Synthesis of Ultra-Wideband Array Antennas

Transcription

1 Synthei of Ultra-Wideband Array Antenna Hamad W. Alawaha Diertation ubmitted to the faculty of Virginia Polytechnic Intitute and State Univerity in partial fulfillment of the requirement for the degree of Doctor of Philoophy In Electrical Engineering Virginia Polytechnic Intitute and State Univerity Ahmad Safaai-Jazi (Chair) Gary Brown William Davi Timothy Pratt Warren Stutzman Werner Kohler December 9, 2013 Blackburg, Virginia Keyword: antenna array, array ynthei, ultra-wideband antenna array Copyright 2013, Hamad W. Alawaha

2 Synthei of Ultra-wideband Array Antenna Hamad W. Alawaha (ABSTRACT) Acquiition of ultra-wideband ignal by mean of array antenna require eentially frequency-independent radiation characteritic over the entire bandwidth of the ignal in order to avoid ditortion. Factor contributing to bandwidth limitation of array include array factor, radiation characteritic of the array element, and inter-element mutual coupling. Strictly peaking, ditortion-free tranmiion or reception of ultra-wideband ignal can be maintained if the magnitude of the radiated field of the array remain contant while it phae varie linearly with frequency over the bandwidth of interet. The exiting wideband-array ynthei method do not account for all factor affecting the array bandwidth and are often limited to conidering the array factor and not the total field of the array in the ynthei proce. The goal of thi tudy i to preent an ultra-wideband array ynthei technique taking into account all frequency-dependent propertie, including array total pattern, phae of the total radiated field, element field, element input impedance, and inter-element mutual coupling. The propoed array ynthei technique i baed on the utilization of frequency-adaptive element excitation in conjunction with expreing the total radiated field of the array a a complex Fourier erie. Uing the propoed method, element excitation current required for achieving a deired radiation pattern, while compenating for frequency variation of the element radiation characteritic and the inter-element mutual coupling, are calculated. An important conideration in the propoed ultra-wideband array deign i that the phae bandwidth, defined a the frequency range over which the phae of the total radiated field of the array varie linearly with frequency, i taken into account a a deign requirement in the ynthei proce. Deign example of linear array with deired radiation pattern that are expected to remain unchanged over the bandwidth of interet are preented and imulated. Two example array, one with a wire dipole a it element and another uing an elliptically-haped

3 dic dipole a the element are tudied. Simulation reult for far-field pattern, magnitude and phae characteritic, and other performance criteria uch a ide-lobe level and canning range are preented. Synthei of two-dimenional planar array i carried out by employing the formulation developed for linear array but generalized to accommodate the geometry of planar rectangular array. A example deign, planar array with wire dipole and elliptical-haped dic dipole are tudied. The imulation reult indicate that ynthei of ultra-wideband array can be accomplihed uccefully uing the technique preented in thi work. The propoed technique i robut and comprehenive, nonethele it i undertood that the achieved performance of a yntheized array and how cloely the deired performance i met alo depend on ome of the choice the array deigner make and other contraint, uch a number of element, type of element, ize, and ultimately cot. iii

4 To my father Waled who paed away on (July 28, 2011) To my mother Fadhela And to my brother Khaled and my iter Rana and Badriya iv

5 Acknowledgement I would like to thank my academic advior and teacher, Dr. Ahmad Safaai-Jazi, for hi upport and guidance through out the year that I have pent on my MS and PhD degree at Virginia Tech. The challenging quetion and valuable dicuion that I had with Dr. Safaai-Jazi were of enormou help in maintaining great quality of education and reearch. I would alo like to thank Dr. Gary Brown for being a great teacher and a member of the adviory committee. I would alo like to thank Dr. William Davi for being a teacher in everal clae and a member of the adviory committee in my MS and PhD degree. I thank Dr. Timothy Pratt for being a teacher, a member of the adviory committee, and for hi feedback on thi diertation. I alo thank Dr. Warren Stutzman for hi acceptance of being a member of my adviory committee and for hi feedback on thi diertation. I would alo like to thank Dr. Werner Kohler for accepting of being a member of my adviory committee. I am grateful to my family for their upport and encouragement during the year of my graduate tudie. I thank my cloe friend, colleague, member of VTAG and member of Wirele@VT for their help and upport. v

6 Table of Content Chapter 1: Introduction BACKGROUND MOTIVATION, SCOPE, AND GOALS OF RESEARCH...2 Chapter 2: Literature Review WIDEBAND ARRAY DESIGN IN SIGNAL-PROCESSING APPLICATIONS ARRAY INTER-ELEMENT MUTUAL COUPLING FREQUENCY-ADAPTIVE ELEMENT EXCITATION ARRAY MAIN-BEAM SCANNING ENERGY-DIRECTING ARRAYS...18 Chapter 3: Synthei of Linear Array with Specified Bandwidth FORMULATION OF THE ARRAY FACTOR SYNTHESIS OF ARRAYS WITH SPECIFIED RADIATION PATTERNS SYNTHESIS OF ARRAYS WITH PRESCRIBED BANDWIDTH REQUIREMENTS DESIGN EXAMPLES FOR LINEAR ARRAYS WITH SPECIFIED BANDWIDTHS...23 Chapter 4: Synthei of Array Uing a Frequency-Adaptive Method FREQUENCY-ADAPTIVE ELEMENT EXCITATION COMPENSATION OF BANDWIDTH REDUCTION DUE TO ELEMENT ANTENNA COMPENSATION OF MUTUAL COUPLING AND ELEMENT INPUT IMPEDANCE Mechanim contributing to inter-element mutual coupling Array modeling with inter-element mutual coupling Array network model Mutual coupling compenation FLOW CHART OF THE SYNTHESIS TECHNIQUE...53 Chapter 5: Deign Implementation and Simulation Reult LINEAR ARRAY OF WIRE DIPOLES ELLIPTICAL DIPOLES AS ELEMENT ANTENNA UWB LINEAR ARRAY...65 Table of Content vi

7 5.4 TRANSIENT RESPONSE...75 Chapter 6: Synthei of Two-Dimenional Planar Array FORMULATION OF PLANAR ARRAY FACTOR PLANAR ARRAY OF WIRE DIPOLES ELLIPTICAL DIPOLE PLANAR ARRAY...89 Chapter 7: Implementation of Array Element Excitation ADAPTIVE ELEMENT EXCITATION USING DIGITAL FILTERS FIR FILTER DESIGN ELEMENT ANTENNA EXCITATION...98 Chapter 8: Concluion and Future Work SUMMARY Element field compenation Inter-element mutual coupling compenation Achieving frequency-invariant total field pattern CONTRIBUTIONS SUGGESTIONS FOR FUTURE WORK Reference Appendix A: Formulation of the Tranient Repone of an Antenna Appendix B: Performance Parameter (HP, D, and SLL) for Array Deign Example Preented in Chapter 5 and Appendix C: Surface Current Denity Ditribution on element of Linear Array of Elliptical Dipole Antenna Table of Content vii

8 Lit of Figure Fig. 3.1: Geometry and coordinate for a linear array of 2N + 1 iotropic element Fig. 3.2: (a) Variation of the magnitude of the element field with frequency, (b) magnitude of the total field of the array a a quare-wave like periodic function Fig. 3.3: Variation of array factor and directivity for linear array of iotropic element with 31 element ( N = 15), element pacing d=15 mm for (a) and (b) and d=13.64 mm for (c) and (d). The deired bandwidth i 9 GHz at θ = 0 with lower and upper frequencie of f = 1GHz, and f = 10GHz Fig. 3.4: Variation of array factor veru frequency for different value of θ and different number of element (2N+1). The element pacing i d=13.64 mm, the deired bandwidth i 9 GHz at θ = 0 with lower and upper frequencie of f 1 1GHz, and f 2 = = 10GHz, and T = 4π ( f + f ) ω 1 2 Fig. 3.5: Element current ditribution required to produce the array factor hown in Fig All parameter ued in calculating thee current are the ame a thoe in Fig Fig. 3.6: Variation of normalized S 11 and realized gain veru frequency at θ = 0 for a wire dipole antenna a array element. The dipole i reonant at f r = 5.5 GHz (length=2.73cm= 0.48λ r ) and ha a radiu of 0.5 mm Fig. 3.7: Variation of the total radiated field of array of wire dipole reonant at 5.5 GHz veru frequency for different value of θ and different number of element (2N+1). The deign parameter are: d=13.64 mm, 0 = =, and T 4π ( f f ) θ =, f GHz, f GHz ω = +. Alo, the total radiated field of an equally-paced uniformly excited array of ame wire dipole with N=100 i hown for comparion Fig. 4.1: Variation of the calculated array factor ( AF ) veru frequency and angle θ for a linear array with 45 iotropic element (N=22) and element pacing d pattern i decribed by AF 1 2 = 1 cm. The deired array 50 = in θ over a frequency range of 1GHz to 10GHz. (a) Three-dimenional view, (b) top view Lit of Figure viii

9 Fig. 4.2: Element current (normalized to I 0, mod max ) required to produce the array factor AF 50 = in θ for the linear array of Fig. 4.1 (N=22, d=1 cm) Fig. 4.3: Variation of the array factor ( AF ) veru frequency and angle θ for the array of Fig. 4.1 but with can angle θ = Fig. 4.4: Illutration of the emergence of grating lobe in the pattern of an arbitrary array factor and it relationhip to element pacing d, ψ1 = ψ o, ψ2 = ψo 2 ( ωh / c) d, f h = 10 GHz. (a) d = 1 cm, (b) d = 2.6 cm Fig. 4.5: Variation of the array factor ( AF ) veru frequency and angle θ for the array of Fig. 4.1 but with element pacing d=5 cm Fig. 4.6: Variation of the array factor ( AF ) veru frequency and angle θ for the array of Fig. 4.1 but with the main beam at θ = 67.5 (can angle: θ = θ 67.5 = = 22.5 ) and element pacing d=5 cm Fig. 4.7: Mechanim of inter-element mutual coupling in a linear array [39] Fig. 4.8: Current ditribution and terminal current of an iolated element antenna [39] Fig. 4.9: Definition of active-element pattern for nth element [39] Fig. 4.10: Repreentation of array a an N-port network with different characteritic impedance Fig : Flow chart illutrating the implementation of the propoed array ynthei technique Fig. 5.1: Linear array of 17 equally paced wire dipole; wire radiu pacing d o a = 0.5 mm, element = 0.4λ r, reonant frequency f r = 5.5 GHz, length of dipole L = 0.48λ r Fig. 5.2: xz-plane cut of the total pattern at different frequencie for the uncompenated deign (frequency variation of the element field and inter-element mutual coupling are not accounted for) of an array with geometry and parameter the ame a thoe given in Fig The deired total array pattern i decribed by f ( θ ) = in θ over the frequency range 4.5GHz to 7GHz Lit of Figure ix

10 Fig. 5.3: xz -plane cut of the total pattern at different frequencie for the compenated deign (frequency variation of the element field and inter-element mutual coupling are accounted for) of an array with geometry and parameter the ame a thoe given in 50 Fig The deired total array pattern i decribed by f ( θ ) = in θ over the frequency range 4.5GHz to 7GHz. All pattern are approximately coincident Fig. 5.4: Three-dimenional view of the magnitude pattern at different frequencie for the compenated array deign decribed in Fig Fig. 5.5: Comparion of group delay for uncompenated and compenated array deign along the direction of maximum radiation, θ = 90. For both deign the array geometry and parameter are the ame a thoe given in Fig. 5.1, and the deired total pattern i 50 decribed by f ( θ ) = in θ over the frequency range 4.5GHz to 7GHz Fig. 5.6: Total pattern in the xz-plane for the compenated array deign decribed in Fig. 5.3 but canned to θ = θ 22.5 = All pattern are approximately coincident o Fig. 5.7: Circular-dic dipole antenna a an ultra-wideband element for frequency-invariant array Fig. 5.8: Variation of (a) VSWR referenced to 100Ω and (b) input impedance veru frequency for the circular-dic dipole antenna hown in Fig Fig. 5.9: Elliptically-tapered dic dipole antenna a an ultra-wideband element with improved input- impedance matching performance compared with the circular-dic dipole hown in Fig Fig. 5.10: Variation of (a) VSWR referenced to 100Ω and (b) input impedance veru frequency for the elliptically-tapered dic dipole antenna hown in Fig Fig. 5.11: Variation of (a) VSWR referenced to 100Ω and (b) input impedance veru frequency for the caled verion of the elliptically-tapered dipole hown in Fig. 5.9 (dipole axial dimenion: 15 mm 20 mm) Fig. 5.12: Illutration of the ditribution of urface current denity at different frequencie for the caled elliptically-tapered dic dipole antenna with performance characteritic hown in Fig Lit of Figure x

11 Fig. 5.13: Ultra-wideband linear array of 17 equally-paced elliptically-tapered dic dipole with element pacing d = 17 mm and element axial dimenion 15 mm 20 mm Fig. 5.14: Variation of (a) total pattern, (b) magnitude of the total field at θ = 90, and (c) phae of the total field at θ = 90 veru frequency for an uncompenated linear array of elliptically-tapered dic dipole with geometry and parameter ame a thoe decribed 50 in Fig The deired total array pattern i decribed by f ( θ ) = in θ over the UWB frequency range of 3.1GHz to 10.6GHz Fig. 5.15: Variation of input impedance veru frequency for half of the element (element 1-8) and the center element (element 0) of the array decribed in Fig Fig. 5.16: Variation of excitation current veru frequency for the element of the array decribed in Fig Solid curve repreent the current calculated uing (5.1) and (5.2), dahed curve repreent the current developed at the input port of element antenna due to mutual coupling effect Fig. 5.17: Illutration of urface current ditribution for half of the element of the array decribed in Fig (center element hown on the left) Fig. 5.18: Variation of (a) total pattern, (b) magnitude of the total field at θ = 90, and (c) phae of the total field at θ = 90 veru frequency for a compenated (baed on iolated element aumption) linear array of elliptically-tapered dic dipole with geometry and parameter ame a thoe decribed in Fig The deired total array pattern i 50 decribed by f ( θ ) = in θ over the UWB frequency range 3.1GHz GHz Fig. 5.19: Variation of excitation current veru frequency for the element of the array decribed in Fig Solid curve repreent the current calculated uing (5.1) and (5.2), dahed curve repreent the current developed at the input port of element antenna Fig. 5.20: Variation of (a) total pattern, (b) magnitude of the total field at θ = 90, and (c) phae of the total field at θ = 90 veru frequency for a compenated (baed on active-element field) linear array of elliptically-tapered dic dipole with geometry and parameter Lit of Figure xi

12 ame a thoe decribed in Fig The deired total array pattern i decribed 50 by f ( θ ) = in θ over the UWB frequency range 3.1GHz GHz Fig. 5.21: Variation of excitation current veru frequency for the element of the array decribed in Fig Solid curve repreent the current calculated uing (5.1) and (5.2), dahed curve repreent the current developed at the input port of element antenna Fig. 5.22: Tranient repone of the wire dipole array configuration of Fig. 5.1; ( a) Input Gauian pule with σ = 0.1 n, (b) Spectrum of the modulated input ignal, (c) Radiated pule of the uncompenated deign of Fig. 5.2 at θ = 90, (d) Radiated pule of the compenated deign of Fig. 5.3 at θ = Fig. 6.1: Planar array of 17x17 wire dipole contructed baed on the linear array of Fig The array parameter are: element pacing in the x-direction d = 0.4λ, element pacing in the y-direction d = 0.6λ, dipole length L = 0.48λ r ; reonant frequency f r = 5.5 GHz 81 y r Fig. 6.2: Three-dimenional magnitude pattern at different frequencie for the total field of a planar array of wire dipole with can angle θ = φ = 0. The geometry and parameter of the array are the ame a thoe decribed in Fig 6.1. The deired total array pattern 50 ued in the deign i given by f ( θ, ϕ) = co θ over the bandwidth of the array, covering the frequency range of 4.5GHz to 7.0GHz. The correponding performance parameter (HP, D, and SLL) are preented in Table B Fig. 6.3: Variation of (a) magnitude of the total field at θ = 0 and (b) phae of the total field at θ = 0 veru frequency for the planar array of wire dipole decribed in Fig Fig. 6.4: Three-dimenional magnitude pattern at different frequencie for the total field of the planar array of wire dipole decribed in Fig. 6.2 but with can angle θ = 45 and φ = 0. The correponding performance parameter (HP, D, and SLL) are given in Table B Fig. 6.5: Variation of the calculated principal-plane array factor veru frequency and angle θ for the planar array of wire dipole decribed in Fig. 6.2 but with can angle θ = 45, φ = 25. (a) AF, (b) x AF y x r Lit of Figure xii

13 Fig. 6.6: Three-dimenional magnitude pattern at different frequencie for the total field of the planar array of wire dipole decribed in Fig. 6.2 but with can angle θ = 45, φ = 25. The correponding performance parameter (HP, D, and SLL) are provided in Table B Fig. 6.7: Wire dipole antenna, ued a an element in the planar array of Fig. 6.1, above an infinite ground plane by a ditance of h = λ c /4 with f c = 5.75 GHz Fig. 6.8: Three-dimenional magnitude pattern at different frequencie for the total field of a ingle wire dipole antenna above an infinite perfectly-conducting ground plane. The dipole ha a length L = λ c /2and i a ditance h = λ c /4 above the ground plane; f = 5.75 GHz c Fig. 6.9: A planar array of wire dipole above an infinite ground plane. The array i aumed to have the ame geometry, parameter and deired total pattern a that decribed in Fig. 6.1 and 6.2. The ground plane and plane of array are parallel and eparated by a ditance of h = λ c /4 at f c = 5.75 GHz Fig. 6.10: Three-dimenional magnitude pattern at different frequencie for the total field of the planar array of wire dipole above ground plane decribed in Fig. 6.9; can angle areθ = 45, φ = 25. The correponding performance parameter (HP, D, and SLL) are preented in Table B Fig. 6.11: Planar array of 15x15 elliptically-tapered dic dipole contructed baed on the linear array of Fig The array parameter are: element pacing in the x-direction d = 17 mm, element pacing in the y-direction d = 43 mm, and element axial x dimenion 15 mm 20 mm. The deired total array pattern i conidered to 50 be f ( θ, ϕ) = co θ Fig. 6.12: Variation of the calculated principal-plane array factor veru frequency and angle θ for the planar array of elliptical dipole decribed in Fig but with can angle θ = 25, φ = 0 (a) AF, (b) x y AF y Lit of Figure xiii

14 Fig. 6.13: Three-dimenional magnitude pattern for the total-field of the planar array decribed in Fig The correponding performance parameter (HP, D, and SLL) are provided in Table B Fig. 6.14: Variation of the magnitude and phae veru frequency for the total field of the planar array of elliptical dipole decribed in Fig at θ = 25, φ = 0. The correponding performance parameter (HP, D, and SLL) for thi array are provided in Table B Fig. 6.15: Variation of the magnitude and phae veru frequency for the total field of the planar array of Fig at θ = 25, φ = 0 after compenating for active-element error. The correponding performance parameter (HP, D, and SLL) for thi cae are preented in Table B Fig. 6.16: Planar array of elliptical dipole with the ame geometry and parameter a thoe decribed in Fig The array i above an infinite ground plane by a ditance of h = λ mid /4 at f mid = 4.25 GHz Fig. 6.17: Three-dimenional radiation pattern at everal frequencie for an iolated elliptical dipole antenna above an infinite ground by a ditance λ mid /4 at f mid = 4.25 GHz Fig. 6.18: Three-dimenional magnitude pattern for the total-field of the planar array decribed in Fig at θ = 25, φ = 0.The correponding performance parameter (HP, D, and SLL) are given in Table B Fig. 6.19: Variation of the magnitude and phae veru frequency for the total field of the planar array of Fig at θ = 25, φ = 0 before compenating for active-element error. The correponding performance parameter (HP, D, and SLL) for thi cae are preented in Table B Fig. 6.20: Variation of the magnitude and phae veru frequency for the total field of the planar array of Fig at θ = 25, φ = 0 after compenating for active-element error. The correponding performance parameter (HP, D, and SLL) for thi cae are preented in Table B Lit of Figure xiv

15 1 Fig. 7.1: Block diagram of a baic FIR filter, W n are real valued weight, and Z are time delay block Fig. 7.2: Block diagram of FIR filter with conjugate ymmetrical tap, W n are real valued 1 weight, Z are time delay block, and ( )* i a conjugation block Fig. 7.3: Block diagram of the FIR filter for element antenna excitation, W n are real valued 1 weight, Z are time delay block, and ( )* i a conjugation block Fig. C1: Illutration of urface current ditribution for half of the element of the array decribed in Fig with center element hown on the left (The geometry and parameter of the array are the ame a thoe given in Fig ) Fig. C2: Illutration of urface current ditribution for half of the element of the array decribed in Fig with center element hown on the left (The geometry and parameter of the array are the ame a thoe given in Fig ) Fig. C3: Illutration of urface current ditribution for half of the element of the array decribed in Fig with center element hown on the left (The geometry and parameter of the array are the ame a thoe given in Fig ) Lit of Figure xv

16 Lit of Table Table 5.1: Summary of performance parameter including, half-power beamwidth (HP), directivity ( D ), and maximum idelobe level ( SLL ) for the array of wire dipole decribed in Fig Table 5.2: Summary of performance parameter including, half-power beamwidth (HP), directivity ( D ), and maximum idelobe level ( SLL ) for the array decribed in Fig Table B1: Summary of performance parameter for the wire dipole array decribed in Fig Table B2: Summary of performance parameter for the elliptically-tapered dic-dipole array decribed in Fig (The geometry and parameter of the array are the ame a thoe given in Fig ) Table B3: Summary of performance parameter for the elliptically-tapered dic-dipole array decribed in Fig (The geometry and parameter of the array are the ame a thoe given in Fig ) Table B4: Summary of performance parameter for the planar array of wire dipole decribed in Fig Table B5: Summary of performance parameter for the planar array of wire dipole decribed in Fig Table B6: Summary of performance parameter for the planar array of wire dipole decribed in Fig Table B7: Summary of performance parameter for the planar array of wire dipole above ground plane decribed in Fig (The geometry and parameter of the array are the ame a thoe given in Fig. 6.9.) Table B8: Summary of performance parameter for the planar array of elliptical dipole decribed in Fig and 6.14 (The geometry and parameter of the array are the ame a thoe given in Fig ) Lit of Table xvi

17 Table B9: Summary of performance parameter for the planar array of elliptical dipole decribed in Fig (The geometry and parameter of the array are the ame a thoe given in Fig ) Table B10: Summary of performance parameter for the planar array of elliptical dipole above ground plane decribed in Fig and 6.19 (The geometry and parameter of the array are the ame a thoe given in Fig and 6.16.) Table B11: Summary of performance parameter for the planar array of elliptical dipole above ground plane decribed in Fig (The geometry and parameter of the array are the ame a thoe given in Fig and 6.16.) Lit of Table xvii

18 Chapter 1: Introduction 1.1 Background Many emerging wirele technologie require highly reconfigurable radiation interface that are capable of operating over wide bandwidth to acquire broadband ignal. Phaed array antenna are found to be the bet candidate for many wideband wirele ytem due to their capability in providing patial filtering with high degree of electivity [1]. Spatial filtering i important epecially when eparation of frequency overlapping ignal originated from different location i needed [2], or when highly-directed, radiated electromagnetic energy i required. Other important function that can be achieved by antenna array include the ability to provide reconfigurable wave polarization, beam canning, and yntheized in power-radiation pattern or beam forming. In recent reearch, frequency independent beam forming or power radiationpattern ynthei ha been of particular interet in application uch a wideband wirele communication ytem, radar ytem, microwave thermotherapy, medical imaging, and tumor detection [3]. Array radiation pattern ynthei, a proce that may be regarded a the oppoite to array analyi, conit of two major tep [4-7]. The firt tep in array ynthei i to define the deired radiation pattern a a function of patial coordinate, while the econd tep i to determine the configuration of the element of the array and their excitation current. Thi conventional approach, however, i limited to narrow-band array. Narrow-band array pattern ynthei i well etablihed and ha been invetigated extenively in the literature of antenna theory, where patial parameter are conidered a the only variable in the array-factor expreion. Wideband array ynthei, on the other hand, ha been motly approached by mean of optimization methodologie. If bandwidth pertaining to radiation characteritic of an array are a concern, frequency in the array factor mut be treated a an additional deign parameter. Many reearcher have invetigated the poibility of providing frequency independent array beam forming; uch tudie have been performed both in the time domain and in the frequency domain. Frequency independent beam forming i an important conideration, particularly if the acquiition of a wideband ignal with minimal ditortion i a performance goal. Introduction 1

19 1.2 Motivation, cope, and goal of reearch Wideband ignal acquiition require wideband array performance with minimized ignal ditortion. In order to tranmit or receive a wideband ignal, the array ha to exhibit an eentially frequency-invariant performance over the bandwidth of the ignal in order to avoid ditortion. All factor contributing to the frequency dependence of the total radiated field of the array have to be conidered and compenated in the deign proce in order for the array to provide a wideband performance. The frequency dependent propertie that are dealt with in thi reearch include: array radiation pattern, array directivity, phae of the total radiated field, element field, element input impedance, and inter-element mutual coupling. Achieving wideband, or to ome extent frequency-invariant, array performance ha been invetigated by reearcher in everal field, a will be dicued in the literature review in Chapter 2. Reearcher [8-26] in the field of ignal proceing have preented everal filter deign technique for compenating only the frequency dependent parameter in the array factor expreion. Mot of thee tudie aume an ideal iotropic antenna a the element of the array, leaving the effect due to frequency-dependent propertie aociated with the element antenna untreated. Incorporating the radiation characteritic of the element antenna into the model of the array allow the effect due to inter-element mutual coupling to be accounted for. Inter-element mutual coupling i an important frequency dependent factor that affect the wideband performance of the array in a complex manner. Frequency-adaptive element excitation i found to be the bet technique for compenating the inter-element mutual coupling. Many reearcher have employed frequency-adaptive element excitation technique, yet none ha conidered all of the frequency-dependent factor mentioned earlier in thi ection. In fact, all of thee factor are inter-related. The input impedance of the element antenna, being a function of frequency, i highly affected by inter-element mutual coupling and i alo a function of the can angle. Becaue of the induced current on a ingle element due to radiation by it urrounding element, the radiated field of the element in a fully-active array i not the ame a that when it i ued in iolation. The radiation pattern and hence the gain of the array are highly impacted by the field of the element. Reearcher working in the area of energy directing array have addreed the importance of uing frequency adaptive excitation to achieve optimal Introduction 2

20 performance for maximum directed energy. However, if the hape of the wideband ignal i a concern, energy maximization technique do not provide the frequency-invariant performance of the array needed to achieve wideband ignal tranmiion and reception with minimal ditortion. In addition to element input impedance and mutual coupling with neighboring element, frequency dependence of the element radiated field alo influence the overall bandwidth of the array. Thi dependence i, in general, two fold, a both the magnitude and the phae of the element field vary with frequency over a wide frequency range. It appear that the impact of phae bandwidth of the element radiated field on the overall array bandwidth ha not been adequately addreed. The goal of thi tudy i to preent an ultra-wideband, array ynthei technique through the utilization of frequency adaptive element excitation in conjunction with expreing the total radiated field of the array a a complex Fourier erie, taking into account all frequencydependent factor mentioned above. Uing the propoed method, array factor with arbitrary patial and frequency dependence can be realized, and compenation for frequency dependence of the element radiation characteritic and mutual coupling are achieved. Chapter 2 of thi diertation preent a literature review and brief dicuion of the related reearch performed over the lat few decade. Attention will be focued on pat work pertaining to wideband array ynthei baed on frequency adaptive technique. In Chapter 3, baic of antenna array analyi and ynthei are preented. Synthei of ultra-wideband linear array that can provide nearly frequency-independent radiated field over the bandwidth of interet but along a pecific direction in pace i addreed. Chapter 4 i devoted to developing an array ynthei technique that allow for compenating the variation of the element field a a function of pace and frequency a well a the effect due to inter-element mutual coupling. Since the aim i to achieve tranmiion or reception of ultra-wideband ignal with minimum ditortion, the phae bandwidth i regarded a being equally important a the magnitude bandwidth. The magnitude bandwidth i defined a the frequency range over which the magnitude of total radiated field of the array (and hence it radiation pattern and gain) remain nearly unchanged, while the phae bandwidth i the frequency range over which the phae of the total radiated field varie linearly with frequency. All compenation are applied to both magnitude and phae. The array element i aumed to be linearly polarized, thu allowing the Introduction 3

21 array factor to be expreed imply are a calar function. The element excitation current required for achieving a deired radiation pattern, while compenating for frequency variation of the element radiation characteritic and the inter-element mutual coupling, are calculated. The dependence of the input impedance of the array element with the can angle i another factor that i addreed in Chapter 4. In Chapter 5, deign example of linear array with deired radiation pattern that are meant to remain unchanged over the bandwidth of interet are preented and imulated. Two array, one with a wire dipole element and another uing an elliptically-haped dic dipole element are tudied. Synthei of two-dimenional planar array i carried out in Chapter 6 by employing the deign technique preented in Chapter 4 but generalized to planar rectangular array. Simulation reult for planar array of wire dipole and elliptical dipole are alo preented in thi chapter. Chapter 7 addree the implementation of element excitation by mean of digital filter. The block diagram of FIR filter needed to contruct the required excitation ignal from the ource ignal i dicued. Chapter 8 ummarize the reult, concluion, and contribution of thi reearch and point out direction for further invetigation. Introduction 4

22 Chapter 2: Literature Review 2.1 Wideband array deign in ignal-proceing application Frequency-invariant beam former have been invetigated theoretically by reearcher in the field of ignal proceing. Mot of thee tudie tart with a mathematical manipulation of the array factor expreion in order to iolate the frequency dependent term and minimize their effect by uing pecific filter deign. An example of uch tudie i decribed by Chan and Pun [8], in which the Park-McClellan filter deign algorithm i ued to achieve the neceary filtering to compenate for frequency dependencie in the array factor. The iolation of the frequency dependent term in [8] i obtained by expanding the complex exponential function in the array factor expreion through the ue of Beel function propertie, followed by an aymptotic approximation of the reulting erie expanion. The approach preented in [8] account for radiation in a ingle plane in pace; that i, one patial variable i aumed to be contant for a two-dimenional planar array. Ghavami [9] recognized the poibility of mapping the patial variable domain of a twodimenional planar array into an auxiliary frequency domain. Thi domain mapping allowed for iolating the frequency dependent patial variable and determining the required element excitation to produce a frequency-invariant radiation pattern in a ingle plane in pace, reulting in a fully-patial wideband beamforming. Baically, the delay needed for temporal filtering are compenated by the relative patial difference between the location of the element. Only a ingle, purely real, weighting coefficient i required for each element, which can be realized by very imple paive or active circuit, eliminating the need for tapped delay line. Thi property arie from the ymmetry of the pattern in the auxiliary domain, and by auming that the direction of arrival (DOA) of the ignal i known. One of the critical aumption in thi method i the direction of arrival of the ignal being in the plane of the planar array. It i aumed that the array form it beam with reaonable maintenance of the deired performance above it plane, which might not be the cae epecially over a wide frequency range. Another drawback aociated with thi method i that, if the electrical ize of the array i large, or approache infinity, the gain of the element that reide in the far edge goe to zero in the direction of the plane of the array. Literature Review 5

23 Uthanakul and Bialkowki [10] noticed that the method developed by Ghavami in [9] practically doe not yield good performance with mall array configuration, becaue the effective aperture area of the array i not large enough to produce adequate directivity. Thi problem, according to Uthanakul and Bialkowki in [10], can be fixed by increaing the denity of the element in the effective aperture area. Wei [11] noticed that in the method developed by Ghavami in [9], the temporal filtering needed in the deign of linear array ha been replaced by patial filtering in the planar deign becaue of the additional dimenion. By extending the deign of the rectangular planar array to a three dimenional array, the problem aociated with large array that wa dicued previouly can be reolved. However, another diadvantage arie, that i the blockage of the radiation of the center element in a large 3-D array environment, which can render the deign impractical. Wei, et al. [12] integrated an adaptive optimization algorithm, that i baed on linearly contrained minimum variance beamforming (LCMV) [13]. Thi adaptive element weighting allow for achieving arbitrary beamforming pattern. Adaptive element excitation i dicued in ection 2.3. Uually DOA etimation method are narrowband, uch a MUSIC [14]. However, by deigning for a frequency-invariant beamformer, narrowband DOA algorithm can be applied at a ingle frequency and the reult can be reditributed over the entire deign bandwidth. An example of wideband DOA etimation algorithm i preented by Tuan, et al. in [15]. Another intereting method of achieving frequency-invariant beamforming i to deign for a narrowband beamformer, oberve how the pattern deviate when the frequency i varied over a wider bandwidth and minimize thi variation. For example, in the cae of achieving inuoidal function for the array factor pattern, caling the frequency parameter will approximately lead to a imilar pattern but with caled width of the main-beam. Yunhong, et al. [16] tudied uch an effect on the tructure of the concentric ring array (CRA). In thi tudy, the weight of the element in a ingle ring are deigned baed on narrowband fat Fourier tranform (FFT). On an inter-ring level, the array factor i realized a a Fourier-Beel erie [5], which allow for recognition of dominant higher order Beel function, and conidering only the dominant term, the analyi become much le complicated. The weight on the inter-ring level are deigned to achieve wideband performance. Literature Review 6

24 At higher frequencie, the array pattern i naturally widened, but if the array i deigned to produce a pre-compreed pattern at higher frequencie, thi increae in the width of the mainbeam i compenated. The inter-ring weight are ued to produce uch pre-compreed pattern. A method imilar to the concept of pattern pre-compreion wa applied by Zhu in the tudy preented in [17] for linear array. In the method preented by Zhu in [17], the weight coefficient are calculated at a reference frequency and linearly contrained minimum variance algorithm i ued to calculate the weight at other frequencie, o that it produce a pattern imilar to that of the reference frequency. Another tudy that applie the technique of tarting the deign proce at a reference frequency and extending it to the ret of the deign bandwidth, i preented by Subramanian in [18]. In [18], the weight are calculated baed on minimizing a quare error function, that i defined a the cot function, in order to produce weight that reult in an optimized pattern cloe to that of the reference frequency. Contraint other than beamwidth can be embedded in uch technique, uch a maximum idelobe level (SLL). Alhammary [19] applied a patial repone variation contraint pattern ynthei algorithm that optimize the pattern of the array around a reference frequency uing a leat mean quare cot function with maximum SLL and beamwidth a contraint in the algorithm of the deign. Near field application may alo benefit from frequency invariant beamforming. Abhayapala et al. [20] have introduced a wideband beamformer deign, coniting of three tage of filtering, that provide the deired beam pattern and the capability of zooming in and out between near-field and far-field region with eentially the ame radiation pattern by expreing the wave equation in term of pherical harmonic. Sometime achieving high directivitie i the primary objective in array deign. Repetto and Trucco [21] have addreed the importance of providing uper-directive radiation pattern, epecially when the dimenion of the array aperture are maller than the wavelength of operation. In [21], a two-tep deign procedure, involving ynthei of the required finite impule repone (FIR) filter, i preented. The author in [21] dicu the limitation of the propoed deign method; one of the major problem being the difficulty in controlling variation of the radiation pattern with frequency over the deired bandwidth, thu reulting in frequency dependent gain. Crocco and Trucco [22] and Travero, et al. [23] introduced tunable method of optimizing the weight of the element of the array to produce an optimal highly directive pattern Literature Review 7

25 or an optimal frequency invariance performance, baed on a trade off between the two propertie. Thee optimization method do not upport impoing a deired arbitrary beam pattern. Ronald [24] ha introduced a methodology that provide an approximately frequencyinvariant broadide pattern for linear array by incorporating a frequency-variant phae parameter into element excitation that minimize the frequency dependence of the array factor. The method introduced in [24] reduce the dependence of the array factor on frequency to ome extent. Exciting different element in the array over limited region of the total bandwidth i defined a aperture frequency tapering method. Thi method, preented by Holmgren et al. [25] and Ouacha et al. [26] provide fairly wideband array radiation pattern. 2.2 Array inter-element mutual coupling In many array deign aimed at achieving wideband performance, for example an array deign intended for a wide can range over a wide bandwidth, tight inter-element pacing i needed. High level of mutual coupling due to tight element pacing can reult in bandwidth degradation and/or undeirable effect uch a alteration in far-field radiation pattern and polarization [4]. One of the major negative effect of inter-element mutual coupling on the performance of an array i variation of the element input impedance with the can angle [27]; thi iue i further dicued in ection 2.4. In ome approache uggeted for deigning wideband array, element mutual coupling i intentionally introduced and i proven to be ueful. Uing the modeling of phaed array by infinite current heet introduced by Wheeler [27] and the concept of frequency elective urface (FSS) [28, 29], Munk et al. [30] noted that the mutual impedance between cloely paced dipole element in an array that i coated with dielectric layer provide wide bandwidth for everal array characteritic. In a more recent tudy, Kaemodel et al. [31] alo preented a wideband array deign employing the concept of current heet. Buxton [32] in her diertation addree the advantage of inter-element mutual coupling in an array that conit of fourquare antenna element, mot notably the increae of impedance bandwidth when enhanced mutual coupling i introduced by tight pacing; the four-quare element antenna wa deigned by Nealy [33]. D. H. Schaubert, et al.[34] and S. Katuri [35], preented a dene array deign that Literature Review 8

26 conit of Vivaldi antenna. In [35], it wa dicovered that cloing the gap along the ide of the element by connecting them improve the impedance matching performance of the array. Schaubert et al. [34] introduced a dual polarized Vivaldi array deign that included a dielectric layer covering the aperture of the array. Thi dielectric layer reulted in an increae of the VSWR bandwidth over a wide canning range, but thi improvement wa obtained at the cot of added diperion. Frequency diperion i a major ource of ditortion of wideband ignal tranmitted or received by array antenna. In the tudie of dene array and tightly coupled element, which are largely baed on numerical analyi technique, the element mutual coupling i not ued a a deign parameter and little or no conideration i given to variation of radiation pattern with frequency, thu the array are relatively narrow bandwidth which, in turn, reult in ditortion for broadband ignal. Element mutual coupling mut be compenated in adaptive array deign intended for wideband ignal acquiition. Extenive tudie have been carried out on modeling the element mutual coupling in phaed array [36-39]. Kelley claim in hi thei [37] that the reult provided in Smith reearch on mutual coupling in microtrip array [36] are not very accurate, largely becaue of ome deficiencie in claical array analyi. In hi reearch, Kelley [37, 38] dicue the concept of active-element pattern a a method of array analyi that account for mutual coupling among the element of the array by taking into conideration the induced current on the urrounding element due to the radiation of the active-element. Becaue radiation characteritic of active-element are difficult to obtain, Kelley and Stutzman [37, 38] introduced a hybrid method that regard the active element pattern in the center of the array a an average active pattern that approximate the pattern of the urrounding element while uing exact activeelement pattern for the edge element. The hybrid active element pattern method [37, 38] wa theoretically teted on an array coniting of even dipole element and wa hown to yield a high degree of accuracy when compared to meaured data. The concept of active-element pattern in a fully active array environment i dicued in Chapter 4 of thi diertation. Uing cattering parameter, Takamizawa [39] introduced a network model that allow to model a fully excited array, accounting for mutual coupling not jut among the element but alo among their ource network. The model preented by Takamizawa [39] encompae not only the array a the device under tet (DUT) but alo the probe antenna ued in the meaurement etup. Tang and Zhou [40] introduced a method of computing mutual impedance matrice by uing multiple bae Literature Review 9

27 function per element uing method of moment to olve for the array problem. Complex curve fitting technique are ued if computing the mutual impedance matrix over the entire bandwidth i impractical, a propoed in [40]. Weighted leat quare optimization algorithm i ued to calculate optimal complex weight of FIR filter in [40] a propoed by the ame author in [41]. A will be dicued in Chapter 4 of thi diertation, mutual impedance i a function not only of frequency, but alo element excitation and can angle. Thi behavior make relying on mutual impedance data to compenate for mutual coupling unreliable. However, uing S-parameter i more reliable for being only a function of frequency and independent of element excitation. The method propoed in [40] doe not take active-element radiation pattern into conideration, which i a parameter that i conidered in the propoed ynthei method of thi reearch. An intereting technique that account for element repone mimatche in achieving frequencyinvariant beamforming i preented by Jing and Huawei in [42, 43]. In [42, 43], the repone of the element i embedded in leat mean quare optimization algorithm a a probability denity function that i identical for all element. Such model if incorporated with adaptive element excitation could be ueful in inter-element mutual coupling compenation. Sörgel et al. [44] introduced a tranient array repone formulation that include the effect of element mutual coupling and the influence of the tranient repone of the ingle element in the array. Such a model i ueful when array analyi i le computationally intenive if carried out in time domain. Mutual coupling between the element and the feed network and alo mutual coupling between the port of the feed network can have ignificant impact on radiation characteritic of wideband array and hould be accounted for. Takamizawa [39] and other reearcher have dicued variou technique for compenating mutual coupling in array ytem including their paive feed network. Mot of the mutual coupling compenation technique preented by thee reearche yield atifactory reult over narrowband, thu are not adequate when dealing with wideband array ytem. Adaptive element excitation technique, dicued in ection 2.3 and alo in the propoed ynthei technique in thi diertation, offer a convenient olution for mutual coupling compenation in wideband array ytem. However, it i not eay to model the adaptive feed network theoretically and meaurement baed method are the mot effective tool in thi cae. Ertel et al. [45] preented a ytematic technique on how to compenate for phae hifting error caued by phae hifter and plitter in the array ytem. The data needed to Literature Review 10

28 determine thee error are obtained through array boreight meaurement. The ame method can alo be ued to compenate for mutual coupling in antenna array and their aociated feed network. A method imilar to that introduced in [45] wa preented by Dehuang et al. [46], but intead of performing meaurement in the frequency domain, a time reveral technique i ued. Time reveral i a proce in which the pectrum of a tranient ignal received by array element i calculated uing Fourier tranform. In both [45] and [46] the received ignal implicitly include information about element mutual coupling and other undeired factor that affect the radiation characteritic of an array. If the array i of a manageable ize, element mutual coupling meaurement can be performed eaily by meauring the cattering matrix of the array ytem by uing a multi-port tet et (MTS) a decribed in the work preented by Tanyer-Tigrek et al. [47]. An overview of reearch activitie aimed at wideband beam forming ha been preented by Gerhman [48]. Some tudie mentioned in [48] have addreed and formulated analyi and deign technique for adaptive beam forming taking into account ignal interference, noie, and other array imperfection. Mutual coupling in paive feed network i ignificant particularly at high frequency band uch a thoe operating in Ka band, i.e GHz-40 GHz. Such effect of mutual coupling may increae inertion lo of feed network. Technique uch a ubtrate integrated waveguide (SIW), introduced by Pukely and Mikulaek in [49], where hole are trategically placed on the conducting layer of a feed network of Vivaldi array in order to provide better iolation. Other technique uch a the ue of electromagnetic band gap tructure (EBG) [29] can be effective in providing iolation between port of feed network at high frequencie. 2.3 Frequency-adaptive element excitation The examination of array factor expreion, that will be dicued in Chapter 3, indicate that in order to achieve wideband array performance, or to provide frequency-invariant beam forming, the ue of frequency-variant element excitation prove ueful. Thi frequency-adaptive element excitation i motly implemented through the ue of adaptive filter that are alo ued in the ignal proceing field a dicued in ection 2.1. In fact, frequency-variant excitation are alo important for operating individual wideband antenna. Pozar [50] ha tudied the effect of pule haping on maximizing the amplitude of the voltage received at the terminal of a receiving antenna a well a maximizing fidelity of the received ignal. The reult in [50] Literature Review 11

29 emphaize the importance of pule haping and waveform optimization at the tranmitter for maximizing the voltage amplitude at the receiving antenna, epecially when adhering to wideband communication tandard uch a FCC UWB ignal regulation which are dicued by Schantz [51]. If pule-haping and waveform optimization i performed on a real-time bai, the reult i an adaptive ytem that yield optimal performance dynamically. Generation of frequency-adaptive element excitation i an important tak in the wideband array deign. The ue of tapped delay line (TDL) i one of the mot common technique in generating frequency varying ignal. The operation of a TDL conit of extracting a ignal at a pecific point on a delay line and poibly caling it to a deired level needed for an element excitation. The benefit of TDL in wideband array deign have been demontrated in many tudie. Marciano et al. [52] preented a technique for linear array pattern ynthei uing a yntheized TDL that i capable of producing an eentially contant radiation pattern over a wide frequency range. Several tudie have alo conidered TDL ynthei a an integral part of adaptive array deign. Mot of the preented reult in thee tudie indicate that the deign are, to a large degree, ucceful [53-56]. One obviou drawback of uing TDL in wideband array deign i the lack of real-time adaptivity which maintain ytem robutne and real time reconfiguration, unle dicrete witching technique are employed. Maintaining real-time hardware reconfiguration require real-time computational capabilitie. Riza [57] introduced a photonic I-Q vector modulator that generate the deired radio frequency (RF) ignal with accurate phae and amplitude value. Field programmable gate array (FPGA) i another popular computer peripheral technology that offer vat hardware reconfiguration capabilitie. Recent reearch in wideband array deign ha recognized the uefulne of FPGA in providing adaptive element excitation that are required to produce wideband array performance. The advantage of real-time hardware reconfiguration are explained very well by Joler et al. [58]. The main idea of the deign preented in [58] i to develop a real-time elf-recoverable thinned array by embedding a genetic algorithm (GA) code in an FPGA to recover the deired radiation pattern. Chiba et al. [59] deigned a digital beam forming (DBF) ytem for mobile atellite communication uing cutom-deigned FPGA Thi deign reulted in a ignificant ize reduction in the DSP engine of uch ytem and eliminated the need for phae hifter. Fichman et al. [60] ued FPGA to deign an electronically teerable Literature Review 12

30 DBF for a ynthetic aperture radar (SAR) and a moving target indication (MTI). Xin-Huai et al. [61] demontrated how real-time reconfiguration can enhance the performance of anti-jamming beam former in GPS ytem. Harter and Zwick [62] have introduced a DBF radar ytem for 3D imaging employing FPGA to provide imultaneouly the required data read-out and ytem control. Fournier et al. [63] achieved electronic canning through the ue of FPGA connected to a Butler matrix feed network that produce narrow orthogonal beam with good directivity in microtrip antenna array. Weber and Huang [64] preented a frequency agile circular array of dipole element for high peed communication between buoy and hip. In [64], the computational capabilitie of the FPGA have been ued to provide real time continuou correction required to compenate for change in the ytem introduced by random motion and interference. It hould be mentioned that while uing certain device may facilitate the production of frequency-adaptive RF ignal needed for element excitation of wideband array, but they may alo introduce more complexity into the ytem and thu higher cot. Alwan, et al. [65] developed an array deign technique that employ code diviion multiplexing (CDM) in order to reduce the complexity and large power requirement uually aociated with beamforming hardware. In the deign preented in [65], before the ignal i digitized, or before the analog to digital converion (ADC) tage, CDM i ued to uniquely identify each ignal aociated with every element antenna in the array. Thi technique allow for the ue of a ingle ADC unit in the beamforming ytem intead of uing one for each element antenna. After the ADC tage, the compoite ignal i decelerated uing the ame CDM. Under ideal condition, if the CDM code are perfectly orthogonal, the cro-correlation between the coded ignal i zero. If the number of element increae, inter-channel interference become more effective, and the performance can be improved by increaing the number of orthogonal code in the ytem. Another important conideration in array beamformer hardware deign i the dynamic range of element excitation. Horrell and du Toit [66] dicued the involvement of the dynamic range of element excitation, the ratio between maximum and minimum voltage level, a a contraint along with beamforming uing population-baed incremental learning (PBIL) optimization algorithm. In an intereting tudy Ward, et al. [67] analyzed the propertie of a filter ued for frequency-invariant enor array. In [67], it wa concluded that the relationhip between filter Literature Review 13

31 at different enor location i a dilation, or frequency caling of a primary filter deign, auming that the element of the array are iotropic. Ihan and Solbach [68] ued the reult provided by Ward, et al. in [67] to deign a UWB array that conit of printed monopole element antenna. In [67], an approximately contant width for the main-beam wa achieved over the UWB frequency range, but accompanied by grating lobe that tend to get cloer to the main-beam at the higher end of the UWB range, an iue that i typical in UWB array. Exploiting the propertie preented by Ward, et al. in [67] with the ue of FIR filter, a primary filter can be ued while the required filter tranfer function for different element can be achieved through multirate ampling technique. Thi method wa introduced by Ward, et al. in [69], where the performance of two frequency-invariant array deign were compared, one deigned baed on the multirate ampling and the other uing a ingle ampling rate. In [69], it wa concluded that the multirate method require fewer filter coefficient, but at the cot of requiring higher ampling rate, which ometime might be too expenive to implement. Forcellini and Kohno [70] applied the ame method preented in [69] but with uing an infinite impule repone filter (IIR), which may reduce the cot of the deign of the frequency-invariant beamformer dratically. Key difference between the FIR and the IIR filter are dicued in Chapter 7 of thi diertation. 2.4 Array main-beam canning Conventionally, main beam canning i achieved through introducing a relative phae difference or relative time delay between the element of an array. Many phae hifter deign have been introduced in the antenna and microwave engineering dicipline [71]. One of the mot notable technique of providing frequency-independent beam canning i that baed on the beam forming matrix. A introduced by Rufenach et al. [72], the beam forming matrix conit of a grid of tranmiion line coupled by capacitor in a way that allow ignal paing for each pre-elected main beam direction over a pecific bandwidth. Photonic device and fiber optic are alo known for their uperb accuracy in providing relative phae hift between element. Frankel et al. [73] introduced an array deign that utilize fiber optic for two dimenional beam teering. Phae hifting can alo be provided by the ue of RF ocillator. Liao and York [74] preented an array deign coniting of coupled ocillator that provide a contant phae progreion between the element excitation of an array. Literature Review 14

32 In large array, time delay unit neceary to achieve main-beam canning may occupy large pace, which may lead to requiring larger real etate than the radiating ection of the array itelf. Haupt in [75] addree the relationhip between the performance of a wideband planar array and ize and ditribution of time-delay unit (digital bit) that are ued to provide the required time delay for the element of the array. In [75], the approach for determining the real etate limitation aociated with the ize of thee time-delay unit i to digitize the time delay range by etimating the maximum delay required and the reolution between the delay of the element. Thi way, time-delay unit can be dealt with a digital bit aociated with certain length of microtrip delay line, then by combining pecific time-delay unit the required amount of time delay for each element of the array i obtained. The mot ignificant bit (MSB) repreent the larget time-delay unit and the longet delay microtrip line, while the leat ignificant bit (LSB) repreent the mallet time-delay unit and the hortet delay microtrip line. Combination of thee bit reult in achieving the time delay needed by a certain element with ome degree of accuracy depending on the quantization error that i directly related to the number of bit ued to repreent a ingle time-delay unit. The ize of a time-delay unit i contrained to a ingle unit cell in the planar array, i.e. the area of the cell that connect four element whoe center are equally-paced along the principal axe of the array. The firt few mot ignificant bit, which repreent longer time delay, may occupy large pace and it would be impractical to ue them eparately for each element. In [75], the placement of large time-delay bit i on a ubarray level and hared between the element of the array. Large idelobe, deignated a quantization lobe, appear with increaed ize of time-delay unit ubarray. Quantization error in time-delay unit can alo be a ource of wideband ignal ditortion. The effect of uch error are dicued by Haupt in [76] and by Corbin and Howard in [77]. Another effective method of minimizing the number of time-delay unit i to divide the array into ubarray, a propoed by Bianchi, et al. [78]. On the ubarray level, the element are in phae, and only one time-delay unit i ued for each ubarray to provide relative phaing between ubarray. If the ubarray are identical, the total array factor i equal to the multiplication of the array factor of the main array configuration and that of the ubarray. The appearance of grating lobe i dependent on the pacing between the contiguou ubarray. In [78], a method of overlapping ubarray i introduced. The overlapping i achieved by haring one or more of the edge element between the ubarray, which lead to inter-ubarray pacing that i maller than Literature Review 15

33 the minimum achievable pacing when the ubarray do not overlap. The ubarray could be randomly overlapped, uing an optimization proce, in order to break the periodicity of the main array and achieve better uppreion of grating lobe. Array main-beam canning over wide bandwidth i eential in many wideband application, uch a etimating DOA of wideband ignal. Technique uch a beampace diviion, in which beamformer are yntheized to cover certain region in pace where the ource ignal i expected to lie, rely on frequency-invariant beamforming and wideband main-beam canning within that pecific region. Reproducing identical radiation pattern within the canning region and the deired bandwidth i eential for ignal proceing in beampace diviion ytem, a explained by Lee in [79]. Uually FIR filter are ued to achieve frequency-invariant beamforming in beampace diviion ytem. Yan and Hou [80] teted achieving frequencyinvariant pattern within the region of the main-beam and not necearily in the idelobe region, while contraining the idelobe level to a certain maximum. The optimization method ued in [80] ha reulted in maller main-beam yntheized error in the frequency-invariant beamforming when the reult are compared to thoe publihed in [79]. Liu et al. [81] extended the concept of beampace diviion to two-dimenional array. Ward et al. [82] preented a technique of beampace proceing uing beamforming filter in time domain, eliminating the need for frequency domain decompoition. The technique in [82] i reliant on the fact that beamforming of the ytem i frequency-invariant, which lead to the avoidance of deigning for beam teering filter weight at each frequency. Beampce diviion technique if combined with adaptive array ytem can be ueful in interference uppreion. Sekiguchi and Karaawa [83] propoed an adaptive beampace diviion technique uing adaptive FIR fan filter. Each frequency-invariant beam i controlled by a FIR fan filter [84] with 2-dimenional adaptive weight, one dimenion correpond to the location of the element antenna in the array and the other correpond to the number of delay unit that precede the weight factor in the FIR filter of a ingle element antenna. The weight of the FIR fan filter are optimized uing LMS algorithm. The technique propoed in [83] wa ucceful in accommodating ignal with up to 100% fractional bandwidth. Liu et al. [85] addreed one more important condition, namely linear independency, for beamforming in beampace diviion technique other than frequencyinvariance. The beam formed by a beampace diviion array hould be linearly independent, otherwie ome of the beam output are jut a linear combination of the other beam, which Literature Review 16

34 reduce the number of effective beam and lead to reduction of the capabilitie of the array in iolating interfering ignal. A it i proven in [85], there i a trade-off between frequencyinvariance and linear independency of beam. Frequency-invariance propertie of beam may need to be acrificed to ome degree in order to improve interference cancellation capabilitie by enhancing linear independencie between different beam. The input impedance of active element of an array are affected by main beam canning according to inter-element mutual coupling, a mentioned in ection 2.2. Mutual coupling compenation through the ue of frequency-adaptive element excitation can help in eliminating the effect of beam canning on the input impedance of active-element. In order to achieve the deired canning range over a wide frequency range, free of grating lobe, uually the element pacing i minimized. However, with the element placed cloe to each other, inter-element mutual coupling become more effective. Inter-element mutual coupling i an important iue epecially in dene array ytem, uch a the fourquare array introduced by Buxton [32] and the Vivaldi array by Katuri [35]. Appearance of grating lobe in the high frequency range of wideband array i largely due to the periodicity of the array factor a a function of pace and frequency. Unequal element pacing or nonperiodic array deign technique, combined with optimization algorithm, might be helpful in uppreing grating lobe and achieving the deired canning range over the deired bandwidth. Gregory and Werner in [86] have preented a method of converting the periodic function of the array factor of a linear array to a power erie by replacing the contant element pacing with a factor raied to an optimized power. Thi factor i alo a function of the location of the element that wa originally placed in an equally paced array. The method preented in [86] wa teted for a linear array with equally excited element. A bandwidth of 20:1 wa achieved with array factor free of grating lobe. A maximum idelobe level criterion wa embedded in the optimization algorithm preented in [86]. However, in [86], the width of the main-beam varie within the deign bandwidth, which i a minor iue if the width of the main beam i very narrow. In a tudy preented by Gregory et al. [87], a emiperiodic technique i ued for deigning circular array. In [87], the location of the element inide a ector of the circular array i optimized and then the circular ector i rotated baed on tandard periodic lattice to fill the circle of the array. The reult in [87] how a ignificant uppreion of grating lobe over the bandwidth of deign. Another common technique in improving the performance of wideband array i the ue of fractal geometry. Gregory et al., [88] Literature Review 17

35 employed uch technique in order to optimize the deign of the array factor of a linear array. The deign proce in the technique preented in [88] tart with a et of generator terminated with connection factor that determine which generator to be connected in the next tage. Progreing to a higher order tage i controlled by optimization algorithm. The deign technique propoed in [88] wa applied to a linear array that conited of 1959 element, and a 28:1 bandwidth wa achieved with a 60º canning angle, free of grating lobe. 2.5 Energy-directing array Radiated electromagnetic energy i mot commonly analyzed uing time domain technique. In ome application, achieving maximum radiation with a deired direction and time duration, in which the energy i radiated, are conidered the only goal of the deign. Of coure, in uch ytem, the iue pertaining to information ignal for wirele communication i not a concern. One of the early tudie on maximizing radiated energy from wire dipole array in a pecific direction and a pecific duration of time wa carried out by Yoon-Won and Pozar [89]. Optimal element excitation ignal were obtained to produce an endfire or a broadide beam with or without idelobe level contraint. The relationhip between the ratio of the radiated energy in a pecified time duration to the total radiated energy and the bandwidth of the ytem ha been invetigated in [89]. Inter-element mutual coupling wa compenated for uing an optimization method. Since the main goal in the tudy preented in [89] wa to provide maximum radiated energy and not wideband ignal tranmiion, ignal ditortion wa not a concern. Ciattaglia and Marrocco [90] introduced a time-domain pattern ynthei technique in which Radon tranform, explained by Johnon and Dudgeon in [91], were ued to calculate element excitation required to produce a pecific array factor. In [90] the array element i regarded a an ideal differentiator, which i not a very accurate repreentation epecially when effect uch a element phae and inter-element mutual coupling i to be compenated. Literature Review 18

36 Chapter 3: Synthei of Linear Array with Specified Bandwidth A a firt tep toward the ultimate goal of deigning ultra-wideband two-dimenional array, thi chapter focue on the ynthei of linear array with bandwidth a the main deign requirement. The array factor i formulated a a Fourier erie expanion in order to facilitate the calculation of element excitation current. The array factor, over a deired frequency range and in a pecified direction in pace, i defined a a function of frequency baed on the bandwidth requirement of the deign. The ynthei approach allow for compenating frequencydependent variation of the array element radiation characteritic. 3.1 Formulation of the array factor Let u conider a linear array coniting of 2N+1 iotropic equally-paced element with jα element current I = I e n ; n= 0, ± 1, ± 2,..., ± N, and element pacing d. The center of array n n i aumed to coincide with the origin. Figure 3.1 illutrate the geometry and coordinate for uch an array. In the far-field region, the array factor i imply um of the field radiated by the jβr array element. Excluding the propagation term e / 4π r, the array factor can be expreed a [4-6], N jn d co ( ωθ, ) AF = I e β θ (3.1) n= N where β = ω μ ε = ω/ i the propagation contant, with c being the peed of light in free o o c pace, and θ i the elevation angle meaured from the line of array. n Fig. 3.1: Geometry and coordinate for a linear array of 2N + 1 iotropic element Synthei of Linear Array with Specified Bandwidth 19

37 Examination of (3.1) reveal that the right hand ide of the array factor equation may be regarded a a truncated Fourier erie expanion. Furthermore, the array factor become purely real if the current of ymmetrically paced element are complex conjugate of each other; that i, I n I = n. In thi cae, the array factor can be written a N (3.2) jnβd ( ) co θ, * jnβd co θ ω θ = + + = + 2 co( β coθ + α ) AF I I e I e I I n d 0 n n 0 n n n= 1 n= 1 Equation (3.2) i the array factor for an odd number of element. If the number of element i even, the array factor may be obtained from (3.2) by etting I 0 = 0 and replacing nd with ( n 0.5) d in it [4]. To facilitate the calculation of excitation coefficient I n, we compare (3.2) with the trigonometric form of the Fourier erie written a ( ω, θ) = + co[ β coθ] + in[ β coθ] 0 N N n n (3.3) n= 1 AF a a n d b n d where a0 = I0, an = 2 In coαn, bn = 2 In inαn. If the element phaing i choen to be α n = 0 orα = κπ ; κ = 0, ± 1, ± 2, ± 3,..., then it i readily noted that b = 0 and (3.3) reduce to n N AF, = I + 2 I co n d co (3.4) ( ω θ) ( β θ) 0 n= 1 n n 3.2 Synthei of array with pecified radiation pattern If in (3.4) the frequency i fixed at ω o, the array factor may be regarded a a truncated Fourier erie expanion of a deired array pattern defined by the function f ( θ ); that i, N d f( θ ) = AF( θω, 0) = I0 + 2 Inco ωoncoθ n= 1 c (3.5) The element excitation current are calculated a follow. π 1 I0 = f ( θ ) inθdθ T (3.6) θ 0 Synthei of Linear Array with Specified Bandwidth 20

38 π 1 2π In = f ( ) co n co in d T θ θ θ θ θ T (3.7) 0 θ where Tθ = 2 πc/ dω0. More detail on the deign of linear array with precribed radiation pattern can be found in [4-6]. It i alo evident from (3.5) that, at a fixed value of θ, the array factor can be regarded a a Fourier erie expanion in the frequency domain with ω a the variable. Thi cae i invetigated in the next ection. 3.3 Synthei of array with precribed bandwidth requirement In a pecific direction, θ o, the array factor of the array in Fig. 3.1 can be expreed a a function of frequency with the following trigonometric Fourier erie expanion. N d AF ( ω) = I0 + 2 Inco ωncoθo n= 1 c (3.8) The element excitation are calculated by uing the following equation. (3.9) ω 2 2 I0 = AF ( ω) dω T (3.9a) ω ω 1 ω2 2 2π In = AF ( ω) co n ω dω T ω T (3.9b) ω1 ω where T ω 2π c = (3.9c) d coθ o To calculate the excitation current I 0 and I n in (3.9a) and (3.9b), the array factor function AF ( ω ) hould be pecified firt. Thi depend on the frequency repone characteritic that the array i deired to have. For ultra-wideband array deign, the array factor i pecified uch that the total radiated field in a given direction θ = θ0, which i the product of the array factor and the radiated field of the array element, remain contant over a deired frequency range ω1 < ω< ω2. The frequency dependence of the array factor within the deired bandwidth can be determined Synthei of Linear Array with Specified Bandwidth 21

39 uch that the effect of the field of the element antenna are compenated. Clearly, uch an array lend itelf to wideband and ultra-wideband deign, depending upon to what extent the bandwidth limitation of the array element can be reaonably compenated. Thi apect will be elaborated upon further in the enuing dicuion. Let u conider an array element whoe radiated field magnitude in a given direction, E ( ω, θ ), varie with frequency ( ω1 < ω< ω2) a an arbitrary function hown in Fig. 3.2a. For e o the total radiated field of the array to remain contant over the frequency range ω1 < ω< ω2, the E ω, θ, that i array factor hould be inverely proportional to ( ) e o AF, o = e, o ( ωθ ) E ( ωθ ) 1 (3.10) Next, a quare-wave like periodic function i formed uch that it i an even function of ω and it period T ω i choen to be larger than 2( ω ω ) +, a hown in Fig. 3.2b. Uing the array factor 1 2 function given by (3.10) in integral expreion in (3.9a) and (3.9b), the element excitation current are readily determined. It i noted from Fig. 3.2b that for a pecified upper frequency of the deired bandwidth ω 2, the minimum value of the period i ( ) min = 2ω2. Moreover, from the expreion for T ω in (3.9c), it i noted that the choice of T ω directly impact the inter-element pacing d. In particular, the maximum inter-element pacing i limited to T ω d max 2π c = (3.11) ( T ω ) min coθ o Alo, from the erie expreion in (3.8), one can notice that the larger the number of element, 2N + 1, the more accurately the truncated Fourier erie can approximate the array factor. In the next ection, the impact of the number of element on the accuracy of the array factor i examined more thoroughly. Synthei of Linear Array with Specified Bandwidth 22

40 (a) (b) Fig. 3.2: (a) Variation of the magnitude of the element field with frequency, (b) magnitude of the total field of the array a a quare-wave like periodic function 3.4 Deign example for linear array with pecified bandwidth To demontrate the application of the formulation developed in the previou ection, deign example for linear array of equally-paced element with deired bandwidth are preented. Initially, the element are aumed to be iotropic. Figure 3.3 how the reult for variation of the array factor and directivity veru frequency for two deign example of linear array of equally-paced iotropic element with N = 15 (31 element) and two value of the inter-element pacing d (d =13.64 mm and d =15 mm). In thee example the deired bandwidth of the array i 9 GHz along the direction θ = θ0 = 0, covering a frequency range of 1 GHz to 10 GHz. Figure 3.3a and 3.3c how three dimenional plot of the magnitude of the normalized array factor veru θ and frequency. A planar top view projection of the 3-D plot of the array factor i alo included in thee figure to how ome of the hidden detail in the lower 3-D plot. Examination of Figure 3.3a and 3.3c indicate that the array factor remain eentially Synthei of Linear Array with Specified Bandwidth 23

41 unchanged over much of the in-band frequencie in the range f 1 =1 GHz to f 2 =10 GHz, particularly over a narrow angular range about θ = 0. It i further noted that a the frequency increae, the angular region over which the array factor i nearly contant become wider, and a the upper frequency of 10 GHZ i approached thi angular region cover 0 θ < 60 Furthermore, for the deign with ( T ) 2 f2. ω = 2ω = 4π = 40π Grad/, the array factor in Fig. min 3.3a ha more idelobe than that of the deign with a larger value of T ω hown in Fig. 3.3c. Alo, comparion of Fig. 3.3a and 3.3c reveal that the ripple, which reult from Fourier erie truncation, are le in the array deigned with ( ) min 40 T ω = π Grad/ and tend to decreae a the frequency increae. Grating lobe are preent over almot the entire deired bandwidth; they emerge when 120 < θ 180 for lower frequencie in the bandwidth and 90 < θ 180 higher frequencie. The emergence of grating lobe at maller value of θ for higher frequencie i due to the fact that the element pacing in term of wavelength i larger at higher frequencie than that at lower frequencie. The grating lobe can be eliminated through multiplication of the array factor and the element pattern. In the next chapter, the elimination of grating lobe by uing frequency adaptive excitation technique i dicued. The directivity at each frequency i calculated uing the following equation [4] for D ( ω) 2 = π AF ( ωθ, ) AF 0 ( ωθ, max ) 2 in θ dθ (3.12) where θ max i the angle at which the magnitude of the array factor function i maximum. Figure 3.3b and 3.3d illutrate variation of the directivity veru frequency for the deign example in Fig. 3.3a and 3.3c, repectively. Both deign exhibit imilar directivity performance, with 3-dB bandwidth of 3.1 GHz to 10.6 GHz which i the frequency range conidered for ultra-wideband (UWB) application. A major limitation of the deign method preented in thi chapter i the retriction on the inter-element pacing, d, a tated in (3.11). If d exceed it maximum value given in (3.11), the frequency performance of the array over the pecified bandwidth i deteriorated. Synthei of Linear Array with Specified Bandwidth 24

42 The number of element of the array ( 2N + 1) i a determining factor on how well the array factor and hence the directivity of the array remain contant over the pecified bandwidth. Figure 3.4 how different patial cut of the magnitude of the array factor for different number of element. It i oberved that a the number of element increae, for a given value of θ, (i) the array factor exhibit le variation with frequency within the deign bandwidth, (ii) the achieved bandwidth become larger while not exceeding the deired deign bandwidth, and (iii) the ripple become maller in amplitude but larger in number. Thee phenomena are all due to truncation of the Fourier erie repreenting the array factor; that i, finite number of term in the erie correponding to finite number of element in the array. Alo, examination of the plot in Fig. 3.4 at angle away fromθ = θ indicate that a θ increae the bandwidth tend to broaden o and hift toward higher frequencie. From a mathematical point of view, thi effect may be attributed to the preence of the term ω coθ 0 in the Fourier erie repreentation of the array factor, a noted in (3.8). Since coθ 0 decreae with θ0,0 θ0 90, for larger value of θ the 0 center frequency of the bandwidth hould occur at higher value, thu a hift of the bandwidth to the right along the frequency axi. The element excitation current required to produce the array factor hown in Fig. 3.4 are given in Fig Uing (3.9a) and (3.9b) it can be hown that, if θ o = 0 and AF ( ω ) i choen to be contant within the frequency range ω1 < ω < ω2, then I n = 0 for odd value of n and I n follow a dicrete inc function for even value of n. The inc-function-like behavior een in Fig. 3.5 i a familiar feature aociated with the Fourier tranform of rectangular pule. Accordingly, amplitude tapering of the element current away from the array center i neceary in order to achieve the required frequency-invariant array factor over the pecified bandwidth. The deciion on limiting the number of element of the array can be baed on the array factor hown in Fig. 3.4 and how well the deired performance i achieved. Another ueful information that help in making reaonable deciion on array truncation i the ignificance of the current excitation hown in Fig. 3.5 and realization of the fact that element cloer to the center have a much tronger impact on the array factor than thoe cloer to the edge of the array. The ynthei example preented above involved iotropic element with no bandwidth limitation (that i infinite element bandwidth). However, in practical application the element Synthei of Linear Array with Specified Bandwidth 25

43 have a limited bandwidth that i often le than the deired bandwidth of the array. One of the key feature of the array ynthei technique preented in thi chapter i the ability to compenate for frequency variation of the field of the element antenna over the pecified bandwidth and achieve an overall array bandwidth larger than the bandwidth of contituent element. To demontrate the effectivene of element bandwidth compenation, the ynthei method i applied to a linear array of half-wave wire dipole, with length L = 0.48λ r, at f r = 5.5 GHz. Figure 3.6 how the reflection coefficient and the realized gain of a wire half-wave dipole antenna made to be reonant approximately at f = 5.5 GHz. The half-power frequencie of the dipole are 4.7 GHz and 7.64 GHz. Figure 3.7 compare the total field radiated by the array of wire dipole with different number of element. The dipole are aumed to be parallel to the y- axi with their center located on the z-axi. It i emphaized that inter-element mutual coupling i not conidered here. Compenation of inter-element mutual coupling i dicued in the next chapter. Since the field of all element are aumed to be identical in the preent dicuion, pattern multiplication principle, tated in (3.13), i applied to produce the total field hown in Fig r where E e ( θ, ω) ( θ, ω) = e ( θ, ω) ( θ, ω) Etot E AF i the field of the dipole element antenna and Etot ( θ, ω) (3.13) i the total field of the array. Alo in Fig. 3.7, the total-field of a uniformly-excited equally-paced linear array (UEESLA) that conit of the ame number of dipole element i included for comparion. For UEESLA, the array factor i obtained from the following equation [4], AF( ωθ, ) = 0 ( N0Ψ ) in ( Ψ / 2) in / 2 N (3.14) where Ψ= ( ωd/ c)(coθ 1) and N0 = 2N + 1 which produce an endfire beam. It i emphaized that the array factor ued in determining the element excitation current for the array of wire dipole in Fig. 3.7 i baed on (3.10). Thi enure that frequency variation of the realized gain of the wire dipole are compenated over the deired frequency range of 4.7 GHz < f < 7.64 GHz. Comparion of plot in Fig. 3.4 with thoe in Fig. 3.7 indicate that, Synthei of Linear Array with Specified Bandwidth 26

44 apart from the ignificant impact of the narrow realized gain bandwidth of the wire dipole on the overall bandwidth of the array, the influence of the number of element on the frequency repone characteritic of the array remain the ame. In particular, ripple with maller amplitude for larger number of element and bandwidth broadening a well a hifting to higher frequencie at larger value of θ0 occur in the ame manner in both cae. Figure 3.7 alo illutrate the total radiated field of a uniformly-excited equally-paced array of wire dipole with N=100. It i noted that the radiated field of thi array generally varie rapidly with frequency, cauing ignificant ditortion on wideband ignal, wherea the array deigned baed on the technique preented here indeed provide a bandwidth much larger than that of the dipole element and will allow radiation of wideband ignal with much le ditortion. Synthei of Linear Array with Specified Bandwidth 27

45 (a) Tω ( Tω) min 4π f2 = =, d max = mm (b) Tω ( Tω) min 4π f2 = =, d max = mm (c) T 4π ( f f ) ω = + and mm 1 2 d = (d) T 4π ( f f ) ω = + and d = mm Fig. 3.3: Variation of array factor and directivity for linear array of iotropic element with 31 element ( N = 15 ), element pacing d=15 mm for (a) and (b) and d=13.64 mm for (c) and (d). The deired bandwidth i 9 GHz at θ = 0 with lower and upper frequencie of f1 = 1GHz, and f2 = 10GHz. 1 2 Synthei of Linear Array with Specified Bandwidth 28

46 1.2 θ o =0 o θ=0 o 1.2 θ o =0 o θ=15 o Normalized AF N=55 N=45 N=35 N=25 N=15 N=5 N=100 Normalized AF N=55 N=45 N=35 N=25 N=15 N=5 N= Frequency [GHz] Frequency [GHz] 1.2 θ o =0 o θ=30 o 1.2 θ o =0 o θ=45 o Normalized AF N=55 N=45 N=35 N=25 N=15 N=5 N=100 Normalized AF N=55 N=45 N=35 N=25 N=15 N=5 N= Frequency [GHz] Frequency [GHz] Fig. 3.4: Variation of array factor veru frequency for different value of θ and different number of element (2N+1). The element pacing i d=13.64 mm, the deired bandwidth i 9 GHz at θ = 0 with lower and upper frequencie of f1 = 1GHz, and f2 = 10GHz, and T = 4π ( f + f ) ω 1 2 Synthei of Linear Array with Specified Bandwidth 29

47 Fig. 3.5: Element current ditribution required to produce the array factor hown in Fig All parameter ued in calculating thee current are the ame a thoe in Fig In ummary, array deigned baed on the method of non-uniform excitation preented in thi chapter provide the pecified 3-dB bandwidth with reaonable accuracy particularly for larger number of element, while the UEESLA deign, with the exception of the cae θ = 0 for which the array factor become unity, provide a poor wideband characteritic. Our ynthei technique will be further developed and expanded in next and ubequent chapter Synthei of Linear Array with Specified Bandwidth 30

48 Fig. 3.6: Variation of normalized S 11 and realized gain veru frequency at θ = 0 for a wire dipole antenna a array element. The dipole i reonant at f r = 5.5 GHz (length=2.73cm= 0.48λ r ) and ha a radiu of 0.5 mm. Synthei of Linear Array with Specified Bandwidth 31

49 Normalized E tot θ=0 o N=45 N=25 N=15 N=5 N=100 N=100 (UEESLA) Normalized E tot θ=15 o N=45 N=25 N=15 N=5 N=100 N=100 (UEESLA) Frequency [GHz] Frequency [GHz] Normalized E tot N=45 N=25 N=15 N=5 N=100 N=100 (UEESLA) θ=30 o Normalized E tot N=45 N=25 N=15 N=5 N=100 N=100 (UEESLA) θ=45 o Frequency [GHz] Frequency [GHz] Fig. 3.7: Variation of the total radiated field of array of wire dipole reonant at 5.5 GHz veru frequency for different value of θ and different number of element = =, and T 4π ( f f ) (2N+1). The deign parameter are: d=13.64 mm, θ = 0, f1 4.7 GHz, f GHz ω 1 2 excited array of ame wire dipole with N=100 i hown for comparion. = +. Alo, the total radiated field of an equally-paced uniformly Synthei of Linear Array with Specified Bandwidth 32

50 Chapter 4: Synthei of Array Uing a Frequency- Adaptive Method In the ynthei technique preented in Chapter 3, bandwidth wa the main deign requirement ued in determining the array element current. In thi chapter, a frequency-adaptive ynthei method i introduced that allow including not only the required bandwidth but alo the deired radiation pattern which in turn may contain uch information a beamwidth, idelobe level, null in certain direction, directivity, etc. It i hown how frequency-adaptive excitation can be employed to provide frequency-invariant total array pattern. It i alo explained how thi method can be ued to compenate for the effect of the field of the element antenna, it input impedance, and inter-element mutual coupling on the overall bandwidth of the array. 4.1 Frequency-adaptive element excitation Let u conider a linear array with the ame geometry, parameter, and ymmetry propertie a that hown in Fig If the array factor in (3.4) i ampled dicretely in the frequency domain, it may be re-written in the following form: where ω ω ( ω) p l p N d AFp( θ ) = I0 p + 2 Inpco ωpncoθ n= 1 c (4.1) = + Δ ; p = 0,1,2,3,..., P ; Δ ω i a contant frequency increment, and ω h ω l define the bandwidth of the array with ω and ω being the upper and lower radian frequencie h (correponding to p=0 and p=p, repectively). It i noted that the right-hand-ide expreion in (4.1) can be regarded a a truncated Fourier erie expanion in the θ -domain at a ingle frequency ample ω p. The element excitation can be calculated at each frequency ample to produce an array factor that cloely approximate a deired function of θ. Thi method, in principle, allow for producing frequency-invariant array factor with arbitrary radiation pattern; AF ( θ ) = f ( θ ) for all p. Introducing the patial period T = 2 πc/ dω, element p excitation current can be calculated a Fourier erie coefficient uing the following equation. l p p Synthei of Array Uing a Frequency Adaptive Method 33

51 u2 1 1 p p T u1 p 0 ( ) ( θ ) I0p = AF u du = f inθdθ T π (4.2) π 1 dω p Inp = f ( θ ) co n coθ inθdθ T p c (4.3) 0 The minimum value of the patial period of the Fourier erie, correponding to the upper frequency of the array bandwidth ω h, i: T πc = (4.4) dω min 2 P It i emphaized that in (4.2) and (4.3) the function f ( θ ) i aumed to be zero outide the range [ 0, π ] θ in order for the bai function of the Fourier erie in (4.1) to remain orthogonal through out the bandwidth of interet. The element pacing hould be adjuted to a value uch that grating lobe are avoided and tolerable idelobe level are maintained. Thi point will be dicued further in the next ection. In order to have a better appreciation of the propoed technique, a deign example for an ultra-wideband array with precribed bandwidth and radiation pattern i preented. The array i deired to produce a frequency-invariant broadide beam over a bandwidth of 1 GHz to 10 GHz (include L-, S-, C-, and part of X-band). In thi example N = 22 i aumed. To calculate the element excitation current, we need to pecify the array factor a a function of θ. We chooe a inuoidal function that provide a highly directive radiation pattern a defined below: ( ) h AF θ, ω = f ( θ) = in m θ, with m = 50 (4.5) The half-power beamwidth for thi pattern i HP For thi array, in order to avoid grating lobe at the highet frequency in the bandwidth ( f h = 10 GHz) it i neceary that, a will be hown later, the element pacing atifie the relationhip d λ h / 2, where λ = c f = = cm; that i, d 1.5 cm. Here d = 1cm i choen. h 8 9 / h 3 10 / Synthei of Array Uing a Frequency Adaptive Method 34

52 The next tep i to calculate the element excitation coefficient uing (4.2) and (4.3). In doing o, integration are carried out numerically. Alo, truncation error due to the finite number of element can be compenated (to ome degree) by modifying the element excitation coefficient in uch a manner that the array factor along the direction of the main beam ( θ = θ ) remain unchanged veru frequency over the bandwidth of the array. Accordingly, the modified excitation coefficient are determined a follow. o I 0p_ mod I np _ mod I 0p = (4.6) AF p I ( θ ) o np = (4.7) AF p ( θ ) o Figure 4.1 how the calculated array factor (AF) uing the current ditribution in (4.6) and (4.7), while Fig. 4.2 illutrate the element excitation coefficient a function of frequency needed to produce the array factor of Fig A cloer examination of Fig. 4.1 reveal that the beamwidth at the lower frequencie exhibit ome broadening accompanied by low-level idelobe that vanih rapidly when approaching the higher frequencie of the bandwidth. Thi beam broadening and ide lobe occurrence are due to reidual truncation error caued by the limited number of element in the array. It i emphaized that thee error were compenated only for the main beam direction, thu occurrence of ide lobe (away from the main beam) and a mall broadening of the main beam hould be expected. It i hown later in thi ection that thee error can be ignificantly reduced by increaing the element pacing. To achieve main beam canning, a phae hift i embedded in the element excitation coefficient in (4.6) and (4.7). The direction of the main beam hift from θ = θ0 to θ = θ if the array factor i modified to the following expreion. N AF ( θ ) = I + 2 I co ( ndω / c) ( coθ coθ + coθ ) + α (4.8) p_ mod 0_ mod np_ mod p 0 np n= 1 where θ i the deired can angle. Accordingly, the following element excitation coefficient will produce the deired array factor in (4.8). It i reminded that the condition I = I, * (-n) p_mod np_mod Synthei of Array Uing a Frequency Adaptive Method 35

53 (a) (b) Fig. 4.1: Variation of the calculated array factor ( AF ) veru frequency and angle θ for a linear array with 45 iotropic element (N=22) and element pacing d = 1 cm. The deired array pattern i decribed by AF 50 = in θ over a frequency range of 1GHz to 10GHz. (a) Three-dimenional view, (b) top view Synthei of Array Uing a Frequency Adaptive Method 36

54 I Fig. 4.2: Element current (normalized to 0, mod max ) required to produce the array factor AF = in 50 θ for the linear array of Fig. 4.1 (N=22, d=1 cm) which wa firt applied in developing the array factor in (3.2), continue to be required here a well. I I j nd c j (4.9) np np_mod = exp ( ωp / )(coθ co θ0) + αnp AFp_ mod( θ) Figure 4.3 how the array pattern for a can angle of θ = 40 with element current calculated by uing (4.9). It i reminded that in thi cae the deired array factor i expreed a Synthei of Array Uing a Frequency Adaptive Method 37

55 ( ) 50 AF θ = in ( θ + 50 ). There i a omewhat broader beamwidth in thi cae compared to the p cae of θ = 90 o hown in Fig Thi i partly due to the Fourier erie truncation error that i larger for can angle away from the broadide direction θ = 90 o. It might be deirable to increae the element pacing a much a poible for variou reaon; for example, to decreae inter-element mutual coupling and reduce the number of element. The major limitation on element pacing i the emergence of grating lobe. The occurrence of uch lobe i directly related to the viible range of the array factor, which may be more conveniently decribed if the array factor in (4.1) i viewed a a function of ψ = dω co θ / c. Clearly, in the ψ -domain, the array factor i periodic with a period of 2π. p The viible range 0 θ π in theψ -domain correpond to ( 2 d p / c) ψ ω ψ ψ, where 0 0 ψ i a contant whoe value depend on the direction of the main beam of the array pattern and 0 the maximum allowed idelobe level. For the deign example of (4.5), ψ 0 = π and to avoid grating lobe and alo have no idelobe over the entire bandwidth the relationhip 2 dω / c 2π hould be atified, a demontrated in Fig. 4.4a. Note that, the viible range, the diameter of the emicircle in Fig. 4.4a i 2 dω / c. Thi relationhip may be expreed a a requirement that the element pacing mut atify; namely, d ( πc/ ω λ /2) idelobe level max h =, where λ = 2 πc / ω. However, if a h h SLL SLL i tolerated, then part of grating lobe can be included in the viible range a hown in Fig. 4.4b. In thi cae, 2 dω / c ( 2π 2ψ ) h + 1, where 1 h h ψ i the point at which the relationhip AFh( ψ1)/ AFh max = SLLmax i atified. Thu, the element pacing can be increaed to a larger value. For element pacing much larger than λ h / 2, grating lobe become unavoidable. Thi i oberved in Fig.4.5 which how the array pattern produced by the element excitation ued in Fig. 4.1 with d = 5cm intead of d 1cm. A expected, grating lobe appear at higher frequencie of the bandwidth and are abent at the lower frequencie. Thi array deign can till be ueful but over a reduced bandwidth of 1 GHz to 5 GHz intead of 1 GHz to 10 GHz. The reduction of bandwidth become even larger for main beam away from the broadide direction h Synthei of Array Uing a Frequency Adaptive Method 38

56 a noted in Fig. 4.6 which how a pattern with it main beam occurring at θ = It i noted in Fig. 4.6 that the bandwidth i reduced to 1 GHz to 4 GHz. Thi reult indicate that if the main beam can the angular region 67.5 θ < (correponding to a full can range of 45 about the broadide direction θ = 90 ) the ueful bandwidth of the array with d = 5 cm will be 1 GHz to 4 GHz. Examination of Fig. 4.5 and 4.6 indicate that a major factor contributing to bandwidth reduction at a certain can angle i the occurrence of grating lobe. Fig. 4.3: Variation of the array factor ( AF ) veru frequency and angle θ for the array of Fig. 4.1 but with can angle θ = 40 Synthei of Array Uing a Frequency Adaptive Method 39

57 (a) (b) Fig. 4.4: Illutration of the emergence of grating lobe in the pattern of an arbitrary array factor and it relationhip to element pacing d, ψ1 ψo =, ψ ψ ( ω ) 2 = o 2 h / c d, f = 10 GHz. (a) d = 1 cm, (b) d = 2.6 cm h Synthei of Array Uing a Frequency Adaptive Method 40

58 Fig. 4.5: Variation of the array factor ( AF ) veru frequency and angle θ for the array of Fig. 4.1 but with element pacing d=5 cm Fig. 4.6: Variation of the array factor ( AF ) veru frequency and angle θ for the array of Fig. 4.1 but with the main beam at θ = 67.5 (can angle: θ = θ 67.5 = = 22.5 ) and element pacing d=5 cm. o Synthei of Array Uing a Frequency Adaptive Method 41

59 4.2 Compenation of bandwidth reduction due to element antenna For array coniting of identical element the total radiated field of the array, baed on the principle of pattern multiplication, i obtained by multiplying the field of the element and the array factor. The element antenna ha it own frequency-dependent radiation characteritic which can ignificantly impact the overall bandwidth of the array. In thi ection, it will be hown how the frequency-adaptive element excitation method may be exploited to mitigate the reduction of array overall bandwidth due to the limited bandwidth of it contituent element. The total field of an array coniting of identical element can be expreed a E E AF ( ω, θ) = ( ω, θ) ( ω, θ) tot p e p p p (4.10) E ω θ where ( ) e p, i the field of an individual element. The radiated electric field of the element in the far-field region can be decompoed into two component written a, E ω θ E ω θ E ω θ θ E ω θ E ω θ φ (, ) = (, ) (, ) (, ) (, ) θ θ + φ φ e p e p e p e p e p (4.11) The polarization of Ee( ω, p θ ) i determined baed on the relationhip between the magnitude and phae of the two component in (4.11) and can alo be characterized by the axial ratio [92]. In order to compenate for the frequency dependence of both the magnitude and phae (or real and imaginary part) of each field component in (4.11) a complex vector array factor i needed. For the ake of reducing the complexity of the deign in thi tudy, the element antenna i aumed to be linearly polarized or circularly polarized [93]. Here we ue linearly polarized element field which can involve one field component only, allowing u to proceed with the ret of the analye dealing with a calar element field, denoted a Ee( ω, p θ ), rather than a vector one. The field of the element antenna, e E, i generally a function of both θ and ϕ, but in the preent dicuion it i aumed to be a function of θ only. Thi retriction will be lifted in Chapter 5 when ynthei of planar array i tudied. Synthei of Array Uing a Frequency Adaptive Method 42

60 In order to demontrate how frequency variation of the element field can be compenated o that the narrowing of the array bandwidth i minimized, we introduce a complex function defined a the invere of the calar element field, (, ) E (, ) 1 Ω ω θ = ω θ (4.12) AF p e p Next, let u aume that the deired total field of the array i decribed by the patial function Etot ( ω, p θ ). The magnitude of thi field repreent the total pattern of the array which, for ditortionle tranmiion of wideband ignal, i required not to vary with frequency over the bandwidth of the ignal. The ame requirement neceitate that the phae of Etot ( ω, p θ ) vary linearly with frequency. The total field can be expreed a ( ω, θ) ( ω, θ) ( ω, θ) E = E AF (4.13) tot p e p p p The element excitation current can be determined in a way that when the array factor i multiplied by the element field, the deired total field at each frequency ample can be produced. Thu, the array factor needed to compenate for the field of the element antenna hould obey the following relationhip: AF ( ω, θ) ( ω, θ) G ( ω, θ) =Ω (4.14) p p AF p p = =, Thi make the total radiated field to be equal to Etot ( ω p, θ) Ee ( ωp, θ) AFp ( ωp, θ) G( ωp, θ) where G( ω, θ) f ( θ) p array and ( p ) 0 = i the deired frequency-independent (UWB) radiation pattern of the G ω, θ = α ωp, ( α 0 = contant), i the phae of the total field which i expreed uch that the deired linear dependence on frequency i maintained. Producing a complex array factor i poible through complex element frequency-adaptive excitation. The complex array factor can be decompoed into two part, a real part and an imaginary part given by the following expreion. { ( )} 0 ( ) N j( nω pd/ c)co θ j( nωpd/ c)coθ { } { } Re AF ω, θ = Re I ω + Re I ( ω ) e + I ( ω ) e (4.15) p p p p np p np p n= 1 Synthei of Array Uing a Frequency Adaptive Method 43

61 { ( )} 0 ( ) N j( nω pd/ c)co θ j( nωpd/ c)coθ { } { } Im AF ω, θ = Im I ω + Im I ( ω ) e + I ( ω ) e (4.16) p p p p np p np p n= 1 In the formulation of the array factor preented earlier in thi chapter, conjugate current for ymmetrically located element, I * np ( ω p ) I+ np ( ω p ) =, were aumed. Here, in order to achieve the deired complex array factor, we conider the current for ymmetrically located element atify the relationhip I np ( ω p ) = Inp ( ω p ). Uing thi current relation in (4.15) and (4.16) yield { ( )} 0 ( ) N { } { ( )} ( ) Re AF ω, θ = Re I ω + 2 Re I ω co nω d / c coθ (4.17) p p p p np p p n= 1 { ( )} 0 ( ) N { } { ( )} ( ) Im AF ω, θ = Im I ω + 2 Im I ω co nω d / c coθ (4.18) p p p p np p p n= 1 Applying the integral relationhip for calculating the Fourier erie coefficient to (3.17) and (3.18), we obtain 1 I ( ω ) = AF ω, θ in θ dθ (4.19) ( ) 0p p p p Tp 0 π π 1 dω p Re { Inp ( ω p )} = Re { AFp ( ωp, θ) } co n coθ in θ dθ T p c 0 (4.20) π 1 dω p Im { Inp ( ω p )} = Im { AFp ( ωp, θ) } co n coθ in θ dθ T p c 0 (4.21) Combining (4.20) and (4.21) yield { } { } I ( ω ) = Re I ( ω ) + jim I ( ω ) np p np p np p 1 = AFp( ωp, θ) co ( ndωp / c)coθ in θ dθ T p 0 1 = G( ω p, θ) / Ee( ωp, θ) co ( ndωp / c)coθ in θ dθ T p π π 0 (4.22) Synthei of Array Uing a Frequency Adaptive Method 44

62 To achieve canning, I ( ω ) in (4.22) hould be replaced with I ( ω ) exp( Ψ ), where * n n p np p 0 np p n Ψ =Ψ = jnd ( ω / c)(coθ co θ ) i an embedded element phaing term a it wa applied and explained in (4.9). The expreion in (4.19) to (4.22) provide exact reult only if N were infinity, correponding to an array with infinite number of element. Clearly, for finite value of N truncation error occur; however, thee error can be reduced to ome degree by modifying the excitation current uch that the error along the direction of maximum radiation ( θ = θ0 ) vanihe and error along other direction ( θ θ0 ) are omewhat reduced. Accordingly, the modified excitation current are expreed a: I 0p_mod I0p( ω p) = (4.23) E ( ω, θ ) tot p o I np _mod Inp ( ω p ) =, n= ± 1, ± 2, ± N (4.24) E ( ω, θ ) tot p o 4.3 Compenation of mutual coupling and element input impedance A it wa dicued in Chapter 2, inter-element mutual coupling i a ignificant factor that cannot be overlooked becaue it may reult in degrading impact on the array radiation characteritic, epecially in the cae of array with tight element pacing. In thi ection, the mitigation of element mutual coupling uing the propoed frequency-adaptive element excitation i dicued Mechanim contributing to inter-element mutual coupling In an antenna array ytem, mutual coupling i caued by the following interaction [39]. Direct pace coupling among array element antenna Indirect coupling caued by near-by catterer Coupling through feed network Synthei of Array Uing a Frequency Adaptive Method 45

63 Fig. 4.7: Mechanim of inter-element mutual coupling in a linear array [39] Figure 4.7 illutrate thee mechanim. In general, mutual coupling deteriorate the performance of the array, epecially when main-beam canning over a wideband i the expected performance. In thi tudy, compenation of mutual coupling i invetigated with the aim of minimizing ditortion on wideband and UWB ignal Array modeling with inter-element mutual coupling A a firt tep in undertanding how the element antenna of the array interact with each other, we hall tart with an ideal cae of iolated element. The current ditribution for an iolated element antenna i given by the following equation. where { } n J r I j r i ( ) = ( ) n n n I are complex valued terminal current and j i n ( r ) (4.25) i the normalized current ditribution of an iolated element antenna, a hown in the example of Fig. 4.8, r i the poition vector ued to decribe patial variation of the current ditribution. Synthei of Array Uing a Frequency Adaptive Method 46

64 Fig. 4.8: Current ditribution and terminal current of an iolated element antenna [39] The radiation pattern in the far-field zone of an ideal array that conit of iolated element antenna, a thoe of Fig. 4.8, i given a follow, where { r n } F g I e g AF (4.26) i N i i i n n n i n= 1 jβor (, ) (, ) ˆ rn θ φ θ φ = = ( θ, φ) ( θ, φ) are vector from a fixed origin to the center of the excitation terminal of the i i element, g ( θ, φ ) i the radiation pattern of the nth element, and AF (, ) n θ φ i the ideal array factor. The upercript i in (4.26) emphaize the fact that the element are aumed to be iolated. In order to model the array ytem more accurately, a new element radiation pattern mut be defined that account for the induced current due to coupling to urrounding element. Thi modified radiation pattern i called the active-element radiation pattern. The following equation repreent the definition of a unit-input voltage active element pattern [37, 38]. 1 au ˆ (, ) ˆ ˆ a jβor r g n θφ = r r jn ( r ) e dv jωε (4.27) v Vgn = 1, Vgj = 0 for all j except n where V gn i the ource voltage, a j n i the current that exit on all element in the array when the nth element i excited with a unit voltage and all of the other element are terminated with their repective ource impedance a hown in Fig The radiation pattern of the array hown in Fig. 4.9 can be computed uing the following equation. F N, = g, V (4.28) au ( θφ) ( θφ) n q gq q= 1 Synthei of Array Uing a Frequency Adaptive Method 47

65 Fig. 4.9: Definition of active-element pattern for nth element [39] au In order to implify the meaurement proce of g ( θφ, ) gq, we aume that the poition of the origin of the array ytem i fixed but the phae reference change a q change. Accordingly, it i ueful to define a new vector quantity for the unit-input active-element pattern that account for thee phae variation baed on the location of the active element inide the array, that i the phae-adjuted unit-input active-element pattern [37, 38]: g =g (4.29) a p a u j orr ˆ n n ( θφ, ) n ( θφ, ) e β It hould be mentioned that the phae term in (4.29) wa implicitly included in (4.28). By ubtituting (4.29) into (4.28), we obtain the following expreion for the radiation pattern of the array. N jβorˆ rq F, = g, e V (4.30) a p ( θφ) ( θφ) q= 1 a p In large array, calculating or meauring ( θ, φ ) q q gq g for each element in the array i a time conuming and mot likely an expenive proce. It ha been hown in [37, 38] that uing the pattern of the center element a an average pattern for the interior element of the array can repreent the total pattern of the array with reaonable accuracy. Thi method i called hybrid method becaue it i a mix of uing an average pattern for the interior element and exact pattern for the edge element of the array. One retriction arie a a reult of uing an average pattern, namely that the interior element have to be equally paced for the aumption of imilar urrounding for all the interior element to be valid. One more reaon that having equally paced interior element i ueful i to be able to apply Fourier erie expanion a it wa Synthei of Array Uing a Frequency Adaptive Method 48

66 dicued in thi chapter. Equation (4.31) how how the hybrid method can be applied to an array configuration with equally paced interior element. i (, ) (, ) e F θφ =g θφ e V g ( θφ, ) e V (4.31) N N av j or rn p jβorˆ rm i gn + m gm n= 1 m= 1 interior element edge element where N i the number of the interior element, i i e N e i the number of edge element, and N = N + N i the total number of element in the array. In [37, 38], the hybrid method wa teted by analyzing an array with even dipole element antenna equally paced by0.4λ. The hybrid method wa hown to be reaonably accurate when compared to the exact method baed on uing (4.30). In (4.13), it wa aumed that the radiated field by individual element of the array are identical regardle of their location in the array and their active-element pattern. The ignificance of the error that reult from auming an iolated element field in (4.13) depend on the geometry of the array and the geometry of the element antenna. Thi active-element error can be compenated by modifying the array factor a follow. 1 (, ) i(, ) ( ) ( ) AF a ω p θ = Ee ωp θ Gp θ Gp θ AF ( ωp, θ) of eq. (4.13) (4.32) where a (, ) ( ) (, ) 1 G ω θ = G θ E ω θ, a p p p tot p p ( ) G θ i the deired complex function of the radiated total field by the array, i e ( p, ) E ω θ i the field of the iolated element which i ued in (4.13), a tot ( p, ) E ω θ i the total field including the error produced by the field of the active-element a Thi tep of compenating for active-element error require knowledge of tot ( p, ) E ω θ, which can be computationally expenive to determine, that i why it i preferred to tart with the aumption of iolated element and oberve whether the error due to the field of the activeelement i ignificant or not. It hould be noted that the modification in (4.32) will reult in new Synthei of Array Uing a Frequency Adaptive Method 49

67 excitation coefficient, which can be calculated by uing (4.22) and replacing AFp( ω, p θ ) with (, p ) AF ω θ. The compenation of error due to an active-element can be integrated into the propoed method of array ynthei through an iterative proce, a it will be hown later in thi chapter Array network model In ub-ection 4.3.2, it wa hown how the total radiation pattern of the array differ from the ideal cae of iolated element becaue of the induced current that reult from inter-element interaction. If the array i viewed a a network, the cattering parameter can be ued to meaure the interaction between the element of the array at the terminal or the port of each element. S- parameter are well known for their convenience in the meaurement proce at radio frequencie. Figure 4.10 how the array repreented a an N-port network withv + and A V n being the incident and reflected voltage wave, repectively, at the port of element n. For convenience, another form of wave amplitude i defined a ummarized below. A n a = V Z (4.33) A+ n n on where b = V Z (4.34) A n n on [ ] V S V A A+ n = n (4.35) [ b] = [ S][ a] (4.36) S ij V Z b = = (4.37) V Z + A i oj i A+ a j oi j V 0 for all ak 0 for all k j k = k j = A V + j denote excitation voltage at the port of the active element when meauring S ij. Synthei of Array Uing a Frequency Adaptive Method 50

68 Fig. 4.10: Repreentation of array a an N-port network with different characteritic impedance The total voltage and current at port n are calculated a follow. (4.38) ( ) V = V + V = Z a + b (4.38a) A A+ A n n n on n n 1 A+ A 1 In = ( Vn Vn ) = ( an bn) (4.38b) Z Z on which can be ued to calculate the impedance at a pecific port in the following equation. on Z A n A Vn = (4.39) I n The reflected voltage at port n can be decompoed a follow. V = S V + S V + + S V A A+ A+ A+ n n1 1 n2 2 nn N V V = + + V A A n A+ n A+ V1 V A+ A+ N 1 aa+ V aa Vk 0 for all k 1 N + = Vk = 0 for all k N (4.40) According to (4.37), if the element antenna i conidered to be a linear time invariant (LTI) ytem, S-parameter do not change with can angle, mainly becaue of S ij being unique to only one active element. However, (4.40) how that A V n varie with can angle ince each A V + n i phaed differently in order to achieve main-beam canning. Thi variation of A V n with can Synthei of Array Uing a Frequency Adaptive Method 51

69 angle make the input impedance at port n in (4.39) alo a function of the can angle. Thi variation of Z A n with can angle make it neceary to adaptively compenate for mutual coupling in order to achieve wideband ignal acquiition with minimal ditortion. Thi variation of element antenna input impedance with can angle according to inter-element mutual coupling make the ue of S-parameter the bet way to decribe the ytem repreentation of the array Mutual coupling compenation The challenge aociated with mutual coupling compenation i to deliver the deired excitation current to the terminal of the element antenna. Here, the implemented compenation account for the collective effect of mutual coupling and element input impedance. Equation (4.38b) can be normalized in the following way. ( ) I = A B (4.41) np np np where A = a / and B = b /. Smith [36] ha hown how the reflected power on on wave, np np Z np np Z B n, can be calculated baed on the deired current, a given by the following equation. 1 B np = I S p S p I np (4.42) where I i the identity matrix, I S np are the deired current, and p i the S-parameter matrix that repreent the array, all at a frequency ω p. From the definition of a np and b np given in (4.33) and (4.34), we can calculate the normalized incident power wave a follow [36]. Then, the required voltage are obtained a 1 where [ W] = [ I] + [ I S] [ S] A = I + B np np np (4.43) 1 ([ ] ) A+ Vnp = Zon I + I S p S p I np = Z on W p I np (4.44) ( ) If the excitation voltage in (4.40) were to be applied to the element of the array, the input impedance of the element antenna would be affected a a conequence of inter-element mutual Synthei of Array Uing a Frequency Adaptive Method 52

70 coupling. However, power reflection at the terminal of the element antenna remain unaffected becaue the cattering propertie do not vary with element excitation voltage. Thi i why S- parameter are mot convenient for analyzing phaed array problem. In thi diertation, S- parameter are calculated uing a commercial full-wave imulation oftware package (FEKO uite 6.3) [94]. 4.4 Flow chart of the ynthei technique The implementation of the propoed technique for ultra-wideband array ynthei i ummarized in thi ection. Figure 4.11 how the flow chart that decribe the tep for implementing the propoed ynthei technique. In the firt tep, the baic deign parameter required for the deign of the array are pecified. Thee parameter include bandwidth, element pacing, and canning range. Step 2 conit of compenating for the effect of the field of the element antenna and calculating the excitation coefficient neceary to produce a deirable total field. In tep 3, the excitation voltage neceary to compenate for inter-element mutual coupling are calculated. An error i produced a a reult of the aumption of iolated element rather than active-element. If the error due to active-element field i intolerable, the deign proce proceed to tep 4, otherwie the deign proce reache it end tate. In tep 4, a modified array factor i calculated in order to compenate for the error produced by the aumption of iolated element. The deign proce then flow from tep 4 back to the lat part of tep 2. The flow of the deign technique preented in Fig. 4.11, may ugget an iterative proce to compenate for the iolated element error; a loop tart from tep 4 and goe through the lat part of tep 2, through tep 3, and back to tep 4. Synthei of Array Uing a Frequency Adaptive Method 53

71 Fig : Flow chart illutrating the implementation of the propoed array ynthei technique Synthei of Array Uing a Frequency Adaptive Method 54

72 Chapter 5: Deign Implementation and Simulation Reult In thi chapter, the array ynthei method preented in the previou chapter i implemented in a tep by tep manner to deign wideband and ultra wideband linear array. Two type of dipole antenna are choen a array element in deign example tudied here. The bandwidth performance of thee array i aeed and the validity of the propoed technique i examined through imulation reult. 5.1 Linear array of wire dipole Figure 5.1 illutrate the geometry of a linear array coniting of 17 equally paced wire dipole antenna. The method of array ynthei dicued in Chapter 3 i ued to calculate the frequency-adaptive element excitation neceary to produce a frequency-invariant total field radiated by the array. Firt, for the ake of comparion, we tart with an uncompenated array deign. In the uncompenated cae, the element excitation are calculated without taking into conideration the effect of the field of the element antenna and inter-element mutual coupling. That i, intead of uing (4.14) for the array factor and (4.44) for the excitation voltage of the element, the following equation are ued. AF ( ω, θ) G ( ω, θ) p p p = (5.1) ( ω) = ( ω ) ; element excitation A V + np Z on I np p where G ( ω, p θ ) i the deired pattern function of the total field, and Ω ( ω θ) (5.2), 1 AF p =. For the 50 example tudied here, the pattern function i choen to be f( θ ) G( ω, θ) in ( θ) = =. The bandwidth of the array i choen to be the frequency range over which S 11 3 db for the iolated element antenna i atified (that i, the input impedance bandwidth of the iolated element). For the example cae of the element antenna ued in the array of Fig. 5.1, the bandwidth cover the frequency range 4.5 GHz to 7.0 GHz. It i emphaized that the propoed array ynthei method doe not compenate for the impedance mimatch between the excitation port and the element antenna, and impedance matching i conidered a part of the deign of the element antenna. The far-field magnitude pattern of the array in Fig. 5.1, with it element p Deign Implementation and Simulation Reult 55

73 excitation calculated uing (5.1) and (5.2), i hown in Fig Examination of thi figure indicate that the total field produced by the array deviate from the deired frequency-invariant performance, with more than 4 db variation in the direction of maximum radiation. Next, for the ame array configuration hown in Fig. 5.1 and the ame deired radiation pattern G ( ω θ) 50 ( θ), in p =, a compenated array ynthei i undertaken. Equation (4.14) i ued to calculate the array factor with the field of the element antenna taken into conideration and (4.44) i ued to calculate the required element excitation to achieve the frequency-invariant total field while compenating for inter-element mutual coupling. A noted in Fig. 5.3, the magnitude pattern of the total field i eentially invariant within the bandwidth of interet; variation are le than 0.3 db in all direction. Comparion of Fig. 5.2 and 5.3 clearly indicate a ignificant improvement toward achieving a frequency-invariant performance. A threedimenional view of the total radiation pattern for the compenated deign of Fig. 5.3 i provided in Fig It i recalled that the array factor ued in the compenated deign i complex, which mean that the phae of the field of the element antenna and the inter-element mutual coupling are compenated a complex quantitie. The bet method to compare the linearity of the phae of the radiated total field of the uncompenated and the compenated deign i to compare their group delay. The group delay i defined a the derivative of the phae of the total-field with repect to radian frequencyω. For the deired performance of minimized ignal ditortion, a linearly varying phae or a contant group delay veru frequency i required. Figure 5.5 compare the group delay of uncompenated and compenated deign of Fig. 5.2 and 5.3. It i oberved that the group delay of the compenated deign i almot contant and exhibit ignificantly le variation compared with the group delay of the uncompenated deign [95]. In order to appreciate thi improvement in group delay, the tranient repone of the uncompenated and the compenated array deign are calculated and compared later in thi chapter. A ummary of performance parameter for the compenated deign decribed in Fig. 5.3, including halfpower beamwidth (HP), directivity (D), and maximum ide-lobe level (SLL), i preented in Table 5.1. Deign Implementation and Simulation Reult 56

74 Fig. 5.1: Linear array of 17 equally paced wire dipole; wire radiu a = 0.5 mm, element pacing d = 0.4λ r, reonant frequency f r = 5.5 GHz, length of dipole L = 0.48λ r Fig. 5.2: xz-plane cut of the total pattern at different frequencie for the uncompenated deign (frequency variation of the element field and inter-element mutual coupling are not accounted for) of an array with geometry and parameter the ame a thoe given in Fig The deired total array pattern i decribed 50 by f ( θ ) = in θ over the frequency range 4.5GHz to 7GHz. Deign Implementation and Simulation Reult 57

75 Fig. 5.3: xz -plane cut of the total pattern at different frequencie for the compenated deign (frequency variation of the element field and inter-element mutual coupling are accounted for) of an array with geometry and parameter the ame a thoe given in Fig The deired total array pattern i decribed 50 by f ( θ ) = in θ over the frequency range 4.5GHz to 7GHz. All pattern are approximately coincident. Fig. 5.4: Three-dimenional view of the magnitude pattern at different frequencie for the compenated array deign decribed in Fig 5.3 Deign Implementation and Simulation Reult 58

76 Fig. 5.5: Comparion of group delay for uncompenated and compenated array deign along the direction of maximum radiation, θ = 90. For both deign the array geometry and parameter are the ame a thoe 50 given in Fig. 5.1, and the deired total pattern i decribed by f ( θ ) = in θ over the frequency range 4.5GHz to 7GHz. Table 5.1: Summary of performance parameter including, half-power beamwidth (HP), directivity ( D ), and maximum idelobe level ( SLL ) for the array of wire dipole decribed in Fig. 5.3 Frequency Performance Parameter (GHz) HP D SLL º 12.2 db db º 12.3 db db º 12.4 db db º 12.5 db < db º 12.6 db < db º 12.7 db < db Deign Implementation and Simulation Reult 59

77 In order to achieve main-beam canning of the total-field of the array decribed in Fig. 5.3, a linear phae-hift i applied to the element excitation a wa implemented in (4.22). Calculating the element excitation uing (4.22) will allow for main-beam canning while preerving the frequency-invariant performance already achieved for the compenated array deign decribed in Fig Figure 5.6 illutrate the canned total pattern at a can angle θ = θ 22.5 = 67.5 o θ = direction of maximum radiation=90, when all the element are in ( o phae), indicating that a 45 canning range around the broadide directionθ o can be provided. The ame linear phae performance that wa achieved in the array deign of Fig. 5.3 i maintained when the main beam i canned in Fig A ummary of performance parameter for the array decribed in Fig. 5.6 i preented in Table B1 in Appendix B. Fig. 5.6: Total pattern in the xz-plane for the compenated array deign decribed in Fig. 5.3 but canned to θ = θ 22.5 = All pattern are approximately coincident. o Deign Implementation and Simulation Reult 60

78 5.2 Elliptical dipole a element antenna The element antenna uitable for ultra-wideband array deign propoed in thi reearch hould provide an acceptable impedance matching performance and far-field radiation pattern over the bandwidth of deign. The field pattern of the element antenna can be compenated through the array factor a dicued in ection 4.2. The element ize i alo an important deign parameter with direct impact on allowable array element pacing. To provide a wide main-beam canning range without the appearance of grating lobe within the deign bandwidth, the element pacing need be minimized. Some of the well known UWB antenna that can provide all of thee requirement are circular- and the elliptical-dic dipole [96-98]. The geometry of a UWB circular-dic dipole antenna [96, 97] i hown in Fig The radiu of both circle i 30 mm and the length of the feed wire i 2 mm. Figure 5.8 illutrate variation of the VSWR (voltage tanding-wave ratio) and the input impedance of thi dipole antenna veru frequency. A noted from Fig. 5.8a, the ize of the antenna can be caled down by a multiplicative factor of 2GHz/3GHz and the antenna would till be adequately impedance matched in the UWB frequency range of 3.1 GHz to 10.6 GHz. It can be alo een in Fig. 5.8b that at lower frequencie, the input reactance of the antenna i highly capacitive. In the tudy preented in [96], the effect of elongating the circle in Fig. 5.7 in the direction perpendicular to the feed line (thu creating an elliptical tructure with the major axi perpendicular to the feed wire line) wa invetigated. The reult wa a wider bandwidth, but at the cot of increaed ize and higher VSWR at lower frequencie. A it wa mentioned in the beginning of thi ection, keeping the ize of the element antenna mall i often one of the important deign requirement. If the antenna with characteritic hown in Fig. 5.8 i modified in a way that provide better impedance matching at the lower frequencie, the ize of the antenna can be caled down even further to hift it bandwidth to the UWB frequency range. Comparion of the input impedance for circular and elliptical dipole in Fig. 5.8b and 5.10b lead one to believe that attaching a narrower tapered tructure to the feed wire intead of a circular one would provide a better impedance matching performance at the lower frequencie. Thi i due to the fact that the input impedance of narrower tapered tructure i le capacitive a een in Fig. 5.10b. Deign Implementation and Simulation Reult 61

79 A modified deign for the elliptically-tapered dipole antenna, a hown in Fig. 5.9, i propoed here which i expected to further improve the impedance matching performance at lower frequencie. A ingle arm of thi elliptically-tapered dipole antenna conit of two half ellipe part with a common center and their major axe perpendicular to each other, a een in Fig A full-wave analyi of thi antenna indeed acertained the better impedance matching performance at lower frequencie, but of coure at the cot of lightly larger ize along the feed line compared to the circular dipole. If the elliptically-tapered dipole i caled to the UWB frequency range, it will yield the ame length a that of a caled circular dipole in the UWB range. The multiplicative caling factor of the elliptically-tapered antenna required to hift it bandwidth to the UWB frequency range i 1.5GHz/3GHz (reult in axial dimenion 15 mm 20 mm for each arm of the dipole), while for the circular dipole thi factor i 2GHz/3GHz (reult in 20 mm diameter for each circular arm). In the direction perpendicular to the feed wire line, the elliptically-tapered dipole i maller in ize than the circular dipole. Thi advantage of the elliptically-tapered dipole over the circular one allow for maller element pacing which, in turn, reult in a wider canning range over the required bandwidth. The VSWR and input impedance characteritic of the caled verion of the elliptically-tapered dipole are hown in Fig Fig. 5.7: Circular-dic dipole antenna a an ultra-wideband element for frequency-invariant array Figure 5.12 illutrate the ditribution of the urface current denity for the caled elliptically-tapered dic dipole antenna with characteritic hown in Fig A oberved in Fig. 5.12, on each arm the current decay moothly from the excitation port all the way to the edge of the dic. In fact, it i thi kind of behavior that provide the UWB performance exhibited by the VSWR characteritic hown in Fig. 5.11a. Deign Implementation and Simulation Reult 62

80 (a) VSWR (b) Input Impedance Fig. 5.8: Variation of (a) VSWR referenced to 100Ω and (b) input impedance veru frequency for the circular-dic dipole antenna hown in Fig. 5.7 Fig. 5.9: Elliptically-tapered dic dipole antenna a an ultra-wideband element with improved inputimpedance matching performance compared with the circular-dic dipole hown in Fig (a) VSWR (b) Input Impedance Fig. 5.10: Variation of (a) VSWR referenced to 100Ω and (b) input impedance veru frequency for the ellipticallytapered dic dipole antenna hown in Fig Deign Implementation and Simulation Reult 63

81 (a) VSWR (b) Input Impedance Fig. 5.11: Variation of (a) VSWR referenced to 100Ω and (b) input impedance veru frequency for the caled verion of the elliptically-tapered dipole hown in Fig. 5.9 (dipole axial dimenion: 15 mm 20 mm) (a) f = 3.0 GHz (b) f = 4.0 GHz (c) f = 5.0 GHz (d) f = 6.0 GHz (e) f = 7.0 GHz (f) f = 8.0 GHz (g) f = 9.0 GHz (h) f = 10.0 GHz (i) f = 11.0 GHz Fig. 5.12: Illutration of the ditribution of urface current denity at different frequencie for the caled elliptically-tapered dic dipole antenna with performance characteritic hown in Fig Deign Implementation and Simulation Reult 64

82 5.3 UWB linear array The UWB elliptically-tapered dic dipole, with VSWR and input impedance characteritic hown in Fig. 5.11, i ued to contruct a 17-element (N=8) linear array with element pacing d=17 mm a depicted in Fig The element are aumed to lie on the yz-plane. The radiation pattern for an uncompenated deign of thi array with the deired total far-field pattern f( θ ) = G ω, θ = in θ i hown in Fig A noted in Fig. 5.14a, the achieved radiation of ( ) 50 p pattern varie ignificantly with frequency over the deired UWB bandwidth. Thi hould be expected a the effect due to inter-element mutual coupling and variation of the element field with frequency are not accounted in an uncompenated array deign. Variation of the input impedance veru frequency for half of the array element including the center element for the uncompenated cae are hown in Fig A noted in Fig the input impedance of the edge element (element 8) differ ignificantly from that of the center element. Thi behavior i largely due to the inter-element mutual coupling effect. A ummary of performance parameter, including HP, D, and SLL i preented in Table 5.2. The excitation current at the input terminal of the element for thi array are hown in Fig Figure 5.16a illutrate variation of the current magnitude veru frequency, while Fig. 5.16b how how current phae vary with frequency. There are two et of curve in each part of Fig. 5.16; the olid curve repreent the current calculated from (5.1) and (5.2), while the dahed curve repreent the current that are actually developed at the element input port under the influence of inter-element mutual coupling and are obtained by mean of a oftware imulation tool (FEKO). Comparion of the olid and dahed curve indicate that for the cae of uncompenated deign the deired current are not delivered to the input port of the element antenna becaue mutual coupling effect are not accounted for in thi deign proce. The ditribution of urface current denitie for the element of the array decribed in Fig are hown in Fig Thee ditribution are hown only at two frequencie of 3 GHz and 11 GHz; the urface current ditribution at other frequencie within the array bandwidth are preented Table C1 in Appendix C. Comparion of current ditribution in Fig, 5.17 with thoe of an iolated element hown in Fig clearly indicate that the ditribution for ame element and at the ame frequency are ignificantly different, particularly for the edge element. Deign Implementation and Simulation Reult 65

83 Fig. 5.13: Ultra-wideband linear array of 17 equally-paced elliptically-tapered dic dipole with element pacing d = 17 mm and element axial dimenion 15 mm 20 mm (a) Polar radiation pattern, 3dB/Div. (b) Magnitude (c) Phae Fig. 5.14: Variation of (a) total pattern, (b) magnitude of the total field at θ = 90, and (c) phae of the total field at θ = 90 veru frequency for an uncompenated linear array of elliptically-tapered dic dipole with geometry and parameter ame a thoe decribed in Fig The deired total array pattern i decribed 50 by f ( θ ) = in θ over the UWB frequency range of 3.1GHz to 10.6GHz. Deign Implementation and Simulation Reult 66

84 (a) Real Part (b) Imaginary Part Fig. 5.15: Variation of input impedance veru frequency for half of the element (element 1-8) and the center element (element 0) of the array decribed in Fig Table 5.2: Summary of performance parameter including, half-power beamwidth (HP), directivity ( D ), and maximum idelobe level ( SLL ) for the array decribed in Fig Frequency Performance Parameter (GHz) HP D SLL º 10.8 db db º 11.9 db db º 12.6 db db º 13.1 db db º 13.4 db < -58 db º 13.5 db < -58 db º 13.2 db < -58 db º 12.3 db < -58 db º 10.6 db < -58 db Deign Implementation and Simulation Reult 67

85 (a) Magnitude [A] (b) Phae [º] Fig. 5.16: Variation of excitation current veru frequency for the element of the array decribed in Fig Solid curve repreent the current calculated uing (5.1) and (5.2), dahed curve repreent the current developed at the input port of element antenna due to mutual coupling effect Deign Implementation and Simulation Reult 68

86 (a) f = 3.0 GHz (b) f = 11.0 GHz Fig. 5.17: Illutration of urface current ditribution for half of the element of the array decribed in Fig (center element hown on the left) After compenation are implemented, radiation characteritic improve ignificantly a oberved in Fig The excitation current for the compenated deign were calculated uing (4.19) (4.21) and (4.44). The phae of the deired total-field of the array, in the range of the half-power beamwidth, i adjuted to the nearet traight line that fit the uncompenated phae characteritic hown in Fig. 5.14c. Comparion of Fig. 5.14a and 5.18a indicate that the radiation pattern of the compenated array varie much le with frequency; however, the deired contant magnitude and linearly varying phae of the total field have not yet been achieved a noted in Fig. 5.18b and 5.18c. A ummary of performance parameter and urface current ditribution of element for the compenated deign of the array decribed in Fig are preented in Table B2 and C2 in Appendice B and C. The current at the input port of the element for half of the array and the center element are hown in Fig It i noted in Fig that after compenating for inter-element mutual coupling, the calculated current and thoe Deign Implementation and Simulation Reult 69

87 actually developed at the input port of the element become nearly the ame in both magnitude and phae. The performance of the array decribed in Fig. 5.18b and 5.18c with regard to magnitude and phae behavior i due to uing the iolated element field in the calculation performed in tep 1-3 of the deign flow chart preented in Fig The reulting error were not ignificant in the cae of the wire dipole array dicued earlier in thi chapter. In tep 4 of the deign proce hown in Fig. 4.11, the error in the total-field due to the active-element field are compenated. Fig how the radiation characteritic of the array deign of Fig after compenation of error due to the active-element field have been made. A oberved in Fig. 5.20, a nearly contant magnitude and a linearly varying phae for the total-field ha been achieved. The reult produced in Fig were obtained after one iteration tarting from tep 4 of the deign proce. However, comparion of pattern hown in Fig. 5.20a and Fig. 5.18a indicate that ide-lobe level have increaed. The maximum SLL i -18 db at 3GHz in Fig. 5.18a, while it i -16dB in Fig. 5.20a. Mot practical UWB ignal have low power pectral denitie at the edge of the UWB band, making the increae in SLL at 3 GHz more tolerable. A ummary of performance parameter and urface current ditribution of element for the compenated deign baed on the active-element field of the array decribed in Fig are preented in Table B3 and C3 in Appendice B and C. The current at the input port of the element for half of the array and the center element are hown in Fig It can be een in Fig that the develped current at the input port of the element match the calculated current after compenation for inter-element mutual coupling uing (4.44). It hould be noted that the current in Fig. 5.21, where (4.14) wa ued to calculate the array factor, differ from thoe of Fig. 5.16, where (4.32) wa ued to calculate the array factor. Deign Implementation and Simulation Reult 70

88 (a) 3dB/Div. (b) (c) Fig. 5.18: Variation of (a) total pattern, (b) magnitude of the total field at θ = 90, and (c) phae of the total field at θ = 90 veru frequency for a compenated (baed on iolated element aumption) linear array of elliptically-tapered dic dipole with geometry and parameter ame a thoe decribed in Fig The 50 deired total array pattern i decribed by f ( θ ) = in θ over the UWB frequency range 3.1GHz GHz. Deign Implementation and Simulation Reult 71

89 (a) Magnitude [A] (b) Phae [º] Fig. 5.19: Variation of excitation current veru frequency for the element of the array decribed in Fig Solid curve repreent the current calculated uing (5.1) and (5.2), dahed curve repreent the current developed at the input port of element antenna. Deign Implementation and Simulation Reult 72

90 (a) 3dB/Div. (b) (c) Fig. 5.20: Variation of (a) total pattern, (b) magnitude of the total field at θ = 90, and (c) phae of the total field at θ = 90 veru frequency for a compenated (baed on active-element field) linear array of ellipticallytapered dic dipole with geometry and parameter ame a thoe decribed in Fig The deired total 50 array pattern i decribed by f ( θ ) = in θ over the UWB frequency range 3.1GHz GHz. Deign Implementation and Simulation Reult 73

91 (a) Magnitude [A] (b) Phae [º] Fig. 5.21: Variation of excitation current veru frequency for the element of the array decribed in Fig Solid curve repreent the current calculated uing (5.1) and (5.2), dahed curve repreent the current developed at the input port of element antenna. Deign Implementation and Simulation Reult 74

92 5.4 Tranient repone In examining the tranient repone, the array antenna i treated a a linear time-invariant (LTI) ytem baed on the fact the underlying governing equation, namely Maxwell equation, are linear equation. The tranient repone of a UWB ytem may be more conveniently evaluated through the impule repone, which i imply the invere Fourier tranform of the tranfer function of the ytem. The radiated total field calculated in the previou ection are taken a the tranfer function of the repective array. Very narrow Gauian pule are ueful model for impule ignal. Typical value for the tandard deviation of UWB Gauian pule are in the order of nano or pico econd, a dicued in [99, 100]. FCC regulation for UWB ignal for medical imaging ytem (FCC part ) and for indoor UWB ytem (FCC part ) [101] tate that the tranmitted power in the frequency range of 3.1 GHz-10.6 GHz hould not exceed 0.5 mw in order to minimize interference. Thi tranlate to a power pectral denity (PSD) of approximately dbm/mhz. Uually thee limit are enforced by applying a making ignal at the UWB tranmitter, which according to (FCC part d) i defined a an intentional radiator that, at any point in time, ha a fractional bandwidth equal to or greater than 0.20 or ha a UWB bandwidth equal to or greater than 500 MHz, regardle of the fractional bandwidth. In thi tudy, un-maked UWB ource ignal are ued to tudy the tranient repone of the array deign that have been preented o far in thi chapter. In ome UWB ytem, uch a thoe ued in medical imaging, the ued ignal utilize the entire UWB frequency range. The general time-domain analyi of an antenna i dicued in [102]. The formulation for the tranient repone i ummarized in Appendix A. In Fig. 5.22, a an example cae, the tranient repone of the wire-dipole antenna array i preented. The radiation characteritic of a wire dipole antenna array in the frequency-domain for both uncompenated and compenated deign were dicued in ection 5.1. To tudy the timedomain repone of the array, a narrow Gauian (approximating an impule) i conidered a the input ignal. Thi input ignal, ( t ), and it pectrum are hown in Fig.5.22a and Fig.5.22b, repectively. It i reminded that the bandwidth of the wire dipole array cover the frequency range 4.5 GHz to 7.0 GHz. From Fig. 5.22b, one can tell that the bandwidth of the ignal i much wider than the bandwidth of the array. The radiated output ignal for the uncompenated and compenated cae are hown in Fig. 5.22c and 5.22d, repectively. Comparion of thee Deign Implementation and Simulation Reult 75

93 figure clearly indicate that the uncompenated deign uffer more ringing effect than the compenated one. (a) (b) (c) Fig. 5.22: Tranient repone of the wire dipole array configuration of Fig. 5.1; ( a) Input Gauian pule with σ = 0.1 n, (b) Spectrum of the modulated input ignal, (c) Radiated pule of the uncompenated deign of Fig. 5.2 at θ = 90, (d) Radiated pule of the compenated deign of Fig. 5.3 at θ = 90 (d) Deign Implementation and Simulation Reult 76

94 Chapter 6: Synthei of Two-Dimenional Planar Array In the previou chapter, a frequency adaptive array ynthei method wa introduced, formulated, and teted for linear array deign. Beam forming and main-beam canning by mean of linear array can be controlled only on a ingle plane in pace that contain the line of array. In other word, the array factor of a linear array, a a patial function, i one dimenional, ay a function of θ only. Thu, the beamwidth in a plane perpendicular to the line of array i determined olely by the beamwidth of the element antenna. In order to provide more directivity and beam canning on two orthogonal plane, planar array whoe array factor are function of two patial coordinate (θ and ϕ) may be ued. In thi chapter, the application of the adaptive array ynthei technique i extended to two-dimenional planar array. Deign example uing the ame element antenna that were ued in linear array deign in Chapter 5 are conidered. 6.1 Formulation of planar array factor Let u conider a planar rectangular array of (2M + 1) (2N+ 1) identical element with center of element reiding on the xy-plane. The entire array may be regarded a a linear array of 2M + 1 element with element pacing dx along the x-axi and array factor AF ( θ x ), where θ x i an angle meaured from the x-axi. The contituting element of uch array itelf i another linear array of 2N + 1 element along the y-axi with element pacing d y and array factor AF ( θ ) and y y, whereθ y i an angle meaured from the y-axi. It can be hown that the angle θ x θ y are related to the patial coordinate θ and ϕ through the relationhip θx θ ϕ 1 1 = co ( in co ) and θy co ( inθ inϕ ) =. The array factor AF ( θ ) erve a the element pattern for the array along the x-axi. Thu, baed on the principle of pattern multiplication, the total array factor for the entire two-dimenional planar array i AF( θ, ϕ ) = AF ( θx) AF ( θ y). x y y y x Auming that the center element of the array i ituated at the origin and element current procee a ymmetrical ditribution decribed by Imn = I mn = Im( n), and I mn = I m I n, then AF x and AF y are expreed a (6.1) Synthei of Two-Dimenional Planar Array 77

95 M AF ( θ, φ) = I + 2 I co ( mωd / c)inθcoϕ (6.1a) x x0 xm x m= 1 [ ] AF ( θ, φ) = I + 2 I co ( nωd / c)inθin ϕ) (6.1b) y y0 yn y n= 1 N The expreion in (6.1a) and (6.1b) are very imilar to (3.4) which i the array factor for a linear array. In fact, if θ in (3.4) i replaced with θ or θ equation (6.1a) or (6.1b) i obtained. Thee x imilaritie allow u to employ much of formulation developed in Chapter 3-5 in analyzing two-dimenional planar array. Accordingly, if the array factor in (6.1) are ampled dicretely in the frequency domain, they may be re-written in the following form: (6.2) y AF ( ω, θφ, ) = I ( ω) + 2 I ( ω)co ( mω d / c)inθcoϕ (6.2a) xp p x0 p p xmp p p x m= 1 M AF ( ω, θφ, ) = I ( ω) + 2 I ( ω)co ( nω d / c)inθin ϕ) (6.2b) yp p y0 p p ynp p p y n= 1 N If the deired frequency-independent total far-field pattern of the array i decribed by the function f ( θ, ϕ ), the excitation current in (6.2a) and (6.2b) are calculated uing the pattern in the principal plane ϕ = 0 in AF xp and ϕ = 90 in yp AF. The reult are ummarized a (6.3) π 1 I ( ω ) = G( ω, θ, ϕ = 0) / E ( ω, θ, ϕ = 0) co ( md ω / c)coθ in θ dθ, M m M xmp p p e p x p Txp 0 (6.3a) π 1 ynp p p e p y p T yp 0 I ( ω ) = G( ω, θ, ϕ = 90 ) / E ( ω, θ, ϕ = 90 ) co ( nd ω / c)coθ in θ dθ, N n N (6.3b) where E ( ω, θϕ, ) i the element field, e p T xp 2π c =, d ω x p T yp 2π c d ω =, and G( ω, θϕ, ) f ( θϕ, ) y p p = i the deired frequency-independent (UWB) radiation pattern of the array and ( p,, ) 0 G ω θϕ = αωp, ( 0 α = contant), i the phae of the total field which i expreed uch that the deired linear dependence on frequency i maintained. Then, the current of the element Synthei of Two-Dimenional Planar Array 78

96 correponding to mth row (in the x-direction) and nth column (in the y-direction) at frequency ω p i obtained from I ( ω ) = I ( ω ) I ( ω ), M m M and N n N (6.4) mnp p xmp p ynp p To achieve canning, the array factor in (6.2a) and (6.2b) hould be modified a, (6.5) M AF ( ω, θφ, ) = I + 2 I ( ω) co ( mω d / c)(inθcoϕ inθ coϕ + inθ coϕ xp p x0 p xmp p p x 0 0 m= 1 (6.5a) N AF ( ω, θφ, ) = I ( ω) + 2 I ( ω) co ( nω d / c)(inθinϕ inθ inϕ + inθ in ϕ) yp p y0 p p ynp p p y 0 0 n= 1 (6.5b) where θ o and φ o define the direction along which the magnitude of the uncanned main beam i maximum and θ and φ are the can angle, defining the direction along which the main beam i deired to occur upon canning. Thi modification amount to replacing I ( ω ) in (6.3a) and I ( ω ) in (6.3b) with I ( ω )exp( Ψ ) and I ( ω )exp( Ψ ), repectively, where (6.6) ynp p x mp p m ynp p n x mp p Ψ =Ψ = jmd ( ω / c)(inθ coϕ inθ co ϕ ) (6.6a) * m m x p 0 0 Ψ =Ψ = jnd ( ω / c)(inθ inϕ inθ in ϕ ) (6.6b) * n n x p 0 0 To reduce truncation error due to uing finite number of term (ideally, infinite number of term hould be ued) in the erie in (6.2) and (6.5), excitation current may be modified in the ame manner a in linear array with the following reult, I I ( ω ) mnp p mnp mod ( ω p ) = (6.7) Etot ( θ0, ϕ0, ωp ) where Etot θ0 ϕ0 ωp = AFxp θ0 ϕ0 ωp AFyp θ0 ϕ0 ωp Ee θ0 ϕ0 ωp (,, ) (,, ) (,, ) (,, ) Finally, taking into account inter-element mutual coupling, a explained in Chapter 3, the excitation voltage of the element can be calculated a follow, Synthei of Two-Dimenional Planar Array 79

97 1 [ ] ( ) ( ) [ ] [ ] [ ] T ( ) ( ) ( ) [ ] ( ) ([ ] 2 ) 2M+ 1 2N+ 1 1 ( 2N+ 1 ) V = Z I + I S S I I (6.8) 2M+ 1 2N+ 1 o m 2M+ 1 1 n where I i the identity matrix, and [ ] T repreent a non-conjugate tranpoe. 6.2 Planar array of wire dipole The linear array of 17 wire dipole element that wa tudied a a deign example in ection 5.1 i ued here to contruct a planar array. Seventeen uch linear array are placed parallel to the x-axi, a hown in Fig. 6.1, to form a 17x17 planar array with element pacing d = 0.4λ and d y = 0.6λ at the reonant frequency of f = 5.5 GHz. Equation (6.3), (6.4) and (6.6) to (6.8) r r are ued to calculate the element excitation current. The deired total pattern of the array i choen to be x r f ( θ, φ) co 50 = θ (6.9) which produce a half-power beamwidth of HP 13.5 The radiation pattern for θ = φ = 0 are hown in Fig It i noted that thee pattern remain eentially unchanged over the frequency rang 4.5 GHz to 7.0 GHz. However, a expected, the directivity of the planar array i ignificantly larger than that of the 17-element linear array. Summarie of performance parameter of the planar array deign preented in thi chapter are provided in Appendix B. The magnitude and phae characteritic of the total radiated field along the direction of maximum radiation ( θ = 0 ) are hown in Fig It i een in Fig. 6.3 that phae varie linearly with frequency and variation of the magnitude are quite mall, validating the application of the propoed technique in yntheizing wideband/uwb planar array. Synthei of Two-Dimenional Planar Array 80

98 Fig. 6.1: Planar array of 17x17 wire dipole contructed baed on the linear array of Fig The array parameter are: element pacing in the x-direction d = 0.4λ, element pacing in the y-direction d = 0.6λ, x r y r dipole length L = 0.48λ r ; reonant frequency f r = 5.5 GHz The main-beam of the total-field produced by the array of Fig. 6.1 i canned to the direction pecified by coordinate θ = 45 and φ = 0. The ame equation ued to calculate the element excitation for the reult in Fig. 6.2 are ued with θ = 45 and φ = 0. The 3-D pattern of the magnitude of the total-field, canned to the mentioned direction, are hown in Fig It i noted from Fig. 6.4e and Fig.6.4f that grating-lobe, or a back-lobe in thi pecific can direction, tart to appear at higher frequencie. The principal array factor contributing to thee grating lobe i AF x. The back-lobe hown in Fig. 6.4f i 15 db below the maximum level of the canned main beam. Next, φ i increaed from zero to φ = 25, while keeping θ = 45. Figure 6.5 how the two principal-plane array factor with applied linear phae-hift to element excitation to achieve main beam canning. It i oberved from Fig. 6.5b that AF y uffer from appearance of grating lobe at the upper frequency limit of the bandwidth. On the other hand, grating lobe in direction farther than that of with θ = 45 AF x hown in Fig. 6.5a do not appear even though it main beam i canned to a AF y, becaue dx < d. The pattern of the total field of the array y and φ = 25 are hown in Fig The maximum ide-lobe level in Fig. 6.6f correponding to 7.0 GHz i about 15 db below the maximum level in the canned direction. Synthei of Two-Dimenional Planar Array 81

99 (a) f=4.5 GHz (b) f=5.0 GHz (c) f=5.5 GHz (d) f=6.0 GHz (e) f=6.5 GHz (f) f=7.0 GHz Fig. 6.2: Three-dimenional magnitude pattern at different frequencie for the total field of a planar array of wire dipole with can angle θ = φ = 0. The geometry and parameter of the array are the ame a thoe 50 decribed in Fig 6.1. The deired total array pattern ued in the deign i given by f ( θ, ϕ) = co θ over the bandwidth of the array, covering the frequency range of 4.5GHz to 7.0GHz. The correponding performance parameter (HP, D, and SLL) are preented in Table B4. Synthei of Two-Dimenional Planar Array 82

100 (a) (b) Fig. 6.3: Variation of (a) magnitude of the total field at θ = 0 and (b) phae of the total field at θ = 0 veru frequency for the planar array of wire dipole decribed in Fig.6.2 In order to produce more directional radiation pattern, a ground plane can be ued a a reflector. The firt conideration i to place the ground plane appropriately for the reflected field to be in-phae with the forward radiated field on an average ene within the deired bandwidth. To achieve thi goal, a ditance equal to λ c /4 between the ground plane and the plane of the array, where λ c = c/ fc with fc being the center frequency of the bandwidth, i choen. For the planar wire dipole array of Fig. 6.1, we have f = 5.75 GHz. We firt examine the field of a c ingle wire dipole element above an infinite ground plane with a ditance of λ c /4, a hown in Fig The three-dimenional radiation pattern of the field of the element antenna of Fig. 6.7 are hown in Fig It i noted that over the frequency range of interet (4.5 GHz to 7.0 GHz), the total pattern of the element and ground plane remain eentially unchanged, indicating that an array of thee element above the ground plane i expected to provide about the ame bandwidth when excitation current are determined baed the ynthei technique introduced in thi work. The planar array of wire dipole above ground plane i hown in Fig The inter-element mutual coupling in Fig. 6.9 i different from that of the array configuration of Fig The reaon for thi difference i the preence of the image of the planar array below the ground plane, a i well known baed on the image theory. Figure 6.10 illutrate the total radiation pattern of the array of Fig. 6.9 at everal frequencie. Examination of Fig indicate that the total pattern remain relatively unchanged over the bandwidth of the array 4.45 GHz 7.0 GHz, although the ide-lobe level at 7.0 GHz (Fig. 6.10f) increaed lightly compared to the cae when ground plane wa not included (Fig. 6.6f). Synthei of Two-Dimenional Planar Array 83

101 (a) f=4.5 GHz (b) f=5.0 GHz (c) f=5.5 GHz (d) f=6.0 GHz (e) f=6.5 GHz (f) f=7.0 GHz Fig. 6.4: Three-dimenional magnitude pattern at different frequencie for the total field of the planar array of wire dipole decribed in Fig. 6.2 but with can angle θ = 45 and φ = 0. The correponding performance parameter (HP, D, and SLL) are given in Table B5. Synthei of Two-Dimenional Planar Array 84

102 (a) AF x (b) AF y Fig. 6.5: Variation of the calculated principal-plane array factor veru frequency and angle θ for the planar array of wire dipole decribed in Fig. 6.2 but with can angle θ = 45, φ = 25. (a) AF, (b) x AF y Synthei of Two-Dimenional Planar Array 85

103 (a) f=4.5 GHz (b) f=5.0 GHz (c) f=5.5 GHz (d) f=6.0 GHz (e) f=6.5 GHz (f) f=7.0 GHz Fig. 6.6: Three-dimenional magnitude pattern at different frequencie for the total field of the planar array of wire dipole decribed in Fig. 6.2 but with can angle θ = 45, φ = 25. The correponding performance parameter (HP, D, and SLL) are provided in Table B6. Synthei of Two-Dimenional Planar Array 86

104 Fig. 6.7: Wire dipole antenna, ued a an element in the planar array of Fig. 6.1, above an infinite ground plane by a ditance of h = λ c /4 with f c = 5.75 GHz (a) f = 4.5 GHz (b) f = 5.0 GHz (c) f = 5.5 GHz (d) f = 6.0 GHz (e) f = 6.5 GHz (f) f = 7.0 GHz Fig. 6.8: Three-dimenional magnitude pattern at different frequencie for the total field of a ingle wire dipole antenna above an infinite perfectly-conducting ground plane. The dipole ha a length L = λ c /2and i a ditance h = λ c /4 above the ground plane; f c = 5.75 GHz. Synthei of Two-Dimenional Planar Array 87

105 Fig. 6.9: A planar array of wire dipole above an infinite ground plane. The array i aumed to have the ame geometry, parameter and deired total pattern a that decribed in Fig. 6.1 and 6.2. The ground plane and plane of array are parallel and eparated by a ditance of h = λ c /4 at f c = 5.75 GHz. (a) f = 4.5 GHz (b) f = 5.0 GHz (c) f = 5.5 GHz (e) f = 6.0 GHz (f) f = 6.5 GHz (g) f = 7.0 GHz Fig. 6.10: Three-dimenional magnitude pattern at different frequencie for the total field of the planar array of wire dipole above ground plane decribed in Fig. 6.9; can angle areθ = 45, φ = 25. The correponding performance parameter (HP, D, and SLL) are preented in Table B7. Synthei of Two-Dimenional Planar Array 88

106 6.3 Elliptical dipole planar array The linear array of elliptically-tapered dic dipole tudied in Chapter 5and illutrated in Fig. 5.12, i ued here to contruct a planar array of 15x15 ( M = 7, N = 7 ) element, a hown in Fig For can angle θ = 25 and φ = 0, the principal-plane array factor are calculated uing the deired total pattern decribed by the function 50 f ( θ, φ) = co ( θ θ ). Thee principalplane array factor are illutrated in Fig A noted in Fig. 6.12b, large grating lobe in the principal array factor AF y begin to emerge when frequency exceed 5.5 GHz, limiting the bandwidth to the frequency range 3.0 GHz to 5.5 GHz. Figure 6.13 how the three-dimenional magnitude total radiation pattern of the array configuration of Fig The maximum ide-lobe level for radiation pattern at all frequencie hown in Fig i around -15dB. A in the cae of the linear array of elliptical dipole tudied in Chapter 5, error due to active-element field are ignificant in thi array deign. The impact of thee error i evident in Fig. 6.14, which how that the magnitude and phae characteritic of the total field of the array, along the direction of can angle, do not exhibit the targeted performance of contant magnitude and linear phae variation with frequency. The active-element error are compenated uing an approach imilar to that decribed in tep 4 of the flow chart in Fig. 4.11, uing a modified array factor given by ( θ, φω, ) ( θ, φ)/ ( θ, φ, ω) ( θ, φ ) / ( θ, φ, ω) a i a G = f E e f E tot (6.10) Fig. 6.11: Planar array of 15x15 elliptically-tapered dic dipole contructed baed on the linear array of Fig The array parameter are: element pacing in the x-direction d x = 17 mm, element pacing in the y- direction d = 43 mm, and element axial dimenion 15 mm 20 mm. The deired total array pattern i y 50 conidered to be f ( θ, ϕ) = co θ. Synthei of Two-Dimenional Planar Array 89

107 (a) AF (b) x AF y Fig. 6.12: Variation of the calculated principal-plane array factor veru frequency and angle θ for the planar array of elliptical dipole decribed in Fig but with can angle θ = 25, φ = 0 (a) AF, (b) x AF y (a) f = 3.0 GHz (b) f = 3.5 GHz (c) f = 4.0 GHz (d) f = 4.5 GHz (e) f = 5.0 GHz (f) f = 5.5 GHz Fig. 6.13: Three-dimenional magnitude pattern for the total-field of the planar array decribed in Fig The correponding performance parameter (HP, D, and SLL) are provided in Table B8. Synthei of Two-Dimenional Planar Array 90

108 The element excitation are calculated uing ( ). The radiation pattern of the array deign of Fig. 6.11, after compenating for active-element error are nearly identical to thoe hown in It can be een in Fig that the magnitude of the total field of the array after compenating for active-element error i ignificantly more uniform than that of Fig. 6.14, and with ignificantly better phae performance. Fig. 6.14: Variation of the magnitude and phae veru frequency for the total field of the planar array of elliptical dipole decribed in Fig at θ = 25, φ = 0. The correponding performance parameter (HP, D, and SLL) for thi array are provided in Table B8. Fig. 6.15: Variation of the magnitude and phae veru frequency for the total field of the planar array of Fig at θ = 25, φ = 0 after compenating for active-element error. The correponding performance parameter (HP, D, and SLL) for thi cae are preented in Table B9. Synthei of Two-Dimenional Planar Array 91

109 In order to provide a more directive total pattern for the planar array of the elliptical dipole, an infinite ground plane i placed below the plane of the array by a ditance equal to λ /4 at f = 4.25 GHz, a hown in Fig The radiation pattern of an iolated elliptical mid mid dipole element antenna above an infinite ground plane i hown in Fig A noted in Fig. 6.17, the radiation pattern of thi element antenna over a ground plane remain largely omnidirectional over the deign bandwidth, indicating that the eparation of λmid / 4 between the ground plane and plane of the array i a reaonable choice. The radiation pattern at everal frequencie for the total field of the array of Fig are hown in Fig The maximum idelobe level, imilar to the cae without ground plane, remain around -15dB. The performance of the array in Fig ha not quite reached the goal of contant magnitude and linear phae variation with frequency a i evident in Fig Thi i of coure due to error reulting from the ue of iolated element field in (6.3). Equation (6.10) i ued to calculate the principal-plane array factor neceary to compenate for uch error, then element current excitation are calculated from (6.4) to (6.6). The radiation pattern for the total field of the array decribed in Fig after compenating for active-element field error remain eentially unchanged. The total field of the array after compenating for active-element error meet the required performance of uniform magnitude and linear phae over the required bandwidth a hown in Fig Fig. 6.16: Planar array of elliptical dipole with the ame geometry and parameter a thoe decribed in Fig The array i above an infinite ground plane by a ditance of h = λ mid /4 at f mid = 4.25 GHz. Synthei of Two-Dimenional Planar Array 92

110 (a) f = 3.0 GHz (b) f = 3.5 GHz (c) f = 4.0 GHz (d) f = 4.5 GHz (e) f = 5.0 GHz (f) f = 5.5 GHz Fig. 6.17: Three-dimenional radiation pattern at everal frequencie for an iolated elliptical dipole antenna above an infinite ground by a ditance λ mid /4 at f mid = 4.25 GHz (a) f = 3.0 GHz (b) f = 3.5 GHz (c) f = 4.0 GHz (d) f = 4.5 GHz (e) f = 5.0 GHz (f) f = 5.5 GHz Fig. 6.18: Three-dimenional magnitude pattern for the total-field of the planar array decribed in Fig at θ = 25, φ = 0.The correponding performance parameter (HP, D, and SLL) are given in Table B9. Synthei of Two-Dimenional Planar Array 93

111 Fig. 6.19: Variation of the magnitude and phae veru frequency for the total field of the planar array of Fig at θ = 25, φ = 0 before compenating for active-element error. The correponding performance parameter (HP, D, and SLL) for thi cae are preented in Table B10. Fig. 6.20: Variation of the magnitude and phae veru frequency for the total field of the planar array of Fig at θ = 25, φ = 0 after compenating for active-element error. The correponding performance parameter (HP, D, and SLL) for thi cae are preented in Table B11. Synthei of Two-Dimenional Planar Array 94

112 Chapter 7: Implementation of Array Element Excitation Real-time adaptive array excitation network, or RF front end, are good candidate for the adaptive array deign introduced in thi work. A dicued in the literature review in Chapter 2, technique baed on the ue of FPGA and Software Defined Radio (SDR) offer the mot convenient olution. SDR ytem allow for real-time reconfiguration, a function that i difficult to achieve with fixed RF hardware. Integrating an adaptive interface with the array tructure i neceary in order to provide frequency-invariant real-time beamforming and main-beam canning. Additionally, an adaptive SDR ytem may extend the capabilitie of the ytem to be integrated with different antenna array and different frequency-adaptive excitation algorithm. Practical contraint impoed by uch factor a bandwidth, dynamic range, reolution, and peed of analog to digital converter (ADC), may limit the performance of real-time adaptive ytem. In thi chapter, the implementation of element excitation for array deign dicued in the previou chapter i addreed. 7.1 Adaptive element excitation uing digital filter Digital filter are a critical part of an adaptive digital ignal proceing (DSP) ytem. The A+ frequency-domain excitation voltage of element antenna, namely Vnp ( ω p ) in (4.44), are regarded a the tranfer function of the required digital filter in the feed network. The common requirement for thee filter are tability and phae linearity. A table filter can be characterized a a bounded-input bounded-output (BIBO) ytem. One way to enure the tability of a digital filter i for the pole of it tranfer function to reide on the left-half of the complex -plane ( = jω ). Other filter tability tet can be found in the literature [103]. The repone of a digital filter can be modeled a a progreive linear um of delayed weighted coefficient. A branch coniting of one delay term and one weight coefficient i deignated a a filter tap. The repone of the filter i the um of the output of thee tap. Two of the mot common digital filter ued in adaptive DSP ytem are the finite impule repone (FIR) and the infinite impule repone (IIR) filter. The main difference between thee two type of digital filter i that the IIR filter include feedback, while the FIR filter involve only forward proceing of the input ignal. The advantage of uing feedback i to reduce memory Implementation of Element Antenna Excitation 95

113 capacity requirement. However, a filter that include feedback i not guaranteed to be table, but a FIR filter i alway table [103, 104]. Another advantage of FIR filter i that it i eaier to achieve linear phae characteritic with thee filter. 7.2 FIR filter deign The block diagram of a FIR filter i hown in Fig The tranfer function of the unittime delay block in Fig. 7.1, repreented by 1 Z, i given by exp( j2π f ) W n are the weight of the filter. The tranfer function of the filter i obtained a ( f ) ( f ) Y H( f ) = = W Z Z = e X, and the coefficient N 1 n n ; j2π ft (7.1) n= 0 From (7.1) it can be een that the impule repone of the FIR filter goe to zero within a finite period of time, hence the name finite impule repone (FIR). Fig. 7.1: Block diagram of a baic FIR filter, W n are real valued weight, and 1 Z are time delay block The required tranfer function of the filter for each element antenna, neceary to achieve the performance of the array deign dicued in previou chapter, i already known. For the nth element, the tranfer function of the aociated filter i defined A+ A+ by H( f) Vn ( f ) Vnp ( ω p) = = in (4.44) which can be divided into two purely real function, one A+ HRe f = Re{ Vn f }, and another repreenting the imaginary repreenting the real part, ( ) ( ) { n } + part, H ( f ) V ( f ) Im = Im A. Each part i realized by a eparate filter with a real tranfer function. A phae hift of 90 between HRe ( f ) and Im ( ) H f mut be applied to form the complex tranfer function. The 90 phae hifter mut operate within the bandwidth of interet; Implementation of Element Antenna Excitation 96

114 phae hifter like thoe ued in [105] can be ued for thi purpoe. It can be een in (7.1) that the weight of the filter, W, can be calculated a Fourier erie coefficient. Higher number of filter n tap allow for generation of more accurate excitation ignal for the element antenna. A FIR filter with purely real tranfer function can be realized by introducing conjugated filter tap to the FIR filter deign of Fig The tructure of the filter with the additional conjugated tap i hown in Fig The tranfer function of the FIR filter with conjugate ymmetrical tap i calculated a Fig. 7.2: Block diagram of FIR filter with conjugate ymmetrical tap, W n are real valued weight, 1 Z are time delay block, and ( )* i a conjugation block N 1 N 1 N 1 (7.2) ( ) = exp( 2π ) + exp( + 2π ) = 2 co( 2π ) H f W j nf W j nf W nf n n n n= 0 n= 0 n= 0 A noted in (7.2), the tranfer function of the FIR filter of Fig. 7.2 i a purely real function that alo aume the form of a truncated Fourier erie. Thi property i neceary for realization of the real and the imaginary part of the excitation of the element antenna a will be hown in the next ection. The cot of FIR filter with conjugate tap might be higher than that of other type of FIR filter deign, but thi cot come with the benefit of providing the capability for realizing complex tranfer function with arbitrary real and imaginary part, and hence arbitrary magnitude and phae function. Implementation of Element Antenna Excitation 97

115 7.3 Element antenna excitation The ignal fed to element of an array each mut poe pecific magnitude and phae pectral characteritic in order to achieve the deired total radiated field over the bandwidth of interet. Thee pectral characteritic and their calculation were tudied in detail in Chapter 3 to 6. Thi ection briefly addree the realization of required pectral characteritic by mean of digital filter. A mentioned earlier, two FIR filter with ymmetrical conjugate tap for the realization of tranfer function Re { A + Vn ( f) } and Im { A + Vn ( f) } are needed. The block diagram of the overall filter tructure, combining the real and imaginary part, i hown in Fig The ource ignal, X ( f ), i equally divided into two tream, one for Re { A + Vn ( f) } for Im { A + Vn ( f) }. The Im { A + Vn ( f) } part include a 90 denoted a Im Wn and Re W n the element excitation ignal i and the other phae-hifter and the filter weight,, are calculated by uing (7.2). The output of the filter which contitute ( ) ( ) { } { } ( ) A+ A+ A+ Y f = X f Re Vn ( f) + jim Vn ( f) = X f Vn ( f) (7.3) The additional phae hifting of the element ignal, neceary to achieve main-beam canning, can be implemented at the output of the filter tructure of Fig The main reaon that digital filter eem to be viable candidate for the implementation of the array deign technique preented in thi reearch i becaue of their flexibility. Digital filter are motly programmable and highly reconfigurable thu allowing real-time reconfiguration of array element excitation. Thi capability facilitate the acquiition of the required magnitude and phae characteritic for element excitation in order to achieve a frequency-invariant antenna array performance. Implementation of Element Antenna Excitation 98

116 Fig. 7.3: Block diagram of the FIR filter for element antenna excitation, W n are real valued weight, 1 Z are time delay block, and ( )* i a conjugation block Implementation of Element Antenna Excitation 99

117 Chapter 8: Concluion and Future Work A robut ultra-wideband array ynthei technique ha been introduced and tudied. The main objective of thi reearch ha been to achieve frequency-invariant radiation characteritic for array antenna that are capable of tranmitting or receiving ultra-wideband ignal with minimum ditortion at boreight or within a pecified patial region. Frequency-adaptive antenna element excitation i employed in order to achieve uch performance for the array. A ummary of the main tak of the propoed ynthei technique and the impact on the wideband performance of the array in each cae i preented below. The main contribution of thi reearch are pointed out and uggetion for further invetigation are given 8.1 Summary Element field compenation The deign technique preented in thi reearch tart with calculating the complex radiated far-field of the iolated element antenna a a function of pace and frequency. The array factor, expreed a a Fourier erie, i contructed uch that it would ideally converge (if the erie had infinite number of term) to a function obtained by multiplying the invere of the element field and the deired total pattern of the array. Preliminary calculation of the array factor with iotropic element provide ueful etimation of maximum element pacing and how it i related to main-beam can angle, appearance of grating-lobe, and achievable bandwidth. Motly, the appearance of grating lobe et the ultimate limit on the main-beam canning range and bandwidth Inter-element mutual coupling compenation Information about inter-element mutual coupling in a fully active array environment along with element field information are ued in compenating the inter-element mutual coupling. Upon carrying out mutual coupling compenation, the adjuted excitation current are delivered to the terminal of the element antenna in order to produce the required array factor. The input impedance of the element antenna i accounted for in the proce of compenation for interelement mutual coupling. Concluion and Future Work 100

118 8.1.3 Achieving frequency-invariant total field pattern At the beginning of the ynthei proce, the field of the element antenna i calculated in iolation. The field of an iolated element might be a good approximation of it actual field in the array environment, a in the cae of the wire dipole array deign preented in Chapter 5 and 6, or might be ignificantly different from it actual field, a in the cae of the array uing elliptical dipole element alo dicued in Chapter 5 and 6. The error reulting from uing the field of an iolated element i related to the current ditribution on the antenna. In a fully active array environment, the current ditribution of the element antenna i affected by the induced current due to radiation by the urrounding element. In the previou ubection, compenation of interelement mutual coupling i implemented at the excitation terminal of the element antenna, which doe not mitigate current ditribution alteration due to mutual coupling. One way to eliminate uch error i to ue active-element pattern at the beginning of the deign proce. Calculating active-element pattern for each element could be impractical epecially for large array uch a the planar array deign preented in Chapter 6. Hybrid active-element method, in which the center element are aumed to have identical active-element pattern, could be helpful in reducing the computational burden. In fact, if Fourier erie or any periodic erie i ued to yntheize the required array factor, the ue of hybrid active-element method i neceary. Another way of eliminating the active-element error i to compenate for it. After producing a total-field baed on the aumption of iolated element field, that contain the activeelement error, the element excitation can be re-calculated iteratively to compenate for thi error. The latter method of compenating for active-element error i adopted in thi reearch. Thi compenation proce could be performed over multiple iteration in order to achieve more accurate reult. For the linear and planar array with elliptical element antenna, one iteration wa enough to obtain atifactory reult. 8.2 Contribution An ultra-wideband array ynthei technique ha been introduced and validated by applying it to deign linear and planar array of dipole element. The main contribution of thi reearch are ummarized a follow: Concluion and Future Work 101

119 The propoed ynthei method i comprehenive and account for all factor contributing to bandwidth limitation of the array. The exiting method often do not addre the frequency dependence of the element antenna or are limited to yntheizing the array factor only. Both magnitude and phae of the total radiated field are conidered in the ynthei proce propoed here. Mot exiting array ynthei method account for the power pattern only and leave the phae untreated. Uing the propoed technique, linear and planar array can be yntheized that, in principle, can produce far-field total radiation pattern with arbitrary patial variation which remain eentially unchanged over a wide frequency range. Alo, the phae of the total radiated field of a yntheized array varie linearly with frequency over the bandwidth. The propoed technique allow yntheizing array that can provide bandwidth ignificantly larger than the bandwidth of their contituent element. Linear and planar array of wire dipole with a minimum bandwidth of 2.5 GHz, which were tudied a example cae in Chapter 5 and 6, are a tetimony to thi fact. 8.3 Suggetion for future work There are a number of iue that require further invetigation in order to improve the performance of the array or optimize the deign with the aim of reducing the cot. Suggetion for future work are ummarized below. Noie and electromagnetic interference can be a ignificant ource of wideband ignal ditortion. Compenation for noie and interference might be integrated into the ynthei method propoed in thi reearch, provided that the modeling of uch ource of ignal ditortion i available. A it wa mentioned in thi tudy, appearance of grating lobe i the main factor limiting the main-beam canning range and bandwidth of the array. The ue of unequal element pacing i an effective method to minimize the limiting effect of grating lobe. Concluion and Future Work 102

120 Combining optimized unequal element pacing and optimized frequency-adaptive element excitation might be a topic of great potential for future reearch. Multi-rate ampling in the DSP tage of the array ytem could help in reducing the number of required filter in order to achieve adaptive element excitation. A it wa dicued in the literature urvey of thi diertation, ome reearcher have applied thi concept and developed frequency-invariant beamformer uing only one reference filter while achieving the required filter repone for the different element of the array through applying multi-rate ampling. Such technique could be ued in the implementation of the adaptive element excitation propoed in thi reearch in order to optimize the cot of the ytem of the wideband array. Having an optimized deign for digital filter required for element excitation, fabricating the array geometrie preented a example deign in thi reearch will be an important tak. Meauring the radiation characteritic of fabricated array and comparing them with imulation reult are indeed neceary tep toward validating the predicted performance of the wideband and ultra-wideband array dicued in thi work. Concluion and Future Work 103

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130 Appendix A: Formulation of the Tranient Repone of an Antenna For an array antenna with it main-beam canned to a direction defined by angle θ and φ, the general expreion of the total field i given a: (8.1)(8.2)(8.3) j( ωt βr) e E( θ, φ, ω) = jωμ P1 ( θ, φ, ω) 4π r P1 ( θ,, ) (,, ) (,, ) ˆ ˆ φ ω = P θ φ ω P θ φ ω r r jβ r rˆ P( θ, φ, ω) = J( r, ω) e dv v (A.1) (A.2) (A.3) where J r ω r r (, ) i the current ditribution of the radiator or the antenna i the poition vector that decribe the geometry of the radiator i the poition vector of that decribe the location of the point of obervation Let S ( ω) be conidered a the frequency-domain repreentation of a wideband input ignal to an LTI ytem uch a an antenna, then if we can define the function (8.4) X P S ( θ, φ, ω) = ( θ, φ, ω) ( ω) 1 the invere Fourier tranform of the field given in (A.1) i: (8.5), (A.4) where x( θ, φ, t) = p ( θ, φ, t) ( t) 1 μ e t x t r c 4π r t ( θ, φ, ) = ( θ, φ, / ). (A.5) NOTE: the formulation above can be ued to calculate the tranient repone of an array in any direction, not jut the direction of the main-beam. Appendix-A 113

131 Appendix B: Performance Parameter (HP, D, and SLL) for Array Deign Example Preented in Chapter 5 and 6 Table B1: Summary of performance parameter for the wire dipole array decribed in Fig. 5.6 (The geometry and parameter of the array are the ame a thoe given in Fig. 5.1.) Frequency Performance Parameter (GHz) HP D SLL º 11.6 db db º 11.7 db db º 11.8 db db º 11.9 db < db º 12.0 db < db º 12.1 db < db Table B2: Summary of performance parameter for the elliptically-tapered dic-dipole array decribed in Fig (The geometry and parameter of the array are the ame a thoe given in Fig ) Frequency Performance Parameter (GHz) HP D SLL º 10.7 db db º 11.9 db db º 12.7 db db º 13.2 db db º 13.5 db db º 13.5 db db º 13.2 db < -58 db º 12.3 db < -58 db º 10.5 db < -58 db Appendix-B 114

132 Table B3: Summary of performance parameter for the elliptically-tapered dic-dipole array decribed in Fig (The geometry and parameter of the array are the ame a thoe given in Fig ) Frequency Performance Parameter (GHz) HP D SLL º 10.8 db db º 12.0 db db º 12.7 db db º 13.0 db db º 13.3 db db º 13.5 db < -58 db º 13.2 db < -58 db º db < -58 db º 10.5 db < -58 db Table B4: Summary of performance parameter for the planar array of wire dipole decribed in Fig. 6.2 (The geometry and parameter of the array are the ame a thoe given in Fig. 6.1.) Frequency Performance Parameter (GHz) HP / HP D SLL x y º/13.5º 20.0 db < -40 db º/13.5º 20.0 db < -40 db º/13.5º 20.0 db < -40 db º/13.5º 20.0 db < -40 db º/13.5º 20.0 db < -40 db º/13.5º 20.0 db < -40 db Appendix-B 115

133 Table B5: Summary of performance parameter for the planar array of wire dipole decribed in Fig. 6.4 (The geometry and parameter of the array are the ame a thoe given in Fig. 6.1.) Frequency Performance parameter (GHz) HP x D SLL º 18.2 db < -40 db º 18.3 db < -40 db º 18.3 db < -40 db º 18.3 db < -40 db º 18.3 db -26 db º 18.0 db db Table B6: Summary of performance parameter for the planar array of wire dipole decribed in Fig. 6.6 (The geometry and parameter of the array are the ame a thoe given in Fig. 6.1.) Frequency Performance Parameter (GHz) HP φ= 25 D SLL º 18.2 db < -40 db º 18.3 db < -40 db º 18.3 db < -40 db º 18.4 db < -40 db º 18.3 db db º 18.2 db db Appendix-B 116

134 Table B7: Summary of performance parameter for the planar array of wire dipole above ground plane decribed in Fig (The geometry and parameter of the array are the ame a thoe given in Fig. 6.9.) Frequency Performance Parameter (GHz) HP φ= 25 D SLL º 21.7 db < -40 db º 21.8 db < -40 db º 21.7 db < -40 db º 21.7 db < -40 db º 21.6 db < -40 db º 21.4 db db Table B8: Summary of performance parameter for the planar array of elliptical dipole decribed in Fig and 6.14 (The geometry and parameter of the array are the ame a thoe given in Fig ) Frequency Performance Parameter (GHz) HP x D SLL º 17.2 db db º 17.8 db db º 18.3 db db º 18.2 db db º 18.2 db db º 18.6 db db Appendix-B 117

135 Table B9: Summary of performance parameter for the planar array of elliptical dipole decribed in Fig (The geometry and parameter of the array are the ame a thoe given in Fig ) Frequency Performance Parameter (GHz) HP x D SLL º 17.2 db db º 17.8 db db º 18.3 db db º 18.2 db db º 18.2 db db º 18.6 db db Table B10: Summary of performance parameter for the planar array of elliptical dipole above ground plane decribed in Fig and 6.19 (The geometry and parameter of the array are the ame a thoe given in Fig and 6.16.) Frequency Performance Parameter (GHz) HP x D SLL º 20.3 db db º 20.8 db db º 21.2 db db º 21.6 db db º 21.2 db db º 21.3 db db Appendix-B 118

136 Table B11: Summary of performance parameter for the planar array of elliptical dipole above ground plane decribed in Fig (The geometry and parameter of the array are the ame a thoe given in Fig and 6.16.) Frequency Performance Parameter (GHz) HP x D SLL º 21.6 db db º 22.3 db db º 22.8 db db º 23.3 db db º 23.0 db db º 23.3 db db Appendix-B 119

137 Appendix C: Surface Current Denity Ditribution on element of Linear Array of Elliptical Dipole Antenna f = 3.0 GHz f = 4.0 GHz f = 5.0 GHz f = 6.0 GHz f = 7.0 GHz f = 8.0 GHz f = 9.0 GHz f = 10.0 GHz f = 11.0 GHz Fig. C1: Illutration of urface current ditribution for half of the element of the array decribed in Fig with center element hown on the left (The geometry and parameter of the array are the ame a thoe given in Fig ) Appendix-C 120

138 f = 3.0 GHz f = 4.0 GHz f = 5.0 GHz f = 6.0 GHz f = 7.0 GHz f = 8.0 GHz f = 9.0 GHz f = 10.0 GHz f = 11.0 GHz Fig. C2: Illutration of urface current ditribution for half of the element of the array decribed in Fig with center element hown on the left (The geometry and parameter of the array are the ame a thoe given in Fig ) Appendix-C 121

139 f = 3.0 GHz f = 4.0 GHz f = 5.0 GHz f = 6.0 GHz f = 7.0 GHz f = 8.0 GHz f = 9.0 GHz f = 10.0 GHz f = 11.0 GHz Fig. C3: Illutration of urface current ditribution for half of the element of the array decribed in Fig with center element hown on the left (The geometry and parameter of the array are the ame a thoe given in Fig ) Appendix-C 122