a self-similar fractal radiation pattern synthesis techniques for reconfigurable multiband array

7/28/2019 A Self-Similar Fractal Radiation Pattern Synthesis Techniques for Reconfigurable Multiband Array

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1486 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 7, JULY 2003

A Self-Similar Fractal Radiation Pattern SynthesisTechnique for Reconfigurable Multiband Arrays

Douglas H. Werner, Senior Member, IEEE, Mark A. Gingrich, and Pingjuan L. Werner

AbstractA novel method for designing reconfigurable multi-band linear and planar antenna arrays is presented in this paper.The technique is based on a generalized Fourier series synthesisapproach that exploits the self-similarity of a specified fractal ra-diation pattern in order to achieve the desired multiband perfor-mance. The fractal radiation patterns are composed of scaled andshifted copies of an appropriately chosen generating window func-tion that exhibits low sidelobe levels and rapid spectral rolloffs inthe transform domain. A newly developed thinning algorithm willbe presented which may be employed to considerably reduce boththe overall physical size as well as the total number of elements ina synthesized multiband array. Finally, a band-switching schemeis introduced that is well-suited for implementation in the form of

a reconfigurable common aperture array.

Index TermsAntenna arrays, fractals, multiband arrays, re-configurable apertures.

I. INTRODUCTION

THERE has been a considerable amount of recent interest in

the radiation characteristics of self-scalable and self-sim-

ilar fractal arrays. For instance, the properties of random fractal

arrays have been investigated in [1] for the purpose of synthe-

sizing a sparse or thinned array with relatively low sidelobes

that is robust with respect to element failure as well as varia-

tions in element location and current excitation. Methods for

the synthesis of Weierstrass fractal radiation patterns using self-scalable linear arrays of discrete elements were first considered

in [2]. Also reported in [2] is a FourierWeierstrass fractal ra-

diation pattern synthesis technique for continuous line sources.

A design methodology for multiband Weierstrass linear arrays

was presented in [3]. Application of fractal concepts to the de-

sign of multiband Koch linear arrays as well as low sidelobe

Cantor linear arrays are discussed in [4].

Planar array configurations based on Sierpinski carpets have

been considered in [5][8]. The properties of concentric circular

Weierstrass arrays and self-similar concentric circular Cantor

arrays have also been investigated in [9] and [10], respectively.

A more comprehensive overview of recent developments in the

area of fractal antenna engineering, with particular emphasis

placed on the theory and design of fractal arrays, can be found

in [11].

A new technique for the design of multiband arrays, based on

the application of fractal geometric concepts to antenna theory,

Manuscript received August 14, 2001; revised May 12, 2002The authors are with The Pennsylvania State University, Department of Elec-

trical Engineeringand Applied Research Laboratory, UniversityPark, PA 16802USA.

Digital Object Identifier 10.1109/TAP.2003.813608

Fig. 1. Linear array geometry.

is presented in this paper. The distinguishing features of the

technique introduced here are its ability to synthesize recon-

figurable multiband arrays that have self-similar fractal radia-

tion patterns with a certain desired (i.e., specified) beamwidth

and sidelobe level as well as no grating lobes in the intended

band or bands of operation. An array thinning procedure and a

novel band-switching scheme have also been developed for use

in conjunction with the multiband fractal radiation pattern syn-

thesis technique. The band-switching scheme introduced in Sec-

tion III-B offers several important advantages over the method

for multiband array design originally proposed in [4]. These in-

clude: 1) a significantly reduced mutual coupling environment;

2) the fact that a minimal amount of element switching is re-quired; and 3) the ability to easily implement in the form of

a reconfigurable multiband aperture. Another noteworthy fea-

ture of the synthesis technique is that all the advantages of the

band-switching scheme are preserved under the array truncation

and thinning operation outlined in Section III-C. The utility of

this new multiband fractal radiation pattern synthesis technique

is demonstrated in Section IV by considering several design ex-

amples for both linear and planar arrays.

II. THEORETICAL DEVELOPMENT

A. Linear Arrays

Suppose we consider the array geometry shown in Fig. 1. Thearray factor for this array may be expressed as [12], [13]

(1)

If we let then

(2)

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WERNER et al.: SELF-SIMILAR FRACTAL RADIATION PATTERN SYNTHESIS TECHNIQUE FOR RECONFIGURABLE MULTIBAND ARRAYS 1487

Fig. 2. Fractal array as a superposition ofP = 4 uniformly-spaced 5-elementsubarrays. Due to symmetry, only half of the array geometry is shown, with oneelement being located at the origin.

If then this represents a Fourier series on the interval

such that

(3)

This fact is well-known and has been used as the basis of a ra-

diation pattern synthesis technique for uniformly spaced linear

arrays [12], [13]. In other words, for some specified radiation

pattern, the current distribution required to produce this desired

pattern may be obtained directly from (3). One of the drawbacks

of this synthesis technique, however, is that it leads to array de-

signs that are characteristically narrow-band, due primarily to

the uniform spacing between elements. Consequently, grating

lobes will begin to appear as the operatingfrequency of the array

is increased beyond that for which it was originally designed.

The main objective of this paper is to develop a radiation pat-

tern synthesis technique that leads to multiband array designs.We will show here that this may be accomplished by general-

izing the fractal radiation pattern synthesis technique for dis-

crete arrays originally introduced in [2]. The first step toward

developing such a multiband synthesis technique is to recog-

nize that a self-similar fractal radiation pattern may be formed

by the superposition of radiation produced by a sequence of

linear arrays whose relative element spacings and current dis-

tributions have been appropriately scaled. For example, Fig. 2

shows a five element linear generating array followed by a se-

quence of three scaled versions of this array. In this case, the

element spacings for each consecutive linear array have been

expanded by a factor of two, while the current distribution has

been reduced by a factor of two. The superposition of these four

uniformly spaced linear arrays (i.e., 1, 2, 3, 4) results in the

nonuniformly spaced array shown at the bottom of Fig. 2. It can

be shown that a generalization of this concept leads to an array

configuration that produces the following composite self-sim-

ilar fractal radiation pattern:

(4)

where

(5)

(6)

and where is the desired generating function, is the

number of stages used in the construction of the fractal radiationpattern, is the scaling or similarity factor, and is an additional

current amplitude scaling parameter. Note that for the first stage

of growth when , expression (4) reduces to

(7)

which is the original linear generating array with elements uni-

formly spaced a distance apart. We also note that if is an

even function, i.e., , then

(8)

From this it follows that , which may be used to show

that (4) reduces to

(9)

Equations (8) and (9) represent an important special case thatfrequently occurs in practice, namely, where the array is as-

sumed to have a symmetric current amplitude distribution.

The standard approach for phasing a uniformly spaced linear

array may be easily generalized to include the particular family

of nonuniformly spaced linear arrays considered here. This is

a direct consequence of the fact that this class of nonuniformly

spaced arrays may be decomposed into a series of consecutively

scaled uniformly spaced arrays as illustrated in Fig. 2. Hence,

the required array element current phases may be obtained from

the formula

(10)
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Fig. 3. Same fractal array as shown in Fig. 2, but as a superposition ofN = 2nonuniformly-spaced subarrays, plus an element at the origin. Due to symmetry,only half of the array geometry is shown.

where and is the desired position angle of the

main beam. By taking into account a current phase distribution

of this type, the array factor expression given in (9) becomes

(11)

Now suppose that we truncate the array to a finite number of

elements and interchange the order of summation found in (11).

This leads to the following expression for the correspondingarray factor:

(12)

where

(13)

The expression given in (12) may be interpreted as representing

the superposition of the radiation produced by a series ofself-scalable arrays as illustrated by the example considered

in Fig. 3. This example shows a sequence of two

nonuniformly spaced eight-element linear arrays.

The element at the origin is considered separately in this case.

As before, the element spacings for each consecutive linear

array have been expanded by a factor of two. In addition to

this, however, the consecutive element spacings within a given

linear subarray are also expanded by a factor of two. This

leads to an array configuration that is doubly self-scalable. The

composite array in this case is shown at the bottom of Fig. 3,

which is exactly the same configuration arrived at in Fig. 2. It

is interesting to note that these self-scalable arrays are closely

related to the FourierWeierstrass arrays previously studied

in [2], [11]. In fact, it can be shown that represent

bandlimited Weierstrass functions provided . The

associated fractal dimension of these Weierstrass functions

as is given by

(14)

Finally, we note that the self-similar fractal radiation patterns

produced by these arrays suggest that they may be used as multi-

band arrays that maintain the same radiation characteristics at

an infinite number of frequencies. To see this, let us consider an

array with a doubly infinite number of stages that has the fol-

lowing normalized expression for the array factor:

(15)

where

(16)

and

(17)

(18)

(19)

where is a unitless parameter. If we assume that this array is

operated at the discrete set of frequencies given by

(20)

where represents the base-band design frequency of thearray,

then it follows from (15) and (16) that

(21)

where

(22)

Hence, the relationship arrived at in (21) suggests that this array

will exhibit multiband performance characteristics. At this point

we note that this multiband property holds for infinite arrays.

However, once these arrays are truncated, the desirable multi-

band properties rapidly degrade. In order to compensate for

these truncation effects, a band-switching methodology that ex-

ploits the inherent self-similarity associated with this family of

arrays will be introduced in Section III-B.
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Fig. 4. Planar array geometry.

B. Planar Arrays

The multiband array synthesis technique introduced for linear

arrays in the previous section will be generalized here to include

planar square arrays. Suppose we consider the uniformly spaced

square planar array configuration shown in Fig. 4. The array

factor for this array may be expressed as [14]

(23)

where

(24)

(25)

(26)

(27)

Now if we assume that

(28)

then the expression for the planar array factor given in (23) may

be decomposed into the product of two linear array factors. The

result is

(29)

where

(30)

(31)

If we further assume that and then (30)

and (31) may be written as

(32)

(33)

At this point we form a sequence of self-scalable planararrays

in a manner directly analogous to the construction procedure in-

troduced in the previous section for linear arrays. The resulting

composite radiation pattern formed by this ensemble of sequen-

tially scaled planar arrays is given by

(34)

where

(35)

for (36)

This synthesis procedure yields nonuniformly spaced planar

array configurations with two-dimensional self-similar fractal

radiation patterns that are based on scaled and translated

versions of a specified generating function .

III. MULTIBAND ARRAY SYNTHESIS

A. Fractal Pattern Synthesis via Window Functions

A technique will be introduced in this section for synthesizing

fractal radiation patterns using window functions that, if prop-

erly chosen, can lead to thinned multiband arrays of minimal

physical size. The window functions that will be considered

here feature low sidelobe levels and rapid spectral rolloffs in

the transform domain. Window functions with these properties

are ideal candidates for use in antenna pattern synthesis because

the resulting element currents rapidly diminish and become in-

significant with increasing distance from the origin. This allows

for the development of very effective thinning techniques thatcan be applied to dramatically reduce the element count and

physical size of the resulting arrays. Three types of window

functions will be investigated in this paper, which include the

Blackman, BlackmanHarris, and KaiserBessel windows [15].

The continuous form of the Blackman window as defined in

[15] is given by

(37)


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where is the lobe width at null (i.e., the first-null beamwidth).

The Blackman window features a transform domain sidelobe

level of 58 dB and a spectral rolloff of 18 dB per octave

compared, for example, with the 13 dB sidelobe level and

6 dB per octave rolloff corresponding to a simple rectangular

window. Hence, the use of this window function would result

in synthesized array current distributions that rapidly taper off

away from the origin. Substituting (37) into (36) and evalu-ating the resulting integral leads to the following expression

for the Fourier cosine series coefficients associated with the

Blackman window:

(38)

Consequently, the excitation currents for a linear array or

for a planar array may be readily calculated using (38) in

conjunction with (5) or (35), respectively.

A four-term BlackmanHarris window is obtained in [15]

as an extension of the above Blackman window such that,

see (39) shown at the bottom of page, where ,

, , and . The

BlackmanHarris window results in a sidelobe level of 92 dB

in the transformed domain and has a 6 dB per octave rolloff.

The Fourier cosine series coefficients for the BlackmanHarris

window are given by

(40)

Fig. 5. Comparison of window functions and their Fourier transforms.

The third window that will be considered is the KaiserBessel

window (also knownas the window)which isdefined

as [15]

(41)

where represents the modified Bessel function of the first

kind of order 0 and argument . The KaiserBessel window

allows an extra degree of freedom over the two previous win-

dows through the parameter . By varying the KaiserBessel

(39)


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(a) (b)

(c) (d)

Fig. 6. Overhead contour plots of a four-band( P = 4 )

planar fractal array radiation pattern withs = 3

andN = 4 5

. Unthinned case with" = 0

synthesizedusing a Blackman window. (a) Band 1, (b) Band 2, (c) Band 3, and (d) Band 4.

window may be adjusted from a rectangular window

of width to a narrow spike approximating a delta func-

tion (as ). In terms of array pattern synthesis, this al-

lows the 3 dB beamwidth to be varied independently of the

first-null beamwidth, which is not possible with the Blackman

or BlackmanHarris windows.

In the case of the KaiserBessel window, closed-form

solutions to the corresponding Fourier coefficient integrals

in (36) are not available. However, a useful approximation to

the Fourier transform of the KaiserBessel window has been

shown to be [15]

(42)
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(a) (b)

(c) (d)

Fig. 7. x z plane ( = 0 ) slice of the synthesized four-band ( P = 4 ) planar array radiation pattern with s = 3 and N = 4 5 . Unthinned case with " = 0synthesized using a Blackman window.

Since the window function vanishes outside the interval

(i.e., has compact support), and using

the fact that the Fourier transform of is a real-valued func-

tion, we find that (36) may be approximated by sampling the

continuous Fourier transform of the KaiserBessel window at

the values . Following this procedure leads to the result

(43)

A comparison is made in Fig. 5 of the three different

window functions (i.e, Blackman, BlackmanHarris, and

KaiserBessel) and their corresponding Fourier transforms.

It is seen that these windows essentially trade off main lobe

width for sidelobe suppression in the transform domain. It is

also obvious that they all three greatly outperform the simple

rectangular window.

B. Band-Switching Scheme

One of the major drawbacks of the multiband array design

approach originally proposed in [4] is the fact that as the arrayis progressively switched from the highest band down through

to the lowest band, the electrical spacing between elements be-

comes closer and closer together. For example, suppose that a

four-band design with a scale factor of is considered.

Further suppose that the minimum electrical separation between

array elements is at the highest band of operation

where . Switchingtothe nextbandwhere

we find that the minimum separation between elements is now

. Likewise, for the third band where

and the fourth band where we find that

and , respectively. Hence, in prac-

tice, these arrays would experience significant mutual coupling
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Fig. 8. Three-dimensional perspective plots of a four-band ( P = 4 ) planar fractal array radiation pattern with s = 3 and N = 4 5 . Thinned case with " = 0 : 1synthesized using a Blackman window. (a) Band 1, (b) Band 2, (c) Band 3, and (d) Band 4.

effects due to the close proximity of the elements, especially

during low-band operation.

In order to circumvent this problem, we introduce a band-

switching scheme that consists of turning off successive subar-

rays as the frequency of operation is increased. This scheme ex-

ploits the unique self-scaling properties of these arrays in such a

way that requires only a minimal number of elements be turned

on or off as the array is switched from one band to the next.

Another advantage of this approach is that it provides an ideal

framework for implementation of these arrays as multiband re-

configurable apertures.

The multiband array design procedure begins by selecting

an appropriate scaling factor and the desired number

of frequency bands . The individual bands of the re-

sulting array (either linear or planar) would be centered at

. For high-band operation (i.e.,

Band 1 where ), all subarrays are excited. However,when the array is switched to the second band (i.e., Band 2

where ) the first subarray corresponding to

is shut off. When the array is switched to the third band (i.e.,

Band 3 where ), the first and second stage subarrays

corresponding to and , respectively, are shut off.

This process is repeated until, for the lowest frequency band

(i.e., Band where ), all subarrays are shut off

except for the last stage.

C. Truncation/Thinning Algorithm

As discussed in Section III-A, the choice of a suitable

window function, from which a self-similar fractal radiation

pattern can be synthesized, is an important consideration in the

design process of multiband arrays. Window functions, such as

those considered in Section III-A, exhibit the highly desirable

properties of having low sidelobes and rapid rolloffs in the

transform domain. These properties can be exploited to develop

thinning algorithms to reduce both the physical size as well as

the number of elements in the synthesized multiband arrays,

while maintaining any associated pattern degradation within

acceptable limits. The steps required to implement an algorithm

useful for thinning the type of multiband arrays considered in

this paper will now be outlined.

One way of interpreting the multiband array factor expres-

sions given in (11) and (34) is that these types of arrays may

be decomposed into a series of uniformly-spaced subarrays

whose current distributions are scaled replicas of one another.

With this fact in mind, the truncation algorithm is then applied

in the same way to each subarray, which has the advantage ofpreserving the desirable self-scaling properties of these sub-

arrays. More specifically, if the element currents on each of

the subarrays (either linear or planar) are normalized by

their respective maximum values, then all normalized element

current magnitudes with values that are less than a specified

tolerance will be set equal to zero. Once this procedure has

been applied, then the current distributions on each of the

thinned subarrays are un-normalized. In other words, the nor-

malized current distributions of the subarrays are rescaled by

the factor for . Finally, the indi-

vidual subarrays are then superimposed to form the resulting

composite thinned linear or planar multiband array. This is


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(a) (b)

(c) (d)

Fig. 9. x z plane ( = 0 ) slices of the synthesized four-band planar array radiation pattern with s = 3 and N = 4 5 . Thinned case with " = 0 : 1 synthesizedusing a Blackman window.

in direct contrast to the algorithms implemented in [4], [16],

[17] where the truncation scheme is applied to the overall

fractal array instead of the individual subarrays. Another im-

portant advantage of the thinning approach developed here is

that it preserves the reconfigurable band-switching scheme in-

troduced in Section III-B.

The choice of the parameter represents a tradeoff between

the accuracy of the synthesized radiation pattern and the overall

size and number of elements in the array. For multiband arrays

synthesized via Blackman, Blackman-Harris, and Kaiser-Bessel

windows, it was found that a practical range for the truncation

tolerance is . The values of closer to 0.1 will lead

to smaller sized arrays with fewer elements. However, they will

also resultin more error between the desired and the synthesized

radiation patterns.

D. Array Elements

The successful implementation in practice of the multiband

array synthesis technique introduced in this paper requires that

the proper choice of antenna elements be made. One possibility

is to use some type of fractal antenna that is designed to have

the same multiband radiation properties as the array (i.e., the

same scaling/similarity factor would be used for designing the

multiband fractal antenna elements and the multiband array).

Also, as previously mentioned, the design methodology devel-

oped in this paper is well suited for application to the syn-

thesis of reconfigurable multiband apertures. For instance, the

reconfigurable aperture could be formed by a grid of electri-

cally small conducting patches with MEMS or other types of

RF switches placed between some or all of the patches. Hence,

switching between the different bands may be accomplished by

opening and closing the required set of connections between

these patches in order to form the appropriate sized radiating

elements.

IV. RESULTS

An example will be presented here in order to illustrate the

multiband fractal radiation pattern synthesis technique intro-

duced in this paper. Suppose we wish to synthesize a multiband

planar array with band center frequencies of , , ,


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(a) (b)

(c) (d)

Fig. 10. x z plane ( = 0 ) slice of the synthesized four-band planar array radiation pattern with s = 3 , N = 4 5 , and the main beam steered to = 4 5 , = 0 . Thinned case with " = 0 : 1 synthesized using a Blackman window.

and . In this case, the scale factor for the design would be

and the number of required bands are . Choosing a

Blackman window function with leads to a first-null

beamwidth (FNBW) of approximately 39 . Consequently, the

3-dB or half-power beamwidth (HPBW) is approximately 11 ,

obtained by numerical solution of

(44)

Strictly speaking, this expression is only valid for a single stage

, but if is greater than about 10 then successive

stages contribute negligibly to the characteristics of the main

lobe and (44) provides a reasonably accurate estimate of the

half-power beamwidth. The parameter is selected in

order to set the sidelobe level at approximately 20 dB down

from the mian beam. Finally, the interelement spacing, ,

is chosen to be a half-wavelength at the highest operating

frequency of the array (i.e., at ). Hence, the ideal

radiation pattern will be composed of a superposition of scaled

and shifted versions of the original Blackman window function

(see Fig. 3.42 of [11]).

Synthesized radiation patterns for this four-band planar array

example with (i.e., unthinned) are shown in Figs. 6 and

7. Overhead contour plots of the radiation patterns are pre-

sented in Fig. 6 for each of the four bands. Fig. 7 contains

a series of four radiation pattern slices, one for each band,

taken in the plane . Note that half-wave spacing

between active array elements is maintained throughout each

of the four bands. This is accomplished by implementing the

band-switching scheme discussed in Section III-B. The syn-

thesized planar array is composed of four subarrays with 91

by 91 elements in each [i.e., the Fourier series coefficients up

to have been retained in (34)], resulting in a com-

posite multiband planar array that requires 30 421 elements to

cover all four bands. However, by implementing the thinning

procedure discussed in Section III-C and choosing ,

the number of elements may be dramatically reduced to only
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(a) (b)

(c) (d)

Fig. 11. Radiation pattern slices synthesized from a thinned ( " = 0 : 1 ) planar array using a KaiserBessel window with (a) = 2 : 5 , (b) = 1 : 8 , (c) = 1 : 5 ,and (d) = 1 : 1 . As the parameter is decreased, the beamwidth increases and the required number of array elements decreases from (a) 409, (b) 281, (c) 249,to (d) 165.

409 to cover all four bands. Three-dimensional views of the

synthesized radiation patterns for the four-band thinned array

are shown in Fig. 8, while Fig. 9 contains the corresponding

pattern slices. This sequence of figures demonstrates

that thinning can lead to arrays with higher average sidelobe

levels. More importantly, however, these figures also show that

the characteristics of the main beam as well as the peak side-

lobe level are essentially preserved by the thinning process.

Finally, Fig. 10 shows a series of four radiation pattern slices

taken in the plane, where the main beam is steered to

, for each band.

The examples of synthesized multiband arrays considered

above were all based on the Blackman window function. How-

ever, more flexibility can be achieved in the design of such ar-

rays by using a KaiserBessel window rather than a Blackman

or BlackmanHarris window. This is due to the additional de-

gree of control offered by the parameter , which may be varied

to tradeoff beamwidth versus sidelobe level in the transform do-

main. The major benefit of this is that it allows more freedom

to adjust the overall size and total number of elements in a syn-

thesized array once the truncation/thinning process has been ap-

plied. Examples of this are provided in Fig. 11, which illustrates

that as the parameter is varied from 2.5 to 1.1, the number

of elements required for the corresponding thinned four-band

array ( , , and ) can be further reduced;

albeit, at the expense of the half-power beamwidth. Note that

only the synthesized radiation patterns for the first band (i.e.,

Band 1) of a four-band array are included in Fig. 11. Fig. 11(a)

shows that a KaiserBessel window with a value of

and will yield an array with 409 elements, which is

comparable to what was achieved previously with a Blackman

window. However, Fig. 11(b) shows the synthesized radiation

pattern that results when the parameters of the KaiserBessel

window are chosen to be and . This choice

of parameters leads to a thinned array having a total of 281 el-

ements (128 less than the previous case). Next we consider the


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Fig. 12. Overhead contour plot of an unthinnedWeierstrass planar arrayfactorwith P = 5 ,s = 3 , and = 0 : 5 . This radiation pattern was synthesized usinga Blackman window with 1 = 2 = 3 .

case shown in Fig. 11(c) where and . In this

case we find that the number of required elements in the syn-

thesized array is reduced to 249. Finally, in the last case shown

in Fig. 11(d), parameter values of and were

used. This resulted in a synthesized four-band array with only

165 elements.

Fig. 12 shows an overhead contour plot of a Weierstrass

fractal radiation pattern synthesized from a planar array with

parameters , , and . This figure nicely con-

veys the rich self-similar structure characteristic of Weierstrass

radiation patterns. This Weierstrass array was synthesized using

a Blackman window function with . The primary

disadvantage of Weierstrass arrays in general is the restriction

on once has been specified; i.e., , which

leads to patterns with relatively poor sidelobe suppression. For

example, the sidelobe level for the radiation pattern shown in

Fig. 12 is approximately dB below the main

lobe.

V. CONCLUSION

It has recently been demonstrated that multiband properties

can be achieved for certain self-scalable arrays, namely those

which produce self-similar fractal radiation patterns in the

limit of infinite array size. More specifically, two types of

multifrequency arrays have been considered in the literature,

one which generated Weierstrass fractal radiation patterns [3],

and the other which generated Koch fractal radiation patterns

[4]. In this paper we have considered a unified approach to the

design of multiband arrays via the synthesis of fractal radiation

patterns. It has been shown that the Weierstrass and Koch

arrays, previously considered independent, are actually special

cases of the more general design methodology introduced

here. Also of equal importance is the fact that this new method

circumvents past limitations associated with the practical

application of fractal radiation pattern synthesis techniques

based on either Weierstrass or Koch arrays. The advantages of

this new multiband array design approach are a significantly

reduced mutual coupling environment, the fact that a minimal

amount of element switching is required, and the ability toeasily implement in the form of a reconfigurable aperture.

A thinning procedure, based on the choice of an appropriate

window function, was also developed that preserves all the

advantages of the bandswitching scheme. Several candidate

window functions were considered including the rectangular,

Blackman, BlackmanHarris, and KaiserBessel. Finally, the

fractal antenna engineering design techniques developed in

this paper have been illustrated by presenting some specific

examples of synthesized multiband planar arrays.

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[2] D. H. Werner and P. L. Werner, On the synthesis of fractal radiationpatterns, Rad. Sci., vol. 30, no. 1, pp. 2945, 1995.

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pp. 17161719.


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Douglas H. Werner (S81M89SM94) receivedtheM.A. degree in mathematics,in 1986, andthe B.S.,M.S., and Ph.D. degrees in electrical engineering allfrom Pennsylvania State University, University Park,in 1983, 1985, and 1989, respectively.

Currently, he is an Associate Professor in theDepartment of Electrical Engineering, PennsylvaniaState University, and is also a Senior ResearchAssociate in the Electromagnetics and Environ-

mental Effects Department, Applied ResearchLaboratory at the same university. He is a Memberof the Communications and Space Sciences Laboratory (CSSL), UniversityPark, PA and is affiliated with the Electromagnetic Communication ResearchLaboratory, University Park. He has published numerous technical papersand proceedings articles and is the author of eight book chapters. He wasthe coeditor of Frontiers in Electromagnetics (Piscataway, NJ: IEEE Press,2000). He has also contributed a chapter for Electromagnetic Optimization byGenetic Algorithms (New York: Wiley, 1999). His research interests includetheoretical and computational electromagnetics with applications to atennatheory and design, microwaves, wireless and personal, communication systems,electromagnetics wave interactions with complex meta-materials, fractal andknot electrodynamics, and genetic algorithms.

Dr. Werner is a Memberof theAmericanGeophysical Union (AGU), Interna-tionalUnionof Radio Science (URSI)Commissions B andG, theApplied Com-putational Electromagnetics Society (ACES), Eta Kappa Nu, Tau Beta Pi, andSigma Xi. He was presented with the Applied Computational Electromagnetics

Society (ACES) Best Paper Award and the International Union of Radio Sci-ence (URSI) Young Scientist Award, both in 1993, the Pennsylvania State Uni-versity Applied Research Laboratory Outstanding Publication Award, in 1994,the College of EngineeringPSES Outstanding Research Awardand OutstandingTeaching Award, in March 2000 and March 2002, respectively, and was also re-centlypresentedwith an IEEE Central Pennsylvania Section MillenniumMedal.He has received several Letters of Commendation from the Department of Elec-trical Engineering, Pennsylvania State University, for outstanding teaching andresearch. He is an Editor of the IEEE ANTENNAS AND PROPAGATION MAGAZINEand a former Associate Editor ofRadio Science.

Mark A. Gingrich was born in Palmyra, PA,in 1971. He received the B.S. and M.S. degreesin electrical engineering from The PennsylvaniaState University (Penn State), University Park, in1999 and 2001, respectively. He currently holds anEducational and Foundational scholarship from theApplied Research Laboratory at Penn State, wherehe is working toward the Ph.D. degree in electricalengineering.

Previously, he spent several years as a technicianfor AT&T Microelectronics. His current research in-terests include the application of fractals and genetic algorithms to antenna andelectromagnetic problems.

Pingjuan L. Werner is an Associate Professorwith the College of Engineering, Pennsylvania StateUniversity, University Park. Her primary researchis in the area of electromagnetics, including fractalantenna engineering and the application of geneticalgorithms in electromagnetics.

She is a Fellow of the Leonhard Center, Collegeof Engineering, The Pennsylvania State University,a member of Tau Beta Pi National EgineeringHonor Society, Eta Kappa Nu National ElectricalEngineering Honor Society, Sigma Xi National

Research Honor Society, and a Senior Member of the IEEE. She received TheBest Paper Award from the Applied Computational Electromagnetics Societyin 1993.

a self-similar fractal radiation pattern synthesis techniques for reconfigurable multiband array

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