two novel approaches to antenna-pattern synthesis

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

[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


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.


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.

ADAPTIVE SAMPLING REDUCES NUMBER OF PATTERN SAMPLES REQUIRED

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PATT

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Average FMGM PatternGM Samples Used


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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.

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.

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.

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.


•THE BASIC IDEA

•SEVERAL EXAMPLES


•SOME BACKGOUND





THE APPROACH IS STRAIGHTFORWARD: 1) A LINEAR-ARRAY GEOMETRY IS CHOSEN --TYPICALLY UNIFORM SPACING IS USED, BUT THIS IS NOT MANDATORY

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


UNIFORM PHASE OF ZERO OR A SMALL POSITIVE ANGLE 3) THE FAR-FIELD PATTERN IS COMPUTED


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


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

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[ ]


WRITTEN AS . . .

OR

!

P "( ) = Snn=1

N

# cos 2n $1( )u[ ]

!

P "( ) = Snn=0

N

# cos 2nu( )


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

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



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[ ]

#

$

% % % %

&

'

( ( ( (

. . . 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

"

#

$ $ $ $

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

"

#

$ $ $ $

. . . 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

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( )

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

"

#

$ $ $ $

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



ELEMENTS --INCREASE OR DECREASE THE ARRAY LENGTH --ADJUST THE PATTERN SPECIFICATION



ELEMENTS --INCREASE OR DECREASE THE ARRAY LENGTH --ADJUST THE PATTERN SPECIFICATION

•IF THE NEAR END-FIRE LOBES BECOME ILL FORMED

--AS ABOVE


•THE BASIC IDEA

•SEVERAL EXAMPLES


•SOME BACKGOUND





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 . . .

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D P

AT

TE

RN

(dB

)

•THE INITIAL PATTERN FOR EQUAL SOURCES


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)

ITERATION #1


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ITERATION #2


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ITERATION #3


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ITERATION #4


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ITERATION #5


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ITERATION #6 AND THE FINAL PATTERN

THE PATTERN DETERIORATES FOR LOWER FREQUENCIES . . .

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D-Left20Right40D0.1to0.8


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THE PATTERN DETERIORATES FOR LOWER FREQUENCIES . . .

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200.2



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ZED

PA

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N (d

B) G-N15L20R40Sep0.1to0.2

. . . WITH SIDELOBES MAINTAINED OVER A NEARLY 2:1 BANDWIDTH . . .

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G-N15L20R40Sep0.3to0.5


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0.4



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PA

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N (d

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N (d

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. . . AND DEVELOPS GRATING LOBES FOR HIGHER FREQUENCIES . . .

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N (d

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N (d

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ZED

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N (d

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A DOLPH-CHEBYSHEV -26 dB PATTERN FOR A 4.5 WAVELENGTH ARRAY IS READILY SYNTHESIZED . . .

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D-L4.5N10DC26to104dB ANGLE FROM ARRAY AXIS (degrees)

NO

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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.

. . . BUT SUCCESIVELY REDUCING THE SIDE-LOBE LEVEL RESULTS IN A WIDENING MAIN LOBE

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N (d

B)


Edmund K. Miller, “Synthesizing Linear-Array Patterns via Matrix Computation of Element Currents,” accepted for publication, IEEE Antennas and Propagation Society Magazine, 2013.

. . . BUT SUCCESIVELY REDUCING THE SIDE-LOBE LEVEL RESULTS IN A WIDENING MAIN LOBE

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-52

-26

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NO

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ALI

ZED

PA

TTER

N (d

B)


MAINTAINING A 0.5 SPACING PRODUCES PATTERNS WITH PROPORTIONATELY MORE LOBES & NARROWER MAIN LOBE

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-26

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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

MAINTAINING A 0.5 SPACING PRODUCES PATTERNS WITH PROPORTIONATELY MORE LOBES & NARROWER MAIN LOBE

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-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

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

THIS PATTERN HAS 15 LOBE MAXIMA INCREASING IN STEPS OF 5 dB •15-ELEMENT ARRAY, 7 WAVELENGTHS LONG

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

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.

SOME EXTENSIONS OF THE BASIC IDEA MIGHT INVOLVE SUCH THINGS AS CONTROLLING: °NULLS

SOME EXTENSIONS OF THE BASIC IDEA MIGHT INVOLVE SUCH THINGS AS CONTROLLING: °NULLS °SIDE-LOBE ANGLES

SOME EXTENSIONS OF THE BASIC IDEA MIGHT INVOLVE SUCH THINGS AS CONTROLLING: °NULLS °SIDE-LOBE ANGLES °MAIN LOBE ANGLE

SOME EXTENSIONS OF THE BASIC IDEA MIGHT INVOLVE SUCH THINGS AS CONTROLLING: °NULLS °SIDE-LOBE ANGLES °MAIN LOBE ANGLE °THE NUMBER OF SIDE LOBES


•THE BASIC IDEA

•SEVERAL EXAMPLES






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

%



!


$ =1

N

%

•. . . THE N SOURCE STRENGTHS S! AND N LOCATIONS z! CAN BE OBTAINED



!


$ =1

N

%


•FOR THE ARRAY TO BE REALIZABLE USING ISOTROPIC SOURCES z! MUST BE PURE IMAGINARY



!


$ =1

N

%


•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 #( )

IMPLEMENTING PRONY’S METHOD FOR PATTERN SYNTHESIS INVOLVES CHOOSING 3 PARAMETERS . . .

•THE ANGLE SAMPLING INTERVAL "cos! --MUST BE SMALL ENOUGH TO AVOID ALIASING


•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

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


90450-45-90-30

-20

-10

0

10

20


COS(ANGLE)x90 ANGLE

FAR

FIE

LD (d


•THIS SHOWS THAT SAMPLING AS A FUNCTION OF COS(!) RATHER THAN ! IS MORE APPROPRIATE


90450-45-90-30

-20

-10

0

10

20


COS(ANGLE)x90 ANGLE

FAR

FIE

LD (d


•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(!)


•THE NUMBER OF POLES OR EXPONENTIALS N --FOR WHICH THE NUMBER OF PATTERN SAMPLES

REQUIRED IS 2N = (W/"cos!) + 1


•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

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





•ROUTINELY COMPUTING THE SVD SPECTRUM OF THE DESIRED PATTERN

•USING A COMPUTE PRECISION OF 24 DIGITS


•THE BASIC IDEA

•SEVERAL EXAMPLES






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

A USEFUL INITIAL TEST IS A MODIFIED PATTERN OF A SINUSOIDAL CURRENT FILAMENT*


!


%

& '

(

) *


•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.

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


18013590450-100

-80

-60

-40

-20

0




NO

RM

AL

IZE

D P

AT

TE

RN

(dB

)


•WINDOWS OF -0.999 TO + 0.999 AND -0.05 TO + 0.05 IN

!

cos" WERE USED


18013590450-100

-80

-60

-40

-20

0




NO

RM

AL

IZE

D P

AT

TE

RN

(dB

)



!

cos" WERE USED •TWO ARROWS INDICATE THE EXTENT OF THE

LATTER


18013590450-100

-80

-60

-40

-20

0




NO

RM

AL

IZE

D P

AT

TE

RN

(dB

)



!

cos" WERE USED •TWO ARROWS INDICATE THE EXTENT OF THE

LATTER •LENGTH OF SCF IS 5 WAVELENGTHS

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


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


•N WAS INCREASED FOR EACH WINDOW UNTIL THE PATTERN CONVERGED


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


•N WAS INCREASED FOR EACH WINDOW UNTIL THE PATTERN CONVERGED

•RESULT IS CONSISTENT WITH 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


(a) (b)


•PHASE IS SHOWN ON A POLAR PLOT


(a) (b)


•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

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

THE CENTER SOURCE DISAPPEARS FOR A MODIFIED SCF 5.5 WAVELENGTHS LONG

•SAMPLED OVER A -0.05 TO +0.05 COS! WINDOW

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


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

. . . BUT THE DERIVED SOURCE DISTRIBUTION IS NOT PHYSICALLY REALIZABLE . . .

•. . . BECAUSE SOME OF SCF 9 SOURCES HAVE REAL COMPONENTS IN THE COMPLEX SPACE PLANE


•THE BASIC IDEA

•SEVERAL EXAMPLES






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.

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


NO

RM

ALI

ZED

PA

TTER

N (d

B)

G-L5N11±UCFPronyPattP1D24

•THE PATTERN FACTOR IS

!

P± = L1" cos #L

$cos%&

' ( )

#L$cos%

&

'

* * *

(

)

+ + +

. . . WHOSE SYNTHESIZED SOURCES ARE NOT UNIFORMLY SPACED

•FOR A 5-WAVELENGTH APERTURE

. . . WHOSE SYNTHESIZED SOURCES ARE NOT UNIFORMLY SPACED

•FOR A 5-WAVELENGTH APERTURE •AND A 11-POLE FITTING MODEL

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


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$

… WHOSE SOURCE DISTRIBUTION IS ALSO NONUNIFORM

•FOR A 5-WAVELENTH APERTURE

… WHOSE SOURCE DISTRIBUTION IS ALSO NONUNIFORM

•FOR A 5-WAVELENTH APERTURE •USING AN 11-TERM FITTING MODEL

A CURRENT FILAMENT OF LENGTH L VARYING AS (sin!)P HAS TAPERED SIDELOBES WITH INCREASING P . . .

18013590450-60

-40

-20

0

G-L5UCFxSIN^XNVarActualPoles


NO

RM

ALI

ZED

PA

TTER

N (d

B)

D-L5UCFxSIN^XNVarActualPoles

P = 0

•ITS PATTERN FACTOR IS

!

PUCF = (sin")P sin(kLcos")kLcos"


1801359 04 50-60

-40

-20

0



NO

RM

ALI

ZED

PA

TTER

N (d

B)


P = 0

1


!



1801359 04 50-60

-40

-20

0



NO

RM

ALI

ZED

PA

TTER

N (d

B)


P = 0

12


!



1801359 04 50-60

-40

-20

0



NO

RM

ALI

ZED

PA

TTER

N (d

B)


P = 0

123


!


. . . ALSO HAS SOURCES THAT ARE NON-UNIFORMLY SPACED

. . . USING 11 EXPONENTIALS IN THE FITTING MODEL

. . . ALSO HAS SOURCES THAT ARE NON-UNIFORMLY SPACED

. . . USING 11 EXPONENTIALS IN THE FITTING MODEL •AND FOR A 5-WAVELENGTH APERTURE

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...


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

A MODIFIED DOLPH-CHEBYSHEV ARRAY

18013590450-100

-80

-60

-40

-20

0

Generating ModelFitting ModelGM Samples Used for FM

D-L7N15-20&-40dBDCPronySyn


NO

RM

ALI

ZED

PA

TTER

N (d

B)

G-L7N15-20&-40dBDCPronySyn

•-20 AND -40 dB SIDELOBES


18013590450-100

-80

-60

-40

-20

0




NO

RM

ALI

ZED

PA

TTER

N (d

B)


•-20 AND -40 dB SIDELOBES •15 ELEMENTS UNIFORMLY SPACED


18013590450-100

-80

-60

-40

-20

0




NO

RM

ALI

ZED

PA

TTER

N (d

B)


•-20 AND -40 dB SIDELOBES •15 ELEMENTS UNIFORMLY SPACED •7 WAVELENGTHS LONG

THIS ARRAY STEPS UP IN 5 dB INCREMENTS FROM LEFT TO RIGHT

18013590450-100

-80

-60

-40

-20

0


D-L7N15-70to0dBPronySyn


NO

RM

ALI

ZED

PA

TTER

N (d

B)

G-L7N15-70to0dBPronySyn

•15 ELEMENTS UNIFORMLY SPACED

THIS ARRAY STEPS UP IN 5 dB INCREMENTS FROM LEFT TO RIGHT

18013590450-100

-80

-60

-40

-20

0


D-L7N15-70to0dBPronySyn


NO

RM

ALI

ZED

PA

TTER

N (d

B)

G-L7N15-70to0dBPronySyn

•15 ELEMENTS UNIFORMLY SPACED •7 WAVELENGTHS LONG

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


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


SIN

GU

LAR

VA

LUES



•THE SPECTRA OF THE CONTINUOUS DISTRIBUTIONS FALL OFF SMOOTHLY


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


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


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


SIN

GU

LAR

VA

LUES



•THE SPECTRA OF THE CONTINUOUS DISTRIBUTIONS FALL OFF SMOOTHLY


•THE BASIC IDEA

•SEVERAL EXAMPLES






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



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


•ARRAY LENGTH IS 4.5 WAVELENGTHS




WHERE

!



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




WHERE

!



1801359 0450-156

-130

-104

-78

-52

-26

0


NO

RM

ALI

ZED

PA

TTER

N (d

B)

G-DC^1to4L4.5to18.5

Exponent M = 1

2




WHERE

!



1801359 0450-156

-130

-104

-78

-52

-26

0


NO

RM

ALI

ZED

PA

TTER

N (d

B)

G-DC^1to4L4.5to18.5

Exponent M = 1

2

3




WHERE

!



1801359 0450-156

-130

-104

-78

-52

-26

0


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

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


18013590450-130

-104

-78

-52

-26

0


NO

RM

ALI

ZED

PA

TTER

N (d

B)


EXPONENT M = 1

•SOLID LINES FOR PRONY FM, DOTS FOR GM


18013590450-130

-104

-78

-52

-26

0


NO

RM

ALI

ZED

PA

TTER

N (d

B)


EXPONENT M = 1

2


18013590450-130

-104

-78

-52

-26

0


NO

RM

ALI

ZED

PA

TTER

N (d

B)


EXPONENT M = 1

2

3


18013590450-130

-104

-78

-52

-26

0


NO

RM

ALI

ZED

PA

TTER

N (d

B)


EXPONENT M = 1

2

3

4


18013590450-130

-104

-78

-52

-26

0


NO

RM

ALI

ZED

PA

TTER

N (d

B)


EXPONENT M = 1

2

3

4

•AT LEVELS ~ ! -110 dB •USING 10, 19, 28, & 37 ELEMENTS

•INITIAL ARRAY LENGTH IS 4.5 WAVELENGTHS

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

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


NO

RM

AL

IZE

D P

AT

TE

RN

(dB

)

G-L7.5DC^xPatternsNewLineEXPONENT M = 1

2

3

4

. . . 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


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


SIN

GU

LAR

VA

LUES

G-SVsL7.5Norm

EXPONENT M = 1

2



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


SIN

GU

LAR

VA

LUES

G-SVsL7.5Norm

EXPONENT M = 1

2 3



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


SIN

GU

LAR

VA

LUES

G-SVsL7.5Norm

EXPONENT M = 1

2 3 4


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

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


9 594939 2919 08988878685-3

-2

-1

0


NO

RM

ALI

ZED

PA

TTER

N (d

B)


EXPONENT M = 1

2



9 594939 2919 08988878685-3

-2

-1

0


NO

RM

ALI

ZED

PA

TTER

N (d

B)


EXPONENT M = 1

2

3



9 594939 2919 08988878685-3

-2

-1

0


NO

RM

ALI

ZED

PA

TTER

N (d

B)


EXPONENT M = 1

2

3

M = 4


. . . AT THE -3 dB LEVEL

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


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


ELEM

ENT C

URRE

NT


M = 2

M = 1


•FOR A 7.5-WAVELENGTH D-C ARRAY NORMALIZED TO CENTER ELEMENTS


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


ELEM

ENT C

URRE

NT


M = 2

M = 3

M = 1


•FOR A 7.5-WAVELENGTH D-C ARRAY NORMALIZED TO CENTER ELEMENTS

•WITH IMPLICATIONS FOR NOISE SENSITIVITY


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


ELEM

ENT C

URRE

NT


EXPONENT M = 4

M = 2

M = 3

M = 1


•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

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

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)


•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)



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



COS(9.*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


•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 “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

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


211815129630-3-6-9-12-15-18-210.5

0.6

0.7


ELEM

ENT

SEPA

RA

TIO

N (w

avel

engt

hs)


M = 2

M = 1



211815129630-3-6-9-12-15-18-210.5

0.6

0.7


ELEM

ENT

SEPA

RA

TIO

N (w

avel

engt

hs)


M = 3

M = 2

M = 1



211815129630-3-6-9-12-15-18-210.5

0.6

0.7


ELEM

ENT

SEPA

RA

TIO

N (w

avel

engt

hs)


M = 4

M = 3

M = 2

M = 1



211815129630-3-6-9-12-15-18-210.5

0.6

0.7


ELEM

ENT

SEPA

RA

TIO

N (w

avel

engt

hs)


EXPONENT M = 5

M = 4

M = 3

M = 2

M = 1


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


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


SIN

GU

LAR

VA

LUES


±0.3

±0.1



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


SIN

GU

LAR

VA

LUES


±0.5

±0.3

±0.1



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


SIN

GU

LAR

VA

LUES


WINDOW ±0.999±0.5

±0.3

±0.1


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



!

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 .



!

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

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( )

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

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 . . .

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

EXPONENTIATED PATTERNS OF A UNIFORM CURRENT FILAMENT ARE NOT SYNTHESIZED AS WELL

•FOR A 5-WAVELENGTH FILAMENT

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


•FOR M = 2, L = 7.5 WAVELENGTHS •WITH A MAXIMUM OF 10% RANDOM VARIATION IN

THE ELEMENT STRENGTHS


•FOR M = 3, L = 5 WAVELENGTHS


•FOR M = 3, L = 5 WAVELENGTHS •WITH A MAXIMUM OF 1% RANDOM VARIATION IN



•FOR M = 4, L = 7.5 WAVELENGTHS •WITH A MAXIMUM OF 1% RANDOM VARIATION IN


PRESENTATION HAS DESCRIBED AN ITERATIVE APPROACH TO PATTERN SYNTHESIS USING A MATRIX BASED ON SPECIFIED LOBE MAXIMA . . .

•THE BASIC IDEA

•SEVERAL EXAMPLES






NORMALIZED RESISTANCE OF A DIPOLE WAS MODELED WITH 6 FMs

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THE FINAL FINELY SAMPLED RESULT

two novel approaches to antenna-pattern synthesis

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