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 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 (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