Network-based positioning using Last Visited Cells report

Transcription

1 Master of Science Thesis in Communication Systems Department of Electrical Engineering, Linköping University, 2016 Network-based positioning using Last Visited Cells report Tor Olofsson

2 Master of Science Thesis in Communication Systems Network-based positioning using Last Visited Cells report Tor Olofsson LiTH-ISY-EX 17/5040 SE Supervisor: Examiner: Trinh van Chien isy, Linköping University Yuxin Zhao Ericsson Research, Ericsson AB Mikael Olofsson isy, Linköping University Communication Systems Department of Electrical Engineering Linköping University SE Linköping, Sweden Copyright 2016 Tor Olofsson

3 Abstract The positioning performance with the lvc (Last Visited Cells) report is evaluated and compared with extended reports with signal strength data. The lvc report contains cell identities and time spent in the last cells listened to. This is an off-line data source and the purpose of the positioning is to extract information about users whereabouts, which for example can be used to optimize the cellular network or vehicular traffic. The positioning evaluation is done in Matlab with a log-distance model, a fingerprinting algorithm, and a new lvc specific algorithm. A particle filter and a particle smoother is used to process simulated lvc reports and extended reports with different amount of information. The results are compared and evaluated with regard to the positioning accuracy and the information density of the reports. iii

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5 Contents Notation xi 1 Introduction Positioning methods Aim Scope Outline of the report Theory Cellular Networks Last Visited Cells Reports Measurement models RSRP and proximity reports LVC report Channel propagation models Received Signal Strength Antenna Gain Path loss Log-distance model Fingerprinting Training Positioning DOD Voronoi model Angle probability Voronoi Diagram Filtering Dynamic Model Bayesian Filtering Particle Filter Particle Smoother Method 39 v

6 vi Contents 3.1 Data set Creating measurements Evaluating performance Results Simulated data LVC report RSRP report Proximity report Report comparison Discussion Results Particle filter and particle smoother Fingerprinting DOD Voronoi Proximity report Report size Method Model performance Reference cells LOS and NLOS distribution Sample time Data sets Matlab In a wider context Online positioning Fingerprinting Privacy Tracking area size Conclusions 63 Bibliography 65

7 List of Figures 2.1 Tracking area Three directional antenna gains Trilateration Fingerprinting Closest fingerprint estimation Antenna gain difference Angle probability Voronoi diagram Voronoi algorithm vectors Voronoi probability Particle filter measurement update Particle filter time update Particle filter resampling Particle smoother transition density Process overview Map with trajectories Log-distance least squares fit Error cdf for the lvc report Comparison of the lvc report log-distance error for PF and PS Error cdf for the rsrp report Comparison of the rsrp report fingerprinting error for PF and PS Error cdf for the proximity report Comparison of cdf for the reports vii

8 List of Tables 2.1 lvc report example Constant simulation parameters Position rmse and cdf for the lvc report Error statistics for the lvc report Position rmse and cdf for the rsrp report using the log-distance model Position rmse and cdf for the rsrp report using fingerprinting Error statistics for the rsrp report Position rmse and cdf for the proximity report Error statistics for the proximity report viii

9 List of Algorithms 2.1 Selecting fingerprint for position Fingerprint evaluation Voronoi values d P E, d SE Bayesian Filter Particle Filter Particle Smoother ix

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11 Notation Abbreviations Notation bs cdf dod fbr ffbsi fp gps hpbw ldm los lte lut lvc mu nlos nan pdf pf ps rmse rsrp rss ta tdoa tu ue wcdma Meaning Base Station Cumulative Density Function Direction of Departure Front to Back Ratio Forward Filter Backward Simulator Fingerprinting Global Positioning System Half Power Beam Width Log-distance model Line of Sight Long Term Evolution Look-up Table Last Visited Cells Measurement Update Non-Line of Sight Not a Number Probability Density Function Particle Filter Particle Smoother Root Mean Square Error Reference Signal Received Power Received Signal Strength Tracking Area Time Difference of Arrival Time Update User Equipment Wideband Code Division Multiple Access xi

12 xii Notation Mathematical notation Notation R n x p(y x) f (x) h(x) y k e h y i c i x k xk i wk i ˆx k G p G A φ σrsrp 2 d P E d SE ĥ i ĥ i j d th P th y th y min N M N s N c N h T s v φ N (x σ 2 ) Φ N (x σ 2 ) x 2 δ(x) O(x) E(x) Meaning Set of real numbers in n x dimensions Posterior probability for y given x Dynamic model function Measurement function Measurement at time k in particle filter Measurement error for h(x) Measurement for cell i Cell identity for cell i System state at time k State of particle i at time k Weight of particle i at time k Estimated state at time k Path loss Antenna gain Horizontal angle between site and a position Shadowing variance Distance from point P to edge E Distance from seed S to edge E Fingerprint vector with index i Fingerprint value of ĥi for cell j Fingerprinting distance threshold Threshold rsrp for serving cell transition Threshold rsrp for a threshold indicator Smallest observable rsrp Number of measurement cells Number of reference cells Number of reference sites Number of cells in the network Number of fingerprints Sample interval ue velocity Normal pdf with zero mean Normal cdf with zero mean Euclidean distance or 2-norm of x Dirac delta function Computational complexity x Expectancy of x

13 1 Introduction Information about the movement of users has potential to be used for many purposes. The obvious use for movement data gathered in cellular networks is optimizing the cellular network and enhancing the user experience. The 3GPP promotes research contributing to the cause of minimization of drive tests. A drive test is an expensive and limited way of analyzing the network. If the results from a drive test can be achieved with other means, considerable costs for network operators can be avoided. Positioning by the network can be done in real time or in off-line mode. Real time positioning is important for locating people in need during emergency calls, and in USA it is an announced goal from the Federal Communications Commission to increase accuracy and capabilities of handsets [1]. Off-line positioning is of course not helpful in emergencies, but can instead be valuable for large data analyses. Analyses on data from positioning can for example be used for traffic planning to reduce the time spent waiting in queues and to avoid traffic jams. Less time in traffic means less pollutants which in the end would prevent many cases of disease, apart from making the saved time available for productive work. It is no surprise that companies seek to use accessible information to produce additional value. For online cell phones, there are many sophisticated methods available for determining position. One example is the uplink time difference of arrival (TDOA). The case with off-line cell phones and information flowing in a single direction has more limited options. The Last Visited Cells report is a way for cell phones to send stored off-line positioning data to the network. When a cell phone is not used for calls or internet connectivity, the network knows only roughly in what area the phone is [2]. While the phone idly moves, it saves a list of the network cells it listens to and 1

14 2 1 Introduction how long it listened to each cell. This information is saved to the Last Visited Cells list which then can be requested from the phone when it actively connects to the network. The Last Visited Cells report and its use in off-line positioning will be the focus of this thesis. 1.1 Positioning methods Positioning in wireless networks is an active research field. Signal strength based positioning with a path loss model is one of the options investigated [3]. The position accuracy indoors using a path loss model has been reported to be mostly the same when using measured signal strength values and when applying a threshold to them to achieve a binary proximity report indicating which network nodes are strong [4]. Along with the increasing computer power available, the particle filter has become a popular solution to non-linear filtering problems. The particle filter achieves good results in many tracking applications, and enables map matching using large databases [5]. The database approach is commonly called fingerprinting, and has been reported useful for positioning handovers to estimate travel time along with crude position [6]. Accuracy of positioning using signal strengths can be significantly improved with fingerprinting compared to using a simple path loss model [7]. Earlier research at Ericsson has investigated the performance when combining a Direction of Departure algorithm with the Time Advance command to get a position estimate in cellular networks [8]. While the Direction of Departure approach is possible to utilize in the models in this thesis, the Time Advance measurement needs to be substituted with another measure of distance that is not dependent on bidirectional communication with the network. 1.2 Aim The aim of this thesis is to evaluate the positioning ability with the Last Visited Cells report and investigate what positioning improvements can be made by using reports with additional signal strength information. This will be done by answering the following questions. What positioning performance can be achieved with the lvc report? How is the positioning improved with additional signal strength information? 1.3 Scope The Last Visited Cells report is defined in a technical specification of LTE release 13, which will be the reference network standard in this thesis. Evaluations of

15 1.4 Outline of the report 3 algorithms will be done in Matlab with generic computer hardware. Models will be based on a simple log-distance model and a fingerprinting algorithm. 1.4 Outline of the report The outline for the remainder of the report is as follows. Chapter 2 starts with necessary definitions for cellular networks and continues with relevant theory. The measurement reports which are investigated are defined with a probabilistic framework. Algorithms for deriving position probabilities from the reports are then presented and explained. Chapter 3 describes the method in which the theory and the algorithms defined in Chapter 2 are used to simulate measurement reports and evaluate the positioning performance of the reports. Chapter 4 presents the results of the evaluation achieved from following the method in Chapter 3. The results are displayed in figures and tables. Chapter 5 discusses the results and the methods used. Chapter 6 summarizes the thesis with conclusions about the work.

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17 2Theory The methods used in this thesis have a theoretical background and make certain assumptions and simplifications of the environment. This section serves as a basis for the theory behind the methods. Models in this section present probability functions p(y x) which are used in a particle filter. The particle filter uses the probabilities generated by the models to estimate the positions x of a simulated cell phone. 2.1 Cellular Networks Following sections in this master thesis requires some knowledge about cellular networks. This section defines some of the crucial concepts. For the average person, what may come first to mind when thinking of wireless networks is the User Equipment (ue), which usually is a cell phone. The ue acts as a client and is served by the network. While the network is static, the ue is dynamic and moves around and is only spuriously connected to the network. The network connection of a ue is granted by a Base Station (bs), which is located on a site. Put simply, each bs has one or more antennas, and each antenna provides connectivity to a cell. A cell comprises the area where ues are served by the same antenna. The ue estimates the signal strength in a cell with the Reference Signal Received Power (rsrp). In LTE, multiple cells are grouped together into a larger Tracking Area (ta) [9]. When a ue is idly loitering, the network is only interested in knowing its ta and does not need to know the specific cell the ue is listening to. In this case, the ue stays silent while moving between cells belonging to the same ta, and only reports to the network when moving to a new ta. This way, the ta is used to reduce the need for the ue to report its serving cell to the network. When the network needs to contact the ue, it sends a paging message from 5

18 6 2 Theory Figure 2.1: Tracking area. Two tracking areas are shown with a trajectory running through them. The trajectory crosses the boundaries of the hexagonal cells multiple times, but only crosses a tracking area boundary once. The tracking area crossing is indicated by an arrow. multiple cells in the ta to be sure that the ue will receive the message. The reduction of upstream data and thus lower power consumption at the ue is paid for by extra downstream capacity use at the bs. An illustration of cells belonging to different tas is shown in Figure Last Visited Cells When the ue moves to a new ta, it may report a list of its Last Visited Cells (lvc) to the network [10]. The lvc report contains entries of cell identities and the time spent in cells while moving through the ta. There is a maximum of 16 entries that can be stored in the list. If a new entry is added when the list is full, the oldest entry is removed from the list. An example of a lvc report is shown in Table 2.1 on the next page. The lvc report can be interpreted as a list of transitions between cells. For example, if the cell visited before has ID 5 and the next cell has ID 4, it can be interpreted as a transition between cells 5 4. Each entry in the lvc report thus represents a transition between two cells in two consecutive entries. The transitions corresponding to a report is shown in the rightmost column in Table 2.1 on the facing page. This cell transition interpretation is used in this thesis to estimate positions for the lvc report. To be able to receive calls, a ue always listens to at least one cell while waiting to be paged by the network. This serving cell being listened to is assumed to be the strongest cell with a signal to the ue, or at least not much weaker than the strongest cell. With the assumption that the ue wants to have the strongest cell as its serving

19 2.3 Reports 7 Table 2.1: lvc report example. An example of what an lvc report could look like is displayed. Each line contains the time spent and what cell was visited during this time. The entries can also be interpreted as transitions between two cells. The same cell can be visited multiple times and therefore occur more than once. The list also does not need to be exactly 16 entries long. Index Time spent Cell ID Transition cell at all times, the ue selects the strongest cell as its new serving cell when it is perceived as stronger than the current serving cell. When this event with a new serving cell occurs, the ue adds a cell transition to its lvc list. To reduce the amount of cell transitions, a threshold for minimum signal strength difference to the current serving cell is used [11]. This threshold acts as a hysteresis and avoids filling the list with numerous transitions between two neighbouring cells with approximately the same signal strength. To further reduce the amount of transitions, there is also a time-to-trigger which defines the minimum time for which the threshold condition must hold before a transition actually is done. 2.3 Reports In this thesis, three different types of reports for serving cell transitions are used. Each time entry in a report is described by a measurement vector y which is used as input to a filter. The Last Visited Cells report is the first report. Two other reports using rsrp values are also defined. These two reports are designed so they might contain more and better information than what is available in the limited lvc report. The measurement vector for the lvc report is defined as y = [c new c old c 1 c M ], (2.1) where c new and c old identifies the old and the new serving cell for a transition, and c i identifies reference cells which should be weaker than c new for i = 1,..., M. The second type of report is the rsrp report. This report describes the N strongest cells with cell identities and an rsrp value for each cell. This variant

20 8 2 Theory with explicit rsrp values for cells is used in [7, 12, 4]. The measurement vector is defined as y = [c 1 y 1 c 2 y 2 c N y N c N+1 c N+M ], (2.2) where c i is the cell identity and y i is the rsrp for cell i in the report. Additionally, M cells which should be weaker than y N are included. The third report is the proximity report. Instead of using explicit values for the measured rsrp of cells, the report contains the N cells which are stronger than or equal to a threshold y th, and M cells which are weaker than the threshold. This report is similar to a report in [4], which used a static threshold. A disadvantage with a static threshold when using the lvc sample times is that occasionally no cells at all are above the threshold, which provides very little accuracy for position prediction. The measurement vector is defined as y = [c 1 c 2 c N y th c N+1 c N+M ], (2.3) where c i as before is the cell identity. The vector is defined so that the cell c N defines the threshold y th, and therefore is equal to the threshold while N 1 cells in the list are stronger than the threshold. While the three reports are described by different vectors, they are all assumed to be sampled at the time of a cell transition as defined by the lvc report. The two rsrp reports are therefore regarded as extensions to the lvc report. They both contain c new as their strongest cell, but c old is not necessarily included. 2.4 Measurement models The reports are used to estimate positions by passing their measurement vectors y through a filter. In the filter, a probability function p(y x) is used to extract information from the vectors. This section presents what this function looks like for the different reports. The probabilities for explicit rsrp measurements will be explored first, and then an adaptation for the cell transitions in the lvc report will be made. A general definition of the probability p(y x) will be made for the reports, and then this will be used to define more precise specifications for a log-distance method and a fingerprinting method which are described later. Assume that a measurement y from a ue can be stated as y = h(x) + e h, (2.4) where h(x) is a deterministic function giving a predicted rsrp value for a position x, and the term e h is the error of the measurement y because of noise and an imperfect prediction function. The definition in Equation (2.4) can be rewritten as e h = y h(x), (2.5) which is an equation to calculate the error e h for a measurement y given a position x. Using a known distribution for the error e h, the equality Equation (2.5) is used to infer probabilities for measurements.

21 2.4 Measurement models RSRP and proximity reports For the measurements in the rsrp report, y describes a numeric rsrp value. The probability p(y x) for a measurement y given a position x is then defined using Equation (2.5) on the preceding page as p(y x) = p(e h = y h(x)), (2.6) which is evaluated with the distribution for e h. Both the rsrp report and the proximity report expects the rsrp of some cells to be on a certain side of a threshold. For the cells c i in the vector y which are related to a threshold and do not have a numeric rsrp value, an implicit threshold indicator y i is used for the probability calculation. Using the notation in Equation (2.4) on the facing page, an rsrp measurement from a cell is converted to a threshold indicator as 0, h(x) + e y = h < y th, (2.7) 1, h(x) + e h y th where e h is the error of the actual rsrp measurement, and y th denotes the rsrp value used as the threshold. A threshold indicator y = 0 infers the inequality which is rewritten as h(x) + e h < y th, (2.8) e h < y th h(x) (2.9) to get an inequality which is used to get the probability p(y x) as p(e p(y x) = h < y th h(x)), y = 0 1 p(e h < y th h(x)), y = 1. (2.10) Now there is a definition for the probability p(y x) for a single measurement y given a position x in both the case when y is a numeric rsrp value and a threshold indicator. The probability for a measurement vector y in the rsrp report or the proximity report can then be stated as p(y x) = N+M i=1 p(y i x), (2.11) where the proper definition of p(y i x) is used depending on if y i is a numeric rsrp value or a threshold indicator. If the distribution for e h is assumed Gaussian, the evaluation of the probability functions in Equation (2.6) and Equation (2.10) can be stated more explicitly. A Gaussian distribution is described by the normal pdf and cdf. For a distribution with a mean of zero, the normal pdf is defined by φ N (x σ 2 1 ) = σ x2 e 2σ 2 (2.12) 2π

22 10 2 Theory and the normal cdf is defined by Φ N (x σ 2 1 ) = σ 2π x e x2 2σ 2 (2.13) where x is the deviation from the mean, and σ 2 is the variance of the distribution. For the measurements y, the error e h acts as the deviation from the mean. Using the normal pdf and cdf, the probability function p(y x) for numeric rsrp values is written as p(y x) = φ N (y h(x) σ 2 e h ), (2.14) and the probability for a threshold indicator is written as Φ p(y x) = N (y th h(x) σe 2 h ), y = 0 1 Φ N (y th h(x) σe 2 h ), y = 1, (2.15) where σ 2 e h is the variance of the error e h. These two definitions in Equation (2.14) and Equation (2.15) are used to calculate the probabilities p(y x) for measurements in the rsrp report and the proximity report, given a prediction function h(x). The actual definitions of h(x) which are used to evaluate the report performance are specified later in the log-distance method and the fingerprinting method. With the probability definitions specified for both numeric rsrp values and threshold indicators, the total probabilities for the measurement vectors in Equation (2.11) on the previous page can be stated in expanded form for the rsrp report and the proximity report. Recall that the rsrp report is represented by a vector containing rsrp values and cell identities for N cells, and only cell identities for M cells which are weaker than the N:th cell. The proximity report is represented by a vector containing M cell identities and a threshold y th which separates all but one of the cells into two groups on either side of the threshold. The probability p(y x) for the rsrp vector can be stated using the equality in Equation (2.14) and the inequality in Equation (2.15) as p(y x) = N φ N (y i = h i (x) σe 2 i ) i=1 N+M i=n+1 and the probability for the proximity report is stated as N 1 p(y x) = 1 Φ N (y th h i (x) σe 2 i ) φ N (y th = h N (x) σe 2 N ) i= LVC report Φ N (y N h i (x) σ 2 e i ), (2.16) N+M i=n+1 Φ N (y th h i (x) σ 2 e i ). (2.17) While the rsrp report and the proximity report uses a single measurement of rsrp to create y, the lvc report is assumed to use two measurements for the

23 2.4 Measurement models 11 equality y new y old = P th, (2.18) which describes the cell transition condition where P th is the rsrp threshold for a transition. Because the lvc report only contains two cell identities c new and c old for the new and the old serving cell, the measurements y new and y old can not directly be used. By substituting y new and y old using Equation (2.4) on page 8, the equation becomes h new (x) + e new h old (x) e old = P th, (2.19) which is rewritten with the error terms on the left-hand side as e new e old = P th h new (x) + h old (x), (2.20) where the error terms are used for the probability function p(c new, c old x) = p(e new e old = 0), (2.21) which determines the probability that the equality in Equation (2.18) is valid. The lvc report is also used to assure that the cell c new is the strongest cell for position x. This can for another cell y i be stated as y i < y new, (2.22) which by substitution using Equation (2.4) on page 8 becomes and results in the probability function e i e new < h new (x) h i (x) (2.23) p(c new, c i x) = p(e i e new < h new (x) h i (x)). (2.24) The probabilities for Equation (2.21) and Equation (2.24) can be stated with the normal pdf and cdf if assuming a Gaussian distribution for the errors e h. Both these equations contains two separate error terms with one distribution each. As the sum of two Gaussian variables is also a Gaussian variable, with a combined variance equal to the sum of the two variances, the probabilities can be interpreted as having a single error using a variance with two parts. The probability for Equation (2.21) for the transition condition can then be stated as p(c new, c old x) = φ N (e new e old σnew 2 + σold 2 ), (2.25) which by using Equation (2.20) for e new e old becomes p(c new, c old x) = φ N (P th h new (x) + h old (x) σnew 2 + σold 2 ). (2.26)

24 12 2 Theory Likewise, the probability for Equation (2.24) on the previous page which compares weaker cells to the new serving cell is stated as p(c new, c i x) = Φ N (h new (x) h i (x) σnew 2 + σi 2 ). (2.27) The total probability p(y x) for the measurement vector representing the lvc report can now be stated using these two equations. The vector contains the cell identities c new and c old for the new and old serving cell, as well as M reference cells which should be weaker than c new. The probability for the lvc report is stated as N p(y x) = φ N (P th h new (x) + h old (x) σnew 2 + σold 2 ) Φ N (h new (x) h i (x) σnew 2 + σi 2 ), where P th as before is the transition threshold. i=1 (2.28) 2.5 Channel propagation models Line of Sight (los) and Non-Line of Sight (nlos) are used to describe propagation situations in radio communication. los indicates that there are no obstacles in the direct path between the transmitting end and the receiving end, while nlos indicates that there are obstacles blocking the signal and that the strongest signal path is not necessarily the shortest. Even if there is los, the signal quality can still be heavily compromised by reflections on surfaces and diffraction effects. Measurements relying on the signal strength and signal propagation time are generally good in los scenarios, while they can be utterly unreliable in a nlos case. To increase robustness of a model against nlos, two separate error distributions for los and nlos can be combined. An example of a two-mode Gaussian mixture is p E = αn (0, σlos 2 ) + (1 α)n (0, σ N 2 LOS ), (2.29) which can be configured for different scenarios by varying the set of parameters [3]. The parameters σlos 2, σ N 2 LOS and α in Equation (2.29) describe the variances for the los and nlos distributions and their weights when combined. 2.6 Received Signal Strength A ue continuously measures the Received Signal Strength (rss) of nearby cells to be able to request a hand-over if the signal of current serving cell becomes too weak. The rss is estimated by calculating the power of specific reference signals, which yield Reference Signal Received Power (rsrp) which will be used as rss in this thesis. The rsrp is dependent on the transmit power P bs of the base station, which is affected by losses in the feeder cable, L c, and antenna gain, G A. During the propagation of the signal from the bs to the ue, a channel and path dependent

25 2.6 Received Signal Strength 13 gain will be present, here notated as G p (d). Using these notations and expressing the gain in db, the received signal strength P r at the ue can be written as a sum of many separate terms P r = P bs + L c + G A + G p db. (2.30) The path loss gain G p in this case comprises attenuation due to signal spread, multipath interference and shadow fading Antenna Gain The typical site has three directional antennas which each cover 120. The gain of a directional antenna can be expressed as a function of polar coordinates, G A (φ, θ), where φ is the horizontal angle and θ is the elevation angle relative to the horizontal plane [8]. To simplify, the antenna gain is often considered to be separable into a horizontal and vertical gain component G A (φ, θ) = g Ah (φ) + g Av (θ) db. (2.31) where g Ah (φ) and g Av (θ) are the horizontal and vertical gain respectively. Information about the antenna gain is typically available as parameters in the antenna data sheet. With a few parameters describing the antenna characteristics, a standardized model can be used to calculate the approximate gain of multiple antennas. Another possibility is to use a look-up table (lut) of the antenna gain values for φ and θ. In this thesis, the elevation angle θ will be neglected. Only the horizontal gain g Ah (φ) in Equation (2.31) will be used, and use the notation G A (φ) for this partial antenna gain. Further, the horizontal angle φ will be referred to as the Direction of Departure (dod). An example of the horizontal antenna gains of a site with three antennas is displayed in Figure 2.2 on the following page. As can be seen in the figure, each antenna dominates its own interval of angles Path loss To be able to use rsrp measurements for estimating a distance, a path loss model is needed. The path loss can potentially be a very complicated function of the signal propagation environment. In the case that there is little information available about the propagation environment, a simple path loss model is needed. A common model for propagation losses of radio signals is the log-distance model G p (d) = 10B log( d d 0 ) + C db. (2.32) The model in Equation (2.32) uses a constant C which corresponds to the path loss at a reference distance d 0. The path loss is then determined by the

26 14 2 Theory 0 5 Antenna 1 Antenna 2 Antenna 3 10 Antenna gain [db] Angle [degree] Figure 2.2: Three directional antenna gains. The antenna gains for three antennas sharing the same site are shown. Their maximum gains are located on three different angles separated by 120 to optimize coverage. distance d and the path loss exponent B, which is dependent on the propagation environment and usually resides in the interval [2, 4]. Path loss prediction is a method used in trilateration when a set of rsrp measurements are used to estimate a position. An example of trilateration is displayed in Figure 2.3 on the facing page. 2.7 Log-distance model The path loss equation in Equation (2.32) on the previous page is able to make a prediction of the path loss component of the rsrp in Equation (2.30) on the preceding page. Only the path loss G p and the antenna gain G A are dependent on the distance or the angle, which together describes a position relative to a bs. The other terms of P r in Equation (2.30) on the previous page are in that sense constant and can be merged into a single constant term. In the following model, the constant is defined as the expected rsrp at distance d 0 and is denoted A 0. By adding terms for the expected rsrp and the antenna gain G A (φ) for the direction φ, a complete model for calculating the expected rsrp for an arbitrary position is acquired as P r (d, φ) = A B log( d d 0 ) + G A (φ) db, (2.33)

27 2.8 Fingerprinting 15 Figure 2.3: Trilateration. The rsrp from a cell can with a path loss function be translated to a distance. In this figure, three circles are created from rsrp values. A position for the measurements can then be estimated by finding the intersection of the three circles, here indicated by an arrow. where d and φ describes a relative position. Equation (2.33) on the facing page will be referred to as the log-distance model. It will be used as a prediction function h(x) introduced in Section 2.3 on page 7 to use for calculating the probabilities p(y x) for measurement vectors y given a position x. This model will be used to make predictions for the rsrp from cells, using one distinct function for each cell to be able to represent different site positions and values for the parameters A 0 and B. 2.8 Fingerprinting An alternative to using an explicit model to predict the rsrp of cells to compare to the reported measurements is using a method called location fingerprinting [7, 13]. Fingerprinting is an algorithm which uses pattern recognition to identify how well new measurements match a database of so called fingerprints from previously collected measurements. While the log-distance model in Equation (2.33) on the facing page does not perform well for significant shadowing variances and nlos conditions, fingerprinting has been shown to perform better for these cases [14]. Using fingerprints, an inverse function can be emulated to estimate a position from measurements of rsrp values. An example of a function being represented by fingerprints is displayed in Figure 2.4 on the next page. This simple form of machine learning requires a first off-line phase of collecting a training data set to create a fingerprint database. In this off-line training phase, an area is selected and fingerprint vectors for positions in the area are sampled. No computations are made with the training data set until the fingerprints are compared to measurements in an on-line positioning phase.

28 16 2 Theory sin(x) 0 sin(x) x [ ] x [ ] (a) Continuous function y = sin x. (b) Map of values from x to y. Figure 2.4: Fingerprinting. A continuous function is shown together with a fingerprinting representation of the same function. Fingerprinting creates a map which links samples from the set x to samples in the set y. Note that the fingerprint map only represents a finite set of mapped value pairs and its precision is limited by the density of samples. While the positions of sites and transmitting power of cells are known to the network operator and are used in the log-distance model, the fingerprinting method does not need such information. Because of this independence of information otherwise needed in an explicit model, a fingerprinting database can be created by virtually anyone and not only the network operator. One example of a crowd sourced database used for location fingerprinting is the Mozilla Location Service [15]. This open service relies on individuals collecting data as well as organizations sharing information about infrastructure, and allows users to get a position from observations of various networks like cellular networks and Wifi networks Training The fingerprints describe features of the training data, which in this thesis will be the rsrp of cells in the network. Each fingerprint vector describing these features is also associated with a position. Both rsrp and position are continuous variables, and the goal of the training phase is to create a database which links positions to vectors of rsrp values. There are different approaches to creating this database. One alternative is that the provider of the service collects accurate estimates for all possible positions and under different weather conditions. This might be expensive and time consuming. Another alternative is to let users with GPS receivers contribute with samples to expand the database, as exemplified with the Mozilla Location Service. However, with users having different measuring equipment, the data might be less accurate. A third option is to let a simulation framework predict samples

29 2.8 Fingerprinting 17 using a model. The last option is used in this thesis. The same simulation model will be used to generate the fingerprint database and the measurements in the model performance evaluation. This will in a sense be similar to the case where the real world is used to create both the database and evaluation measurements. A square grid of equidistant points will be assumed for the reference positions of the fingerprints which describe the rsrp features of nearby cells. Each grid point is then associated with a fingerprint vector of rsrp values. The distance between the grid points will be one of the limits for how well a position can be estimated. As each fingerprint in practice will represent a square area and not just a specific position, the fingerprint should be created from an average of multiple measurements in the area. Additionally, the fingerprints should also be averaged from multiple measurements over time to avoid large influence of temporary fading effects. Assume like before in Equation (2.4) on page 8 that the observed rsrp y from a cell depend on the position x as y = h(x) + e h (2.34) where h(x) represents a deterministic path loss and e h represents noise from shadowing. An average measurement ĥ(x) is then defined as E(y), y y ĥ = min, (2.35) NaN, y < y min where E(y) is the expectancy of y, which here is interpreted as the average of y in the local area around the position x over a period of time. To model the case when a signal is too weak to be measured by a non-ideal receiver, a threshold y min is used to set a lower bound for the fingerprint measurements. The fingerprint ĥ is then set to NaN, not a number, as an indicator if its rsrp is below the threshold y min. The fingerprint vector ĥ(x) is defined as the collection of averaged measurements ĥ(x) = [ĥ1(x) ĥ 2 (x) ĥ Nc (x)], (2.36) where ĥj(x) is the averaged measurement for cell j and N C is the number of cells in the network. When using a grid with a finite set of points with fingerprints, each fingerprint position can instead be indexed with a number. Let p i be the position for the fingerprint at the grid point indexed by i. The fingerprint vector in Equation (2.36) can then be stated as ĥi for grid point i with the elements ĥ i = [ĥi 1 ĥ i 2 ĥ i N c ], (2.37) where ĥi j is the averaged measurement for cell j at the grid position pi.

30 18 2 Theory Positioning When the training phase is completed, new measurements of rsrp values can be compared to the fingerprint vectors. The fingerprints will like the log-distance model be used to provide a prediction function h(x) as defined in Section 2.3 on page 7. Unlike the log-distance model, the fingerprints only covers a discrete set of positions x, and also sometimes does not always have a numeric value for each cell. This section describes how these issues are handled. Because the location fingerprinting method works with a finite set of reference positions and reference vectors of rsrp values, some kind of conversion has to take place to translate a continuous position x to a fingerprint. This allows the use of the probability p(y x) for a vector y as defined in Section 2.4 on page 8. In the solution chosen here, any position x will be mapped to the closest reference position p i and use its fingerprint vector ĥi. To restrict the possible positions to the training area, a position will be assigned zero probability if it its distance to the closest reference position is more than d th. This mapping to the closest reference point can be compared to using a set of step functions around the reference positions to generate a partially continuous function. It also has the effect that the integration of the probability over x R 2 does not equal unity, which luckily is ignored in a particle filter. An example of using the closest fingerprint like this is illustrated in Figure 2.5 on the facing page. The algorithm to select a fingerprint vector based on a position x is described in Algorithm 2.1. Algorithm 2.1: Selecting fingerprint for position Position estimation: Using fingerprints ĥi on positions p i, the posterior probability p(y x) for a measurement vector y given a position x is determined as following. 1. Calculate the distance of the position x to each of the grid points as 2. Find the grid point closest to x as d i = x p i 2 (2.38) î = arg min d i (2.39) i 3. Calculate the probability p(y x) as p(y x) = p(y ĥî), dî d th (2.40) 0, dî > d th where d th is a threshold which limits the positioning to the training area. Having defined the translation from continuous positions to the finite set of reference positions in the fingerprinting database, only reference positions will

31 2.8 Fingerprinting sin(x) 0 sin(x) x [ ] x [ ] (a) Fingerprinted function. (b) Partially continuous function. Figure 2.5: Closest fingerprint estimation. A partially continuous function is created from the fingerprint map by assigning each value of the continuous variable x to the fingerprint of the closest sampled x. The resulting function is constructed by a sequence of step functions. be considered in the algorithm for comparing fingerprints. The probability p(y x) will be calculated as in Section 2.4 on page 8, with the fingerprint h i j as the value of the prediction function h(x), where i is the index closest to the position x and j is the cell that y represents. This is correct for when h i j has a numeric value. For the case when h i j is NaN and only known to be below the threshold y min, a new definition for p(y x) is needed. When the fingerprint value is below the threshold y min, so that ĥi j = NaN, the probability for a measurement y is given by which by using Bayes rule can be rewritten as p(y ĥi j ) = p(y ĥi j = NaN), (2.41) p(y ĥi j ) = p(ĥi j = NaN y)p(y) p(ĥi j = NaN), (2.42) which depends on the prior probability p(y) and the probability p(ĥi j = NaN), which both are unknown and hard to find. In [7], another approach is suggested to produce an appropriate likelihood for this case which is possible to calculate using only known variables and has provided satisfactory results. The value of p(y ĥi j ) when ĥi j = NaN is defined so that it satisfies the inequality p(y ĥi j ) p(y ĥi 2 j ), (2.43) where the fingerprint value on the right-hand side ĥi 2 j NaN is above the threshold, while ĥi j = NaN is not. The inequality in Equation (2.43) is a feature to avoid

32 20 2 Theory locations where the difference between y and ĥi j could be large, while preferring locations where the difference is known to be limited, which in this case is ĥi 2 j. The proposed method to achieve this for y with a numeric rsrp value is where the set M j is defined as p(y ĥi j ) = min m M p(y ĥm j ), (2.44) j M j = { i ĥi j NaN}. (2.45) This method assures that the probability in Equation (2.44) is valid for the inequality in Equation (2.43). For a threshold indicator y meaning h(x) + e h < y th, the value for h(x) can simply be set to y min so that p(y ĥi j ) = p(y h(x) = y min). (2.46) While a numeric y j does not produce a different probability depending on the sign of its difference to ĥi j, the threshold indicator does. This means that it is not unreasonable to use y min for the threshold indicator, while it is for the numeric measurement y which may yield the same probability for a fingerprint ĥi j with a very high value and a very low value. The algorithm described above is summarized in Algorithm 2.2 on the next page. 2.9 DOD Voronoi model In this section, a new method for estimating positions from the lvc report is presented. The method uses two separate models, a Direction of Departure (dod) model to restrict positions with regard to angles and a nearest neighbour (Voronoi) model to limit distances to sites. Only the information in y = [y new y old ] about the new and the old serving cell is used for signal strength assumptions. The combination of the dod model and the Voronoi model is combined to a new measurement model used for the lvc report in addition to the log-distance model and fingerprinting. This combined dod and Voronoi model is simply defined by the product of their two individual probabilities, here denoted by p DOD (y φ) and p V (y x). Using the position x to calculate the angle φ between x and the site, the dod and Voronoi model probability function is stated as Angle probability p(y x) = p DOD (y φ)p V (y x). (2.51) With measurements from two cells on the same site, an approximation of the Direction of Departure (dod) can be made. The following method uses the difference in the antenna directions for the cells and consequently the antenna gains to derive an angle from rsrp measurements.

33 2.9 DOD Voronoi model 21 Algorithm 2.2: Fingerprint evaluation Training: Collect a training data set of fingerprint vectors ĥi for i = 1,..., N h defined by their position p i and their rsrp values ĥi j for j = 1,..., N c, where N c is the number of cells in the reference network and N h is the number of fingerprint vectors. Let NaN be the value of ĥi j when the observed rsrp is below the threshold y min. Select fingerprint: Select a fingerprint ĥi for position x using Algorithm 2.1 on page 18. Probability evaluation: The probability p(y x) for a measurement vector y given a fingerprint ĥi is determined as in Section 2.4 on page 8 with the probability p(y x) defined as following, with ĥi j as the fingerprint for the cell j that y represents. If ĥi j is a numeric value above the threshold y min, use If ĥi j is NaN, below y min, use when y is a threshold indicator, and p(y x) = p(y ĥi j ). (2.47) p(y x) = p(y h(x) = y min ), (2.48) p(y ĥi j ) = min m M p(y ĥm j ), (2.49) j when y is a numeric rsrp and the set M j is defined as M j = { i ĥi j NaN}. (2.50)

34 22 2 Theory The two measurements from two cells i and j can each be written in the form of Equation (2.30) on page 13, giving P r,i = P bs,i + L c,i + G A,i + G p,i db, (2.52) P r,j = P bs,j + L c,j + G A,j + G p,j db. (2.53) With two antennas sharing virtually the same position, the complicated and typically inaccessible path loss G p is assumed to be the same for both rsrp measurements at the ue. This assumption simplifies the problem by disregarding scenarios where a different nlos paths may be stronger than the los path. In these cases, the actual rsrp measurements by the ue could represent different signal paths with different departure angles, and invalidate the assumption of equal G p. Still, it is the best assumption possible to make without knowledge of the environment in which the signal propagates. By assuming equal path loss G p from the two cell antennas, the difference between Equation (2.52) and Equation (2.53) can be put as P r,i P r,j = G A,i G A,j + P bs,i P bs,j + L c,i L c,j db. (2.54) The transmit power P bs and the feeder loss L c are easily available for a bs, and their differences between cells i and j might from now on be assumed to cancel out to zero. Remaining unknown in the equation, is the difference of the antenna gains H ij (φ) = G A,i (φ) G A,j (φ) db. (2.55) The probability for an angle φ can be derived from the difference of antenna gain. With information about the direction and the directional gain of each antenna, a relative antenna gain function can be constructed as the difference of antenna gain for each φ. A diagram of the relative antenna gain H ij for two of the antennas in Figure 2.2 on page 14 is illustrated in Figure 2.6 on the facing page. The diagram in Figure 2.6 on the next page shows that the dod can be extracted from where the relative gain intersects the measured rsrp difference. As the relative gain function is continuous, there will be an even number of intersections for any rsrp difference. Two intersections will be found in the normal case. However, it is apparent that one angle is unlikely as it is in the territory of the third antenna, which was not strong enough to be one of the two measured. By comparing the antenna gains of the two measured antennas with the third, it is easy to select the correct angle. The vector y = [y new y old ] contains information about the cell transition. This information indicates whether the transition was between cells on the same site or if the cells belonged to different sites. Let G A,i be the expected perceived antenna gain for cell i, and N be the number of cells on the site, where the cells are ordered by decreasing signal strength so that the strongest cell is first. The strongest cell is then the new serving cell y new which is indexed by 1. For a transition between cells on the