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, r

C.TDR-63-131

NEW METHODS FOR SYNTHESIS OFNONONIFORMLY SPACED ANTENNA ARRAYS

TECHNICAL DOCUMENTARY REPORT NO. RADC.TDR.63-131

May 1963

Electronic Warfare Laboratory

V • Rome Air Development Center

Research and Technology Division

Air force Syutems Command

Oriffiss Air Force gaos, New York

Project No. 4540, Task No. 454001

(Prepared under Contract No. AF30(602).2711 by the Antenna Laborototy,Department of Electrlcel Engineering, Ohio Stote University Reseoarch

F- ,,.ndation, Columbus, Ohio. Author: H. Una)

When Government drawings, specifications, or other data areused for any purpose other than in connection with a definitelyrelated: Government procurement operation, the United StatesGovernment thereby incurs no responsibility nor any obligationwhatsoever, and the fact that the Government may have formulated,furnished,, or in any way supplied, the said drawings, specifications,'or other data, is not to be regarded by implication or otherwise as,in any manner licensing the holder or any other person or corpora-

tion, or conveying any rights or permission to manufacture, use, orsell any patented invention that may in any way be related thereto.

The Government has the right to reproduce, use, and distribute.this: report for governmental purposes in accordance with the contractunder which the report Was produced. To protect the proprietaryinterests of the contractor and to avoid jeopardy of its obligationsto the Government, the report may not be released for non-governmentaluse such as might constitute general publication without the expressprior consent of The Ohio State University Research Foundation.

Qualified requesters may obtain copies of this report from theASTIA Document Service Center, Arlington Hall Station, Arlington 12,Virginia. Department of Defense contractors must be established forASTIA services, or have their "need-to-know" certified by the cogni-zant military agency of their project or contract.

Do not return this copy. Retain or destroy..

FOREWORD

The research reported herein was performed in the AntennaLaboratory, Department of Electrical Engineering, The Ohio StateUniversity, under Contract No. AF30(602)-2711 between The OhioState University Research Foundation and Rome Air Development Center,Air Force Systems Command, USAF, Griffiss AFB, N. Y. Mr. Karl Kirkof Rome Air Development Center was the contract initiator.

This report (Contractor's No. 1423-1) was written by Dr. H. Unzof The Ohio State University Antenna Laboratory.

Much of the research work reported in this paper was done whilethe author was a consultant for the Research and Development Depart-ment, Apparatus Division, Texas Instruments, Incorpo'rated, Dallas,Texas, and it is published with their permission. The author is es-pecially grateful to Mr. John B. Travis from Texas Instruments, Inc,for initiating this research program and for his support and help.

M & W luc., Syracwe, New Yorl102. S114/63

ABSTRACT

Four new methods of synthesis of nonuniformly spaced antennaarrays are given:

IL The mechanical quadratures method, developed recently byBruce and Unz and independently by Lo.

I1. The eigenvalues method.

III. The expansion method.

IV. The orthogonalization method using the Schmidt's procedure.

An exponential-decay directive pattern is suggested in order toavoid numerical integrations. A summary of the research work in non-uniformly spaced antenna arrays is given.

PUBLICATION REVIEW

This report has been reviewed and is approved.

Approved: -K, 1 (1 .EDWARD N. MUNZERChief, Electronic Warfare LaboratoryDirectorate of Intelligence & Electronic Warfare

Approved:0 J QUINN, , Col, USAF

Di~rect r of, Intellig-ence & Electronic Warfare

TABLE OF CONTENTS

Page

1. INTRODUCTION 1

2. THE EXPONENTIAL PATTERN 5

3. MECHANICAL QUADRATURES MET-HOD 6

4. EIGENVALUES METlHOD 12

5. EXPANSION METHOD 16

6. ORTHOGONALIZATION METHOD 18

7. SUMMARY 22

APPENDIX 25

REFERENC ES 28

V

NEW METHODS FOR SYNTHESIS OFNONUNIFORMLY SPACED ANTENNA ARRAYS

1. INTRODUCTION

The importance of directive, antennas was realized in the early daysof radio communications. The principles of wave interference, on

which systems of directive radio are based, has been known probablyfor several centuries. However, the first thorough treatment of thissubject was conducted by Huygens and by Fresnel, who established the,wave theory of light in the early part of the nineteenth century.

During the decade 1920-1930, a concentrated effort on directiveproperties of antenna arrays was started. During this period short-wave adio communications were becoming more popular, taking the

place of long radio waves, and the use of antenna arrays of reasonablesize became more feasible.

In 1937 Wolff' published his method of synthesizing any arbitraryfar-zone circularly symmetric pattern with radiators uniformly distribu-ted along an array axis. His theory was based upon comparison of the

far-zone field of a pair of radiators to a term of the Fourier series ex-

pansion of the prescribed pattern.

During the second world war, the invention and the use of radarincreased the interest in directive arrays. In 1943 SchelkunoffZ util-ized the correspondence between nulls of the pattern of a linear arrayhaving equidistant elements and the roots of a polynomial in the complex

plane. Different types of: pattern variations were derived by choosingdifferent distributions of the zeros of the polynomial.

In 1946 Dolph 3 devised a method of synthesizing an optimum patternof isotropic elements equally spaced in a uniform broadside linear array,in which the elements are fed in phase and are symmetrically arranged

about the center of the array. The resultant current distribution acrossthe array is based upon the properties of the Tchebyscheff polynomials

and offers, from the design standpoint, much greater control of thepattern than previous theories.

Woodward 4 and Lawson . gave a method for calculating the field

over a plane aperture to produce a given polar diagramh and discussed

the theoretical precision with which an arbitrary radiation pattern may

be obtained from an array of finite size. This method i.s ba~sed on theidea of matching the radiation pattern of the aperture at a finite number

of directions with the prescribed radiation pattern.

Van der .Maas 6 gave the ideal Tchebyscheff space factor corres-ponding to a continuous: source of finite length, in the following formn:

"(la) F:(O) cos (r uz -A 3 )

2a(Ib) u = - sine

where 2a in the length of the line source,. X is the wavelength, and 0 isthe angle from -the normal to the antenna axis. From the above it can

be found that:

(1c) ch ( w A) = side lobe ratio.

By using Fourier integrals, Taylor7 showed that the aperture currentdistribution required in order to produce the ideal radiation pattern inEq. (1) is:

I,(A wrZ _pZ)

( 2 ) AA w2~ - p2

1 1p -+-, &(p - r) +-• -(p. + w) , p2 < r

g(p) 0 , pZ> Z

Z

where p = T ,, is the -Drar. delta function, and I is the modified

Bessel function. This pattern was considered later by Taylor 8 fromthe complex function theory point of view. Because of theoretical limi-tations, the above pattern cannot be obtained from a physically re-ilizab-leantennabut its ideal characteristics can be approached arbitrarilyclosely.

Cheng and Maýa proposed a new approach to the uniformly spacedlinear-array analysis. They considered the current distribution inthe discrete elements of the linear array as the sampled data values ofa continuous, function, and used known relations in, Z transforms devel-oped for sarnp-Ied-data systems in order to express the array poly-nomial in a closed form, where the array properties are easily found.

In all, of the above cases, the spacings of the array elements areas sumed to be uniform. In 1956 Unz 9 ' O1" 1" first introduced lineararrays with arbitrarily distributed elements and developed a theory forthem. A relationship between the currents in the elements of the array,thleir distribution along the axisa of the array, and the coefficients of thecomplex Fourier expansion of the radiation pattern were given in amatrix form which involved Bessel functions. It was pointed out that anonun.'frmly spaced antenna array has -more degrees of freedom than

a similar uniform array with equally spaced elements, and as a resultits performance should be better.

Further numerical work on the subject has been done by King,,Packard and Thomas' 2 , who showed that nonuniform antenna arrayswith unequally spaced elements are also more broadband for differentsource, frequencies. This is especially useful because of the earlierdevelopment of frequency-independent antenna elements by Rumsey,DuHamel, Isbeil, and others' . Sandler'' showed. some equivalencesbetween e, -illy and unequally spaced arrays.

Andreaseni 5 and several of his associates did extensive numericalwork on digital and analog computers in order to find the general be-havior of the radiation patterns of nonuniformly spaced arrays, in par-ticular when the average spacing is larger than one wa ;elength. Swensonand Lo' 6 considered the use of nonuniformly spaced arrays for largeradio telescopes. Harr;ngtonl 7 used perturbational procedures forreducing the side-lobe level of a nonuniformly spaced array with uni-form, excitation. Extensions of the previous theory to dipole elements

3

in the Fresnel zonel 8, and to nonuniform spacings larger than onewavelength 9, were given by Unz. Bruce and Unz 2 0 gave a possiblecondition for broadbanding. Maffett 2 1 discussed array factors witha nonuniform spacing parameter by using well-known numerical, inte-gration techniques (trapezoidal, Simpson), and considered a statistical

theory. Yen and Chow2 1 a discussed the possibility of expressing, the

radiation pattern of large nonuniformly spaced arrays in closed formby using integration by stationary phase technique.

Pokrovskii 2 2 designed a 4- and 6- element array having nonuni-form spacing and uniform excitation of the elements, with improvedpatterns over the Dolph-Tchebyscheff array for the same length and, thesame number of elements. Whil.e his method of solution is general,the transcendental equations become more and more involved for largerarrays. Additional discussion has been given by Brown3 4 .

Nonuniformly spaced arrays have several advantages, over uni-formly spaced arrays; namely, their performance can be better, fewerelements can be used, and they are more broadband. However, non-uniformly spaced arrays are seldom used in practice at present. Be-sides being latecomers, the great difficulty in synthesizing such, arraysseem to be the main reason for the hesitancy to, use them.

Recently a new method of synthe-sizing radiation patterns usingnonuniformly spaced arrays has been developed by Bruce and Unz2 s' 2 4

They applied a mechanical quadratures formulation in order to trans-form a continuous aperture distribution to a nonuniformly spaced array..Because of its importance, a summary of this work will be given inSecti~ori 3. of this paper. Some work along similar lines has been doneindependently by Lo2 5.

The aim of this paper is to give new methods for synthesis of non-uniformly spaced antenna arrays. It is hoped that these new methodswill be found useful in the design of nonuniformly spaced arrays andhelpful in achieving, the general use of their full potentialities.

4

2. THE EXPONENTIAL PATTERN

One of the possible directive radiation patterns with exponentialdecay, which will be used later on, can be written in, the form:

2az sinz e"(3). F(9) e cos (2b sin 0)

where 8 is the angle of the radius vector from the normal to the arrayaxis, and "a", "b" are arbitrary constants. The maximum value of thepattern in the direction normal to the array axis is normalized, F(O) 1.

The first null of the radiation pattern will be at 0 = 80 such that

ir(4a) Zb sin o =

and the beamwidth w between the first nulls will be wo 0 eo ; oneobtains, therefore,

ir(4b) b =

4 sin 7

Using Eq. (4b), the constant "b" can be determined if the beamwidthbetween the nulls wo is specified.

Assuming the first side-lobe maximum of the radiation pattern tobe at e = 61 , then the sideý-lobe level M, of the. radiation pattern will bedefined by:

F(O) 1 ea sinz 6(5a) M-= ....

tF(8e) l fF(e01) Icos (Zb sin9I

The side-lobe level in decibels. Mdb will be then:

(5b) Mdb = 20 log M =

- 20 (a2 sinz 9, loge - log cos (2b sin.01) I.

5

In order to find tlke position of the first maximum, one should solveF'(91) = 0. Using Eq. (3) one obtains-

b sin(2bsin 1 ) + az sin e 1 co's (2b sin e1 ) = 0.

Using the notation a = 2b sin GI one gets:

(6) tga a 2•, :• 2b z

Using Eq. (6) and the same notation, Eq. (5b) becomes:

(7) Mdb= - 0.Zl7atga- log cosal20

Equations (6) and (7) may be used in order to find the relationshipbetween the constants "a"', "bI", and the side-lobe level Mdb, and theresults are shown graphically in Figures 1, 2, 3.

The required radiation pattern is given usually in terms of itsbeamwidth w and the side-lobe level Mdb. From Eq. (4b) the constant"b" of the radiation pattern can be found from the beamwidth w 0 . For

a given side-lobe level Mdb,the transcendental equation (7) can besolved for a. This could be accomplished simply by using Figure 2.Then Eq. (6) and the value of the constant "b" should be used in, orderto find the constant "a". Figure 1 could be helpful in this case. Figure3, which is a combination of the previous figures, gives a direct rela-tionship, in a graphical form, between the side-lobe level Mdb and the

a2

constant Z-Z

W6 have seen thkt if the beamwidth wo and the side-lobe level Mdbare given, the constants "a", "b" may be found and the directive, ex-

ponential-decay radiation pattern in Eq. (3) will be completely specified.

3. MECHANICAL QUADRATURES METHOD

Recently a new method of synthesis of nonuniformly spaced lineararrays has been developed by Bruce and Unz; 3 ,Z 4 and independentlyby Loz 5. They applied mechanical quadratures 2 6 in order to transform

6

71

tga aa 2b?

5

b 2

3

2

o l900 120c) 1500 1800

a i.

Fig. 1.

"80C-

70-

60

50

40

M(db)

30

20:

,10 •.

-- 2.00 2.502 I900 1200 1506 1800

a-]Fig. 2.

8

.80

70

60 .

50 -

t 40

M (db):

30

20

10

0 I2 o2 4 5

2b2

Fig. 3.

a continuous aperture distribution to a nonuniformly spaced array.Because of its importance, a summary of this new method z 3, 24 ispresented here.

Using weight functions,, the general form ,of mechanical quadra-

tures s is given by:

+1I n

(8) w(x) f'(x) dx, H- f(a-j)-l1 ji=l

The n values of aj are real, distinct, within the interval (-1, +1), and

are the roots of a polynomial orthogonal with respect to the weightfunction w(x) on the interval (-1, +1). The n values of H. are real,positive, and can be determined by solving the set of linear equationsgenerated by letting f(x) = xk for k = 0, 1, 2 ... 2n - 1. The only re-quirements on the weight function are that it does not vanish within theinterval of integration and that it is integrable.

For a finite aperture with symmetric excitation, the radiation

pattern can be written as:

+1

(9) F(u) g .! ) g(x) cosuxdx

-l

where the aperture length has been normalized, u = Tr sin@, and x isthe distance from the origin measured in half wavelengths.

Comparing Eqs. (8) and (9), and taking the weight function to beequal to the aperture distribution w(x) g(x), one obtains:

n

(1,0) F(u) L Hj cos (ua j )j=l

Equation (10) is identified as the nonuniform symmetric array radiationpattern, where the element positions are given by the n values of a. and

the current excitation coefficients are given by n values of H..

10

In case of no weight function, W(x) = 1, Eq, (8): becomes:ý

+1 n

(11) f(x) dx = * Hj; f(aj):Q

I -i j=l

and as a result, Eq. (10) becomesr:'

n

.(12) F(u) - j g(aj.)cos(ua.)

j=l

Bruce and UnzZ 3, Z 4 applied this method to the exponential direc-tive radiation pattern described in Section 2, as well as to the Dolph-Tchebyscheff patte'rnwith very good results. In the case of the ex-ponential pattern,, by taking the weight function to be w(x) = e-xz one

finds the values of a. and Hj to be tabulatedz 6

The basic requirement for the mechanical quadratures method isthe knowledge of the aperture distribution g(x) in Eq. (9). If the radia-tion pattern F(u) is prescribed, g(x) could be found sometimes by usingthe inverse Fourier transform, as for the case of the exponential pat-tern?. 3, Z 4 in Section. 2. Sometimes the aperture distribution g,(x) maybe found by taking the envelope 2 7 of an already designed uniformly-spaced array, as for the case of the Dolph-Tchebyscheff patternIn both case-s the current at the aperture edge will give difficulty', ascan be seen in Eq. (2), for example, and should be discussed separ-ately

8 , 28

For the design of a general prescribed radiation pattern, the diffi-culties of this method are twofold: (a) To find the aperture excitationfrom the prescribed radiation pathern by a Fourier integral; (b) Approxi-

mating the aperture excitation by a polynomial, a process which willinvolve the inversion of a matrix. However, for directive patternscertain short cuts in the process have been shown above,.

.11

4. EIGENVALUES METHOD

A symmetric, nonuniformly spaced antenna array will give the

S radiation pattern:

L

('l3) F(u) A Icos,(ux1 )

A-= 0

where u = 7sin 9, R being the angle with the normal to the array axis,and x is the distarce from the center, measured in half wavelengths.In Eq. (13) the radiation pattern F(u) is given and the array elementdistribution x, and the element amplitudes, A, have to be determined.

Let us discuss the integral:

(14a) I(xf;xm) cos. (uxt)cos(uxm)du.

When x, = xm one obtains:

sin Z xf -r"

(14b) II =(Ixg;xc =! 7r[ 1 + Z I "

By using trigonometric identities Eq. (14a) becomes:

sin (xi +x m)7 sin (x, - x)T7rI(xI;Xm) xL+xm + MX, - xm

Equating the last expression to zero leads to:

(x m) sin(x, Xm 7Tr+ (x. +xm sin (x, x 7r 0.

(x-x+ ' m X ir 0

; 12

Rearranging,

x1[ sin (x• + xm)7Tr+ sinix Xm) r] =

= X[ sin(x1 : + Xm)T - sin(xi, -X

Using trigonometric identities, one obtains finally:

S(I 4c) (•• 7T) tg (XI 7T) =(XM 70 tg (Xm 7T) •

It is found that for the eigenvalues of Eq. (1 4c), the integral I(xl X)

in Eq. (14a) will have orthogonality properties such that:

sin 2x1 7r(14d) I(x,;xm) 7T[ 1+ ] 6m

S.Zx•r l m

where

1 M.61'Pm ={0 I Xm.

is the Kronecker delta.

From Eq. ( f', or the eigenvalues one can see that all the posi-tions of the radiating elements are determined by the position of thefirst element with iespect to the middle of the array. In order to in-clude the first eigenfunction in the complete set, the first elementposition should be 0 < xo < 1 ; in other words, the distance betweenthe two innermost elements should be smaller than one wavelength. Theeigenvalues in general are unequally spaced for small values and arealmost equally spaced, a half wavelength apart, for large values.

Using the orthogonality properties in Eqs. (14) one may obtainfrom Eq. (13):

1 sin 2 x7-I +7T(5) [l+ ]r F(u)cos(ux )duu

13

S_,,

Thus, one could get a nonuniformly spaced array which will produceany required radiation pattern. Its element distribution will be deter-mined by Eq. (14c) and the currents will be determined by Eq. (15).

For the particular case of the exponential pattern, Eq. (3) may berewritten:

-a2z

2b116) F(u)=e •' Z c'os (ý -- u).;

where u = 7T sine. Substituting Eq. (16) into Eq. (15) and using trigo-nometric identities, one, may obtain definite integrals, which may beevaluated approximately2 for a > 2:

az'7T -T-U2

(17) e '"T cos (Zpu)

00 0az 7Tz P2 4

"e cos (Z p u) du Za

0

Using Eq. (17) one may obtain for the exponential pattern:7t 7Tb+-r2 x b-- x

sin~x T ("a 1)2 (2 a

(18a) A1 =I ['LE1+Tx ]- 1 [ e +e.,2 a

or in alternative form:

_in 2x -l -T. a- •7T X1(18b) A1 = 7[1+ Yx,, , I e e b x,

14

Equations (18) give the current' distribution of an antenna array whichf produces the exponential radiation pattern in Eq. (3) or (16), and theelements of the array are dist~ributed according to the eigenvalues ofEq. (14c),

From Eq., (18a) one can see that for

(19) i > a +b2i

the value of'A quickly becomes very small, and the contributions ofthese terms to the radiation pattern are so small that they could beneglected. Equation (19) gives an-indication of the number of elementsrequired to produce the exponential pattern within a certain approxi-mation. -

For the particular condition of uniformly spacedarrays, one coulddistinguish between two cases:

A. Odd number of elements:

In this case x^ = 0. and from Eq, (14c) one obtains x,= . Equation15) becomes then:

(20a) A - F(u) cos (A u)*du I > 07[

1 •+ 7(20b) A F(u)du

"7T"

B.. Even number of elements;

Ln this case x0 = . and from Eq. (1 4c) one obtains x + 1

Equation (15) becomes then:

(21) A = F(u) cos( +) udu27[

15

and the relationships, for the exponential pattern should be changedaccordinglyin Eqs. (18).

In the last two methodsý of Sections 3 and 4 the positions ,of thearray elements have been determined during the synthesis procedures.

These methods will fail in general when the element positions are, pre-assigned. In the next two sections we will dis~cuss cases of this kind.

5.. EXPANSION METHOD

When the radiation pattern F (u)) is given and the nonuniformly dis-tributed antenna array element positions, are preassigned, the determi-nation ýof the currents A in Eq. (13) is relatively difficult. The reasonis that one fs required to expand a given function in terms .of a non-orthogonal set, of functions. This problem has been discussed by Kan-torovich and Krylov°3 0 .

One possible method of evaluating the coefficients A in Eq. (13)

will be to take any complete set of functions {?Pm (u)}, multiply bothsides of Eq. (13) by ?pm(u) and -integrate. One willl Obtain then a systemof equations:

L

(22a) : c-A, b ... m 0, 1- 2" Lý=0

where:

+ 7'(22b) c ,Im 5 cos (u xA)pml(u)du

7r

+7T

(ZZc) brn= 5 F (u) 1/m (u) d u

where c ) and b are known and A is to be evalnated by solving

(L + 1) linear equations with L + 1 unknowns.

16

I.

It will be useful to indicate also another method of obtaining sys-tern (Z2a), which is doubtless even more convenient. Let us expandEq. (13) in terms of any orthono~rmal system /m(U) Let:

Co•:r .00

(2 3b) cos (uxj) C M m(u)

m=O

............. where bm is given by Eq. (22c) and-c is given by Eq. ('22b). Sub-mm

stituting Eqs. (23) into Eq. (13) and equating coefficients of V/ru) in bothsides, one obtains Eq. (22a). We note that the second method of ob-taining system (22a) presupposes, as distinct from the first, a systemof functions i.n(u) that is orthogonal and normalized.

One possible such set is lPm(U) = cos mu. This set will have adefinite advantage, as Eq. (22a) will be simplified considerably if someof the: elements, of the array are uniformly distributed at multiples of ahalf wavelength~apart. Also, the integrals (22b) are readily available:.

The above method could be extended to the case of asymmetricnonuniformly spaced arrays:

iuxI

(24) F(u) = A e

1=0imu

One could take then *m,(u) = e Repeating the, above process, one

will obtain:

(2 5a) ct mAt= brn 0n=0, 1, 2 ... L1). m AI M

i=0

(2 5b) +Tr iux e imu 2 sin 7 (x + m)

T-r 7T (X + m)

17

(28c) b F(U) i mU d u,.

In this case, in general, b and A will be complex and one should takeboth +:(m) and - (mi). The theory Aeveloped by Unz 9 1 11 originally is a

particular case of the method described in this section.,

"6. ORTHOGONALIZATION METHOD

In order to expand, an arbitrary, function in terms of a nonorthogonal

set of functions, the orthogonalization procedure by Schmidt may be

used. This proc-edure has been discuss:ed 'in detail by Kantorovich and

Krylov30, and its details for complex functions may be found in Appen-

d dixA of the present paper.

It may be shown3 0 th.t from an infinite set of nonorthogonal complex,

functions:

¢I(u), ý2 (u)..

an ,orthonorinal set of functions

7P, (u), *z (u) ""

-may be derived by using the relationships:

r -(u)

(26b) 4., (u) j1a=

(u)'1 (u) d(u) du

n

4)n.÷l (u) -ý ( a ¢ýn l +1 d' u) (u).(26b ) *n+l (u) - r I

' [¢~l~u" ( on+, ?Pýd u) 7p.(u) 'd ,

ta nl a iu

j=l

where AA* = A[ . The orthonormal functions Wz (u), ?p3 (u), .. n+l(u)may be found in succession.

18

The orthonorrina set V/n(U) obeys:

b, M(2 71 a (u) xv)d u 6 1 =

a f.0 A

Let us assume that a set of nonorthogonal functions is given by:

(28) .(, .n'nu n - 1,.e 2., L

where x1 < x < < < xL. We would like to orthogonalize this

set of functions, in the region, ( -7r, +7T), where the orthono:x.rnaLsoet.willbe denoted by ?pin(u). From (28) and (26a). one obtains:

(2 9a) /1(u) = 1 i

Let us define the following, function:

+• -Tr i X m (xI - Xm) u + 7T

(29b) •i(xt-Xm)Uu+7T

= Z~r sin(x1 - Xm) Tr

(XI - Xm) 7r

where S,,..,

Using Eqs. (28), (29), and Eq., (Z6b),for, n 1, (Qone.obtains:

,1 x2 u ,i x1 u6 • -2 SI 4 X

(29c) 4i2(u) 1 S,2

'9

Simiilarly, one may find in Eq. (26b) after rearranging:

(Z9C) *f3 (U)4

1, (1 S 22 )e'x3u ($3.2 -$, s 3 )Xe1 XZU (1Sl -ZSl , 3 ) elXu

By continuing the above process in Eq. (Z6b) one may find formally:

(Zge)' (4 ()u) = c )ei (4)ý X2 U X4 eU (4) IX4 U

or in general:

n

(Z9f) P n(u) = I e

(n)where the constants cn may be found 'by the Schmidt orthogonalizationprocedure described above.

In the case of a general, asymmetric nonuniformly spaced array,one has::

L

(30) F(u) = A A1ixfu• =1

Taking the orthogonal set. of functions ?Pn(U) described in Eqs. ( 49 )onecould expand:

N

(31a) F (u) = Z Bn~nlu)n=l

20

where, using the orthogonality relationship in Eq. (27), one can find:

+7rS'(31b) B3n = F (u) • ('U)! du,J~nl

-7r

Since F(u) is prescribed, B may be evaluated from Eq. (31b), usingi nEqs. (29).

Substituting Eq. (29f) into Eq. (31a) one obtains, after rearranging:N n

(32) F(u)= e nn XU .n1 ~=nS" n=l I =1

Comparing, Eqs. ( 32 ) and ( 30 ) one obtains:

(33a) A, B' c1 + B2 Ci ~+B c ( B (~N)

(33) =(2)(3) (N)(33b) Az B2 c 2 +B 3 cz + BNNc 2

(33c) A3 B 3 C3 +" ( N)

(33d) A= B c(N)N N N

By using the above procedure one can see that in principle the non-uniformly spaced array , P.n be designed as well as uniformly spacedarrays without using the process of inversion of large matrices. Theprocess of finding the orthogonal set of functions in general is ratherinvolved algebraically, but they could be found with the help of a digitalcomputer using the successivq process described by Eqs. (26). In gen-eral the result will correspond to the radiation pattern within a certainapproximation, rather than be equal to it.

The great importance of this process of solution of nonuniformlyspaced arrays is that it is especially designated foi asymmetric non-uniformly spaced arrays, which have more degrees of freedom thansymmetric nonuniformly spaced arrays. Of course, symmetric arraysalso can be designed by using this method.

2-1

For the case of an exponential pattern described in Section 2, oneis able to use Eqs. (16) and (17) in order to integrate in Eq. (31b) andfind the corresponding coefficients .1n,

For the particular case of uniformly spaced arrays with • spacings,

one Will have in Eq. (29b):

(34a) xI X - m Sr 6 4n m

.-and Eqs. (29) become:

1 iui,, (34b, •hP lu) -". "" Z €

(34c) ?PZ (u) 1 u

(34d) */3 (U) - e

I inu(34e) 7n- ( -

and Eqs. (31), (32 ) and (,33 ) will reduce to the standard Fourierseries analysis of uniformly spaced arrays.

7. SUMMARY

In this paper the published research work in, nonuniformly spacedantenna arrays has been summarized to date. Four new methods for

Z2

synthesis of nonuniformly spaced antenna arrays for a given radiation

pattern have been suggested:

I. The mechanical quadratures method, developed recently by

Bruce and UnzZ z 4 and independently by Loz 5 is summarized.

I." The eigenvalues method, where the positions of the elements

are given as eigenvalues of a transcendental equation.

III. The: expansion method, which involves the inversion of ma-

trices in order to find the currents of the elements.

IV. The orthogonalization method by Schmidt's procedure.

In methods I and I1 the positions of the elements are predetermined

by the weight function involved (method I), or by the position of the

innermost element (method II). In methods III and IV the array ele-

ments are first arbitrarily distributed (in accordance with the pre-

requisites, e, g.., broadbanding) and then the current distributions in

the elements are determined. Methods I and III will involve in general

the inversion of a matrix, while in methods II and IV this is avoided.

An exponential-decay radiation pattern is suggested and used as an

example in the above methods. Its main advantage in the present syn-

thesis is that the definite integrals .rnvolved may be approximatelyevaluated explicitly, and thus simplify the numerical work. Thisdirective exponential pattern could also be used for electronic scanning 31

problems in the form:

35 F(0 -e sin- -Y )cos [2 b sin (,e- -y)]

where -y = y(t) is the angle direction in which the radiation pattern willhave a maximum and y(t) is a function of time. The above methodscould be modified slightly in order to calculate the required currentdistribution by numerical integration. The linear -'.ase delay case forscanni'ng may be6found as a particular case under certain approximations.

Some numerical work using the above methods has been done, andit will 'be published in the future together with additional numerical workand comparisons between the four methods of synthesis, with regard tothe best approximations.

23

Sharp3U and Willey 3 suggested methods of design of linear andplanar arrays in order to reduce the number of the elements required

for a specific radiation pattern. Ishimaru 3 5 has recently suggested

the use of the Poisson's sum formula for the. design of nonuniformly

spaced arrays. His method is useful in treating nonuniform arrayswith large number of elements and unequally spaced arrays on curvedsurfaces.

It is hoped that by using the synthesis methods suggested by the

author in the present paper and others, the design of the nonunilormly

spaced arrays, will become simpler and their use more generally

accepted.,

24

"APPENDIX

In the following: we will extend the orthogonalization process ofSchmidt, as given by Kantorovich and Krylov, 30 to a set of complexfunctions.

Let there be given an infinite system of complex functions,

U. (A- 1) 01 (u), ý2 (U), "

defined and continuous in the interval (a, b.). We can exclude from thegiven system of functions those that represent linear combinations ofthe preceding ones, since by their nature they do not extend the sys-tem. Let us now carry through the. orthqgonalization of the system..(A-I), i.e., let us now construct the orthonormal system of functions

(A - 2) 7p (u), Vý.' (u), ..

such -that

",(A- 3) 7p,(u) **r (u) d u 8,

where 6 ip the Kronecker delta, and ,m(u) is the complex conju-I'm m

gate function. Each function of Eq. (A-Z) is to represent some linearcombination of the functions of system (A-i), i.e. , 1n(u) will have theform:

(A-) n)(n) (n)(A- 4) ?Pnu=a(1n) 1 (u) + a 2 42 (u) + - a(u)n n ¢n-u-)

Let us carry out this orthogonalization step by step. The first functiont/i (u) must have the form c , (u).

Determining the constant c from the condition

(5 b

i• 25

we find

S'€ (u)

Let us assume that the first n functions ?Pj (u), tz (u), "' n(u) have

been determined. The function n + 1 (u) must be a linear combination

of these and the function on+ 1(u) in the form:

" (A-6) ?pn l(u) = c1 ,(u) +c z V(u)+ + Cnn(u)+-.+n +(U"

We determine the constant c. (i 1, 2, n) from the conditions of1orthogonality of Pn + i(u) to ?P (u), Pz (u) "'*mn(u); multiplying Eq. (A-6)

by ?P* (u) and integrating we obtain, using the orthogonality and normalityconditions:

b(Ai-7a) c, + c O n+1(U) ?-(u)du 0.

a

Similarly, by multiplying by 4,* (u) and integrating:

b2

(A-7b) c. + cS0a (n+i(U) (u) du= 0

and so on.: Finally we obtain

b(A-7c) n+ c+ Cn+ l(u)n * (u) du 0

Substituting c, c2 ," c- from Eqs. (A-7) into Eq., (A-6) one obtains:

n

.(A-8) *n+l(u) c n+I(u)" b (u)•(14du) *jlu}

26

The constant c.can be determined from the normalization S 'n÷I 'a+I dul.

Using this, the final resuilt will be:

•,: n

j=1 *'a(A-ý9) *jj+ lu) r n b

i, S n+l(U) n+l• du)Wj(U): ' ~a SnlU " < a

J=1

where JA 14 AA*.

By means of Eq. (A-9) the, functions ?p2z (,u), ip3 (u) ... n+(u) may Ve found

in succession by cascade procedure.

After having constructed the orthonormal system,, one can write,for any arbitrary function f(u), its series:

(A-lOa) f(u) An in(u)

1 n=l1

where the coefficients An may be found by using Eq. (A-3) to be

b(A-10b) A= n f)(u)*n*lU) dLU

a

It is, of course, generally impossible to guarantee the convergence of theseries in (A-Oa). It would be, therefore, more correct to say that thisseries corresponds to the function f(u.i) rather than is equal to it.

Substituting Eq. (A-4) into Eq. (A-l0a) one obtains:

("an) (n),(A-11) f(j,)- a/n) (u) 2)u;+*a 0(u)

n=l

f(u) is written in terms of combination of the original nonorthogonal set,

' nlU) •

: 27

REFERENCES

.1. Wolff, I., "Determination of the Radiating System which willProduce a Specified Directive Characteristic, " Proc. I.R.E.,Vol. 25, pp. 630-643, May, 1937.

2. Schelkunoff, S. A., "A Mathematical Theory of Linear Arrays,"Bell Syst. Tech. Jour., Vol. 22, pp. 80:-107, January, 1943.

3. Dolph, C. L., "A Current Distribution for Broadside, Arrayswhich Optimizes the Relationship Between Beamwidth and Side-Lobe Level," Proc. I.R.E., Vol. 34, pp. 335-348, June, 1946.

4. Woodward, P. M., "A Method of Calculating the Field Over aPlane Aperture Required to Produce a. Given Polar Diagram,"J.I.E.E. , (London), Vol. 93, Part IHia, pp. 1554-1558, 1946.

5. Woodward, P. M., and Lawson, J. D., "The Theoretical Pre-cision with which an Arbitrary Radiation Pattern May be Obtainedfrom a Surface of Finite Size," J. I.E.E., (London), Vol. 95,Pt. III, pp. 363-370, 1948.

6. Van der Maax, G. J., "A Simplified Calculation for Dolph-Tchebyscheff Arrays," Jour. Appl. Phys., Vol. 25, pp. 121,I. 24,January, 1954.

7. Taylor, T. T., "Dolph Arrays of Many Elements, " Hughes AircraftCompany, Research and Development Laboratories, Tech. Memo.No. 320, Contract No. AF 19 (604)-Z62-F-12, August 18, 1953.

8. Taylor, T. T., "Design of Line Source Antennas for NarrowBeamwidth and Low Side-Lobe Level, " Trans. I.R. E., Vol. AP-3,pp. 16-28, January, 1955,

8a. Cheng, D. K., and Ma, M. T., "A New Mathematical Approachin Linear Array Analysis," Trans. I.R.E., Vol. AP-8, No. 3,pp. 255-259, May, 1960.

9. Unz. , H., "Linear Arrays with Arbitrarily Distributed Elements,"University of California, Berkeley, Electronics Research Labor-tory, Report Series No. 60, Issue No. 168, November 2, 1956.

28

S10 Unz., H. , "Multi-Dimensional Lattice Arrays with ArbitrarilyDistributed Elements," University of California, Berkeley,Electronics Research Laboratory, Report Series No. 60, IssueNo. 172, December 19, 1956.

11. Unz. , H. , "Linear Arrays with Arbitrarily Distributed Elements,"Trans. I.R.EW, Vol. AP-8, pp. 222-223, March, 1960.

"12. King, D. D., Packard, R. F., and Thomas, R. K., "UnequallySpaced, Broadband Antenna Arrays,!' Trans. I.R,E., Vol. AP-8,pp. 380-384, July, 1960.

13. Jasik, H., Antenna Engineering Handbook, McGraw-Hill., 1961,Chapter 18: Frequency Independent Antennas.

14. Sandler, S. S., "Some Equivalences between Equally and UnequallySpaced Arrays," Trans. I.R,E., Vol. AP-8, pp. 496-500, Sept-ember, 1960.

15. Andreasen, M. GA., "Linear Arrays with Variable InterelementSpacings, " Trans. I.R.E., Vol. AP-l0, pp. 137-1.43, March,1962.

16. Swenson, G. W., and Lo, Y. T., "The University of Illinois RadioTelescope," Trans. I.R.E., Vol. AP-9, pp. 9-16, January, 1961.

17. Harrington, R. F., "Sidelobe Reduction of Nonuniform Element

Spacing," Trans. I. A.E., Vol. AP-9, pp. 187-192, March, 1961.

18. Unz., H., "Matrix Relations for a Linear Array with Dipol6ElemrXents in the Fresnel Zone, " Trans. I. R. E., Vol. AP-9, p.220, March, 1961.

19. Unz, H., "Nonuniform Arrays with Spacings Larger than OneWavelength," Trans. i.R.E.., Vol. AP-l.0, pp. 647-"48,September 1962.

20. Bruce, J. D., and Unz., H., "Broadband Nonuniformly SpacedArrays," Proc. I.RI.E., Vol. 50, p. 228, February, 1962..

21. Maffett, A. L., "Array Factors with Nonuniform Spacing Para-meters," Trans. I.R.E., Vol. AP-10, pp. 13 31136, March, 1962.

29

2-a. Yen, Z. L., and Chow, J. L., "On Large Nonuniformly SpacedArrays," Institute of Radio Engineers, PGAP, Symposium onElectromagnetic Theory and Antennas, Copenhagen, Denmark,June 25-30, 1962. To be published by Pergamon Press.

22. Pokrovikii, U. L., "General Method of Seeking the OptimumDistribution for Linear Antennas, " Doklady AN USSR, Vol. 138,No. 3, pp. 584-586, 1961. (English translation - Air ForceCambridge Research Laboratories, August, 1961).

23. Bruce, J. D. , and Unz,, H., "Mechanical Quadratures to Syn-thesize Nonuniformly Spaced Antenna Arrays," Proc. I. R. E.,VVal, 50, pp. Zl.Z8, October. 9,62..,

24. Bruce, J. D., and Unz, H., "Synthesis of Nonuniformly SpacedAntenna Arrays using Mechanical Quadratures:"a) Exponential

Pattern; b) Dolph-Tchebyscheff Pattern, ' (to be published).

25. Lo, Y. T. , "A Spacing Weighted Antenna Array,"abstract of atechnical paper, Proc. I -R.E., Vol. 50, No. 3, p. 353, March,1962.

26. Kopal, Z., Numerical Analysis, Chapman and Hall, London, 1955.

27. Ksienski, A. , "Equivalence between Continuous and DiscreteRadiating Arrays., " Canad. Journ. Phys., Vol. 39, pp. 335-349,1961.

28. Bailin, L. L., Wehner, R. S., and Kaminow, I. P., "EmpiricalApproximations to the Current Values for Large Dolph-Tcheby-scheff Arrays, " Hughes Aircraft Company, Research and Devel-opment Laboratories, I.R.E. Convention Record, Part 2, 1954.

29. Dwight, H. B. , Tables of Integrals and Other Mathematical Data,McMillan, 1947, p. 201, 863. 3.

30. Kantorovich, L. V., and Krylov, V. I., Approximate Methods ofHigher Analysis, Interscience Publishers, N. Y., 1958, Chap-ter 1. § 3 ,

31. Von Aulock, WI H., "Properties of Phased Arrays," Proc. I.R. E.,Vol. 48, No. l0, pp. 1715-1727, October, 1960.

30

3Z. Sharp, E. D., "A Triangular Arrangement of Planar-ArrayElements that Reduces the Number Needed, " Trans. I. R. E.,Vol. AP-9, pp. 126-129, March 1961.

33. Willey, R. E., "Space. Tapering of Linear and Planar Arrays,"Trans. I.R.E., Vol. AP-10, pp. 369-377, July 1962.

34, Brown, F. W., "Note on Nonuniformly Spaced Arrays•," Trans.

I.IR.E., Vol. AP-10, pp. 639-640, September 1962.

35. Ishimaru, A., "Theory of Unequally Spaced Arrays, " University

of Washington, Electrical Engineering Department, TechnicalReport No. 63, January 1962, Prepared under contractAF 19 (604)-4098. To be published in the Trans. I. R. E., PGAP.

31

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