NORMALIZED PATTERN
TWO NOVEL APPROACHES TO ANTENNA - PATTERN SYNTHESIS Edmund K. Miller Los Alamos National Laboratory (Retired) 597 Rustic Ranch Lane Lincoln, CA 95648 916-408-0915 [email protected]
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
ANTENNA PATTERN SYNTHESIS REMAINS A TOPIC OF INTEREST . . . 1. W. W. Hansen and J. R. Woodyard, “A New Principle in Directional Antenna Design,” Proc. IE, 26, 3, March, 1938, pp. 333-345. 2. S. A. Schelkunoff, “A Mathematical Theory of Linear Arrays,” Bell System Technical Journal, 22, pp. 80-107, 1943. 3. 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. 4. 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. 5. 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. 6. Robert S. Elliott, “On Discretizing Continuous Aperture Distributions,” IEEE Transactions on Antennas and Propagation, AP-25, 5, pp. 617-621, September 1977.
ANTENNA PATTERN SYNTHESIS REMAINS A TOPIC OF INTEREST . . . 7. Robert S. Elliott, Antenna Theory and Design, Englewood Cliffs, NJ, Prentice-Hall, 1981. 8. 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. 9. 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. 10. D. Marcano, and F. Duran, Synthesis of antenna arrays using genetic algorithms, IEEE Antennas and Propagation Magazine, 42, 3, pp. 12-20, June 2000. 11. 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. 12. 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.
ANTENNA PATTERN SYNTHESIS REMAINS A TOPIC OF INTEREST . . . 13. 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. 14. 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. 15. N. G. Gomez, J. J. Rodriguez, K. L. Melde, and K. M. McNeill, Design of LowSidelobe Linear Arrays With High Aperture Efficiency and Interference Nulls, IEEE Antennas and Wireless Propagation Letters, 8, pp. 607-610, 2009. 16. 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. 17. 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.
ANTENNA PATTERN SYNTHESIS REMAINS A TOPIC OF INTEREST . . . 18. P. S. Apostolov, Linear Equidistant Antenna Array With Improved Selectivity, IEEE Transactions on Antennas and Propagation, 59, 10, pp. 3940-3943, October 2011. 19. 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. 20. 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. 21. 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.
VARIOUS SOURCE DISTRIBUTIONS AND/OR PATTERNS FROM THE FOLLOWING SOURCES WERE USED 22. AHMAD SAFAAI-JAZI, “A NEW FORMULATION FOR THE DESIGN OF CHEBYSHEV ARRAYS,” IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, AP42, 3, PP. 349-443, MARCH 1994. C. A. BALANIS, “ANTENNA THEORY, ANALYSIS AND DESIGN,” HARPER AND ROWE, 1982. E. V. JULL, “RADIATION FROM APERTURES,” IN ANTENNA HANDBOOK, ed. Y. T. LO AND S. W. LEE, VAN NOSTRAND REINHOLD CO., 1988.
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
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 ARE OBTAINED USING THESE ANGLES AND THE DESIRED VALUES OF THE LOBE MAXIMA 5) RETURNING TO 2) THESE NEW CURRENTS ARE USED TO COMPUTE A NEW PATTERN & THE PROCESS CONINUES UNTIL THE PATTERN CONVERGES
EVEN AND ODD NUMBERS OF ELEMENTS WERE USED FOR SYMMETRIC ARRAYS •FOR SYMMETRIC ARRAYS THE PATTERN CAN BE WRITTEN AS . . . N
P(! ) = " Sn cos[( 2n # 1) u] n=1
OR N
P(! ) = " Sn cos( 2nu) n=0
FOR AN EVEN OR ODD NUMBER OF ELEMENTS RESPECTIVELY, WHERE *# !d & u = ,% ( cos) / +$ " ' .
LOBE MAXIMA GENERATE A MATRIX . . . 1) The initial pattern P1(θ) is sampled finely enough in θ to accurately locate its positive and negative maxima at the angles θ1,n, n = 1,…,N with the corresponding pattern maxima denoted by P1(θ1,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
" cos( u11) $ cos( u12 ) $ [ M1, N ] = $ M $ # cos( u1N )
cos( 3u11) cos( 3u12 ) M cos( 3u1N )
L cos[( 2N ! 1) u11] % ' L cos[( 2N ! 1) u12 ] ' ' O M ' L cos[( 2N ! 1) u1N ]&
. . . WHICH IS THEN INVERTED TO SOLVE FOR A NEW SET OF CURRENTS S1,n FROM S1,1 ! % cos( u11) # ' S1, 2 # ' cos( u12 ) = M # ' M # ' S1, N " &cos( u1N )
cos( 3u11) L cos[( 2N $ 1) u11] ! L1 ! # # cos( 3u12 ) L cos[( 2N $ 1) u12 ] # L2 # # M# M O M # # cos( 3u1N ) L cos[( 2N $ 1) u1N ]" LN " $1
. . . WHERE THE Ln ARE THE MAXIMUM VALUES DESIRED FOR THE LOBES OF THE SYNTHESIZED PATTERN
A SECOND SET OF PATTERN MAXIMA P2(θ2,n) AND MATRIX [M2,N] ARE COMPUTED TO OBTAIN AN UPDATED SET OF CURRENTS . . . S2,1 ! % cos( u21 ) cos( 3u21 ) ' S2,2 # ' cos( u22 ) cos( 3u22 ) #= M # ' M M ' S2,N #" &cos( u2N ) cos( 3u2N )
L cos[(2N $ 1) u21 ] ! L1 ! # L cos[(2N $ 1) u22 ] # L2 # # # M # O M # L cos[(2N $ 1) u2N ]" LN #" $1
. . . WHICH RESULTS IN A THIRD SET OF PATTERN MAXIMA P3(θ3,n), etc., UNTIL THE PATTERN CONVERGES ACCEPTABLY --ITERATION IS NECESSARY BECAUSE THE ANGLES AT WHICH MAXIMA OCCUR DEPEND SLIGHTLY ON THE CURRENT
FOR THE MORE GENERAL CASE OF A NON-SYMMETRIC ARRAY THE PATTERN CAN BE WRITTEN . . . N
P(! ) = " Sn exp
i ( kx n cos! + # n )
n=1
. . . WHICH LEADS TO A CURRENT COMPUTATION OF THE FORM . . . Si,1 ! % exp(ikx1 cos$ i1 ) exp(ikx 2 cos $ i1 ) # ' Si,2 # 'exp(ikx1 cos$ i2 ) exp(ikx 2 cos $ i2 ) = M # ' M M # ' Si,N " &exp(ikx1 cos$ iN ) exp(ikx 2 cos $ iN )
L L O L
. . . FOR THE i’th ITERATION
(1
exp(ikx N cos $ i1 ) ! # exp(ikx N cos $ i2 )# # M # exp(ikx N cos $ iN )"
L1 ! # L2 # M# # LN "
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
•IF THE NEAR END-FIRE LOBES BECOME ILL FORMED --AS ABOVE
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
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
A SEQUENCE OF PATTERNS . . . 0
NORMALIZED PATTERN (dB)
D-L7N15L20R40dB7PassesUpdate
-20
-40
-60
-80
-100
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
•ITERATION #1
180
A SEQUENCE OF PATTERNS . . . 0
NORMALIZED PATTERN (dB)
D-L7N15L20R40dB7PassesUpdate
-20
-40
-60
-80
-100
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #2
180
A SEQUENCE OF PATTERNS . . . 0 D-L7N15L20R40dB7PassesUpdate
NORMALIZED PATTERN (dB)
-10 -20 -30 -40 -50 -60 -70 -80 -90 -100
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #3
180
A SEQUENCE OF PATTERNS . . . 0
NORMALIZED PATTERN (dB)
D-L7N15L20R40dB7PassesUpdate
-20
-40
-60
-80
-100
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #4
180
A SEQUENCE OF PATTERNS . . . 0
NORMALIZED PATTERN (dB)
D-L7N15L20R40dB7PassesUpdate
-20
-40
-60
-80
-100
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #5
180
A SEQUENCE OF PATTERNS . . . 0
NORMALIZED PATTERN (dB)
D-L7N15L20R40dB7PassesUpdate
-20
-40
-60
-80
-100
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #6
180
A SEQUENCE OF PATTERNS . . . 20
NORMALIZED PATTERN (dB)
D-L7N15L20R40dB7PassesUpdate
0
-20
-40
-60
-80
-100
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
180
ITERATION #7 AND THE FINAL PATTERN
THE PATTERN DETERIORATES FOR LOWER FREQUENCIES . . . NORMALIZED PATTERN (dB)
20
D-Left20Right40D0.1to0.8
Separation 0.1 wavelengths
G-N15L20R40Sep0.1to0.2
0 -20 -40 -60 -80
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
THE PATTERN DETERIORATES FOR LOWER FREQUENCIES . . . NORMALIZED PATTERN (dB)
20
D-Left20Right40D0.1to0.8
0.2
G-N15L20R40Sep0.1to0.2
0 -20 -40 -60 -80
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
. . . WITH SIDELOBES MAINTAINED OVER A NEARLY 2:1 BANDWIDTH . . . 0
D-Left20Right40D0.1to0.8
NORMALIZED PATTERN (dB)
G-N15L20R40Sep0.3to0.5
-20
-40
-60
-80
Separation 0.3 wavelengths
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
. . . WITH SIDELOBES MAINTAINED OVER A NEARLY 2:1 BANDWIDTH . . . 0
D-Left20Right40D0.1to0.8
NORMALIZED PATTERN (dB)
G-N15L20R40Sep0.3to0.5
-20
-40
-60
-80
0.4
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
. . . WITH SIDELOBES MAINTAINED OVER A NEARLY 2:1 BANDWIDTH . . . 0
D-Left20Right40D0.1to0.8
NORMALIZED PATTERN (dB)
G-N15L20R40Sep0.3to0.5
-20
-40
-60
-80
0.5
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
. . . AND DEVELOPS GRATING LOBES FOR HIGHER FREQUENCIES . . . NORMALIZED PATTERN (dB)
20
Separation 0.6 wavelengths
D-Left20Right40D0.1to0.8 G-N15L20R40Sep0.6to0.8
0 -20 -40 -60 -80
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
. . . AND DEVELOPS GRATING LOBES FOR HIGHER FREQUENCIES . . . NORMALIZED PATTERN (dB)
20
0.7
D-Left20Right40D0.1to0.8 G-N15L20R40Sep0.6to0.8
0 -20 -40 -60 -80
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
. . . AND DEVELOPS GRATING LOBES FOR HIGHER FREQUENCIES . . . NORMALIZED PATTERN (dB)
20
0.8
D-Left20Right40D0.1to0.8 G-N15L20R40Sep0.6to0.8
0 -20 -40 -60 -80
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
A STANDARD DOLPH-CHEBYSHEV PATTERN IS READILY GENERATED . . .
•9-ELEMENT ARRAY 4 WAVELENGTHS LONG
VARIATIONS ON THE DOLPH-CHEBYSHEV DESIGN ARE EASY TO DEVELOP . . .
•15-ELEMENT ARRAY, 7 WAVELENGTHS LONG
VARIATIONS ON THE DOLPH-CHEBYSHEV DESIGN ARE EASY TO DEVELOP . . .
•15-ELEMENT ARRAY, 7 WAVELENGTHS LONG
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#1
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
STARTING PATTERN
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#2
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #1
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#3
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #2
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#4
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #3
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#5
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #4
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#6
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #5
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#7
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #6
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#8
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #7
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#9
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #8
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#10
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #9
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#11
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #10
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#12
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #11
•15-ELEMENT ARRAY, 7 WAVELENGTHS LONG
180
THE DOLPH-CHEBYSHEV PATTERN DOES NOT REQUIRE UNIFORM SPACING NORMALIZED PATTERN (dB)
0
-20
-40
-60
-80
0
45
90
ANGLE FROM ARRAY
135
AXIS (degrees)
180
•VARIABLE SPACINGS OF 0.4 AND 0.6 WAVELENGTHS
THE DOLPH-CHEBYSHEV PATTERN DOES NOT REQUIRE UNIFORM SPACING NORMALIZED PATTERN (dB)
0
-20
-40
-60
-80
0
45
90
ANGLE FROM ARRAY
135
AXIS (degrees)
180
•VARIABLE SPACINGS OF 0.4 TO 0.7 WAVELENGTHS
THE DOLPH-CHEBYSHEV PATTERN DOES NOT REQUIRE UNIFORM SPACING NORMALIZED PATTERN (dB)
0
-20
-40
-60
-80
0
45
90
ANGLE FROM ARRAY
135
AXIS (degrees)
180
•ARRAY LENGTHS OF 4 (BLACK) AND 5 (BLUE) WAVELENGTHS RESPECTIVELY
THE DOLPH-CHEBYSHEV PATTERN DOES NOT REQUIRE UNIFORM SPACING NORMALIZED PATTERN (dB)
0
-20
-40
-60
-80
0
45
90
ANGLE FROM ARRAY
135
AXIS (degrees)
180
•VARIABLE SPACING (BLACK) AND UNIFORM SPACING (RED) RESPECITVELY
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 for the first two iterations.
SOME EXTENSIONS OF THE BASIC IDEA MIGHT INVOLVE SUCH THINGS AS CONTROLLING: °NULLS °SIDE-LOBE ANGLES °MAIN LOBE ANGLE °THE NUMBER OF SIDE LOBES
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
PRONY’S METHOD OR ITS EQUIVALENT PROVIDES THE ARRAY PARAMETERS FROM PATTERN SAMPLES •GIVEN A DESIRED PATTERN Pdesired(θ) . . . N
Pdesired (! ) " PDSA (! ) = $ S# e kz# cos(! ) # =1
•. . . THE N SOURCE STRENGTHS Sα AND N LOCATIONS zα CAN BE OBTAINED •FOR THE ARRAY TO BE REALIZABLE USING ISOTROPIC SOURCES zα MUST BE PURE IMAGINARY •OTHERWISE A SOURCE DIRECTIVITY WOULD BE REQUIRED AS GIVEN BY D! = e kz! ,real cos(" )
IMPLEMENTING PRONY’S METHOD FOR PATTERN SYNTHESIS INVOLVES CHOOSING 3 PARAMETERS . . . •THE ANGLE SAMPLING INTERVAL Δcosθ --MUST BE SMALL ENOUGH TO AVOID ALIASING
•THE TOTAL ANGLE OBSERVATION WINDOW W MEASURED IN UNITS OF cosθ --MUST BE WIDE ENOUGH TO AVOID ILL CONDITIONING OF THE DATA MATRIX
THE LOBES OF A LINEAR ARRAY ARE SPACED UNIFORMLY IN COS(θ) 30
D-L20UCFPattAdaptNew G-L20UCFvsCosang,angle
FAR FIELD (dB)
20
10
0
-10
-20
-30 -90
-45
COS(ANGLE)x90
0
45
ANGLE
90
•THIS SHOWS THAT SAMPLING AS A FUNCTION OF COS(θ) RATHER THAN θ IS MORE APPROPRIATE •BESIDES WHICH PRONY’S METHOD REQUIRES THAT SAMPLING USE EQUAL STEPS IN COS(θ)
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/Δcosθ) + 1
. . . WHICH RESULTS IN REQUIRING THAT N BE THE LARGER OF N ≥ WL + 1 AND N≥R WITH L THE SOURCE SIZE IN WAVELENGTHS, R THE PATTERN RANK AND W THE WINDOW WIDTH
THE RESULTS THAT FOLLOW WERE GENERALLY OBTAINED USING THE FOLLOWING STEPS: •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
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
A USEFUL INITIAL TEST IS A MODIFIED PATTERN OF A SINUSOIDAL CURRENT FILAMENT •ITS FAR-FIELD PATTERN IS GIVEN BY $ e(ikL / 2)cos! + e #(ikL / 2)cos! # 2cos(kL /2) ' PMSCF (! ) = sin ! " PSCF (! ) = sin ! & ) sin ! % (
•PMSCF(θ) 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
TWO DIFFERENT WINDOW WIDTHS PRODUCE ESSENTIALLY IDENTICAL PATTERN MATCHES NORMALIZED PATTERN (dB)
0
-20
-40
-60
GM -0.05to0.05 FM -0.05to0.05 GM -0.999to0.999 FM -0.999to0.999
-80
-100
D-L5SCFxSINVarCOSANG G-L5DelCos0.1&2x0.999Patts
0
45
90
135
180
ANGLE FROM CURRENT AXIS (degrees)
•LENGTH OF SCF IS 5 WAVELENGTHS •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
SINGULAR-VALUE SPECTRA
SINGULAR-VALUE SPECTRA FOR SEVERAL WINDOW WIDTHS EXHIBIT A PATTERN RANK OF 3 FOR PMSCF . . . 10 2 10 1 10 0 10 - 1 -2 10 10 - 3 10 - 4 10 - 5 -6 10 10 - 7 10 - 8 10 - 9 -10 10 10 -11 10 -12 -13 10 -14 10 10 -15 -16 10 -17 10 10 -18 10 -19 -20 10 -21 10 10 -22 -23 10 -24 10
0
G-L5CosangVarSingValuesw/0IDs
2
4
6
SINGULAR VALUES
8
10
•N WAS INCREASED FOR EACH WINDOW UNTIL THE PATTERN CONVERGED •RESULT IS CONSISTENT WITH A 3 POINT SOURCES
. . . 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
THE NUMBER OF FITTING MODELS NEEDED FOR A CONVERGED PATTERN INCREASES SYSTEMATICALLY WITH WINDOW WIDTH NUMBER OF FITTINGS MODELS
12
10
8
6
4 G-L5FMsVsCosangw/oIDs
2 0.0
0.5
1.0
1.5
WIDTH OF OBSERVATION WINDOW
•TO AVOID ALIASING
2.0
THE CENTER SOURCE DISAPPEARS FOR A SCF 5.5 WAVELENGTHS LONG
•SAMPLED OVER A -0.05 TO +0.05 COSθ WINDOW
AN 11-TERM FITTING MODEL MATCHES THE ACTUAL PATTERN OF A 5WAVELENGTH SCF DOWN TO -60 dB . . . 0
NORMALIZED PATTERN (dB)
-10 -20 -30 -40 -50 -60 -70 -80
GENERATING MODEL FITTING MODEL
-90
-100
D-PronyN11L5SCFPattern
SAMPLES USED FOR FITTING MODEL
0
45
90
G-PronyN11L5SCFPatterm
135
180
ANGLE FROM CURRENT AXIS (degrees)
•THE BLACK DOTS DENOTE THE GENERATINGMODEL SAMPLES USED TO COMPUTE THE 11POLE FITTING MODEL
. . . BUT THE DERIVED SOURCE DISTRIBUTION IS NOT PHYSICALLY REALIZABLE . . .
•. . . BECAUSE SOME OF SCF 9 SOURCES HAVE REAL COMPONENTS IN THE COMPLEX SPACE PLANE
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
THE PATTERN OF A ±1 SQUARE-WAVE APERTURE IS GRAPHICALLY INDISTINGUISHABLE FROM AN 11-TERM FM . . . NORMALIZED PATTERN (dB)
20
GENERATING MODEL FITTING MODEL SAMPLES USED FOR FITTING MODEL
0 -20
-40 -60 -80
-100
D-L5N11±UCFPronyPattP1D24
G-L5N11±UCFPronyPattP1D24
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
•THE PATTERN FACTOR IS
% % "L '' 1! cos cos$ ) &# (* P± = L) * "L cos$ ) * # & (
180
. . . WHOSE SYNTHESIZED SOURCES ARE NOT UNIFORMLY SPACED
•FOR A 5-WAVELENGTH APERTURE •AND AN 11-POLE FITTING MODEL
THE PATTERN OF AN APERTURE 2 VARYING AS cos (π/L) IS ALSO GRAPHICALLY IDENTICAL TO ITS PRONY FM . . . 0 D-PronySynL5Cos^2N11
NORMALIZED PATTERN (dB)
G-PronySynL5Cos^2N11
-20
-40
-60 GENERATING MODEL FITTING MODEL SAMPLES USED FOR FITTING MODEL
-80
-100
0
45
90
135
180
ANGLE FROM CURRENT AXIS (degrees)
•ITS PATTERN FACTOR IS GIVEN BY Pcos2
sin(u) # ! 2 & = !L sin # % 2 2( u = WHERE u $! " u ' "
(
)
… WHOSE SOURCE DISTRIBUTION IS ALSO NONUNIFORM
•FOR A 5-WAVELENTH APERTURE •USING AN 11-TERM FITTING MODEL
PATTERN OF UNIFORM CURRENT OF LENGTH L TIMES (sinθ)P HAS TAPERED SIDELOBES WITH INCREASING P NORMALIZED PATTERN (dB)
0
P = 0
-20
1 2 3
-40
-60
0
45
D-L5UCFxSIN^XNVarActualPoles
90
135
ANGLE FROM ARRAY AXIS (degrees)
G-L5UCFxSIN^XNVarActualPoles
•ITS PATTERN FACTOR IS PUCF = (sin ! ) P
sin(kL cos! ) kL cos!
180
ITS SOURCES ARE ALSO NONUNIFORMLY SPACED
. . . USING 11 EXPONENTIALS IN THE FITTING MODEL •AND FOR A 5-WAVELENGTH APERTURE
A DOLPH-CHEBYSHEV ARRAY IS READILY SYNTHESIZED 0
D-DC^1VarL4.5...&N10...dB26...
NORMALIZED PATTERN (dB)
G-DC^1tL4.5dB26w/oPts
-26
-52
Generating Model Samples Used for FItting Model Fitting Model -78
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
180
•5-WAVELENGTHS LONG WITH -26 dB SIDELOBES AND 10 ELEMENTS
A MODIFIED DOLPH-CHEBYSHEV ARRAY IS ALSO SYNTHESIZED 0
D-L7N15-20&-40dBDCPronySyn
NORMALIZED PATTERN (dB)
G-L7N15-20&-40dBDCPronySyn
-20
-40 -60 Generating Model
-80
-100
Fitting Model GM Samples Used for FM
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
180
•-20 AND -40 dB SIDELOBES •15 ELEMENTS UNIFORMLY SPACED •7 WAVELENGTHS LONG
THIS ARRAY STEPS UP IN 5 dB INCREMENTS FROM LEFT TO RIGHT 0
D-L7N15-70to0dBPronySyn
NORMALIZED PATTERN (dB)
G-L7N15-70to0dBPronySyn
-20
-40
-60 Generating Model
-80
-100
Fitting Model GM Samples Used for FM
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
180
•15 ELEMENTS UNIFORMLY SPACED •7 WAVELENGTHS LONG
SINGULAR VALUES
SVD SPECTRA FOR SEVERAL ARRAYS ILLUSTRATE THEIR DIFFERENCES 10 0 10 - 1 10 - 2 -3 10 -4 10 10 - 5 10 - 6 -7 10 10 - 8 10 - 9 -10 10 10 -11 -12 10 -13 10 -14 10 10 -15 10 -16 10 -17 -18 10 10 -19 10 -20 10 -21 -22 10 10 -23 10 -24 -25 10
1
D-SVsVariousSources
7.5 Wavelength 10-Element DC Array 5-Wavelength Sinusoid 5-Wavelength COS^2 Aperture 5-Wavelength Uniform Current Filament
3
5
7
9
11
13
15
SINGULAR-VALUE ORDER
G-PronySynVarSrcsSVD
•THE DOLPH-CHEBYSHEV ARRAY CLEARLY SHOWS THE NUMBER OF ELEMENTS IT CONTAINS •THE SPECTRA OF THE CONTINUOUS DISTRIBUTIONS FALL OFF MORE SMOOTHLY
THIS PATTERN FROM ELLIOTT WAS REPLICATED USING PRONYS’ METHOD 0 G-N12L5.5ElliottPronySyn
NORMALIZED PATTERN (dB)
D-N12L5.5ElliottPronySyn
-20
-40
-60
GENERATING MODEL
-80
-100
SAMPLES USED FOR FITTING MODEL FITTING MODEL
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
180
Robert S. Elliott, “On Discretizing Continuous Aperture Distributions,” IEEE Transactions on Antennas and Propagation, AP-25, 5, pp. 617-621, September 1977. •12 ELEMENTS IN 5.5 WAVELENGTHS.
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
CONSIDER EXPONENTIATING A PATTERN: •FOR EXAMPLE THE PATTERN OF A 10-ELEMENT DOLPH-CHEBYSHEV ARRAY AS GIVEN BY . . . P10 (! ) = 2.798 cos(D) + 2.496 cos(3D) + 1.974 cos(5D) + 1.357 cos(7D) + cos(9D) WITH D = [(!d / ") cos# ] AND d THE ELEMENT SPACING
. . . TO YIELD SUCCESSIVELY LOWER SIDELOBES NORMALIZED PATTERN (dB)
0
Exponent M = 1
-26
2
-52
3
-78
4
-104 -130
-156
0
G-DC^1to4L4.5to18.5 D-DC^1VarL4.5...&N10...dB26...
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
180
•INITIAL ARRAY LENGTH IS 4.5 WAVELENGTHS
REFINING THE D-C -26 dB PATTERN USING * MATRIX-SYNTHESIS APPROACH YIELDS A WIDENING MAIN LOBE FOR FIXED L NORMALIZED PATTERN (dB)
0
-26
-52
-78
-104
-130
-156
0
G-L4.5N10DC26to104dB D-L4.5N10DC26to104dB
45
90
135
180
ANGLE FROM ARRAY AXIS (degrees)
Edmund K. Miller, “Synthesizing Linear-Array Patterns via Matrix Computation of Element Currents,” IEEE Antennas and Propagation Society Magazine, October, 2013, 55 (5), pp. 85-96.
SYNTHESIZING THE D-C -26 dB D-C PATTERN USING PRONY’S METHOD PROVIDES A 0.1 Db OR BETTER MATCH NORMALIZED PATTERN (dB)
0 EXPONENT M = 1
-26
2
-52
3
-78
4
-104
-130
0
G-PronyNewdB-26N10PVarDVar D-PronyNewdB-26N10PVarDVar
45
90
135
180
ANGLE FROM ARRAY AXIS (degrees)
•AT LEVELS ~ ≥ -110 dB •USING 10, 19, 28, & 37 POLES 4.5, 8.5, 13.5 AND 18.5 WAVELENGTHS LONG
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
PATTERNS WITH SAME SIDELOBE LEVELS WERE GENERATED WITH THE MATRIX APPROACH NORMALIZED PATTERN (dB)
0
-26
-52
-78
-104
-130
-156
0
G-VarL,N,dB26to104
45
90
135
180
ANGLE FROM ARRAY AXIS (degrees)
D-26,52,etc.dBArraysWithVariousL,N
. . . 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
SIMILAR RESULTS ARE OBTAINED WHEN THE D-C ARRAY IS 7.5 WAVELENGTHS LONG . . . 0
D-NewDC^x~180to0 G-L7.5DC^xPatternsNewLine
NORMALIZED PATTERN (dB)
EXPONENT M = 1 -26
2
-52
3
-78
4
-104
-130
-156
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
180
SINGULAR VALUES
. . . FOR WHICH THE SINGULAR-VALUE SPECTRA INDICATE THE NUMBER OF ARRAY ELEMENTS 10 0 10 - 1 10 - 2 10 - 3 10 - 4 10 - 5 10 - 6 10 - 7 10 - 8 10 - 9 10 -10 10 -11 10 -12 10 -13 10 -14 10 -15 10 -16 10 -17 10 -18 10 -19 10 -20 10 -21 10 -22 10 -23 10 -24 10 -25 10 -26
0
3
2
4
EXPONENT M = 1 G-SVsL7.5Norm
10 D-L7.5DC^xMBPE
20
30
SINGULAR-VALUE ORDER
40
50
•FOR EXPONENT M = 1 TO 4 ARE 10, 19, 28, 37 RESPECTIVELY
THE NUMBER OF SINGULAR VALUES INCREASES LINEARLY WITH THE EXPONENT M MAXIMUM SINGULAR VALUE
40
30
20
10
D-L7.5DC^xMBPE G-MaxSVvsDC^x
0
1
2
3
4
EXPONENT OF DOLPH-CHEBYSHEV ARRAY (M)
•FOR A 7.5-WAVLENGTH, 10-ELEMENT DOLPHCHEBYSHEV ARRAY
THE MAIN BEAMWIDTH DECREASES FROM ABOUT 7.4 TO 3.6 DEGREES FOR AN EXPONENT PARAMETER VALUE OF 4 . . . NORMALIZED PATTERN (dB)
0
EXPONENT M = 1 -1
3
-2 2
M = 4 -3 85
86
G-DC^xRadPattGMMainLobe
87
88
89
90
91
92
93
ANGLE FROM ARRAY AXIS (degrees)
94
D-L7.5DC^xMBPE
•FOR THE 7.5 WAVELENGTH ARRAY
95
. . . AT THE -3 dB LEVEL
THE PRONY-DERIVED ARRAYS CAN HAVE WIDELY VARYING SOURCE STRENGTHS: 10 0
M = 1 10 -
1
M = 2 - 2
M = 3
10 -
3
10 -
4
10 -
5
15.006
13.339
11.672
10.005
8.338
6.671
5.004
3.337
1.670
0.003
-1.664
-3.331
-4.998
-6.665
-8.332
-9.999
-11.666
-13.333
EXPONENT M = 4
-15.000
ELEMENT CURRENT
10
ELEMENT POSITION (wavelengths) D-DC^xImagPoleVsRealRes
G-PolesVsResiduesDC^xVert
•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
. . . WITH EACH ARRAY SIZE VARYING LINEARLY WITH INCREASING EXPONENT 30
ARRAY WIDTH (wavelengths)
Initial Width 7.5 Wavelengths Initial Width 5 Wavelengths 20
10
D-Poles&ResiduesDC^4 G-DC^xWidthVsExponent
0
1
2
3
EXPONENT OF DOLPH-CHEBYSHEV ARRAY (M)
•AS MxINITIAL ARRAY WIDTH
4
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.
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))^2
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))^3
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 (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))^4
ABOVE PRONY-SYNTHESIZED ARRAYS ARE UNIFORMLY SPACED FOR M ≤ 3 BUT EXHIBIT A TAPERED SPACING FOR M ≥ 4 ELEMENT SEPARATION (wavelengths)
0.7 EXPONENT M = 5 M = 4
0.6 M = 2 M = 1
0.5 -21 -18 -15 -12 -9 D-N10,L5,DC10VarFMsVarP2
-6
-3
0
M = 3
3
6
9
12 15 18 21
ELEMENT NUMBERS
G-PronySynDC^1to5Spacing
•THE RESPECTIVE NUMBER OF ARRAY ELEMENTS ARE 10, 19, 28, 35, AND 42 FOR AN INITIAL ARRAY 5-WAVLENGTHS LONG
THE DOLPH-CHEBYSHEV SVD SPECTRUM ROLLS OFF SLOWER WITH INCREASING WINDOW WIDTH
SINGULAR VALUES
10 2 10 1 10 0 10 - 1
±0.5
10 - 2 10 - 3 10 - 4 10 - 5
WINDOW ±0.999
±0.3
10 - 6 10 - 7 10 - 8 10 - 9
±0.1
10 -10 10 -11 10 -12 10 -13 10 -14 10 -15 10 -16 10 -17 10 -18 10 -19 10 -20 10 -21 10 -22 10 -23 10 -24
1
3
G-PronyL10N10dB-26P1SVC D-PronyL10N10DB-26P1SVD
5
7
9
SINGULAR-VALUE ORDER
11
13
•FOR A 10-WAVELENGTH, 10-ELEMENT ARRAY
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 P4 = [ A1 cos(u ) + A 2 cos(3u)] M
M .
• FOR M = 2 THIS BECOMES 2 2 2 2 $ A1 + A2 ! A1 A2 P4 = +# + A1 A2 & cos(2u) + A1 A2 cos( 4u) + cos(6u) . 2 2 " 2 % 2
*
G. J. BURKE, PRIVATE COMMUNICATION, 2013 VIA MATHEMATICA
ITS PATTERNS FOR M = 3 AND M = 4 ARE GIVEN BY [
]
[
]
3 3 1 3 2 2 2 3 A1 + A1 A2 + 2A1 A2 cos( u) + A1 + 6A1 A2 + 3A2 cos( 3u) 4 4 3 2 3 1 3 2 2 + A1 A2 + A1 A2 cos(5u) + A1 A2 cos( 7u) + A2 cos(9u) 4 4 4 P4 = 3
[
]
AND 3 !1 4 1 3 1 4 $ 3 !1 4 2 2 3 2 2 3$ P4 = # A1 + A1 A2 + A1 A2 + A2 & + # A1 + A1 A2 + A1 A2 + A1 A2 & cos(2u) % 2 "3 % 2 "4 3 4 3! 1 4 1 2 2 3 !1 3 1 4$ 3 3$ 2 2 + # A1 + A1 A2 + A1 A2 + A1 A2 & cos( 4u) + # A1 A2 + A1 A2 + A2 & cos(6u) % % 2 "12 2 2 "3 3 3 !1 2 2 1 1 1 4 3$ 3 + # A1 A2 + A1 A2 & cos(8u) + A1 A2 cos(10u) + A2 cos(12u). % 2 "2 3 2 8 4
RESPECTIVELY
PRONY-SYNTHESIZED AND ANALYTIC PATTERNS FROM THE PREVIOUS FORMULAS AGREE TO WITHIN 0.1 dB . . .
. . . FOR A 2-WAVELENGTH ARRAY . . .
. . . WHOSE ELEMENT STRENGTHS ARE FOUND TO BE . . . Element Number 1 2 3 4 5 6 7
Basic Array 4 Elements 0.8794 1
TABLE 1. M=2 7 Elements 1.773 2.532 1.759 1
M=3 10 Elements 9.640 8.320 4.958 2.638 1
M=4 13 Elements 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
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
SYNTHESIZED EXPONENTIATED PATTERNS FOR A TRIANGLE CURRENT FILAMENT ARE IMPROVED OVER THE UCF
.
•FOR A 5-WAVELENGTH CURRENT FILAMENT
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
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
WIDE DYNAMIC RANGE OF SOURCE STRENGTHS CAN MAKE PATTERNS NOISE SENSITIVE . . .
•FOR M = 3, L = 7.5 WAVELENGTHS •WITH A MAXIMUM OF 1% RANDOM VARIATION IN THE ELEMENT STRENGTHS
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
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
ANTENNA PATTERN SYNTHESIS REMAINS A TOPIC OF INTEREST . . . 1. W. W. Hansen and J. R. Woodyard, “A New Principle in Directional Antenna Design,” Proc. IE, 26, 3, March, 1938, pp. 333-345. 2. S. A. Schelkunoff, “A Mathematical Theory of Linear Arrays,” Bell System Technical Journal, 22, pp. 80-107, 1943. 3. 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. 4. 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. 5. 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. 6. Robert S. Elliott, “On Discretizing Continuous Aperture Distributions,” IEEE Transactions on Antennas and Propagation, AP-25, 5, pp. 617-621, September 1977.
ANTENNA PATTERN SYNTHESIS REMAINS A TOPIC OF INTEREST . . . 7. Robert S. Elliott, Antenna Theory and Design, Englewood Cliffs, NJ, Prentice-Hall, 1981. 8. 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. 9. 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. 10. D. Marcano, and F. Duran, Synthesis of antenna arrays using genetic algorithms, IEEE Antennas and Propagation Magazine, 42, 3, pp. 12-20, June 2000. 11. 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. 12. 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.
ANTENNA PATTERN SYNTHESIS REMAINS A TOPIC OF INTEREST . . . 13. 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. 14. 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. 15. N. G. Gomez, J. J. Rodriguez, K. L. Melde, and K. M. McNeill, Design of LowSidelobe Linear Arrays With High Aperture Efficiency and Interference Nulls, IEEE Antennas and Wireless Propagation Letters, 8, pp. 607-610, 2009. 16. 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. 17. 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.
ANTENNA PATTERN SYNTHESIS REMAINS A TOPIC OF INTEREST . . . 18. P. S. Apostolov, Linear Equidistant Antenna Array With Improved Selectivity, IEEE Transactions on Antennas and Propagation, 59, 10, pp. 3940-3943, October 2011. 19. 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. 20. 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. 21. 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.
VARIOUS SOURCE DISTRIBUTIONS AND/OR PATTERNS FROM THE FOLLOWING SOURCES WERE USED 22. AHMAD SAFAAI-JAZI, “A NEW FORMULATION FOR THE DESIGN OF CHEBYSHEV ARRAYS,” IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, AP42, 3, PP. 349-443, MARCH 1994. C. A. BALANIS, “ANTENNA THEORY, ANALYSIS AND DESIGN,” HARPER AND ROWE, 1982. E. V. JULL, “RADIATION FROM APERTURES,” IN ANTENNA HANDBOOK, ed. Y. T. LO AND S. W. LEE, VAN NOSTRAND REINHOLD CO., 1988.
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
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 ARE OBTAINED USING THESE ANGLES AND THE DESIRED VALUES OF THE LOBE MAXIMA 5) RETURNING TO 2) THESE NEW CURRENTS ARE USED TO COMPUTE A NEW PATTERN & THE PROCESS CONINUES UNTIL THE PATTERN CONVERGES
EVEN AND ODD NUMBERS OF ELEMENTS WERE USED FOR SYMMETRIC ARRAYS •FOR SYMMETRIC ARRAYS THE PATTERN CAN BE WRITTEN AS . . . N
P(! ) = " Sn cos[( 2n # 1) u] n=1
OR N
P(! ) = " Sn cos( 2nu) n=0
FOR AN EVEN OR ODD NUMBER OF ELEMENTS RESPECTIVELY, WHERE *# !d & u = ,% ( cos) / +$ " ' .
LOBE MAXIMA GENERATE A MATRIX . . . 1) The initial pattern P1(θ) is sampled finely enough in θ to accurately locate its positive and negative maxima at the angles θ1,n, n = 1,…,N with the corresponding pattern maxima denoted by P1(θ1,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
" cos( u11) $ cos( u12 ) $ [ M1, N ] = $ M $ # cos( u1N )
cos( 3u11) cos( 3u12 ) M cos( 3u1N )
L cos[( 2N ! 1) u11] % ' L cos[( 2N ! 1) u12 ] ' ' O M ' L cos[( 2N ! 1) u1N ]&
. . . WHICH IS THEN INVERTED TO SOLVE FOR A NEW SET OF CURRENTS S1,n FROM S1,1 ! % cos( u11) # ' S1, 2 # ' cos( u12 ) = M # ' M # ' S1, N " &cos( u1N )
cos( 3u11) L cos[( 2N $ 1) u11] ! L1 ! # # cos( 3u12 ) L cos[( 2N $ 1) u12 ] # L2 # # M# M O M # # cos( 3u1N ) L cos[( 2N $ 1) u1N ]" LN " $1
. . . WHERE THE Ln ARE THE MAXIMUM VALUES DESIRED FOR THE LOBES OF THE SYNTHESIZED PATTERN
A SECOND SET OF PATTERN MAXIMA P2(θ2,n) AND MATRIX [M2,N] ARE COMPUTED TO OBTAIN AN UPDATED SET OF CURRENTS . . . S2,1 ! % cos( u21 ) cos( 3u21 ) ' S2,2 # ' cos( u22 ) cos( 3u22 ) #= M # ' M M ' S2,N #" &cos( u2N ) cos( 3u2N )
L cos[(2N $ 1) u21 ] ! L1 ! # L cos[(2N $ 1) u22 ] # L2 # # # M # O M # L cos[(2N $ 1) u2N ]" LN #" $1
. . . WHICH RESULTS IN A THIRD SET OF PATTERN MAXIMA P3(θ3,n), etc., UNTIL THE PATTERN CONVERGES ACCEPTABLY --ITERATION IS NECESSARY BECAUSE THE ANGLES AT WHICH MAXIMA OCCUR DEPEND SLIGHTLY ON THE CURRENT
FOR THE MORE GENERAL CASE OF A NON-SYMMETRIC ARRAY THE PATTERN CAN BE WRITTEN . . . N
P(! ) = " Sn exp
i ( kx n cos! + # n )
n=1
. . . WHICH LEADS TO A CURRENT COMPUTATION OF THE FORM . . . Si,1 ! % exp(ikx1 cos$ i1 ) exp(ikx 2 cos $ i1 ) # ' Si,2 # 'exp(ikx1 cos$ i2 ) exp(ikx 2 cos $ i2 ) = M # ' M M # ' Si,N " &exp(ikx1 cos$ iN ) exp(ikx 2 cos $ iN )
L L O L
. . . FOR THE i’th ITERATION
(1
exp(ikx N cos $ i1 ) ! # exp(ikx N cos $ i2 )# # M # exp(ikx N cos $ iN )"
L1 ! # L2 # M# # LN "
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
•IF THE NEAR END-FIRE LOBES BECOME ILL FORMED --AS ABOVE
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
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
A SEQUENCE OF PATTERNS . . . 0
NORMALIZED PATTERN (dB)
D-L7N15L20R40dB7PassesUpdate
-20
-40
-60
-80
-100
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
•ITERATION #1
180
A SEQUENCE OF PATTERNS . . . 0
NORMALIZED PATTERN (dB)
D-L7N15L20R40dB7PassesUpdate
-20
-40
-60
-80
-100
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #2
180
A SEQUENCE OF PATTERNS . . . 0 D-L7N15L20R40dB7PassesUpdate
NORMALIZED PATTERN (dB)
-10 -20 -30 -40 -50 -60 -70 -80 -90 -100
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #3
180
A SEQUENCE OF PATTERNS . . . 0
NORMALIZED PATTERN (dB)
D-L7N15L20R40dB7PassesUpdate
-20
-40
-60
-80
-100
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #4
180
A SEQUENCE OF PATTERNS . . . 0
NORMALIZED PATTERN (dB)
D-L7N15L20R40dB7PassesUpdate
-20
-40
-60
-80
-100
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #5
180
A SEQUENCE OF PATTERNS . . . 0
NORMALIZED PATTERN (dB)
D-L7N15L20R40dB7PassesUpdate
-20
-40
-60
-80
-100
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #6
180
A SEQUENCE OF PATTERNS . . . 20
NORMALIZED PATTERN (dB)
D-L7N15L20R40dB7PassesUpdate
0
-20
-40
-60
-80
-100
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
180
ITERATION #7 AND THE FINAL PATTERN
THE PATTERN DETERIORATES FOR LOWER FREQUENCIES . . . NORMALIZED PATTERN (dB)
20
D-Left20Right40D0.1to0.8
Separation 0.1 wavelengths
G-N15L20R40Sep0.1to0.2
0 -20 -40 -60 -80
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
THE PATTERN DETERIORATES FOR LOWER FREQUENCIES . . . NORMALIZED PATTERN (dB)
20
D-Left20Right40D0.1to0.8
0.2
G-N15L20R40Sep0.1to0.2
0 -20 -40 -60 -80
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
. . . WITH SIDELOBES MAINTAINED OVER A NEARLY 2:1 BANDWIDTH . . . 0
D-Left20Right40D0.1to0.8
NORMALIZED PATTERN (dB)
G-N15L20R40Sep0.3to0.5
-20
-40
-60
-80
Separation 0.3 wavelengths
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
. . . WITH SIDELOBES MAINTAINED OVER A NEARLY 2:1 BANDWIDTH . . . 0
D-Left20Right40D0.1to0.8
NORMALIZED PATTERN (dB)
G-N15L20R40Sep0.3to0.5
-20
-40
-60
-80
0.4
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
. . . WITH SIDELOBES MAINTAINED OVER A NEARLY 2:1 BANDWIDTH . . . 0
D-Left20Right40D0.1to0.8
NORMALIZED PATTERN (dB)
G-N15L20R40Sep0.3to0.5
-20
-40
-60
-80
0.5
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
. . . AND DEVELOPS GRATING LOBES FOR HIGHER FREQUENCIES . . . NORMALIZED PATTERN (dB)
20
Separation 0.6 wavelengths
D-Left20Right40D0.1to0.8 G-N15L20R40Sep0.6to0.8
0 -20 -40 -60 -80
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
. . . AND DEVELOPS GRATING LOBES FOR HIGHER FREQUENCIES . . . NORMALIZED PATTERN (dB)
20
0.7
D-Left20Right40D0.1to0.8 G-N15L20R40Sep0.6to0.8
0 -20 -40 -60 -80
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
. . . AND DEVELOPS GRATING LOBES FOR HIGHER FREQUENCIES . . . NORMALIZED PATTERN (dB)
20
0.8
D-Left20Right40D0.1to0.8 G-N15L20R40Sep0.6to0.8
0 -20 -40 -60 -80
0
45 90 135 ANGLE FROM ARRAY AXIS (degrees)
180
A STANDARD DOLPH-CHEBYSHEV PATTERN IS READILY GENERATED . . .
•9-ELEMENT ARRAY 4 WAVELENGTHS LONG
VARIATIONS ON THE DOLPH-CHEBYSHEV DESIGN ARE EASY TO DEVELOP . . .
•15-ELEMENT ARRAY, 7 WAVELENGTHS LONG
VARIATIONS ON THE DOLPH-CHEBYSHEV DESIGN ARE EASY TO DEVELOP . . .
•15-ELEMENT ARRAY, 7 WAVELENGTHS LONG
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#1
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
STARTING PATTERN
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#2
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #1
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#3
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #2
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#4
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #3
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#5
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #4
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#6
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #5
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#7
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #6
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#8
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #7
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#9
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #8
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#10
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #9
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#11
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #10
180
A PATTERN DESIGNED WITH 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB 20
D-L7N15dB70to0 G-L7N15dB70to0#12
NORMALIZED PATTERN (dB)
0 -20 -40 -60 -80 -100 -120
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
ITERATION #11
•15-ELEMENT ARRAY, 7 WAVELENGTHS LONG
180
THE DOLPH-CHEBYSHEV PATTERN DOES NOT REQUIRE UNIFORM SPACING NORMALIZED PATTERN (dB)
0
-20
-40
-60
-80
0
45
90
ANGLE FROM ARRAY
135
AXIS (degrees)
180
•VARIABLE SPACINGS OF 0.4 AND 0.6 WAVELENGTHS
THE DOLPH-CHEBYSHEV PATTERN DOES NOT REQUIRE UNIFORM SPACING NORMALIZED PATTERN (dB)
0
-20
-40
-60
-80
0
45
90
ANGLE FROM ARRAY
135
AXIS (degrees)
180
•VARIABLE SPACINGS OF 0.4 TO 0.7 WAVELENGTHS
THE DOLPH-CHEBYSHEV PATTERN DOES NOT REQUIRE UNIFORM SPACING NORMALIZED PATTERN (dB)
0
-20
-40
-60
-80
0
45
90
ANGLE FROM ARRAY
135
AXIS (degrees)
180
•ARRAY LENGTHS OF 4 (BLACK) AND 5 (BLUE) WAVELENGTHS RESPECTIVELY
THE DOLPH-CHEBYSHEV PATTERN DOES NOT REQUIRE UNIFORM SPACING NORMALIZED PATTERN (dB)
0
-20
-40
-60
-80
0
45
90
ANGLE FROM ARRAY
135
AXIS (degrees)
180
•VARIABLE SPACING (BLACK) AND UNIFORM SPACING (RED) RESPECITVELY
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 for the first two iterations.
SOME EXTENSIONS OF THE BASIC IDEA MIGHT INVOLVE SUCH THINGS AS CONTROLLING: °NULLS °SIDE-LOBE ANGLES °MAIN LOBE ANGLE °THE NUMBER OF SIDE LOBES
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
PRONY’S METHOD OR ITS EQUIVALENT PROVIDES THE ARRAY PARAMETERS FROM PATTERN SAMPLES •GIVEN A DESIRED PATTERN Pdesired(θ) . . . N
Pdesired (! ) " PDSA (! ) = $ S# e kz# cos(! ) # =1
•. . . THE N SOURCE STRENGTHS Sα AND N LOCATIONS zα CAN BE OBTAINED •FOR THE ARRAY TO BE REALIZABLE USING ISOTROPIC SOURCES zα MUST BE PURE IMAGINARY •OTHERWISE A SOURCE DIRECTIVITY WOULD BE REQUIRED AS GIVEN BY D! = e kz! ,real cos(" )
IMPLEMENTING PRONY’S METHOD FOR PATTERN SYNTHESIS INVOLVES CHOOSING 3 PARAMETERS . . . •THE ANGLE SAMPLING INTERVAL Δcosθ --MUST BE SMALL ENOUGH TO AVOID ALIASING
•THE TOTAL ANGLE OBSERVATION WINDOW W MEASURED IN UNITS OF cosθ --MUST BE WIDE ENOUGH TO AVOID ILL CONDITIONING OF THE DATA MATRIX
THE LOBES OF A LINEAR ARRAY ARE SPACED UNIFORMLY IN COS(θ) 30
D-L20UCFPattAdaptNew G-L20UCFvsCosang,angle
FAR FIELD (dB)
20
10
0
-10
-20
-30 -90
-45
COS(ANGLE)x90
0
45
ANGLE
90
•THIS SHOWS THAT SAMPLING AS A FUNCTION OF COS(θ) RATHER THAN θ IS MORE APPROPRIATE •BESIDES WHICH PRONY’S METHOD REQUIRES THAT SAMPLING USE EQUAL STEPS IN COS(θ)
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/Δcosθ) + 1
. . . WHICH RESULTS IN REQUIRING THAT N BE THE LARGER OF N ≥ WL + 1 AND N≥R WITH L THE SOURCE SIZE IN WAVELENGTHS, R THE PATTERN RANK AND W THE WINDOW WIDTH
THE RESULTS THAT FOLLOW WERE GENERALLY OBTAINED USING THE FOLLOWING STEPS: •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
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
A USEFUL INITIAL TEST IS A MODIFIED PATTERN OF A SINUSOIDAL CURRENT FILAMENT •ITS FAR-FIELD PATTERN IS GIVEN BY $ e(ikL / 2)cos! + e #(ikL / 2)cos! # 2cos(kL /2) ' PMSCF (! ) = sin ! " PSCF (! ) = sin ! & ) sin ! % (
•PMSCF(θ) 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
TWO DIFFERENT WINDOW WIDTHS PRODUCE ESSENTIALLY IDENTICAL PATTERN MATCHES NORMALIZED PATTERN (dB)
0
-20
-40
-60
GM -0.05to0.05 FM -0.05to0.05 GM -0.999to0.999 FM -0.999to0.999
-80
-100
D-L5SCFxSINVarCOSANG G-L5DelCos0.1&2x0.999Patts
0
45
90
135
180
ANGLE FROM CURRENT AXIS (degrees)
•LENGTH OF SCF IS 5 WAVELENGTHS •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
SINGULAR-VALUE SPECTRA
SINGULAR-VALUE SPECTRA FOR SEVERAL WINDOW WIDTHS EXHIBIT A PATTERN RANK OF 3 FOR PMSCF . . . 10 2 10 1 10 0 10 - 1 -2 10 10 - 3 10 - 4 10 - 5 -6 10 10 - 7 10 - 8 10 - 9 -10 10 10 -11 10 -12 -13 10 -14 10 10 -15 -16 10 -17 10 10 -18 10 -19 -20 10 -21 10 10 -22 -23 10 -24 10
0
G-L5CosangVarSingValuesw/0IDs
2
4
6
SINGULAR VALUES
8
10
•N WAS INCREASED FOR EACH WINDOW UNTIL THE PATTERN CONVERGED •RESULT IS CONSISTENT WITH A 3 POINT SOURCES
. . . 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
THE NUMBER OF FITTING MODELS NEEDED FOR A CONVERGED PATTERN INCREASES SYSTEMATICALLY WITH WINDOW WIDTH NUMBER OF FITTINGS MODELS
12
10
8
6
4 G-L5FMsVsCosangw/oIDs
2 0.0
0.5
1.0
1.5
WIDTH OF OBSERVATION WINDOW
•TO AVOID ALIASING
2.0
THE CENTER SOURCE DISAPPEARS FOR A SCF 5.5 WAVELENGTHS LONG
•SAMPLED OVER A -0.05 TO +0.05 COSθ WINDOW
AN 11-TERM FITTING MODEL MATCHES THE ACTUAL PATTERN OF A 5WAVELENGTH SCF DOWN TO -60 dB . . . 0
NORMALIZED PATTERN (dB)
-10 -20 -30 -40 -50 -60 -70 -80
GENERATING MODEL FITTING MODEL
-90
-100
D-PronyN11L5SCFPattern
SAMPLES USED FOR FITTING MODEL
0
45
90
G-PronyN11L5SCFPatterm
135
180
ANGLE FROM CURRENT AXIS (degrees)
•THE BLACK DOTS DENOTE THE GENERATINGMODEL SAMPLES USED TO COMPUTE THE 11POLE FITTING MODEL
. . . BUT THE DERIVED SOURCE DISTRIBUTION IS NOT PHYSICALLY REALIZABLE . . .
•. . . BECAUSE SOME OF SCF 9 SOURCES HAVE REAL COMPONENTS IN THE COMPLEX SPACE PLANE
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
THE PATTERN OF A ±1 SQUARE-WAVE APERTURE IS GRAPHICALLY INDISTINGUISHABLE FROM AN 11-TERM FM . . . NORMALIZED PATTERN (dB)
20
GENERATING MODEL FITTING MODEL SAMPLES USED FOR FITTING MODEL
0 -20
-40 -60 -80
-100
D-L5N11±UCFPronyPattP1D24
G-L5N11±UCFPronyPattP1D24
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
•THE PATTERN FACTOR IS
% % "L '' 1! cos cos$ ) &# (* P± = L) * "L cos$ ) * # & (
180
. . . WHOSE SYNTHESIZED SOURCES ARE NOT UNIFORMLY SPACED
•FOR A 5-WAVELENGTH APERTURE •AND AN 11-POLE FITTING MODEL
THE PATTERN OF AN APERTURE 2 VARYING AS cos (π/L) IS ALSO GRAPHICALLY IDENTICAL TO ITS PRONY FM . . . 0 D-PronySynL5Cos^2N11
NORMALIZED PATTERN (dB)
G-PronySynL5Cos^2N11
-20
-40
-60 GENERATING MODEL FITTING MODEL SAMPLES USED FOR FITTING MODEL
-80
-100
0
45
90
135
180
ANGLE FROM CURRENT AXIS (degrees)
•ITS PATTERN FACTOR IS GIVEN BY Pcos2
sin(u) # ! 2 & = !L sin # % 2 2( u = WHERE u $! " u ' "
(
)
… WHOSE SOURCE DISTRIBUTION IS ALSO NONUNIFORM
•FOR A 5-WAVELENTH APERTURE •USING AN 11-TERM FITTING MODEL
PATTERN OF UNIFORM CURRENT OF LENGTH L TIMES (sinθ)P HAS TAPERED SIDELOBES WITH INCREASING P NORMALIZED PATTERN (dB)
0
P = 0
-20
1 2 3
-40
-60
0
45
D-L5UCFxSIN^XNVarActualPoles
90
135
ANGLE FROM ARRAY AXIS (degrees)
G-L5UCFxSIN^XNVarActualPoles
•ITS PATTERN FACTOR IS PUCF = (sin ! ) P
sin(kL cos! ) kL cos!
180
ITS SOURCES ARE ALSO NONUNIFORMLY SPACED
. . . USING 11 EXPONENTIALS IN THE FITTING MODEL •AND FOR A 5-WAVELENGTH APERTURE
A DOLPH-CHEBYSHEV ARRAY IS READILY SYNTHESIZED 0
D-DC^1VarL4.5...&N10...dB26...
NORMALIZED PATTERN (dB)
G-DC^1tL4.5dB26w/oPts
-26
-52
Generating Model Samples Used for FItting Model Fitting Model -78
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
180
•5-WAVELENGTHS LONG WITH -26 dB SIDELOBES AND 10 ELEMENTS
A MODIFIED DOLPH-CHEBYSHEV ARRAY IS ALSO SYNTHESIZED 0
D-L7N15-20&-40dBDCPronySyn
NORMALIZED PATTERN (dB)
G-L7N15-20&-40dBDCPronySyn
-20
-40 -60 Generating Model
-80
-100
Fitting Model GM Samples Used for FM
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
180
•-20 AND -40 dB SIDELOBES •15 ELEMENTS UNIFORMLY SPACED •7 WAVELENGTHS LONG
THIS ARRAY STEPS UP IN 5 dB INCREMENTS FROM LEFT TO RIGHT 0
D-L7N15-70to0dBPronySyn
NORMALIZED PATTERN (dB)
G-L7N15-70to0dBPronySyn
-20
-40
-60 Generating Model
-80
-100
Fitting Model GM Samples Used for FM
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
180
•15 ELEMENTS UNIFORMLY SPACED •7 WAVELENGTHS LONG
SINGULAR VALUES
SVD SPECTRA FOR SEVERAL ARRAYS ILLUSTRATE THEIR DIFFERENCES 10 0 10 - 1 10 - 2 -3 10 -4 10 10 - 5 10 - 6 -7 10 10 - 8 10 - 9 -10 10 10 -11 -12 10 -13 10 -14 10 10 -15 10 -16 10 -17 -18 10 10 -19 10 -20 10 -21 -22 10 10 -23 10 -24 -25 10
1
D-SVsVariousSources
7.5 Wavelength 10-Element DC Array 5-Wavelength Sinusoid 5-Wavelength COS^2 Aperture 5-Wavelength Uniform Current Filament
3
5
7
9
11
13
15
SINGULAR-VALUE ORDER
G-PronySynVarSrcsSVD
•THE DOLPH-CHEBYSHEV ARRAY CLEARLY SHOWS THE NUMBER OF ELEMENTS IT CONTAINS •THE SPECTRA OF THE CONTINUOUS DISTRIBUTIONS FALL OFF MORE SMOOTHLY
THIS PATTERN FROM ELLIOTT WAS REPLICATED USING PRONYS’ METHOD 0 G-N12L5.5ElliottPronySyn
NORMALIZED PATTERN (dB)
D-N12L5.5ElliottPronySyn
-20
-40
-60
GENERATING MODEL
-80
-100
SAMPLES USED FOR FITTING MODEL FITTING MODEL
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
180
Robert S. Elliott, “On Discretizing Continuous Aperture Distributions,” IEEE Transactions on Antennas and Propagation, AP-25, 5, pp. 617-621, September 1977. •12 ELEMENTS IN 5.5 WAVELENGTHS.
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
CONSIDER EXPONENTIATING A PATTERN: •FOR EXAMPLE THE PATTERN OF A 10-ELEMENT DOLPH-CHEBYSHEV ARRAY AS GIVEN BY . . . P10 (! ) = 2.798 cos(D) + 2.496 cos(3D) + 1.974 cos(5D) + 1.357 cos(7D) + cos(9D) WITH D = [(!d / ") cos# ] AND d THE ELEMENT SPACING
. . . TO YIELD SUCCESSIVELY LOWER SIDELOBES NORMALIZED PATTERN (dB)
0
Exponent M = 1
-26
2
-52
3
-78
4
-104 -130
-156
0
G-DC^1to4L4.5to18.5 D-DC^1VarL4.5...&N10...dB26...
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
180
•INITIAL ARRAY LENGTH IS 4.5 WAVELENGTHS
REFINING THE D-C -26 dB PATTERN USING * MATRIX-SYNTHESIS APPROACH YIELDS A WIDENING MAIN LOBE FOR FIXED L NORMALIZED PATTERN (dB)
0
-26
-52
-78
-104
-130
-156
0
G-L4.5N10DC26to104dB D-L4.5N10DC26to104dB
45
90
135
180
ANGLE FROM ARRAY AXIS (degrees)
Edmund K. Miller, “Synthesizing Linear-Array Patterns via Matrix Computation of Element Currents,” IEEE Antennas and Propagation Society Magazine, October, 2013, 55 (5), pp. 85-96.
SYNTHESIZING THE D-C -26 dB D-C PATTERN USING PRONY’S METHOD PROVIDES A 0.1 Db OR BETTER MATCH NORMALIZED PATTERN (dB)
0 EXPONENT M = 1
-26
2
-52
3
-78
4
-104
-130
0
G-PronyNewdB-26N10PVarDVar D-PronyNewdB-26N10PVarDVar
45
90
135
180
ANGLE FROM ARRAY AXIS (degrees)
•AT LEVELS ~ ≥ -110 dB •USING 10, 19, 28, & 37 POLES 4.5, 8.5, 13.5 AND 18.5 WAVELENGTHS LONG
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
PATTERNS WITH SAME SIDELOBE LEVELS WERE GENERATED WITH THE MATRIX APPROACH NORMALIZED PATTERN (dB)
0
-26
-52
-78
-104
-130
-156
0
G-VarL,N,dB26to104
45
90
135
180
ANGLE FROM ARRAY AXIS (degrees)
D-26,52,etc.dBArraysWithVariousL,N
. . . 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
SIMILAR RESULTS ARE OBTAINED WHEN THE D-C ARRAY IS 7.5 WAVELENGTHS LONG . . . 0
D-NewDC^x~180to0 G-L7.5DC^xPatternsNewLine
NORMALIZED PATTERN (dB)
EXPONENT M = 1 -26
2
-52
3
-78
4
-104
-130
-156
0
45
90
135
ANGLE FROM ARRAY AXIS (degrees)
180
SINGULAR VALUES
. . . FOR WHICH THE SINGULAR-VALUE SPECTRA INDICATE THE NUMBER OF ARRAY ELEMENTS 10 0 10 - 1 10 - 2 10 - 3 10 - 4 10 - 5 10 - 6 10 - 7 10 - 8 10 - 9 10 -10 10 -11 10 -12 10 -13 10 -14 10 -15 10 -16 10 -17 10 -18 10 -19 10 -20 10 -21 10 -22 10 -23 10 -24 10 -25 10 -26
0
3
2
4
EXPONENT M = 1 G-SVsL7.5Norm
10 D-L7.5DC^xMBPE
20
30
SINGULAR-VALUE ORDER
40
50
•FOR EXPONENT M = 1 TO 4 ARE 10, 19, 28, 37 RESPECTIVELY
THE NUMBER OF SINGULAR VALUES INCREASES LINEARLY WITH THE EXPONENT M MAXIMUM SINGULAR VALUE
40
30
20
10
D-L7.5DC^xMBPE G-MaxSVvsDC^x
0
1
2
3
4
EXPONENT OF DOLPH-CHEBYSHEV ARRAY (M)
•FOR A 7.5-WAVLENGTH, 10-ELEMENT DOLPHCHEBYSHEV ARRAY
THE MAIN BEAMWIDTH DECREASES FROM ABOUT 7.4 TO 3.6 DEGREES FOR AN EXPONENT PARAMETER VALUE OF 4 . . . NORMALIZED PATTERN (dB)
0
EXPONENT M = 1 -1
3
-2 2
M = 4 -3 85
86
G-DC^xRadPattGMMainLobe
87
88
89
90
91
92
93
ANGLE FROM ARRAY AXIS (degrees)
94
D-L7.5DC^xMBPE
•FOR THE 7.5 WAVELENGTH ARRAY
95
. . . AT THE -3 dB LEVEL
THE PRONY-DERIVED ARRAYS CAN HAVE WIDELY VARYING SOURCE STRENGTHS: 10 0
M = 1 10 -
1
M = 2 - 2
M = 3
10 -
3
10 -
4
10 -
5
15.006
13.339
11.672
10.005
8.338
6.671
5.004
3.337
1.670
0.003
-1.664
-3.331
-4.998
-6.665
-8.332
-9.999
-11.666
-13.333
EXPONENT M = 4
-15.000
ELEMENT CURRENT
10
ELEMENT POSITION (wavelengths) D-DC^xImagPoleVsRealRes
G-PolesVsResiduesDC^xVert
•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
. . . WITH EACH ARRAY SIZE VARYING LINEARLY WITH INCREASING EXPONENT 30
ARRAY WIDTH (wavelengths)
Initial Width 7.5 Wavelengths Initial Width 5 Wavelengths 20
10
D-Poles&ResiduesDC^4 G-DC^xWidthVsExponent
0
1
2
3
EXPONENT OF DOLPH-CHEBYSHEV ARRAY (M)
•AS MxINITIAL ARRAY WIDTH
4
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.
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))^2
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))^3
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 (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))^4
ABOVE PRONY-SYNTHESIZED ARRAYS ARE UNIFORMLY SPACED FOR M ≤ 3 BUT EXHIBIT A TAPERED SPACING FOR M ≥ 4 ELEMENT SEPARATION (wavelengths)
0.7 EXPONENT M = 5 M = 4
0.6 M = 2 M = 1
0.5 -21 -18 -15 -12 -9 D-N10,L5,DC10VarFMsVarP2
-6
-3
0
M = 3
3
6
9
12 15 18 21
ELEMENT NUMBERS
G-PronySynDC^1to5Spacing
•THE RESPECTIVE NUMBER OF ARRAY ELEMENTS ARE 10, 19, 28, 35, AND 42 FOR AN INITIAL ARRAY 5-WAVLENGTHS LONG
THE DOLPH-CHEBYSHEV SVD SPECTRUM ROLLS OFF SLOWER WITH INCREASING WINDOW WIDTH
SINGULAR VALUES
10 2 10 1 10 0 10 - 1
±0.5
10 - 2 10 - 3 10 - 4 10 - 5
WINDOW ±0.999
±0.3
10 - 6 10 - 7 10 - 8 10 - 9
±0.1
10 -10 10 -11 10 -12 10 -13 10 -14 10 -15 10 -16 10 -17 10 -18 10 -19 10 -20 10 -21 10 -22 10 -23 10 -24
1
3
G-PronyL10N10dB-26P1SVC D-PronyL10N10DB-26P1SVD
5
7
9
SINGULAR-VALUE ORDER
11
13
•FOR A 10-WAVELENGTH, 10-ELEMENT ARRAY
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 P4 = [ A1 cos(u ) + A 2 cos(3u)] M
M .
• FOR M = 2 THIS BECOMES 2 2 2 2 $ A1 + A2 ! A1 A2 P4 = +# + A1 A2 & cos(2u) + A1 A2 cos( 4u) + cos(6u) . 2 2 " 2 % 2
*
G. J. BURKE, PRIVATE COMMUNICATION, 2013 VIA MATHEMATICA
ITS PATTERNS FOR M = 3 AND M = 4 ARE GIVEN BY [
]
[
]
3 3 1 3 2 2 2 3 A1 + A1 A2 + 2A1 A2 cos( u) + A1 + 6A1 A2 + 3A2 cos( 3u) 4 4 3 2 3 1 3 2 2 + A1 A2 + A1 A2 cos(5u) + A1 A2 cos( 7u) + A2 cos(9u) 4 4 4 P4 = 3
[
]
AND 3 !1 4 1 3 1 4 $ 3 !1 4 2 2 3 2 2 3$ P4 = # A1 + A1 A2 + A1 A2 + A2 & + # A1 + A1 A2 + A1 A2 + A1 A2 & cos(2u) % 2 "3 % 2 "4 3 4 3! 1 4 1 2 2 3 !1 3 1 4$ 3 3$ 2 2 + # A1 + A1 A2 + A1 A2 + A1 A2 & cos( 4u) + # A1 A2 + A1 A2 + A2 & cos(6u) % % 2 "12 2 2 "3 3 3 !1 2 2 1 1 1 4 3$ 3 + # A1 A2 + A1 A2 & cos(8u) + A1 A2 cos(10u) + A2 cos(12u). % 2 "2 3 2 8 4
RESPECTIVELY
PRONY-SYNTHESIZED AND ANALYTIC PATTERNS FROM THE PREVIOUS FORMULAS AGREE TO WITHIN 0.1 dB . . .
. . . FOR A 2-WAVELENGTH ARRAY . . .
. . . WHOSE ELEMENT STRENGTHS ARE FOUND TO BE . . . Element Number 1 2 3 4 5 6 7
Basic Array 4 Elements 0.8794 1
TABLE 1. M=2 7 Elements 1.773 2.532 1.759 1
M=3 10 Elements 9.640 8.320 4.958 2.638 1
M=4 13 Elements 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
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
SYNTHESIZED EXPONENTIATED PATTERNS FOR A TRIANGLE CURRENT FILAMENT ARE IMPROVED OVER THE UCF
.
•FOR A 5-WAVELENGTH CURRENT FILAMENT
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
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
WIDE DYNAMIC RANGE OF SOURCE STRENGTHS CAN MAKE PATTERNS NOISE SENSITIVE . . .
•FOR M = 3, L = 7.5 WAVELENGTHS •WITH A MAXIMUM OF 1% RANDOM VARIATION IN THE ELEMENT STRENGTHS
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
