DEGREE PROJECT, IN OPTIMIZATION AND SYSTEMS THEORY , SECOND LEVEL STOCKHOLM, SWEDEN 2015
Calibration of Multilateration Positioning Systems via Nonlinear Optimization
SEBASTIAN BREMBERG
KTH ROYAL INSTITUTE OF TECHNOLOGY
SCI SCHOOL OF ENGINEERING SCIENCES
Calibration of Multilateration Positioning Systems via Nonlinear Optimization
SEBASTIAN BREMBERG
Master’s Thesis in Optimization and Systems Theory (30 ECTS credits) Master Programme in Applied and Computational Mathematics (120 credits) Royal Institute of Technology year 2015 Supervisor at Ericsson: Daniel Henriksson Supervisor at KTH: Johan Karlsson Examiner: Johan Karlsson
TRITA-MAT-E 2015:62 ISRN-KTH/MAT/E--15/62--SE
Royal Institute of Technology SCI School of Engineering Sciences
KTH SCI SE-100 44 Stockholm, Sweden
URL: www.kth.se/sci
Kalibrering av System f¨or Multilaterations Positionerssystem genom Icke-linj¨ar Optimering
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Sammanfattning
I denna masteruppsats utv¨arderas en metod syftande till att f¨orb¨attra noggran- nheten i den funktion som positionerar sensorer i ett tr˚adl¨ost transmissionsn¨atverk. Den positioneringsmetod som har legat till grund f¨or analysen ¨ar TDOA (Time Dif- ference of Arrival), en multilaterations-teknik som baseras p˚am¨atning av tidsskillnaden av en radiosignal fr˚an tv˚arumsligt separerade och synkrona transmittorer till en mottagande sensor. Metoden syftar till att reducera positioneringsfel som orsakats av att de ursprungliga positionsangivelserna varit felaktiga samt synkroniserings- fel i n¨atet. F¨or rekalibrering av transmissionsn¨atet anv¨ands redan k¨anda sensor- positioner. Detta uppn˚as genom minimering av skillnaden mellan signalbaserade TDOA-m¨atningar fr˚an systemet och uppskattade TDOA-m˚att vilka erh˚allits genom ber¨akningar av en given sensorposition baserat p˚aoptimering via en ickelinj¨ar minstakvadratanpassning. Genom ett antal simuleringar testas sedan den f¨oreslagna metoden med olika grundinst¨allningar och olika grad av m¨atbrus samt ett varier- ande antal sensorer och transmittorer. Denna metod ger tydlig f¨orb¨attring f¨or estimering av systemparametrar och klarar ¨aven av att hantera multipla felk¨allor f¨orutsatt att antalet m¨atningar ¨ar tillr¨ackligt stort.
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Abstract
This master thesis presents an evaluation of a method for improving performance of sensor positioning in a network of emitters. The positioning method used for the analysis is Time Di↵erence of Arrival, TDOA, a multilateration technique based on measurements of di↵erences in signal travel time between a pair of synchronous and spatially separated pairs of emitters and a sensor. The method in question aims at reducing positioning errors caused by errors in initially reported emitter positions as well as network synchronization errors by using already known sensor positions to re-calibrate the network of emitters. This is done by minimizing the di↵erence between signal based TDOA measurements from the system and estimated TDOA measurements made by calculations based on given sensor positions by means of nonlinear least squares optimization. Alterations of the method with di↵erent settings and error contributions and with varying amount of sensors and emitters are tested throughout several simulations. The proposed method shows apparent results of improving the system parameters and also copes well with contributing errors provided that the amount of measurements is su ciently large.
Acknowledgements
I would like to extend my greatest gratitude to Daniel Henriksson at Ericsson, who with his knowledge and enthusiasm has given invaluable support and guidance throughout this project. I would also like to thank my supervisor at KTH, Johan Karlsson, who has with his experience been a much appreciated support and dedicated advisor throughout the project.
Contents
1 Introduction 2 1.1 Signal Based Positioning ...... 2 1.2 ThesisOutline ...... 3
2 Time Di↵erence of Arrival, TDOA 4 2.1 Method ...... 4 2.2 Factors influencing accuracy ...... 7 2.2.1 Synchronisation errors and time delays ...... 8 2.2.2 Emitter position errors ...... 8 2.2.3 Geometry...... 9 2.2.4 Altitude...... 9 2.2.5 Multipath ...... 10
3 Optimization and Method of Estimation 11 3.1 Unconstrained Optimization ...... 11 3.2 Nonlinear Least Squares ...... 12 3.2.1 Trust-Region-Reflective Least Squares Algorithm ...... 12 3.2.2 Nonlinear Least Square in Larger Scale ...... 14 3.2.3 Weighted Nonlinear Least Squares Minimisation ...... 15 3.3 Cram´er-Rao Lower Bound ...... 15
4 Modelling and Optimizing Antenna Position Errors in Cell Network 19 4.1 Formulation ...... 20 4.1.1 System of Insu cientRank ...... 21 4.2 Determination of Weights ...... 22 4.3 Cram´er Rao Lower Bound Analysis ...... 23
5 Simulations 26 5.1 Method ...... 26
6 Results 28 6.0.1 Weighted and Non-Weighted Least Squares ...... 28 6.0.2 No error in sensor positions ...... 29 6.0.3 Small error in sensor positions ...... 30 6.0.4 Large error in sensor positions ...... 31
7 Discussion 32 7.1 Futurework...... 33
5 8 Conclusion 35
Appendix A TDOA Positioning of Sensors 36 A.1 TDOA Positioning by Sum of Least Squares ...... 36
Appendix B Positioning with height parameter 37 Glossary
TDOA Time Di↵erence of Arrival
ETDOA Expected Time Di↵erence of Arrival
GPS Global Positioning System
NLLS Nonlinear Least Squares
TTFF Time to First Fix
TOA Time of Arrival
CRLB Cram´er-Rao Lower Bound
GDOP Geometric Dilution of Precision
FIM Fischer Information Matrix
Nomenclature e Emitter position eest Estimated emitter position s Sensor position sest Estimated sensor position
˜s Estimated sensor position g Emitter position error
✏ TDOA Measurement noise
Time synchronization delay
⌧ TDOA value
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1 Introduction
This aim of this thesis is to evaluate an algorithm for improving positioning accuracy in wireless networks. Primarily, aspects of improving Time Di↵erence of Arrival (TDOA) positioning are considered. However, some of the methods used are also applicable to other positioning methods in wireless networks and other similar signal based positioning methods. More specifically, the algorithm to be evaluated will address certain common error contributions to see whether an adjusting calibration can decrease systematical errors and their impact on positioning accuracy.
1.1 Signal Based Positioning
There are many examples of wireless and signal based systems that feature the possibil- ity of locating the position of an emitting or receiving unit. Global Positioning System, GPS, is a widely adopted positioning method using several time synchronized satellites making signal time measurements and multilateration to locate a receiving GPS signal unit [1].
It has become increasingly important to be able to position a mobile device accur- ately in a cellular network. Positioning a mobile device in a mobile network can ,e.g., enable localization of emergency calls, advanced location based services and possible also aid network tra c optimization [2]. As mobile network coverage is increasing and the technology is becoming increasingly advanced, the methods of positioning are under constant improvement [2].
Both GPS positioning methods and cellular network positioning methods are based on measurements on signals from distant satellites and antennas respectively. An obstacle free environment, where signals can travel without disturbance between the transmitting and receiving unit is to be preferred, however that is in most environments of applica- tion not possible. For example, these disturbances lead to multipath errors which will be described further in Section 2.2. These methods are not ideal for ,e.g., urban and indoor environments. A more recent and evolving positioning feature is that of locating units with local wireless networks, such as WiFi or Wireless Sensor Networks, WSN [4][5]. An example of application is indoor robot localisation, as described by Cheng et al. [4].
2 As the computational capacity of network related equipment increases, it is possible to combine several positioning methods to increase accuracy. These are known as Hybrid Positioning systems. By combining results from multiple positioning methods, knowing the limitations and sources of errors for each method, it is possible to achieve a higher accuracy than when applying one single method [7]. A common method of combining di↵erent methods is for a GPS to achieve assistance data from a cellular networks to improve start up speed and decrease the Time-to-First-Fix, TTFF, known as the time required for a GPS receiver to acquire necessary GPS signals and fix a first position [7]. The assistance date sent from the cellular network could for example be in what cell sector the unit is situated whereby the GPS can limit its scope of search significantly [7].
The above mentioned positioning methods are examples of methods using known in- formation from emitters to locate unknown positions of sensors. However, the opposite can also be true. An example is when trying to locate from where radar signals are sent from. By measuring the signal and their characteristics from di↵erent known locations it is possible to estimate where the signals are transmitted from [6] [3]. One such method is known as Passive Radar Localization where positioning by means of TDOA is one of the most fundamental and popular localization methods [6]. This method will be described in more detail later in this thesis. However, the emphasis in this thesis will be on sensor positioning in a network of emitters.
1.2 Thesis Outline
The proposed method aims to improve accuracy of a TDOA sensor positioning system prone to systematic errors. More specifically the algorithm addresses errors caused by faulty position information and time synchronization errors in the positioning network. The general idea is to use already known sensor positions to refine emitter system para- meters and synchronization.
What will follow is an introduction to TDOA positioning including general theory and a description of common error contributions and how these a↵ect accuracy. Thereafter, the proposed method and alterations of it will be described. A description of the op- timization algorithms used are presented followed by a simulations results from testing the proposed method with di↵erent settings and parameters.
This thesis aims at evaluating the proposed method to see whether it proves to be e cient in increasing the accuracy in a positioning network. The method is evaluated through a theoretical model and if the method proves to be successful, the thesis can be used as a foundation when implementing in a physical network.
3 2 Time Di↵erence of Arrival, TDOA
There are numerous methods of estimating a stationary sensor position from meas- urements on for example signal arrival times, angle of arrival and Doppler shifts of electromagnetic waves at di↵erent sites [11]. This thesis focuses on multilateration, a hyperbolic location system, more commonly known as Time Di↵erence of Arrival (TDOA), which exploits the di↵erence in arrival time between transmitted signal from emitters to a sensor that is to be positioned. TDOA is not to be confused with Time of Arrival (TOA), a trilateration positioning method. TOA measures the absolute time it takes for the signal to travel from the emitters to the sensor. When measuring TOA it is important that both the emitters and the sensors are time synchronized. If they are not synchronized, TDOA is to prefer, eliminating such errors [12].
2.1 Method
Let s Rd be a sensor position and e Rd, j =1, 2, 3, be the positions of three 2 j 2 emitters. These emitters are synchronized and the sensor which is to be positioned receives a signal from each of these emitters. The sensor calculates the TDOA for each unique pair of emitters ⌧12,⌧13,⌧23. Each TDOA measurement corresponds to an equation:
⌧ = s e s e (1) ij || i|| || j|| where denotes the euclidean distance in Rd. k·k
In the two dimensional case, at least three emitters are needed to locate a sensor. To give a simple example let us look at a situation with two emitters and one sensor. Let e =( D/2, 0) and e =(D/2, 0) and s =(x, y). The measured TDOA values for the 1 2 pair of emitters are given by:
t = s e = (x e )2 +(y e )= y2 (x + D/2)2 4 1 k 1k 1,x 1,y q t = s e = (x e )2 +(y e )=py2 (x D/2)2 (2) 4 2 k 2k 2,x 2,y q ⌧ = t t = y2 (x + D/2)2 yp2 (x D/2)2 . 12 4 1 4 2 ⇣p p ⌘ The last equation can also be written as
x2 y2 =1. (3) ⌧ 2 /4 (D2 ⌧ )/4 12 12
4 This can be recognized as a hyperbolic equation where the solution has asymptotes along
2 D ⌧12)/4 y = 2 x. (4) ±s ⌧12/4 The solution to the equation corresponds to the sensor being somewhere along a hyper- bola as can be seen in Figure 1 for di↵erent values of ⌧12.
3
2
1
τ = 0.4 τ = 0.8 0 τ = 1.2 τ = 1.4
-1
-2
-3 -3 -2 -1 0 1 2 3
Figure 1: Hyperbolas as a result of di↵erent TDOA measurements
The measurements made in the systems are prone to errors. Each TDOA measurement has an uncertainty ✏ij and stated as:
⌧ = s e s e + ✏ 1 i 3 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3 Figure 2: TDOA measurement with an uncertainty relating to measurement noise 5 With three emitters there is a set of three TDOA values: ⌧ = s e s e + ✏ 12 || 1|| || 2|| 12 ⌧13 = s e1 s e3 + ✏13 || || || || (6) ⌧ = s e s e + ✏ . 23 || 2|| || 3|| 23 To locate s we will want to solve the following system of equations for the unknown variable sest, the estimated position of s: sest e sest e + ✏ ⌧ =0 || 1|| || 2|| 12 12 est est s e1 s e3 + ✏13 ⌧13 =0 || || || || (7) sest e sest e + ✏ ⌧ =0. || 2|| || 3|| 23 23 The three equations in (7) correspond to three hyperbolic areas with a certain width corresponding to the uncertainty ✏ in the measurements, as seen in Figure 3. One ex- pects to find the sensor in question where these hyperbolas intersect. x . x x Figure 3 With m available emitters there will be a total of K TDOA measurements and equations where K is given by all pairwise unique enumerations of emitters m m(m 1) K = = . (8) 2 2 ✓ ◆ If we disregard any measurement noise, the problem will be to solve an over determined system when m d +1 where d once again is dimensions of the space in which the positions that are to be determined. However in the presence of measurement noise, the hyperbolas will almost certainly intersect in several di↵erent points. The results is an estimation problem where one seeks to find the most probably solution. There is a large amount of research in finding as accurate and e cient methods of solving these estimation problems (e.g. [19][20][21]). In this thesis a non-linear least squares method 6 will be implemented, similar to what is done in [12]. In short one then defines F (sest) to be a vectorized function consisting of all equations corresponding to each TDOA measurement v =1, 2,...,K f (sest)= sest e sest e + ✏ ⌧ for 1 j 2.2 Factors influencing accuracy In addition to the signal measurement noise there are numerous other factors that a↵ect the accuracy of TDOA positioning. Below are some examples such of factors of which some will be addressed further in this thesis. • Synchronisation errors and time delays • Emitter position errors • Geometry • Altitude • Multipath 100 e2 80 60 40 20 0 s -20 -40 -60 -80 e1 e2 -100 -80 -60 -40 -20 0 20 40 60 80 100 Figure 4: Sensor and emitters of a TDOA measurement 7 2.2.1 Synchronisation errors and time delays If the emitters are not perfectly time-synchronized or if there are signal path time delays, the measured distance time values will contain errors that will a↵ect the accuracy. An example of how synchronisation errors a↵ect the positioning of a sensor in a emitter network is shown in Figure 5. The sensor is positioned in the middle of three emitters as shown in Figure 4 and one of the emitters is given a time delay. 8 7 6 5 4 3 Sensor Positioning Error 2 1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Time Delay of one emitter [s] ×10-8 Figure 5: Sensor positioning error with unknown time delay If we know for a fact that the two emitters have perfect synchronization it can be the case that one, or both, signals sent from the two emitters are subject to path delays. The emitters are said to emit the signals at a certain time, t, however the signals are for some reason, delayed and actually can be considered sent out a slightly later, t + t.This 4 could for example be caused by obstacles in the environment. This in combination with synchronization errors will have a considerable impact on the accuracy and a method of reducing these contributions will be addressed in the proposed method. 2.2.2 Emitter position errors When estimating positions from TDOA measurements it is of great importance to have accurate emitter positions in the system. When building a wireless network that is to be used for positioning of sensors, each emitter position is determined and noted in the system. It is in some cases di cult to be precise when determining these positions and often it is a time consuming process to achieve high accuracy. These positions have a significant role in the estimation of sensor positions and if the actual emitter positions di↵er from the positions known to the system a systematic error in positioning accuracy will be apparent [13]. An example of how errors in given emitter positions a↵ect the positioning of a mobile unit in a mobile network is shown in Figure 6. The sensor is once again located in the middle of three emitters as shown in Figure 4 and Figure 6 8 12 10 8 6 4 2 Maximum Sensor Positioning Error [m] 0 0 2 4 6 8 10 12 14 16 18 20 Emitter displacement [m] Figure 6: Sensor positioning error with emitter position errors shows the maximum positioning error when one of the emitters actual positions di↵ers from the known position to the system. 2.2.3 Geometry The geometry of where the measurements are taken have an impact on the accuracy. Figure 7 shows an example of how the geometric dilution a↵ects the estimation when the sensor is moved further way from a pair of emitters. As can be seen, the area of uncertainty increases when the sensor is placed further away from the two emitters. x x x x . . Figure 7: Example of Geometric Dilution A common method of analyzing to what extent is the geometry a↵ects is to look at the Geometrical Dilution of Precision, GDOP. The GDOP is a result of The Cram´er-Rao Lower bound, CRLB, which is a well known result of mathematical statistics. It gives a lower bound of the covariance of estimated parameters. For more information on the Cram´er-Rao Lower bound is found in Section 3.3. 2.2.4 Altitude For this thesis, the focus is on horizontal sensor positioning in two dimensions. However it is possible that the emitters and sensors have vertical components. This will have an impact if disregarded in the estimation problem. Figure 8 illustrates the maximum position error when the vertical component is disregarded in the calculations. Once 9 again, we have the same setup as in Figure 4. If one of the emitters were to have an ver- tical component, h, and the sensor having zero vertical component then the maximum positioning error is shown in Figure 8. As mentioned previously, the evaluation made in this thesis will regard a two-dimensional setting although the theory is applicable in general dimension. However, in Section B presents some short results on how height a↵ects the proposed method. Figure 8: Sensor positioning error when disregarding height parameter 2.2.5 Multipath When signals propagate between an emitter and a sensor it is often the case that there are obstacles in the environment. The signal measurements are made on nonline-of- sight paths which will di↵er from the line-of-sight signal travel distance [10]. These errors generally have a large contribution to the overall accuracy and are especially prominent in indoor settings. To dampen the magnitude of this error contribution, the receiver of the di↵erent signals searches for the signal for the signal which has travelled the shortest time. This signal is likely to give the most accurate measurement [10]. 10 3 Optimization and Method of Estimation What follows is an introduction to the optimization method used for the analysis of the proposed method and the algorithm behind Matlab’s solver lsqnonlin, later used in the simulations. This is followed by theory concerning estimation and the Cram´er-Rao Lower Bound. 3.1 Unconstrained Optimization We will be interested in minimizing a function where no constraints are placed on the variables. This is known as unconstrained optimization and what follows is general theory of optimality conditions concerning unconstrained optimization. Let y(x)bea function of the variables x =(x1,..,xn) that we wish to minimize, min y(x). (10) x To define optimality conditions we denote x⇤ to be a possible solution to the problem. A global optimum to (10) is a point x⇤ for which y(x⇤) y(x) for all x, (11) whereas a local optimum is a point x⇤ for which y(x⇤) y(x) for all x such that x x⇤ <✏, (12) k k , for some ✏>0 [25]. For unconstrained problems it is very di cult or impossible to evaluate global optimality. In general, to determine a global optimum, one needs in- formation about the function in question for each and every point. The algorithms used to solve these types of problems only has information about a finite set of points of the function. However it is possible to state conditions for local optimality [25]. Consider the Taylor series expansion of y around the point x⇤: T 1 T 2 y(x⇤ + p)=y(x⇤)+ y(x⇤) p + p y(⇠)p (13) 5 2 5 where ⇠ is a point between x and x⇤ and p is a nonzero vector. Suppose that x⇤ is a local optimum then there are no feasible descent directions from this point. This means that the gradient at this point must be zero [25], y(x⇤)=0. (14) 5 11 This is known as the first-order necessary condition for a local minimum [25]. However this condition is not enough to state the point being a local minimum. To be further confirm that a point is local minimum we need to look at the second derivatives. Consider the Taylor series expansion for a point for which the first-order necessary condition is satisfied ( y(x ) = 0): 5 ⇤ 1 T 2 y(x⇤ + p)=y(x⇤)+ p y(⇠)p. (15) 2 5 2 The second-order necessary condition states that y(x⇤) must be positive semidefin- 5 T 2 ite. If it is not semidefinite then b y(x⇤)b<0 for some b. This also means that 5 T 2 b y(⇠)b<0if ⇠ x⇤ is su ciently small. Thus, if p is chosen as a small multiple 5 k k of b, the point ⇠ will be a close approximate to x⇤ such that by the Taylor expansion it can be concluded that y(x) Furthermore, there is a second-order su cient condition. A condition that is su cient to ensure a local minimum and consists of the first-order necessary condition together 2 whith the condition that y(x⇤) is positive definite. Such an x⇤ is known as a strict 5 minimizer [25]. 3.2 Nonlinear Least Squares The model evaluated in this thesis will involve fitting a set of variables to a large number of data points. Because of measurement noise and other contributing errors it can not be expected that it is possible to find an exact solution. We instead want to find an solution that fits to the collected data by means of reducing the residual error. Let F (x) be a vector-valued function with Fv = fv(x). Each fv(x) is a function of the residual of data point v =1,...,n and x is a vector of variables. The nonlinear least squares, NLLS, problem is then defined as: 2 2 min fv (x)=min F (x) 2. x x k k (16) v X A simple yet powerful method of solving (16), is a method based on trust regions. It is an iterative method that searches for points that fulfill second-order necessary condition of a local optimum. In this section follows a description of the method used and underlying theory. 3.2.1 Trust-Region-Reflective Least Squares Algorithm Consider a scalar valued nonlinear function f(x), with vector arguments x, that is subject to minimization. Let x0 be an initial point and one wishes to move from this 12 point to decrease the current function value. The idea is to, in the neighbourhood N of x, approximate f with a more simple function q. The neighbourhood, N, for which q is a su ciently good approximation of f is called the trust region. A step, s, is calculated by minimizing q(s)overN, min q(s),s N . (17) x { 2 } If f(x + s) A standard trust-region method is described in [14]. The idea is to approximate f by letting q be defined as the first two terms of the Taylor expansion of f. The subproblem is thus stated as: 1 T T 2 min s Hs + s g such that Ds 2 . (18) s 2 k| k| 4 ⇢ where H is the Hessian and g is the gradient of f at x, D is scaling matrix with diag- onal entries and a positive scalar. There are a good number of algorithms for solving 4 (18) and many of these are mentioned in [14]. However these algorithms require time that is proportional to several factorizations of the Hessian, H, which becomes costly process [16][17]. In trust-region algorithms the most computationally costly process is to evaluate a step at the current iterate [15]. The algorithm used in this thesis is the so called ”Indefinite Dogleg Algorithm” first described in by Schultz et. al. [18] and later evaluated and analyzed by Byrd et al [15] and Branch et. al. [16]. It makes a di↵erent approach to approximating (18) by restricting the trust-region subproblem to a subspace S in two dimensions. By choosing two suitable directions to span a space S it is possible to with high accuracy and good convergence optimize a function f Rn 2 [15]. To explain the di↵erence from and the computational advantages of the used Infinite Dogleg Algorithm, what follows is an introduction to important aspects of algorithms used to solve (18). One such important concept is the so called ⌧-optimality. ⌧-optimal methods take steps which are at least a ⌧ fraction of the true optimal value. We define a more formal formulation of ⌧-optimality by the following definitions: n n n Definition 1 (Step function). The function q is called a step function if: R R ⇥ ⇥ ⇥ (0, inf) Rn. Here denoted as q(g, H, ). ! 4 13 n n n Definition 2 (Optimal step computing function). For any g R , H R ⇥ and 2 2 > 0. We denote the solution to (18) to be q (g, H, ). An optimal step computing 4 ⇤ 4 function is thus refered to as q . ⇤ Definition 3. For q Rn,letpred(q, g, H)= qT g 1 qT Hq. 2 2 Definition 4 (⌧-Optimal). For ⌧ (0, 1], a step computing function q is ⌧ optimal 2 if for any symmetric H Rnxn,s Rn, and > 0: 2 2 4 pred(q(s, H, ),s,H) ⌧pred(q (s, H, ),s,H). 4 ⇤ 4 Byrd et al. [22] show that if a trust-region step function satisfies three conditions it has the same global and local convergence as a ⌧-optimal step computing function. However, the conditions presented below are compared to ⌧-optimality somewhat weaker [22]. n nxn 1. There are c1, c¯1 such that for all > 0, s R and H R . 4 s2 2 pred(q(s, H, ),s,H) c1 s min , c¯1 kHk . 4 k k {4 k k } 2. Let be the smallest eigenvalue of H.Thereisac > 0 s.t. for all s Rn, > 0 2 2 4 n n and H R ⇥ 2 pred(q(s, H, ),s,H) c ( (H)) 2. 4 2 4 1 n n 1 3. If H s and H R ⇥ is positive definite, then q(g, H, )= H s. 4 2 4 The proposed ”Indefinite Dogleg Step Computing Function” satisfies the above condi- tions.The subspace S is spanned by s1 and s2 where s1 is the direction of the gradient, g, and s2 is defined to be an approximate Newton direction or, if the Hessian is indefinite, a direction of negative curvature. H s2 = g if H positive definite s2 = · (19) 8 sT H s < 0ifHindefinite. < 2 · · 2 With these vectors one aims: to encourage global convergence with steepest descent or negative curvature direction and also enable local convergence with the Newton step when possible. For more details about this algorithm see Schultz. et al. [18]. 3.2.2 Nonlinear Least Square in Larger Scale The solver used in the simulations is lsqnonlin implemented in Matlab which solves non- linear least squares using Trust-Region-Reflective Least Squares Algorithm as described in the previous section. However, when the system increases in size, the solver in Matlab makes an adjustment to the algorithm to improve e ciency. To avoid finding second derivatives of all f(x) an approximate Gauss Newton direction is found instead to define the subspace S. Let J be the Jacobian of F (x). s is found by solving min Js+ F 2. (20) || ||2 14 The method of preconditioned conjugate gradients is used in every iteration to find an approximate solution to the normal equations [24]. 3.2.3 Weighted Nonlinear Least Squares Minimisation A small error in supplied information to an optimization model can, due to the formu- lation of the equations and the optimization, have a significant impact on the accuracy of the estimation. One method to decrease to what extent these errors e↵ect the model is to introduce a weighted NLLS. It is when dealing with heteroscedasticity problems, problems where the error vari- ance of each measurements is di↵erent, introducing a weighted optimization can be of benefit [9]. A general Weighted Nonlinear least squares minimization is defined as 2 2 min wvfv (x)=min wF(x) 2, x x k k (21) v X where w consists of all weights wv for each function fv for all v. wp is often chosen to be 1/ p,where p is the variance of the error contributing to the p:th measurement [26]. However, it is not always possible to know the variance or the parent distribution of each measurement. In many cases it is not the variance that is of importance , instead it is to find outliers that deviate from the majority of measurements and to dampen their a↵ect on the results [27]. In many practical settings where one knows the general behaviour it is suitable to approximate how and to what extent the weights are set [26]. The weighs used in this thesis are described in Section 4.2 and evaluated in the results in Section 6.0.1. 3.3 Cram´er-Rao Lower Bound When working with estimation problems it is of interest to know the lower bound on the variance for the estimated parameter. Such a bound is known as a Cram´er-Rao Lower bound or sometimes Cram´er-Rao inequality. In short, the lowest possible variance of an unbiased estimator is at least as large as the inverse of the Fischer information matrix [8]. To be able to state the Cram´er-Rao bound we will begin with a small example. Assume that we observe the signal x and are interested of estimating the parameter A, x = A + w, (22) where w N(o, 2) is measurement noise. A good unbiased estimate of A would be ⇠ A˜ = x. Since all information needed for the estimation is given in the observed parameter 15 x we can expect that the PDF, probability density function, of x has a strong dependence on the accuracy of the estimation. We can expect that the estimation will be better if the variance 2 of x is small. The likelihood function for the estimation is given by: 1 1 p(x A)= exp (x A)2 . (23) | p 2 2 2 2⇡ It can be assumed that the steeper the curvature is, the better the estimation of A will be. The ’steepness’ of the curvature can measured by the negative second derivative of the logarithm of the likelihood function at the peak. Taking the logarithm yields: 1 ln p(x A)= ln p2⇡ 2 (x A)2. (24) | 2 2 The first derivative is given by: @ln p(x A) 1 | = (x A). (25) @A 2 The negative of the second derivative is given by: @2ln p(x A) 1 | = . (26) @A2 2 As can bee seen; the curvature increases as the variance decreases. For the given case we know that the estimation A˜ = a has variance 2. With these results the following is true: ˜ 1 var(A)= @2ln p(x A) . (27) | @A2 However, in many cases the second derivative will depend on x. In this case it a more suitable measure of the curvature is given by @2ln p(x A) E | , (28) @A2 thus measuring the average curvature of the log-likelihood function [23]. We are now ready to state the Cram´er-Rao Lower Bound theorem [23]. Theorem 1 (Cram´er-Rao Lower Bound for a Scalar Parameter). If the PDF satisfies @p(x ✓) E @✓| =0 for all ✓. Thenh for anyi unbiased estimator ✓˜ the variance must satisfy: ˜ 1 var(✓) @2ln p(x ✓) The derivative is evaluated at the true ✓ value. E | @✓2 h i The denominator is known as the Fischer Information and denoted I(✓) [23]. @2ln p(x ✓) I(✓)= E | . (29) @✓2 16 In this thesis, the point of interest is vector parameter estimations made on signals in white Gaussian noise. What follows is a derivation of the CRLB for an estimation of a scalar parameter. Later we will extend this to a vector parameter estimation in white Gaussian noise. Assume we observe a signal xn in white gaussian noise, with an unknown parameter ✓, N times where n =1,...,N. x = f (✓)+w n =0, 1,...,N w N(0, 2). (30) n n n 2 The likelihood function of the estimation is defined as: 1 1 N p(x ✓)= exp (x f (✓))2 . (31) | (2⇡ 2)(N+1)/2 2 2 n n ( n=0 ) X Di↵erentiating the log-likelihood function: @lnp(x ✓) 1 N @f (✓) | = (x f (✓)) n (32) @✓ n n @✓ n=0 X @2lnp(x ✓) 1 N @2f (✓) @f (✓) 2 | = (x f (✓)) n n . (33) @✓2 n n @✓2 @✓ n=0 ( ) X ✓ ◆ Evaluating the expected value yields: @2lnp(x ✓) 1 N @f (✓) 2 E | = n . (34) @✓2 2 @✓ n=0 ✓ ◆ X ✓ ◆ We can now state our CRLB as [23]: 2 ˜ var(✓) 2 . (35) N @fn(✓) n=0 @✓ An interesting remark that can be madeP is how⇣ the signal⌘ dependence on the parameter, ✓, a↵ects the bound. If a signal has a good response to a change in parameter the es- timation can be made more accurate. The above example will now be extended to estimation of a vector parameter. Assume we want to estimate ✓ =[✓1,✓2,...,✓p]. As derived in [23] the CRLB for each parameter, i, is given by the [i, i] component of the inverse p p Fisher Information matrix. ⇥ 1 var(✓˜ ) [I (✓)] . (36) i ii The components of the Fisher Information matrix is given by 17 @2lnp(x ✓) [I(✓)] = E | . (37) ij @✓ @✓ i j for i =1,...,p and j =1,...,p. Once again (37) is evaluated at the true value of ✓.With signals in white Gaussian noise the above can be stated as [23]: 1 N @f (✓) @f (✓) [I(✓)] = n n . (38) ij 2 @✓ @✓ n=0 i j X 18 4 Modelling and Optimizing Antenna Position Errors in Cell Network As described in Section 2.1, both systematic time delay errors and errors in the reported emitter positions to the system have a significant impact on the accuracy of TDOA sensor positioning. The proposed method addresses these error sources by utilizing a plurality of known sensor positions to improve system information. Emitter positions in system True emitter positions Estimated emitter positions Sensor positions Figure 9: Example of results after implementing the proposed method. The errors in the system emitter positions are corrected by use of known sensor positions. As described earlier; for each sensor one can calculate TDOA values, the di↵erence in time it takes for a signal to travel from one emitter to the sensor and from a second emitter to the sensor, for all pairs of emitters. If the positions of the sensors involved in these measurements are known, by e.g GPS positioning, it is possible to compare the measured TDOA values with an Expected Time Di↵erence of Arrival (ETDOA) derived from the supplied sensor positions. When setting the emitter positions and time delay as variables in calculation of ETDOA it is of interest so see whether, by minimizing the di↵erence between the TDOA and ETDOA, one can improve system emitter positions and to decrease unknown time delay. 19 4.1 Formulation The mathematical formulation of the problem aims at a large extent to resemble actual conditions and error contributions. To not take focus from the question at hand other known deviations and error contributions that are not directly related to the question at hand have been left out. The model consists of n sensors placed in positions s Rd, i =1,...,n and m emitters, i 2 placed in positions e Rd, j =1,..,m. Each sensor is given a TDOA value for each j 2 pair of emitters. In the model this value will be defined by use of the real positions with an additive time delay for each emitter, R1, and unknown additive Gaussian noise, j 2 ✏ R1, for each measurement value. ijk 2 Let ⌧ R1 be the given TDOA measurements and defined as: ijk 2 ⌧ = s e s e +( est est)+✏ . (39) ijk k i jk k i kk j k ijk As described, the objective is to minimize the di↵erence between geven TDOA measure- ments and calculated ETDOA values. Let ˜si be a the supplied sensor position of sensor i, by means of e.g., a GPS. Both cases when ˜si matches the exact position si and when contributing errors, gi, are present, making ˜si = si + gi, will be considered. To estimate the emitter positions and corresponding time delays, we define the following function Rd Rd Rd R: ⇥ ⇥ ! f (eest, eest,x)= ˜s eest + ˜s eest x. (40) i j k k i j k k i k k In the sections that follow are two possible formulations of the optimization problem and their respective performance are later analyzed and compared. Method 1: Pairwise Systems of Equations A set of equations are set up for each successive pair of emitters 1, 2 , 3, 4 ,..., m { } { } { 1,m and the systems of equations are considered separately. This method assumes that } m is an even number. The result is m/2 sets of n equations. The di↵erence between TDOA and ETDOA is minimized by means of NLLS and the optimization is defined as follows: 20 n est est est 2 min fi(ej , ek ,⌧ijk j,k ) eest, est j j,k i=1 X (41) e est Rd j 2 est R1, j,k 2 for i =1,...,n and k = j + 1. The above is repeated for j =1, 3,...,m 1. Here = . The reason for this will be described further in Section 4.1.1. Each set j,j+1 j k of equations consists of n equations, one for every sensor. Each set of equations has two unknown emitter positions in Rd and one time error in R1 corresponding to the pair of emitters and resulting in 2d + 1 unknown variables. Method 2: All Unique Pairs of Emitters In comparison to Method 1 this method will consider and solve for all unknown variables in one larger system of equations. Furthermore, this method enables the unknown time delays for each emitter to be estimated separately relative to a reference emitter. The di↵erence between TDOA and ETDOA is once again minimized by means of NLLS and the optimization is defined as follows: n est est est est 2 min fi(ej , ek ,⌧ijk ( j k )) eest, est j j i=1 1 j m nm(m 1) n = equations. 2 2 ✓ ◆ The system of equations has m unknown emitter positions e Rd,j =1,...,m and j 2 m 1 unknown time delays (see section 4.1.1), in R1. The result is a system of a j total of m(d + 1) 1 unknown variables. 4.1.1 System of Insu cient Rank With no measurement errors and sensor position errors the problem in Method 2 is to solve the set of equations f (eest, eest,⌧ ( est est)) = 0 for all i =1,...,n and i j k ijk j k 1 j 21 are left to solve ( ) ( est est)=0. (43) j k j k For three emitters we can write this on matrix form, 1 10 est 1 2 1 0 1 = 010 11 0 est1 . (44) 1 3 2 est B 2 3C B01 1C B 3 C B C B C B C @ A @ A A @ A The matrix A has rank 2 and 3 unknown| variables.{z In} general, with m emitters the rank in this case will be (m 1) which is insu cient to find an exact solution. However, for TDOA positioning it is su cient to know the di↵erence in time delays for each measure- ment. Therefore we can set an arbitrary emitter delay to be a point of reference. In this study the first emitter’s time delay is set to 1 = 0, decreasing the number of unknowns est by one. Thus making the rank equal to the number of unknowns. The remaining j , j=2,3,...,m are estimated with reference to the first emitter and thereby the di↵erence in time delays, used in TDOA measurements, will be estimated correctly. Since Method 1 only takes into account two emitters in each estimation it will only estimate the di↵erence between the two emitters in question and it is not possible to draw any further conclusions on the time delays relative to other emitters. That is the reason why in Method 1 one only seeks for the pairwise time delay. 4.2 Determination of Weights To dampen the error contributions from inaccurate supplied sensor positions it is of interest to see whether introducing a weighted model, as described in Section 3.2.3, is advantageous. However, in this case one can not assume that one knows the variance of each measurement. One can however assume that one is able to know how accurate a given sensor position, from e.g a GPS, is. By knowing if the given information can be classified as accurate, less accurate and not accurate it is possible to scale with a weight accordingly to dampen the errors. To resemble for examples a GPS’ indication whether a sensor position measurement is accurate, less accurate and not accurate the weighting method used in this thesis de- termines the weights for each sensor on a relative scale. Let the maximal sensor position error norm be , the weights for each sensor are set as: 22 2 accurate: if s ˜s w= || || 3 1 2 less accurate: if s ˜s and < w= (45) || || 3 3 2 2 not accurate: if s ˜s < w=1. || || 3 The above weight determination ensures that GPS positions with a high certainty have a more significant part in the calibration and optimization of the system whereas a less accurate supplied sensor position does not. 4.3 Cram´er Rao Lower Bound Analysis It is of interest to understand better how the geometrical set-up will a↵ect the error variance of estimated emitter positions. We will analyze a case with two emitters, e1 and e R2 and four supplied sensor positions in R2. In this case we are only interested 2 2 in the geometrical aspects and we will therefore disregard the time delay. It is also assumed that all supplied sensor positions are accurate. The sensors, sk, k =1, 2, 3, 4, correspond to the following set of equations for which the solution is to be estimated: fk(e1, e2)=⌧k12 sk e1 + sk e2 + ✏k k k k k (46) for k =1, 2, 3, 4, where ✏ N(0, 2). To find the lower bound on the variance of estimated parameters k ⇠ the Fisher information matrix needs to be defined as described in Section 3.3. To simplify notation, let x be the vector consisting of all four parameters that are to be estimated, x1 e11 0x21 0e121 x = = . (47) Bx3C Be21C B C B C Bx C Be C B 4C B 22C @ A @ A The function of estimation is thus given by f (x)=⌧ (s x )2 +(s x )2 + (s x )2 +(s x )2. (48) k k12 k1 1 k2 2 k1 3 k2 4 p p The entries of the 4 4 Fisher information matrix are stated as: ⇥ 1 k @f (x) @f (x) [I(x] = k n . (49) ij 2 @x @x n=1 i j X Each derivative is given by: 23 @f (x) (s x ) k = k1 1 @x1 (s x )2 +(s x )2 k1 1 k2 2 @f (x) (s x ) k = p k2 2 2 2 @x2 (sk1 x1) +(sk2 x2) (50) @f (x) (x s ) k = p 3 k1 @x3 (s x )2 +(s x )2 k1 3 k2 4 @f (x) (x s ) k = p 4 k2 . @x4 (s x )2 +(s x )2 k1 3 k2 4 p The Cram´er Rao Lower Bound matrix is given by the inverse of the the above Fisher Information matrix. Each diagonal entry of the CRLB matrix corresponds to the estim- ation variance for one of the estimated parameters. As an example, let four sensors be placed in the corners of a square: s = (10 0), s = (0 10), s =( 10 0), s =(0 10). 1 2 3 4 The first emitter is placed in e1 = (0 0) and e2 is admitted to vary. To visualize the res- ults, all diagonal entries of the CRLB matrix are added for each point, thus evaluating trace(CRLB). The variance is set to 2 = 1. The result can be seen in Figure 10. Figure 10: Visualization of the Cram´er Rao Lower Bound Figure 11 shows an example of when the positions of the four sensors are placed at random in a set grid. It is clear that the geometry of where the emitters that are to be estimated are placed does have an impact on the estimation accuracy and that it is possible to choose positions to minimize the estimation error variance. 24 Figure 11: Visualization of the Cram´er Rao Lower Bound when sensors are placed at random 5 Simulations The two methods mentioned in Section 4.1 have been tested with varying parameters and error contributions. The methods used and corresponding results will be presented in this section. 5.1 Method Unless stated otherwise, each set-up is run 100 times, each time the positions and con- tributing errors are randomized and the results from the simulations are aggregated. For each iteration the emitter and sensor positions, sensor position errors, time delays and measurement errors and have been chosen from random distributions to resemble actual errors in a sensors network. All sensor and emitters are randomized in a set region. The initial errors of the emitters are assumed to follow a two dimensional uniform distribu- tion. The measurement noise is assumed to follow a a zero mean normal distribution. The sensor positions errors have a multivariate Gaussian distribution representing an equal Gaussian noise in both coordinates which could be expected from example GPS positioning. The corresponding variance is varied for the di↵erent simulations. The output of variables for each iteration can be summarized as follows: 25 Distribution Notation Parameters Amount of entity s Uniform ( (a, b), (a, b)) a=-2000, b=2000 [10,150] U U stest Uniform ( (a, b), (a, b)) 2 =0, 1.5, 4 100 U U e Uniform ( (a, b), (a, b)) a=-3000, b=3000 4, 10, 20 U U { } eerr Uniform ( (a, b), (a, b)) a=-5, b=5 4, 10, 20 U U { } 2 2 8 ✏ Gaussian (0, ) = 10 One for each TDOA value N 2 g Multivariate Gaussian (µ, ⌃) µ = (0 0)T , ⌃ = 0 2 = 0, 1.5, 4 [10,150] N 0 2 { } 8 8 Uniform (a, b) a= 10 b=10 4, 10, 20 j U { } Table 1: Output of variables for each simulation iteration Let esyst = e + eerr be the positions of the emitters initially known to the system and let eest be the estimated emitter positions after the optimization. For each simulation iteration the following procedure is executed: 1. Calculate initial emitter position errors e esyst . || || 2. Estimate positions of all test sensors using the initially available emitter positions esyst. Estimation of sensor positioning as described in Appendix A. The results is test 100 estimated positions of the test sensors, sest1. 3. Calculate positioning errors s stest || est1|| 4. Run optimization to improve emitter positions and synchronization. The initial values are set to be the ones known to the system currently, esyst. 5. Calculate di↵erence between estimated positions and actual emitter positions e eest . || || 6. Position all test sensors using the new estimated emitter positions eest and estimated time delays. Positioning sensors as described in Appendix A. The results is a second test set of 100 estimated positions of the test sensors, sest2. 7. Calculate new positioning errors s stest || est2|| The results are aggregated for the 100 simulations and presented in the next section. 26 6 Results Results of simulations with di↵erent variations and settings of the mentioned methods are presented below. 6.0.1 Weighted and Non-Weighted Least Squares Below is a results showing the di↵erence between a weighted and a non weighted non linear least squares optimization, as described in 3.2.2 using Method 2. Figure 12: Method 1 The forthcoming results are from simulations that are subject to weighted least squares. 27 6.0.2 No error in sensor positions Figure 13: Method 1 with no sensor error Figure 14: Method 2 with no sensor error 28 6.0.3 Small error in sensor positions Figure 15: Method 1 with small sensor error Figure 16: Method 2 with small sensor error 29 6.0.4 Large error in sensor positions Figure 17: Method 1 with large sensor error Figure 18: Method 2 with large sensor error 30 7 Discussion The weighted NLLS shows a significantly better result than the non-weighted NLLS as seen in Figure 12 when 10 emitter positions are to be improved by use of sensors with a small error contribution. The weighted NLLS provides improvement of emitter pos- itions already when utilizing 10 sensors for the calibration, however the non-weighted requires more than 60 sensors in order to achieve an improvement. As described in Section 3.2.2 it is not possible to know the corresponding variance for each measure- ment and the weighting is made on a relative scale. Although the weights introduced in these tests have proven to be successful and providing a better estimation; it is of interest to test other weights and their determination. It has been assumed that one only knows the magnitude errors corresponding to three categories: high, low or me- dium error contribution and the weights have been set accordingly on a relative scale. It is possible that other values or classifications will further improve the e↵ect of weighting. With no contributing errors in sensor positions we in general see that we are able to improve the emitter position errors significantly even with as little as five sensors used for calibration as can be seen in Figure 13. Method 1 shows no clear correlation between number of emitters and improvements which can be expected since the emitters’ pos- itions are adjusted in emitter pairs. Method 1 also shows a clear correlation between the amount of emitters used for sensor positioning. Method 2 does not perform as well as Method 1 for emitter positioning. Although the results from the sensor positioning suggest otherwise. Since the emitter positions are in comparison better in Method 1, it suggests that the additional available information about relative time delay utilized in Method 2 has a significant impact on the performance of sensor positioning. An interesting observation is that the performance using 20 emitters is inferior to that of 10 emitters for emitter position optimization. It can be assumed that this increase of system size impairs the performance. However, as can be seen the additional emitters contribute to an improved sensor positioning making the positioning results very similar for 10 and 20 emitters. Another interesting observation is the positioning of sensors when using 4 emitters. When comparing to the other results it is only when we are using Method 2 and do not have errors in supplied sensor positions the positioning algorithm can produce good estimates. The estimation of time delay seems to have a significant impact in this case. With a small contributing error in the supplied sensor positions we can see that the amount of emitters in Method 2 now has a much larger contribution to the accuracy as can be seen in Figure 16. With 10 emitters the result is significantly better when the amount of supplied sensors are more than 20. Once again, these results are not as 31 apparent when it refers to the resulting sensor positioning accuracy where the additional emitters almost outweigh the e↵ect of the residual emitter’s position errors. Method 1 does have comparable emitter position optimization to Method 2 but as in the previous example it is clear that it lacks the relative time delay and the sensor positioning accur- acy su↵ers. With a large error in supplied sensor positions the di↵erences become more appar- ent. By comparing the two methods in Figure 17 it is clear that Method 2 outperforms Method 1 to a much larger extent than the previous cases. The result show that in this case we are not able to improve the emitter positions with less than 45 supplied sensor positions. An interesting remark is that when the average emitter position error is, wrongfully, increased after the optimization the information provided after optimiz- ation proves to be better than the original information known before the optimization. It is plausible that with larger errors in supplied sensor positions, the amount of local minima increase. The local minima found may not be the one searched for but non the less the results prove to improve the calibration of the system values used for sensor positioning. 7.1 Future work As described earlier the focus of this thesis has been to make an initial analysis of the performance of the proposed methods in a general setting, taking it to account the most apparent error contributions. There are several other aspects that would be of interest to introduce to the model. The results from above simulations show that it is possible to make improvements even in the case with high error in supplied sensor positions. Provided that the sensor position error can be kept su ciently small it is plausible that a system with su ciently many calibration sensors can manage with other error contri- butions. One already mentioned is the error contribution from signal multipath. These have a large impact on the signal measurements and it would be of interest to analyze whether these errors can be estimated similarly to the time delay errors. If one disregards the height parameter and when one or more sensors or emitters has a height di↵erent from others this will have an impact. A small study of this impact is presented briefly in Appendix B. A possible next step would be to expand the current two-dimensional simulation setting to be three-dimensional. Multialteration positioning techniques generally do not have limitations to two dimensions and it would in many cases be beneficial to estimate positions in three dimensions enabling both more insight- ful positions and furthermore reducing the error contributions from the variations in height. 32 In this study the simulations have been made assuming that all emitters and sensors are visible to each other. It would be of interest to see whether limiting the search scope could further improve the calibration. For example, by limiting the emitters used for calibration depending on known terrain. This could possibly enhance the performance if one introduces error contribution from multipath. It would also be interesting to investigate how the performance is a↵ected if it was possible to assume that a handful of the emitters had an exact known position and time synchronization. It would then be possible to utilize these as fixed reference nodes and hence reducing the amount of variables to be determined in the calibration. 33 8 Conclusion The proposed algorithm has proven to a large extent that it enables to improve calib- ration of emitter networks and thereby also performance of TDOA position estimation. Without contributing errors on the supplied sensor positions the results are significantly improved after running the algorithm. When errors on supplied sensor positions are in- troduced similar results can be attained provided that su ciently many sensor positions are supplied. Furthermore, the weighted NLLS has been proven to be superior to classic NLLS, reducing the necessary amount of supplied sensor positions. Two di↵erent methods have been evaluated and compared. The main di↵erence be- ing that one takes into account separate pair of emitters to the optimization whereas the other takes all available emitters into account in the optimization. Both methods have to a large extent proven to be e cient in correcting emitter positions. However, the method taking in to account all emitters at once is able to a much larger extent correct the time synchronization errors for the system of emitters thus outperforming when comparing performance in sensor positioning after calibration. The downside of the second method is that it is noticeably more computer intensive. There are numerous ways of expanding the current model. There are various error contributions that can be introduced depending on the underlying technology of the positioning system and setting at hand. One common error that would be of interest to expand the model with is to also take into account a height parameter for both emitters and sensors. The calculations in this thesis have been made on theoretical and fictive values and an interesting next step would be to deploy the algorithm in a physical positioning sys- tem in a emitter network. As mentioned earlier, there are various error contributions that become apparent when deploying in a real setting. However, the algorithm has proven to reduce the impact of systematical errors in an emitter network. With that said; the work done in this thesis may be used as a foundation for further development and implementation in a physical emitter network. 34 A TDOA Positioning of Sensors In order to evaluate the proposed algorithms of the thesis, a TDOA positioning model is used. From the available data the sensors are located both before and after the antenna position and time delay improvements have been made. By comparing the accuracy in sensor positioning in both these cases it is possible to see at large if or to what extent the proposed algorithm improves the sensor positioning accuracy. A description of how this TDOA positioning has been done in this thesis is described in this Appendix. As mentioned in Section 3.3, there is plenty of research on the methods of solving these systems as accurately, e ciently and while keeping the computer intensity low. What is common for all methods is that their accuracy relies on the supplied information in form of emitter positions and signal time measurements. The positioning method used in this thesis is not to been seen as the most accurate method but more a tool of reference to compare the results before and after antenna and time syncronisation optimization. A.1 TDOA Positioning by Sum of Least Squares The positioning is locates each sensor separately. First of all available TDOA values are computed as follows: ⌧ = s e s e + (51) jk k jk k kk j k here using the exact values of all variables. Note that in these equations the additive Gaussian measurement noise has been left out. Since this is only for reference all unne- cessary noise has been left out to make the enable a better comparison. The objective is to minimize the di↵erence between the these TDOA values and an ETDOA from what is known in the system and by keeping the sensors positions as variables. Define F (sest) est to be a vector consisting of the all functions fr(s ) for r=1,...,K where K is the number of possible TDOA measurements. In this case a TDOA value is calculated for all unique m pairs of antennas resulting in K = 2 equations. f (sest)= e sest e sest + ⌧ (52) r 1 j i k 1 j k jk Before the optimization of antenna positions and time synchronization is made it is as- sumed that no information about the time delays, , are known and they are all set to be zero. However the emitters relative time delays are estimated in Method 2 (see Section 4.1). 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