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NOTICE: When government or other drawings, speci-fications or other data are used for any purposeother than in connection with a definitely relatedgovernment procurement operation, the U. S.Government thereby incurs no responsibility, nor anyobligation whatsoever; and the fact that the Govern-ment may have formulated, furnished, or in any waysupplied the said drawings, specifications, or otherdata is not to be regarded by implication or other-wise as in any manner licensing the holder or anyother person or corporation, or conveying any rightsor permission to manufacture, use or sell anypatented invention that may in any way be relatedthereto.

THE

ANTENNAL'A B 0 R A TOR Y

R?'EARCH ACTIVITIES in ---I If/oPh, I/t, ( ')Hiii /I t/I l h' i I 13'( ho I r'P /l/t'

.11011, 11,',i (tiiot A1/ron,/1 1, 1 1 FL'h/b Th7./eo,

5 1'aw p q 11a/ioi S iII'mPillief'r lIpphla/iol.I

cmL I

I-

ON THE OPTIMUM GAIN OF UNIFORMLYSPACED ARRAYS OF ISOTROPIC

SOURCES OR DIPOLES

by

C. T. Tai

Contract Number N123(953)-31663APrepared for

U. S. Navy Purchasing OfficeLos Angeles 55, California

1522-1 31 March 1963

Department of ELECTRICAL ENGINEERING

THE OHIO STATE UNIVERSITYqRESEARCH FOUNDATION

Columbus, Ohio

NOTICES

When Government drawings, specifications, or other data are usedfor any purpose other than in connection with a definitely related Gov-ernment procurement operation, the United States Government therebyincurs no responsibility nor any obligation whatsoever, and the fact thatthe Government may have formulated, furnished, or in any way suppliedthe said drawings, specifications, or other data, is not to be regarded byimplication or otherwise as in any manner licensing the holder or anyother person or corporation, or conveying any rights or permission tomanufacture, use, or sell any patented invention that may in any way berelated thereto.

The Government has the right to reproduce, use, and distribute thisreport for governmental purposes in accordance with the contract underwhich the report was produced. To protect the proprietary interests ofthe contractor and to avoid jeopardy of its obligations to the Government,the report may not be released for non-governmental use such as mightconstitute general publication without the express prior consent of TheOhio State University Research Foundation.

Qualified requesters may obtain copies of this report from theASTIADocument Service Center, Arlington Hall Station, Arlington 12, Virginia.Department of Defense contractors must be established for ASTIA serv-ices, or have their "need-to-know' certified by the cognizant militaryagency of their project or contract.

I'l E, P 0 J,

L y

THE OHIO1 STATE UNIVERSITY RESEARCM FOIJNJDATIONCOLUMBUS 12, OHIO

Coopeiator U. S, Na-vy Puichasiig Otfice(4 Soi Br oadwkavB~ox 50%( Ivltrol-ollan Stati*ooLos Angeles~ 5 Cajifoinia

Con r act 'Numbe r NQ091 10W 3Ai3l

investigatior of S~udv Prog ram Related !o Shipboard AntennaSvsoani Er' '.ormeri

Subje~-ct W~ lRpoyt On -he OA inmjrria of Unifoin-ily SpacedA-1iEr.\ of Noyiopic So>,rcys or Dipoles

Subrni'ied W~ C. TI- FaiAr.jterna Laborato!'

D, p a rtm vr~ oDf E:. i E r)-,iir, i in

Dan A X1arwh 19'.

1 2-': I

TABLE OF CONTENTS

Page

INTRODUCTION 1

FORMULATION 2

POWER PATTERN AND THE CORRESPONDING ARRAYMATRIX FOR VARIOUS TYPES OF ARRAYS 6

NUMERICAL COMPUTATION 12

THE LIMITING VALUE OF GN FOR BROADSIDEARRAYS OF ISOTROPIC SOURCES 13

COMPARISON OF GN WITH THE GAIN OF AUNIFORMLY EXCITED ARRAY 16

ACKNOWLEDGEMENT 16

REFERENCES 317

1522-1 ii

ABSTRACT

The optimum gain of uniformly spaced arrays of isotropic sourcesor dipoles is investigated theoretically in this paper. The formulationis processed with the aid of an array matrix. The optimum gain and thecorresponding excitation are expressed directly in terms of the elementsof the array matrix.

1522-1 iii

Page 1 of 37

ON THE OPTIMUM GAIN OF UNIFORMLY SPACEDARRAYS OF ISOTROPIC SOURCES OR DIPOLES

INTRODUCTION

For a uniformly spaced array consisting of a given number ofisotropic sources, the renowned synthesis of Dolph,' including the ex-tension by Riblet, 2 offers the minimum beamwidth for a prescribedside-lobe level. It is not an optimum design from the point of view ofmaximizing the directivity or the gain. Ma and Cheng3 have recentlypresented another synthesis which optimizes the gain under the con-dition of a prescribed side-lobe level. Because of the polynomial formu-lation, neither of these two methods can conveniently be applied to arraysof directive sources, such as those consisting of dipoles. The problemthat deals solely with the maximum or the optimum gain of uniformlyspaced arrays was first investigated by Uzkov 4 in a very elegent formu-lation. By means of an orthogonal transformation in vector space, heobtained some important results concerning the optimum gain of anarray of isotropic sources. In particular, he showed that the optimumgain of an end-fire array of isotropic sources as the separation approacheszero is numerically equal to N , where N denotes the number of sources.He also showed that when the separation is equal to X /2, the optimumgain is numerically equal to N. Although he indicated that the methodcan be applied to arrays of directive sources, he did not elaborate. Theexcitation of the arrays to produce the optimum gain was not discussedin his work. Several years later, Block, Medhurst and Pool 5 proposedan optimization method based upon the impedance matrix defined for theelements of an array. Only a very limited calculation was reported intheir work. Recently, Stearn6 made some extensive calculations on thegain of an end-fire array of half-wave dipoles, and its correspondingexcitation for various separations based upon their method.

In this work, we shall formulate the problem with the aid of anarray matrix. The method, in certain respects, is equivalent to theimpedance method except that all the essential results, namely, thegain and the corresponding excitations, can be expressed directly interms of the elements of an array matrix. The formulation is particularlyeffective in dealing with arrays of short dipoles as well as isotropicsources. In the present report, we shall consider only broadside arrays,and ordinary end-fire arrays. The latter are characterized by a restraint

1 522-1 1

that the progressive phase shift between adjacent elements is assumedto be equal to the electrical distance between them. This type of end-firearray does not yield the optimum gain, because of the restraint on thephase shift. We treat these arrays here because the formulation is verymuch like that for broadside arrays. In a subsequent report, a generaltreatment of end-fire arrays will be given without imposing the above-mentioned restraint on the progressive phase shift.

FORMULATION

As an illustration of the method, the simplest case of a broadsidearray consisting of Zn isotropic sources as shown in Fig. l(a) will betreated here. The field pattern of such an array is given by

(1) F(0) = Ai cost~ D cos ei=l

where

D = Zrd/k

d = separation between adjacent elements

0 = angle of inclination measured from the axisof the array

Ai = amplitude of the i-th pair of elements

Zn = N = number of elements.

The amplitude Ai is assumed to be real, but it may be either positiveor negative. The corresponding power pattern can then be written inthe form

(2) 5(O) =[F(e) 2 n n, AiAj cos[(i - cos

i=l j=l

cos[(j - OD cos el

1522-1 2

The directivity or the gain of the array, designed to be broadside, isdetermined by

s(2) S~j(3) gN W

4 S(e)) d]Q S(O) sine: de

(A i)2

=1

n n

i=i j=i

where

(4) Tri = cos(i - D cos cos[(- D)cos jsin 0 dG

= I [si_j(D) + si+j (D)

The Sr(D) function appearing in (4) is defined by

sr(D) = sin rD r 0;rD ' rO

(5) and

So(D)= , r= 0.

To optimize the gain, we set

(6) a8gN

8Ap

1522-1 3

which yields the following set of equations:

(7n n n

i=l j=l i=l j=l

p = 1, 2, .. n;

or

n n n /

j=l i=l j=l i=l

p = 1, 2, "'" n.

Since the quantity on the right side of (8) is independent of p, we maydenote it by K, and (8) can be written, in the matrix form,

(9) [P] {A} = fK} .

The elements of the square matrix [P], which is symmetric, are givenby (4). For convenience, [P] will be designated as the array matrix.Denoting the adjoint of [1P] by [B], one has

(10) {A} = [P]" ' {K} = 1j "j [B] (K}

where 1P denotes the determinant of the array matrix. From (10),we conclude that for optimum gain, the amplitude distribution mustsatisfy the following relations:

n n n

(11) A, :A2: ... An= 7Bj: ' Bzj:" _ Bnj.

j=1 j=l j=l

1522-1 4

In view of (8), Eq. (3) may be written as:n n(12) G N Ai pjAj; p = 1, or Z, ... or n

i=l l

where the capital letter GN signifies the optimum value of gN. Bymaking use of (11), and a multiplication rule in determinant theory,(1 2) can be converted into the following expression:

n n

(13) GN =1 7 Bij/ I Pi=l j=l

Equations (11) and (13) constitute the principal result of this formulation.The simplicity of these formulae is manifest since only the elements ofthe array matrix are contained in tnese expressions. For the case of abroadside array with an odd number of elements equal to 2n-l, Fig. l(b),Eqs. (11) and (13), are still valid, except that the elements of thearray matrix are changed to

(14) ij = -I (si-j + si+j-z) , for N = 2n - 1.

The method described here can be applied equally well to end-fire arrays,or arrays of directive sources. In the following section we list the ex-pression for the power pattern and the corresponding array matrix forseveral types of arrays composed of isotropic sources or of short dipoles.It is assumed that the individual short dipoles have a figure-eight patterndescribed by sin Od, where ed denotes the polar angle measured fromthe axis of the dipole. For completeness, the cases discussed previouslyare also included.

1522-1 5

POWER PATTERN AND THE CORRESPONDING ARRAY

MATRIX FOR VARIOUS TYPES OF ARRAYS

(1) Broadside array of isotropic sources (Fig. 1)

a) N = Znn

Se(8= Ai cos ( D cos 0i=1

Pij [si_j(D) + si+j-I(D)]2

where

so(D) =1, Sr(D) = sin rD, D =.Tr d.rD

b) N = Zn-I

So(( ) [ Ai cos(i-l)D cos j

Pij -2 [si_j(D) + si+j_zlD).

de

An A2 Al Al A2 An

(a)

d9

An A2 2A I A2 An

(b)Fig. I. Broadside arrays of isotropic sources (a) N = 2n, (b) N = Zn-I.

The designation for the elements of an array with odd numberof elements also applies to other types of arrays sketched inthe subsequent figures.

1522-1 6

In the following tabulation, the functions Se() and So(9) have the samemeaning as those defined for broadside arrays of isotropic sources.

(II) Broadside array of parallel dipoles (Fig. 2)

a) N = Zn

S(0,0) = (1 - sin? a cos 2 41) Se(()1

Pij = 1 [Pi-j(D) + Pi+j.l(D)]

where

Po(D) = r( 1 sin rD+ cos rD3o , Pr75) = 1 - rD rT-D- -

b) N = Zn -I

S(O,4) (1 - sin z 0 cos2 c$) SO()

Pij = - [pij(D) + Pi+j-z(D)].

d

Y

Fig. 2. A broadside array of parallel dipoles.

The angles 9 and appearing in the

text correspond to the ones shown inthis figure.

1522-1 7

(III) Broadside array of collinear dipoles (Fig. 3)

a) N = Zn

S(O) = sin2 0 Se()

Pij = li-j(D) + li+j-l(D)

where

S(D) r(D) sin rD cos rD3 = r(, r 3 D3 r 2 D2

b) N = Zn- 1

S(O) = sin2 2 So(0)

Pij = li-j(D) + li+j-z(D).

d

An A2 A, A, A2 An

Fig. 3. A broadside array of collinear dipoles.

(IV) Broadside array of crossed dipoles (Fig. 4)

In this case the two crossed dipoles are assumed to be of equalamplitude but excited 900 out of phase. The field due to a single unit inthe broadside direction (8 = /2,' = T/2) is therefore circularly polari-zed.

a) N = 2n

S(9,4) 1 (1 + sin? c sin' 6) Se(8)

I [C i j(D) + Ci+jI(D)]1ij -- 4 -

1522-1 8

where

0°(D) = 4 Cr(D)=(1 + l-- sin rD cos rD

3 r 2 D'J rD

b) N =Zn- 1

S(, 1 (1 + sin? 0 sin? ())SoO)

Pij = I [cij(D) + Ci+j_2 (D)].

4x

d

An A2 A, Al A2 An

Fig. 4. A broadside array of crossed dipoles;the excitation of the first pair is:Alx = A1 andAz = jA1 , etc. The

angles 0 and are shown in Fig. 2.

(V) End-fire array of isotropic sources (Fig. 5)

As explained in the introduction, the end-fire arrays consideredhere are of the ordinary type, with a progressive phase shift betweenadjacent elements numerically equal to D.

a) N = Zn {nSo(e) Ai cos-(i D(l - Cos 0)

lil

Pij [sZ(i_j)(D) + sZ(i+jl)(D)]

where the sr(D) function is the same as that defined in Case (1). Wemay remark that s2r(D) can be decomposed as follows:

1522-1 9

s 2 r(D) = cos rD sin rD - cos rD sr(D).rD

b) N = 2n-i

So = A i cos[(i-l)D(1-cos 0)]

i=l

I [ s (D) + (D)]Pij = ZG ~ij) s(i+j - )(

d6

3D D .D .3D .2n-I

AneJ2 A2eJT A2ej2 Ale- AzeJT Ane

Fig. 5. An end-fire array of isotropicsources, D = ZTr(d/X ).

(VI) End-fire array of parallel dipole (Fig. 6)

a) N = Zn

S(O, 4) = (1 - sin? 6 cos z 4)) Se'(()

_ij I [qi1 j(D) + qi+j_l(D)]ij 2

where Se1(8) is the same as that defined in Case (V-a) and

q(D) cos rD - sin r + cos

(I-rz D z rD r z D2

cos rD Pr(D).

1522-1 10

The function pr(D) appears in Case (II).

b) N -Zn-1

S(G,4) = (1 - sin? 0 cos 2 4) So 0 e)

: 1 [qi (D) + qi+j (D)]

where Sole) is defined in Case (V-b).

d

AneJ20' 3D .D A e2D_2n-I 3 D DAne ~~A2e 2 Ae 2 Aie J A26e n

Fig. 6. An end-fire array of parallel dipoles;the angles 0 and 4 are shown in Fig. 2.

(VII) End-fire array of crossed dipoles (Fig. 7)

The crossed dipoles are assumed to be of equal amplitude but

excited 900 out of phase.

a) N = Zn

S(O) =1 ( + cos E) S2 e) ())

Pij = [qij + qi+j-1]

Although the pattern for this case is different from that of an end-fire

array of parallel dipoles, the array matrix, and hence the optimumgain, is the same for both cases.

1522-1 11

b) N = 2n-i

S(O) = -(1 + cos 2 0) SOW8)2

2. =1 [qi-j + ]

2n- I 3D .D .3D 2n-IAne D Ae A' ," e Ale ' A 2eJ - Anel T D

Fig. 7. An end-fire array of crossed dipoles; the excitationof the first unit althe right is:

Aix = Ae , Aly = jAje , etc.

The angle 0 and t are shown in Fig. 2.

NUMERICAL COMPUTATION

The numerical computations for the amplitude distribution and theoptimum gain based upon (11) and (13) seem to be straightforward.This is indeed the case if the order of the array matrix is equal to two,or if D, the separation between the elements, is not too small. Whenn is greater than two, and D is less than Tr for the broadside arrays orless than Tr/2 for the ordinary end-fire arrays, the computation becomesrather difficult even with the aid of an IBM-7090 computer. The factthat such a giant machine could not evaluate accurately a 3 x 3 determi-nant or invert the associated matrix without going into multiple precisionprogramming is truly unexpected. It should be mentioned that D = Trfor the broadside cases, and D = Tr/2 for the ordinary end-fire cases,correspond to the demarcations of the Dolph synthesis and the Riblet'sextension. For convenience, we shall call the region of D lying belowthese two values the "super-gain" region. Pending a more accurateevaluation, we shall present here only the data of GN for the values ofD where the accuracy was certain. These results are plotted inFig. 8-13. In regard to the corresponding amplitude distributions, the

1522-1 12

data are too numerous. Figures 14-19 give a few sample curves for sometypical distributions. In the case of broadside arrays of isotropic sources,as discussed in the following section, it is possible to find the limitingvalues of GN as d - 0. The dotted lines of the GN curves in the super-gainregion correspond to the interpolated values between these limiting valuesand the accurately computed values. For other cases, where theselimiting values have not yet been found, only the accurately computedvalues are plotted. Each set of curves contain also a plot of Gmax/Gcand GminiGoo where Gmax and Gmin denote, respectively, the first maxi-mum and the first minimum of the GN curves. Go denotes the asymptoticvalue of GN as d -oo. Numerically, G. is equal to N for arrays of iso-tropic sources, and is equal to 3/Z N for arrays of dipoles. In the caseof broadside arrays of isotropic sources, we also plotted a curve,Fig. 8(e), labelled Go/G. where Go denotes the limiting value of GN asd - 0. A discussion on the values of Go is given in the next section.

THE LIMITING VALUE OF GN FOR BROADSIDEARRAYS OF ISOTROPIC SOURCES

For N = 3 or 4, it is relatively simple to evaluate the limiting valueof GN as D approaches zero. This can be done by expanding the relevantfunction sn(D) in a series of D2 and retaining the leading terms of

I P and ZBij.

The result gives

f 2

(15) lim G 3 = lim G4 = 2.25.D- 0 D-*0

When N = 5 or 6, the same procedure calls for seven terms in the seriesexpansion of sn(D) which already involves a very tedious algebraicmanipulation. The result yields

(16) lim G5 =lim G 6 5 3.515625D-0 D-0 24/

To obtain this simple result we must deal with many clumsy numberssuch as (6)14/15! . It should be mentioned here that these calculationswere performed before we had access to Uzkov's article. In viewof the orderly figures contained in (15) and (16), we then proposed a

1522-1 13

conjecture that for arrays with odd number of elements N or those with

even number of elements N + 1 the limiting value must be given by

(17) lir G N 5 4 7 ... N

D-04 6 N-

After reading Uzkov's paper, we realized that as far as these limitingvalues are concerned they can most conveniently be obtained from hismethod. Although Uzkov gave these values only for end-fire arrays, itcan be shown by means of the orthogonal transformation which he out-lined that, in general,

N-12

(18) lim GN(Oo) = (2n+l) Pn (cos 00)D -0 n=

where Pn(cos 0 o) denotes the Legendre polynomial of order n, and 00denotes the angle, measured with respect to the axis of the array, inwhich direction the array was designed for the optimum gain. The valuesof

lim Gn(6o)D-0

are plotted, on db scale, in Fig. 20(a) and (b). For end-fire arrays,without the phase restraint mentioned previously in the introduction,one has

N-1

(19) lim GN(0) = (2n+l) = N 2

D- 0n= 0

This formula was originally enunciated by Uzkov. For the broadside

array, (18) reduces to

N-1

(20) lim GN = (2n+l) Pn(1).D-0

n=0

1522-1 14

II

The same formula applies to the ordinary end-fire arrays as D - 0.It can be shown that the sum given by (20) is indeed identical to the onegiven in product form by (17). Looking back at the hard work which wedid in arriving at (17), one has to admire Uzkov's ingenious method of

successive orthogonal transformations in providing the answer for theselimiting values of GN. The only consolation for our innocent hard workis that without it, the alternative expression for (20) as given by (17)probably would not be recognized at first glance. In regard to thenumerical values of the limiting values of GN.(1r/2) as given by (17) or(20), it is of some interest to point out that these can be approximatedquite accurately by an even simpler formula. From the Peirce-FosterTable, 7 one finds that

2

(21) 4n > 3 5 n- > 2(2n-l)T ( 2 4 2n-2J Tr

hence the mean value of 4n/Tr and (4n-2/ITr) may be used an an approxi-mate value of GN(Tr/2), i.e.,

(22) 2(2n--) _ 2N+l(22)N T T

The expression given in (22) is certainly the asymptotic value ofGN(7r /2) for large values of N. The fact that it may be used for allvalues of N in approximating (17) is clear from the following table:

TABLE OF( - ... - AND ZN+IZ 4 _1Tr

( 3 5 N 2N+I2 4 21N-I iT

3 2.250 2.2285 3.515 3.5017 4.785 4.7759 6.056 6.048

11 7.328 7.32113 8.600 8.594

15 9.873 9.86817 11.14 11.14

19 12.42 12.42

1522-1 15

Finally, we wish to emphasize that the result found here also agrees withthe conclusion drawn by Bouwkamp and Bruijn 8 that for a continuouslydistributed line source, N - o, the optimum gain is without bound. How-ever, for discrete arrays there is a unique solution.

COMPARISON OF GN WITH THE GAIN OF AUNIFORMLY EXCITED ARRAY

Before we conclude this report, it is important to point out thatthe gain of a uniformly excited array with the same number of elementsand spacing, which will be denoted by g(), is, in general, slightly lessthan the optimum gain except inth$e super-gain region. Figures 21-25show the comparison between g." and G for some typical cases. In

vfiewy of plth e prata valcue of nfr excte varrys, es hfarae. copied1N

data will soon be published as a separate report.

ACKNOWLEDGEMENT

The author gratefully acknowledges many valuable discussionswhich he had with Dr. H.C. Ko, Dr. E. Kennaugh, and Mr. T. Compton.The assistance of Mr. JohnHughes of The Ohio State University Numerical

Computation Laboratory is also appreciated.

1522-1 16

I.: LJ -LI . . - ---70 I . . .

d

(a) N =3-6

2--

10 N= -1

1(b2- N 1

31 124----

G N

00,d

(c) N z11-16

0C7

2'~9 -+--

GN L ,

00 50

(d) N 0 d 175 20

1522-1 18

1.84

GN 1.2-- - -

1.0 -

0.8- - Min/ g~

5 10 15 20

N

(e) gmnax/gcc, gmin/g., and og

1522-1 19

__0

-4

0

4'

00

1522-1. 20

I0

1 0

0

I ';4

.0

41

I K40

0 OD 0 I C\j OC) a, -C\j - -

1522-1 21

24 - - - - --- I I

22-------------- - - -- - - -- - -

20--

1 - -- /-- -

- -0000-- - 00 - 0

G N 12 - - - 0-, 8

152- 22-

0

- -4-

0

k02

0- - -- - +4

0

4

0

44

152- 23

InX

0

44

04.

-- 4

0.0-

Z4

I

I I 1 - ' ' 1

1.2-1.0- --

0.6 -

-0.2

-0.4 Broadside Array Of

Isotropic Sources

.3 ;O.8---------------

IA

-12--- A3I ~-1.4 ---

0 0.5 1.0 d 1.5 2.0

Fig. 14. Amplitude distribution of a broadside array

II

of six isotropic sources.

11522-1 25

II

2.0 --

1.6-- ----- - I-1.2 .

I-0.8 - - - *- -

0 _ _ _ 1 4

I

1.8 - Broadside Array Of Parallel Dipoles

2.0 '!JA 4 A3 A2 A I AI A2 A3 A4

2.4 AA4T- A,/A"2. - A3/A 4

0 0.5 1.0 1.5 2.0

dx

Fig. 15. Amplitude distribution of a broadside arrayof eight parallel dipoles.

1522-1 26

1.2- i

0---1-

0.8

0.6

0.---

0.2

CI

-0.2 - -- Broadside Array Of Collinear Dipoles

-A 4 A3 A2 A, A, A2 A3 A4-0.4 -- - -/A4

-0. -

-1.2-0.8--- -

0 0.5 1.0 1.5 2.0

dx

Fig. 16. Amplitude distribution of a broadside array

of eight collinear dipoles.

1522-1 27

1.4 - - - - .-

1.2

0.2----

0.6

0.4 ,

0.2

1oI

00

-0.2 Broadside Array Of Crossed Dipoles

-0. 4-- Io2 I 3

-0.4- A4 A-3 A2 A A A3 A 4

-0.6 - - --

-0.8- - - - A /A 4

-1.0 1 --- -0 0.5 1.0 d 1.5 2.0

Fig. 17. Amplitude distribution of a broadside arrayof eight crossed dipoles.

1522-1 28

1.4

1.2 - - - - - --

1.0----

0.8 -- or

0.6

0.4 /Ordinary End-Fire Array OfA / Isotropic Sources

AA 0.2-

A2 Al Al A2

-0.2 --

-0.4 I

-0.6-

-0.8

-1.0

-0 0.5 1.0

dx

Fig. 18. Amplitude distribution of an ordinary end-fire

array of four isotropic sources.

1522-1 29

00

co co- - - - - - a 40

152 -1 30

90 80 70 60 50 40

0 5 1015 2025

(a)

90 80 70 60 50 40

0 5 1015 2025

(b)

Fig. 20. Polar plot of 10 log[ Lim G(6 0 ).,D-0

(a) N = even, (b) N =odd.

1522-1 31

0

0N

000

4 0

o 0 0 '4-.0

00

4-40 00

0

,

U,)

N 0 CD CD

1522-1 32

0

00

0

*00

-~ _ 0

000

0-4

002'

U d

NN

10~

1522-1 33

U)

0

0 0

~ "0 P4r.. -

0

04) 0

0= k

0 0cl0

1522- 34-

00

0ou00

w ra 00 0

o0 - 4-1I If 00 t

- 4-0 0 I0 Cd

0.~

1522-1 35

00

REFERENCES

I. C.L. Dolph, "A current distribution for broadside arrayswhich optimizes the relationship between beam width andside-lobe level, " Proc. I.R.E. Vol. 34, p. 335 (June 1946).

2. R.J. Riblet, "Discussion of Dolph's paper, " Proc. I.R.E.Vol. 35, p. 489 (May 1947).

3. M.T. Ma and D.K. Cheng, "A critical study of linear arrayswith equal side lobes, "I.R.E. Convention Record, Part I,p. 110 (1961).

4. A.I. Uzkov, "An approach to the problem of optimum directiveantenna design, " Comptes Rendus (Doklady) de l'Academie deSciences des l'URSS, Vol. LIII, No. 1, p. 35 (1946).

5. A. Bloch, R.G. Medhurst and S.D. Pool, "A new approach tothe design of super-directive aerial arrays," J.I.E.E., 100,Part III, p. 303 (1953).

6. C.O. Stearns, "Computed performance of moderate size, super-gain antennas, " Report No. 6797, Boulder Laboratories,National Bureau of Standards, U.S. Department of Commerce(September 1961).

7. B.O. Peirce and R.M. Foster, "A short Table of Integrals,"Ginn and Company, Boston, p. 107 (1956).

8. C.J. Bouwkamp and N.G. deBruijn, "The problems of optimumantenna current distribution, " Philips Research Report 1, p. 135(1946).

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