on the synthesis of fractal radiation patterns

7/28/2019 On the Synthesis of Fractal Radiation Patterns

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Radio Science, Volume 30, Number 1, Pages29-45, January-February 1995

On the synthesisof fractal radiation patternsD. H. WernerApplied ResearchLaboratory, Pennsylvania tate University, State CollegeP. L. WernerCollegeof Engineering,PennsylvaniaState University, DuBois

Abstract. The fundamental elationshipbetween self-similar, hat is, fractal, arraysand their ability to generate adiationpatternswhichpossessractal features sexamined n this paper. The theoretical oundationand designprocedures redevelopedor using ractal arrays o synthesizeractalradiationpatternshavingcertaindesiredcharacteristics.A family of functions,known as generalizedWeierstrassfunctions,are shown o play a pivotal role in the theory of fractal radiationpatternsynthesis. hese unctions re everywherecontinuous ut nowheredifferentiable ndexhibit fractal behavior at all scales. It will be demonstrated that the array factor for anonuniformly ut symmetrically paced inear array can be expressedn terms of aWeierstrass artial sum band-limitedWeierstrassunction) or an appropriate hoiceofarray elementspacings nd excitations.The resulting ractal radiationpatterns romthesearrayspossess tructureover a finite rangeof scales.This rangeof scalescan becontrolledby the numberof elements n the array. For a fixed array geometry, hefractal dimensionof the radiation pattern may be varied by changing he array currentdistribution.A generaland highly lexiblesynthesisechnique s introducedwhich sbasedon the theory of Fourier-Weierstrass xpansions.One of the appealing ttributesof this synthesisechnique s that it provides he freedom o selectan appropriategeneratingunction, n addition o the dimension, or a desired ractal radiationpattern.It is shown hat this synthesis rocedure esults n fractal arrays which are composedof a sequence f self-similar niformlyspaced inear subarrays. inally, a synthesistechnique or application o continuousine sourcess presentedwhich also makesuseof Fourier-Weierstrass expansions.1. Introduction

Mandelbrot [1983] observed that many naturalobjects possessan inherent self-similarity n theirgeometrical structure. In order to quantify thisbehavior Mandelbrot coined the term fractal andintroduced the concept of fractal geometry. Sincethe pioneering work of Mandelbrot, fractals havebeen finding ncreasingapplications n the fields ofengineeringand science. Of particular interest inthis paper is the research area known as fractalelectrodynamics.The term fractal electrodynamicswas first suggested y Jaggard [ 1990] o identify thenewly emergingbranch of researchwhich combinesfractal geometry with Maxwell's theory of electro-magnetism.Many natural objectsare known to exhibit fractalCopyright 1995by the American GeophysicalUnion.Paper number 94RS02315.0048-6604/95/94R S-02315 $08.00

29

behavior. These objects typically possessstructureat several scale lengths. This feature can be attrib-uted to the fact that natural fractals are often theresult of regular but nonperiodic orces which giverise to complex structures through repetitive ac-tions [Jaggard, 1990]. Some important examplesofnatural fractal structures are vegetation canopies,irregular terrain, subterranean geological forma-tions, coastline and seafloor topography, oceansurfaces, cloud boundaries, and atmospheric andionospheric layers [Mandelbrot, 1983; Barnsley,1988; Mareschal, 1989; Turcotte, 1992]. It has alsobeen demonstrated that many turbulent processesare fractal in character because of their inherentlymultiscale structure. This fractal property has beenobserved n oceanic, atmospheric, onospheric, andmagnetospheric urbulence [Burlaga and Klein,1986; Kim and Jaggard, 1988b; Collins and Ras-togi, 1989; Voros, 1990; Bhattacharyya, 1990;Baker et al., 1990]. Techniques for the remote


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30 WERNER AND WERNER: FRACTAL RADIATION PATTERNS

sensingand classificationof fractally rough objectsas well as the identification of fractally turbulentprocesses are just beginning to be developed andapplied by the scientific community [Grassbergerand Procaccia, 1983a, b; Kim and Jaggard, 1988b;Jaggard and Sun, 1990b; Jaggard, 1990].The scattering of electromagnetic waves fromcorrugated random surfaceswith fractal slopeswasconsidered by Jakeman [1982a, b]. A generalizedRayleigh solution [Jaggard and Sun, 1990a] as wellas a Kirchhoff solution [Jaggard and Sun, 1990b]have been obtained for scattering from fractallyrough surfaces. Other areas of research include thestudy of diffraction by band-limited fractal phasescreens [Jaggard and Kim, 1987], optical beampropagation in a band-limited fractal medium [Kimand Jaggard, 1988a], wave transmission through aone-dimensional Cantor-like fractal medium [Kono-top et al., 1990], and reflection from fractal multi-layer media [Jaggard and Sun, 1990c; Sun andJaggard, 1991].In addition to the fractal electrodynamics re-search noted above, there has also been some workdone in the area of fractal antennas, arrays, andapertures. The application of fractals to the disci-pline of antenna array theory was first reported byKim and Jaggard [1986]. They made use of theunderlying order in fractal geometry to develop aprocedure for the design of low sidelobe randomarrays. This procedure combines the virtues ofperiodic subarray generators with those of randomarray initiators to form a quasi-random linear arraycomposed of self-similar subarrays. Allain andCloitre [1987] discussproperties associatedwith thespatial spectrum of a general family of self-similardeterministic arrays which are constructed recur-sively by a certain inflation method. The problemsof diffraction by fractally serrated apertures andtriadic Cantor targets have also been investigated[Kim et al., 1991; Jaggard and Spielman, 1992].A technique is developed in this paper for thesynthesisof fractal radiation patterns from a specialclass of nonuniformly but symmetrically spacedlinear arrays, which we refer to as Weierstrassarrays. The goal of this synthesis technique is todetermine the required array element current exci-tations and spacings hat would result in the real-ization of a radiation pattern which exhibits certaindesired fractal features. The fractal dimension ofthe radiation pattern, that is, the degree to whichthe radiation pattern fills space, may be controlled

by the array element excitation currents. In addi-tion to this the scale size of the fractal structure inthe radiation pattern may be controlled by thenumber of elements contained in the array. Thesynthesis technique developed for Weierstrass ar-rays is generalized to include not only the freedomto choose a desired fractal dimension, but also theability to select a suitable generating function. Thismore general class of fractal arrays has been calledFourier-Weierstrass arrays. Also addressed in thispaper is the more fundamental issue of how theunderlying self-similar geometrical structure andcorrespondingcurrent distributions associatedwiththese fractal arrays are linked to their ability tosuccessfullyproduce fractal radiation patterns. It isdemonstrated, for example, that Fourier-Weier-strass arrays may be decomposed into a sequenceof self-similar uniformly spaced linear subarrays.In addition to linear arrays of discrete elements, afractal radiation pattern synthesis echnique is alsodeveloped which is applicable to continuous linesources. It is shown that in the case of the linesource, the desired fractal radiation pattern and thecorresponding current distribution are relatedthrough a Fourier transform pair. The properties ofinfinite fractal arrays and line sources are firstinvestigated, with further consideration given to theeffects of truncation on the synthesis procedure.These techniques of radiation pattern synthesisare strictly concerned with the creation of fractalradiation patterns and therefore are not intended toaddress traditional concerns such as minimizationof sidelobe evels. In fact, in order to produce a truefractal radiation pattern, the sidelobe evels in manycasescan be rather high. For this reason, we chooseto call them fractal lobes so as to differentiate themfrom ordinary sidelobes. Another important at-tribute which sets these synthesis techniques apartfrom the more traditional approaches is the richstructure of the radiation patterns which result.This is a direct consequence of the unique fractalproperties associated with these radiation patterns.The synthesis techniques developed in this papercan be used to realize radiation patterns whichpossess structure at arbitrarily small scales,whereas traditional methods generally strive to syn-thesize smooth radiation patterns. These propertiesallow an entirely new regime of fractal electrody-namics problems to be explored, namely those inwhich it would be desirable to create radiation


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WERNER AND WERNER: FRACTAL RADIATION PATTERNS 31

patterns containing many scale sizes of structurewith the ability to vary their fractal dimension.There are several applications n which the abilityto create fractal radiation patterns might be desir-able. This appears to be especially true when deal-ing with the interaction of electromagnetic waveswith natural structures or systems. For example,since it is possible to vary the fractal dimension oftheir radiation patterns, fractal arrays may be ben-eficial in certain remote sensingapplications. Theremay also be advantages to using fractal arrays fromthe signal-processing point of view due to theunique fractal characteristics of their radiation pat-terns. Another potential application of fractal ar-rays is in the area of plasma physics. The signatureof fractal radiation patterns could be impressedon aplasma in an effort to artificially produce irregular-ities. This would be advantageous because of themany ranges of scales t is possible o achieve in thestructure of fractal radiation patterns. Ionosphericmodification experiments in which high-powerground-based transmitters are used to induce localchanges in the ionosphere represent one possibleexample of where this technique might be applied[Ferraro et al., 1982, 1984; Barr and Stubbe, 1984;Rietveld et al., 1987; Wong and Brandt, 1990].Section 2 provides a brief introduction to theconcept of fractal dimension, which is central to thetheme of this paper. The properties of generalizedWeierstrass functions and the key role they play inthe synthesis of fractal radiation patterns are alsodiscussed in section 2. Section 3 introduces fractalline sources and discusseshow they may be used tosynthesize fractal radiation patterns from a speci-fied generating function and dimension. An expres-sion is derived for the line source current distribu-tion required in order to synthesize a desired fractalradiation pattern. The general theory of Fourier-Weierstrass fractal arrays is developed in section 4.Several examples of synthesized fractal radiationpatterns are presented and their properties dis-cussed.

2. Weierstrass Fractal ArraysFractals can be quantified and compared by using

certain numbers which are related to their behavior.These numbers are commonly called fractal dimen-sions. Fractal dimensions provide a measure of thedegree to which a fractal fills the metric space it iscontained in. There are several definitions of fractal

dimension in use [Barnsley, 1988; Falconer, 1990].However, the box-counting definition or box defi-nition is usually used for the computational orempirical determination of fractal dimensions. For agiven fractal F, the box-counting fractal dimension,denotedby dimB(F), is defined as [Falconer, 1990]In N (F)dim (F) -- lim-, o In (1/) (1)

where No represents he smallestnumber of sets ofdiameter at most required to cover the fractal F.The connection between the box-counting defini-tion of fractal dimension and the intuitive Euclideanconcept of dimension is discussed by Voss [1988]and Jaggard [1990].

The class of functions known as generalizedWeierstrass functions are represented by

f (x) - Z (D-2)n](x)n=l

(2)

where 1 < D < 2, /> 1, and 7 s a suitable boundedperiodic function [Berry and Lewis, 1980;Falconer,1990]. These generalized Weierstrass functionshave the property that they are everywhere contin-uous but nowhere differentiable and exhibit fractalbehavior at all scales. Suppose that F = graph(f);then it can be shown that the parameter D repre-sents the box-counting fractal dimension of F, thatis, dimB(F) = D, provided / is sufficiently arge[Falconer, 1990]. The fractal dimension D in thiscase is a fractional dimension which lies betweenthe integer dimensions of one and two. Of course,we expect the fractal dimension of F to be at leastone because F is the graph of a continuous functionwith one-dimensional domain. On the other hand,we expect the fractal dimension of F to be no morethan two because F is contained in the plane, atwo-dimensional space. Generalized Weierstrassfunctions are the foundation on which the theory offractal radiation pattern synthesis is based. In theremainder of this section a relationship betweenthese functions and classical antenna theory will beestablished and used to investigate the geometricalproperties as well as radiation characteristicsof theresulting fractal arrays.The subject of nonuniformly spaced antenna ar-rays has received considerable attention over theyears and continues o be a topic of interest. This is


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To Far Field Point

/v-% /2 -"2 /e;' //oX,/e, /e %d d

d2 d2

J%

Figure 1. A uniformly but symmetrically spaced inear array of 2N elements with a conjugate symmetric currentdistribution.

primarily because the element spacingsprovide athird variable, in addition to the amplitude andphase of the array excitation currents, with whichto control the radiation pattern. The array factor forthe nonuniformly but symmetrically spaced lineararray of 2N elements illustrated in Figure 1 may beexpressed in the form [Ma, 1974]

where

Nf(O)= 2E In COSkdn os + an)n=l

(3)

f (u)= 2E n=l

(D-2)n OSalntt an) (6)also represents a Weierstrass function with a box-counting fractal dimension of D. This particularWeierstrass function may be interpreted as repre-senting the array factor for a certain nonuniformlybut symmetrically spaced linear array consisting ofinfinitely many elements. This is easily verified bycomparing (6) with (3) and recognizing that therequired current amplitudes and element spacingsare

271'k (4) in= /(D-2)n (7a)is the free-space wavenumber and A is the corre-sponding ree-space wavelength. The array ampli-tude and phase excitations are representedby I nand an, respectively, while dn represents he arrayelement locations. These current excitations areconjugate symmetric as indicated in Figure 1. Now,suppose the following Weierstrass function is con-sidered

f(u)= E */(D-2)nOSa*lnU)n=l

(5)

where 1 < D < 2, r/> 1, a is a constant, and 9 hasbeen chosen to be the cosine function. The proper-ties of (5) are preserved when the series is multi-plied by a constant or arbitrary phasesare added toeach term in the series. Therefore

kdn = a,qn (7b)with u = cos 0, r/> 1, and 1 < D < 2. A fractalradiation pattern resulting from (6) would possessstructure over an infinite range of scales. It isinstructive to study the properties of infinite arraysof elements from the theoretical point of view.However, physical arrays necessarily consist of afinite number of elements. The infinite series in (6)may be truncated to yield

Nfv(u) 2 *ln=l

(o-2)n os at/ntt an) (8)This Weierstrass partial sum represents the arrayfactor for a nonuniform linear array of 2N elementswith current amplitudes and spacingsgiven by (7a)and (7b), respectively. The Weierstrass partial sum


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of (8) may be classified as band-limited since theresulting radiation pattern only exhibits fractal be-havior over a finite range of scales. It should bepointed out that the box-counting dimension ofband-limited Weierstrass functions, such as (8), nolonger necessarilyyield the number D. However, ithas been demonstrated, through numerical experi-ment, that it approachesD for certain values of r/[Jaggard and Sun, 1989]. The lower bound on thescale size for which the radiation pattern remainsfractal s 2rdarl . This suggestshat the rangeofscalesmay be controlled by the number of elementsin the array. That is, the addition of two or moreelements to the array has the effect of enhancing hefine structure in the radiation pattern. In fact, thestructure of the radiation pattern becomes iner andmore detailed as the number of array elements isincreased.

Suppose hat it is desirable to have (8) attain itsmaximumvalue at somespecified ngle 00. This canbe accomplishedby choosing he excitation currentphases according toan = -anuo (9)

whereUo = cos 0o (10)

The maximum value of (8) under these conditions isthenNfN(uO)2 q

n=l(D-2)n (11)

The series in (11) represents a geometric progres-sion which, since r/ > 1 and 1 < D < 2, can besummed to give

1/(D-2)N]N(uO)2-2)- '('-; (12)A normalized orm of the Weierstrassarray factor canbe obtainedby dividing 8) by its maximumvalue (12).The expression or this normalizedarray factor isgN(U)1 --O---27v]nCOSanuOln) (13)where

in = /(O-2)(n-1) (14)

represent the normalized excitation current ampli-tudes. Equations (7a) and (14) indicate that thefractal dimension of the radiation pattern can becontrolledby the array element current distribution.Equation (7b) may be used to show that theseparation between any two consecutive array ele-ments is given by

dn+lna(*/1)*/n= A n=l, 2,...,N-1 (15)2rrSince r/> 1, it follows that r/n > for n > 1. Thisinequality can be used to prove thatdn+l -dn > d2 -dl n = 2, 3, ''' , N- 1 (16)Let r be a constraint which is imposed on theminimum separation between any two consecutiveelements in the array. There are two possible casesin which this minimum spacing constraint may besatisfied. These cases are as follows:Case 1 d2 -dl = r and d -> r/2 (17a)

Case 2 d =- and d2 - d -> r (17b)2Note that if either condition (17a) or condition (17b)is satisfied, then the spacing between all other pairsof consecutive array elements will automaticallysatisfy the minimum separation criterion r. Thisproperty is a direct consequenceof inequality (16).An expression or a as a function of r and r/can bederived by using (7b) in conjunction with conditions(17a) and (17b). The result is

kTa= 1


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0.9

0.8

0.70.60.50.40.30.20.1

, ...............................................................................,.........................................................................................................................=.8 ................................................

D .1.......................................................................................................................................................................................................................--.5X ..............................................

-1 -03 o 03 1

Figure 2. The normalized Weierstrassarray factor for a16-elementarray with u0 = 0.0, r = 0.5A, r/= 2.34, andD= 1.1.

In = rl(O-2)In-1 I1 = ,./(0-2) (19a)Ot -- Otn_1 Oil = -auo (19b)

dn = rldn_1 d 1 = art/k (19c)where n = 2, 3, , N in all three cases. From thiswe can conclude that Weierstrass arrays are indeedfractal arrays since their element spacingsand cur-rent distributions, amplitudes as well as phases,obey certain power laws. This fact strongly sug-gests that there may be a connection between theunderlying self-similarity properties of fractal ar-rays and their ability to generate fractal radiationpatterns. In other words, fractal arrays give rise tofractal radiation patterns.Suppose we consider a Weierstrass array inwhich the minimum separation between any twoadjacent array elements is restricted to a half wave-length. The minimum separationrequirement of r =;t/2 and a choice of r/ = 2.34, taken together with(18) implies that the array designparameter a = 1.0.Consequently, these values of r/= 2.34 and a = 1.0may be used in conjunction with (7b) to determinethe element spacings or this array. Figure 2 showsthe radiation pattern which would result from aWeierstrass array consisting of 16 elements with aD = 1.1 current distribution. The current distribu-tion on this same 16-element array may be appro-

priately adjusted to produce radiation patternswhich have any desired fractal dimension betweenthe integer dimensions of one and two. For exam-ple, radiation patterns having a desired fractal di-mension of 1.5 and 1.9 are illustrated in Figures 3and 4, respectively. One conclusion which may bedrawn from these Figures is that the higher thefractal dimension, the more irregular the radiationpattern becomes and the more it tries to fill space.This space-fillingproperty of Weierstrass arrays ismost likely characteristic of other, perhaps all,fractal arrays. The radiation pattern shown in Fig-ure 5 corresponds o a 16-elementWeierstrassarrayin which the current excitations were selected toproduce a desired fractal dimension of D = 1.5 anda radiation maximum at 00 = 60 (u0 = 0.5).

Figure 6 indicates the normalized current ampli-tude distributions which lead to the band-limitedfractal radiation patterns shown in Figures 2through 5. This figure illustrates that a high degreeof current amplitude tapering is necessary in orderto synthesize radiation patterns which have lowerfractal dimensions associated with them. Table 1contains a listing of the element locations for a16-element Weierstrass array whose geometry isprescribedby a value of r/= 2.34 and a - 1.0.The maximum directive gain or directivity asso-ciated with a nonuniform linear Weierstrass array of

0.9

0.8

0.4.30.20.1

i i , , i-1 -0.5 0 0.5 1

Figure 3. The normalizedWeierstrassarray factor for a16-element rray with Uo = 0.0, r = 0.5A, r/= 2.34, andD= 1.5.


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0.7o.s t0.4

t ID= I ; i I

o -$o-1 0 O 1 1 2 3 4 5 6 7 8

Figure 4. The normalizedWeierstrassarray factor for a16-element rray with u0 = 0.0, r = 0.5A, r/= 2.34, andD= 1.9.

Element Number, NFigure 6. The normalizedcurrent distributions equlreoto produce radiation patterns with fractal dimensionsof1.1, 1.3, 1.5, 1.7, and 1.9 for a symmetric 16-elementWeierstrass array.

isotropic sources may be determined using theband-limited Weierstrass array factor (8) and itsmaximum value (12). The expression or directivityunder these circumstances is2f}(uo)G(uo) = ' ' (20)

l fN(u)uiin which1 N N0.9 N=834.....................................................2N(U)4 Z rlD-2'(m+n'osarlmuam).8 ....................................

0.7 cos at/nu an)0.6 -- - and050.4 Table1. Element eometryor a Symmetric03 16-ElementWeierstrass rray

,

02. Element,,

0.13o -1 -05 o 0.5 1 45

u 6Figure 5. The normalizedWeierstrassarray factor for a16-elementarray with u0 = 0.5, r = 0.5A, r/= 2.34, andD= 1.5. ? = 0.5A, *t = 2.34, and a = 1.0.

(21)

, , ,

Element Location, dn/A0.3720.8722.0394.772

11.16626.12961.141

143.069


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36 WERNER AND WERNER: FRACTAL RADIATION PATTERNS14

12

11

10

I -.---6 i

.... i .... i .... i .... i .... i i .... i .... ] .... i ....I 1.1 1.2 1.3 1A 1.5 1.6 1.? 1.8 1.9 2D

Figure 7. The directivity versus fractal dimension fordifferent sized inear Weierstrassarrays of isotropic radi-ators.

f2(Uo) 4,/ (D-2)N'1-/ (D-2)l-r/ (22)Integration of (21) yields

N N: u)u=4r/1 m=l n=l (D-2)(m+n) mn(ar/, UO (23)where

Amn(a,,Uo)l COS ar/m(u -- U0)1cos [ar/n(u - u0) du = cos [a(r/m + r/n)u0]sinc a(r/m + r/n)] + COSa(r/m -- r/n)u0]sinc a(r/m -- r/n)] (24)

andsin (x)sinc (x)= (25)

Finally, substituting 22) and (23) into (20) results na directivity given by

fl[ 1--r/(D-2)]= /E Z r/(D-2)(n+m-2)Amn(a,/, Uom=l n=l

-1

(26)

Note that the particular case when D = 2 corre-sponds to an array with uniform excitation. Theexpression or directivity in this case becomes

G(uo) = (27)1 N N2N /mn(, /,Uo)m=l n=lIt can be shown that (27) reduces to the well-knownresult

G(0) = 2N (28)for a half wavelength uniformly spaced broadsidelinear array when ,/n is replacedby n,r/a and u0 =0.

The directivity as a function of fractal dimensionD is plotted in Figure 7 for several different arraysizes. A uniform current distribution is obtained ona Weierstrassarray when D = 2. This representsanupper bound on the fractal dimension. Figure 7demonstrates that as the fractal dimension of theradiation pattern is decreased, the main beambroadens and the corresponding directivity alsodecreases.

3. Fractal Line SourcesIn this section we investigate the use of long,

straight current-carrying antennas, known as linesources, for the synthesis of fractal radiation pat-terns. The geometry for a line source of length Lcentered symmetrically about the origin of the zaxis with a current distribution of I(z) is shown inFigure 8. For a line source of infinite length, theradiation pattern F(u) and the current distributionI(s) are related by the following Fourier transformpair [Stutzman and Thiele, 1981]'

F(u)-o(s)eJ2uss (29a)


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I(s)-o(u)e j2sudu (29b)where

u = cos 0 (30a)

s = z/A (30b)Any fractal function may be constructed by therecursive application of an appropriate generatingfunction. In particular, suppose that the radiationpattern of an infinite line sourcemay be representedas a band-limited generalized Weierstrass function

of the formN-1F(u) Z */(D-2)n(11u)n=0

(31)

where D is the fractal dimension and O(u) is agenerating function. Here we assume that the gen-erating function 9(u) is periodic and even, that is,9(u + 2) = 9(u) and 9(-u) = 9(u). Hence, 9(u) maybe expanded in a Fourier cosine series as

g(u)--+ amosmru)m=l (32)where the Fourier coefficients are determined from

am2 O(u)OSmru)u (33)Substituting (32) into (31) leads to an expression orthe radiation pattern given by

(D-2)N _ 10 /F(u)= -- (D-2)2 *t -1oc N-1+ Y am(D-2)nOSmrlnu) (34)

m=l n=0

Without loss of generality we replace u by u + 1 in(34), effectively mapping the interval [-1, 1] to theinterval [0, 2]. This results in

To Far Field Point

L L

Figure 8. Geometry of a continuous line source oflength L oriented along the z axis.

(D-2)N _ 10 /(u) - (o-2)+71 m /(D-2)nOSm'rln(U1)] (35)Ln=

where

am 9(u- 1) cos (mru) du (36)

with the requirements that r/> 1 and 1 < D < 2.Equation (35) represents a Fourier decompositionof the fractal radiation pattern F(u) in which thebasis functions are band-limited Weierstrass cosinefunctions. We call such a representation a Fourier-Weierstrass expansion. Other examples of Fourier-Weierstrass expansions may be found in work byKim [1987].The line source current distribution required inorder to produce the desired fractal radiation pat-terns may be obtained by evaluating the Fourierintegral (29b)

I(s) F(u)ea2'suu (37)where F(u) is defined n (35). Performing the neces-sary integration results in an expression for thecurrent distribution given byI/(D-2)N 1 o N-1sinc2rs) y y am*lD-2)nI(s) ao (D-2)_lI/ m=l n=0

e mrr*t"sinc2rs m,rr )+ e jm'*t"inc 2rs+ m'rln)] (38)


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Equation (38) represents he required current distri-bution for a line source of infinite extent. However,an approximation may be obtained for the currentdistribution on a finite length line source by trun-cating (38) in the following way:

Lls) (s) Isl-


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WERNER AND WERNER: FRACTAL RADIATION PATTERNS 39The Fourier coefficients associated with this trian-gular generating unction may be found from (36).That is,

u cos (m*ru) du = 1

1

0.90.8

m=0 '' 0.7 13.6(47)0.5.4 0.:50.2

m=2, 4, 6,.-- 0.10 -1 -0.5 o 0.5 1

am: 2 u cos (m*ru) du = -

m= 1,3,5,---

am2d osm*ru)u0Substituting he Fourier coefficients 47) into (35)and choosing value of r/= 2 leads o the result

2D-2)D-2)2}(u) - 209_2)_ -lul Figure9. Synthesizedractal adiation attern normal-ized) for a line sourcewith D - 1.3, r/= 2, N = 14, anda triangular generating unction.= = (2m- )cos(2m )2n*ru]48)

which represents he desired line source fractalradiationpattern,with fractaldimension , formedby the recursiveapplicationof a triangulargenerat-ing function.The fact that F(-+ 1) = 0 and F(0) = 1may be easily verified using 48). Expression 48)may also be used to show hat when N = 1, theradiation pattern F(u) reduces to the triangulargeneratingunction (u) - 1 - lul.The line sourcecurrentdistribution equired n order o produce hefractal radiationpatternsof (48) is given by

which may be used o normalize 49). The resultingnormalized current distribution is

2(D-2)ND-2)'(s) 2 -2)N- sinc (2,rs)+L--3/-_l'Jnc*rs) L -----57v'_

o N-1 2(D-2)n'7n2m--1)[sine2*rS- 2m-1)2n*r]+ sinc [2*rs + (2m - 1)2n*r]} (51)

2(D-2)N2 D-2)]I(s) 2-ff=- ] sine2*rs)sine*rs)()o -2D-2)nYn2m1) sine2*rs-2m1)2n-r]=l =

+ sinc 2*rs+ (2m - 1)2n-r]} (49)The maximum value of the current distribution (49)occurs when s = 0, hence

2 (D-2)N -- 1max I(s)]= I(0)=14 o 2 - 1 (50)

It is interestingo notehere hat, except or an "endeffect," the sinc function is self-similar with asimilarity actor of two [Schroeder, 1991].The line source current distributions of (49) or(51) may be used to synthesize adiation patternswith any desired ractaldimension,whichare basedon the triangular generating unction of (46). Forexample, he synthesizedadiation attern or a linesource with a desired fractal dimension of D - 1.3is shown n Figure 9. Figure 10 illustrates he linesource current distribution required in order toproduce he fractal adiation atternshown n Fig-ure 9. Likewise, Figure 11 shows he synthesizedline sourceradiation pattern which results from a


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0.8

0.6

0.4

0.2

-0.2

.............................i.....................................'"'"'":............................

..... I .... I ........ I .... I,,,,i,,,,j,,,,-20 -15 -1_0 -5 0 5 1_0 20

0.75

0.5

O25

D=l.7................................................................/=2....................................--26........

..

-20 -15 -10 -5 0 5 10 15 20

Figure 10. The normalized line source current distribu-tion required in order to produce the fractal radiationpattern shown in Figure 9.Figure 12. The normalized line source current distribu-tion required in order to produce the fractal radiationpattern shown in Figure 11.

specified fractal dimension of D = 1.7. The currentdistributioncorrespondingo a value of D = 1.7 isshown n Figure 12. Once again, we see that theirregularityof the radiationpattern ncreaseswithincreasing ractal dimension.

If we considera line sourceof finite lengthL,then the current distribution (49) is truncated ac-cording o (39) and the resultingsynthesizedractalradiationpattern may be expressed s1 [2 D-2)N-2(D-2)if(u)--[-_ ]{Sir(L/A)(1u)]

0.9 ..............................................................................Si ,r(L/A)(1u)]} --{(1 u)Si ,r(L/A)(1u)]0.1t i0.7 i i ... + 1u) i,r(L/A)(1u)]2u i(L/A)u]

1 2 N-] 2(D-2)n

0.2 ' S2,n-n(U)sin (2m 1)2nu]C2m_ln(U)}0a where0 S2m-ln(U= Si [(L/A-(2m-1)2n)(1 +u)]+Si [(L/A-1 5 0 05 1

Figure 11. Synthesized ractal radiation pattern (nor-malized) or a line sourcewith D - 1.7, r/= 2, N - 26,and a triangulargenerating unction.

(52)

+ (2m 1)2n)(1 u)] + Si [,r(L/A (2m 1)2n)(1 u)]+ Si [,r(L/A + (2m - 1)2n)(1 u)] -1 -< u -< 1 (53a)

C2m-ln(lt= Ci [rlL/A (2m - 1)2hi(1 u)]


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- Ci [-(L/A + (2m - 1)2n)(1 u)] - Ci [-[L/A- (2m - 1)2nl(1 u)] + Ci [r(L/A + (2m - 1)2n) 1 - u)] -1 < u < 1 (2m - 1)2n-L/A

C2m-ln(U = In -+ Ci [2-(L/A)(1 - u)]

Ci [2-(L/A)(1 + u)]

-1 < u < 1 (2m - 1)2n = L/A

C2m-ln()+_{Ci2-[L/A(2m1)2nil- Ci [2'(L/A + (2m - 1)2n)]

2

(53b)

(53c) 0.5

0-1 -0.5 0 0.5

- In LIA - (2m - 1)2nL/A + (2m- 1)2n (2m- 1)2nL/A

C2m_ln(+' ) = -+'{3 In [4r(L/A)]

(53d)uFigure 13. The first eight stages (N = 1-8) in theconstruction of a fractal radiation pattern based on aquadratic generating unction with D = 1.1 and /- 2.

- Ci [4-(L/A)]} (2m - 1)2n = L/A (53e)Note that the term in (52) which contains sin[L/2A)] vanishes if the half-length of the linesourceL/2 is an integer multiple of the wavelengthA. As our next example, we consider an initialpattern of the form

9(0) = sin20 0 --<


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am (D-2)nIron- - 1/ (58b)kdmn - mrrqn (58C)Otmn - mrr,1n (58d)

and the Fourier coefficientsa m corresponding o aparticular generating function may be obtainedthrough the use of (36). Equation (57) represents theFourier-Weierstrass array factor for a discrete arraycontaining infinitely many elements. The array ele-ment locations with respect to the origin are dmn,while the element current amplitudes and phasesare Imn and Otmnespectively. A useful representa-tion of the array factor for a Fourier-Weierstrassarray with a finite number of elements may beobtained by simply truncating he outer summationin (57) and interchanging the order of summation.This leads to an approximate expression for thedesired fractal radiation pattern given by

N-1 MF"M)Io+ 2 ' ' Iron OSkdmnuOtmn)n=0 m=l

(59)

At this point, we recognize that the double summa-tion appearing in (59) may be interpreted as repre-senting he superpositionof the radiation producedby a sequence of N uniformly spaced M elementlinear arrays. It follows directly from (58c) that theelement spacings for each of the M element uni-formly spaced linear subarrays areAn = dm+ n -- dmn - I1n__2 n=0, 1,2,...,N-1

(60)which may be used to derive the recurrence relation

An+l = r/An A 0 =- (61)2Hence, an iterative procedure may be used todetermine the element spacings or each consecu-tive linear subarray. This unique property of Fou-rier-Weierstrass arrays reveals their underlyingfractal structure by suggesting hat they are com-posed of a sequence of self-similar uniformlyspaced linear subarrays. Recurrence relations forthe excitation current amplitudes and phases may

be found in a similar way using (58b) and (58d),respectively. These recurrence relations areamIron+-' 'rl D-2)Imn ImO (62a)2

Otmn+l - 110tmn OtmO mrr (62b)Finally, an expression for the normalized arraycurrent excitation amplitudes may be obtained bydividing (58b) by (58a), which yields

i0 = 1 (63a)

lmn- 00 -D---'V1D-2)n63b)Suppose we wish to use a Fourier-Weierstrassarray to synthesizea fractal radiation pattern basedon the triangular generating function (46) with rt =2. The Fourier coefficientsa m associatedwith thetriangular generating function were found in (47)and may be used to show that

1 1 - 2 (D-2)NI0= 1 2 0-2) (64a)Imn-'- (2m- ) (64b)

Otmn- (2m - 1)2nrr (64c)dmn- (2m - 1)2n-lA (64d)

Various stages n the construction of the Fourier-Weierstrass array corresponding to a triangulargenerating function are illustrated in Figure 14. Inthis case, since rt = 2, (61) becomesAn+l = 2An A 0 = A (65)

which suggests hat the element spacings or eachconsecutive subarray may be obtained by doublingthe spacing of the previous subarray. This self-similarity property of the subarrays s clearly iden-tifiable in Figure 14. It is evident from Figure 14 thatthe decompositionof the Fourier-Weierstrass arrayinto a sequenceof self-similar linear subarrays alsoincludes an array element located at the origin.Figure 15 shows a synthesized radiation patternformed by a triangular generating unction with rt -2 and a desired fractal dimension of D = 1.1. AFourier-Weierstrass array with M = 4 and N = 8


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ii

llnlllt i I i i i I I I I I I I i I i i I I I I I I I I I I I I I I I I I I i i I+

II II II II IIE E E E EStageO i_,_l_,__l(n=O) ,-I-I-I-,-I I I I I I I I I 04 o-) 1. u') +ii Ii ii ii iiE E E E E$tsel I I I ! I I I ! ! I I I I I I I I I I I I I I I I I I I I I I I I(n-- 1) i i I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

CXl Cv'J r +1.II II II II IIE E E E E$tSe9 I I I I I I I I I ! ! I I I I I I I I I I I I I I I I I I I I(n=9) i i i i i i i I T I I I I I I I I I I I I I I I I I I I I ! ! I I

+ CXl Cv'J r II II II II IIE E E E EStage I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I J(n=3) I I I I T I I I I I I I T I , I I I I I T I I I I I ! I T I I I I I I I r

Array T'T"'!'-T-T-T I I I 1' I I I I I' I I I I I I I I I I I I I I GeometryFigure 14. The construction of a Fourier-Weierstrass fractal array from a sequence of four (N = 4) self-similaruniformly spaced ive-element M = 5) linear subarrays.The subarrayelement spacings re scaledaccording o thoseassociatedwith the triangulargenerating unctionwith = 2, that is, An+l = 2An where A0 = A.was used to synthesize his radiation pattern. Table2 contains a listing of Fourier-Weierstrass arrayelement spacingsand excitation currents requiredin order to produce the fractal radiation patternshown in Figure 15.5. Conclusion

A connectionetweenhegeometricalropertiesof fractal arrays and their ability to generate ractalradiation patterns has been established n this pa-per. This relationship was used as the foundationfor the developmentof several synthesis echniquesapplicable to the subset of radiation problems nwhich fractal characteristics re desired. In partic-ular, Weierstrass and Fourier-Weierstrass arrayswere introduced as two examples of linear fractalarrays which may be used for the synthesisoffractal radiation patterns. The range of scalesover

0.9

0.80.70.650.4

0.30.20.1

I

-1 -0.5 o 0.5 I

Figure 5. SYnthesizedractal adiationatternnor-which givenadiationatternxhibitsractalmalized)ora Fourier-Weierstrassrray ith = 1.1,behavior as ound o be directly elated o the n = 2,M = 4,and = 8,andwith riangulareneratingnumber of elements contained in the array. The function.


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Table 2. Fourier-Weierstrass Array Element Spacingsand Excitation Currentsp dt,/A it, a, deg0 0.0 1.000E + 00 01 0.5 -7.068E - 01 1802 1.0 -3.788E - 01 03 1.5 -7.853E - 02 1804 2.0 -2.030E - 01 05 2.5 -2.827E - 02 1806 3.0 -4.209E - 02 07 3.5 -7.730E - 03 1808 4.0 - 1.088E - 01 09 5.0 -1.515E - 02 0

10 6.0 -2.255E - 02 011 7.0 -4.142E - 03 012 8.0 -5.829E - 02 013 10.0 -8.119E - 03 014 12.0 - 1.209E - 02 015 14.0 -2.220E - 03 016 16.0 -3.124E - 02 017 20.0 -4.351E - 03 018 24.0 -6.477E - 03 019 28.0 - 1.190E - 03 020 32.0 - 1.674E - 02 021 40.0 -2.332E - 03 022 48.0 -3.471E - 03 023 56.0 -6.375E - 04 024 64.0 -8.970E - 03 025 80.0 - 1.249E - 03 026 96.0 - 1.860E - 03 027 112.0 -3.416E - 04 028 160.0 -6.696E - 04 029 192.0 -9.967E - 04 030 224.0 -1.831E - 04 031 320.0 -3.588E - 04 032 448.0 -9.811E - 05 0

Values shown are those required in order to produce thefractal radiation pattern shown in Figure 15, with D = 1.1, r/=2, M = 4, and N = 8, and with a triangular generating unction.

more elements used in the array, the finer and moredetailed the structure will be until ultimately theradiation pattern will possessstructure at arbitrarilysmall scales n the limit of infinite array size. It wasalso shown that the fractal dimension of a radiationpattern is governed exclusively by the current dis-tribution on the array. In addition to the synthesistechniques developed for discrete fractal arrays, asynthesis echnique is presentedwhich may be usedin conjunction with continuous ine sources. Exam-ples have been included which illustrate the appli-cation of each fractal radiation pattern synthesistechnique.

Acknowledgments. The authors are greatly indebtedto Anthony J. Ferraro of The Pennsylvania State Univer-sity, Department of Electrical and Computer Engineeringfor his insightful discussions egarding potential applica-

tions of fractal arrays. The computational assistanceprovided by Jenn-Ren Lien and Joseph Jiacinto is grate-fully acknowledged.

ReferencesAliain, C., and M. Cloitre, Spatial spectrum of a generalfamily of self-similar arrays, Phys. Rev. A, 36, 5751-5757, 1987.Andrews, L. C., Special Functions for Engineers andApplied Mathematicians, Macmillan, New York, 1985.Baker, D. N., A. J. Klimas, R. J. McPherron, and J.

Buchner, The evolution from weak to strong geomag-netic activity: An interpretation in terms of determinis-tic chaos, Geophys. Res. Lett., 17(1), 41-44, 1990.Barnsley, M. F., Fractals Everywhere, Academic, NewYork, 1988.Barr, R., and P. Stubbe, ELF and VLF radiation from the

polar electrojet antenna, Radio Sci., 19, 1111-1122,1984.

Berry, M. V., and Z. V. Lewis, On the Weierstrass-Mandelbrot fractal function, Proc. R. Soc. London A,370, 459-484, 1980.

Bhattacharyya, A., Chaotic behavior of ionospheric tur-bulence from scintillation measurements, Geophys.Res. Lett., 17(6), 733-736, 1990.Burlaga, L. F., and L. W. Klein, Fractal structure of theinterplanetarymagnetic ield, J. Geophys.Res., 91(A1),

347-350, 1986.Collins, R. L., and P. K. Rastogi, Fractal analysis of

gravity wave spectra in the middle atmosphere, J.Atmos. Terr. Phys., 51(11/12), 997-1002, 1989.Falconer, K., Fractal Geometry, Wiley, New York, 1990.Ferraro, A. J., H. S. Lee, R. Allshouse, K. Carroll, A. A.Tomko, F. J. Kelly, and R. C. Joiner, VLF/ELFradiation from the ionospheric dynamo current systemmodulated by powerful HF signals, J. Atmos. Terr.Phys., 44, 1113-1122, 1982.Ferraro, A. J., H. S. Lee, R. Allshouse, K. Carroll, andR. Lunnen, Characteristics of ionospheric ELF radia-tion generated by HF heating, J. Atmos. Terr. Phys.,46, 855-865, 1984.

Grassberger, P., and I. Procaccia, Characterization ofstrange attractors, Phys. Rev. Lett., 50,346-348, 1983a.Grassberger, P., and I. Procaccia, Estimation of theKolmogorov entropy from a chaotic signal, Phys. Rev.A, 28, 2591-2593, 1983b.Jaggard, D. L., On fractal electrodynamics, in RecentAdvances in Electromagnetic Theory, edited by H. N.Kritikos and D. L. Jaggard, pp. 183-224, Springer-Verlag, New York, 1990.Jaggard, D. L., and Y. Kim, Diffraction by bandlimitedfractal screens, J. Opt. Soc. Am. A Opt. Image Sci., 4,1055-1062, 1987.Jaggard, D. L., and T. Spielman, Triadic Cantor target


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on the synthesis of fractal radiation patterns

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