Optimization of some aperture antenna performance indiceswith and without pattern constraintsCitation for published version (APA):Worm, S. C. J. (1980). Optimization of some aperture antenna performance indices with and without patternconstraints. (EUT report. E, Fac. of Electrical Engineering; Vol. 80-E-112). Technische Hogeschool Eindhoven.

Document status and date:Published: 01/01/1980

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Download date: 27. Apr. 2022

Optimization of some aperture antenna

performance indices with and without

pattern contraints

by

S.C.J.Worm

• ....

E I N D H 0 V E NUN I V E R SIT Y 0 F T E C H N 0 LOG Y

Department of Electrical Engineering

Eindhoven The Netherlands

OPTIMIZATION OF SOME APERTURE ANTENNA

PERFORMANCE INDICES WITH AND WITHOUT

PATTERN CONSTRAINTS

by

S.C.J. Worm

TH-Report 80-E-112

ISBN 90-6144-112-9

Eindhoven

August 1980

Contents

Abstract.

Acknowledgements.

I. Introduction.

-0.1-

2. Fields and related parameters.

3. The optimization procedure.

4. Some examples of circular and annular apertures.

4. I. The circular aperture.

4.1.1. Aperture fields with f (I) z 0 and f' (I)

4. I. 2. Aperture fields with f (I) i' 0 and f' (I)

4.1.3. Aperture fields with f(l) z 0 and f'(I)

4.2. The annular aperture.

4.2.1. Zero-edge aperture distributions.

References.

Appendix.

Figures.

i' o. z o.

o.

0.2

0.3

1.1

2. I

3. I

4. I

4.2

4.2

4. IS

4.19

4.27

4.27

RI

Al

AS

-0.2-

ABSTRACT

For circular and annular apertures an investigation is carried out of

the optimization of some antenna performance indices. These are the

well-known efficiency n, the less well-known spread of radiated power

02 and a new index, namely the ratio n/0 2. For the purposes of this

investigation some new Bessel-function integrals have been evaluated.

The performance indices are written as ratios of two Hermitian quadratic

forms. The relative extremes of these ratios are the roots of characteristic

equations. Optimizations have been carried out for several types of

aperture distribution. Analytical, numerical and graphical results are

given for the optimization of unconstrained radiation patterns. Limit

. values of the indi~es, corresponding modal excitation coefficients,

aperture fields and far fields are determined.

Furthermore, computations are made when pattern constraints are introduced.

These constraints are introduced in such a way that the indices remain

ratios of two Hermitian quadratic forms.

The constrained optimizations result in normal pencil beams, flat-top

beams or patterns with constant 3dB beam width while, at the same time,

the side lobe extremes are prescribed and a parameter is optimized. Results

of these constrained optimizations are given in tables and graphs.

Worm, S.C.J. OPTIMIZATION OF SOME APERTURE ANTENNA PERFORMANCE INDICES WITH AND WITHOUT PATTERN CONSTRAINTS. Department of Electrical Engineering, Eindhoven University of Technology, 1980. TH-Report 80-E-112

Address of the author:

ir. S.C . .). Worm, Group Electromagnetism and Circuit Theory, Department of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB EINDHOVEN, The Netherlands

-0.3-

ACKNOWLEDGEMENTS

The author wishes to express his thanks to the following persons:

Dr. M. Jeuken and Dr. V. Vokurka of the Department of Electrical

Engineering for stimulating discussions,

Prof.Dr. J. Boersma and drs. P. de Doelder of the Department of

Mathematics for the evaluation of some Bessel-function integrals.

-1. 1-

I. INTRODUCTION

In practical antenna applications there can be a need to tailor a

radiation pattern to a specified shape and to improve a performance

index of the antenna system at the same time. In this way the influence

of unwanted signal sources can be reduced to an acceptable level or

totally eliminated with the smallest possible index deterioration. This

differs from synthesis techniques in which only the pattern shape is

modified [12], [13], [20], [26], or from optimization techniques without

pattern constraints [3], [18], [22].

Many performance indices of antennas can be expressed as ratios of two

Hermitian quadratic forms. Commonly used indices are the specific

directivity or efficiency and the ratio of the directivity to the

antenna noise temperature. The relative extremes of such a ratio of

quadratic forms are the roots of a characteristic equation [lOa].

The pattern constraints can be applied to sidelobe peak levels [23], [24],

directions of pattern nulls or sidelobes [9] and to the main-lobe beam

width [15].

Techniques for index optimization with pattern constraints are the method

of Lagrange multipliers, which results in rather complicated numerical

procedures, and a matrix method in which the dimensions of vectors and

matrices reduce, depending on the number of constraints. This matrix

method has been used for the maximization of the directive gain of an

aerial array while specifying the directions of sidelobes and/or pattern

nulls [9]. However, the method is applicable whenever a performance index

can be written as a ratio of two Hermitian quadratic forms or a product

of two independent ratios. For instance the constrained efficiency optimi­

zation of a rectangular aperture results in the independent efficiency

optimization in the two principal planes if the aperture distribution

consists of the product of two independent distributions. Furthermore,

the method is not restricted to direction constraints but can be applied

to level constraints as well.

In this report the above-mentioned matrix method is used for circular and

annular apertures with several types of aperture distribution. The optimized

performance indices are the efficiency n, the spread of radiated power a2

2 and a new performance index n/a .

-1.2-

In section 2 the indices are written as ratios of two Hermitian

quadratic forms. The variables are the modal excitation coefficients

of the aperture field expansions. The elements of the Hermitian matrices

depend on the index to be optimized and the type of aperture distribution.

In section 3 the optimization theorem for a ratio of two Hermitian

quadratic forms is outlined and pattern constraints are formally introduced.

The theorem provides the optimum value of an index and the corresponding

modal excitation coefficients.

In section 4 the aperture distributions are described and the elements of

the Hermitian matrices are determined. Analytical and numerical optimizations

are then carried out. Results are given in formulas, tables and graphs. At

the end of the subsections summarizing conclusions are drawn.

In the appendix, results are given of the evaluation of some Bessel-function

integrals. A full treatment can be found in [2], in which the private

communications [I], [7] are brought together.

-2.1-

2. FIELDS AND RELATED PARAMETERS

In this section we define fields and parameters relevant to our

optimization problems.

Assume a circular or annular aperture with outer diameter D. The

aperture points can be described with polar coordinates p and $, with

o :: p :: D/2 and 0 :: $ < 2lT. Instead of p we will use the normalized

variable r = 2p/D.

We restrict ourselves to $-independent aperture distributions fer) which

can be written as series of N terms as follows,

N fer) = L a e if r

l < r < 1 ,

n=1 n n

(2. 1 )

= 0 if r > I.

For circular apertures rl

= 0 and for annular apertures 0 < rl

< I. The

excitation coefficients a and the elementary functions e are taken to n n

be real. In section 4 we will introduce functions en such that gn(u),

eq. (2.2a), exists.

The far field g(u) and the partial far field g (u) are taken as n

1 g(u) = f fer) J (ur)rdr

o

and

g (u) n

1 f e J (ur)rdr

n 0 r

l

(2.2)

(2.2a)

In these expressions

zeroth order and u

J is the Bessel function of the first kind and o

sin(8)lTD/A , (A is the wavelength and 8 is the angle o 0

with the main beam axis).

The far-field power pattern p(u) is

p(u) = g2(u).

The power Pr radiated by the aperture is

1 2 Pr f f (r)rdr.

rl

(2.3)

(2.4)

-2.2-

The second moment m2

of the far-field radiated power with respect to 2 the axis u = 0 is found by integrating u p(u). This results in

1TD IA 2 m

2 = J u p(u)udu

o

1TD/A 3 J u p(u)du. o

(2.5)

In accordance with [16] the upper limit of the integral will be replaced

by 00 if the aperture illumination is continuous and approaches zero at

the edges. This means that the reactive power is neglected and that the

radiated power is set equal to the total power.

The performance indices which we want to optimize separately are the

efficiency n, the normalized second moment a2 and the efficiency power­

spread ratio nlaZ defined respectively by

n = 2 p(O)/Pr (Z.6)

Z m/Pr a = , (Z.7)

2 2p(O)/mz· nla = (2.8)

The formula for n can be deduced from the expression for the maximum gain

GM

= nG with G the maximum gain of the uniformly illuminated aperture o 0

[ Z5].

The normalized second moment of the far-field radiated power with respect

to the beam axis is a measure of the spread of radiated power from the

same axis.

We redefine the equations (Z.I) - (2.8) in terms of vectors and matrices

in order to determine the optimum values of the performance indices

and the related excitation coefficients. The optimum value of a performance

index is an eigenvalue of an optimization problem and the excitation coeffi­

cients are the elements of the corresponding eigenvector.

The aperture field can be written as an inner product

f(r) = <a,e> if rl

< r < I , (Z.9)

0 if r > I ,

with e> aN element column vector having elements e n' and with <a a N

element row vector having elements a n

The far field is

g(u) = <a, T (e» I

-2.3-

(2.10)

with TI(e» = TI(e» a N element column vector with elements TI(en

) when

T I is defined by

I T f = f fJ (ur)rdr.

I 0 r

l

The far-field power pattern is

<a,V a>,

with V a NxN matrix wit: elements

V .. 1J

(2.1 I)

(2.12)

(2.13)

whj C~l are functions of u. Expressions like <a, Va> are quadratic forms in

the variables a • n

The total power radiated by the aperture is

The

The

with

p = <a,A a>. r

NxN matrix A has

I A .• = f e.e.rdr.

1J 1 J rl

second moment is

m2

= <a,W a>,

elements

W a NxN matrix with elements

00

J 3 w .. u V •• du. 1J

0 1J

The efficiency expressed in vectors and matrices is

n = 2<a,V(o)a> <a,Aa>

(2.14)

(2.15)

(2.16)

(2.17)

(2.18)

-2.4-

The normalized second moment of the far-field radiated power with

respect to the axis u = 0 is

<a,Wa> <a,Aa>

(2.19)

The combined performance index of eq. (2.8) is

2 n/o 2<a,V(o)a>

<a,Wa> (2.20)

In this way we have expressed each of the performance indices as a ratio

of two quadratic forms. The real matrices v, A and Ware Hermitian because

v .. = V .. , A.. = A.. and 1J J1 1J J1

equations (2.13), (2.15)

W •• 1J

and

= W .. , which 1S immediately evident from the J1

(2. 17). If the denominator, <a, A a> or <a, Wa>,

is positive definite, which means that the denominator is greater than 0

for any a 1 0, the optimization is easily performed by making use of a

theorem on the properties of a function of a vector [lOa]. This theorem

is given in the next section.

"._'"

-3.1-

3. THE OPTIMIZATION PROCEDURE

Assume that we want to optimize the performance index PI given by

PI <8, H a> <a, L a> '

(3. I)

in which <a is a real N element row vector, a> is a real N element column

vector and Hand L are NxN real square matrices. The optimization theorem

[lOa] states that with Hand L Hermitian and L positive definite, all the

relative maxima and minima of the performance index are given by the

eigenvalues determined from

H a> = A L a> , (3.2)

where A denotes the eigenvalues which must satisfy

det(H-AL) = O. (3.3)

Equation (3.3) is a characteristic equation with N roots AI ~ A2 ~ ..• ~ AN

which are all real. The performance index is bounded by the smallest and

the largest of these roots,

a> satisfies eq. (3.2) with

AI' A bound is achieved when the vector

or A = AI' The restriction to real

vectors and matrices is not necessary for the theorem.

If ln addition H is a one-term dyad which means that it can be expressed

as "he product of a Nxl vector h> and a IxN vector <h, then there is only

one nonzero eigenvalue. This is the maximum value of PI and is equal to [5]

-I AI = <h, L h>. (3.4)

The corresponding eigenvector is

a> L-Ih> (3.5)

In the case of optimization of eq. (2.18) and (2.20) H is of the above­

mentioned form,

H = 2

-3.2-

Another special case occurs when both Hand L are diagonal matrices. Then

the eigenvalues are derived from eq. (3.3) as

A. = H .. /L .. 1 11 11

i = 1,2, ... ,N, (3.6)

and the corresponding eigenvectors are (O ... Oa.O ... O) with a. = I. This 1 1

type of solution occurs when the normalized second moment is optimized

(without constraints) for a circular aperture with a special type of

aperture expansion function e as will be outlined in section 4. n

First, however, we will show the technique which we have employed when

there is a constrained optimization problem. The problem of optimizing

a quantity subject to constraints can in general be approached by the method

of Lagrange multipliers [17], [24]. This technique is characterized by the

fact that as the number of applied constraints increases, so also does the

number of manipulations required for the solution. The existence of

constraints, however, reduces the number of independent variables which can

be adjusted for the optimization. Therefore a method which reduces the

vector and matrix dimensions by the number of applied constraints is more

appropriate for our optimization problems.

The constraints on the far field in M points V can be expressed as m

m 1,2, .•. , M, (3.7)

if 1 is the prescribed relative value in V . In a shortened notation m m

eq. (3.7) becomes

<a, q > = 0 , m

m=I,2, ... ,M, (3.8)

with q > the independent constraint vectors. The N dimensional vector <a m

is, according to eq. (3.8), required to be orthogonal to the M dimensional

subspace of the constraint vectors. So the required vector <a lies in a

subspace of dimension N-M and has at least M elements equal to zero if an

appropriate set of spanning vectors can be constructed. This is possible

with the aid of the Gram-Schmidt procedure [6], [9], [II]. Form a NxN

constraint matrix C having as the first M rows the vectors <~ and in the

remaining N-M rows any arbitrary collection of N element independent vectors.

From C we construct with the Gram-Schmidt procedure a matrix C orthogonal o

-3.3-

by rows and columns, i.e. -,

C ,where T means transposition. The o

matrix C o

is the transformation matrix. Applied to a> satisfying eq. (3.8)

it generates a N element vector a > which has zeros as the first M elements c

because each of the first M rows of C is a linear combination of the o

constraint vectors <q . The vector a > is m c

a > C a>. c 0

The first element of a > is c

The first row ~f Co is <c, = «q,)/<q"q,>!

<q"a>/<q"q,>'. The second row of Co is

<c = {«q ) - «q ,c »«c )}/{<q ,q > - <q2,c,>2}! and the second element 2 2 2' , 22

of a > is the inner product of a> and c

matrix C o

the first

are deduced in this way and

M elements of a > are zero. c

<c2

. Expressions

if the vector a>

for the rows <c. of 1

satisfies eq. (3.8)

Rewriting the performance index of eq. (3.') we get a ratio of two quadratic

forms in which the constraints are incorporated as follows

PI = <a,H a>

<a,L a>

<a,

<a,

<C a, C H CT C a>

o 0 0 0

<C a, C L CT C a> o 0 0 0

<a ,H a > c c c

<a ,L a > c c c

where H c

T = C H C and L

o 0 c a

c are zero one can reduce

C L o

the

(3.9)

T C . As the first M elements of the vector

o dimensions of a to N-M and of Hand L

c c c to (N-M)x(N-M)

(3.9) becomes

resulting in a vector ad and matrices Hd and Ld

. Equation

<ad' Hd ad> PI = -:--=--"7''---=:-­

<ad' Ld ad> (3.'0)

The problem of optimizing eq. (3.') with the constraints of eq. (3.8) is

transformed into the problem of optimizing the unconstrained eq. (3.9).

-3.4-

It can be shown [9] that if H is Hermitian Hc and Hd are also Hermitian.

The same applies to L, Lc and Ld . The quadratic form <a,L a> is positive

definite as it represents the radiated power or the spread of the radiated

power. The quadratic forms associated with Lc and Ld represent the

radiated power or its spread in the constrained case, so that they must

also be positive definite. The optimization of eq. (3.10) is similar to

that of eq. (3.1) except for the reduced dimensions. The relative maxima

and minima of eq. (3.10) are given by the eigenvalues determined from

(3.11 )

with AC the constrained eigenvalues satisfying

(3.12)

c c c Eq. (3.12) has F-M roots Al : A2 : ... : AN_M which are all real. Ordered

according to numerical size as with the unconstrained eigenvalues it can

be shown that A~ lies between A. and A. M [11, p.174]. This implies that 1. 1. 1. +

the constrained minimum is always equal to or greater than the unconstrained

minimum and that the constrained maximum is always equal to or less than

the unconstrained maximum.

If H is a one-term dyad it can be shown [9] that Hc and Hd are also one­

term dyads, namely

and

with

H c

h > <h , c c

h > <h d d

h > = C h> c 0

(3.13)

(3.14)

(3.15)

and with hd> formed from hc > by deleting the first M entries. In this case

the constrained eigenvalue and the constrained eigenvector are respectively

AC <hd

, -1 hd> (3.16) 1 Ld ,

and

ad> = -1

Ld hd> (3.17)

-3.5-

After ad> has been determined from eq. (3.11) or (3.17) one can form the

vector ac

> by adding M zeros to ad>' The vector a> can then be computed

with

a> a >. c

The same result is obtained if

formed by deleting the first M

0.18)

ad> is multiplied by the matrix which is T

columns of C . o

In this section we have given the formulas for the (constrained)

optimization, for instance of the normalized second moment of the far­

field radiated power, the efficiency of an antenna or the efficiency

power-spread ratio. In the next section we will be more specific in the

treatment of some examples.

-4.1-

4. SOME EXAMPLES OF CIRCULAR AND ANNULAR APERTURES

I . 4 . 11 .. h f . d' 2 d / 2 n sect~on we Wl optltnlze t e per ormance 1U lees n,o an n a

defined respectively by the equations (2.18), (2.19) and (2.20). We

will start with the circular aperture having successively three different

types of functions e . We will then treat the annular aperture with one n

type of functions e . n

It is found [16] that the aperture illumination function F(x,y) satisfying

the minimum condition for 02 is the lowest mode solution of the Helmholtz

equation for the given planar aperture,

a2F --+

ax2 a2

F 2 -+kF 0, (4.1) ai

2 where k 2 = 0 and F(x,y) is continuous and approaches zero at the closed

boundary. This illumination is called the optimum illumination for the

given aperture with regard to 02

F of the circular aperture, in polar coordinates defined by 0 < r < opt

and 0 < ~ < 2n, 1S the lowest mode of

00 00

F(r,~) = L [C sin n~ + D cos n~]J (k r), urn nm nnm

n=o m=o

with C and D constants and k nrn nrn urn

solution of J (k) n

annular aperture, in polar coordinates defined by 0 <

o < ~ < 2n, is the lowest mode of

00 00

(4.2)

= 0. F of the opt

r 1 ::: r < 1 and

F(r,~) = I L [C sin n~ + D cos n~][J (k r)-nrn nm n nrn n=o m=o

with k the solutions of nm

J (k) n

O. (4.4)

(4.3)

For our optimization (constrained or unconstrained) we use, for instance,

the elements of the ~-independent form'of eq. (4.2) in the circular aperture

case and of eq. (4.3) in the annular aperture case.

-4.2-

4. I. The circular aperture

As the first aperture illumination we take fer) consisting of the above­

mentioned elements of eq. (4.2). This can be represented by a series in

the following form

N fer) = L

n=1

o

a J (u r) non if 0< r < I,

if r > I,

with un the solutions of Jo(u) = 0, which means that u l = 2.4048,

u2

= 5.5201, etc. In Fig. I the lowest mode is shown.

o

Fig. I. Lowest mode of eq. (4.5).

The far field g(u), eq. (2.2) with eqs. (AI) and (AZ), is

N 2 2 g(u) = J (u) L au JI(u )/(u -u ) if u l' u ,

0 n=1 n n n n n

2 if !aJI(u) u u . n n n

The vector <e, eq. (2.9), can be written as

The vector <TI(e), eq. (2.10), is

2 2 2 2 <TI(e) = Jo(u) {u1J1(ul)/(ul-u ) ..• ~Jl(~)/(~-u )} .

(4.5)

(4.6)

(4.7)

(4.8)

-4.3-

The vector <TI{e(o)} needed for the optimization of nand n/a2 is

(4.9)

The elements of the matrix A, eq. (2.15), are computed with eqs. (AI)

and (A2) as

A .. = 0 1J

if i f j (4.10)

A .. jJ~(ui) 1J if i j.

2.4048, u2

= 5.520 etc., matrix A is a positive definite

diagonal matrix. The aperture radiated power is

N

~ L i= I

2 2 . a. J I (u.).

1 1 (4.11)

The elements of the matrix w, eq. (2.17), are

00 u3J 2(u) w .. = u.u·JI(u·)JI(u.) f 0 duo (4.12)

1J 1 J 1 J 2 2 2 2 0 (u.-u )(u.-u )

1 J

Evaluation of the integrals results in [I] , [2]

w .. 0 if i f j, 1J (4.13)

w .. !u~J~(u.) if i j. 1J 1 1

(See appendix A2) .

Matrix W is also a positive definite diagonal matrix. The second moment

of the far-field power pattern is N \' 222 m2 = ! L a.u.J I (u.) i= 1 1. 1. 1.

(4.14)

We are now able to derive some simple results for the unconstrained

performance indices. The unconstrained maximum value of n and the

corresponding eigenvector are determined with eqs. (3.4) and (3.5).

With N modes in the aperture field this results in

-2 -2 -2 4(u

l +u

2 + ••. +un ), (4.15)

<a (4. 16)

-4.4-

If N is allowed to become infinite, the value of ~,N becomes I,

[Z7,p.50Z], which corresponds to a uniformly illuminated aperture. The

elements of <a are, apart from a constant, equal to the coefficients of

expansion of a constant function C on 0 < r < I in a series of the form

[10,p.319] 00

C = L n=1

s J (u r) non

with u the solutions of J (u) = O. n 0

The coefficients s are [27, p. 580] n

l s = ZJI-Z(u ) J rCJ (u r)dr = 2C/{u JI(u )}. n non n n

o

The aperture illumination for optimum n is with N modes

N fer) = I J (u r)/{u JI(u )}

n=1 0 n n n

= 0

The corresponding far field is

PI (u )/u n n

The second moment is then

m = jN 2

and the normalized second moment is

N

= NO n=1

This means that

and

2 nMAX N • a ,

-Z -I u }

n

4N

if

if

if

if

0< r < I,

r> I.

u #-

u = u • n

(4.17)

(4.18)

(4.19)

(4.Z0)

(4.21)

(4.22)

(4.23)

(4.24)

-4.5-

As the number of modes grows, the aperture illumination tends to

uniform illumination with n = I having 02 = 00 if the upper limit

integration in eq. (4.12) is taken to be infinite. If, however,

the

of 2 • o 1S

calculated for the uniform illumination with the upper limit of integration

t = ITD/A it follows that

t 2 2 J uJ 1 (u)du (4.25)

o

With D/A 100 this is approximately equal to 200.

In Table 1 we give nMAX,N' the 3dB beam width u3dB ' 02

and the first 5

side lobe levels (denoted by sll) in dB for some values of N

N

2

3

4

5

6

7

00

2 s11 I s11 2 s11 3 s11 4 s11 5 nMAX,N u3dB 0

0.6917 4.15 5.7831 27.5 36.4 42.6 47.3 51.2

0.8229 3.62 9.7212 16. I 30.5 37.3 42.4 46.4

0.8764 3.47 13.6930 17.0 21.5 32.8 38.6 42.9

0.9051 3.41 17.6771 17.3 22.9 25.2 34.8 39.8

0.9231 3.37 21. 6668 17.4 23.3 26.7 27.9 36.5

0.9353 3.34 25.6597 17.4 23.5 27.2 29.6 30. I

0.9442 3.33 29.6544 17.5 23.6 27.4 30. I 31.8

3.20 17.6 23.8 28.0 31.1 33.6

Table I. Unconstrained optimum n, corresponding 3dB beam width,

02 and side lobe levels with N modes in the aperture

field of eq. (4.5).

Data in the first and last rows of Table I are reference data representing

the fields of the lowest mode Jo(ulr) and of the uniform illumination.

With only one mode there is no optimization possible, all the parameters

are completely determined.

In Fig. 2 we show the normalized aperture illuminations which maximize n

for N is 3,5 and 7 while the illumination for N = I is added for reference.

The normalized numerical values of the first 7 components of the optimum

excitation vector are

{I; -0.6647; 0.5315; -0.4555; 0.4048; -0.3680; 0.3397}.

-4.6-

Fig. 2. Normalized aperture illuminations

for unconstra'ined optimum 11 with N

modes in the aperture field of eq. (4.5).

2 The unconstrained minimum value of 0 and the corresponding eigenvector

<a can be computed with eq. (3.6) because both Wand A are diagonal

matrices. The relative extremes A. of the unconstrained 02 follow from

det (W-AA) = o.

This yields

A 1

1

The minimum value of 02 is therefore

02 MIN,N u~= 5.7831

The corresponding eigenvector <a is

<a=(IO ••• O).

(4.26)

(4.27)

(4.28)

(4.29)

-4.7-

The aperture illumination needed for minimum 02 is Jo(utr). This is in

accordance with the results of [16]. The far field having O~IN is

Some numbers associated

-2 11 4u

I = 0.6917,

2 2 4 110 110M1N

= ,

11/02 2 -4 11/oM1N 4u

I =

if

if

with this far

0.1196.

field are

(4.30)

(4.31)

(4.32)

(4.33)

2 Using eqs. (3.4) and (3.5) the unconstrained optimization of 11/0 results,

with N terms 1n the aperture field of eq. (4.5), in

2 (11/0 )MAX N ,

-4 -4 -4 4 (u I + u2 +... + uN ), (4.34)

which has a limi t value of 1/8 if N + 00 [27,p.502].

The eigenvector <a with eq. (4.34) is

(4.35)

Tite normalized numerical values of the excitation coefficients are

{I; -0.12615; 0.04104; -0.01894; 0.01050; -0.00652; 0.00437; ••. }.

The efficiency is now N -4 2 N -6

11 = 40: u. } / I u. i=1 1 i=1 1

which for N + 00 becomes 3/4.

And the normalized second moment is

2 N o = U

i=1

N

I i=1

-6 u.

1

which for N + 00 becomes 6.

(4.36)

(4.37)

-4.8-

The product of and 2 is n 0 now

2 N -4 3 N -6 2

no = 40: u. } / {I u. } , i=1 1 i=1 1

(4.38)

which is 41 2 for N -+ 00.

The aperture field for optimum n/02 with N terms is

N 3 f(r) = L J (u r) / {u JI(u )}.

n=i 0 n n n (4.39)

The far field corresponding to this f(r) is

N -2 2 2-1 g(u) J (u) I u (u -u ) if u '"

u n' 0 n=1 n n (4.40)

-3 if = PI (u )u u u n n n

2 In Table 2 we show the results of the unconstrained optimization of n/o

for Some values of N, while N = is added for reference. Convergence

towards the end values J/8, 3/4 and 6 is fast.

N

2

3

4

5

6

7

2 (n/o )MAX,N

0.1196

0.1239

o. 1246

O. 1248

0.1249

0.1250

0.1250

n

0.6917

0.7374

0.7455

0.7480

0.7489

0.7494

0.7496

2 (J

5.7831

5.9507

5.9822

5.9917

5.9955

5.9972

5.9982

sl1 sll 2 sll 3 sll 4 sll 5

4.15 27.5 36.4 42.6 47.3 51.2

4.01

3.99

3.99

3.99

3.98

3.98

24.0 34.7

24.5 32.9

24.6 33.4

24.6 33.5

24.6 33.5

24.6 33.6

41.1

40.3

39. I

39.5

39.6

39.7

46.0

45.3

44.8

43.8

44.3

44.4

49.9

49.3

48.9

48.5

47.8

48. I

Table 2. Results of unconstrained optimization of n/02 with N terms

in the aperture field of eq. (4.5).

In Fig. AI we show for N = 7 the aperture fields and the far fields for

optimum n/02 and n. Also, the lowest mode with optimum 0 2 is added.

Patterns with optimum n/02 have envelopes which lie between the envelopes

f . h .. 2 d . o patterns Wlt minimum a an maXimum n.

-4.9-

Bl'CZIlLSt' or tilL' v<lLue of I}, the sidelobe extremes and the 3dll beam width

InvcBtigatc' till" relationship of the

. 1 1 . 1 .. 2 W1t 1 t 1e well-known 1 lunllnatlon l-r

we wprc led to 2

optimum q/o

illumination for

[25,p.195]. Expand 2

i-r for 0 < r < 1 in a series [27,p.580] as follows

00

2 I-r l: sJ(ur),

non (4.41)

n=1

with II the n-th positive zero of J (u). Then the coefficients s are non

given by

.)

s n

r( I-r-).J (u r)dr o n

(4.42)

whicll results in

s n

Apart from the constant 8, tIle

a of eq. (4.35). The aperture

coefficient s of eq. (4.43) is n

field of eq. (4.39) for optimum

(4.43)

equal 2

r,/a ,

to

~ .. 2 Wltl! N ->- co, 1S equal to j-r apart from a constant multiplier. Stated

I . 2 . / 2 ot 1eYWl se, )-r has maXllTIUm fl a .

The constrained optimization of 11, 0 2 or n/a2 can easily be performed

using the technique outlined in section 3. The constrained optimization

is done while prescribing for instance a level (relative to the level

In u = 0) at a point of the main beam and/or the extremal values of a

numilcr of sidelohe peaks. The prescribed level in a point of the main

beam can be used, for instance, to get a certain 3dB beam width or a

socallcd flat-top beam. As the positions of the side lobe peaks are not

foreknown we must choose stnrting values for them. Suitable starting

points ;Irl' midwny iwtw(,l'n tilL' zerOR of J (u). If the computed levels of o

tilL' t'xlrl'lIu.'s nrc then found not tu be in accordance with the prescribed

values we repcilt the computation with new locations midway between the

old starting points and the computed positions of the extremes. We can

thus, in a number of iterations, obtain the prescribed sidelobe levels

to any desired degree of accuracy. In our computations the iterative

prOCCRS is stopped when the differences between the computed and prescribed

levels are less than O.OSdB.

-4.10-

If no parameter 1S to be optimized we need N Q P + 1 source terms if

the levels at P points are prescribed. If a parameter is optimized

while at the same time the levels at P points are prescribed, we need

N > P+2 source terms.

In Tables 3 and 4 we show some results of the optimization of the

efficiency when side lobe values are specified. In Table 3 we have two

side lobes of -25dB and in Table 4 four sidelobes of -30dB. With

increasing N the efficiency and the 3dB beam width come close to a limit

which depends on the requirements. Adding more modes in the aperture field

will have less and less effect upon nand u3dB • The efficiencies shown in

Tables 3 and 4 are less than the unconstrained efficiencies in Table 1 for

the

for

N

3

4

5

6

7

N

5

6

7

8

9

same values of N. In Fig. A2 far fields and aperture fields are shown

two cases of Table 4"

nOPT 0 2 u3dB sl1 3 sl1 4 sl1 5

0.8040 7.9210 3.81 36.0 41.6 45.9

0.8495 14.6343 3.64 24.6 35.4 40.6

0.8705 19.0847 3.58 26.2 27.4 36.7

0.8838 23.2823 3.55 26.7 29.2 29.7

0.8934 27.1914 3.53 27.0 29.8 31.5

Table 3. Results of constrained efficiency optimization with N

modes in the aperture field of eq.

side lobes prescribed at -25dB.

nOPT 0 2 u3dB sl1 5

0.8033 11.3297 3.80 39.4

0.8236 19. 1341 3.72 29.2

0.8345 23.7835 3.69 31.1

0.8423 28.0795 3.67 31.7

0.8480 32.4256 3.65 32.0

Table 4. Results of constrained efficiency

optimization with N modes in the

aperture field of eq. (4.5). First

4 side lobes prescribed at -30dB.

(4.5). First 2

-4.11-

In 1'.1111('s 5-8 we givC' results of the optimization of 02

with N modes in

eq. (4.5) when sidelobe values arc prescribed. These are -25, -30, -40 2

and -SOdB. With increasing N, a and u3dB

diminish to limit values while

tlte change in Tl towards a limit value depends on the prescribed sidelobe

level. In Table 5 n decreases and in Tables 6-8 '1 increases with increasing

N. The values of O~PT N are greater than the minimum possible value 5.7831. , In Fig. A3 two examples of Table 6 are shown.

N

2

3

4

N

2

3

4

5

6

N

3

4

5

6

7

5.8623

5.8493

5.8478

0.7240 4.05

0.7124 4.09

0.7110 4.09

sl1 2

35.2

37.0

36.6

sl1 3

41.6

42.4

43.1

sl1 4

46.4

47. I

47.3

sl1 5

50.3

50.9

51.1

2 Table,S. Results of constrained 0 optimization with N

modes in the aperture field of eq. (4.5). First

sidelobe prescribed at -25dB.

5.8311

5.8192

5.8185

5.8182

5.8182

0.6640 4.23

0.6746 4.20

0.6756 4.19

0.6759 4.19

0.6761 4.19

sl1 2

37.5

35.9

36.2

36.2

36.2

sl1 3

43.5

42.7

42.2

42.4

42.4

sl1 4

48.2

47.5

47.3

47.0

47.1

sl1 5

52.0

51.4

51.2

51.1

50.6

Table 6. Results of constrained 0 2 optimization with N

modes in the aperture field of eq. (4.5).

First side lobe prescribed at -30dB.

6.2894

6.1961

6.1852

6.1822

6.1813

0.5968 4.44

0.6227 4.35

0.6255 4.34

0.6267 4.33

0.6268 4.33

sl1 3

45.9

40.0

40.9

41.1

41.2

sl1 4

50.4

47.2

45.3

46.0

46.2

sl1 5

54.1

51.5

50.7

49.4

50.0

Table 7. Results of constrained 0 2 optimization

with N modes in the aperture field of

eq. (4.5). First 2 sidelobes prescribed

at -40dB.

-4.12-

N 2

sl1 7 °OPT n u3dB

7 6.7919 0.5621 4.55 56.3

8 6.7819 0.5662 4.54 53.4

9 6.7774 0.5677 4.53 54.5

Table 8. Results of constrained 0 2 optimization

with N modes in the aperture field of

eq. (4.5). First 6 sidelobes prescribed

at -50dB.

In Tables 9 and 10 we show optimization results for 0 2 when there is

also a prescription for the main beam of the radiation pattern. This

yields flat-top beams with prescribed sidelobes and optimum 0 2 or

beams with a prescribed 3dB beam width,prescribed sidelob~s and optimum 2 ° . These kinds of pattern can even be synthesized while another

parameter is optimized. The patterns in Figs. A4 and A5 are flat-top in

type and in Fig. A6 a pattern with prescribed 3dB beam width is shown.

N 2

°OPT n u3dB sl1 4 sl1 5

a)6 20.0990 0.0760 12.28 32.6 39.2

b)6 15.6062 0.1765 9.08 37.8 44.8

Table 9. Constrained o~PT with N modes of eq. (4.5).

Requirements: a) Three side lobes of -30dB, and for u I the

relative power is 0.5dB.

b) Three side lobes of -35dB, and for u I the

relative power is OdB.

N 2

sl1 2 s11 3 s11 4 s11 5 °OPT n u3dB

4 6.1083 0.6129 4.40 42.4 52.9 53.4 56.4

5 6.1024 0.6130 4.40 41.7 49.0 57. I 57.6

6 6.0996 0.6131 4.40 41.5 48.5 54.0 60.5

Table 10. Constrained o~PT with N modes of eq. (4.5).

Requirements: First side lobe prescribed at -30dB, and for

u = 2.20 the relative power is -3dB.

-4.13-

In Tables II-I) results of the optimization of

are given when side lobe values are specified.

n/er 2 with N modes of eq. 2 The values of 'I/O are,

with constraints, less than the maximum unconstrained value 1/8. The

(4.5)

2 values of n and a , however, can be less or greater than the unconstrained

limit values 3/4 and 6. This depends on the constraints chosen. In Fig. A7

two patterns and corresponding aperture fields of Table 13 are shown.

N

2 0.1226

3 0.1229

4 0.1230

5 0.1230

6 0.1231

7 0.12'31

n

O.75H3 6.1773

0.7647 6.2219

0.7668 6.2346

0.7676 6.2395

0.7683 6.2411

0.7685 6.2423

3.95

3.93

3.93

3.93

3.92

3.92

511 2

34.0

33. I

33.4

33.5

33.5

33.5

sl1 3

40.5

40.0

39.3

39.6

39.6

39.7

sl1 4

45.4

45.0

44.7

44. I

44.3

44.4

Table II. Constrained (n/er2)OPT with N modes of eq. (4.5). First

sidelobe prescribed at -22.5dB.

N

2

3

4

5

6

7

0.1139

0.1188

0.1194

0.1196

O. I 197

0.1197

TJ

0.6640

0.7064

0.7109

0.7130

0.7139

0.7144

5.8311

5.9455

5.9550

5.9623

5.9656

5.9673

4.23

4.10

4.09

4.08

4.08

4.08

sl1 2

37.5

31.8

32.7

32.8

32.9

32.9

sl1 3

43.5

40.5

38.4

39. I

39.2

39.3

sl1 4

48.2

45.6

44.7

43.2

43.9

44.0

Table 12. Constrained (n/er2)OPT with N modes of eq. (4.5). First

sidelobe prescribed at -30dB.

N

4

5

6

7

8

2 (n/er ) OPT

0.1005

O. 1027

D.1032

0.1034

0.1035

0.1036

n

0.6227

0.6418

0.6461

0.6476

0.6486

0.6492

6.1952

6.2478

6.2587

6.2628

6.2658

6.2678

4.35

4.28

4.27

4.26

4.26

4.26

sl1 4

47.2

41.3

42.5

42.9

43.0

43.0

sl1 5

51.5

48.4

45.6

46.6

46.9

47.0

Table 13. Constrained (n/er2)OPT with N modes of eq. (4.5).

First 3 side lobes prescribed at -40dB.

sl1 5

49.3

49.0

48.7

48.5

47.9

48.2

sl1 5

52.0

49.7

49.0

48.3

47.2

47.8

-4.14-

Conclusions

In section 4.1. I we have analytically and numerically optimized three

performance indices for aperture illuminations consisting of a series

of Bessel functions J (u r) with u positive solutions o n n

Results of unconstrained and constrained optimizations

of J (u) = O. o

are given.

Unconstrained optimization of the efficiency results in excitation

coefficients which are equal to the coefficients of expansion of a

uniform illumination into a Fourier-Bessel series. The unconstrained

optimum efficiency is a function of the zeros of a Bessel function and

has I as its limit value. The unconstrained optimum of the spread of

radiated power occurs

'II ' , h 2 1 UID1nat10n as a =

with the aperture illumination J (2.4048r). This 2 0

5.7831, n = 0.6917 and n/o = 0.1196. The uncon-

strained optimization of the efficiency power-spread ratio results in

the aperture illumination I - r2. This distribution has limit values 2 2 n/o= 1/8, n = 3/4 and 0 = 6.

The applied constraints on the values of the far fields are various.

This finds expression in

- flat-top patterns with prescribed sidelobes and an optimized index,

- patterns with prescribed 3dB beam width, prescribed sidelobes and

an optimized index,

- patterns with prescribed sidelobes and an optimized index.

The values of the performance indices depend on the constraints and the

nL@ber of modes. The influence on an index is greatest for the lowest

modes. 2 The advantages of n/o optimizations are that, with smooth aperture

distributions, efficiencies can be attained which are higher than those

of 02

optimizations and that the non-prescribed sidelobes decay faster

than with n optimizations.

-4.15-

Other aperture functions e which can be used for optimization purposes n

are Bessel functions .f (u r) where o n

u is a positive solution of JI (u) = 0

n The aperture distribution consists

N f(r) ~ I

n=1

o

a J (u r) non

of a truncated series

if o < r < 1,

(4.44)

if r > I.

The values of un are UI

~ 0, U2

~ 3.8317, u3

the modes with u1

and u2

are shown.

7.0156 etc. In Fig. 3

1 k-------:-

• •

Fig. 3. J (u.r) for ul o 1

1

r -

The far field g(u) is, wi til eg. (AI) and (A2),

g(tI) N

u.i I (tI) l: n= I

2 ! a .1 (u ) non

The vector <e 18 now

2 2 a J (u ) / (u -u ) non n

if

if

J(ur)}. o N

u ;& u , n

u = u • n

3.8317.

(4.45)

(4.46)

-4.16-

The vector <TI(e) is

(4.47)

In u = 0 this becomes

<T I {e (O)} = l {I 0 ... O} • (4.48)

The elements of the matrix A can be computed with the eqs. (AI) and (A2)

which results in

A .. 0 if i I< J • 1J (4.49)

A .. p2(u. ) if i j. 1J. o 1

Matrix A is a positive definite diagonal matrix. The'power radiated by

the aperture is N

p = 1 L r i=1

2 2 a.J (u.).

1 I 1

The elements of the matrix Ware

W •• 1J

J (u.)J (u.) J o 1 0 J

5 2 u J I (u)

( u 2 _'--u-co~c-)-(-u72-_u-2=-. )- d u • 1 J

(4.50)

(4.51)

Because of the discontinuous aperture distributions the reactive power

cannot be neglected. The integration limits in eq. (4.51) must therefore

be equal to 0 and nD/A.

After partial fractioning in eq. (4.51) there is one integral which

diverges. the others are convergent for upper limit 00. (see appendix A3).

If 02 is involved in an optimization the elements of W with an upper

limit of nD/A must be evaluated numerically.

The unconstrained optimi~ation of n with the aperture field of eq. (4.44)

results in nMAX •N <a = (1.0 ..... 0).

= nMAX = I. The corresponding eigenvector is

and the aperture field

well-known uniform illumination with far

is fer) = I. This

field J I (u) /u.

is the

-4.17-

The unconstrained optimizations of 02 or n/0 2 do not yield simple

results due to the complexity of the elements of matrix W.

The constrained optimization of the efficiency is interesting as we

can compare results with those in the literature [14), [19). To be

able to make these comparisons we have made computations with N modes

when N-I sidelobes are prescribed (which in fact is not an optimization)

and when less than N-I sidelobes are prescribed. Our results are

summarized in Tables 14-17. In Table 17 results for the combinations

(N,N-I) = (4,3), (6,5) etc. are given for -40dB sidelobe& Computations

for the combinations (N,N-2) showed that N should be ~ 14 to get the

first not-prescribed sidelobe below the -40dB level. For instance with

(N,N-2)

(N,N-2)

(13,11) the twelfth sidelobe was -39.8dB and with

(14,12) the thirteenth sidelobe was -4I.IdB. However, the first

not-prescribed sidelobe will decrease somewhat if N is increased while

the number of prescribed side lobes is unchanged.

Comparing our efficiencies with [19) shows that they are greater than

those for the Taylor distributions. This is because the prescribed side­

lobes do not decay as in the Taylor distributions, and because we are

not restricted to N modes and N-I prescribed sidelobes.

If we compare our results with those in [14) it appears that the

efficiencies are almost the same. In some cases we have computed higher

efficiencies. The optimum distributions in [14), however, are determined

in another way. In Fig. A8 the aperture illumination of two cases are

shown.

Furthermore it should be noted that the lowest 3dB beam width is not

achieved with the highest efficiency.

nOPT u3dB

~ 3 4 5 6 3 4 5 6

2 0.9434 0.9482 0.9487 0.9489 3.463 3.420 3.415 3.413

3 - 0.9455 0.9456 0.9456 - 3.390 3.394 3.394

4 - - 0.9263 0.9301 - - 3.336 3.361

5 - - - 0.8919 - - - 3.296

Table 14. Results of efficiency optimization with N modes in eq.

(4.44) and with K sidelobes prescribed at -25dB.

-4.18-

TlOPT u3dB

~ 4 5 6 7 10 4 5 6 7

3 0.8780 0.8882 0.8894 0.8901 0.8906 3.631 3.569 3.559 3.557

4 - 0.8872 0.8899 0.8902 0.8906 - 3.585 3.561 3.557

5 - - 0.8886 0.8888 0.8890 - - 3.548 3.546

6 - - - 0.8832 0.8839 - - - 3.520

9 - - - - 0.8383 - - - -

Table 15. Results of efficiency optimization with N modes in eq. (4.44)

and with K sidelobes prescribed at -30dB.

Additional results for -25 and -30dB sidelobe levels are given in

Table 16 for N = 15 modes.

Number of equal side lobe level (dB) TlOPT u

3dB side lobes

2 -25 0.9491 3.411

2 -30 0.8907 3.553

Table 16. Results of efficiency optimization with

15 modes in eq. (4.44).

N 4 6 8 10 12 14

0.7200 0.7347 0.7475 0.7554 0.7594 0.7604 Tl

4.037 3.993 3.951 3.920 3.898 3.881 u3dB

Table 17. Efficiencies and corresponding 3dB beam widths

when,with N modes in eq. (4.44), N-I side lobes

are prescribed to have the level of -40dB.

Conclusions -----------

In section 4.1.2 we have investigated aperture illuminations consisting

of a series of Bessel functions J (u r) with u positive solutions of o n n

JI(u) = O. We have restricted the optimization to the efficiency index.

Comparison of our results with those in the literature shows that the

computational method which we have employed provides efficiencies which

are higher than those of the Taylor distributions [19] and which are

much the same as those of the optimum distributions derived in [14].

10

3.553

3.553

3.543

3.531

3.463

-4.19-

If the side lobe extremes of a far-field pattern must lie below a

certain level it is favourable from the efficiency point of view

to keep the number of prescribed side lobe extremes as low as

possible.

In this section we will investigate aperture distributions which have

low and rapidly decaying sidelobes. The decay rate of the sidelobes

is dependent on the behaviour of the aperture distribution at the edge

of the aperture [26].

The general expression for the aperture illumination f(r) is

N f(r) I a {J (u r) - J (u )} if o < r < I ,

n=1 n o n o n (4.52)

0 if r > I.

The values of un are ul

= 3.8317, u2

= 7.0156 etc., that is the positive

zeros of JI(u)/u. This type of series has been used for synthesis

purposes before in [28]. The lowest mode is shown in Fig. 4.

Z1 o ~ .... ::s ... ~

i ... '" ::s !< ~ -"" .. .. ,. ~

~ .. - r j 'I:--~-~-~-~-"'-: o 1

Fig. 4. Lowest mode of eq. (4.52),

normalized to I.

The far. field is, with eqs. (AI) and (A2),

J I (u) N g(u) = -'-- I

u n=1

2 ja J (u ) non

2 2 2 a u J (u )/(u -u ) n non n if

if u = u • n

(4.53)

-4.20-

For the lowest mode the far field is given in Fig. A9 while the

numerical values of the side lobe levels are summarized in the

first row of Table 18.

The vector <e is

<e = {J (u r) - J (u ) o I 0 I

The vector

<TI (e) =

u

In u = 0 we have

(4.54)

(4.55)

(4.56)

The elements of the matrix A can be computed with eqs. (AI) and (A2),

which yields

A .. lJ

A •. lJ

lJ (u.)J (u.) o 1 0 J

2 J (u.) o 1

if i :f j,

(4.57)

if i j.

As <a,Aa> represents the aperture-radiated power, matrix A is positive

definite. The power radiated by the aperture is

N

P = HI r i=1

2 2 a.J (u.) 101

N

+ I i=1

The elements of matrix Ware

N I a.a.J (u.)J (u.)}. . I 1 J 0 1 0 J J=

W •• lJ

2 2 ooJ UJ~(U) = u. u. J (u.)J (u.) -""2-;;'2-""2"----;2;- duo

1 J 0 1 0 J 0 (u -u.)(u -u.) 1 J

Evaluation of these integral yields, see appendix A4,

W .. = 0 lJ

if i " j ,

2 2 j . W .. = ju.J (u.) if i

lJ 101

(4.58)

(4.59)

(4.60)

The second moment is

N

= ! I n=1

{auJ(u)}2. n non

-4.21-

(4.61)

The unconstrained optimum efficiency with N modes can be derived from

the equation 2V(O) a> = AAa>. By inspection it is found for nMAX,N that,

due to the special nature of the matrices involved,

nMAX N = N/(N+I). , (4.62)

The corresponding eigenvector is

< a = (4.63)

If N is allowed to become infinite, the value of nMAX N becomes I, which , corresponds to the efficiency of a uniformly illuminated aperture.

The aperture illumination for optimum efficiency with N modes is

N fer) = I

n=1 {J (u r) - J (u )}/J (u )

o non 0 n if

= 0, if

The corresponding far field is

g(u) J 1 (u) N

L u n=1

2( 2 2)-1 u u-u n n

if

= jJ (u ) o n

if

With optimum nthe normalized second moment is

2 a 2 u

n

as the radiated power power p = jN(N+I).

0< r < I,

r > I.

u = u • n

(4.64)

(4.65)

(4.66)

2 r 2 Formulas for nMAX,N a and nMAX,N/a are easy to derive with eq. (4.62).

In Table 18 we give some results of the unconstrained optimization of

the efficiency with N modes of eq. (4.52).

-4.22-

N 2 sl1 I sl1 2 sl1 3 sl1 4 sl1 5 I1MAX ,N 0 u3dB

0.5000 7.3410 4.85 35. I 46. I 53.7 60.0 65. I

2 0.6667 10.6501 4.02 16.2 36.9 45.9 52.4 57.7

3 0.7500 13.9501 3.76 17. I 21.6 38.4 46.2 51.9

4 0.8000 17.2461 3.62 17.3 22.9 25. I 39.8 46.7

5 0.8333 20.5401 3.55 17.4 23.3 26.7 27.8 41.0

6 0.8571 23.8330 3.49 17.5 23.5 27.2 29.5 29.9

7 0.8750 27.1251 3.46 17.5 23.5 27.4 30. I 31.7

Table 18. Unconstrained optimum n, corresponding 3dB beam width,

02 and side lobe levels with N modes of eq. (4.52).

The far fields and aperture fields with N = I and N = 7 are shown in

Fig. A9 •.

The normalized numerical values of the first 7 components of the

optimum excitation vector are:

{I; -1.3420; 1.6129; -1.8445; 2.0500; -2.2368; 2.4091}.

The unconstrained optimum of 02 gives no simple results with the aperture

functions of eq. (4.52). Using eq. (3.4) the unconstrained optimization

of 11/02 with N modes yields

2 (11/0 )MAX,N

N

L n=1

(4.67)

which for N ~ 00 becomes 1/8 [27,p.502].

The maximum value of eq. (4.67) is reached if the excitation vector equals

<a

where use has been made of eq. (3.5).

The efficiency is now

N 11 = q:

n=1

-4 N un + n

n=1

which is 3/4 if N ~ 00 [27,502].

(4.68)

(4.69)

-4.23-

The normalized second moment is

N

0: n=1

N u-4 + n

n n=1

which for N ~ 00 equals 6 [27,p.502].

The aperture field for optimum n/02 with N modes is

N

fer) = I {J (u r) - J (u )}/{u2 J (u )}.

on on non n=1

The far field is then

J I (u) N (2 2)-1 g(u) I u -u if u f u n=1 n

-2 if jJ (u )u u = o n n

u n

u n

(4.70)

(4.71)

(4.72)

In Table 2

19 we show the results of the unconstrained optimization of

11/0 for some values of N, while N is added for reference.

Convergence to the limit values 1/8, 3/4 and 6 is not as fast as in

Table 2.

N

10

20

30

40

50

0.0681 0.5000

0.1156 0.7169

0.1201 0.7348

0.1217 0.7399

0.1225 0.7424

0.1230 0.7439

7.3410

6.2259

6.1171

6.0791

6.0597

6.0479

4.85

4.06

4.02

4.01

4.00

4.00

s11 I

35. I

24.6

24.6

24.6

24.6

24.6

s11 2

46. I

33.4

33.5

33.6

33.6

33.6

s11 3

53.7

39.5

39.7

39.7

39.7

39.7

s11 4

60.0

44.0

44.4

44.4

44.5

44.5

s11 5

65. I

47.7

48.2

48.3

48.3

48.3

Table 19. Results of the unconstrained optimization of n/a2 with N modes

in eq. (4.52).

Fig. AID gives the aperture fields for N = I and N = 50. The values of

the parameters 2 2 n, 0 and n/o which result from the optimization of the

efficiency power-spread ratio have led us to the assumption that for

N ~ 00 the aperture field equals I - r2, apart from a constant multiplier.

-4.24-

This means that, with c a constant,

00

I n=1

{J (u r) - J (u )}/{u2J (u )} on on non

2 c (I-r ).

Integrating both sides, for the far field u = 0, yields

00

-! I n=1

-2 u

n c/4.

(4.73

(4.74)

with [27,p.502] it follows that c = -1/4. Substituting c and r = 0 in

eq. (4.73) gives

00

I -1/8. n=1

It should be noted here that eq. (4.73) can be proved exactly by

expanding l_r2 in the Dini series with H = V =:0 [27,§18.·3].

(4.75)

In Table 20 results are given of the optimization of the efficiency when

sidelobe values are prescribed. In Fig. All two cases are shown.

N 110PT a2 u3dB sll 3 sll 4 sll 5

3 0.6880 8.2977 4.12 41.5 49.2 54.9

5 0.7856 18.2239 3.77 26.2 27.4 41.3

10 0.8614 34.9761 3.59 27.3 30.3 32.6

15 0.8891 51.8375 3.53 27.4 30.5 33.0

Table 20. Results of constrained efficiency optimization

with N modes in eq. (4.52). First 2 sidelobes

prescribed at -25dB.

Tables 21 and 22 summarize the results of the constrained optimization

of a2 with N modes of eq. (4.52) and with a number of prescribed side­

lobes. In Figs. AI2 and AI3 some cases are shown graphically.

-4.25-

N 2 °OPT 11 u

3dB 811 2 811 3 sl1 4 sl1 5

2 6.8954 0.5376 4.69 43.9 52. I 58.3 63.4

3 6.5172 0.5905 4.48 33.7 47. I 54.2 59.6

4 6.3537 0.6114 4.41 35. I 39.5 50.7 56.9

5 6.2534 0.6242 4.36 35.5 41.0 43.9 53.8

6 6.1853 0.6327 4.33 35.7 41.5 45.5 47.5

10 6.0440 0.6501 4.28 36.0 42.1 46.7 50.3

15 5.9702 0.6594 4.25 36.0 42.2 46.9 50.7

Table 21. Results of constrained 2 optimization with 0

N modes in eq. (4.52). First sidelobe prescribed

3 7.2476

4 6.8414

5 6.6943

6 6.6032

7 6.5408

10 6.4312

15 6.3462

at -30dB.

0.5095

0.5570

0.5731

0.5831

0.5898

0.6015

0.6104

4.80

4.59

4.53

4.50

4.47

4.42

4.39

sl1 3

51.4

38.3

39.8

40.3

40.6

40.9

41.1

sl1 4

58. I

51.2

43. I

44.7

45.3

45.9

46.2

sl1 5

63.3

57.6

53.9

46.7

48.5

49.6

50.0

2 Table 22. Results of constrained 0 optimization

with N modes in eq. (4.52). First 2

side lobes prescribed at -40dB.

In Tables 23 and 24 results of the optimization of 11/02 with N modes

of eq. (4.52) are given when a number of sidelobe values is prescribed.

In Figs. A14 and A15 some of these results are shown in graphical form.

-4.26-

2 2 s11 2 s11 3 s11 4 N (n/a )OPT n a u3dB s11 5

2 0.0780 0.5376 6.8954 4.69 43.9 52.1 58.3 63.4

3 0.0923 0.6126 6.6372 4.40 30.6 45.3 52.7 58.2

4 0.0985 0.6398 6.4952 4.31 31.9 36.5 48.6 54.9

5 0.1023 0.6549 6.3999 4.26 32.3 38.0 41.0 51.5

6 O. 1050 0.6652 6.3353 4.23 32.5 38.5 42.6 44.5

10 O. 1106 0.6857 6. 1993 4.16 32.8 39.1 43.7 47.4

15 O. 1136 0.6956 6.1261 4.13 32.9 39.2 44.0 47.7

Table 23. Constrained optimization of n/a 2 with N modes of eq. (4.52).

First side lobe prescribed at -30dB.

N 2 2 sl1 4 sl1 5 (n/a )OPT en a u3dB

4 0.0801 0.5494 6.8548 4.63 52.2 58.4

5 0.0864 0.5829 6.7490 4.49 40.0 52.2

6 0.0895 0.5969 6.6720 4.44 41.6 43.7

7 0.0916 0.6061 6.6174 4.41 42.1 45.4

8 0.0931 0.6123 6.5739 4.38 42.4 46. 1

9 0.0943 0.6173 6.5438 4.37 42.6 46.3

10 0.0953 0.6206 6.5131 4.35 42.8 46.6

15 0.0981 0.6311 6.4331 4.32 43. I 47.0

Table 24. Constrained optimization of n/a 2 with N modes

of eq. (4.52). First 3 sidelobes prescribed

at -40dB.

Conclusions -----------

In section 4.1.3 we have optimized the performance indices n, 2 a 2 n/a . The aperture fields comprise series of functions with both

field and its derivative equal to zero at the aperture edge r =

This ensures lower and more rapidly decaying sidelobes than with

functions having only one of them equal to zero.

and

the

1.

Unconstrained optimization of the efficiency with N modes results in

n = N/(N+1) with I as limit value. The unconstrained optimum of n/a2

is a function of the zeros of a Bessel function. The limit value is 1/8

and occurs with the aperture illumination 1_r 2 having n = 3/4 and a2 = 6.

The convergence towards the limit values is slower than with the functions

of section 4.1.1.

-4.27-

The values of the constrained performance indices depend on the constraints

and the number of modes. The influence on an index is greatest for the

lowest modes. Several more modes, compared with section 4.1.1., are

nevertheless needed in order to achieve certain values for the optimized

indices. However, it is not known whether or not the limit values of an

index with constraints is the same for both types of aperture functions.

4.2. The annular aperture

In section 4. I we have treated optimization for an unblocked aperture.

Now we will introduce blocking of the central part of the aperture. The

¢-independent form of eq. (4.3) will be used to generate aperture

distributions. The aperture illumination can be written as

N J (u rl)

f(r) L a {J (u r) - o n Y (u r)} if o < rl

< I , Y (u r

l)

r < n=1 n 0 n o n

o n

0 if 0 < r < r . r > I '

Y is the Bessel function of the second kind and zeroth order. With o

u positive solutions of n

(4.76)

I.

J (u) -o

Y (u) = 0, o

(4.77)

we have illuminations which are zero at the aperture edges r = rl

and

r = I. In Fig. 5 we show the lowest mode for rl

= O. I having ul

= 3.31394.

z gl 5 '" ~ to ~ .... .. '" ~QS' '" ~ ~ ... ~

~ ~

~o --• o.S f

Fig. 5. Lowest mode of eq. (4.76) normalized to I.

-4.28-

The far field 's, with eqs. (A') and (A2),

g(u)

and

g(u) =

N

I n='

a.

a u {J (u)a n non

2 2 + J (ur,)8 }/(u -u )

o n n

2' {J,(ui)ai + r,J,(uir,)Si} +

if

N + I au {J (u.)a + J (u.r,)8 }/(u~_u2)

n n 0 ~ n 0 1 n 1 n n=' nfoi

u fo u , n

(4.78)

if u = u.. (4.79) ,

In these equations an and Sn are introduced for convenience. They are

respectively

a ='-J (u ) n 'n

.~ (4.80)

(4.8' )

Here we have used the Wronskian relation ['0,p.252]

J ,(x)Y (x) - J (x)Y +, (x) = 2/(rrx). n+ n n n

(4.82)

From eqs. (4.78) and (4.79) it is evident that the partial far-field

patterns are not orthogonal. Each mode of the aperture field contributes

t·) every far-field point u .. The far field of the lowest mode of eq. (4.76) , with r, = 0.' is given in Fig. A'6. It clearly shows the irregular behaviour.

The vector <e is

<e = {J (u r) -a ,

Jo(u,r,)

Yo(u,r,)

The vector <T,(e) is

u, [2 2 {Jo(u)a, + J o (ur,)8,} ..• u -u,

and <T,{e(O)} is

Jo(uNr,)

Yo(uNr,)

(4.85)

(4.83)

(4.84)

-4.29-

The elements of the matrix A are

A .. 0 if i # J , 1J 2 S~) i A .. Ha. if = J. 1J 1 1

(4.8b)

The power radiated by the aperture is

N p =

r l I a 2(a2 _ n n

(2). n

(4.87) n=1

The elements of matrix Ware

W •• 1J

00 3

u.u. u 2 = f -2::--='''''2:''.J--,2;:---''''2- {a.a.J (u) + (a.S.+S.a.)J (u)J (ur,) +

o (u -u.)( u -u.) 1 J 0 1 J 1 J 0 0 1 J

2 + S.S.J (ur,)}du. 1 J 0

(4.8€)

Evaluation for i # J yields, see appendix A5,

W •• 1J

I.f i

w .. 1J

11 2 2 2 2 = u.u. -2 p.a.{u.J (u.)Y (u.) - u.J (u.)Y (u.)}!(u.-u.) +

1J 1JJOJOJ 10101 1J

IS.s.{(u.r,)2J (u.r,)Y (u.r,) - (u.r l )2J (u.rl)Y (u.r,)}!{(U.Tj)2_(u.r,)2}+

1J J oJ OJ 10101 1 J

2 2 2 (a.S.+S.a.){u.Y (u.)J (u.r,) - u.Y (u.)J (u.rl)}/(u. 1J1J JOJOJ 10101 1

2 - u.)].

J

(4.89)

J we get, see appendix AS,

2112 u. -4[a.{u.J (u.)Y,(u.) + u.JI(u.)Y (u.) - 2J (u.)Y (u.)} +

1. 11011. 1. 101 0101

2 + S.{u.rIJ (u.rl)Y,(u.rl)+u.rIJI(u.r,)Y (u.r l )-2J (u.rl)Y (u.r l )} +

1. 1. 0 1. 1. 1. 1. 0 1. 0 1. 0 1.

+ 2a.S.{u.Y,(u.)J (u.r,)+u.r,Y (u.)J I (u.r,)-2Y (u.)J (u.r l )}]. (4.90) 11.11011. 011. 0101

Using eqs. (3.4) and (3.5) the unconstrained optimization of n with N

modes yields N

I i='

2 (a.+S.)/{u. (a.-S.)}. 1. 1. 1. 1. 1.

(4.9')

-4.30-

The corresponding excitation vector is

(4.92)

2 The value of the efficiency is I-rl

and the aperture illumination is

uniform if N + 00. The elements of <a, eq. (4.92), are the expansion

coefficients sn if the uniform illumination for r 1 < r < I is written as

[10,p.397) 00

1 = L n=1

s {J (u r) -non

J (u rl

) on}

Y ( ) Y (u r) , u rl

0 n o n

Jo(url

) with u solutions of J (u) - Y (u) O.

n 0 Yo(url) 0

Multiply both sides of eq. (4.93) by

J (u rl

) o m

r{J (u r) - Y ( ) o m u rl o m

Y (u r)} o m

(4.93)

(4.94)

and assume that the series expansion 1S integrable termwise from rl

to I.

Every integral on the right vanishes except when n = m. The value of this

non-vanishing term is

J (u rl

) 1 [{J ( ) _ ,,-o,.....:.n'-'.., Y. (u )}2 ,sn 1 Un - Y (u r

l) 1 n

o n

Integration of the left-hand side yields

(4.

(4.96)

Introducing an and Sn as defined in the eqs. (4.80) and (4.81) into eqs.

(4.95) and (4.96) gives for s n

s = 2/{u (S -a )}. n n n n

(4.97)

Apart from the constant -2, a of eq. (4.92) equals s • If a more general n n

function fer) with rl

< r < 1 is introduced into eq. (4.93) we have

s n

J (u rl

) {

On f (r) J (u r) - Y ( ) o n u r

l rIo n Y (u r)}rdr.

o n (4.98)

-4.31-

The unconstrained optimizations of 02 and n/02 yield no simple results

because W is not a diagonal matrix as in the case of the circular

aperture in section 4.1.1.

Constrained optimization with the aperture illuminations of eq. (4.76)

is done for 02

. Now it is not easy, as in the circular aperture

optimizations, to control the levels of the sidelobes because the patterns

do not have the sampling property. Some examples are shown in Figs. AI7

and AlB.

Conclusions -----------In section 4.2.1. the blocked aperture with zero edge-fields is

investigated. We have analytically optimized the efficiency. With N

modes in the aperture field the excitation coefficients for unconstrained

optimum efficiency are equal to the coefficients of expansion of a

uniform illumination into a series of Bessel functions. The efficiency

is then a function of the zeros of J (u) - J (url)Y (u)/Y (urI) = O. o 0 0 0

Constrained optimizations have been carried out numerically only for the

index 02

. In these optimizations it is more difficult to achieve patterns

fulfilling certain requirements than it is with unblocked apertures. It

is therefore more appropriate to use another optimization method [21].

-RI-

REFERENCES

[I] Boersma, J. and P.J. de Doelder, private communication, September 1979.

[2] Boersma, J. and P.J. de Doelder, ON SOME BESSEL-FUNCTION INTEGRALS ARISING IN A TELECOMMUNICATION PROBLEM. Memorandum 1979-13, Department of Mathematics, Eindhoven University of Technology, December 1979.

[3] Borgiotti, G. DESIGN OF CIRCULAR APERTURES FOR HIGH BEAM EFFICIENCY AND LOW SIDELOBES. Alta Frequenza, Vol. 40 (1971), No.8, p. 652-657.

[4] Cheng, D.K. and F.r. Tseng, GAIN OPTIMIZATION FOR ARBITRARY ANTENNA ARRAYS. IEEE Trans. on Antennas and Propagation, Vol. AP-13 (1965), No.6, p. 973-974.

[5] Cheng, D.K. and F.I. Tseng, MAXIMISATION OF DIRECTIVE GAIN FOR CIRCULAR AND ELLIPTICAL ARRAYS. Proc. lEE, Vol. 114 (1967), No.5, p. 589-594.

[6] Cheng, D.K. OPTIMIZATION TECHNIQUES FOR ANTENNA ARRAYS. Proc. IEEE, Vol. 59, (1971), No. 12, p. 1664-1674.

[7] Doelder, P.J. de private communications.

[8] Dorr, J. UNTERSUCHUNG EINIGER INTEGRALE MIT BESSEL-FUNKTIONEN, DIE FOR DIE ELASTIZITATSTHEORIE VON BEDEUTUNG SIND. Zeitschrift fur angewandte Mathematik und Physik, Vol. 4(1953), p. 122-127.

[9] Drane, Jr., C. and J. McIlvenna GAIN MAXIMIZATION AND CONTROLLED NULL PLACEMENT SIMULTANEOUSLY ACHIEVED IN AERIAL ARRAY PATTERNS. The Radio and Electronic Engineer, Vol. 39 (1970), No. I, p. 49-57.

[10] Farrel, O.J. and B. Ross, SOLVED PROBLEMS: GAMMA AND BETA FUNCTIONS, LEGENDRE POLYNOMIALS, BESSEL FUNCTIONS. The Macmillan Company, New York, 1963.

[IDa] Gantmacher, F.R. THE THEORY OF MATRICES. Chelsea Publishing Co., New York, 1959, Vol. I.

[II] Guillemin, E.A. THE MATHEMATICS OF CIRCUIT ANALYSIS. Technology Press, Cambridge, Mass. 1949.

[12] Ishimaru, A. and G. Held, ANALYSIS AND SYNTHESIS OF RADIATION PATTERNS FROM CIRCULAR APERTURES. Canadian Journal of Physics, Vol. 38 (1960), No. I, p. 78-99.

-R2-

[13] Kouznetsov, V.D. SIDELOBE REDUCTION IN CIRCULAR APERTURE ANTENNAS. In: Int. Conf. on Antennas and Propagation; London, 28-30 Nov. 1978. lEE Conf. Publication, No. 169, London: Institution·of El~ctrical Engineers, 1978. Part I: Antennas. P. 422-427.

[14] Kritskiy, S.V. and M.T. Novosartov,

[ 15]

[ 16]

[ I 7]

[ 18]

DERIVATION OF THE OPTIMUM FIELD DISTRIBUTIONS FOR ANTENNAS WITH A CIRCULAR APERTURE. Radio Engineering and Electronic Physics, Vol. 19 (1974), No.5, p. 23-30.

Kurth, R.R. OPTIMIZATION OF ARRAY PERFORMANCE SUBJECT TO MULTIPLE POWER PATTERN CONSTRAINTS. IEEE Trans. on Antennas and Propagation, Vol. AP-22(1974), No. I, p. 103-105.

Ling, C., E. Lefferts, D. Lee and J. Potenza, RADIATION PATTERN OF PLANAR ANTENNAS WITH OPTIMUM AND ARBITRARY ILLUMINATION. IEEE International Convention Record, 1964, Pt. 2, p •. 111-lt4

Lo, Y.T., S.W. Lee and Q.H. Lee, OPTIMIZATION OF DIRECTIVITY AND SIGNAL-TO-NOISE RATIO OF AN ARBITRARY ANTENNA ARRAY. Proc. IEEE, Vol. 54 (1966), No.8, p. 1033-1045.

MIRONENKO, I.G. SYNTHESIS OF A FINITE-APERTURE ANTENNA MAXIMIZING THE FRACTION OF POWER RADIATED IN A PRESCRIBED SOLID ANGLE. Telecommunications and Radio Engineering, Vol. 21-22 (1967), No.4, p. 99-104.

[19] Rudduck, R.C., D.C.F. Wu and R.F. Hyneman, DIRECTIVE GAIN OF CIRCULAR TAYLOR PATTERNS.

[ 20]

Radio Science, Vol. 6 (1971), No. 12, p. 1117-1121.

Ruze, J. CIRCULAR APERTURE SYNTHESIS. Vol. AP-12 (1964), No.6, p.

IEEE Trans. on Antennas and Propagation, 691-694.

[21] Sachidananda, M. and S. Ramakrishna, OPTIMUM APERTURE DISTRIBUTIONS IN THE PRESENCE OF BLOCKAGE. Electronics Letters, Vol. 14 (1978), No. 17, p. 555-558.

[22] Safak, M. and P.P. Delogne, OPTIMISATION OF CIRCULAR-APERTURE DISTRIBUTIONS FOR INTERFERENCE REDUCTION. Proc. lEE, Vol. 125 (1978), No.8, p. 717-723.

[23] Sanzgiri, S.M. and J.K. Butler, CONSTRAINED OPTIMIZATION OF THE PERFORMANCE INDICES OF ARBITRARY ARRAY ANTENNAS. IEEE Trans. on Antennas and Propagation, Vol. AP-19 (1971), No.4, p. 493-498.

[24] Sanzgiri, S.M. , J.K. Butler and R.C. Voges, OPTIMUM APERTURE MONOPULSE EXCITATIONS. IEEE Trans. On Antennas and Propagation, Vol. AP-20 (1972), No.3, p. 275-280.

-R3-

12S1 Silver, S.(c'd.) MICROWAVE ANTENNA THEORY AND DESIGN. McGraw-Hi 11 Book Company, Inc. New York, 1949. MIT Radiation Laboratory Series, Vol. 12.

[26) Taylor, T.T. DESIGN OF CIRCULAR APERTURES FOR NARROW BEAMWIDTH AND LOW SIDELOBES. IRE Trans. Antennas and Propagation, Vol. AP-8 (1960), No.1, p. 17-22.

[27) Watson, G.N.

[28)

A TREATISE ON THE THEORY OF BESSEL FUNCTIONS, Second Edition. Cambridge University Press, 1966.

Worm, S.C.J. RADIATION PATTERNS OF CIRCULAR APERTURES WITH PRESCRIBED SIDELOBE LEVELS. TH-Report 79-E-97. Department of Electrical Engineering, Eindhoven University of Technology, August 1979.

-AI-

APPENDIX.

AI. The following integrals of a product of Bessel functions C and D

apply [27,p.134]. If k f 1

z J zC (kz)D (lz)dz = z{kC l(kz)D (lz)-lC (kz)D l(lz)}/(k2-l2). (AI)

n n n+ n n n+

If k = 1

z J zC (kz)D (kz)dz =

n n 2

lz {2C (kz)D (kz)-C I (kz)D I (kz)-C I (kz)D I (kz)}. n n n- n+ n+ n-

(A2)

A2. Evaluation of the integrals resulting frOID eq. (4.12). Omitting the

constant we have integrals of the type

I3

(u. ,u.) 1 J

I3(u.,u.)

1 1

00 u3J 2(u) J -2~o'-;;2~-;2'---;;2-o (u -u.)(u -u.)

1 J

du, (A3)

(A4)

in which u. and u. are positive zeros of the Bessel function 1 J

J (u). a

The integrals (A3) and (A4) are special cases of the general integral

00 uIDJ 2 (u)

f 0 I (a, b) du a and b real, m 0 (u2_a2)(u2_b 2) ,

(AS)

where f denotes that the Cauchy principal value of the integral 1S

to be taken. The integral (AS) has been evaluated by J. Boersma and

P.J. de Doelder for m = 0,1,2,3 [I], [2]. For instance, it is found

that 13 is expressible in terms of Bessel functions J o and Yo as

follows

(Al)

Applying (A6) and (A7) to the integrals of (A3) and (A4) yields

further simplification

the Wronskian relation

because J (u.) o 1

J (u.)YI(u.) -o 1 1

= J (u.) = 0 and because of o J

JI(u.)Y (u.) = -JI(u.)Y (u.) 10110 1

Finally we have

and

I3

(U.,u.) 1 J

I3(u.,u.)

1 1

= 0,

I ; .

-A2-

(AS)

(A9)

A3. Evaluation of the integrals resulting from eq. (4.51). Omitting the

constants, the integral reads with upper limit t

t u5J~(u) J 2 ~ 2 2 du o (u -u.)(u -u.)

1 J

where uiand uj

are solutions of JI(u) = o.

Par'tial fraction decomposition if u. " u., gives J J

2 I I u.{-- + --}j

1. u+u. u-u. 1 1

2 2 2 u./{2(u.-u.)} 1 1 J

- [2u + u~ {_I _ + _I _}j J u+u. u-u.

J J

2 2 2 u ./{2(u.-u.)}. J 1 J

Integral (AIO) is composed of integrals of the type

t 2 fUJI (u)du , o

which is divergent for t ~ 00, and

(AIO)

(All)

(AI2)

t I I 2 J {- + -} JI (u)du, (Al3)

u+a u-a o

where a is u. or u .• The integral (AI3) with t ~ 00 can be found in 1 J

a paper by Dorr [8] and equals -TIJI(a)YI(a) which is 0 if, for

instance JI(a) = O.

If in (AIO) u. u. we have the following partial fractioning 1 J

2 I I u.[--+--1 u+u. u-u.

1 1

u i I 4{ 2

(u+u. ) 1

_ I } 1 2

(u-u. ) 1

(A14 )

Besides integrals like (AI2) and (AI3) we have now the integral

t

J { o

2 (u+u. ) 1

2 --'--;;-2} J I (u)du (u-u. )

1

(AI5)

1TJ (u.)YI(u.) 011

2 U.

1

-A3-

0, see [B], equals

(AI6)

A4. Evaluation of the integrals resulting from eq. (4.59). Omitting the

constants we have

00 2 uJ I (u)

f -"-2 -'-;2'---2~-;;2- du , o (u -u.)(u -u.)

1 ]

where u. and u. are solutions of JI(u) = o. 1 ]

For u. # u. the partial fractioning is 1 ]

u 2 2 2 2 (u -u.)(u -u.)

1 J

And for u. = u. 1 J

u = {

I H--+-----­u+u. u-u. u+u.

1 1 ]

we have

I -(u2_u~)2

1 (u-u. ) 2

(u+u. ) 2}/(4ui )·

1 1

I 2 2 --}/(u.-u.) . u-u. 1 ]

]

(AI7)

(AlB)

(A19 )

Integral (AI7) is composed of integrals (AI3) and (AI5) with t ~ 00_

2 -I It is found that (AI7) equals 0 if u. # u. and -1TJ (u.)Y

l(u.)/(4u.)=(2u.)

1. J 0 1. 1. 1. 1.

if u. u .. 1 J

A5. Evaluation of eq. (4.BB). The integrals occurring in this equation

are

00 3F f 2 u2 2 2 du

o (u -u.)(u -u.) 1 J

(A.20)

where F (omitting constants) equals J 2(u), J 2(ur l ) and J (u)J (urI) 2 2 0 0 0 0

with 0 < r l < I. For Jo(u) and Jo(ur l ) evaluation is possible by

using the results of appendix A2. Simplifications do not take place

as u. and u. are positive solutions of J (u)-J (url)Y (u)/Y (urI) O. 1. J 0 0 0 0

For F = J (u)J (urI) evaluation of (A20) has been done in [2] and [7] o 0

for u3

as well as u in the nominator. These integrals are special

cases of

Km(a,b;r l ) 00 umJ (u)J (rlu) ~ 0 0 T 2 2 2 2 du, o (u -a )(u -b )

(A21)

-A4-

with a and b real, and 0 < r J ~ J. Evaluation for m = I and m = 3

results in

= - 21:.2 {J (arJ)Y (Ial)-J (brJ)Y (lbl)}/(a2-b

2),

o 0 0 0 a '" b, (A22)

Another integral which has been evaluated in [2] and [7] is

'. mum! J (u)J 0 (r J ,,) Lm(a,b;r J) = f 2 2 2' 2 du (A26)

o (u -a )(u -b )

with real a and b, and 0 : rJ

: J. The results for m = 2 and m = 4 are

These expressions are valid for 0 : r J < I. The case rJ

= can be

found in [2].

o

-/0

~

~ uJ -'fQ ,. .... ~ ~

u.I

at!' -so

1

z 0 ~ .... => ~

~ « ::;; <:: 6.5 ll! :::s :;c uJ ~

-< ., uJ

'" ~ ~

<C

!i 0 z 0

o s

\ " \

\ " ./."

/ \~ . ......'\:

".j "

"

o

-A5-

10

/- ...... . \. . I \ -r-" . .

/ . , \ 1\. {

{

\ \

\ \

\ \

1\ . \

I . I . \

, \

\ \ \ \ \ \ \ I

\ \ \ \ i .r. ~

Fig. AI. . Ids d far he . fields an 2 Aperture . d 0

unconstralne Optimum

" " " "

with n n/o 2 "

1S

--+ U

/'-', . \ 1 .

\ \

N = 7.

"

o

-10

-10

d>

3-JO

"" .... 3 o I>- -YO uJ >"

S ~

~ -SO

<> '" --....

-60

1 \

\ \ \ , ~ \

~ ~

~ \ ~

5

-A6-

10

\ - , \(\ ,\ r:vr. \, 1/ \ 1/ \ { ,I \ I' \ f \ \ I \

\

\

'" L~~~-, :l o 0 D.S z

o

\ \ \

r

15

-, / \

-u.

I \

/f\

Fig. A2. d far fields. of -30dB. fields an .delobes Aperture . n with 4 91

. constral.ned Opt1mum N = 9. -N = 5

-A7-

6 C""~C-~--__ -c~ __ --____ --__ ~--__________ ~I!-) ____________ -= .. 10 --ll­

S o

-10

-6D

11_~ ...

Fig. A3.

'{\

1

d far fields_ fields an 2 . I sidelobe Aperture . a with

constra~ned Optimum

-- N = 2 - - N = 6.

of -30dB.

s o

-70

;0--0 --' c< -JO ... 3 o .... 0;; -~O .... 0-

'" -' ... "" -50

-60

z ., .... 0-

'" "" .... .., ~ ..... <> M

'" "" '" 0-

"" .... 0-

-<

'" .... ~

~ ~ .., :E l!5 0 :z

o

0

Fig. M. Aperture field and far

A . . d 2 pt1mum constra1ne a

-A8-

10 1S

r --+

1

field with N = 6.

with 3 sidelobes of -30dB

and in u = 1 the relative power is O.SdB.

.to

__ IL

-10

-20

;0 -:s ·30

'" ... ~ :r ... -~o ::. .. .... '" ~ .... "'-50

.... ~ :oJ <­w: ~ .... -<

'" .... -~ ~ c: s: ..

-60

~ 0

o

o

Fig. A5.

-A9-

10 1$ ).0

--+ u.

(

r -o.s 1

Aperture field and far field with N = 6. . . d 2 Opt1ffium constra1ne a with 3 sidelobes of -35dB

and in u = 1 the relative power is OdB.

-AIO-

o S 1D IS ./.0

o --16

-26

-60

1

Fig. A6. field with N = 6. field and far2 . delobe of -30dB Aperture with I s'

constrained a. is -3dB. Optimum h relat,ve power 2.20 t e and in u

0

0

-I'

-to

;a -0 -30 -.J

"" ... 3 ~ -~o ~

>-~

:c ~ -50

-60

1

'" " " "Z .. -':i '" .... '" I-

:::D.5 .. ... ... :3 I-0<. oJ ~

4: .. ... g ... ~ ~ 0

0

Fig. A7.

-II 1 1-

S 10 1$

, , ~ ~

~

~ \ \ \ \ \

/\'-' n I \ I ' 'f' I \

f \ I (\

\ I \ I \ \ ( \ I \ \ I I I

"

- r

O.S 1

and far fields. Aper /0 with

t ure fields 2 3 constrained n Optimum

-N = 4 - - - N = 9. f -40dB. sidelobes a

,l0

-- U.

"", / , /1\

1 "­

"-

" " , " " " " "

o D.S

" " \ ,

\

\

-AI2-

\./

/ (

I I I

I

Fig. A8. Aperture fields for constrained optimum n. --- N = 10, with 5 sidelobes of -30dB.

- - N = 14, with 12 sidelobes of -40dB.

-70

-16

~

"" ~-jO a:: ... ~ .. .... -~o ~

> H .... c ~

~-SO

1

o

\ \ \ \ \ \

..... ,

\ I -

'\ \

\ \

-A 13-

10

\ / -,

\

I I

"' f \

\ I

\ / \ I \ I ,

\ \ \

-, /, ...... -,

/ \ / \ I \ f \ / \ I \

1\, \/ \

1\ I \

'/

~

" I \ f \ I \

I \ I , I \ / , ~/ .... ,

I , \ , \ \ \ I \ \

ol-~~-- 0.5 o \ -+r 1

Fig. A9. d far fields. fields an . h N modes. Aperture . d n Wlt

unconstra1ne Optimum __ N = 7.

N = I

1S 10

_ tJ,

'-, / \

I \ I \

" \ /

-..... , / \

I \ I \ I ,

I

:z: o .... I­:l ... ....

1

0< .... ". H

"" ... 0. ..t ::J

~ ... -...

o

---..... "-

"-

" " " "-" "-

"-, , , , \ ,

\ \

-AI4-

\ \

\

'.f 1

Fig. AIO. Aperture fields with N modes.

Unconstrained optimum n/o2•

- N = I - - - N = 50.

0 0

-/0

-10

-;; " -JO ~

'" '" 3 ~-lfO ... '" H -:'5 uJ "" -so

-60

1

% .. .. I-

'" .. .... .. .... $5

'" uS .c

'" I-.., ... .=! P-.. .. ~ ~

4: :E .. ~o

0

s

" , \

\ \

\ \ \ \ \ I I I I I'

/' \ \ '-, ....

\ \

'-, \J

o.S

-AI5-

10

I I

I I

r' I ' I ,

(' I I / \ I I \.1

I , I I

, \ -+

1

10

--+ II.

/ ,,-I" / ....

/ \ I' / ...... I \ / \ I \

1/ I I \ I \

1\\~\I ' 1\

r

Fig. All. Aperture fields and far fields.

911.t;~mum constrained 11 with 2 sidelobes of -25dB. I';;" '\

, -~<N = 3 N = 15.

-10

-/.0

dJ

~ -30

:. 3 12 -\'0 "' > ~ I-

5 ~-so

1

z: '" ..... ... =I <D ..... ,.. ~o.s H P

W

'" '" :;. ~ -< Q

"' ~ ~ < z: "" ~ 0

0

~ ~

\ \

D.f

\ /

\ , \ , \1

, , , ,

-AI6-

--- ...... / , I \ I

, \

\ \

\ .... 1"

1

Fig. A12. Aperture fields and far fields.

-, / ,

I \ f \

/' -, I ,

f \

f\\ I \

I \

(\

Optimum constrained 02 with 1 sidelobe of -30dB.

-N = 2 N = 6.

_10

-10

dl -,:)

-.J -10

""' '" 3

~ -'I ... ~ .... j ~ .. s

"Zo. o ... .-::0 ... ~ :;; ... A 0.' '" w:

'" ... .. ... -.., "" ... "" ~ ~

o

~""

, \

'\ ,

io~~_~ __ o

Fig. A13.

s

'\ \

\ \

\ \

\

\ \

-AI7-

10

\ I \ I \ I

15

-,

'\ \

\ '\

\

D.~

" " ''-'-"

" -r ,

Aperture fields an 0 2 with 2 sidelobes d far fields. of -40dB. constrained

Optimum __ N = 7. -N = 3

:1.0

'" ... \ \

\

(\

-111

-10

~

4>

3 -JO 0<

'" ~ ""-f0

'" :> .... >--<

~-so

z: .. ~ ... ... ~ .... oft ....

1

""0.5

o

" " '\ \

\ \ \ \ \ \ \ \ (

-AIB-

10

1

'\ \

\

Fig. A14. Aperture fields and far fields.

15

'\ \

\ I

f

r\

- I/.

/ --, / \

I I \ I

Optimum constrained n/a2 with sidelobe of -30dB.

-N = 2 N = 6.

Fig. AIS. Aperture fields and far fields.

Optimum constrained n/a2 with 3 sidelobes of -40dB.

--N 4 N = 9.

-A20-

o 5 10 15 zo

o - Ii.

-1/)

~

cO ~-30

"" .u ~ 0 ... -~o w - "'1.S > ..... ~ ~

~ -SO

_rq.3

-60 n -70

1

Fig. A16. Aperture field and far field.

-10

-60

-70

z <> ):: ::J .. .. .. :;; ;:: D.S

~ " I-.. .oJ -..,

o

Fig. A17.

-A21-

Aperture field and far

O . . d 2 ptlIDum constra1ne 0

field with 6 modes.

with the relative power

prescribed as follows: u = 2.15, -3dB; u = 7, -27dB;

u = 9.5, -30dB; u = 13, -27dB.

- Ii.

-10

-10

co-30 3

"" '" ~ g.-'Io ... > -:c ~

=:-so

-60

'" ~ " :. ~, -c

:;l '" !:l

5

1

~o o

Fig. AlB.

-A22-

_.1.0.0

-+r

os 1

Aperture field and far

O . . d 2 pt~mum constra1ne a

v: o"U,3

(1"1: f&.IS9~-

field with 5 modes.

with the relative power

prescription -30dB in u = 6.

EINUIIOVEN UNIVEI{SITY OF TECIINOLO(;Y THE NETltERLANOS DEPARTMENT OF ELECTRICAL ENGINEEIUNG

Reports:

93) Duin, C.A. van DIPOLE SCATTERING OF I::LECTROMAGNI::'l'IC WAVES PROPAGATION THROUGH A RAIN MEDIUM. Til-Report 79-E-93. 1979. ISBN 90-6144-093-9

94) Kuijper, A.Il. de and L.K.J. Vandamme CHARTS OF SPATIAL NOISE DISTRIBUTION IN PLANAR RESISTORS WITH FINITE CONTACrS. Til-Report 79-E-94. 1979. ISBN 90-6144-094-7

95) Hajdasinski, A.K. and A.A.H. Damen REALIZATION OF THE MARKOV PARAMETER SEQUENCES USING THE SINGULAR VALUE DECOMPOSITION OF THE HANKEL MATRIX. TH-Report 79-E-95. 1979. ISBN 90-6144-095-5

96) Stdanov, B. ELECTRON MOMENTUM TMNSFER CROSS-SECTION IN CESIUM AND RELATED CALCULATIONS OF THE LOCAL PARAMETERS OF Cs + Ar MHD PLASMAS. TH-Report 79-E-96. 1979. ISBN 90-6144-096-3

97) Worm, S.C.J. RADIA'l'ION PA'l''I'ERNS OF CIRCULAR APERTURES WITH PRESCRIBED SIDELOBE LEVELS. TH-Report 79-E-97. 19'10. ISBN 90-6144-097-1

98) Kroezen, P.H.C. A SERIES REPRESENTA'l'ION METHOD F'O~ 'I'HE FAR FIELD OF AN OFFSET Rtf'LECl'OR ANTENNA. 'I'H-Report 79-E-98. 1979. ISBN 90-6144-098-X

99) Koonen, A.M.J. ERROR PROBABILITY IN DIGITAL FIBER OPTIC COMMUNICATION SYSTEMS. TH-Report 79-E-99. 1979. ISBN 90-6144-099-8

100) Naidu, M.S.

101 )

STUDIES ON THE DECAY OF SURFACE CHARGES ON DIELECl'RICS. TH-Report 79-E-l00. 1979. ISBN 90-6144-100-5

Verstappen, H.L. A SHAPED CYLINDRICAL DOUBLE-REFLECTOR ANTEN:IA. 'I'H-Report 79-E-101.

SYSTEM FOR A BROADCAST-SATELLITE 1979. ISBN 90-6144-101-3

102) Etten, w.e. van

THE THEORY OF' NONLINI::AR DISCRETE-TIME SYSTEMS AND ITS APPLICATION 'l'HE EQUALIZATION OF NONLINEAR DIGITAL COMMUNICATION CHANNELS. 'I'H-Report 79-E-102. 1979. ISBN 90-6144-102-1

103) Roer, Th.G. van de

ANALYTICAL THEORY OF PUNCH-'l'HROUGH DIODES. 'I'H-Report 79-E-103. 1979. ISBN 90-6144-103-X

104) Herben, M.H.A.J.

DESIGNING A CONTOURED BEAM ANTENNA. TH-Report 79-E-104. 1979. ISBN 90-6144-104-8

'1'0

EINDHOVEN UNIVERSITY OF TECHNOLOGY THE NETHERLANDS DEPARTMENT OF ELECTRICAL ENGINEERING

Reports:

105) Videc, M.F. STRALINGSVERSCH1JNSELEN ] N PLASMA'S EN BE1.JEGENnE MED] A: Een ~eomet risch­optische en een ~olf7.onehenaderinR. TH-Report 80-E-105. 1980. ISBN 90-6144-105-6

106) Hajdasinski, A.K. LINEAR MULTIVARIABLE SYSTEMS: Preliminary prohlems 1n mathematical description, modelling and identification. TH-Report 80-E-106. 1980. ISBN 90-6144-106-4

107) Heuvel, W.M.C. van den CURRENT CHOPPING IN SF6' TH-Report 80-E-l07. 1980. ISBN 90-6144-107-2

108) Etten, W.C. van and T.M. Lammers TRANSMISSION OF FM-MODULATED AUDIOSIGNALS IN THE 87.5 - 108 MHz BROADCAST BAND OVER A FIBER OPTIC SYSTEM. TH-Report 80-E-l08. 1980. ISBN 90-6144-108-0

109) Krause, J.C. SHORT-CURRENT LIMITERS: Literature survey 1973-1979. TH-Report 80-E-l09. 1980. ISBN 90-6144-109-9

110) Matacz, J.S. UNTERSUCHUNGEN AN GYRATORFILTERSCHALTUNGEN. TH-Report 80-E-I10. 1980. ISBN 90-6144-110-2

III) Otten, R.H.J.M. STRUCTURED LAYOUT DESIGN. Til-Report 80-E-Ill. 1980. ISBN 90-6144-111-0 (in preparation)

112) Worm, S.C.J. OPTIMIZATION OF SOME APERTURE ANTENNA PERFORMANCE INDICES WITH AND WITHOUT PATTERN CONSTRAINTS. TH-Report 80-E-112. 1980. ISBN 90-6144-112-9