Realizing φ-Symmetric Radiation Patterns of Circular Apertures Using Circular Square-Grid Arrays...

IEEE Antennas and Propagation Magazine, Vol. 54, No. 3, June 2012 135

Antenna Designer’s Notebook Founded by Hal Shrank

Tom Milligan8204 West Polk PlaceLittleton, CO 80123 USATel: +1 (303) 977-7268;Fax: +1 (303) 977-8853E-mail: [email protected]://www.antennadesigner.org

Realizing φ -Symmetric Radiation Patterns of Circular Apertures

Using Circular Square-Grid Arrays

R. Eirey-Pérez, J. A. Rodríguez-González, and F. J. Ares-Pena

Department of Applied Physics, Faculty of PhysicsUniversity of Santiago de Compostela15782 Santiago de Compostela, Spain

E-mail: [email protected]; [email protected]; [email protected]

Abstract

The radiation patterns generated by circular apertures can be approximated quite accurately using circular square-grid arrays, if the pattern is sampled with suffi cient density.

Keywords: Antenna arrays; circular arrays; planar arrays; antenna radiation patterns

1. Introduction

The symmetry of the continuous circular aperture makes it relatively easy to calculate the aperture-excitation

distri bution that generates a desired φ -symmetric power pattern. Distributions have thus been found for optimal pencil beams accompanied by a given number of inner sidelobes of heights almost equal to a given limit [1], and for shaped beams with completely controlled sidelobes [2, 3]. Actually, engineering such aperture distributions is harder: in practice, the desired radiation pattern is usually directly or indirectly approximated by means of an antenna array with element excitations that may or may not be related in some way to the distribution of the corresponding continuous circular aperture.

Often, the array employed is a planar array, with elements laid out on a square or rectangular grid within a circular contour. Simply sampling the corresponding continuous circular aper-ture distribution onto this array affords poor results. It is better to calculate the array excitations that fi x the pattern in certain critical directions [4], or give a least-squares fi t to pattern samples taken in the neighborhood of sidelobe peaks [5], or to feed the naïve samples as starting values to algorithms that minimize deviation from desired pattern quality and/or distri-bution parameters [6]. Another approach is to consider the circular aperture as composed of squares centered on the array elements, and to assign each element the integrated excitation of its corresponding square [7].

David R. Jackson, Chair, Distinguished Lecturer Program, Department of ECE, N308 Engineering Bldg. 1, University of Houston, Houston, TX 77204-4005, USA; Tel.:+1 713-743-4426; Fax: +1 713-743-4444; E-mail: [email protected].

The full abstracts of the Distinguished Lecturers’ talks as well as the biographies of the Distinguished Lecturers appeared in the April 2011 issue of the Magazine, with an update in the April 2012 issue. An updated listing of this mate rial may also be found on the AP-S Web site by vis iting http:// www.ieeeaps.org/distlectureres.html.

Distinguished Lecturers and Their Lectures

“Floquet Modal Based Analysis of Finite and Infi nite Phased Array Antennas,” “Effi cient Shaped Beam Synthesis in Phased Arrays and Refl ectors,” “Advanced Horn structures for Refl ectors and Phased Arrays,” Arun K. Bhattacharyya, Northrop Grum man Corporation, Redondo Beach, CA 90278 USA; E-mail: [email protected].

“Scattering by Load-Modulated Antennas: Background and Sens ing Applications,” “Near Field and Very-Near-Field Techniques: Current Status and Future Trends,” “Overview of Electromagnetic Scattering-Based Imaging Techniques,” Jean-Charles Bolomey, Supelec, Electromagnetic Research Department, Univer sity Paris-Sud XI, France; E-mail: [email protected].

“Computational Electromagnetics in the Frequency Domain,” “Higher-Order Modeling for Computational Electromagnet ics,” Roberto D. Graglia, Politecnico di Torino, Dipartimento di Elet tronica, Corso Duca degli Abruzzi 24, 10129 Torino, Italy; E-mail: [email protected].

“Hierarchical Parallelization of the Multilevel Fast Multipole Algorithm (MLFMA),” “Solution of World’s Largest Integral-Equation Problems,” “Novel and Effective Preconditioners for Iterative Solvers,” Levent Gürel, Dept. of Electrical and Elec tronics Engineering, Bilkent University, TR-06800 Bilkent, Ankara, Turkey; E-mail: [email protected].

“Gap Waveguides and Antennas up to Terahertz,” “OTA-MIMO Measurements of Wireless Stations in Reverberation Chamber,” “Low Noise Decade-Bandwidth Eleven Feed for Future Radio Telescopes,” Per-Simon Kildal, Antenna Group, Depart ment of Signals and Systems, 41296 Gothen-burg, Sweden; E-mail: [email protected].

“CEM Algorithms for EMC/EMI Modeling: Electrically Large (Antenna Placements on Platforms) and Small (SI in ICs and Pack aging) Problems,” “Science and Applications of CEMs and Multi-Physics Computations,” Jin-Fa Lee, Elec-troScience Laboratory, Department of Electrical and Com-puter Engineering, The Ohio State University, 1320 Kinnear Rd., Columbus, OH 43212 USA; E-mail: [email protected].

“Development of the Dielectric Resonator Antenna,” “Trans-parent Antennas: From 2D to 3D,” “Analyses of Spherical Antennas,” Kwok Wa (Ben) Leung, Department of Elec-tronic Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Kowloon, Hong Kong; E-mail:[email protected]

“Wideband and Low-Loss Metamaterials for Microwave and RF: Fast Algorithm and Applications,” “Fast Solvers Based on Hybrid FFT-Integral Equations for Design and Analysis of Large Scaled Systems,” Joshua Le-Wei Li, School of Elec-tronic Engi neering, University of Electronic Science and Technology of China, 2006 Xi-Yuan Avenue, Western High-Tech District, Chengdu, China 611731; E-mail: [email protected].

“Design, Analysis, and Applications of Waveguide-Fed Slot Arrays,” Sembiam R. Rengarajan, Department of Electrical and Computer Engineering, California State Univer sity, North ridge, CA 91330 USA; E-mail: [email protected].

“Physics of Multi-Antenna Systems and their Impacts on Wireless Systems,” “An Overview of Ultra-Wideband Anten-nas,” “Who Is the Father of Electrical Engineering?,” “Solving Challenging Electromagnetic Problems using MoM and a Par-allel Out-Of-Core Solver,” “An Exposition on the Choice of the Proper S-Parameters in Characterizing Devices Including Transmission Lines with Complex Reference Impedances and a General Methodology to Compute them,” Tapan K. Sarkar, Depart ment of Electrical Engi neering and Computer Science, Syra cuse University, 323 Link Hall, Syracuse, NY 13244-1240 USA; E-mail: [email protected].

“Challenges in Practical Design of Planar Arrays,” “An Over view of European Cooperation on Antenna Research,” “Ter minal Antenna Design: Practical Considerations,” Marta Martínez Vázquez, Department of Antennas and EM Mod eling, IMST GmbH, Carl-Friedrich-Gauss-Str. 2-4, 47475 Kamp-Lintfort, Ger many; E-mail: [email protected].

AP-S Distinguished Lecturer Program for 2011-2012

AP_Mag_Jun_2012_Final.indd 135 7/30/2012 6:55:05 PM

136 IEEE Antennas and Propagation Magazine, Vol. 54, No. 3, June 2012

In general, the methods proposed by Elliott [2, 3] and Park [5] both require repeated application, with inspection of the pattern produced at each stage. This is because they only use information for the pattern at or in the neighborhood of critical points identifi ed in terms of pattern features (usually, sidelobe peaks). Here, we extend Park’s method by sampling the pattern systematically throughout the hemisphere. With an adequate sampling density, this removes the need for repeated application.

2. Method

We are given a planar array with isotropic elements laid out on a square grid, within a circular contour of radius a. The array has symmetry by quadrants, with no element at the ori gin. The distance between adjacent elements in any row or column is 2λ , where λ is the radiation wavelength. Figure 1 shows an example of this type of array for 5a λ= .

The fi eld generated by the array is

( ) ( ) ( )t1 1

, 4 cos sin cos cos sin sinmNM

mn mn mnm n

F I kx kyθ φ θ φ θ φ= =

= ∑ ∑

(1)

where k is the wavenumber, M is the number of rows in each quadrant, mN is the number of elements in the mth row of the fi rst quadrant, and mnI , mnx , and mny are respectively the excitation and x and y coordinates of the nth element of the mth row.

We aim to approximate a Taylor fi eld pattern, or a φ-symmetrically shaped or modifi ed Taylor pattern that has been obtained as per Elliott and Stern [2, 3], i.e., by shifting the zeros of the pattern of a uniformly excited circular aper ture. The profi le, ( )dF t , of any such pattern is given by

( )dF t =

( ) ( ) ( )

2 2 2

2 2 21 11

2

21 1

1 1 1

2

1

N N P

n n N nn n n nN N P

n n

t t ttt js t jsJ t

t t

ε

ε

ππ

γ

+

= = +

+ +

=

− − − + −

−

∏ ∏

∏

(2)

where ( )2 sint a λ θ= , 1nπγ is the nth zero of 1J (the Bessel function of the fi rst kind and order 1), N or 2N is the number of nulls of ( )1J t tπ π that are fi lled to create a shaped beam (if the fi eld must be real, 1ε = , this number is twice the num ber N that is required if it need not be, 0ε = ), and P is the number of additional nulls that must be moved to achieve specifi ed sidelobe levels or to make the roll-off steeper.

The fi eld dF is sampled, and the fi eld, tF , generated by the array (Equation (1)) is fi tted to the samples by least-squares optimization of the excitations, mnI .

If ε is zero in Equation (1), and ( )dF t is therefore com-

plex, the power pattern, ( ) ( )d dF t F t∗ , corresponding to the set

( ){ },n nt s , is also the pattern corresponding to any set ( ){ },n nt s′

, where .n ns s′ = ± In general, this means that there are 2N

different values of F ( 12N − if solutions related by a trivial phase symmetry are identifi ed [2]) and the same num ber of solutions { }mnI . From among these, it is possible to choose the one with the best combination of sidelobe level, ripple, and/or dynamic-range ratio, max minI I , where maxI and minI are the excitations of maximum and minimum ampli tude, respectively.

If necessary, the element excitations, iI , can be tweaked as follows, so as to reduce the dynamic-range ratio of the array:

• An excitation-amplitude threshold 0 1 dT DRR< is defi ned, where dDRR is the desired dynamic-range ratio;

• Elements i with excitations iI such that

0i maxI I T< are eliminated; and

• Those of elements i such that 0 1i max f dT I I T DRR≤ < = are set to max fI T , so

ensuring a dynamic-range ratio of dDRR .

Figure 1. One quadrant of a circular square-grid array with 2λ element spacing.

3. Examples

The examples that follow were calculated for the 300-element array, one quadrant of which is shown in Figure 1. In all cases, the starting pattern, dF , was sampled at 1° intervals between 0θ = ° and 90θ = ° in each of 90 φ cuts, taken at 1° intervals between 0φ = ° and 89φ = ° . With the proposed method implemented in MATLAB (R2009b), in no case did it take more than 2.2 s to calculate a solution on a desktop com-puter with an Intel Core i7 processor running at 3.2 GHz.

The patterns obtained were compared with those obtained by simply sampling onto the array the continuous aperture distribution corresponding to dF , which is given by [2, 3]

( ) ( ) ( )( )

1 0 12 2

0 0 1

2 N N Pd m m

n m

F J pg p

J

ε γ γ

π γ π

+ +

== ∑ , (3)

where p was aπ times the distance from the center of the aperture. The comparison was in terms of the defi ning char-acteristics of the starting pattern, dF , the dynamic-range ratio of the array, and the rms error of tP (the power pattern corre-sponding to the fi eld solution tF ) with respect to the power pattern, dP , corresponding to dF . This mean was taken over the entire hemisphere (rms error totRMSE ), over the cone with controlled sidelobes (out to θ corresponding to N N Pt t ε+ += ;

contRMSE ), and, for shaped patterns with fi lled-in nulls, over the shaped zone (out to N Nt t ε+= ; shRMSE ).

3.1 A Circular Taylor Pattern with 25SLL = −dB and 5n =

Figure 2 shows the power pattern, tP , synthesized by the proposed method, starting from a 25− dB, 5n = circular Taylor pattern, dF [1]. The maximum sidelobe level was

25.56− dB, the dynamic-range ratio of the array generating the pattern was 14.47, 5.81totRMSE = dB, and 1.70contRMSE =dB. Simply sampling ( )g p onto the array afforded a power pattern sP with a sidelobe level (SLL) of 24.03− dB, an excitation distribution with a dynamic-range ratio of 2.22,

13.36totRMSE = dB, and 9.53contRMSE = dB. Figure 3 compares the profi les of tP , sP , and dP for 0φ = ° , 15°, 30°, and 45°. tP approximated dP very closely except with regard to the last sidelobe in the direction 15φ = ° , and clearly behaved better than sP everywhere. By tweaking the excitations generating tP as described in the fi nal paragraph of Section 2, with 0 1 14T = and 1 12fT = , a dDRR of 12 was achieved (and two elements per quadrant were lost), at the cost of increasing totRMSE to 6.61 dB and contRMSE to 1.79 dB. The maximum sidelobe level, 25.39− dB, still com plied with specifi cations.

Figure 2. A power pattern approximating a –25 dB, 5n = circular Taylor pattern (Example 1): sin cosu θ φ= ,

sin sinv θ φ= .

3.2 A φ -Symmetric Pencil-Beam Pattern with the Innermost Sidelobe at –35 dB and

the Next Four at –25 dB

In this example (case 1 of [2]), the proposed method marginally failed to achieve its sidelobe-level goals, the maximum sidelobe level of tP being 24.75− dB (Figures 4 and 5). The dynamic-range ratio was 12.99, 5.77totRMSE = dB, and 2.32contRMSE = dB. The sidelobes of the pattern produced by simple sampling, dP , had a higher maximum ( 21.20SLL = −dB) and were much more uneven ( 12.77totRMSE = dB,

9.51contRMSE = dB), but the corre sponding dynamic range ratio was only 5.75. Tweaking the excitations generating tP , with 0 1 12T = and 1 8fT = , elimi nated just one element per quadrant, and lowered the dynamic-range ratio to eight in exchange for small increases in sidelobe level (to –24.57 dB),

totRMSE (to 6.28 dB), and contRMSE (to 2.85 dB).

3.3.1 A φ -Symmetric Flat-Topped Beam with the Three Innermost Sidelobes at

–25 dB: Complex Field

This is case 3 of [2], and as in that paper, we fi lled two nulls to achieve the fl at-topped beam (Figures 6 and 7). Of the two essentially distinct solutions tF , that with the better associ-ated dynamic-range ratio was the one in which the two complex roots had imaginary parts with the same sign. The proposed



In general, the methods proposed by Elliott [2, 3] and Park [5] both require repeated application, with inspection of the pattern produced at each stage. This is because they only use information for the pattern at or in the neighborhood of critical points identifi ed in terms of pattern features (usually, sidelobe peaks). Here, we extend Park’s method by sampling the pattern systematically throughout the hemisphere. With an adequate sampling density, this removes the need for repeated application.

2. Method

We are given a planar array with isotropic elements laid out on a square grid, within a circular contour of radius a. The array has symmetry by quadrants, with no element at the ori gin. The distance between adjacent elements in any row or column is 2λ , where λ is the radiation wavelength. Figure 1 shows an example of this type of array for 5a λ= .

The fi eld generated by the array is

( ) ( ) ( )t1 1

, 4 cos sin cos cos sin sinmNM

mn mn mnm n

F I kx kyθ φ θ φ θ φ= =

= ∑ ∑

(1)

where k is the wavenumber, M is the number of rows in each quadrant, mN is the number of elements in the mth row of the fi rst quadrant, and mnI , mnx , and mny are respectively the excitation and x and y coordinates of the nth element of the mth row.

We aim to approximate a Taylor fi eld pattern, or a φ-symmetrically shaped or modifi ed Taylor pattern that has been obtained as per Elliott and Stern [2, 3], i.e., by shifting the zeros of the pattern of a uniformly excited circular aper ture. The profi le, ( )dF t , of any such pattern is given by

( )dF t =

( ) ( ) ( )

2 2 2

2 2 21 11

2

21 1

1 1 1

2

1

N N P

n n N nn n n nN N P

n n

t t ttt js t jsJ t

t t

ε

ε

ππ

γ

+

= = +

+ +

=

− − − + −

−

∏ ∏

∏

(2)

where ( )2 sint a λ θ= , 1nπγ is the nth zero of 1J (the Bessel function of the fi rst kind and order 1), N or 2N is the number of nulls of ( )1J t tπ π that are fi lled to create a shaped beam (if the fi eld must be real, 1ε = , this number is twice the num ber N that is required if it need not be, 0ε = ), and P is the number of additional nulls that must be moved to achieve specifi ed sidelobe levels or to make the roll-off steeper.

The fi eld dF is sampled, and the fi eld, tF , generated by the array (Equation (1)) is fi tted to the samples by least-squares optimization of the excitations, mnI .

If ε is zero in Equation (1), and ( )dF t is therefore com-

plex, the power pattern, ( ) ( )d dF t F t∗ , corresponding to the set

( ){ },n nt s , is also the pattern corresponding to any set ( ){ },n nt s′

, where .n ns s′ = ± In general, this means that there are 2N

different values of F ( 12N − if solutions related by a trivial phase symmetry are identifi ed [2]) and the same num ber of solutions { }mnI . From among these, it is possible to choose the one with the best combination of sidelobe level, ripple, and/or dynamic-range ratio, max minI I , where maxI and minI are the excitations of maximum and minimum ampli tude, respectively.

If necessary, the element excitations, iI , can be tweaked as follows, so as to reduce the dynamic-range ratio of the array:

• An excitation-amplitude threshold 0 1 dT DRR< is defi ned, where dDRR is the desired dynamic-range ratio;

• Elements i with excitations iI such that

0i maxI I T< are eliminated; and

• Those of elements i such that 0 1i max f dT I I T DRR≤ < = are set to max fI T , so

ensuring a dynamic-range ratio of dDRR .

Figure 1. One quadrant of a circular square-grid array with 2λ element spacing.

3. Examples

The examples that follow were calculated for the 300-element array, one quadrant of which is shown in Figure 1. In all cases, the starting pattern, dF , was sampled at 1° intervals between 0θ = ° and 90θ = ° in each of 90 φ cuts, taken at 1° intervals between 0φ = ° and 89φ = ° . With the proposed method implemented in MATLAB (R2009b), in no case did it take more than 2.2 s to calculate a solution on a desktop com-puter with an Intel Core i7 processor running at 3.2 GHz.

The patterns obtained were compared with those obtained by simply sampling onto the array the continuous aperture distribution corresponding to dF , which is given by [2, 3]

( ) ( ) ( )( )

1 0 12 2

0 0 1

2 N N Pd m m

n m

F J pg p

J

ε γ γ

π γ π

+ +

== ∑ , (3)

where p was aπ times the distance from the center of the aperture. The comparison was in terms of the defi ning char-acteristics of the starting pattern, dF , the dynamic-range ratio of the array, and the rms error of tP (the power pattern corre-sponding to the fi eld solution tF ) with respect to the power pattern, dP , corresponding to dF . This mean was taken over the entire hemisphere (rms error totRMSE ), over the cone with controlled sidelobes (out to θ corresponding to N N Pt t ε+ += ;

contRMSE ), and, for shaped patterns with fi lled-in nulls, over the shaped zone (out to N Nt t ε+= ; shRMSE ).

3.1 A Circular Taylor Pattern with 25SLL = −dB and 5n =

Figure 2 shows the power pattern, tP , synthesized by the proposed method, starting from a 25− dB, 5n = circular Taylor pattern, dF [1]. The maximum sidelobe level was

25.56− dB, the dynamic-range ratio of the array generating the pattern was 14.47, 5.81totRMSE = dB, and 1.70contRMSE =dB. Simply sampling ( )g p onto the array afforded a power pattern sP with a sidelobe level (SLL) of 24.03− dB, an excitation distribution with a dynamic-range ratio of 2.22,

13.36totRMSE = dB, and 9.53contRMSE = dB. Figure 3 compares the profi les of tP , sP , and dP for 0φ = ° , 15°, 30°, and 45°. tP approximated dP very closely except with regard to the last sidelobe in the direction 15φ = ° , and clearly behaved better than sP everywhere. By tweaking the excitations generating tP as described in the fi nal paragraph of Section 2, with 0 1 14T = and 1 12fT = , a dDRR of 12 was achieved (and two elements per quadrant were lost), at the cost of increasing totRMSE to 6.61 dB and contRMSE to 1.79 dB. The maximum sidelobe level, 25.39− dB, still com plied with specifi cations.

Figure 2. A power pattern approximating a –25 dB, 5n = circular Taylor pattern (Example 1): sin cosu θ φ= ,

sin sinv θ φ= .

3.2 A φ -Symmetric Pencil-Beam Pattern with the Innermost Sidelobe at –35 dB and

the Next Four at –25 dB

In this example (case 1 of [2]), the proposed method marginally failed to achieve its sidelobe-level goals, the maximum sidelobe level of tP being 24.75− dB (Figures 4 and 5). The dynamic-range ratio was 12.99, 5.77totRMSE = dB, and 2.32contRMSE = dB. The sidelobes of the pattern produced by simple sampling, dP , had a higher maximum ( 21.20SLL = −dB) and were much more uneven ( 12.77totRMSE = dB,

9.51contRMSE = dB), but the corre sponding dynamic range ratio was only 5.75. Tweaking the excitations generating tP , with 0 1 12T = and 1 8fT = , elimi nated just one element per quadrant, and lowered the dynamic-range ratio to eight in exchange for small increases in sidelobe level (to –24.57 dB),

totRMSE (to 6.28 dB), and contRMSE (to 2.85 dB).


–25 dB: Complex Field

This is case 3 of [2], and as in that paper, we fi lled two nulls to achieve the fl at-topped beam (Figures 6 and 7). Of the two essentially distinct solutions tF , that with the better associ-ated dynamic-range ratio was the one in which the two complex roots had imaginary parts with the same sign. The proposed



Figure 3. tP : Sections of the power pattern of Figure 2 in the directions 0φ = ° (A), 15φ = ° (B), 30φ = ° , (C) and 45φ = ° (D). dP and iP are respectively the corresponding sections of the desired pattern and the pattern obtained by naïve sampling of the continuous aperture distribution onto the array.

A B

C D

Figure 4. The power pattern approximating a φ -symmet ric pencil-beam pattern with the innermost sidelobe at –35 dB and the next four sidelobes at –25 dB (Example 2):

sin cosu θ φ= , sin sinv θ φ= .

A B

C D

Figure 5. tP : Sections of the power pattern of Figure 4 in the directions 0φ = ° (A), 15φ = ° (B), 30φ = ° , (C) and 45φ = ° (D). dP and iP are as in Figure 3, with the neces sary changes made.

Figure 6. The power pattern approximating a φ -symmet ric fl at-topped beam with the three innermost sidelobes at –25 dB (Example 3.1): sin cosu θ φ= , sin sinv θ φ= .



Figure 3. tP : Sections of the power pattern of Figure 2 in the directions 0φ = ° (A), 15φ = ° (B), 30φ = ° , (C) and 45φ = ° (D). dP and iP are respectively the corresponding sections of the desired pattern and the pattern obtained by naïve sampling of the continuous aperture distribution onto the array.

A B

C D

Figure 4. The power pattern approximating a φ -symmet ric pencil-beam pattern with the innermost sidelobe at –35 dB and the next four sidelobes at –25 dB (Example 2):

sin cosu θ φ= , sin sinv θ φ= .

A B

C D





A B

C D



A B

C D




5. Acknowledgement

This work was supported by the Spanish Ministry of Edu-cation and Science under Project TEC2008-04485, and by the Xunta de Galicia under Project 09TIC006206PR.

6. References

1. T. T. Taylor, “Design of Circular Apertures for Narrow Beamwidth and Low Side Lobes,” IRE Transactions on Antennas and Propagation, AP-8, 1960, pp. 17-22.

2. R. S. Elliott and G. J. Stern, “Shaped Patterns from a Con-tinuous Planar Aperture,” IEE Microwaves, Antennas and Propagation Proceedings H, 135, 6, 1988, pp. 366-370.

3. R. S. Elliott and G. J. Stern, “Footprint Patterns Obtained by Planar Arrays,” IEE Microwaves, Antennas and Propagation Proceedings H, 137, 2, 1990, pp. 108-112.

4. R. S. Elliott, Antenna Theory and Design, Revised Edition, New York, John Wiley & Sons, Inc., 2003.

5. P. K. Park, “Discretizing Technique for a Rectangular Grid Array with Arbitrary Shape Aperture,” 1988 IEEE Interna tional Symposium on Antennas and Propagation, 1, Syracuse, NY, 1988, pp. 208-211.

6. F. Ares-Pena, “Application of Genetic Algorithms and Simulated Annealing to Some Antenna Problems,” in Y. Rahmat-Samii and E. Michielssen (eds.), Electromagnetic Optimization by Genetic Algorithms, New York, Wiley and Sons, 1999, Chapter 5, pp. 119-156.

7. R. E. Hodges and Y. Rahmat-Samii, “On Sampling Con-tinuous Aperture Distributions for Discrete Planar Arrays,” IEEE Transactions Antennas and Propagation, AP-44, 11, November 1996, pp. 1499-1508.

method again just failed to achieve its goals: the maximum sidelobe level of tP was –24.81 dB, and the peak-to-trough ripple in the shaped region was 1.15 dB. totRMSE was 5.75 dB,

contRMSE was 1.91 dB, shRMSE was 0.02 dB, and the dynamic-range ratio was 37.78. Tweaking the array excitations to achieve a dynamic-range ratio of 15 with 0 1 25T = eliminated four elements per quadrant, raised the maximum sidelobe level to –23.90 dB, and increased ripple to 1.23 dB,

totRMSE to 6.88 dB, contRMSE to 2.84 dB, and shRMSE to 0.07 dB. Naïve sampling of the continuous aper ture distribution onto the array resulted in a dynamic-range ratio of 5.08 dB and a pattern with 3.01 dB of ripple, a sidelobe level of 20.09− dB, dB, 7.89contRMSE = dB, and 1.13shRMSE = dB.


–25 dB: Real Field

When the pattern specifi cations of the previous example had to be satisfi ed by a real-fi eld pattern [3], both the shaped beam and the surrounding annulus with controlled sidelobes were necessarily broader than with a complex fi eld (Figures 8 and 9). Once more, the proposed method just failed to achieve the goals set, although the greater pattern breadth apparently favored slightly better performance: the maximum sidelobe level of tP was –24.39 dB, and the peak-to-trough ripple in the shaped region was 1.09 dB. totRMSE was 3.85 dB, contRMSE was 2.73 dB, shRMSE was 0.05 dB, and the dynamic-range ratio was 183.81. Tweaking the array excita tions to achieve a dynamic-range ratio of 50 with 0 1 90T = eliminated six elements per quadrant, raised the maximum sidelobe level to –22.06 dB, and increased ripple to 1.34 dB, totRMSE to 5.27 dB, contRMSE to 4.33 dB, and shRMSE to 0.13 dB. Naïve sampling of the continuous aperture distribu tion onto the array resulted in a dynamic-range ratio of 27,000 and a pattern with 3.83 dB of ripple, a sidelobe level of –23.02 dB, 9.87totRMSE =dB, 8.31contRMSE = dB, and 1.86shRMSE = dB. The extremely high dynamic-range ratio arose because the continuous aperture distribution had a zero for this pattern.

4. Conclusions

φ -symmetric radiation patterns, generated by circular apertures, can be quite accurately approximated using circular square-grid arrays. The required element excitations are cal-culated by a least-squares fi t of the pattern generated by the array to a sample of the desired pattern. If the desired pattern is sampled with suffi cient density, there is no need to repeat the synthesis to improve the results.

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