Reflectarray Antenna with Reduced Crosspolar Radiation Pattern
TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS
Transcript of TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS
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TWO NOVEL APPROACHES TWO NOVEL APPROACHES TO ANTENNATO ANTENNA -- PATTERN PATTERN
SYNTHESISSYNTHESIS
Edmund K. Miller Los Alamos National Laboratory (Retired)
597 Rustic Ranch Lane
Lincoln, CA 95648 916-408-0915
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PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
•SOME BACKGOUND
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 3: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/3.jpg)
PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
•SOME BACKGOUND
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 4: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/4.jpg)
PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
•SOME BACKGOUND
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 5: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/5.jpg)
PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
•SOME BACKGOUND
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 6: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/6.jpg)
PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
•SOME BACKGOUND
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 7: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/7.jpg)
PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
•SOME BACKGOUND
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 8: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/8.jpg)
PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
•SOME BACKGOUND
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 9: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/9.jpg)
PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
•SOME BACKGOUND
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 10: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/10.jpg)
PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
•SOME BACKGOUND
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 11: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/11.jpg)
PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
Edmund K. Miller, “Synthesizing Linear-Array Patterns via Matrix Computation of Element Currents, IEEE Antennas and Propagaation Magazine, Vol. 55, No. 5, October 2013, pp. 85-96.
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
Edmund K. Miller, “Using Prony’s Method to Synthesize Discrete Arrays for Prescribed Source Distributions and Exponentiated Patterns, IEEE Antennas and Propagation Society Magazine, Vol. 56, No. 1, February 2015, pp. 147-163.
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PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
Edmund K. Miller, “Synthesizing Linear-Array Patterns via Matrix Computation of Element Currents, IEEE Antennas and Propagaation Magazine, Vol. 55, No. 5, October 2013, pp. 85-96.
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
Edmund K. Miller, “Using Prony’s Method to Synthesize Discrete Arrays for Prescribed Source Distributions and Exponentiated Patterns, IEEE Antennas and Propagation Society Magazine, Vol. 56, No. 1, February 2015, pp. 147-163.
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ANGLE FROM ARRAY AXIS (degrees)
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G-DC20/40FM#1
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ANGLE FROM ARRAY AXIS (degrees)
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ANGLE FROM ARRAY AXIS (degrees)
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ANGLE FROM ARRAY AXIS (degrees)
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ANGLE FROM ARRAY AXIS (degrees)
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G-DC20/40FM#9
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ANGLE FROM ARRAY AXIS (degrees)
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G-DC20/40FM#10
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ANGLE FROM ARRAY AXIS (degrees)
PATT
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G-DC20/40FM#11
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
18013590450-100
-80
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ANGLE FROM ARRAY AXIS (degrees)
PATT
ERN
(dB)
G-DC20/40FM#12
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ANGLE FROM ARRAY AXIS (degrees)
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ANGLE FROM ARRAY AXIS (degrees)
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ANGLE FROM ARRAY AXIS (degrees)
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G-DC20/40FM#19
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ANGLE FROM ARRAY AXIS (degrees)
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ANGLE FROM ARRAY AXIS (degrees)
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G-DC20/40FM#22
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ANGLE FROM ARRAY AXIS (degrees)
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G-DC20/40FM#23
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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ANGLE FROM ARRAY AXIS (degrees)
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G-DC20/40FM#24
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ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED
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Average FMGM PatternGM Samples Used
ANGLE FROM ARRAY AXIS (degrees)
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G-DC20/40FM#25
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ANTENNA PATTERN SYNTHESIS REMAINS A TOPIC OF INTEREST . . . 1. S. A. Schelkunoff, “A Mathematical Theory of Linear Arrays,” Bell System Technical Journal, 22, pp. 80-107, 1943. 2. C. L. Dolph, “A Current Distribution for Broadside Arrays Which Optimizes the Relationship Between Beam Width and Side-lobe Level,” Proceedings of the IRE, 34, 7, pp. 335-348, 1946. 3. P. M. Woodward, “A Method of Calculating the Field Over a Plane Aperture Required to Produce a Given Polar Diagram,” Journal Institute of Electrical Engineering, (London), Pt. III A, 93, pp. 1554-1558, 1946. 4. T. T. Taylor, “Design of Line-Source Antennas for Narrow Beamwidth and Low Sidelobes,” IRE Transactions on Antennas and Propagation, 7, pp. 16-28, 1955. 5. Robert S. Elliott, “On Discretizing Continuous Aperture Distributions,” IEEE Transactions on Antennas and Propagation, AP-25, 5, pp. 617-621, September 1977. 6. Robert S. Elliott, Antenna Theory and Design, Englewood Cliffs, NJ, Prentice-Hall, 1981.
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ANTENNA PATTERN SYNTHESIS REMAINS A TOPIC OF INTEREST . . . 7. Hsien-Peng Chang, T. K. Sarkar, and O. M. C. Pereira-Filho, Antenna pattern synthesis utilizing spherical Bessel functions, IEEE Transactions on Antennas and Propagation, AP-48, 6, pp. 853-859, June 2000. 8. M. Durr, A. Trastoy, and F. Ares, Multiple-pattern linear antenna arrays with single pre-fixed amplitude distributions: modified Woodward-Lawson synthesis, Electronics Letters, 36, 16, pp. 1345-1346, 2000. 9. D. Marcano, and F. Duran, Synthesis of antenna arrays using genetic algorithms, IEEE Antennas and Propagation Magazine, 42, 3, pp. 12-20, June 2000. 10. K. L. Virga, and M. L. Taylor, Transmit patterns for active linear arrays with peak amplitude and radiated voltage distribution constraints, IEEE Transactions on Antennas and Propagation, 49, 5, pp. 732-730, May 2001. 11. O. M. Bucci, M. D'Urso, and T. Isernia, Optimal synthesis of difference patterns subject to arbitrary sidelobe bounds by using arbitrary array antennas, Microwaves, Antennas and Propagation, IEE Proceedings, 152 , 3, pp. 129-137, 2005.
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ANTENNA PATTERN SYNTHESIS REMAINS A TOPIC OF INTEREST . . . 12. R. Vescovo, Consistency of Constraints on Nulls and on Dynamic Range Ratio in Pattern Synthesis for Antenna Arrays, IEEE Transactions on Antennas and Propagation, AP-55, 10, pp. 2662-2670, October 2007. 13. Yanhui Liu, Zaiping Nie, and Qing Huo Liu, Reducing the Number of Elements in a Linear Antenna Array by the Matrix Pencil Method, IEEE Transactions on Antennas and Propagation, 56, 9, pp. 2955-2962, September 2008. 14. N. G. Gomez, J. J. Rodriguez, K. L. Melde, and K. M. McNeill, Design of Low-Sidelobe Linear Arrays With High Aperture Efficiency and Interference Nulls, IEEE Antennas and Wireless Propagation Letters, 8, pp. 607-610, 2009. 15. M. Comisso, and R. Vescovo, Fast Iterative Method of Power Synthesis for Antenna Arrays, IEEE Transactions on Antennas and Propagation, 57, 7, pp. 1952-1962, July 2009. 16. A. M. H. Wong, and G. V. Eleftheriades, G. V., Adaptation of Schelkunoff's Superdirective Antenna Theory for the Realization of Superoscillatory Antenna Arrays, IEEE Antennas and Wireless Propagation Letters, 9, pp. 315-318, 2010.
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ANTENNA PATTERN SYNTHESIS REMAINS A TOPIC OF INTEREST . . . 17. P. S. Apostolov, Linear Equidistant Antenna Array With Improved Selectivity, IEEE Transactions on Antennas and Propagation, 59, 10, pp. 3940-3943, October 2011. 18. J. S. Petko, D. H. Werner, Pareto Optimization of Thinned Planar Arrays With Elliptical Mainbeams and Low Side-lobe Levels, IEEE Transactions on Antennas and Propagation, 59, 5, pp. 1748-1751, May 2011. 19. R. Eirey-Perez, J. A. Rodriguez-Gonzalez, and F. J. Ares-Pena, Synthesis of Array Radiation Pattern Footprints Using Radial Stretching, Fourier Analysis, and Hankel Transformation, IEEE Transactions on Antennas and Propagation, 60, 4, pp. 2106-2109, April 2012. 20. M. Garcia-Vigueras, J. L. Gomez-Tornero, G. Goussetis, A. R. Weily, and Y. J. Guo, Efficient Synthesis of 1-D Fabry-Perot Antennas With Low Sidelobe Levels, IEEE Antennas and Wireless Propagation Letters, 11, pp. 869-872, 2012. 21. Ahmad Safaai-Jazi, “A New Formulation for the Design of Chebyshev Arrays,” IEEE Transactions on Antennas and Propagation, AP-42, 3, pp. 439-443, March 1994.
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PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
•SOME BACKGOUND
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 43: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/43.jpg)
THE APPROACH IS STRAIGHTFORWARD: 1) A LINEAR-ARRAY GEOMETRY IS CHOSEN --TYPICALLY UNIFORM SPACING IS USED, BUT THIS IS NOT MANDATORY
![Page 44: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/44.jpg)
THE APPROACH IS STRAIGHTFORWARD: 1) A LINEAR-ARRAY GEOMETRY IS CHOSEN --TYPICALLY UNIFORM SPACING IS USED, BUT THIS IS NOT MANDATORY 2) AN INITIAL SET OF ELEMENT CURRENTS IS SPECIFIED --IT’S CONVENIENT TO USE UNIT-AMPLITUDE CURRENTS WITH A
UNIFORM PHASE OF ZERO OR A SMALL POSITIVE ANGLE
![Page 45: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/45.jpg)
THE APPROACH IS STRAIGHTFORWARD: 1) A LINEAR-ARRAY GEOMETRY IS CHOSEN --TYPICALLY UNIFORM SPACING IS USED, BUT THIS IS NOT MANDATORY 2) AN INITIAL SET OF ELEMENT CURRENTS IS SPECIFIED --IT’S CONVENIENT TO USE UNIT-AMPLITUDE CURRENTS WITH A
UNIFORM PHASE OF ZERO OR A SMALL POSITIVE ANGLE 3) THE FAR-FIELD PATTERN IS COMPUTED
![Page 46: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/46.jpg)
THE APPROACH IS STRAIGHTFORWARD: 1) A LINEAR-ARRAY GEOMETRY IS CHOSEN --TYPICALLY UNIFORM SPACING IS USED, BUT THIS IS NOT MANDATORY 2) AN INITIAL SET OF ELEMENT CURRENTS IS SPECIFIED --IT’S CONVENIENT TO USE UNIT-AMPLITUDE CURRENTS WITH A
UNIFORM PHASE OF ZERO OR A SMALL POSITIVE ANGLE 3) THE FAR-FIELD PATTERN IS COMPUTED 4) THE ANGLES AT WHICH THE PATTERN MAXIMA OCCUR
ARE LOCATED AND A NEW SET OF ELEMENT CURRENTS IS OBTAINED USING THESE ANGLES AND THE DESIRED VALUES OF THE LOBE MAXIMA
![Page 47: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/47.jpg)
THE APPROACH IS STRAIGHTFORWARD: 1) A LINEAR-ARRAY GEOMETRY IS CHOSEN --TYPICALLY UNIFORM SPACING IS USED, BUT THIS IS NOT MANDATORY 2) AN INITIAL SET OF ELEMENT CURRENTS IS SPECIFIED --IT’S CONVENIENT TO USE UNIT-AMPLITUDE CURRENTS WITH A
UNIFORM PHASE OF ZERO OR A SMALL POSITIVE ANGLE 3) THE FAR-FIELD PATTERN IS COMPUTED 4) THE ANGLES AT WHICH THE PATTERN MAXIMA OCCUR
ARE LOCATED AND A NEW SET OF ELEMENT CURRENTS IS OBTAINED USING THESE ANGLES AND THE DESIRED VALUES OF THE LOBE MAXIMA
5) RETURNING TO 2) THE NEW CURRENTS ARE USED TO
COMPUTE A NEW PATTERN & THE PROCESS CONINUES UNTIL THE PATTERN CONVERGES
![Page 48: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/48.jpg)
EVEN AND ODD NUMBERS OF ELEMENTS WERE USED FOR SYMMETRIC ARRAYS •FOR SYMMETRIC ARRAYS THE PATTERN CAN BE
WRITTEN AS . . .
OR
!
P "( ) = Snn=1
N
# cos 2n $1( )u[ ]
![Page 49: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/49.jpg)
EVEN AND ODD NUMBERS OF ELEMENTS WERE USED FOR SYMMETRIC ARRAYS •FOR SYMMETRIC ARRAYS THE PATTERN CAN BE
WRITTEN AS . . .
OR
!
P "( ) = Snn=1
N
# cos 2n $1( )u[ ]
!
P "( ) = Snn=0
N
# cos 2nu( )
![Page 50: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/50.jpg)
EVEN AND ODD NUMBERS OF ELEMENTS WERE USED FOR SYMMETRIC ARRAYS •FOR SYMMETRIC ARRAYS THE PATTERN CAN BE
WRITTEN AS . . .
OR
FOR AN EVEN OR ODD NUMBER OF ELEMENTS RESPECTIVELY, WHERE
!
P "( ) = Snn=1
N
# cos 2n $1( )u[ ]
!
P "( ) = Snn=0
N
# cos 2nu( )
!
u ="d#
$
% &
'
( ) cos*
+
, -
.
/ 0
![Page 51: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/51.jpg)
LOBE MAXIMA GENERATE A MATRIX . . . 1) The initial pattern P1(!) is sampled finely enough in ! to
accurately locate its positive and negative maxima at the angles !1,n, n = 1,…,N with the corresponding pattern maxima denoted by P1(!1,n).
![Page 52: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/52.jpg)
LOBE MAXIMA GENERATE A MATRIX . . . 1) The initial pattern P1(!) is sampled finely enough in ! to
accurately locate its positive and negative maxima at the angles !1,n, n = 1,…,N with the corresponding pattern maxima denoted by P1(!1,n).
2) A matrix is then developed from the cosines of the angles where
the maxima are found, since these multiply the source currents in Equation (1), to determine the lobe maxima from
![Page 53: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/53.jpg)
LOBE MAXIMA GENERATE A MATRIX . . . 1) The initial pattern P1(!) is sampled finely enough in ! to
accurately locate its positive and negative maxima at the angles !1,n, n = 1,…,N with the corresponding pattern maxima denoted by P1(!1,n).
2) A matrix is then developed from the cosines of the angles where
the maxima are found, since these multiply the source currents in Equation (1), to determine the lobe maxima from
!
M1,N[ ] =
cos u11( ) cos 3u11( ) ! cos 2N "1( )u11[ ]cos u12( ) cos 3u12( ) ! cos 2N "1( )u12[ ]" " # "
cos u1N( ) cos 3u1N( ) ! cos 2N "1( )u1N[ ]
#
$
% % % %
&
'
( ( ( (
![Page 54: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/54.jpg)
. . . WHICH IS THEN INVERTED TO SOLVE FOR A NEW SET OF CURRENTS S1,n FROM . .
. . . WHERE THE Ln ARE THE MAXIMUM VALUES DESIRED FOR THE LOBES OF THE SYNTHESIZED PATTERN
!
S1,1S1, 2!S1,N
"
#
$ $ $ $
=
cos u11( ) cos 3u11( ) " cos 2N %1( )u11[ ]cos u12( ) cos 3u12( ) " cos 2N %1( )u12[ ]! ! # !
cos u1N( ) cos 3u1N( ) " cos 2N %1( )u1N[ ]
&
'
( ( ( (
"
#
$ $ $ $
%1L1L2!LN
"
#
$ $ $ $
![Page 55: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/55.jpg)
. . . WHICH IS THEN INVERTED TO SOLVE FOR A NEW SET OF CURRENTS S1,n FROM . .
. . . WHERE THE Ln ARE THE MAXIMUM VALUES DESIRED FOR THE LOBES OF THE SYNTHESIZED PATTERN
!
S1,1S1, 2!S1,N
"
#
$ $ $ $
=
cos u11( ) cos 3u11( ) " cos 2N %1( )u11[ ]cos u12( ) cos 3u12( ) " cos 2N %1( )u12[ ]! ! # !
cos u1N( ) cos 3u1N( ) " cos 2N %1( )u1N[ ]
&
'
( ( ( (
"
#
$ $ $ $
%1L1L2!LN
"
#
$ $ $ $
![Page 56: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/56.jpg)
A SECOND SET OF PATTERN MAXIMA P2(!2,n) AND MATRIX [M2,N] ARE COMPUTED TO OBTAIN AN UPDATED SET OF CURRENTS . . .
!
S2,1S2,2!S2,N
"
#
$ $ $ $
=
cos u21( ) cos 3u21( ) " cos 2N %1( )u21[ ]cos u22( ) cos 3u22( ) " cos 2N %1( )u22[ ]! ! # !
cos u2N( ) cos 3u2N( ) " cos 2N %1( )u2N[ ]
&
'
( ( ( (
"
#
$ $ $ $
%1L1L2!LN
"
#
$ $ $ $
![Page 57: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/57.jpg)
. . . WHICH RESULTS IN A THIRD SET OF PATTERN MAXIMA P3(!3,n), etc., UNTIL THE PATTERN CONVERGES ACCEPTABLY --ITERATION IS NECESSARY BECAUSE THE ANGLES
AT WHICH MAXIMA OCCUR DEPEND SLIGHTLY ON THE CURRENT
![Page 58: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/58.jpg)
FOR THE MORE GENERAL CASE OF A NON-SYMMETRIC ARRAY THE PATTERN CAN BE WRITTEN . . .
. . . WHICH LEADS TO A CURRENT COMPUTATION OF THE FORM . . . !
P "( ) = Snn=1
N
# expi kxn cos" + $ n( )
![Page 59: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/59.jpg)
FOR THE MORE GENERAL CASE OF A NON-SYMMETRIC ARRAY THE PATTERN CAN BE WRITTEN . . .
. . . WHICH LEADS TO A CURRENT COMPUTATION OF THE FORM . . .
. . . FOR THE i’th ITERATION
!
P "( ) = Snn=1
N
# expi kxn cos" + $ n( )
!
Si,1Si,2!Si,N
"
#
$ $ $ $
=
exp ikx1 cos%i1( ) exp ikx2 cos%i1( ) " exp ikxN cos%i1( )exp ikx1 cos%i2( ) exp ikx2 cos%i2( ) " exp ikxN cos%i2( )
! ! # !exp ikx1 cos%iN( ) exp ikx2 cos%iN( ) " exp ikxN cos%iN( )
&
'
( ( ( (
"
#
$ $ $ $
)1L1L2!LN
"
#
$ $ $ $
![Page 60: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/60.jpg)
SOME ADJUSTMENT MAY BE NEEDED DURING THE ITERATION PROCESS
•IF THE NUMBER OF LOBES CHANGES --INCREASE OR DECREASE THE NUMBER OF ARRAY
ELEMENTS --INCRESE OR DECREASE THE ARRAY LENGTH --ADJUST THE PATTERN SPECIFICATION
![Page 61: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/61.jpg)
SOME ADJUSTMENT MAY BE NEEDED DURING THE ITERATION PROCESS
•IF THE NUMBER OF LOBES CHANGES --INCREASE OR DECREASE THE NUMBER OF ARRAY
ELEMENTS --INCREASE OR DECREASE THE ARRAY LENGTH --ADJUST THE PATTERN SPECIFICATION
![Page 62: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/62.jpg)
SOME ADJUSTMENT MAY BE NEEDED DURING THE ITERATION PROCESS
•IF THE NUMBER OF LOBES CHANGES --INCREASE OR DECREASE THE NUMBER OF ARRAY
ELEMENTS --INCREASE OR DECREASE THE ARRAY LENGTH --ADJUST THE PATTERN SPECIFICATION
![Page 63: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/63.jpg)
SOME ADJUSTMENT MAY BE NEEDED DURING THE ITERATION PROCESS
•IF THE NUMBER OF LOBES CHANGES --INCREASE OR DECREASE THE NUMBER OF ARRAY
ELEMENTS --INCREASE OR DECREASE THE ARRAY LENGTH --ADJUST THE PATTERN SPECIFICATION
•IF THE NEAR END-FIRE LOBES BECOME ILL FORMED
--AS ABOVE
![Page 64: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/64.jpg)
PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
•SOME BACKGOUND
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 65: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/65.jpg)
A SEQUENCE OF PATTERNS THAT CONVERGES TO ONE HAVING -20 dB & -40 dB SIDELOBES ON THE LEFT AND RIGHT ILLUSTRATES THE APPROACH
•15 ELEMENTS, 0.5 WAVELENGTHS APART
![Page 66: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/66.jpg)
A SEQUENCE OF PATTERNS . . .
18013590450-100
-80
-60
-40
-20
0
D-L7N15L20R40dB7PassesUpdateANGLE FROM ARRAY AXIS (degrees)
NO
RM
AL
IZE
D P
AT
TE
RN
(dB
)
•THE INITIAL PATTERN FOR EQUAL SOURCES
![Page 67: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/67.jpg)
A SEQUENCE OF PATTERNS . . .
18013590450-100
-80
-60
-40
-20
0D-L7N15L20R40dB7PassesUpdate
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
AL
IZE
D P
AT
TE
RN
(dB
)
ITERATION #1
![Page 68: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/68.jpg)
A SEQUENCE OF PATTERNS . . .
18013590450-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0D-L7N15L20R40dB7PassesUpdate
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
AL
IZE
D P
AT
TE
RN
(dB
)
ITERATION #2
![Page 69: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/69.jpg)
A SEQUENCE OF PATTERNS . . .
18013590450-100
-80
-60
-40
-20
0D-L7N15L20R40dB7PassesUpdate
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
AL
IZE
D P
AT
TE
RN
(dB
)
ITERATION #3
![Page 70: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/70.jpg)
A SEQUENCE OF PATTERNS . . .
18013590450-100
-80
-60
-40
-20
0D-L7N15L20R40dB7PassesUpdate
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
AL
IZE
D P
AT
TE
RN
(dB
)
ITERATION #4
![Page 71: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/71.jpg)
A SEQUENCE OF PATTERNS . . .
18013590450-100
-80
-60
-40
-20
0D-L7N15L20R40dB7PassesUpdate
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
AL
IZE
D P
AT
TE
RN
(dB
)
ITERATION #5
![Page 72: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/72.jpg)
A SEQUENCE OF PATTERNS . . .
18013590450-100
-80
-60
-40
-20
0
20D-L7N15L20R40dB7PassesUpdate
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
AL
IZE
D P
AT
TE
RN
(dB
)
ITERATION #6 AND THE FINAL PATTERN
![Page 73: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/73.jpg)
THE PATTERN DETERIORATES FOR LOWER FREQUENCIES . . .
18013590450-80
-60
-40
-20
0
20Separation 0.1 wavelengths
D-Left20Right40D0.1to0.8
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B) G-N15L20R40Sep0.1to0.2
![Page 74: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/74.jpg)
THE PATTERN DETERIORATES FOR LOWER FREQUENCIES . . .
18013590450-80
-60
-40
-20
0
200.2
D-Left20Right40D0.1to0.8
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B) G-N15L20R40Sep0.1to0.2
![Page 75: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/75.jpg)
. . . WITH SIDELOBES MAINTAINED OVER A NEARLY 2:1 BANDWIDTH . . .
18013590450-80
-60
-40
-20
0
Separation 0.3 wavelengths
D-Left20Right40D0.1to0.8
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-N15L20R40Sep0.3to0.5
![Page 76: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/76.jpg)
. . . WITH SIDELOBES MAINTAINED OVER A NEARLY 2:1 BANDWIDTH . . .
18013590450-80
-60
-40
-20
0
0.4
D-Left20Right40D0.1to0.8
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-N15L20R40Sep0.3to0.5
![Page 77: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/77.jpg)
. . . WITH SIDELOBES MAINTAINED OVER A NEARLY 2:1 BANDWIDTH . . .
18013590450-80
-60
-40
-20
0
0.5
D-Left20Right40D0.1to0.8
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-N15L20R40Sep0.3to0.5
![Page 78: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/78.jpg)
. . . AND DEVELOPS GRATING LOBES FOR HIGHER FREQUENCIES . . .
18013590450-80
-60
-40
-20
0
20Separation 0.6 wavelengths D-Left20Right40D0.1to0.8
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-N15L20R40Sep0.6to0.8
![Page 79: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/79.jpg)
. . . AND DEVELOPS GRATING LOBES FOR HIGHER FREQUENCIES . . .
18013590450-80
-60
-40
-20
0
200.7 D-Left20Right40D0.1to0.8
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-N15L20R40Sep0.6to0.8
![Page 80: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/80.jpg)
. . . AND DEVELOPS GRATING LOBES FOR HIGHER FREQUENCIES . . .
18013590450-80
-60
-40
-20
0
200.8 D-Left20Right40D0.1to0.8
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-N15L20R40Sep0.6to0.8
![Page 81: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/81.jpg)
A DOLPH-CHEBYSHEV -26 dB PATTERN FOR A 4.5 WAVELENGTH ARRAY IS READILY SYNTHESIZED . . .
18013590450-156
-130
-104
-78
-52
-26
0
D-L4.5N10DC26to104dB ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-L4.5N10DC26to104dB
Edmund K. Miller, “Synthesizing Linear-Array Patterns via Matrix Computation of Element Currents,” accepted for publication, IEEE Antennas and Propagation Society Magazine, 2013.
![Page 82: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/82.jpg)
. . . BUT SUCCESIVELY REDUCING THE SIDE-LOBE LEVEL RESULTS IN A WIDENING MAIN LOBE
18013590450-156
-130
-104
-78
-52
-26
0
D-L4.5N10DC26to104dB ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-L4.5N10DC26to104dB
Edmund K. Miller, “Synthesizing Linear-Array Patterns via Matrix Computation of Element Currents,” accepted for publication, IEEE Antennas and Propagation Society Magazine, 2013.
![Page 83: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/83.jpg)
. . . BUT SUCCESIVELY REDUCING THE SIDE-LOBE LEVEL RESULTS IN A WIDENING MAIN LOBE
18013590450-156
-130
-104
-78
-52
-26
0
D-L4.5N10DC26to104dB ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-L4.5N10DC26to104dB
Edmund K. Miller, “Synthesizing Linear-Array Patterns via Matrix Computation of Element Currents,” accepted for publication, IEEE Antennas and Propagation Society Magazine, 2013.
![Page 84: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/84.jpg)
. . . BUT SUCCESIVELY REDUCING THE SIDE-LOBE LEVEL RESULTS IN A WIDENING MAIN LOBE
18013590450-156
-130
-104
-78
-52
-26
0
D-L4.5N10DC26to104dB ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-L4.5N10DC26to104dB
![Page 85: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/85.jpg)
MAINTAINING A 0.5 SPACING PRODUCES PATTERNS WITH PROPORTIONATELY MORE LOBES & NARROWER MAIN LOBE
18013590450-156
-130
-104
-78
-52
-26
0
D-26,52,etc.dBArraysWithVariousL,NANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-VarL,N,dB26to104
•FOR THE 4.5 WAVELENGTH D-C ARRAY •USING 10, 19, 28, 37 ELEMENTS
![Page 86: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/86.jpg)
MAINTAINING A 0.5 SPACING PRODUCES PATTERNS WITH PROPORTIONATELY MORE LOBES & NARROWER MAIN LOBE
18013590450-156
-130
-104
-78
-52
-26
0
D-26,52,etc.dBArraysWithVariousL,NANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-VarL,N,dB26to104
•FOR THE 4.5 WAVELENGTH D-C ARRAY •USING 10, 19, 28, 37 ELEMENTS
![Page 87: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/87.jpg)
MAINTAINING A 0.5 SPACING PRODUCES PATTERNS WITH PROPORTIONATELY MORE LOBES & NARROWER MAIN LOBE
18013590450-156
-130
-104
-78
-52
-26
0
D-26,52,etc.dBArraysWithVariousL,NANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-VarL,N,dB26to104
•FOR THE 4.5 WAVELENGTH D-C ARRAY •USING 10, 19, 28, 37 ELEMENTS
![Page 88: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/88.jpg)
MAINTAINING A 0.5 SPACING PRODUCES PATTERNS WITH PROPORTIONATELY MORE LOBES & NARROWER MAIN LOBE
18013590450-156
-130
-104
-78
-52
-26
0
D-26,52,etc.dBArraysWithVariousL,NANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-VarL,N,dB26to104
. . . Using 10, 18, 28 and 38 elements and array lengths of 4.5, 8.5, 13.5 and 18.5 wavelengths with 0.5 WL spacing
![Page 89: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/89.jpg)
VARIATIONS ON THE DOLPH-CHEBYSHEV DESIGN ARE EASY TO DEVELOP . . .
•15-ELEMENT ARRAY, 7 WAVELENGTHS LONG
![Page 90: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/90.jpg)
VARIATIONS ON THE DOLPH-CHEBYSHEV DESIGN ARE EASY TO DEVELOP . . . •15-ELEMENT ARRAY, 7 WAVELENGTHS LONG
![Page 91: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/91.jpg)
THIS PATTERN HAS 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB •15-ELEMENT ARRAY, 7 WAVELENGTHS LONG
![Page 92: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/92.jpg)
THE DOLPH-CHEBYSHEV PATTERN DOES NOT REQUIRE UNIFORM SPACING •VARIABLE SPACINGS OF 0.4 AND 0.6 WAVELENGTHS AND 0.4 TO 0.7 WAVELENGTHS RESPECTIVELY
![Page 93: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/93.jpg)
NON-UNIFORM STARTING CURRENTS CAN BE USED The pattern for the -20 dB and -40 dB array when the initial element currents are all zero except for unit-amplitude currents on elements 1 and 15, and the first two iterations.
![Page 94: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/94.jpg)
SOME EXTENSIONS OF THE BASIC IDEA MIGHT INVOLVE SUCH THINGS AS CONTROLLING: °NULLS
![Page 95: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/95.jpg)
SOME EXTENSIONS OF THE BASIC IDEA MIGHT INVOLVE SUCH THINGS AS CONTROLLING: °NULLS °SIDE-LOBE ANGLES
![Page 96: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/96.jpg)
SOME EXTENSIONS OF THE BASIC IDEA MIGHT INVOLVE SUCH THINGS AS CONTROLLING: °NULLS °SIDE-LOBE ANGLES °MAIN LOBE ANGLE
![Page 97: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/97.jpg)
SOME EXTENSIONS OF THE BASIC IDEA MIGHT INVOLVE SUCH THINGS AS CONTROLLING: °NULLS °SIDE-LOBE ANGLES °MAIN LOBE ANGLE °THE NUMBER OF SIDE LOBES
![Page 98: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/98.jpg)
PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 99: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/99.jpg)
PRONY’S METHOD OR ITS EQUIVALENT PROVIDES THE ARRAY PARAMETERS FROM PATTERN SAMPLES
•GIVEN A DESIRED PATTERN Pdesired(!) . . .
!
Pdesired (") # PDSA "( ) = S$ekz$ cos "( )
$ =1
N
%
![Page 100: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/100.jpg)
PRONY’S METHOD OR ITS EQUIVALENT PROVIDES THE ARRAY PARAMETERS FROM PATTERN SAMPLES
•GIVEN A DESIRED PATTERN Pdesired(!) . . .
!
Pdesired (") # PDSA "( ) = S$ekz$ cos "( )
$ =1
N
%
•. . . THE N SOURCE STRENGTHS S! AND N LOCATIONS z! CAN BE OBTAINED
![Page 101: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/101.jpg)
PRONY’S METHOD OR ITS EQUIVALENT PROVIDES THE ARRAY PARAMETERS FROM PATTERN SAMPLES
•GIVEN A DESIRED PATTERN Pdesired(!) . . .
!
Pdesired (") # PDSA "( ) = S$ekz$ cos "( )
$ =1
N
%
•. . . THE N SOURCE STRENGTHS S! AND N LOCATIONS z! CAN BE OBTAINED
•FOR THE ARRAY TO BE REALIZABLE USING ISOTROPIC SOURCES z! MUST BE PURE IMAGINARY
![Page 102: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/102.jpg)
PRONY’S METHOD OR ITS EQUIVALENT PROVIDES THE ARRAY PARAMETERS FROM PATTERN SAMPLES
•GIVEN A DESIRED PATTERN Pdesired(!) . . .
!
Pdesired (") # PDSA "( ) = S$ekz$ cos "( )
$ =1
N
%
•. . . THE N SOURCE STRENGTHS S! AND N LOCATIONS z! CAN BE OBTAINED
•FOR THE ARRAY TO BE REALIZABLE USING ISOTROPIC SOURCES z! MUST BE PURE IMAGINARY
•OTHERWISE A SOURCE DIRECTIVITY WOULD BE REQUIRED AS GIVEN BY
!
D" = ekz" ,real cos #( )
![Page 103: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/103.jpg)
IMPLEMENTING PRONY’S METHOD FOR PATTERN SYNTHESIS INVOLVES CHOOSING 3 PARAMETERS . . .
•THE ANGLE SAMPLING INTERVAL "cos! --MUST BE SMALL ENOUGH TO AVOID ALIASING
![Page 104: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/104.jpg)
IMPLEMENTING PRONY’S METHOD FOR PATTERN SYNTHESIS INVOLVES CHOOSING 3 PARAMETERS . . .
•THE ANGLE SAMPLING INTERVAL "cos! --MUST BE SMALL ENOUGH TO AVOID ALIASING •THE TOTAL ANGLE OBSERVATION WINDOW W MEASURED IN UNITS OF cos! --MUST BE WIDE ENOUGH TO AVOID ILL
CONDITIONING OF THE DATA MATRIX
![Page 105: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/105.jpg)
THE LOBES OF A LINEAR ARRAY ARE SPACED UNIFORMLY IN COS(!)
90450-45-90-30
-20
-10
0
10
20
30D-L20UCFPattAdaptNew
COS(ANGLE)x90 ANGLE
FAR
FIE
LD (d
B)G-L20UCFvsCosang,angle
![Page 106: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/106.jpg)
THE LOBES OF A LINEAR ARRAY ARE SPACED UNIFORMLY IN COS(!)
90450-45-90-30
-20
-10
0
10
20
30D-L20UCFPattAdaptNew
COS(ANGLE)x90 ANGLE
FAR
FIE
LD (d
B)G-L20UCFvsCosang,angle
•THIS SHOWS THAT SAMPLING AS A FUNCTION OF COS(!) RATHER THAN ! IS MORE APPROPRIATE
![Page 107: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/107.jpg)
THE LOBES OF A LINEAR ARRAY ARE SPACED UNIFORMLY IN COS(!)
90450-45-90-30
-20
-10
0
10
20
30D-L20UCFPattAdaptNew
COS(ANGLE)x90 ANGLE
FAR
FIE
LD (d
B)G-L20UCFvsCosang,angle
•THIS SHOWS THAT SAMPLING AS A FUNCTION OF COS(!) RATHER THAN ! IS MORE APPROPRIATE
•BESIDES WHICH PRONY’S METHOD REQUIRES THIS BE DONE IN EQUAL STEPS OF COS(!)
![Page 108: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/108.jpg)
IMPLEMENTING PRONY’S METHOD FOR PATTERN SYNTHESIS INVOLVES CHOOSING 3 PARAMETERS . . .
•THE NUMBER OF POLES OR EXPONENTIALS N --FOR WHICH THE NUMBER OF PATTERN SAMPLES
REQUIRED IS 2N = (W/"cos!) + 1
![Page 109: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/109.jpg)
IMPLEMENTING PRONY’S METHOD FOR PATTERN SYNTHESIS INVOLVES CHOOSING 3 PARAMETERS . . .
•THE NUMBER OF POLES OR EXPONENTIALS N --FOR WHICH THE NUMBER OF PATTERN SAMPLES
REQUIRED IS 2N = (W/"cos!) + 1
. . . WHICH RESULTS IN REQUIRING THAT N BE THE LARGER OF
N ! WL + 1 AND
N ! R
WITH L THE SOURCE SIZE IN WAVELENGTHS AND R THE PATTERN RANK
![Page 110: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/110.jpg)
THE RESULTS THAT FOLLOW WERE GENERALLY OBTAINED USING THE FOLLOWING:
•BEGINNING THE FITTING-MODEL COMPUTATION USING A SLIGHTLY SMALLER VALUE FOR N THAN GIVEN ABOVE
![Page 111: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/111.jpg)
THE RESULTS THAT FOLLOW WERE GENERALLY OBTAINED USING THE FOLLOWING:
•BEGINNING THE FITTING-MODEL COMPUTATION USING A SLIGHTLY SMALLER VALUE FOR N THAN GIVEN ABOVE
•SUCCESSIVELY INCREASING N UNTIL THE FITTING MODEL CONVERGES TO WITHIN 0.1 dB (UNLESS OTHERWISE NOTED) OF THE GENERATING- MODEL PATTERN
![Page 112: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/112.jpg)
THE RESULTS THAT FOLLOW WERE GENERALLY OBTAINED USING THE FOLLOWING:
•BEGINNING THE FITTING-MODEL COMPUTATION USING A SLIGHTLY SMALLER VALUE FOR N THAN GIVEN ABOVE
•SUCCESSIVELY INCREASING N UNTIL THE FITTING MODEL CONVERGES TO WITHIN 0.1 dB (UNLESS OTHERWISE NOTED) OF THE GENERATING- MODEL PATTERN
•SOMETIMES VARYING THE WIDTH OF THE OBSERVATION WINDOW
![Page 113: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/113.jpg)
THE RESULTS THAT FOLLOW WERE GENERALLY OBTAINED USING THE FOLLOWING:
•BEGINNING THE FITTING-MODEL COMPUTATION USING A SLIGHTLY SMALLER VALUE FOR N THAN GIVEN ABOVE
•SUCCESSIVELY INCREASING N UNTIL THE FITTING MODEL CONVERGES TO WITHIN 0.1 dB (UNLESS OTHERWISE NOTED) OF THE GENERATING- MODEL PATTERN
•SOMETIMES VARYING THE WIDTH OF THE OBSERVATION WINDOW
•ROUTINELY COMPUTING THE SVD SPECTRUM OF THE DESIRED PATTERN
![Page 114: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/114.jpg)
THE RESULTS THAT FOLLOW WERE GENERALLY OBTAINED USING THE FOLLOWING:
•BEGINNING THE FITTING-MODEL COMPUTATION USING A SLIGHTLY SMALLER VALUE FOR N THAN GIVEN ABOVE
•SUCCESSIVELY INCREASING N UNTIL THE FITTING MODEL CONVERGES TO WITHIN 0.1 dB (UNLESS OTHERWISE NOTED) OF THE GENERATING- MODEL PATTERN
•SOMETIMES VARYING THE WIDTH OF THE OBSERVATION WINDOW
•ROUTINELY COMPUTING THE SVD SPECTRUM OF THE DESIRED PATTERN
•USING A COMPUTE PRECISION OF 24 DIGITS
![Page 115: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/115.jpg)
PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 116: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/116.jpg)
A USEFUL INITIAL TEST IS A MODIFIED PATTERN OF A SINUSOIDAL CURRENT FILAMENT
•ITS FAR-FIELD PATTERN IS GIVEN BY
!
PMSCF (") = sin" # PSCF (") = sin" e( ikL / 2)cos" + e$(ikL / 2)cos" $ 2cos(kL /2)sin"
%
& '
(
) *
![Page 117: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/117.jpg)
A USEFUL INITIAL TEST IS A MODIFIED PATTERN OF A SINUSOIDAL CURRENT FILAMENT
•ITS FAR-FIELD PATTERN IS GIVEN BY
!
PMSCF (") = sin" # PSCF (") = sin" e( ikL / 2)cos" + e$(ikL / 2)cos" $ 2cos(kL /2)sin"
%
& '
(
) *
•PMSCF(!) IS SEEN TO BE THE SUM OF THREE POINT SOURCES
![Page 118: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/118.jpg)
A USEFUL INITIAL TEST IS A MODIFIED PATTERN OF A SINUSOIDAL CURRENT FILAMENT
•ITS FAR-FIELD PATTERN IS GIVEN BY
!
PMSCF (") = sin" # PSCF (") = sin" e( ikL / 2)cos" + e$(ikL / 2)cos" $ 2cos(kL /2)sin"
%
& '
(
) *
•PMSCF(!) IS SEEN TO BE THE SUM OF THREE POINT SOURCES
•THE FIRST TWO TERMS ARE DUE TO THE ENDS OF THE FILAMENT
![Page 119: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/119.jpg)
A USEFUL INITIAL TEST IS A MODIFIED PATTERN OF A SINUSOIDAL CURRENT FILAMENT*
•ITS FAR-FIELD PATTERN IS GIVEN BY
!
PMSCF (") = sin" # PSCF (") = sin" e( ikL / 2)cos" + e$(ikL / 2)cos" $ 2cos(kL /2)sin"
%
& '
(
) *
•PMSCF(!) IS SEEN TO BE THE SUM OF THREE POINT SOURCES
•THE FIRST TWO TERMS ARE DUE TO THE ENDS OF THE FILAMENT
•THE LAST IS A LENGTH-DEPENDENT CONTRIBUTION DUE TO A CURRENT-SLOPE DISCONTINUITY AT THE CENTER
*E. K. Miller, “The Incremental Far Field and Degrees of Freedom of the Sinusoidal Current Filament,”IEEE AP-S Magazine, 49 (4), August 2007.
![Page 120: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/120.jpg)
TWO DIFFERENT WINDOW WIDTHS PRODUCE ESSENTIALLY IDENTICAL PATTERN MATCHES
18013590450-100
-80
-60
-40
-20
0
GM -0.05to0.05FM -0.05to0.05GM -0.999to0.999FM -0.999to0.999
D-L5SCFxSINVarCOSANG
ANGLE FROM CURRENT AXIS (degrees)
NO
RM
AL
IZE
D P
AT
TE
RN
(dB
)
G-L5DelCos0.1&2x0.999Patts
![Page 121: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/121.jpg)
TWO DIFFERENT WINDOW WIDTHS PRODUCE ESSENTIALLY IDENTICAL PATTERN MATCHES
18013590450-100
-80
-60
-40
-20
0
GM -0.05to0.05FM -0.05to0.05GM -0.999to0.999FM -0.999to0.999
D-L5SCFxSINVarCOSANG
ANGLE FROM CURRENT AXIS (degrees)
NO
RM
AL
IZE
D P
AT
TE
RN
(dB
)
G-L5DelCos0.1&2x0.999Patts
•WINDOWS OF -0.999 TO + 0.999 AND -0.05 TO + 0.05 IN
!
cos" WERE USED
![Page 122: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/122.jpg)
TWO DIFFERENT WINDOW WIDTHS PRODUCE ESSENTIALLY IDENTICAL PATTERN MATCHES
18013590450-100
-80
-60
-40
-20
0
GM -0.05to0.05FM -0.05to0.05GM -0.999to0.999FM -0.999to0.999
D-L5SCFxSINVarCOSANG
ANGLE FROM CURRENT AXIS (degrees)
NO
RM
AL
IZE
D P
AT
TE
RN
(dB
)
G-L5DelCos0.1&2x0.999Patts
•WINDOWS OF -0.999 TO + 0.999 AND -0.05 TO + 0.05 IN
!
cos" WERE USED •TWO ARROWS INDICATE THE EXTENT OF THE
LATTER
![Page 123: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/123.jpg)
TWO DIFFERENT WINDOW WIDTHS PRODUCE ESSENTIALLY IDENTICAL PATTERN MATCHES
18013590450-100
-80
-60
-40
-20
0
GM -0.05to0.05FM -0.05to0.05GM -0.999to0.999FM -0.999to0.999
D-L5SCFxSINVarCOSANG
ANGLE FROM CURRENT AXIS (degrees)
NO
RM
AL
IZE
D P
AT
TE
RN
(dB
)
G-L5DelCos0.1&2x0.999Patts
•WINDOWS OF -0.999 TO + 0.999 AND -0.05 TO + 0.05 IN
!
cos" WERE USED •TWO ARROWS INDICATE THE EXTENT OF THE
LATTER •LENGTH OF SCF IS 5 WAVELENGTHS
![Page 124: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/124.jpg)
SINGULAR-VALUE SPECTRA FOR SEVERAL WINDOW WIDTHS DEMONSTRATE A PATTERN RANK OF 3 FOR PMSCF . . .
1086420010-2410 -2310 -2210 -2110 -2010 -1910 -1810 -1710 -1610 -1510 -1410 -1310 -1210 -1110 -1010 -910 -810 -710 -610 -510 -410 -310 -210 -1100101102
SINGULAR VALUES
SIN
GU
LA
R-V
AL
UE
SPE
CT
RA
G-L5CosangVarSingValuesw/0IDs
![Page 125: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/125.jpg)
SINGULAR-VALUE SPECTRA FOR SEVERAL WINDOW WIDTHS DEMONSTRATE A PATTERN RANK OF 3 FOR PMSCF . . .
1086420010-2410 -2310 -2210 -2110 -2010 -1910 -1810 -1710 -1610 -1510 -1410 -1310 -1210 -1110 -1010 -910 -810 -710 -610 -510 -410 -310 -210 -1100101102
SINGULAR VALUES
SIN
GU
LA
R-V
AL
UE
SPE
CT
RA
G-L5CosangVarSingValuesw/0IDs
•N WAS INCREASED FOR EACH WINDOW UNTIL THE PATTERN CONVERGED
![Page 126: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/126.jpg)
SINGULAR-VALUE SPECTRA FOR SEVERAL WINDOW WIDTHS DEMONSTRATE A PATTERN RANK OF 3 FOR PMSCF . . .
1086420010-2410 -2310 -2210 -2110 -2010 -1910 -1810 -1710 -1610 -1510 -1410 -1310 -1210 -1110 -1010 -910 -810 -710 -610 -510 -410 -310 -210 -1100101102
SINGULAR VALUES
SIN
GU
LA
R-V
AL
UE
SPE
CT
RA
G-L5CosangVarSingValuesw/0IDs
•N WAS INCREASED FOR EACH WINDOW UNTIL THE PATTERN CONVERGED
•RESULT IS CONSISTENT WITH 3 POINT SOURCES
![Page 127: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/127.jpg)
. . . AS IS REVEALED BY A PLOT OF THE PRONY-DERIVED SOURCES
(a) (b)
•SOURCE STRENGTHS ARE PLOTTED AS ARROWS ON 3-DECADE LOGARITHMIC SCALE
![Page 128: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/128.jpg)
. . . AS IS REVEALED BY A PLOT OF THE PRONY-DERIVED SOURCES
(a) (b)
•SOURCE STRENGTHS ARE PLOTTED AS ARROWS ON 3-DECADE LOGARITHMIC SCALE
•PHASE IS SHOWN ON A POLAR PLOT
![Page 129: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/129.jpg)
. . . AS IS REVEALED BY A PLOT OF THE PRONY-DERIVED SOURCES
(a) (b)
•SOURCE STRENGTHS ARE PLOTTED AS ARROWS ON 3-DECADE LOGARITHMIC SCALE
•PHASE IS SHOWN ON A POLAR PLOT •THE X’s DENOTE THE PHYSICAL SCF EXTENT
![Page 130: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/130.jpg)
THE NUMBER OF FITTING MODELS NEEDED FOR A CONVERGED PATTERN INCREASES SYSTEMATICALLY WITH WINDOW WIDTH
2.01.51.00.50.02
4
6
8
10
12
WIDTH OF OBSERVATION WINDOW
NU
MBE
R O
F FI
TTIN
GS
MO
DEL
S
G-L5FMsVsCosangw/oIDs
•TO AVOID ALIASING
![Page 131: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/131.jpg)
THE CENTER SOURCE DISAPPEARS FOR A MODIFIED SCF 5.5 WAVELENGTHS LONG
•SAMPLED OVER A -0.05 TO +0.05 COS! WINDOW
![Page 132: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/132.jpg)
THE ACTUAL PATTERN OF A 5-WAVELENGTH SCF IS MATCHED DOWN TO -60 dB BY AN 11-TERM FITTING MODEL . . .
18013590450-100
-90-80
-70-60
-50
-40-30
-20-10
0
GENERATING MODELFITTING MODELSAMPLES USED FOR FITTING MODEL
D-PronyN11L5SCFPattern
ANGLE FROM CURRENT AXIS (degrees)
NO
RM
AL
IZE
D P
AT
TE
RN
(dB
)
G-PronyN11L5SCFPatterm
•THE BLACK DOTS DENOTE THE GENERATING- MODEL SAMPLES USED TO COMPUTE THE 11- POLE FITTING MODEL
![Page 133: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/133.jpg)
. . . BUT THE DERIVED SOURCE DISTRIBUTION IS NOT PHYSICALLY REALIZABLE . . .
•. . . BECAUSE SOME OF SCF 9 SOURCES HAVE REAL COMPONENTS IN THE COMPLEX SPACE PLANE
![Page 134: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/134.jpg)
PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 135: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/135.jpg)
PRONY SYNTHESIS WAS FIRST TESTED FOR THIS PATTERN
•SYNTHESIZED USING 15 ELEMENTS (BY R. S ELLIOTT, “On Discretizing Continuous Aperture Distributions,” IEEE, AP-S Trans., AP-25 (5) September 1977). •PRONY APPROACH REQUIRED 12 ELEMENTS.
![Page 136: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/136.jpg)
THE PATTERN OF A ±1 SQUARE-WAVE APERTURE IS GRAPHICALLY INDISTING- UISHABLE FROM AN 11-TERM FM . . .
180135904 50-100
-80
-60
-40
-20
0
20 GENERATING MODELFITTING MODELSAMPLES USED FOR FITTING MODEL
D-L5N11±UCFPronyPattP1D24
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-L5N11±UCFPronyPattP1D24
•THE PATTERN FACTOR IS
!
P± = L1" cos #L
$cos%&
' ( )
#L$cos%
&
'
* * *
(
)
+ + +
![Page 137: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/137.jpg)
. . . WHOSE SYNTHESIZED SOURCES ARE NOT UNIFORMLY SPACED
•FOR A 5-WAVELENGTH APERTURE
![Page 138: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/138.jpg)
. . . WHOSE SYNTHESIZED SOURCES ARE NOT UNIFORMLY SPACED
•FOR A 5-WAVELENGTH APERTURE •AND A 11-POLE FITTING MODEL
![Page 139: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/139.jpg)
THE PATTERN OF AN APERTURE VARYING AS cos2("/L) IS ALSO GRAPHICALLY IDENTICAL TO ITS PRONY FM . . .
18013590450-100
-80
-60
-40
-20
0
GENERATING MODELFITTING MODELSAMPLES USED FOR FITTING MODEL
D-PronySynL5Cos^2N11
ANGLE FROM CURRENT AXIS (degrees)
NO
RM
AL
IZE
D P
AT
TE
RN
(dB
) G-PronySynL5Cos^2N11
•ITS PATTERN FACTOR IS GIVEN BY
!
Pcos2 =sin(u)u
" 2
" 2 # u2$
% &
'
( )
WHERE
!
u = "L#( )sin$
![Page 140: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/140.jpg)
… WHOSE SOURCE DISTRIBUTION IS ALSO NONUNIFORM
•FOR A 5-WAVELENTH APERTURE
![Page 141: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/141.jpg)
… WHOSE SOURCE DISTRIBUTION IS ALSO NONUNIFORM
•FOR A 5-WAVELENTH APERTURE •USING AN 11-TERM FITTING MODEL
![Page 142: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/142.jpg)
A CURRENT FILAMENT OF LENGTH L VARYING AS (sin!)P HAS TAPERED SIDELOBES WITH INCREASING P . . .
18013590450-60
-40
-20
0
G-L5UCFxSIN^XNVarActualPoles
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
D-L5UCFxSIN^XNVarActualPoles
P = 0
•ITS PATTERN FACTOR IS
!
PUCF = (sin")P sin(kLcos")kLcos"
![Page 143: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/143.jpg)
A CURRENT FILAMENT OF LENGTH L VARYING AS (sin!)P HAS TAPERED SIDELOBES WITH INCREASING P . . .
1801359 04 50-60
-40
-20
0
G-L5UCFxSIN^XNVarActualPoles
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
D-L5UCFxSIN^XNVarActualPoles
P = 0
1
•ITS PATTERN FACTOR IS
!
PUCF = (sin")P sin(kLcos")kLcos"
![Page 144: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/144.jpg)
A CURRENT FILAMENT OF LENGTH L VARYING AS (sin!)P HAS TAPERED SIDELOBES WITH INCREASING P . . .
1801359 04 50-60
-40
-20
0
G-L5UCFxSIN^XNVarActualPoles
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
D-L5UCFxSIN^XNVarActualPoles
P = 0
12
•ITS PATTERN FACTOR IS
!
PUCF = (sin")P sin(kLcos")kLcos"
![Page 145: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/145.jpg)
A CURRENT FILAMENT OF LENGTH L VARYING AS (sin!)P HAS TAPERED SIDELOBES WITH INCREASING P . . .
1801359 04 50-60
-40
-20
0
G-L5UCFxSIN^XNVarActualPoles
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
D-L5UCFxSIN^XNVarActualPoles
P = 0
123
•ITS PATTERN FACTOR IS
!
PUCF = (sin")P sin(kLcos")kLcos"
![Page 146: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/146.jpg)
. . . ALSO HAS SOURCES THAT ARE NON-UNIFORMLY SPACED
. . . USING 11 EXPONENTIALS IN THE FITTING MODEL
![Page 147: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/147.jpg)
. . . ALSO HAS SOURCES THAT ARE NON-UNIFORMLY SPACED
. . . USING 11 EXPONENTIALS IN THE FITTING MODEL •AND FOR A 5-WAVELENGTH APERTURE
![Page 148: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/148.jpg)
A DOLPH-CHEBYSHEV ARRAY IS READILY SYNTHESIZED
18013590450-78
-52
-26
0
Generating ModelSamples Used for FItting ModelFitting Model
D-DC^1VarL4.5...&N10...dB26...
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
AL
IZE
D P
AT
TE
RN
(dB
)
G-DC^1tL4.5dB26w/oPts
•5-WAVELENGTHS LONG WITH -26 dB SIDELOBES AND 10 ELEMENTS
![Page 149: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/149.jpg)
A MODIFIED DOLPH-CHEBYSHEV ARRAY
18013590450-100
-80
-60
-40
-20
0
Generating ModelFitting ModelGM Samples Used for FM
D-L7N15-20&-40dBDCPronySyn
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-L7N15-20&-40dBDCPronySyn
•-20 AND -40 dB SIDELOBES
![Page 150: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/150.jpg)
A MODIFIED DOLPH-CHEBYSHEV ARRAY
18013590450-100
-80
-60
-40
-20
0
Generating ModelFitting ModelGM Samples Used for FM
D-L7N15-20&-40dBDCPronySyn
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-L7N15-20&-40dBDCPronySyn
•-20 AND -40 dB SIDELOBES •15 ELEMENTS UNIFORMLY SPACED
![Page 151: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/151.jpg)
A MODIFIED DOLPH-CHEBYSHEV ARRAY
18013590450-100
-80
-60
-40
-20
0
Generating ModelFitting ModelGM Samples Used for FM
D-L7N15-20&-40dBDCPronySyn
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-L7N15-20&-40dBDCPronySyn
•-20 AND -40 dB SIDELOBES •15 ELEMENTS UNIFORMLY SPACED •7 WAVELENGTHS LONG
![Page 152: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/152.jpg)
THIS ARRAY STEPS UP IN 5 dB INCREMENTS FROM LEFT TO RIGHT
18013590450-100
-80
-60
-40
-20
0
Generating ModelFitting ModelGM Samples Used for FM
D-L7N15-70to0dBPronySyn
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-L7N15-70to0dBPronySyn
•15 ELEMENTS UNIFORMLY SPACED
![Page 153: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/153.jpg)
THIS ARRAY STEPS UP IN 5 dB INCREMENTS FROM LEFT TO RIGHT
18013590450-100
-80
-60
-40
-20
0
Generating ModelFitting ModelGM Samples Used for FM
D-L7N15-70to0dBPronySyn
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-L7N15-70to0dBPronySyn
•15 ELEMENTS UNIFORMLY SPACED •7 WAVELENGTHS LONG
![Page 154: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/154.jpg)
SINGULAR-VALUE SPECTRA FOR SEVERAL ARRAYS ILLUSTRATE THEIR DIFFERENCES
15131 19753110 -2510 -2410 -2310 -2210 -2110 -2010 -1910 -1810 -1710 -1610 -1510 -1410 -1310 -1210 -1110 -1010 - 910 - 810 - 710 - 610 - 510 - 410 - 310 - 210 - 110 0
7.5 Wavelength 10-Element DC Array
D-SVsVariousSources SINGULAR-VALUE ORDER
SIN
GU
LAR
VA
LUES
G-PronySynVarSrcsSVD
•THE DOLPH-CHEBYSHEV ARRAY CLEARLY SHOWS THE NUMBER OF ELEMENTS IT CONTAINS
![Page 155: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/155.jpg)
SINGULAR-VALUE SPECTRA FOR SEVERAL ARRAYS ILLUSTRATE THEIR DIFFERENCES
15131 19753110 -2510 -2410 -2310 -2210 -2110 -2010 -1910 -1810 -1710 -1610 -1510 -1410 -1310 -1210 -1110 -1010 - 910 - 810 - 710 - 610 - 510 - 410 - 310 - 210 - 110 0
7.5 Wavelength 10-Element DC Array5-Wavelength Sinusoid
D-SVsVariousSources SINGULAR-VALUE ORDER
SIN
GU
LAR
VA
LUES
G-PronySynVarSrcsSVD
•THE DOLPH-CHEBYSHEV ARRAY CLEARLY SHOWS THE NUMBER OF ELEMENTS IT CONTAINS
•THE SPECTRA OF THE CONTINUOUS DISTRIBUTIONS FALL OFF SMOOTHLY
![Page 156: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/156.jpg)
SINGULAR-VALUE SPECTRA FOR SEVERAL ARRAYS ILLUSTRATE THEIR DIFFERENCES
15131 19753110 -2510 -2410 -2310 -2210 -2110 -2010 -1910 -1810 -1710 -1610 -1510 -1410 -1310 -1210 -1110 -1010 - 910 - 810 - 710 - 610 - 510 - 410 - 310 - 210 - 110 0
7.5 Wavelength 10-Element DC Array5-Wavelength Sinusoid5-Wavelength COS^2 Aperture
D-SVsVariousSources SINGULAR-VALUE ORDER
SIN
GU
LAR
VA
LUES
G-PronySynVarSrcsSVD •THE DOLPH-CHEBYSHEV ARRAY CLEARLY SHOWS
THE NUMBER OF ELEMENTS IT CONTAINS •THE SPECTRA OF THE CONTINUOUS DISTRIBUTIONS FALL OFF
SMOOTHLY
![Page 157: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/157.jpg)
SINGULAR-VALUE SPECTRA FOR SEVERAL ARRAYS ILLUSTRATE THEIR DIFFERENCES
15131 19753110 -2510 -2410 -2310 -2210 -2110 -2010 -1910 -1810 -1710 -1610 -1510 -1410 -1310 -1210 -1110 -1010 - 910 - 810 - 710 - 610 - 510 - 410 - 310 - 210 - 110 0
7.5 Wavelength 10-Element DC Array5-Wavelength Sinusoid5-Wavelength COS^2 Aperture5-Wavelength Uniform Current Filament
D-SVsVariousSources SINGULAR-VALUE ORDER
SIN
GU
LAR
VA
LUES
G-PronySynVarSrcsSVD
•THE DOLPH-CHEBYSHEV ARRAY CLEARLY SHOWS THE NUMBER OF ELEMENTS IT CONTAINS
•THE SPECTRA OF THE CONTINUOUS DISTRIBUTIONS FALL OFF SMOOTHLY
![Page 158: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/158.jpg)
PRESENTATION DESCRIBES AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 159: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/159.jpg)
CONSIDER EXPONENTIATING A PATTERN: •FOR EXAMPLE THE PATTERN OF A 10-ELEMENT
DOLPH-CHEBYSHEV ARRAY AS GIVEN BY . . .
WHERE WITH d THE ELEMENT SPACING
. . . WHICH YIELDS SUCCESSIVELY LOWER SIDELOBES
•ARRAY LENGTH IS 4.5 WAVELENGTHS
![Page 160: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/160.jpg)
CONSIDER EXPONENTIATING A PATTERN: •FOR EXAMPLE THE PATTERN OF A 10-ELEMENT
DOLPH-CHEBYSHEV ARRAY AS GIVEN BY . . .
P10M (! ) = [2.798cos(D)+ 2.496cos(3D)+1.974cos(5D)+1.357cos(7D)+ cos(9D)]M
WHERE
!
D = "d /#( )cos$[ ] WITH d THE ELEMENT SPACING
. . . WHICH YIELDS SUCCESSIVELY LOWER SIDELOBES
•ARRAY LENGTH IS 4.5 WAVELENGTHS
![Page 161: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/161.jpg)
CONSIDER EXPONENTIATING A PATTERN: •FOR EXAMPLE THE PATTERN OF A 10-ELEMENT
DOLPH-CHEBYSHEV ARRAY AS GIVEN BY . . .
P10M (! ) = [2.798cos(D)+ 2.496cos(3D)+1.974cos(5D)+1.357cos(7D)+ cos(9D)]M
WHERE
!
D = "d /#( )cos$[ ] WITH d THE ELEMENT SPACING
. . . WHICH YIELDS SUCCESSIVELY LOWER SIDELOBES
1801359 0450-156
-130
-104
-78
-52
-26
0
D-DC^1VarL4.5...&N10...dB26...ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-DC^1to4L4.5to18.5
Exponent M = 1
![Page 162: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/162.jpg)
CONSIDER EXPONENTIATING A PATTERN: •FOR EXAMPLE THE PATTERN OF A 10-ELEMENT
DOLPH-CHEBYSHEV ARRAY AS GIVEN BY . . .
P10M (! ) = [2.798cos(D)+ 2.496cos(3D)+1.974cos(5D)+1.357cos(7D)+ cos(9D)]M
WHERE
!
D = "d /#( )cos$[ ] WITH d THE ELEMENT SPACING
. . . WHICH YIELDS SUCCESSIVELY LOWER SIDELOBES
1801359 0450-156
-130
-104
-78
-52
-26
0
D-DC^1VarL4.5...&N10...dB26...ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-DC^1to4L4.5to18.5
Exponent M = 1
2
![Page 163: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/163.jpg)
CONSIDER EXPONENTIATING A PATTERN: •FOR EXAMPLE THE PATTERN OF A 10-ELEMENT
DOLPH-CHEBYSHEV ARRAY AS GIVEN BY . . .
P10M (! ) = [2.798cos(D)+ 2.496cos(3D)+1.974cos(5D)+1.357cos(7D)+ cos(9D)]M
WHERE
!
D = "d /#( )cos$[ ] WITH d THE ELEMENT SPACING
. . . WHICH YIELDS SUCCESSIVELY LOWER SIDELOBES
1801359 0450-156
-130
-104
-78
-52
-26
0
D-DC^1VarL4.5...&N10...dB26...ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-DC^1to4L4.5to18.5
Exponent M = 1
2
3
![Page 164: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/164.jpg)
CONSIDER EXPONENTIATING A PATTERN: •FOR EXAMPLE THE PATTERN OF A 10-ELEMENT
DOLPH-CHEBYSHEV ARRAY AS GIVEN BY . . .
P10M (! ) = [2.798cos(D)+ 2.496cos(3D)+1.974cos(5D)+1.357cos(7D)+ cos(9D)]M
WHERE
!
D = "d /#( )cos$[ ] WITH d THE ELEMENT SPACING
. . . WHICH YIELDS SUCCESSIVELY LOWER SIDELOBES
1801359 0450-156
-130
-104
-78
-52
-26
0
D-DC^1VarL4.5...&N10...dB26...ANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-DC^1to4L4.5to18.5
Exponent M = 1
2
3
4
•INITIAL PATTERN FOR 4.5 WAVELENGTH ARRAY
![Page 165: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/165.jpg)
SYNTHESIZING THE EXPONENTIATED -26 dB D-C PATTERN USING PRONY’S METHOD PROVIDES A 0.1 Db OR BETTER MATCH
18013590450-130
-104
-78
-52
-26
0
D-PronyNewdB-26N10PVarDVarANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-PronyNewdB-26N10PVarDVar
EXPONENT M = 1
•AT LEVELS ~ ! -110 dB •USING 10, 19, 28, & 37 ELEMENTS •SOLID LINES FOR PRONY FM, DOTS FOR GM
![Page 166: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/166.jpg)
SYNTHESIZING THE EXPONENTIATED -26 dB D-C PATTERN USING PRONY’S METHOD PROVIDES A 0.1 Db OR BETTER MATCH
18013590450-130
-104
-78
-52
-26
0
D-PronyNewdB-26N10PVarDVarANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-PronyNewdB-26N10PVarDVar
EXPONENT M = 1
•SOLID LINES FOR PRONY FM, DOTS FOR GM
![Page 167: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/167.jpg)
SYNTHESIZING THE EXPONENTIATED -26 dB D-C PATTERN USING PRONY’S METHOD PROVIDES A 0.1 Db OR BETTER MATCH
18013590450-130
-104
-78
-52
-26
0
D-PronyNewdB-26N10PVarDVarANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-PronyNewdB-26N10PVarDVar
EXPONENT M = 1
2
![Page 168: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/168.jpg)
SYNTHESIZING THE EXPONENTIATED -26 dB D-C PATTERN USING PRONY’S METHOD PROVIDES A 0.1 Db OR BETTER MATCH
18013590450-130
-104
-78
-52
-26
0
D-PronyNewdB-26N10PVarDVarANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-PronyNewdB-26N10PVarDVar
EXPONENT M = 1
2
•SOLID LINES FOR PRONY FM, DOTS FOR GM
![Page 169: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/169.jpg)
SYNTHESIZING THE EXPONENTIATED -26 dB D-C PATTERN USING PRONY’S METHOD PROVIDES A 0.1 Db OR BETTER MATCH
18013590450-130
-104
-78
-52
-26
0
D-PronyNewdB-26N10PVarDVarANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-PronyNewdB-26N10PVarDVar
EXPONENT M = 1
2
3
![Page 170: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/170.jpg)
SYNTHESIZING THE EXPONENTIATED -26 dB D-C PATTERN USING PRONY’S METHOD PROVIDES A 0.1 Db OR BETTER MATCH
18013590450-130
-104
-78
-52
-26
0
D-PronyNewdB-26N10PVarDVarANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-PronyNewdB-26N10PVarDVar
EXPONENT M = 1
2
3
•SOLID LINES FOR PRONY FM, DOTS FOR GM
![Page 171: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/171.jpg)
SYNTHESIZING THE EXPONENTIATED -26 dB D-C PATTERN USING PRONY’S METHOD PROVIDES A 0.1 Db OR BETTER MATCH
18013590450-130
-104
-78
-52
-26
0
D-PronyNewdB-26N10PVarDVarANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-PronyNewdB-26N10PVarDVar
EXPONENT M = 1
2
3
4
![Page 172: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/172.jpg)
SYNTHESIZING THE EXPONENTIATED -26 dB D-C PATTERN USING PRONY’S METHOD PROVIDES A 0.1 Db OR BETTER MATCH
18013590450-130
-104
-78
-52
-26
0
D-PronyNewdB-26N10PVarDVarANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-PronyNewdB-26N10PVarDVar
EXPONENT M = 1
2
3
4
•AT LEVELS ~ ! -110 dB •USING 10, 19, 28, & 37 ELEMENTS
•INITIAL ARRAY LENGTH IS 4.5 WAVELENGTHS
![Page 173: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/173.jpg)
ARRAYS FOR SUCCESSIVELY LOWER SIDE LOBES EXPAND PROPORTIONATELY IN SIZE
M = 2 M = 3 M = 4 . . . WHILE RETAINING UNIFORM SPACING AND THE SAME NUMBER OF SIDE LOBES
![Page 174: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/174.jpg)
SIMILAR RESULTS ARE OBTAINED WHEN THE D-C ARRAY IS 7.5 WAVELENGTHS LONG . . .
18013590450-156
-130
-104
-78
-52
-26
0D-NewDC^x~180to0
ANGLE FROM ARRAY AXIS (degrees)
NO
RM
AL
IZE
D P
AT
TE
RN
(dB
)
G-L7.5DC^xPatternsNewLineEXPONENT M = 1
2
3
4
![Page 175: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/175.jpg)
. . . FOR WHICH THE SINGULAR-VALUE SPECTRA INDICATE THE NUMBER OF ARRAY ELEMENTS
5 04030201 0010 -2510 -2410 -2310 -2210 -2110 -2010 -1910 -1810 -1710 -1610 -1510 -1410 -1310 -1210 -1110 -1010 - 910 - 810 - 710 - 610 - 510 - 410 - 310 - 210 - 110 0
D-L7.5DC^xMBPESINGULAR-VALUE ORDER
SIN
GU
LAR
VA
LUES
G-SVsL7.5Norm
EXPONENT M = 1
•FOR EXPONENT M = 1 TO 4 ARE 10, 19, 28, 37 RESPECTIVELY
![Page 176: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/176.jpg)
. . . FOR WHICH THE SINGULAR-VALUE SPECTRA INDICATE THE NUMBER OF ARRAY ELEMENTS
5 04030201 0010 -2510 -2410 -2310 -2210 -2110 -2010 -1910 -1810 -1710 -1610 -1510 -1410 -1310 -1210 -1110 -1010 - 910 - 810 - 710 - 610 - 510 - 410 - 310 - 210 - 110 0
D-L7.5DC^xMBPESINGULAR-VALUE ORDER
SIN
GU
LAR
VA
LUES
G-SVsL7.5Norm
EXPONENT M = 1
2
•FOR EXPONENT M = 1 TO 4 ARE 10, 19, 28, 37 RESPECTIVELY
![Page 177: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/177.jpg)
. . . FOR WHICH THE SINGULAR-VALUE SPECTRA INDICATE THE NUMBER OF ARRAY ELEMENTS
5 04030201 0010 -2610 -2510 -2410 -2310 -2210 -2110 -2010 -1910 -1810 -1710 -1610 -1510 -1410 -1310 -1210 -1110 -1010 - 910 - 810 - 710 - 610 - 510 - 410 - 310 - 210 - 110 0
D-L7.5DC^xMBPESINGULAR-VALUE ORDER
SIN
GU
LAR
VA
LUES
G-SVsL7.5Norm
EXPONENT M = 1
2 3
•FOR EXPONENT M = 1 TO 4 ARE 10, 19, 28, 37 RESPECTIVELY
![Page 178: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/178.jpg)
. . . FOR WHICH THE SINGULAR-VALUE SPECTRA INDICATE THE NUMBER OF ARRAY ELEMENTS
5 04030201 0010 -2610 -2510 -2410 -2310 -2210 -2110 -2010 -1910 -1810 -1710 -1610 -1510 -1410 -1310 -1210 -1110 -1010 - 910 - 810 - 710 - 610 - 510 - 410 - 310 - 210 - 110 0
D-L7.5DC^xMBPESINGULAR-VALUE ORDER
SIN
GU
LAR
VA
LUES
G-SVsL7.5Norm
EXPONENT M = 1
2 3 4
•FOR EXPONENT M = 1 TO 4 ARE 10, 19, 28, 37 RESPECTIVELY
![Page 179: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/179.jpg)
THE NUMBER OF SINGULAR VALUES INCREASES LINEARLY WITH THE EXPONENT M
43210
10
20
30
40
D-L7.5DC^xMBPE
EXPONENT OF DOLPH-CHEBYSHEV ARRAY (M)
MA
XIM
UM
SIN
GU
LA
R V
AL
UE
G-MaxSVvsDC^x
•FOR A 7.5-WAVLENGTH, 10-ELEMENT DOLPH- CHEBYSHEV ARRAY
![Page 180: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/180.jpg)
THE NUMBER OF SINGULAR VALUES INCREASES LINEARLY WITH THE EXPONENT M
43210
10
20
30
40
D-L7.5DC^xMBPE
EXPONENT OF DOLPH-CHEBYSHEV ARRAY (M)
MA
XIM
UM
SIN
GU
LA
R V
AL
UE
G-MaxSVvsDC^x
•FOR A 7.5-WAVLENGTH, 10-ELEMENT DOLPH- CHEBYSHEV ARRAY
![Page 181: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/181.jpg)
THE MAIN BEAMWIDTH DECREASES FROM ABOUT 7.4 TO 3.6 DEGREES FOR AN EXPONENT PARAMETER VALUE OF 4 . . .
9 594939 2919 08988878685-3
-2
-1
0
D-L7.5DC^xMBPEANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-DC^xRadPattGMMainLobe
EXPONENT M = 1
•FOR THE 7.5 WAVELENGTH ARRAY
![Page 182: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/182.jpg)
THE MAIN BEAMWIDTH DECREASES FROM ABOUT 7.4 TO 3.6 DEGREES FOR AN EXPONENT PARAMETER VALUE OF 4 . . .
9 594939 2919 08988878685-3
-2
-1
0
D-L7.5DC^xMBPEANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-DC^xRadPattGMMainLobe
EXPONENT M = 1
2
•FOR THE 7.5 WAVELENGTH ARRAY
![Page 183: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/183.jpg)
THE MAIN BEAMWIDTH DECREASES FROM ABOUT 7.4 TO 3.6 DEGREES FOR AN EXPONENT PARAMETER VALUE OF 4 . . .
9 594939 2919 08988878685-3
-2
-1
0
D-L7.5DC^xMBPEANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-DC^xRadPattGMMainLobe
EXPONENT M = 1
2
3
•FOR THE 7.5 WAVELENGTH ARRAY
![Page 184: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/184.jpg)
THE MAIN BEAMWIDTH DECREASES FROM ABOUT 7.4 TO 3.6 DEGREES FOR AN EXPONENT PARAMETER VALUE OF 4 . . .
9 594939 2919 08988878685-3
-2
-1
0
D-L7.5DC^xMBPEANGLE FROM ARRAY AXIS (degrees)
NO
RM
ALI
ZED
PA
TTER
N (d
B)
G-DC^xRadPattGMMainLobe
EXPONENT M = 1
2
3
M = 4
•FOR THE 7.5 WAVELENGTH ARRAY
![Page 185: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/185.jpg)
. . . AT THE -3 dB LEVEL
![Page 186: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/186.jpg)
THE PRONY-DERIVED ARRAYS CAN HAVE WIDELY VARYING SOURCE STRENGTHS . . .
15.00
6
13.33
9
11.67
2
10.00
5
8.338
6.671
5.004
3.337
1.670
0.003
-1.66
4
-3.33
1
-4.99
8
-6.66
5
-8.33
2
-9.99
9
-11.66
6
-13.33
3
-15.00
010 - 5
10 - 4
10 - 3
10 - 2
10 - 1
10 0
D-DC^xImagPoleVsRealResELEMENT POSITION (wavelengths)
ELEM
ENT C
URRE
NT
G-PolesVsResiduesDC^xVert
M = 1
•THE NUMBER OF SOURCES VARIES FROM 10, 19, 28 TO 35 FOR M VARYING 1 TO 4
![Page 187: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/187.jpg)
THE PRONY-DERIVED ARRAYS CAN HAVE WIDELY VARYING SOURCE STRENGTHS . . .
15.00
6
13.33
9
11.67
2
10.00
5
8.338
6.671
5.004
3.337
1.670
0.003
-1.66
4
-3.33
1
-4.99
8
-6.66
5
-8.33
2
-9.99
9
-11.66
6
-13.33
3
-15.00
010 - 5
10 - 4
10 - 3
10 - 2
10 - 1
10 0
D-DC^xImagPoleVsRealResELEMENT POSITION (wavelengths)
ELEM
ENT C
URRE
NT
G-PolesVsResiduesDC^xVert
M = 2
M = 1
•THE NUMBER OF SOURCES VARIES FROM 10, 19, 28 TO 35 FOR M VARYING 1 TO 4
•FOR A 7.5-WAVELENGTH D-C ARRAY NORMALIZED TO CENTER ELEMENTS
![Page 188: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/188.jpg)
THE PRONY-DERIVED ARRAYS CAN HAVE WIDELY VARYING SOURCE STRENGTHS . . .
15.00
6
13.33
9
11.67
2
10.00
5
8.338
6.671
5.004
3.337
1.670
0.003
-1.66
4
-3.33
1
-4.99
8
-6.66
5
-8.33
2
-9.99
9
-11.66
6
-13.33
3
-15.00
010 - 5
10 - 4
10 - 3
10 - 2
10 - 1
10 0
D-DC^xImagPoleVsRealResELEMENT POSITION (wavelengths)
ELEM
ENT C
URRE
NT
G-PolesVsResiduesDC^xVert
M = 2
M = 3
M = 1
•THE NUMBER OF SOURCES VARIES FROM 10, 19, 28 TO 35 FOR M VARYING 1 TO 4
•FOR A 7.5-WAVELENGTH D-C ARRAY NORMALIZED TO CENTER ELEMENTS
•WITH IMPLICATIONS FOR NOISE SENSITIVITY
![Page 189: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/189.jpg)
THE PRONY-DERIVED ARRAYS CAN HAVE WIDELY VARYING SOURCE STRENGTHS . . .
15.00
6
13.33
9
11.67
2
10.00
5
8.338
6.671
5.004
3.337
1.670
0.003
-1.66
4
-3.33
1
-4.99
8
-6.66
5
-8.33
2
-9.99
9
-11.66
6
-13.33
3
-15.00
010 - 5
10 - 4
10 - 3
10 - 2
10 - 1
10 0
D-DC^xImagPoleVsRealResELEMENT POSITION (wavelengths)
ELEM
ENT C
URRE
NT
G-PolesVsResiduesDC^xVert
EXPONENT M = 4
M = 2
M = 3
M = 1
•THE NUMBER OF SOURCES VARIES FROM 10, 19, 28 TO 35 FOR M VARYING 1 TO 4
•FOR A 5-WAVELENGTH D-C ARRAY NORMALIZED TO END ELEMENTS
•WITH IMPLICATIONS FOR NOISE SENSITIVITY
![Page 190: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/190.jpg)
. . . WITH EACH ARRAY SIZE VARYING LINEARLY WITH INCREASING EXPONENT
43210
10
20
30
Initial Width 7.5 WavelengthsInitial Width 5 Wavelengths
D-Poles&ResiduesDC^4
EXPONENT OF DOLPH-CHEBYSHEV ARRAY (M)
AR
RA
Y W
IDT
H (
wav
elen
gths
)
G-DC^xWidthVsExponent
•AS MxINITIAL ARRAY WIDTH
![Page 191: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/191.jpg)
. . . WITH EACH ARRAY SIZE VARYING LINEARLY WITH INCREASING EXPONENT
43210
10
20
30
Initial Width 7.5 WavelengthsInitial Width 5 Wavelengths
D-Poles&ResiduesDC^4
EXPONENT OF DOLPH-CHEBYSHEV ARRAY (M)
AR
RA
Y W
IDT
H (
wav
elen
gths
)
G-DC^xWidthVsExponent
•AS MxINITIAL ARRAY WIDTH
![Page 192: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/192.jpg)
THE PRONY ARRAY MATCHES A “STANDARD” D-C, -10 dB VERSION* . . .
![Page 193: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/193.jpg)
THE PRONY ARRAY MATCHES A “STANDARD” D-C, -10 dB VERSION* . . .
•FOR A 10-ELEMENT PATTERN GIVEN BY 0.4463*COS(U) + 0.4306*COS(3.*U) + 0.4003*COS(5.*U)
+ 0.3576*COS(7.*U) + COS(9.*U)
![Page 194: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/194.jpg)
THE PRONY ARRAY MATCHES A “STANDARD” D-C, -10 dB VERSION* . . .
•FOR A 10-ELEMENT PATTERN GIVEN BY 0.4463*COS(U) + 0.4306*COS(3.*U) + 0.4003*COS(5.*U)
+ 0.3576*COS(7.*U) + COS(9.*U)
*Ahmad Safaai-Jazi, “A New Formulation for the Design of Chebyshev Arrays,” IEEE Transactions on Antennas and Propagation, AP-42, 3, pp. 439-443, March 1994.
![Page 195: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/195.jpg)
THE EXPONENTIATED PATTERN MAIN BEAMWIDTH SUCCESSIVELY DECREASES
•THE “STANDARD” -20 dB PATTERN (RED) IS GIVEN BY 1.5585*COS(U) + 1.4360*COS(3.*U) + 1.2125*COS(5.*U) + 0.9264*COS(7.*U) +
COS(9.*U)
![Page 196: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/196.jpg)
THE EXPONENTIATED PATTERN MAIN BEAMWIDTH SUCCESSIVELY DECREASES
•THE “STANDARD” -20 dB PATTERN (RED) IS GIVEN BY 1.5585*COS(U) + 1.4360*COS(3.*U) + 1.2125*COS(5.*U) + 0.9264*COS(7.*U) +
COS(9.*U) •THE 19-ELEMENT PRONY PATTERN (BLACK) COMES FROM
(0.4463*COS(U) + 0.4306*COS(3.*U) + 0.4003*COS(5.*U) + 0.3576*COS(7.*U) + COS(9.*U))^22
![Page 197: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/197.jpg)
THE EXPONENTIATED PATTERN MAIN BEAMWIDTH SUCCESSIVELY DECREASES
•THE “STANDARD” -30 dB PATTERN (RED) IS GIVEN BY 3.8830*COS(U) + 3.4095*COS(3.*U) + 2.5986*COS(5.*U) + 1.6695*COS(7.*U) +
COS(9.*U)
![Page 198: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/198.jpg)
THE EXPONENTIATED PATTERN MAIN BEAMWIDTH SUCCESSIVELY DECREASES
•THE “STANDARD” -30 dB PATTERN (RED) IS GIVEN BY 3.8830*COS(U) + 3.4095*COS(3.*U) + 2.5986*COS(5.*U) + 1.6695*COS(7.*U) +
COS(9.*U) •THE 28-ELEMENT PRONY PATTERN (BLACK) COMES FROM
(0.4463*COS(U) + 0.4306*COS(3.*U) + 0.4003*COS(5.*U) + 0.3576*COS(7.*U) + COS(9.*U))^33
![Page 199: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/199.jpg)
THE EXPONENTIATED PATTERN MAIN BEAMWIDTH SUCCESSIVELY DECREASES
•THE “STANDARD” -40 dB PATTERN (RED) IS GIVEN BY
7.9837*COS(U) + 6.6982*COS(3.*U) + 4.6319*COS(5.*U) + 2.5182*COS(7.*U) + COS(9.*U)
![Page 200: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/200.jpg)
THE EXPONENTIATED PATTERN MAIN BEAMWIDTH SUCCESSIVELY DECREASES
•THE “STANDARD” -40 dB PATTERN (RED) IS GIVEN BY
7.9837*COS(U) + 6.6982*COS(3.*U) + 4.6319*COS(5.*U) + 2.5182*COS(7.*U) + COS(9.*U)
•THE 35-ELEMENT PRONY PATTERN COMES FROM (0.4463*COS(U) + 0.4306*COS(3.*U) + 0.4003*COS(5.*U)
+ 0.3576*COS(7.*U) + COS(9.*U))^44
![Page 201: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/201.jpg)
ABOVE PRONY-SYNTHESIZED ARRAYS ARE UNIFORMLY SPACED FOR M " 3 BUT EXHIBIT A TAPERED SPACING FOR M ! 4
211815129630-3-6-9-12-15-18-210.5
0.6
0.7
D-N10,L5,DC10VarFMsVarP2 ELEMENT NUMBERS
ELEM
ENT
SEPA
RA
TIO
N (w
avel
engt
hs)
G-PronySynDC^1to5Spacing
M = 1
•THE RESPECTIVE NUMBER OF ARRAY ELEMENTS ARE 10, 19, 28, 35, AND 42 FOR AN INITIAL ARRAY 5-WAVLENGTHS LONG
![Page 202: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/202.jpg)
ABOVE PRONY-SYNTHESIZED ARRAYS ARE UNIFORMLY SPACED FOR M " 3 BUT EXHIBIT A TAPERED SPACING FOR M ! 4
211815129630-3-6-9-12-15-18-210.5
0.6
0.7
D-N10,L5,DC10VarFMsVarP2 ELEMENT NUMBERS
ELEM
ENT
SEPA
RA
TIO
N (w
avel
engt
hs)
G-PronySynDC^1to5Spacing
M = 2
M = 1
•THE RESPECTIVE NUMBER OF ARRAY ELEMENTS ARE 10, 19, 28, 35, AND 42 FOR AN INITIAL ARRAY 5-WAVLENGTHS LONG
![Page 203: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/203.jpg)
ABOVE PRONY-SYNTHESIZED ARRAYS ARE UNIFORMLY SPACED FOR M " 3 BUT EXHIBIT A TAPERED SPACING FOR M ! 4
211815129630-3-6-9-12-15-18-210.5
0.6
0.7
D-N10,L5,DC10VarFMsVarP2 ELEMENT NUMBERS
ELEM
ENT
SEPA
RA
TIO
N (w
avel
engt
hs)
G-PronySynDC^1to5Spacing
M = 3
M = 2
M = 1
•THE RESPECTIVE NUMBER OF ARRAY ELEMENTS ARE 10, 19, 28, 35, AND 42 FOR AN INITIAL ARRAY 5-WAVLENGTHS LONG
![Page 204: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/204.jpg)
ABOVE PRONY-SYNTHESIZED ARRAYS ARE UNIFORMLY SPACED FOR M " 3 BUT EXHIBIT A TAPERED SPACING FOR M ! 4
211815129630-3-6-9-12-15-18-210.5
0.6
0.7
D-N10,L5,DC10VarFMsVarP2 ELEMENT NUMBERS
ELEM
ENT
SEPA
RA
TIO
N (w
avel
engt
hs)
G-PronySynDC^1to5Spacing
M = 4
M = 3
M = 2
M = 1
•THE RESPECTIVE NUMBER OF ARRAY ELEMENTS ARE 10, 19, 28, 35, AND 42 FOR AN INITIAL ARRAY 5-WAVLENGTHS LONG
![Page 205: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/205.jpg)
ABOVE PRONY-SYNTHESIZED ARRAYS ARE UNIFORMLY SPACED FOR M " 3 BUT EXHIBIT A TAPERED SPACING FOR M ! 4
211815129630-3-6-9-12-15-18-210.5
0.6
0.7
D-N10,L5,DC10VarFMsVarP2 ELEMENT NUMBERS
ELEM
ENT
SEPA
RA
TIO
N (w
avel
engt
hs)
G-PronySynDC^1to5Spacing
EXPONENT M = 5
M = 4
M = 3
M = 2
M = 1
•THE RESPECTIVE NUMBER OF ARRAY ELEMENTS ARE 10, 19, 28, 35, AND 42 FOR AN INITIAL ARRAY 5-WAVLENGTHS LONG
![Page 206: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/206.jpg)
THE DOLPH-CHEBYSHEV SVD SPECTRUM ROLLS OFF SLOWER WITH INCREASING WINDOW WIDTH
13119753110 -2410 -2310 -2210 -2110 -2010 -1910 -1810 -1710 -1610 -1510 -1410 -1310 -1210 -1110 -1010 -910 -810 -710 -610 -510 -410 -310 -210 -110 010 110 2
D-PronyL10N10DB-26P1SVDSINGULAR-VALUE ORDER
SIN
GU
LAR
VA
LUES
G-PronyL10N10dB-26P1SVC
±0.1
•FOR A 10-WAVELENGTH, 10-ELEMENT ARRAY
![Page 207: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/207.jpg)
THE DOLPH-CHEBYSHEV SVD SPECTRUM ROLLS OFF SLOWER WITH INCREASING WINDOW WIDTH
13119753110 -2410 -2310 -2210 -2110 -2010 -1910 -1810 -1710 -1610 -1510 -1410 -1310 -1210 -1110 -1010 -910 -810 -710 -610 -510 -410 -310 -210 -110 010 110 2
D-PronyL10N10DB-26P1SVDSINGULAR-VALUE ORDER
SIN
GU
LAR
VA
LUES
G-PronyL10N10dB-26P1SVC
±0.3
±0.1
•FOR A 10-WAVELENGTH, 10-ELEMENT ARRAY
![Page 208: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/208.jpg)
THE DOLPH-CHEBYSHEV SVD SPECTRUM ROLLS OFF SLOWER WITH INCREASING WINDOW WIDTH
13119753110 -2410 -2310 -2210 -2110 -2010 -1910 -1810 -1710 -1610 -1510 -1410 -1310 -1210 -1110 -1010 -910 -810 -710 -610 -510 -410 -310 -210 -110 010 110 2
D-PronyL10N10DB-26P1SVDSINGULAR-VALUE ORDER
SIN
GU
LAR
VA
LUES
G-PronyL10N10dB-26P1SVC
±0.5
±0.3
±0.1
•FOR A 10-WAVELENGTH, 10-ELEMENT ARRAY
![Page 209: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/209.jpg)
THE DOLPH-CHEBYSHEV SVD SPECTRUM ROLLS OFF SLOWER WITH INCREASING WINDOW WIDTH
13119753110 -2410 -2310 -2210 -2110 -2010 -1910 -1810 -1710 -1610 -1510 -1410 -1310 -1210 -1110 -1010 - 910 - 810 - 710 - 610 - 510 - 410 - 310 - 210 - 110 010 110 2
D-PronyL10N10DB-26P1SVDSINGULAR-VALUE ORDER
SIN
GU
LAR
VA
LUES
G-PronyL10N10dB-26P1SVC
WINDOW ±0.999±0.5
±0.3
±0.1
•FOR A 10-WAVELENGTH, 10-ELEMENT ARRAY
![Page 210: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/210.jpg)
ANALYTIC EXPRESSIONS FOR THE EXPONENTIATED PATTERNS CAN BE DERIVED*
•CONSIDER THE 4-ELEMENT D-C ARRAY WHOSE PATTERN IS
!
P4 = A1 cos u( ) + A2 cos 3u( )
WHERE A1 = 0.8794 and A2 = 1
![Page 211: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/211.jpg)
ANALYTIC EXPRESSIONS FOR THE EXPONENTIATED PATTERNS CAN BE DERIVED*
•CONSIDER THE 4-ELEMENT D-C ARRAY WHOSE PATTERN IS
!
P4 = A1 cos u( ) + A2 cos 3u( )
WHERE A1 = 0.8794 and A2 = 1
•ITS EXPONENTIATED PATTERN IS THEN
!
P4M = A1 cos u( ) +A2 cos 3u( )[ ] M .
![Page 212: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/212.jpg)
ANALYTIC EXPRESSIONS FOR THE EXPONENTIATED PATTERNS CAN BE DERIVED*
•CONSIDER THE 4-ELEMENT D-C ARRAY WHOSE PATTERN IS
!
P4 = A1 cos u( ) + A2 cos 3u( )
WHERE A1 = 0.8794 and A2 = 1
•ITS EXPONENTIATED PATTERN IS THEN
!
P4M = A1 cos u( ) +A2 cos 3u( )[ ] M .
• FOR M = 2 THIS BECOMES
!
P42 =
A12 + A2
2
2+
A12
2+ A1A2
"
# $
%
& ' cos 2u( ) + A1A2 cos 4u( ) +
A22
2cos 6u( ) .
*G. J. BURKE, PRIVATE COMMUNICATION, 2013 VIA MATHEMATICA
![Page 213: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/213.jpg)
ITS PATTERNS FOR M = 3 AND M = 4 ARE GIVEN BY
!
P43 =
34A13 + A1
2A2 + 2A1A22[ ]cos u( ) +
14A13 + 6A1
2A2 + 3A23[ ]cos 3u( )
+34A12A2 + A1A2
2[ ]cos 5u( ) +34A1A2
2 cos 7u( ) +14A2
3 cos 9u( )
![Page 214: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/214.jpg)
ITS PATTERNS FOR M = 3 AND M = 4 ARE GIVEN BY
!
P43 =
34A13 + A1
2A2 + 2A1A22[ ]cos u( ) +
14A13 + 6A1
2A2 + 3A23[ ]cos 3u( )
+34A12A2 + A1A2
2[ ]cos 5u( ) +34A1A2
2 cos 7u( ) +14A2
3 cos 9u( )
AND
!
P44 =
3214A14 +13A13A2 + A1
2A22 +14A2
4"
# $ %
& ' +3213A14 + A1
3A2 + A12A2
2 + A1A23"
# $ %
& ' cos 2u( )
+32112
A14 + A1
3A2 +12A12A2
2 + A1A23"
# $ %
& ' cos 4u( ) +
3213A13A2 + A1
2A22 +13A2
4"
# $ %
& ' cos 6u( )
+3212A12A2
2 +13A1A2
3"
# $ %
& ' cos 8u( ) +
12A1A2
3 cos 10u( ) +18A2
4 cos 12u( ).
RESPECTIVELY
![Page 215: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/215.jpg)
PRONY-SYNTHESIZED AND ANALYTIC PATTERNS FROM THE PREVIOUS FORMULAS AGREE TO WITHIN 0.1 dB . . .
. . . FOR A 2-WAVELENGTH ARRAY . . .
![Page 216: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/216.jpg)
. . . WHOSE ELEMENT STRENGTHS ARE FOUND TO BE . . .
TABLE 1. Element Number
Basic Array 4 Elements
M = 2 7 Elements
M = 3 10 Elements
M = 4 13 Elements
1 2 3 4 5 6 7
0.8794 1
1.773 2.532 1.759
1
9.640 8.320 4.958 2.638
1
16.79 30.38 23.95 16.00 8.158 3.518
1
WITH THEIR DYNAMIC RANGE INCREASING FROM 1.14:1 TO 30.4:1
![Page 217: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/217.jpg)
EXPONENTIATED PATTERNS OF A UNIFORM CURRENT FILAMENT ARE NOT SYNTHESIZED AS WELL
•FOR A 5-WAVELENGTH FILAMENT
![Page 218: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/218.jpg)
EXPONENTIATED PATTERNS OF A UNIFORM CURRENT FILAMENT ARE NOT SYNTHESIZED AS WELL
•FOR A 5-WAVELENGTH FILAMENT
•DIFFERENCES BETWEEN SYNTHESIZED AND ACTUAL PATTERNS BECOME SIGNIFICANT AT LEVELS " -50 TO -60 dB
![Page 219: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/219.jpg)
SYNTHESIZED EXPONENTIATED PATTERNS FOR A TRIANGLE CURRENT FILAMENT ARE IMPROVED OVER THE UCF
.
•FOR A 5-WAVELENGTH CURRENT FILAMENT
![Page 220: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/220.jpg)
WIDE DYNAMIC RANGE OF SOURCE STRENGTHS CAN MAKE PATTERNS NOISE SENSITIVE . . .
•FOR M = 2, L = 7.5 WAVELENGTHS
![Page 221: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/221.jpg)
WIDE DYNAMIC RANGE OF SOURCE STRENGTHS CAN MAKE PATTERNS NOISE SENSITIVE . . .
•FOR M = 2, L = 7.5 WAVELENGTHS •WITH A MAXIMUM OF 10% RANDOM VARIATION IN
THE ELEMENT STRENGTHS
![Page 222: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/222.jpg)
WIDE DYNAMIC RANGE OF SOURCE STRENGTHS CAN MAKE PATTERNS NOISE SENSITIVE . . .
•FOR M = 3, L = 5 WAVELENGTHS
![Page 223: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/223.jpg)
WIDE DYNAMIC RANGE OF SOURCE STRENGTHS CAN MAKE PATTERNS NOISE SENSITIVE . . .
•FOR M = 3, L = 5 WAVELENGTHS •WITH A MAXIMUM OF 1% RANDOM VARIATION IN
THE ELEMENT STRENGTHS
![Page 224: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/224.jpg)
WIDE DYNAMIC RANGE OF SOURCE STRENGTHS CAN MAKE PATTERNS NOISE SENSITIVE . . .
•FOR M = 4, L = 7.5 WAVELENGTHS •WITH A MAXIMUM OF 1% RANDOM VARIATION IN
THE ELEMENT STRENGTHS
![Page 225: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/225.jpg)
PRESENTATION HAS DESCRIBED AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .
•THE BASIC IDEA
•SEVERAL EXAMPLES
. . . AND PATTERN SYNTHESIS USING SPATIAL POLES
• PRONY’S METHOD AS A WAY TO DETERMINE SOURCE LOCATIONS AND STRENGTHS FOR SPECIFIED PATTERNS
•THE SINUSOIDAL CURRENT FILAMENT
•SEVERAL EXAMPLES OF PRONY SYNTHESIS
•SYNTHESIZING EXPONENTIATED PATTERNS
![Page 226: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/226.jpg)
NORMALIZED RESISTANCE OF A DIPOLE WAS MODELED WITH 6 FMs
2826242220181614121086420-0.5
0.0
0.5
1.0
FREQUENCY (MHz)
NORM
ALIZ
ED D
IPOL
E RE
SIST
ANCE
Dipole#0
USING 18 INITIAL GM SAMPLES
![Page 227: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/227.jpg)
NORMALIZED RESISTANCE OF A DIPOLE WAS MODELED WITH 6 FMs
2826242220181614121086420-0.5
0.0
0.5
1.0
FREQUENCY (MHz)
NORM
ALIZ
ED D
IPOL
E RE
SIST
ANCE
Dipole#1
19 GM SAMPLES
![Page 228: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/228.jpg)
NORMALIZED RESISTANCE OF A DIPOLE WAS MODELED WITH 6 FMs
2826242220181614121086420-0.5
0.0
0.5
1.0
FREQUENCY (MHz)
NORM
ALIZ
ED D
IPOL
E RE
SIST
ANCE
Dipole#2
20 GM SAMPLES
![Page 229: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/229.jpg)
NORMALIZED RESISTANCE OF A DIPOLE WAS MODELED WITH 6 FMs
2826242220181614121086420-0.5
0.0
0.5
1.0
FREQUENCY (MHz)
NORM
ALIZ
ED D
IPOL
E RE
SIST
ANCE
Dipole#3
21 GM SAMPLES
![Page 230: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/230.jpg)
NORMALIZED RESISTANCE OF A DIPOLE WAS MODELED WITH 6 FMs
2826242220181614121086420-0.5
0.0
0.5
1.0
FREQUENCY (MHz)
NORM
ALIZ
ED D
IPOL
E RE
SIST
ANCE
Dipole#4
22 GM SAMPLES
![Page 231: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/231.jpg)
NORMALIZED RESISTANCE OF A DIPOLE WAS MODELED WITH 6 FMs
2826242220181614121086420-0.5
0.0
0.5
1.0
FREQUENCY (MHz)
NORM
ALIZ
ED D
IPOL
E RE
SIST
ANCE
Dipole#5
23 GM SAMPLES
![Page 232: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/232.jpg)
NORMALIZED RESISTANCE OF A DIPOLE WAS MODELED WITH 6 FMs
2826242220181614121086420-0.5
0.0
0.5
1.0
FREQUENCY (MHz)
NORM
ALIZ
ED D
IPOL
E RE
SIST
ANCE
Dipole#6
24 GM SAMPLES
![Page 233: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/233.jpg)
NORMALIZED RESISTANCE OF A DIPOLE WAS MODELED WITH 6 FMs
2826242220181614121086420-0.5
0.0
0.5
1.0
FREQUENCY (MHz)
NORM
ALIZ
ED D
IPOL
E RE
SIST
ANCE
Dipole#7
25 GM SAMPLES
![Page 234: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/234.jpg)
NORMALIZED RESISTANCE OF A DIPOLE WAS MODELED WITH 6 FMs
2826242220181614121086420-0.5
0.0
0.5
1.0
FREQUENCY (MHz)
NORM
ALIZ
ED D
IPOL
E RE
SIST
ANCE
Dipole#8
26 GM SAMPLES
![Page 235: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/235.jpg)
NORMALIZED RESISTANCE OF A DIPOLE WAS MODELED WITH 6 FMs
2826242220181614121086420-0.5
0.0
0.5
1.0
FREQUENCY (MHz)
NORM
ALIZ
ED D
IPOL
E RE
SIST
ANCE
Dipole#9
27 GM SAMPLES
![Page 236: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/236.jpg)
NORMALIZED RESISTANCE OF A DIPOLE WAS MODELED WITH 6 FMs
2826242220181614121086420-0.5
0.0
0.5
1.0
FREQUENCY (MHz)
NORM
ALIZ
ED D
IPOL
E RE
SIST
ANCE
Dipole#10
28 GM SAMPLES
![Page 237: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/237.jpg)
NORMALIZED RESISTANCE OF A DIPOLE WAS MODELED WITH 6 FMs
2826242220181614121086420-0.5
0.0
0.5
1.0
FREQUENCY (MHz)
NORM
ALIZ
ED D
IPOL
E RE
SIST
ANCE
Dipole#11
29 GM SAMPLES
![Page 238: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/238.jpg)
NORMALIZED RESISTANCE OF A DIPOLE WAS MODELED WITH 6 FMs
2826242220181614121086420-0.5
0.0
0.5
1.0
FREQUENCY (MHz)
NORM
ALIZ
ED D
IPOL
E RE
SIST
ANCE
Dipole#12
30 GM SAMPLES
![Page 239: TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESIS](https://reader031.fdocuments.us/reader031/viewer/2022022802/621af9a8cbe6aa031710fe69/html5/thumbnails/239.jpg)
NORMALIZED RESISTANCE OF A DIPOLE WAS MODELED WITH 6 FMs
2826242220181614121086420-0.5
0.0
0.5
1.0
ACTUAL RESISTANCEMODELED RESISTANCE
FREQUENCY (MHz)
NORM
ALIZ
ED D
IPOL
E RE
SIST
ANCE
Dipole#13
THE FINAL FINELY SAMPLED RESULT