nodes 5 nodes. — —. 7 nodes 8nodes 9 nodes 10 nodes 11 nodes 12 nodes 13 nodes — 14 nodes - 15 nodes 16 nodes 17 nodes 18 nodes 19 nodes 20 nodes 21 nodes. (. / ;,/ / / '; /' '/ '' / i 'i /i ' /'' '• '1 1/ / 1000 xmelers. Figure 4.23: M£>[50]. 1200. 1400. 1600. 1800. 2000. 6 ~ C7[-2.5, 2.5].. In the Gaussian distributed angle, we cannot use the Gamma approximation for angles above 30° unless we use the equations which describe the Gamma Parameters for angles above 35° (see Appendix F), this is because there is a degradation in its parameters, and it cannot be accurate enough for angles above 30°. Considering smart antennas in the network, the mean angle estimation should be under 30°, and the Gamma distribution function will be accurate enough.. — l n — — —. 2 nodes 3 nodes 4 nodes. —. 6 nodes. 7n —. —. 80. 100. 8nodes 9 nodes 0 nodes 1 nodes 2 nodes 3 nodes 4 nodes 5 nodes 6 nodes 7 nodes. 120. xmelers. Figure 4.24: MD[5] 0 ~ G[5]..

(67) 4.4. NUMERICAL. CONCLUSIONS. 49. In Figure 4.24 we can see an environment in which the mean error distance value is 5 meters and the Gaussian mean angle distribution of 5°. If we consider the FCC requirements, only 8 nodes can be involved in the position estimation. If there are more they will not be under these requirements, but maybe the position information can be used for other position-based services.. 1 ' 1 ' 1. — — — — — — —. '. 1 ' /. -/ , , ; 1. I I. ' 1 ' I. '. 9 nodes to nodes 11 nodes. ' ' '. :. — 12 nodes. /". / / /. / / '. / i. 0. 200. 400. mode 2 nodes 3 nodes 4 nodes 5 nodes erodes 7 nodes. .. i. 600. 800. i 1000. i 1200. i UOO. i. — 13 nodes — H nodes 15 nodes 16 nodes 17 nodes 18 nodes 19 nodes 20 nodes 21 nodes. 161. x meters. Figure 4.25: MD[50]. 0 ~ G[5]..

(68) 50. CHAPTERS NUMERICAL. RESULTS.

(69) Chapter 5 Conclusions and Further Research We can see from the simulation that the error is additive when more nodes are involved. This means that the position of the estimated node is far away from the trusted node, and for the environmental conditions can make the error big enough to make an incorrect position estimation.. 5.1. Conclusions. If we try to fill the FCC requirements, some environmental conditions will not allow us to completely fill these requirements and in some cases only a few nodes can be involved to keep the error distance according to these requirements. But for position-based services we do not need a highly accurate position estimation, so the range will increase and let more nodes become involved. In Figure 4.24 only 8 nodes will be involved for an accurate position estimation (according to the FCC), but 21 (maybe more nodes) will give position based information if the range is changed to 300 meters, therefore a wide range of nodes will receive this information, like weather news, marketing on demand, publicity according to it profile, etc. Considering A as the mean value of the error distance and N the number of the nodes, the Gamma random variable represents the position location error. Then the Gamma equation will be. 51.

(70) 52. CHAPTER 5. CONCLUSIONS AND FURTHER RESEARCH. Gamma Random Variable. Sx = (0, +00). fx(x]. =. r(N}. * > 0 and N > 0, A > 0. (5.1). />oo. where T(N) is the gamma function. T(N) = I. xN~le~xdx. N >0. Jo. (5 2). '. Considering smart antennas in the network, the mean angle estimation can be under 30°, and the Gamma distribution function will be accurate enough.. 5.2. Further Research. For further research more angles will be considered (even to 120°) with the same Uniform or Gaussian distributed angles, to see what happens with the Gamma parameter for higher angles in both cases.. • Research with more angles will be considered (even to 120°) with the same Uniform or Gaussian distributed angles, to see what happens with the Gamma parameter for higher angles in both cases. • It would be desirable to find a relationship between the angle degradation in the Gaussian angle distribution case and the sum of the exponential with a different parameter, which leads us to a similar Gamma but imitated according to the exponential parameters. • We can check if the method is used from different references or trusted nodes. The final position will be more accurate if we consider the mean value of these measurements. • We can perform an analisis of the method using random walk and/or Brownian motion..

(71) Appendix A Exponential Sum with Different Parameters This appendix shows the sum of exponential distribution functions with different A parameters, in which n means the number of exponential functions involved and An its parameter and En means the equation n of the array. We use partial fraction decomposition and convolution solutions.. fx(x) =. •*- x v "^ /. \. X— jw. Using partial fraction decomposition forn = 2 If we multiply the first characteristic function by the second and after that we obtain its inverse Fourier transform, we will obtain f2(x).. i- w. (A.I). z-. From equation (A.I) and using partial fraction decomposition we will need to find A\ and. (Ai - JW)(\2 - JW). \1~JW. \2~JW. Reducing the right side of equation (A.2). - AI\I — A2jw + A2\i - jw)(\2 - jw) 53. (A-3).

(72) 54. APPENDIX A. EXPONENTIAL. SUM WITH DIFFERENT. PARAMETERS. Comparing the left side of equation (A.2) and equation (A.3). we have:. (A.4) (A.5) Using matrix form:. -i -i A2. (A.6). A!. This is. -1 -1 \2. (A.7). AiA 2. AI. If we multiply the first line (Ri) for (-1) and multiply RI for AI and add the result to the second line (R->] -1 -1 0 (A.8) A2 AI Equation (A.7) beeomes, 1 1 0 AI - A2. x. (A.9). Multipling R2 by 1 1 0 AI - A2. (A.10). Equation (A.9) becomes, 1 1 0 1. 0. (A.ll). Adding —R2 to 1 1 0 1. (A.12). Equation (A.11) becomes, 1. 0. Ai-A2. (A.13) 0 1. Ai-A 2. ..

(73) 55. Therefore we have, -. (A.14). A 2 — AI. AI — A 2. Once we found AI and A^ we use them on (A.I). \2-JW. fx(x). With the inverse Fourier transform. _ Ax. $x(w) 1 A—jw. we have,. for n = 3, and having an equal n = 2 pursuit. i — jw. \2 — jw. A3 — jw. AiA 2 A 3 - jw)(X2 - jw)(X3 - jw). AI - jw. A 2 - jw. we have the equations: AI + A2 + A3 = 0. -(A 2 + A3).4i - (A! + X3)Ai - (Ai + X2)A3 = 0. A3 -.

(74) 56. APPENDIX A. EXPONENTIAL SUM WITH DIFFERENT PARAMETERS. Matrix form:. — A2 — AS — AI — A3 — AI — A2. AiA 2 A 3. 0 r>. 0. A2 — AI. 0 Rs. A 3 (A,-A 2 ). 0 AjA 3 — A 2 A 3 A^A 2 — A 2 A 3 A!A 2 A 3 .. 1 1 0 1 n. U. I. 1. A 2 (Ai-A 3 ) A 3 (Ai-A 2 ). AiA 2 Ai-A 2. I I A 3 (Ai-A 2 ). 0 1 n u. I. n. u. I. (Ai-A 3 )(A 2 -A 3 ) A 3 (Ai-A 2 ). AiA 2 Ai-A 2. 1. -R3 0 1 I Aj—AQ. 0 0. 1. (Ai-A 3 )(A 2 -A 3 ) J.

(75) 57. • 1. i n. —AIA2A3 (Ai-A 3 )(A 2 -A 3 ). o i o. A1 (AlrA 2 )^A3). 1. L. i. U. A A. n n i. U. i aA 3. U. (Ai-A 3 )(A 2 -A 3 ) J. AA A. • i n n 1. L. U. i23. U. (A 1 -A 2 )(A,-A 3 ). n. —~ A ' A 2 A 3—. nU i. U. 0 u. 1. 0 u. A. 4 ^2. 3. 1A2A3. A A A. i23. U. (A 2 -A,)(A 3 -A 1 ). 0 U. 1 0 U. A A2A ' 3 (Ai-A 2 )(A 3 -A 2 ). 0 U. 0 U. AlA2A 3 (x,-\3)(\2-\3). L1. J. AI A 2 A 3 (A 2 -A,)(A 3 -A 1 ). 1. A. >. (A 1 -A 3 )(A 2 -A 3 ) J. U. L. •. (Ai-A 2 )(A 2 -A 3 ). "i n n. 4. -RI + RI. AiA 2 A 3 (Al_A2)(A3_A2) __. AI A 2 A 3 (Ai-A 3 )(A 2 -A 3 ). { using fj \ / 3xl J. Ai A 2 A 3 (A 2 -A 1 )(A3-A 1 ). —: : AI - jw. fx(x). &x(w). e-Xx. ^~. ,. T~. AiA 2 A 3 ( A i- A 2)( A 3~ A 2). ; : A 2 - jw. ,. "l~. AI A 2 A 3 "\ (Ai-A 3 )(A 2 -A 3 ) I. ^ : A 3 - jw. r J. AlA2A3 AlA2As AlA2As =\ }e-^+\ l e - A ^+[ le-*3* L(A 2 - AO(A 3 - AOJ ^ [ ( A i - A 2 )(A 3 - A 2 )J ^ [ ( X , - A 3 )(A 2 - A 3 )J.

(76) 58. APPENDIX A. EXPONENTIAL. SUM WITH DIFFERENT. PARAMETERS. for n = 4, and having an equal n = 2 pursuit. o-i / AI = 3 L\--. A2. A3. A4. -T-— A 3 — ]W. lAi— J. Al \\-jw. AiA 2 A 3 A 4. - jw)(X2 - jw}(\3 - jw)(X4 - jw). A2 1. ». A3 .. i. i. .. i. A4 - jw. (A2 + A3 + A 4 )(ju;) 2 + (-A 2 A 3 - A 2 A 4 - \3X4)(jw} + (A 2 A 3 A 4 - jw)(X3 - ju»)(A 4 - jw). A3 + A 4 )(ju;) 2 - jw)(\2 - j. - A 3 A 4 )(» + (AiA 3 A 4 )] - jw). (AiA 2 A 4 )] (Ai - JW)(\2 - JW)(\3 - JW)(\4 - jw). (-AiA 2 - AiA 3 - A 2 A 3 )(jt») + (AiA 2 A 3 )]. - jw)(\2 - jw)(\3 - jtu)(A 4 - jw). I + A-2 + A3 + A4 = 0 (A 2 + A3. i + A3. (A 2 A 3 + A 2 A 4 + A 3 A 4 )^i + (AiA 3 AiA 3 + A 2 A 3 )A 4 = 0 (A 2 A 3 A 4 )/ii + (A 1 A 3 A 4 ),4 2 Matrix form:. A2. i + A2 + A 3 )A 4 = 0. A3A4),42 + (AiA 2 -f AiA 4 + A 2 A 4 ),4 3 + (AiA 2.

(77) 59. 1. 1. 1. 1. 0 -(A 2 + A3 + A 4 )fli + R2. A 2 + A3 + A 4 AI + A3 + A 4 AI + A2 + A4 AI + A 2 + A3. 0 —(A 2 A 3 + A 2 A 4 + A 3 A 4 ).Ri. A 2 A 3 -\- A 2 A 4 +A 3 A 4. A^A 3 -|- A^A 4 +A 3 A 4. AjA 2 -f- A^A 4 4-A 2 A 4. AjA 2 -1- AjA 3 +A 2 A 3. U. A2A3A4. AjA 3 A 4. A^A 2 A 4. A^A 2 A 3. AjA 2 A 3 A 4. '. 1 (J. — (A 2 A 3 A 4 )/?i 4- /?4. 1. 1. 1. 0. \i — A 2. AI — A3. AI — A4. 0. 1. f i 2. Ai-A 2. /j, (Ai-A 2 )(A 3 +A 4 ). 0 ( A ! - A 2 ) ( A 3 + A4). (A 2 + A 4 ) ( A ! - A 3 ). (A 2 + A 3 )(A! - A 4 ). 0. R4. A a A 4 ( A i — A2). 0. A^A/i(Ai — Ao). AoA^fAi — A^i. " 1 1 nU ii. 0. AI-AS •* *. Ai-A \ »4. nVJ. n u. AI—A2. 1. ( A 2+ A 4)(Ai-A 3 ) (Ai-A 2 )(A 3 +A 4 ). (A 2 +A 3 )(A 1 -A.i) (A,-A2)(A3+A 4 ). n. i. A2(Ai-A3) A 3 (A,-A 2 ). A2(Ai-A4) A4(Ai-A2). ' 1 1. .. 1. A I — \2. .. Q. 1. 0 U. 0. A o A ^ f A i — A/I ). 1. 1 A. 1~ A 3 AI—A2. 0. 1-A-1 AI—A2. 0. U. U. A. (Ai-A 3 )(A 2 -A 3 ) (Ai-A 2 )(A 3 +A 4 ). (Ai-A4)(A2-A4) (A 1 -A 2 )(A 3 +A 4 ). n. n. (Ai-A 3 )(A 2 -A 3 ) A 3 (Ai-A 2 ). (A!-A 4 )(A 2 -A 4 ) A 4 (Ai-A 2 ). — R-2 + RZ —R2 + R*. AiA2 A,-A 2 .. 1 A. AiAoA^A/i. (Ai-A2)(A3+A4) p (Ai-A 3 )(A 2 -A 3 )- r t 3 A3(Aj—A2). U. fi. AtA2 Ai-A 2 .. (A. A )(A. A. r-> )-«4.

(78) APPENDIX A. EXPONENTIAL SUM WITH DIFFERENT. 60. PARAMETERS. 1 1. 1 1 0. 0 1. Ai-A 2. Ai-A 2. 0 0. 1. (Ai-A 4 )(A 2 -A 4 ) (Ai-A 3 )(A 2 -A 3 ). 0 0. 1. A 3 (A!-A 4 )(A 2 -A 4 ) A4(Ai-A 3 )(A2-A 3 ). 1. 0. -R3 0. AiA 2 A 3 (A,-A 3 )(A 2 -A 3 ) J. 0. Ai-A2. Ai-A2. 0. 0. 1. (Ai-A4)(A2-A4) (A,-A 3 )(A 2 -A 3 ). 0. 0 0. 0. (A 1 -A 4 )(A 2 -A 4 )(A 3 -A 4 ) A 4 (Ai-A 3 )(A 2 -A 3 ). 0. A 4 (Ai-A 3 )(A 2 -A 3 ) p (A,-A 4 )(A 2 -A 4 )(A 3 -A 4 ) / l 4. AiA 2 A 3. (Ai-A 3 )(A 2 -A 3 ) J. 1 1 -R4. Ai-A4 Ai-A 2. 0 0. 1. -(AI— A 4 ) (Ai-A 2 ). (Ai-A 4 )(A 2 -A 4 ) (A 1 -A 3 )(A 2 -A 3 ). — (Ai— ~(Ai-A ~7\ -rT7\ )(A -A )\ ~ " 3. 0 0. 0. 11. (Ai-A 4 )(A 2 -A 4 )(A 3 -A 4 ) .. 1. 0. (A,-A 4 )(A 2 -A 4 )(A 3 -A 4 ). 0 1 =+^ 0 Ai-A 2. _ _ (Ai-A 2 )(A 2 -A 4 )(A 3 -A 4 ). 0 0. 1. 0. _ — AI A 2 A 3 A 4 _ (Ai-A 3 )(A 2 -A 3 )(A 3 -A 4 ). 0 0. 0. 1. _ AI A2 Ag A^ _ (Ai-A 4 )(A 2 -A 4 )(A 3 -A 4 ) J. -(Ai-A 3 ) (A,-A 2 ). 2. 3. _i r.

(79) m —. m. -. (E Y -^)(c Y -E Y )(e Y -i Y ). m. -. (71 f —. (i Y _» Y )(i Y -e Y )(i Y -;. 8AV 90UQ. =. £„. *. '. 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1. ( t 'Y- E Y)('-Y- z Y)(''Y- 1 Y) ^Y^Y^Y ^Y. C-Y- E Y)( E Y- S Y)( E Y- I Y). 1 0 0 0 0 I 0 0 0 0 I 0 0 0 I. I. 19.

(80) 62. APPENDIX A. EXPONENTIAL. using. fx(x]. 3>x(w]. e-Ax. ^~. SUM WITH DIFFERENT. f ( \_ \ AjA 2 A 3 A 4 1 _AIX L (A 2 — Ai)(A 3 — Ai)(A 4 — A j j J. PARAMETERS. r AjA 2 A 3 A 4 i _ A2X L(Ai — A 2 )(A 3 — A 2 )(A 4 — A 2 )J. (A: - A 3 )(A 2 - A 3 )(A 4 - A 3 )J 6 ** + [(X, - A 4 )(A 2 - A 4 )(A 3 - A 4 )J. Using convolution: for n = 2. AiA 2 e- Air. =. e-. A2(x T). -. dr. AiA2_e_^r e _ T ( A l _ A a ) j a : (A2 — AI) L J T=O. A2 — AI. (A2-A. AI — A. for n = 3. 1 A ^~ A2X 1. AI — A2 J.

(81) (£Y-SY)(£Y-TY). |. (TY-£Y)(TY-SY). +. 7". (TY-£Y)(TY-. ( g Y - £ Y)( S Y - T Y) ,. (z\ -. ~ T Y). -9 y. [I. (TY-£Y)(TY(E Y _I Y )x- 9 J. sv — T Y. lv. —. XJ. )[-.

Figure

Figure 1.1: Error in the new estimated position
Figure 2.3: Direction finding Position Location Solution.
Figure 2.4: Smart antenna systems with a different beam for each subscriber.
Figure 3.1: Difference between actual position and estimated position, as (r,6) function.
+7

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