Building and Evaluating a 3D Scanning System for Measurementsand Estimation of Antennas and Propagation Channels

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Master Thesis Building and Evaluating a 3D Scanning System for Measurements and Estimation of Antennas and Propagation Channels

Erik Johannes Aagaard Fransson

Tobias Wall-Horgen

Stockholm, Sweden 2012

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Building and Evaluating a 3D Scanning System

for Measurements and Estimation of Antennas

and Propagation Channels

ERIK AAGAARD FRANSSON AND TOBIAS WALL-HORGEN

Master’s Thesis in Electrical Engineering Supervisors: Andr´es Alayon Glazunov and Peter Fuks

Examiner: Martin Norgren

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Abstract

Wireless communications rely, among other things, on the understanding of the properties of the radio propagation channel, the antennas and their interplay. Adequate mea-surements are required to verify theoretical models and to gain knowledge of the channel behavior and antenna perfor-mance. As a result of this master thesis we built a 3D field scanner measurement system to predict multipath propa-gation and to measure antenna characteristics. The 3D scanner allows measuring a signal at the point of interest along a line, on a surface or within a volume in space. In or-der to evaluate the system, we have performed narrowband channel sounding measurements of the spatial distribution of waves impinging at an imaginary spherical sector. Data was used to estimate the Angle-of-Arrivals (AoA) and am-plitude of the waves. An estimation method is presented to solve the resulting inverse problem by means of regulariza-tion with truncated singular value decomposiregulariza-tion. The reg-ularized solution was then further improved with the help of a successive interference cancellation algorithm. Before applying the method to measurement data, it was tested on synthetic data to evaluate its performance as a function of the noise level and the number of impinging waves. In order to minimize estimation errors it was also required to find the phase center of the horn antenna used in the chan-nel measurements. The task was accomplished by direct measurements and by the regularization method, both re-sults being in good agreement.

Key words: 3D field scanner, radio propagation

chan-nel, virtual array, MIMO, Angle-of-Arrival, regularization, inverse problem, L-curve, truncated singular value decom-positions, successive interference cancellation, horn antenna, phase center.

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Acknowledgments

We would like to acknowledge a few people that have made the completion of this thesis possible. First and foremost we would like to express our gratitude to Andr´es Alayon Glazunov who has guided and helped us a lot along the way with the theoretical parts of the thesis. Without his help this thesis would never have been made. We would also like to give a special thanks to Peter Fuks. Without his help on some practical points and logistics the measurements for this thesis would never have happened. We would also like to thank Julia Eriksson, even if we never met her, her MATLAB programs for Taxen helped us a long the way. We are grateful for Jim C from Robotforum.com for all his help and input in the repairs of the RX90L robot, without his help the robot would not be moving today.

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B.7.1 move.m . . . 94 B.7.2 move.v2 . . . 94 B.7.3 check.m . . . 94 B.7.4 check.v2 . . . 95 B.7.5 drive.m . . . 95 B.7.6 drive.v2 . . . 96 B.7.7 tinysphere.m . . . 96 B.7.8 ContTinysphere.m . . . 97 B.7.9 position.v2 . . . 97 B.7.10 init VNA.m . . . 97 B.7.11 init serial.m . . . 98 B.7.12 get SDATA.m . . . 98 B.7.13 tooltrans.m . . . 98 B.7.14 mollyplot.m . . . 98 B.8 Troubleshooting . . . 99

B.8.1 After Startup MOLLY Could Not Calibrate Properly . . . . 99

B.8.2 MOLLY Displays a Motor Error . . . 99

B.8.3 When in READY position the joint markings do not align . . 99

B.8.4 I moved MOLLY with the Teach Pendant but now the pro-grams won’t make her move . . . 100

C Some pictures 101

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List of Figures

2.1 Illustration of a multi path. TX is the transmitting antenna and RX is

the receiving antenna. . . 4

2.2 Circular array definitions [7, p.365] . . . 6

2.3 Incident plane waves on an arbitrary field pattern rotating around φ . . . 8

2.4 The normalized singular values of B showing no clear gap. . . 11

2.5 Plot of the Fourier Coefficients and singular values illustrating an unful-filled condition, a, and plot of it’s corresponding regularized solution,b. Taken from a simulation using 360 singular values and an additive white Gaussian noise corresponding to 10dB SNR. . . 12

2.6 L-curve sketch . . . 13

2.7 Base functions from the SVD. . . 14

2.8 Comparison between solutions with normalization and without. Dashed lines are solutions without normalization. Different lines represent dif-ferent SNR levels . . . 16

2.9 Example of the Successive Interference Cancellation. ’X’ represent pk, ’O’ represents ˆpk,peaks,i . . . 17

3.1 Example of an estimation of a realization. m = 360. 20dB SNR. . . 21

3.2 Mean errors plotted against SNR for one scatterer, dashed lines represent the naive solutions. . . 22

3.3 L-curves for 1 scatterer case. Comparison of m = 90 and m =720. k for SNR 20 dB . . . 23

3.4 Mean power error for m = 90. . . 23

3.5 Example of an estimation of a realization using 2 scatterers. m = 360. 20dB SNR. . . 24

3.6 Mean errors plotted against SNR for the larger scatterer. . . 25

3.7 Mean errors plotted against SNR for the smaller scatterer. . . 27

3.8 Mean absolute angular errors plotted against SNR. . . 28

3.9 Different errors v.s. SNR for 5 and 10 scatterers. . . 30

3.10 Angular errors plotted against SNR for 5 and 10 scatterers. . . 31

3.11 Example of estimations of a realization using 10 and 5 scatterers. m = 360. . . 32

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4.2 Joint, length and angle definitions on MOLLY . . . 37 4.3 Visual representation of the spherical sector. [mm]. . . 38 4.4 Visual representation of the largest perimeter error on a circle. . . 39 4.5 Taxen an one-dimensional stepper with the dipole mounted on top of it. . . . 40 4.6 The VNA. . . 41 4.7 The two different antennas . . . 44 4.8 Horn dimensions . . . 45 4.9 Full simulated normalized antenna pattern in dB as seen from bore sight. φ

represents the H-plane and θ the E-plane. . . . 45 4.10 Normalized relative power pattern of the horn in H-plane. . . 46 5.1 Dynamic range. ∆dB represents the difference between the two highest peaks. 51 5.2 Phase center measure 6mm steps. . . 51 5.3 Phase center measure 2mm steps. . . 52 5.4 Graph showing Friis transmission formula compared to measured data

with and without phase center compensation. . . 53 6.1 Sketch over the room for setup 1. . . 56 6.2 Sketch over the room for setup 2. . . 57 6.3 Different plots for setup 1 TX position 1. Plot c is obtained by

integrat-ing over all elevations with SNR levels above 5dB. The SNR levels can be viewed in figure 6.9 . . . 58 6.4 The two elevations with highest, a, and lowest, b, SNR from setup 1 TX

position 1. . . 59 6.5 Different plots for setup 1 TX position 10. Plot c is obtained by

inte-grating over all elevations with SNR levels above 5dB. The SNR levels can be viewed in figure 6.9 . . . 60 6.6 The two elevations with highest, a, and lowest, b, SNR from setup 1 TX

position 10. . . 61 6.7 Different plots for setup 1 TX position 20. Plot c is obtained by

inte-grating over all elevations with SNR levels above 5dB. The SNR levels can be viewed in figure 6.9 . . . 62 6.8 The two elevations with highest, a, and lowest, b, SNR from setup 1 TX

position 20. . . 63 6.9 SNR levels at different elevations for setup 1. . . 63 6.10 Measured and estimated plots for setup 1 TX position 10 using 90

mea-surement points in φ and every fourth degree in θ. . . 64 6.11 Different plots for setup 2 TX position 1. Plot c is obtained by

integrat-ing over all elevations with SNR levels above 30dB. The SNR levels can be viewed in figure 6.17 . . . 65 6.12 The two elevations with highest, a, and lowest, b, SNR from setup 2 TX

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6.13 Different plots for setup 2 TX position 10. Plot c is obtained by inte-grating over all elevations with SNR levels above 30dB. The SNR levels

can be viewed in figure 6.17 . . . 67

6.14 The two elevations with highest, a, and lowest, b, SNR from setup 2 TX position 10. . . 68

6.15 Different plots for setup 2 TX position 20. Plot c is obtained by inte-grating over all elevations with SNR levels above 30dB. The SNR levels can be viewed in figure 6.17 . . . 69

6.16 The two elevations with highest, a, and lowest, b, SNR from setup 2 TX position 20. . . 70

6.17 SNR levels at different elevations for setup 2. . . 70

6.18 Measured and estimated plots for setup 2 TX position 10 using 90 mea-surement points in φ and every fourth degree in θ. . . 71

A.1 Sketch of the safety board. . . 77

A.2 The safety chain. . . 79

A.3 Picture of the two connectors J11 and J12 with the jumpers. . . 80

B.1 Joint definitions . . . 88

B.2 Chart over the custom interface . . . 93

C.1 Ladder hanging in the roof. . . 101

C.2 Metal bench 1.5m from RX. . . 102

List of Tables

3.1 Condition values for different m. . . 22

3.2 εφ in degrees for different SNR and m. . . 23

3.3 Period time of oscillation taken from figure 3.6 and compared to ones calculated using (2.34) . . . 26

3.4 Portion of scatterers possible to estimate taken from figure 3.10. Ns = 10/Ns= 5 . . . 29

4.1 Table over the different lengths and angles on MOLLY as presented in [1] . . . 38

4.2 Table over the angular resolutions for Joint 5 & 6 . . . 39

4.3 Table over the different times for different parameters, MS = Monitor Speed, PS = Program Speed, ∆α = Azimuthal Steps in Degree, MTPP = Measurement Time Per Point. The times are averaged over, 360 for ∆α = 1 and 720 for ∆α = 0.5, measurements with the stated settings. . . 42

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4.4 Table over the different times measurement times, PT = Points Total, PE = Points Elevation, PA = Points Azimuth, TPS = Time Per Sphere . . . 43 4.5 Table over the different relative power levels from figure 4.10 . . . 47

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Chapter 1

Introduction

The amount of data we are sending over wireless communication networks is grow-ing larger in size. This increase in size leads to problems with sendgrow-ing information over the available spectrum. To tackle this problem the usage of one single antenna is slowly transitioning over to the usage of so called Multiple-Input Multiple-Output (MIMO) systems that exploit the properties of the channel to boost data rates. This means that the data can be divided into several parallel streams and transmitted. The different algorithms used on MIMO-systems depends on the properties of the radio propagation channel. These channel properties are complicated to predict in a real world environment since they are easily disturbed by objects and move-ments. Another important factor to consider are the properties of the receiving and transmitting antennas.

Propagation models can be described through parametric estimation from mea-surements. Field scanner measurements together with an estimation method can be used to get a understanding of the propagation channel.

1.1 Problem statement

The objective of this master thesis is to build a 3D field scanner measurement sys-tem to predict multipath propagation.

The goals of this thesis are:

• To estimate the complex incident electrical field and Angle-of-Arrival (AoA) as an evaluation for a radio channel propagation measurement system. • To simulate the estimation model over a set of known conditions to see what

to expect from the model in a measurement situation.

• To build a measurement system for radio channel propagation channel mea-surements.

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CHAPTER 1. INTRODUCTION

1.2 Thesis Outline

This thesis is outlined as follows.

Chapter 2: The theoretical background used in the later chapters is presented.

The propagation model and a mathematical formulation of the problem is also in-troduced .

Chapter 3: Presents simulations of the estimation model with the aim to see

how the model behaves.

Chapter 4: A description of the measurement system is provided.

Chapter 5: The phase center estimation problem is presented and discussed. Chapter 6: A discussion about the measurement and channel propagation

re-sults is presented.

Chapter 7: The final conclusions of the thesis are provided as well as some ideas

for future work.

This thesis has been written by two authors. We would like to point out that both have been a part of everything in this thesis however we have divided the responsibility for writing each chapter as follows:

• Chapter 1: Written by Erik and Tobias.

• Chapter 2: Erik has written sections 2.1 and 2.6-2.13. Tobias has written the sections 2.2-2.5.

• Chapter 3: Written by Erik. • Chapter 4: Written by Tobias. • Chapter 5: Written by Tobias.

• Chapter 6: Erik has written sections 6.2-6.3. Tobias has written the sections 6.1 and 6.3.

• Chapter 7: Written by Erik and Tobias. • Appendix A: Written by Erik and Tobias. • Appendix B: Written by Erik and Tobias.

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Chapter 2

The Radio Propagation Channel

Using a vector network analyzer (VNA) we can measure the forward voltage gain scattering parameter (S21) from all directions in the room of interest. We can do this by having an antenna rotating mechanically around an axial point. For example, by measuring the S21-parameter on the surface of a sphere. If the antenna radiation pattern is known the room’s spatial properties can be extracted from the measurement. This approach will give us a direct estimation of Angle-of-Arrival (AoA). Hence, when we point the antenna in the direction of an incoming wave the estimation from that direction directly gives us the scattered wave.

There are many ways to approach the problem. For example measuring the S-parameters on some other surface or volume than a sphere. We have also decided to limit the test to estimate the AoA and complex amplitude property. The main motivation for this is time and also we feel that this will provide enough data to discuss within the scope of this master thesis.

2.1 Propagation Channel Model

When an electromagnetic signal is sent out from a transmitting antenna it enters the propagation channel. The signal during it’s path undergoes different propagation mechanisms, such as scattering, reflection and diffraction. Hence, the received signal will arrive from different paths. By superposition, the received signal will be a sum of all versions of the transmitted signal impinging from different directions. The receiving angle is named the AoA. The angle of the transmitted signal is called Angle-of-Departure, (AoD). Assuming that the waves are scattered from objects in the far-field region of the receiving antenna, the waves can be approximated as plane waves. The incident field thus can be written as a sum of plane waves incoming from different directions

Einc(θ, φ) = Nwaves

X

n=1

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CHAPTER 2. THE RADIO PROPAGATION CHANNEL

where φ is the spherical azimuth angle, θ the spherical elevation angle, En is the

complex amplitude and kn· rn builds up a constant phase front that propagates in

the direction of kn.

Since the waves have taken different paths they will imping at the receiver with different amplitudes, phases, polarizations and time delays. This leads to a constructive and destructive interference called multi path fading.

Figure 2.1: Illustration of a multi path. TX is the transmitting antenna and RX is the receiving antenna.

2.2 Far-Field Region

In the far-field region of an antenna, also called the Fraunhofer region, the electro-magnetic field is built up by plane waves. The radiation pattern does not change it’s shape as a function of the distance. There are three criteria[7, p.167] that are fulfilled in the far-field:

R >> D (2.2) R >> λ (2.3) R >(2D2)/λ, (2.4)

where D is the largest dimension of the antennas and λ is the wave length.

2.3 Basic Antenna Properties

2.3.1 Radiation Pattern

The antenna radiation pattern is described as the variation of the radiation intensity of an antenna in the far-field region. The radiation pattern can be presented in terms of either the power or the field pattern[7, p.28]. Throughout this thesis the radiation

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2.4. FREE SPACE PATH LOSS

pattern used is the complex field pattern in far-field, F(θ, φ) [v/m]. The radiation pattern can be written as the angular components of the far-field region electrical field intensity of the antenna as

F(θ, φ) = Fφ(θ, φ) + Fθ(θ, φ), (2.5)

where Fφ(θ, φ) is the azimuthal components and Fθ(θ, φ) the elevation components.

2.3.2 Gain

The gain of an antenna is defined as ”the ratio of the intensity, in a given direction, to the intensity that would be obtained if the power accepted by the antenna were radiated isotropically. The radiation intensity corresponding to the isotropically radiated power is equal to the power accepted (input) by the antenna divided by 4π.”[23] Which can be expressed as

G(θ, φ) = 4π| F(θ, φ) |

2

ηPin

, (2.6)

where η is the intrinsic impedance of the medium and Pin is total input power. We

can also define the bore sight gain as

G0= G(θ, φ) |max (2.7)

2.3.3 Half-Power Beam Width

The half-power beam width is defined as ”In a plane containing the direction of the maximum of a beam, the angle between the two directions in which the radiation intensity is one-half value of the maximum of the beam.”[23]

2.4 Free Space Path Loss

In free-space, where no obstacles disturbs the signal, between a receiving- and a transmitting antenna the signal power at the receiving antenna is given by Friis transmission formula[7, p.95]. Pr= Pt _λ 4πR 2 G0tG0r, (2.8)

where Pr is the received signal power, Ptthe transmitted signal power, λ the

wave-length, R the distance between the antennas and G0t, G0r the gain of the

transmit-ting respectively the receiving antenna. In decibel scale, dB, this formula becomes:

Pr,dB = Pt,dB+ G0t,dB+ G0r,dB+ 20log10 _λ 4πR , (2.9) where 20log10 λ 4πR

is commonly called the free space path loss.

For this formula to be valid the antennas must be placed in the far-field region of each other.

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2.5 Circular Array

As mentioned earlier we will be using a mechanically rotating antenna as a receiver to measure the radio propagation channel. In a static environment this can be viewed as a circular array if the antenna is kept on a fixed radius, see figure 2.2.

If the rotational spacing is equal between the antenna placements we can view the rotational antenna as N identical elements with equal angular spacing. If they are all placed on the radius a and the distance from the rotational center to the observation point noted r the phase difference can be written in the same way as with a circular array [7, p.365-356].

If r >> a Rncan be written as

Rn= r − a sin θ cos (φ − φn), (2.10)

which would mean that, assuming that for small variations in amplitude Rn' r

(2.5) can now be rewritten as

F(θ, φ) =       F(θ₁0, φ0₁)ej∆Φ1,1 _F(θ0 1, φ02)ej∆Φ1,2 . . . F(θ01, φ0n)ej∆Φ1,n F(θ₂0, φ0₁)ej∆Φ2,1 _F(θ0 2, φ02)ej∆Φ2,2 . . . F(θ02, φ0n)ej∆Φ2,n ... ... ... ... F(θ0_m, φ0₁)ej∆Φm,1 _F(θ0 m, φ02)ej∆Φm,2 . . . F(θm0 , φ0n)ej∆Φm,n       , (2.11) where ∆Φm,n= ka sin (θ − θm) cos (φ − φn) and F(θm0 , φ0n) = F(θ − θm, φ − φn).

Figure 2.2: Circular array definitions [7, p.365]

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2.6. THE MEASURED VOLTAGE AND THE NORMALIZED INCIDENT FIELD

2.6 The Measured Voltage and the Normalized Incident

Field

Now, the problem is to estimate the incident electrical field, Einc on the surface of

a sphere. The incident field from the direction (θ0, φ0) will induce a voltage over

the receiving antenna’s terminals as

v= cF(θ0, φ0) · Einc(θ0, φ0), (2.12)

where c is an unknown constant.

By superposition, the total induced voltage will be the sum of all the impinging fields on the surface for all directions. If the number of directions forms a continuum the sum can be written as an surface integral

v= c Z π −π Z π 0 F(θ, φ) · Einc(θ, φ) sin θ dθdφ (2.13)

In order to estimate the incident field we must know the induced voltage v, from all directions. Consider now that the receiving antenna is mechanically rotated around on the surface of a sphere of radius a. The measured voltage will then be the convolution of F and Einc, where θ and φ are the direction the antenna is pointed

at. v(θ, φ) = c Z π −π Z π 0 F(θ − θ0, φ − φ0) · Einc(θ0, φ0) sin θ0dθ0dφ0 (2.14)

Since the constant c is unknown, what we will be estimating is the normalized incident field, defined as

p(θ, φ) = cEinc(θ, φ) (2.15)

From now and on we specialize our analysis to extracting the plane wave parame-ters on a single plane. Hence, the problem reduces to a one-dimensional convolution integral. In the one-dimensional case the equation (2.14) is reduced to

v(φ) =

Z π

−π

F(φ − φ0) · p(φ0) dφ0, (2.16)

where we have integrated over the dependences on the θ angle.

We then estimate the incident field in only one elevation angle θ at a time. The motivation for this is that this will simplify the problem. The downside is that this will degrade the resolution in the θ dimension of the estimations.

Solving (2.16) for p is an inverse problem. Inverse problems arises, for example, when we want to extract information hidden in measurement data.

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2.7 A Discrete Inverse Problem

In a real measurement scenario we are restricted to a discrete set of data points. We will start by transforming the convolution (2.16) to a discrete convolution problem

v(φj) = n

X

i=1

F(φj− βi)p(βi), (2.17)

where n is the number of discretization points and βi = 2(i − 1)π/n − π. Writing

this in matrix form, for the measured angles {φj}mj=1 and the integration angles

cor-responding to the antenna field pattern,{βi}ni=1, we get a system of linear equations

v= Bp, (2.18)

where B is the convolution matrix for a fixed θ angle. The convolution matrix can be defined as

B=       F(φ1− β1) F(φ1− β2) . . . F(φ1− βn) F(φ2− β1) F(φ2− β2) . . . F(φ2− βn) ... ... ... ... F(φm− β1) F(φm− β2) . . . F(φm− βn)       (2.19) and p becomes p= p(β1, β2, . . . , βn) (2.20)

For simplicity we have specialized our analysis to m=n.

Figure 2.3: Incident plane waves on an arbitrary field pattern rotating around φ

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2.7. A DISCRETE INVERSE PROBLEM

We are now faced with a linear inverse problem. Inverse problems arise when one tries to extract hidden information contained in measurement data. Thus, it is a very important and ˜A¡ common mathematical problem. Here, we are dealing with a specific type of inverse problem, namely a deconvolution problem.

There are several potential difficulties involved with solving a deconvolution problem

• Noise in the measurement data v or inaccuracies in the matrix B. • The matrix B is ill-conditioned.

• The problem is ill-posed.

In our channel model the noise vector, e, is modeled as an additive white Gaus-sian noise(AWGN), e. Which leads to that (2.18) becomes

v= Bp + e, (2.21)

The condition number of B gives an indication of how sensitive the solution is to perturbations in B or in the measurement data v. A problem with a large condition number is called ill-conditioned, [19].

cond(B) =k B k2k B−1 k2, (2.22)

When the condition number is large, even if the noise level is low, the noise will have a huge impact on the resulting estimation ˆp. Using (2.22) we can calculate the condition number for the matrix B. For example our model B of the size 720 × 720 the condition value then is 1.1097 × 106_{. Therefore, we can say that we have an}

ill-conditioned problem.

Many inverse problems are ill-posed. A problem is said to be ill-posed if it is not well-posed. A well-posed problem has to fulfill the following requirements

• A solution must exist.

• The solution has to be unique.

• A small change in the initial condition should lead to a small change in the solution.

A direction solution of (2.18) cannot rely on the direct inversion since the actual problem is given by (2.21). Hence, the standard approach is to find the solution ˆp that satisfies the following conditions

min k p k2subject to min k B · p − v k2, (2.23)

where k p k2 is the euclidean norm of the solution and k B · p − v k2 is the residual

norm.

In the next section we explain how to solve (2.21) by the use of singular value decomposition.

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2.8 Singular Value Decomposition (SVD)

The singular value decomposition is a factorization form of a matrix into two unitary matrices and one diagonal matrix of singular values.

Let B be the matrix to decompose, then we can write

B= UΣV†, (2.24)

where † denotes the Hermitian-conjugate. The left and right matrices, U ∈ Cmxm

and V ∈ Cnxn _{, satisfies the unitary condition U}†_U _{= I, V}†_V_{= I and Σ ∈ R}mxn

is a diagonal matrix. The diagonal matrix contains the elements {σ}n

i=1 which are

given in a descending order. These elements are called the singular values of the matrix B.

The inverse of B can be obtained by inverting the singular values and writing the SVD factors in reverse order [10]. Note that if any of the singular values σi is

zero the inverse will be undefined.

B−1 = VΣ−1U† (2.25)

We refer to this inverse as the naive inverse of B.

(2.25) may also be written as sums of n rank-1 matrices [10]

B−1= n X i=1 1 σi νiµ†i, (2.26)

where νi is the column vectors of V and µi is the column vectors of U.

Before we continue, let’s study the singular values of B. As mentioned pre-viously, the singular values are given in a descending order. Depending on the structure of matrix B many ill-conditioned problems can be categorized into two groups: rank-deficient- and ill-posed- problems [19].

In rank-deficient problems there exist a gap between the size of the larger singu-lar values and the smaller. The smaller singusingu-lar values reflects the approximation errors in the model of B. In these cases the problem can be transformed to a well conditioned problem by identifying the matrix’s numerical rank. The rank corre-sponds to the singular value before the gap. The other major group is the case where there is no clear gap. In this case it’s not as easy to identify the rank of the matrix. This case is called ill-posed. Here we instead have to find a middle ground between the residual norm and the size of the solution, see (2.23). Let us plot the singular values from the matrix B. It can be seen in figure 2.4 that identifying a clear gap is not possible. Instead there is a gradual decay. We therefore can say that we have a problem that is ill-conditioned and ill-posed.

Using (2.25) or (2.26) the naive solution to (2.23) is given by: ˆpn= VΣ+U†v= n X i=1 µ†_iv σi νi (2.27) 10

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2.9. TRUNCATED SINGULAR VALUE DECOMPOSITION

Figure 2.4: The normalized singular values of B showing no clear gap. The elements of µ†

iv are called the Fourier coefficients. A test to see if the

so-lution is meaningful is to look at the decay rate of the Fourier coefficients. The Discrete Picard Condition [16] says that the solution is meaningful if the Fourier Coefficients decrease (on average) at a higher rate then the singular values. This also means that the quotient vector µ†_iv

σi must not increase. This is rarely the case

in real measurements because of measurement errors in v and round off errors in the matrix B. Here the Fourier coefficients will settle on a level due to the errors in v and the singular values will settle on a level due to the errors in the matrix

B, as seen in figure 2.5a. However, if the underlying exact problem satisfies the

Discrete Picard Condition it is often possible to find a meaningful approximation to the solution. In ill-conditioned problems there is a need for some kind of regu-larization. The objective is to modify the system so that the solution becomes less sensitive to the influence of measurement noise and/or inaccuracies in the matrix B. If the regularization of the problem is successful the system for regularized solution satisfies the Discrete Picard Condition. An example of this can be viewed in figure 2.5b.

2.9 Truncated Singular Value Decomposition

There are many different methods and variants to regularize the problem (2.23). Two methods of wide-spread use are the Tikhonov regularization and the Truncated Singular Values Decomposition (TSVD). The advantage of TSVD is that it is very intuitive because you just remove the singular values that amplifies the noise in the solution. The solution for different regularization parameters can easily be calculated. The downside to TSVD is that we first have to compute the SVD. This is for large problems very computationally heavy. In those cases other methods can perhaps be a better choice, such as the Tikhonov regularization. In our case the problem is not large. The size of the problem is limited by the measurement time.

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(a) Unfulfilled Discrete Picard Condition (b) Fulfilled Discrete Picard Condition

Figure 2.5: Plot of the Fourier Coefficients and singular values illustrating an un-fulfilled condition, a, and plot of it’s corresponding regularized solution,b. Taken from a simulation using 360 singular values and an additive white Gaussian noise corresponding to 10dB SNR.

Since the purpose of the thesis is not to compare different regularization methods we have chosen to only use the TSVD. The choice is made mainly because of it’s simplicity.

In the Truncated Singular Value Decomposition (TSVD) the problem is filtered by setting the smaller singular values of B equal to zero. Truncating the SVD will lower the condition value of Bkand the solution will be less contaminated by noise.

The condition number of Bk is given by κ(k) = σ1/σk.

Let us divide the singular values into two parts: one containing the values that will lead us to the solution of the problem (values 1 to k) and one part containing the values that will distort the solution (the last n−k values). The resulting matrix, called Bk, is defined as

B_k= UΣ_kV†, Σk= diag(σ1, . . . , σk,0, . . . , 0) ∈ Rmxn (2.28)

The psuedoinverse, B+

k, also called the Moon-Penrose inverse of B:

B+_k = VΣ+_kU†, Σ_k+= diag(σ₁−1, . . . , σ−1_k ,0, . . . , 0) ∈ Rnxm (2.29)

Σ+ is defined as as Σ−1. Any elements of Σ = 0 are set to 0 in Σ+.

The TSVD solution to the problem, (2.23) , can now be calculated by ˆpk= B+kv= k X i=1 µ†_iv σi νi (2.30)

The problem now is to select the regularization parameter, k. In other words, choose the number of singular values to use in the pseudoinverse of B that satisfies (2.23).

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2.10. CHOOSING A REGULARIZATION PARAMETER

2.10 Choosing a Regularization Parameter

Truncating the SVD will directly improve the condition number of B due to the definition of SVD but, on the other hand, too much truncation will remove wanted information from the solution. There has to be a trade off between these two. There are different ways to approach this problem. The most simple way is perhaps to filter the solution so that the residual norm (2.23) equals an upper bound set by the norm of the error in the measurement data, k e k2, where e is the additive noise

from v. This is called the discrepancy principle [20]. A problem with this is that we don’t know that error, at least not with enough accuracy. It also tends to filter to much from the solution [18]. A common method to graphically determine this trade off is called the L-curve method. This method extracts information from the noise contaminated measurement data. This is the method we have chosen to use.

Figure 2.6: L-curve sketch

The idea was first introduced by Lawson and Hanson[11]. Hansen and O’Leary investigated the L-curve to be used together with the TSVD method [12]. In the L-curve method the size of the regularized solution is plotted against the size of the residual of the error. They are plotted in logarithmic scale for different values of the regularization parameter. For ill-posed problems this often lead to a plot with an L-shaped curve. The solution is found by identifying the parameter, k, corresponding to the curve’s corner. This can be seen in the figure 2.6, which is a example from one of the simulations in the simulation section. If the solution is calculated using a regularization parameter located to the left of the L-curve the error in the solution will be dominated by perturbation errors. This case is sometimes called under-smoothing. On the other side, to the right of the corner, the error in the solution will be dominated by a regularization error. This case can

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be called over-smoothing. It is convenient to plot the L-curve in logarithmic scale since this emphasizes the corner. This method is quite intuitive since this leads to a trade off between minimizing the size of the error and minimizing the size of the regularized solution, ˆp. A problem with the L-curve method is that it is not easy to implement an algorithm to find the corner, even though it is often easy to spot it with the human eye. We will go into this a bit in section 2.13.

2.11 Dealing with the Solution to the Regularized Problem

We have now described the problem as a mathematical problem. We have also presented tools to find an approximated solution to the problem. How good will this approximation be? That will depend mainly on the noise level, provided that the model itself is sufficiently good. The noise level basically determines how much too filter out. In figure 2.7 we see estimations of a single discrete spike (at φ = 0) using different number of singular values (regularization parameter). As can be seen in figure 2.7, the side lobe levels of each solution, calculated using a different number of singular values, will be reduced when the number of singular values used is higher. Also, the oscillation frequency of the side lobes will be higher. Of course, these two facts will effect the resolution in the solution, both in the angular dimension and in the amplitude dimension.

Figure 2.7: Base functions from the SVD.

We have assumed that the incident field is composed of scatterings from dis-crete directions in the far-field region, (2.4). To produce a good estimation of the complex amplitudes of the scatterers in the presence of noise the solution has to be normalized.

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2.11. DEALING WITH THE SOLUTION TO THE REGULARIZED PROBLEM

For a case without noise the correct amplitude is given when all parts of the solution are used, k = n in (2.30). If we assume that each term in (2.30) contributes an equal amount at the estimation angle of interest, ˆφk, such that

Γ = µ † iv σi νi, for i = 1, 2, . . . , n, (2.31) this leads to nΓ( ˆφk) = p, (2.32)

for a case with noise, say that the appropriate regularization parameter is k. For both sides to be equal that leads to the proportional factor K

nΓ( ˆφk) = KkΓ( ˆφk), (2.33)

the normalization used on the solution to the regularized problem (2.30) is therefore

K = n

k, (2.34)

where k is the number of singular values used in the filtered solution. The normal-ization equation was derived through experimentations on simulations. As can be seen in figure 2.8. The figure illustrates a simulation of one scatterer performed as in the simulation section of the mean value of N = 2000 realizations. The amplitude of the scatterer is 1. The picture shows that for the noiseless non normalized case (i.e. the purple dotted line) that it increases linearly with the number of singular values

kused. For the normalized noiseless case (the filled purple line) the amplitude stays

constant at 1. Looking at the other traces, they correspond to different SNR levels. When k is increased, at a certain point the solution will become instable. One can note that the normalization constant with n = 360 also corresponds to the φ angle between two zeros of the side lobes in figure 2.7.

Due to the plane wave assumption we are only interested in estimations from discrete angles. The other angles are seen as artifacts. Therefore the solutions we pick out from ˆpk, are instead the complex peak values and their AoA

ˆpk,peaks(φ) =

(

0 for φ 6= ˆφk

ˆpk( ˆφk) for φ = ˆφk (2.35)

where ˆφk are the estimated AoA of the Ns number of scatterers in ˆpk.

For the general case, with multiple directions of the incident field, choosing the AoA, φ = ˆφk corresponding to the largest peak values will not be enough. For low

SNR levels the filtering used to produce ˆpk is so large that the resolution becomes

to low. The technique we have used to improve the estimation is called successive interference cancellation. It is an iterative process where each peak is dealt with one AoA at a time. When largest peak has been estimated from ˆpkthe contribution

from that peak is subtracted. The algorithm then moves on to the second largest peak, and so on. We will present the algorithm in more detail in the next section.

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Figure 2.8: Comparison between solutions with normalization and without. Dashed lines are solutions without normalization. Different lines represent different SNR levels

2.12 Successive Interference Cancellation

The truncated SVD solution described above gives an estimation of the normalized incident electrical field. However when there are multiple scatterers in the incident field the solution can be greatly degraded by noise. This is because the solution is then built up by few singular values and this leads to an increased interference from the other scatterers, due to the pattern seen in figure 2.7. A way to reduce the interference is to remove the effect from each scatterer.

• Start with the largest peak in the solution given by i=1. ˆpk,peaks( ˆφk,i).

• Subtract a simulated peak from the solution. ˆpk,redi = ˆpk− B

+

kBˆpk,peaks( ˆφk,i)

• Now a new maximum peak will appear. Repeat the steps above for i = i + 1 and ˆpk= ˆpk,redi and iterate until i = Ns.

This technique can improve the estimation greatly. As can be seen in figure 2.9. In ˆp41the smallest scatterer is not easy to pick out. After using three iterations of

the algorithm in ˆp41,red3 the accuracy of the estimation is increased. Even if you

just want to estimate two scatterers the solution might, depending on noise levels, not be satisfactory. The technique assumes that the contribution to the peak’s maximum value are from just one scatterer. If this is not true it will cause an error.

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2.13. IMPLEMENTATION IN MATLAB −100 0 100 −40 −30 −20 −10 0 φ ˆp41 , [d B ] −100 0 100 −40 −30 −20 −10 0 φ ˆp41 ,r e d1 , [d B ] −100 0 100 −40 −30 −20 −10 0 φ ˆp41 ,r e d2 , [d B ] −100 0 100 −40 −30 −20 −10 0 φ ˆp41 ,r e d3 , [d B ]

Figure 2.9: Example of the Successive Interference Cancellation. ’X’ represent pk,

’O’ represents ˆpk,peaks,i

2.13 Implementation in MATLAB

The model described earlier has been programmed by the authors in MATLAB. The convolution matrix, B, described in (2.19) is a circulant Toeplitz matrix. It is implemented effectively using the MATLAB function Toeplitz. The problem to select the regularization parameter from the L-curve has been approached by using an algorithm written by Hansen et. al. as well as using a graphical approach by looking at L-curves. The algorithm is based on an adaptive pruning algorithm. It rescales the L-curve in difference scales with the purpose of finding a global corner. More information on how the algorithm works can be read in [17]. The truncation of the SVD components is performed using a MATLAB function, TSVD, written by Hansen [22].

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Chapter 3

Simulation

The next step is to simulate the propagation model. The simulations will let us examine the model’s possibilities under clear and known conditions before using the model on the measurements.

Various simulation scenarios are considered. The varied parameters are • The number of scatterers, Ns = 1, 2, 5 and 10.

• SNR = 50, 40, 30, 20, 10, 0 and -10 dB.

• The number of measurement points, m = 720, 360, 180 and 90. Ns = 1

For the case with one scatterer we have the most simple case. Here we don’t have any other scatterers that can affect the main scatterer. All energy is concentrated to a single AoA. For our planned measurements this will not be the case. Still, the scenario is important for a number of reasons. We will see how well our model can estimate the power and AoA from one scatterer, depending on noise level and number of measurement points. We should also be able to error check our code by seeing that the model behaves as expected. The expected results are that the errors should increase when the noise level rises and when the number of measurement points decreases.

Ns = 2

We then test a case with two scatterers. This will provide us insight into how scat-terers can affect each other and how this changes the solution. The two scatscat-terers are produced with an AoA of different fixed separation distances. This setup will give us a good idea of the achievable resolution of scatterers at different noise levels.

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CHAPTER 3. SIMULATION

Ns = 5 and 10

To test how the model can handle multiple scatterers we simulate the method for five and ten scatterers. The scatterers have a random AoA, taken from a uniform distribution.

Simulation Programs

The simulation programs are written in MATLAB. The programs are based on the implementation in section 2.13, ’Implementation in MATLAB’.

The noise is implemented as an additive white Gaussian noise to the simulated measurement data. The signal-to-noise level is calculated as

SN RdB = SdB− EdB, (3.1)

where EdB is the average power of the noise component e. SdB is the average power

of the noise free signal v and SNRdB is the wanted signal-to-noise ratio.

All scatterers are produced with a random complex amplitude. Random accord-ing to a uniform distribution. The total power of the simulated scatterer vectors,

p are normalized to 1, k p k2₂= 1. The simulated measurement data was calculated

using (2.18). Instead of solving the inverse problem we use the forward computation of the problem.

Due to the randomness of noise, the random complex amplitude and AoA the simulations are repeated many times. Each repetition is called a realization. The number of repetitions depends on the number of varied variables. Before errors are calculated some obvious outliers among the realizations are discarded. These outliers were selected by looking at the singular values picked out by the L-curve corner algorithm. Realizations with non-typical values were discarded.

To know how well the method works we have defined different measures of errors. Three different errors were calculated from the simulations. The first error is the mean norm error.

εp = 1 N N X i=1 k p_i(φ0) − ˆp_k,peaks,i( ˆφ_k) k₂, (3.2)

where N is the number of realizations, i the realization index and φ0 _{is the exact}

AoA of the scatterers. The second error is the mean power error.

εkpk = 1 N N X i=1 |k p_i(φ0) k2₂ − k ˆp_k,peaks,i( ˆφ_k) k2₂| (3.3) The third error is the mean absolute angular error of AoA.

εφ= 1 N N X i=1 | φ0− ˆφk | (3.4) 20

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3.1. SIMULATION RESULTS

3.1 Simulation Results

3.1.1 One Scatterer Case

Figure 3.1: Example of an estimation of a realization. m = 360. 20dB SNR. This simulation was performed for 7 different signal-to-noise ratios, 50-, 40- ,30-, 20-, 0- and -10 dB. Four different number of measurement points were compared: 720, 360, 180 and 90. This simulation was based on N = 1400 realizations for each noise level. An example of a realization can be seen in figure 3.1.

Figure 3.2 shows the mean errors. The dashed lines in figure 3.2 are the errors from the naive solutions calculated (without any filtering) using (2.27). The angular errors are presented in table 3.2. The general trend of the two non-naive amplitude errors behaves as expected. Using more measurement points does reduce the errors and with increasing noise level the errors increase. The relative power error at 20dB SNR between the number of measurements points is approximately 44%, 124% and 740% larger for the 360, 180 and 90 cases compared to 720 measurement points. Looking at the three different errors in, figure 3.2 and table 3.2, we can conclude that it is not meaningful to estimate at noise levels corresponding to -10dB SNR using 90 measurement points. Those estimations also, as expected, do not fulfill the Discrete Picard Condition discussed in section 2.8.

The trend of the naive solution curves behave as expected. The order of the curves follows the condition number which, as discussed in section 2.7, provides an insight on how sensitive the solutions are to noise. A unexpected result was that the condition number greatly depends on the phase shift matrix. The numbers depends on the measurement system and is discussed in chapter 5. Using the phase center adjustment in our measurement setup (radius a = 217) the condition number are presented in table 3.1.

The basic general trend behaves as expected. When the amount of measurement points, m, decrease the size of the errors increase. This can be explained by how the resolution in the L-curves decreases along with it. This leads to a less distinct corner in the L-curve. Figure 3.3 illustrates this for a set noise level. There is, however, an oddity in the simulation. The error level, in figure 3.2, for SNR 40dB and 10dB using m=90, are slightly deviant from the general trend. This could be

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CHAPTER 3. SIMULATION −10 0 10 20 30 40 50 10−2 100 102 104 SNR [dB] εkpk m = 720 m = 360 m = 180 m = 90

(a) Mean Power Error

−10 0 10 20 30 40 50 10−2 100 102 SNR [dB] εp m = 720 m = 360 m = 180 m = 90

(b) Mean Norm Error

Figure 3.2: Mean errors plotted against SNR for one scatterer, dashed lines represent the naive solutions.

m= 720 m= 360 m= 180 m= 90

Condition number 1.11e+006 7.05e+005 1.74e+006 3.9e+004

Table 3.1: Condition values for different m.

an effect of our normalization factor (2.34). If we plot the power error for these SNR levels for different number of singular values, k, we can see an explanation of why the error can go down.

The arrows in figure 3.4 corresponds to the median number of singular values chosen at each SNR with the L-curve method. Looking at figure 3.4 one could say that the L-curve method does not find the optimal solution (the minimum error here is actually around k = 13). However the L-curve method does not look at ˆpk,peaks but on the L2-norm of ˆpk. Also, it looks at the original solution, not the

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3.1. SIMULATION RESULTS m \dB 50 40 30 20 10 0 -10 720 0 0 0 0.001 0.02 0.42 1.63 360 0.27 0.27 0.27 0.27 0.27 0.60 2.13 180 0.49 0.49 0.49 0.49 0.50 0.86 3.08 90 1.03 1.03 1.03 1.04 1.05 1.35 87.05

Table 3.2: εφin degrees for different SNR and m.

Figure 3.3: L-curves for 1 scatterer case. Comparison of m = 90 and m =720. k for SNR 20 dB

Figure 3.4: Mean power error for m = 90.

normalized solution. In the original TSVD solution the error will decrease when k is increased until a certain point set by the noise level where the solution will be unstable. Note also that this is a simulation using one scatterer, referring to the table 3.2, we can see that the angular resolution in the solution does not matter

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that much. This will not be the case when there are more scatterers involved. • For measurements where there essentially is one scatterer in the room the

number of measurement points to use depends on the needed accuracy in the solution. Using m = 720, m = 360 and m = 180 provides a relative small error even when the SNR is small.

• m = 90 can produce bad results for SNR lower then 10dB.

• It is clear from this simulation that the regularized solutions are a huge im-provement over the naive solutions.

3.1.2 Two Scatterer Case

Figure 3.5: Example of an estimation of a realization using 2 scatterers. m = 360. 20dB SNR.

The second scenario uses two scatterers. The simulation was performed using

m = 360. The scatterers AoA separation was shifted towards each other. Starting

with ∆φ = 50o _{separation and ending on ∆φ = 2}o _{with one degree steps. The}

start separation was chosen so that the two peaks still were distinguished in the measurement data. 2o _{was chosen as a stop separation from the number of}

mea-surement points. Figure 3.5 shows an example where ∆φ = 50o_{. The simulation}

was performed with N = 2400 different realizations.

In figure 3.6 we can see oscillations in the errors for the larger scatterer. Looking closely we can see that these oscillations have different frequencies at different SNR. This is due to the fact that the side lobes in ˆpk interfere differently depending on

∆φ. Looking at figure 3.5 we can see that the side lobes from ˆp41 do not influence

the scatterer amplitudes to any large extent. Looking at figures 3.6 and 3.7, εkpk

and εp both have a local minimum for SNR = 20dB and ∆φ = 50o .

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(b) Mean Norm Error

Figure 3.6: Mean errors plotted against SNR for the larger scatterer.

As discussed earlier, the level of filtering is essentially set by the SNR. In other words, the number of singular values used in the solutions differs for each SNR. Referring back to section 2.11, we discussed that the φ angle between two zero crossings of the side lobes is set by 360/k. Table 3.3 compares angles between two zeros calculated using 360/k with figure 3.6. For k we used the mean value of all realizations for each SNR. As seen in the table 3.3, they correspond quite well.

The size of the errors in figures 3.6 and 3.7 rises, on average, when the two scat-terers are closing. This is because the contribution, in ˆpk, from the two scatterers,

interfere more with each other. Figure 3.7 shows the errors of the smaller scatterer. We can see that the oscillations are still present. They have lower amplitude com-pared to the larger scatterer. This is because the interference cancellation does not

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50 40 30 20 10 0 -10

360/k 4.92 5.14 5.48 8.85 17.00 24.86 27.72

graph 5.00 5.00 5.25 9.00 17.00 24.00 24.00

Table 3.3: Period time of oscillation taken from figure 3.6 and compared to ones calculated using (2.34)

manage to remove the effects of the larger scatterer perfectly.

The error εφcan be seen in figure 3.8. We can see that the error oscillates much

like the amplitude errors. For the larger scatterer the error is lower when the two scatterers get close. The case is reversed for the smaller scatterer. The explanation is that the solution, ˆpk for the larger scatterer is essentially one main peak from

a combination of the two scatterers at small ∆φ’s. This means that the smaller scatterer will not affect the larger one on an angular level. However, the amplitude level will be greatly disturbed. For the smaller scatterer the successive interference cancellation will not work very well since the amplitude for the larger scatterer is badly estimated. This leads to that the interference cancellation can remove both scatterers in one iteration, leaving the estimation of the smaller scatterer to fail. The success of cancellation, of course, also depends on the SNR. As can be seen in figure 3.8, the SNR has to be higher then 20 dB for the angular error to be less then 10o_{, when the separation is ∆φ = 2}o_.

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(b) Mean Norm Error

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(a) Mean absolute angular error for the larger scatterer

(b) Mean absolute angular error for the smaller scatterer

Figure 3.8: Mean absolute angular errors plotted against SNR.

• The regularization method still works for the two scatterer case. If the SNR is too low and, or the scatterers are too close to each other, the regularization method fails to estimate the two scatterers. The solution corresponding to the smaller scatterer will then incorrectly be an estimation of a side lobe. • The AoA estimation for the larger scatterer can always be estimated well for

all the separations and noise levels in this simulation.

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3.1.3 Five and Ten Scatterer Cases

To examine how the model handles multiple scatterers we have simulated a case with five scatterers and a case with ten scatterers. Both were simulated from N = 60000 realizations and with a uniformly distributed random AoA. The measurement points used were m = 360, 180 and 90 for both cases.

The errors εp and εkpk can be seen in figure 3.9. It shows the error in Ns = 10

to be larger then Ns = 5. This is expected since the average distance between the

five scatterers are larger. This leads to less interference between the scatterers. The error levels at SNR -10dB for m = 90 suggests that it is not meaningful to estimate neither Ns= 5 or 10. However, these shows the sum of all error from each individual

scatterer in according to (3.2 and (3.3). It might be possible to estimate some of these scatterers. If we take a look at the angular errors shown figure 3.10 we can see that with a maximum upper angular error of 10o _{per scatterer the portion of}

scatterers within this limit is given in table 3.4. We can also note that the portion is approximately the same for both Ns = 10 and Ns = 5. Figure 3.11 shows two

random realizations, for Ns = 5 and for Ns = 10 relevant for the measurement in

chapter 6.

m/SNR [dB] inf→10 10→0 0→-10

360 80%/80% 50%/60% 40%/60% 180 80%/80% 50%/60% 40%/40% 90 70%/60% 50%/60% 30%/20%

Table 3.4: Portion of scatterers possible to estimate taken from figure 3.10. Ns =

10/Ns= 5

ε0_φ can be seen in figure 3.9c. For Ns = 10 the probability for two scatterers to

interfere in ˆpk is greater than for Ns= 5. Thus ε

0

φ for Ns= 10 becomes larger. At

SNR -10dB the resolution in ˆpk is too low to get a reliable result. This leads to

that the two curves converge.

Figure 3.9 shows one realization for four different SNR levels that are relevant to the measurements in chapter 6. It shows that with Ns = 5 it is usually possible

to estimate down to SNR= 10dB. For 0dB SNR m = 90 does not provide suffi-cient resolution. When the amount of scatterers is increased to Ns = 10 the errors

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CHAPTER 3. SIMULATION −10 0 10 20 30 40 50 10−2 10−1 100 101 102 SNR [dB] εkpk m=360 m=180 m=90 (a) Ns = 5 −10 0 10 20 30 40 50 10−2 10−1 100 101 102 SNR [dB] εkpk m=360 m=180 m=90 (b) Ns = 10 −10 0 10 20 30 40 50 10−1 100 101 102 SNR [dB] εp m=360 m=180 m=90 (c) Ns = 5 −10 0 10 20 30 40 50 10−1 100 101 102 SNR [dB] εp m=360 m=180 m=90 (d) Ns = 10

Figure 3.9: Different errors v.s. SNR for 5 and 10 scatterers.

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3.1. SIMULATION RESULTS −10 0 10 20 30 40 50 10−2 10−1 100 101 102 SNR [dB] εφ 360 1st 360 2nd 360 3rd 360 4th 360 5th 180 1st 180 2nd 180 3rd 180 4th 180 5th 90 1st 90 2nd 90 3rd 90 4th 90 5th

(a) Angular Error, Ns=5

−10 0 10 20 30 40 50 10−2 100 102 SNR [dB] εφ 360 1st 360 2nd 360 3rd 360 4th 360 5th 360 6th 360 7th 360 8th 360 9th 360 10th (b) Angular Error, Ns=10, m=360 −10 0 10 20 30 40 50 10−2 100 102 SNR [dB] εφ 180 1st 180 2nd 180 3rd 180 4th 180 5th 180 6th 180 7th 180 8th 180 9th 180 10th (c) Angular Error, Ns=10, m=180 −10 0 10 20 30 40 50 10−1 100 101 102 103 SNR [dB] εφ 90 1st 90 2nd 90 3rd 90 4th 90 5th 90 6th 90 7th 90 8th 90 9th 90 10th (d) Angular Error, Ns=10, m=90

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CHAPTER 3. SIMULATION −150 −100 −50 0 50 100 150 −35 −30 −25 −20 −15 −10 −5 0 φ dB (a) Ns=10 at SNR 0dB. −150 −100 −50 0 50 100 150 −35 −30 −25 −20 −15 −10 −5 0 φ d B v, ˆ ppeaks,m = 720 p ˆ ppeaks,m = 360 ˆ ppeaks,m = 180 ˆ ppeaks,m = 90 (b) Ns=5 at SNR 0dB. −150 −100 −50 0 50 100 150 −35 −30 −25 −20 −15 −10 −5 0 φ dB (c) Ns=10 at SNR 10dB. −150 −100 −50 0 50 100 150 −35 −30 −25 −20 −15 −10 −5 0 φ d B v, ˆ ppeaks,m = 720 p ˆ ppeaks,m = 180 ˆ ppeaks,m = 90 ˆ ppeaks,m = 360 (d) Ns=5 at SNR 10dB. −150 −100 −50 0 50 100 150 −35 −30 −25 −20 −15 −10 −5 0 φ dB (e) Ns=10 at SNR 20dB. −150 −100 −50 0 50 100 150 −35 −30 −25 −20 −15 −10 −5 0 φ d B v, p ˆ ppeaks,m = 720 ˆ ppeaks,m = 360 ˆ ppeaks,m = 180 ˆ ppeaks,m = 90 (f) Ns=5 at SNR 20dB. −150 −100 −50 0 50 100 150 −35 −30 −25 −20 −15 −10 −5 0 φ dB v, p ˆ ppeaks,m = 720 ˆ ppeaks,m = 360 ˆ ppeaks,m = 180 ˆ ppeaks,m = 90 (g) Ns=10 at SNR 40dB. −150 −100 −50 0 50 100 150 −35 −30 −25 −20 −15 −10 −5 0 φ d B v p ˆ ppeaks,m = 720 ˆ ppeaks,m = 360 ˆ ppeaks,m = 180 ˆ ppeaks,m = 90 (h) Ns=5 at SNR 40dB.

Figure 3.11: Example of estimations of a realization using 10 and 5 scatterers. m = 360.

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• The case shows that the larger scatterers are easier to estimate. The scatterers with lower amplitude will be more difficult not only because of the larger relative noise level but also because of high disturbances from the estimation of the larger scatterers . While the SIC algorithm reduces these disturbance to a degree it is not perfect.

• From these simulations we extracted a guideline (table 3.4) of how to estimate the measurement data to be within 10o _{AoA error.}

3.1.4 Simulation Comments

The three simulation cases provides a good view of what the model is capable of. It also shows some interesting behaviors that, although most of them only will show up in a simulation setting, might be good to know in a measurement setting.

• The model could not be simulated in the above manner for m = 720 points. For the one scatterer case it works but when the number of realizations increases, which is needed when also the amplitude is random, the simulations became too large for our computer to handle.

• We have chosen to perform the measurement using m = 360. The motivation is partly that m = 720 takes almost twice the amount of time to measure and the time available was limited. Secondly m = 360 provided good simulation results for SNR 30dB to 0dB, which is about the noise level present in our measurements presented in chapter 6.

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Chapter 4

The Measurement System

4.1 Introduction to The Measurement System

To perform measurements to use with the model we needed to build a measurement system.

The whole measurement system is built up from several hardwares, a PC, a RX90L St¨aubli Robot (MOLLY), a Vector Network Analyser (VNA), an one dimensional stepper (Taxen), a horn antenna and a half-wavelength dipole. The frequency cho-sen was 4.5GHz. To our disposal we had two identical horn antennas and one dipole that were matched at 4.5GHz to a 50Ω transmission line. The horns had a data sheet that was lacking in details so we had to simulate our horns to be able to get a field pattern that we could work with.

With the measurement system there was a few parts that needed to be handled: • The RX90L robot, MOLLY.

• The one dimensional stepper, Taxen. • The antennas.

• Measurement times.

For the different parts there was different restrictions that had to be taken into consideration. These restrictions are presented at their respective topic.

4.2 MOLLY - A Stäubli RX90L Robot

The first part handles the robot MOLLY. The task is to find out what kind of movement that would be most suitable for the chosen propagation model. MOLLY has 6 different rotational centers that can all be controlled separately. As discussed in chapter 1, we are interested in measuring the field strength over a spherical surface area. The problem at hand is to make MOLLY create a spherical sector that is as close too a whole sphere as possible. There are some restrictions present:

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CHAPTER 4. THE MEASUREMENT SYSTEM

Figure 4.1: MOLLY a RX90L robot.

• The arm has to be located under the antenna.

• The spherical sector needs to be as close to a whole sphere as possible and with as small radius as possible.

• The movements need to be fully controlled to not endanger the measurement equipment.

• The spherical sector can’t have any large spatial errors, i.e., repeated mea-surements should not deviate in spatial positions.

• The measurement needs to be fast.

The first restriction is needed to reduce the interference from MOLLY while perform-ing the measurements. The second restriction is to be able to account for additive signals errors from other directions than the measured, the measurement system would need to be able to create a perfect sphere. If this is not possible the spherical sector needs to be as big as possible to account for as many errors as possible. The third restriction is straightforward, MOLLY should not put any other equipment or people in danger during a measurement. The fourth restriction is there to see if

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4.2. MOLLY - A ST¨AUBLI RX90L ROBOT

there is a way to eliminate spatial errors or if it is something that need to be taken into consideration. Spatial errors will be present if the robot does not have good enough precision in it’s movements. The fifth is to fulfill another model restriction, the room needs to be kept unchanged during measurements. If there are people moving around it can alter the propagation channel and bring disturbances into the measurements. To minimize this problem the measurements needed to be held at night and weekends. This puts a time limit on the measurements. The first four will be handled in this section, 4.2, while the last one will be handled in the section 4.6.

4.2.1 MOLLY definitions

In this section all the definitions for MOLLY will be presented. First all the dimen-sions and joints will be defined.

Building and Evaluating a 3D Scanning System for Measurementsand Estimation of Antennas and Propagation Channels

Erik Johannes Aagaard Fransson

Tobias Wall-Horgen

Building and Evaluating a 3D Scanning System

for Measurements and Estimation of Antennas

and Propagation Channels

Abstract

Acknowledgments

Contents

List of Figures

List of Tables

Chapter 1

Introduction

1.1

Problem statement

1.2

Thesis Outline

Chapter 2

The Radio Propagation Channel

2.1

Propagation Channel Model

2.2

Far-Field Region

2.3

Basic Antenna Properties

2.4

Free Space Path Loss

2.5

Circular Array

2.6

The Measured Voltage and the Normalized Incident

Field

2.7

A Discrete Inverse Problem

2.8

Singular Value Decomposition (SVD)

2.9

Truncated Singular Value Decomposition

2.10

Choosing a Regularization Parameter

2.11

Dealing with the Solution to the Regularized Problem

2.12

Successive Interference Cancellation

2.13

Implementation in MATLAB

Chapter 3

Simulation

3.1

Simulation Results

Chapter 4

The Measurement System

4.1

Introduction to The Measurement System

4.2

MOLLY - A Stäubli RX90L Robot