JOINT-100.2
FRACTAL CONSTRUCTIONS OF LINEAR AND PLANAR ARRAYS
Randy L. Haupt
HQ USAFA/DFEE
2354 Fairchild Dr ., Suite 2F6
USAF Academy, CO 80840-6236
Douglas H. Werner
Applied Research Laboratory
The Pennsylvania State University
P.O. Box 30
State College, PA 16804-0030
1.
INTRODUCTION
The term fractal, which means broken or irregular fragments, was originally
coined by Mandelbrot [l] to describe complex shapes that can’t be defined by classical
Euclidean geometry. Since the pioneering work of Mandelbrot and others, fractal
applications have appeared in many branches of engineering and science. One such area
is fractal electrodynamicsin which fractal geometry is combined with electromagnetics.
An introductionto the subject of fractal electrodynamicsmay be found in the excellent
review by Jaggard [2]. More recent developments in the field are summarized by Jaggard
[3] and Werner [4]. This paper focuses on the application of fractal geometric concepts to
the analysis and design of thinned fractal linear as well as planar arrays. There are many
applications where it is advantageous to apply thinning techniques to the design of antenna
arrays [5]. In this paper, we approach thinning with the sole purpose of arranging the
elements in a fractal pattem to investigate the usefulness of fractal array designs. In
particular, the multiscaling of the fractal makes them attractive for wideband applications
[6,7]. Another advantage of these fractal arrays is that the self-similarityin their
geometrical structure may be exploited in order to develop algorithms for rapid
computation of radiation patterns. These algorithms are based on convenient product
representations for the array factors and are much quicker to calculate than the discrete
Fourier transform approach. Results for both the Cantor linear array and the Sierpinski
carpet planar array are presented.
2.
THINNED FRACTAL LINEAR ARRAYS
The array factors for an arbitrary row of symmetric elements along the z-axis are
given by
I
M
a
.
+
2
U,,,
cos [my]. odd number of elements (
m=l
where
N = number of elements
am =
Y!
=
{ 01
=
=
element turned on or connected to feed network
element turned off or connected to matched load
kdu,
0-7803-4178-3/97/$10.00 0 1997 IEEE
1968
2 + 1)
~
k = -2n
1
1 = wavelength
d = constant spacing between elements
= case - -e,
e = angle of incidence of electromagneticplane wave
8, = direction of mainbeam
Element weights are either 1 or 0 and indicate whether or not the individual elements
contribute to the total radiation pattem of the array.
The easiest way to generate the Cantor array is by starting with a three element
generating subarray then applying the generator to P scales. The generation subarray for
the Cantor array has three elements with the center element turned off. An expression for
this generating subarray may be easily obtained from (1) using the fact that M= 1, q = O ,
and a, = 1 in order to arrive at
G A P ) = &s['F]
3
Applying this rule a total of P times results in the following array factor for the Cantor
array:
P P
AF'Cp) =
cos(Y-''F)
(3)
p=l
(3) rI
An expression for the directive gain of a Cantor array of isotropic point sources may be
derived from (3). Suppose that the spacing between array elements is a quarterwavelength (i.e., d=1/4) and that 8,=90", then the directive gain for this Cantor array is
P
P
GJu) =
n
[l
+
cos(3P-*n.u)] = 2p
p=l
n
cos2(
7
nu)
(4)
p=l
Finally, it is easily demonstrated from (4) that the directivity (maximum directive gain) for
this Cantor array is given by
Dp
=
Gp(0) = 2;'
P
= 1,2,
...
(5)
or
Dp(dB) = 3.01P; P
=
1,2,..,
One of the interesting properties of this Cantor array is the ability to quickly
calculate the array factor. If (1) is used to calculate the array factor for an odd number of
elements, then M cosine functions must be evaluated and M additions performed for each
angle. Using (3), only P cosine function evaluations and P-1 multiplications are required.
For the case of an 81 element array, the fractal array factor is at least M/P=40/4= 10
times faster.
1969
3.
THINNED FRACTAL PLANAR ARRAYS
A symmetric planar array with an odd number of elements in the x and y
directions has an array factor given by
&re
M = number of elements in the xdirection
N = number of elements in the y-direction
%, = excitation current amplitude at element (m,n)
4 and 4- are the element spacings in the x and y directions
= Sin8d-sie.d.
t = sin0sin@-sinO,cm$,
(e.,@,)
'px
= steering angles
= L4Y,
YY = kdyulr
A Sierpinski carpet is a two-dimensional version of the Cantor set and can be
applied to thinning planar arrays. Consider the simple generator array
111
101
111
This array is generated by the rule of turning off the center element of a nine element
square as a pattern. The array factor for &=%=Ansimplifiw to
'4qUll,"J
=
2cos(nuJ + 2cos(mJ + 4cos(TtuJw(nuJ
The array fictor at scale P is
P
AF&,u,,
=
I-I[2cos(3p-'Tru,, + 2 c o s ( Y - ' T c u ~
p=l
+ 4cos(Y-,TruJcos(Y-ln#J]
The directive gain of the Sierphki carpet array for the case in which 8,=0 may be
expressed in the form
(10)
1970
+ 4 cos(3p-l x sinecos +)cos(3p-1x
sine sin+)?
follows directly from (9). Tbe product representation of the array factor (1 1) may also be
used to h w that the maxi"value of directive gain (directivity) for the Sierpimki
carpet is
The double integral which appears in the denominator of (10)and (12) does not have a
closed form solution in this case and, therefore, must be evaluated numerically.
REFERENCES
U1
PI
r31
M
PI
r61
r7l
B. B. Mandelbrot, lire Fractal Geometry of Nature, New York: W.H.Freeman
and company, 1977.
D. L.Jaggard, 'On fractal electrodynamics," in Recent Advances in
Eleunmtagnetic Iheory, D. L. Jaggard and H.N. Kritikos, Eds., New York
Springer-Verlag, 1990,pp. 183-224.
D.L.Jaggard, 'Fractal electrodynamics: Wave interactiom with discretely selfsimilar stnrctures,nin Electmmagnetic@nametry, C. E.Baum and H.N.
Kritikoe, Eds., Bristol, PA: Taylor, 1995,pp. 231-280.
D.H.Werner, 'An overview of fractal electrodynamics research," Proceedings
of bre 1 lth Annual Review of Progress in Applied Computational
Electromagnetics,Naval Postgraduate School, Monterey, CA, March 20-25,
1995,pp. %4-969.
Y.T. Lo, "Aperiodic arrays," in Antenna H W : Theory, AppliCatiOnr, and
Design, Y . T . Lo and S. W.Lee, Eds., New York Van Nostrand Reinhold,
1988, pp. 14.1-14.37.
C. Puente-Bdiarda and R. POUS,'Fractal design of multiband and low side-lobe
arrays," B E E Tnms. Antemm Pmpagat., Vol. 44,No. 5,pp. 730-739,May
1996.
D.H.Werner and P. L. Werner, "Frequency-independent features of self-similar
fractal an&nnas," Radio Science, Vol. 31, No. 6,pp. 1331-1343,Nov.-Dec.
1996.
1971