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Master of Science Thesis in Electrical Engineering

Department of Electrical Engineering, Linköping University, 2020

Time Synchronization of

TDOA Sensors Using a

Local Reference Signal

Alfred Hult

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Master of Science Thesis in Electrical Engineering

Time Synchronization of TDOA Sensors Using a Local Reference Signal:

Alfred Hult

LiTH-ISY-EX--20/5318--SE Supervisor: Johannes Lindblom

Swedish Defense Research Agency (FOI)

Unnikrishnan Kunnath Ganesan

isy, Linköping University

Examiner: Mikael Olofsson

isy, Linköping University

Division of Communication Systems Department of Electrical Engineering

Linköping University SE-581 83 Linköping, Sweden Copyright © 2020 Alfred Hult

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Sammanfattning

Synkronisering av ett distribuerat nätverk av sensorer för tidsdifferensmätning (TDOA-sensorer) kan göras med referenssignaler från GPS-satelliter. Den meto-den ger hög noggrannhet, men är känslig för störning, och är inte tillräckligt på-litlig för att användas i militära tillämpningar. En lösning som inte beror på någ-ra signaler utsända från externa aktörer är att förednåg-ra. Ett sätt att åstadkomma detta är att använda referenssignaler som sänds ut från en UAV. En UAV är lämp-lig eftersom endast lokal synkronisering för ett geografiskt begränsat område är nödvändig. Den lokala synkroniseringen görs genom estimering av tidsfördröj-ningen mellan sändtidsfördröj-ningen och mottagandet av en referenssignal. Den estimerade tidsfördröjningen kan användas för att upptäcka drivningar i TDOA-sensorernas klockor. Detta arbete analyserar vanliga referenssignaler, för att utvärdera vilka som ger tillräckligt hög noggrannhet vid estimering av tidsfördröjningar, och vil-ka egensvil-kaper hos signalerna som påvervil-kar noggrannheten hos estimeringarna mest. Simuleringarna visar att prestandan kan uppnå samma noggrannhet som synkronisering gjord mot GPS för flera typer av signaler. Det är viktigare att öka bandbredden än att öka signallängden eller signal-till-brus-förhållandet för att höja noggrannheten hos estimeringen.

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Abstract

Synchronization of distributed time difference of arrival (TDOA) sensor networks can be performed using reference signals from GPS satellites. This method pro-vides high accuracy, but is vulnerable to jamming, and is not reliable enough to be used in military applications. A solution that does not depend on any sig-nals transmitted from external actors is preferred. One way to achieve this is to use reference signals transmitted from a UAV. A UAV is suitable since only lo-cal synchronization for a geographilo-cally restricted area is necessary. The lolo-cal synchronization is achieved by estimating the time-delay between the transmis-sion and reception of a reference signal. The estimated time-delay can be used to detect drifts in the clocks of the TDOA sensors. This thesis analyzes com-mon reference signals, to evaluate which provide high accuracy for time-delay estimation, and what properties of the signals influence the estimation accuracy the most. The simulations show that the time-delay estimation performance can reach the same accuracy as synchronization against GPS for different types of sig-nals. An increased bandwidth is more important than an increased signal length or signal-to-noise ratio to improve the estimation accuracy.

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Acknowledgments

Thanks

to my supervisor at FOI, Johannes, who showed patience while explaining things I should already know.

Thanks

to Per at FOI, for helping with the laboratory tests, and Tommy, for the code used to generate the Gold sequences.

Thanks

to the colleagues at FOI, employees and other thesis students, for making it such a joy to be at the office.

Thanks

to my supervisor Unnikrishnan and examiner Mikael at the university, for feed-back on the report, and help along the way.

Linköping, 2020 Alfred Hult

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Contents

List of Figures xii

List of Tables xv Notation xvii 1 Introduction 1 1.1 Purpose . . . 3 1.2 Problem Statements . . . 3 1.3 Limitations . . . 4 2 Theoretical Background 5 2.1 Positioning . . . 5 2.1.1 TOA . . . 6 2.1.2 TDOA . . . 7

2.2 Methods for Synchronization . . . 8

2.2.1 GPS . . . 8

2.2.2 DCF77 . . . 9

2.2.3 Wired Synchronization . . . 10

2.3 Signal Model . . . 10

2.3.1 Time-delay in Frequency Domain . . . 11

2.4 Free-space Propagation . . . 11

2.5 Time-delay Estimation . . . 12

2.5.1 Correlation . . . 12

2.5.2 Cramér-Rao Bound for Time-delay Estimation . . . 13

2.5.3 Maximum Likelihood Estimator . . . 14

2.5.4 Sub-sample Accuracy . . . 15

2.6 Reference Signals . . . 15

2.6.1 Frequency Band . . . 16

2.6.2 Signal-to-noise Ratio . . . 16

2.6.3 Ambiguity Function . . . 16

2.6.4 General Expression for the Cramér-Rao Bound for Time-delay Estimation . . . 17

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x Contents

2.6.5 Pulse . . . 18

2.6.6 Chirp . . . 19

2.6.7 White Gaussian Noise . . . 21

2.6.8 Gold Sequences . . . 22 3 Method 25 3.1 Setup . . . 25 3.1.1 Parameter Restrictions . . . 26 3.2 Analytical Evaluation . . . 26 3.3 Simulations . . . 26

3.3.1 Time-delay Estimation in the Simulations . . . 27

3.3.2 Generation of Gold Sequences . . . 27

3.4 Laboratory Tests . . . 28

4 Results 31 4.1 Cramér-Rao Bounds for Time-delay Estimation . . . 31

4.2 Simulations . . . 33

4.2.1 Varied Bandwidth . . . 33

4.2.2 Varied Signal Length . . . 35

4.2.3 Varied Signal-to-noise Ratio . . . 37

4.3 Tests in Laboratory . . . 39

5 Discussion 43 5.1 Results . . . 43

5.1.1 Estimation Accuracy of the Waveforms . . . 43

5.1.2 Effect of the Signal Properties . . . 44

5.2 Method . . . 46 5.2.1 Chosen Waveforms . . . 46 5.2.2 Theoretical Tools . . . 47 5.2.3 Simulations . . . 48 5.2.4 Laboratory Tests . . . 48 5.2.5 Sources . . . 49

5.3 The Thesis in a Larger Perspective . . . 49

6 Conclusions 51 6.1 Problem Statements . . . 51

6.1.1 Type of Waveform . . . 51

6.1.2 Effect of Signal Properties . . . 52

6.1.3 Performance Compared to GNSS Synchronization . . . 52

6.2 Future Improvements . . . 52

A Calculations for Cramér-Rao Bounds 53

B Cramér-Rao Bounds for Time-delay Estimation Compared to

Simula-tion Errors 57

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Contents xi

D Additional Laboratory Results 65

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List of Figures

2.1 The two possible locations for the target, TA and TB, using two TOA sensors, R1 and R2. Since no unambiguous estimate is given, the target can be at either TA or TB. . . 6 2.2 The estimated location for the target using three TDOA sensors R1,

R2, and R3. Three pairs of receivers generate three hyperbolas, for which the intersection is the estimated position of the target, TA. . 7 2.3 Example showing auto-correlation of 20 000 samples of white

Gaus-sian noise, w ∼ N (0, 1). Since there is a distinct peak at lag 0, it can be concluded that none of the signals are time-delayed relative to the other. If one of the signals was time-delayed, the peak would have been at a lag corresponding to the delay measured in samples. 13 2.4 Ambiguity function of a pulse with A = 1, T = 100 µs and fs = 2

MHz. . . 18 2.5 Spectrum of a baseband pulse signal with A = 1, T = 100 µs and fs

= 2 MHz. . . 19 2.6 Ambiguity function of a chirp with B = 1 MHz, T = 100 µs and fs

= 2B. . . 20 2.7 Spectrum of a baseband chirp signal with bandwidth B = 1 MHz,

T = 100 µs and fs= 2B. . . 20

2.8 Ambiguity function of complex WGN with B = 1 MHz, T = 100 µs, fs= 2B and variance σ2= 1. . . 21

2.9 Spectrum of one realization of complex WGN with bandwidth B = 1 MHz, T = 100 µs, fs= 2B and variance σ2= 1. . . 22

2.10 Ambiguity function of a Gold sequence with B = 1 MHz, T = 100 µs and fs= 2B. . . 23

2.11 Spectrum of a Gold sequence with bandwidth B = 1 MHz, T = 100 µs and fs= 2B. . . 23

3.1 Schematic drawing of the setup of the TDOA receivers and the UAV used in this thesis. . . 25 4.1 The CRBs for the time-delay estimation for different bandwidths,

signal lengths, and SNRs. If the parameter is not varied, it is set to B = 10 MHz, T = 1 ms, and SNR = 0. The sampling frequency is set to fs= 2B for all three plots. . . . 32

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LIST OF FIGURES xiii 4.2 The effect of different bandwidths on the synchronization error,

with T = 1 ms, SNR = 0 dB, and fs= 2B. . . 33

4.3 The effect of different bandwidths on the synchronization error, with T = 1 ms, SNR = 0 dB, and fs= 4B. . . 34

4.4 The effect of different bandwidths on the synchronization error, with T = 1 ms, SNR = 0 dB, and fs= 2B. The distance between the

UAV and the receiver for channel 1 is 5 km and 15 km for channel 2. . . 35 4.5 The effect of different signal lengths on the synchronization error,

with B = 10 MHz, SNR = 0 dB and, fs = 2B. . . 35

4.6 The effect of different signal lengths on the synchronization error, with B = 10 MHz, SNR = 0 dB and, fs = 4B. . . 36

4.7 The effect of different signal lengths on the synchronization error, with B = 1 MHz, SNR = 0 dB and, fs= 2B. The distance between the

UAV and the receiver for channel 1 is 5 km and 15 km for channel 2. . . 37 4.8 The effect of different SNRs on the synchronization error, with B =

10 MHz, T = 1 ms, and fs= 2B. . . 38

4.9 The effect of different signal-to-noise ratios on the synchronization error, with B = 10 MHz, T = 1 ms, and fs= 4B. . . 38

4.10 The effect of different signal-to-noise ratios on the synchronization error, with B = 10 MHz, T = 1 ms, and fs = 2B. The distance

be-tween the UAV and the receiver for channel 1 is 5 km and 15 km for channel 2. . . 39 4.11 Resulting differences in error from the laboratory test for

band-widths 500 kHz and 1 MHz, with T = 1 ms and Pt= 10 dBm. . . . 40

4.12 Resulting differences in error from the laboratory test for different signal lengths, with B = 1 MHz and Pt= 10 dBm. . . 40

4.13 Resulting differences in error from the laboratory test for different output powers, with B = 1 MHz and T = 1 ms. . . 41 B.1 Comparison between the square root of the Cramér-Rao bound for

the time-delay estimation and the synchronization error for three reference signals, with T = 1 ms, SNR = 0, fs = 2B, and varying

bandwidth. . . 58 B.2 Comparison between the square root of the Cramér-Rao bound for

the time-delay estimation and the synchronization error for three reference signals, with B = 10 MHz, SNR = 0, fs= 2B, and varying

signal length. . . 59 B.3 Comparison between the square root of the Cramér-Rao bound for

the time-delay estimation and the synchronization error for three reference signals, with B = 1 MHz, T = 1 ms, fs = 2B, and varying

SNR. . . 60 C.1 The effect of different bandwidths on the synchronization error,

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xiv LIST OF FIGURES

C.2 The effect of different signal lengths on the synchronization error, with B = 1 MHz, SNR = 0 dB, and fs= 2B. . . 62

C.3 The effect of different SNRs on the synchronization error, with B = 1 MHz, T = 1 ms, and fs= 2B. . . 62

C.4 The effect of different SNRs on the synchronization error, with B = 10 MHz, T = 100 µs, and fs= 2B. . . 63

D.1 Resulting differences in error from the laboratory test for different output powers, with B = 1 MHz and T = 10 µs. . . 65 D.2 Resulting differences in error from the laboratory test for different

output powers, with B = 1 MHz and T = 100 µs. . . 66 D.3 Resulting differences in error from the laboratory test for different

output powers, with B = 500 kHz and T = 10 µs. . . 66 D.4 Resulting differences in error from the laboratory test for different

output powers, with B = 500 kHz and T = 100 µs. . . 67 D.5 Resulting differences in error from the laboratory test for different

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List of Tables

3.1 Table showing the maximal length linear feedback shift register polynomials used to generate the Gold codes. . . 28 4.1 Table showing the ratios between the errors for doubling of the

bandwidth. . . 34 4.2 Table showing the ratios between errors for doubling of the signal

length. . . 36

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Notation

Defined parameters Parameter Description A Amplitude Cx Covariance matrix c Speed of light

d Distance between transmitter and receiver

f0 Instantaneous frequency

fc Carrier frequency fs Sampling frequency

I(θ) Fisher information

IN Identity matrix of size N × N

LB Path loss

Np Noise power

Pr Received signal power Pt Transmitted signal power

s Reference signal vector

w Additive white Gaussian noise vector

x Received signal vector

Time-delayed signal vector

 Rate of frequency increase for chirp signal

θ Estimation parameter ˆ θ Estimated value µ Mean value σ2 Variance τ Estimated delay xvii

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xviii Notation

Mathematical Operations

Operator Description ( · )∗ Conjugate operator diag( · ) Diagonal matrix

F { · } Discrete Fourier transform E{ · } Expectation

( · )H Hermitian operator

Re[ · ] Real part operator tr( · ) Trace operator Abbreviations

Abbreviation Description

awgn Additive white Gaussian noise comint Communications intelligence

crb Cramér-Rao bound

dft Discrete Fourier transform fft Fast Fourier transform

fi Fisher information

foi Totalförsvarets forskningsinstitut (Swedish Defense Research Agency)

gnss Global navigation satellite system gps Global positioning system

los Line-of-sight

mle Maximum likelihood estimator rf Radio frequency

snr Signal-to-noise ratio tde Time-delay estimation tdoa Time difference of arrival

toa Time of arrival

uav Unmanned aerial vehicle utc Coordinated universal time wgn White Gaussian noise

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1

Introduction

In military applications, it is important to have compromising information about the enemy. On the battlefield, for example, that can mean knowing the coordi-nates of enemy forces or vehicles. The process of determining the coordicoordi-nates of a target is called positioning. One common positioning method is time of arrival (TOA). A known signal is emitted at a known time. The positioning is performed by reckoning the time difference from emission to reception of the signal, so that the distance to the target can be calculated. If two sensors are performing TOA, the two resulting time differences can be compared. This process is called time difference of arrival (TDOA), and gives information about how much closer the target is to one of the sensors than the other. A difference from TOA is that TDOA does not have to have prior knowledge about the signal, or when it was transmit-ted. TDOA can achieve high positioning accuracy, and is, therefore, suitable to be used in military applications, where incorrect information can lead to catas-trophic consequences.

For TDOA to achieve accurate positioning of an object at range, a widely dis-tributed network of positioning sensors is needed. However, as is the case for most sensor networks, synchronization is an issue. The sensors need to work ac-cording to a common time, meaning that the sensor clocks have to have the same perception of when measurements should be made. If the sensors in a TDOA sen-sor network are not synchronized, the measurements will be inaccurate, because the measurements will not give coherent information about where the target was located at a certain time. There are several methods used to achieve synchroniza-tion for a network of TDOA sensors. For example, the timing signals transmitted from global navigation satellite systems (GNSS) like global positioning system (GPS) can be used as a reference, so that all the sensors in the network work ac-cording to a common time. The reference signals transmitted from GPS satellites contain universal coordinated time (UTC) timestamps [19]. UTC is seen as the

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2 1 Introduction

official time scale of the world [2], and via the GPS satellites, it can be distributed to the positioning sensors on the ground.

But communication systems intended for civilian or military use impose dif-ferent challenges related to security. The need for stability and security increases from preferable, for systems intended for civilian use, to absolutely crucial. This means that the systems used regularly by civilians might be unusable by the mil-itary. For the military forces of a nation it is also important with self-sufficiency, and it is preferable not to rely on technology that is sustained or operated by other nations or organizations. Moreover, it is undesirable to use technical solu-tions for which the specificasolu-tions are common knowledge. Such systems can be easily jammed by external forces.

GPS reference signals suffer from the above-mentioned vulnerabilities [9]. The reference signals are vulnerable to jamming, or might disclose the position of the user. GPS might even be shut off completely. Ideally, it would be possi-ble to achieve the same synchronization accuracy with a solution that does not depend on any external system. Additionally, it would be of great benefit if the signals only reached sensors in a geographically restricted area. One solution to this problem is to mount a clock on an unmanned aerial vehicle (UAV). The clock on the UAV would act as the reference clock for all the sensors in the net-work. The benefit of having the reference signal being transmitted from a UAV is that the height can make line-of-sight (LOS) transmission possible, making the connection more precise. The positioning sensors could use the reference signals from the UAV to achieve local synchronization. Local synchronization means three things:

• Only the relative time between the receivers is of importance. The TDOA receivers do not need to know the correct time according to UTC to work, only synchronization with the other sensors in the network is required. • The reference signal from the UAV only need to reach receivers in the

geo-graphically restricted area in which the TDOA sensors are located.

• No external systems are needed. The sensor should not need any more in-put besides the reference signals from the UAV to achieve synchronization. To achieve synchronization between the TDOA sensors, the UAV will with pre-determined intervals transmit reference signals to the positioning sensors. The UAV and the TDOA receivers will be in fixed positions, so that the position of the UAV is known at the receivers. To perform positioning, it is a requirement that every receiver knows the position of all the other sensors in the network, includ-ing their own. Naturally, there will be a delay between the transmission of the reference signal and the reception at the receiver. However, given the positions of the UAV and the receivers, it is possible to estimate the time-delay for a signal transmitted from the UAV, since the distance between the UAV and receiver can be calculated.

The estimation is performed using correlation of the original reference sig-nal and the received sigsig-nal. If a sigsig-nal is correlated with itself (so-called auto-correlation), the correlation reaches its maximum when the two signals are

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com-1.1 Purpose 3

pletely overlapping. This peak is said to occur when the lag is zero, since the signals are not shifted in time relative to each other [11, Ch. 4]. But if the orig-inal reference signal is correlated with a time-delayed version of itself, the peak occurs at a lag equal to the corresponding delay, measured in number of samples. If the delay is larger or smaller than what is expected according to the distance between the UAV and the receiver, the receiver clock is no longer synchronized with the reference clock on the UAV. Given the estimated time-delay, and the knowledge of the expected delay, the receiver clock can be adjusted. If the TDOA receivers (independently of each other) are synchronized with the UAV, conse-quently, they will be synchronized with each other.

For this method of estimation to be accurate and unambiguous, the peak that occurs at maximum correlation has to be narrow, meaning that it has a large amplitude relative to the rest of the correlation function. The width of the peak also has to be small, in terms of samples. Some signals and waveforms yield a more narrow correlation peak than others, and are therefore more well-suited for time-delay estimation.

This thesis looks into how well different signals work when used for time-delay estimation. The goal is to find what type of signals give the best estima-tion accuracy, and how different parameters affect the estimaestima-tion accuracy. The accuracy of the estimation is measured as how close the estimation is to the ex-pected delay. The signals are analyzed theoretically, to enable explanations for the behavior. The estimation accuracy is evaluated by simulations performed in Matlab, where the effect of different bandwidths, sampling frequencies, signal lengths, and signal-to-noise ratios are studied. To verify whether the simulation results would hold for a real communication channel, tests were carried out in a laboratory.

1.1

Purpose

The overall purpose of this thesis is to compare signals and waveforms that have good auto-correlation properties. The influence of different signal parameters on the behavior of the time-delay estimation is studied. The goal is to find out which signal gives the lowest error when used for time-delay estimation for a certain set of time-bandwidth restrictions. The thesis also aims to highlight the practical aspects of using the analyzed waveforms as synchronization signals.

1.2

Problem Statements

The problems that this thesis investigates can be summarized in three problem statements. The thesis aims at answering these questions.

• What kind of signal should be used to synchronize two TDOA sensors as accurately as possible, using a local reference signal from a UAV?

• What signal properties are of the highest importance when performing time-delay estimation using correlation (bandwidth, signal length, etc.)?

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4 1 Introduction

• Can two TDOA receivers be locally synchronized as accurately as if GNSS would have been used?

1.3

Limitations

Signal theory and signal processing are large subjects, and covering every aspect of the synchronization process in this thesis would be impossible. Therefore, to keep the scope of the project at a reasonable size, this thesis focuses on the design and correlation properties of the reference signal which is transmitted from the UAV. This means that subjects like timing recovery algorithms are left out. To make the comparisons of the signals meaningful, only properties that are com-mon for all the signals are varied, such as bandwidth and signal length.

One reason to not use external reference signals is to achieve robustness against jamming. However, this thesis will not investigate what can be done to achieve higher resistance against such disturbances.

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2

Theoretical Background

This chapter will provide the underlying theory of the topics in this thesis. In Section 2.1, the positioning methods TDOA and TOA will be explained. Since the synchronization signals should replace reference signals from external sys-tems, Section 2.2 describes a few common systems used today. Section 2.3 gives a mathematical description of the signal model, and Section 2.4 explains the free-space propagation channel model. In Section 2.5, the theory of correlation and time-delay estimation (TDE) is explained. Lastly, in Section 2.6, important sig-nal properties when performing correlation are described, as well as the chosen reference signals examined in this thesis.

2.1

Positioning

Positioning is the process of estimating the position of a target, in relation to one’s own position. In military applications this could for example be detection of approaching enemy surveillance drones or determining the coordinates of an artillery target. Positioning can be made in either two dimensions (coordinates on a plane) or three dimensions (exact position). In the following sections, only positioning in terms of determining coordinates in two dimensions is considered. There are several approaches to the problem of positioning, based on what property of the received signal the estimator utilizes. Depending on what sig-nal property is used, the approaches are typically divided into three observation models: waveform observations, timing observations, and power observations [7, Ch. 4]. An observation model can be viewed as a function that takes the measure-ments (the observations) as input and yield the relevant positioning information as output. The information the chosen observation yields, is usually categorized as either range or bearing measurements. This thesis studies two types of range measurements using timing observations: TOA and TDOA. Both TOA and TDOA

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6 2 Theoretical Background

measures the timing of the arrival of the signal at the receiver (timing observa-tions) and use that information to give information about the distance to the tar-get (range measurements).

2.1.1

TOA

One method for estimating range is time of arrival (TOA). This is the basic prin-ciple behind radar [17, Ch. 1]. TOA measurements utilize the fact that radio frequency (RF) signals travel through space with a known velocity, the speed of light. If a transmitter emits a pulse, that pulse will be reflected off of the target object. The reflected version of the pulse returns to a receiver placed in the same place as the transmitter. Then, the distance from the transmitter and the receiver to the target can be calculated. Mathematically, this distance to the target is cal-culated as:

d = ct

2, (2.1)

where c is the speed of light and t is the propagation time from the transmitter to the object. Note that the transmitted signal has to be known, so that its echo can be recognized. To get accurate estimates of the TOA, a characteristic reference signal is usually embedded into the signal [7, Ch. 4]. This known reference is correlated with the received echo of the pulse at the receiver, to enable a more accurate estimation of d. More information about correlation can be found in Section 2.5.1. R1 R2 TA TB d2 d1

Figure 2.1: The two possible locations for the target, TA and TB, using two TOA sensors, R1 and R2. Since no unambiguous estimate is given, the target can be at either TA or TB.

A setup with two TOA sensors is shown in Figure 2.1. The estimated distances

d1 and d2 do not contain any information about the direction of the target is,

which is different from radar, where direction is also measured. Since the pulse is emitted in every direction at the same time, the constraint of possible locations for the target looks like a circle around the receiver. Therefore, two sensors are not enough to get an unambiguous estimate of the position of the target, since the circles which mark the possible locations for the target intersect at two points.

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2.1 Positioning 7

Therefore, at least three TOA measurements by three separate sensors are needed. Then, the three circles would only intersect at one point.

2.1.2

TDOA

Instead of calculating the absolute distance to the target, TDOA utilizes the dif-ference between two TOA measurements from a pair of sensors [7, Ch. 4]. This means that it is not the actual TOAs that are of interest, the measure that matters is the difference between them. Hence, one TDOA measurement means the dif-ference between two TOA measurements from two different sensors. For regular TOA to work, the transmitted signal has to be known, as well as the the time of transmission. However, TDOA need only to calculate the difference between at what time a signal was received at two receivers, no further knowledge of the signal is needed. Hence, TDOA can be performed with signals not transmitted from the TDOA sensors.

The constraint of possible locations for the target for one TDOA measure-ment is a hyperbola, marking all the positions where the difference in distance to the two sensors is the same. Thus, three TDOA measurements (from at least three sensors) are required to give an unambiguous estimate of the position of the target, because, in that scenario, three hyperbolas of possible locations are calculated. The estimated target position is the intersection between the three hyperbolas, as shown in Figure 2.2.

R1 R2 R3 TA R1-R2 R1-R3 R2-R3

Figure 2.2:The estimated location for the target using three TDOA sensors R1, R2, and R3. Three pairs of receivers generate three hyperbolas, for which the intersection is the estimated position of the target, TA.

TDOA can achieve very high positioning accuracy, but with strict requirements on the synchronicity of the sensors. Moreover, the shorter the distance between the sensors, the more accurate the synchronization needs to be [9]. For example,

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8 2 Theoretical Background

with a distance of 10 m between the sensors, a synchronization error of only 10 ns can lead to a deviation from the correct target angle of up to ±27◦. If the sensors are placed 10 km apart, the angle error decreases to only ±0.0027◦for the same synchronization error.

2.2

Methods for Synchronization

Synchronization of widely distributed sensor networks is a well-known problem, and several methods have been used to solve it. Usually, an oscillator is used to generate a given frequency which in turn is used to provide a timing signal [18]. In the following section, a few common solutions are described. Solutions which incorporate individual oscillators in each sensor are omitted, since those are either not accurate enough to be used for TDOA, or very expensive [9]. The methods described are divided into two categories: common-view time transfer models and master-slave models. For common-view time transfer, each sensor individually receives the same synchronization signal from a common transmit-ter [4]. Mastransmit-ter-slave models have one sensor (the mastransmit-ter) that controls the other sensors in the network, i.e., either forwards a received synchronization signal or transmits a signal generated by the clock in the master device.

2.2.1

GPS

Today, there are several global navigation satellite systems (GNSS) in use. GNSS is the term used for any navigation system which uses satellites and has global or regional coverage. Examples of GNSS systems are GLONASS, which is owned and operated by Russia, and Galileo, operated by the European Union [13]. All GNSS systems could be used for time synchronization, but this thesis will focus on GPS, which is operated by the United States. The method for synchronization against GPS signals is a common-view time transfer method.

GPS satellites keep time using atomic clocks, which are made out of cesium and rubidium oscillators [3]. Though very accurate, the clocks can drift, and are therefore regularly controlled and adjusted by monitor stations on Earth. The clocks follow universal coordinated time (UTC). According to the current speci-fication for GPS, the time GPS transmits to receivers on Earth has a maximum error of 40 ns 95% of the time [19], relative to UTC.

GPS satellites transmit carrier signals to GPS receivers on two frequency bands: 1575.42 MHz (called L1), and at 1227.6 MHz (L2) [19]. Both of these carrier signals are encoded with the current time and the position of the transmit-ting satellite. In total GPS satellites transmit three codes:

• P code on both the L1 and L2 channel, • C/A code on the L1 channel,

• the navigation message on the L1 channel.

The P code, or precision code, is restricted to be used only by the United States military and NATO. The C/A (coarse/acquisition) code is available to the general

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2.2 Methods for Synchronization 9

public. The navigation message is also publicly available, and contains general information about the satellite [3].

The publicly available C/A code is a pseudo-random noise code, specifically, a Gold sequence, which is unique for every GPS satellite. Gold sequences are described in Section 2.6.8. The signal includes a timestamp for when the signal is generated in the satellite [3]. To synchronize a receiver against a GPS satel-lite, the received Gold sequence should be correlated with a copy of the same Gold sequence, unspoiled by noise and delay. Since the signals for each satel-lite are unique, the receiver can determine what signal it should correlate with, knowing from which satellite the received signal came from. When correlation is performed between the received signal and the copy, the delay from generation in the satellite to reception on Earth is estimated. This means that the receiver cal-culates the TOA for the received signal using t = dc. This expression is similar to Equation (2.1), but without the factor 12since the signal is only transmitted from the satellite to the sensor. The estimated delay multiplied with the speed of light is sometimes called the pseudo-range, implying that it is not the true range. The pseudo-range is longer than the true range, because it is derived from the total time from generation of the signal to finished estimation in the receiver. However, since the position of the satellite is known, the true range can be calculated, and subtracted from the range. Knowing what delay is caused by the pseudo-range, any undesirable clock deviations can be calculated and accounted for. For this kind of common-view time transfers, the accuracy is roughly 1 - 10 ns [4]. However, the common-view method is most effective when the distance from the receivers to the transmitter is very large [15].

As mentioned in the introduction, it is preferable not to depend on a system operated by foreign states. This is the main drawback of using GPS for synchro-nization in military applications. Apart from that, the accuracy is reasonable for the intended purpose of synchronizing a network of TDOA sensors. The cost can be kept low, as only GPS receivers are needed, rather than expensive atomic clocks for each reciever [9].

2.2.2

DCF77

DCF77 is a long-wave (77.5 kHz) transmitter, which can be used as a time ref-erence in most of Europe [5]. It transmits a signal that has a reach of roughly 1500 - 2000 km from Frankfurt am Main, where it is located. That means that the most northern parts of Sweden are not covered. To keep time according to UTC, the DCF77 uses cesium and rubidium atomic clocks. It is maintained by the Physikalisch-Technische Bundesanstalt (PTB), the national physics laboratory of Germany. It is commonly used to synchronize regular consumer clocks, like alarm clocks and wristwatches.

A major advantage of using the time signal and standard frequency of the DCF77 is that the long wavelength (≈3868.3 m) of the emissions makes them invulnerable to obstacles. This is of great benefit, and is a reason why it is used by clocks indoors. In the case of GPS, the receiver needs LOS communication with the satellite [12].

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10 2 Theoretical Background

The transmitted signal deviates from UTC by less than 2 · 10−12 s in one day on average [5]. However, since the distance to the receiver is very long, reflections in the ionosphere make the DCF77 an unreliable distributor of time reference, es-pecially when the sensors are widely separated geographically [9]. The accuracy is not high enough to motivate the use of DCF77 as an external reference. It is also geographically restricted to central Europe. Furthermore, it has one of the same drawbacks as GPS, namely that it is controlled by another state.

2.2.3

Wired Synchronization

One way to achieve robust time distribution between the sensors is to physically connect them, using either coaxial or fiber optic cables. Using this solution, very high accuracy can be achieved, e.g., 10 - 50 ps for optical fiber [2]. However, the system becomes very inflexible, and almost unmovable if the distance between the sensors is large.

Either a master/slave or a common-view model can be applied. For the common-view model, a separate control station is needed, which contains the oscillator. For a master/slave model, one of the TDOA receivers hold the oscil-lator. One benefit, which applies both models, is that since sensors in a TDOA network only need to be synchronized relative to each other. This means that cheaper oscillators can be used, since the oscillator does not need to keep UTC for long periods.

2.3

Signal Model

This section describes the notation in the thesis. It also provides the assumptions made about the signals. Section 2.3.1 describes how time-shifts are applied, as well as an additional method used to express the time-delay of a signal. The complex baseband representation of signals is used.

The signal transmitted from the UAV is denoted s, and the signal received by the TDOA sensor is denoted x. The received signal x is both time-shifted and affected by noise. The noise is modeled as complex additive white Gaussian noise (AWGN), denoted w. The noise samples in w are independent and identically distributed according to w ∼ CN (0, σ2), where σ2is the noise variance.

In this thesis, the generated signals are time-discrete, and are also sampled at the reciever. However, the delay due to the propagation is continuous. If the transmitted signal is denoted s, the received version might be delayed by τ, which is continuous. The notation used to describe a continuous delay in a time-discrete signal is x(fn

s

τ) = xτ[n], where n = −N

2, −N2 + 1, . . . ,N2 −1 with N = fsT . Here, fsis the sampling frequency, and T is the duration of the signal in seconds. Thus,

the received time-shifted and noisy signal is denoted xτ = sτ+ w, where sτis a

time-delayed version of the original reference signal.

In the simulations, the number of samples to generate was calculated ac-cording to N = fsT . However, for the WGN signal and the Gold sequences, BT

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2.4 Free-space Propagation 11

upsampled to fs. This way of generating the WGN and Gold sequences is made

because the symbol rate is equal to the bandwidth for these signals.

2.3.1

Time-delay in Frequency Domain

Since many fast Fourier transform (FFT) algorithms are very computationally ef-ficient, it can be beneficial to do certain operations in the frequency domain [8, Ch. 2]. For a time-discrete signal x[n], −N2, −N2 + 1, . . . ,N21, a delay τ in the time domain equals to

F {x[n − τfs]} = X[k]e

j2πτ fsk

N, (2.2)

where fs is the sampling frequency andF { · } denotes the discrete Fourier

trans-form (DFT). The time-shift pertrans-formed in 2.2 is circular. That means that if the signal is shifted n samples to the right, the n samples farthest to the right will end up outside of the signal length. However, those are appended to the signal as the

n samples farthest to the left. Therefore, to avoid unwanted samples added to the

left of the signal (for a shift to the right), zeros have to be added at the beginning and end of the signal.

If the signal cannot be expressed as a function of time, derivation with re-spect to the time-delay might be difficult. To make the derivation easier, a method described in [20] can be used to multiply the signal with a delay factor. This method also performs the time-shift in the frequency domain, but is only ap-proximately equal to the time-shift performed in Equation 2.2. The benefit is that the time-delay τ is introduced as a factor multiplied with the signal x, mak-ing derivation with respect to the time-delay easy. The formula is, with n = h −N 2, −N2 + 1, . . . ,N2 −1, iT , ≈ FHDτF x (2.3) where F = √1 Nej NnnT, (2.4) and = diagej2πτ fsNn, (2.5)

where diag( · ) creates a diagonal matrix, with the argument vector as the diag-onal. In the expressions above, F is the unitary N × N DFT matrix, while Dτ

can be compared to the delay factor in (2.2). This method of expressing the time-delay makes derivation of the signal with respect to the time-time-delay much easier, especially for signals not easily expressed by mathematical formulas.

2.4

Free-space Propagation

There are many methods used to model the effects of a wireless channel on a signal. The energy the signal loses during propagation is called the path loss [1, Ch. 2]. The path loss used for the simulations in this thesis is free-space prop-agation. Free-space propagation path loss disregards any possible obstacles in

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12 2 Theoretical Background

the propagation path, and also neglects effects of multi-path propagation. It is therefore suitable to model LOS communications and estimate received signal strengths. Mathematically it is formulated as Lb= 4πfcd c !2 , (2.6)

where fcis the carrier frequency, and d is the distance of the propagation path.

The received signal power is

Sp =E{|x| 2} Lb = E{|x| 2} c 4πfcd !2 , (2.7)

where E{|x|2}denotes the expectation of |x|2.

2.5

Time-delay Estimation

This thesis focuses on estimation of one parameter, namely time-delay. When electromagnetic waves propagate through air, they propagate at the speed of light. Knowing the propagation speed means that if the position of both the trans-mitter and the receiver (or a least the distance between them) is known, the TOA can be calculated. Additionally, as always for TOA, the signal must be known. If the clocks of the transmitter and receiver are synchronized, the signal will be time-shifted according to the propagation time. However, if the clocks are out of sync, the time-shift will be either larger or smaller than what is expected. Usually what is meant by time-delay estimation, is the estimation of the total time-shift from generation of the signal to the arrival at the receiver. Knowing the total time-shift, any deviation from the expected can be detected. To reach very high accuracy, other sources of delay, such as the time from generation of the signal to the actual time of transmission, must be accounted for. The following sections describe the fundamental concepts of TDE, such as correlation, the Cramér-Rao bound for TDE, and maximum likelihood estimators.

2.5.1

Correlation

In signal processing applications, correlation means finding the similarity be-tween two signals. The higher the similarity, the higher the correlation. Math-ematically, this is described as the expectation of the product of two stochastic processes X(t) and Y (t) [14, Ch. 4]. For complex signals, the expression is

E{X(t1)Y

(t2)}, (2.8)

where ( · )∗ denotes the complex conjugate. This is called cross-correlation. Cor-relation between a signal with itself is called auto-corCor-relation. More accurately,

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2.5 Time-delay Estimation 13

auto-correlation means comparing two realizations of a single stochastic process at different time instances, and it is described by

E{X(t1)X

(t2)}, (2.9)

In this thesis, cross-correlation between the transmitted and received version of the signal will be performed.

Figure 2.3 visualizes why TDE can be performed using correlation. The fig-ure shows auto-correlation of 20 000 samples of white Gaussian noise with vari-ance σ2 = 1, but in the same time instance t1= t2. Maximum correlation occurs

when the two signals completely overlap, meaning that the first sample of one signal is correlated with the first sample of the other signal, etc. In Figure 2.3, maximum correlation happens at lag 0. Lag means how many samples one of the signals is time-shifted compared to the other. If instead, one of the WGN signals had been time-delayed, the peak would have been located at a lag corresponding to the number of samples equal to the delay.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Lag [samples] 104

Correlation

Figure 2.3: Example showing auto-correlation of 20 000 samples of white Gaussian noise, w ∼ N (0, 1). Since there is a distinct peak at lag 0, it can be concluded that none of the signals are time-delayed relative to the other. If one of the signals was time-delayed, the peak would have been at a lag corresponding to the delay measured in samples.

2.5.2

Cramér-Rao Bound for Time-delay Estimation

Because of the additive white Gaussin noise w, the received signal can be seen as a realization of a random process. To map this realization to an estimated value for the time-delay, an estimator is used. An estimator gives a value for

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14 2 Theoretical Background

the delay for each realization of the received signal [10, Ch. 1]. If, on average, an estimator yields the true value for the estimated parameter, it is said to be unbiased [10, Ch. 3]. If the true value of the estimated parameter ˆθ is θ, then the

chosen estimator is unbiased if

E{ ˆθ} = θ. (2.10)

In estimation theory, a theoretical lower bound on the variance of an unbiased estimator is called the Cramér-Rao bound, or CRB for short. It is one of the most interesting and useful metrics when performing estimation. If, for a certain set of parameter values, the CRB for the TDE is var( ˆθ), the root mean squared error

(RMSE) on the estimate will not be lower than q

var( ˆθ). The CRB is very useful

when different estimators are compared, but can also be used as a benchmark to evaluate how close a chosen estimator is to the theoretical limit of its variance. The goal is to find the minimum variance unbiased estimator (MVU) and see if it satisfies the CRB. If it does, the MVU estimator is said to be efficient, meaning the estimator is unbiased and attains the CRB for all possible values of the estimation parameter. For an estimator to be efficient, it is required that the signal length tends to an infinite number of samples, N → ∞. It is also required that the noise variance σ2→0.

One way to describe the CRB is as the inverse of the Fisher information (FI) I(θ). If θ is the real-valued parameter that should be estimated, then

var( ˆθ) ≥ 1

I(θ). (2.11)

The CRB depends on the probability density function (PDF) of the samples of the signal. For samples distributed according to a complex Gaussian PDF, as in this thesis because of the complex AWGN, the expression for the FI is

I(θ) = tr " Cx−1(θ)∂Cx ∂θ C1 x (θ) ∂Cx ∂θ # + 2Re" ∂µ H(θ) ∂θ C1 x (θ) ∂µ(θ) ∂θ # (2.12) where Cxis the covariance matrix of the data and µ(θ) is the mean of the received

signal xτ. The operator tr[ · ] is the trace operator, ( · )His the Hermitian operator,

and Re[ · ] outputs the real part of the argument. In this thesis, the time-delays of received signals are estimated. Therefore, θ = τ.

2.5.3

Maximum Likelihood Estimator

Estimators that attain the theoretically lowest possible variance of the estimation parameter according to the CRB are called minimum variance unbiased (MVU) estimators [10, Ch. 2]. However, MVU estimators can be very hard to derive, or in some cases, do not exist. Preferably, there would then exist an estimator which is almost as accurate, but more practical to find. Often, that estimator is the maximum likelihood estimator (MLE). The principle behind the MLE is to find for which value of the parameter θ the PDF for xτ is maximized. The MLE is

asymptotically efficient for large data sets, and is in some cases easier find for complicated estimation problems [10, Ch. 7].

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2.6 Reference Signals 15

Let the original reference signal be denoted s[n], n = 0, 1, . . . , N − 1, and let the received time-shifted version be denoted x[n − n0]. The delay n0is also

time-discrete. Then, the MLE for time-delay estimation is ˆ n0= argmax n0 n0+N −1 X n=n0 s[n]x[n − n0], (2.13)

where ˆn0 is the estimated values for the true delay n0 [10, Ch. 7]. The sum in

(2.13) does describe cross-correlation between s and x. This means that the MLE is equal to finding the delay for which the correlation between the reference sig-nal with the received sigsig-nal is maximized.

2.5.4

Sub-sample Accuracy

There is a limit to how accurately it is possible to estimate the delay for a signal sampled with a certain sampling frequency fs, if the MLE is used. The sampled

signal is time-discrete, while the delay is continuous. No matter how large fs is,

the delay might not be a multiple of the sampling period T . It is necessary to be able to find values between samples to find an accurate estimate of the delay.

Interpolation is a well-known method used for the problem of finding new data points in the interval between already known data points. Following is a description of an interpolation method that can be used to achieve sub-sample accuracy for estimation. The idea is to choose the sample for which the correla-tion is maximized, ni, as well as the two adjacent samples, ni−1 and ni+1. The

three samples are then used to calculate the coefficients of the quadratic polyno-mial that fits the samples best in a least-squares sense [16, Ch. 16].

Assume that the polynomial that is to be fitted to the samples is f (t) = at2+

bt + c. Then the maximum of that polynomial (assuming that the middle sample ni is larger than the other two, and thus a < 0) will be at t = −2ab. Since f (t) is

known for three values of t = t1, t2 and t3, it is possible to calculate a, b and c

using the matrix equation          t12 t1 1 t22 t2 1 t32 t3 1                  a b c         =         f (t1) f (t2) f (t3)         . (2.14)

This is a system of linear equations with three unknown variables a, b and c, so only one solution exists. The resulting values for the coefficients a and b are then used to calculate t = −2ab.

2.6

Reference Signals

In this thesis, different reference signals used for synchronization are compared. This section will describe those signals, as well as signal properties which affect the estimation accuracy. The choice of what waveforms to compare is made based on the knowledge presented here.

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16 2 Theoretical Background

The signals in this section are baseband signals. A signal which is not modu-lated up to a carrier frequency fc, and therefore has its energy contents centered

around f = 0, is called a baseband signal [11, Ch. 2]. A signal centered around a carrier frequency is instead called narrow-band, or radio frequency (RF). The effect of different modulation techniques are not accounted for in this thesis.

2.6.1

Frequency Band

The frequency band which is suitable for a signal to be transmitted over is deter-mined by the application. Certain frequency bands are suitable for long-range communications, while other frequency bands are suitable to achieve high data rates. The path loss (2.6) used to model free-space propagation depends on the carrier frequency fc. In the simulations, fc = 300 MHz was used. This frequency

belongs to the very high frequency band (30 - 300 MHz) or the ultra high fre-quency band (300 - 3000 MHz). These frefre-quency bands are usually used for transmission with directional antennas and fixed point-to-point communications over large distances [1, Ch. 1].

2.6.2

Signal-to-noise Ratio

As the name suggests, the signal-to-noise ratio (SNR) is the ratio between the total received signal power and the noise power [1, Ch. 3]. In this thesis, it is formulated as

SNR = Pr

Np =

E{|s|2}

σ2 (2.15)

where, Pris the total signal power, and Npis the noise power at the receiver. It is

often expressed in dB, for which an SNR > 0 dB means that the signal energy is greater than the noise energy for the signal. It is desirable to maximize the SNR. If the SNR is small, noise will introduce a lot of disturbance to the signal. The maximum achievable SNR depends only on the signal energy, so the SNR has no theoretical upper limit.

2.6.3

Ambiguity Function

The width of the peak at lag 0 in Figure 2.3 is very small, only a few samples wide, and the amplitude of the peak is large compared to the rest of the values. If the auto-correlation of a signal has those properties, the signal is suitable to be used for time-delay estimation. But not all signals have an auto-correlation curve with a narrow peak and a large amplitude for the peak sample. Rather, it is only for certain types of signals that the peak becomes distinct when performing correlation. Since this property is desirable when estimating time-delay, the auto-correlation curve is used as a performance metric, and is called the ambiguity function. In literature, the ambiguity function sometimes is formulated so that it also measures the Doppler resolution [17], but that version of the ambiguity function is omitted from this thesis, as only time-delay resolution is of interest.

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2.6 Reference Signals 17

It is a well-known fact that a waveform that has an ambiguity function with a single, narrow peak is well-suited for time-delay estimation [11, Ch. 4]. In the time-continuous case, the ambiguity function is defined as

rx(τ) =

Z

−∞

x(t − τ)x(t)dt. (2.16)

This expression is analogous to the auto-correlation of a function, which for some signals produces a peak with a small width and a large amplitude for signals suitable for TDE. The narrower the peak, and the higher the amplitude, the better the signal is suited for TDE. Therefore, the ambiguity function is an intuitive way of evaluating if a certain signal is appropriate to use for TDE.

2.6.4

General Expression for the Cramér-Rao Bound for

Time-delay Estimation

For some signals, deriving analytical closed-form expressions for the CRB for TDE can be difficult. For random signals such as white Gaussian noise (WGN), it’s impossible, since the signal cannot explicitly be expressed analytically. A general expression for the CRB for TDE for the signals examined in this thesis is however possible, given that the expression depends on the transmitted signal s. Using Equation (2.12) and the approximation in (2.3), the result is

var(τ) ≥ N

2σ2

2f2

s ResHFHGHGF s

, (2.17)

where G = diag(n) with n =h−N

2, −N2 + 1, . . . ,N2 −1,

iT

, and var(τ) is the CRB of the estimated delay τ. The approximation in (2.3) is used so that τ is canceled out from the expression. Full calculations can be found in Appendix A.

It is hard to see how (2.17) depends on the bandwidth B and signal length T . Therefore, one more approximation is made. The full calculations for this step can also be found in Appendix A. The approximation is

h

sHFHGHGF si≈ B

3N3P

12fs3

(2.18) where P = |Sk|2 for |k| ≤ 2fBsN , S being the discrete Fourier transform (DFT) of

the signal. Combining (2.17) and (2.18), the result is var(τ) ≥

2

2B3T π2P. (2.19)

This expression conveniently describes the relation between bandwidth, signal length and the CRB.

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18 2 Theoretical Background

2.6.5

Pulse

Mathematically, a pulse is expressed as

x(t) =        A, 0 ≤ t < T 0, otherwise (2.20)

where A is the amplitude of the pulse. As described in Section 2.6.3, it is prefer-able to have an ambiguity function which gives a peak with a small width and large amplitude when performing time-delay estimation. As showed in Figure 2.4, the ambiguity function of a pulse has the shape of a triangle. The shorter the pulse, the shorter the base of the triangle will be.

-250 -200 -150 -100 -50 0 50 100 150 200 250

Lag [samples]

Correlation

Figure 2.4: Ambiguity function of a pulse with A = 1, T = 100 µs and fs= 2

MHz.

The CRB for TDE for an ideal pulse exists, but is zero, which is a result that give no meaningful information [17, Ch. 7]. The CRB for TDE is zero because an ideal rectangular pulse has an infinite effective bandwidth, which makes (2.19) tend to zero. This means that to be able to calculate an expression for the CRB for TDE with a pulse, the pulse has to be filtered with a low-pass filter. Low-pass filtering will spread the pulse in the time-domain, which will decrease the bandwidth. Figure 2.5 shows the spectrum for an ideal pulse of length T = 100 µs and sampling frequency fs = 2 MHz.

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2.6 Reference Signals 19 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Frequency [MHz] 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 Power [W]

Figure 2.5:Spectrum of a baseband pulse signal with A = 1, T = 100 µs and fs= 2 MHz.

2.6.6

Chirp

A chirp waveform is a pulse whose frequency increase linearly with time. It is given by

x(t) = ej2π(f0t+2t2). (2.21)

In this thesis, the chirp sweeps over a frequency interval as large as the signal bandwidth. With t = fn

s, n = −

N

2, −N2 + 1, . . . ,N2 −1, assume that the instantaneous

frequency of the sweep f0+2t is set to 0 at t = 0. The reason why  is divided by

2 in (2.21) is because then the term 2 denotes the rate of frequency increase per unit time. At t = 2fN

s

− 1

fs the instantaneous frequency is set to B. Consequently,

the rate of frequency increase per unit time is set to2 =2TB .

In Figure 2.6, the ambiguity function for a chirp signal is presented. The width of the ambiguity function for the chirp signal is very small. This can be understood intuitively, by realizing that since the frequency constantly differs, the correlated signals will not have matching frequencies, except for lag 0.

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20 2 Theoretical Background

-250 -200 -150 -100 -50 0 50 100 150 200 250

Lag [samples]

Correlation

Figure 2.6: Ambiguity function of a chirp with B = 1 MHz, T = 100 µs and fs= 2B.

Figure 2.7 shows the spectrum of a baseband chirp signal with bandwidth B = 1 MHz. It is clear that the power is approximately the same for the whole frequency range that the chirp signal sweeps over.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Frequency [MHz] 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 Power [W]

Figure 2.7:Spectrum of a baseband chirp signal with bandwidth B = 1 MHz, T = 100 µs and fs= 2B.

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2.6 Reference Signals 21

2.6.7

White Gaussian Noise

Signals that are random, or appear random, are usually called noise, or noise-like waveforms. Noise-like waveforms has good correlation properties [11, Ch. 4]. The effect of the randomness on the ambiguity function is that the signal does not contain repeating intervals, which means that the correlation will only be large at lag 0. The result is an ambiguity function with a peak which is very narrow, which can be seen in Figure 2.8, which shows the ambiguity function for 200 samples of WGN. The complex WGN signal is generated from a complex Gaussian distribution p(x) = 1 πσ2e −1 σ 2|x| 2 . (2.22) -250 -200 -150 -100 -50 0 50 100 150 200 250 Lag [samples] Correlation

Figure 2.8:Ambiguity function of complex WGN with B = 1 MHz, T = 100 µs, fs = 2B and variance σ2= 1.

True WGN can not be generated in reality, because the large and rapid changes in amplitude would require unreasonably precise and powerful amplifiers. It has only a practical meaning if it has been filtered [11, Ch. 3]. The randomness of WGN can be seen in Figure 2.9, where the spectrum for 200 samples of complex WGN is plotted. The spectrum do not follow any pattern, and the frequency contents are random.

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22 2 Theoretical Background -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Frequency [MHz] 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 Power [W]

Figure 2.9:Spectrum of one realization of complex WGN with bandwidth B = 1 MHz, T = 100 µs, fs= 2B and variance σ2= 1.

2.6.8

Gold Sequences

The correlation properties of random noise signals are favorable, but WGN sig-nals are impossible to generate in practice. Therefore, pseudo-random sequences are easier to use. One set of pseudo-random sequences is Gold sequences [6]. Gold sequences are designed to resemble noise, but contains only binary values. However, since Gold sequences are only pseudo-random, they can be generated in practice. Every GPS satellite has its own Gold sequence for its C/A code, which makes it possible for the GPS receivers on Earth to uniquely identify each satellite [3]. Gold sequences are generated by using maximal linear-feedback state regis-ters. The maximal linear-feedback state register generates two binary pseudo-random number sequences. Those sequences are then used as input to an XOR gate, and the output is a Gold code. An in-depth description of generation of Gold codes can be found in [6].

The Gold sequences are tested for different bandwidths and signal lengths in this thesis. But Gold sequences have lengths described by 2n1, where n is a positive integer (2n1 = 255, 511, etc.). This introduces a dilemma: should a shorter Gold sequence be generated, and then be extended with a part of the same sequence to achieve the desired length, or should a part of a longer sequence be used instead? The solution used in this thesis is the latter; for a time-bandwidth product BT , n was calculated as n = blog2(BT )c + 1, so that a larger sequence than needed always was generated.

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2.6 Reference Signals 23

Figure 2.10 shows the ambiguity function for a Gold sequence of length 200 samples. The behavior is similar to that of the WGN signal in Figure 2.8.

-250 -200 -150 -100 -50 0 50 100 150 200 250

Lag [samples]

Correlation

Figure 2.10: Ambiguity function of a Gold sequence with B = 1 MHz, T = 100 µs and fs= 2B. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Frequency [MHz] 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 Power [W]

Figure 2.11:Spectrum of a Gold sequence with bandwidth B = 1 MHz, T = 100 µs and fs= 2B.

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24 2 Theoretical Background

Figure 2.11 shows the spectrum for a Gold sequence of length 200 samples. The spectrum is symmetrical around f = 0, since the signal is real-valued (binary). Except for the symmetry, the spectrum is random.

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3

Method

In this chapter the method used to answer the problem statements in Section 1.2 is described. Section 3.1 provides the assumed setup for the UAV and the TDOA receivers, as well as the specifications set by the Swedish defense research agency (FOI). The analytical evaluation of the signals is described in Section 3.2. Section 3.3, explains the Matlab simulations, and Section 3.4 contains a description of the laboratory tests.

3.1

Setup

Figure 3.1 presents the assumed setup of the TDOA receivers and the UAV. In this thesis, only two TDOA receivers are considered. However, for TDOA to work, three receivers are required.

 max 20 km

TDOA receiver TDOA receiver

UAV

Reference signal Reference signal

Figure 3.1: Schematic drawing of the setup of the TDOA receivers and the UAV used in this thesis.

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26 3 Method

The main focus is the fundamental theory of TDE, so implementation and scala-bility were not accounted for. The maximal distance between the TDOA receivers were set to 20 km, with the UAV in the intermediate area. However, the effect of different distances between the sensors and the UAV was also tested.

3.1.1

Parameter Restrictions

The restrictions set on the parameter values are listed below. The restrictions were set based on how current communications intelligence (COMINT) systems work. For the tested parameters, the limits were:

• maximum signal bandwidth: 10 MHz, • maximum signal length: 1 ms,

• maximum output power: 5 W (≈ 37 dBm).

3.2

Analytical Evaluation

The four chosen signals were analyzed theoretically by deriving the Cramér-Rao bound (CRB) for the time-delay estimation. Based on the CRBs, predictions about the TDE accuracy could be made. The signal which would yield the lowest CRB is in theory the most suitable for estimating time-delays. Matlab was used to compute the expression in (2.17). The TDE performance was evaluated from the CRB for the TDE, and the ambiguity function.

From (2.19), it can be found that the CRB decreases with a factor B13 and T1. Since the CRB is a measure of variance, the actual error (the standard deviation) would therefore decrease with a factor √1

B3 and

1 √

T. This fact was used to verify

the simulations. If the bandwidth is doubled, for example, from 1 MHz to 2 MHz, the error is multiplied with a factor√1

23 ≈0.35. Using the same method, the error is multiplied with a factor √1

2 ≈0.71 for each doubling of the signal length. As

mentioned in Section 2.6.5, the CRB for TDE for a rectangular pulse does not give any meaningful information, so the CRB for TDE for the pulse was not derived.

3.3

Simulations

To verify the TDE performance of the reference signals, simulations were per-formed in Matlab. Four signal parameters were varied: the bandwidth, the sam-pling frequency, the signal length, and the SNR. The parameters were varied ac-cording to the restrictions in Section 3.1.1. Simulations were also performed for different UAV positions.

In the simulations, the sampling frequency is a multiple of the bandwidth. Two levels of oversampling were tested, most simulations were performed with

fs = 2B, and a few were performed with fs = 4B. The speed of light was set to

(45)

3.3 Simulations 27

First, the reference signals were generated for every set of parameter values. For WGN, 1000 realizations were generated for each parameter configuration. After the signals were generated, fsT

10 zeros were added to the beginning and end

of every signal, as the signals could not be assumed to be strictly time-limited. The next step was pulse-shaping, where the signals were filtered with a raised-cosine filter. All the simulations were made over two channels, so the next step was to simulate transmission over the channels. First, the signals were delayed in the frequency domain, as explained in Section 2.3.1. The size of the delay corresponded to the distance between the UAV and the receivers. Then the output signal power Pt was normalized to Pt = 1 W. The next step was to divide the

signal with the path loss√Lb, as described in Section 2.4. The carrier signal

was set to fc = 300 MHz, as it is a reasonable carrier frequency for signals with

bandwidths in the simulated interval. The path loss also depends on the distance between transmitter and receiver, d. To model different positions of the UAV, the simulated distances between the sensors and the UAV were varied. Finally, WGN was added, with a variance set to achieve the specified SNR. If the UAV was not placed with equal distance to the receivers, the noise variance was calculated based on the received signal power for the corresponding distance, with the SNR fixed for both channels.

3.3.1

Time-delay Estimation in the Simulations

The estimation of the time-delay in the simulations was performed as described in Section 2.5. The received signal was correlated with the original signal. To get sub-sample accuracy, interpolation with a quadratic polynomial was made (see Section 2.5.4). The error was calculated as the deviation from the expected delay with regards to the distance between UAV and receiver. For example, if the distance between the receiver and the UAV was 10 km, the expected delay was

τ = 10000c33 µs. If then the estimated delay was, e.g., 33.4 µs, the error was 0.4

µs. If the estimated delay was, e.g., 32.6 µs, the error was still considered to be 0.4 µs, i.e., no consideration was taken to if the delay was positive or negative.

For WGN, the deviation from the expected delay was calculated for each realization, and the final error was set to the mean error for the 1000 delays.

3.3.2

Generation of Gold Sequences

In Table 3.1, the maximum length polynomials used to generate the Gold se-quences are listed. The generation was performed using code provided by FOI. Due to restrictions in the generation code, sequences of lengths where n is divisi-ble with 4 were not generated. Instead, a longer code of length 2nwas generated and truncated. Example 3.1 explains how to read Table 3.1.

References

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