38530668 Fractal Antenna Engineering the Theory and Design of Fractal Antenna Arrays by Douglas H Werner Et Al October 1999

Fractal Antenna Engineering: The Theory andDesign of Fractal Antenna Arrays

Douglas H. Werner], Randy L. Haupf,and Pingjuan L. Werne~

lCommunications and Space Sciences LaboratoryThe Pennsylvania State University .Department of Electrical Engineering211A Electrical Engineering EastUniversity Park, PA 16802E-mail: [email protected]

2Department of Electrical EngineeringUtah State UniversityLogan, UT 84322-4120Tel: (435) 797-2840Fax: (435) 797-3054E-mail: [email protected] or [email protected]

3The Pennsylvania State UniversityCollege of EngineeringDuBois, PA 15801E-mail: [email protected]

Keywords: Fractals; antenna arrays; antenna theory; antennaradiation patterns; frequency-independent antennas; log-periodicantennas; low-sidelobe antennas; array thinning; array signalprocessing

1. Abstract

A fractal is a recursively generated object having a fractionaldimension. Many objects, including antennas, can be designedusing the recursive nature of a fractal. In this article, we provide acomprehensive overview of recent developments in the field offractal antenna engineering, with particular emphasis placed on thetheory and design of fractal arrays. We introduce some importantproperties of fractal arrays, including the frequency-independentmulti-band characteristics, schemes for realizing low-sidelobedesigns, systematic approaches to thinning, and the ability todevelop rapid beam-forming algorithms by exploiting the recursivenature of fractals. These arrays have fractional dimensions that arefound from the generating subarray used to recursively create thefractal array. Our research is in its infancy, but the results so far areintriguing, and may have future practical applications.

2. Introduction

The term fractal, which means broken or irregular fragments,was originally coined by Mandelbrot [1] to describe a family

of complex shapes that possess an inherent self-similarity in theirgeometrical structure. Since the pioneering work of Mandelbrotand others, a wide variety of applications for fractals has beenfound in many branches of science and engineering. One such areais fractal electrodynamics 'r2-6], in which fractal geometry is com­bined with electromagnetic theory for the purpose of investigating

a new class of radiation, propagation, and scattering problems. Oneof the most promising areas of fractal electrodynamics research isin its application to antenna theory and design.

Traditional approaches to the analysis and design of antennasystems have their foundation in Euclidean geometry. There hasbeen a considerable amount of recent interest, however, in the pos­sibility of developing new types of antennas that employ fractalrather than Euclidean geometric concepts in their design. We referto this new and rapidly growing field of research as fractal antennaengineering. There are primarily two active areas of research infractal antenna engineering, which include the study of fractal­shaped antenna elements, as well as the use of fractals in antennaarrays. The purpose of this article is to provide an overview ofrecent developments in the theory and design of fractal antennaarrays.

The first application of fractals to the field of antenna theorywas reported by Kim and Jaggard [7]. They introduced a method­ology for designing low-sidelobe arrays that is based on the theoryof random fractals. The subject of time-harmonic and time­dependent radiation by bifractal dipole arrays was addressed in [8].It was shown that, whereas the time-harmonic far-field response ofa bifractal array of Hertzian dipoles is also a bifractal, its time­dependent far-field response is a unifractal. Lakhtakia et al. [9]demonstrated that the diffracted field of a self-similar fractal screenalso exhibits self-similarity. This finding was based on resultsobtained using a particular example of a fractal screen, constructedfrom a Sierpinski carpet. Diffraction from Sierpinski-carpet aper­tures has also been considered in [6], [10], and [11]. The relatedproblems of diffraction by fractally serrated apertures and Cantortargets have been investigated in [12-17].

IEEE Antennas and Propagation MagaZine, Vol. 41, No.5, October 1999 1045-9243/99/$10.00©1999 IEEE 37

The fact that self-scaling arrays can produce fractal radiationpatterns was first established in [18]. This was accomplished bystudying the properties of a special type of nonunifonn lineararray, called a Weierstrass array, which has self-scaling elementspacings and current distributions. It was later shown in [19] how asynthesis technique could be developed for Weierstrass arrays thatwould yield radiation patterns having a certain desired fractaldimension. This work was later extended to the case of concentric­ring arrays by Liang et al. [20]. Applications of fractal concepts tothe design of multi-band Koch arrays, as well as to low-sidelobeCantor arrays, are discussed in [21]. A more general fractal geo­metric interpretation of classical frequency-independent antennatheory has been offered in [22]. Also introduced in [22] is a designmethodology for multi-band Weierstrass fractal arrays. Other typesof fractal array configurations that have been considered includeplanar Sierpinski carpets [23-25] and conceritric-ring Cantor arrays[26].

The theoretical foundation for the study of detenninisticfractal arrays is developed in Section 3 of this article. In particular,a specialized pattern-multiplication theorem for fractal arrays isintroduced. Various types of fractal array configuration are alsoconsidered in Section 3, including Cantor linear arrays andSierpinski carpet planar arrays. Finally, a more general and sys­tematic approach to the design of detenninistic fractal arrays isoutlined in Section 4. This generalized approach is then used toshow that a wide variety of practical array designs may be recur­sively constructed using a concentric-ring circular subarray gen­erator.

-_---4l--.....--¥--_-_-..J'--<_~z

Figure 1. The geometry for a Iiuear array of uniformly spacedisotropic sources.

3.1 Cantor linear arrays

A linear array of isotropic elements, unifonnly spaced a dis­tance d apart along the z axis, is shown in Figure I. The array fac­tor corresponding to this linear array may be expressed in the fonn[27,28]

jIO+2i/ncOS(nlj/], for 2N+I elements

AF(Ij/) = N n=1 (2)

2~/nCOs[(n-I/2)1j/ J, for 2N elements

3. Deterministic fractal arrayswhere

The array factor for a fractal array of this type may beexpressed in the general fonn [23-25]

A rich class of fractal arrays exists that can be fonned recur­sively through the repetitive application of a generating subarray.A generating subarray is a small array at scale one (P =I ) used toconstruct larger arrays at higher scales (i.e., P > 1). In many cases,the generating subarray has elements that are turned on and off in acertain pattern. A set fonnula for copying, scaling, and translationof the generating subarray is then followed in order to produce thefractal array. Hence, fractal arrays that are created in this mannerwill be composed of a sequence of self-similar subarrays. In otherwords, they may be conveniently thought of as arrays of arrays [6].

where GA (Ij/) represents the array factor associated with the gen­

erating subarray. The parameter £5 is a scale or expansion factorthat governs how large the array grows with each recursive appli­cation of the generating subarray. The expression for the fractalarray factor given in Equation (I) is simply the product of scaledversions of a generating subarray factor. Therefore, we may regardEquation (I) as representing a fonnal statement of the pattern­multiplication theorem for fractal arrays. Applications of this spe­cialized pattern-multiplication theorem to the analysis and designof linear as well as planar fractal arrays will be considered in thefollowing sections.

These arrays become fractal-like when appropriate elements areturned off or removed, such that

(5)

(3)

(4)

if element n is turned on

if element n is turned off.{I,

I =n 0,

The basic triadic Cantor array may be created by starting witha three-element generating subarray, and then applying it repeat­edly over P scales of growth. The generating subarray in this casehas three unifonnly spaced elements, with the center elementturned off or removed, i.e., 101. The triadic Cantor array is gener­ated recursively by replacing I by 101 and 0 by 000 at each stageof the construction. For example, at the second stage of construc­tion (P = 2 ), the array pattern would look like

Hence, fractal arrays produced by following this procedure belongto a special category of thinned arrays.

Ij/ =kd(cose -coseo]

k = 2;r / A..

and

One of the simplest schemes for constructing a fractal lineararray follows the recipe for the Cantor set [29]. Cantor linear arrayswere first proposed and studied in [21] for their potential use in thedesign of low-sidelobe arrays. Some other aspects of Cantor arrayshave been investigated more recently in [23-25].

(I)p

AFp(lj/) =11 GA(£5 P-11j/),

p=1

38 IEEE Antennas and Propagation MagaZine, Vol. 41, No.5, October 1999

101000101,

and at the third stage ( P = 3 ), we would have

1 0 1 000 I 0,1 000000000 1 0 1 000 1 0 1. O.G ----...---+----1

The array factor of the three-element generating subarray with therepresentation 101 is

which may be derived from Equation (2) by setting N = 1, 10 = 0,

and II = 1, Substituting Equation (6) into Equation (I) and choos­

ing an expansion factor of three (i,e., 0 = 3) results in an expres­sion for the Cantor array factor given by

21.50.5a·0.5·1·1.5

.06 L_--..l.__..L.-_--.l__-L__.l-_--l__...L-_----l

-2

·o,~

Figure 2c. A plot of the triadic Cantor fractal array factor forthe third stage of growth, P = 3. The array factor is

cos (If/ )cos (31f/) cos{91f/).

0.4

-0.4

(7)

(6)

1'0 P 1'0 p

AFp{lf/) =TI GA(3P-11f/) =TI cos(3P-11f/),p~l p~1

GA{If/) = 2cos{If/),

Figure 2a. A plot of the triadic Cantor fractal array factor for

the first stage of growth, P ='1. The array factor is cos{If/).

(8)

--_ _--- -_._.._--

---'-+·~-trHt-.--_j·--'--- .-_.f--

o4 f-.---... - ..-.---r----.--.-. -- ..-.-

·0.4 r------ ----

Figure 2d. A, plot of the triadic Cantor fractal array factor forthe fourth stage of growth, P = 4. The array factor is

cos(1f/ )cos(31f/ )cos (91f/ )cos (271f/).

-06l---L__L..._-L__.L.._-l.:__.L-_---:'::--~

·2 ·1.5 -1 ·0.5 0 0.5 1.5

o21---'--~ \+_.- -- +-- - --A- ''A''~-- .--.--Or--- - - -- _1_~ .. -- wJ~ --f'V-v---

IV v·0.2 -t---. -,- ---. · ..f-- -'-'''-' -----

o Bf---- --.-f-.---f------ ----- ------ ...-..-........ ----

o61----- --- f------. ---- -_. --'''-' ----..- ..._--

Suppose that the spacing between array elements is a quarter­

wavelength (i.e., d = AI4 ), and that eo = 90° . Then, an expression

for the directivity 'of a Cantor array of isotropic point sources maybe derived from

where the hat notation indicates that the quantities have been nor­malized. Figure 2 contains plots of Equation (7) for the first fourstages in the growth of a Cantor array.

21.5

1.5

0.5

0.5

o

o-0.5-,-1.5

,/ ~

/ [\

/ \~6

/ 1--- t--1/ ,5

I1---- t\4

I \2

/ \ ----II -\1 ----

) 1\

0.3

0.7

o·2

o.

o.

0.9

o.

o.

0.8

0.8

0.6

0.4

0.2

0

·0.2

·0.4

·0.6·2 ·1.5 ·1 ·0,5

Figure 2b. A plot of the triadic Cantor fractal array factor forthe second stage of growth, P = 2. The array factor is

cos{1f/)cos(31f/).where If/ =!!...u, with u = cose, and

2

IEEE Antennas and Propagation Magazine, Vol. 41, No, 5, October 1999 39

Substituting Equation (9) into Equation (8) and using the fact that

leads to the following convenient representation for the directivity:

For n =1 (5 =3 ), the generating subarray has a pattern 101 suchthat Equation (18) reduces to the result for the standard triadicCantor array found in Equation (7). The generating subarray pat­tern for the next case, in which n =2 (5 =5 ), is 10101 and, like­wise, when II =3 (5 =7), the array pattern is 1010101. The fractaldimension D of these uniforn1 Cantor arrays can be calculated as[21 ]

(19)IOg(T)

log(5)D

(10)

(9)" (tr) P (3 P-

1 JAFp 2u =::DCOS -2-1T:U .

This suggests that D =0.6309 for n = 1, D =0.6826 for IJ =2,and D=0.7124 for 11=3.

Finally, it is easily demonstrated from Equation (11) that themaximum value of directivity for the Cantor array is

As before, if it is assumed that d = A/4 and 80 = 90° , then the

directivity for these uniform Cantor arrays may be expressed as[23-25]

or

Dp =:: Dp(O) =:: 2P, where P = 1,2, ...

Dp(dB) =3.01P, where P =1,2, ....

(12)

(13)

(20)

Locations of nulls in the radiation pattern are easy to computefrom the product form of the array factor, Equation (1). Forinstance, at a given scale P, the nulls in the radiation pattern ofEquation (9) occur when

[3P

-1

)cos -2-1T:U = 0 . (14)

where use has been made of the fact that

The corresponding expression for maximum directivity is

Solving Equation (14) for U yields

uk = ±(2k-1)(l/3)P-I, where k = 1,2, ... ,(3P- 1+ 1)/2. (15) (0 + l)P

Dp=Dp(O)= -2- where P = 1,2, ... , (22)

Hence, from this we may easily conclude that the radiation patterns

produced by triadic Cantor arrays will have a total of 3P-

1 + 1nulls.

20

The generating subarray for the triadic Cantor array discussedabove is actually a special case of a more general family of uni­form Cantor arrays. The generating subarray factor for this generalclass of uniform Cantor arrays may be expressed in the form

'0

·10

(16)

-20

-30

-40

where'50

8=2n+l and n=I,2, .... (17) -60

'80

_80L.-_-'--_-1__-'-_--'-__-'-_--'-__~_ ___'__ ___'o 20 40 60 80 100 120 140 160

Theta (degreE:!s)

-70

Hence, by substituting Equation (16) into Equation (1), it followsthat these uniform Cantor arrays have fractal array factor repre­sentations given by [21,23-25]

. . [(5 +1)5 p-I ]" ( 2 )P P sm -2- If/

AFp(lf/) = - f1 .5+1 p=l sin[oP-Ilf/]

(18)

Figure 3. A directivity plot for a uniform Cantor fractal arraywith P =1 and 5 =7. The spacing between elements of thearray is d = A / 4. The maximum directivity for stage 1 isD1 = 6.02 dB.

40 IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999

(26)

(25)

(24)

= fI GA(5n

-JIf/)=AF(If/),

n=-oo

AF(If/)= fI GA(5P-11f/).p=-CfJ

AF(5±qlf/)= TI GA(5P±q-llf/)p=-CfJ

which is valid provided q is a positive integer (i.e., q = 0,1,2, ... ).

The parameter 5±Q, introduced in Equation (25), may be inter­preted as a frequency shift that obeys the relation

Next, by making use of Equation (24), we find that

18016014012080 100Theta (degrees)

604020

(\AA(\

A (\ AA(\

r n·70

·80o

·40

·30

·20

10

20

·50

·10

·60

·70

AFp(If/) for q = 1,2, ... , (27)

P+Q

I1 GA(5H

If/)/=P+I

One of the more intriguing attributes of fractal arrays is thepossibility for developing algorithms, based on the compact prod­uct representation of Equation (1), which are capable of performingextremely rapid radiation pattern computations [23]. For example,if Equation (2) is used to calculate the array factor for an odd num­ber of elements, then N cosine functions must be evaluated, and Nadditions performed, for each angle. On the other hand, however,using Equation (7) only requires P cosine-function evaluations andP -1 multiplications. In the case of an 81 element triadic Cantor

array, the fractal array factor is at least N/ P =40/4 =10 timesfaster to calculate than the conventional discrete Fourier transform.

where fo is the original design frequency. Finally, for a finite size

array, Equation (1) may be used in order to show that

where the bracketed term in Equation (27) clearly represents the"end-effects" introduced by truncation.

180160140.80 LC-'--'.LUJ.._--U-'L-"----"lW-'-L----'-LL..L-Lli.il.-L.LU--.JL.....L-ll-_-W.l..l..-L.J...J

a 20 40 60 80 100 120Theta (degrees)

·40

10

·50

20

·60

·10

~ -20

:;'B -30~

<5

Figure 4. A directivity plot for a uniform Cantor fractal arraywith P =2 and 5 =7. The spacing between elements of thearray isd = ,,1,/4. The maximum directivity for stage 2 is

D2 = 12.04 dB.

Figure 5. A directivity plot for a uniform Cantor fractal arraywith P = 3 and 5 = 7. The spacing between elements of thearray is d = A /4. The maximum directivity for stage 3 is

D3 = 18.06 dB.

or

Dp (dB) = IOPIOg(5

; 1), where P =1,2,.... (23)

A plot of the directivity for a uniform Cantor array, with

d = ,,1,/4, eo = 90°, 5 = 7, and P = 1, is' shown in Figure 3. Fig­

ures 4 and 5 show the directivity plots that correspond to stages ofgrowth for this array of P = 2 and P = 3, respectively.

The multi-band characteristics of an infinite fractal array may bedemonstrated by following a procedure similar to that outlined in[21] and [22]. In other words, suppose that we consider the arrayfactor for a doubly infinite linear fractal (uniform Cantor) array,which may be defined as

z

-+--ff-_ff-......-ff--4f---+--.... Y

d-:r­---1Lx ~-~'----1I(---j...--~-'1I--~

x

Figure 6. The geometry for a symmetric planar array of iso­

tropic sources with elements uniformly spaced a distance dx

and dy apart in the x and y directions, respectively.

IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999 41

3.2 Sierpinski carpet arrays factor of is = 3 yields the following expression for the fractal arrayfactor at stage P:

The previous section presented an application of fractal geo­metric concepts to the analysis and design of thinned linear arrays.In this section, these techniques are extended to include the moregeneral case of fractal planar arrays. A symmetric planar array of

isotropic sources, with elements uniformly spaced a distance d x

and dy apart in the x and y directions, respectively, is shown in

Figure 6. It is well known that the array factor for this type of pla­nar array configuration may be expressed in the following way[28]:

M

III + 2 L {Iml cos[mlfx] + 11m cos[mlfy]}m=2

AFp (ux,uy) = --;,fr[cos(3P-I7rUx ) + cos(3

p-

l7ruy )

4 P=!

+2cos(3P-I7rux)cos(3P-

I 7ru y ) J. (35)

The geometry for this Sierpinski carpet fractal array at variousstages of growth is illustrated in Figure 7, along with a plot of thecorresponding array factor. A comparison of the array factors forthe first four stages of construction shown in Figure 7 reveals theself-similar nature of the radiation patterns.

As before, these arrays can be made fractal-like by following asystematic thinning procedure, where

Array Factor Plot

')@@@ (§:)

''''M!!!II.",'''''''''",,''''''''=~

Fractal Array

4

3

2

Scale(28)

(29)

(30)

(31)

for (2M)2 elements

for (2M - 1)2 elements

M M+ 4L L I mil cos [mlfx] cos[ nlfyJ,

1l=2m=2

M M

4L L IlIlil cos[(m -1/2)1fx]cos[(n -1/2)1fy],1l=11ll=1

Ifx = kdx [sinBcosrp - sinBocosrpo],

{I, if element (m, n) is turned on

I mn = 0, if element (m, n) is turned off'

GA(ux'uy )

=Hcos (7rUx) + cos(7rU y ) + 2cos (7rUx)cos(7rU y )J, (32)

1 1

o 1 .

1 1

A Sierpinski carpet is a two-dimensional version of theCantor set [29], and can similarly be applied to thinning planararrays. Consider, for example, the simple generating subarray

where

The normalized array factor associated with this generating subar­

ray, for d x = d y = A/2, is given by

Substituting Equation (32) into Equation (1) with an expansion

where

Ux = sinBcosrp - sinBocostPo,

U y = sinBsinrp - sinBosintPo'

(33)

(34)

Figure 7. The geometry for the Sierpinski carpet fractal arrayat various stages of growth. Scale 1 is the generator subarray.Column 2 is the geometrical configuration of the Sierpinskicarpet array: white blocks represent elements that are turnedon, and black blocks represent elements that are turned off.Column 3 is the corresponding array factor, where the anglephi is measured around the circumference of the plot, and theangle theta is measured radially from the origin at the lowerleft.

42 IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999

An expression for the directivity of the Sierpinski carpet array,

for the case in which eo = 0° , may be obtained from

AFf, (e,¢)I 2n n A

4 I IAF} (e,¢)sineded¢Jr 0 0

AF}(e,¢)1 l( l( A

- I IAFJ(e,¢)sineded¢2Jr 0 0

(36)

which follows directly from Equation (35). The double integralthat appears in the denominator of Equation (36) does not have aclosed-form solution in this case, and therefore must be evaluatednumerically. However, this technique for evaluating the directivityis much more computationally efficient than the alternativeapproach, which involves making use of the Fourier-series repre-.sentation for the Sierpinski carpet array factor given by Equa­tions (28) and (31), with

A p

AFJ(e,¢) =~TI[COs(3P-1Jrsinecos¢)16 p~1

+cos(3P- 1Jrsinesin ¢)

+2cos(3P-1Jrsin ecos¢)cos(y-l Jrsinesin¢)J, (37)

Figure 9. The complement of the Sierpinski carpet array forP=4.

The geometry for a stage-four (P =4) Sierpinski carpet arrayand its complement are shown in Figures 8 and 9, respectively.These figures demonstrate that the full 81 x 81 element planararray may be decomposed into two subarrays, namely, a Sierpinskicarpet array and its complement. This relationship may be repre­sented formally by the equation

(40)3

P +1M=--.

2

I11-

(38)

(39)If/y =Jr sinesin¢,

If/x =Jr sin ecos¢,

where

(41)

Figure 8. A Sierpinski carpet array for P = 4.

where

P

AFp (ux,uy ) =2PTI[cos(3P- 1Jrux )+cos(3P- 1Jruy )p~l

+2cos(y-1Jrux )coS(3P-J7TUy)J (42)

is the Sierpinski carpet array factor, AFp is the array factor asso­ciated with its complement, and AF denotes the array factor of thefull planar array. It can be shown that the array factor for a uni­

formly excited square planar array (i.e., I mn = 1 for all values of m

and n) may be expressed in the form

where

(44)

IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999 43

The multi-band nature of the planar Sierpinski carpet arrays maybe easily demonstrated by generalizing the argument presented inthe previous section for linear Cantor arrays. Hence, for doublyinfinite carpets, we have

Finally, by using Equation (43) together with Equation (42), an

expression for the complementary array factor AFp may beobtained directly from Equation (41). Plots of the directivity for aP == 4 Sierpinski carpet array and its complement are shown inFigures 10 and 11, respectively.

806040-20 0 20Theta (degrees)

-40-60-80

1\ A

\ /\ A rrn i~~. I~ I

·40

·30

10

20

-10

-20

30

40

Figure 11. A directivity plot for the complement to theSierpinski carpet array with P == 4. The spacing between ele­ments of the array is d == A/2, with ¢ = 0°. The maximum

directivity for stage 4 is D4 = 34.41 dB.

(45)AF(lf/x,lf/y)== fI GA(5 P- 1If/x,O"P-llf/y),p=-oo

4.1 Theory

4. The concentric circular ring subarray generator

from which we conclude that

An alternative design methodology for the mathematical con­struction of fractal arrays wiH be introduced in this section. Thetechnique is very general, and consequently provides much moreflexibility in the design of fractal arrays when compared to otherapproaches previously considered in the literature [21, 23-26]. Thisis primarily due to the fact that the generator, in this case, is basedon a concentric circular ring array.

where

If/mn (B,r/J) =krmsinBcos(q) - q)mn) + an1/1

k == 2iT/A

M = Total number of concentric rings

(48)

(49)

(50)The generating array factor for the concentric circular ring

array may be expressed in the form [30] N m = Total number of elements on the mth ring (51 )

Imn = Excitation current amplitude ofthe nth element

on the mthring located at r/J =q)mn (53)

M N m

GA(B,q))== L Llmne)¥Jm,,(B,¢),m=ln=1

(47)rm = Radius of the mth ring (52)

30

amn = Excitation current phase of the nth element on

the mth ring located at r/J =r/Jmn' (54)

20

1Ol\ f\

A wide variety of interesting as weH as practical fractal arraydesigns may be constructed using a generating subarray of the formgiven in Equation (47). The fractal array factor for a particularstage of growth P may be derived directly from Equation (47) byfoHowing a procedure similar to that outlined in the previous sec­tion. The resulting expression for the array factor was found to be

·10

-20 (55)

·30

where 0" represents the scaling or expansion factor associatedwith the fractal array.

·80 ·80 ·40 ·20 0 20Theta (degrees)

40 60 80

Figure 10. A directivity plot for a Sierpinski carpet array withP == 4. The spacing between elements of the array is d =A / 2,with r/J = 0°. The maximum directivity for stage 4 is

D4 = 37.01dB.

A graphical procedure, which is embodied in Equation (55),can be used to conveniently illustrate the construction process forfractal arrays. For example, suppose we consider the simple four­element circular array of radius r, shown in Figure 12a. If weregard this as the generator (stage 1) for a fractal array, then the

44 IEEE Antennas and Propagation MagaZine, Vol. 41, No.5, October 1999

(57)

which has a corresponding representation in terms of decibelsgiven by

Another unique property of Equation (56) is the fact that the con­ventional co-phasal excitation [30],

(60)

(59)

. (58)

M Nm

LLlmnm=\n=\

M N P"" ~ I e}If/(B,¢)L..J £.....J mnm=ln=l

For the special case when c5 =1, we note that Equations (57) and(58) reduce to

Stage 1

Stage 2 (61)

Figure 12. An illustration of the graphical construction proce­dure for fractal arrays based on a four-element circular subar­ray generator: (a) Stage 1 (P =I) and (b) stage 2 (P =2).

where Bo and ¢o are the desired main-beam steering angles, can

be applied at the generating subarray level. To see this, we recog-

y

Taking the magnitude of both sides of Equation (56) leads to

It is convenient for analysis purposes to express the fractalarray factor, Equation (55), in the following normalized form:

next stage of growth (stage 2) for the array would have a geometri­cal configuration of the form shown in Figure 12b. Hence the firststep in the construction process, as depicted in Figure 12, is toexpand the four-element generator array by a factor of c5 •This isfollowed by replacing each of the elements of the expanded arrayby an exact copy of the original unsealed four-element circularsubarray generator. The entire process is then repeated in a recur­sive fashion, until the desired stage of growth for the fractal arrayis reached.

---tll-------l'"------.--~_+x

Figure 13. The geometry for a two-element circular subarraygenerator with a radius of r =A. / 4.

1

(56)

1

M Nm

. p-l ( )1"" "" I eJo If/mn B,¢A P ~~ mn

AFp(B,¢) =n m=ln=l N .M m

p=1 2: 2:Imn

m=ln=l

IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999 45

1\ 1 p 2 j8p-

1[ '!..cos(¢-¢" )+a"J

(62)AFp(¢) 0= pTI'Le 2 ,

<p 2 p=l n=1

1 ) x where, without loss of generality, we have set () =90° and

¢n =(n-l):r, (63)

an =-%cos(¢o-¢n)' (64)<p

1 ) x Substituting Equations (63) and (64) into Equation (62) results in asimplified expression for the fractal array factor, given by

1\ P [:r ]AFp (¢) =!]cos e5P-''2(cos¢-cos¢o) . (65)

<p If we choose an expansion factor equal to one (i.e., e5 '= 1),

1 then Equation (65) may be written as) x

ipp (¢) =cosP [%(cosr/J-cosr/Jo)] . (66)

. ~ I ~ .I+- 2, ~I+-- 2, ~I+- 2,-+1222

Stage 3

y

y

-------~.....+---1---+-..~--------'--....2

Stage 1

y

__0_'1_""'-0---'----+

I+- 2, --+I+- 2, -+/2 2

Stage 2

Figure 14. The first four stages in the construction process of auniformly spaced binomial array.

nize that the position of the main beam produced by Equation (56)

is independent of the stage of growth P, since it corresponds to a

value of 'l'mn '= 0 (i.e., when (J 0= (Jo and ¢ '= ¢o)' In other words,

once the position of the main beam is determined for the generat­ing subarray, it will remain invariant at all higher stages of growth.

y

. : 61 : .I+- 2, --+I+-- 2,~I+- 2, -+I+- 2,-----+1222 2

Stage 4

q>

1 ) x

This represents the array factor for a uniformly spaced (d '= A12)

linear array with a binomial current distribution, where the totalnumber of elements, Np, for a given stage of growth, P, is

N p =P + 1. The first four stages in the construction process ofthese binomial arrays are illustrated in Figure 14. The general rulein the case of overlapping array elements is to replace each ofthoseelements by a single element that has a total excitation-currentamplitude equal to the sum of all the individual excitation-currentamplitudes. For example, in going from stage 1 to stage 2 in thebinomial-array construction process illustrated in Figure 14, wefind that two elements will share a common location at the centerof the resulting three-element array. Since each of these two arrayelements has one unit of current, they may be replaced by a singleequivalent element that is excited by two units of current.

Next, we will consider the family of arrays that are generatedwhen t5 0= 2. The expression for the array factor in this case is

4.2 Examples y

Several different examples of recursively generated arrayswill be presented and discussed in this section. These arrays havein common the fact that they may be constructed via a concentriccircular ring subarray generator of the type considered in the previ­ous section. Hence, the mathematical expressions that describe theradiation patterns of these arrays are all special cases of Equation(56).

4.2.1 Linear arrays

Various configurations of linear arrays may be constructedusing a degenerate form of the concentric circular ring subarraygenerator introduced in Section 4.1. For instance, suppose we con­sider the two-element circular subarray generator with a radius of

r 0= :i14; shown in Figure 13. If the excitation-current amplitudesfor this two-element generating subarray are assumed to be unity,then the general fractal array factor expression given in Equation(56) will reduce to the form

1

- ..-------f--------1t----'---.... x1

1

Figure 15. The geometry for a four-element circular subarray

generator with a radius of r '= A. !(2J2).

46 IEEE Antennas and Propagation MagaZine, Vol. 41, No.5, October 1999

Stage 1

"1 "1

"1 "1

Stage 2

"1 "2 "1

"2 "4 "2

"1 "2 "1

(67)

which results from a sequence of uniformly excited, equally spaced

(d == .:t/2) arrays. Hence, for a given stage of growth P, these

arrays will contain a total of N p == 2 P elements, spaced a half­wavelength apart, with uniform current excitations. Finally, the lastcase that will be considered in this section corresponds to a choiceof 5 == 3 . This particular choice for the expansion factor gives riseto the family of triadic Cantor arrays, which have already been dis-

cussed in Section 3.1. These arrays contain a total of N P == 2P

elements, and have current excitations which follow a uniformdistribution. However, the resulting arrays in this case are non­uniformly spaced. This can be interpreted as being the result of athinning process, in which certain elements have been systemati­cally removed from a uniformly spaced array in accordance withthe standard Cantor construction procedure. The Cantor array fac­tor may be expressed in the form

(68)

Stage 3

"1 °3 "3 "1

°3 "9 "9 "3

"3 "9 "9 "3

"1 "3 "3 "1

which follows directly from Equation (65) when 5 == 3.

4.2.2 Planar square arrays

In this section, we will consider three examples of planarsquare arrays that can be constructed using the uniformly excitedfour-element circular subarray generator shown in Figure 15. Thissubarray generator can also be viewed as a four-element squarearray. The radius of the circular array was chosen to be

r == .:tl(2.fi), in order to insure that the spacing between the ele­

ments of the circumscribed square array would be a half-wave­

length (i.e., d == .:t/2). In this case, it can be shown that the generalexpression for the fractal array factor given in Equation (56)reduces to

/\ 1 P 4 }OP-'[,jzsinOcOS(¢-¢n)+an]AFp (8,¢)==pTIIe 2 , (69)

4 p=ln=1

Stage 4

"1 "4 "6 "4 "1

"4 "16 "24 "16 "4

"6 "24 "36 "24 "6

"4 "16 "24 "16 "4

"1 "4 "6 "4 "1

Figure 16. The first four stages of growth for a family of squarearrays with uniformly spaced elements and binomially distrib­uted currents.

where

¢n ==(n-1)!:,2

Ifwe define

then Equation (69) may be written in the convenient form

(70)

(71)

(73)

IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999 47

where Stage 10

lfIn(Oo,¢O)=O.

iii'Now suppose we consider the case where the expansion fac- ~-20

tor 8 =1. Substituting this value of 8 into Equation (73) leads toQ)"0.a'c

AFp(O,¢)=[~~eJVln(B'¢)r~-40

(74) ~

-60The first four stages of growth for this array are illustrated in Fig- 0 20 40 60 80ure 16. The pattern that emerges clearly shows that this construc- thetation process yields a family of square arrays, with uniformly

. spaced elements (d = A/2) and binomially distributed currents. Fora given stage of growth P, the corresponding array will have a total Stage 2

of Np =(P+ 1)2 elements. Figure 17 contains plots of the far-field 0

radiation patterns that are produced by the four arrays shown inFigure 16. These plots were generated using Equation (74) for val- iii'ues of P =1, 2, 3, and 4. It is evident from Figure 17 that the ~-20

Q)radiation patterns for these arrays have no sidelobes, which is a "0

::::lfeature characteristic ofbinomial arrays [27]. :g

~-40The next case that will be considered is when the expansion ~

factor 0 =2. This particular choice of 0 results in a family of uni-formly excited and equally spaced (d =A/2) planar square arrays, -60which increase in size according to Np =22P . The recursive array a 20 40 60 80

factor representation in. this case is given by theta

AFp(O,¢) =~ fIi:ej2P-IVln(B,¢) . (75) Stage 34 p=ln=l a

For our final example of this section, we return to ·the squareSierpinski carpet array, previously discussed in Section 3.2. The iii'generating subarray for this Sierpinski carpet consisted of a uni- ~-20

Q)

formly excited and equally spaced (d =A/2) 3 x 3 planar array "0::::l

with the center element removed. However, we note here that this ""2Cl .

generating subarray may also be represented by two concentric til -40four-element circular arrays. By adopting this interpretation, it is

~

easily shown that the Sierpinski carpet array factor may beexpressed in the form -60

0 20 40 60 80AlP 2 4 pI theta

AFp(O,¢) =p nL ~::ej3 - Vlmn(B,¢) , (76)8 p=lm=ln=!

where Stage 4a

lfImn (o,¢) =.;;;;1l'[sinOcos(¢ - ¢mn) - sin 00 cos(¢o - ¢mn)] (77) iii'~-20Q)

11 =( mn-l)!:"0

(78) .amn m 2' 'c

~-40~

Np =23P . (79)

-60a 20 40 60 804.2.3 Planar triangular arrays theta

A class of planar arrays will be introduced in this section that Figure 17. Plots of the far-field radiation patterns produced byhave the property that their elements are arranged in some type of the four arrays shown in Figure 16.

48 IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999

y

Ar=--2{3

-+-----r-----l"------..::;e.----'-~x

Stage 1

Stage 2

Stage 3

Figure 18. The geometry of a three-element circular subarray

generator with a radius of r == A 1(2J3).

recursively generated triangular lattice. The first category of trian­gular arrays that will be studied consists of those arrays that can beconstructed from the uniformly excited three-element circularsubarray generator shown in Figure 18. This three-elemerit circular

array of radius r == AI(2J3) can also be interpreted as an equilat­

eral triangular array, with half-wavelength spacing on a side (i.e.,

d == ,1,/2). The fractal array factor associated with this triangulargenerating subarray is

where

Stage 4

Figure 19. The geometry for a uniformly excited Sierpinskigasket array. The elements for this array are assumed to corre­spond to the centers of the shaded triangles.

(81)y

where-'+:-----+----:-t"---i'''-------'''~---+_---l--+x

(82)

(83)

The compact form of Equation (80) is then

If it is assumed that <5 == 1, then we find that Equation (83)can be expressed in the simplified form

(85)

This represents the array factor for a stage-P binomial triangular

Figure 20. The geometry for a six-element generating subarraythat consists of two three-element concentric circular arrays,

with radii rj =AI(2J3) and r2=?./(J3).

IEEE Antennas and Propagation MagaZine, Vol. 41, No.5, October 1999 49

Stage 3

Stage 1

Stage 2

Stage 4

Figure 21. The first four stages in the construction process of atriangular array via the generating subarray illustrated in Fig­ure 20, with an expansion factor of 5 =2. The element loca­tions correspond to the vertices of the triangles.

array. The total number of elements contained in this array may bedetermined from the following formula:

This array factor corresponds to uniformly excited Sierpinski gas­ket arrays, of the type shown in Figure 19. The construction proc-

which has been derived by counting overlapping elements onlyonce. On the other hand, if we choose J =2, then Equation (83)becomes

ess illustrated in Figure 19 assumes that the array elements arelocated at the center of the shaded triangles. Hence, these arrays

have a growth rate that is characterized by N p = 3P.

The second category of triangular arrays that will be exploredin this section is produced by the six-element generating subarrayshown in Figure 20. This generating subarray consists of two three-

element concentric circular arrays, with radii lj =A/(2J3) and

rz =A/(J3). The excitation current amplitudes on the inner three­

element array are twice as large as those on the outer three-elementarray. The dimensions of this generating subarray were chosen insuch a way that it forms a non-uniformly excited six-element trian­gular array, with half-wavelength spacing between its elements

(i.e., d =A/2). If we treat the generating subarray as a pair ofthree-element concentric circular ring arrays, then it follows from

(86)

(87)

(P+1)(P+2)2

50 IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999

Equation (56) that the fractal array factor in this case may beexpressed as .

(94)

AFp

(B,tP) =~ IT±±I/lllIejl",-1 [kr,,, sinOcos(¢-¢m,'}ta;;;j:'til§)""" ".":""":W~!rwiII next consider two special cases of Equati~n (94),

9 I'~I iJl~lll~l ' namely when 0 = 1 and 0 = 2. In the first case, EquatIOn (94)reduces to

where

I/lill = 21m,

mJrkr,1I = Jj'

JrtPiJllI =(2n+m-3)3'

al1l11 = -kr,,, sin Bocos (tPo - tPl1I11 ) .

Ifwe define

(89)

(90)

(91)

(92)

where

2P+!

Np = L p=(P+l)(2P+I).p~1

In the second case, Equation (94) reduces to

(95)

(96)

(97)

'1/111" (B,¢) = Ji [sinBcos(tP -tPl1IlI) - sinBocos(¢o - tPl1IlI)] (93) where

such that 'I/"/I/(Bo,¢o) =0, then Equation (88) may be written inthe form

2P+1-1

Np = L p=2P (2 P+

1-I).p~1

(98)

Stage 1 Stage 3 Stage 2

·1 ·1 ·1

·2 ·2 ·2 ·2 ·2 ·2

·1 ·2 ·1 ·3 ·2 ·3 ·3 ·2 ·3

·4 ·4 ·4 ·4 ·4 ·4 ·4 ·4

·s ·4 ·6 ·4 ·s ·3 ·4 ·6 ·4 ·3

·6 ·6 ·4 ·4 ·6 ·6 ·2 ·2 ·4 ·4 ·2 ·2

·7 ·6 ·9 ·4 ·9 ·6 ·7 ·1 ·2 ·3 ·4 ·3 ·2 ·1

·S ·S ·S ·S ·S ·S ·S ·S

·7 ·S ·12 ·S ·14 ·S ·12 ·S ·7

·6 ·6 ·S ·S ·12 ·12 ·S ·S ·6 ·6

·s ·6 ·9 ·S ·14 ·12 ·14 ·S ·9 ·6 ·s·4 ·4 ·4 ·4 ·S ·S ·S ·S ·4 ·4 ·4 ·4

·3 ·4 ·6 ·4 ·9 ·S ·12 ·S ·9 ·4 ·6 ·4 ·3

·2 ·2 ·4 ·4 ·6 °6 ·S ·S °6 °6 °4 °4 ·2 °2

·1 ·2 ·3 °4 ·5 ·6 ·7 ·S ·7 ·6 Os ·4 °3 ·2 ·1

Figure 22. The current distributions for the first three trian-gular arrays shown in Figure 21.

IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999 51

52

y

--.----------I~-------..._-...L-_. X

Figure 25. The geometry for a uniformly excited six-elementcircular subarray generator of radius r = ..1,/2.

These are both examples of fully-populated uniformly spaced(d = ..1,/2) and non-uniformly excited triangular arrays. Hence, thistype of construction scheme can be exploited in order to realizelow-sidelobe-array designs. To further exemplify this importantproperty, we will focus here on the special case where 15 = 2. Thefirst four stages in the construction process of this triangular arrayare illustrated in Figure 21. The triangular arrays shown in Fig­ure 21 are made up of many small triangular subarrays that haveelements located at each of their vertices. Some of these verticesoverlap and, consequently, produce a nonuniform current distribu­tion across the array, as shown in Figure 22. Figure 23 contains asequence of far-field radiation-pattern plots, calculated usingEquation (97) with ¢ = 0° , which correspond to the four triangulararrays depicted in Figure 21. It is evident from the plots shown inFigure 23 that a sidelobe level of at least -20 dB can be achievedby higher-order versions of these arrays. Color contour plots ofthese radiation patterns are also shown in Figure 24. This sequenceof contour plots clearly reveals the underlying self-scalability,which is manifested in the radiation patterns of these triangulararrays.

4.2.4 Hexagonal arrays

Another type of planar array configuration in common use isthe hexagonal array. These arrays are becoming increasinglypopular, especially for their applications in the area of wirelesscommunications. The standard hexagonal arrays are formed byplacing elements in an equilateral triangular grid with spacings d(see, for example, Figure 2.8 of [31 D. These arrays can also beviewed as consisting of a single element located at the center, sur­rounded by several concentric six-element circular arrays of differ­ent radii. This property has been used to derive an expression forthe hexagonal array factor [31]. The resulting expression, in nor­malized form, is given by

IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999

Stage 1 Stage 2 Stage 1 Stage 2

Stage 3 Stage 4 Stage 3 Stage 4

Figure 24. Color contour plots of the far-field radiation pat­terns produced by the four triangular arrays shown in Figure21. The angle phi varies azimuthally from 0° to 360°, and theangle theta varies radially from 0° to 90°.

Figure 29. Color contour plots of the far-field radiation pat­terns produced by the four hexagonal arrays shown in Figures26 and 27. The angle phi varies azimuthally from 0° to 360°,and the angle theta varies radially from 0° to 90°.

Stage 1 Stage 2

Stage 1 Stage 2

Stage 3 Stage 4

Figure 32. Color contour plots of the far-field radiation pat­terns produced by a series of four (P = 1,2,3,4) fully-populatedhexagonal arrays generated with an expansion factor of 0 = 2.The angle phi varies azimuthally from 0° to 360°, and the angletheta varies radially from 0° to 90°.

Figure 33. Color contour plots of the far-field radiation pat­terns produced by a series of four (P = 1,2,3,4) fully-populatedhexagonal arrays generated with an expansion factor of 0 = 2.In this case, the phasing of the generating subarray was chosensuch that the maximum radiation intensity would occur ateo =45° and ¢o =90° . The angle phi varies azimuthally from0° to 360°, and the angle theta varies radially from 0° to 90°.

Stage 4Stage 3

IEEE Antennas and Propagation MagaZine, Vol. 41, No.5, October 1999 53

Stage 1

o

Stage 3

Stage 2 Stage 4

Figure 26. A schematic representation of the first four stages in the construction of a hexagonal array. The element locationscorrespond to the vertices of the hexagons.

Stage 1 Stage 2 Stage 4

., ., ., .,., -2 .,

., -2 ., ., -2 .,., ., ., .,

·1 ·2 ·1 ·1 ·2 ·1

·1 ·2 ·1., ., ., .,

Stage 3

., ., ., ., ., ., ., .,~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~., -2 -3 -2 ., -3 -3 ., -2 -3 -2 .,

~ ~ ~ ~ ~ ~ ~., -2 ., -2 -3 -2 ., ., -2 -3 -2 ., -2 .,

~ ~ ~ ~ ~ ~ ., ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ 1~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ 1., ., ., ., ., ., ., .,

~ ~ ., ~ ~ ., ~ ., 1 ., 1 ~ 1 1 1 ~

~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ 1 1 ~ ~ 1 ., ~ ~ 1 1 ~ ~ 1 1 ~ 1~ ~ ~ ~ ~ ~ ~ ~ ~ ~

., ~ ~ ~ 1 ~ ~ 1 ~ ~ ~ ~ 1 ~ ~ 1 ~ ~ ~ 1~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

1 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~.~ ~ ~ ~ ~

1 ~ ~ ~ ~ ~ ~ ~'~ ~ ~ ~ ~ ~ 1~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

1 1 1 1 1 1 1 1 1 1 ., 1 1 1 1 ~

1 ~ 1 ~ ~ ~ 1 ~ ~ ~ ~ ~ ~ ~ 1 ~ ~ ~ ~ ~ ~ ~ ~ 1 ~ ~ ~ 1 ~ 1~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

1 ~ 1 ~ ~ ~ ~ ~ ~ ~ 1 ~ ~ ~ ~ 1 ~ ~ ~ ~ ~ ~ ~ 1 ~ 1~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

1 ~ ~ ~ ~ ~ ~ ~ 1 ~ ~ ~ ~ ~ ~ 1 ~ ~ ~ ~ ~ ~ ~ 1~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

1 ~ 1 ~ ~ ~ 1 1 ~ ~ ~ ~ ~ ~ 1 1 ~ ~ ~ 1 ~ 1~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

1 ~ ~ ~ 1 ~ ~ 1 ~ ~ ~ ~ 1 ~ ~ 1 ~ ~ ~ 1~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

1 ~ ~ ~ ~ ~ ~ ~ 1~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Figure 27. The element locations and associated current distri·butions for each of the four hexagonal arrays shown in Figure 26.

54 IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999

P P 5

10 + I I Il plIlllp==! m=1 n=O

where

(99)

(100)

Stage 1or--==::-..---~--~-~---'

~-20CO"CCD"C -40.a'c~-60

::2:

[2 0 2 0 ( )2]'_ -1 rplIl +d-p -d- m-l 1l1r

¢PIIlIl - COS +-,2rpIIldp 3

(101)

-80

o 20 40 60theta

80

a 1'11I" = -krplII sin Bocos(¢o - ¢1'11I,,) , (102)Stage 2

and P is the number of concentric hexagons in the array. Hence,the total number of elements contained in an array with P hexagonsis

At this point, we investigate the possibility that usefuldesigns for hexagonal arrays may be realized via a constructionprocess based on the recursive application of a generating subarray.To demonstrate this, suppose we consider the uniformly excitedsix-element circular generating subarray of radius r = :t12, shownin Figure 25. This particular value of radius was chosen so that thesix elements in the array correspond to the vertices of a hexagon

with half-wavelength sides (i.e., d = :t12). Consequently, the arrayfactor associated with this six-element generating subarray may beshown to have the following representation:

8040 60theta

Stage 3

20o

-80

~-20CO"CCD"C -40.a'c~-60

::2:

(103)Np =3P(P+1)+I.

where

a" = -1rsinBocos(¢o - ¢Il)'

(104)

(105)

(106)

~-20CO"CCD"C -40.a'c~-60

::2:

-80

o 20 40 60theta

80

The array factor expression given in Equation (104) may also bewritten in the form

Stage 4

where

v/" (B,¢) =1r[sinBcos(¢ - ¢,,) -sinBocos(¢o -¢n)]. (108)

8040 60theta

20o

-80

~-20CO"CCD"C -40.a'c~-60

::2:

Figure 28. Plots of the far-field radiation patterns produced bythe four hexagonal arrays shown in Figures 26 and 27.

(109)1\ [1 G ]1'AFp(B,¢) = 6'L e )\f/n(8,¢)

,,~I

We will first examine the special case where the expansionfactor of the recursive hexagonal array is assumed to be unity (i.e.,8 = 1). Under these circumstances, Equation (107) reduces to

IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999 55

Stage 1

iii'~-20Q)"0::l

~~-40~

Stage 1

iii'~-20Q)"02·2~-40~

8020 40 60Theta (Degrees)

80-60 L..-:_~__~_~__'-------'

o 20 40 60Theta (Degrees)

Stage 2 Stage 2or-=-~--~--~-~-----,

8020 40 60Theta (Degrees)

-60 ,--_~_.....u.~_~__~----'

o

iii'~-20Q)"02·2~-40~

8020 40 60Theta (Degrees)

-60'---~--~-~--~--'o

iii'~-20Q)

"02·2~-40~

Stage 3 Stage 3

20 40 60 80Theta (Degrees)

iii'~-20Q)

"02·2~-40~

-60 '--_--'--_~_~-+-_~__1o

iii'~-20Q)"02·2~-40~

20 40 60Theta (Degrees)

80

Stage 4 Stage 4

8020 40 60Theta (Degrees)

-60 '--"'---.............l..l.-.....u...............-'----'-'--'--_--'-----'

o

iii'~-20Q)

"02·2~-40~

8020 40 60Theta (Degrees)

-60'---~--~_l.....o..-_-'-------'

o

iii'~-20Q)"0::l

~~-40~

Figure 30. Plots of the far-field radiation patterns produced bya series of four (P = 1,2,3,4) fully-populated hexagonal arraysgenerated with an expansion factor of 0 = 1.

Figure 31. Plots of the far-field radiation patterns produced bya series of four (P = 1,2,3,4) fully-populated hexagonal arraysgenerated with an expansion factor of 0 = 2.

56 IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999

These arrays increase in size at a rate that obeys the relationship

where 0PI represents the Kronecker delta function, defined by

{I,

OPI=0,

(110)

(Ill)

These plots indicate that a further reduction in sidelobe levels maybe achieved by including a central element in the generating subar­ray of Figure 25. Figure 32 contains a series of color contour plotsthat show how the radiation pattern intensity evolves for a choiceof 0 =2. Finally, a series of color contour plots of the radiationintensity for this array are shown in Figure 33, where the phasingof the generating subarray has been chosen so as to produce a

main-beam maximum at eo =45 0 and ¢o =900•

In other words, every time this fractal array evolves from one stageto the next, the number of concentric hexagonal subarrays con­tained in it increases by one.

The second special case of interest to be considered in thissection results when a choice of 0 =2 is made. Substituting thisvalue of 0 into Equation (107) yields an expression for the recur­sive hexagonal array factor given by

(112)

where

5. Conclusions

Fractal antenna engineering represents a relatively new fieldof research that combines attributes of fractal geometry withantenna theory. Research in this area has recently yielded a richclass of new designs for antenna elements as well as arrays. Theoverall objective of this article has been to develop the theoreticalfoundation required for the analysis and design of fractal arrays. Ithas been demonstrated here that there are several desirable proper­ties of fractal arrays, including frequency-independent multi-bandbehavior, schemes for realizing low-sidelobe designs, systematicapproaches to thinning, and the ability to develop rapid beam­forming algorithms by exploiting the recursive nature of fractals.

(113)6. Acknowledgments

Clearly, by comparing Equation (113) with Equation (110), weconclude that these recursive arrays will grow at a much faster ratethan those generated by a choice of 0 = 1. Schematic representa­tions of the first four stages in the construction process of thesearrays are illustrated in Figure 26, where the element locations cor­respond to the vertices of the hexagons. Figure 27 shows the ele­ment locations and associated current distributions for each of thefour hexagonal arrays depicted in Figure 26. Figures 26 and 27indicate that the hexagonal arrays that result from the recursiveconstruction process with 0 = 2 have some elements missing, i.e.,they are thinned. This is a potential advantage of these arrays fromthe design point of view, since they may be realized with fewerelements. Another advantage of these arrays are that they possesslow sidelobe levels, as indicated by the set of radiation patternslices for ¢ = 900

, shown in Figure 28. Full color contour plots ofthese radiation patterns have also been included in Figure 29.Finally, we note that the compact product form of the array factorgiven in Equation (112) offers a significant advantage in terms ofcomputational efficiency, when compared to the conventional hex­agonal array factor representation of Equation (99), especially forlarge arrays. This is a direct consequence of the recursive nature ofthese arrays, and may be exploited to develop rapid beam-formingalgorithms.

It is interesting to look at what happens to these arrays whenan element with two units of current is added to the center of thehexagonal generating subarray shown in Figure 25. Under thesecircumstances, the expression for the array factor given in Equa­tion (107) must be modified in the following way:

The authors gratefully acknowledge the assistance providedby Waroth Kuhirun, David Jones, and Rodney Martin.

7. References

I. B. B. Mandelbrot, The Fractal Geometry ofNature, New York,W. H. Freeman, 1983.

2. D. L. Jaggard, "On Fractal Electrodynamics," in H. N. Kritikosand D. L. Jaggard (eds.), Recent Advances in ElectromagneticTheory, New York, Springer-Verlag, 1990, pp. 183-224.

3. D. L. Jaggard, "Fractal Electrodynamics and Modeling," in H. L.Bertoni and L. B. Felson (eds.), Directions in ElectromagneticWave Modeling, New York, Plenum Publishing Co., 1991, pp.435-446.

4. D. L. Jaggard, "Fractal Electrodynamics: Wave InteractionsWith Discretely Self-Similar Structures," in C. Baurn and H.Kritikos (eds.), Electromagnetic Symmetry, Washington DC,Taylor and Francis Publishers, 1995, pp. 231-281.

5. D. H. Werner, "An Overview of Fractal ElectrodynamicsResearch," in Proceedings of the 11 th Annual Review ofProgressin Applied Computational Electromagnetics (ACES), Volume II,(Naval Postgraduate School, Monterey, CA, March, 1995), pp.964-969.

(114)

6. D. L. Jaggard, "Fractal Electrodynamics: From Super Antennasto Superlattices," in J. L. Vehel, E. Lutton, and C. Tricot (eds.),Fractals in Engineering, New York, Springer-Verlag, 1997, pp.204-221.

Plots of several radiation patterns calculated from Equation (114)with 0 = 1 and 0 = 2 are shown in Figures 30 and 31, respectively.

7. Y. Kim and D. L. Jaggard, "The Fractal Random Array," Pro-ceedings ofthe IEEE, 74, 9,1986, pp.1278-1280. .

IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999 57

8. A. Lakhtakia, V. K. Varadan and V. V. Varadan, "Time­harmonic and Time-dependent Radiation by Bifractal DipoleArrays," Int. J Electronics, 63, 6,1987, pp. 819-824.

9. A. Lakhtakia, N. S. Holter, and V. K. Varadan, "Self-similarityin Diffraction by a Self-similar Fractal Screen," IEEE Transactionson Antennas and Propagation, AP-35, 2, February 1987, pp. 236­239.

10. C. Allain and M. Cloitre, "Spatial Spectrum of a General Fam­ily of Self-similar Arrays," Phys. Rev. A, 36, 1987, pp. 5751-5757.

11. D. L. Jaggard and A. D. Jaggard, "Fractal Apertures: TheEffect of Lacunarity," 1997 IEEE International Symposium onAntennas and Propagation and North American Radio ScienceMeeting URSI Abstracts, (Montreal, Canada, July 1997), p. 728.

12. M. M. Beal and N. George, "Features in the Optical Transformof Serrated Apertures and Disks," J Opt. Soc. Am., A6, 1989, pp.1815-1826.

13. D. L. Jaggard, T. Spielman and X. Sun, "Fractal Electrody­namics and Diffraction by Cantor Targets," 1991 IEEE Interna­tional Symposium on Antennas and Propagation and North Ameri­can Radio Science Meeting URSI Abstracts, (London, Ontario,Canada, June 1991), p. 333.

14. Y. Kim, H. Grebel and D. L. Jaggard, "Diffraction by FractallySerrated Apertures," J Opt. Soc. Am., AS, 1991, pp. 20-26.

15. T. Spielman and D, L. Jaggard, "Diffraction by Cantor Targets:Theory and Experiments," 1992 IEEE International Symposium onAntennas and Propagation and USNCIURSI National Radio Sci­ence Meeting URSI Abstracts, (Chicago, Illinois), p. 225.

16. D. L. Jaggard and T. Spielman, "Diffraction From TriadicCantor Targets," Microwave and Optical Technology Letters, 5,

1992, pp. 460-466.

17. D. L. Jaggard, T. Spielman and M. Dempsey, "Diffraction byTwo-Dimensional Cantor Apertures," 1993 IEEE InternationalSymposium on Antennas and Propagation and USNCIURSINational Radio Science Meeting URSI Abstracts, (Ann Arbor,Michigan, June/July 1993), p. 314.

18. D. H. Werner and P. L. Werner, "Fractal Radiation PatternSynthesis," DRSI National Radio Science Meeting Abstracts,(Boulder, Colorado, January, 1992), p. 66.

19. D. H. Werner and P. L. Werner, "On the Synthesis of FractalRadiation Patterns," Radio Science, 30,1, 1995, pp. 29-45.

20. X. Liang, W. Zhensen, and W. Wenbing, "Synthesis of FractalPatterns From Concentric-Ring Arrays," Electronics Letters (lEE),32,21, October 1996, pp. 1940-1941.

21. C. Puente Baliarda and R. Pous, "Fractal Design of Multibandand Low Side-lobe Arrays," IEEE Transactions on Antennas andPropagation, AP-44, 5, May 1996, pp.730-739.

22. D. H. Werner and P. L. Werner, "Frequency-independent Fea­tures of Self-similar Fractal Antennas," Radio Science, 31, 6, 1996,pp.1331-1343.

23. R. L. Haupt and D. H. Werner, "Fast Array Factor Calculationsfor Fractal Arrays," Proceedings of the 13th Annual Review ofProgress in Applied Computational Electromagnetics (ACES),Volume I, (Naval Postgraduate School, Monterey, California,March, 1997), pp. 291-296.

24. D. H. Werner and R. L. Haupt, "Fractal Constructions of Lin­ear and Planar Arrays," 1997 IEEE International Symposium onAntennas and Propagation Digest. Volume 3, (Montreal, Canada,July 1997), pp. 1968-1971.

25. R. L. Haupt and D. H. Werner, "Fractal Constructions andDecompositions of Linear and Planar Arrays," submitted to IEEETransactions 'on Antennas and Propagation.

26. D. L. Jaggard and A. D. Jaggard, "Cantor Ring Arrays," 1998IEEE International Symposium on Antennas and PropagationDigest, Volume 2, (Atlanta, Georgia, June, 1998), pp. 866-869.

27. W. L. Stutzman and G. A. Thiele, Antenna Theory and Design,New York, Wiley, 1981.

28. C. A. Balanis, Antenna Theory: Analysis and Design, NewYork, Wiley, 1997.

29. H. O. Peitgen, H. Jurgens and D. Saupe, Chaos and Fractals:New Frontiers ofScience, New York, Springer-Verlag, 1992.

30. M. T. Ma, Theory and Application of Antenna Arrays, NewYork, Wiley, 1974.

31. 1. Litva and T. K. Y. Lo, Digital Beamforming in WirelessCommunications, Boston, Massachusetts, Artech House, 1996.

Introducing Feature Article Authors

Douglas H. Werner is an Associate Professor in the Penn­sylvania State University Department of Electrical Engineering. Heis a member of the Communications and Space Sciences Lab(CSSL) and is affiliated with the Electromagnetic CommunicationsLab. He is also a Senior Research Associate in the Intelligence andInformation Operations Department of the Applied Research Labo­ratory at Penn State. Dr. Werner received the BS, MS, and PhDdegrees in Electrical Engineering from the Pennsylvania StateUniversity in 1983, 1985, and 1989, respectively. He also receivedthe MA degree in Mathematics there in 1986. Dr. Werner was pre­sented with the 1993 Applied Computational ElectromagneticsSos;iety (ACES) Best Paper Award, and was also the recipient of a1993 International Union of Radio Science (DRSI) Young Scien­tist Award. In 1994, Dr. Werner received the Pennsylvania State

58 IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999

University Applied Research Laboratory Outstanding PublicationAward. He has also received several Letters of Commendationfrom the Pennsylvania State University Department of ElectricalEngineering for outstanding teaching and research. Dr.':Werner ISan Associate Editor of Radio Science, a Senior Member of theIEEE, a member of the American Geophysical Union,USNCIURSI Commissions Band G, the Applied ComputationalElectromagnetics Society (ACES), Eta Kappa Nu, Tau Beta Pi, andSigma Xi. He has published numerous technical papers and pro­ceedings articles, and is the author of six book chapters. Hisresearch interests include theoretical and computational electro­magnetics with applications to antenna theory and design, micro­waves, wireless and personal communication systems, electromag­netic wave interactions with complex materials, fractal and knotelectrodynamics, and genetic algorithms.

Randy Haupt is Professor and Department Head in theElectrical and Computer Enginee~ing Department at Utah StateUniversity, Logan, Utah. He has a PhD in Electrical Engineeringfrom the University of Michigan, an MS in Electrical Engineeringfrom Northeastern University, an MS in Engineering Managementfrom Western New England College, and a BS in Electrical Engi­neeririg from the USAF Academy. Dr. Haupt was a project engi­neer for the OTH-B radar and a research antenna engineer forRome Air Development Center. Prior to coming to Utah State, hewas Chair and a Professor in the Electrical Engineering Depart­ment of the University of Nevada, Reno, and a Professor of Elec­trical Engineering at the USAF Academy. His research interestsinclude genetic algorithms, antennas, radar, numerical methods,signal processing, fractals, and chaos. He was the Federal Engineerof the Year in 1993, and is a member of Tau Beta Pi, Eta KappaNu, USNCIURSI Commission B, and the Electromagnetics Acad­emy. He has eight patents, and is co-author of the book PracticalGenetic Algorithms (John Wiley & Sons, January 1998).

Pingjuan L. Werner received her PhD degree from PennState University in 1991. She is currently an Associate Professorwith the College of Engineering, Penn State. Her research interestsinclude antennas, wave propagation, genetic-algorithm applicationsin electromagnetics, and fractal electrodynamics. She is a memberof the IEEE, Eta Kappa Nu, Tau Beta Pi, and Sigma Xi.

1IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIJIIIllllllllllUIIIIIIIIIIIIIIIIIIIIIIIIIII

Juan Mosig Named EPFL"Ex~raordinaryProfessor"

The Council of the Federal Institutes of Technology, Swit­zerland, have appointed Juan R. Mosig as Extraordinary Professorin electromagnetism in the Department of Electrical Engineering ofthe Lausanne Federal Institute of Technology (EPFL). Hisappointment will become effective January 1,2000.

Born in Cadix, Spain, Juan Mosig obtained his Diploma ofTelecommunications Engineer in 1973 at the Polytechnic Univer­sity of Madrid. In 1975, he received a Fellowship from the SwissConfederation to canoy out advanced studies at the then Electro­magnetism and Microwaves Chair of EPFL. He worked on thedesign and analysis of microwave printed structures and, supportedby the Hasler Foundation and the Swiss National Fund for Scien­tific Research, originated a new direction for research that is stillbeing strongly pursued.

Under the direction of Prof. Fred Gardiol, Mr. Mosig com­pleted his doctoral thesis at the Laboratory of Electromagnetismand Acoustics (LEMA) of EPFL in 1983, on the topic "MicrostripStructures: Analysis by Means of Integral Equations." In 1985, healso received the Doctor in Engineering from the Polyteclmic Uni­versity of Madrid. That same year, Dr. Mosig became director of aEuropean Space Agency project to develop an optimization processfor the computation of planar antenna arrays. This project was thestart of long and fruitful cooperation between LEMA and theEuropean Space Agency (ESA), leading to nine doctoral theses anda dozen research projects in collaboration with aerospace compa­nies and other Swiss and European Universities.

Between 1984 and 1991, Dr. Mosig was an invited researcherat the Rochester Institute of Technology and at the University ofSyracuse, New York, where he worked on spurious electromag­netic radiation from high-speed computer circuits. He was then anInvited Professor at the Universities of Rennes, France (1985);Nice, France (I986); Boulder, Colorado, USA (1987); and at theTechnical University of Delm1ark in Lyngby (1990). He developedtechniques to study electromagnetic fields radiated by printed cir­cuits and antennas, some of which led to commercial software.Since 1978, Dr. Mosig has taught electromagnetism and antennatheory at EPFL. He became a Professor in 1991. He is the author ofnumerous monographs and publications on planar antennas, andtwice received the Best Paper Award at the JINA conferences onantennas (1988 and 1998). He has been the Swiss delegate for theEuropean COST-Telecommunications projects since 1986. He isalso a member of the Political Technology Committee of the SwissFederal Commission on Space Matters. He became a Fellow of theIEEE in 1999. At EPFL, Prof. Mosig will pursue his teachingactivities in the Electrical Engineering and Communications Sys­tems sections. He research activities will be in the fields of propa­gation and electromagnetic radiation from very-high-frequencyantennas and circuits.

[The above item was taken from a press release by EPFL.]

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IEEE Antennas and Propagation Magazine, Vol. 41, No.5, October 1999 59