Adaptive Antenna Array Beamforming and Power Management

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ADAPTIVE ANTENNA ARRAY BEAMFORMING AND POWER MANAGEMENT

WITH ANTENNA ELEMENT SELECTION

Sajjad Abazari Aghdam

A Dissertation Submitted to the Faculty of

The College of Engineering and Computer Science

In Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

Florida Atlantic University

Boca Raton, FL

December 2016

ACKNOWLEDGEMENTS

I am using this opportunity to express my gratitude to everyone who supported me in the throughput of this research and made the way of research smooth for me. First of all,

I would like to convey my sincere gratitude to my advisor and committee chair, Dr.

Jonathan Bagby, Department of Computer and Electrical Engineering and Computer

Science, who provided me with his aspiring guidance, invaluable constructive criticism and friendly advices. He always supported me throughout my research with dedication, patience and encouragement.

I also gratefully thank my committee members; Dr. Lorenzo Poncedeleon,

Distinguished Member of the Technical Staff at Motorola, for his exquisite comments towards the perfection of my dissertation and his unique support in my academic career;

Dr. Vichate Ungvichian, Department of Computer and Electrical Engineering and

Computer Science, for his precious recommendations and suggestions; and De Groff

Dolore, Department of Computer and Electrical Engineering and Computer Science, for always being ready to passionately answer my questions.

I would like also to thank from AntennaWorld Inc. Technology Company for providing us the opportunity to work on this extensive research project. Also providing us the opportunity to use their anechoic chamber facility for Antenna measurement analysis and National Science Foundation, under Grant No. 01141, for the support of this work.

I personally would like to extend my sincerst thanks to Dr. Ali Zilouchian,

Associate Dean for Academic Affairs of the College of Engineering for his key advises, direction, and assistant when I needed it. A special thanks to Dr. Nurgun Erdol, Chair of

Department of Computer and Electrical Engineering and Computer Science for supporting entire my PhD program. I would also like to send a heartfelt acknowledgment to my friends at Florida Atlantic University who made all these years vey joyful and memorable.

Finally, and most importantly, I could not have accomplished this without the support of my family. To mom and dad you have been supreme, you have nurtured my learning, supported my dreams. Reza and Somayeh my dear brother and sister, receive my deepest gratitude and love for their dedication and the many years of support during my education.

ABSTRACT

Author: Sajjad Abazari Aghdam

Title: Effects of Adaptive Antenna Array Beamforming and Power Management with Antenna Element Selection

Institution: Florida Atlantic University

Dissertation Advisor: Dr. Jonathan Bagby

Degree: Doctor of Philosophy

Year: 2016

This research is the array processing help wireless communication techniques to increase the signal accuracy. This technique has an important part of prevalent applications.

The wireless communication system, radar, and sonar. Beamforming is one of methods in array processing that filters signals based on their capture time at each element in an array of antennas spatially. Numerous studies in adaptive array processing have been proposed in the last several decades, which are divided in two parts. The first one related to non- adaptive beamforming techniques and the next one related to digitally adaptive

Beamforming methods. The trade-off between computational complexity and performance make them different. In this thesis, we concentrate on the expansion of array processing algorithms in both non-adaptive and adaptive ones with application of beamforming in 4G mobile antenna and radar systems. The conventional and generalized side-lobe canceller

(GSC) structures beamforming algorithms were employed with a phase array antenna that

changed the phase of arrivals in array antenna with common phased array structure antennas. An eight-element uniform linear array (ULA), consisting of di-pole antennas, represented as the antenna array. An anechoic chamber measures the operation of beamforming algorithms performance. An extended modified Kaiser weighting function is proposed to make a semi-adaptive structure in phased array beamforming. This technique is extended to low complexity functions like hyperbolic cosine and exponential functions.

Furthermore, these algorithms are used in GSC beamforming. The side-lobe levels were so lower than other algorithms in conventional beamforming around -10 dB.

On the other hand, a uniform linear arrays for smart antenna purposes designed to utilize in implementing and testing the proposed algorithms. In this thesis, performance of smart antenna with rectangular aperture coupled microstrip linear array which experimental investigations carried out for obtaining X-band operation of rectangular microstrip antenna by using aperture coupled feeding technique. Frequency range set at approximately 8.6 to 10.9 GHz, by incorporating frequency range of the antenna resonates for single wideband with an impedance bandwidth of 23%. The enhancement of impedance bandwidth and gain does not affect the nature of broadside radiation characteristics. This thesis describes the design, operation, and realization of the beamforming such as Sidelobe level (SLL) control and null forming array antenna are examined with the prototype. An antenna radiation pattern beam maximum can be simultaneously placed towards the intended user or Signal of interest (SOl), and, ideally nulls can be positioned towards directions of interfering signals or signals not of interest (SNOIs).

Finally, we focused on the adaptive digitally algorithms in compact antenna that faces with mutual coupling. The variable step-size normalized lease mean square (VS-

vii

NLMS) algorithm is implemented in beamforming. This algorithm utilizes continuous adaptation. The weights are attuned that the final weight vector to the most satisfied result.

The gradient vector can be achieved by iterative beamforming algorithm from the available data. This algorithm is compared with LMS, NLMS, VSS-NLMS algorithms, it is determined that the VSS-NLMS algorithm is better performance to other algorithms.

Finally, we introduced novel adaptive IP-NNLMS beamformer. This beamformer reaches to faster convergence and lower error floor than the previous adaptive beamformers even at low SNRs in presence of mutual coupling. The experimental results verified the simulation results that the proposed technique has better performance than other algorithms in various situations.

viii DEDICATION

To my parents

To my brother, Reza

To my Sister, Somayeh ADAPTIVE ANTENNA ARRAY BEAMFORMING AND POWER

MANAGEMENT WITH ANTENNA ELEMENT SELECTION

LIST OF FIGURES ...... xv

Fundamentals Of Antenna Arrays And Beamforming ...... 1

1.1 Introduction ...... 1

1.1.1 Pattern of a Generalized Array ...... 2

1.1.2 Array factor ...... 3

1.1.3 Array Pattern ...... 4

1.1.4 Phase and Time Scanning ...... 4

1.1.5 Introduction of Beamforming ...... 5

1.1.6 Array Response Vector ...... 7

1.1.7 Spatial Polarization Signature ...... 7

1.1.8 Spatial Polarization Signature Matrix ...... 7

1.1.9 Signals and Noise ...... 8

1.1.10 Optimum Weights ...... 9

1.1.11 Adaptive Algorithms ...... 9

1.1.12 Least Mean Squares (LMS)...... 10

1.1.13 Direct Sample Covariance Matrix Inversion (DMI) ...... 10

1.1.14 Recursive Least Squares (RLS)...... 11

1.1.15 Other Techniques ...... 12

1.2 Fundamentals of Antenna Arrays ...... 12

x 1.2.1 Maxwell’s Equation ...... 13

1.2.2 Pointing Vector ...... 14

1.2.3 Radiation Attributes ...... 16

1.3 Array Antenna ...... 18

1.3.1 Linear Array Antennas ...... 21

1.3.2 Rectangular Array Antennas ...... 24

1.3.3 Circular Array Antennas ...... 26

1.3.4 Conformal Array Antennas ...... 27

Design and Development of Rectangular Aperture Coupled Microstrip

Antennas with Application to Beamforming ...... 29

2.1 Introduction ...... 29

2.2 Microstrip Antenna Configuration ...... 30

2.2.1 Advantages of Microstrip Antenna ...... 31

2.2.2 Microstrip Radiators ...... 32

2.2.3 Excitation Techniques ...... 35

2.2.3.1 Transmission Line Feed...... 35

2.2.3.2 Coaxial Feed ...... 36

2.2.3.3 Aperture Coupling ...... 36

2.2.4 Equivalent Circuit ...... 37

2.3 Proposed Array Antenna Configuration ...... 39

2.3.1 Proposed Eight Antenna Configuration ...... 39

2.4 Experimental Comparison for Proposed Antenna ...... 44

2.5 Array Antenna Beamforming ...... 47

xi 2.5.1 Array Element Design ...... 48

Analysis of an Adjustable Modified Kaiser Weighting Function and Its

Application To Linear Array Antenna Beamforming ...... 59

3.1 Introduction ...... 59

3.2 Derivation of Proposed Weighting Function ...... 62

3.2.1 Kaiser Array Weighting Function ...... 62

3.2.2 Proposed Weighting Function ...... 62

3.2.3 Mainlobe Width Controlling Technique ...... 68

3.3 Theoretical and Characteristic Results ...... 70

3.4 Spectrum Comparisons ...... 74

3.5 Application to Linear Array Beamformer Design ...... 76

3.6 Conclusion ...... 78

Flexible Exponential Weighting Function Toward Sidelobe

Suppression In Antenna Arrays ...... 80

4.1 Introduction ...... 80

4.2 Derivation of Proposed Weighting Function ...... 83

4.2.1 Exponential Array Weighting Function ...... 83

4.2.2 Proposed Weighting Function ...... 83

4.2.3 Mainlobe Width Controlling Technique ...... 89

4.3 Theoretical and Characteristic Results ...... 91

4.4 Spectrum Comparisons ...... 92

4.5 Application to Linear Array High Side Lobe Suppression Design ...... 93

4.6 Conclusion ...... 97

xii Malleable Hyperbolic Cosine Weighting Function For Side Lobe

Suppression In Medical Image Beamforming ...... 98

5.1 Introduction ...... 98

5.2 Derivation of Proposed Weighting Function ...... 101

5.2.1 Hyperbolic Cosine Array Weighting Function ...... 101

5.2.2 Proposed Weighting Function ...... 102

5.2.3 Mainlobe Width Controlling Technique ...... 107

5.3 Theoretical and Characteristic Results ...... 108

5.4 Spectrum Comparison ...... 110

5.5 High Side Lobe Suppression in Medical Imaging Beamforming Design ...... 111

Adaptive Antenna Array Beamforming Using Variable-Step-Size

Normalized Least Mean Square ...... 116

6.1 Introduction ...... 116

6.2 System Model ...... 117

6.2.1 Adaptive Beamformer Structure ...... 117

6.2.2 Optimum Beamformer and Adaptive Beamformer ...... 119

6.2.3 Numerical Results ...... 121

6.2.3.1 Beam Pattern...... 122

6.2.3.2 Number of Array elements and error floor ...... 122

6.2.3.3 Experimental Result ...... 123

Two-Normalized Least Mean Square Using In Adaptive Array

Beamforming In Medical Imaging ...... 125

7.1 Introduction ...... 125

xiii 7.2 System Model ...... 127

7.2.1 Adaptive Beamformer Structure ...... 127

7.2.2 Optimum Beamformer and adaptive Beamformer ...... 128

7.3 Numerical Results ...... 130

7.3.1 Beam Pattern ...... 131

7.3.2 Error Vector Magnitude ...... 131

7.4 Biomedical Application and Experimental Result ...... 132

Adaptive Antenna Array Beamforming Using Inversely-Proportionate

Non Negative Least Mean Square ...... 135

8.1 Introduction ...... 135

8.2 System Model ...... 137

8.2.1 Adaptive beamformer Structure ...... 137

8.2.2 Optimum Beamformer and adaptive Beamformer ...... 139

8.3 Inversely-Proportionate Non Negative LMS algorithm for beamforming...... 140

8.3.1 Convergence in Mean Sense Analysis ...... 143

8.3.2 Transient excess mean-square error model ...... 143

8.4 The Beamformer Experimental Parameters ...... 146

8.5 Numerical Results ...... 147

8.6 Conclusion ...... 151

References ...... 153

xiv LIST OF FIGURES

FIGURE 1-1 AN ARBITRARY M-ELEMENT TRI-DIMENSIONAL ARRAY ...... 3

FIGURE 1-2 (A) A PHASE SCANNED LINEAR ARRAY (B) A TIME SCANNED LINEAR

ARRAY ...... 5

FIGURE 1-3 ADAPTIVE ARRAY ANTENNA BLOCK DIAGRAM ...... 6

FIGURE 1-4 ANTENNA FIELD AREA1.2.2 ANTENNA FIELD REGIONS ...... 15

FIGURE 1-5 WHERE PLOSS IS THE ANTENNA FATALITY, PLUS THE ANTENNA

DIRECTIVITY IS THE RATION WITH ANTENNA RADIATION INTENSITY AND THE

RADIATED ENERGY DIVIDED TO THE 4Π: ...... 17

FIGURE 1-6 UNIFORM LINEAR ARRAY ...... 22

FIGURE 1-7 ARRAY GAIN OF A ULA FOR N = 5 AND ΡL= 0.5Λ...... 24

FIGURE 1-8 UNIFORM RECTANGULAR ARRAY...... 25

FIGURE 1-9 UNIFORM CIRCULAR ARRAY ...... 27

FIGURE 2-1 MICROSTRIP ANTENNA CONFIGURATION ...... 31

FIGURE 2-2 RECTANGULAR PATCH ANTENNA ...... 31

FIGURE 2-3 FIELD CONFIGURATION OF (A) MICROSTRIP PATCH ANTENNA (B) SIDE

VIEW (C) TOP VIEW ...... 33

FIGURE 2-4 FAR FIELD BY REPLACING SLOT MODEL ...... 34

FIGURE 2-5 EXCITATION TECHNIQUES (A) DIRECT FEED, (B) COAXIAL FEED ...... 35

FIGURE 2-6 APERTURE COUPLED PATCH ANTENNA ...... 36

FIGURE 2-7 CONFIGURATION OF APERTURE COUPLED MICROSTRIP ANTENNA. (A)

SIDE VIEW. (B) TOP VIEW...... 38 xv FIGURE 2-8 GENERAL APERTURE COUPLED MICROSTRIP ANTENNA NETWORK MODEL .. 38

FIGURE 2-9 EQUIVALENT CIRCUIT FOR AN APERTURE COUPLED MICROSTRIP

ANTENNA...... 39

FIGURE 2-10 3-D SCHEMATIC AND FEEDING NETWORK OF ERMSA APERTURE OUPLED

MICROSTRIP LINEAR ARRAY ANTENNA ...... 40

FIGURE 2-11 COMPARISON BETWEEN MEASURED AND SIMULATION RETURN LOSS

DIAGRAM OF THE PROPOSED ANTENNA FOR ERMSA ...... 40

FIGURE 2-12 COMPARISON MEASURED RADIATION (POLAR PLOT) PATTERNS OF THE

FOR ERMSA. A) PHI=0 DEG. B) PHI=90 DEG...... 42

FIGURE 2-13 PROTOTYPE OF PROPOSED ERMSA ANTENNA ...... 42

FIGURE 2-14 GEOMETRY OF 8-WAY T-JUNCTION POWER DIVIDER.SIMULATED

PERFORMANCE OF 8-WAY TJUNCTION POWER ...... 43

FIGURE 2-15 RETURN LOSS DIAGRAM OF THE PROPOSED ANTENNA FOR SINGLE

ELEMENT, 4 (1X4)-ELEMENT AND 8 (1X8)-ELEMENT ...... 45

FIGURE 2-16 COMPARISON RADIATION (POLAR PLOT) PATTERNS OF THE SINGLE

ELEMENT, 4 (1X4)-ELEMENT AND 8 (1X8)-ELEMENT, PHI=0 DEG ...... 45

FIGURE 2-17 COMPARISON RADIATION (POLAR PLOT) PATTERNS OF THE FOR SINGLE

ELEMENT, 4 (1X4)-ELEMENT AND 8 (1X8)-ELEMENT , PHI=90 DEG ...... 46

FIGURE 2-18 COMPARISON PROPOSED ANTENNA GAINS OF SINGLE ELEMENT, 4 (1X4)-

ELEMENT AND 8 (1X8)-ELEMENT AS A FUNCTION OF FREQUENCY ...... 46

FIGURE 2-19 MEASURING FRONT TO BACK RATIO WITH NETWORK ANALYZER ...... 48

FIGURE 2-20 THE ANTENNA USES PHASE SHIFTERS TO STEER THE RADAR BEAM

ELECTRONICALLY OVER THE SCAN SECTOR. THE RADIO-FREQUEN ...... 49

FIGURE 2-21 AN IDEALIZED RADIATION PATTERN FROM A SINGLE ANTENNA

ELEMENT COVERS THE SCAN SECTOR, WITH SIGNAL STRENGTH DROP ...... 49

xvi FIGURE 2-22 WHEN ALL THE PHASE SHIFTERS OF THE ARRAY ARE PROPERLY

ALIGNED, THE ARRAY PRODUCES A MAIN ...... 50

FIGURE 2-23 CONCEPTUAL IMAGES OF BLIND-SPOT OCCURRENCE IN A PHASED-

ARRAY ANTENNA. THESE RESULTS ARE TYPICAL OF AN ARRAY DESIGNED

WITHOUT REGARD FOR ARRAY MUTUAL-COUPLING EFFECTS. A BLIND SPOT

OCCURS WHEN EITHER THE ARRAY ELEMENT PATTERN HAS A NULL ...... 51

FIGURE 2-24 CONCEPTUAL IMAGES OF BLIND-SPOT OCCURRENCE IN A PHASED ...... 51

FIGURE 2-25 THE RADIATION PATTERNS OF (A) A CONVENTIONAL BROADSIDE PEAK

RADIATING ELEMENT (DIPOLE OR WAVEGUIDE)AND (B) A BROADSIDE NULL

RADIATING ELEMENT (MONOPOLE). A BROAD SIDE NULL ELEMENT PLACES A

BLIND SPOT IN DIRECTIONS WHERE IT IS UNDESIRABLE TO TRANSMIT OR RECIVE

RADAR ENERGY ...... 53

FIGURE 2-26 PROPOSED PROTOTYPE PHASED ARRAY, 8X1 APERTURE COUPLED

MICROSTRIP LINEAR ANTENNA ...... 53

FIGURE 2-27 RADIATION PATTERN AND NULL FORMING ...... 56

FIGURE 2-28 RADIATION PATTERN OF SIDE LOBE CANCELATION 8-ELEMENT

ANTENNA AT 9.5 GHZ WITH APPLYING 0 PF TO 2 PF TO THE VOLTAGE

CONTROLLING ...... 56

FIGURE 3-1 A TYPICAL WEIGHTING FUNCTION NORMALIZED AMPLITUDE

SPECTRUM ...... 61

FIGURE 3-2 RELATION BETWEEN Β ̅ AND Ψ FOR CONSTANT MAINLOBE WIDTH ...... 70

FIGURE 3-3 ELLIPTIC CENTER CHANGED WITH INSERTING Ψ_K IN COMPARISON TO

PROPOSED WEIGHTING FUNCTION AND KAISER WEIGHTING FUNCTION ...... 71

FIGURE 3-4 RELATION BETWEEN Α AND R WITH APPROXIMATION MODEL FOR THE

COSH, KAISER AND PROPOSED WEIGHTING FUNCTIONS WITH N = 51 ...... 72

FIGURE 3-5 ERROR CURVE OF APPROXIMATED Α GIVEN BY (31) VERSUS R FOR N = 51 .... 72

xvii FIGURE 3-6 RELATION BETWEEN D AND R WITH APPROXIMATION MODEL FOR THE

COSH, KAISER AND PROPOSED WEIGHTING FUNCTION WITH N = 51 ...... 73

FIGURE 3-7 RELATIVE ERROR OF APPROXIMATED D GIVEN BY (32) IN PERCENT

VERSUS R FOR N = 51 ...... 74

FIGURE 3-8 COMPARISON OF THE PROPOSED, COSH AND KAISER WEIGHTING

FUNCTIONS FOR N=51 AND Β=6 ...... 75

FIGURE 3-9 COMPARISON OF THE PROPOSED, COSH AND KAISER WEIGHTING

FUNCTIONS FOR N=31 AND Β=2 ...... 75

FIGURE 3-10 RELATION BETWEEN Α AND THE MINIMUM STOPBAND ATTENUATION

WITH APPROXIMATED MODEL FOR THE PROPOSED WEIGHTING FUNCTION WITH

N =51 ...... 77

FIGURE 3-11 ERROR CURVE OF APPROXIMATED Α GIVEN BY (33) VERSUS AS FOR N =

51 ...... 77

FIGURE 3-12 THE FILTERS DESIGNED BY THE PROPOSED AND KAISER WEIGHTING

FUNCTIONS FOR WCT = 0.4 Π , ∆W = 0.2 RAD /SAMPLE WITH N = 51 ...... 78

FIGURE 4-1 A TYPICAL WEIGHTING FUNCTION NORMALIZED AMPLITUDE SPECTRUM

[4] ...... 82

FIGURE 4-2 LINEAR ANTENNA ARRAY WITH N RADIATING ELEMENTS ...... 84

FIGURE 4-3 RELATION BETWEEN Β ̅ AND Ψ FOR CONSTANT MAINLOBE WIDTH ...... 91

FIGURE 4-4 ELLIPTIC CENTER CHANGED WITH INSERTING Ψ_K IN COMPARISON TO

PROPOSED WEIGHTING FUNCTION AND KAISER WEIGHTING FUNCTION ...... 92

FIGURE 4-5 COMPARISON OF THE PROPOSED, COSH AND KAISER WEIGHTING

FUNCTIONS FOR N=17 AND Β=5...... 92

FIGURE 4-6 A) ARRAY ANTENNA WITH THEIR IMPEDANCE OPTIMIZATION. B) ARRAY

ANTENNA DESIGN IMPLEMENTATION WITH FEEDING NETWORK...... 96

xviii FIGURE 4-7 THE RADIATION PATTERN OF EXPERIMENTAL TEST OF ANTENNA ARRAY

BASED ON PROPOSED METHOD...... 97

FIGURE 5-1 A TYPICAL WEIGHTING FUNCTION NORMALIZED AMPLITUDE SPECTRUM

[4] ...... 100

FIGURE 5-2 RELATION BETWEEN Β ̅ AND Ψ FOR CONSTANT MAINLOBE WIDTH ...... 109

FIGURE 5-3 ELLIPTIC CENTER CHANGED WITH INSERTING Ψ K IN COMPARISON TO

PROPOSED WEIGHTING FUNCTION AND KAISER WEIGHTING FUNCTION ...... 109

FIGURE 5-4 COMPARISON OF THE PROPOSED, COSH AND KAISER WEIGHTING

FUNCTIONS FOR N=51 AND Β=6 ...... 110

FIGURE 5-5 COMPARISON OF THE PROPOSED, COSH AND KAISER WEIGHTING

FUNCTIONS FOR N=31 AND Β=8 ...... 111

FIGURE 5-6 A) ARRAY ANTENNA WITH THEIR IMPEDANCE OPTIMIZATION. B) ARRAY

ANTENNA DESIGN IMPLEMENTATION WITH FEEDING NETWOR ...... 114

FIGURE 5-7 THE RADIATION PATTERN OF EXPERIMENTAL TEST OF ANTENNA ARRAY

BASED ON PROPOSED METHOD...... 115

FIGURE 6-1 AN ADAPTIVE BEAMFORMER MODEL FOR NARROW BAND SIGNALS ...... 118

FIGURE 6-2 THE BEAM PATTERNS ACHIEVED WITH THE LMS, NLMS AND VSS-NLMS

ALGORITHMS WHEN THE REFERENCE SIGNAL CONTAMINATED BY AWGN WHEN

SNR=-10 DB...... 121

FIGURE 6-3 EFFECT OF NUMBER OF ARRAY ELEMENTS ON MEAN SQUARE ERROR

FLOOR IN KLMS BEAMFORMER WHEN SNR=0 DB...... 122

FIGURE 6-4 MODEL OF PROPOSED ALGORITHM ON A ARRAY ANTENNA

IMPLEMENTATION...... 123

FIGURE 6-5 THE BEAM PATTERNS ACHIEVED BY IMPLEMENTATION OF VSS-NLMS

ALGORITHMS WHEN THE REFERENCE SIGNAL CONTAMINATED BY AWGN ...... 124

xix FIGURE 7-1 AN ADAPTIVE BEAMFORMER MODEL FOR NARROW BAND SIGNALS...... 127

FIGURE 7-2 THE BEAM PATTERNS USING THE LMS, NLMS AND TWO-NLMS

ALGORITHMS WHEN THE REFERENCE SIGNAL POLLUTED BY AWGN WHEN

SNR=10 DB...... 131

FIGURE 7-3 THE EVM VALUES OBTAINED WITH TWO-NLMS, LMS, AND NLMS

ALGORITHMS AT DIFFERENT SNR ...... 132

FIGURE 7-4 ADAPTIVE ARRAY HYPERTHERMIA SYSTEM ...... 133

FIGURE 7-5 MODEL OF PROPOSED PROCEDURE ON A ARRAY ANTENNA

IMPLEMENTATION...... 134

FIGURE 7-6 THE BEAM PATTERNS ACHIEVED BY IMPLEMENTATION OF VSS-NLMS

ALGORITHMS WHEN THE REFERENCE SIGNAL CONTAMINATED BY AWGN ...... 134

FIGURE 8-1 AN ADAPTIVE BEAMFORMER MODEL FOR NARROW BAND SIGNALS ...... 137

FIGURE 8-2 THE IMPLEMENTATION SETUP WITH 8-ELEMENT RECTANGULAR

APERTURE MICROSTRIP ARRAY ANTENNA WITH TWO HORN ANTENNAS, ONE FOR

DESIRED SIGNAL AND ANOTHER FOR INTERFERENCE SIGNAL IN THE ANECHOIC

CHAMBER...... 145

FIGURE 8-3 PROTOTYPE OF PROPOSED 8X1 APERTURE COUPLED MICROSTRIP LINEAR

ANTENNA BASE ON THE MUTUAL COUPLING ...... 146

FIGURE 8-4 THE CONVERGENCE OF IP-NNLMS, LMS, AND NLMS ALGORITHMS FOR

DIFFERENT VALUES OF INPUT SNR. (A) SNR = 5DB, (B) SNR = 15DB...... 148

FIGURE 8-5 THE BEAM PATTERNS ATTAINED WITH THE IP-NNLMS, LMS, AND NLMS

ALGORITHMS WHEN THE REFERENCE SIGNAL CORRUPTED BY AWGN WITH

SNR=10 DB ...... 149

FIGURE 8-6 EFFECT OF NUMBER OF ARRAY ELEMENTS ON MEAN SQUARE ERROR

FLOOR IN IP-NNLMS BEAMFORMER IN COMPARISON WITH LMS, AND NLMS

BEAMFORMERS AT DIFFERENT SNR...... 150

FIGURE 8-7 THE EVM VALUES OBTAINED WITH IP-NNLMS, LMS, AND NLMS

ALGORITHMS AT DIFFERENT SNR IN AWGN CHANNEL...... 151

FIGURE 8-8 THE EVM VALUES ACHIEVED IN THE PRESENCE OF RAYLEIGH FADING

FOR IP-NNLMS, LMS, AND NLMS ALGORITHMS AT DIFFERENT SNRS...... 151

xxi Fundamentals Of Antenna Arrays And Beamforming

1.1 Introduction

Array antenna beamforming methods can perform several, concurrent actions. The beam pattern could be have high gain and low side lobes, or adjustable beam width.

Adaptive array antenna beamforming methods dynamically control the array antenna pattern to optimize certain aspects of the received signal. In beam pattern scanning, a single main beam pattern of an array antenna lead scanning either continuously or in the small discrete steps.

Array antenna systems with adaptive beamforming techniques can avoid interfering signals having a direction of arrival (DOA) different from that of desired signal. Multi- polarized array antennas may also reject interfering signals having various polarization modes from the desired signal, even though if the signals have the equal direction of arrival

(DOA). These abilities could be used to modify the valence of wireless radio communication systems.

An array antenna includes of two or more radiation antenna elements that are spatially serialized and electrically interconnected to generate a directional radiation pattern. The interconnection among array elements, called the array feed network, could prepare fixed phase to each element or may figure a phased array. In adaptive array beamforming, the phases of the feed array network are regulated to optimize the received

1 signal. The geometry of an array antenna and the beam patterns, orientations, and polarizations of the elements influence the performance of the array.

1.1.1 Pattern of a Generalized Array

A three dimensional array with an arbitrary geometry is shown in Figure 1-1. In spherical coordinates, the vector from the origin to the nth element of the array is specified

̂ by 푟⃑⃑푚⃑⃑ = (휌푚, 휃푚, ∅푚) and −푘 = (1, 휃, 휑) is the vector in the orientation of the basis of an incident wave. It is assumes that the source of the wave is in the far field of the array and the incident wave can be approximates as a plane wave. To find out the array factor, it is impotent to know the relative phase of the received plane wave at each array element. The phase of the array is referred to the phase of the plane wave at the source. Therefore, the phase of the received plane wave at the nth array element is the phase constant β=2π ̸ λ

̂ multiplied by the element location ⃑푟⃑⃑푚⃑ on to the plane wave arrival vector −푘 . This is

̂ determined by −푘 • 푟̂푚 with the dot yield in rectangular coordinates. In rectangular coordinates, −푘̂ = sin 휃푐표푠∅ 푥̂+ sin 휃푠푖푛∅ 푦̂+ cos 휃 푧̂ = 푟̂ ,

푟 푚 = 휌푚 sin 휃푚푐표푠∅푚푥̂+ 휌푚푠푖푛∅푚푦̂+ 휌푚푐표푠휃푚푧̂, and the relative phase of the

2 Figure 1-1 An arbitrary m-element tri-dimensional array incident wave at the nth element is

⃑ 휁푚 = – 푘 • 푟 푚

= βρm(sinθcosφsinθcos∅m +sinθsinφsinθmsin∅m +cosθcos∅m) (1.1)

= β(xmsinθcos∅ + ymsinθsin∅ + zmcosθ)

1.1.2 Array factor

For an array of M elements, the array factor is determined by:

푀 푗(휁푚+훿푚) 퐴퐹(휃, 휙) = ∑푚=1 퐼푚푒 (1.2)

th Were 퐼푚 is the magnitude and 휁푚 is the phase of the weighting of the m element.

The normalized array factor is given by

퐴퐹(휃, 휙) 푓 (휃, 휙) = 푚푎푥 {|퐴퐹(휃, 휙)|} (1.3)

3 This is the same as the array pattern of the array antenna if they are ideal isotropic elements.

1.1.3 Array Pattern

Each element of the array antenna has a pattern 혨푚(휃, 휙) , which may be different for each element, and the normalized total array pattern is determined by:

∑푀 퐼 혨 (휃, ∅)푒푗(휁푚+ 훿푚) 퐹(휃, ∅) = 푚=1 푚 푚 푚푎푥{|∑푀 퐼 혨 (휃, ∅)푒푗(휁푚+ 훿푚)|} 푚=1 푚 푚 (1.4)

1.1.4 Phase and Time Scanning

Beamforming and beam controlling are commonly accomplished by phasing the feed to each element of an array antenna so that signals received or transmitted from/to all elements would be in phase in a specific direction ok is the direction of a maximum of the beam pattern. Beamforming and beam controlling methods are basically used with linear, circular, or planar arrays but some approaches are enforceable in any array geometry.

Adaptive array techniques applies to forming fixed beams, scanning directional beam patterns and to reject interfering signals. Array beamforming could be shaped or scanned using either phase shift or time delay systems. Each of these techniques has advantages and disadvantages. Which, both can be used for other geometries, the following study refers to symmetrically spaced linear arrays such as shown in Figure 1-2. In the phase scanning techniques the interelement phase shift α is changed to scan the beam pattern.

4 Figure 1-2 (A) A Phase Scanned Linear Array (B) A Time Scanned Linear Array 1.1.5 Introduction of Beamforming

Complex weighting function for each array element could be computed to optimize some characteristics of the received signal. Beamforming does not result in an array pattern having a beam pattern in the desired signals orientation but does yield the optimal array output signal. Oftentimes this is achieved by steering nulls in the direction of interfering unwanted signals. Adaptive array beamforming is an iterative approximation of optimum weighting function beam pattern.

A typical array processing with variable element weights is shown in Figure 1-3.

The throughput of the array y(t) is the weighted sum of the received signals si(t) at the array elements with having the patterns 혨푚(휃, 휙) and the noise n(t) from receivers connected to each array element.

5 Figure 1-3 Adaptive Array Antenna Block Diagram

As shown in Figure 1-3, s1(t) is the desired signal, and L signals assumed as interferes signals. In an adaptive array beamforming system, the weights wm are iteratively specified based on the array throughout y(t), a reference signal d(t) which is an estimate of the desired signal, and prior weight w*. The reference signal is supposed to be similar to the desired signal. Optimum weights that minimize the mean squared error 휀(t) between the output signal and the reference signal. A desired signal s1(t), L interfering signals, and increasable white Gaussian noise are used thought-out the work. The array element beam pattern need not necessarily be the same for all elements. The array throughput is determined by:

푦 (푡) = 푤퐻푥(푡) (1.5)

wH is the complex conjugate transpose of the weight vector w.

6 1.1.6 Array Response Vector

The array response vector for a signal with direction of arrival (휃, 휙) and polarization state P can be written as follows:

푗휁1 푒 혨1(휃, ∅, 훲) 푗휁2 푎(휃, ∅, Ρ) = 푒 혨2(휃, ∅, 훲) ⋮ 푗휁푀 (1.6) [푒 혨푀(휃, ∅, 훲)]

The phase shifts 휁푀 represent the spatial phase delay of an incoming plane wave arriving from angle (휃, ∅).The factor 혨푀(휃, ∅, 훲) is the antenna pattern of the mth element.

1.1.7 Spatial Polarization Signature

The spatial polarization signature is the repose of the entire array antenna to a signal with

N multipath components and is determined by:

푁

풗 = ∑ 훼푛 푎(휃푛 , 휙푛, 훲푛) 푛=1 (1.7)

Were 훼푛is the amplitude and phase of the nth component. The angle of arrival and polarization state of the nth component are given by 휃푛,휙푛, and 훲푛 .

1.1.8 Spatial Polarization Signature Matrix

The response of the array to multiple signals such as desired signal and L interfering signals can be expand is a spatial polarization signature matrix. The columns of matrix spatial polarization signatures of the individual signals is

푈 = [푣1‖ 푣2 … 푣 퐿 + 1 ]

= [푈푑 ‖ 푈푖 ] (1.8)

Here Ud is the response to the desired signal s1(t) and Ui is the response to the interfering signals. The output of the M receivers prior to weighting is

푥(푡) = 푈푠(푡) + 푛(푡) (1.9)

1.1.9 Signals and Noise

The incident signals are determined by:

T T s(t)=[s1(t) s2(t) ….. sL+1(t) ] = [ sd(t) ‖ si(t) ] ( 1 . 10 )

where sd(t) = s1(t) is the desired signal and si(t) consists of the remaining, interfering signals. In this case all signals are considered to be uncorrected and to have the form

jω0t 푠푘(푡) = √푆푘푢푘(푡)e where √푆푘is the amplitude of the signal and uk(t) is a normalized baseband modulating signal. The noise in the M receivers is given by

T n (t) = [n1(t) n2(t) … nm(t)] (1.11)

And the noise in different receiver branches is assumed uncorrelated.

1.1.10 Optimum Weights

To optimize the element weights, we seek to minimize the mean squared error between the array output and the reference signal d (t). Optimizing signal-to-interference- plus-noise ratio (SINR) will lead to weights that differ by a scalar multiplier from the weights shown:

α = -2πd cosφ0 / λ0 (1.12)

The derivation proceeds as for the case of omnidirectional elements, and the solution for the optimum weights is

Wopt = rxd /Rxx (1.13)

H Here Rxx = x(t) x (t) is the signal covariance matrix and rxd =d*(t)x(t). This is the same as the expression for the optimum weights for an array with isotropic elements in

(1.12). In this case, however, Rxx ,rxd , and hence wopt are functions of the angles of arrival of the L+1 signals, and of the element patterns.

1.1.11 Adaptive Algorithms

Adaptive array beamforming algorithms iteratively estimate the optimum weighting function. Adaptive array beamforming began with the Howells and Applebaum introduced by

푀−1 2휋푑 푗푚( 푐표푠휙+ 휔Δ푡) 퐴퐹(휙) = ∑ 퐴푚푒 휆 푚=0 (1.14)

푑 Δ푡 = − 푐표푠휙 푐 0 (1.15)

Since then many beamforming algorithm have been developed. Several algorithms are briefly described below.

1.1.12 Least Mean Squares (LMS)

Least mean squares algorithm uses a steepest-descent technique and calculates the weight vector recursively using the equation

H w (n+1) = w(n) +μx(n) [d*(n) –x (n)w(n)] (1.16)

Here μ is a gain constant and controls the rate of adaptation. The LMS algorithm needs knowledge of the beamforming desired signal. This may be accomplished in a digital system by periodically transmitting a training sequence that is well known to the receiver, or exploit the spreading code in the case of a direct sequence CDMA system. LMS algorithm will converges slowly if the eigenvector spread of Rxx is large.

1.1.13 Direct Sample Covariance Matrix Inversion (DMI)

Direct sample covariance matrix algorithm presented in (1.10) can be used to achieve the weights, but with Rxx and rxd approximated from data sampled over a finite interval. The approximations are determined by:

푁2 퐻 푅̂푥푐 = ∑ 푥(푖)푥 (푖) 푖=푁1 (1.17)

푁2 ∗ 푟푥푑̂ = ∑ 푑 (푖)푥(푖) 푖=푁1 (1.18)

The direct sample covariance matrix inversion algorithm converges more quickly than the Least mean square algorithm (LMS) but it is requires more complex calculation.

This algorithms so requires the reference signal.

1.1.14 Recursive Least Squares (RLS)

The recursive least squares algorithm approximates Rxx and rxd using weighed sums so that

푁 푛−1 퐻 푹̃풙풄 = ∑ 훾 푥(푖)푥 (푖) 푖=1 (1.19)

푁 (1.20) 푛−1 ∗ 푟풙풅̃ = ∑ 훾 푑 (푖)푥(푖) 푖=1

The inverse of the covariance matrix could be achieved recursively, and this leads to the following equation

푤̂(푛) = 푤̂(푛 − 1) + 푞(푛)[푑∗(푛) − 푤̂퐻(푛 − 1)푥(푛)] (1.21)

Here

−1 −1 훾 푅푥푥 (푛 − 1)푥(푛) 푞(푛) = −1 퐻 −1 (1.22) 1 + 훾 푥 (푛)푅푥푥 (푛 − 1)푥(푛)

And

−1 −1 −1 −1 푅푥푥 (푛) = 훾 [푅푥푥 (푛 − 1) − 푞(푛)푥(푛)푅푥푥 (푛 − 1)] (1.23)

The recursive least squares algorithm converges approximately an order of magnitude more quicker than the least mean square algorithm if signal to interference plus noise ratio

-1 (SINR) is high. It needs a primary approximation of Rxx and a reference signal.

1.1.15 Other Techniques

Other adaptive beamforming achievements include spectral self-coherence restoral a blind adaptive array algorithm that uses the cyclostationary property of a signal. Neural networks and maximum likelihood sequence approximations may also be utilize to execute adaptive array beamforming. In partially adaptive arrays, only few of the elements are weighted adaptively. This technique is beneficial for large arrays. Partial adaptively utilizes an array to neutralize the interference signals but needs less complex calculating than considering all the element weights.

1.2 Fundamentals of Antenna Arrays

The following is a short fundamental description of the fundamental principles of antennas and a description of the mechanism of wireless electromagnetic propagation. We briefly summarize antenna and electromagnetic propagation theory are recalled from the literature and an preface to antenna arrays is provided together with a definition of the most common array geometries.

1.2.1 Maxwell’s Equation

Wireless communication are supported by the transmission and reception of electromagnetic signals from antennas. Electromagnetic propagation depends on

Maxwell’s equations, which for an isotropic and homogeneous medium are given by

휌 (푟, 푡) ∇ • E(r. t) = 푣표푙 휀 (1.24)

∇ • H(r, t) = 0 (1.25)

휕퐇(푟, 푡) ∇ × E(r, t) = −휇 휕푡 (1.26)

휕퐄(푟, 푡) ∇ × H(r, t) = 휀 + J(푟, 푡) 휕푡 (1.27)

Here r defines the spatial position vector, t is the time change, E(r,t) is the electric field intensity vector , μ is the permeability of the medium, H(r,t) is the magnetic field intensity vector, ε is the permittivity of the medium, J (r,t) is the current density vector and

ρvol (r,t) is the volume charge density. Conforming to the mathematical sense of the divergence factor, which measures the amount of the vector field’s source could be exhibit as a flux density, and to the meaning of the curl operation, which perfumes the vector field’s rate of rotation, these Maxwell’s equations determine the principles of electrodynamics.

1.2.2 Pointing Vector

Radiation fields E(r,t) and H(r,t) can be explain in terms of the complex fields Es(r) and Hs( r) as :

jwt E(r,t) = Re { Es ( r) e } (1.28)

jwt H(r,t) = Re {Hs(r) e } (1.29)

Here j= √-1 is the imaginary unit and w is the angular frequency. The components of the complex field produced by an antenna are directly dependent to the physical feature of the antenna shape and could be derived by solving Maxwell’s equations using numerical methods. The quantity used to determine the power related to an electromagnetic field is the pointing vector, while be describe as a sinusoidal

1 푤⃑⃑ (푟) ≜ 푬 (푟) × 퐇∗ (r), 2 푠 퐬 (1.30)

Here (.)* is the complex conjugate. The real part of 푊⃑⃑⃑ (r) is the average power density of the field produced by the antenna, while the imaginary part of 푊⃑⃑⃑ (r) represents the reactive power density of this field. Hence, the mean power radiated over a closed surface ℳ surrounding the antenna can be found as:

1 푃 = ∯ 푅푒 (푤⃑⃑ (푟)) • ds = ∯ Re{퐸 (푟) × 퐇∗ (r)} • ds 푟푎푑 2 푠 퐬 푀 푀 (1.31)

Figure 1-4 Antenna Field Area1.2.2 Antenna Field Regions The produced electromagnetic field and the radiation attributes of an antenna depend on the physical shape itself and the wavelength. The space surrounding an antenna is generally subdivided into three regions, which are describes by appropriate boundaries as shown in Figure 1-4. The first sector, denotes the reactive near-field region, is limited

3 by the antenna surface and by the spherical surface with 0.62√퐷푚푎푥 /휆 where 퐷max is the largest dimension of the antenna and λ is the wavelength. This sector have the reactive power congested in the proximity of the antenna, which the active radiated energy in negligible. The second sector, denotes the radiating near- field or Fresnel region, is

3 2 determine by spheres by radius 0.62√퐷푚푎푥 /휆 and 2퐷푚푎푥 /λ. In this sector the radiation of active energy overcomes, but the angular field diffusion related on the space from the

2 antenna. The far-field or Fraunhofer sector occurs outside the sphere by radius 2퐷푚푎푥/λ and describes the principle sector of radiating for most of the antennas. In the far-field the imaginary of the pointing vector approximates zero much quicker than its real part and so the reactive energy is insignificant. Electrical field and magnetic field components are

15 perpendicular to each other and transverse to the radial direction of propagation r. Thus, in the far-field sector, Es(r) and Hs(r) from a Transverse Electromagnetic (TEM) wave.

퐄푠(푟) = 휂0퐇푠(푟) × 퐫̂, (1.32)

1 (1.33) 퐇푠(푟) = 퐫̂ × 퐄푠(푟) 휂0

Here 휂0 = √휇/휀 ≅ 377Ω is the intrinsic impedance of free space.

1.2.3 Radiation Attributes

The radiation intensity 푊푢(휗, φ) is a far-field characteristic that describes the radiated power per unit solid angle in a known orientation determined by the zenith angle

θ and the azimuth angle φ. It an be pound by multiplying the component of the pointing vector in the orientation of propagation by r2:

2 푊푢(휗, 휑) = 푟 푅푒{푊⃑⃑⃑ (푟) • 퐫̂} (1.34)

It is used to determine the antenna power gain, ratio of the radiation intensity and the energy at the input of the antenna, P input, divided by 4π:

4휋푊푢(휗, 휑) 풢(휗, 휑) ≜ 푃푖푛푝푢푡 (1.35)

The antenna gain is determined by the outcome between the efficiency ecd and the directivity D (휗, φ). Therefore, antenna efficiency is used for the conduction and dielectric losses, and is describe as ratio of the radiated energy to the input power:

푃푟푎푑 푃푟푎푑 푒푐푑 ≜ = 푃푖푛푝푢푡 푃푟푎푑 + 푃푙표푠푠 (1.36)

Figure 1-5 Where Ploss is the antenna fatality, plus the antenna directivity is the ration with antenna radiation intensity and the radiated energy divided to the 4π:

4휋푊푢(휗, 휑) 퐷(휗, 휑) ≜ 푃 푟푎푑 (1.37)

The antenna radiation features are as a function of the spatial coordinates and are determined by the antenna radiation pattern. The radiation pattern determines several antenna characteristics, like antenna radiation intensity, field strength, directivity and gain as shown in the Figure 1-5 defined as like below:

 Null : The smallest values of the pattern assumes; ideally zero

 Lobe: The space angular restricted between two nulls

 Main Lobe: The lobe that contains the direction of maximum of radiation pattern

 Minor Lobe: All the other lobes except the main lobe. If purposely maximum radiation value in undesirable directions exterior to the main lobe, the lobes comprising these directions are called grating lobe.

 Side Lobe: A minor lobe that is mostly next to the main lobe and employs the similar hemisphere in the orientation of maximum gain.

 Half Power BeamWidth (HPBW) or 3 dB BeamWidth: Space between the directions of maximum radiation that occurs among the two directions in which the radiation pattern is one-half of the maximum values. As shown in Figure 1-5 휑3푑퐵demonstrate the HPBW in the azimuth plane.

 First null Beamwidth (FNBW): Total space angle that contains the main lobe. The FNBW can be related to the capability of an antenna to avoid interference. As shown in the Fig 휑퐹푁 demonstrate the FNBW in the azimuth plane.

 Side Lobe Level( SLL) : Maximum related radiation pattern of the supreme side lobe with regards to the maximum radiation of the main lobe.

1.3 Array Antenna

Basically the beamwidth of a single element antenna radiation pattern is very wide for wireless applications; however, in many other applications, it is necessary to have high gain in certain directions [1] . In additional to a narrower HPBW we could be required to have spatial filtering and to increase the gain in suitable directions and also to decrease it in undesirable ones. In this second category, the suitable radiation pattern for assumed application all radiators of the array are typically selected as identical even if, for significant practical purposes, array antennas consisting of different elements can be produced. By pursuing multi element antenna infrastructure, it is possible to produce 18 radiation patterns with numerous main beams, while all with expected beam width, with desirable SLL and null circumscription. The beam pattern synthesizing procedure could be proceed by properly choosing the number of the array antenna elements, elements geometrical formation and the spacing between the elements. In addition, the radiation pattern could be electrically altered by changing the phase and amplitude of the current distributions.

The entire radiation field pattern of an antenna array is found as the vector addition of the particular elements. The far field radiation function of an antenna array can be computationally summed up by the array steering vector:

2휋푣푐 2휋푣푐 푗 휏1(휗,휑) 푗 휏푁(휗,휑) 푇 헮(휗, 휑) ≜ [휀1(휗, 휑)푒 휆 , … , 휀푁(휗, 휑)푒 휆 ] (1.38)

T Where (.) denotes the transpose of a vector, 휀푘(휗, 휑) = √풢(휗, 휑) is the k-th individual element field pattern, 푣푐 is the speed of light in free space and 휏푘(휗, 휑) denotes the delay of a plane wave arriving from direction (ϑ, 휑) and measured from the kth element of the array with considered to the source. The array antenna pattern steering vector contains both the radiation characteristics specification of every element and the geometrical properties of the antenna array. These define the delay, which can be written as:

퐢 퐫푘 • 퐫̂ 휏푘(휗, 휑) = 휐푐 (1.39)

푇 th Here 푟푘 = (푟푘푠푖푛 휗푘푐표푠 휑푘, 푟푘푠푖푛 휗푘 푠푖푛 휑푘, 푟푘푐표푠 휗푘) is the location of the k array element in spherical coordinates, also:

퐫̂퐢 = (sin 휗 푐표푠 휑 , sin 휗 푠푖푛 휑 , 푐표푠 휗)T (1.40)

is the unit vector of the arriving wave. Based on the maxim of pattern proliferation, if the individual elements are equal, the pattern generated by an array can be found as the product of the pattern of the individual element, considering to a certain reference point, and the array factor:

푁 2휋푣푐 푗 휏푘(휗,휑) 퐴퐹(휗, 휑) ≜ ∑ 퓌푘푒 휆 푘=1 (1.41)

th Here 퓌푘 is the complex excitation of the k array antenna element. The AF (ϑ,휑) can be regulated by adjusting the excitations, the number of array antenna elements, the spacing between the elements and the geometrical formation of the array. Whenever antenna array elements are adequately close, the field radiated by single element couples into adjacent elements one other and contrariwise. This phenomenon is called and propagation mutual coupling and relies on the spacing between the elements, the radiation pattern features and the direction of the single element. This interior mutate of energy complexes the array antennas design, since predicting mutual coupling is difficult to predict by analytical techniques, but, in several scenarios, it might be taken into account because of its remarkable contribution [2]. Actually, mutual coupling distorts the individual element

20 radiation pattern, since impress the throughout pattern of the overall antenna array and may guide to unprecise navigating of the main lobe and the nulls. A spacing between the elements larger than or equal to half the wavelength is generally sufficient to avoid mutual coupling effects, which, for lower spacing between the elements, distorts element patterns which can be aimed with numerical methods.

1.3.1 Linear Array Antennas

In generally, for designing the array antenna, some restrictions, for example SLL,

FNBW, HPBW and the directivity, are determined. However, others, for example phase and amplitude of the antenna elements, the number of radiators and the physical geometry of the antenna array, might be appropriately derived to meet the design specifications. The most typical and most analyzed geometry in the literature is the linear antenna array, which consists of N antenna elements placed on a straight line. If the contiguous elements are similarly spaced as shown in Figure 1-6, the antenna array is designed as a Uniform Linear

Array (ULA). If, moreover, all excitations have similar amplitudes (|퓌1| = |퓌1| = ... = |퓌푁|) and the phase of the kth element is excessed by a constant oncoming phase ι with respect to the (k − 1) th element, array accords to as a uniform linear array. In this significant case, the normalized array factor in the azimuth plane may be shown to be:

2휋휌 푁 1 푐표푠 휑+휄 sin [ ( 휆 )] 휋 1 2 퐴퐹푛표푟푚 (휗 = , 휑) = 2휋휌 2 푁 1 1 푐표푠 휑+휄 sin [ ( 휆 )] (1.42) 2

Here 휌1 is the distance between the elements. The main lobe could be adjusted towards the required direction by appropriately adding a progressive phase shift, since a narrow HPBW could be achieved by increasing N.

Figure 1-6 Uniform Linear Array Considering the uniform array it can be shown that independently of the number of elements, the ratio between the maximum of the main lobe and the uppermost of the first side lobe is almost equal to 13.5 dB. This level might not be desirable for significant applications, where a more precise restriction should be applied the level of the secondary lobes. Actually, to decrease the transmitted or received power beyond the main lobe, the minor lobes must be as low as feasible. Usually, the matter of low side lobe level array antenna design is considerably equivalent to the difficulty of designing Finite Impulse

Response (FIR) digital filters in a Digital Signal Processor (DSP). Accordingly, most of methods have been taken from digital signal processing to generate radiation patterns with low side lobes. These methods analyze weighting of the array elements by adopting appropriate subsidiary like as Gaussian, Kaiser-Bessel and Blackman windows [3].

Extensively adopted synthesis methods for ULAs are the Dolph-Tschebyscheff and the binomial techniques, which use a nonuniform distributing of the excitation amplitudes.

Comparing the features of uniform, Tschebyscheff and binomial arrays, it could be apperceived that the uniform arrays introduce an HPBW with the smallest amount and a

SLL with highest amount, since the binomial arrays produce the largest HPBW and the lowest SLL as shown in Figure 1-4 In addition, a binomial array antenna with spacing between the element less than or equal to λ/2 has no side lobes. The HPBW and the SLL of a Tschebyscheff array stand among those of binomial and uniform arrays. In addition, for a given SLL, the Dolph-Tschebyscheff arrays produce the smallest FNBW and, bilaterally, for a given FNBW, the Dolph-Tschebyscheff arrays produce the lowest feasible

SLL [4]. Even if ULAs are broadly excited, there are several scenarios in which the linear geometry is not suitable. Other configurations may be adopted to meet some exclusive obstacle necessity on buildings or vehicles and to produce higher performance in terms of beam scanning.

Figure 1-7 ARRAY GAIN OF A ULA FOR N = 5 AND Ρl= 0.5λ. ………Uniform array ------Binomial array Dolph-Tschebyscheff array

1.3.2 Rectangular Array Antennas

Rectangular arrays antenna are more versatile compared to linear arrays and can be employed to obtain patterns with a lower SLL. Similarly to ULAs, proper weighting methods can be considered to further reduce the side-lobe level and the beam width.

Besides, a rectangular configuration has the capability to perform the beam scanning in both the azimuth and the elevation planes. As shown in Figure 1-8, a rectangular array antenna is formed when the radiating elements are placed along a rectangular region. If the contiguous elements are identically spaced the array is called a Uniform Rectangular Array

(URA). This arrangement finds application in a broad range of communication systems, such as remote sensing and tracking radars. An Nx .Ny URA can

Figure 1-8 Uniform Rectangular Array. y Also be viewed as Ny linear arrays positioned space ρr , where each array is arranged by

x Nx elements with spacing between the elements ρr . Therefore, for an oncoming phase

URA with uniform amplitudes, the normalized array factor can be determined by:

2휋휌푥 푁 푟 푠푖푛 휗 푐표푠 휑+휄 푠푖푛 [ 푥 ( 휆 푥)] 1 2 퐴퐹 (휗, 휑) = . 푛표푟푚 2휋휌푥 푁푥 1 푟 푠푖푛 휗푐표푠 휑+휄 푠푖푛 [ ( 휆 푥)] { 2 }

푦 (1.43) 2휋휌 푁 푟 푠푖푛 휗 푐표푠 휑+휄 sin [ 푦 ( 휆 푦)] 1 2 푦 2휋휌 푁푦 1 푟 푠푖푛 휗푐표푠 휑+휄 sin [ ( 휆 푦)] { 2 }

Here 휄푥 and 휄푦 are the oncoming phase shifts belong the x and the y axis, respectively.

Whenever the spacing between the elements is greater than or equal to half the wavelength, grating lobes can be developed in the radiation pattern because of the in phase feature excess of the radiated pattern fields to the directions different from that of the main lobe.

1.3.3 Circular Array Antennas

Another typically employed arrangement is the Uniform Circular Array (UCA), in which the elements are uniformly situated orderly formed on a circular ring as shown in

Figure 1-9. The UCA is of very high interest and is frequently adopted in radar, sonar systems and cellular base stations. Considering a UCA of radius ρc, the normalized array factor can be determined as:

푁 2휋 푗 휌푐 sin 휗 cos(휑−휉푘) 퐴퐹(휗, 휑) = ∑ 퓌푘푒 휆 푘=1 (1.44)

th Here 휉푘is the angular location of the k element. Because of numerous degree of proportion the UCA is capable of forming an identical HPBW in all azimuth directions. In addition, brief comparison between a UCA and a ULA indicates that, adopting the similar number of elements and the same interelement spacing between contiguous radiators, the circular array antenna provides narrower main beams with regard to the corresponding linear array antenna. The pattern of the circular array can be improved by using various rings in order to achieve a Concentric Ring Array (CRA). Planar array configuration provides the scanning in both azimuth and elevation planes [5] [6].

Figure 1-9 Uniform Circular Array

1.3.4 Conformal Array Antennas

ULAs, URAs and UCAs are broadly adopted configurations, relying on a regular and symmetrical design. These formations are appropriate for applications in which we need to meet mounting requirements to maintain the structure and size and geometry restrictions are not present. Note, in most practical scenarios, like spacecraft and missile applications, the mounting surface is irregular and the existing space is restricted. In these conditions the array configuration might be designed to meet significant requirements of the supporting surface and then a conformal array is wanted. The antenna radiating elements may be arranged on an arc, on an ellipse, or can be positioned conforming to additional intricate three dimensional topologies, generally circularly symmetric surfaces, for example cones, cylinders or spheres [7]. Typically, the patterns of the array elements could be oriented to different paths, due to the irregularity of the configuration, and so the individual element pattern cannot be factored out from the overall array pattern [8].

Appropriate antenna elements may be needed for conformal arrays, since the generic radiator may have to adhere to aerodynamic profiles. In addition, significant individual element radiation patterns could be essential to meet the design specifications for dedicated array topologies [9].

Design and Development of Rectangular Aperture Coupled

Microstrip Antennas with Application to Beamforming

2.1 Introduction

Microstrip array antennas are receiving much attention due to their small size, low- cost fabrication, and easy positioning. Array antennas are used in many operations ranging from defense technologies such as unmanned aircraft systems, or in commercial, airborne wireless communication devices [10]. These low profile low-weight antennas can also be used to implement an array based on a single antenna element to boost directivity.

Increased directivity will naturally produce a higher gain which will allow for greater scope communication, while broad matching bandwidth offers higher data transfer rate. Beam steering is an excellent property to be able to scan the major lobe of the antenna to be able to pick up signals from multiple angles. This is achieved by applying a progressive phase shift β between the array elements. The applications for array antenna beam steering are nearly limitless, again ranging from defense applications to very day needs. Also X-band radar frequency sub-bands are used in civil, military, and government institutions for weather monitoring, air traffic control, maritime vessel traffic control, defense tracking, and vehicle speed detection for law enforcement

[11],[12],[13],[14].

Such a system utilizes the concept of adaptive array antennas for mobile communication. The antenna pattern is continuously adapted to the actual situation of the antenna pattern requirements utilizing an algorithmic controlled phased array.

This section, introducs part of our work on adaptive array antenna systems. The designed antenna operates in X-band at 9.5 GHz, where the required bandwidth is 230

MHz The broad bandwidth is concluded by using multiple stacked radiation patches with different lengths [15] or adapting the aperture shape, which results in enhanced coupling for an aperture-coupled patch antennas [16].

Since microstrip patch antennas have a low bandwidth, a special feeding technique, aperture coupling, was adopted. For the design of the antenna, a simulation tool based on the Methods of Moments was used. After optimizing and developing a single element antenna that achieved desired measured results, an eight element rectangular microstrip array antenna was designed. This results in excellent final results for the antenna array with a bandwidth of 230 MHz, a front-to-back ratio larger than 17dB, and a maximum mutual below -14.5 dB as a demonstration that aperture coupled microstrip patch antennas are suitable for the operation in an X-band communication system.

2.2 Microstrip Antenna Configuration

A microstrip antenna consists of a good metallic conductor bonded to good grounded dielectric substrate [17]. Figure 2-1, shown the microstrip antenna in its ordinary configuration [18], [19]. The antenna includes a radiation patch on one side of a dielectric sheet and a ground plane on the other side. The patch, generally of copper, may have different geometries, such as rectangular, or circular patches covers most of the previous designed in terms of beam pattern, beam width, gain and polarization [20]. We will focus

30 on the rectangular configuration (Figure 2-2) because its simplicity for analysis and performance prediction make it easy to handle it [21] [22].

Figure 2-1 Microstrip Antenna Configuration

Figure 2.1: Micr ostrip antenna con_gur ati on.

Figure 2-2 Rectangular Patch Antenna 2.2.1 Advantages of Microstrip Antenna

There are a number of advantages, A most the most significate ones [23] [24]:

 Low Profile: Printed circuits are slim and needed less volume than waveguide and coaxial cable components.  Polarization: With the variety of the antenna patch shapes any polarization can be obtained. It is possible to design antennas with multi polarization

capability with either single port or multiple ports. These characteristics could be performance for dual  Doul Frequency antennas are possible  Excitation technique: Patches lets a lot of variant excitation techniques to be exploit, compatible with any new technology of the active circuitry and beamforming systems.  Appropriate technology with microwave integrated circuits

In addition, there are economic advantages to microstrip antennas, such as low cost manufacturing. These advantages lead microstrip antenna applications in many systems , for example, mobile communications, satellite communications, remote sensing, Doppler radar, automotive radar, etc [22].

2.2.2 Microstrip Radiators

The significant part of the microstrip antenna is the placed conductor on the top of finite dimensions [21] [25]. This patch can be considered to be an open ended transmission line of length Lp and the width Wp. The amplitude of the surface current distribution peaks whenever the required frequency is close to resonance frequency. However with replacing the fundamental mode in to the formula, the resonant frequency f0 can be represents by:

퐶0 풇ퟎ = 2(퐿푝 + 2∆퐿푝)√휀푒푓푓 (2.1)

Here ΔLp is the equivalent t length that takes in to account the fringing fields at the two open ends, and εeff is the effective relative permittivity given by

휀 + 1 휀 − 1 퐻 휀 = 푟 + 푟 (1 + 10 )−푎푏 푒푓푓 2 2 푊 (2.2)

Electric field configuration represents the patch radiates. Therefore assuming the patch with spacing a small fraction of wavelength over the ground plane electric field indicates is approximately uniform along the width of the microstrip. Radiation can be attributed to the fringing fields (Figure 2-3) of the open circuited edges of the antenna patch

[26].

Figure 2-3 Field configuration of (a) microstrip patch antenna (b) Side view (c) Top view The radiation fields at the end of the patch may be split in to the tangential and normal components with respect to the ground plane[27][28][29]. The normal field

33 components are out of phase when the length of the radiator is near λ/2. Then the performance to the far field in broadside orientation cancel each other. The tangential fields are in the phase, giving the topmost radiated field to the face of the patch. This configuration may modeled two parallel slots, positioned half a wavelength at the edges of patch. This plan model the far field pattern F(θ) in the H-plane may be analytically represented by [23]:

휋푊 sin( 푝 sin 휃) 휆0 푭(휽) = 휋푊 cos 휃 푝 sin 휃 휆0 (2.3)

This approximate pattern is valid for the overhead hemisphere (│θ│≤ 90 degree).

Figure 2-4 indicates F (θ) for a patch width Wp = 0.4 λ0.

Figure 2-4 Far Field By Replacing Slot Model

2.2.3 Excitation Techniques

Ther e are a l ot of di _erent feedi ng techniques, but I will discuss onl y the most i mportant ones. Ther e are a l ot of di _erent feedi ng techniques, but I will discuss onl y the most i mportant There are a several feeding techniques such as microstrip line and coaxial line as shown in

Figure 2-5.

Figure 2-5 Excitation techniques (a) Direct feed, (b) Coaxial Feed 2.2.3.1 Transmission Line Feed

The easiest way to feed the microstrip patch is to connect a microstrip line straightly to the end of the patch. This way of feeding is named direct feed (Figure 2-5(a)). The benefit of this technique is simplicity in terms of manufacturing and analysis. Impedance matching may be achieved by choosing a suitable inset cut length. Microstrip properties with the transmission line and the patch on the same substrate plane cannot be optimized simultaneously as an antenna component and as a transmission line, thus the characteristic requirements are contradictory. Therefore a compromise has to be made and this leads to spurious radiation which results in higher sidelobes and higher cross polarization level.

2.2.3.2 Coaxial Feed

A second way as shown in Figure 2-5(b) to feed the radiators is coaxial line that penetrates the ground and substrate and is conjoined to the patch. The input impedance depends on the location of the feed. Thus, the antenna is matched by selecting the appropriate feed position. In this technique the patch and the feeding network are shielded by the ground plane. The dielectric substrate may be chosen autonomously to optimize both the patch and feeding network.

Figure 2-6 Aperture Coupled Patch Antenna 2.2.3.3 Aperture Coupling

This technique is eschews a direct connection between the microstrip patch and the feed line as shown in Figure 2-6. There is a feed line consisting of an open ended microstrip line that is positioned on a second dielectric layer below the ground plane. The two lines are electromagnetically coupled via an electrically thin slot in the ground plane between them. This slot is called an aperture and will not resonate at the required frequency band of the antenna. Therefore this would generate radiation towards the rear of the antenna.

The patch is shielded from the feed because of the ground plan. Another significant benefit of this technique is the freedom to choose two different substrates.

2.2.4 Equivalent Circuit

It’s important to know which design specifications to screw to modify antenna performance, and how to change a particular design is properties to achieve the required antenna efficiency. As long as aperture coupled antenna is a great wrapped electromagnetic properties which explains simply some of the design specs may be given on estimate influence to increase antenna efficiency [30].

The basic equivalent circuit for an aperture coupled microstrip antenna is shown in

Figure 2-7. The feed line is modeled by a transmission line with characteristic impedance of Z0,f . At one end the input port of the antenna is conjoined, and at the other end is left open circuits. The feed line does not terminate abruptly under the aperture. Instead, it is lengthenes by the stub length Lstubas indicated in Figure 2-7. Stub length has significate influence on the input impedance of the antenna. At the first look with regarding the coupling between the feed line and the aperture as a typical three port and the aperture itself as a usual two port. Figure 2-8 indicates that the patch is viewed as a common one port.

The aperture oneself can be considered as a parallel tank circuit with a inductance Lap and a capacitance Cap. This is because of the resonant length of a half wavelength aperture.

When the aperture is thinner it acts inductively.

Figure 2-7 Configuration of aperture coupled microstrip antenna. (a) Side view. (b) Top view.

Figure 2-8 General Aperture Coupled Microstrip Antenna Network Model The patch can act as an open ended transmission line. To compute for the fringing field at each open end a capacity or Cfring and a resistor Rrad is embedded. The resistors models the radiation from the patch antenna. As a replacement for modeling the patch by a transmission line, it can also be considered as a series circuit [31].

The characteristic impedance of the transmission line that models the patch, Z0,P , is different than Z0, f . The transformer is repositioned at the middle of the aperture and the patch and has a turns ratio of N. Figure 2-9 indicates the equivalent circuit.

Figure 2-9 Equivalent circuit for an aperture coupled microstrip antenna.

2.3 Proposed Array Antenna Configuration

2.3.1 Proposed Eight Antenna Configuration

The proposed eight element rectangular microstrip array antenna (ERMSA) is designed using an FR4 substrate having a dielectric constant ɛr= 4.4 and thickness h= 0.5 mm. The 3-D schematic and geometry of ERMSA is shown in Figure 2-10 . The size of the patch is 8×8 mm2. The distance between the centers of two adjacent patch antenna is 6 mm. Elements of the array are designed for a 9.5 GHz frequency of with dimensions

118×100 mm2.

The feeding scheme for aperture-coupled patch antennas, which were first proposed by Pozar [12], dispose of the soldering process needed to protect a pin-connected feed circuit to the patch, whichever is unavoidable in fabricating the conventional probe-fed patch antenna [32][33]. In some cases using the adaptive algorithms for tapering in phased

39 array antennas cause better results in beamforming [34] [35]. The feeding design can enormously cut down the complexity in constructing large patch arrays consisting of many elements[36].

Figure 2-10 3-D Schematic And Feeding Network Of Ermsa Aperture Oupled Microstrip Linear Array Antenna A simulated impedance bandwidth of the proposed antenna element covers 8.6 to

10.9 GHz for less than 10 dB as shown in Figure 2-11.

Figure 2-11 Comparison Between Measured and Simulation Return Loss Diagram Of The Proposed Antenna For Ermsa

The radiation patterns of the proposed array antenna were measured in an anechoic chamber at the AntennaWorld Inc. Technology lab in Miami, Florida with an Orbit/FR

FR959 Antenna Measurement system [37] [38] as shown in Figure 2-12. Measured radiation patterns at 9.5 GHz are shown in Figure 2-12(a) for phi equal 0 degrees and

Figure 2-12(b), for phi equal 90 degrees. It can be observed that good direction of the proposed antenna in the radiation patterns in the operating band are less than -17 dB.

The array antenna has been simulated and optimized using Ansoft HFSS 2016 software [39]. Figure 2-13 shows the simulated, prototype and measured two layer eight element rectangular microstrip array antenna[40].

(a)

(b)

Figure 2-12 Comparison Measured Radiation (Polar Plot) Patterns Of The For ERMSA. A) PHI=0 DEG. B) PHI=90 DEG.

Figure 2-13 Prototype of Proposed ERMSA Antenna

A wideband T-junction power divider is needed to feed this aperture coupled antenna array. According to the position of the array element’s port and the operation bandwidth of antenna array, a three section T-junction power divider, which provides 42 signals with balanced amplitudes and phases from output ports, is used to design the 8-way feed network. The total section number of impedance transformers from Rin to Rout is 14 according to the theory of the Quarter-wave transformer [41] [42]. The length of each section of impedance transformers is set approximately λg/4, where λg the guided wavelength at center frequency. The widths are firstly optimized by Advanced Design

System (ADS). Optimal parameters set as follow: R1=91.7Ω, R2=87.5Ω, R3=76.9Ω,

R4=65.9Ω, R5=59.4Ω, R6=50.3Ω, R7=40.9Ω, R8=34.3Ω, R9= 59.7Ω, R10=53.0Ω,

R11=45.2Ω, R12= 36.8Ω, R13= 62.9Ω, R14=56.8Ω. The simulated results of return loss

S11 obtained by the HFSS which was lower than -13.7 dB from 8.6 to 12.8 GHz. The geometry of microstrip fed along the sides is shown in Figure 2-14.

Figure 2-14 Geometry of 8-way T-junction power divider.Simulated performance of 8-way Tjunction power These designed, we propose a Broadband stacked, low-cost low profile, based on an aperture coupled feeding with linear array. Paper presents the experimental investigations carried out for obtaining X-band operation of rectangular microstrip antenna by using aperture coupled feeding technique. Frequency range set at approximately 8.6 to

10.9 GHz, by incorporating frequency range of the antenna resonates for single wideband with an impedance bandwidth of 23%. The optimum design dimensions are used to achieve 43 the compact aspect and high radiation efficiency. In this design, the enhancement of impedance bandwidth and gain does not disturb the humor of broadside radiation characteristics. The design concepts of antennas are presented and experimental results are discussed. This letter describes the design, operation, and realization of the beamforming array antenna[43][44].

2.4 Experimental Comparison for Proposed Antenna

The proposed ERMSA antenna array are designed using FR4 substrate having dielectric constant ɛr= 4.4 and thickness h= 0.5 mm. The 3-D schematic and geometry of

ERMSA is shown in Figure 2-10 . The size of the patch is 8×8 mm2. Distance between the centers of two adjacent patch antenna 6 mm. elements of array are designed for 9.5 GHz frequency with dimensions 160×88 mm2 and four element rectangular microstrip array antenna (FRMSA) antenna with 80 × 44 mm2.

Comparison impedance bandwidth of the proposed single, FRMSA and ERMSA antenna element covers 8.6 to 10.9 GHz for less than 10 dB as shown in Figure 2-15. The proposed antennas delivers about 10, 11 and 14 dBi gains at 10.4 GHz, respectively for single element, FRMSA and ERMSA shows in Figure 2-18. It can be seen that the proposed antenna array’s gains are increasing 10 to 15 dBi

The radiation patterns comparisons of the single, FRMSA and ERMSA antennas were measured in an anechoic chamber at the AntennaWorld Inc. Technology lab in

Miami, Florida with Orbit/FR FR959 Antenna Measurement system [38] [37][5-6] as shown in Figure 2-16 and Figure 2-17. Measurement radiation patterns at 9.5 GHz are shown in Figure 2-16 for phi equal 0 degree and Figure 2-17, for phi equal 90 degree.

Figure 2-15 Return loss diagram of the proposed antenna for single element, 4 (1x4)-element and 8 (1x8)-element

Figure 2-16 Comparison Radiation (polar plot) patterns of the single element, 4 (1x4)- element and 8 (1x8)-element, Phi=0 deg

Figure 2-17 Comparison Radiation (polar plot) patterns of the for single element, 4 (1x4)- element and 8 (1x8)-element , Phi=90 deg

Figure 2-18 Comparison proposed antenna gains of single element, 4 (1x4)-element and 8 (1x8)-element as a function of frequency

2.5 Array Antenna Beamforming

Continuously adapted to the actual situation of the antenna pattern requires an algorithmic controlled phased array which will explain in the future chapterswith adaptive array antenna systems. The designed antenna operating in X-band at 9.5 GHz, where the required bandwidth is 230 MHz Since microstip patch antenna have a low bandwidth, a special feeding technique aperture coupling was accomplished For the design of antenna a simulation tool based on Methods of Moments was used. After optimizing and implementing a single element antenna that achieved satisfactory measurement results, the antenna array four element rectangular microstrip array antenna (FRMSA) and eight element rectangular microstrip array antenna (ERMSA), was worked out. The excellent final results for the antenna array a bandwidth of 230 MHz, a front to back Ration larger than 17dB as shown in Figure 2-19 , and a maximum mutual coupling below -14.5 dB is another proof that aperture coupled microstrip patch antenna are suitable for the operation in a X-band communication system. In this thesis, eight-element array antenna prototype of the microwave beamforming structure is constructed and tested. The prototype is composed of the microwave hardware and control hardware. A new compact switch is designed and utilized in the structure. The control circuits is an eight branch belong with the Varactor diodes. Different beamforming examples such as beam steering, sidelobe level

(SLL) control and null forming are examined with the prototype.

Figure 2-19 Measuring front to back ratio with network analyzer

2.5.1 Array Element Design

One of the substantial difficulty in modeling a phased array antenna is that considerable sections of the microwave power transmitted by one element of the array can be received by the circumambient array antenna elements. This efficacy, which is acknowledged as array mutual coupling, can outcome in a remarkable or whole waste of transmitted or received signal, pertaining on the coherent composition of all of the mutual- coupling signals in the array. The amplitudes and phases of the array antenna mutual coupling signals appertain preliminary on the formation of the radiating antenna elements, the distancing among the array elements, and the number of the radiating array antenna elements. There are as plenty several design contingencies for phased arrays antenna as there are lots of different radiating array antenna elements to select from them. Therefore, distancing between the array elements and number of radiating array antenna elements can vary extensively, related to the searching operational requirements. Inherently, demands to comprehend fully the mutual coupling features of whatever reason was to choose the

48 radiating elements. A notable challenge in modeling antenna arrays is required specification of scan mass and bandwidth while steering clear of blind positions and retaining low sidelobes [18] [30] [45] [46] [47] [48][49] [50] As shown in Figure 2-20.

Significance of Block diagram of the transmitter corporate fed phased array antenna system that uses phase shifters to electronically steer the beam pattern over the scan segment. The antenna input is a RF source generates a waveform that is distributed into individual route denominated element channels, each comprising a phase shifter and power distribution network.

Figure 2-20 The antenna uses phase shifters to steer the radar beam electronically over the scan sector. The radio-frequen

Figure 2-21 An idealized radiation pattern from a single antenna element covers the scan sector, with signal strength drop

Figure 2-22 When all the phase shifters of the array are properly aligned, the array produces a main Figure 2-21 displays an idealized univalent element radiation pattern that coverages the scan segment, with signal stability dropping outside of the segment. When all the phase shifters of the array antenna are accurately lined up, the array antenna generates a main beam in the desired punctuate aim, as shown in Figure 2-23.

Typically, the power distribution network is designed with minimal crosstalk amongst the channels. Whenever the signals have attained the radiating antenna array elements, although, a remarkable bulk of crosstalk such as antenna array mutual coupling befalls. The amplitudes and phases of these antenna array mutual coupling signals can earnestly strikes the proficiency of the smart array antenna. Array antenna mutual coupling generally can happens when spacing between the elements of array around one half wavelength.

This mutual coupling outwards itself in often detrimental changes in the element’s radiation of pattern and its return loss. Unless care is taken in the design of the array, blind positions in the scan segment can happen. These blind positions are angles where the beam pattern has a null and the return loss of the array antenna has a maximum around to univalent, as displayed in Figure 2-23. At these blind positions the total signal is notably

50 degraded in amplitude. Based on need, in some application blind position would place in directions where it is unwanted to transmit or receive in communication energy (e.g. jammers).

Figure 2-23 Conceptual images of blind-spot occurrence in a phased-array antenna. These results are typical of an array designed without regard for array mutual-coupling effects. A blind spot occurs when either the array element pattern has a null

Figure 2-24 Conceptual images of blind-spot occurrence in a phased Conceptual images of blind-spot occurrence in a phased-array antenna. These results are typical of an array designed without regard for array mutual-coupling effects. A blind spot occurs when either Figure 2-24 the element reflection coefficient has unity magnitude. The blind spot is often caused by array mutual coupling, which tends to direct

51 the radiation in the plane of the array as a surface wave, rather than as a wave propagating away from the array. Careful design of the array element shape, size, and spacing can prevent the occurrence of blind spots.

For example, Figure 5 compares a broadside-peak radiator (dipole or waveguide aperture) and a broadside-null radiator (monopole antenna). The latter element is useful when broadside radiation is undesirable, such as in reducing broadside clutter and jamming. As the radar beam is steered away from 0° (broadside) toward 60°, the conventional broadside-peak-type element radiation pattern drops off, but the broadside- null-type element radiation pattern increases to a peak at about 45° to 50°.

Early developments of phased-array radiating element technology were conducted at

Lincoln Laboratory during the period from 1959 to 1967. Beginning in 1959, the

Laboratory contributed to the theoretical understanding of phased arrays, particularly the effects of array mutual coupling on the performance of various configurations of dipole arrays; for example, the reports by Allen et al. [51][52][53][54][55]

[56][57][58][59][60][61]. Figure 2-26 shows one of the proposed X-band aperture coupled phased array antenna test beds used in measuring array element patterns, mutual coupling, null forming, pattern synthesis and array active scan impedance.

Figure 2-25 The radiation patterns of (a) a conventional broadside peak radiating element (dipole or waveguide)and (b) a broadside null radiating element (monopole). A broad side null element places a blind spot in directions where it is undesirable to transmit or recive radar energy

Figure 2-26 Proposed prototype phased array, 8x1 aperture coupled microstrip linear antenna

I this thesis we presents the design, development and experimental characteristics of phase array antenna integrated on a microwave laminate. Proposed eight element rectangular mutual coupled microstrip array antenna has been simulated & optimized on

ANSYS Electromagnetics Suite 16.2 [39]. A eight element linear antenna array is realized by cofabrication the corporate feed network, DC blocks, varactor diode on the aperture coupled stacked microstrip array antenna. All fabrication processes employed here can be performed at a good printed circuit manufacturing facility. The development of this system involves several electromagnetic design challenges. The design of the power distribution network is crucial in a phased array antenna to excite the array elements with the required amplitude and phase. As this involves several branching networks, a power distribution network can be implemented easily using microstrip or stripline transmission lines in a corporate feed arrangement as showed in Figure 2-14. In addition, DC bias voltage is required for actuating these electrostatic beams. This requires addition of DC block and RF feed through in the circuit. The transition on the input side includes the DC-block to prevent the bias voltage at a varactor diode from reaching the RF source and other varactor diodes.

Radial stub is included at the other end of the varactor diode to avoid leakage of RF energy through the DC bias terminals. This distributed variable capacitance acts in parallel with the capacitance per unit length of the distributed model of a transmission line. Therefore the change in capacitance affects the characteristic impedance and phase constant of the line [62][63].

Varactor diodes can be explained based on the changes in the voltage. These diodes structured, whose height can be varied by applying a DC bias. of At the had embedded

54 the control hardware which generates electrically different value of capacity. In each branch with embedding the varactor diode which shows in Figure 2-26.

As results of sidelobe reduction, first null steering on the phased array system at

9.5 GHz, by applying 0 V and 25 V to the voltage controlling the Voltage of the Varactor.

As can be seen in Figure 2-28, the SLL with applying 2 PF is lower than the 0 PF by more than 3 dB and the pattern is broadened. Figure 2-28 indicate the phase shift obtained by changing the beam height. High DC voltage needs to be applied to the varactor and beam geometry to move these beams. Wired are soldered to the radial stubs and to a common ground conductor to apply the DC bias voltage to actuate the varactor diode. In this demonstration eight DC power supply of 0-300 V are used to provide the DC bias to the varactor diodes. The outputs of this divider are approximately at 0-5V- 15V-25V , where is the DC supply voltage. It may be noted that the beam deflection and hence capacitance change due to actuation are not linearly proportional to the voltage applied.

The beam steering measurement of the phased array is performed in an anechoic chamber. The radiation pattern is recorded at 9.5 GHz in the azimuth plane by applying 25

V to the voltage of the varactor diodes. The Null-forming capability of novel techniques investigated. As can see, the first null in the -30 and 30 degrees directions. The null depths for -30, 30 degree directions for 0, 0.3, 0.7, 1.5, 2 PF are respectively desired null depth are generated 0,-22, -19, -18, -17 dB. As noted before, as various layers are assembled, there could be minor differences in the thickness from expected values. This may also have caused the pattern to show a higher than expected sidelobe level. On the application of the bias, the null shifts by 50 degree as shown in Figure 2-28. However, the beam scan angle can be increased by increasing the voltage[64]. As the null shifts, there is a major change

55 in beam width. Furthermore, first null of beam width (FNBW) has increased with increasing the applied voltage to the varactor diode as shown in Figure 2-28.

Figure 2-27 Radiation pattern and null forming

Figure 2-28 Radiation pattern of side lobe cancelation 8-element antenna at 9.5 GHz with applying 0 PF to 2 PF to the voltage controlling

Concerning beamforming, the uniform linear array antenna shown to have excellent beamforming capabilities in directing the maximum towards the direction of the signal of interest and deep nulls towards the directions of signals not of interest. Therefore, some deficiency with beamforming along elevation using uniform linear arrays antenna was observed, this can be justified by the fact that the uniform linear array in our simulations was located on the x-y plane. Apparently, better results should be expected with an array possessing additional degree of freedom along elevation, a three dimensional such as a cylindrical or spherical array.

These designed antennas are very simple, cost effective and high efficiency for the applications in 8.6-10.9 GHz frequency ranges. We presented a 9.5 GHz band with side lob cancelation antenna using varactor diodes for the first time. We fabricated 8 (1x8)- element planar rectangular aperture coupled microstrip linear array developed for 9.5 GHz applications. Moreover, we examined the relationship between antenna performance and the arrangement of radiation elements. Antenna gain of more than 14 dBi and bandwidth of about 2 GHz were achieved for ERMSA antenna. Furthermore, a side lobe cancelation and null steering ERMS Antenna using varactor diodes was demonstrated. Results shows that a Sidelobe level control and null forming pattern radiation. Since the phase shifters of this antenna consist of varactor diodes, the smart antenna can be fabricated at low cost.

This simple approach of fabricating various components can be readily extended for large phased arrays required in radar and space communication applications. With the increased use of radio frequency terminals capable of directing the radiated beam adaptively, possibility for beam steering have gained much attention in communication and radar applications. In radar systems, this feature is required to determine the direction of targets.

Modern radar systems require adaptive control of beam direction, beam shape, beam width, sidelobe level and directivity to improve the coverage and tracking resolution. The amplitude and phase of radiated signal at each antenna element is controlled to adjust side lobe levels, conveniently locate nulls, and to shape the main beam of radiation.

Analysis of an Adjustable Modified Kaiser Weighting Function

and Its Application To Linear Array Antenna Beamforming

3.1 Introduction

Weighting functions are widely used in digital signal processing for application in signal analysis and estimation, digital filter design, speech processing and especially in array antenna beamforming [65] [66] [67]. Due to their flexibility properties, the adaptable weighting functions such as Kaiser and Dolph-Chebyshev weighting functions are very smart in signal processing applications [68]. Since the weighting functions are suboptimal solutions, the best weighting function depends on the applications. The basic idea behind the weighting function is choosing a proper ideal frequency selective beamformer which always has a non-causal, infinite-duration impulse response and then truncate or weighting function its impulse response hd[n] to achieve a linear-phase and causal FIR filter [65].

푓(푛) 0 ≤ 푛 ≤ 푀 h[n]=h [n] w[n] ; w[n]={ } d 0 표푡ℎ푒푟푤푖푠푒 (3.1)

The Kaiser weighting function is a kind of two parameter weighting function that has maximum energy concentration in the main lobe, which controls the mainlobe width and ripple ratio [69]. A weighting function, w (nT), with a length of N is a time domain function which is defined by:

푛표푛푧푒푟표|푛| ≤ (푁 − 1)/2 W (nT) ={ 0 표푡ℎ푒푟 (3.2)

Beamforming weighting functions are generally compared and classified in terms of their spectral characteristics. The frequency spectrum of W (nT) can be introduced as

[70].

푗푤(푁−1)푇 푗푤푇 푗푤푇 − W (푒 ) =푊0(푒 )푒 2 ( 3 .3)

Where W (푒푗푤푇) is called the amplitude function, N is the weighting function length, and

T is the space of time between samples. Two parameters of the weighting functions in common are the null to null width BN and the main-lobe width BR. These quantities are defined as BN = 2ωN and BR = 2ωR, where ωN and ωR are the half null-to-null and half main-lobe widths, respectively, as shown in Figure 3-1.

There are two important performance measures for the weighting function which are called [68]:

R= - (the ripple ratio in dB)

S = the side-lobe roll-off ratio

Figure 3-1 A Typical Weighting Function Normalized Amplitude Spectrum Convex minimization has many applications in engineering such as signal processing, communication networks, estimation and detection, data processing and modeling [71] [72]. With contemporary developments in computations and implementation, various optimization problems can be considered as convex and nonconvex minimization problems [72].

In this chapter we propose a modified Kaiser weighting function based on minimized sidelobes that have a better ripple ratio and lower sidelobe (at least 7 dB) compared to the Kaiser and Cosine hyperbolic weighting functions having an equal mainlobe width. Also, its sidelobe level can be controlled by coefficients which are very important for array beamforming in any angular of radiation pattern.

3.2 Derivation of Proposed Weighting Function

3.2.1 Kaiser Array Weighting Function

In discrete time domain, Kaiser Array weighting function is defined by [73] [69]

[69] .

퐼0(훽푥) 푁 − 1 ( ) |푛| ≤ 푤푘 푛 = { 퐼0(훽) 2 0 표푡ℎ푒푟 (3.4)

Where β is the shape parameter, N is the length of array weighting function and

I0(x) is the modified Bessel function of the first kind of order zero [68].

2푛 2 푥 = √1 − ( ) 푁 − 1 (3.5)

The final expression of 푤푘 is expressed as:

2푛 2 퐼 (훽√1 − ( ) ) 0 푁−1 푁 − 1 푤푘(푛) = |푛| ≤ 퐼 (훽) 2 0 (3.6) { 0 표푡ℎ푒푟

3.2.2 Proposed Weighting Function

A linear array which has uniform distribution with N isotropic branches antenna at the same space with distance d is considered. The received signals x (n) at the sample n is an N×1 vector given as:

푥(푛) = 푏(푢푡)푠(푛) + 푣(푛) (3.7)

푗2휋푑푢⁄ 푗2휋(푁−1)푑푢⁄ 푏(푢푡) = [1, 푒 휆, … , 푒 휆] (3.8)

s (n) is the desired signal waveform, and b (푢푡) vector of target signal impinging array from direction θt, v(t) represents the sum of undesired interferences and noise. The output of beamform method is given as:

푦(푛) = 푤퐻푥(푛) (3.9)

Where w is the N×1 complex weight vector and 푤퐻 shows the Hermitian transpose of w. The aim of an adaptive beamformers to cancel the interferences and noise and prepare the desired signal. Minimum variance distortionless response beamformer investigates the optimal weight vector w by solving the subsequent problem

min 퐻 퐻 푤푤 푅푣푤 , |푤 푏(푢푖)| < 휀 ( 3 . 10)

Where Rv is the interference and noise covariance matrix which contains the sidelobe region statistics. Steering vector error can be neglected by compelling the gain of

63 array over a varied direction range more than constant and low sidelobes in contrast to unexpected interferences. Rv can be estimate by [74] [75].

푁 푏(푢 )푏퐻(푢 ) 푅̂ = ∑ 푖 푖 ∆푢 푣 −1 퐻 푖=1 푏(푢푖)푅̂푥 푏 (푢푖) (3.11)

Where 푅̂푥 denotes the sample covariance matrix which is assumed by:

푁 1 퐻 푅̂푥 = ∑ 푥(푘)푥 (푘) 푁 푘=1 (3.12)

By inserting the estimated covariance matrix 푅̂푣 into the (2.2), it can be obtained:

푁 |푤퐻푏(푢 )|2 푤퐻푅̂ 푤 = ∑ 푖 ∆푢 푣 −1 퐻 푖=1 푏(푢푖)푅̂푥 푏 (푢푖) (3.13)

퐻 2 The proposed weighting function based on minimization of this term |푤 푏(푢푖)| is made by inserting new parameter which called 휓 which gives one degree of freedom in

(3.5). Optimal weight vector w can be given by solving the minimization problem (3.12).

min 2 min푤퐻푅̂ 푤 ≈ |푤퐻푏(푢 )| 푤 푣 푤 훾 푖 (3.14)

By keeping (3.12) in (3.13):

min 2 휓푛 2 퐼 (훽√1 − ( ) ) 0 푁−1 min|푤퐻푏(푢 )|2 = | 푏(푢 )| 푤 푖 | 퐼 (훽) 푖 | 0 (3.15) 푤

This problem can be assumed as nonconvex form [76] [77], then:

휕|푤퐻푏(푢 )|2 푖 = 2푤푏(푢 )퐻 ≈ 0 휕푤 푖 (3.16)

Now the results can rewrite to scalar form

푁−1 2 푛 퐻 ∑ 푤푛푏(푢푖 ) ≈ 0 푁−1 푛=− (3.17) 2

With combination of (3.15), (3.17):

푁−1 휓푛 2 2 퐼 (훽√1 − ( ) ) 0 푁−1 ∑ 푏(푢푛)퐻 ≈ 0 퐼 (훽) 푖 푁−1 0 푛=− (3.18) 2

푁−1 휓푛 2 2 퐼 (훽√1 − ( ) ) 0 푁−1 −푗2휋푛푑푢푖⁄ ∑ 푒 휆 ≈ 0 퐼 (훽) 푁−1 0 푛=− (3.19) 2

2휋푑푢푖 The ∅푖 ≜ 휆 definition it can be reduced to

푁−1 휓푛 2 2 퐼 (훽√1 − ( ) ) 0 푁−1 ∑ 푒−푗푛∅푖 ≈ 0 퐼 (훽) 푁−1 0 푛=− (3.20) 2

Solving this symmetric summation helps to achieve desire result, then extending this term:

푁/2 ⏞ 2 푁−1 휓 푁 − 1 휓 2 2 퐼 훽√1 − ( 2 ) 푐표푠( ∅ ) + ⋯ + 퐼 (훽√1 − ( ) ) 푐표푠(∅ ) ) 0 푁 − 1 2 푖 0 푁 − 1 푖

( ) ( ) (3.21)

+ 퐼0(훽) = 0

For solving this equation, assume that any 푁/2 term has the uniform answer corresponding (3.22)

휓 2 −퐼 (훽) √ 0 퐼0 (훽 1 − ( ) ) cos(∅푖) = 푁 − 1 푁

2휓 2 −퐼 (훽) √ 0 퐼0 (훽 1 − ( ) ) cos(2∅푖) = 푁 − 1 푁

⋮ (3.22) 2 푁−1 휓 푁 − 1 −퐼 (훽) 퐼 훽√1 − ( 2 ) cos ( ∅ ) = 0 0 푁 − 1 2 푖 푁 { ( )

Finally an answer set of 휓 vector can be achieved by (3.22)

휓 = 1 0 2 −1 −퐼0(훽) 퐼0 ( ) √ 푁 cos(∅푖) 휓1 = (푁 − 1) 1 − 2 훽

2 −1 −퐼0(훽) 퐼0 ( ) (푁 − 1) √ 푁 cos(2∅푖) 휓 = 1 − 2 2 훽2 (3.23) ⋮ 2 −1 −퐼0(훽) 퐼0 ( (푁−1) ) √ 푁 cos( ∅ ) (푁 − 1) 2 푖 휓(푁−1) = (푁−1) 1 − 2 2 훽 { 2

As mentioned the 퐼0 is even function then 훙 vector should be symmetric

휓푘 = 휓−푘 (3.24)

휓 (푁−1) − 2 ⋮ 휓−푘 ⋮ 훙 = 휓0 ⋮ (3.25) 휓 푘 휓(푁−1) [ 2 ]

3.2.3 Mainlobe Width Controlling Technique

In the Kaiser Weighting function, when 훙 vector and the previous algorithm is used for decreasing the sidelobes, the mainlobe width will change, therefore, we have introduced parameters for controlling these alterations of the proposed method. First, the Fourier transform of the Kaiser window is given where time is treated as continuous:

2 2 휓푘푛 2 푀휔 퐼0 (훽√1 − ( ) ) sinh (√훽 − ( ) ) 푁−1 푭 푀 휓푘 ⇒ 퐼0(훽) 퐼0(훽) 푀휔 2 √훽2 − ( ) (3.26) 휓푘

Where 푀 = (푁 − 1)푇 continues the time form. Then, the first null after the main lobe of W0 (휔) occurs at:

2휓 휔 = 푘 √훽2 + 휋2 0 푀 푘 (3.27)

Now by solving (3.27) for 훽푘

푀휔 2 훽 = √( 0) − 휋2 푘 2휓 푘 (3.28)

In the proposed Kaiser weighting function, 휓푘 is less than 2 and this decreases the mainlobe width. By increasing the 훽푘 quantity the mainlobe becomes constant. The RMS value of

훽푘 vector is equal to the desire width:

(푁−1) 2 1 훽̅ = √ ( ∑ 훽2) 푁 + 1 푘 (푁−1) 푘=− (3.29) 2

̅ The relation between 훽 and 휓푘 can shows by:

(푁−1) 2 1 푀휔 2 훽̅ = √ ( ∑ (( 0) − 휋2)) 푁 + 1 2휓 (푁−1) 푘 푘=− (3.30) 2

3.3 Theoretical and Characteristic Results

In this section theoretical results and characteristic behaviors of the previous

̅ section are presented. Figure 3-2 illustrates the relation between 훽 and 휓푘 for the constant mainlobe width.

By substituting (3.25) in (3.26), the elliptic function center of x will change to greater than one which helps in reducing the first sidelobe to lower the amount and have an equiripple form lower than 휀 which is shown in (3.10). Figure 3-3 illustrates this process.

Figure 3-2 Relation between β ̅ and ψ for constant mainlobe width

Figure 3-3 Elliptic center changed with inserting ψ_k in comparison to Proposed Weighting function and Kaiser Weighting function Figure 3-4 shows the relationships between the adjustable shape parameter and ripple ratio for the proposed Kaiser and Cosh weighting function with N=51. It is shown that by increasing R, the proposed weighting function has lower than -6 dB in comparison with the Kaiser weighting function and also in comparison with the Cosh weighting function. However, for R larger than -17.5 dB the Cosh weighting function has a lower quantity. In Figure 3-5 an approximate relationship can be found by using a curve fitting method as shown below:

0, 푅 > − 13.26 2 −0.001912푅 − 0.2508푅 훼 = −2.759, − 50 < 푅 ≤ − 13.26 ( 3 . 31 ) −5 3 2 −1.073 × 10 푅 − 0.002407푅 { −0.3179푅 − 6.217, − 120 < 푅 ≤ −50

The model which is plotted in Figure 3-4 provides a good approximation with an error designed in Figure 3-5. The largest deviation in alpha is 0.18 which corresponds to an error of 0.3dB in the actual ripple ratio. As for the Kaiser and Cosh models, the largest

71 deviation in alpha is 0.07 and 0.1 respectively, but these correspond to an error of 0.44dB and 0.4 dB respectively, in the actual ripple ratio [69]. The second design equation is the relation between the weighting function length and the ripple ratio.

Figure 3-4 Relation between α and R with approximation model for the Cosh, Kaiser and Proposed Weighting functions with N = 51

Figure 3-5 Error curve of approximated α given by (31) versus R for N = 51

To envisage the weighting function length (N) for given quantities of the ripple ratio(R) and half mainlobe width (WR), the normalized width Dw = 2wR (N − 1) is used

[73]. The relation between the normalized width and the ripple ratio for the proposed weighting function with N = 51 is given in Figure 3-6. By using the curve fitting method for Figure 3-6, an approximate design relationship between the normalized width (Dw) and the ripple ratio(R) can be established as:

0, 푅 > − 15 2 −0.002076푅 − 0.5663푅 w= +2.088, − 50 < 푅 ≤ − 15 ( 3 . 32 ) 2 0.000171푅 − 0.4453푅 {+2.53, − 120 < 푅 ≤ −50

The approximation model for the normalized width given by Eq. (3.32) is plotted in Figure 3-6. The relative error for the approximated normalized width in percent versus the ripple ratio for N =51 is plotted in Figure 3-7. The percentage error in the model is between 0.042 and −0.028. This error variety satisfies the error criterion [68] which states that the predicted error in the normalized width must be smaller than 0.1%. The digit value of the weighting function length (N) can be predicted from [68].

Figure 3-6 Relation between D and R with approximation model for the Cosh, Kaiser and Proposed weighting function with N = 51

Figure 3-7 Relative error of approximated D given by (32) in percent versus R for N = 51

3.4 Spectrum Comparisons

Figure 3-8 and Figure 3-9 show the frequency domain plots of Proposed, Cosh and Kaiser weighting functions with N=31 and N= 51 respectively. In these figures it is easily observed that as N increases the mainlobe width decreases as the sidelobe increases.

In addition, the sidelobe has a -7 dB lower than the others and has an equiripple form.

Figure 3-8 Comparison of the proposed, Cosh and Kaiser weighting functions for N=51 and β=6

Figure 3-9 Comparison of the proposed, cosh and kaiser weighting functions for N=31 And Β=2

3.5 Application to Linear Array Beamformer Design

FIR filter design is almost entirely restricted to discrete time implementations. The design techniques for FIR filters are based on directly approximating the desired frequency response of the discrete time system [78] [79].

To find a suitable array weighting function which satisfies the given prescribed beamformer specification, it is necessary to obtain the relation between the weighting function coefficients and beamforming parameters. The near stopband attenuation also gives the minimum stopband attenuation (As). The attenuation of the near stopband is important for some applications [80]. Figure 3-10 shows the relation between the minimum stopband attenuation (As) for N = 51. It is seen that as the weighting function parameter increases the minimum stopband attenuation also increases. By using the curve fitting method, an approximate expression of the first filter design equation can be expressed as:

0, 퐴푠 < 20.8 2 −0.002076퐴푠 − 0.5663퐴푠 훼 = +2.088, − 50 < 퐴푠 ≤ − 15 ( 3 . 33 ) 2 0.000171퐴푠 − 0.4453퐴푠 {+2.53, − 120 < 퐴푠 ≤ −50

The approximation model for the given by (3.33) is plotted in Figure 3-10. The model provides a good approximation with an error plotted in Figure 3-11.

Figure 3-10 Relation between α and the minimum stopband attenuation with approximated model for the Proposed weighting function with N =51

Figure 3-11 Error curve of approximated α given by (33) versus As for N = 51

Figure 3-12 The filters designed by the Proposed and Kaiser weighting functions for wct = 0.4 π , ∆w = 0.2 rad /sample with N = 51 For another comparison with the Kaiser weighting function, the filters were designed to achieve the cutoff frequency wct= 0.4π rad/sample and transition width ∆w =

0.2 rad/sample with N = 51 which is shown near the same pass band. The comparison result is shown in Figure 3-12. It shows that the filter designed by the proposed weighting function performs well As., whereas the filter designed by the Kaiser weighting function performs similar in far-end stopband attenuation.

3.6 Conclusion

This research proposes an adaptive modified Kaiser weighting function with beam pattern modeling which converts the beamforming problem into a nonconvex pattern problem. A parameter was inserted to minimize the sidelobe level and another algorithm was used for controlling the mainlobe width. This technique improved the ripple ratio and reduced the fluctuations of that. Furthermore it has equiripple form in sidelobes for the same antenna array length and mainlobe width compared with Kaiser Weighting function.

There was an improvement in terms of minimum stopband attenuation, filter length and near stopband attenuation with choosing the optimum adjustable parameters.

Flexible Exponential Weighting Function Toward Sidelobe

Suppression In Antenna Arrays

4.1 Introduction

Most communication systems, for example wireless LANs, and also radar microwave and millimeter-wave systems need antennas with better performance.

Especially, they have low weight, huge reproducibility and the ability of mixing with other microwave devices. The array antennas have all these benefits unlike previous aged antenna systems. However, the side lobe suppression (SLS) is the key drawback in these antennas [81]. Also, this problem is critical for evaluation radar systems. This is expected because unsettled SLS which can reflect signal by side lobe out in the radar antenna and makes errors in results. Weighting functions are broadly utilized in digital signal processing for application in signal analysis and estimation, and particularly in array antenna SLS [82]

[65] [66] [83] [84] [85]. Due to their flexibility properties, the adaptable weighting functions such as Kaiser and Dolph-Chebyshev weighting functions are very shrewd in various applications [86] [87]. Since the weighting functions are suboptimal solutions, the best weighting function depends on the applications. The basic idea behind the weighting function is choosing a proper ideal frequency-selective beamformer which always has a non causal, infinite-duration impulse response and then truncate or weighting function its impulse response ℎ푑[n] to achieve a linear-phase and causal FIR filter [65].

푓(푛) 0 ≤ 푛 ≤ 푀 h[n]=h [n] w[n] ; w[n]={ } d 0 표푡ℎ푒푟푤푖푠푒 (4.1)

Two enviable specifications for a weighting function are producing smaller main lobe width and good side lobe rejection. However, these two requirements are incongruous, since for a given length, a weighting function with a narrow main lobe has a poor side lobe rejection and contrariwise. By tapering the weighting function effortlessly to zero, the side lobes are greatly reduced in amplitude [66]. The Kaiser and Cosh weighting function have two parameters in weighting function which made maximum energy concentration in the main lobe, controls the mainlobe width and ripple ratio [69] [88] [87]. A weighting function, w(nT), with a length of N is a time domain function which is defined by:

푛표푛푧푒푟표|푛| ≤ (푁 − 1)/2 푤(푛푇) = { 0 표푡ℎ푒푟 (4.2)

Beamforming weighting functions are generally compared and classified in terms of their spectral characteristics. The frequency spectrum of w (nT) can be introduced as

[89][90].

푗푤(푁−1)푇 푗푤푇 푗푤푇 − 푊(푒 ) = 푊0(푒 )푒 2 (4.3)

Where W(푒푗푤푇) is called the amplitude function, N is the weighting function length, and T is the space of time between samples. Two parameters of the weighting functions in common are the null to null width BN and the main-lobe width BR. These quantities are defined as BN = 2ωN and BR = 2ωR, where ωN and ωR are the half null-to-

81 null and half main-lobe widths, respectively, as shown in Figure 4-1. There are two important performance measures for the weighting function which are called [68] [91][91]:

R= minus (-) the ripple ratio in dB.

S = the side-lobe roll-off ratio

Figure 4-1 A Typical Weighting Function Normalized Amplitude Spectrum [4] Convex minimization has many applications in engineering such as signal processing, communication networks, estimation and detection, data processing and modeling [71][92][72] With contemporary developments in computations and implementation, various optimization problems can be considered as convex and nonconvex minimization problems [93] .

In this chapter we propose a flexible exponential weighting function based on minimized sidelobes that have a better ripple ratio and lower sidelobe (at least 7 dB) compared to the Kaiser and Cosine hyperbolic weighting functions having an equal mainlobe width. Also, its sidelobe suppression can be controlled by coefficients which are very important for array in any angular of radiation pattern [81] .

4.2 Derivation of Proposed Weighting Function

4.2.1 Exponential Array Weighting Function

In discrete time domain, exponential array weighting function is defined by [94] [73][69].

푒푥푝(훽푥) 푁 − 1 |푛| ≤ 푤푘(푛) = { 푒푥푝(훽) 2 0 표푡ℎ푒푟 (4.4)

Where β is the shape parameter, N is the length of array weighting function and x is

2푛 2 푥 = √1 − ( ) 푁 − 1 (4.5)

The final expression of 푤푘 is expressed as:

2푛 2 푒푥푝 (훽√1 − ( ) ) 푁−1 푁 − 1 푤푘(푛) = |푛| ≤ 푒푥푝(훽) 2 (4.6) { 0 표푡ℎ푒푟

4.2.2 Proposed Weighting Function

A linear array which has uniform distribution with N isotropic branches antenna at the same space with distance d is shown in Figure 4-2.

Figure 4-2 Linear antenna array with N radiating elements The received signals x(n) at the sample n is an N×1 vector given as

푥(푛) = 푏(푢푡)푠(푛) + 푣(푛) (4.7)

푗2휋푑푢푐표푠(휃) 푗2휋(푁−1)푑푢푐표푠(휃) 푏(푢푡) = [1, 푒 , … , 푒 ] (4.8)

s(n) is the desired signal waveform, and b(푢푡) vector of target signal impinging array from direction θt, v(t) represents the sum of undesired interferences and noise. The output of beam form method is given as

퐻 푦(푛) = 푤 푥(푛) (4.9)

Where w is the N×1 complex weight vector and 푤퐻 shows the Hermitian transpose of w. The aim of an adaptive beamformer is to cancel the interferences and noise and prepare the desired signal. Minimum variance distortion less response beamformer investigates the optimal weight vector w by solving the subsequent problem

min 퐻 퐻 푤푤 푅푣푤 , |푤 푏(푢푖)| < 휀 ( 4 . 10)

Where Rv is the interference and noise covariance matrix which contains the sidelobe region statistics. Steering vector error can be neglected by compelling the gain of array over a varied direction range more than constant and low sidelobes in contrast to unexpected interferences. Rv can be estimate by [95] [96] [81].

푁 푏(푢 )푏퐻(푢 ) 푅̂ = ∑ 푖 푖 ∆푢 푣 −1 퐻 푖=1 푏(푢푖)푅̂푥 푏 (푢푖) (4.11)

Where 푅̂푥 denotes the sample covariance matrix which is assumed by

푁 1 퐻 푅̂푥 = ∑ 푥(푘)푥 (푘) 푁 푘=1 (4.12)

by inserting the estimated covariance matrix 푅̂푣 into the (4.2), it can be obtained

푁 |푤퐻푏(푢 )|2 푤퐻푅̂ 푤 = ∑ 푖 ∆푢 푣 −1 퐻 푖=1 푏(푢푖)푅̂푥 푏 (푢푖) (4.13)

퐻 2 The proposed weighting function based on minimization of this term |푤 푏(푢푖)| is made by inserting new parameter which called 휓 which gives one degree of freedom in

(4.5). Optimal weight vector w can be given by solving the minimization problem (4.12).

min 2 min푤퐻푅̂ 푤 ≈ |푤퐻푏(푢 )| 푤 푣 푤 훾 푖 (4.14)

By keeping (4.6) in (4.12)

min 2 휓푛 2 푒푥푝 (훽√1 − ( ) ) 푁−1 min|푤퐻푏(푢 )|2 = | 푏(푢 )| 푤 푖 | 푒푥푝(훽) 푖 | (4.15) 푤

This problem can be assumed as nonconvex form [97] [98] then

휕|푤퐻푏(푢 )|2 푖 = 2푤푏(푢 )퐻 ≈ 0 휕푤 푖 (4.16)

Now the results can rewrite to scalar form

푁−1 2 푛 퐻 ∑ 푤푛푏(푢푖 ) ≈ 0 푁−1 푛=− (4.17) 2

With combination of (4.15), (4.17)

푁−1 휓푛 2 2 푒푥푝 (훽√1 − ( ) ) 푁−1 ∑ 푏(푢푛)퐻 ≈ 0 푒푥푝(훽) 푖 푁−1 푛=− (4.18) 2

푁−1 휓푛 2 2 푒푥푝 (훽√1 − ( ) ) 푁−1 ∑ 푒−푗2휋푑푢푖푐표푠(휃) ≈ 0 푒푥푝(훽) 푁−1 푛=− (4.19) 2

The ∅푖 ≜ 2휋푑푢푖푐표푠(휃) definition it can be reduced to

푁−1 휓푛 2 2 푒푥푝 (훽√1 − ( ) ) 푁−1 ∑ 푒−푗푛∅푖 ≈ 0 푒푥푝(훽) 푁−1 푛=− (4.20) 2

Solving this symmetric summation helps to achieve desire result, then extending this term

푁/2 ⏞ 푁−1 2 휓 푁−1 휓 2 2 푒푥푝 (훽√1 − ( 2 ) ) 푐표푠( ∅ ) + ⋯ + 푒푥푝 (훽√1 − ( ) ) 푐표푠(∅ ) + 푁−1 2 푖 푁−1 푖 (4.21) ( )

푒푥푝(훽) = 0

For solving this equation, assume that any 푁/2 term has the uniform answer corresponding

휓 2 −푒푥푝(훽) √ 푒푥푝 (훽 1 − ( ) ) cos(∅푖) = 푁 − 1 푁

2휓 2 −푒푥푝(훽) √ 푒푥푝 (훽 1 − ( ) ) cos(2∅푖) = 푁 − 1 푁

⋮ (4.22) 2 푁−1 휓 푁 − 1 −푒푥푝(훽) 푒푥푝 훽√1 − ( 2 ) cos ( ∅ ) = 푁 − 1 2 푖 푁 { ( )

Finally an answer set of 휓 vector can be achieved by (4.22)

휓 = 1 0 −푒푥푝(훽) 2 푙푛 ( ) √ 푁 cos(∅푖) 휓1 = (푁 − 1) 1 − 2 훽

−푒푥푝(훽) 2 푙푛 ( ) (푁 − 1) √ 푁 cos(2∅푖) 휓 = 1 − 2 2 훽2 (4.23) ⋮ 2 −푒푥푝(훽) 푙푛 ( (푁−1) ) √ 푁 cos( ∅ ) (푁 − 1) 2 푖 휓(푁−1) = (푁−1) 1 − 2 2 훽 { 2

As mentioned the 푒푥푝(훽푥) is even function then 훙 vector should be symmetric

휓푘 = 휓−푘 (4.24)

휓 (푁−1) − 2 ⋮ 휓−푘 ⋮

휓0 ⋮ (4.25) 휓 푘 휓(푁−1) [ 2 ]

4.2.3 Mainlobe Width Controlling Technique

In the proposed Weighting function, when 훙 vector and the previous algorithm are used for decreasing the sidelobes, the mainlobe width will change, therefore, we have introduced parameters for controlling these alterations of the proposed method. First, the Fourier transform of the exponential window is given where time is treated as continuous:

휓 푛 2 푀휔 2 푒푥푝 (훽√1 − ( 푘 ) ) exp (√훽2 − ( ) ) 푁−1 푭 푀 휓푘 ⇒ 푒푥푝(훽) 푒푥푝(훽) 푀휔 2 √훽2 − ( ) (4.26) 휓푘

Where 푀 = (푁 − 1)푇 continues the time form. Then, the first null after the main lobe of W0(휔) occurs at:

2휓 휔 = 푘 √훽2 + 휋2 0 푀 푘 (4.27)

Now by solving (4.27) for 훽푘

푀휔 2 훽 = √( 0) − 휋2 푘 2휓 푘 (4.28)

In the proposed exponential weighting function, 휓푘 is less than 2 and this decreases the mainlobe width. By increasing the 훽푘 quantity the mainlobe becomes constant. The RMS value of 훽푘 vector is equal to the desire width:

(푁−1) 2 1 훽̅ = √ ( ∑ 훽2) 푁 + 1 푘 (푁−1) 푘=− (4.29) 2

̅ The relation between 훽 and 휓푘 can shows by:

(푁−1) 2 1 푀휔 2 훽̅ = √ ( ∑ (( 0) − 휋2)) 푁 + 1 2휓 (푁−1) 푘 푘=− (4.30) 2

4.3 Theoretical and Characteristic Results

In this section theoretical results and characteristic behaviors of the previous section are

̅ presented. Figure 4-3 illustrates the relation between 훽 and 휓푘 for the constant mainlobe width.

By substituting (4.25) in (4.26), the elliptic function center of x will change to greater than one which helps in reducing the first sidelobe to lower the amount and have an equiripple form lower than 휀 which is shown in (4.10). Figure 4-4 illustrates this process.

Figure 4-3 Relation between β ̅ and ψ for constant mainlobe width

Figure 4-4 Elliptic center changed with inserting ψ_k in comparison to Proposed Weighting function and Kaiser Weighting function

Figure 4-5 Comparison of the proposed, cosh and kaiser weighting functions for n=17 and β=5.

4.4 Spectrum Comparisons

Figure 4-5 shows the frequency domain plots of Proposed, Cosh and Kaiser weighting functions with N=51. In these figures it is easily observed that as N increases

92 the mainlobe width decreases as the sidelobe increases. In addition, the sidelobe has a -6 dB lower than the Kaiser and -11 dB lower than Cosh and has aequiripple form [81].

4.5 Application to Linear Array High Side Lobe Suppression Design

In order to using the weighting function in the SLS, first, we should find 푢푛, as the kth element excitation of the of the array branches based on the following expression [99]

[35]:

푁−1

퐴퐹(휃) = 퐴 ∑ 푢푛푤푛 푛=1 (4.31)

Where 푤푛 = exp (휓푛). The array factor AF (θ) of the linear array antenna depends on the

θ and 푢푛 is written as:

푁−1 푗2휋푛푑푐표푠(휃) 퐴퐹(휃) = 퐴 ∑ 푢푛푒 푛=1 (4.32)

The number of weights must be one order fewer than the number of elements of the array. The weights are calculated using proposed modified exponential weighting method.

So, the feeding network in array antenna has three steps. A BAL-UN form is used for symmetrical microstrip structure next to the coaxial connector. In the first step, a T-junction considered for feeding network, and then in the second step, there are two T-junctions, and finally, four T junctions for last step. A relationship between the first and the second steps

93 are existed. Besides the second and the third steps have similar relationship. Thus, there is a linear taper with the purpose of transforming the impedance from 100Ω to 50Ω are existed. The array antenna design is investigates in Figure 4-6(a). Figure 4-6(b) shows the implemental feeding network for a designed rectangular aperture coupled microstrip linear array antenna with varactor diode in order to fix the impedance of the microstrip line.

A balanced microstrip method with λ0/4 impedance transformers is used for the feeding network. Zd is assumed as ends impedances at any microstrip lines. It can analyze relative power of array elements Pi based on TEM analysis, where i=1, 2, 3, 4.

2 푃푖 = |푢푖| (4.33)

The values 푢푖 are taken from proposed technique. The ratio between relative feeding power of rectangular aperture 1 and rectangular aperture 2 is:

2 푃2 |푢2| = 2 푃1 |푢1| (4.34)

2 |푢2| The 2 can be named as훾21. Similarly, |푢1|

2 푃4 |푢4| = 2 푃3 |푢3| (4.35)

2 | 4| The 2 can be named as 훾43. On the other hand, the input impedance ratio of |푢3| transformer Z1, Z2, and Z3, Z4, corresponding to rectangular antenna is:

푍2 푍4 = 훾21 , = 훾43 푍1 푍3 (4.36)

In the point A of feeding lines the rectangular 1 and 2 divides. If the ZS impedance is the A point, then the 푍푖 can be achieved by:

푍푠(훾21+1) 푍푠(훾43+1) 푍2 = , 푍4 = 훾21 훾43 (4.37)

and

푍1 = 푍푠(훾21 + 1) , 푍3 = 푍푠(훾43 + 1) ( 4 .38 )

Consequently,

푃퐵 푃4 + 푃3 = = 훾퐵퐴 푃퐴 푃2 + 푃1 (4.39)

So, the 푍퐴, 푍퐵 can be achieved by

푍푠(훾퐵퐴+1) 푍퐵 = , 푍퐴 = 푍푠(훾퐵퐴 + 1) 훾퐵퐴 (4.40)

The Zd is load impedance at the output of transformers in ideal case. So the푍푠 =

푍푑 = 50 Ω.

Figure 4-6 a) Array antenna with their impedance optimization. b) Array antenna design implementation with feeding network.

The radiation pattern of experimental test of antenna array with is presented in

Figure 4-7. The gain is about 14 dB in 10.4 GHz. The side lobe suppression varies in

different tests of weighting. When the rectangular antenna array are fed by generators with

varactor diode in their centers, the side lobe suppression is about 8 dB [81][100][101].

Figure 4-7 The radiation pattern of experimental test of antenna array based on proposed method.

4.6 Conclusion

This research proposes an adaptive flexible exponential weighting function to side lobe suppression which converts the SLS problem into a nonconvex pattern problem. A parameter was inserted to minimize the sidelobe level and another algorithm was used for controlling the mainlobe width. This technique improved the ripple ratio and reduced the fluctuations of that. Furthermore it has equiripple form in sidelobes for the same antenna array length and mainlobe width compared with Kaiser Weighting function which has lower complexity. There is an improvement in terms of maximum SLS attenuation, filter length. Finally the experimental results of a designed rectangular antenna array proofed the idea[81].

Malleable Hyperbolic Cosine Weighting Function For Side Lobe

Suppression In Medical Image Beamforming

5.1 Introduction

Most communication systems, for example wireless LANs, and radar microwave and millimeter-waves systems, need antennas with better performance. Especially, they have low weight, huge reproducibility and the ability of mixing with other microwave devices. Array antennas have these benefits unlike previous aged antenna systems.

However, the side lobe suppression (SLS) is the key drawback in these antennas. Also, this problem is critical for evaluation radar systems. This is expected because unsettled SLS which can reflect signal by side lobe out on the radar antenna and makes errors in results.

Weighting functions are broadly utilized in digital signal processing for application in signal analysis and estimation, and particularly in array antenna SLS [65][66][85][102].

Due to their flexibility properties, adaptable weighting functions such as Kaiser and Dolph-

Chebyshev weighting functions are very common in various applications [94] [86] [103].

Since the weighting functions are suboptimal solutions, the best weighting function depends on the application. The basic idea behind a weighting function is choosing a proper ideal frequency-selective beamformer which has a non-causal, infinite-duration impulse response and then truncate the weighting function to its impulse response ℎ푑[n] to achieve a linear-phase and causal FIR filter [104] [35][105].

푓(푛) 0 ≤ 푛 ≤ 푀 h[n]=h [n] w[n] ; w[n]={ } d 0 표푡ℎ푒푟푤푖푠푒 (5.1)

Two designable specifications for a weighting function are producing smaller main lobe width and good side lobe rejection. However, these two requirements are incompanble, since for a given length, a weighting function with a narrow main lobe has a poor side lobe rejection and contrariwise. By tapering the weighting function to zero, the side lobes are greatly reduced in amplitude [106] [66]. The Kaiser and Cosh weighting functions have two parameter which made maximum energy concentration in the main lobe and control the mainlobe width and ripple ratio [69] [87]. A weighting function, w(nT), with a length of N is a time domain function which is defined by:

푛표푛푧푒푟표 |푛| ≤ (푁 − 1)/2 푤(푛푇) = { 0 표푡ℎ푒푟푤푖푠푒 (5.2)

Beamforming weighting functions are generally compared and classified in terms of their spectral characteristics. The frequency spectrum of w(nT) can be introduced as

[107][108]

푗푤(푁−1)푇 푗푤푇 푗푤푇 − 푊(푒 ) = 푊0(푒 )푒 2 (5.3)

푗푤푇 Where W0 (푒 ) is called the amplitude function, N is the weighting function length, and T is the time between samples. Two parameters of the weighting function that are in common are the null to null width, BN, and the main-lobe width, BR. These quantities

99 are defined as BN = 2ωN and BR = 2ωR, where ωN and ωR are the half null-to-null and half main-lobe widths, respectively, as shown in Figure 5-1. There are two important performance measures for the weighting function which are [96] [109].

R= minus (-) the ripple ratio in dB. S = the side-lobe roll-off ratio

Figure 5-1 A typical weighting function normalized amplitude spectrum [4] Convex minimization has many applications in engineering, such as signal processing, communication networks, estimation and detection, data processing and modeling [110] [111]. With contemporary developments in computations and their implementation, various optimization problems can be considered as convex and nonconvex minimization problems [112][72].

In this chapter we propose a flexible hyperbolic cosine weighting function having an equal mainlobe width based on minimized sidelobes that have a better ripple ratio and lower sidelobe (at least 7 dB) compared to the Kaiser and Cosine hyperbolic weighting

100 functions. Also, its sidelobe level can be controlled by coefficients which are very important for array beamforming in any angular radiation pattern [35].

5.2 Derivation of Proposed Weighting Function

5.2.1 Hyperbolic Cosine Array Weighting Function

In a discrete time domain, the hyperbolic cosine array weighting function is defined by

[35][113].

푒푥푝(훽푥) 푁−1 |푛| ≤ 푤푘(푛) = { 푒푥푝(훽) 2 ( 5 . 4 ) 0 표푡ℎ푒푟푤푖푠푒

Here β is the shape parameter, N is the length of the array weighting function and x is

2푛 2 푥 = √1 − ( ) 푁 − 1 (5.5)

The final expression of 푤푘 is expressed as:

2푛 2 푒푥푝 (훽√1 − ( ) ) 푁−1 푁 − 1 푤푘(푛) = |푛| ≤ 푒푥푝(훽) 2 (5.6) { 0 표푡ℎ푒푟푤푖푠푒

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5.2.2 Proposed Weighting Function

A linear array which has uniform distribution with N isotropic branch antennas at the same space with distance d is considered. The received signals x(n) at the sample n is an N×1 vector given as:

푥(푛) = 푏(푢푡)푠(푛) + 푣(푛) (5.7)

푗2휋푑푢⁄ 푗2휋(푁−1)푑푢⁄ 푏(푢푡) = [1, 푒 휆, … , 푒 휆] (5.8)

s(n) is the desired signal waveform, and b(푢푡) is the vector of the target signal’s impinging on the array from direction θt, and v(t) represents the sum of undesired interferences and noise. The output of the beamforming method is given as:

퐻 푦(푛) = 푤 푥(푛) (5.9)

Here w is the N×1 complex weight vector and 푤퐻 is the Hermitian transpose of w.

The aim of an adaptive beamformer is to cancel the interference and noise and prepare the desired signal. A minimum variance distortionless response beamformer considers the optimal weight vector w by solving the following problem:

min 퐻 퐻 푤푤 푅푣푤 , |푤 푏(푢푖)| < 휀 ( 5 . 10)

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Here Rv is the interference and noise covariance matrix which contains the sidelobe region statistics. The steering vector error can be omitted by imposing the gain of the array over a varied direction range. Rv can be estimated by [114] [115]

푁 푏(푢 )푏퐻(푢 ) 푅̂ = ∑ 푖 푖 ∆푢 푣 −1 퐻 푖=1 푏(푢푖)푅̂푥 푏 (푢푖) (5.11)

Here 푅̂푥 denotes the sample covariance matrix which is given by

푁 1 퐻 푅̂푥 = ∑ 푥(푘)푥 (푘) 푁 푘=1 (5.12)

By inserting the estimated covariance matrix 푅̂푣 into (5.2), we obtain

푁 |푤퐻푏(푢 )|2 푤퐻푅̂ 푤 = ∑ 푖 ∆푢 푣 −1 퐻 푖=1 푏(푢푖)푅̂푥 푏 (푢푖) (5.13)

퐻 2 The proposed weighting function based on minimization of this term |푤 푏(푢푖)| is made by inserting a new parameter 휓 which gives one degree of freedom in (5.5). The optimal weight vector w can be given by solving the minimization problem (5.12):

min 2 min푤퐻푅̂ 푤 ≈ |푤퐻푏(푢 )| 푤 푣 푤 훾 푖 (5.14)

By using (4.6) in (4.12)

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min 2 휓푛 2 퐼 (훽√1 − ( ) ) 0 푁−1 min|푤퐻푏(푢 )|2 = | 푏(푢 )| 푤 푖 | 퐼 (훽) 푖 | 0 (5.15) 푤

This problem can be assumed as a non-convex form [114] [76], yielding:

휕|푤퐻푏(푢 )|2 푖 = 2푤푏(푢 )퐻 ≈ 0 휕푤 푖 (5.16)

Now the results can be rewritten in scalar form

푁−1 2 푛 퐻 ∑ 푤푛푏(푢푖 ) ≈ 0 푁−1 푛=− (5.17) 2

With a combination of (5.15), (5.17) we obtain

푁−1 휓푛 2 2 푒푥푝 (훽√1 − ( ) ) 푁−1 ∑ 푏(푢푛)퐻 ≈ 0 푒푥푝(훽) 푖 푁−1 푛=− (5.18) 2

104

푁−1 휓푛 2 2 푒푥푝 (훽√1 − ( ) ) 푁−1 −푗2휋푛푑푢푖⁄ ∑ 푒 휆 ≈ 0 푒푥푝(훽) 푁−1 푛=− (5.19) 2

2휋푑푢푖 Defining ∅푖 ≜ 휆 this can be reduced to

푁−1 휓푛 2 2 푒푥푝 (훽√1 − ( ) ) 푁−1 ∑ 푒−푗푛∅푖 ≈ 0 푒푥푝(훽) 푁−1 푛=− (5.20) 2

Solving this symmetric summation helps to achieve the desired result, then expanding this term gives

푁/2 ⏞ 2 푁−1 휓 푁 − 1 휓 2 2 푒푥푝 훽√1 − ( 2 ) 푐표푠( ∅ ) + ⋯ + 푒푥푝 (훽√1 − ( ) ) 푐표푠(∅ ) 푁 − 1 2 푖 푁 − 1 푖

( ) (5.21) ( )

+/푒푥푝(훽) = 0

For solving this equation, assume that any 푁/2 term has the uniform answer corresponding to (5.22)

105

푁 푛−1 퐻 푹̃풙 = ∑ 훾 푥(푖)푥 (푖) 푖=1 (5.22)

Finally an answer set for the 휓 vector can be achieved by (5.22):

휓 = 1 0 −푒푥푝(훽) 2 푙푛 ( ) √ 푁 cos(∅푖) 휓1 = (푁 − 1) 1 − 2 훽

−푒푥푝(훽) 2 푙푛 ( ) (푁 − 1) √ 푁 cos(2∅푖) 휓 = 1 − 2 2 훽2 (5.23) ⋮ 2 −푒푥푝(훽) 푙푛 ( (푁−1) ) √ 푁 cos( ∅ ) (푁 − 1) 2 푖 휓(푁−1) = (푁−1) 1 − 2 2 훽 { 2

As mentioned above, the 푒푥푝(훽푥) is an even function and the 훙 vector should be symmetric:

휓푘 = 휓−푘 (5.24)

106

휓 (푁−1) − 2 ⋮ 휓−푘 (5.25) ⋮ 훙 = 휓0 ⋮ 휓 푘 휓(푁−1) [ 2 ]

5.2.3 Mainlobe Width Controlling Technique

In the proposed weighting function, when 훙 vector and the previous algorithm are used for decreasing the sidelobes, the mainlobe width will change, therefore, we have introduced parameters for controlling these alterations of the proposed method. First, the

Fourier transformation of the hyperbolic cosine window is given where time is treated as continuous:

휓 푛 2 푀휔 2 푒푥푝 (훽√1 − ( 푘 ) ) exp (√훽2 − ( ) ) 푁−1 푭 푀 휓푘 ⇒ 푒푥푝(훽) 푒푥푝(훽) 푀휔 2 √훽2 − ( ) (5.26) 휓푘

Here 푀 = (푁 − 1)푇 continues the time form. Then, the first null after the main lobe of w0

(휔) occurs at:

2휓 휔 = 푘 √훽2 + 휋2 0 푀 푘 (5.27)

107

Now by solving (5.27) for 훽푘 :

푀휔 2 훽 = √( 0) − 휋2 푘 2휓 푘 (5.28)

In the proposed hyperbolic cosine weighting function, 휓푘 is less than 2 and this decreases the mainlobe width. By increasing 훽푘 the mainlobe becomes constant. The RMS value of 훽푘 vector is equal to the desired width:

(푁−1) 2 1 훽̅ = √ ( ∑ 훽2) 푁 + 1 푘 (푁−1) 푘=− (5.29) 2

̅ The relation between 훽 and 휓푘 is given by:

(푁−1) 2 1 푀휔 2 훽̅ = √ ( ∑ (( 0) − 휋2)) 푁 + 1 2휓 (푁−1) 푘 푘=− (5.30) 2

5.3 Theoretical and Characteristic Results

In this section theoretical results and characteristic behaviors of the previous section

̅ are presented. Figure 5-2 illustrates the relation between 훽 and 휓푘 for constant mainlobe width. 108

By substituting (5.25) in (5.26), the elliptic function center of x will change to greater than one which helps in reducing the first sidelobe and generats an equiripple form lower than 휀 as shown in (5.10). Figure 5-3 illustrates this process.

Figure 5-2 Relation between β ̅ and ψ for constant mainlobe width

Figure 5-3 Elliptic center changed with inserting ψ k in comparison to proposed weighting function and Kaiser weighting function

109

5.4 Spectrum Comparison

Figure 5-4 and Figure 5-5 show the frequency domain plots of the proposed Cosh and Kaiser weighting functions with N=31 and N= 51, respectively. The diagrams below clearly demonstrate that as N increases, the mainlobe width decreases as the sidelobe increases. In addition, the sidelobe is -7 dB lower than the others and has an equiripple form.

Figure 5-4 Comparison of the proposed, Cosh and Kaiser weighting functions for N=51 and β=6

110

Figure 5-5 Comparison of the proposed, Cosh and Kaiser weighting functions for n=31 and β=8

5.5 High Side Lobe Suppression in Medical Imaging Beamforming Design

Biomedical applications are an important area in medicine science that can benefit from the advantage of array antennas [116] [117]. Magnetic resonance imaging (MRI) is one of the most usaful of these adaptive antennas to detect breast tumors. The radio frequency (RF) antenna array is major component in forming the MRI signal that is transmitted and received, and therefore is one of the most important components of medical imaging systems. The design of adaptive RF array antennas is important in order to attain the best clinical decision [118] [119]. In this technique, an antenna array transmits wideband impulses at the breast surface, and then the receiver array antenna utilizes adaptive beam-forming to focus the reflected signal on the harmful tumor and recompense the other distractive propagation influence [81] [120]. In order to using the weighting

th function in the SLS, first, we should find 푢푛, as the k element excitation of the of the array branches based on the following expression:

푁−1

퐴퐹 (휃) = 퐴 ∑ 푢푛푤푛 푛=1 (5.31)

휓 Here 푤푛 = exp ( 푛). The array factor AF (θ) of the linear array antenna depends on θ and

푢푛 is written as:

푁−1 푗2휋푛푑푐표푠(휃) 퐴퐹 (휃) = 퐴 ∑ 푢푛푒 (5.32) 푛=1

111

The number of weights must be one order less than the number of elements of the array [81]. The weights are calculated using proposed modified exponential weighting method. So, the feeding network in array antenna has three stages. A BALUN form is used for symmetrical microstrip structure next to the coaxial connector [35][121]. In the first stage, a T-junction is used for feeding network, and then in the second stage, there are two

T-junctions, and finally, four T junctions for last stage. A relationship between the first and the second stages exist. Besides the second and the third stages have similar relationships.

Thus, there is a linear taper with the purpose of transforming the impedance from 100Ω to

50Ω are utilized. The array antenna design is shown in Figure 5-6 ( a). Figure 5-6 (b) shows the implemental feeding network for a designed rectangular aperture coupled microstrip linear array antenna with varactor diode in order to fix the impedance of the microstrip line.

A balanced microstrip method with λ0/4 impedance transformers is used for the feeding network. Zd is assumed as termination impedances of any microstrip lines. We can analyze the relative power of array elements Pi based on TEM analysis, where i=1, 2, 3, 4:

2 푃푖 = |푢푖| (5.33)

The values 푢푖are taken from the proposed technique. The ratio between relative feeding power of rectangular aperture 1 and rectangular aperture 2 is:

112

2 푃2 |푢2| = 2 푃1 |푢1| (5.34)

2 |푢2| Define 2 as 훾21 Similarly, |푢1|

2 푃4 |푢4| = 2 푃3 |푢3| (5.35)

2 |푢4| Define 2 as 훾43. On the other hand, the input impedance ratio of transformer Z1,Z2, and |푢3|

Z3,Z4, corresponding to rectangular antenna is:

푍2 푍4 = 훾21 , = 훾43 푍1 푍3 (5.36)

In the point A of feeding lines the rectangular 1 and 2 divides. If the ZS impedance is the A point, then the 푍푖 can be found as:

푍푠(훾21+1) 푍푠(훾43+1) 푍2 = , 푍4 = 훾21 훾43 (5.37)

And

푍1 = 푍푠(훾21 + 1) , 푍3 = 푍푠(훾43 + 1) ( 5 . 38 )

Consequently,

113

푃퐵 푃4 + 푃3 = = 훾퐵퐴 푃퐴 푃2 + 푃1 (5.39)

(A)

Figure 5-6 a) Array antenna with their impedance optimization. B) Array antenna design implementation with feeding networ (B)

114

Figure 5-7 The radiation pattern of experimental test of antenna array based on proposed method.

So, the 푍퐴, 푍퐵 can be found as

푍푠(훾퐵퐴+1) 푍퐵 = , 푍퐴 = 푍푠(훾퐵퐴 + 1) 훾퐵퐴 (5.40)

Zd is the load impedance at the output of transformers in the ideal case. So 푍푠 =

푍푑 = 50 Ω. The radiation pattern of the experimental test of antenna array is shown in

Figure 5-7. The gain is about 14 dB at 10.4 GHz. The side lobe suppression varies in different tests of weighting functions. When the rectangular antenna array is fed by generators with varactor diode in their centers, the side lobe suppression is about 8dB.The radiation patterns of the array were measured in an anechoic chamber at the Antenna World

Inc. Technology lab in Miami, Florida with an Orbit/FRFR959 Antenna Measurement system.

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Adaptive Antenna Array Beamforming Using Variable-Step-Size

Normalized Least Mean Square

6.1 Introduction

Adaptive antenna beamforming have become an important factor for wireless communication applications [122]. They are so useful accurate detection especially in mobile cellular communication systems [48]. These antennas are an example of using spatial filters to diminish the undesired signals coming in any directions. At the same time, they maximize the receiving desired signals deriving from any directions. Beamforming is basis of modern systems like adaptive phase array radars, optical communications, wireless communications, Ad-hoc systems, imaging, position system [123] [124][125][121]. A beamforming technique is presented for audio processing in [126] [127]. An overview of signal processing techniques used for adaptive antenna array beamforming is described in[94].

The optimum beam formers requires information of second order statistics that are unknown. But, with the assumption of ergodicity, they can be calculated using accessible data. Furthermore, statistics may change over time. This problem can be solved by adaptive weights based on some algorithms. Adaptive beamforming wants to be achieved in real time; consequently, low cost procedures should be used. The LMS has low complexity and including adaptive antenna array application. However, with this algorithm, it is unbearable to improve both the convergence speed and reduce the steady state error floor

116

concurrently. Moreover, when step size is large, the algorithm converges faster; however, the residual error has a larger floor level. On the other hand, using a smaller step creates a slower convergence and lower steady state error floor.

Lately, some changes have been proposed in the literature to try to overcome the cooperation between convergence speed and error floor using the conventional LMS algorithm. Most of these modified LMS algorithms use various criteria in the step size value. The Normalized Mean Square (NLMS) algorithm [128][120][125]. Is a variant of the NLMS algorithm that helps to select a variable-step-size that guarantees stability of the algorithm by (VSS) NLMS with the input power [128].

6.2 System Model

6.2.1 Adaptive Beamformer Structure

Consider 푁 uniform linear array sensors with distance Δ, as shown in Figure 6-1

This array, which samples the propagating wave field in space, is typically used for processing narrow band signals. The output at time푘,푦(푘) is given by a linear combination of the data at the 푁 sensors at time 푘:

푁 ∗ 퐻 푦(푘) = ∑ 푤푛(푘)푥푛(푘) = 풘 (푘)풙(푘) 푛=1 (6.1)

Where

퐻 ∗ ∗ ∗ 푇 풘 (푘) = [푤1(푘) 푤2(푘) ⋯ 푤푁(푘)]and풙(푘) = [푥1(푘) 푥2(푘) ⋯ 푥푁(푘)] are weight and signal vector at time 푘, respectively.

117

Adaptive algorithm

Figure 6-1 An adaptive beamformer model for narrow band signals

Beamformer response is represents as the amplitude and phase presented because the amplitude and section given to a posh plane wave as an operate of location and frequency. Location generally may be a 3-dimensional amount; however, usually we have a tendency to area unit solely involved with one or two dimensional direction of arrival

(DOA). Assume that array sensors area unit isotropous and also the signal s (t), that may be a complicated plane wave, arrives at direction θ and frequency ω. For convenience, let

−푗휔Δ푛(휃) the section be zero at the primary detector. This implies 푥푛(푘) = 푠(푘)푒 , where

2휋(푛−1)Δ Δ (휃) = sin(휃) signifies the time delay due to propagation from the first to 푛′푡ℎ 푛 휆 sensor. We refer to

−푗휔Δ (휃) 푇 풂(휃, 휔) = [1 푒 2 푒−푗휔Δ3(휃) ⋯ 푒−푗휔Δ푁(휃)] (6.2)

118

It's conjointly called the steering vector or direction vector. The output of array sensors, that received the required signal 푠푑(푡) with DOA 휃푑 and 푀 interfering signals

푀 푀 {푠 (푡)} with DOAs of{휃 } , can be shown as 푖푚 푚=1 푖푚 푚=1

푀

풙(푘) = 푠푑(푘)풂(휃푑, 휔) + ∑ 푠푖푚(푘)풂(휃푖푚 , 휔) + 풏(푘) 푚=1 (6.3)

Where 풏(푘) is additive white Gaussian noise with zero mean and variance of 휎푛.

6.2.2 Optimum Beamformer and Adaptive Beamformer

The weights in a very statistically optimum beamformer are chosen supported the statistics of the array information to optimize the array response. In general, the statistically optimum beamformer places nulls within the directions of the meddling sources in a shot to maximize the signal to noise quantitative relation at the beamformer output. The array information statistics aren't sometimes acknowledged and will amendment over time therefore reconciling algorithms are generally used to work out the weights. The reconciling rule is intended therefore the beamformer response converges to a statistically optimum resolution. In reconciling algorithms the weights are chosen to estimate the required signal d(k) a linear combination of the weather of the information vector x(k). We incline to choose w to attenuate MSE

2 퐻 2 2 퐻 퐻 퐻 퐽(푘) = 퐸{|푒| } = 퐸{|푦 − 풘 풙| } = 휎푑 − 풘 풓푥푑 − 풓푥푑풘 + 풘 푹푥풘 ( 6 . 4 )

2 2 ∗ 퐻 Where 휎푑 = 퐸{|푑(푘)| }, 풓푥푑 = 퐸{풙푑 } and 푹푥 = 퐸{풙풙 }. (6.4) is minimized by

119

−1 풘표푝푡 = 푹푥 풓푥푑 (6.5)

Note that J (k) may be a quadratic error surface. Since Rx is that the variance matrix of rackety information, it's positive definite. This means that the error surface includes a minimum worth. The optimum weight corresponds to the current minimum worth. One approach to adaptation filtering is to contemplate some extent on the error surface that corresponds to the current weight vector w (k). We tend to choose a brand new weight vector w (k+1) therefore on descend on the error surface. The gradient vector

휕 ∇풘(푘)= 퐽(푘)| = −2풓푥푑 + 2푹푥풘(푘) 휕풘 풘=풘(푘) (6.6)

Steepest descent, i.e., adjustment in the VSS-NLMS, leads to the widely used least mean square (LMS) adaptive algorithm. The LMS algorithm replaces

∗ 퐻 ∇̂풘(푘)= −2[풙(푘)푑 (푘) − 풙(푘)풙 (푘)풘(푘)]. Denoting푒(푘) = 푑(푘) − 푦(푘) = 푑(푘) −

풘퐻(푘)풙(푘), we have

∗ 풘(푘 + 1) = 풘(푘) + 2휇풙(푘)푒 (푘) ( 6 . 7 )

푇 2 2 −2 푻 푷(푘 + 1) = 퐹푘푷(푘)퐹푘 + 휇 휎푛 풙(푘)(풙(푘)풙(푘)) 풙 (푘) ( 6 . 8 )

−2 푻 Where 퐹푘 = 푰 − 2휇풙(푘)(풙(푘)풙(푘)) 풙 (푘) and thus their covariance 푷(푘) is given by in

푇 the following recursive form. The 푷(0) = 퐸(푤0푤0 ) and the step size μ controls the

120 convergence characteristics of the rule. The first virtue of the LMS rule is its simplicity. Its performance is appropriate in several applications, however, its convergence characteristics rely on the form of the error surface and therefore the eigenvalues of Rx. once eigenvalues are wide unfold, convergence is slow and alternative adaptive algorithms with higher convergence characteristics ought to be thought of.

6.2.3 Numerical Results

The ability of the proposed VSS-NLMS algorithm has been calculated by means of simulations. Results are gotten with the conventional LMS, and NLMS algorithms. For simulations, the subsequent parameters are used. A preferred binary phase shift keying

° (BPSK) arrives at direction of 휃퐷 = 0 , if definite at other direction. An interference signal

° reaches at direction of 휃퐼 = 30 with the similar amplitude as the desired signal. A uniform linear array (ULA) with 8 isotopic elements.

Figure 6-2 The beam patterns achieved with the LMS, NLMS and VSS-NLMS algorithms when the reference signal contaminated by AWGN when SNR=-10 dB.

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6.2.3.1 Beam Pattern

Figure 6-2. Shows the resulting beam patterns achieved with VSS-NLMS, LMS, and NLMS algorithms once the reference signal is corrupted by AWGN. The required

° ° signal attains at θd = −30 and the interfering signal arrives at θi = 45 . These figures show that the proper beam pattern will still be achieved victimization the VSS-NLMS rule with the assorted SNRs. It's obvious the VSS-LMS beamformers have a lower sidelobe compare to different adaptive algorithms once the reference signal is distracted with noise.

6.2.3.2 Number of Array elements and error floor

Beamformer performance in a high number of interferences is shown Figure 6-3.

° ° ° ° In this result, 4 BPSK interference signals come at directions of 휃푖 = 30 , −30 , 60 , −60 with the similar amplitude as the desired signal. The SIR is 0 dB. It is clear that the MSE converges so faster than the others and has a minor error.

Figure 6-3 Effect of number of array elements on mean square error floor in KLMS beamformer when SNR=0 dB. 122

6.2.3.3 Experimental Result

The proposed VSS-NLMS algorithm are tested using an antenna array with FR4 substrate having dielectric constant ɛr= 4.4 and thickness h= 0.5 mm. The 3-D schematic is shown in Figure 6-4. Distance between the centers of two adjacent patch antenna 6 mm. elements of array are designed for 9.5 GHz frequency with dimensions 118×100 mm2. The feeding scheme for aperture-coupled patch antennas, which were first suggested by [10].

Throw away the joining process wanted to keep a pin-connected feed circuit to the patch, either is unavoidable in fabricating the conventional probe-fed patch antenna. The radiation patterns of the array were measured in an anechoic chamber at the AntennaWorld

Inc. Technology lab in Miami, Florida with Orbit/FR FR959 Antenna Measurement system.

Figure 6-4 Model of proposed algorithm on a array antenna Implementation.

123 Figure 6-5 The beam patterns achieved by implementation of vss-nlms algorithms when the reference signal contaminated by AWGN Figure 6-5 shows the beam patterns attained with the implementation of VSS-

NLMS algorithm when the reference signal distracted by AWGN.

124

Two-Normalized Least Mean Square Using In Adaptive Array

Beamforming In Medical Imaging

In this This Chapter new Beamforming technique is suggested for array antenna.

This technique controls electrically the beam-pattern which reduced the noise, position of the pattern, and signal superposition. The proposed two-NLMS algorithm is utilized to decline the unwelcome interference. The first NLMS has a large step size and the other one small. The combinations of two outputs are related with a mixing parameter. This algorithm can control the suitable initial convergence and a minor steady state error adaptation. The weights are adjusted that the ending weight vector to the most contented result. This beamformer is compared with LMS, NLMS methods, it is achieved that the two-NLMS algorithm is better performance to other methods and the experimental results are confirmed by simulations

7.1 Introduction

Adaptive antenna beamforming is one of the most important factors in wireless communication applications[122]. These methods are so beneficial accurate detection specifically in mobile cellular communication [48]. Unwelcome signals can be diminished by these kinds of spatial filters in any directions. On the other hand, they enlarge the received desired signals coming from any angles. Adaptive phase array radars, Ad-hoc systems, imaging, wireless communications, position system and optical communications use the beamforming technique [98][117]. An audio beamforming technique is proposed

125

in a summary of adaptive array beamforming in signal processing methods is designated in.

The optimal beamformers necessitates information of second order statistics that are unidentified. But, with the assumption of ergodicity, they can be achieved by available data. Additionally, statistics may modify over time. This problem can be resolved through adaptive weights based upon some procedures. Adaptive beamforming needs to be achieved in real time; therefore, low cost processes should be utilized. The LMS is one of the simple methods that has low complexity in adaptive antenna array application. But, it is unattainable to improve both the steady state error floor and fast convergence concurrently. Also, when the step size is large, the residual error has a greater floor level besides LMS converges faster. Instead, using a minor step makes a slower convergence and lower steady state error floor.

Today, some improvements change has been proposed in the studies to control the convergence speed and error floor by the conventional LMS method. Variable step size adaptive methods suggest a possible solution allowing achieving both together fast convergence and small steady state error floor. Utmost these improved LMS algorithms utilize various norms in the step size value. The Normalized Mean Square (NLMS) algorithm is a different of the two-NLMS method that makes a variable-step-size based on the combination of two NLMS adaptive algorithms that assurances stability of the algorithm by two- NLMS with the input power [128][120].

126

7.2 System Model

7.2.1 Adaptive Beamformer Structure

N uniform linear array branches with distance푑, is exposed in Figure 7-1. This array is usually used for process narrow-band signals which is sampled the propagating wave in the area. The output at time k, y(k) is achieved by a linear combination of the information at the N branches at time k:

Figure 7-1 An adaptive beamformer model for narrow band signals. 푁 ∗ 퐻 푦(푘) = ∑ 푤푛(푘)푥푛(푘) = 풘 (푘)풙(푘) 푛=1 (7.1)

퐻 ∗ ∗ ∗ 푇 Where 풘 (푘) = [푤1(푘) 푤2(푘) ⋯ 푤푁(푘)] and 풙(푘) [푥1(푘) 푥2(푘) ⋯ 푥푁(푘)] are weight and signal vector at time 푘, correspondingly. Beamformer response is given because the amplitude given to posh plane wave as operation of position and frequency. The position normally may be a three dimensional amount; but, usually we have want to reach to area unit only involved with multiple dimensional direction of arrival

(DOA). It can be assumed that array branches has isotropic pattern and also the signal 127 s(t), that might be a complex plane wave, attains at direction θ and frequency ω. For ease,

−푗휔dn(휃) let the section be zero at the primary detector. This implies 푥푛(푘) = 푠(푘)푒 , where

2휋(푛−1)d d (휃) = sin(휃) indicates the time delay owing to propagation from the first to 푛 휆

푛푡ℎ branch. We refer to

−푗휔d (휃) 푇 풂(휃, 휔) = [1 푒 2 푒−푗휔d3(휃) ⋯ 푒−푗휔푑(휃)] ( 7 . 2 )

The (7.2) is steering vector. The output of array branches, that received the

푀 necessary signal 푠 (푡) with DOA 휃 and 푀 interfering signals {푠 (푡)} with DOAs 푑 푑 푖푚 푚=1

푀 of{휃 } , can be shown as: 푖푚 푚=1

푀

풙(푘) = 푠푑(푘)풂(휃푑, 휔) + ∑ 푠푖푚(푘)풂(휃푖푚 , 휔) + 풏(푘) 푚=1 (7.3)

Where 풏(푘) is an additive white Gaussian noise (AWGN) with 0 mean and variance of 휎푛.

7.2.2 Optimum Beamformer and adaptive Beamformer

The weights in optimal beamforming are selected and supported the statistics of the array info to optimize the beam array response; statistically. As a whole, the statistically optimal beamformer seats nulls within the directions of the interfering resources in a shot to enhance the signal to noise relation at the output. Sometimes, the array info statistics are not recognized and will change over time therefore merging algorithms are generally utilized to work out the weights. The merging rule is proposed therefore the beamformer 128 response converges to a statistically optimal resolution. The required signal d(k) a linear combination of the information vector x(k). The vector w choose to weaken the MSE,

2 퐻 2 2 퐻 퐻 퐻 퐽(푘) = 퐸{|푒| } = 퐸{|푑 − 풘 풙| } = 휎푑 − 풘 풓푥푑 − 풓푥푑풘 풘 푹푥풘 (7.4)

2 2 ∗ 퐻 Where 휎푑 = 퐸{|푑(푘)| }, 풓푥푑 = 퐸{풙푑 } and 푹푥 = 퐸{풙풙 }. (4) is minimized by

−1 풘표푝푡 = 푹푥 풓푥푑 (7.5)

Note that J(k) should be a quadratic error surface. Since Rx is that the variance matrix of information, it's positive definite. We incline to choose a kind of new weight vector w (k+1) therefore on descend on the error surface. The gradient vector

휕 ∇풘(푘)= 퐽(푘)| = −2풓푥푑 + 2푹푥풘(푘) 휕풘 풘=풘(푘) (7.6)

In this chapter, two adaptive NLMS is considered. Any of NLMS adaptation are using update rule by inserting a parameter. The LMS algorithm replaces ∇풘(푘) with the instantaneous gradient estimate

∗ 퐻 ∇̂풘( )= −2[풙(푘)푑 (푘) − 풙(푘)풙 (푘)풘(푘)]. Denoting푒(푘) = 푑(푘) − 푦(푘) = 푑(푘) −

풘퐻(푘)풙(푘), we have

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∗ 풘(푘 + 1) = 풘(푘) + 2휇풙(푘)푒 (푘) |풙(푘)|ퟐ (7.7)

퐻 푒(푘) = 푑(푘) − 푦(푘) = 푑(푘) − 풘 (푘)풙(푘) ( 7 . 8 )

In the above, w (k) is the N vector of coefficients of the kth adaptive filter. The true weight vector is the vector wo. The input process has a zero mean wide sense stationary

(WSS) Gaussian process. Sum of the output of the beamformer makes d (n) that has zero mean independent an identically distributed and the Gaussian distribution. In this chapter, the step sizes selects as μ1 > μ2. The outputs of the two adaptive filters are combined according to

푦(푛) = 휆(푛)푦1(푛) + (1 − 휆(푛))푦2(푛) (7.9)

To find 휆(푛), the minimization of the MSE of the a priori system error. The MSE derivate with respect to 휆(푛) reads

퐸[푅푒{(푑(푛)−푦2(푛))(푦1(푛)−푦2(푛))}] 휆(푛) = ퟐ 퐸[|(푦1(푛)−푦2(푛))| ] (7.10)

7.3 Numerical Results

The proposed two-NLMS algorithm for beamforming has been achieved by means of several simulations. Results are caught with the main LMS, and NLMS procedures. For

° simulations, A binary phase shift keying (BPSK) attains at direction of 휃퐷 = 0 , which can be changed at other directions. An interference unwanted signal comes at direction of 휃퐼 =

130

30° with the same amplitude as the desired signal amplitude. An ULA with 8 isotopic elements is assumed for beamforming

7.3.1 Beam Pattern

Beam patterns of two-NLMS, LMS, and NLMS methods are shown in Figure 7-2; which is the reference signal is corrupted by AWGN. These figures display that the better beam pattern will still be achieved by the two-NLMS rule with the same SNRs. It's obvious the two-LMS beamformer has minor sidelobe compare to LMS and NLMS algorithms once the reference signal is corrupted with noise.

Figure 7-2 The beam patterns using the lms, nlms and two-nlms algorithms when the reference signal polluted by awgn when SNR=10 db. 7.3.2 Error Vector Magnitude

Often, performance comparison between different adaptive beamforming schemes is made in terms of the convergence errors and resultant beam patterns. For a digitally 131 modulated signal, it is also suitable to make use of the error vector magnitude (EVM) as an accurate measure of any distortion introduced by the adaptive scheme at a given SNR

[128][129].

Figure 7-3 The evm values obtained with two-nlms, lms, and nlms algorithms at different SNR 7.4 Biomedical Application and Experimental Result

Biomedical applications are so important area in medicine sciences that use the advantage of array antenna [116] [130], Microwave imaging is one of the most usages of these adaptive antennas to detect breast tumor [131][132]. In this technique, an antenna array transmits some kinds of wideband impulses at the breast surface, and then receiver array antenna utilizes adaptive beam-forming to focus the reflected signal on the harmful tumor and recompense the other distractive propagation influence. Figure 7-4 shows the adaptive array hyperthermia system. To develop the electromagnetic field spreading in a clinical adaptive array, receiving sensors are located as closely as possible to the tumor section and where high temperature are to be eschewed, like close to the spinal cord and scar tissue. The adaptive array weights would control by the either the NNLMS algorithm

132 or two-NLMS algorithm to construct the rapidly before a remarkable amount of heating happens [128][120][35].

Figure 7-4 Adaptive array hyperthermia system The proposed two-NLMS algorithm is verified by an antenna array with these characteristics: substrate FR4 with dielectric constant ɛr= 4.4, and h= 0.5 mm thickness.

Figure 7-5 shows the Distance between the centers of two adjacent patch antenna 6 mm which is working as ultra-wide band (UWB) impulse in a wide frequency band with dimensions 118×100 mm2. A pin-connected feed the circuit to the patch with the conventional probe-fed patch antenna. An anechoic chamber at the Antenna World Inc.

Technology tests the radiation patterns of the array. Figure 7-6 shows the beam patterns achieved with the experimental implementation of two-NLMS technique when the reference signal corrupted by AWGN.

133

Figure 7-5 Model of proposed procedure on a array antenna implementation.

Figure 7-6 The beam patterns achieved by implementation of vss-nlms algorithms when the reference signal contaminated by AWGN

134

Adaptive Antenna Array Beamforming Using Inversely-

Proportionate Non Negative Least Mean Square

8.1 Introduction

One of the important components wireless applications like mobile communications, radar, cognitive radio, and worldwide interoperability for microwave access (WiMAX) is using array antenna[122] [133][134][135] .The array processing help wireless communication techniques to increase the signal accuracy. This technique has an important part of prevalent applications. The wireless communication system, radar, and sonar. Beamforming is one of methods in array processing that filters signals based on their capture time at each element in an array of antennas spatially[136] [137][44] . Numerous studies in adaptive array processing have been proposed in the last several decades, which are divided in two parts. The first one related to non-adaptive beamforming techniques and the next one related to digitally adaptive Beamforming methods. The trade-off between computational complexity and performance make them different. Beamforming is dominant in all antenna arrays. A summary of adaptive antenna array beamforming in signal processing techniques are introduced in[138]. It’s application can be found in radar

[139][111],wireless communications [140][141],optical communications [142][143],mesh networks [144][121] imaging in [145][146][147]. Another approaches that uses adaptive beamforming is audio and speech processing[126][108][96], and finally in beamforming of array antenna [94][120][81][35][128].

135

Optimum beamforming needs information about second order statistics, which are typically not recognized. Based on this assumption, the ergodic mood should be estimated from available knowledge. Moreover, the channel changes the statistics because of moving interferers over time. Using adaptive algorithms in determining weights solves this problem [96].Real time adaptive beamforming is so important in these areas which require low complexity techniques. The least mean square which is called LMS is so popular in various applications because of its robustness and low complexity. Authors in [148] used the LMS for adaptive antenna arrays. Unfortunately, LMS algorithm cannot improve the convergence speed and decrease the steady state error floor concurrently [110].When the step size is big, the algorithm converges faster; on the other hand, residual error becomes large. Instead, when we are using a smaller step, this makes slower convergence besides minor steady state error floor.

Since then, several changes have been suggested in the various studies to reduce both convergence speed and error floor with improving the conventional LMS algorithm.

Someone worked on modification of LMS algorithm with controlling the step size value

[149] [99] [150].The most popular version of LMS is Normalized Mean Square (NLMS) algorithm[129].The authors in [151], proposed a variant of LMS algorithm that guarantees stability by normalizing the step size with power of input.

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8.2 System Model

8.2.1 Adaptive beamformer Structure

A uniform linear array antenna is considered with푁 branches with distanceΔ.

Displays Figure 8-1 An adaptive beamformer model for narrow band signals a uniform linear array antenna. The sampling of the wave propagation is done in space. This process is usually utilized for narrow band signal processing. The output at time 푘, 푦(푘)is achieved by

푁 ∗ 퐻 푦(푘) = ∑ 푤푛(푘)푥푛(푘) = 풘 (푘)풙(푘) 푛=1 (8.1)

푇 퐻 In Eq. (1) the 풙(푘) = [푥1(푘) 푥2(푘) ⋯ 푥푁(푘)] and 풘 (푘) =

∗ ∗ ∗ [푤1(푘) 푤2(푘) ⋯ 푤푁(푘)] at time 푘.

Adaptive algorithm

Figure 8-1 An adaptive beamformer model for narrow band signals

137

Amplitude and phase response is the main key in designing a beamformer. These characteristics can be shown as position and frequency. Position means finding dimensional direction of arrival (DOA) in one or two dimensions. It can be assumed the signal 푠(푡) comes from direction of 휃 and frequencyω. Therefore, the received signal can

2휋(푛−1)Δ be represented as푥 (푘) = 푠(푘)푒−푗휔Δ푛(휃), where Δ (휃) = sin(휃) represents the 푛 푛 휆 time delay of determined branch. We refer to

−푗휔Δ (휃) 푇 풂(휃, 휔) = [1 푒 2 푒−푗휔Δ3(휃) ⋯ 푒−푗휔Δ푁(휃)] (8.2)

The steering vector can be known as direction vector. Then output of array antenna

푀 with 푀 interfering signals {푠 (푡)} and received desired signal 푠 (푡) can be represent 푖푚 푚=1 푑 as

푀

풙(푘) = 푑(푘)풂(휃푑, 휔) + ∑ 푠푖푚(푘)풂(휃푖푚 , 휔) + 풏(푘) 푚=1 (8.3)

In above relation, the풏(푘) is additive white Gaussian noise with zero mean and variance of 휎푛.

138 8.2.2 Optimum Beamformer and adaptive Beamformer

Generally, the statistically optimization of a beamformer makes some nulls in the place of interference signals in order to maximize the signal to noise ratio (SNR) at the output. The data statistics changes over the time, so, the adaptive algorithms require that works to update the weights. In adaptive algorithms, the weight vector is selected to focus on the desired direction. So, the select 풘to minimize MSE

2 퐻 2 2 퐻 퐻 퐻 퐽(푘) = 퐸{|푒| } = 퐸{|푦 − 풘 풙| } = 휎푑 − 풘 풓푥푑 − 풓푥푑풘 + 풘 푹푥풘 (8.4)

2 2 ∗ 퐻 where 휎푑 = 퐸{|푑(푘)| }, 풓푥푑 = 퐸{풙푑 }and 푹푥 = 퐸{풙풙 }.(4) is minimalized by selecting

−1 풘표푝푡 = 푹푥 풓푥푑 (8.5)

The quadratic error function is 퐽(푘). Wherethe covariance matrix of noisy data is

푹푥that is positive definite. Assume that this error function has a minimum, therefore, the optimum weight achieves by a iteration. The gradient vector

휕 ∇풘(푘)= 퐽(푘)| = −2풓푥푑 + 2푹푥풘(푘) 휕풘 풘=풘(푘) (8.6)

The general least mean square (LMS) adaptive algorithm substitutes ∇풘(푘) with the

∗ 퐻 immediate gradient estimate ∇̂풘(푘)= −2[풙(푘)푑 (푘) − 풙(푘)풙 (푘)풘(푘)] By the way,

푒(푘) = 푑(푘) − 푦(푘) = 푑(푘) − 풘퐻(푘)풙(푘), we have

139

∗ 풘(푘 + 1) = 풘(푘) + 2휇풙(푘)푒 (푘) ( 8 . 7 )

The step size 휇 changes the convergence speed of algorithm. The simplicity of

LMS is the main key in applications, however, the eigenvalues of 푹푥 is the shape of error surface. When eigenvalues are various, the convergence can be slow; therefore we should find other adaptive algorithms. There are several algorithm developed based on LMS algorithm such as NLMS.

8.3 Inversely-Proportionate Non Negative LMS algorithm for beamforming

By a linear model that characterized the relation between the input-output, we have

∗푇 푦(푛) = 푤 푥(푛) + 푧(푛) (8.8)

∗ ∗ ∗ ∗ 푇 w = [푤1, 푤2, … , 푤푁] is an unknown weight vector, and 푥(푛) = [푥(푛), 푥(푛 −

푇 1), … , 푥(푛 − 푁 + 1) is the regresses vector that is positive definite matrix 푅푥 > 0 and the desired output signal 푦(푛) are considered zero-mean stationary. The modeling error 푧(푛) is assumed zero-mean stationary, that is independent and identically distributed (iid) with

2 variance 휎푧 . To minimize the MSE condition:

휊 푤 = 푎푟푔 푚푖푛 퐽(푤) (8.9)

subject to 푤푖 ≥ 0, ∀푖, and 퐽(푤) is the MSE condition

⊺ 2 퐽(푤) = 퐸{[푦(푛) − 푤 푥(푛)] } (8.10)

140

The Lagrange function related with this problem, is given by 퐿(푤, 휆) = 퐽(푤) −

휆푇푤 to optimize (2) with 휆 the nonnegative Lagrange vector. Therefore, based on the conditions in [152] [153][154][155], we have

표 표 푤푖 [−훻푤퐽(푤 )]푖 = 0 (8.11)

where 훻푤 shows the gradient operator based on 푤. The component-wise gradient descent algorithm in [156] gives

푤푖(푛 + 1) = 푤푖(푛) + 휂(푛)푓푖(푤(푛))푤푖(푛)[−훻푤퐽(푤(푁))] ( 8 . 12 )

In above relation, the 휂(푛) is the positive step size that determines and controls the convergence rate. This iteration is like in the expectation maximization (EM) algorithm

[157]. By using this condition 푓푖(푤(푛)) = 1, it gives LMS algorithm. Indeed with rewriting the update equation, it gives the NNLMS algorithm [156]:

푤(푛 + 1) = 푤(푛) + 휂(푛)퐷푤(푛)푥(푛)푒(푛) ( 8 . 13 )

where 퐷푤(푛) becomes 푑푖푎푔 {푤(푛)} and 푒(푛) is the estimation error at time 푛:

푒(푛) = 푦(푛) − 푤⊺(푛)푥(푛) (8.14)

If the step size assure (15) the 푤(푛) becomes nonnegative at time 푛.

141

1 0 < 휂(푛) ≤ min 푖 −푒(푛)푥푖(푛) (8.15)

Another form of NNLMS algorithm is exponential NNLMS form in [153]. In this

훾 훾 form, 푤푖 (푛) = 푠푔푛 (푤푖(푛))|푤푖(푛)| . In order to decrease the slow-down effect in beamforming sidelobe in (13) , the 푤푖(푛) is used in 푓푖(푤(푛)), then we have

1 푓푖(푤(푛)) = 푓푖(푛) = |푤푖(푛)| + 휖 (8.16)

By choosing 휖 a small positive parameter, the gradient correction term leads to the subsequent algorithm,

푤(푛 + 1) = 푤(푛) + 휂퐷푓(푛)퐷푤(푛)푥(푛)푒(푛) ( 8 . 17 )

The new weight vector becomes

푤푖(푛) 퐰i(푛) = 푓푖(푛)푤푖(푛) = |푤푖(푛)| + 휖 (8.18)

This algorithm is called inversely-proportionate NNLMS (IP-NNLMS) in [158], which is the base of the antenna array beamforming. Now, we are summarized the

142 convergence of proposed technique in mean sense and transient MSE in order to prove the stability of using this algorithm as beamforming.

8.3.1 Convergence in Mean Sense Analysis

Define The weight error vector υ(푛) is define as distance to optimized weight vector. Therefore,

υ(푛) = 푤(푛) − 푤∗ (8.19)

In [27] analysis the mean sense convergence and showed that is satisfied. The mean weight error behavior is defined by

퐸{휐(푛 + 1)}

= 퐸{휐(푛)}

∗ ∗ 퐸{휐1(푛) + 푤1} 퐸{휐푁(푛) + 푤푁} (8.20) − 휂퐷 { ∗ , … , ∗ } 푅푥퐸{휐(푛)} |퐸{휐1(푛) + 푤1 }| + 휖 |퐸{휐푁(푛) + 푤푁}| + 휖

8.3.2 Transient excess mean-square error model

The transient MSE behavior is another test that is verified the stability of IP-

NNLMS algorithm. The objective of this section is to derive a model for of the algorithm.

Authors in [158] proved this conditions and this can be modeled by 푒(푛) = 푧(푛) −

휐푇(푛)푥(푛). It makes MSE,

퐸{푒2(푛)} = 퐸{(푧(푛) − 휐푇(푛)푥(푛))(푧(푛) − 휐푇(푛)푥(푛))} (8.21) 2 = 휎푧 + 푡푟푎푐푒{푹푥푲(푛)}

143

0.97 0.33 -0.21 -0.02 0.24 -0.35 0.34 -0.24 +j0.03 -j0.19 +j0.39 -j0.47 +j0.42 -j0.27 +j0.1

0.33 0.99 -0.36 0.13 0.15 -0.35 0.4 -0.32 +j0.03 -j0.3 +j0.61 -j0.84 +j0.83 -j0.59 +j0.26

-0.21 -0.36 1.01 -0.26 0.05 0.16 -0.29 0.3 -j0.19 -j0.3 -j0.78 +j1.02 -j1.07 +j0.82 -j0.41

-0.02 0.13 -0.26 1.01 -0.22 0.1 0.04 -0.16 +j0.39 +j0.61 -j0.78 -j0.99 +j0.97 -j0.78 +j0.44

0.24 0.15 0.05 -0.22 1.02 -0.35 0.24 -0.06 -j0.47 -j0.84 +j1.02 -j0.99 -j0.69 +j0.52 -j0.34

-0.35 -0.35 0.16 0.1 -0.35 1.01 -0.44 0.26 +j0.42 +j0.83 -j1.07 +j0.97 -j0.69 -j0.19 +j0.14

0.34 0.4 -0.29 0.04 0.24 -0.44 1 -0.35 -j0.27 -j0.59 +j0.82 -j0.78 +j0.52 -j0.19 +j0.03

-0.24 -0.32 0.3 -0.16 -0.06 0.26 -0.35 0.98 +j0.1 +j0.26 -j0.41 +j0.44 -j0.34 +j0.14 +j0.03

Table 8-1 The Zij received mutual coupling impedance of the linear array antenna shown in Figure 8-2.

144 Figure 8-2 The implementation setup with 8-element rectangular aperture microstrip array antenna with two horn antennas, one for desired signal and another for interference signal in the anechoic chamber.

145 Figure 8-3 Prototype of proposed 8x1 aperture coupled microstrip linear antenna base on the mutual coupling 8.4 The Beamformer Experimental Parameters

An experimental setup with 8-elements rectangular aperture microstrip array antenna shown in Figure 8-2, total size of antenna as you seen Figure 8-3 in 322x 56 mm2 in two layer thickness 0.4 mm with FR4 substrate is utilized to calculate the performance of the IP-NNLMS beamformer. The mutual coupling impedances of array antenna were achieved in the antenna terminal voltages. The antenna gain is measured in

146

(0,π) angles in working frequency 2.4 GHz. The mutual coupling impedance is normalized

th by 50 Ω and collected in Table I. In this table, zij is the mutual impedance between i and jth branches of array antenna. The very low distance of elements in designing antenna caused high mutual coupling in antenna. Two horn antennas transmit signal to array antenna in different angles inside of anechoic chamber. One of the horn antennas works as the desired signal and another used for interference signal.

8.5 Numerical Results

In this section, the received signal by array antenna that is contaminated by AWGN noise in order to consider the performance of the proposed IP-NNLMS beamformer by means of simulations. For comparing the performance, two conventional LMS, and NLMS algorithms are used for this purpose. For simulations, a binary phase shift keying (BPSK)

° attains at direction of 휃퐷 = 0 . An interference signal with similar desired signal attains at

° direction of 휃퐼 = 20 . At the outset, the MSE convergence performance of the mentioned algorithms is shown for different SNR in Figure 8-4(a)-(b). It can be observed the proposed

IP-NNLMS has superior performance than other algorithms.

147 a)

Figure 8-4 The convergence of IP-NNLMS, LMS, and NLMS algorithms for different values of input SNR. (a) SNR = 5dB, (b) SNR = 15dB.

Figure 8-5 displays the output beam patterns related to IP-NNLMS, LMS, and

NLMS algorithms after corrupting the reference signal by AWGN. The desired signal

148

° ° attains at θD = −10 besides interfering signal at θI = 30 . It is apparent that IP-NNLMS beam pattern has minor sidelobe in comparison to LMS and NLMS.

Figure 8-5 The beam patterns attained with the IP-NNLMS, LMS, and NLMS algorithms when the reference signal corrupted by AWGN with SNR=10 dB Figure 8-6 investigates the performance of arriving plenty of some interference signal. In this result, four BPSK interference signals attain at 휃퐼 =

45°, −45°, 90°, −90°with similar amplitude that makes the as signal to interference ratio

(SIR) is 0 dB. It can be seen proposed algorithm has lower MSE than others.

149

Figure 8-6 Effect of number of array elements on mean square error floor in IP-NNLMS beamformer in comparison with LMS, and NLMS beamformers at different SNR. Besides the previous achieved results, the error vector magnitude (EVM) as a truthful measure to determine the error made by the adaptive algorithm in various SNRs for a digitally modulation signal [159]. EVM is defined as

퐾 1 2 퐸푉푀푟푚푠 = √ ∑|푦(푘) − 푠푡(푘)| 퐾푃0 푘=1 (8.22)

In ((8.22)), 푦(푘) is the processed output of the beamformer and 푠푡(푘) is the desired transmit symbol, K is the total number of symbols and 푃0 is the average power. Figure 8-7

& Figure 8-8 show that IP-NNLMS algorithm attains the lowest EVM values in comparison to LMS and NLMS algorithms when the signal is suffered by AWGN channel and Rayleigh channel, respectively.

150

Figure 8-7 The EVM values obtained with IP-NNLMS, LMS, and NLMS algorithms at different SNR in AWGN channel.

Figure 8-8 The EVM values achieved in the presence of Rayleigh fading for IP-NNLMS, LMS, and NLMS algorithms at different SNRs. 8.6 Conclusion

In this paper, a new applicable beamforming technique is proposed to adaptive beamforming. This method is evaluated by a mutually coupled array antenna. Over experimental results, the proposed algorithm attains better performance and lower MSE

151 floor than the LMS and NLMS beamformers in various SNRs. It was also validated that

IP-NNLMS beamformer has better performance in the high number of interferer situations.

Finally, it should be mentioned the EVM of received symbols that is a useful metric to measure the total error than BER. It investigated the IP-NNLMS beamforming has very low error than other algorithms.

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