CALIBRATION OF UNIFORM CIRCULAR ARRAYS

A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY

BY

BUKET AYKANAT

IN PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN ELECTRICAL AND ELECTRONICS ENGINEERING

SEPTEMBER 2010

Approval of the Thesis:

CALIBRATION OF UNIFORM CIRCULAR ARRAYS

submitted by BUKET AYKANAT in partial fulfillment of the requirements for the degree of Master of Science in Electrical and Electronics Engineering Department, Middle East Technical University by, Prof. Dr. Canan Özgen Dean, Gradute School of Natural and Applied Sciences Prof. Dr. Đsmet Erkmen Head of Department, Electrical and Electronics Engineering Dept. Assoc. Prof. Dr. Sencer Koç Supervisor, Electrical and Engineering Dept., METU

Electronics

Examining Committee Members

Prof. Dr. Yalçın TANIK Electrical and Electronics Engineering Dept., METU Assoc. Prof. Dr. Sencer KOÇ Electrical and Electronics Engineering Dept., METU Prof. Dr. Fatih CANATAN Electrical and Electronics Engineering Dept., METU Assoc. Prof. Dr. Lale ALATAN Electrical and Electronics Engineering Dept., METU MSc. EE. Caner TEZEL REHĐS Division, ASELSAN

Date :

I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work.

Name, Last name

: BUKET AYKANAT

Signature

:

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ABSTRACT

CALIBRATION OF UNIFORM CIRCULAR ARRAYS

Aykanat, Buket

M.S., Department of Electrical and Electronics Engineering Supervisor

: Assoc. Prof. Dr. Sencer Koç

September 2010, 97 pages

In practice, there exist many error sources which distort the antenna array pattern. For example, elements of antenna arrays influence each other (known as mutual coupling), mismatches in cables and element positions affects the antenna radiation pattern and also unequal gain and phase characteristics of RF receiver distorts the received signal. These effects generally degrade the array performance. They cause an increase in sidelobe levels with an accompanying decrease in gain. Also, these errors limit the performance of direction finding (DF) algorithms. So, in order to have low sidelobe level, good performance in direction finding and beamforming, calibration is necessary. In the literature, there exist many algorithms proposed for the calibration of errors. Calibration method used in this thesis assumes that there is a linear transformation between ideal signal and measurements. Calibration matrix C is formed by using measurements. In this work, we look for the adequate number of measurements for successful calibration. Performance of calibration method may depend on the angle

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interval from which measurements are taken. So, the width of the data collection angle interval is also studied. Moreover, in real life, measurements can be collected from equally or randomly spaced angles. Does it affect the performance of calibration? The answer of this question is also inspected in this thesis. Additionally, the performance of algorithm under noise is studied. Performance evaluation is done for both elevation and azimuth sectors. Simulations are carried out on MATLABTM and Ansoft HFSS software package. Keywords : Online calibration, uniform circular array

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ÖZ

DAĐRESEL DĐZĐ ANTENLERDE KALĐBRASYON

Aykanat, Buket

Yüksek Lisans, Elektrik Elektronik Mühendisliği Bölümü Tez Yöneticisi

:Doç. Dr. Sencer Koç

Eylül 2010, 97 sayfa

Pratik hayatta oluşan anten örüntüsü çeşitli hatalardan dolayı ideal anten örüntüsünden sapmaktadır. Bunun nedenleri olarak anten elemanlarının birbirlerine olan

etkileri,

kablo

uyumsuzlukları,

anten

elemanlarının

pozisyonundan

kaynaklanabilecek hatalar ve anten RF kanalları arasındaki faz ve genlik dengesizliğini sayabiliriz. Tüm bu hatalar anten performansında düşüşe sebep olmaktadır. Bu hatalar yüzünden anten yan huzme seviyesinde artış, kazançta azalma ve yön bulma algoritmalarının performansında düşüş gözlemlenmektedir. Anten performansını arttırmak için bu hataların kalibre edilmesi gerekmektedir. Literatürde bu konu ile ilgili bir çok kalibrasyon metodu bulunmaktadır. Bu tezde kullanılan kalibrasyon metodunda ideal sinyal ile ölçümler arasında lineer bir transformasyon olduğu varsayılmıştır. Alınan ölçümler ile kalibrasyon matrisi C oluşturulmuş ve başarılı kalibrasyon için gerekli ölçüm sayısı araştırılmıştır.

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Kalibrasyon metodunun performansı ölçüm toplanan açı aralığına bağlı olarak değişebilmektedir. Bu nedenle ölçüm toplanan açı aralığının genişliği incelenmiştir. Gerçek hayatta ölçümler eşit aralıklarla veya rastgele açılardan toplanabilir. Bu durumun kalibrasyon metodunun performansı üzerine olan etkileri gösterilmiştir. Bunlara ek olarak, performans algoritmasının gürültü altındaki performansı değerlendirilmiştir. Analizler hem yancada hem de yükselişte yapılmıştır. Simülasyonlarda MATLABTM ve Ansoft HFSS programları kullanılmıştır. Anahtar Kelimeler: Online kalibrasyon, düzgün dairesel dizi anten

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ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my supervisor Assoc. Prof Sencer Koç for his guidance, advice and help that he provided throughout this study. Also, I would like thank to my manager, my colleagues and friends who contributed this thesis by their understanding, patience and encouragement. I wish to express my appreciation to ASELSAN Inc. for all facilities provided. Mostly, I am grateful to my parents, my brother for their love, confidence, patience and encouragement they provided me not only throughout this thesis but throughout my life. Special thanks to my cousin Aslı for her patient and help during documentary.

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TABLE OF CONTENTS

ABSTRACT ............................................................................................................. iv CALIBRATION OF UNIFORM CIRCULAR ARRAYs ................................... iv ÖZ ............................................................................................................................. vi ACKNOWLEDGEMENTS..................................................................................viii TABLE OF CONTENTS........................................................................................ ix LIST OF TABLES .................................................................................................. xi LIST OF FIGURES ............................................................................................... xii CHAPTER 1 INTRODUCTION ....................................................................................... 1 1.1 Motivation and the objective of Thesis.................................................. 3 1.2 Thesis Organization ............................................................................... 4 2 ARRAY REPRESENTATION and SIGNAL MODEL........................... 5 2.1 Uniform Circular Array Representation ................................................ 5 2.2 Signal Model.......................................................................................... 8 3 ERROR SOURCES

ıN ANTENNA ARRAY and

MUTUAL

COUPLING................................................................................................... 11 3.1 Error Sources in Antenna Arrays ......................................................... 12 3.2 Mutual Coupling .................................................................................. 12 3.3 HFSS Simulation ................................................................................. 15 4 CALIBRATION METHOD ..................................................................... 23

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4.1 Online and Offline Calibration Techniques ......................................... 23 4.2 Calibration Method .............................................................................. 24 5 SIMULATION RESULTS........................................................................ 29 5.1 Elevation Pattern Calibration.............................................................. 31 5.1.1 Calibration with Equally Spaced Measurements, Noiseless Case ............................................................................................................ 31 5.1.2 Calibration with Randomly Spaced Measurements, Noiseless Case..................................................................................................... 54 5.1.3 Calibration under Noise ............................................................. 59 5.2 Azimuth Pattern Calibration ............................................................... 68 5.2.1 Calibration with Equally Spaced Measurements, Noiseless Case ............................................................................................................ 69 5.2.2 Calibration with Randomly Spaced Measurements, Noiseless Case..................................................................................................... 84 5.2.3 Calibration under Noise ............................................................. 87 6 CONCLUSIONS ........................................................................................ 94 REFERENCES....................................................................................................... 96

x

LIST OF TABLES

TABLES Table 1 Parameters of HFFS simulation .................................................................. 16 Table 2 Percentage error between calibrated and ideal patterns for 10° calibration data collection interval width at φ = 0 ................................................................ 62 Table 3 Percentage error between calibrated and ideal patterns for 10° calibration data collection interval width at φ = 30 .............................................................. 63 Table 4 Percentage error between calibrated and ideal patterns for 10° calibration data collection interval width at φ = 31 .............................................................. 64 Table 5 Percentage error between calibrated and ideal patterns for different calibration data collection interval widths at φ = 31 for equally spaced measurements ...................................................................................................... 68 Table 6 Percentage error between calibrated and ideal patterns for different calibration data collection interval widths at φ = 31 for randomly spaced measurements ...................................................................................................... 68 Table 7 Percentage error between calibrated and ideal patterns for different calibration data collection interval widths at θ = 40 for equally spaced measurements ...................................................................................................... 88 Table 8 Percentage error between calibrated and ideal patterns for different calibration data collection interval widths at θ = 40 for randomly spaced measurements ...................................................................................................... 88

xi

LIST OF FIGURES

FIGURES Figure 3-1 Mutual coupling mechanism in receiving mode .................................... 13 Figure 3-2 HFSS model ........................................................................................... 16 Figure 3-3 Isolated element pattern of dipole placed to x = 8 / k ............................ 17 Figure 3-4 Isolated element pattern of dipole placed to x = 4 / k ............................ 17 Figure 3-5 Active element pattern when a = 8 / k ................................................... 18 Figure 3-6 Active element pattern when a = 4 / k ................................................... 18 Figure 3-7 The active array pattern when a = 8 / k ................................................ 19 Figure 3-8 The active array pattern when a = 4 / k ................................................. 20 Figure 3-9 Antenna elevation patterns with and without mutual coupling for a = 8 / k ............................................................................................................... 21

Figure 3-10 Antenna elevation patterns with and without mutual coupling for a = 4 / k ............................................................................................................... 21

Figure 3-11 Antenna azimuth patterns at azimuth with and without mutual coupling for a = 8 / k ......................................................................................................... 22 Figure 3-12 Antenna azimuth patterns at azimuth with and without mutual coupling for a = 4 / k ......................................................................................................... 22 Figure 5-1 Block diagram of matlab program.......................................................... 30 Figure 5-2 Comparison of calibrated, distorted and ideal antenna patterns at φ = 14° ............................................................................................................................. 33 Figure 5-3 Residual errors after and before calibration at φ = 14° .......................... 33 Figure 5-4 Comparison of calibrated, distorted and ideal antenna patterns at φ = 16° ............................................................................................................................. 34 Figure 5-5 Residual errors after and before calibration at φ = 16° .......................... 34 Figure 5-6 Comparison of calibrated, distorted and ideal antenna patterns at φ = 30° ............................................................................................................................. 35

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Figure 5-7 Residual errors after and before calibration at φ = 30° ......................... 35 Figure 5-8 Comparison of calibrated, distorted and ideal antenna patterns at φ = 31° ............................................................................................................................. 36 Figure 5-9 Residual errors after and before calibration φ = 31° ............................. 36 Figure 5-10 Effect of the number of measurements when data is collected from 0°30° interval .......................................................................................................... 39 Figure 5-11 Effect of the number of measurements when data is collected from 30°60° interval .......................................................................................................... 39 Figure 5-12 Effect of the number of measurements when data is collected from 60°90° interval .......................................................................................................... 40 Figure 5-13 Effect of the number of measurements when data is collected from 90°120° interval ........................................................................................................ 40 Figure 5-14 Effect of the number of measurements when data is collected from 120°- 150° interval .............................................................................................. 41 Figure 5-15 Effect the number of measurements when data is collected from 150°180° interval ........................................................................................................ 41 Figure 5-16 Difference between ideal and distorted array patterns ......................... 42 Figure 5-17 Comparison of patterns after and before calibration for φ = 16° ......... 42 Figure 5-18 Error after calibration for increasing calibration data collection interval at θ = 10° ............................................................................................................ 45 Figure 5-19 Error after calibration for increasing calibration data collection interval at θ = 45° ............................................................................................................ 45 Figure 5-20 Error after calibration for increasing calibration data collection interval at θ = 90° ............................................................................................................ 46 Figure 5-21 Error after calibration for increasing calibration data collection interval width at θ = 170° ................................................................................................ 46 Figure 5-22 Comparison of calibrated, distorted and ideal antenna patterns for different number of measurements...................................................................... 47 Figure 5-23 Effect of number of measurements when data is collected from 0°- 180° interval................................................................................................................. 47

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Figure 5-24 Error after calibration for 10°calibration data collection interval width at θ = 10° ............................................................................................................ 48 Figure 5-25 Error after calibration for 10°calibration data collection interval width at θ = 45° ............................................................................................................ 48 Figure 5-26 Error after calibration for 10°calibration data collection interval width at θ = 90° ............................................................................................................ 49 Figure 5-27 Error after calibration for 10°calibration data collection interval width at θ = 170° .......................................................................................................... 49 Figure 5-28 Error after calibration for 30° calibration data collection interval width at θ = 10° ............................................................................................................ 50 Figure 5-29 Comparison of array patterns after and before sector by sector calibration for 10° calibration data collection interval width.............................. 51 Figure 5-30 Error after sector by sector calibration for 10° calibration data collection interval width...................................................................................... 51 Figure 5-31 Comparison of array patterns after and before sector by sector calibration for 30° calibration data collection interval width.............................. 52 Figure 5-32 Error after sector by sector calibration for 30° calibration data collection interval width...................................................................................... 52 Figure 5-33 Comparison of array patterns after and before sector by sector calibration for 180° calibration data collection interval width............................ 53 Figure 5-34 Error after sector by sector calibration for 180° calibration data collection interval width...................................................................................... 53 Figure 5-35 Effect of number of measurements when data is collected from 0°- 30° interval................................................................................................................. 55 Figure 5-36 Effect of number of measurements when data is collected from 30°- 60° interval................................................................................................................. 55 Figure 5-37 Effect of number of measurements when data is collected from 60°- 90° interval................................................................................................................. 56 Figure 5-38 Effect of number of measurements when data is collected from 90°120° interval ........................................................................................................ 56

xiv

Figure 5-39 Effect of number of measurements when data is collected from 120°150° interval ........................................................................................................ 57 Figure 5-40 Effect of number of measurements when data is collected from 150°180° interval ........................................................................................................ 57 Figure 5-41 Comparison of array patterns after and before sector by sector calibration for 30° calibration data collection interval width.............................. 58 Figure 5-42 Error after sector by sector calibration for 30° calibration data collection interval width...................................................................................... 58 Figure 5-43 Power patterns at φ = 0 , φ = 30 and φ = 31 ........................................ 61 Figure 5-44 Comparison of array patterns after and before sector by sector calibration for 30° calibration data collection interval width at φ = 0 ................ 62 Figure 5-45 Comparison of array patterns after and before sector by sector calibration for 30° calibration data collection interval width at φ = 30 .............. 63 Figure 5-46 Comparison of array patterns after and before sector by sector calibration for 10° calibration data collection interval width at φ = 31 .............. 64 Figure 5-47 Comparison of array patterns after and before sector by sector calibration for 30° calibration data collection interval width at φ = 31 .............. 65 Figure 5-48 Comparison of array patterns after and before sector by sector calibration for 60° calibration data collection interval width at φ = 31 .............. 66 Figure 5-49 Comparison of array patterns after and before sector by sector calibration for 90° calibration data collection interval width at φ = 31 .............. 67 Figure 5-50 Comparison of patterns after and before calibration at θ = 40° ......... 71 Figure 5-51 Residual errors after and before calibration at θ = 40° ....................... 71 Figure 5-52 Comparison of patterns after and before calibration at θ = 50° .......... 72 Figure 5-53 Residual errors after and before calibration at θ = 50° ....................... 72 Figure 5-54 Comparison of patterns after and before calibration at θ = 90° .......... 73 Figure 5-55 Residual errors after and before calibration at θ = 90° ....................... 73 Figure 5-56 Comparison of patterns after and before calibration at θ = 40° ......... 74 Figure 5-57 Comparison of patterns after and before calibration at θ = 50° .......... 74

xv

Figure 5-58 Effect of number of measurements when data is collected from 0°- 30° interval................................................................................................................. 76 Figure 5-59 Effect of number of measurements when data is collected from 0°- 90° interval................................................................................................................. 76 Figure 5-60 Effect of number of measurements when data is collected from 0°- 180° interval................................................................................................................. 77 Figure 5-61 Error after calibration for increasing calibration data collection interval at φ = 10° ............................................................................................................. 77 Figure 5-62 Error after calibration for increasing calibration data collection interval at φ = 210° .......................................................................................................... 78 Figure 5-63 Error after calibration for increasing calibration data collection interval at φ = 310° .......................................................................................................... 78 Figure 5-64 Comparison of array patterns after and before sector by sector calibration for 10° calibration data collection interval width.............................. 80 Figure 5-65 Residual errors after calibration for 10° calibration data collection interval width....................................................................................................... 80 Figure 5-66 Comparison of array patterns after and before sector by sector calibration for 20° calibration data collection interval width.............................. 81 Figure 5-67 Residual errors after calibration for 20° calibration data collection interval width....................................................................................................... 81 Figure 5-68 Comparison of array patterns after and before sector by sector calibration for 90° calibration data collection interval width.............................. 82 Figure 5-69 Residual errors after calibration for 90° calibration data collection interval width....................................................................................................... 82 Figure 5-70 Comparison of array patterns after and before sector by sector calibration for 180° calibration data collection interval width............................ 83 Figure 5-71 Residual errors after calibration for 180° calibration data collection interval width....................................................................................................... 83 Figure 5-72 Effect of number of measurements when data is collected from 0°- 30° interval................................................................................................................. 84

xvi

Figure 5-73 Effect of number of measurements when data is collected from 0°- 90° interval................................................................................................................. 85 Figure 5-74 Effect of number of measurements when data is collected from 0°- 180° interval................................................................................................................. 85 Figure 5-75 Comparison of array patterns after and before sector by sector calibration for 20° calibration data collection interval width.............................. 86 Figure 5-76 Residual errors after calibration for 20° calibration data collection interval width....................................................................................................... 86 Figure 5-77 Power patterns at θ = 40 , θ = 50 and θ = 90 ..................................... 88 Figure 5-78 Comparison of array patterns after and before sector by sector calibration for 30° calibration data collection interval width at θ = 90 .............. 89 Figure 5-79 Residual errors after calibration for 20° calibration data collection interval width at θ = 90 ...................................................................................... 89 Figure 5-80 Comparison of array patterns after and before sector by sector calibration for 30° calibration data collection interval width at θ = 50 .............. 90 Figure 5-81 Residual errors after calibration for 20° calibration data collection interval width at θ = 50 ...................................................................................... 90 Figure 5-82 Comparison of array patterns after and before sector by sector calibration for 30° calibration data collection interval width at θ = 40 ............. 91 Figure 5-83 Residual errors after calibration for 30° calibration data collection interval width at θ = 40 ...................................................................................... 91 Figure 5-84 Comparison of array patterns after and before sector by sector calibration for 30° calibration data collection interval width at θ = 40 ............. 92 Figure 5-85 Comparison of array patterns after and before sector by sector calibration for 30° calibration data collection interval width at θ = 40 ............. 92 Figure 5-86 Comparison of array patterns after and before sector by sector calibration for 90° calibration data collection interval width at θ = 40 ............. 93

xvii

CHAPTER 1

INTRODUCTION

With development of antenna and microwave technology, array antennas have been popular in many applications such as radars, wireless communication systems etc. However, the use of antenna arrays brings many difficulties. Elements of antenna arrays influence each other (known as mutual coupling), mismatches in cables and element positions affect the antenna radiation pattern and also unequal gain and phase characteristics of RF receiver distorts the received signal. These effects generally degrade the array performance. They cause an increase in sidelobe levels with an accompanying decrease in gain, [1]. Also these errors limit the performance of direction finding (DF) algorithms. So, in order to have low sidelobe level, good performance in direction finding and beamforming, calibration is necessary. Calibration of antenna arrays has been discussed for several decades and many algorithms are proposed. However, it is hard to implement most of the calibration algorithms in analog form, it needs digital array data for each antenna element which was the problem to deal with for earlier decades. Nowadays, with the help of technology digital antenna arrays can be implemented and these algorithms can be realized in real systems. In the literature many calibration techniques have been studied to decrease array errors. H. Steyskal and J. S. Herd [2] deal with the mutual coupling error and treat mutual coupling coefficients as Fourier transform coefficients of array element 1

patterns. Then, they try to obtain desired signal from received signal by using these coefficients. It is an offline calibration technique and it requires isolated element patterns. H.G.Park and S.C.Bang [10] improved a subspace optimization method and proposed a cost function which minimizes array errors. Type of error is not important for this method. It does not need element pattern (hard to know in practice). It obtains the element pattern however it assumes multipath is negligible. Also C.A. Balanis and Z. Huang have many studies which aim to calibrate the mutual coupling effects, [6], [11], [12]. These works use eight element circular dipole arrays as in this work. In these works, signal of interest and signal of not interest are generated. It is tried to obtain desired signal and eliminate the undesired ones by using LMS and MMSE algorithms. Although these works do not deal with the signals received from different angles to calibrate the array, they give different look angles and useful results. In addition to these works Hung E. K. L. [7] suggested a matrix construction calibration method developed for receive arrays. This method constructs a calibration matrix which is resistant to mutual coupling errors, positioning errors (due to temperature etc.) and unequal gain and phase responses of antenna array. This method is also tolerant to multipath or interference signals. It is also successful when calibration data is collected from different angle intervals. For these reasons, this calibration method is selected and used in this work. To observe the performance of this algorithm we need the distorted signal and ideal (desired) signal and we must give these signals as input to the calibration algorithm. Ideal signal is generated by using mathematical model. Distorted signal is obtained from HFSS tool since we do not have any experimental set-up to collect measurements. Then, we exported the distorted signal to MATLAB and observed the performance of calibration. In this work, only the mutual coupling effects are included in the simulations since there are many mutual coupling models in the literature that can be used and it severely affects the array pattern.

2

1.1 Motivation and the objective of Thesis Motivation of the thesis is to determine if it is possible to calibrate an antenna array on the field without resorting to lengthy measurements. For this purpose we may assume that there are one or more known radiation sources. The radiation from these sources will be intercepted whenever the antenna of concern gets close to these sources. The question is whether it would be possible or not to calibrate the antenna by using the signals received from these known sources. In this thesis we will investigate one calibration method developed by Eric. K. L. Hung, [7]. An important assumption of this method is that there is a linear transformation between ideal and actual received signal. This assumption can be justified if the non-ideal effects are small since the small changes can be linearized about the design point. Throughout the thesis we try to answer the following questions. •

How many measurements are necessary for a successful calibration?

•

Whether it makes any difference to take measurements from equally or unequally spaced angles?

•

How the performance of calibration method is affected by the signal to noise ratio (SNR)?

•

Is it possible to calibrate the array over all angles from data received from a restricted interval?

•

If not, how does the calibration performance depend on the data collection interval?

3

1.2 Thesis Organization

First chapter of the thesis is the introduction part in which scope, objective and organization of the thesis is given. In Chapter 2 antenna array representation and signal models are presented. Chapter 3 contains error sources in antenna array and HFSS simulation results. Chapter 4 considers the method of mutual coupling compensation. In chapter 5 simulation results carried out on MATLABTM are given. Finally, a conclusion is provided in Chapter 6.

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CHAPTER 2

ARRAY REPRESENTATION AND SIGNAL MODEL

In this chapter, we focus on the array configuration and signal models that are used for problem formulation. In the first part of this chapter, uniform circular array geometry and mathematical formulation of array factor and element factor are explained. In this formulation mutual coupling and other non-ideal effects are all neglected. In the second part, signal model that is used throughout the text is defined and formulation is presented.

2.1 Uniform Circular Array Representation Array configuration consists of two parts. First part is array geometry. Array geometry defines the physical locations of the antenna elements: It can be linear, circular, planar or volumetric. Within each geometry, array elements can be distributed uniformly or non-uniformly. Second part of array configuration is the array pattern. In general, it is assumed that all elements in the array have isotropic element patterns. When the combination of these elements is considered, they form the array pattern which is a function of array geometry.

5

In this thesis, circular array geometry is chosen because uniform circular arrays have better coverage, especially in azimuth beamforming than uniform linear arrays, when mutual coupling is not considered, [6]. The geometry of circular array is shown in Figure 2-2, [6].

Figure 2-1 Geometry of N element dipole uniform circular array When N isotropic λ / 2 length dipole elements are equally spaced on x − y plane along a circular ring of radius a , and if r >> a is assumed, the normalized electric field can be expressed as, [5]:

E n (r ,θ ,φ ) =

e − jkr r

N

∑w e n

+ jka sin θ cos(φ −φn )

(2-1)

n =1

where wn is the excitation coefficient of the n th element with α n being the phase of the excitation and I n being its amplitude, i.e.,

6

wn = I n e jα n .

(2-2)

φn is the angular position of the n th element defined as φ n = 2π

n , N

n = 0,1,K N − 1 .

(2-3)

[AF (θ ,φ )]

(2-4)

When (2-1) and (2-2) are combined we get e E n (r ,θ ,φ ) =

− jkr

r

where the array factor can be expressed as N

AF (θ , φ ) = ∑ I n e + j [ka sin θ cos(φ −φn )+α n ] .

(2-5)

n =1

If the coefficients α n are chosen as

α n = −ka sin θ 0 cos(φ0 − φn )

(2-6)

the array factor will be maximized in the direction, (θ 0 , φ0 ) . Therefore, after some manipulations the corresponding array factor can be written as N

AF (θ , φ ) = ∑ I n e + jkρ0 (cos(φn −δ ) )

(2-7)

n =1

where

 sin θ sin φ − sin θ 0 sin φ0    sin θ cos φ − sin θ 0 cos φ0 

δ = tan −1 

[

ρ 0 = a (sin θ cos φ − sin θ 0 cos φ0 )2 + (sin θ sin φ − sin θ 0 sin φ0 )2

7

(2-8)

]

1

2

.

(2-9)

When the array factor is obtained, total far field of array is found by multiplication of array factor and field of single antenna element at the reference point, usually at the origin. For half-wavelength dipole, electric field of single antenna element can be expressed as, [5]:  π  cos cosθ    I e 2  ,   Eθ = jη 0 2πr  sin θ      − jkr

(2-10)

and normalized electric field of single antenna element will be expressed as:  π cos cosθ  . 2 Eθ n = sin θ

(2-11)

So,

E ( total) = [E (single element at reference point )]× [array factor ]

(2-12)

Note that the mutual coupling and other non-ideal effects are not included in this derivation.

2.2 Signal Model The second issue is the structure of the signal. In this particular work, signals are assumed to come from different directions. If mutual coupling and other non-ideal effects do not exist, they are known signals. The other issue is the structure of the noise. A noise component is included as a white Gaussian noise process, statistically independent from array element to element.

8

When mutual coupling and other non-ideal effects are ignored, the signal received by the N element ideal array can be written as, [7]

x(θ ,φ ) = [x1 (θ ,φ ) x2 (θ ,φ ) L x N (θ ,φ )]

T

(2-13)

where θ , φ denotes the angle of arrival (AoA) (elevation and azimuth) of the signal. For signal arriving from angle θ m ,φ m , the received signal is x(θ m, φ m ) which will be denoted by x m . This signal can be written as

x(θ m , φ m ) = α m a(θ m , φm )

(2-14)

where α m is the signal amplitude and a(θ m ,φm ) is a vector defined by array geometry. For N element uniform circular array an (θ m , φm ) can be written as a n (θ m , φ m ) = e + jka sin θ m cos(φm −φn ) .

(2-15)

In an actual array, the received signal can be expressed as

y M = Bx M + e M

(2-16)

where B is an N × N matrix which relates the ideal received array signal and actual received array signal. B can depend on relative element gain and phase responses of elements, mismatches in the cables, mutual coupling among the array elements, and errors in element locations. In addition to these factors, in practice, B may depend on the direction of arrival(s) of the signal. However, to have a solution of problem, we assume that B is independent of AoA at least over a sector. This assumption is justified by measurements as indicated in [7]. Although many nonideal effects can be included in matrix B , in this work, only mutual coupling effects are considered because there exist many models for mutual coupling in the literature, [6], [9], [13].

9

Once B is determined and non-singular, the ideal signal x M can be estimated and the AoA of the signal can be determined by using one of the many DoA algorithms that can be found in the literature, [3], [7], [14]. Also, noise and interference can be included in the analysis. In equation (2-16) e M denotes the sum of noise and interference. For this particular work, we consider uniform circular array geometry with signals received from known directions and additive, zero mean, complex, Gaussian noise. Interference is neglected. When the signal model and array representation of field are combined as in equation (2-14), a(θ m ,φm ) will represent array manifold, given in equation (2-7), which is maximized at θ 0 = 0 and φ0 = 0 and α m will represent the signal amplitude which is determined by element factor given in equation (2-7) and the excitation amplitude I n given in equations (2-11) and (2-2). In this work, analyses are carried out in both azimuth and elevation directions. Further assumptions are stated in the discussions wherever they are necessary.

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CHAPTER 3

ERROR SOURCES IN ANTENNA ARRAY AND MUTUAL COUPLING

In the first part of this chapter, we will give a brief explanation about the error sources in antenna array systems. In practice, there exist many non-ideal effects that distort the received signal, such as relative element gain and phase responses at the elements, mismatches in the cables, mutual coupling among the array elements, and errors in element locations, etc. In this work we will assume that such effects can be modeled by a linear transformation as in equation (2-16). This assumption can be justified if the non-ideal effects are small, since the small changes can be linearized about the design point. Although many non-ideal effects can be included in matrix B , in this work, only mutual coupling effects are included, since there exist many surveys that model mutual coupling effects as a matrix, [6], [9], [13] and mutual coupling also severely affects the array pattern. In the second part, we are going to explain the mechanism of mutual coupling and define the active and isolated element patterns. Lastly, we will present the simulations done by Ansoft HFSS software package. HFSS is used to obtain the active element patterns under mutual coupling. Then we

11

will compare active array pattern obtained by HFSS with the ideal array pattern and observe the distortion on the array pattern.

3.1 Error Sources in Antenna Arrays In practical systems, uncertainties of array may arise from following factors, [7], [11]:

o Mutual coupling between elements o Unequal gain and phase characteristics of RF receiver (channel error) o Mismatches in the cables o Error in array element locations o Environmental effects This means that in an actual implementation of an array system, the actual received signal is different from ideal one due to these error sources. This will result in an increased sidelobe level, distorted beam shape and DF errors. Therefore, the methods of measuring, modeling and compensation of these errors are important, [3]. With the help of these methods, estimate of ideal received signal will be obtained before processing the array data. If all these effects can be assumed to be linear, they can be modeled by a matrix. In this work, only the mutual coupling effects are included in the simulations because there are many mutual coupling models in the literature that can be used.

3.2 Mutual Coupling When two or more antennas are located close to each other or when there exists an obstacle, such as body of airplane etc., the current distribution, the radiated field and in turn the input impedance of antenna element is affected by the field radiated 12

by other elements or obstacles. Thus, the antenna performance does not depend only on its own current but also depends on the currents on other radiating elements. This effect is named as mutual coupling. Mutual coupling is more complicated in small arrays since each element have different pattern due to edge effects. On the other hand, in large arrays, each element sees essentially the same environment, and mutual coupling can be taken into account by modifying the element pattern, [2]. The mechanism of mutual coupling in receiving mode is shown in Figure 3-1. Suppose that a plane wave (0) is incident, and it strikes antenna m first where it induces a current. Part of the incident wave will be scattered into the space (2), the other will be directed towards antenna n (3) where it will add with the incident wave (0), and part will travel into its feed (1). It is evident that the amount of energy received by each element of an antenna array is the vector sum of the directed waves and those that are coupled to it parasitically from the other elements, [6].

Figure 3-1 Mutual coupling mechanism in receiving mode

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The effect of mutual coupling on the performance of array depends on the following, [6]:

o Antenna type and its design parameters o Relative positioning of the elements in the array o Feed of array elements o Scan direction of the array These parameters influence the performance of the array by varying its impedance, reflection coefficients and overall antenna pattern. Although it is hard to quantify for physical systems, mutual coupling can be quantified by the following measurable quantities, [8]:

o Mutual impedance o Mutual admittance o Scattering parameters. The above measurable quantities are related to each other by simple algebraic relations. When the effects of mutual coupling are compensated, knowledge about the array will be improved. Low sidelobe level, deterministic null pattern can be obtained. Also, more accurate DF calculations can be done. If mutual coupling is neglected, radiation pattern of an antenna array, composed of identical elements, is the product of element factor and array factor. The antenna element pattern, when antenna element stands alone (means all environmental effects are ignored), is named as isolated element pattern. Unfortunately in practice, environmental effects such as mutual coupling may cause major distortions in element patterns. The element pattern in a real array environment is called as an

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active element pattern. If the isolated element patterns are used instead of active element patterns, significant degradation can be observed in array performance, [13]. Therefore, to properly account for the mutual coupling effects, knowledge of active element pattern is important. There are many surveys in the literature which aim to relate the isolated element pattern and active element pattern, [6], [9] and [13]. These surveys state that the relationship between isolated element pattern and active element pattern can be expressed as a matrix. In this work, we obtained active element patterns from HFSS software. In the next section, HFSS simulation results are discussed and array patterns with and without mutual coupling will be compared.

3.3 HFSS Simulation Firstly, 8 element uniform circular array is created in HFSS as shown in Figure 3-2 and its parameters are set as in Table 1. Radiation boundary conditions should be chosen such as to satisfy far field conditions, kr >> 1 and r >> D

(3-1)

where D is the dipole length, k is the wave number and r is the far field distance. In HFSS simulation these radiation boundaries are set as in Table 1 since it is observed that the more increase does not provide any improvement. Also, dipole radius is chosen as to satisfy the thin dipole assumption a