Emitter geolocation estimation using power difference of ... · the geolocation estimation accuracy and computational complexity of these algorithms. A MATLAB–Google Earth testbed

Emitter geolocation estimation using power difference of arrival

An algorithm comparison for non-cooperative emitters B.R. Jackson, S. Wang and R. Inkol

Defence R&D Canada – Ottawa

Technical Report DRDC Ottawa TR 2011-040

May 2011

Emitter geolocation estimation usingpower difference of arrivalAn algorithm comparison for non-cooperative emitters

B.R. Jackson


S. Wang


R. Inkol


Defence R&D Canada – OttawaTechnical Report

DRDC Ottawa TR 2011-040

May 2011

Principal Author

Original signed by B.R. Jackson

B.R. Jackson

Approved by

Original signed by M.W. Katsube

M.W. KatsubeHead/CNEW Section

Approved for release by

Original signed by C. McMillan

C. McMillanHead/Document Review Panel

c© Her Majesty the Queen in Right of Canada as represented by the Minister of NationalDefence, 2011

c© Sa Majeste la Reine (en droit du Canada), telle que representee par le ministre de laDefense nationale, 2011

Abstract

The performance of several power difference of arrival (PDOA), also known as receivedsignal strength difference (RSSD), emitter geolocation algorithms, is explored. First, ageneral discussion of radio frequency emitter geolocation techniques is presented. After-wards, the motivation for employing PDOA geolocation techniques is discussed, followedby a review of several path loss models and a description of algorithms for obtaining geolo-cation solutions from PDOA measurement data: Non-Linear Least Squares, the DiscreteProbability Density method, Maximum Likelihood, the Intersection Density method, andthree Least Squares techniques. Both simulated and measured data are used to comparethe geolocation estimation accuracy and computational complexity of these algorithms. AMATLAB–Google Earth testbed was developed to aid in the visualization and interpre-tation of the results of the experimentation. Finally, conclusions and recommendationsregarding the selection of the most appropriate PDOA algorithms for a given applicationare given.

Resume

Le present document traite des performances de plusieurs algorithmes de geolocalisationd’emetteurs pour la difference de puissance incidente (PDOA), aussi appelee differenced’intensite du signal recu (RSSD). Nous presentons d’abord une analyse generale des tech-niques de geolocalisation d’emetteurs radiofrequences (RF). Nous examinons ensuite lesraisons qui motivent l’utilisation de techniques de geolocalisation PDOA, puis plusieursmodeles d’affaiblissement sur le trajet, ainsi qu’une description des algorithmes qui per-mettent d’obtenir des solutions de geolocalisation a partir des resultats des mesures de laPDOA : methode des moindres carres non lineaire, methode de la densite de probabilitediscrete, methode du maximum de vraisemblance, methode de la densite d’intersection ettrois methodes des moindres carres. De plus, nous utilisons des donnees simulees et desdonnees mesurees pour comparer la precision des estimations de position geographique etla complexite des calculs de ces algorithmes. Un banc d’essai MATLAB–Google Earth aen outre ete developpe pour aider a visualiser et a interpreter les resultats de l’experience.Finalement, nous presentons des conclusions et des recommandations sur la selection desmeilleurs algorithmes PDOA pour une application donnee.

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ii DRDC Ottawa TR 2011-040

Executive summary

Emitter geolocation estimation usingpower difference of arrival: An algorithm comparisonfor non-cooperative emitters

B.R. Jackson, S. Wang, R. Inkol; DRDC Ottawa TR 2011-040; Defence R&D

Canada – Ottawa; May 2011.

Background: Modern military operations depend on the timely acquisition of intelligenceinformation by electro-optic, infra-red, and radio-frequency (RF) sensors. Specialized RFsensors have long been used to geolocate radio frequency transmitters used by opposingforces. However, these systems require a substantial investment in expensive equipmentand highly trained personnel. Consequently, concepts for exploiting technical advancesto network relatively low-cost autonomous RF sensors for geolocating radio frequencyemitters are of practical interest and are being explored at Defence Research & Develop-ment Canada (DRDC) through the Klondike Technology Demonstration Project (TDP).Approaches based on the use of power measurements, are of particular interest due to thesimplicity of the sensors. The basic concept of the power difference of arrival (PDOA) ge-olocation technique is that, using a suitable path loss model, a geolocation estimate can beobtained from measurements of the signal power level from an array of sensors. This Tech-nical Report compares the accuracy of several algorithms for evaluating PDOA geolocationsolutions using both simulated and measured data.

Principal results: The results presented show that the choice of PDOA algorithm dependson the sensor-emitter geometry and computational capabilities available for processing.Measured data was collected simultaneously for two different emitter positions (each on adifferent channel). In one case where the emitter was very near the edge of the measurementcollection area, the statistical PDOA algorithms (Non-Linear Least Squares, Discrete Prob-ability Density, and Maximum Likelihood methods) outperformed the Intersection Densitymethod and the Least Squares approaches. However, when the emitter is in a more centralposition relative to the sensor array, the results are reversed due primarily to a bias in theLeast Squares algorithms. Simulated experiments confirm these results. However, in theabsence of a priori knowledge of the transmitter position, a measure of the overall accu-racy for the possible sensor-emitter configurations is a more useful performance measure.For simulations performed with random emitter positions, the statistical methods provedto give the best accuracy. However, their computation time was approximately two ordersof magnitude larger than that of the Least Squares methods, and therefore, with limitedcomputational capabilities the latter may be preferable.

DRDC Ottawa TR 2011-040 iii

Significance of results: The results presented clearly demonstrate the relative advantagesand disadvantages of various PDOA algorithms and are useful for the selection of geolo-cation algorithms in future EW systems. Furthermore, the experiments carried out usingFRS emitters provide a general sense of the accuracy that can be achieved using PDOAtechniques as a function of the number of sensors. The results indicate that the techniquehas some limitations and that, to achieve adequate accuracy for some military applications,other geolocation techniques or a combination of geolocation techniques may be needed.The computation time required for each method was also measured; this information canbe a consideration in the selection of algorithms or used to determine computation require-ments.

Future work: There are several areas that could be explored in the future to augment andextend the results presented in this report. For instance, airborne sensor platforms couldbe employed using unmanned aerial vehicles (UAVs) in addition to ground-based sensorplatforms. Since the airborne platforms would, in many scenarios, have line of sight withthe emitter, the statistical stability of the power measurements would be improved with aconsequent benefit to the geolocation accuracy. Furthermore, future research on combiningdifferent types of sensor data (e.g., PDOA and angle of arrival information) could be veryvaluable. Since modern systems can produce many measurements per second, research isalso needed on the best ways of processing this plethora of data and extracting the importantinformation to enable the linking of sensors via capacity constrained networks. Finally, theinvestigation of ways of optimizing the path loss models or adaptively estimating the pathloss exponent could produce a significant improvement in the overall accuracy that can beobtained using PDOA techniques.

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Sommaire

Emitter geolocation estimation usingpower difference of arrival: An algorithm comparisonfor non-cooperative emitters

B.R. Jackson, S. Wang, R. Inkol ; DRDC Ottawa TR 2011-040 ; R & D pour la

defense Canada – Ottawa ; mai 2011.

Introduction : Les operations militaires modernes dependent de l’acquisition opportunede renseignements de securite par des capteurs electro-optiques, infrarouges et radio freq-uences (RF). Des capteurs RF specialises servent depuis longtemps a geolocaliser desemetteurs RF utilises par des forces ennemies. Ces systemes necessitent toutefois un in-vestissement considerable en equipement et en personnel bien entraıne. Par consequent, lesconcepts lies a l’utilisation des progres techniques pour mettre en reseau des capteurs RFautonomes relativement peu couteux afin de geolocaliser des emetteurs RF sont d’un interetpratique et sont etudies a Recherche et developpement pour la defense Canada (RDDC)dans le cadre du projet de demonstration de technologies (PDT) Klondike. Les methodesaxees sur l’utilisation des mesures de puissance sont particulierement interessantes en rai-son de la simplicite des capteurs. La technique de geolocalisation au moyen de la differencede puissance incidente (PDOA) a pour concept de base que, grace a un bon modele d’affai-blissement sur le trajet, on peut estimer la position geographique a partir des mesures du ni-veau de puissance du signal produit par un reseau de capteurs. Le present rapport techniquecompare la precision de plusieurs algorithmes pour evaluer des solutions de geolocalisationPDOA au moyen de donnees simulees et de donnees mesurees.

Resultats : Les resultats presentes montrent que le choix d’un algorithme PDOA dependde la geometrie capteur-emetteur et des capacites de calcul disponibles pour le traitement.Les donnees mesurees ont ete recueillies simultanement pour deux positions distinctesd’emetteurs (chacun sur un canal FRS different). Dans l’un des cas, ou l’emetteur etaita proximite de la limite de la zone de collecte de mesures, les algorithmes statistiquesPDOA (methode des moindres carres non lineaire, methode de la densite de probabilitediscrete et methode du maximum de vraisemblance) ont donne de meilleurs resultats quela methode de la densite d’intersection et les methodes des moindres carres. Par contre,lorsque l’emetteur est plus pres du centre du reseau de capteurs, les resultats sont in-verses en raison d’une erreur systematique dans les algorithmes des moindres carres. Lesexperiences simulees confirment ces resultats. Cependant, si on ne connaıt pas a priori laposition des emetteurs, une mesure de la precision globale pour les configurations capteur-emetteur possibles constitue une meilleure indication des performances. Pour les simu-lations effectuees avec des positions aleatoires d’emetteurs, les methodes statistiques ont

DRDC Ottawa TR 2011-040 v

donne la meilleure precision. Le temps de calcul de ces methodes etait toutefois deux foisplus long que celui des methodes des moindres carres, et, en consequence, lorsque les ca-pacites de calcul sont limitees, le dernier choix est sans doute preferable.

Portee : Les resultats presentes montrent clairement les avantages et les inconvenients liesa divers algorithmes PDOA et sont utiles pour la selection des algorithmes de geolocalisationdes futurs systemes de guerre electronique (GE). De plus, les experiences effectuees avecdes emetteurs FRS donnent une idee generale de la precision qui peut etre obtenue a l’aidedes techniques PDOA en fonction du nombre de capteurs. D’apres les resultats, ces tech-niques ont des limitations et l’obtention d’une bonne precision pour certaines applicationsmilitaires peut necessiter l’utilisation d’autres techniques de geolocalisation ou une com-binaison de techniques de geolocalisation. Par ailleurs, le temps de calcul necessaire pourchaque methode a ete mesure, et cette information peut etre prise en consideration pourselectionner des algorithmes ou etre utilisee pour determiner les calculs necessaires.

Recherches futures : Plusieurs aspects pourraient faire l’objet d’etudes ulterieures en vued’augmenter et d’etendre les resultats presentes dans le rapport. Par exemple, on pourraitutiliser comme plates-formes de capteurs des vehicules aeriens sans pilote (UAV) en plusdes plates-formes de capteurs au sol. Puisque, dans de nombreux scenarios, les capteursaeroportes seraient en visibilite directe avec l’emetteur, la stabilite statistique des mesuresde puissance serait amelioree et augmenterait ainsi la precision de la geolocalisation. Desurcroıt, les recherches futures sur la combinaison de divers types de donnees de cap-teurs (comme des informations sur la PDOA et l’angle d’incidence) pourraient etre tresutiles. Comme des systemes modernes peuvent produire de nombreuses mesures par se-conde, il faut aussi realiser une recherche sur les meilleures facons de traiter cette sur-abondance de donnees et d’extraire les informations importantes pour permettre la liaisondes capteurs aux reseaux a capacite limitee. Enfin, l’etude des methodes d’optimisationdes modeles d’affaiblissement sur le trajet ou d’evaluation adaptative de la composanted’affaiblissement sur le trajet pourrait ameliorer beaucoup la precision globale obtenue aumoyen des techniques PDOA.

vi DRDC Ottawa TR 2011-040

Table of contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Executive summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Sommaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Emitter Geolocation Techniques . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Angle of Arrival (AOA) . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Time of Arrival (TOA)/Time Difference of Arrival (TDOA) . . . 1

1.1.3 Frequency Difference of Arrival (FDOA) . . . . . . . . . . . . . 2

1.1.4 Received Signal Strength (RSS)/Power Difference of Arrival(PDOA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Motivation for the Study of PDOA Algorithms . . . . . . . . . . . . . . . 4

1.3 MATLAB–Google Earth Testbed . . . . . . . . . . . . . . . . . . . . . . 5

2 Path Loss Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 PDOA Algorithms for Non-Cooperative Emitters . . . . . . . . . . . . . . . . . 11

3.1 Non-Linear Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Maximum Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 The Discrete Probability Density Method . . . . . . . . . . . . . . . . . . 15

3.4 Intersection Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.5 Modified Y.T. Chan’s Least Squares Method . . . . . . . . . . . . . . . . 21

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3.6 S. Wang’s Least Squares Method . . . . . . . . . . . . . . . . . . . . . . 22

3.7 L. Zhu’s Least Squares Method . . . . . . . . . . . . . . . . . . . . . . . 24

4 Results and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1 Measured Data Collection & Use in PDOA Algorithms . . . . . . . . . . 27

4.2 Simulated Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Results from Emitter Position #1 . . . . . . . . . . . . . . . . . . . . . . 31

4.4 Results from Emitter Position #2 . . . . . . . . . . . . . . . . . . . . . . 34

4.5 Random Emitter Position Simulations . . . . . . . . . . . . . . . . . . . . 37

4.6 Path Loss Exponent and Exclusion Zone Effects . . . . . . . . . . . . . . 39

4.7 Algorithm Computation Time Comparison . . . . . . . . . . . . . . . . . 43

5 Conclusions and Recommendations . . . . . . . . . . . . . . . . . . . . . . . . 45

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Annex A: List of FRS Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Annex B: Discussion of Issues Affecting PDOA Geolocation . . . . . . . . . . . . 51

List of Abbreviations/Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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List of figures

Figure 1: Example of the Google Earth display resulting from theMATLAB–Google Earth testbed. . . . . . . . . . . . . . . . . . . . . . 6

Figure 2: Grid used in the DPD PDOA emitter geolocation algorithm. . . . . . . . 16

Figure 3: Three sensors using PDOA to calculate an emitter location (sensorsshown as diamonds, estimates shown as dots). . . . . . . . . . . . . . . 18

Figure 4: Histogram of the DPD PDOA emitter location accuracy for thegeometry shown in Figure 3 (1,000 runs). . . . . . . . . . . . . . . . . . 18

Figure 5: Surface generated from multiplying individual probability densityfunctions in the DPD method. . . . . . . . . . . . . . . . . . . . . . . . 19

Figure 6: Illustration of the Intersection Density method. . . . . . . . . . . . . . . 21

Figure 7: Satellite image of DRDC Ottawa Shirley’s Bay campus wheremeasured PDOA data was collected. . . . . . . . . . . . . . . . . . . . . 28

Figure 8: Mobile data collection vehicle used to measure power data in andaround the DRDC Ottawa Shirley’s Bay campus. . . . . . . . . . . . . . 29

Figure 9: Receiver and processing equipment inside mobile data collection vehicle. 29

Figure 10: Location of the transmitter for FRS Channel 1 measured data. . . . . . . 32

Figure 11: Average error distance obtained for 1,000 iterations using randomlyselected measured data for Emitter Position #1 on FRS Channel 1 withα = 4 and an exclusion radius of 250 m. . . . . . . . . . . . . . . . . . . 32

Figure 12: Average error distance for 1,000 iterations using randomly selectedsensor positions from the measured data along with simulated powerlevels (zero-mean Gaussian noise with σ = 12 dB) for Emitter Position#1 (α = 4 and exclusion radius of 250 m). . . . . . . . . . . . . . . . . . 33

Figure 13: Average error distance for 1,000 iterations using random sensorpositions and simulated power levels (zero-mean Gaussian noise withσ = 12 dB) for Emitter Position #1 (α = 4 and exclusion radius of 250 m). 34

Figure 14: Location of the transmitter for FRS Channel 3 measured data. . . . . . . 35

DRDC Ottawa TR 2011-040 ix

Figure 15: Average error distance for 1,000 iterations using randomly selectedmeasured data for Emitter Position #2 on FRS Channel 3 with α = 4and an exclusion radius of 250 m. . . . . . . . . . . . . . . . . . . . . . 36

Figure 16: Average error distance for 1,000 iterations using randomly selectedsensor positions from the measured data, but with simulated powerlevels (zero-mean Gaussian noise with σ = 12 dB) for Emitter Position#2 (α = 4 and exclusion radius of 250 m). . . . . . . . . . . . . . . . . . 36

Figure 17: Average error distance for 1,000 iterations using random sensorpositions and simulated power levels (zero-mean Gaussian noise withσ = 12 dB) for Emitter Position #2 (α = 4 and exclusion radius of 250 m). 37

Figure 18: Simulated results for average error distance with the emitter randomlylocated along one edge of the AOI (10,000 iterations, 250 m exclusionradius). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Figure 19: Simulated results with the emitter randomly located in a500 m × 500 m square area in the centre of the AOI (10,000 iterations,250 m exclusion radius). . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Figure 20: Simulated results for average error distance with the emitter randomlypositioned over the entire AOI (10,000 iterations, 250 m exclusion radius). 40

Figure 21: Average error distance for 1,000 iterations with 10 sensors usingrandomly selected measured data for Emitter Position #1 on FRSChannel 1 for path loss exponent values of α = 2 to α = 5 (250 mexclusion radius). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Figure 22: Average error distance for 1,000 iterations with 10 sensors usingrandomly selected measured data for Emitter Position #2 on FRSChannel 3 for path loss exponent values of α = 2 to α = 5 (250 mexclusion radius). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41



Figure 25: Total computation time for each algorithm (combined for 1,000iterations). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

x DRDC Ottawa TR 2011-040

Figure B.1: Sensor and emitter geometry with the emitter moving parallel to thex-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Figure B.2: x and y bias errors plotted as a function of emitter x-coordinate(y-coordinate fixed to -600). . . . . . . . . . . . . . . . . . . . . . . . . 52

Figure B.3: Sensor and emitter geometry with the emitter moving parallel to they-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Figure B.4: x and y bias errors plotted as a function of emitter y-coordinate(x-coordinate fixed to 100). . . . . . . . . . . . . . . . . . . . . . . . . 54

Figure B.5: Sensor and emitter configuration used to generate distributions of the xand y bias errors shown in Figures B.6 to B.9. . . . . . . . . . . . . . . . 55

Figure B.6: Histogram of NLS emitter estimate (x-coordinate). . . . . . . . . . . . . 55

Figure B.7: Histogram of NLS emitter estimate (y-coordinate). . . . . . . . . . . . . 56

Figure B.8: Histogram of LS emitter estimate (x-coordinate). . . . . . . . . . . . . . 56

Figure B.9: Histogram of LS emitter estimate (y-coordinate). . . . . . . . . . . . . . 57

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Acknowledgements

The author’s would like to thank the members of the Tactical Electronic Warfare Systemsgroup for their support of this work. In particular, many members of the group contributedtoward obtaining the experimental results included in this report. The author’s would alsolike to specifically acknowledge Dr. Shanzeng Guo for useful discussions and participationin the PDOA algorithm development.

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1 Introduction

The ability to accurately locate the position of a radio emitter is often of vital importance.A civilian example would be geolocating an emergency 911 caller. In the military domain,knowledge of the location of enemy radio emitters contributes to situational awareness. Inthe military context, the problem is complicated by the fact that the emitters of interest areuncooperative (i.e., there is generally very limited a priori knowledge of the transmitterpower, frequency, modulation, etc.). Various emitter geolocation techniques are possible,each of which has specific advantages and disadvantages. The more important ones aresummarized in the following text.

1.1 Emitter Geolocation TechniquesIn general, emitter geolocation techniques are based upon the measurement of one or moreof the following properties of the signal of interest:

◦ The direction of arrival (angle of arrival).◦ The signal’s propagation time (time of arrival/time difference of arrival).◦ Relative shifts in the signal frequency observed at pairs of spatially separated sensors,

at least one of which is moving (frequency difference of arrival).◦ The received signal power (received signal strength/power difference of arrival).

1.1.1 Angle of Arrival (AOA)

Angle of arrival (AOA) techniques directly measure the propagation direction of the wavefront of the incoming signal. They generally require an antenna array, which is inherentlybandwidth-limited and can be physically large, particularly if frequency coverage extend-ing into the low VHF and HF bands is required. Furthermore, AOA systems usually requirecalibration to suppress systemic errors and are adversely affected by multipath propagationarising from reflections in the vicinity of the antenna array. The more sophisticated im-plementations require a coherent multi-channel receiver, which has a significant impact onsystem cost and power consumption. Consequently, the design of AOA systems involvesa host of compromises involving size, frequency coverage, sensitivity, accuracy and cost.AOA techniques do have the advantage that a line of position can be obtained using onlya single sensor. In addition, they are well understood and conceptually simple to use forgeolocation.

1.1.2 Time of Arrival (TOA)/Time Difference of Arrival (TDOA)

Emitter geolocation techniques based on the measurement of propagation time are referredto as time of arrival (TOA) or time difference of arrival (TDOA) depending on whether

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the geolocation algorithm uses the absolute time of propagation from the transmitter toreceiver or the time differences between the versions of the signal received at pairs of spa-tially distinct receiver locations. Since electromagnetic waves travel at the speed of lightand the refractive index of air is very close to unity under any practical conditions, theirvelocity can be treated as a constant. Consequently, the distance between two locations canbe directly obtained from a measurement of the time required for a signal to propagate fromone location to another. TOA techniques are generally limited to cooperative transmitters,whereas TDOA techniques can be used for uncooperative emitters. For TDOA geolocation,the versions of the signal received at the sensor locations are relayed to a common site andthe relative time delays between each pair of signals is measured using signal processingtechniques such as cross-correlation. Since each measured time delay defines a hyperbolicline of position, the position of the emitter can then be determined by finding the centroidof the intersections of the lines of position. Time-based techniques require careful engi-neering if accuracy is not to be degraded by systemic errors. Other disadvantages concernthe need for high capacity data links and the requirement for a minimum of three sensors tolocate an emitter in a 2 dimensional space. However, well implemented systems can pro-vide very good geolocation accuracy. Closely related ideas are widely used in navigationsystems (e.g., the global positioning system (GPS) and LORAN (LOng RAnge Navigationsystem)).

1.1.3 Frequency Difference of Arrival (FDOA)

The Doppler shift describes the apparent change in frequency of signal when an observeris moving towards or away from the signal source. Given knowledge of the velocity anddirection of travel of the sensor, knowledge of the Doppler shift can be used to define a lineof position that passes through the sensor. However, in addition to a precise measurementof the signal frequency, the determination of the Doppler shift would require accurate apriori knowledge of the transmitter frequency. In practice this is unlikely to be available;even if the nominal frequency of the transmitter is known, the uncertainty due to normalfrequency variations would be a significant problem. A more practical approach is to use asecond sensor and relay the signals received by both sensors to a common location. Withsuitable signal processing, the differential Doppler shift of the two received signals can bemeasured and used to define a line of position. The achievable accuracy will depend onseveral factors, including the signal frequency, the signal-to-noise ratios at the sensors, andtheir velocity(s). Since the technical requirements for implementing FDOA geolocationare similar to those of TDOA geolocation, the two techniques can be combined if either orboth of the sensors are moving. The use of both techniques in this way has the advantagethat a pair of sensors is sufficient to geolocate a signal source.

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1.1.4 Received Signal Strength (RSS)/Power Difference of Arrival(PDOA)

The final emitter geolocation technique discussed in this section uses measurements of thereceived signal strength (RSS) obtained from the individual sensors in the sensor constel-lation in combination with a path loss model that relates the path loss to the propagationdistance. A particularly simple case occurs for free space propagation; here the path loss ofan electromagnetic wave is proportional to 1/d2, where d is the distance from the emitterto the sensor. Therefore, given a priori knowledge of the transmitter power, the distancebetween the transmitter and sensor can be determined from properly calibrated measure-ments of the received signal power. This distance directly defines the equation of a lineof position having the form of a circle centred on the sensor position. If two sensors areused, the result will be a pair of circles that normally intersect at two points, one of whichcorresponds to the transmitter location. Consequently, using N sensors (N > 2), the emitterposition will correspond to a cluster of N(N −1)/2 intersections. Of course, this approachimplies a cooperative transmitter. Furthermore, in a terrestrial environment, the receivedsignal power will depend on various factors, such as the heights of the transmit and re-ceive antennas and the nature of the terrain. The terrain effects are particularly important;in practice the path loss will not be a monotonic function of distance and the underlyingdependence on distance can vary between 1/r2 and 1/r4.

The power difference of arrival (PDOA) technique is more complicated, but has importantadvantages. Aside from being usable with uncooperative transmitters, the effects of sometypes of measurement errors are eliminated if they are common to all of the sensors. Thistechnique, also known as RSS difference geolocation, is the focus of this report and isdescribed in more detail in the following section.

Emitter geolocation using RSS techniques has been the subject of numerous studies (e.g.,[1–3]). A general discussion of RSS and other emitter geolocation techniques was pre-sented in [1]. Well-known algorithms for RSS geolocation include the Min-Max [4], Mul-tilateration [5], and Maximum Likelihood [6] methods. The Min-Max method computesthe overlap of squares that are centred at each sensor location and calculates the positionfix to be at the middle of the resulting rectangle. The Multilateration technique uses thetransmitter power in conjunction with a suitable propagation path loss model to draw aseries of circles centred at each sensor with the transmitter indicated by the intersectionof the circles. In their usual implementations both the Min-Max and the Multilaterationtechniques assume knowledge of the transmitted power level and are therefore most suit-able for use with cooperative emitters. The Maximum Likelihood (ML) estimation methodcomputes the probability that the transmitter is at each potential position given the receivedsignal power at the sensor position and gives the location at which this probability is max-imized as the emitter location estimate. The ML method generally outperforms the othertwo methods if enough sensors are present [7,8] (approximately 7 or more from the exper-imental results in [7]). However, with fewer sensors, non-statistical techniques appear to


have potential advantages in terms of both accuracy and computational cost [8].

The primary advantage of RSS and PDOA techniques is the simplicity of the hardwarerequired. Virtually any receiver can provide power measurements, and thus could be usedas part of a geolocation system. However, these techniques generally rely on statisticalmodels for propagation loss that represent the distance dependence of the path loss as apower of the path length. In a terrestrial propagation environment, the path loss is highlyvariable. Shadowing and multipath propagation (slow and fast fading) caused by obstruc-tions such as trees, buildings, and other geographical features can significantly affect thereceived signal power [9]. For this and other reasons, RSS geolocation techniques are gen-erally considered to be less accurate than alternative approaches, even though a comparisonof time of arrival (TOA) and RSS techniques in [3] indicated that RSS techniques can becompetitive. Nevertheless, since power readings can be obtained very easily with low-costhardware, RSS techniques can be attractive, even when many sensors are needed to achievethe required accuracy.

1.2 Motivation for the Study of PDOA AlgorithmsThe Tactical Electronic Warfare Systems (TEWS) group, within the Communication andNavigation Electronic Warfare (CNEW) section at DRDC Ottawa, is exploring innovativeapproaches for enhancing the effectiveness of CF EW capabilities. The Klondike Tech-nology Demonstration Project (TDP) has the goal of demonstrating the feasibility of ge-olocating radio frequency emitters using a single Mobile Electronic Warfare Technologies(MEWT) platform supplemented by a network of simple autonomous sensors on non-EWvehicles. The MEWT sensor is referred to as a “thick” sensor in the Klondike paradigmsince it uses large specialized antennas and very high performance sensor components,which would be prohibitively expensive for installation on many non-EW land forces ve-hicles. A “thin” sensor generally refers to a relatively low-cost sensor with a small sizeand power footprint and limited capabilities. The thin sensors must meet stringent require-ments concerning size, weight, cost, and power consumption since they must be installedon non-EW vehicles without adversely impacting the usability of the vehicles for their pri-mary functions or inconveniencing their operators. The basic idea is that the thick sensorin the MEWT will task the thin sensors to provide geolocation information on signals ofinterest.

Thin sensors based on PDOA techniques have several attractive advantages:

◦ Virtually any radio frequency receiver can be used as a sensor to provide the mea-surements of the received signal power used to geolocate the transmitter of a signal ofinterest;

◦ PDOA based sensors can use simple omnidirectional antennas which can be relativelycompact, particularly if active antenna techniques are used;


◦ The demanding data link requirements associated with the TDOA and FDOA tech-niques are avoided.

The accuracy of a PDOA system depends on many factors such as the accuracy of the pathloss model, the tolerance of the sensor’s power readings, the measurement environment,and the antenna radiation pattern. In addition, an important consideration in the practicalimplementation of a PDOA geolocation system concerns the choice of algorithm to beused to compute the position fix. This Technical Report explores several different PDOAalgorithms and presents a comparison using sensor power measurement data obtained usingsimulation and field experiments. The results of this report are directly applicable to theKlondike TDP and include conclusions and recommendations for the algorithms to be usedin the Analyst software package for computing the emitter location estimate, as well as theparadigms that are possible (e.g., number of sensors and their locations, etc.) and theapproximate accuracy that can be expected in real-world applications.

1.3 MATLAB–Google Earth TestbedTo support the evaluation of PDOA algorithms and develop a physical and intuitive un-derstanding of the factors affecting the practically achievable performance of PDOA ge-olocation systems, a MATLAB–Google Earth Testbed was developed. Although intendedprimarily for the analysis of experimental data, this testbed can also be used with sim-ulated power measurement data and is easily scaled to include new algorithms. Variousparameters can be set manually, or read from a simulation or measurement data file:

◦ Locations of the sensors and the emitter specified as absolute (latitude and longitude)or relative positions;

◦ Power measurement data or statistics depending on whether power data is obtainedfrom actual measurements or simulations.

Using this information, a candidate algorithm can be executed and the corresponding po-sition fixes obtained. Using the Google Earth Toolbox for MATLAB [10], a KML file(the native file type for Google Earth) is created. Figure 1 shows a sample result of theMATLAB–Google Earth testbed with five sensors, each represented as a light armouredvehicle (LAV), in and around the DRDC Ottawa campus. The emitter location is repre-sented by an antenna symbol and the estimates of the emitter locations are represented ascoloured spheres (one colour for each algorithm). The received signal strength at each sen-sor is displayed as well as the miss distance for each algorithm (i.e., the distance betweenthe estimate and the actual emitter location).

The use of Google Earth to display the results provides a very intuitive understanding ofthe geolocation accuracy, particularly when data from actual field experiments is used. Thepositions and signal power measurements associated with the sensors and the locations ofthe sensors with respect to the emitter and local obstructions, such as trees and buildings,


or other obstructions are all easily discerned. Inferences can be readily drawn as to theeffect of terrain features on the power measurements at the sensors and how these affectthe geolocation estimates. Furthermore, given Google Earth’s global map coverage, resultsfrom sensor measurements taken anywhere in the world can be easily viewed by simplytagging the data with latitude and longitude information.

Figure 1: Example of the Google Earth display resulting from the MATLAB–Google Earthtestbed.


2 Path Loss Models

An electromagnetic signal propagating through any medium will undergo attenuation, alsoknown as path loss. RSS geolocation techniques inherently rely upon a path loss model toprovide a statistical relationship between the emitter to sensor path loss and distance foreach sensor. PDOA geolocation techniques also involve the implicit assumption that anyvariation in power measurements observed across the sensor population can be attributedto differences in the emitter to sensor path losses. The problem of modeling path loss in aterrestrial environment is complicated by dependencies on environmental factors, many ofwhich involve significant uncertainties. Since various path loss models have been devel-oped for the analysis of communications system performance and other applications, thechoice of path loss model in implementing PDOA geolocation techniques is a problem thatdeserves careful consideration. In this section, several different path loss models will beexplored as background for the description of the PDOA algorithms.

In free space, the Friis equation models the propagation of an electromagnetic wave:

Pr =PtGtGrλ2

(4π)2d2L, (1)

where Pr is the received signal power at a distance d from the emitter, Pt is the transmittedpower, Gt and Gr are the transmitter and receiver antenna gains, respectively, λ is thewavelength of the signal (λ = c/ f where c is the speed of light and f is the frequency), andL represents the system losses not related to propagation. Using this model, the path lossis given by:

PL(d) (dB) = 10log10Pt

Pr=−10log10

(GtGrλ2

(4π)2d2L

). (2)

An alternative formulation of Equation (1) can be presented by using a reference powerlevel at distance d0, Pr, which could be predicted from Equation (1) or measured. In thiscase,

Pr = P0

(d0

d

)2

, d ≥ d0 ≥ d f , (3)

where P0 is the received signal power at the reference distance d0 and d f is the bound-ary of the far-field region (or Fraunhofer region) of the transmitting antenna. Using thisformulation, the path loss is given by:

PL(d) (dB) = PL(d0)+20log10

(dd0

). (4)

In terrestrial applications, the ground reflection can significantly affect the received power,depending on the transmitter and receiver separation and antenna heights. If the heights of


the transmitting and receiving antennas are given by ht and hr, then at distances

d >20πhthr

3λ(5)

the 2-ray model [11] can be used,

Pr = PtGtGrh2

t h2r

d4L. (6)

Note from Equation (6) that the power decreases with the distance to the fourth power, or arate of 40 dB/decade (compared to the distance squared, or 20 dB/decade in the free-spacecase of Equation (1)). Also note that the received power and path loss become independentof frequency.

In most path loss models, the received signal power is proportional to d−α where α isdefined as the path loss exponent. For the free space model, α = 2 (Equation (1)), and inthe 2-ray model, α = 4 (Equations (6)). In practice, typical path loss exponent values rangebetween α = 2 and α = 6, depending on the environment [11]. Therefore, a more generalpath loss expression is given by:

PL(d) = PL(d0)+10α log10

(dd0

), (7)

and the received signal power in dBm is given by:

Pr = P0 −10α log10

(dd0

). (8)

The Egli model [12], a well-known RF propagation model, can be regarded as a modifica-tion of the 2-ray model to include a frequency dependent correction term. The latter wasobtained by fitting measured VHF and UHF data obtained in various urban environments.The Egli model is given by:

PL(d) = GtGr

(hthr

d2L

)2(40f

)2

, 40 MHz ≤ f ≤ 1000 MHz (9)

where the frequency, f , is in MHz. The Egli model is applicable for environments withirregular terrain, but does not take into account fading caused by vegetation.

More complex path loss models have been developed to cover extended frequency rangesor better account for environmental factors. For example, the Okumura model [13], widelyused for urban areas, is applicable to frequencies in the range of 150 MHz to 1920 MHz(and often extrapolated up to 3000 MHz). The Hata model [14], which is an empirical


formulation of Okumura’s model, is valid between 150 MHz and 1500 MHz, and for urbanareas is given by:

PL (dB) = 69.55+26.16log10( f )−13.82log10(ht)−a(hr)

+(44.9−6.55log10(ht)) log10(d),(10)

where a(hr) is an antenna height correction factor.

These models generally express the distance dependence of the path loss measured in dB asa product of the distance and a path loss exponent. The solution of the PDOA problem canbe simplified by assuming that the factors affecting the path loss are the same for all of thereceivers so that only the distance dependent term needs to be considered. In this case, theonly variables of significance are the distances between the sensors and the emitter and thepath loss exponent, which is typically between 2 and 6. In practice, a further simplificationcan be obtained by assigning a fixed value to the latter based on a priori knowledge.

A further issue in the modeling of the path loss concerns the statistics of the path lossprobability distribution. Measurements have shown [11] that a log-normal distributioncan be used to describe the random shadowing effects caused by the terrain in the areasurrounding the transmitter and receiver. Therefore, the path loss model of Equation (7)can be modified as follows:

PL(d) (dB) = PL(d0)+10α log10

(dd0

)+Xσ, (11)

where Xσ is a zero-mean Gaussian random variable with a standard deviation σ (both ofwhich are in dB). Depending on the environment, σ can vary from 1.5 dB to more than16 dB [11] and is generally determined by collecting measurements in the area of interest.This is the model that was used to obtain the results presented in this report.




3 PDOA Algorithms for Non-CooperativeEmitters

In this section, several PDOA algorithms are described for the geolocation estimation ofnon-cooperative emitters. First, three statistical techniques are explored, followed by amethod that computes the highest density of circle intersections which represent potentialemitter locations. Finally, three least squares approximations will be presented.

3.1 Non-Linear Least SquaresConsider two sensors located at (x1,y1) and (x2,y2). If the emitter is located at (x,y), thedistances from the emitter to the sensors are given by

d1 =√

(x− x1)2 +(y− y1)2, (12)

d2 =√

(x− x2)2 +(y− y2)2, (13)

and the measured signal power in dBm at each sensor is given by

P1 = P0 −10α log10

(d1

d0

), (14)

P2 = P0 −10α log10

(d2

d0

), (15)

assuming that the path loss exponent is the same for both sensors. The reference powerlevel, P0, is equal for both sensors because they are both receiving a signal from the sametransmitter. The power difference of these two sensors is therefore,

P12 = P1 −P2 = 10α log10

(d2

d1

). (16)

Using Equations (12),(13) along with Equation (16),

P12 = 5α log10

[(x− x2)

2 +(y− y2)2

(x− x1)2 +(y− y1)2

]. (17)

In general, if there are N sensors, and 1 ≤ k < l ≤ N, then

Pkl = Pk −Pl = 5α log10

[(x− xl)

2 +(y− yl)2

(x− xk)2 +(y− yk)2

]. (18)


If the actual measured power difference between the receivers at Rk = (xk,yk) and Rl =(xl,yl) is Pkl , the optimal Non-Linear Least Squares method finds the (x,y) that minimizethe sum of the squares of the differences between the actual measured received signalstrengths and the theoretical received signal strengths given by Equation (18),

Q(x,y) =∑k<l

[Pkl −5α log10

[(x− xl)

2 +(y− yl)2

(x− xk)2 +(y− yk)2

]]2

, (19)

for all combinations of receiver pairs. The objective function Q is non-linear and the onlymethod to find its minimum is to define a grid over which a search is conducted with Qevaluated at each point on the grid. Ultimately, the grid point that minimizes Q is selectedas the position fix for the transmitter. Since a search grid must be defined, there is a directcorrelation between computation time and emitter geolocation accuracy.

3.2 Maximum LikelihoodAssume the transmitter is located at (x,y) and there are N receivers located at Rk = (xk,yk),k = 1,2, · · · ,N. Let the received signal strength (RSS) at Rk be denoted by Pk, 1 ≤ k ≤ N.Then we can write

Pk =C−10α log10(dk)+ηk, 1 ≤ k ≤ N (20)

where C is a constant, α is the path loss exponent,

dk =√

(x− xk)2 +(y− yk)2, 1 ≤ k ≤ N

and ηk, 1 ≤ k ≤ N, are i.i.d. zero mean Gaussian random variables with equal varianceE(|ηk|2) = σ2 > 0. Define Pkl = Pk −Pl , 1 ≤ k, l ≤ N. We can write

Pkl = 10α log10(dl/dk)+ηkl (21)

where ηkl = ηk −ηl = −ηlk are zero mean Gaussian random variables with equal vari-ance 2σ2 if k �= l. The random variables ηkl , k �= l, however, are no longer statisticallyindependent. For example, we have

E(ηklηk(l+p)) = σ2, p > 0, k < l. (22)

Moreover, it is easy to see that

ηkl = η1l −η1k, k �= l. (23)

In other words, each of the random variables ηkl , 2 ≤ k < l ≤ N, can be represented as thedifference between two terms in the small subsequence η1k, 2 ≤ k ≤ N. As a consequencethe random vector v defined by

v = (η12, · · · ,η1N ,η23, · · · ,η2N ,η34, · · · ,η(N−1)N)T


has a singular covariance matrix. This implies that v has a singular distribution and nomaximum likelihood estimator of the transmitter location (x,y) can be directly derivedfrom the N(N − 1)/2 RSS differences Pkl , 1 ≤ k < l ≤ N. However, we can indirectlyderive a maximum likelihood estimate of the transmitter location from the N(N − 1)/2RSS differences Pkl , 1 ≤ k < l ≤ N, as follows. First consider the RSS differences P1k,2 ≤ k ≤ N. Let Σ be the (N − 1)× (N − 1) covariance matrix of the random vector wdefined by

w = (η12,η13, · · ·η1k,η1(k+1), · · · ,η1N)T

i.e.,

Σ = E(wwT ). (24)

It can be verified that

Σ =

⎡⎢⎢⎢⎢⎣

2σ2 σ2 σ2 · · · σ2

σ2 2σ2 σ2 · · · σ2

σ2 σ2 2σ2 · · · σ2

σ2 σ2 · · · 2σ2 σ2

σ2 σ2 · · · σ2 2σ2

⎤⎥⎥⎥⎥⎦ (25)

and Σ is non-singular with its inverse Σ−1 given by

Σ−1 =1

σ2

⎡⎢⎢⎢⎢⎣

N−1N − 1

N − 1N · · · − 1

N− 1

NN−1

N − 1N · · · − 1

N− 1

N − 1N

N−1N · · · − 1

N− 1

N − 1N · · · N−1

N − 1N

− 1N − 1

N · · · − 1N

N−1N

⎤⎥⎥⎥⎥⎦ (26)

The probability density function of the Gaussian random variables P12,P13, · · · ,P1N , de-noted by p(P12,P13, · · · ,P1N), is then computed by:

p(P12,P13, · · · ,P1N) =1

(2π)(N−1)/2 detΣexp

[−1

2(P−P0)

T Σ−1(P−P0)

](27)

where detΣ is the determinant of Σ and P and P0 are (N −1)-dimensional column vectorsdefined respectively by

P= (P12,P13, · · · ,P1N)T (28)

and

P0 =−10α[

log10

[d2

d1

], log10

[d3

d1

]· · · , log10

[dN

d1

]]T

(29)


A maximum likelihood estimate of the transmitter location (x,y) is then obtained by mini-mizing the objective function Q1(x,y) defined by

Q1(x,y) = Nσ2(P−P0)T Σ−1(P−P0)

= (N −1)N∑

k=1

{P1k −10α log10

[dk

d1

]}2

−∑k �=l

{P1k −10α log10

[dk

d1

]}×{

P1l −10α log10

[dl

d1

]}. (30)

In general, for each 1 ≤ k ≤ N, a maximum likelihood estimate of the transmitter location(x,y) can be obtained by minimizing the objective function Qk(x,y) defined by

Qk(x,y) =

= (N −1)N∑

l=1

{Pkl −10α log10

[dl

dk

]}2

−∑p�=q

{Pkp −10α log10

[dp

dk

]}×{

Pkq −10α log10

[dq

dk

]}. (31)

Finally, a maximum likelihood estimate of the transmitter location (x,y) can be obtainedfrom the N(N − 1)/2 RSS differences Pkl , 1 ≤ k < l ≤ N, by minimizing the sum of theobjective functions Qk(x,y), 1 ≤ k ≤ N, defined by Q =

∑Nk=1 Qk and computed by

Q(x,y) =N∑

k=1

Qk(x,y)

= (N −1)N∑

k=1

N∑l=1

{Pkl −10α log10

[dl

dk

]}2

−N∑

k=1

∑p�=q

{Pkp −10α log10

[dp

dk

]}×{

Pkq −10α log10

[dq

dk

]}. (32)

When N is relatively large, it is reasonable to expect that the objective function Q definedby (32) is dominated by the expression

Q∗(x,y) = (N −1)N∑

k=1

N∑l=1

{Pkl −10α log10

[dl

dk

]}2

= (N −1)∑k �=l

{Pkl −10α log10

[dl

dk

]}2

= 2(N −1)∑k<l

{Pkl −10α log10

[dl

dk

]}2

. (33)


and this implies that the maximum likelihood estimator and the NLS estimator for thetransmitter location (x,y) can be expected to have very similar performance.

3.3 The Discrete Probability Density MethodThe discrete probability density (DPD) method assigns probability density functions to thesensor data and obtains an estimate of the emitter location through a joint probability dis-tribution. The DPD method has been claimed to be robust in the presence of anomalousor wild measurements [15]. This technique allows considerable flexibility and sophisti-cation in the weighting of information from the sensors by the assignment of appropriateprobability density functions (PDFs). The DPD method is broadly similar to the Maxi-mum Likelihood method in that it attempts to find the position estimate with the highestprobability. It involves the following steps:

1. The general idea that geolocation techniques involve the generation of lines of po-sition is modified by convolving each line of position with an estimated or assumedprobability distribution to represent the geolocation information as a 3-dimensionalsurface whose height at any given position corresponds to the probability that thetransmitter is at that location;

2. The joint probability estimate for the various lines of position can be obtained bycomputing the product of the spatial probability densities associated with each lineof position obtained in Step 1;

3. The geolocation estimate is the position where the value of the joint probability den-sity defined in Step 2 is maximized. In a practical implementation, the probabilitydistributions are computed for a 2-dimensional grid.

Similar to the Non-Linear Least Squares and Maximum Likelihood methods describedpreviously, an area of interest (AOI) that includes the potential locations of the emitter isdefined with dimensions X ×Y . A 2D set of grid points is overlaid with a grid spacing of Δxand Δy, resulting in N ×M grid points where N = X/Δx and M = Y/Δy. For each sensor,a PDF is calculated for each grid point, after which the product of each of the individualPDFs is computed [16], assuming the measurements for each sensor are independent,

P′XY (n,m) =

S∏s=1

FXY (s,n,m), n = 1...N,m = 1...M (34)

where FX ,Y (s,n,m) is the PDF for a specific sensor at a specific grid point and S is thenumber of sensors. This value is then normalized by

C =N∑

n=1

M∑m=1

P′XY (n,m) (35)


resulting in the product of individual PDFs,

PXY (n,m) =1C

P′XY (n,m). (36)

The estimate of the target location can be determined from the peak of PXY (n,m) or bycomputing the “centre of mass” using probability mass functions.

The DPD method can be applied to the RSS difference emitter geolocation as follows. Asdescribed in Section 2, assuming a log-normal distribution of the signal strength measure-ments [17, 18], the path loss measured on a logarithmic scale for the received power levelsis given by:

PL(d) = PL(d0)+10α log10

(dd0

)+Xσ, (37)

where, as discussed in Section 2, PL(d0) is the path loss at a close-in reference distanced0 from the transmitter, α is the path loss exponent (typically between 2 and 6), and Xσis a zero-mean Gaussian random variable with standard deviation of σ, which representsthe received signal strength fluctuations. From Figure 2, the distances between a potential(but ultimately incorrect) emitter location at (x,y) and the sensors S1 and S2 are d1 and d2,respectively, and the received powers are P1 and P2. The noise components of P1 and P2 aregiven by X1 and X2, which are independent and identically distributed (i.i.d.) zero-meanGaussian random variables with variance σ2. Using Equations (14) and (15), the differencein power between P1 and P2 is given by:

P12 = P1 −P2 = 10α log10

(d2

d1

)+X , (38)

d 1

d2

Tx

p1

(x,y)

p2

S 1

S 2

Figure 2: Grid used in the DPD PDOA emitter geolocation algorithm.


where X = X1 − X2. Since the difference between two i.i.d. zero-mean Gaussian ran-dom variables is taken, X is a zero-mean Gaussian random variable with variance equalto 2σ2. From Equation (38), a Gaussian PDF can then be computed using a mean of10α log10(d2/d1) and a variance of 2σ2,

FXY =1

2σ√

πe−(P12−10α log10(d2/d1))

2

4σ2 . (39)

When a Gaussian distribution is used for the PDF in the DPD method, it becomes equiva-lent to the Non-Linear Least Squares method described previously.

The standard deviation of each individual sensor power measurement is given by σ, whichis typically between 4 dB and 12 dB. The PDF in Equation (39) is computed for each gridpoint and for every possible pair of sensor combinations. Note that the computational costincreases rapidly with the grid resolution and number of sensors and it may be necessary tomake some trade-offs between accuracy and computational cost in a practical implementa-tion. The joint PDF is obtained by multiplying the PDFs together for each grid point; theresult is analogous to a 3D surface where the peak represents the most likely location ofthe emitter.

To demonstrate the DPD method, a basic simulation experiment was performed in a MAT-LAB environment with a path loss exponent of α = 4. The emitter was placed in the centreof a 2 km × 2 km area of interest with three sensors denoted by diamonds as shown in Fig-ure 3. In this figure, 1,000 emitter position estimates are plotted as circles, most of whichoverlay each other. The grid used in this simulation was 50 m × 50 m. From the histogramof the errors plotted in Figure 4, the majority of the fixes lie within 50 m of the transmitter.

To illustrate the result from the multiplication of PDFs, a 3D surface is shown in Figure 5for a single iteration of the previous configuration with the standard deviation increased to4 dB. As expected, the peak occurs around the emitter location at (1000,1000).

3.4 Intersection DensityThe Intersection Density method [19] utilizes pairs of receivers to generate circles, uponwhich the transmitter lies. By using multiple different pairs of sensors, multiple circles canbe derived each of which intersect at the emitter in the absence of noise and measurementerrors. Of course, the effects of fast and slow fading, in addition to any measurement errorsresult in circles which do not all intersect at a single point. However, the Intersection Den-sity method assumes that the number of circle intersections will be the highest in the areasurrounding the transmitter of interest and therefore calculates the density of intersectionsusing a grid.


0 250 500 750 1000 1250 1500 1750 20000

250

500

750

1000

1250

1500

1750

2000

x (m)

y (m

)

SensorsPF EstimatesEmitter

Figure 3: Three sensors using PDOA to calculate an emitter location (sensors shown asdiamonds, estimates shown as dots).

0 50 100 150 200 2500

50

100

150

200

250

300

350

400

450

Miss Distance (m)

Tria

ls

Figure 4: Histogram of the DPD PDOA emitter location accuracy for the geometry shownin Figure 3 (1,000 runs).


0500

10001500

2000

0500

10001500

20000

0.5

1

1.5

2x 10

−3

x (m)y (m)

P(x

,y)

Figure 5: Surface generated from multiplying individual probability density functions inthe DPD method.

To explain this method, consider three sensors, where the signal power measured by eachsensor is given by Pn, and their respective unknown distances from the emitter are given bydn where n = 1,2,3,

Pn = P0 −10α log10

(dn

d0

). (40)

Since each sensor is receiving power from the same transmitter, the received power at theclose-in reference distance should be equal for each sensor. Therefore,

P12 = P1 −P2 =−10α log10

(d1

d2

)(41)

P23 = P2 −P3 =−10α log10

(d2

d3

)(42)

P31 = P3 −P1 =−10α log10

(d3

d1

). (43)

Rearranging these equations to obtain the distance ratios,

d12 =d1

d2= 10−

P1−P210α (44)

d23 =d2

d3= 10−

P2−P310α (45)


d31 =d3

d1= 10−

P3−P110α . (46)

If the transmitter of interest is located at point (x,y), then the distance from each sensor tothe emitter is given by:

d21 = (x− x1)

2 +(y− y1)2 (47)

d22 = (x− x2)

2 +(y− y2)2 (48)

d23 = (x− x3)

2 +(y− y3)2. (49)

Using Equations (44),(45),(46) along with Equations (47),(48),(49), the unknown distancescan be eliminated. After some straight-forward mathematical manipulations, three circleequations can be derived with centre points located at:

c12 =

(x2d2

12 − x1

d212 −1

,y2d2

12 − y1

d212 −1

)(50)

c23 =

(x3d2

23 − x2

d223 −1

,y3d2

23 − y2

d223 −1

)(51)

c31 =

(x1d2

31 − x3

d231 −1

,y1d2

31 − y3

d231 −1

), (52)

and radii equal to:

r12 =

√√√√(x1 − x2d2

12

d212 −1

)2

+

(y1 − y2d2

12

d212 −1

)2

− d212x2

2 +d212y2

2 − x21 − y2

1

d212 −1

(53)

r23 =

√√√√(x2 − x3d2

23

d223 −1

)2

+

(y2 − y3d2

23

d223 −1

)2

− d223x2

3 +d223y2

3 − x22 − y2

2

d223 −1

(54)

r31 =

√√√√(x3 − x1d2

31

d231 −1

)2

+

(y3 − y1d2

31

d231 −1

)2

− d231x2

1 +d231y2

1 − x23 − y2

3

d231 −1

. (55)

The intersection of these three circles can be found through conventional means. As de-rived, with three sensors there are three circles, each of which can intersect with anothercircle a maximum of two times. Therefore there are up to six intersections using threesensors, half of which are incorrect estimates of the emitter location.

In the unlikely scenario where a pair of circles intersect each other at precisely one locationthen this location is used as the intersection. If there are two intersections between the twocircles then both are considered potential emitter positions because it is not possible toknow which one is the more accurate estimate. Finally, in the case where two circles do


not intersect due to the presence of noise, shadowing, and multipath, the emitter estimatechosen is the mid-point between the two circles where they are closest to intersecting.

A minimum of three sensors are required for this method. Adding more sensors results inthe generation of more circles and thus more intersections that can be used for the determi-nation of the emitter position.

Once all of the intersections have been determined, the question remains of the methodby which the position of the transmitter will be found since many of the estimates (up tohalf) are invalid since they were produced by ambiguous intersections. In the IntersectionDensity method, a search is conducted over the area of interest using a grid. The centreof the grid cell with the highest intersection density is used as the estimate of the emitterlocation. Figure 6 shows an example of three sensors used to generate three circles thatintersect in six different positions (denoted in red). The cell with the most intersectionsin it (three) is selected as the highest intersection density cell and the centre of the cell isused as the estimate for the emitter location (denoted by an ×). To refine this technique,a sliding search cell can be employed as opposed to a static grid. The selection of the cellsize is crucial to this technique and methods for automatic optimization of this parameterare currently under investigation and show promise, however, a rudimentary version of thisalgorithm with a static grid was used to generate the results in this report.

c12c 23

c 31

r 23r 12

r 31

Figure 6: Illustration of the Intersection Density method.

3.5 Modified Y.T. Chan’s Least Squares MethodIn order to improve computational complexity relative to the Non-Linear Least Squarestechnique, the non-linear system in Equation (18) can be linearized, yielding a linear least


squares solution. Such an approach was formulated by Y.T. Chan at the Royal MilitaryCollege of Canada [20] using RSS, and has been extended in this work to operate with RSSdifference measurement data. The development of this solution is described as follows:

Pkl = 5α log10

[(x− xl)

2 +(y− yl)2

(x− xk)2 +(y− yk)2

],1 ≤ k < l ≤ N (56)

If a constant is defined,βkl = 10

Pkl5α , (57)

then Equation (56) can be rewritten as

(x− xl)2 +(y− yl)

2 = βkl[(x− xk)2 +(y− yk)

2], (58)

or,

(1−βkl)(x2 + y2)−2(xl −βklxk)x−2(yl −βklyk)y = wkl, 1 ≤ k < l ≤ N, (59)

where N is the number of sensors, and wkl = βklr2k − r2

l (rk =√

x2k + y2

k , rl =√

x2l + y2

l ). In

order to linearize this system, a new parameter, c = x2 + y2 is introduced and is treated asif it is independent of x and y. With this new parameter,

(1−βkl)c−2(xl −βklxk)x−2(yl −βklyk)y = wkl, 1 ≤ k < l ≤ N. (60)

If a matrix, A is defined to be the (N −1)×3 matrix consisting of the row vectors,

(1−β1k,2(β1kx1 − xk),2(β1ky1 − yk)), 2 ≤ k ≤ N, (61)

and b is an (N −1)×1 matrix consisting of the (N −1) terms of w1k,2 ≤ k ≤ N, then themodified Y.T. Chan’s Least Squares estimate of the emitter position is obtained from thefollowing:

(c,x,y)T = (ATA)−1ATb (62)

where T denotes a matrix transpose.

3.6 S. Wang’s Least Squares MethodAn alternative approach to linearizing the non-linear system described in Equation (18)was introduced by S. Wang at DRDC Ottawa [21]. Equation (59), reproduced here,

(1−βkl)(x2 + y2)−2(xl −βklxk)x−2(yl −βklyk)y = wkl, 1 ≤ k < l ≤ N, (63)

can also be rewritten as

x2 + y2 +aklx+bkly = ckl, 1 ≤ k < l ≤ N, (64)


where N is the number of sensors, and

akl = 2βklxk − xl

1−βkl(65)

bkl = 2βklyk − yl

1−βkl(66)

ckl =βklr2

k − r2l

1−βkl(67)

with rk =√

x2k + y2

k and rl =√

x2l + y2

l . Consider ((p,q),(s, t)) ∈ ΓN where ΓN is definedby:

ΓN = {((p,q),(s, t))|(p,q)≺ (s, t),(p,q),(s, t) ∈ ΔN} (68)

with ≺ denoting the lexical order defined on the set of integer pairs ΔN , where,

ΔN = {(p,q)|p < q,1 ≤ p,q ≤ N}. (69)

By subtracting the following equations,

x2 + y2 +apqx+bpqy = cpq (70)

x2 + y2 +astx+bsty = cst (71)

the non-linear x and y terms are eliminated, yielding the following linear equation:

μ(p,q,s, t)x+ν(p,q,s, t)y = τ(p,q,s, t) (72)

whereμ(p,q,s, t) = apq −ast (73)

ν(p,q,s, t) = bpq −bst (74)

τ(p,q,s, t) = cpq − cst . (75)

Let Γ ⊂ ΓN be a subset of ΓN . From the linear system associated with the set Γ, S. Wang’sLeast Squares estimate of the emitter position is given by:

(x,y) =

(TννTμτ −TμνTντ

TμμTνν −T 2μν

,TμμTντ −TμνTμτ

TμμTνν −T 2μν

), (76)

whereTμμ =

∑((p,q),(s,t)∈Γ

μ(p,q,s, t)2 (77)


Tνν =∑

((p,q),(s,t)∈Γ

ν(p,q,s, t)2 (78)

Tμν =∑

((p,q),(s,t)∈Γ

μ(p,q,s, t)ν(p,q,s, t) (79)

Tμτ =∑

((p,q),(s,t)∈Γ

μ(p,q,s, t)τ(p,q,s, t) (80)

Tντ =∑

((p,q),(s,t)∈Γ

ν(p,q,s, t)τ(p,q,s, t). (81)

3.7 L. Zhu’s Least Squares MethodThe final Least Squares method that will be discussed in this Technical Report is a methoddeveloped by L. Zhu [22]. As opposed to the previous algorithms which were derivedusing a logarithmic form of the received signal strength, all powers, P, denoted in thissection represent the linear form of received power given by:

P = kPtd−α (82)

where Pt is the transmitted power, α is the path loss exponent, and k includes all otherfactors (discussed in Section 2). Solving for the distance, d yields

d =

(P

kPt

)− 1α

(83)

Each sensor must lie on a circle that is centred at the emitter with a radius of di for the ith

receiver,(x− xi)

2 +(y− yi)2 = d2

i (84)

or,

(x− xi)2 +(y− yi)

2 =

(Pi

kPt

)− 2α. (85)

If a constant, C, is defined as follows

C =

(1

kPt

)− 2α

(86)

then(x− xi)

2 +(y− yi)2 = P

− 2α

i C. (87)


If the difference between the kth and lth receivers is used,

(P− 2

αk −P

− 2α

l )C = (x− xk)2 +(y− yk)

2 − (x− xl)2 − (y− yl)

2. (88)

This subtraction eliminates the non-linear terms and, with k = 1,

2(xl − x1)x+2(yl − y1)y+(

P− 2

αl −P

− 2α

1

)z = r2

l − r21, 1 < l ≤ N, (89)

where N is the number of sensors and r1 =√

x21 + y2

1 and rl =√

x2l + y2

l . A matrix, A, canbe defined as the (N −1)×3 matrix consisting of the row vectors,(

2(xl − x1) , 2(yl − y1) ,

(P− 2

αl −P

− 2α

1

)),2 ≤ l ≤ N (90)

and b is an (N − 1)× 1 matrix consisting of the (N − 1) terms of r2l − r2

1. L. Zhu’s LeastSquares estimate of the emitter location can then be obtained from the following:

(x,y,z)T = (ATA)−1ATb, (91)

where T denotes a matrix transpose.




4 Results and Comments

This section presents results obtained using the PDOA algorithms discussed in Section 3using both measured and simulated data. Note that several assumptions were used. First,each algorithm was implemented using the path loss model given by Equation (11) withthe same path loss exponent. Initially, a path loss exponent of α = 4 is used as this is acommonly assumed value for terrestrial wireless communications links. To assess whetherthe choice of path loss exponent had any significant effect, a further simulation experi-ment was performed to explore the dependence of each algorithm on α. For the statisticalmethods (Non-Linear Least Squares, the Discrete Probability Density method, MaximumLikelihood), a search grid must be defined. For all the results presented in this section, thegrid cell size was 100 m × 100 m. Further reductions in the grid size were found to providea very marginal improvement in estimation accuracy while greatly increasing the requiredcomputation time. For the Intersection Density method, a search cell is used that was de-fined to be 100 m × 100 m and moved in 50 m increments. In the cases where a grid mustbe defined, a selection of the area of interest (AOI) is necessary, larger than the area overwhich measurements were collected (since the emitter could lie outside the measurementarea, and likely will in EW applications).

The measured data used to produce the results in this section covered an area of approxi-mately 2 km × 2 km. An AOI of 3 km × 3 km was used with the centre of the measurementcollection area set to the centre of the AOI. This implicitly assumes that the emitter of inter-est is located within the AOI, which is not an issue in this case (since the emitter positionswere known to be within the AOI), but the selection of AOI size is a serious practical issuewhen such a priori knowledge is not available. By defining an AOI, it is not possible toobtain solutions outside the area of interest for grid-based search algorithms (and thus verylarge errors are excluded for the scenarios considered with these algorithms). The furtheroptimization of the path loss model, path loss exponent, or grid size is beyond the scopeof this report, which is focused on the relative performance of various PDOA algorithms.The number of sensors considered in producing the results in this section ranges from threeto 12. For the three Least Squares techniques, numerical instabilities are present when thenumber of sensors is less than five. Therefore, for the Least Squares methods, a minimumof five sensors was used.

4.1 Measured Data Collection & Use in PDOAAlgorithms

The measurements used to produce the results described in this section were acquired May4, 2010 from experiments conducted on the DRDC Ottawa Shirley’s Bay campus, shownin Figure 7.


Figure 7: Satellite image of DRDC Ottawa Shirley’s Bay campus where measured PDOAdata was collected.

For these experiments, the data collection vehicle, shown in Figure 8, was configured with aCRC Spectrum Explorer spectrum monitoring system (see Figure 9) based on the followingcomponents:

◦ DRS SI-9144 Tuner◦ DRS SI-9475 A/D Converter◦ Laptop computer running Spectrum Explorer software ver. 1.9.12◦ 4-element Butler Matrix Direction finding Antenna (PR11), 400 MHz – 2.4 GHz

The receive antenna on the mobile data collection vehicle was approximately 3.8 m abovethe ground and approximately 0.5 m above the vehicle superstructure. While the receiveantenna theoretically provides an omnidirectional radiation pattern, its interactions withthe mobile data collection vehicle will cause a non-uniform antenna gain pattern. As canbe seen in Figure 8, the roof of the vehicle is cluttered with a railing and other equipment,which will affect the radiation pattern. This is particularly problematic for power-basedgeolocation methods since the received signal strength (and hence geolocation estimate) isaffected by the orientation of the vehicle relative to the transmitter. In fact, it was foundthat power measurement recorded from the same location varied by up to 10 dB when thevehicle was travelling in opposite directions. This issue will be addressed in future researchincluding electromagnetic simulations to select optimal antenna placements.


Figure 8: Mobile data collection vehicle used to measure power data in and around theDRDC Ottawa Shirley’s Bay campus.

Figure 9: Receiver and processing equipment inside mobile data collection vehicle.


The signal power estimates were obtained using frequency domain techniques to measurethe power contained in a FRS radio channel at the sum output of the Butler Matrix an-tenna. Blocks of digitized time domain in-phase and quadrature (I&Q) signal data fromthe SI-9475 digitizer are transformed to the frequency domain using a Fast Fourier Trans-form (FFT). Since the FFT frequency bins have a width of slightly more than 1.7 kHz, apower measurement corresponding to each 12.5 kHz bandwidth channel in the FRS radioband (see Annex A) is obtained by selecting the 7 contiguous FFT frequency bins corre-sponding to the channel and summing their spectral power estimates. The blocks of dataprocessed by the FFT are windowed with a Blackman time domain window to control spec-tral sidelobe levels. The noise bandwidth of slightly greater than 12.5 kHz resulting fromthis signal processing implementation is a reasonable choice for processing narrow bandfrequency modulation signals. Note that spectral leakage resulting from a strong signal inan adjacent channel will introduce a significant positive bias in the channel power mea-surement. Another source of a positive bias in power measurement data occurs for veryweak signals near the system noise floor (observed to be approximately −120 dBm for thesystem configuration used).

Two transmitters were set up at different locations to transmit a continuous wave (CW) pilottone on two channels in the Family Radio Service (FRS) band (Channel 1: 462.5625 MHzand Channel 3: 462.6125 MHz) with a power of 0.5 W (27 dBm). Discone antennas with aheight of approximately 2 m were used to transmit the signals. The mobile data collectionvehicle was driven on the roads in and around the DRDC Ottawa campus while measuringreceived signal strength in both FRS bands and a log file of this data was recorded alongwith the GPS coordinates for each measurement and several other parameters. This logfile was then imported using the MATLAB–Google Earth testbed to visualize the results ofvarious PDOA algorithms and produce the results described in this section.

A large number of measurements were recorded (∼15 measurements per second) and themobile collection vehicle came within 10 m of both emitters at some point in the datacollection process. To improve the relevance of the results for the proposed applicationin EW, only a small fraction of the measurements were used and all measured data thatwas within a certain distance from the actual emitter position was discarded (an exclusionradius of 250 m was used unless otherwise noted). In order to compare the various PDOAalgorithms, the methodology was as follows:

◦ Randomly choose a data point in the measurement log file.◦ Ensure that this sensor is separated from the transmitter by at least 250 m.

� If not, then randomly choose another point.◦ Randomly choose another data point in the measurement log file.◦ Ensure a separation of at least 100 m from the locations of the other selected powermeasurements.

� If not, then randomly choose another point.◦ Ensure a separation of at least 250 m away from the emitter.


� If not, then randomly choose another point and check separation distance again.◦ And so on, up to the number of sensors under consideration.

It should be noted that anywhere in this section where a “random” sensor distribution isreferred to it is actually subject to the constraints listed above. The error distance (or missdistance), defined as the distance from the actual emitter position to the estimated emitterposition for a given algorithm, was calculated for each iteration using this methodology.The average miss distance for a specified number of iterations was calculated for each al-gorithm. The same data points (i.e., sensor power levels and locations) for a given iterationwere used for each algorithm, and therefore provides a fair comparison for that specificmeasurement scenario (i.e., the specific transmitter position, measurement collection area,and equipment used).

A more accurate estimation of the received signal strength, and ultimately a more accurateemitter location estimate, could be obtained by performing preprocessing on the receiveddata. For example, the multiple measurements per second recorded as the vehicle wasmoving could be averaged. This spatial diversity can be exploited to substantially reducethe spread of the measured power values contributed by multipath propagation. Researchis currently ongoing to optimize the preprocessing methodology, however, this is outsidethe scope of this study, which was to provide a relative comparison of the various PDOAalgorithms and not on the processing optimizations needed to achieve the absolute lowestpossible PDOA miss distance.

4.2 Simulated DataSome of the results presented in this section also use simulated data. There were twovariants of simulated data that will be noted throughout this discussion: (1) the sensorpositions were acquired from the measured data but the power measurements were replacedwith a simulated power measurements, and (2) both the sensor positions and power levelswere simulated. To simulate the power levels for a given sensor, a mean path loss wascalculated using the distance from the transmitter to the sensor using a path loss model ofthe type discussed in Section 2. Random zero-mean Gaussian noise was then added to thispower level with a specific standard deviation to represent the power fluctuations due tofast fading, slow fading, and other effects (e.g., sensor measurement errors).

4.3 Results from Emitter Position #1The first of the two test emitters was located at the position shown in Figure 10. Thisposition is near the east edge of the measurement collection area.

Shown in Figure 11 is the result of 1,000 iterations of randomly selected measured datapoints (restricted by the methodology outlined in Section 4.1).


Figure 10: Location of the transmitter for FRS Channel 1 measured data.

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Figure 11: Average error distance obtained for 1,000 iterations using randomly selectedmeasured data for Emitter Position #1 on FRS Channel 1 with α = 4 and an exclusionradius of 250 m.


As noted previously, using a Gaussian distribution (which was the case here), the Non-Linear Least Squares (NLS), Maximum Likelihood (ML), and the Discrete ProbabilityDensity (DPD) method produce equivalent results, and thus are shown as one line in Figure11. From this figure, it is clear that the statistical methods (NLS, ML, DPD) outperformthe Least Squares (LS) methods as well as the Intersection Density method which was thesecond-most accurate. Note that the gain in accuracy obtained by increasing the numberof sensors falls off after 6-8 sensors. The difference between the most accurate and leastaccurate algorithms is approximately 200 m.

A simulation experiment was next performed that was identical to the one described previ-ously with one exception: the actual measured power was replaced with simulated power.The sensor positions were still obtained from the measured data (i.e., the sensor positionswere restricted to the roads on which the mobile data collection vehicle travelled). Theresults, when zero-mean Gaussian noise with a standard deviation of 12 dB is added to themean path loss is shown in Figure 12.

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Figure 12: Average error distance for 1,000 iterations using randomly selected sensorpositions from the measured data along with simulated power levels (zero-mean Gaussiannoise with σ = 12 dB) for Emitter Position #1 (α = 4 and exclusion radius of 250 m).

Comparing Figures 11 and 12, it is apparent that the results are extremely close to eachother. The significance of this, and the purpose of this simulation, was to determine howaccurately Gaussian noise added to a mean path loss could model the results from actualmeasurements. Since they match very well with a standard deviation of 12 dB it impliesthat future simulations using this type of simulated power can provide an accurate depiction


of the results that would be obtained if measured data was collected (without the significantexpenditure of resources that is required for a field exercise). Since these simulated powerresults still used the sensor positions from the measurement data, it is interesting to inves-tigate a random sensor placement along with simulated power levels (again with a 12 dBstandard deviation in the Gaussian noise added to the mean path loss). The result of thissimulation is shown in Figure 13.

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Figure 13: Average error distance for 1,000 iterations using random sensor positions andsimulated power levels (zero-mean Gaussian noise with σ = 12 dB) for Emitter Position#1 (α = 4 and exclusion radius of 250 m).

A comparison of Figures 11 and 13 reveals minor differences, however, the results are sim-ilar enough that the same conclusions and approximate estimation accuracies can be ob-tained. Therefore, it is a reasonable assumption that future potential measurement locationscan be reasonably approximated using a random sensor distribution given the availabilityof a variety of measurement positions that is similar to the ones explored here (i.e., theavailable roads in and around the DRDC Ottawa campus).

4.4 Results from Emitter Position #2The second emitter used for experimentation was positioned as shown in Figure 14. Thisposition is more centrally located in the measurement collection area compared with theposition of the first test emitter previously discussed.


Figure 14: Location of the transmitter for FRS Channel 3 measured data.

A similar approach to Emitter Position #1 was used and the results of 1,000 iterations ofrandomly selected measurement data points are shown in Figure 15.

For the data shown in this figure, the Least Squares algorithms and the Intersection Densitymethod outperform the statistical approaches. Since the Least Squares methods are anapproximation to the Non-Linear Least Squares algorithm, it is interesting to observe theapproximation producing a more accurate result than the exact solution. This behaviourseems to be a result of estimation biases, which in this case are towards the centre of theAOI. In fact, the results are almost reversed compared to the previous emitter location.Note that the change in position is the most significant factor affecting the results shown inFigures 11 and 15. The 50 kHz difference in frequency is insignificant with respect to thepath loss model statistics.

Annex B provides results from simulation experiments that provide some insights into thebias errors observed with PDOA geolocation algorithms.

To determine if simulated power levels using a mean path loss along with zero-mean Gaus-sian noise with a standard deviation of 12 dB would yield similar results, as was shownfor Emitter Position #1, a similar test was conducted with the Channel 3 data. The results,presented in Figure 16, show the results are similar to the measured power levels used togenerate Figure 15 (somewhat of a change in the statistical results, but the trend of lowestaccuracy remains).


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Figure 15: Average error distance for 1,000 iterations using randomly selected measureddata for Emitter Position #2 on FRS Channel 3 with α = 4 and an exclusion radius of250 m.

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Figure 16: Average error distance for 1,000 iterations using randomly selected sensorpositions from the measured data, but with simulated power levels (zero-mean Gaussiannoise with σ = 12 dB) for Emitter Position #2 (α = 4 and exclusion radius of 250 m).


Finally, as was conducted for Emitter Position #1, a simulated random sensor distributionwas used along with simulated power levels. The results, which are shown in Figure 17,give the same general conclusions as the measured data, and therefore provide a secondconfirmation on the viability of using a random sensor distribution along with Gaussiannoise (zero-mean, σ = 12 dB) corrupted power levels to model actual measurement results.

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Figure 17: Average error distance for 1,000 iterations using random sensor positions andsimulated power levels (zero-mean Gaussian noise with σ = 12 dB) for Emitter Position#2 (α = 4 and exclusion radius of 250 m).

4.5 Random Emitter Position SimulationsThe previous results indicate that the relative PDOA algorithm accuracy is highly depen-dent upon the emitter position. Since the simulation experiments were shown to provideapproximately the same results as those obtained in field experiments, further simulationexperiments were conducted to investigate the performance dependence of the algorithmson the relative positions of the emitter and sensors.

The first simulation was with a transmitter randomly positioned along the east edge ofa 2 km × 2 km AOI. This emitter position was used to further investigate the results ofEmitter Position #1, which was near the edge of the AOI. The results are shown in Figure 18for 10,000 iterations. Similar to the measured results for Emitter Position #1, the statisticalalgorithms outperform the Least Squares methods, and the Intersection Density method isthe second most accurate.


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Figure 18: Simulated results for average error distance with the emitter randomly locatedalong one edge of the AOI (10,000 iterations, 250 m exclusion radius).

Next, a simulation was conducted where the transmitter was randomly located within a500 m × 500 m square area at the centre of the AOI. The aim was to investigate the effectof having an emitter close to the centre of the measurement area, similar to Emitter Posi-tion #2. The results, shown in Figure 19 for 10,000 iterations, show that the Least Squarestechniques outperform the statistical methods as was the case with the measured resultsfor Emitter Position #2. A potential explanation for these results is that the Least Squaresmethods produce estimates that are biased towards the centre of the AOI. To provide addi-tional insights, a simulation experiment was conducted with the sensors evenly distributedon a circle with the transmitter located at the centre. In this rather arbitrary scenario, it wasfound that the Least Squares methods always produce an estimate at the exact centre ofthe circle regardless of the sensor power levels. This result confirms that the Least Squaresmethods are biased and helps explain the previous results where the Least Squares methodsoutperform the others. Furthermore, the relatively poor accuracy of the other algorithms inthis scenario leads to the conclusion that this seemingly favourable emitter-sensor geome-try is suboptimal. Other simulations and tests have demonstrated that the best performanceis achieved when at least some of the sensors are close to the emitter and that the sensorsthat are relatively distant from the emitter generally contribute little useful information.Consequently, for optimal results, the sensors should be uniformly distributed in the AOI.

Since significant a priori knowledge of the emitter location is seldom available in for EWapplications, the question of obtaining an overall performance measure arises. To addressthis question, the position of the transmitter was randomly changed over the entire AOI


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Figure 19: Simulated results with the emitter randomly located in a 500 m × 500 m squarearea in the centre of the AOI (10,000 iterations, 250 m exclusion radius).

for 10,000 iterations. Figure 20 shows that on average, if there is no knowledge of thetransmitter position, the statistical methods outperform the Least Squares techniques andthe Intersection Density method. While this result is somewhat expected, there is a cleartradeoff to obtain this increase in accuracy with increased computation time, which will bediscussed next.

4.6 Path Loss Exponent and Exclusion Zone EffectsIn order to determine the sensitivity of each algorithm to the path loss exponent used in thepath loss model, calculations were performed using randomly selected measured data, asdescribed previously, but for 10 sensors with values of α ranging from α = 2 to α = 5. Theresults, shown in Figure 21, indicate that large changes in α have a relatively small effecton the performance of the Least Squares algorithms and the Intersection Density method(with the exception of L. Zhu’s method at α = 2 to α = 2.5). The NLS, ML, and the DPDmethods have a stronger dependence on the path loss exponent, and interestingly are mostaccurate with the free space path loss exponent of α = 2.

The same calculations were performed for Emitter Position #2 on FRS Channel 3 and theresults are shown in Figure 22. These results show a similar trend to those in Figure 21with a strong accuracy dependence on α for the statistical methods (with low values of αbeing the most accurate) and a much weaker dependence for the other algorithms.


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Figure 21: Average error distance for 1,000 iterations with 10 sensors using randomlyselected measured data for Emitter Position #1 on FRS Channel 1 for path loss exponentvalues of α = 2 to α = 5 (250 m exclusion radius).


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Figure 22: Average error distance for 1,000 iterations with 10 sensors using randomlyselected measured data for Emitter Position #2 on FRS Channel 3 for path loss exponentvalues of α = 2 to α = 5 (250 m exclusion radius).

To determine the effects of varying the path loss exponent or decreasing the transmitterexclusion radius (i.e., the minimum separation between the emitter and sensors) additionalcalculation were performed. Increasing the path loss exponent to α = 3 (from α = 4 previ-ously) and reducing the exclusion radius from 250 m to 100 m yielded the results shown inFigures 23 and 24 for randomly selected measured data. As expected, the accuracy of eachalgorithm has improved, due in large part to the inclusion of sensors that are closer to theemitter, but also due to a more suitable path loss exponent (see Figure 21). The accuracy ofthe NLS/DPD/ML methods in Figure 23 is improved by approximately 150 m comparedto the previous results shown in Figure 11. A comparison of the Emitter Position #2 resultsfrom Figures 15 and 24 shows an improvement of over 200 m.

It is also apparent from Figures 15 and 24 that the NLS/DPD/ML methods have signif-icantly improved relative to the other algorithms. Further investigation revealed that themajority of this relative improvement is due to the reduction in the exclusion radius (asopposed to the change in α). The results presented in this subsection reaffirm the generalconclusion drawn previously that one of the statistical methods should be selected if abso-lute accuracy is the only concern and computation time (to be discussed next), for example,is not an issue.

Note that these results do not indicate the ultimate level of accuracy that can be achievedusing PDOA techniques. The scenarios considered involve the operation of the sensors


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at fixed locations; a much more favourable situation arises if one or more of the sensorsperform multiple measurements while moving and the resulting data sets are preprocessedto provide power estimates having better statistical stability. Also, approaches involvingthe automatic optimization of α may be worthwhile and deserve study.

4.7 Algorithm Computation Time ComparisonTo compare the computational complexity of the various PDOA algorithms, 1,000 iter-ations were performed and the total computation time required by each algorithm wascalculated. The computer used to obtain these results had the following specifications:

◦ Processor: Intel Xeon W3580 processor with a 3.33 GHz clock speed◦ Memory: 8 GB of 1333 MHz DDR3 RAM◦ Operating System: 64-bit Windows 7◦ Numerical Computing Environment: MATLAB R2010a

The results, shown in Figure 25, clearly distinguish the Least Squares techniques from theother algorithms. Whereas the computation time exhibits an exponential dependence onthe number of sensors with the statistical methods and the Intersection Density method, thedependence is weaker for the Least Squares methods. Figure 25 gives another perspectiveon the results shown in Figure 20, and the price that is paid for increased accuracy with theNLS, ML, or DPD methods. For example, with 10 sensors, the Non-Linear Least Squares

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Figure 25: Total computation time for each algorithm (combined for 1,000 iterations).


method achieves an average miss distance that is approximately 200 m better than that ofL. Zhu’s Least Squares method (a ∼40% improvement). However, the computation timewith the Non-Linear Least Squares method is ∼ 150× longer than with L. Zhu’s LeastSquares method. This, of course, is primarily determined by the fixed grid resolution inthe statistical methods, but regardless, Least Squares methods have a significant advantagewhen computational cost is an issue.


5 Conclusions and Recommendations

This report has explored several PDOA algorithms and their performance using both mea-sured and simulated sensor data. A new MATLAB–Google Earth testbed was developedand described in this report that provides a very intuitive and clear understanding of thesensor/emitter environment and its affect on the geolocation estimate of each algorithm.

In general, the statistical methods (Non-Linear Least Squares, the Discrete ProbabilityDensity method, and Maximum Likelihood) outperform the other algorithms consideredif the emitter of interest is near the edge of the sensor measurement collection area. Fur-thermore, if there is no a priori knowledge of the transmitter position and it could equallylikely be located anywhere in the AOI (i.e., a random emitter position), the statistical meth-ods outperform the other methods considered in this report. However, if the transmitteris located within the sensor constellation, the Least Squares and Intersection Density ap-proaches are the most accurate, which, in the case of the Least Squares algorithms, mayreflect a bias toward the centre of the AOI. In many electronic warfare applications, sce-narios where the emitter is surrounded by sensors are unlikely and one of the statisticalmethods should be chosen if the most accurate estimation is required (if a Gaussian distri-bution is used, the three statistical methods are equivalent).

A further conclusion of this report concerns the computation time required for each algo-rithm. Since the Least Squares techniques require approximately two orders of magnitudeless computation time that the statistical methods, they may be preferable if computationalresources are limited. Alternatively, there are possibilities for reducing the computationalcost of the statistical approaches. In particular, they can be implemented in a two stageapproach where the first stage uses a relatively coarse grid to provide an initial geoloca-tion estimate which is further refined in the second stage using interpolation techniques.Finally, for very small numbers of sensors (e.g., three or four), the Intersection Densityapproach may be considered as it generally outperformed the other algorithms when asmany as five sensors were used and was comparable to the statistical methods with respectto computation time.




References

[1] Roxin, A., Gaber, J., Wack, M., and Nait-Sidi-Moh, A. (2007), Survey of wirelessgeolocation techniques, In IEEE Globecom Workshops.

[2] Chen, W.-Y. and Miller, S.L. (2009), Distributed linear combination estimators forlocalization based on received signal strength in wireless networks, In 43rd AnnualConference on Information Sciences and Systems (CISS).

[3] Patwari, N., Hero, A.O., Perkins, M., Correalnd, N.S., and O’Dea, R.J. (2003),Relative location estimation in wireless sensor networks, IEEE Transactions onSignal Processing, 51(8), 2137–2148.

[4] Langendoen, K. and Reijers, N. (2003), Distributed localization in wireless sensornetworks: a quantitative comparison, Computer Networks, 43(4), 499–518.

[5] Kuruoglu, G.S., Erol, M., and Oktug, S. (2009), Localization in wireless sensornetworks with range measurement errors, In Fifth Advanced InternationalConference on Telecommunications (AICT 09).

[6] Namik, S. and U.J. Ferner, K.W. Sowerby (2008), Localization in harsh propagationenvironments, In Australian Communications Theory Workshop (AusCTW 2008).

[7] Goldoni, E., Savioli, A., Risi, M., and Gamba, P. (2010), Experimental analysis ofRSSI-based indoor localization with IEEE 802.15.4, In 2010 IEEE EuropeanWireless Conference (EW).

[8] Zanca, G., Zorzi, F., Zanella, A., and Zorzi, M. (2008), Experimental comparison ofRSSI-based localization algorithms for indoor wireless sensor networks, InWorkshop on Real-World Wireless Sensor Networks (REALWSN08).

[9] Pugh, J. A., Bultitude, R. J. C., and Vigneron, P. J. (2006), Path loss measurementswith low antennas for segmented wideband communications at VHF, In MilitaryCommunications Conference (MILCOM).

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[11] Rappaport, T. (1999), Wireless Communications Principles and Practice, PrenticeHall.

[12] Egli, John J. (1957), Radio Propagation above 40 MC over Irregular Terrain,Proceedings of the IRE (IEEE), 45(10), 1383–1391.

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[15] Elsaesser, D. (2006), The Discrete Probability Density Method for emittergeolocation, In IEEE Canadian Conference on Electrical and ComputerEngineering (CCECE).

[16] Elsaesser, D. (2006), The Discrete Probability Density Method for electronic warfaresensor data fusion, (DRDC Ottawa TR 2006-242) Defence R&D Canada – Ottawa.

[17] Cox, D.C., Murray, R., and Norris, A. (1984), 800 MHz attenuation measured in andaround suburban houses, Technical Report AT&T Bell Laboratory.

[18] Bernhardt, R.C. (1987), Macroscopic diversity in frequency reuse systems, IEEEJournal on Selected Areas in Communications, 5, 862–867.

[19] Guo, S., Jackson, B.R., Wang, S., Arnold, W., and Inkol, R. (2011), A noveldensity-based geolocation algorithm for a non-cooperative radio emitter using powerdifference of arrival, In SPIE Defence, Security, and Sensing.

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Annex A: List of FRS ChannelsFRS Channel Frequency (MHz)

1 462.56252 462.58753 462.61254 462.63755 462.66256 462.68757 462.71258 467.56259 467.5875

10 467.612511 467.637512 467.662513 467.687514 467.7125




Annex B: Discussion of Issues AffectingPDOA Geolocation

The physical interpretation of the PDOA geometric solution in terms of circle intersec-tions provides important insights. Since the pair of circles corresponding to a pair of re-ceivers will usually intersect at two points, RSS geolocation involves inherent possibilitiesfor ambiguities that normally do not arise with the corresponding geolocation problem in-volving angle of arrival techniques. Symmetry considerations suggest that a linear (i.e.,one-dimensional) array of sensors will yield a pair of solutions on opposite sides of the linecontaining the sensor array. A two-dimensional sensor array is needed if an unambigu-ous geolocation estimate is to be obtained. However, depending on the array geometry,the ability to resolve ambiguous solutions may be unreliable with the result that the statis-tical properties of the geolocation solution can become complicated in ways that are notintuitively apparent.

A further observation is that an examination of geolocation data obtained with the variousgeolocation algorithms suggests that the statistical properties of the results obtained withthe least squares and non-linear least squares algorithms can differ significantly.

Two simulation experiments were designed to further explore aspects of PDOA geoloca-tion. These experiments involved a simple array configuration where all of the sensors,except for one, were distributed along a straight line. The specific sensor configurationconsists of 10 sensors, 9 of which lie on the line defined by y = −400 with x-coordinatesgiven by {-800, -600, -400, -200, 0, 200, 400, 600, 800}. The 10th sensor was positionedat (-1000, 0). Note that, without the last sensor, RSS measurement data from the sensorarray would not provide any information that could determine whether the transmitter y-coordinate is greater or less than the y-coordinates of the other sensors (y = −600). Bothexperiments used a signal model with path loss exponent of 4 and RSS standard deviationof 6 dB. For each emitter position, 200 runs were performed and the geolocation solutionscomputed using the NLS and a modification of Y.T. Chan’s LS algorithms.. The area ofinterest selected for the NLS algorithm was an 8,000×8,000 square centred on (0, 0). Allpositions are in units of meters.

The first simulation experiment involved an emitter moving parallel to the x-axis in 100 mincrements from (-2000, -600) to (2000, -600). Figure B.1 shows the emitter-sensor ge-ometry. Figure B.2 plots the bias errors in the x and y-coordinates as a function of theemitter displacement from x = 0 for the NLS and LS algorithms. The behavior of the twoalgorithms differs significantly. For the LS algorithm, the x component of the bias error isapproximately equal to x where x is the displacement along the x-axis i.e., the x componentof the geolocation solution estimate is relatively insensitive to the displacement along thex-axis. The y component of the bias error shows a weaker dependence on the displacementalong the x-axis, ranging from -250 to -500 meters. For the NLS algorithm, the x compo-


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nent of the bias errors is better than that of the LS algorithm for large displacements, butthe y component of the bias error tends to be larger, particularly for emitter positions thatare relatively distant from the sensor at (-1000, -600).

The second simulation experiment involved an emitter moving along the line defined byx = 100 in increments of 100 from (100, -2000) to (100, 2000). Figure B.3 shows theemitter-sensor geometry. Figure B.4 plots the bias errors in the x and y-coordinates as afunction of the emitter displacement from y = 0 for the NLS and LS algorithms. For bothmethods, the y bias error component has a strong dependence on the emitter displacement.The x component of the bias error is relatively small, particularly for the LS solution.

It is apparent that there is a complex interrelationship between the emitter-sensor geometryand the bias errors for the LS and NLS methods. To provide some further insight into thestatistical aspects of the PDOA geolocation estimates, histograms of the x and y compo-nents of the geolocation solution data were obtained using the two methods for an emitterat (100, -600), as shown in Figure B.5. The results, plotted in Figures B.6 to B.9, wereobtained with 2000 simulation runs. Several observations are worth noting. First, the ge-olocation data obtained with the LS method has a smaller spread than that obtained withthe NLS method. Second, the y-coordinates of the NLS solution data display a distinct bi-modal distribution, thereby demonstrating that the sensor array geometry has only a weakability to obtain a stable solution that correctly resolves the ambiguous solutions; in factit appears that a failure to correctly resolve the ambiguity in a proportion of the measure-ments is at least partly responsible for the y component bias error 1. Finally, the finite sizeof the selected AOI shows up as peaks at the values of x and y corresponding to the AOIboundaries.

These results demonstrate how the statistical properties of the LS and NLS geolocationsolutions have a complex dependence on the emitter-sensor geometry and can differ sub-stantially.

1. Although not shown here, increasing the RSS standard deviation to 12 dB results in the y-coordinatedata following a broad unimodal distribution roughly centred about y = 400.


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Figure B.5: Sensor and emitter configuration used to generate distributions of the x and ybias errors shown in Figures B.6 to B.9.

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List of Abbreviations/Acronyms

AOA Angle of Arrival

AOI Area of Interest

CNEW Communication and Navigation Electronic Warfare

CW Continuous Wave

DF Direction Finding

DND Department of National Defence

DPD Discrete Probability Density (method)

DRDC Defence Research and Development Canada

EW Electronic Warfare

FDOA Frequency Difference of Arrival

FRS Family Radio Service

GPS Global Positioning System

LAV Light Armoured Vehicle

LOB Line of Bearing

LS Least Squares

MEWT Mobile Electronic Warfare Technologies

MiDAS Military Digital Analysis System

ML Maximum Likelihood

NLS Non-Linear Least Squares

PDF Probability Density Function

PDOA Power Difference of Arrival

PF Position Fix

RF Radio Frequency

RSS Received Signal Strength

TDP Technology Demonstration Project

TDOA Time Difference of Arrival

TEWS Tactical Electronic Warfare Systems

TOA Time of Arrival

UAV Unmanned Aerial Vehicle




DOCUMENT CONTROL DATA(Security classification of title, body of abstract and indexing annotation must be entered when document is classified)

1. ORIGINATOR (The name and address of the organization preparing thedocument. Organizations for whom the document was prepared, e.g. Centresponsoring a contractor’s report, or tasking agency, are entered in section 8.)


3701 Carling Avenue, Ottawa, Ontario, Canada

K1A 0Z4

2. SECURITY CLASSIFICATION (Overallsecurity classification of the documentincluding special warning terms if applicable.)

UNCLASSIFIED

3. TITLE (The complete document title as indicated on the title page. Its classification should be indicated by the appropriateabbreviation (S, C or U) in parentheses after the title.)

Emitter geolocation estimation using power difference of arrival: An algorithm comparison for

non-cooperative emitters

4. AUTHORS (Last name, followed by initials – ranks, titles, etc. not to be used.)

Jackson, B.R.; Wang, S.; Inkol, R.

5. DATE OF PUBLICATION (Month and year of publication ofdocument.)

May 2011

6a. NO. OF PAGES (Totalcontaining information.Include Annexes,Appendices, etc.)

78

6b. NO. OF REFS (Totalcited in document.)

22

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Technical Report

8. SPONSORING ACTIVITY (The name of the department project office or laboratory sponsoring the research and development –include address.)


3701 Carling Avenue, Ottawa, Ontario, Canada K1A 0Z4

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12po

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10a. ORIGINATOR’S DOCUMENT NUMBER (The officialdocument number by which the document is identified by theoriginating activity. This number must be unique to thisdocument.)

DRDC Ottawa TR 2011-040

10b. OTHER DOCUMENT NO(s). (Any other numbers which maybe assigned this document either by the originator or by thesponsor.)

11. DOCUMENT AVAILABILITY (Any limitations on further dissemination of the document, other than those imposed by securityclassification.)

( X ) Unlimited distribution( ) Defence departments and defence contractors; further distribution only as approved( ) Defence departments and Canadian defence contractors; further distribution only as approved( ) Government departments and agencies; further distribution only as approved( ) Defence departments; further distribution only as approved( ) Other (please specify):

12. DOCUMENT ANNOUNCEMENT (Any limitation to the bibliographic announcement of this document. This will normally correspondto the Document Availability (11). However, where further distribution (beyond the audience specified in (11)) is possible, a widerannouncement audience may be selected.)

Full unlimited announcement

13. ABSTRACT (A brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highlydesirable that the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication of thesecurity classification of the information in the paragraph (unless the document itself is unclassified) represented as (S), (C), (R), or (U).It is not necessary to include here abstracts in both official languages unless the text is bilingual.)

The performance of several power difference of arrival (PDOA), also known as received signal

strength difference (RSSD), emitter geolocation algorithms, is explored. First, a general discus-

sion of radio frequency emitter geolocation techniques is presented. Afterwards, the motivation

for employing PDOA geolocation techniques is discussed, followed by a review of several path

loss models and a description of algorithms for obtaining geolocation solutions from PDOA mea-

surement data: Non-Linear Least Squares, the Discrete Probability Density method, Maximum

Likelihood, the Intersection Density method, and three Least Squares techniques. Both simu-

lated and measured data are used to compare the geolocation estimation accuracy and compu-

tational complexity of these algorithms. A MATLAB–Google Earth testbed was developed to aid

in the visualization and interpretation of the results of the experimentation. Finally, conclusions

and recommendations regarding the selection of the most appropriate PDOA algorithms for a

given application are given.

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (Technically meaningful terms or short phrases that characterize a document and couldbe helpful in cataloguing the document. They should be selected so that no security classification is required. Identifiers, such asequipment model designation, trade name, military project code name, geographic location may also be included. If possible keywordsshould be selected from a published thesaurus. e.g. Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus identified.If it is not possible to select indexing terms which are Unclassified, the classification of each should be indicated as with the title.)

Tactical Electronic Warfare

Emitter Geolocation

Geolocation Algorithms

Power Difference of Arrival (PDOA)

Received Signal Strength (RSS)

Non-Linear Least Squares

Discrete Probability Density Method

Maximum Likelihood

Least Squares

Documents

Emitter geolocation estimation using power difference of ... · the geolocation estimation accuracy and computational complexity of these algorithms. A MATLAB–Google Earth testbed