A Generalized Fractal Radiation Pattern SynthesisTechnique for the Design of

Multiband Arrays

A Generalized Fractal Radiation Pattern SynthesisTechnique for the Design of

Multiband ArraysD.H. Werner, M.A. Gingrich, and P.L. Werner

The Pennsylvania State UniversityDepartment of Electrical Engineering

D.H. Werner, M.A. Gingrich, and P.L. WernerD.H. Werner, M.A. Gingrich, and P.L. WernerThe Pennsylvania State UniversityThe Pennsylvania State University

Department of Electrical EngineeringDepartment of Electrical Engineering

SelfSelf--Similar Radiation Patterns for the Design of Similar Radiation Patterns for the Design of Multiband ArraysMultiband Arrays

IFS(stage 8)

IFS(stage 8)

IFS(stage 8)

Previous ResearchPrevious Research• Weierstrass Arrays

D.H. Werner and P.L. Werner, “On the Synthesis of Fractal Radiation Patterns,” Radio Science, Vol. 30, No. 1, pp. 29-45, 1995.

D.H. Werner and P.L. Werner, “ Frequency-Independent Features of Self-Similar Fractal Antennas,” Radio Science, Vol. 31, No. 6, pp. 1331-1343, 1996.

• Koch Arrays

C. Puente Baliarda and R. Pous, “ Fractal Design of Multiband and Low Sidelobe Arrays,” IEEE Transactions on Antennas and Propagation, Vol. 44, No. 5, pp. 730-739, 1996.

Suppose we consider the periodic functionSuppose we consider the periodic function

d23λ

−d2

λ−

d2λ

d23λ

ww

g(g(ww)) f(f(ww))

This function may be expressed mathematically as:This function may be expressed mathematically as:

∑ ∑∞

−∞=

∞

−∞=

−=−=n n

nwwfnwwwfwg )()(*)()( 00δ

where where ww0 0 = = λ /λ /d d = 2= 2π π //kdkd

Construction of a Fractal Array FactorConstruction of a Fractal Array Factor

Construction of a Fractal Array FactorConstruction of a Fractal Array FactorWe may construct a selfWe may construct a self--similar (i.e., fractal) array factor bysimilar (i.e., fractal) array factor byappropriately scaling and shifting a generating array factor appropriately scaling and shifting a generating array factor of the formof the form

∑∞

−∞=

−=n

nwwwfAF )(*)( 01 δ

The resulting fractal array factor is given byThe resulting fractal array factor is given by

−

∞

−∞==

= −

−

−

∑∑ 101

1

*)(

1

1p

pp

snww

n

ws

P

p

fAFP δδγ

where where P = P = stage of growthstage of growths = s = scaling or similarity factorscaling or similarity factorγ γ = = amplitude of scaling parameteramplitude of scaling parameter

Fractal Array Current DistributionFractal Array Current DistributionThe fractal array factor and current distribution are The fractal array factor and current distribution are related via a Fourier transform pairrelated via a Fourier transform pair

)()( wAFuI ↔Hence taking the Fourier transform of the fractal arrayHence taking the Fourier transform of the fractal arrayfactor yieldsfactor yields

( ) ( ) ( )∑∑∞

−∞=

−

=

= −

−

n

nkdsunkd

P

p

kdu p

p

FI 1

1

1

12

δδγπ

where the element where the element spacingsspacings are given byare given by11 −− =⇒= p

n

p

n ndsznkdsu

Koch Fractal Array FactorKoch Fractal Array Factor

w

w

( ) ( )∑ ∑=

∞

−∞=−−

−

−∗=

3

111

13 33

13p n

po

pp nwwwfwAF δ

( )wf( )wf 3

31 ( )wf 9

91

Comparison of Several Window Comparison of Several Window FunctionsFunctions

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

w

mag

nitu

de

Blackman (dot-dash), Blackman-Harris (dotted), and Kaiser-Bessel (solid)

Fourier Transforms of Several Fourier Transforms of Several Window FunctionsWindow Functions

100 101 102 103-120

-100

-80

-60

-40

-20

0Rec tangular w indow

S LL = -13 dB

100 101 102 103-120

-100

-80

-60

-40

-20

0B lac k m an w indow

S LL = -58 dB

100 101 102 103-120

-100

-80

-60

-40

-20

0B lac k m an-Harris w indow

S LL = -92 dB

100 101 102 103-120

-100

-80

-60

-40

-20

0K ais er-B es s e l w indow

S LL = -82 dB

Successively Scaled and Shifted Successively Scaled and Shifted Copies of a Blackman WindowCopies of a Blackman Window

Array Factor Synthesized Via A Array Factor Synthesized Via A Blackman WindowBlackman Window

Array Factor Synthesized Via A Array Factor Synthesized Via A BlackmanBlackman--Harris WindowHarris Window

Array Factor Synthesized Via A Array Factor Synthesized Via A KaiserKaiser--Bessel WindowBessel Window

The Linear Array Factor as a Fourier SeriesThe Linear Array Factor as a Fourier Series

Suppose we consider the following array geometry:

θ

Z

d d d d

I-N I-2 I-1 I0 I1 I 2 IN

The array factor may be express as:where w = cos θ.

( ) ( )∑−=

=N

Nn

wdnjneIwAF λπ2

If N is infinite then this represents a Fourier Series on the interval -λ/2d < w < λ/2d such that:

( ) ( )∫−

−

=

d

d

wdnjn dwewAFdI

2

2

2λ

λ

λπ

λ

A Generalized Fourier Series Radiation Pattern A Generalized Fourier Series Radiation Pattern Synthesis Technique For Multiband Linear ArraysSynthesis Technique For Multiband Linear Arrays

wsdnjP

p npnP

p

eIwAF1)/(2

1)(

−

∑ ∑=

∞

−∞=

= λπ

n

p

pn Is

I1

1−

=

γ

dwewfdId

d

wdnjn ∫

−

−

=

2/

2/

)/(2)(λ

λ

λπ

λ

wherewhere

andand

Special CasesSpecial CasesNote that for the first stage of growth when P = 1, we have

If f(w) is an even function such that f(-w) = f(w) then I-n = In where

and therefore we may write

wdnj

nneIwAF )/(2

1 )( λπ∑∞

−∞=

=

[ ]dwdnwfdId

n ∫

=

2/

0

)/2(cos)(2λ

λπλ

( )( ) [ ]wsdnIssIwAF p

P

p npn

P

P1

1 10 )/2(cos2

/11/11)( −

=

∞

=∑∑+

−−

= λπγ

γ

Multiband Linear Phased ArraysMultiband Linear Phased Arrays

The required array element current phases for a multibandlinear array are obtained from the formula

where wo = cosθo and θo is the desired position angle of the main beam.

By taking into account a current phase distribution of this type, the multiband array factor becomes

01wnkds ppn

−−=α

( )( ) [ ])(cos2

/11/11

01

1 10)( wwnkdsI

ssI p

P

p

N

npn

P

P wAF −+−

−= −

= =∑∑γ

γ

Fractal Radiation PatternsFractal Radiation PatternsThere are two possible geometrical interpretations for the arrayswhich result from this fractal radiation pattern synthesis technique

Case 1: A Series of Self-Scalable Uniformly Spaced Subarrays

∑ ∑= =

−

−−

+

−−

=

P

p

N

n

p

p

s

Ps wnkdsnkdFs

kdkd FwAF1 1

1

1

1

11

)cos()(12)(1

)(1)0(

2)( γππ γ

γ

Case 2: A Series of Self-Scalable Nonuniformly Spaced Subarrays

∑ ∑= =

−−−

π+

−−

π=

N

n

P

p

pp

sγ

Psγ wnkdssγnkdkdkd FFAF(w)

1 1

11

1

11

)(cos1)(2)(1

)(1(0)

2

Fractal Array as a Superposition of P = 4 Fractal Array as a Superposition of P = 4 Uniformly Spaced Subarrays (N = 5)Uniformly Spaced Subarrays (N = 5)

p = 1

p = 3

p = 2

p = 4

Total

n

n

n

n

n

Fractal Array as a Superposition of N = 5 Fractal Array as a Superposition of N = 5 NonuniformlyNonuniformly Spaced Subarrays (P = 4)Spaced Subarrays (P = 4)

p

p

p

p

p

p

n = 1

n = 2

n = 4

n = 3

n = 5

Total

Special Case: Weierstrass ArraysSpecial Case: Weierstrass Arrays

Let )(cos)()( 1

1

11 xasxW pP

p

psP

−

=

−∑= γ

then ∑∞

=

−−

∞→==

1

111 )cos()()()(p

ppsPP

xasxWLimxW γ

represents a Weierstrass function provided the following condition is met:

1/s < γ < 1 where s > 1

The fractal dimension of this Weierstrass function is

D = 1 – ln(γ) / ln (s )

Extension to Multiband Extension to Multiband Planar ArraysPlanar Arrays

A composite fractal radiation pattern may be formed by an ensembA composite fractal radiation pattern may be formed by an ensemble of le of sequentially scaled planar arrays. The expression for the sequentially scaled planar arrays. The expression for the multibandmultiband array array factor in this case is given byfactor in this case is given by

wherewhere

u = u = sinsinθθ coscosΦΦ, , v = v = sinsinθθ sinsinΦΦ, , uuoo = = sinsinθθoo coscosΦΦoo, , vvoo = = sinsinθθoo sinsinΦΦoo

[ ] [ ])(cos)(cos),( 01

01

1 0 0

vvnkdsuumkdsIvuAF ppP

ppmnn

N

m

N

nmP −−= −−

= = =∑∑∑ εε

nm

p

mn

p

pmn IIs

Is

I)1()1(

11−−

=

=

γγ

∫

=

kd

q dwqkdwwfkdI/

0

][cos)(π

πfor q = m or nfor q = m or n

Multiband (Multiband (ss = 3) = 3) UnthinnedUnthinned Planar Planar Fractal Array Current DistributionFractal Array Current Distribution

30,421 elements, ε = 0

Multiband Multiband UnthinnedUnthinned Planar Fractal ArrayPlanar Fractal Array

30,421 elements, ε = 0

Multiband Multiband UnthinnedUnthinned Planar Fractal ArrayPlanar Fractal Array

30,421 elements, ε = 0

Multiband (Multiband (ss = 3) Planar Array= 3) Planar Arrayφ = 0° pattern sliceφ = 0° pattern slice

- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0

u

- 8 0

- 6 0

- 4 0

- 2 0

0

Mag

nitu

de (d

B)

B a n d 1

- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0

u

- 8 0

- 6 0

- 4 0

- 2 0

0

Mag

nitu

de (d

B)

B a n d 2

- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0

u

-8 0

-6 0

-4 0

-2 0

0

Mag

nitu

de (d

B)

B a n d 3

- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0

u

-8 0

-6 0

-4 0

-2 0

0

Mag

nitu

de (d

B)

B a n d 4

30,421 elements, ε = 0

Multiband Thinned (s=3Multiband Thinned (s=3) Planar Fractal ) Planar Fractal Array Current DistributionArray Current Distribution

3,412 elements, ε = 0.0001

Multiband Thinned (s=3Multiband Thinned (s=3) Planar Fractal ) Planar Fractal Array Current DistributionArray Current Distribution

3,412 elements, ε = 0.0001

Multiband Thinned (s=3Multiband Thinned (s=3) Planar ) Planar Fractal Array Radiation PatternsFractal Array Radiation Patterns

3,412 elements, ε = 0.0001

Multiband Thinned (s=3Multiband Thinned (s=3) Planar Fractal ) Planar Fractal Array Radiation PatternsArray Radiation Patterns

- 1 - 0 . 5 0 0 . 5 1- 8 0

- 7 0

- 6 0

- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

0B a n d 1

u

Mag

nitu

de (d

B)

- 1 - 0 . 5 0 0 . 5 1- 8 0

- 7 0

- 6 0

- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

0B a n d 2

u

Mag

nitu

de (d

B)

- 1 - 0 . 5 0 0 . 5 1- 8 0

- 7 0

- 6 0

- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

0B a n d 3

u

Mag

nitu

de (d

B)

- 1 - 0 . 5 0 0 . 5 1- 8 0

- 7 0

- 6 0

- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

0B a n d 4

u

Mag

nitu

de (d

B)

3,412 elements, ε = 0.0001

Multiband (Multiband (ss = 3) Thinned Planar = 3) Thinned Planar Fractal Array Current DistributionFractal Array Current Distribution

409 elements, ε = 0.1

Multiband Thinned Planar Fractal ArrayMultiband Thinned Planar Fractal Array

409 elements, ε = 0.1

Multiband (Multiband (ss = 3) Thinned Planar Array= 3) Thinned Planar Arrayφ = 0° pattern sliceφ = 0° pattern slice

- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0

u

-8 0

-6 0

-4 0

-2 0

0

Mag

nitu

de (d

B)

B a n d 1

- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0

u

-8 0

-6 0

-4 0

-2 0

0

Mag

nitu

de (d

B)

B a n d 2

-1 . 0 -0 . 5 0 . 0 0 . 5 1 . 0

u

-8 0

-6 0

-4 0

-2 0

0

Mag

nitu

de (d

B)

B a n d 3

-1 . 0 -0 . 5 0 . 0 0 . 5 1 . 0

u

-8 0

-6 0

-4 0

-2 0

0

Mag

nitu

de (d

B)

B a n d 4

409 elements, ε = 0.1

BeamsteeringBeamsteering ExampleExampleθθ00 = 60= 60°°, , φφ00 = 45 = 45 °°

- 1 - 0 . 5 0 0 . 5 1- 8 0

- 7 0

- 6 0

- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

0B a n d 0

u

Mag

nitu

de (d

B)

- 1 - 0 . 5 0 0 . 5 1- 8 0

- 7 0

- 6 0

- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

0B a n d 1

u

Mag

nitu

de (d

B)

- 1 - 0 . 5 0 0 . 5 1- 8 0

- 7 0

- 6 0

- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

0B a n d 2

u

Mag

nitu

de (d

B)

- 1 - 0 . 5 0 0 . 5 1- 8 0

- 7 0

- 6 0

- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

0B a n d 3

u

Mag

nitu

de (d

B)

φ = 0° slice

Pattern Slices Synthesized from a Pattern Slices Synthesized from a KaiserKaiser--Bessel WindowBessel Window

Weierstrass Array FactorWeierstrass Array Factor

γγ

Weierstrass Array FactorWeierstrass Array Factorγγ