TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESISs3.amazonaws.com/sdieee/1794-EKMiller-PatternSynthesis.pdf•THE BASIC IDEA •SEVERAL EXAMPLES ... “A New Principle in Directional

TWO NOVEL TWO NOVEL APPROACHES TO APPROACHES TO

ANTENNAANTENNA--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

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 Low-Sidelobe 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, AP-42, 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.


•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 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 . . .

OR

FOR AN EVEN OR ODD NUMBER OF ELEMENTS RESPECTIVELY, WHERE

P !( ) = Sn

n=1

N

" cos 2n #1( )u[ ]

P !( ) = Sn

n=0

N

" cos 2nu( )

u =!d"

#

$ %

&

' ( 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

M1,N[ ] =

cos u11( ) cos 3u

11( ) L cos 2N !1( )u11[ ]cos u

12( ) cos 3u12( ) L cos 2N !1( )u12[ ]

M M O M

cos u1N( ) cos 3u

1N( ) L 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

S1,1

S1, 2

M

S1,N

!

"

# # # #

=

cos u11( ) cos 3u

11( ) L cos 2N $1( )u11[ ]cos u

12( ) cos 3u12( ) L cos 2N $1( )u12[ ]

M M O M

cos u1N( ) cos 3u

1N( ) L cos 2N $1( )u1N[ ]

%

&

' ' ' '

!

"

# # # #

$1L1

L2

M

LN

!

"

# # # #

A SECOND SET OF PATTERN MAXIMA P2(θ2,n) AND MATRIX [M2,N] ARE COMPUTED TO OBTAIN AN UPDATED SET OF CURRENTS . . .

S2,1

S2,2

M

S2,N

!

"

# # # #

=

cos u21( ) cos 3u

21( ) L cos 2N $1( )u21[ ]

cos u22( ) cos 3u

22( ) L cos 2N $1( )u22[ ]

M M O M

cos u2N( ) cos 3u

2N( ) L cos 2N $1( )u2N[ ]

%

&

' ' ' '

!

"

# # # #

$1L1

L2

M

LN

!

"

# # # #

. . . 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 . . .

. . . WHICH LEADS TO A CURRENT COMPUTATION OF THE FORM . . .

. . . FOR THE i’th ITERATION

P !( ) = Sn

n=1

N

" expi kxn cos! + # n( )

Si,1

Si,2

M

Si,N

!

"

# # # #

=

exp ikx1 cos$i1( ) exp ikx2 cos$i1( ) L exp ikxNcos$

i1( )

exp ikx1 cos$i2( ) exp ikx2 cos$i2( ) L exp ikxNcos$

i2( )

M M O M

exp ikx1 cos$iN( ) exp ikx2 cos$iN( ) L exp ikxNcos$

iN( )

%

&

' ' ' '

!

"

# # # #

(1L1

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


•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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•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


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ITERATION #7 AND THE FINAL PATTERN

THE PATTERN DETERIORATES FOR LOWER FREQUENCIES . . .

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Separation 0.1 wavelengthsD-Left20Right40D0.1to0.8


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THE PATTERN DETERIORATES FOR LOWER FREQUENCIES . . .

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

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D-Left20Right40D0.1to0.8


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

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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

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G-L7N15dB70to0#1

STARTING PATTERN


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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


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ITERATION #7


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ITERATION #8


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ITERATION #9


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ITERATION #10


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ITERATION #11 •15-ELEMENT ARRAY, 7 WAVELENGTHS LONG

THE DOLPH-CHEBYSHEV PATTERN DOES NOT REQUIRE UNIFORM SPACING

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•VARIABLE SPACINGS OF 0.4 AND 0.6 WAVELENGTHS


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•VARIABLE SPACINGS OF 0.4 TO 0.7 WAVELENGTHS


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•ARRAY LENGTHS OF 4 (BLACK) AND 5 (BLUE)

WAVELENGTHS RESPECTIVELY


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•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


•THE BASIC IDEA

•SEVERAL EXAMPLES






PRONY’S METHOD OR ITS EQUIVALENT PROVIDES THE ARRAY PARAMETERS FROM PATTERN SAMPLES

•GIVEN A DESIRED PATTERN Pdesired(θ) . . .

Pdesired

(!) " PDSA

!( ) = S#ekz# cos !( )

# =1

N

$

•. . . 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!

= ekz! ,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)

90450-45-90-30

-20

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0

10

20

30D-L20UCFPattAdaptNew

COS(ANGLE)x90 ANGLE

FA

R F

IEL

D (

dB

)G-L20UCFvsCosang,angle

•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


•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(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

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D-L5SCFxSINVarCOSANG

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G-L5DelCos0.1&2x0.999Patts

•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 FOR SEVERAL WINDOW WIDTHS EXHIBIT 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

SP

EC

TR

A

G-L5CosangVarSingValuesw/0IDs

•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

2.01.51.00.50.02

4

6

8

10

12

WIDTH OF OBSERVATION WINDOW

NU

MB

ER

OF

FIT

TIN

GS

MO

DE

LS

G-L5FMsVsCosangw/oIDs

•TO AVOID ALIASING

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 5-WAVELENGTH SCF DOWN TO -60 dB . . .

18013590450-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

GENERATING MODEL

FITTING MODEL

SAMPLES USED FOR FITTING MODEL

D-PronyN11L5SCFPattern


NO

RM

AL

IZE

D P

AT

TE

RN

(d

B)

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






THE PATTERN OF A ±1 SQUARE-WAVE APERTURE IS GRAPHICALLY INDISTING- UISHABLE FROM AN 11-TERM FM . . .

18013590450-100

-80

-60

-40

-20

0

20GENERATING MODELFITTING MODELSAMPLES USED FOR FITTING MODEL

D-L5N11±UCFPronyPattP1D24


NO

RM

AL

IZE

D P

AT

TE

RN

(d

B)

G-L5N11±UCFPronyPattP1D24

•THE PATTERN FACTOR IS

P±

= L

1! cos"L#cos$%

&

'

(

"L#cos$

%

&

)

)

)

'

(

*

*

*

. . . WHOSE SYNTHESIZED SOURCES ARE NOT UNIFORMLY SPACED

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

THE PATTERN OF AN APERTURE VARYING AS cos2(π/L) IS ALSO GRAPHICALLY IDENTICAL TO ITS PRONY FM . . .

18013590450-100

-80

-60

-40

-20

0

GENERATING MODEL

FITTING MODEL


D-PronySynL5Cos^2N11


NO

RM

AL

IZE

D P

AT

TE

RN

(d

B)

G-PronySynL5Cos^2N11

•ITS PATTERN FACTOR IS GIVEN BY

Pcos

2 =sin(u)

u

! 2

! 2 " u2#

$ %

&

' (

WHERE

u = !L"( )sin#

… 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

18013590450-60

-40

-20

0

G-L5UCFxSIN^XNVarActualPoles


NO

RM

AL

IZE

D P

AT

TE

RN

(d

B)

D-L5UCFxSIN^XNVarActualPoles

P = 0

1

2

3

•ITS PATTERN FACTOR IS

PUCF

= (sin!)P sin(kLcos!)

kLcos!

ITS SOURCES ARE ALSO 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 Model

Samples Used for FItting Model

Fitting Model

D-DC^1VarL4.5...&N10...dB26...


NO

RM

AL

IZE

D P

AT

TE

RN

(d

B)

G-DC^1tL4.5dB26w/oPts

•5-WAVELENGTHS LONG WITH -26 dB SIDELOBES AND 10 ELEMENTS

A MODIFIED DOLPH-CHEBYSHEV ARRAY IS ALSO SYNTHESIZED

18013590450-100

-80

-60

-40

-20

0

Generating Model

Fitting Model

GM Samples Used for FM

D-L7N15-20&-40dBDCPronySyn


NO

RM

AL

IZE

D P

AT

TE

RN

(d

B)

G-L7N15-20&-40dBDCPronySyn

•-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

Generating Model

Fitting Model

GM Samples Used for FM

D-L7N15-70to0dBPronySyn


NO

RM

AL

IZE

D P

AT

TE

RN

(d

B)

G-L7N15-70to0dBPronySyn

•15 ELEMENTS UNIFORMLY SPACED •7 WAVELENGTHS LONG

SVD SPECTRA FOR SEVERAL ARRAYS ILLUSTRATE THEIR DIFFERENCES

1513119753110 -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

5-Wavelength Sinusoid

5-Wavelength COS^2 Aperture

5-Wavelength Uniform Current Filament

D-SVsVariousSources SINGULAR-VALUE ORDER

SIN

GU

LA

R V

AL

UE

S

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

18013590450-100

-80

-60

-40

-20

0

GENERATING MODEL


FITTING MODEL

D-N12L5.5ElliottPronySyn


NO

RM

AL

IZE

D P

AT

TE

RN

(d

B)

G-N12L5.5ElliottPronySyn

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.


•THE BASIC IDEA

•SEVERAL EXAMPLES






CONSIDER EXPONENTIATING A PATTERN: •FOR EXAMPLE THE PATTERN OF A 10-ELEMENT

DOLPH-CHEBYSHEV ARRAY AS GIVEN BY . . .

P10 (! ) = 2.798cos(D)+ 2.496cos(3D) +1.974 cos(5D)+1.357cos(7D)+ cos(9D)

WITH

D = !d /"( )cos#[ ] AND d THE ELEMENT SPACING

. . . TO YIELD SUCCESSIVELY LOWER SIDELOBES

18013590450-156

-130

-104

-78

-52

-26

0

D-DC^1VarL4.5...&N10...dB26...


NO

RM

AL

IZE

D P

AT

TE

RN

(d

B)

G-DC^1to4L4.5to18.5

Exponent M = 1

2

3

4

•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

18013590450-156

-130

-104

-78

-52

-26

0

D-L4.5N10DC26to104dBANGLE FROM ARRAY AXIS (degrees)

NO

RM

AL

IZE

D P

AT

TE

RN

(d

B)

G-L4.5N10DC26to104dB

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

18013590450-130

-104

-78

-52

-26

0

D-PronyNewdB-26N10PVarDVar


NO

RM

AL

IZE

D P

AT

TE

RN

(d

B)

G-PronyNewdB-26N10PVarDVar

EXPONENT M = 1

2

3

4

•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

18013590450-156

-130

-104

-78

-52

-26

0

D-26,52,etc.dBArraysWithVariousL,N


NO

RM

AL

IZE

D P

AT

TE

RN

(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

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

(d

B)

G-L7.5DC^xPatternsNewLineEXPONENT M = 1

2

3

4

. . . FOR WHICH THE SINGULAR-VALUE SPECTRA INDICATE THE NUMBER OF ARRAY ELEMENTS

50403020100

10-26

10-25

10-24

10-23

10-22

10-21

10-20

10-19

10-18

10-17

10-16

10-15

10-14

10-13

10-12

10-11

10-10

10- 9

10- 8

10- 7

10- 6

10- 5

10- 4

10- 3

10- 2

10- 1

100

D-L7.5DC^xMBPE

SINGULAR-VALUE ORDER

SIN

GU

LA

R V

AL

UE

S

G-SVsL7.5Norm

EXPONENT M = 1

2 3 4

•FOR EXPONENT M = 1 TO 4 ARE 10, 19, 28, 37 RESPECTIVELY

THE NUMBER OF SINGULAR VALUES INCREASES LINEARLY WITH THE EXPONENT M

4321

0

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 . . .

9594939291908988878685-3

-2

-1

0

D-L7.5DC^xMBPE


NO

RM

AL

IZE

D P

AT

TE

RN

(d

B)

G-DC^xRadPattGMMainLobe

EXPONENT M = 1

2

3

M = 4

•FOR THE 7.5 WAVELENGTH ARRAY

. . . AT THE -3 dB LEVEL

THE PRONY-DERIVED ARRAYS CAN HAVE WIDELY VARYING SOURCE STRENGTHS:

15.0

06

13.3

39

11.6

72

10.0

05

8.33

8

6.67

1

5.00

4

3.33

7

1.67

0

0.00

3

-1.6

64

-3.3

31

-4.9

98

-6.6

65

-8.3

32

-9.9

99

-11

.66

6

-13

.33

3

-15

.00

0

10 - 5

10 - 4

10 - 3

10 - 2

10 - 1

10 0

D-DC^xImagPoleVsRealRes

ELEMENT POSITION (wavelengths)

EL

EM

EN

T C

UR

RE

NT

G-PolesVsResiduesDC^xVert

EXPONENT M = 4

M = 2

M = 3

M = 1

•THE NUMBER OF SOURCES VARIES FROM 10, 19, 28 TO 35 FOR M VARYING 1 TO 4

•FOR A 5-WAVELENGTH D-C ARRAY NORMALIZED TO END ELEMENTS

•WITH IMPLICATIONS FOR NOISE SENSITIVITY

. . . WITH EACH ARRAY SIZE VARYING LINEARLY WITH INCREASING EXPONENT

43210

10

20

30

Initial Width 7.5 Wavelengths

Initial Width 5 Wavelengths

D-Poles&ResiduesDC^4

EXPONENT OF DOLPH-CHEBYSHEV ARRAY (M)

AR

RA

Y W

IDT

H

(wa

vel

eng

ths)

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)

*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))^22


•THE “STANDARD” -30 dB PATTERN (RED) IS GIVEN BY


•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


•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))^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

EL

EM

EN

T S

EP

AR

AT

ION

(w

avel

ength

s)

G-PronySynDC^1to5Spacing

EXPONENT M = 5

M = 4

M = 3

M = 2

M = 1

•THE RESPECTIVE NUMBER OF ARRAY ELEMENTS ARE 10, 19, 28, 35, AND 42 FOR AN INITIAL ARRAY 5-WAVLENGTHS LONG

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

LA

R V

AL

UE

S

G-PronyL10N10dB-26P1SVC

WINDOW ±0.999±0.5

±0.3

±0.1

•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

= A1cos u( ) + A

2cos 3u( )

WHERE A1 = 0.8794 and A2 = 1

•ITS EXPONENTIATED PATTERN IS THEN

P4

M

= A1cos u( ) +A

2cos 3u( )[ ]

M

.

• FOR M = 2 THIS BECOMES

P4

2

=A1

2+ A

2

2

2+

A1

2

2+ A

1A2

!

" #

$

% & cos 2u( ) + A

1A2cos 4u( ) +

A2

2

2cos 6u( ) .

*G. J. BURKE, PRIVATE COMMUNICATION, 2013 VIA MATHEMATICA

ITS PATTERNS FOR M = 3 AND M = 4 ARE GIVEN BY

P4

3=3

4A1

3+ A

1

2A2

+ 2A1A2

2[ ]cos u( ) +1

4A1

3+ 6A

1

2A2

+ 3A2

3[ ]cos 3u( )

+3

4A1

2A2

+ A1A2

2[ ]cos 5u( ) +3

4A1A2

2cos 7u( ) +

1

4A2

3cos 9u( )

AND

P4

4

=3

2

1

4A1

4

+1

3A1

3

A2

+ A1

2

A2

2

+1

4A2

4!

" # $

% & +3

2

1

3A1

4

+ A1

3

A2

+ A1

2

A2

2

+ A1A2

3!

" # $

% & cos 2u( )

+3

2

1

12A1

4

+ A1

3

A2

+1

2A1

2

A2

2

+ A1A2

3!

" # $

% & cos 4u( ) +3

2

1

3A1

3

A2

+ A1

2

A2

2

+1

3A2

4!

" # $

% & cos 6u( )

+3

2

1

2A1

2

A2

2

+1

3A1A2

3!

" # $

% & cos 8u( ) +1

2A1A2

3

cos 10u( ) +1

8A2

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

•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


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

THE ELEMENT STRENGTHS


•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






Documents

TWO NOVEL APPROACHES TO ANTENNA-PATTERN SYNTHESISs3.amazonaws.com/sdieee/1794-EKMiller-PatternSynthesis.pdf•THE BASIC IDEA •SEVERAL EXAMPLES ... “A New Principle in Directional