Direction Finding: Determine the direction to a transmitter with randomly placed sensors

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STOCKHOLM SWEDEN 2019,

Direction Finding

Determine the direction to a transmitter with randomly placed sensors

FERNANDO FRANZÉN

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE

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Determine the direction to a transmitter with randomly placed ground-based sensors

by

Fernando Franz´ en March 2019

Master of Science Thesis TRITA-EECS-EX 2019:XXX School of Electrical Engineering and Computer Science

KTH Royal Institute of Technology SE-100 44 STOCKHOLM

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Radiopejling

Best¨ am riktningen till en s¨ andare med slumpm¨ assig placering av mottagare

av

Fernando Franz´ en Mars 2019

Examensarbete TRITA-EECS-EX 2019:xx Skolan f¨or Elektronik och Datavetenskap

KTH Kungliga Tekniska H¨ogskolan SE-100 44 STOCKHOLM

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Direction Finding

Determine the direction to a transmitter with randomly placed sensors

Fernando Franz´en

Approved: Examiner:

2019-03-XX Daniel M˚ansson

Abstract

There are a lot of stand-alone and mobile platforms using transmitters today. Some want to be found while others do not. In our modern society there is a great demand of mobility and communication abilities. This means that several mobile platforms could potentially carry a sensor to record incoming signals to be used in Direction Finding.

This thesis identifies the possibility to determine the direction to a transmitter with randomly placed sensors. By conducting a literature review well-known methods such as Time Difference Of Arrival (TDOA) and MUltiple SIgnal Classification (MUSIC) where chosen as methods in this analysis. The methods are applied on two antenna arrays, an Uniform Circular Array (UCA) and a Random Circular Ar- ray (RCA). The RCA is generated with randomly placed sensors. The performance in the Direction Of Arrival (DOA) is investigated in presence of time synchronization error and with different numbers of elements, radius and Signal to Noise Ratio (SNR). The ambiguity in the arrays is also investigated to insure a ambiguity-free DOA estimation.

The results from this analysis identifies that the accuracy in the DOA estimation is dependent on the number of elements, SNR, the elements positions and the radius of the DF array. Furthermore, the accuracy of a UCA is greater than a RCA when the elements are randomly distributed within the area of a circle with radius R.

Finally, it has shown that if time synchronization error occurs between the sensors, then the MUSIC method the accuracy will decrease greatly.

Keywords:

Direction Finding, TDOA, MUSIC, UCA, Random Array, DOA

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Examensarbete TRITA-EECS-EX 2018:XX Radiopejling

Bestäm riktningen till en sändare med slumpmässig placering av mottagare

Fernando Franz´en

Godk¨ant: Examinator:

2019-03-xx Daniel M˚ansson

Sammanfattning

Det finns m˚anga individer och mobila platformar som använder sändare idag. Vissa vill bli hittade, andra inte. I v˚arat moderna samhälle är det en stor efterfr˚agan p˚a rörlighet och kommunikationsmöjligheter. Detta innebär att m˚anga mobila plattfor- mar skulle kunna spela in signaler för att användas i radiopejling.

Denna uppsats identifierar möjligheten att bestämma riktningen till en sändare med slumpmässigt placerade sensorer. Genom litteraturstudien identifierades de välkända riktningsmetoder som Time Difference Of Arrivial (TDOA) och MUlti- ple SIgnal Classification (MUSIC) som vidare valdes som metoder i denna analys.

Tv˚a antennstrukturer används i analyserna. Den ena är en Uniform Circular Ar- ray (UCA) och den andra är en Random Circular Array (RCA). RCA är gener- erad med slumpmässigt utplacerade sensorer. Prestandan i riktningsuppskattnin- gen undersöks när det existerar ett tidssynkroniseringsfel, olika antal sensorer i an- tennstrukturerna, varierande radier och olika signal- och brusförh˚allanden. Även tvetydigheter undersöks i strukturerna för att säkerställa att en entydig riktnings- bestämning kan utföras.

Resultaten implicerar att noggrannheten i riktningsbestämningen är beroende av antalet element, SNR, elementens position och radien i antennmatrisen. Utöver detta visar resultaten att en UCA har högre noggrannhet än en RCA d˚a elementen

¨

ar slumpmässigt utplacerade inom en cirkelradie, R. Slutligen, om tidssynkroniser- ings fel uppst˚ar mellan sensorerna kommer detta resultera i minskad noggrannhet när MUSIC metoden tillämpas.

Nyckelord:

Direction Finding, TDOA, MUSIC, UCA, Random Array, DOA

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I would like to start with thanking the Department of Electromagnetic Engineering for good support during the working progress of this thesis.

My time at KTH has been an interesting journey that will soon come to an end. I’d like to thank the institution for helping me in gathering knowledge that I can put to good use in the future, developing products that will help people in their daily lives. This also would not have been possible without support from my classmates, to whom I send my deepest gratitude.

Finally, I would like to thank my love who has been understanding and supporting throughout the working progress of this thesis.

Fernando Franz´en

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List of Figures

1 Hopping frequency illustration. . . 2

2 Multipath illustration. . . 3

3 Illustration of the triangulation method. . . 8

4 Uncertainty ellipse illustration. . . 8

5 Illustration of the localization method TDOA. . . 9

6 Hyperbola geometry. . . 10

7 Accuracy of the localization method TDOA. . . 12

8 An hyperbolic equation with its asymptotic line. . . 14

9 Illustration of the Watson-Watt method and an Adcock antenna of four dipoles. . . 15

10 Illustration of the Phase interferometer . . . 17

11 An illustration of the sub-matrices of a ULA which the method ES- PRIT is using. . . 22

12 Two sensors located on the x-axis separated by the distance d and the incident angle of the plane wave is θ. . . 26

13 When there is a large distance between the two sensors, the time synchronization error has less effect on the DOA estimation when using the method TDOA. . . 27

14 When there is a large distance between the two sensors, the time synchronization error has less effect on the DOA estimation when using the method TDOA. In this case the frequency is 300 MHz and λ ≈ 1m. . . 28

15 When increasing the distance between the two sensors, the time synchronization error has less effect on the DOA estimation when using the MUSIC method. In this case, the frequency is 300 MHz and λ ≈ 1m. . . 29

16 When the time synchronization error increases, the effect on the DOA estimation has a periodic behavior when using the MUSIC method. In this case, the frequency is 300 MHz and λ ≈ 1m. . . 29

18 An increased distance between the elements increase the number of peaks in the MUSIC spectrum. . . 31

19 A UCA where φ_i is the angle to the i-th element, R is the radius and DOA of the plane wave is θ. . . 32

20 UCA with different numbers of elements and their different radii versus the accuracy in the DOA estimation. . . 33

21 UCA with seven elements and different radii versus SNR and the accuracy in RMS. . . 33

22 MUSIC ”Pseudo Spectrum” with a 4-element UCA with radius λ and where the DOA is 0 degrees. . . 34

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25 MUSIC ”Pseudo Spectrum” with a 5-element UCA. . . 36

28 Random element placement where Ri and φi is the radius and angle to each element in the random array. . . 38

29 RCA with different numbers of elements and their different radii versus the accuracy in the DOA estimation. . . 39

30 SNR VS R_max. . . 39

31 MUSIC ”Pseudo Spectrum” with a 3-element RCA. . . 40

36 An RCA and a UCA, both with 7 elements. . . 43

37 The accuracy of a UCA and an RCA. . . 44

41 The difference in the DOA between the senors when the transmitter is located at different distances from the senors. . . 46

42 Difference in DOA of two sensors separated with different distances. . 46

43 The positions of 4 randomly placed sensors and the MUSIC ”Pseudo Spectrum”. . . 47

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Nomenclature

ADF Automatic Direction Finder AoA Angle of Arrival

COM IN T Communication Intelligence DF Direction Finding

DOA Direction Of Arrival

ESP RIT Estimation of Signal Parameters Via Rotational Invariant Techniques F HSS Frequency-Hopping Spread Spectrum

GN SS Global Navigation Satellite System HF High Frequency

LOB Line Of Bearing

LOCA Localization on the Conic Axis LOS Line Of Sight

M U SIC MUltiple SIgnal Classification N LOS Non Line Of Sight

RDL Range Difference Location SIGIN T Signals Intelligence SN R Signal to Noise Ratio T DOA Time Difference of Arrival T OA Time Of Arrival

U HF Ultra High Frequency V HF Very High Frequency PPS Precise Positioning Service RCA Random Circular Array UCA Uniform Circular Array ULA Uniform Linear Array

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1 Introduction

This section will start with an background and introduction to Direction Finding and its complexity. Thereafter the problematization will be described. Finally the research questions will be outlined and the delimitations and limitations of this analysis be stated.

1.1 Background

In 1901 the first transatlantic signal was transmitted by Guglielmo Marconi and was the start of our modern communication society [17]. Finding the Direction Of Arrival (DOA) to transmitters is a challenge as old as the first transatlantic signal.

In 1903 Bellini and Tosi developed the first radio Direction Finding (DF) system using crossed magnetic loops. This system was searching for the DOA in the High Frequency (HF) band. This resulted in a problem with sky waves [4] but which pushed the development of the Adcock array using the DF method Watson-Watt.

Throughout history different DF methods and localization methods have been developed such as Time Difference Of Arrival (TDOA), Triangulation, Homing, Watson- Watt, MUltiple SIgnal Classification (MUSIC) and many more [4]. Before the Global Navigation Satellite System (GNSS) with its application Global Positioning System (GPS), land-based navigation systems such as Loran, Decca and Omega were more common. These land-based systems were based on the method ”fixing” which is measuring the distance to two known locations [32].

Furthermore, the application of the above mentioned methods can not only be found within the civil context but also within the military. In electronic warfare, systems such as radar, jamming, DF and radio localization is used [13]. These four aspects affects the duration of the OODA-loop which stands for observation, orientation, decision and action [27]. The loop is also used by civilians [7].

Another aspect regarding the military application is the crucial need of robust communication that is non-sensitive to jamming [25]. An anti-jamming invention is Hopping Frequency and was invented by actress Hedy Kiesler Markey and pianist George Antchil in 1941 [1]. The reason for the invention was to secure the mobile communication between the torpedo and the ship during WWII. At this time the enemy ship had the ability to detect and jam the carrier frequency of the torpedo communication. With the Hopping Frequency method the carrier frequency was continuously changed, see Figure 1, and made the communication between the torpedo and the ship difficult to interfere with [1]. In Hopping Frequency systems the antenna response and the propagation losses for the Frequency-Hopping Spread Spectrum (FHSS) has to be considered [14]. Furthermore, there is a minimum power

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Introduction 1.1

required of the received signal to make the Hopping Frequency work [14].

Figure 1: Hopping frequency illustration.

As mentioned, the Direction Finding systems are applied in several contexts, both military and civilian. The DF system is a complex system in a complex environment. There are several aspects to consider when developing such a system. One of the first and most essential aspect being what frequency to apply. For instance, the Very High Frequency (VHF) and Ultra High Frequency (UHF) bands have Line of Sight (LoS) limitation which means that the propagation range is approximately 100 km [4]. If the scenario is applied in an environment such as forest the range will decrease due to losses in the path of propagation [36]. The propagation losses in forest has been investigated for decades, especially in the HF and VHF band [22].

Furthermore, if Direction Finding is applied in urban or mountain environments, multipath directions need to be considered. Buildings and mountains act as reflec- tors and create multipath of the electromagnetic propagation. This means that DF systems can receive more than one DOA from a single source, see Figure 2. In the VHF and UHF bands are usually narrow band signals transmitted [14], which is considered to be beneficial in DF systems. On the other hand if the signal is wideband, common DF methods such as MUSIC will have a decreased performance in the DOA estimation instead [9].

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Figure 2: Multipath illustration.

The frequency is the determining factor which decides the size of the antenna arrays in the DF system. Because the antenna elements are dependent on the wavelength this means that low frequencies results in very large wavelengths and, thus, also requires big antennas. The most common antenna arrays in DF are Uniform Linear Array (ULA) and Uniform Circular Array (UCA). Another array with randomly spaced elements and has been investigated were it has been shown that the direc- tivity is proportional to the number of elements [23]. Random antenna arrays also requires fewer elements than uniform antenna arrays [3]. A random placement of the elements in antenna array can be seen as mobile platforms such as cars and drones [5].

Something that must be taken into account when constructing and using antenna arrays for DF is the mutual coupling between the elements. This because of how it affects the phase and amplitude in each element [11], which decrease the performance in the DF methods.

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Introduction 1.2

1.2 Problematization and Purpose

There are a lot of individuals and mobile platforms using transmitters today. Some want to be found while others do not. Our modern society demand mobility and communication abilities. This means that several mobile platforms could potentially carry a sensor to record incoming signals. The recorded data could be used for Direction Finding (DF) and localization of transmitters. For a scenario where this concept could be applied is to find a person in a natural disaster zone using drones with sensors scanning the area. One could state that the telecommunication companies have the possibility to use their cell towers to locate the person, however this requires legislative machinery to put in place. Having drones with this ability as a resource for rescue teams could potentially save lives by quickly identifying and locating individuals in an emergency. Within the framework of such scenarios the drones would continue to move randomly within a subsequently defined radius until an incoming signal is detected, after which they land. When this process is done, the drones start measuring the signal at the same time, which needs high precision time synchronization and data sharing. Thus, the areas of application are plentiful and could be the difference between life and death for many. Therefore, this thesis will identify the possibility to determine the direction to a transmitter with randomly placed sensors by using the Direction Finding method MUltiple SIgnal Classification (MUSIC).

A detailed description of the considered scenario in this thesis now follows. There will be 10 drones in this scenario and before they can collect any data by measuring the incoming signal the drones have to land on the ground. Some of the drones may not be able to land on the ground but a minimum of 5 drones will be able to land to conduct the measurement. The distance between the drones after they land will only be a few wavelengths. The wavelength is determined by the transmitter’s frequency which is set at 300 MHz in this scenario. The Signal-to-Noise-Ratio (SNR) is considered to be 20 dB when the measurements are conducted. Every drone has one sensor which is omnidirectional in the xy-plane and is equipped with a receiver.

The receivers are limited to a sampling frequency of 1.2 GHz and can only collect 3000 samples for every new measurement. Before a measurement can be performed the drones have to do a time synchronization to make sure that all drones start their measurements at the same time.

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1.3 Research Question

Main Research Question: To identify the possibility to determine the direction to a transmitter with randomly placed sensors by using the Direction Finding method MUltiple SIgnal Classification (MUSIC).

RQ1: How does the time synchronization error between sensors impact the direction finding methods Time Difference Of Arrivial (TDOA) and MUltiple SIgnal Classi- fication (MUSIC) in a two element ULA?

RQ2: How does the number of elements and maximum radius of a DF array affect the DOA estimation with the Direction Finding method MUltiple SIgnal Classifica- tion (MUSIC)?

1.4 Delimitations

• No coupling

• One transmitter with known frequency

• The narrowband assumption [17]

• The signal is continuous

• No Hopping Frequency

• Only ideal omnidirectional sensors in the xy-plane are considered

• The data is shared between the sensors without any interruption

• The position of the sensors are known

• The environment is free space, thus not taking forest, urban or mountain- environment into consideration. Which means no multipath of the incident wave.

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Introduction 1.5

1.5 Disposition

In Section 2, known localization and direction finding methods will be described and illustrated. Section 3 will give insights into how accurate time synchronization and positioning are today. The choice of method which is applied in this thesis is discussed in Section 4. The result is presented in Section 5 were the accuracy of UCA and Random Circular Array (RCA) is investigated. Section 5 also presents the impact of time synchronization error in DF methods. Finally will a real scenario of randomly placed sensors with presence of time synchronization error be analyzed.

The discussion and conclusion of the results are presented in Section 6.

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2 Methods

This section will first outline the localization methods, Triangulation, TDOA and Homing. This will then be followed by a description of Direction Finding (DF) methods, such as TDOA, Watson-Watt, Phase Interferometer, MUSIC and ESPRIT.

2.1 Localization Methods

Throughout history, different localization methods have been developed in order to determine the position of transmitters. In this Section three localization methods will be described, with the first method being triangulation. This method is considered to be one of the oldest and most frequently used. Secondly, the Time Difference Of Arrival (TDOA) method will be outlined. Finally, this section will describe the Homing method which differs from the previously mentioned methods by continuously providing the user with a DOA. This guides the user towards the transmitters position instead of providing the specific coordinate of the transmitters [4].

2.1.1 Triangulation

Triangulation determines the position of a transmitter with an unknown location. A Direction Of Arrival (DOA) is needed from two locations. As of today the DOA has no articulated definition and is often synonymously described as Angle of Arrival (AoA) [29]. As a result of this, this study will view the DOA as the direction vector to the transmitter which the AoA (θ) can be obtained from.

A DOA is obtained from a direction finding (DF) method which will be described later in Section 2.2. As an example, let’s consider two DF systems and a transmitter in two-dimensional space. In this space two DF systems are located at the following positions (x₁, y₁) and (x₂, y₂) while the transmitter is located at an unknown position (x₀, y₀). The DOA of the transmitter and the AoA relative to the systems are denoted as θ₁ and θ₂, which is depicted in Figure 3. To determine the unknown location of the transmitter, the linear equation system (1) has to be solved.







y₀− y₁ = tan(θ₁)(x₀− x₁)

y₀− y₂ = tan(θ₂)(x₀− x₂) (1) With i number of DF systems in the ideal case, an overdetermined system of equations 2 can be obtained [29].











y₀− y₁ = tan(θ₁)(x₀− x₁) ...

y₀− y_i = tan(θ_i)(x₀ − x_i)

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Methods 2.1

Figure 3: The directions to a transmitter from two direction finding systems and their DOAs, θ₁ and θ₂.

DF systems are not given an exact DOA to the transmitter because of measurement errors. This means that different DF systems DOAs will not intersect in the same point. They will provide with multiple of intersection points which are possible locations of the transmitter [26]. This problem can be handled by an uncertainty ellipse which was first introduced by Stansfield [35]. Taking all intersection points into account they can be covered with the area of a ellipse that gives the transmitters possible location, see Figure 4.

Figure 4: Uncertainty ellipse illustration.

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2.1.2 Time Difference Of Arrival (TDOA)

The localization method Time Difference Of Arrival (TDOA) is considered to be very useful in SIGINT when a emitter is transmitting short pulses [42]. This method needs at least three sensors in order to determine an ambiguity free position of the transmitter. However, this is only for the ideal case when there are no measurement errors in time or in the positioning of the sensors [15].

The incident signal from the transmitter arrives at different times to the sensors.

This implies that the localization system using TDOA need to be up and running before a continuous time signal is transmitted. In theory, however, it is possible that the signal is arriving at the same time to all three sensors but it would be depend on their positions. Given the placement of the sensors and the Time Of Arrival (TOA), the TDOA can be measured. Furthermore, with three sensors, three TDOAs and three hyperbolic curves can be provided. However, it is only in the ideal case the three hyperbolic curves intersect in the position of the transmitter, see Figure 5.

Figure 5: Three hyperbolic curves are obtained from three sensors and intersect in the position of the emitter.

The ideal case, with no time measurements or positioning errors occurring at the sensors, can be mathematically derived. Consider two sensors located at positions (x₁, y₁) and (x₂, y₂) which is the f oci of a hyperbola. The sensors are placed on the x-axis which means y₁ = y₂ = 0. The distance from the origin to both sensors are d and the distances from origin to the vertex are a. In Figure 6 the sensors and hyperbola geometry are illustrated.

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Methods 2.1

Figure 6: The sensors and hyperbola geometry.

The distance to the unknown location of the transmitter (x, y), which is assumed to be on the conic, from the sensors are

r₁ =p

y²+ (x − x₁)² (3a)

r₂ =p

y²+ (x − x₂)² (3b)

The difference in distance to the transmitter is

∆r = r₂− r₁ = (d + a) − (d − a) = 2a (4)

py²+ (x − x₂)²−p

y²+ (x − x₁)² = 2a. (5)

y²+ (x − x₂)² = (2a +p

y²+ (x − x₁)²)². (6) This can be simplified to the following expression

x²(d²− a²) − a²y² = a²(d²− a²). (7) Divide Eq.7 with a²(d²− a²) gives

x²

a² − y²

(d²− a²) = x² a² − y²

b² = 1, (8)

were b² = d² − a² which is the equation of a hyperbola. Recall that the difference in distance to the unknown transmitter is 2a, which is equal to c∆t were c is the

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speed of the propagating wave and ∆t is the TDOA. This gives a = c∆t

2 (9)

and

b² = d²− a² = d²−c∆t 2

2

(10) Now the equation of a hyperbola can be written as

x²

c∆t 2

2 − y² d²−

c∆t 2

2 = 1 (11)

In a scenario where the two sensors are not lined up on the x-axis it is preferable to use a local coordinate system to facilitate the calculation. This was a demonstration of the two-dimensional case but it can also be applied in the three-dimensional case.

When there is limited access to computer power, a simpler TDOA algorithm for the two-dimensional case is proposed by Fang [10].

The accuracy of the TDOA method depends on the placement of the sensors when measurement errors are present. The best geometric placement is where the sensors are placed around the transmitter [4]. Outside the area surrounded by the sensors the accuracy quickly decreases, which is illustrated in Figure 7. Thus, it is preferable to have knowledge of the transmitters location. However this is rarely the case.

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Methods 2.1

Figure 7: The accuracy of the TDOA method is depending on the geometric placement around the transmitter. The sensors are represented by the blue dots, while the accuracy is represented by the red marked areas inside and outside the preferable placement.

2.1.3 Homing

The Homing method use a directive antenna to scan the direction in which it re- ceives the strongest signal. This is considered an approximation of the DOA. By following the approximation of the true DOA the method guides the user towards the location of the transmitter. When the user is approaching the transmitter’s location the error in the DOA decrease [4]. This method requires that the transmitter continuously transmit a signal that is known to the user.

Instead of manually scanning for the DOA, an Automatic Direction Finding (ADF) is preferable. This method was used by airplanes during WWII, which gave the pilot the direction to a friendly land-based beacon tower. The concept is similar to the ordinary compass. By keeping the needle in the ADF compass at 0 degrees the pilot was homing on the transmitting station. The zero degrees indicator in the ADF compass is the DOA to the friendly beacon tower.

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2.2 Direction Finding Methods

Today there are several DF methods used to determine the DOA. Due to the amount of techniques a limit must be drawn to focus on the best known and used methods, therefore this study will only investigate Time difference of arrival (TDOA), Watson-Watt, Phase Interferometer and the subspace based methods MUltiple SIg- nal Classification (MUSIC) and Estimation of Signal Parameters via Rotational In- variant Techniques (ESPRIT). These methods will be outlined and further analyzed in this Section.

2.2.1 Time Difference Of Arrival (TDOA)

With two sensors the Time Difference Of Arrival (TDOA) method can be used to approximately calculate the AoA. This under the same conditions described in Section 2.1.2 and if the distance between the receivers is small compared to the distance to the transmitter [9]. The hyperbolic curve calculated from the TDOA with two sensors has an asymptotic line, illustrated in Figure 8. From the asymptotic line the AoA can be calculated according to equation 12.

sin(θ) = c∆t

d (12)

Were c is the speed of the light, ∆t is the TDOA between the two receivers and d is the distance between the receivers in meters.

If the sensors are separated by long distances, there will be a problem with the frequency error when the TDOA method is used. The reason being that each sensor has its own receiver and with its own oscillator. The difference between the oscil- lators increase the frequency error with increasing distance. If the sensors are at a close distance they can share the same receiver and avoid frequency error. [8]

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Methods 2.2

Figure 8: An hyperbolic equation with its asymptotic line.

As described in Section 2.1.2 the TDOA method can also be used for localization.

The method is assuming that the position of the transmitter is on the hyperbolic curve. Another approach to determine the location of the transmitter is using the TDOA method and assuming that the curves of position not have to be hyperbolic instead being a straight line. This straight line can be used for DOA. The conventional TDOA method put the sensors at the f oci and in this approach the sensors are on the hyperbolic curve. Three sensors with known location in the xy-plane can obtain the location of the transmitter under the circumstance where the transmitter is in the Line of Sight (LoS) from each of the three sensors. This new approach generate a conic axis and the focal point (x₀, y₀) can be the transmitters position but not necessarily. With a new combination of three sensors another conic axis can be calculated. Using the localization method triangulation, the position of the transmitter can be determined. [32]

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2.2.2 Watson-Watt

The method is named after the inventor R. A. Watson Watt, which he developed and used for atmospheric direction finding [40]. The Watson-Watt method uses an Adcock antenna array to determine the DOA. The Adcock antenna array consists of at least four antenna elements in pairs who are faced 90 degrees to each other.

Having eight elements instead of four gives a greater frequency coverage range [38]

and mitigated the effect of element spacing error [39]. This method was used during WWII to determine the DOA to enemy submarines [29].

Figure 9: Illustration of the Watson-Watt method and an Adcock antenna of four dipoles.

In Figure 9 the method illustrated with a four dipole Adcock antenna. The blue dots represent the omnidirectionality of the dipoles in the xy-plane. The basics of the Watson-Watt method is measuring the voltage over the north-south dipole pair and the west-east dipole pair to determine the DOA according to the derivation below [24][2].

Consider an incoming linear polarized plane wave s(t) with a DOA (θ). North is the y direction and east is the x direction in the xy-plane. If the origin is placed in the center of the array then the distance to each sensor is d. The voltages at each sensor are

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Methods 2.2

UN(t) = s(t)e^{jkd sin(θ)} U_S(t) = s(t)e−jkd sin(θ)

U_E(t) = s(t)e^{jkd cos(θ)} U_W(t) = s(t)e−jkd cos(θ)

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were k is the wavenumber. The voltage difference between north and south sensors are as follows

UN(t) − US(t) = s(t)(e^{jkd sin(θ)}− e−jkd sin(θ)

) (14)

and using the identity below

sin(x) = 1

2j(e^jx− e^−jx) (15)

the expression can be simplified to

U_N(t) − U_S(t) = s(t)2j sin(kd sin(θ)) (16) When the value of the wavenumber is low, i.e. k << 1, then sin(x) ≈ x and the expression can be simplified to

U_N(t) − U_S(t) = s(t)2jkd sin(θ). (17) Analogously the voltage difference between east and west sensor can be simplified to

U_E(t) − U_W(t) = s(t)2jkd cos(θ) (18) The DOA can be calculated according to the equation below

U_{N S}

U_{W E} = U_N(t) − U_S(t)

U_E(t) − U_W(t) = s(t)2jkd sin(θ)

s(t)2jkd cos(θ) = sin(θ)

cos(θ) (19)

tan θ = U_{N S}

U_{W E} (20)

Due to how the trigonometric function in Equation 20 is defined, an ambiguity problem arises. The aforementioned problem is solved by having a sensor in the origin to determine which quadrant the DOA is located.

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2.2.3 Phase Interferometer

In the Phase Interferometer method the phase difference between the sensors is measured [19]. This can be put in contrast to the TDOA which measure the time difference.

Figure 10: Illustration of the Phase interferometer were d is the distance between the sensors, θ is the DOA and the dotted lines represent the phase difference ∆Φ.

Considering two antennas separated with a distance d from each other, see Figure 10. The AoA of the incident wave is θ and can be calculated according to

θ = cos⁻¹(λ∆Φ

2dπ ) (21)

were ∆Φ is the phase difference and λ is the wavelength of the incident wave.

If the distance d is greater than λ/2 there will be an ambiguity problem due to the periodicity in equation 21. The equation will provide unlimited solutions because it is periodic. To avoid ambiguity the distance d have to be < λ/2 and the phase measurement have to be in the interval of (0, 2π) or (−π, +π). This under the condition on the DOA being in the interval of (−π/2, +π/2). [6]

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Methods 2.2

2.2.4 MUltiple SIgnal Classification (MUSIC)

The MUSIC method is a generic term for theoretical and experimental methods to determine the parameters of receiving signals in an antenna array [31]. MUSIC can provide information about a number of received signals, DOA, amplitude, correlation between the different waveforms, polarization and Signal to Noise Ratio (SNR).

Furthermore, the MUSIC algoritm is also applicable to identify information such as conventional interferometry, monopulse DF and multiple frequency estimation [31].

The advantage with this method is that it can be applied on arbitrary antenna arrays [21]. When a randomly spaced antenna array is applied the MUSIC method is preferable [5]. However, the disadvantage and limitation occurs when the MUSIC method for example is applied on a Uniform Circular Array (UCA) and its elements have a small displacement [16]. This leads to decreased performance in the DOA estimation [18]. Another disadvantage with the method is the ambiguity in the DOA estimation when the signal from the transmitter has low power and wideband [9]. This is because the MUSIC method is based on the narrowband assumption [21].

Before the MUSIC method can be explained a signal model has to be described first and follows. A signal, noted as s(t), generated by a transmitter far away is represented by the equation below.

s(t) = αe^jωt (22)

were α is the amplitude, ω is angular frequency and t is the time.

A time delay in the signal is a phase shift in the time domain, which can be mathematically described as

s(t − t₀) = αe^jω(t−t⁰⁾ = e^−jωt⁰s(t) = e^−jΦ⁰s(t) (23) were t₀ is the time delay and Φ is the phase shift.

Consider an arbitrary antenna array with N antenna elements and an incoming signal with θ as the AoA relative to the array. Assume that the first antenna element x₁(t) is receiving the signal s₁(t). The elements position are represented in the position vector ~ri and the AoA is represented in the DOA vector ˆr0 in polar coordinates. The phase in each element is defined in the equation below.

Φ_i = k~r_i · ˆr₀ (24)







~r_i = R_icos(φ_i) ˆx + R_isin(φ_i) ˆy

ˆr₀ = cos(θ) ˆx + sin(θ) ˆy (25)

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were k is the wavenumber. The received signal at the N elements are

xi(t) = gi(θ)e^−jΦⁱs(t) = gi(θ)e^−jk~rⁱ^·ˆ^r⁰s(t) (26) were g_i(θ) is the embedded element pattern and is equal to 1 because of the assumption of no coupling. All signals received in each element is represented in the vector below





 x₁(t) x₂(t) ...

xN(t)







=







g₁(θ)e^−jk~r¹^·ˆ^r⁰s(t) g₂(θ)e^−jk~r²^·ˆ^r⁰s(t)

...

gN(θ)e^−jk~r^N^·ˆ^r⁰s(t)







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which can be written in vector form were a(θ) is the steering vector

x(t) = a(θ)s(t) (28)

In this case the M number of arriving signals to the antenna array with different frequencies are assumed to be uncorrelated. Furthermore, the N elements are assumed to have background noise n_i(t) in each signal. The total signal model can be written as

x(t) = As(t) + n(t) (29)

were n(t) is the noise vector and A is a N × M matrix with steering factors a(θ) for all N elements and for each source.

A =[a(θ₁), ..., a(θ_M)]

s(t) =[s₁(t), ..., s_M(t)]^T (30) The MUSIC algorithm now follows [31], the first step is to evaluate the covariance matrix R_x. For practical reasons the sample covariance matrix is evaluated.

R_x = 1 K

K

X

t=1

x(t)x^T(t) (31)

R_x can then be expressed as

Rx = E[x(t)x^T(t)]

= {E[X + Y ] = E[X] + E[Y ]}

= E[As(t)] + E[n(t)]

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Methods 2.2

The expression can then be rewritten as

R_x = E[As(t)] + E[n(t)]

= AR_sA^H + σ₀²I (33)

were R_s = E[s(t)s^H(t)] is the covariance matrix of the source vector and σ₀²I = E[n(t)n^H(t)] is covariance matrix of the noise vector. When there is more elements than sources, N > M , the matrix AR_sA^T is singular which means

det(AR_sA^H) = det(R_x− σ₀²I) = 0 (34) The dimension of the null space of ARsA^H is given by N − M . Rx and ARsA^H are defined as non-negative. For M incoming signals there are M other eigenvalues σ²_i and as σ₀ is the minimum eigenvalue it implies that σ_i² > σ²₂ > σ₁² > σ₀² > 0. The eigenvectors corresponding to the eigenvalues σ²_i are defined as e_i. The correlation matrix can therefore be written as

R_xe_i= (AR_sA^H + σ²₀I)e_i = σ_i²Ie_i (35) which implies that

ARsA^Hei =







(σ²_i − σ₀²)Ie_i; i = 1, 2, ..., M

0; i = M + 1, ..., N (36)

When i ≥ M the correlation matrix becomes zero, as a result of the signal subspace and the noise subspace is orthogonal to each other. If E_N is a N × M matrix were the columns are the noise eigenvectors and the points along a(θ) can be plotted as a function of the incident angle, θ. The function is called the MUSIC ”Pseudo Spectrum” and is given by

P_{M U SIC}(θ) = 1

a^H(θ)E_NE^H_Na(θ) (37)

Ambiguities in DF Arrays with the MUSIC method

Consider an arbitrary antenna array containing N elements and J elements whose positions that are collinear, then there will be at least N − J + 2 directions that creates sets of steering vectors that are linear dependent [12]. This means that if such an array is applied with the MUSIC method, there will be an ambiguity in those directions in the MUSIC ”Pseudo Spectrum” [12].

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If a UCA is applied and its elements are linear independent, the MUSIC method can provide an ambiguity free DOA estimation [21]. A UCA with 2, 3, 4 and 6 elements will have a rank-1 in ambiguity [41]. An antenna array with rank-1 ambiguity means that there exist 1 pair of linear dependent steering vectors [37]. To have an ambiguity free UCA for DOA estimations the number of elements have to be 5 or ≥ 7 [41].

In presence of noise, the cause of ambiguity is not only linear dependent steering vectors. In ambiguity-free DF systems using MUSIC, ambiguities can appear and that is because of the noise. [20]

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Methods 2.2

2.2.5 Estimation of Signal Parameters via Rotational Invariance Tech- niques (ESPRIT)

The method Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) can briefly be described as an similar approach to the MUSIC method [21]. ESPRITE is only applicable with a ULA and using the benefits of the array structure by dividing it in sub-matrices [28].

Figure 11: An illustration of the sub-matrices of a ULA which the method ESPRIT is using.

Assuming a ULA, A, with N elements than the first sub-matrices A₁ will span from the first element to the N − 1 element. The second sub-matrices A₂ span from the second element to the last element N , see Figure 11. By having the ULA divided in to sub-matrices makes the method robust against imperfections in the antenna array.

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3 State of the Art Technology

This section will outline two state of the art technologies which have been com- mercialized and tried by the market. These are GNSS, a global navigation satellite system, and Rhode&Schwarz which offers solutions within radiomonitoring and radiolocation.

3.1 GNSS

GNSS stands for global navigation satellite system with the most used application being the global positioning system (GPS). It was developed during the Cold War by the United States Air Force [29]. Since then other satellite based navigation systems have been developed, such as the Russians Global Navigation Satellite System (GLONASS) and the European Galileo [29]. The GPS system is primarily a military application and for non-civilian users the system provides a precise positioning service (PPS) [30].

At the United States Air Force Base Schriever in Colorado the master control station (MCS) is located and controls all GPS satellites. One important control factor is the synchronization of the satellites atomic clocks [29]. The control factor is considered to be crucial due to the fact that the GPS system is using the time of arrival (TOA) method.

3.2 Rhode & Schwarz

Rhode & Schwarz is a multinational corporation established in Germany 1933 with one expertise being radiomonitoring and radiolocation. In DF systems using TDOA method for DOA with Rhode&Schwarz products a minimum of three receivers are needed to achieve an unambiguous result of the transmitters position [33]. The ability to measure time precisely is crucial for the TDOA method. The R&S receivers synchronize the internal clock with the GPS clock and the internal reference frequency is adjusted with a pulse per second signal. The GPS clock and PPS, which is received from the GPS, synchronization gives the system the ability to have an accuracy in the picoseconds range. A R&S receiver can provide time synchronization accuracy of 100 ns and a positioning accuracy of 2m using GPS with a RF receiver [34].

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Methodology 4.2

4 Methodology

In this section, the choice of localization and direction finding method is discussed and determined. The methodology is based on the considered scenario described in Section 1.2.

4.1 Localization Methods

Homing is a time consuming method and will put the operator in potential danger when approaching the transmitter’s location. The unknown transmitter’s location may be a hostile environment. This scenario requires a rapid localization method to minimize the duration of the OODA loop.

TDOA is most optimal when the location of an unknown transmitter is geometri- cally surrounded by sensors. In the considered scenario of this study, the transmitter is more likely not to be surrounded by the sensors. It can still occur but the position of the transmitter will more frequently be outside the high accuracy area of the TDOA method as shown in Figure 7. The TDOA method also require high accuracy in time synchronization and positioning of the sensors. The required accuracy can be obtained from GNSS if state of the art technology is used, which can provide a time synchronization accuracy of 100 ns which was described in Section 3.2.

Triangulation is considered to be a simple and robust method. The performance limitations are dependent on the DOA estimations from the DF systems and their accuracy in positioning. Hence, when the transmitter is not surrounded by sensors, triangulation is more preferable than TDOA, which is considered to be the most frequent scenario. In the considered scenario only one group of drones is considered but using another group of drones triangulation can easily be applied. Therefore, in this scenario the localization method triangulation will be applied.

4.2 DF Methods

Watson-Watt is a well developed method with moderate accuracy [4]. The Adcock antenna array, which this method is using can also be used by the MUSIC algorithm.

The Watson-Watt is more suitable in the HF band while MUSIC is more preferable in VHF and UHF bands [4]. The Watson-Watt method do not consider multiple sources and do not consider noise affecting the signal and the measurement. The method is limited to the two-dimensional plane and the structure of the Adcock antenna array. In this scenario will the positions of the sensors be randomly placed and therefore is the method Watson-Watt not applicable in this scenario.

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Phase Interferometer can easily handle the estimation of the DOA when the incident signal is continuous. The method doesn’t take into account the surrounding noise and that affects the DOA estimation negatively. When the distance between the sensors is greater than λ/2 an ambiguity in the DOA estimation arises. In this scenario the elements will be separated with a distance greater than λ/2. The sensors also need to be positioned on the same baseline and in this scenario are the sensors randomly placed. Therefore, is the method Phase Interferometer not applicable in this scenario.

TDOA needs high accuracy in time synchronization. This method is very suitable when short pulses are received. When the signal is continuous, it is more difficult to determine the DOA. A continuously transmitted signal is considered in this scenario but if the drones have landed and can conduct a measurement of the incoming signal’s first peak then is it possible to determine the DOA to the transmitter. TDOA also have more potential in estimating a ambiguity free DOA when the signal is low power and wideband than the methods Watson-Watt and MUSIC [9]. The method only needs two sensors which can be randomly placed and this is applicable with the considered scenario. Therefore will the affect of time synchronization error with the method TDOA using two sensors be analyzed.

ESPRIT is using the benefits of the structure of a ULA by dividing it into two sub-matrices. In this scenario are randomly placed sensors considered. It’s unlikely that the randomly placed sensors will form a ULA. Therefore this method isn’t applicable with the considered scenario.

MUSIC can handle multiple sources and estimate different signal parameters such as source frequency. The method is also handling interference from noise. Noise will be present in the path of propagation and in the measurement equipment. MUSIC is not limited to one array structure and can be applied in arbitrary antenna arrays, which is the case in the considered scenario. The MUSIC method is robust against interference from noise and can be applied to arbitrary antenna arrays, which also can be described as randomly placed sensors. MUSIC will be the main DF method applied in this thesis.

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Results 5.1

5 Results

In this section four scenarios will be analyzed by using Matlab and the fundamental code in this section can be found in Appendix A and B. The analysis will start with a ULA containing two sensors with a time synchronization error. Secondly, the accuracy of a UCA with different numbers of elements will be analyzed. Thereafter the accuracy of a Random Circular Array (RCA) with a different number of elements is investigated. Finally, a real scenario of randomly placed sensors with a time synchronization error will be analyzed.

5.1 ULA with time Synchronization Error

In this section a ULA with two sensors located on the x-axis is considered. The sensors are omnidirectional in the xy-plane and the incident wavefront from a transmitter is a plane wave in a noise-free environment. Sensor 1 will be located in the origin and Sensor 2 with a distance d from origin on the positive side of the x-axis.

The DOA of the plane wave is θ and is shown in Figure 12.

Figure 12: Two sensors located on the x-axis separated by the distance d and the incident angle of the plane wave is θ.

The DF methods TDOA and MUSIC will be analyzed with a time synchronization error between the two sensors and with different values of d. There are no other errors in this scenario and the DOA is 45 degrees throughout the analysis.

The DOA estimation using the DF method TDOA with two sensors is cos(θ) = c(∆_t+ ε)

d (38)

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where ε is the time synchronization error between the two sensors. As described in Section 1.2, the time synchronization error make the sensors start measuring at different times and that affects the DOA estimation.

When using the TDOA method and a time synchronization error between the sensors occurs, this results in an error in the DOA estimation. The negative effect of the time synchronization error is decreased when the distance between the elements is increased, which is also shown in Figure 13. In this case the error is represented as an absolute error. The distance between the elements is represented in meters rather than λ, due to the fact that the TDOA method is independent from the wavelength.

Figure 13: When there is a large distance between the two sensors, the time synchronization error has less effect on the DOA estimation when using the method TDOA.

In Figure 14 the distance between the sensors is represented in wavelengths. In this scenario the frequency is 300 MHz and λ ≈ 1m.

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Results 5.1

Figure 14: When there is a large distance between the two sensors, the time synchronization error has less effect on the DOA estimation when using the method TDOA. In this case the frequency is 300 MHz and λ ≈ 1m.

To estimate the DOA to a transmitter with the MUSIC method a synthetic signal has to be generated from the transmitter. In this scenario the frequency is f = 300 MHz, the number of samples is 3000, and the sample rate is fs = 2f . The time synchronization error ε is added to the synthetic signal at Sensor 2 according to the equations below.

x₂(t) = e−j(2πf (t+ε)+k~r·ˆr0) (39)







~r = d(cos(φ)ˆx + sin(φ)ˆy) ˆ

r₀ = cos(θ) ˆx + sin(θ) ˆy (40) In this case f is the transmitters operating frequency, k is the wavenumber of that frequency, d is the distance between the sensors and φ is the angle to the element relative the x-axis. φ is equal to zero in this ULA.

In MUSIC the time synchronization error has the same effect on the DOA estimation as it has on TDOA. However, the error in the DOA estimation is considered to be greater, especially at small distances which is depicted in Figure 15 and 16.

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Figure 15: When increasing the distance between the two sensors, the time synchronization error has less effect on the DOA estimation when using the MUSIC method. In this case, the frequency is 300 MHz and λ ≈ 1m.

Figure 16: When the time synchronization error increases, the effect on the DOA estimation has a periodic behavior when using the MUSIC method. In this case, the frequency is 300 MHz and λ ≈ 1m.

Furthermore, if the transmitter’s frequency is higher, say 3 GHz, the DOA estimation becomes more sensitive to time synchronization error, which is shown in Figure 17.

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Results 5.1

Figure 17: When the distance between the two sensors is increased the time synchronization error has less effect on the DOA estimation using the method MUSIC.

Here the frequency is 3 GHz and λ ≈ 1m.

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5.2 Ambiguities in a 2-Element ULA

The true DOA is 45 degrees but MUSIC provides two peaks in the spectrum even if the distance between the elements is < 0.5λ. This is due to the symmetry problem, cos(θ) = cos(−θ), which gives the same signal in those directions according to the equation described in Section 5.1. When the distance between the elements is larger than 0.5λ the MUSIC method provides two additional peaks in the spectrum, which is shown in Figure 18. The two additional peaks are described in Section 2.2.3 and it stems from the distance between the sensors being ≥ 0.5λ.

Figure 18: An increased distance between the elements increase the number of peaks in the MUSIC spectrum.

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Results 5.3

5.3 UCA with Noise

In this section, a UCA with N sensors is considered and the DF method is MUSIC.

The sensors are omnidirectional in the xy-plane and the incident wavefront from a transmitter is a plane wave. The sensors are in a noisy environment where the SNR is chosen to be 20 dB. The DOA of the plane wave is θ, R is the radius of the UCA and φ_i is the angle to the i-th sensor, see Figure 19.

Figure 19: A UCA where φ_i is the angle to the i-th element, R is the radius and DOA of the plane wave is θ.

The synthetic signal is generated with a frequency of f = 300 MHz with 3000 samples and the sample rate is fs = 2f . The signals at each sensor are generated according to the equations below.

x_i(t) = e−j(2πf t+k~r_i·ˆr0)+ n(t) (41)







~r_i = R cos(φ_i) ˆx + R sin(φ_i) ˆy

ˆr₀ = cos(θ) ˆx + sin(θ) ˆy (42) In the analysis the DOA is 45 degrees, while the number of elements and the radius of the UCA is changed. For each number of elements, a radius sweep is conducted from 0.1λ to λ, and for each new radius, an additional 1000 DOA estimations are calculated with a added random white Gaussian noise for each iteration. The difference between the estimated DOAs and the true DOA are noted and the accuracy is calculated with Root Mean Square (RMS). The number of elements and their different radii versus the accuracy in RMS is shown in Figure 20.

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In Figure 21 is a UCA with seven elements with different radii versus SNR is depicted.

Figure 20: UCA with different numbers of elements and their different radii versus the accuracy in the DOA estimation.

Figure 21: UCA with seven elements and different radii versus SNR and the accuracy in RMS.

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Results 5.4

5.4 Ambiguities in a UCA

A UCA with 3, 4 or 6 elements will have a rank-1 ambiguity, which was also described in Section 2.2.4. In Figure 22, a rank-1 ambiguity is illustrated with a 4-element UCA with radius λ and where the DOA is 0 degrees. The ambiguity stems from the two sensors being placed on the x-axis, which makes them collinear and therefore they have linear dependent steering vectors in the directions 0 and 180 degrees.

This also appplies to the two sensors placed on the y-axis giving an ambiguity in the directions 90 and 270 degrees. If the radius of a 4-element UCA is < 0.5λ then the DOA estimation is ambiguity-free. In a 6-element UCA it is the directions 0, 60, 120, 180, 240 and 300 degrees which gives an ambiguity in the DOA estimation.

If the radius of a 6-element UCA is < λ then the DOA estimation is ambiguity-free.

This can be verified by using the code in Appendix A.

Figure 22: MUSIC ”Pseudo Spectrum” with a 4-element UCA with radius λ and where the DOA is 0 degrees.

If the radius is a couple of λ, which is the considered scenario described in Section 1.2, then it is easy to determine the true DOA and is depicted in Figures 23, 24 and 26. This under the condition that the DOA is not the collinear directions (e.g. 90 and 60 degrees) which were described before with a 4-element and a 6-element UCA.

To have an ambiguity-free UCA, the number of elements should either be 5 or

≥ 7 according to Section 2.2.4. A 5-element and 7-element ambiguity-free UCA can determine an ambiguity free DOA in all directions. If the radius is a couple of λ then it is easy to determine the DOA and is depicted in Figures 25 and 27. The true DOA in all figures is 45 degrees and is the largest peak in the MUSIC ”Pseudo Spectrum”.

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Figure 23: MUSIC ”Pseudo Spectrum” with a 3-element UCA.

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Results 5.4

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Results 5.5

5.5 Random Circular Array (RCA)

In this section two scenarios will be analyzed with the DF method MUSIC. The first scenario is exactly the same as the UCA scenario, except that the array is constructed with randomly placed elements. The elements are randomly distributed in an area of a circle with radius R. The positions of every element in polar coordinates are generated by the Matlab random number generation function randi, where φ_i = randi([0 360], 1, 7) is the angle and R_i = randi([0 1000], 1, 7)λ/1000 is the radius to the i-th element. In Figure 28 the geometry of the scenario is illustrated.

Figure 28: Random element placement where Ri and φi is the radius and angle to each element in the random array.

In the synthetic signal R is changed to R_i, as shown in the equations below.

xi(t) = e−j(2πf t+k~ri·ˆr0)

+ n(t) (43)







~r_i = R_icos(φ_i) ˆx + R_isin(φ_i) ˆy

ˆr₀ = cos(θ) ˆx + sin(θ) ˆy (44)

In the analysis the DOA is 45 degrees, while the number of elements and the maximum radius (R_max) of the RCA is changed. For each number of elements a radius sweep is conducted from 0.1λ to λ, and for each new radius an additional 1000 RCA is generated and the DOA estimations are calculated with an added random white Gaussian noise for each iteration. The difference between the estimated DOAs and

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the true DOA are noted and the accuracy is calculated with RMS. The number of elements and their different maximum radii (R_max) versus the accuracy in RMS is shown in Figure 29.

Figure 29: RCA with different numbers of elements and their different radii versus the accuracy in the DOA estimation.

The second scenario has the same base as the first scenario. The difference is that only 7-elements RCAs with different maximum radii (R_max) are generated and the accuracy in RMS for different SNR are calculated. In Figure 30 the accuracy of the RCA depending on the maximum radius versus SNR is illustrated.

Figure 30: SNR VS R_max.

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Results 5.6

5.6 Ambiguities in an RCA

As described in Section 2.2.4, a 3-element, 4-element and 6-element UCA is a rank-1 ambiguity array. An RCA, with the same number of elements do not form a UCA and will therefore not have a rank-1 ambiguity. If the radius is a couple of λ then it is easy to determine the DOA which is depicted in Figures 31, 32 and 34. An RCA with 5 or 7 elements are ambiguity-free in the same way as a UCA with 5 or 7 elements are ambiguity-free. This also means that the same applies for RCAs with elements ≥ 7 as long the steering vectors are linearly independent, which is depicted in Figures 33 and 35. The RCAs in the figures are randomly generated and it is possible to generate an RCA with linear dependent steering vectors. Thus, it is preferable to also plot the MUSIC ”Pseudo Spectrum” and visually inspect it. The SNR is 20 dB in all figures.

Figure 31: MUSIC ”Pseudo Spectrum” with a 3-element RCA.

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Results 5.6

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5.7 RCA and UCA Comparison

A UCA with 7 elements and a radius R is compared to an RCA similar to the UCA but whose elements have a small displacement, see Figure 36. Each position of the elements in the RCA are known and the displacement is the same for every new radius. Both arrays are iterated 1000 times for each new radius with white Gaussian noise and the SNR is 20 dB. The difference between the estimated DOAs and the true DOA are noted and the accuracy is calculated with RMS.

Figure 36: An RCA and a UCA, both with 7 elements.

First generated displacement has an average radius of 1R and a maximum radius of 1.08R. The accuracy versus radius for both a UCA and an RCA is depicted in Figure 37. The RCA average RMS is 4.16 × 10⁻⁴ degrees more accurate than the average RMS of a UCA.

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Results 5.7

Figure 37: The accuracy of a UCA and an RCA.

The second generated displacement has an average radius of 0.98R and a maximum radius of 1.04R. The accuracy versus radius for both a UCA and an RCA is depicted in Figure 38. The RCA average RMS is 1.86 × 10⁻⁴ degrees more accurate than the average RMS of a UCA.

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The third generated RCA have its elements distributed in the area of a circle with radius R, see Figure 39. The average radius of the RCA is 0.79R and the maximum radius is R. The accuracy versus radius for both a UCA and an RCA is depicted in Figure 40. The RCA average RMS is 3.50 × 10⁻³ degrees less accurate than the average RMS of a UCA. Thus, it is considered better to have a UCA than an RCA in this case, if good accuracy is desired.

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Results 5.8

5.8 Plane Wave Approximation

In DF methods a plane wave is considered. This means that every sensor has the same DOA but the wave arrives at different times. This approximation can only be used when a transmitter is located at large distances from the sensors. When the location of the transmitter is closer to the sensors, they will have a difference in their DOAs which is depicted in Figure 41, where d is the distance between the sensors, R is the distance to the transmitter, θ₁ is DOA of Sensor 1 and θ₂ is the DOA of Sensor 2. In Figure 42 the difference in the DOA (θ₂− θ₁) between the two sensors is illustrated with different R and d.

Figure 41: The difference in the DOA between the senors when the transmitter is located at different distances from the senors.

Figure 42: Difference in DOA of two sensors separated with different distances.

Direction Finding: Determine the direction to a transmitter with randomly placed sensors