f

i,

a. O o

(/)

ESD-TR 70-195

ESD ACCESSION LIST ESTI Call No. W? J P

Copy No. i__ Dl / R5.

Report No. 7

June 1970

LINEARLY POLARIZED ARRAYS WITH ALMOST ISOTROPIC RADIATION PATTERNS

ESD RECORD COPY RETURN TO

SCIF.NTiriC & TECHNICAL INFORMATION DIVISION (CSTIV BUILDING 1211

H. S. CABAYAN G. A. DESCHAMPS

Prepared by

Antenna Laboratory Department of Electrical Engineering

University of Illinois Urbana, Illinois

For

Massachusetts Institute of Technology Lincoln Laboratory

I Rfc.Cfc.IVtD

AUG 2rM970

niSTRiBinxw

This document has been approved for public release and sale; its distribution is unlimited.

16

Report No. 70-10

June 1970

LINEARLY POLARIZED ARRAYS WITH ALMOST ISOTROPIC RADIATION PATTERNS

H. S. CABAYAN

G. A. DESCHAMPS

Prepared by

Antenna Laboratory Department of Electrical Engineering

University of Illinois Urbana, Illinois

For

Massachusetts Institute of Technology Lincoln Laboratory

Under

Purchase Order No. CC-401 Prime Contract No. AF 19(628)-5167

This work was partially supported by NSF Grant GK 2120.

This document has been approved for public release and sale; its distribution is unlimited.

UILU-ENG-70-308

iii

ABSTRACT

A linearly polarized antenna cannot radiate power uniformly in

all directions. However, by controlling the aperture excitation, as

is done in an array, it is possible to reduce the maximum gain and

approach conditions for isotropic radiation. A numerical method is

presented which generates a family of designs which depend on a param-

eter a. As a approaches zero, the radiation pattern tends to become

more isotropic. However, the efficiency is reduced and the sensi-

tivity of the pattern to errors in the aperture function is increased.

In some cases this sensitivity is so high as to make the result worth-

less. The design, therefore, must be a compromise between closeness

to conditions of isotropic radiation on one hand, and losses and sen-

sitivity to errors on the other.

The sensitivity to errors, expressed by the pattern deteriora-

tion for a given level of error, has been evaluated by a numerical

experiment (Monte Carlo method). A general relation has also been

established between the sensitivity and the losses in the antenna.

It agrees with the numerical experiment and can be used as a guide

to choose the regular!zation parameter a.

Accepted for the Air Force Franklin C. Hudson Chief, Lincoln Laboratory Office

iv

TABLE OF CONTENTS

Chapter Page

1. INTRODUCTION 1

2. STATEMENT OF THE PROBLEM 5

3. METHOD OF SOLUTION 8

4. NUMERICAL RESULTS 11

5. COMMENTS ON THE NUMERICAL RESULTS AND THE CHOICE OF a 31

6. STATISTICAL ESTIMATION OF 1/Q AND E2 33 v v

7 . CONCLUSION. 38

REFERENCES 39

LIST OF ILLUSTRATIONS

Figure Page

1. Schematic representation of array and its far-field gain and power patterns 3

2. Schematic representation of the quantities e , e , and e 15

— 2 —7 3. Far-field power intensity patterns |g | and |g | for the

case N = 5 and a = 0 , 16

1- 12 - 12 4. Far-field power intensity patterns g and |g ' for the

case N = 5 and a = 10~3 17

1- 12 1- 12 5. Far-field power intensity patterns |g |" and |g | for the

case N = 5 and a = 10"2 ? ?? 18

— 2 — 2 6. Far-field power intensity patterns |g | and |g |" for the

case N = 7 and a = 0 . ? 19

— 2 — 2 7. Far-field power intensity patterns |g I and |g | for the

c Ct CtV case N = 7 and a = 10_;j 20

— 12 1 — 2 8. Far-field power intensity patterns |g ' and g ' for the

case N = 7 and a = 10 4. , 21

1- 12 1- ,2 9. Far-field power intensity patterns g and g for the

xi -1 J -in-3 a ' av1

case N = 7 and a = 10 J 22

1 - 12 1 - 2 10. Far-field power intensity patterns g " and g for the XT -7 J in-2 a a^ case N = 7 and a = 10 ^ 23

— 2 — 2 11. Far-field power intensity patterns |g | and |g |" for the

case N = 7 and a = 10-1 ? ?V 24

2 2 ~~2 12. Error parameters e , e , and e , and Q vs. regularization

ctv t parameter a for the five-element array 25

2 2 ~2 13. Error parameters c , e , and c , and Q vs. regularization parameter ot for the seven-element array 26

14. Gain vs. solid angle for the five-element array 28

15. Gain vs. solid angle for the seven-element array 29

16. Plot of 1/QV vs. 1/Q for both five- and seven-element arrays.. 30

vi

Figure Page

17. Schematic representation of the relationship between the pattern functions and the error parameters 35

2 18. Plot of the theoretical and numerical values of e and

I/O vs. the Q-factor V. 36 v

1. INTRODUCTION

A linearly polarized antenna cannot radiate power uniformly in all

directions. This condition of isotropic radiation, or unity gain, is

also impossible if one specifies the axial ratio of the polarization

ellipse. However, accepting the existence of nulls in the radiation

pattern, one may ask if it is possible to reduce the total solid angle

in which the gain falls below a certain level. This is equivalent to

reducing the maximum gain.

It will be shown that it is, indeed, possible to make the maximum

gain as close to unity as one wishes. However, the practical difficul-

ties in doing this are the same as those encountered in trying to increase

the gain of an antenna indefinitely. To reach a certain level of per-

formance (gain or lack of gain), one must have an antenna of sufficient

aperture. If, for a given aperture, we try to improve the performance

above a certain level, we have to increase drastically the antenna cur-

rents (or the near field). As a consequence, both the losses and the

sensitivity to construction errors are increased. This is character-

istic of the supergain problem. It occurs more generally anytime one

tries to shape a radiation pattern too accurately. In particular, it

occurs in the "minimum gain" problem considered here.

The need for an isotropic radiator arises when one disposes of a

certain amount of power and one wishes to produce a radiation intensity

higher than a given level in a solid angle as large as possible. This

may be for the purpose of reaching a target whose direction is not known

a priori. Conversely, in the receiving case, one may want to receive a

sufficient signal from a source of given power and unknown direction.

2

Although the method used could be generalized, we shall consider

in this report a colinear array of dipoles as shown in Figure la. From

the far-field power pattern of this antenna, which is shown in Figure lb,

and some gain level G , one can read the total solid angle Q in which m

the gain falls below G . The plot of G(fi) versus ft is shown in Figure m

lc. In this plot, (4TT-f2) is the total solid angle in which the gain

exceeds G . m

Given the curve G(ft), the total available power P watts, and the

desired radiation intensity I watts/steradian, the useful angle (4TT-Q)

is defined by

^- G(4Tr-f2) = I

or

4TTI G(4ff-ft) - 2%r - G . P m

If the ratio I/P is small, ft will be small for any design. However,

when I/P is large, then special designs become necessary in order to

produce an intensity I in as large a solid angle as possible.

In Figure.lc, the shaded portion under the gain curve has an area

equal to 4TT. The total area of the rectangles ABCD and DEFH exceeds

that of the shaded portion. Accordingly, the following inequality may

be written,

(47T-ft)Gw + aG > 4TT M m —

or

Q <_ 4ff (GM-D/CGm-g),

-a +a

(b)

Solid Angle'&oCf

(C)

Figure 1. Schematic representation of array and its far-field gain and power patterns. (a) Array of colinear dipoles. (b) Far-field power pattern. (c) Curve of gain versus solid angle.

where G is the maximum gain. This last inequality implies that for

ft •*• 0, the maximum gain G must approach unity.

To obtain an indication of what can be achieved practically with a

moderately short antenna (1 wavelength), this report considers the syn-

thesis of linear antenna arrays made of dipoles aligned in some direc-

tion, Ox, as shown in Figure la. The resulting pattern has nulls in the

directions +x. The problem is to reduce the cones where the gain is

below a certain level. A general method based on the concept of regular-

(2 3) ization ' was developed in the Antenna Laboratory under NSF Grant

GK 2120. It will be applied to the problem.

2. STATEMENT OF THE PROBLEM

The general case of a pattern function gn(5) defined over the vis-

ible region[-k ^ { _< k] will be first considered, and the numerical

results will be specialized to the case

g0(C) = 1 [-kiC£k] (1)

where the gain is equal to one over the whole visible region. A source

distribution f(x) representing an array of N elements over the aperture

[-a <_ x _< a] is of the form

f(x) = I f 6(x-x ). (2) i n n n=l

The problem is to determine the {f } such that the pattern function

of the array comes close to g_(£).

The far-field pattern of a linearly polarized source f(x) is given

by

3 i£x g<0 = Tf = e<0 / f(x)eli;x dx [-k<5<k] (3) -a

where C = k sin 9. The first factor e(0 is the element pattern.

For a source distribution described in Equation (2), the far-field

pattern is given by,

N g(0 - Tf - I f E(C) (4)

n-1 n n

where

gn(C) = e(OeUXn [-k<?< k]

6

The function g(£), therefore, belongs to a finite dimensional vector

N space 0 spanned by the N elements g (£) (1 <_ n <_ N) . In general, gn(S)

does not belong to this subspace and the problem is to determine the

complex coefficients {f } such that the pattern given by (4) comes close

to g0(O-

In one method proposed in the next Chapter, the coefficients {f }

are determined such that the corresponding pattern g(0 is the orthogonal

N projection of gn(C) on the finite dimensional subspace $ • This gives

the least-squares approximation to gn(£)«

This straightforward projection, however, may result in a solution

with a high Q factor, that is, excitation currents {f } that are very

large for a given power radiated. These currents vary widely from ele-

ment to element, and the resulting far-field pattern becomes very sen-

sitive to errors in construction.

(2) To avoid this difficulty in problems of the same type, Tihonov

has proposed a method of regularization which consists of imposing some

restrictions on the acceptable solutions, that is,on the aperture functions f.

The regularization algorithm depends on a parameter a such that for

N a = 0, we have the direct projection on the subspace t . As a increases,

a set of solutions is obtained with decreasing values of the Q factor.

This set of solutions invariably produces far-field patterns that are

further away from the desired pattern but are also less subject to

ohmic losses and less sensitive to errors in construction.

To give a precise meaning to the concept of projection and of Q

factor, one has to introduce appropriate norms in the space Crof source

functions f • {f } and the space CXof far-field patterns g(5)« In both

spaces, it is natural to choose the L„ norm which can be interpreted

in the space Q as the power radiated by the antenna. Thus, 'J 1,2 N *

f - 2 f f (5) 11 n n n n=l

and

+k ||g||2 = \ ; g*(0 g(0 de. (6)

K -k

The Q factor for a given design is represented by

Q- llfll2 / llgll2. (7)

Because of the element pattern, this factor is not always larger than

one.

3. METHOD OF SOLUTION

Tihonov's procedure consists of minimizing the functional

Ja(f;gj = l|g-g0ll2 + a||f||2

. k N N - = k i I \ fnV*ol d? + a Z lfJ (8)

-k n=l n=l

where a is a positive constant referred to as the regularization param-

eter.

By setting the first variation of J (f;g) with respect to f - {f }

equal to zero, the following equation for f • {f }, the minimizing set n

of excitations, is obtained,

\ f. £ ' «nCd5 + afa =I^0'^d^ (1 < n, m < N) n=l n -k m -k

where, since g. is symmetrical, the excitations {f } are assumed real,

In matrix notation,

(G+aI)fa = GQg0 (9)

where G is an (N x N) square matrix with terms G given by

l k * i^^xr,~xm^ G =£ f e(^)e(C)e n m dCt, (10) mn k . -k

and I is the identity matrix of dimension N.

The column matrix gQ = {gQ } (1 <_ j _<• M) represents the sam- j

pled values of the function g at the M equally spaced points over the

9 visible region [-k _< £ <_ k], and G_ is a (N x M) quadrature matrix derived

to perform numerically the integral

-k

for any given function gn(£).

Equation (10) shows that the G are the Fourier Transforms H(x) of n mn 2

h(0 • |e(£)| evaluated for x - x .

G = H(x -x ). mn n m

When the elements are dipoles in the x direction,

Jl 1/2 e(0 = cos 6 = (1 - \ ) (11)

k

and

J~/9(kx) H(x) = 2/(2-) ' 3/2

(kx)

4 r sin(kx) ,, . , ,. „. 9 I (C\ cos(kx)]. (12) (kxr K '

To derive the elements Gn of the Quadrature matrix G_, the func-

tion gn is sampled at equally spaced points over the visible region

[k <_ £ <_ k] and approximated by straight lines over each interval. For

the range,

5J-I -5 - 'i

g0(O "£ [gjU-5;1_1)-gJ_1(5-eJ)]

10 where h is the length of each interval. For the range,

'i -5 - Vi

!0(« =h tSj+l^-^'-Sj^-^+l^

and the terms G-. are given by ij

1 1 5j £2 1/2 H%±

j+1 £2 1/2 Uxi -/ (£-£ Jd-ijre \u>. 6jj k

Since these integrals cannot be evaluated in closed form, they were eval-

uated by Simpson's three-point quadrature. From (9),

fa = (G+al)"1 GQg0. (13)

Once a choice for a is made, Equation (13) gives a unique solution

f for a given g_. This solution f has a far-field pattern g , given a 0 a a

by

g = Tf (14) a a

and a corresponding Q factor. The function f has the property that,

for this value of Q,its far-field pattern g yields the least-squares

fit to gQ.

4. NUMERICAL RESULTS 11

In order to study the relationship of a to the sensitivity of the

design to errors, Equation (13) was solved for a number of values of a

ranging from 0 to 10

The magnitudes of the element excitations for arrays of five and

seven elements are reported in Tables I and II,respectively, for various

values of the regularization parameter. The element pattern e(£) is

given in Equation (11).

TABLE I. 'MAGNITUDE OF ELEMENT EXCITATIONS FOR THE FIVE-ELEMENT ARRAY.

Element Number

a = 0 a = 10 -3

a = 10 -2

a = 10 -1

1 .64 .35

2 -2.25 -1.47

3 4.28 3.26

4 -2.25 -1.47

5 ,64 .35

-.15 -.28

-.11 .335

1.47 .86

-.11 .336

-.15 -.28

The far-field pattern g corresponding to each solution f is eval-

uated using Equation (4). Since it is assumed that only power

i i2 P = is available to the array for radiation, the excitations

{f } are normalized such that a n

Let f = {f } and g be the normalized sources and far-field patterns, a a a

n 2 respectively. The relative error e due to regularization is expressed

_ o by the norm of the error IIg -g_I " divided by that of the desired

' • a U '

12

TABLE II. MAGNITUDE OF ELEMENT EXCITATION FOR THE SEVEN-ELEMENT ARRAY.

Element Number

a - 0 a = 10~6 a = 10"5 a - 10~4

1 -6.71 -1.58 .57 .87

2 35.07 10.19 -.29 -1.80

3 -81.46 -26.53 -3.36 .07

4 107.20 36.88 7.22 2.79

5 -81.46 -26.53 -3.36 .07

6 35.07 10.19 -.29 -1.80

7 -6.71 -1.58 .57 .87

Element Number

a = 10 -2

a - 10 -1

1

2

3

4

5

6

7

.05

-.55

.43

1.08

.43

.55

.05

-.23

-.01

.41

.62

.41

-.01

-.23

pattern ||g.

£a = IIIa-S0M2 ' ll80112

1 r r I2 AC I I I k_£ lg-gol d5 / Hgol (15)

13

Next, in order to study the sensitivity of each design to errors in

construction, a random relative error _v is introduced to the source func-

tion, and these excitations with noise are denoted by f and given by n

f - f (1+v ) (16) av a n

n n

where v = {v } are random complex numbers with a Gaussian distribution — n

E(v ) = 0 n

* 2 E(v v ) = v 6

n m nm

The far-field pattern corresponding to sources {f } is denoted by g av °av

n 2 2 The excitations {f } are normalized such that llg I = IIg_II .

av '' av" '' 0'' n

Let f = {f } and g be the normalized sources and far-field pattern, av av av

n 2 respectively. The relative error e , due to the intro4uction of random

noise, is expressed by the norm of the error llg -g " divided by that

of the desired pattern.

2 , i- - i , 2 i,2 ev = I 18av-*aI I I I Ig01 I , (17)

and the corresponding relative errors due to regularization and the intro-

^2 duction of random noise are expressed by the norm of the error ||g -g.||

2 divided by ||gn|| :

2 IT II2 / i I II2

et • Il8ov-80M / I 1801 I •

For each design, the Q-factor is calculated with the presence of random

errors and denoted by Q , where

Q - II* II2 / Mi.JI2- US) 1L oiv ' ' ' ' av

14

In each instance, 25 sets of random errors were introduced and the aver-

2 2 age of the quantities e , e , and (1/Q ) were calculated over the 25 trials,

where

2 2 <e > v. av

~2 et =

<c2> t av

1/Qv- <1/Q > xv av

(19)

The quantities e , e , and e are illustrated schematically in Figure 2.

In this diagram, vectors are understood to represent functions and straight

lines, distances in the sense of the L„ norm. The curved lines represent

operators mapping the space of source functions into the space of far-

field pattern functions. The three vectors g», g , and g are drawn 0 o av

with their tips on the same circle to indicate that their L_ norms are

equal.

- o i— 2 The far-field power patterns ||g II and g "of the arrays for

a ' av' '

various values of the regularization parameter a are shown in Figures 3 to 11.

Also in the same figures, the power pattern intensity of an elementary

dipole is shown for comparison. The horizontal line drawn for a radia-

tion intensity = 1 is that of an ideal isotropic radiator. It is

pointed out again that all pattern functions are normalized such that

their L„ norms are equal. This makes their radiated power, represented

by the area under the curve, equal in all cases. In Figures 12 and 13,

2 2 ~2 curves for the quantities e , e , e and Q are shown as a function of the

a v t

regularization parameter a for the arrays of five elements and seven ele-

ments, respectively.

15

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Number of elements:5 Aperture size : 1 X

10 10 10 10

REGULARIZATION a

2 2 ~2 Figure 12. Error parameters e , ev, and e , and Q vs. regularization parameter

a for the five-element array.

26

Number of elements:7

10 10 10 10 10

REGULARIZATION a

10 10

2 2 ~~2 Figure 13. Error parameters e , ev, and e , and Q vs. regularization parameter

a for the seven-element array.

27

From the far-field power pattern intensity |g | " in Figures 3 to 11,

curves for the gain vs. solid angle, as described in the introduction,

were derived. These curves are shown in Figures 14 and 15 for the five-

-2 and seven-element array, respectively, for the cases a = 0 and a • 10

The gain curve for an elementary dipole is also shown for comparison.

For example, in Figure 14, for a directive gain level of 0.8, the total

solid angle in which the gain of each of the cases considered exceeds

the 0.8 gain level is as follows:

Ideal isotropic radiator

Five element array (a • 0)

Five element array (a = 10 *)

Elementary dipole

,-2,

4 TT

3.82TT

3.36IT

2.72IT.

A plot of 1/Q vs. 1/Q is shown in Figure 16.

28

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•H

If) I,

ro/i)

31

5. COMMENTS ON THE NUMERICAL RESULTS AND THE CHOICE OF a.

The first observation is that using an array does,indeed, improve

the low gain performance. The gain curves in Figures 14 and 15 for

a = 0 come closest to that of an ideal isotropic radiator among all

designs considered. Comparing the gain curves of the five-element to

the seven-element array, only a slight improvement in low-gain perform-

ance is observed in the latter. The aperture size in both cases, how-

ever, is the same.

The performance of those designs corresponding to relatively small

values of a looks, indeed, promising. However, in the presence of random

errors in the currents, the performance of these designs deteriorates

drastically, as shown in Figures 3 to 4 and 6 to 9. In such instances,

a single element has a better performance than the array!

In deciding which value of a to use, it becomes imperative to study

2 2 the behavior of the quantities z , e , and Q. For a constant power radi-

ated, the Q-factor is directly proportional to ohmic losses while the

2 error e is an indication of the sensitivity of the design to error.

2 In Figures 12 and 13, as a approaches zero, e decreases, and both Q and

2 e increase. For the larger values of a, these trends are reversed. v

2 2 The design then must be a compromise between e , e , and Q. A good com-

—2 promise is that value of a which minimizes the total error e . For the

_3 five-element array, this corresponds to a = 3.54 x 10 , Q = 2.25, and

2 -3 e = .008 and, for the seven-element array, to a = 1.6 x 10 , Q = 4.0,

2 and e • .0045. For these particular values of a, the seven-element

v

array has larger ohmic losses, but is less sensitive to errors than the

five-element array.

32

It is observed in Figure 16 that the introduction of random errors

in the element excitations produces a decrease of the Q-factor, that

is, an increase of the power radiated for a given norm of the sources.

For values of Q less than 10, the reduction is very small. For values

of Q larger than 10, the reduction becomes appreciable,and for values

4 of Q larger than 10 Q ,becomes constant, equal to 270.

33

6. STATISTICAL ESTIMATION OF 1/Q AND e2

2 The numerical determination of 1/Q and e , as outlined in the pre-

v v

vious section, can be time consuming since the averages have to be taken

over a large number of trials. It would be advantageous, therefore, if

2 the quantities 1/Q and e could be estimated analytically. From the

expression for f in Equation (16) and the definition of the L„ norm av 2

in the space of aperture source functions given in Equation (5), the L

norm of f is evaluated: av

I I f I [2 = 2 f f * (1+v ) (1+v *), II av1* , a a . n n n=l n n

and the expected value of this quantity is found to be

i, ,2 ii- ,.2 2 E{ f = f (1+v )

av ' ' ' ' a

where,for v << 1

E{||f ||2} = ||f ||2. (20)

The far-field pattern g , corresponding to an aperture source distribu-

tion f ,is given by Equation (4):

n=l n

From Equations (6) and (10), the L norm of g is evaluated,

9 N N - *- * g |r= E E 6 f f (1+v )(l+v ), 6av'' n , nm a a n m ' n=l m=l n m

34

and the expectation is found to be

E<lkJ|2>-||I||2 + v2G„n||fj|2. (21) 'av'1 '|ea'' 00

The expected value of 1/Q is defined to be

E{l/Q } = E{||g ||2 / ||f ||2}. ^v ' '°av • ' ' ' av' '

However, Equation (20) shows that when the mean square level of error

v is small, the expected value ||f || may be approximated by ||f ||

and the following relation for 1/Q is obtained from Equation (21):

1/Qv = E{l/Qv) =1/Q + GQOv2. (22)

2 Before computing the relative error e due to random errors, g

should be normalized so that

E<HgJI2>-lliJI2.

From Equation (21), the normalization factor F is given by

F = [l+v2G00Q]"1/2 = [Q.E { -^ } ] 1/2. (23) v_

Let g denote the normalized far-field pattern of the aperture sources av

in the presence of random error.

g = F g . av av

2 The relative error e is given by

E2 = EU2} = E{||I -i ||2} / ||g ||2 v v ' ' av "a" ' ' °a' '

35

HM^o

Figure 17. Schematic representation of the relationship between the pattern functions and the error parameters.

36

o-

en

s >

U

c

9 id u

M o QJ

4J u <u H-l

j= i

4-1 OJ o x. u u o •

!-H en PM >

oo

a 00

•H

37

where

N g " g = I f g [F(l+v )-l], av a n a n n n=l n

Taking the norm and then the expectation, the following relationship

is obtained for e : v

e* = 2(1-F). (24)

2 This equation relates e simply to the drop in the Q-factor when random

errors are introduced in the sources.

Equation (24) can be rederived from the schematic representation

in Figure 17. Equation (21) shows that the quantities |g ||, v||f | | /G ~,

— i2 1/2 and [E{||g | }] are related by the Pythagorean theorem. Therefore,

the tangent of the angle 9 shown in Figure 17 is given by

tan 6 = v/QGQ0 (25)

and

e = 2 sin f , (26) v 2

which can be rewritten in a form similar to that of Equation (24):

e2 = 2(l-cos 9). v

2 Plots of 1/Q and e as a function of Q calculated from Equations

(22) arid (24) are shown in Figure 18. The values of these quantities

evaluated numerically are also shown for comparison. The agreement is

very good.

38

7. CONCLUSION

A method has been outlined to design a linear array whose gain is

close to unity and, therefore, whose pattern is nearly isotropic. Tihonov's

regularization algorithm is used to produce a family of designs which

depend on a parameter a.

2 It is found that as a approaches 0, the error e from the actual to

2 the desired pattern decreases. But the norm ||f| " of the aperture

function, which for a unit power radiated is also the supergain ratio

Q introduced by Taylor, increases. This is accompanied by an increase

2 in the sensitivity of the design to errors represented by e . An accept-

able design has to be a compromise between these three quantities.

Although the sensitivity to error varies in the same direction as

Q, it is not directly proportional to it. A general relation has been

established between these two quantities. This relation has been veri-

fied numerically and it provides a useful guide for choosing the amount

of regularization.

The regularization method proves to be effective in designing nearly

isotropic arrays of reasonable sizes.

39

REFERENCES

W. G. Scott and R. M. SooHoo, "A theorem on the polarization of null-free antennas," IEEE Trans. Antennas and Propagation, Vol. AP-14, pp. 587-590, September 1966.

A N. Tihonov, "Solution of incorrectly formulated problems and the regularization method," Soviet Math., Dokl. 4, pp. 1035-1038, 1964.

H. S. Cabayan, G. A. Deschamps, and P. E. Mayes, "Stable numerical solutions to antenna synthesis problems," URSI 1969 fall meeting, pp. 38-39.

UNCLASSIFIED Security Classification

DOCUMENT CONTROL DATA - R&D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified)

I. ORIGINATING ACTIVITY (Corporate author) Antenna Laboratory, Department of Electrical Engineering, Univ. of Illinois under Purchase Order No. CC-401 to M.I.T. Lincoln Laboratory

2a. REPORT SECURITY CLASSIFICATION

Unclassified 20. GROUP

None 1 REPORT TITLE

LINEAELY POLARIZED ARRAYS WITH ALMOST ISOTROPIC RADIATION PATTERNS

4. DESCRIPTIVE NOTES (Type of report and inclusive dates)

Technical Report S. AUTHOR/SI (Last name, first name, initial)

H. S. Cabayan G. A. Deschamps

TK NO. OF REFS

1 6. REPORT DATE

TiinP 1Q70

7a TOTAL NO. OF PAGES

45 ea CONTRACT OR GRANT NO.

AF 19(628)-5167 6. PROJECT AND TASK NO. 62 7A

Purchase Order No. CC-401

SO. ORIGINATOR'S REPORT NUMBERS)

Antenna Laboratory Report No. 70-10

si. OTHER JIEPORT HO(S)(Any other numbers that may be assigned tfus report)

ESD-TR-70-195 UILU-ENG-70-308

10. AVAILABILITY/LIMITATION NOTICES

This document has been approved for public release and sale; its distribution is unlimited. II. SUPPLEMENTARY NOTES

This work was partially supported by NSF Grant GK 2120

12 SPONSORING MILITARY ACTIVITY

Air Force Systems Command, USAF

13. ABSTRACT

A linearly polarized antenna cannot radiate power uniformly in all directions. However, by controlling the aperture excitation, as is done in an array, it is possible to reduce the maximum gain and approach con- ditions for isotropic radiation. A numerical method is presented which generates a family of designs which depend on a parameter a. As a approaches zero, the radiation pattern tends to become more isotropic. However, the efficiency is reduced and the sensitivity of the pattern to errors in the aperture function is increased. In some cases this sensi- tivity is so high as to make the result worthless. The design, therefor* , must be a compromise between closeness to conditions of isotropic radia- tion on one hand, and losses and sensitivity to errors on the other. The sensitivity to errors, expressed by the pattern deterioration for a givei i level of error, has been evaluated by a numerical experiment (Monte Carl method). A general relation has also been established between the sensi- tivity and the losses in the antenna. It agrees with the numerical exper- iment and can be used as a guide to choose the regularization parameter

14. Key Words Unity Gain Antenna Isotropic Radiation

Linearly Polarized Arrays Regularization

Sensitivity to Error Synthesis Arrays

DO ronu 1473 """ I JAN •«

UNCLASSIFIED Security Classification