networks with acoustic range...

Institutionen för systemteknikDepartment of Electrical Engineering

Examensarbete

Sensor Localization Calibration of Ground SensorNetworks with Acoustic Range Measurements

Examensarbete utfört i Reglerteknikvid Tekniska högskolan vid Linköpings universitet

av

Viktor Deleskog

LiTH-ISY-EX--12/4624--SE

Linköping 2012

Department of Electrical Engineering Linköpings tekniska högskolaLinköpings universitet Linköpings universitetSE-581 83 Linköping, Sweden 581 83 Linköping

Sensor Localization Calibration of Ground SensorNetworks with Acoustic Range Measurements

Examensarbete utfört i Reglerteknikvid Tekniska högskolan vid Linköpings universitet

av

Viktor Deleskog


Handledare: David Lindgrenfoi

Hans Habberstadfoi

Niklas Wahlströmisy, Linköpings universitet

Examinator: Fredrik Gustafssonisy, Linköpings universitet

Linköping, 18 september 2012

Avdelning, InstitutionDivision, Department

Avdelningen för ReglerteknikDepartment of Electrical EngineeringSE-581 83 Linköping

DatumDate

2012-09-18

SpråkLanguage

� Svenska/Swedish

� Engelska/English

�

�

RapporttypReport category

� Licentiatavhandling

� Examensarbete

� C-uppsats

� D-uppsats

� Övrig rapport

�

�

URL för elektronisk version

http://www.ep.liu.se

ISBN

—

ISRN


Serietitel och serienummerTitle of series, numbering

ISSN

—

TitelTitle

Kalibrering av Sensorpositioner i Sensornätverk med Akustiska Avståndsmätningar

Sensor Localization Calibration of Ground Sensor Networks with Acoustic Range Measure-ments

FörfattareAuthor

Viktor Deleskog

SammanfattningAbstract

Advances in the development of simple and cheap sensors give new possibilities with largesensor network deployments in monitoring and surveillance applications. Commonly, thesensor positions are not known, specifically, when sensors are randomly spread in a big area.Low cost sensors are constructed with as few components as possible to keep price and en-ergy consumption down. This implies that self-positioning and communication capabilitiesare low. So the question: “How do you localize such sensors with good precision with a feasi-ble approach?” is central. When no information is available a stable and robust localizationalgorithm is needed.

In this thesis an acoustic sensor network is considered. With a movable acoustic source awell-defined and audible signal is transmitted at different spots. The sensors measure thetime of arrival which corresponds to distance. A two-step sensor localization approach isapplied that utilizes the estimated distances.

A novel approach in the first step is presented to incorporate more measurements and gainmore position information. Localization and ranging performance is evaluated with simu-lations and data collected at field trials. The results show that the novel approach attainshigher accuracy and robustness.

NyckelordKeywords sensor network, calibration, sensor localization, multidimensional scaling, ranging, time de-

lay estimation

http://www.ep.liu.se

Abstract

Advances in the development of simple and cheap sensors give new possibilitieswith large sensor network deployments in monitoring and surveillance applica-tions. Commonly, the sensor positions are not known, specifically, when sensorsare randomly spread in a big area. Low cost sensors are constructed with asfew components as possible to keep price and energy consumption down. Thisimplies that self-positioning and communication capabilities are low. So the ques-tion: “How do you localize such sensors with good precision with a feasible ap-proach?” is central. When no information is available a stable and robust local-ization algorithm is needed.

In this thesis an acoustic sensor network is considered. With a movable acousticsource a well-defined and audible signal is transmitted at different spots. Thesensors measure the time of arrival which corresponds to distance. A two-stepsensor localization approach is applied that utilizes the estimated distances.

A novel approach in the first step is presented to incorporate more measurementsand gain more position information. Localization and ranging performance isevaluated with simulations and data collected at field trials. The results showthat the novel approach attains higher accuracy and robustness.

iii

Acknowledgments

First I would like to give a big thanks to my supervisors at foi: David Lindgrenand Hans Habberstad, for all insightful comments and help during my work withthis thesis. With David’s expertise in sensor networks and signal processing andHans’s knowledge and experience in acoustics made no problems invincible.

For my supervisor Niklas Wahlström and examinator Prof. Fredrik Gustafsson atLinköping University I give thanks for the weekly feedback during my work withideas and further improvements.

I would like to thank David, Niklas, Gustaf Hendeby (foi), and Fredrik for valu-able comments on structure, language, and technical descriptions during the edit-ing of this thesis.

Lastly, I stand in great gratitude to my family and friends who gave me endlessjoy and support during my studies. Big up!

Linköping, September 2012Viktor Deleskog

v

Contents

Notation ix

I Background and Theory

1 Introduction 31.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Thesis Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Concepts 72.1 Sensor Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Sensor Localization Calibration . . . . . . . . . . . . . . . . . . . . 72.3 Multidimensional Scaling (mds) . . . . . . . . . . . . . . . . . . . . 82.4 mds and Sensor Localization . . . . . . . . . . . . . . . . . . . . . . 82.5 Ranging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Theory 113.1 Sound Propagation in Air . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Classical Multidimensional Scaling . . . . . . . . . . . . . . . . . . 133.3 Least Squres Fitting of Two Point Sets . . . . . . . . . . . . . . . . . 15

II Models and Estimation

4 Models 214.1 Signal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Time Delay Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 Sensor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Estimation 25

vii

viii CONTENTS

5.1 Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Fundamental Performance Bounds . . . . . . . . . . . . . . . . . . 265.3 Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.4 Sensor Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Signals 416.1 ofdm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.2 Chirp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.3 Birdsong: Black-Throated Loon . . . . . . . . . . . . . . . . . . . . 44

III Results and Discussion

7 Data Collection 497.1 Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.2 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8 Results 558.1 Ranging Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558.2 Sensor Localization Scenario . . . . . . . . . . . . . . . . . . . . . . 55

9 Concluding Remarks 679.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

IV Appendices

A Scenario Configurations 73A.1 Indoors Configuration . . . . . . . . . . . . . . . . . . . . . . . . . 73A.2 Randomized Configuration . . . . . . . . . . . . . . . . . . . . . . . 74A.3 Benchmark Configuration . . . . . . . . . . . . . . . . . . . . . . . 74A.4 Outdoors Configuration . . . . . . . . . . . . . . . . . . . . . . . . 75

B Evaluation of Signal parameters 79

Bibliography 83

Notation

Symbols

Symbol Description

sn Position vector of the n:th sensorxk Position vector of the k:th sourceSN Set of N sensorsXK Set of K sourcesy ik Single toameasurement at sensor i from source k

yijk

Single tdoa measurement between sensors i and jfrom source k

yik Measurement vector with M toa measurements atsensor i from source k

yijk Measurement vector with M tdoa measurements be-tween sensors i and j from source k

Rxx(t) Autocorrelation function (acf)Ryx(t) Cross correlation function (ccf)Φxx(ω) Power spectral density (psd)Φyx(ω) Cross spectral density (csd)

E Expectation operator~ Convolution in time domainx[n] Discrete signal sampled with frequency fs

DFT{ · } The dft of a discrete signal|| · || Euclidean 2-norm· T Matrix transpose

h(xk ; θ) Nonlinear function of state vector xk parameterized bythe parameter vector θ

ix

x Notation

Parameters

Parameter Description

c Propagation speed[

ms

]fs Sampling frequency [Hz]B Bandwidth [Hz]fc Centrum freqency [Hz]Ts Signal duration [s]br toa range measurement bias [m]N Number of sensorsK Number of sourcesM Number of multiple measurements

Definitions

Definition Description

E =∫ T0 |s(t)|2dt

Energy of signal s(t)

SNR = E/σ2e snr of a signal s(t) observed inwgn with variance σ2eSNRdB =

10 log10 SNRsnr in logarithmic decibel (dB) scale

e ∼ N (0, σ2) e is Gaussian distributed with zero mean and covari-ance σ2

x̄ =1N

∑Ni=0 x[i]

Sample mean estimator of data vector x

Notation xi

Abbreviations

Abbreviation Description

acf Autocorrelation Functionccf Cross Correlation Functioncmds Classical Multidimensional Scalingcrlb Cramér-Rao Lower Boundcsd Cross Spectral Densitydft Discrete Fourier Transformfoi Swedish Defence Research Agencygcc Generalized Cross Correlationgps Global Positioning Systemlplse Linear Phase Least Squares Estimatorls Least Squareslse Least Squares Estimatormc Monte Carlomds Multidimensional Scalingnls Nonlinear Least Squaresofdm Orthogonal Frequency Division Multiplexingpsd Power Spectral Densityrmse Root Mean Square Errorscc Simple Cross Correlatorsnr Signal to Noise Ratiosvd Singular Value Decompositiontde Time Delay Estimationtdoa Time Difference of Arrivaltoa Time of Arrivaltpi Triple Parabolic Interpolationwgn White Gaussian Noise

Part I

Background and Theory

1Introduction

A sensor network is a set of connected and cooperating sensors. A deployed sen-sor network can be used in various applications, often in the area of surveillanceand monitoring activity around buildings and objects worthy of protection. Asensor can be any type of device that can measure signals in their environment,e.g. sound waves, ground vibrations, or light. With the development of simpleand inexpensive sensors expands the possibilities for large networks with manysparsely spread sensors to achieve better coverage and precision.

If an unknown object, e.g. humans or vehicles, is detected by the sensors mea-suring the signals generated by the object. With these signals the object can belocated by combining the measurements and the positions of the sensors. So theposition of the sensors is central to locate the target. Different applications havedifferent accuracy requirements on the positions. Regardless accuracy require-ment, the sensor network should provide a process to determine the positionsthat focuses on simplicity and effectiveness.

This chapter presents the problem formulation, assumptions, related work andthe outline of the thesis.

1.1 Problem Formulation

In this thesis the problem of calibrating sensor positions in acoustic ground sen-sor networks is studied. The main goal is to develop a calibration method thatfrom erroneous range measurements estimates the positions of the sensor nodesindependent of any prior knowledge of the positions. The problem can thereforebe divided into two sub-problems:

3

4 1 Introduction

(i) Estimate range at far distances from time delay measurements based onacoustic wave propagation.

(ii) Determine the sensor positions, without any prior information of the net-work topology, by using erroneous range estimates.

Both algorithms should be evaluated by simulations and by data collected at fieldtrials.

1.2 Assumptions

To keep the work into the time frame of the thesis some assumptions have beenmade to fulfil the time constraint:

• The sensor nodes are synchronized.

• The atmosphere is considered as a homogeneous acoustic channel with knownpropagation speed.

• The acoustic source is considered to be stationary while emitting the acousticsignal.

• An emitted signal from the source can be sensed by all sensor nodes in thenetwork.

1.3 Related Work

The problem of sensor localization have been investigated since early 2000’s. In[Patwari et al., 2005], IEEE Signal Processing Magazine, various algorithms andfundamental issues in the area are summarized and discussed.

In [Moore et al., 2004] the concept of robust quadliterals is introduced to localizesensors with noisy sensor-to-sensor distances using trilateration. Sensors are di-vided into clusters and are able to be localized up to a global rotation and transla-tion which fits in our problem setting. The localization algorithm is divided intomultiple phases. This approach is showed to be sound where the first phase findsrough positions and the second refines them to reduce effects of measurementnoise using numerical optimization. In [Shang et al., 2004] a algorithm basedon Multidimensional scaling (mds) is applied to localize the sensors instead oftrilateration. The problem is formulated as least squares problem and solved bySingular value decomposition (svd). Finding the svd is a fast operation com-pared to ordinary iterative least squares solvers.

The mds approach is used in this thesis but a novel extension is presented tolower the influence of measurement noise by incorporating more measurementsthat are available in our problem setting.

In this thesis the acoustic propagation delay is used to estimate range. The con-cept of time delay estimation is introduced in [Knapp and Carter, 1976] which

1.4 Thesis Work 5

formulates the problem and presents and derives commonly applied time delayestimation algorithms. In [Gustafsson et al., 2010] a Orthogonal frequency divi-sion multiplexing (ofdm) modulated signal is used by a time delay estimationalgorithm that exploit the connection between phase and delay to estimate thetime delay with least squares theory. Benefits gained with the algorithm pre-sented in [Gustafsson et al., 2010] are sub-sample resolution and quality metric.In [Ahlström and Falk, 2001] Triple parabolic interpolation (tpi) is presentedto achieve sub-sample resolution when integer sample estimation algorithms areused. Experiments of outdoors acoustic time delay estimation is presented in[Ash and Moses, 2005] where interesting signal parameters and their impact onestimation performance are evaluated.

Sound propagation is covered in [Jönsson and Nilsson, 2007] and [Blackstock,2000] where fundamental concepts of sound waves and physical acoustics arepresented and derived.

1.4 Thesis Work

This report is a part of a master’s thesis at the department of Electrical Engineer-ing (isy) at Linköping University under Prof. Fredrik Gustafsson at the divisionof Automatic Control. The work has been performed at the Swedish DefenceResearch Agency (foi) under supervision of Dr. David Lindgren, and Hans Hab-berstad.

This thesis work is a part of foi Centre for Advanced Sensors, Multisensors andSensor Networks (focus) funded by the Swedish Governmental Agency for Inno-vation Systems (vinnova), and the Knowledge Foundation (KK-stiftelsen).

1.5 Thesis Outline

This thesis is divided into different levels where the top levels:

• Background and Theory

– Introduction

– Concepts

– Theory

• Models and Estimation

– Models

– Estimation

– Signals

• Results and Discussion

– Data Collection

6 1 Introduction

– Results

– Summary

are the three main cornerstones. This structure is used to give a better overviewof the distinct parts of thesis. Background and Theory a brief introduction tothe problem, concepts, and the interesting theory behind existing methods andderivations are covered. Later in Models and Estimation the model and sensordefinitions, and derived estimators are covered. In Results and Discussion thedata collection, results, and summary are covered.

Two appendices are enclosed in this thesis covering sensor network configura-tions and evaluation of signal parameters.

2Concepts

This chapter introduces some important concepts, e.g sensor networks, sensorlocalization, mds, and ranging, that are applied and investigated in the thesis.

2.1 Sensor Network

A Sensor Network (sn) consists in general of multiple connected nodes [Gustafs-son, 2010]. The nodes can communicate either by wire, or wireless communica-tion channels dependent on application. A node can either be a sensor (receiver)or a target (transmitter). When operational, the sensors detect a target and givesome type of observation of it, a measurement.

A measurement can be some sort of range (toa, tdoa), bearing (aoa), or energy(RSS) related to the target. A common application of a sensor network is to esti-mate a position of an observed target by utilizing the measurements, [Lindgrenet al., 2010].

2.2 Sensor Localization Calibration

Calibration is the problem of finding or estimating the unknown configurationparameters of the sensor nodes in a sensor network [Gustafsson, 2010]. With sen-sor localization calibration, the unknown position parameters of the sensor nodesare estimated. Calibration is important in a sensor network application becauseit has to be performed before it can be used for its proposed task. Calibrationoften involves dedicated experiments with known premisses where parametersare estimated or measured.

7

8 2 Concepts

Sensor localization calibration can be performed in numerous ways where sensorcapabilities and resources define the limits. By equipping sensors with a simpleGlobal Positioning System (gps) receiver the absolute position can be determinedwith accuracy up to 5 - 15 m. To equip all sensors with such receiver, is a bothcostly and energy consuming way to achieve accurate positions. Other scenariosinclude reference nodes which are nodes with known positions [Patwari et al.,2005]. If the other sensors can relate their measurements to the available refer-ence nodes they can estimate their positions using the known absolute positions.In [Moore et al., 2004, Sottile and Spirito, 2008] methods are presented that relyon inter-sensor communication where each sensor can measure the distance be-tween neighbouring sensors and with trigonometric relations calculate their po-sition.

The positions that are estimated with a localizing algorithm can either be definedin a world fixed coordinate system or in a locally fixed coordinate system. De-pending on prior knowledge of the sensor network configuration the estimatedpositions can be correct up to a arbitrary point transformation (translation androtation/reflection).

2.3 Multidimensional Scaling (MDS)

Multidimensional Scaling (mds) is a family of data analysis methods. Thesemethods calculate a low-dimensional representation of high-dimensional data byusing a defined model. It started in the 1950s with Torgerson who introduced thegeneral topic of mds applied in psychometrics1 [Young, 1987]. His method wasthe first systematic algorithm that provided a multidimensional map of pointsfrom erroneous inter point Euclidean distances, known as metric mds.

A mdsmodel can be defined in different spaces. In general the model is a simplealgebraic equation with some geometric interpretation, e.g. Euclidean distance ina Euclidean space. The essence of mds is to visualize complex data in a spatiallyrepresentation. The data can be represented by a matrix whose elements indicatesome relation to each other, e.g. correlations, distances, and similarities.

2.4 MDS and Sensor Localization

In this thesis the derivedmdsmethod Classical Multidimensional Scaling (cmds)[Torgerson, 1965] will be used as basis to estimate sensor positions utlilzing erre-nous range measurements [Shang et al., 2004]. Localization problems are oftenformulated as a standard optimization problem. To solve such problems itera-tive solvers are generally used where the initial start parameters are crucial toguarantee convergent solutions. With cmds it can instead be solved in closedform.

1Field of study with the theory and technique of psychological measurements.

2.5 Ranging 9

2.5 Ranging

Ranging is the problem of estimating the distance between two points, e.g. be-tween a transmitter and a receiver. Here we will estimate the time delay when anacoustic signal is transmitted from an acoustic source and received by an acous-tic sensor through air. With the estimated time delay the range can be foundby multiplying with the propagation speed of sound in air. Various methods onTime Delay Estimation (tde) exist where [Knapp and Carter, 1976] covers themost used variants which is based on estimating the Cross-Correlation Function(ccf) between the transmitted and the received signal. In [Gustafsson et al., 2010,Chan et al., 1978] the estimated Cross Spectral Density (csd) between the trans-mitted and received signals is used to exploit the relationship between phase shiftand time delay.

3Theory

This chapter presents and describes the theory and methods behind the devel-oped algorithms. By studying the theory it will be easier to comprehend thealgorithms and to discover potential improvements and constraints.

3.1 Sound Propagation in Air

Sound is longitudinal waves that propagate through a material medium, often agas. The speed of sound, denoted c, that we are interested in is the speed at whichthis wave propagates through air. From the acoustic wave equation [Jönsson andNilsson, 2007, p. 96]

∂2s

∂t2=

1κρ

∂2s

∂x2(3.1)

we can see from the general wave equation that the propagating speed is actually

c2 =1κρ

=⇒ c = 1√κρ

(3.2)

where κ is the compressibility factor and ρ is the density of the gas. The com-pressibility factor can be defined by assuming that sound waves propagate asan adiabatic process. During an adiabatic process the entropy in the mediumremains constant, isentropic. If we assume that the energy losses are negligiblein air, which is common in acoustics, the entropy is constant [Blackstock, 2000,p. 32-33]. For an adiabatic process it follows that

P V γ = k (3.3)

11

12 3 Theory

Table 3.1: Coefficients for air

Coefficient Valueγ 1.40

M 0.029 kg/mol

R R0M = 287 J/kgK

where k is a constant, P is pressure, V is molar volume and γ is the ratio ofspecific heats. By writing (3.3) as

P = k ·V −γ

we can take the derivative of P with respect to V as

∂P∂V

= −γ · k ·V −γ−1

and replace the constant gives

∂P∂V

= −γ · k ·V −γ−1 = −γ · P V γ ·V −γ−1 = −γP V −1 (3.4)By definition the compressibility factor is defined as [Jönsson and Nilsson, 2007,p. 97]

κ ≡ − 1V∂V∂P

= − 1V

(1

−γP V −1)

=1γP

(3.5)

where the derivative ∂V /∂P is replaced with the one in (3.4). So the sound speedin a gas is according to (3.2)

c =1√κρ

=

√γP

ρ(3.6)

Through experiments the expression in (3.6) holds so the assumptions above arecorrect. By applying the perfect gas law

P = RρT (3.7)

where T is the absolute temperature and R is the universal gas constant dividedby the molecular weight of air the speed of sound in air can be simplified

c =

√γP

ρ=

/Pρ

= RT/

=√γRT . (3.8)

In this application the gas is air so we know the constant values of γ and R, seeTable 3.1. One interesting detail is that sound speed in a gas varies as the squareroot of absolute temperature.

When sound propagates through air it is attenuated both by the square-distance

3.2 Classical Multidimensional Scaling 13

101

102

103

104

105

10−4

10−2

100

102

104

Frequency [Hz]

Sou

nd A

bsor

ptio

n [d

B/1

00*m

]

Figure 3.1: Sound absorption in air as a function of frequency at 20 ◦C per100 m, [Blackstock, 2000]

and by the absorption of sound in the atmosphere. In [Blackstock, 2000, p. 513-516] the absorption coefficient α [nepers/m] in the atmosphere is described to bedependent on frequency. In Figure 3.1 the absorption coefficient α [dB/100 · m]is showed as a function of frequency. So signals with high frequency contentgets attenuated because of sound absorption in the atmosphere. If you are near astrike of a lightning during a thunderstorm you both hear a low-frequency boomand high-frequency sound but at far distances you just hear the boom. So thehigh-frequency sound is gone because of different sound absorptions.

3.2 Classical Multidimensional Scaling

In the cmds framework we have distance data which we want to visualize ina low-dimensional geometric representation, often 2-D or 3-D. The data is Eu-clidean distances D ∈ RN×N amongst nodes X = (x1, . . . , xN )T . The nodes areCartesian points xi = (x1, . . . , xm)T ∈ Rm in a m-dimensional Euclidean space.The information from the distances D is sufficient to exactly reconstruct thenodes X up to translation, rotation, and reflection [Torgerson, 1965].

First assume that the nodes in X are known, then the outer product, B ∈ Rm×m,of X is given by

B = XXT (3.9)

which can be connected to the distances in D as

d2ij = (xi − xj )T (xi − xj ) = xTi xi − 2xTi xj + xTj xj = bii − 2bij + bjj (3.10)where the matrix elements are defined as

dij := ||xi − xj ||bij := x

Ti xj .

(3.11)

14 3 Theory

So if the outer product matrix B can be calculated from the distances D, the nodescould be found by factoring B as in (3.9). In [Torgerson, 1965] it is shown that Bcan be calculated by double-centering D2, where D2 is the element-by-elementsquared distance matrix of D. A matrix is double-centered as follows: for eachelement subtract column and row mean, adding the grand mean, and divide by-2. Each element bij in B is calculated as

bij = −12

d2ij −

1N

N∑k=1

d2ik︸︷︷︸Row mean

− 1N

N∑k=1

d2kj︸︷︷︸Column mean

+1N2

N∑g=1

N∑h=1

d2gh︸︷︷︸Grand mean

. (3.12)

The nodes X can then be recreated through the (svd) of B.

B = UΣV T = UΣU T

X̂ = U√Σ

(3.13)

where U = V as B is symmetric. U is an orthogonal unitary matrix and Σ isa diagonal matrix containing the singular values Σ = diag(σ1, σ2, . . . , σN ). Thesingular values are ordered as σ1 ≥ . . . ≥ σN . In (3.13) the solution X̂ have dimen-sion N because of the high-dimensional distance data in D. From the beginningwe know that the distances are generated from points in a m-dimensional space.The best low-rank approximation, in Least squares (ls) sense, of the original N -dimenional solution is given by using the m first singular values and left singularvectors in B as

Bm = UmΣmUTm

X̂ = Um√Σm

(3.14)

where Um contains the m first left singular vectors and Σm = diag(σ1, . . . , σm)contains the m first singular values. The quality of the low-dimensional approxi-mation can be calculated with

gof =∑mi=1 |σi |∑Ni=1 |σi |

(3.15)

which is the Goodness of fit (gof). The solution X̂ in (3.14) gives the best approx-imation

Bm = X̂X̂T (3.16)

of B in ls-sense [Torgerson, 1965]. The method is summarized in Algorithm 1.

Example 3.1: cmds and Swedish citiesOne simple example to demonstrate possible applications of cmds is to findthe locations of cities by the distance between them. Here we will apply thison the distances between six cities in Sweden: Stockholm, Gothenburg, Malmö,Linköping, Jönköping, and Kiruna. Table 3.2 shows the distances between thecities and with cmds their 2-D/3-D locations can be calculated.

3.3 Least Squres Fitting of Two Point Sets 15

Algorithm 1 Classical Multidimensional Scaling, cmds

1: Form distance matrix D from known or estimated distances between nodes.2: Calculate the squared double centered matrix B from D.3: Calculate the svd of B, B = UΣU T .4: Keep the m largest singular values and their singular vectors and define them

as Σm and Um.5: X̂ = Um

√Σm are the recreated nodes with dimension m.

Figure 3.2 shows the recreated map of the cities and the goodness of fit whichgives the best fit when m ≥ 2 (2-D points), so two dimensions is sufficient torepresent the distances as Euclidean points, which is intuitive because the differ-ence in elevation between the cities is much lower than the distances amongstthem. Note that a one-dimenional solution would also represent a rather good fitbecause Sweden is more elongated than wide.

The recreated map has been rotated to a consistent view of the true map due tothe nature of cmds.

Table 3.2: Distances between cities in kilometers, calculated from decimaldegrees (wgs84)

Stockholm Gothenburg Malmö Linköping Jönköping KirunaStockholm 0 397.9 513.3 174.8 284.7 955.8

Gothenburg • 0 242.2 228.8 131.2 1204.1Malmö • • 0 349.0 252.0 1414.5

Linköping • • • 0 109.9 1078.2Jönköping • • • • 0 1163.1

Kiruna • • • • • 0

3.3 Least Squres Fitting of Two Point Sets

Consider two point sets X = (x1, . . . , xN ) and Y = (y1, . . . , yN ) where the pointsx and y are two-dimensional Cartesian points x, y ∈ R2×1. The relation betweenthe points can be described by

yi = Rxi + T + ei (3.17)

where R ∈ R2×2 is a transformation matrix (rotation or reflection), T ∈ R2×1 istranslation vector, and ei ∈ R2×1 is a noise vector. The goal is find the Leastsquares estimator (lse) of R and T

(R, T ) = arg minR,T

N∑i=1

||yi − (Rxi + T )||2 . (3.18)

16 3 Theory

Stockholm

Gothenburg

Malmö

LinköpingJönköping

Kiruna

(a) Recreated map

1 2 3 4 5 60.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

1.005

Dimensions

GO

F

(b) Goodness of fit

Figure 3.2: Recreated map of Swedish cities and gof

The lse is often found by iterative algorithms but instead a closed-form solutioninvolving the svd can be used [Arun et al., 1987].

If R̂ and T̂ are the lse to (3.18) then Y and Ŷ = (ŷ1, . . . , ŷN ), where ŷi = R̂xi , havethe same centroid, i.e

ȳ = ¯̂y (3.19)

where

ȳ :=1N

N∑i=1

yi (3.20)

¯̂y :=1N

N∑i=1

ŷi = R̂x̄ + T̂ (3.21)

x̄ :=1N

N∑i=1

xi . (3.22)

Denote

x̃i := xi − x̄ (3.23)ỹi := yi − ȳ . (3.24)

3.3 Least Squres Fitting of Two Point Sets 17

Algorithm 2 Finding R̂ using the svd

1: Calculate the "covariance" matrix H ∈ R2×2

H =:N∑i=1

x̃i ỹTi (3.27)

2: Find the svd of HH = UΣV T (3.28)

3: Calculate the lse R̂ asR̂ = VU T (3.29)

The minimum in (3.18) can instead be reformulated as

R̂ = arg minR

N∑i=1

||ỹi − Rx̃i ||2 (3.25)

where R̂ is the optimal transformation matrix in ls sense. [Arun et al., 1987]. Theoriginal ls problem is therefore divided into two parts:

1) Find the lse R̂ that minimizes (3.25) with Algorithm 2.2) The translation T̂ is found by

T̂ = ȳ − R̂x̄ . (3.26)A closed-form solution to the problem in (3.18) is presented involving the svd.The found orthogonal matrix R̂ can potentially contain reflections in addition torotations. If a correct rotation matrix is required some changes in Algorithm 2can be implemented, see [Arun et al., 1987], but is not considered in this thesis.

Part II

Models and Estimation

4Models

With a model, we intend to find a mathematical relationship that describes aprocess that we want to investigate or predict its future behavior. Sensors receivesignals that are emitted from an acoustic source causing a time delay of the signal.From these delayed signals we intend to find the propagation delay to estimatethe travelled distance. Therefore it is required that three types of models needsto be defined to simulate the processes:

Signal model Describes the properties of the emitted signal from the acousticsource.

Time delay models Describes how the transmitted signal is delayed when trav-eling from source to sensor.

Sensor models Describes how a sensor interprets time delay measurements intodistances in the presence of simulated measurement noise.

4.1 Signal model

A signal, s(t), is emitted from an acoustic source with specific parameters. Thefrequency content of the signal is defined in the frequency set

Ω ∈ [fc − B/2, fc + B/2] (4.1)and non-zero in the interval [0, Ts] where

• B: Bandwidth, [Hz]• fc: Center frequency, [Hz]• Ts: Signal duration, [s]

21

22 4 Models

The energy E of a signal is finite and is calculated as

E =Ts∫

0

|s(t)|2dt (4.2)

where the physical unit is discarded. The sampled version of the signal s(t) isdefined as s[n], where it is sampled with the sampling frequency fs ≥ 2(fc + B/2)(Nyquist rate).

4.2 Time Delay Models

An acoustic signal, s(t), is emitted from a remote source and observed by a spa-tially separated sensor in the presence of white Gaussian noise (wgn) e(t) duringthe time T = Ts + Tmax where Tmax is the maximum considered time delay. Thedelayed signal can then be modelled as

y(t) = as(t − τ) + e(t), 0 ≤ t ≤ T (4.3)where τ corresponds to the time delay and a is a possible attenuation of the signal.The signal s(t) is assumed to be uncorrelated with the noise e(t), E s(t)e(t) ≡ 0.We can also extend this model to cover the case when a signal is received at twospatially separated sensors as

y1(t) = s(t − τ) + e1(t), 0 ≤ t ≤ T (4.4a)y2(t) = s

(t − (τ + δ)

)+ e2(t), 0 ≤ t ≤ T (4.4b)

where δ is the difference in time arrival between the sensors during the timeT = Ts + Tmax. The signal s(t) is assumed to be uncorrelated with the two noiseterms, e1(t) and e2(t).

4.3 Sensor Models

A sensor gives some type of observation of a target xk which is interesting tous. A typical sensor output yk is modelled by a function h( · ) of a target xk andparameterized by a set of parameters θ in the presence of noise ek which iswgnwith zero mean and covariance σ2e

yk = h(xk ; θ) + ek , e ∼ N (0, σ2e ) (4.5)where the parameters can consist of necessary properties of the sensor, e.g sensorposition and measurement bias. The target xk can generate multiple measure-ments at the sensor which will be denoted with boldface yk = {yk}M1 ∈ R1×M ,i.e xk generates M sensor measurements stored in the measurement vector yk .

4.3 Sensor Models 23

4.3.1 Time of Arrival (TOA)

Using the time delay model in (4.3) an acoustic signal from source xk is modeledto reach sensor si at time

hT OA(xk ; si , br ) =

1c||xk − si || + br (4.6)

where the model is parameterized by sensor position si and measurement bias brand c is the known propagation speed. A toa sensor then gives timing measure-ments as

τ ik = hT OA(xk ; si , br ) + e

ik , e

ik ∼ N (0, Rik) (4.7)

where eik is independentwgn. This toa sensor can be converted to a range sensorby multiplying the timing τ ik with the propagation speed c

r ik = c · τik = c · hT OA(xk ; s

i , br ) + vik , v

ik ∼ N (0, c2Rik) (4.8)

where vik is wgn. Figure 4.1a shows the ranging measurements rik and r

jk from

two toa sensors.

4.3.2 Time Difference of Arrival (TDOA)

Using the time delay model in (4.4) the difference in time arrival between sensorssi and sj when a signal is transmitted from source xk will be on the form

hT DOA(xk ; si , sj ) =

1c

(||xk − si || − ||xk − sj ||

)(4.9)

where the model is parameterized by the two sensor positions si and sj . A tdoasensor then gives timing measurements as

τijk = hT DOA(xk ; s

i , sj ) + eijk , eijk ∼ N (0, R

ijk ) (4.10)

where eijk = eik − e

jk is wgn. This tdoa sensor can also be converted to a range

sensor by multiplying the timing τ ijk with the propagation speed c

rijk = c · τ

ijk = c · hT DOA(xk ; s

i , sj ) + vijk , vijk ∼ N (0, c2R

ijk ) (4.11)

where vijk is wgn. Figure 4.1b shows the ranging measurement rijk from a tdoa

sensor.

24 4 Models

xk

si

sj

r ik

rjk

(a) Two toa sensors outputs the

range measurements r ik and rjk .

xk

si

sj

rijk

(b) A tdoa sensor outputs the

range measurement rijk which is

the time-difference of arrival be-tween sensors si and sj .

Figure 4.1: Range measurements from two toa sensors and one tdoa sensorgenerated by target xk .

5Estimation

This chapter presents in detail the developed and applied estimation algorithms;ranging and sensor localization, and a brief introduction to the least squares (ls)estimation framework and fundamental estimation bounds.

5.1 Least Squares

When to find the parameters of a mathematical model the theory of system iden-tification can be applied. System identification is to find the model parameterswhich best describes the observed data [Gustafsson et al., 2010, chapter 6]. Avery common model structure is the linear Gaussian model

yk = φTk θ + ek , ek ∼ N (0, R), k = 1, 2, . . . ,N (5.1)

where yk is the observed data at time instant k, φk is the known regression vector,θ is the unknown model parameters and ek iswgnwith variance R. So the goal isto estimate the unknown parameters θ given the observed data. The least squaresmethod minimizes the cost function

V (θ) =1N

N∑k=1

(yk − φTk θ)2 =1N

N∑k=1

�k(θ)2 (5.2)

where �k(θ) is the residual or model error at time instant k. The lse θ̂ can beformulated as

θ̂ = arg minθ

V (θ) (5.3)

where the operator arg min is the minimizing argument. So θ̂ is the estimate ofθ that minimizes the cost function in (5.2). To keep the problem compact it can

25

26 5 Estimation

instead be formulated with vector notation

YN =

y1y2...yN

, ΦN =φT1φT2...φTN

, EN =e1e2...eN

(5.4)so (5.1) can be written as

YN = ΦNθ + EN (5.5)

With vector notation the ls cost function takes the form of

V ls(θ) =N∑k=1

(yk − φTk θ)T (yk − φTk θ) = (YN −ΦNθ)T (YN −ΦNθ) (5.6)

and the lse

θ̂ls

= arg minθ

V ls(θ) = (ΦTNΦN )−1ΦTNYN (5.7)

which is the minimizing argument of (5.6). ls problems can efficiently be solvedusing numerical methods discussed in [Gustafsson et al., 2010, p.224-225] wherethe QR factorization is suggested instead of calculating the matrix inverse in(5.7).

5.2 Fundamental Performance Bounds

In estimation theory the Cramér-Rao Lower Bound (crlb) expresses a lowerbound on the variance of any unbiased estimator [Kay, 1993, p. 27]. An unbi-ased estimator θ̂ has the property

E[θ̂] = θ0 θ0 ∈ Θ (5.8)where Θ denotes the range of possible values of the true parameters θ0. In prac-tice the crlb provides a benchmark on the theoretical lower bound of any unbi-ased estimator.

The crlb is given by the Fisher Information Matrix (fim) I (θ) [Gustafsson, 2010,p. 81]. With the assumption of Gaussian measurement errors e ∼ N

(0,R

)the

fim equals

I (θ) = JT (θ)R−1J(θ)J(θ) = ∇θh(θ)

(5.9)

where ∇θh(θ) is the Jacobian evaluated at θ. The crlb is given byCov(θ̂) = E(θ0 − θ̂)(θ0 − θ̂)T ≥ I−1(θ0) (5.10)

When investigating localization performance the Root Mean Square Error (rmse)gives a practical measure of the mean position error in meters. In two dimensions

5.3 Time Delay 27

the achievable position rmse is given by

rmse =√

E[(θ01 − θ̂1)2 + (θ02 − θ̂2)2

]=

√tr Cov(θ̂) ≥

√trI−1(θ0) (5.11)

which is implied from the crlb.

When multiple independent measurements are available their corresponding in-formation matrices can be added because information is additive [Gustafsson,2010, p. 81]

I1:M = I1 + I2 + . . . + IM =M∑k=1

Ik

Cov(θ̂) ≥ I−11:M(5.12)

and with more information the better estimation accuracy.

5.3 Time Delay

From the background on ranging in Section 2.5 two commonly and easy to im-plement estimation methods are chosen to estimate the time delay parameter inthe models defined in Section 4.2. The first method is based on the Cross Corre-lation Function(ccf) which is the Simple Cross Correlator (scc) and the secondis based on the Cross Spectral Density (csd) which is the Linear Phase LeastSquares Estimator (lplse).

5.3.1 Cross Correlation Method - SCC

With the time delay model (4.3) we can compute the ccf

Rys(t) = E y(l)s(l − t) = E[[as(l − τ) + e(l)] s(l − t)

]= (5.13)

aRss(t − τ) + Eindependent︷︸︸︷e(l)s(l − t)︸︷︷︸

=0

= aRss(t − τ) = aRss(t) ~ δ(t − τ) (5.14)

where δ(t) is the Dirac delta function and ~ denotes convolution in time domain.So it turns out that the ccf is a time shifted attenuated copy of the Autocorrelac-tion function (acf) Rss(t). At time t = τ the ccf should take its maximum value.Due to sampled signals and finite observation time we must estimate the ccf[Gustafsson et al., 2010, p. 96]

R̂ys[k] =1N

N−k−1∑m=0

y[m + k]s[m], k = 0, 1, . . . , N − 1 (5.15)

28 5 Estimation

−5000 0 5000−1

−0.5

0

0.5

1

Lag [samples]

Nor

mal

ized

cor

rela

tion

0 50 100 150 200−1

−0.5

0

0.5

1

Lag [samples]

Nor

mal

ized

cor

rela

tion

Figure 5.1: Estimated ccf of the transmitted and received signal. Left: Dur-ing the whole duration of received signal. Right: Zoomed in around themaximum value at τ = 100.

where we observe N samples sampled with frequency fs. We can then form thetime delay estimate τ̂ as

∆̂ = arg maxk

R̂ys[k]

τ̂ = fs∆̂(5.16)

where ∆̂ is the estimated delay in samples. (5.16) is the Simple Cross Correlator(scc) estimate. In theory thewgn term disappears due to independence w.r.t. thetransmitted signal but in a real application that is not the case. A band pass filterapplied on the received signal y would increase the snr and just keep the signalinformation in the frequency set Ω.

Example 5.1: scc EstimatorA signal s(t) with bandwidth B = 500 Hz and center frequency fc = 1 kHz is

transmitted to a receiver. The receiver observes the signal with snr 10 dB as theBP-filtered signal y(t). The true delay between the transmitter and the receiveris 100 samples. With scc the delay is estimated to τ̂ = 100 samples which is thetrue delay. Figure 5.1 shows the ccf R̂ys[k].

Sub-sample Resolution

In some cases the maximum value of the ccf is not attained at the true delaysince the resolution of the delay estimate τ̂ in (5.16) is limited to multiples of 1fs .In [Ahlström and Falk, 2001] the method Triple Parabolic Interpolation (tpi) ispresented to interpolate the ccf around the maximum peak. The method fits aquadratic polynomial with three points, the peak (τ̂) and the surrounding twopoints. The interpolated ccf around the found peak at ∆̂ is modelled as

R̂ys,tpi[k] = θ1 + θ2k + θ3k2, k = ∆̂ − 1, ∆̂, ∆̂ + 1 (5.17)

5.3 Time Delay 29

98 98.5 99 99.5 100 100.5 101 101.5 1020.92

0.93

0.94

0.95

0.96

0.97

0.98

Lag [samples]

Nor

mal

ized

cor

rela

tion

Figure 5.2: Interpolated ccf with tpi. Solid: ccf. Dotted: ccf +tpi

where the unknown parameters in θ are estimated using ls. The sub-sampleestimate ∆̂tpi can is estimated by setting the derivative of R̂ys,tpi w.r.t. the samplelag k to zero

ddkR̂ys,tpi[k]

∣∣∣∣∣k=∆̂tpi

= θ̂2 + 2θ̂3∆̂tpi = 0 =⇒ ∆̂tpi = − θ̂22θ̂3

τ̂tpi = fs∆̂tpi

(5.18)

where τ̂tpi is the scc + tpi time delay estimator.

Example 5.2: Sub-sample ResolutionConsider the same signal parameters as in Example 5.1 but with 100 13 samplesdelay. The standard scc estimates the delay to ∆̂ = 100 samples but with scc+ tpi we get the correct delay ∆̂tpi = 100

13 samples. Figure 5.2 illustrates the ccf

(solid) where circles represent integer samples and the tpi interpolated version(dotted) where the black circle is the estimated sub-sample delay ∆̂tpi.

5.3.2 Linear Phase Method - LPLSE

Still consider the time delay model (4.3). By taking the Fourier transform of theccf in (5.13) we get the theoretical Cross Spectral Density (csd)

Φys(ω) = ae−jωτΦss(ω) (5.19)

between the signals y(t) and s(t). With the csd, an estimate of the time delaycan be provided from relations in the frequency domain. The method is based onthe description made in [Gustafsson et al., 2010, p. 111,232-235]. The phase ofthe csd Φys(ω) should be a straight line with constant slope τ . From the signalmodel described in Chapter 6 which are defined in a bounded frequency set Ωwith center frequency fc and bandwidth B the phase of the csd can be modeled

30 5 Estimation

as

ϕ(ω) = arg(Φys(ω)) =

φ0 − ωτ + eφ(ω) if ω ∈ Ωeφ(ω) otherwise (5.20)where φ0 is the random initial phase and eφ is additive phase noise which comesfrom the noise term in (4.3). The model describes a linear phase curve in the fre-quency set Ω with an initial offset and random outside. The signals are sampledso instead we use a discrete phase model

ϕk = arg(Φys[ωk]

)=

φ0 − ωk∆ + eφ[ωk] if ωk ∈ Ωeφ[ωk] otherwise (5.21)where ∆ is the time delay in samples and

Φys[ωk] = Y [ωk]S∗[ωk]

Y [ωk] = DFT{y[n]

}S[ωk] = DFT

{s[n]

}ωk =

2πfsN

k , k = 0, 1, . . . , N − 1

(5.22)

from N signal observations. With the linear model the lse of the parameters canbe found using the linear estimation framework in Section 5.1

YN = ΦNθ

YN =

ϕ1ϕ1...ϕN

, ΦN =1 −ω11 −ω2...

...1 −ωN

, θ =[φ0∆

](5.23)

where ω1, . . . , ωN ∈ Ω. The Linear Phase Least Squares Estimator (lplse) canthen be calculated as

θ̂ = (ΦTNΦN )−1ΦTNYN

τ̂ = fs∆̂ .(5.24)

The variance of the time delay estimate τ̂ can be calculated from ls theory [Gustafs-son et al., 2010, p. 233-234]

Var(τ̂) ≈ Bσ2e /2

aB4/12=

6σ2eaB3

(5.25)

where a is possible signal attenuation or amplitude.

Example 5.3: lplse EstimatorA signal s(t) with bandwidth B = 500 Hz and center frequency fc = 1 kHz istransmitted to a receiver. The receiver observes the signal with snr 10 dB as thesignal y(t). The true delay between the transmitter and the receiver is 100 sam-ples. With lplse the delay is estimated to ∆̂ = 99.77 ± 0.41 samples. Figure 5.3

5.3 Time Delay 31

0 1000 2000 3000 4000 5000−80

−60

−40

−20

0

20

40

Frequency [Hz]

Pha

se [r

ad]

800 900 1000 1100 1200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]

Figure 5.3: Phase of the csd. Left: Φys[ωk] phase. Right: Measured (thin)and estimated linear phase (thick) in frequency set Ω.

shows the measured and the estimated linear phase of the csd.

5.3.3 Estimation Performance

To study the efficiency of the three estimators: scc, scc + tpi, and lplse theywill be compared against the derived crlb in (5.26) when estimating a unknownparameter of a deterministic signal observed inwgn [Kay, 1993, p. 53-56]

Var(τ̂) ≥ σ2e

EF2=

1

SNR · F2(5.26)

where

SNR :=Eσ2e

(5.27)

is the signal to noise ratio (snr) defined as the quotient of signal energy and noisevariance. F2 is the Mean square bandwidth (msb)

F2 :=

∫∞−∞(2πf )

2 |S(f )|2 df∫∞−∞ |S(f )|

2 df(5.28)

of the signal s(t). S(f ) is the Fourier transform of s(t) and f is continuous-timefrequency. The expression in (5.26) can be rewritten as [Gustafsson, 2010, p. 400]

Var(τ̂) ≥ 18π2BTsf

2c · SNR

(5.29)

which is easier to interpret by the signal parameters. From (5.29) we see that thelower bound is inversely proportional to snr and the signal parameters of s(t).

To compare the theoretical variance of the lplse in (5.25) with crlb in terms ofsignal snr one can recall that the energy of a spectrum with constant amplitude

32 5 Estimation

0 10 20 30 40 5010

−20

10−15

10−10

10−5

100

SNR [dB]

Var

ianc

e [s

2 ]

CRLBLPLSE

Figure 5.4: crlb and calculated lplse variance with respect to snr. Signal:ofdm with parameters: fc = 1000 Hz, B = 500 Hz, and Ts = 0.3 s.

a and bandwidth B is E = aB. The signal snr can then be calculated as

SNR =Eσ2e

=aB

σ2e, (5.30)

so it turns out that (5.25) can be approximated as

Var(τ̂) ≥ 6B2 · SNR

(5.31)

where the amplitude of signal s(t) is considered as constant within its definedfrequency set Ω.

Figure 5.4 illustrates the crlb and the approximative variance of the lplse in(5.25) with respect to snr. We see that the lplse does not attain the crlb despitehigh snr.

5.4 Sensor Localization

Consider a sensor network with N sensors SN = {sn}N1 , where sn = (x1, x2)T ∈ R2,spread over an area A. We also have K acoustic sources XK = {xk}K1 , where xk =(x1, x2)T ∈ R2, around the area A. Neither the positions of sensors nor sourcesare known.

The sensors measure range to a source, toa timing, and range difference betweentwo sensors, tdoa timing, as described in Section 4.3.

The parameters that are of highest interest to estimate are the sensor positions SN

while the source positions Xk and the toa measurement bias br are consideredas nuisance parameters.

5.4 Sensor Localization 33

r ik

rijk

toa

tdoa

ŜN

ŜN X̂K

X̂K b̂r

initialstep

refinementstep

Figure 5.5: Localization algorithm workflow from measurements to esti-mated parameters.

Due to the nature of the problem structure with many parameters and nonlin-ear relationships the algorithm needs to be divided into two steps. Commonlyapplied localizing algorithms relies on nonlinear minimization which is oftensolved by numerical iterative solvers, e.g. Gauss-Newton and Newton-Raphson.Iterative solvers requires a initial guess of the unknown parameters. If no param-eters are known the solver tends to diverge or find a local minima. But if theinitial guess is close enough to the true solution a iterative solver would in thebest case find the global minima, which corresponds to the true solution. Thesensor localization algorithm will therefore be divided into two steps:

Initial step Calculates an estimate that is close to true solution.

Refinement step Refines the initial estimate with nonlinear optimization.

The initial step will estimate positions that are in a sufficient neighbourhood,initial estimate, of the true solution by applying a extended approach using thecmds algorithm. The refinement step uses the initial estimate as prior informa-tion and with numerical optimization find the global solution, refined estimate.Because of the nature of the problem the final solution is defined in a locally co-ordinate system which is correct up to arbitrary transformation and translation.

Instead of applying the original approach of cmds, described in [Shang et al.,2004], a novel approach will be presented, called the novel cmds approach, whichis a own contribution. The novel approach result in more accurate estimates ofthe sensors positions and provides more information to integrate by refinementstep estimation algorithms.

In Figure 5.5 the whole sensor localization workflow can be studied which inte-grates both toa and tdoameasurements divided into two localization steps.

34 5 Estimation

5.4.1 Initial step

With cmds based localization methods, see Section 3.2, the sensors are localizedgiven that the Euclidean distances

dij = ||si − sj || (5.32)are known [Shang et al., 2004]. With the novel approach we also want to incor-porate the knowledge that we measure the distances between each source andsensors, toa measurements. This will probably give more accurate sensors posi-tion estimates and sources position estimates as a bonus. The idea is to augmentthe distance matrix D with source to sensor distances

Dkaug =

d1k

D ...dNk

d1k . . . dNk 0

d ik = ||xk − si ||

(5.33)

which r K unique distance matrices, {Dkaug}K1 . By applying cmds, Algorithm 1,on each extended distance matrix Dkaug will therefore give estimates of the sensorpositions ŜNk and source position x̂k . So when all unique estimates are calculatedwe get a total of K estimates of the sensor positions which are stacked in the set,{ŜNk }K1 , and one source positions estimate, X̂k .

In the nature of cmds these estimates are centered in arbitrary origins. By calcu-lating a sequential mean value of the sensor positions set {ŜNk }K1 0 for each sourcexk , will result in a mean value of the sensor positions defined as

¯̂SN . For each k:thsensor positions estimate transform it to fit the sequential mean, see Section 3.3with a transformation T( · ) that includes both rotation/reflection and translation.Using a sequential mean separate origins of the K sensor positions estimates areavoided and possible influences of measurement noise is lowered.

Unfortunately the available sensors do not give inter-sensor distance measure-ments dij instead they must be approximated. One approach is to estimate thedistances utilizing the tdoa range measurements in (4.11) and by applying thetriangle inequality, see Figure 5.6, we get an underestimated estimate d̂ij of the

distance dij . First we define the noise-free and zero bias range variants r̃ik and r̃

ijk

of the range measurements.

According to the triangle inequality we can state the following

0 ≤ r̃ ik − r̃jk = r̃

ijk ≤ ||si − sj || . (5.34)

Assume that infinitely many sources are spread evenly around the network

limK→∞

maxk∈K

r̃ijk = ||si − sj || (5.35)


xk

si sj

r ik rjk

dij

Figure 5.6: Illustration of the triangle inequality,∣∣∣||xk − si || − ||xk − sj ||∣∣∣ ≤

||si − sj ||

where the limit approaches the true distance dij . So

d̂ij = maxk∈K

r̃ijk (5.36)

is a plausible approximation of the true distance dij between sensors si and sj .

In Section 4.3 it is stated the each target xk generates M range measurements:rijk and r

ik . Instead of using one single range measurement the sample mean r̄

ijk

and r̄ ik of the measurement vectors would lower the influence of the noise whenestimating the distances

d̂ij = maxk∈K

r̄ijk

d̂ ik = r̄ik

(5.37)

With the approximated distances the augmented distance matrices {D̂kaug}K1 canbe formed and the novel localization method can be applied, as described in Al-gorithm 3. Compared with the original cmds approach the novel approach havesome advantages:

1) Lower the influence of measurement noise.2) Higher precision of sensor position parameters.3) Ability to estimate source position parameters.

Example 5.4: Sensor localization and cmdsConsider a sensor network with 8 sensors and 10 sources distributed in an area of10 m2. The measurement noise is set to σe = 0.1 m. With the novel approach thesensors localization rmse is lower than with the original approach, see Figure 5.7.Note that sources are estimated with localization rmse 0.289 m.

36 5 Estimation

Algorithm 3 Novel approach to estimate 2-D positions of sensors and sourceswith cmds utilizing tdoa and toameasurements

for k = 1→ K doForm the augmented distance matrix DkaugCalculate estimates SNk , xk by applying cmds in Algorithm 1 with Dkaugif k=1 then . Initiate sequential mean with estimates from first source

¯̂SN := ŜN1

Store source positions in vector X̂1 ← x̂1else

Update sequential mean ¯̂SN ← ¯̂SN + 1k+1(T(ŜNk ) − ¯̂SN

)Store transformed source positions in vector X̂k ← T(x̂k)

end ifend for

5.4.2 Refinement step

The second step in the sensor localization algorithm is to minimize the differencebetween model output and measurements to achieve possibly better accuracy ofthe sensor positions. With the given range measurements and the sensor mod-els in Section 4.3 we can formulate an optimization problem that will minimizethe residuals by finding the lse of the unknown parameters. With the novelapproach on cmds localization above both estimates of the sensors and sourcesposition are available. Therefore the toa sensor model will be considered, de-scribed in Section 4.3.1. To keep the structure compact for each target xk themeasurements and model outputs will stacked by sensors in vectors as

Yk = h(xk ; SN , br ) + Vk , Cov(Vk) = Rk

Yk =

r1kr2k...

rNk

, h(xk ; SN , br ) = c ·

htoa(xk ; s1, br )htoa(xk ; s2, br )

...htoa(xk ; sN , br )

, Vk =

v1kv2k...

vNk

(5.38)

where the measurement noise iswgnwith known covariance, Rk . The covariancematrix is a diagonal matrix with the estimated variances of the measurementvectors rik in the diagonal

Rk =

Var(r1k) 0 0

0. . . 0

0 0 Var(rNk )

. (5.39)


−10 −5 0 5 10

−5

0

5

x1 [m]

x 2 [m

]

(a) Original cmds approach, Sensorsrmse 0.057 m

−10 −5 0 5 10

−5

0

5

x1 [m]

x 2 [m

]

(b) Novel cmds approach, Sensorsrmse 0.045 m, Sources rmse 0.289 m

Figure 5.7: Original and novel cmds approach. Ground truth are denotedwith cross, sensor position estimates with circles and source position esti-mates with squares.

To integrate measurements from all sources XK we introduce the stacked struc-ture by sources to drop the index k on measurement and noise vectors

Y = H(XK ; SN , br ) + V , Cov(V) = R

Y =

Y1Y2...

YK

, H(XK ; SN , br ) =

h(x1; SN , br )h(x2; SN , br )

...h(xK ; SN , br )

, V =

v1v2...

vK

(5.40)

where the covariance R is a diagonal block matrix

R =

R1 0 0

0. . . 0

0 0 RK

. (5.41)From now on the stacked structure by sources will be used to formulate the opti-mization problem.

From the structure in (5.40) we want to formulate optimization problem whosegoal is to find the estimates of sensor positions, source positions and measure-ment bias that minimize the cost function

V (SN ,Xk , br ) =12

(Y −H(XK ; SN , br )

)T (Y −H(XK ; SN , br )

)=

12εT ε (5.42)

in ls sense where ε = Y − H(XK ; SN , br ) is the residual vector. The lse of theunknown parameters are then given by(Xk ,SN , br) = arg min

Xk ,SN ,br

V (SN ,Xk , br ) = arg minXk ,SN ,br

12εT ε . (5.43)

38 5 Estimation

Algorithm 4 General Gauss-Newton

1: Given initial value x̂(0), the function h(x) and its gradient J(x) = −∂hT (x)∂x . Seti = 0.

2: Set α(i) = 1.3: Compute

x̂(i+1) = x̂(i) + α(i)(J(x)JT (x)

)−1J(x)

(y − h(x)

)4: If the cost V (x̂(i+1)) > V (x̂(i)), set α(i) = α(i)/2 and repeat from step 3.5: Terminate if the change in cost, the change in estimate, or the size of the

gradient is small enough, or if the number of iterations has reached an upperlimit.

6: Otherwise, set i := i + 1 and repeat from step 2.

The linear approach in Section 5.1 can not be applied because of the nonlinearrelationships in the cost function. To find the lse we must introduce the Non-linear Least Squares (nls) framework. nls problems are usually solved by usingiterative numerical optimization methods. Generally these methods updates theestimates iteratively as

x̂(i+1) = x̂(i) + α(i)f (i) (5.44)

where α(i) is the step length and f (i) is the search direction in the i:th itera-tion. Such methods are characterized by their calculation of the search direction.Here the basic Gauss-Newton algorithm will be used [Gustafsson, 2010, p. 49-50]which works as in Algorithm 4.

The Jacobian J(x) is central in Gauss-Newton and is constructed by stacking thegradients of all residuals ε on each other and defined as

J(Xk ,SN , br ) =

∂εT

∂(Xk ,SN , br ), J(Xk ,S

N , br ) ∈ R[2(K+N )+1]×MNK (5.45)

which can be calculated symbolically or numerically by approximating the deriva-tives using Finite-difference methods (fdm). Here it will be approximated byapplying central differences as

J(i, :) ≈(H(θ + �ei) −H(θ − �ei)

)/2� (5.46)

where the parameters are stored in the vector θ = [XK SN br ]T for simplification,ei denotes the i:th column of the identity matrix, and � is the step size. The stepsize should be small enough to find the linear region of H(Xk ,SN , br ), but stillsufficiently large to avoid numerical ill-conditioning.

5.4.3 Localization Performance

The theoretical performance of the refinement step will be investigated by study-ing the sensor localization rmse using crlb theory. In Section 5.2 the FIM for an


unbiased estimator with Gaussian measurement errors is calculated as

I (θ) = JT (θ)R−1J(θ) (5.47)for the sensor positions θ = SN . The crlb implies the following bound on thermse

rmse ≥√

tr I−11:M(θ0

)(5.48)

where θ0 are the true parameters consideringM independent measurements. Fig-ure 5.8 shows the crlb of the localization rmse of the refinement step evaluatedwith the benchmark configuration, see Section A.3. The localization rmse in-creases log-log linear with measurement noise variance. The rmse decreases asmore independent measurements are generated from each source, which is intu-itive because more information are available.

40 5 Estimation

10−4

10−3

10−2

10−1

100

101

10−6

10−4

10−2

100

102

σe [m]

RM

SE

[m]

(a) Localization rmse with respect to simu-lated measurement noise, M = 10

1 5 10 15 2010

−2

10−1

100

Measurements

RM

SE

[m]

(b) Localization rmse with respect to gen-erated measurements from each source withsimulated measurement noise, σe = 0.1 m.

Figure 5.8: Refinement step localization performance crlb.

6Signals

This chapter presents the signals that are implemented to evaluate how the esti-mation algorithms perform in terms of ranging and localization. Three signalsare described where two are artificially generated and the third is a recordedbirdsong [P2-fågeln, 2012].

The signals are implemented as discrete data sequences sampled with 48 kHzsampling rate, which is the sampling rate applied during data collection.

6.1 OFDM

This signal is a waveform that is a modulated code sequence by the Orthogo-nal Frequency Division Multiplexing (ofdm) principle. ofdm is widely usedin data modems and in the Long Term Evolution (lte) wireless communicationstandard.

The signal is generated in a given frequency interval, Ω, during the time intervalTs. Map a given real-valued code sequence c = (c1, ..., cM ) to the frequenciesf = (f1, ..., fM ) ∈ Ω. The signal can then be written as [Gustafsson et al., 2010,p. 66]

s(t) =M∑m=1

cm cos(2πfmt) . (6.1)

Instead of generating the signal in the time domain it can be generated in the

41

42 6 Signals

Table 6.1: ofdmmodulated signal parameters

Parameter Value

Signal length Ts = 0.3 s

Bandwidth B = 500 Hz

Center frequency fc = 1000 Hz

0 0.05 0.1 0.15 0.2 0.25−1

0

1

Time [s]

s(t)

0 500 1000 1500 2000 2500

0.050.1

0.150.2

0.25

Frequency [Hz]

Tim

e [s

]

(a) Signal waveform and spectrogram

0 20 40 6010

−7

10−4

100

103

SNR [dB]

RM

SE

[m]

CRLBCalculated LPLSELPLSESCCSCC+TPI

(b) Range estimation performance

Figure 6.1: ofdmmodulated signal

frequency domain

S[fm] = c, fm = (f1, ..., fM ) (6.2)

s[n] = dft−1{S} (6.3)where S is the dft of s(t). So the dft offers a powerful way of directly generatingthe signal samples s[n] = s(n/fs) from the code sequence c. The frequency inter-val Ω is divided in a regular frequency grid and is a subset of the dft frequenciesdefined by the sampling frequency fs. In this application no information is sentso the code sequence is best generated by a random code, ±1, to achieve lowestsaturation effects when transmitting the modulated signal sequence s[n] [Gustafs-son et al., 2010, p. 66].

Figure 6.1a shows the ofdm modulated signal generated with the parametersin Table 6.1 both in time domain and in frequency domain as a spectrogram.The energy is constant over the whole frequency and time interval. Figure 6.1bshows the performance of range estimation with the three estimators and thecorresponding lower bound.

6.2 Chirp 43

Table 6.2: Chirp signal parameters

Parameter Value




0.005 0.01 0.015 0.02 0.025 0.03−1

0

1

Time [s]

s(t)

0 500 1000 1500 2000 2500

0.050.1

0.150.2

0.25

Frequency [Hz]

Tim

e [s

]


−10 0 10 20 30 40 5010

−7

10−4

100

103

SNR [dB]

RM

SE

[m]



Figure 6.2: Chirp signal

6.2 Chirp

A time continuous chirp signal is a sinusoid signal with time varying frequencycontent [Gustafsson et al., 2010, p. 10-11]

s(t) = sin(2πf (t)t

)(6.4)

where the frequency varies as an increasing linear function

f (t) = f0 + f∆t (6.5)

with constant slope f∆ and initial frequency f0. The signal is generated with theparameters in Table 6.2 where f0 and f∆ are defined by bandwidth, center fre-quency, and signal duration. Figure 6.2a shows how the frequency content variesas a linear function as predicted and in Figure 6.2b shows the range estimationperformance when the Chirp signal is used.

44 6 Signals

6.3 Birdsong: Black-Throated Loon

Figure 6.3: Illustration of the Black-Throated Loon (source: P2-fågeln,Sveriges Radio)

The Black-Throated Loon (Gavia Artctica)is a bird in the family loons first de-scribed by Carl Linnaeus in the 18th cen-tury. It breeds in Scotland, Scandinavia,the Baltics, and in the eastern throughSiberia. In Sweden it is vary familiar ex-cept on the islands of Öland and Gotlandand in the south.The song of the bird isgenerally known in Sweden and is alsoused in many Swedish movies to intensifya dramatic scene, where the call is charac-terized by the high-pitched wail.

When using the birdsong as a signal s(t) the parameters are identified by thespectrogram in Figure 6.4a summarized in Table 6.3. When looking at the spec-trogram it looks like a chirp signal with increasing frequency and amplitude gen-erating the high-pitch wail. The parameters are identified in the region wheremost energy is concentrated.

Table 6.3: Black-Throated Loon signal parameters

Parameter Value




6.3 Birdsong: Black-Throated Loon 45


0 20 40 6010

−7

10−4

100

103

SNR [dB]

RM

SE

[m]



Figure 6.4: Black-Throated Loon signal

Part III

Results and Discussion

7Data Collection

This chapter presents the collected data used in the evaluation. Two scenarios;ranging and sensor localization, are introduced that will be used to specify whichdata are needed. The datasets are collected indoors, outdoors, and with simula-tions.

7.1 Scenarios

This section introduces two evaluation scenarios. The Ranging scenario defineshow ranging performance is evaluated and the Sensor localization scenario de-fines how the sensor localization performance is evaluated.

7.1.1 Ranging

The purpose of this scenario is to evaluate how the estimators perform duringhigh and low snr conditions. The scenario will be evaluated with data fromthree datasets: indoors, outdoors, and simulations. In the indoors and outdoorsdatasets the performance is evaluated against distance, where far distance resultsin lower snr. The simulation dataset alters the snr by varying the measurementnoise variance σ2e [m

2].

7.1.2 Sensor Localization

The purpose of this scenario is to evaluate how the localization algorithm per-forms based on specific parameters, such as: number of sources, deploymentarea, and measurement noise. The scenario will be evaluated with data from twodatasets: outdoors and simulations. In the outdoors dataset the performance isevaluated by two sensor network configurations (dense and sparse) where num-

49

50 7 Data Collection

Table 7.1: Anechoic room technical specificationsCut-off frequency 100 HzISO 3745 100 Hz - 10 kHzDimensions (LxWxH) 6.40x4.00x4.10 m

ber of sources can be varied. In the simulated dataset the performance is eval-uated by a benchmark sensor network configuration and a randomized sensornetwork configuration.

7.2 Datasets

This section summarizes how the data in the datasets are collected. The scenariosare evaluated by data from three datasets: simulations, indoors, and outdoors.

7.2.1 Simulations

This dataset is collected by running written code snippets in matlab®. The con-cept of Monte Carlo (mc) simulation are applied to give statistically reliable mea-sures of mean and variance of the realizations.

7.2.2 Indoors

The indoors dataset is collected in the anechoic room at foi which is iso 37451

certified. In Table 7.1 the technical specification of the anechoic room is listed.The collected data are used to evaluate the ranging scenario and evaluation ofsignal parameters, see Appendix B. The dimension of the anechoic room limitsdata to be collected at ranges up to maximum range 6 m. The data are collectedwith the parameters in Table 7.2 using the indoors configuration specified in Ap-pendix A.1.

The data are collected with the setup in Figure 7.1 which consist of five micro-phones mounted on a bar, one microphone acting as a timing reference, and aloudspeaker. The microphones are connected to two separate field recorders andthe signals are played from a computer connected to the loudspeaker. Becausethe field recorders are separate some sort of sync is needed. The sync is createdby connecting a gps receiver to the recorders and record the Pulse Per Second(pps) signal. This signal is later decoded with high sample accuracy.

7.2.3 Outdoors

The outdoors dataset is collected at foi test site Lilla Gåra located just south ofLinköping, see Figure 7.2. At this test site a gravel yard with dimensions 100x200m is available. This gives the opportunity to collect data at far distances and

1iso 3745 - Acoustics – Determination of sound power levels and sound energy levels of noisesources using sound pressure – Precision methods for anechoic rooms and hemi-anechoic rooms.

7.2 Datasets 51

Table 7.2: Indoors dataset parameters

Parameter Value

Signals ofdm, Chirp, Birdsong

Air temperature 20◦ C

Sampling frequency 48 kHz

(a) Recording parameters

Parameter Value

Bandwidth, B 500 Hz, 1 kHz

Center frequency, fc 500 Hz, 1 kHz, 10 kHz

Signal duration, Ts 0.5 s, 1 s

(b) ofdm and Chirp signal parameters

deploy sparse and big sensor network configurations which is of high interestwhen evaluating both scenarios.

The data are collected with the configurations specified in Appendix A.4. Notethat only sensor network configurations are used to collect data so the rangingscenario will also use this data. When collecting data a mobile wagon with a com-puter, loudspeaker, and a microphone is used. The microphone acts as a timereference to find the time when the signal is emitted from the loudspeaker andtherefore placed in front of the loudspeaker element. This wagon is then movedto spots that encircles the sensor network. For each spot the three signals are emit-ted one after the other and the position is acquired just behind the loudspeaker.This position is the position of the k:th source xk .

Figure 7.1: Indoors dataset recording setup in anechoic room

(a) Loudspeaker with timing referencemicrophone

(b) Five microphones mounted on abar

52 7 Data Collection

Figure 7.2: Aerial view of Lilla Gåra test site.

xk

si

br

||xk − si ||r ik

Figure 7.3: Illustration of the wagon and one sensor in the outdoors dataset.From this the toameasurement bias br can be set.

The toa range is estimated by measuring the time delay between the referencemicrophone and the sensors SN while the sources position are measured behindthe loudspeaker. Therefore measurement bias exists in all toa measurements,see Figure 7.3. Ground truth positions are measured with a high-accuracy gpsreceiver (± 10 mm). In Table 7.3 the parameters of the dataset are listed.

7.2 Datasets 53

Table 7.3: Outdoors dataset parameters

Parameter Value

Signals ofdm, Chirp, Birdsong

toa bias, br 0.248 m

Air temperature 13.13◦ C

Rel. humidity 86 %

Air pressure 1007.41 hPa

Wind speed 0.25 m/s

ofdm sound level 102 dB SPL

Chirp sound level 109 dB SPL

Birdsong sound level 106 dB SPL

Sampling frequency 48 kHz

(a) Recording parameters

Parameter Value

Bandwidth, B 500 Hz

Center frequency, fc 1 kHz

Signal duration, Ts 0.5 s

(b) ofdm and Chirp signal param-eters

8Results

This chapter presents the results when the estimation algorithms presented inChapter 5 are evaluated. The two scenarios are evaluated separately to becausetheir evaluation parameters and performance metrics are different.

8.1 Ranging Scenario

The ranching scenario, see Section 7.1.1, is evaluated by the ranging rmse:

rmse :=

√σ̂2 (8.1)

where σ̂2 is the estimated variance of ranging measurements. The figures aregrouped by scenario and dataset where all signals are showed but with differentline styles. Table 8.1 and Table 8.2 contain references to figures that are used inthe evaluation of the estimators and signals.

We see that the Chirp signal generally performs better than the ofdmmodulatedsignal. The birdsong signal performs worst in all cases. The estimator to useis the scc estimator, predicted by the crlb analysis. The indoors and outdoorsdatasets are of good quality. The rmse levels are almost constant despite dis-tances up to 100 m. The scc estimator combined with Chirp signals achievesbest performance with rmse levels of 10−2 m magnitude, Figure 8.1f.

8.2 Sensor Localization Scenario

The evaluation of the sensor localization scenario is divided into two parts wherethe first, Section 8.2.2, covers the localization performance of the initial and re-

55

56 8 Results

Table 8.1: Line styles of the signals used in the evaluation

Signal Style

ofdm Solid

Chirp Dashed

Birdsong Dotted

Table 8.2: Evaluation of the ranging scenario

Dataset Estimator Figure

Simulated lplse Figure 8.1a

scc Figure 8.1b

Indoors lplse Figure 8.1c

scc Figure 8.1d

Outdoors lplse Figure 8.1e

scc Figure 8.1f

finement step. The second part, Section 8.2.3, covers the sensor localization re-sults when the outdoors dataset is considered.

8.2.1 Performance metric

To evaluate the localization algorithm the rmse of the estimated sensor positionsis calculated. Because the source positions are considered to be nuisance parame-ters they are not studied. The localization rmse is calculated by finding the lseof a transformation T( · ) that fits the estimated sensor positions to ground truth,see Section 3.3. By this transformation the localization rmse is calculated as

rmse :=

√1N

(SN − T(ŜN )

)2. (8.2)

This performance metric is derived from a least squares approximation of a trans-formation which tends to give low localization rmse but is necessary to overcomethe issue that the presented localization algorithm finds a solution up to arbitrarytranslation, rotation, or rotation of the unknown sensor positions due to the factthat only distances are available.

8.2.2 Simulated performance

The performance of the two sensor localization steps are evaluated with simu-lated data from the benchmark configuration and the randomized configuration.

8.2 Sensor Localization Scenario 57

Table 8.3: Initial step evaluation parameters

Parameter Description Figure

Measurement noise, σ2e Range measurement error variance Figure 8.2a

Sensors, N Number of sensors in network Figure 8.2b

Sources, K Number of sources around network Figure 8.2c

Range measurements, M Times source emits signal Figure 8.3

Initial step

The initial step performance will be evaluated by how the novel cmds approachperforms compared to the original cmds approach in terms of the key parametersin Table 8.3.

From the evaluation figures we see that the novel cmds performs better than theoriginal approach in all cases which is expected due to more available range infor-mation. In Figure 8.2a we see that the novel approach attains lower localizationrmse than the original approach up to σe ≈ 1 m where they coincide. In Fig-ure 8.2b we see that the localization rmse grows with number of sensors, whichcan be explained by the degraded capability of estimating the distances betweensensors as the number of sources is constant. In Figure 8.2c the localization rmseis lowered as more sources are added, which is intuitive.

In Figure 8.3 the localization rmse is illustrated with respect to number of rangemeasurements generated by each source in three noise configurations. The rmseis lower with the novel approach and the number of generated measurementsbecomes more important when the noise level is increased.

Refinement step

The performance of the refinement step is evaluated by the localization rmsewith different initial solutions, see Table 8.4. They will be compared against thelocalization rmsewhen ground truth positions are initial parameters. The refine-ment step does not attain zero localization rmse which is predicted by the crlbin Figure 5.8. With the novel approach as initial estimate the localization rmsecoincides with same rmse as when ground truth estimate is used.

The localization sensitivity with respect to the quality of the initial estimate isevaluated by perturbing each true sensor position with position noise

ŝi ∼ N (si , R) (8.3)

R =(σ2pert 0

0 σ2pert

)(8.4)

and with ground truth positions of the sources. In Figure 8.5 the localizationrmse remains constant until the true positions are perturbed with ±1 m in eachdimension x1 and x2 where the estimate starts to diverge.

58 8 Results

Table 8.4: Initial solutions studied in the evaluation of the refinement step

Initial solution Estimates FigureSensors Sources

Original cmds • – Figure 8.4a

Novel cmds • • Figure 8.4b

Randomized – – Figure 8.4c

8.2.3 Results

Figure 8.6 and Figure 8.7 show the sensor localization rmse with respect to num-ber of sources and emitted signal for the dense and sparse outdoors configura-tions. With ≥ 5 sources a noticeable knee in the localization rmse for the re-finement step is present which is common in all cases. No noticeable differencein localization rmse when comparing different signals. Instead of using a signalthat sounds strange out in the nature the birdsong can be applied which increasesthe ability to configure the sensor networks without disturbing the surroundings.


0 10 20 30 4010

−4

10−2

100

102

SNR [dB]

RM

SE

[m]

(a) Dataset: Simulated, Estimator:lplse

0 10 20 30 4010

−6

10−4

10−2

100

102

SNR [dB]

RM

SE

[m]

(b) Dataset: Simulated, Estimator:scc

4 5 610

−3

10−2

Distance [m]

RM

SE

[m]

(c) Dataset: Indoors, Estimator:lplse

4 5 610

−3

10−2

Distance [m]

RM

SE

[m]

(d) Dataset: Indoors, Estimator: scc

101

102

10−2

100

Distance [m]

RM

SE

[m]

(e) Dataset: Outdoors, Estimator:lplse

101

102

10−2

100

Distance [m]

RM

SE

[m]

(f) Dataset: Outdoors, Estimator: scc

Figure 8.1: Evaluation of the ranging scenario with three datasets, Legend:solid - OFDM, dashed - Chirp, and dotted - Birdsong

60 8 Results

10−2

100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

σe [m]

RM

SE

[m]

Novel approachOriginal approach

(a) Standard deviation of range measurementerror

101

102

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Number of Sensors

RM

SE

[m]

(b) Number of sensors

100

101

102

10−3

10−2

10−1

100

101

Number of Sources

RM

SE

[m]

(c) Number of sources

Figure 8.2: Initial step localization performance comparing original andnovel cmds approaches with randomized configuration with respect to threenetwork parameters.


5 10 15 200.042

0.043

0.044

0.045

0.046

0.047

Measurements

RM

SE

[m]

σe= 10−3 m

σe= 10−2 m

σe= 10−1 m

(a) Novel cmds approach

5 10 15 200.051

0.052

0.053

0.054

0.055

0.056

Measurements

RM

SE

[m]

(b) Original cmds approach

Figure 8.3: Initial step localization performance with respect to number ofmeasurements generated by each source in three noise configurations, Con-figuration: Benchmark

62 8 Results

10−4

10−2

100

10−6

10−4

10−2

100

102

σe [m]

RM

SE

[m]

RefinedGround truth refinement

(a) Original cmds initial estimate

10−4

10−2

100

10−5

10−4

10−3

10−2

10−1

100

σe [m]

RM

SE

[m]

(b) Novel cmds initial estimate

10−4

10−2

100

10−6

10−4

10−2

100

102

σe [m]

RM

SE

[m]

(c) Randomized initial estimate

Figure 8.4: Refinement step localization performance with respect to threeinitial estimates and increasing measurement error. With the novel cmdsapproach same results is achieved as when ground truth parameters areused.


10−4

10−2

100

102

10−2

10−1

100

σpert

[m]

RM

SE

[m]

Figure 8.5: Refinement step localization sensitivity with respect to the accu-racy of the initial estimate. The rmse starts to increase when ground truthis perturbed with ±1 m position error in each dimension.

64 8 Results

2 4 6 8 1010

−1

100

101

Sources

RM

SE

[m]

(a) Signal: OFDM

2 4 6 8 1010

−1

100

101

Sources

RM

SE

[m]

(b) Signal: Chirp

2 4 6 8 1010

−1

100

101

Sources

RM

SE

[m]

(c) Signal: Birdsong

Figure 8.6: Localization performance of the dense network configurationwith outdoors dataset. Legend: solid - Refinement step, dashed: Initial step


5 10 15 20 25 3010

−1

100

101

Sources

RM

SE

[m]

(a) Signal: OFDM

5 10 15 20 25 3010

−1

100

101

Sources

RM

SE

[m]

(b) Signal: Chirp

5 10 15 20 25 3010

−1

100

101

Sources

RM

SE

[m]

(c) Signal: Birdsong

Figure 8.7: Localization performance of the sparse network configurationwith outdoors dataset. Legend: solid - Refinement step, dashed: Initial step

9Concluding Remarks

Here the work is summarized with aspects on achieved results and methods withreflections on possible and interesting work which can further be done.

9.1 Conclusions

In this thesis a sensor localization calibration algorithm of a ground sensor net-work has been developed. The algorithm e

networks with acoustic range...

Documents