Radiation patterns of circular apertures with prescribed sidelobe levels

Radiation patterns of circular apertures with prescribed

sidelobe levels

Citation for published version (APA):

Worm, S. C. J. (1979). Radiation patterns of circular apertures with prescribed sidelobe levels. (EUT report. E, Fac. of Electrical Engineering; Vol. 79-E-097). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1979

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

providing details and we will investigate your claim.

with prescribed sidelobe levels

RADIA'IION PA'!'!ERNS OF CIRCULAR APERrURES WI'IH PRESCRIBED SIDELOBE LEVELS

by

S.C.J. Worm

'IH-Report 79-E-97 ISBN 90-6144-097-1

Department of Electrical Engineering

Eindhoven The Netherlands

RADIATION PATTERNS OF CIRCULAR APERTURES WITH PRESCRIBED SIDELOBE LEVELS by S.C.J. Worm TH-Report 79-E-97 ISBN 90-6144-097-1 Eindhoven August 1979

Contents Abstract

Acknowledgement Introduction

I. The Taylor distribution for a circular aperture 1.1. Some observations

1.2. Results and conclusions

2. The method of Ishimaru and Held for the synthesis of radiation patterns from circular apertures

3. The modified methods with other source functions 3.1. Source functions with a zero-edge field and a nonzero

first derivative of the edge fieid 3.2. Results and conclusions

3.3. Source functions with both the field and the first derivative of the--[{eld equal--to zero at r ~ I

3.4. Results and conclusions References

Appendix A: Derivation of equation (2.7) Figures -0.2- -0.3- -1.1--I. 1- -1.3- -1.5- -2.1--3. \- -3.1- -3.5- -3.9- -3.11- -4.-A.

1-Abstract

Some possibilities of controlling the side lobes of circular apertures are investigated in this report. Special attention

is paid to modifications of a method published by Ishimaru and Held. Source functions with different behaviour at the aperture edge are used to synthesize radiation patterns. The differences concern the value of the edge field and the value of the first edge-field derivative. Aperture distributions comprise series of Bessel functions, namely

I

a J (u r) and

I

a {J (u r)-J (u )}

n n o n n o n o n

with un solution of J1(u)

=

0, and

I

a J (u nr ) with u solution

n n o n n

of Jo(u) = O. The synthesis methods which are described, compared and modified allow a number of equal sidelobes, or a number of

dif-ferent sidelobe extrema over a certain region of space beyond the main beam to be prescribed. The computed patterns are compared with copolar specifications for satellite transmitting antennas and earth-station antennas. The patterns are computed to show the influence of the types of series expansions for the aperture distributions, and to show the possibilities of the synthesis methods.

~, S.C.J.

RADIATION PATTERNS OF CIRCULAR APERTURES WITH PRESCRIBED SIDELOBE

LEVELS. TH-Report 79-E-97.

Eindhoven University of Technology, Department of Electrical Engineering, Eindhoven, The Netherlands. August 1979.

Address of the author: Ir. S.C.J. Worm,

Group Electromagnetism and Circuit Theory, Department of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513,

5600 MB EINDHOVEN, The Netherlands

Acknowledgement

The author wishes to express his thanks to dr. V. Vokurka and dr. M. Jeuken for suggesting the problem.

Introduction

Controlling the sidelobes of radiating apertures 1S acquiring ever greater importance in view of the growth of satellite communications. The reduction of sidelobe levels or sidelobe power reduces the interference between systems and allows more efficient use of the RF spectrum and the geostationary orbit.

The radiation patterns which can be realized theoretically depend on the type of source functions used in the expansion of the aperture illumination and the relative strength of the source functions.

There are synthesis methods which maximize the fraction of power radiated in a prescribed solid angle. As a result the fraction of power contained in the region outside this solid angle is mini-mized. Examples of these methods are given by Borgiotti [I] who uses hyper spheroidal functions and by Mironenko [8] who uses Legendre polynomials. Both types of source functions yield aperture fields with a finite value at the aperture edge.

Methods to synthesize a radiation pattern with a prescribed side-lobe level are given for instance by Taylor [10], 1shimaru and Held [5], and Kritskiy [7]. They are concerned merely with patterns having a number of equal sidelobes, so as to achieve minimum beam width for a given sidelobe level. The aperture illumi-nations consist of a series of Bessel functions of the first kind and the zeroth order. The aperture fields have a finite value at the aperture edge, except for one type of illumination used by Kritskiy, which has a zero value there.

A method developed by Kouznetsov [6] who uses power series 1n r, the normalized radial variable in the aperture, and reduces the

level of the first sidelobe to a prescribed value is worth mentioning. This can be achieved with aperture fields having a finite or a zero edge value. For the Same level of the first sidelobe the finite value edge fields result in a higher aperture efficiency and a lower decay rate of the remote sidelobes.

In this report, patterns are computed by a number of methods and a comparison is made with certain antenna gain specifications. In section 1 the Taylor distribution for a circular aperture [10] is investigated in connection with satellite antenna gain specifications [3]. In section 2 a synthesis method described by Ishimaru and Held [5] is briefly treated. Modified versions of this method are presented in section 3. The aperture fields consist of series of Bessel

functions of the first kind and zeroth order. By judiciously choosing these functions one can generate aperture illuminations with a zero value and a nonzero first derivative of the edge field, or aperture illuminations with at the edge both the value and the first derivative of the field equal to zero. With the modified methods it is possible to prescribe individual sidelobe levels, allowing several different types of envelopes to be realized. The computed radiation patterns are compared with existing [2] and new specifications for the gain of earth-station antennas. The

I. The Taylor distribution for a circular aperture

The formulas derived in [4], [9], and [10] are used to compare the radiation patterns of the Taylor distribution with the satellite transmitting antenna reference pattern which, for the copolar com-ponent, reads

[3]

for -30 for

o

<i ~ <i I. 58 ~ o I • 58 ~ < ~ <i 3. 16 ~ o 0 (I. I)

after intersection with the minus on-axis gain, as the minus on-axis gain.

The angle measured from the main beam axis is ~ and ~ is the 3 dB o

beam width. The reference pattern is shown in fig. I.

o.---~~---~ -10

~

.. - 30

...

!

_...

z c '" - 40

~

0,2 0,3

I'I1MU' THE ON-~tIS - - - GAlI\

0,5 3 5 10 20 30 50

RmTm A~GL~

(1'/<p.J

Fig. I. Copolar component of satellite transmitting

antenna reference pattern.

For easy reference the formulas for the aperture distributions, the far field [4], [10] and the efficiency [9] are summarized below.

The aperture distribution g(p) 1S

g(p)

n-I

2

L

1T2 m=o

where F(u ,A,n) = 1 if m 0, o -J (1TU ) o m n-I { u 2 m II 1- 2 2 2 n=1 a [A +(n-j)

1

n-I {

ufu}

II 1 -2 n=1 u n nfm (]. 2) if m > 0,

p = TI pia is a normalized radial variable if a is the aperture radius,

A is related to the design sidelobe level n by n = cosh(1TA), a is the beam-broadening factor which equals u_/VA2+(n-j)2,

n

n is a parameter for

n -

1 equal sidelobes, 1Tu. is a zero of

1

J

1(1TU) = 0 so that the aperture field has a finite value at the aperture edge.

The point 1TU- separates the region of equal side lobes from that of n

decaying sidelobes. As n is increased more sidelobes are forced towards the level n and the beam width is decreased.

The radiation pattern F(z,A,;;) is

F(z,A,n) = 2 2J 1(1TZ) n-I {I- 2

~

2} __{~_ II a [A +(n-D )} 1TZ n=1

{I_Z~}

u n (]. 3) where z is related the angle S by z = to 2a A

the wavelength A, the aperture radius a, and sinS.

With aperture distributions the efficiency n

T 1S =

r

+ n-I

V(U

_{n ,A,n)(2]-1}

nT

~ ~=I /J

_{o(1TUn) (} (I. 4)

The efficiency 1S defined by the ratio of the maximum gain from the

1.1. Some observations

a. The normalized radiation pattern of the ideal Taylor distribution for

and

a circular aperture is [9]

2 2!

F (z ,A) = .=c.=o;:.s h={ 11,,-,,( A-=--..::z,--,,)_'.;..}

n

=

cos{1I(z2_A2)!}

F (z ,A)

n

for the main beam, (1.5)

for the sidelobes. (1.6)

This optimum pattern is shown in fig. 2, it is optimum in its beam width sidelobe relationship.

(.()sh

t

IT (A'-l' J'I,}

"l

MAIN f,EAM

-10 (.o~ \ IT (l'-A') 'Iz }

"l

-10

-so

Fig. 2. An optimum pattern.

An approximation to the ideal pattern is realized by setting the first

n-I

sidelobes equal to each other while the remaining side-lobes decay as (sin8)-3/2. For higher values of

n

the approximation is closer and the beam-broadening factor tends to unity.

b. The zeros of the approximating pattern are "VA2+(n-02 with n = 1,2, .... , n-l, n

if

z ~ u_. These are the zeros of

~

2 2 n

cos{n (~) -A }. If z ~ u- the zeros of the pattern are equal to

" n

the zeros of JI(TIz). The beam-broadening factor a coordinates the nth zero of

cos{TI~(~)2-A2}

and the nth zero of JI(TIZ), so that it

a TIZ

can be computed from

(I. 7)

c. Taylor [II] uses a series of J -functions with a finite value at o

the aperture edge and adjusts the first zeros of the radiation pattern to the zeros of

cos{n~(~)2-A2}.

Perhaps this kind of zeros

"

.

can be employed if a values. In that case

series of J -functions is

o used with zero-edge

F(z,A,ii) =

The aperture field is ii-I the radiation

n-I

II n=1 g(p) =

I

a J (u p) , m=O mom (I .8) (I .9)

where TIU is a zero of J (TIu) = 0 and a is an excitation coefficient m o m which equals 2 F(z,A,ii)!z=u (1.10) 2 m J I (TIU m)

Similar reasoning applies if the aperture field consists of the series g(p) n-I

L

a {J (u p) m=1 mom - J o (rru )} m where TIU

m is a zero of JI(TIu) = O. In this case the zero Uo be dropped. The F(z,A,n) = radiation pattern is 2 z 2J I (rrz) 2 z TIz(l- - ) 2 u l n-I II n=2 I - -'2--~---;;2"" a [A+(n-j) ] 2 z 1

-2

u n (1.11) = 0 can (J. 12)

and the excitation coefficient a equals m

( 1 • 13)

The equations (1.8) and (1.12) have not been investigated further, because they don't allow individual sidelobes to be prescribed.

1.2. Results and conclusions

Equations (1.2), (1.3) and (1.4) have been programmed with n = 3,4,5 and 6, a design sidelobe level of -30 dB, and a design 3 dB beam width of 3 degrees.

In figures 6 - 9 the radiation patterns are shown together with the specifications (1.1). The computed sidelobe extrema, 3 dB beam width, aperture size and efficiency are also indicated. In figures 10 and II

the accompanying aperture fields are shown.

The first n-I computed sidelobes are not exactly equal to each other. They decrease slightly with increasing distance from the main beam axis. The difference between the level of the first sidelobe and the design level is at least 0.4 dB.

One can notice that the decay rate of the far-out sidelobes is low. The efficiencies are high and range from 0.8377 to 0.8735.

With a prescribed sidelobe level of -30 dB and with

n

=

3,4,5 the

computed copolar patterns fulfil the requirements imposed by eq. (1.1). If

n

=

6 the requirements are not fulfilled.

2. The method of Ishimaru and Held for the synthesis of radiation patterns from circular apertures

In order to be able to show the analogy between the method of Ishimaru and Held [5] and the further-developed methods and because we believe that the Ishimaru and Held paper has not been widely available we will deal with the relevant part of it briefly now.

Ishimaru and Held [5, part I] describe a method to synthesize a radiation pattern having a number of equal sidelobes. They want to realize a pattern approximating the optimum beam width sidelobe-Ievel relationship. The optimum pattern has a minimum beam width for a given sidelobe level employing a given number of terms in the series

expansion of the source field. After determining the level and the number N of the equal sidelobes, the first N zeros of the radiation pattern are computed.

The radiation pattern g(u) from a circular aperture with radius a ~s

given by where g(u) I =

f

f(r)J (ur)rdr o o

r is the normalized radial variable,

(2. I)

f(r) is the aperture distribution which is r-dependent only,

1 2n .

e .

b d '

u equa s T a s~n w~th

e

the angle from roa s~de.

With N+I terms in the aperture-field series expansion one can control N sidelobe levels. Assume that the field can be expanded in a series of Bessel functions, f(r) N =

I

a J (u r) n=O n o n

=

0 if

o

~ r ~ I (2.2) if r > I .

The possible values of u are derived from the homogeneous boundary n

condition at r = 1

u J'(u ) + hJ (u ) = 0

n o n 0 n (2.3)

An acceptable approximation of the optimum pattern can be obtained if the following requirements are fulfilled.

a. The zeros of the radiation pattern are real. Nonreal zeros tend to produce higher sidelobe levels and broader beam width.

b. The constant h is zero. Then the first zero of the radiation pattern is as near to the origin as possible.

c. The number N is finite. The lower limit follows from the desired level and the number of sidelobes higher than that level. The upper limit is mainly bounded by the supergain consideration.

With h=O eq. (2.3) reduces to J'(u ) = -J (u ) = 0 o n I n (2.4) so u o r = I

=

0 _• u _I

=

the value 3.8317, u 2 = of the field 7.0156 etc. up is nonzero and

to and including uN' At that of the first

derivative of the field is zero. A few source functions are shown in fig. 3.

~s

1 IL, ' 0

.r,

11.,'

~.S';I'1'

3 IL,:

1-°156

Fig. 3. Source functions J (u.r).

o 1

Substituting eq. (2.2) in the integral for the far field and employing eq. (2.4) yields

I

{-anu g(u) = n=O 2 = !a J (u ) n o n J (u ) o n 2 2 u -u n J I (u) } i f u

1

u n if u = u n (2.5) (2.6)

After rewriting and normalizing eq. (2.5) the following form is obtained for the far field (see ap endix A),

1- u2 2J I (u) N (u +t ) 2 g(u} IT n n (2.7) u n=1 2 1-

₂

u u n

The first N zeros of the radiation pattern (u<u

N+I) are un + tn with

n = 1,2, ..• ,N. If u ~ u

N+1 the zeros of g(u} are the zeros of 2J I (u)

u

In the region u

l < U < uN+1 there are N sidelobes. Assume that the

extrema occur in

u = M with n = 1,2, ••.

,N.

n

The conditions for the optimum pattern are

(2.8)

(2.9)

where b is the desired sidelobe level and g is given by eq. (2.7).

A relationship sidelobe level

between the unknowns t , n =

n 1,2, ••.

,N

and the desired

is determined as follows. If the variables t

n

can be deduced from

are small compared with un an expression for g(Mn) eq. (2.7) which, together with eq. (2.9), yields N linear equations for N unknowns. Rewrite eq. (2.7) as

t2 g(u}

=

n=1 2un 1+ 2 2 u -u n 2 1+ - t u n n I tn + 2 2 u -u n + _1_ t2 2 n u n n (2.10) 2

Assume that the terms with t can be neglected and

n that the points

M

n are located at about the middle of u and n products in nominator and denominator can be

so that N 2u 1+

_L

n 2 2 t 2J I (M_i ) n=1 u -M. n g(M.) "" _{1 M. N} n 1 2 1 1+

_I

t n=1 u n n with i = 1,2, ... ,N. u n+l . The remaining approximated by series, (2. II)

Assume that the points M are independent of t , n n U +u M

=

n n n+J 2 (2.12)

Then eqs. (2.11) and (2.9) yield N linear equations for N unknowns t . If it is found that the variables t are not small enough to

n n

make eq. (2.11) valid and the optimum pattern is not approximated sufficiently closely, another set of

Assume that the first N zeros of the

variables t' can be computed. n

pattern closer to the optimum

are u +t +t'. The pattern representation is

n n n _u2 1-(u +t +t,)2 2J I (u) N g2(u) = II ₂ n n n u n=1 _1- u

2"

u n 2 2 1- u 1- u 2J I (u) N (u +t +t,)2 N (u +t ) 2 II n n n II n n = u n=1 _1- u 2 n=1 2 1- u (u +t ) 2

2"

_u n n n 2 1_ u N gl (u) II n=1 (u +t +t,)2 n n n 2 (2.13) 1- u (u +t ) 2 n n

if the first approximation to the optimum pattern is 2 1_ u . 2J I (u) N (u +t ) 2 II n n u n=1 u2 1-

2

u n (2. 14)

The eqs. (2.13) and (2.7) are similar. The procedure for computing t can also be used for t'. In eqs. (2.7) - (2.12) replace 2JI(u)

n n by gl(u), u (+t ) by u +t (+t'), t by t' M n n n n n n n' n M'

=

n u +t +u +t 2J (M ) n n n+l n+l 1 i 2 and M. by gl (M;.>. 1 by

The adjusted equation for geM. ) and the optimum condition

1

sidelobes) yield N linear equations for N unknowns t' • As n

a pattern is synthesized with three -25 dB sidelobes. The

u

(N equal an example, zeros of

JI(U) are assumed to be U

o

=

0, ul

=

3.83, U

z

=

7.016 and u3

=

10.173. The sidelobe extrema of 2Jl(u) are taken to be in Ml = 5.33,

u -I

M2

=

8.53 and M3

=

11.70. The -25 dB level means b = 17.78 • Solving for t , the first approximation is

n

2J I (u)

(2. 15) u

The first sidelobe of gl (u) is still higher than -25 dB so that it is necessary to compute t'. The improved pattern is

n

g2(u) =

2J I (u) u

The coefficients a are computed with

n g2 (un)

a = 2 n = 1,2, ..• ,N.

n i(u )

o n

The aperture field which generates g2(u) is

(2.16)

(2.17)

(2.18)

The coefficients we have computed differ from those given by Ishimaru and Held [51.

If the method of Taylor is used in synthesizing a radiation pattern with three -25 dB sidelobes one gets

(2.19)

The patterns gl (u), g2(u) and gT(u) are shown in figs. 12 and 13. The first zero of g2(u) is nearer to the origin than the first zero of gT(u) so that g2(u) has the minimum beam width. For g2(u) and gT(u) the deviations from the required levels are of the same magnitude. The maximum gain G is given by

I 2

If

f(r)rdrl

Substituting for f(r) yields 81T2 a 2

::---,g,,--2~(~O.!-)

_ _ G=---;ZN 2

L

!a J (u ) n=O n o n (2.21 ) Using 2a 2g(u ) = D and a = n n i ( u ) o n results 1n (2.22)

3. The modified methods with other source functions

3.1.

~~~E£~_£~~£~!~~~_~!~~_~_~~E~_~~g~_£!~!~_~~~_~_~~~~~E~_£!E~~

~~E!Y~~!Y~_£f_~~~_~~g~_f!~!~

In this section we will employ the source functions which arise on the assumption h

=

~ in eq. (2.3). The aperture field f(r) consists of a series of Bessel functions

f(r) = N

I

a J (u r) O n 0 n n=

=

0 O " r < l , (3. I) r >

r

If h

=

~ the homogeneous boundary condition, eq. (2.3), reduces to

J ( u ) = o . (3.2)

o n

The possible values of un are now U

o

=

2.4048, ul

=

5.5201 etc. up to and including uN'

A

few source functions are shown in fig.

4.

1 ~i =- ZM~8

~

\.I.,'

S.,l.01

Fig.

4.

Source functions J (u.r).

o ~

N sidelobe levels can be controlled with N+I source functions. The far field is

I

g(u)

=

J

f(r)J (ur)rdr

o o

and with eq. (3.1) and eq. (3.2) g(u) is computed as N g(u) =

L

n=O {a n if u

f

u n (2. I ) (3.3)

g(u) =!aJ2

1(u)

n n if u u • n (3.4)

Rewriting and normalizing eq. (3.3) in the same way as eq. (2.5) gives g(u)

=

J (u) o

~

{I_U:}

n=O u n (3.5) If u ~ u

N+1 the zeros of g(u) are the zeros of Jo(u) = O. If u

<

~+I the zeros of g(u) are u +t. n =

n n 1.2 •••.• N. To show the analogy with the far field where h = O. eq. (3.5) is written as

g(u)

=

J (u) N o II 2 1- ~ n=1 2 u o 2 u 1- - - - ' " (u +t ) 2 n n 2 1- ~ 2 u n J (u) o

The factor ~-"'2 is proportional to the field of

I-~

2 fig. 14. The first sidelobe is

(3.6)

J (u r) shown in o 0

-27.5 dB and the

u

o second is -36.4 dB down. The zeros. the efficiency and the 3 dB beam width are also indicated. The changes in the

aperture field can be small in order to lower the first side lobe to -30 dB. for instance.

Where h = 0 the expression for the far 2 2J 1 (u) N g(u) = - ' - - - II u n=1 u 1-(u +t ) 2 n n 2 I-~ 2 u n field is (2.7) 2J 1 (u)

The factor is proportional to the field of the uniformly

u

illuminated aperture. The first sidelobes are -17.6 dB and -23.8 dB down. Write eq. (3.6) as g(u)

=

J (u) N o II 2 1- u n=1

2

u o 2u n 1+ 2 2 tn + u -u n

I+~t

u n n 1 2 + - t 2 n u n 1 t2 2 2 n u -u n (3.7)

In order to determine a relationship between the variables t and n the prescribed sidelobe extrema we proceed in the same way as in section Z. Assume that the extrema occur at the points M. at about

1

the middle of u. and u. I. Assume t is small compared to u so

1 2 1+ n n

that terms with t can be dropped. The products in nominator and n

denominator can be replaced by series and

N Zu 1+

_I

n t J (M.) n=1 uZ-M~ n g(M.) :::! o 1 n 1 (3.8) M( N 1 ~t 1

-

1 1+

I

2

n=1 u n n u 0 with i = I.Z ••..• N.

We introduce the following conditions for the side lobe levels

(3.9)

N equal sidelobes is a special case of eq. (3.9). The points Mare n taken to be independent of t n

M

=

n u +u _{n n+} 1 2 (3.10)

Then the equations (3.8) and (3.9) provide

N

linear equations for

N

unknowns tn. The first approximation gl(u) to the desired pattern is

J (u) N o II Z u n=1

1-2

u o Z I __ -'u"---..,,. (u +t ) Z n n 2 I-~ Z u n

If this approximation deviates too much one can introduce the variables t~ to describe the better pattern gZ(u)

J (u)

N

o II uZ n=1 1-Z u o Z 1_ u (u +t +t')Z n n n Z I-~ Z u n (3.11)

2 1_ u (u +t +t,)2 n n n 2 I-~ 2 u n (3.12)

The computation of t' follows the same lines as the computation of n

tn due to the similarity of eq. (3.12) and eq. (3.6).

2 In eqs. (3.6) - (3.11) replace J o(u)/(I- u2) by gl(u), u u (+t ) by u +t (+t') n n n n n t by t' n n'

M

n = u +u _{n n+} 1 2 by

M'

n

=

o u +t +u +t n n n+1 n+1 2 and

M~

J (M.)/(l-

2.

2) o 1. u o Again N linear equations are obtained for ~N- variable~-and-the t' are

n easily solved.

Instead of eq. (3.10) one can also use

M

= n u +d u n n n+1 2 (3.13) where d n with the < I. "-real

Then the points M can be n

positions of the sidelobe

cho·sen in better agreement extrema. The weight factors d can be estimated from the pattern of J (u r).

n 0 0

The excitation coefficients a are computed with n

g(u )

= 2 n

an 2 n = O,1,2, ... ,N.

J I (un)

The maximum gain G is derived from

1 2

II

f(r)rdrl G = 8 22 IT a ~o ________ __ ).2 1

I

If(r)12rdr o (3. 14) (2.20)

Substituting the source functions we are concerned with, yields

G

=

(3. 15)

G = (TID)2 g2(0) A N _g2(u

n

)

L

₂ n=O J I (un) (3.16)

The difference ~g(dB) between the normalized power patterns of J (u r)

N 0 0

and

L

a J (u r) for large values of u is estimated as n=O n o n

~g(dB) ~ 40 N 10

L

log(l+ - ) tn u

n=1 n

(3.17)

With suitable values of the variables b. in eq. (3.9) the pattern

1-envelopes can be made to fulfil, for instance, the specifications for earth-station antennas [2] . The following requirements for G. ,

1-the gain relative to isotropic, are represented in fig. 15 :

G. 32 0 (3.18a) I = - 25 * log (S ) dB 1-32 0 (3.18b) II G. = - 30*10g(S) dB 1-G. 32 0 (3.18c) III = - 35 * logeS ) dB 1-0 (3.18d) IV G. = 25 - 25 * logeS ) dB

1-These formulas are assumed to apply to the region beyond the first A

sidelobe peak, that is at and beyond S(degr) ~ 100

D'

until -10 dB relative to isotropic is reached.

The method described in section 3.1. is used to compute a number of radiation patterns. In figs. 16-20 we represent computed patterns and aperture distributions if only the first sidelobe is prescribed at a certain level. Furthermore, some zeros, the efficiency, the gain reduction relative to the uniformly illuminated aperture, the 3 dB beam width and the excitation coefficients are indicated. If we compare these 5 patterns with each other and with the 'reference' pattern of fig. 14 (no sidelobes prescribed) we note that:

- The deviation between the prescribed and computed level increases if the difference between the required level and the level of the first sidelobe of the 'reference' pattern is increased. The devia-tions in the first approximation patterns are less than 0.9 dB. - Increasing the first sidelobe, relative to the 'reference' pattern,

results in a shift of the first zero ul+t

l towards the origin, an increase of the efficiency and a decrease of the 3 dB beam width. The shift of the first zero results in opposite signs for the excitation coefficients a

o and al.

- Lowering the first sidelobe, relative to the 'reference' pattern, results in a shift of the first zero ul+t

l away from the origin, a decrease of the efficiency and an increase of the 3 dB beam width. The shift of the first zero now results in a

o and al having the same sign.

- For greater values of the difference between the prescribed level and the level of the first sidelobe of the ,'reference' pattern, the shift of the first zero as well as the ratio of the excitation coefficients 1:11 are greater.

o

- The difference between the first and the second side lobe level of a pattern decreases if the level of the first sidelobe is decreased. The estimated differences ~g, eq. (3.17), range from -4.05 dB to 2.11 dB. These values are computed for the -17.0 dB and the -35.1 dB sidelobe patterns. The differences between the computed and the

'reference' pattern range from -4.4 dB to 2.2 dB for the fifth side-lobe. Hence the value of ~g is almost reached at u ~ 20.

- The decay of the sidelobe extrema is greater than the decay of G.,

1

eq. (3.18a-d). These specifications are therefore fulfilled if the peak of the first sidelobe is below the curves I - IV of fig. 15 when the gain reduction is taken into account.

- If the first sidelobe is prescribed to have a higher (lower) value than the first lobe of the 'reference' pattern, the normalized aperture field for 0 < r < I is higher (lower) than the aperture field generating the 'reference' pattern.

In figs. 21-25 we represent computed patterns and aperture distri-butions if two or more sidelobes are prescribed to have equal levels. In most cases it is necessary to compute a second approximation

because the levels in the first approximation deviate more than dB (otherwise an arbitrarily chosen value) from the required levels. The results of each approximation are indicated: the computed levels, the zeros, the excitation coefficients, the efficiency, the gain re-duction relative to a uniformly illuminated aperture and the 3 dB beam width. If we compare the pattern with three -25 dB sidelobes of fig. 21 with those of figs. 12 and 13 we observe that the sidelobe

levels not prescribed are lower and decay faster, and that the 3 dB beam width has increased (the first zero of the pattern is further away from the origin). These effects are due to the fact that we are now dealing with zero-edge aperture fields. If we compare the patterns of figs. 22-25,having two or more sidelobes prescribed to have the -30 dB level, with the Taylor patterns of figs. 6-9 we note that the required levels are now reached more accurately, that the 3 dB beam widths are greater and that the sidelobes not prescribed are lower and decay faster. The efficiencies are lower than in the case of the Taylor patterns. If we look at the accompanying aperture distributions we see that the Taylor distributions (figs. 10 and II) are smoother, which is imputed to the different source functions and to the fact

that the sidelobes of the Taylor patterns are actually decaying instead of being equal. If we compare the patterns in figs. 22-25 with each other we see that with an increasing number of equal

side-lobes the zeros have shifted more and more towards the origin, the 3 dB beam width is decreased and efficiency is increased. With N-I prescribed sidelobes the width of the lobes I up to and including N-I increases. For instance in fig. 25 these widths are successively 2.22, 2.80, 3.34, 3.57 and 4.26. The width of sidelobes not prescribed is always equal to the distance between the zeros of J (u), which is

o

approximately 3.14.

The requirements set by eq. (3.18a-d) are not all fulfilled. The patterns of figs. 22-24 meet a-d, a-c and a-b, respectively, while the pattern of fig. 25 meets none of these requirements.

In figs. 26-33 we show computed patterns, aperture distributions and other relevant data when, starting at different levels for the first sidelobe, the decay rate of the first 3 or 4 extrema is prescribed. Once again, if there is a deviation of more than I dB between a re-quired and a computed level a second approximation 1S made. Decay

rates of 4,5,6,7 and 8 dB are required and the starting levels are -30, -33, -36 and -48 dB. However, not all the combinations of rates and levels are investigated. At the cost of the efficiency and with increasing 3 dB beam width the starting level can be decreased and the decay rate can be increased.

All the diagrams shown meet the requirements of eq. (3.18a-d).

The computed differences ~g according to eq. (3.17) for the patterns of fig. 26 and fig. 33 are, respectively, 0.2 dB and 7.8 dB. These final values are almost reached at the fifth sidelobe extrema (u ~ 20) because the actual differences (compare with fig. 14) are 0.3 dB and 9.4 dB.

The normalized aperture distributions for the described patterns are smooth functions which, for 0 < r < I, are lower than the distributions for the pattern with none of the sidelobes prescribed (fig. 14).

Finally, 1n figs. 34-36 some cases are illustrated in which the first sidelobe 1S required to be lower than the second. We observe that the required levels are reached within the 1 dB limit in the second

approximation, except the -50 dB level in fig. 36.

The 3 dB beam widths are increased and the efficiencies are decreased compared with the pattern of fig. 14. The differences ~g according to eq. (3.17) and the actual differences to the pattern of fig. 14 at u ~ 20 are equal for these three patterns. They are respectively -0.3, 0.2 and 1.6 dB.

The patterns meet the specifications imposed by the equations (3.18a-d).

The normalized aperture distributions for these patterns are smooth functions having lower values for 0 < r < 1 than the distribution J (u r) shown in fig. 14.

o 0

In conclusion we can say that the method described in section 3.1. 1S flexible, accurate and capable of handling several types of side-lobe envelopes. The source functions having zero-edge values ensure a more rapid decay of the side lobes not prescribed than do the distributions with non-zero edge values.

3.3. ~~~E£~_f~~£~i~~~_~i~~_~~~~_~~~_f!~!~_~~~_~~~_f!E~~_~~E!~~~!~~

~f_~~~_f!~!~_~~~~!_~~_~~E~_~~_E_=_l

The aperture field is taken to be a series of the form N

f(r) =

I

a {J (u r)-J (u )}

n=l n o n 0 n

o

.cS r ~ ] ,

= 0 r > I ,

where the values of u are determined by n

(3.19)

(3.20)

The normalized source functions with u

1 = 3.8317 and u2 = 7.0156 are shown in fig. 5.

1

J.(UL'I.)-

J.(",1

1

1 -

J.("d

Fig. 5. Source functions.

1

11.,.

3.8~11

,. IA.t-1

01Sb

With N source functions, N-I sidelobe levels can be controlled.

Now the radiation pattern of a circular aperture is

N

I-a u

2

g(u) =

I

~ n₂ Jo(un)J1(u) if u <I u

n (3.21)

n=1 (un-u )u

=!aJ_{n o n} 2(u) if u = u

n (3.22)

Eq. (3.21) can be written in a form similar to eq. (2.7) and eq. (3.6), namely

2 1- u 2J I (u) N (u +t )2 g(u) IT n n (3.23) 2 2 u n=2 _1- u u(l- - ) 2 2 u l u n 2 u

The factor 2J I

(u)/u(I-Z ')

is proportional to the field of the aperture

u l ;

distribution Jo(ulr) - Jo(u_l), see fig. 37. The first sidelobe is -35.1 dB down.

Assume that the variables lobe extrema occur in the

t

n are small compared to u n and the side-M. at about the middle of u. and

1 1

points

u. I. Suppose further _,+ that the points M. are independent of the

1

be evident from the foregoing treatment in variables t . It will

n

sections 2 and 3.1. that N-I linear equations for N-I unknowns t n are deduced from the approximation

N 2u 1+

_L

n

u2-M~

t 2J I (M_i) n=2 n g(M.) ~ n 1 1 _M.2 N 2 1 1+

L

t Mi(l- -2-) _N=2 u _n n (3.24) u l

with i = 1,2, •.. ,N-I and the condition

(3.25)

The variables bl-b

N_I are the prescribed sidelobe levels. The first approximation gl(u) to the desired pattern is

2J I (u) 2 u u( 1- - ) . 2 u l N II 2 1- _u=-_o 2 (u +t ) n n n=2 1- u 2

Z

u n (3.26)

If the sidelobe levels of gl(u) differ too much from the desired levels, the variables t' are introduced to make the computation of

n

1- u N (u +t +t,)2 g2(u) g I (u) IT n n n ₂ (3.27) n=2 u 1-(u +t ) 2 n n

Eq. (3.27) and eq. (3.23) are similar so that the scheme used to relate linearly the sidelobe levels and the variables t can be

n employed to do the same for the levels and the variables t'.

n

After the desired pattern has been obtained the excitation coefficients a are computed from

n

a

n = 2g(u) lu=u _{i ( u ) n}

o n

with n = 1,2, ... ,N. (3.28 )

The maximum gain G for the aperture with radius a is

I 2

If

f(r)rdrl G = 8 22 1l a ).2 o (2.20)

With the applied source functions, 2a = D and eq. (3.28) G becomes 2 G = (1lD)2 g (0) A

{N

_g(Un

)}2

N g2(u_p)

I

J (u) +

I

':"'2,---LL. n= I o n n= I J (u ) o n

An estimate for the difference 6g(dB) between the normalized power patterns of {Jo(ulr)-Jo(u

l)} and is for great values of u

L\g(dB) "" 40

I

N 10 10g(l+ - ) tn n=2 Un (3.29) extrema of the N f a {J (u r)-J (u )} ~=l n o n 0 n (3.30)

The method of section 3.3. is used in the numerical computation of several radiation patterns.

In figs. 38 and 39 we show the patterns having the first sidelobe prescribed at -30 dB and -33 dB, respectively. These levels are higher than the -35.1 dB level of the pattern of Jo(ulr) - Jo(u

l), fig. 37, which for the moment, is the 'reference' pattern. The first zero of each of the patterns shifts towards the origin, the 3 dB beam width

decreases and the efficiency increases relative to the 'reference' pattern. Because of the shift towards the origin, the excitation coefficients a

l and a2 have opposite signs.

The differences ~g according to eq. (3.30) are for the patterns of fig. 38 and fig. 39, -1.6 dB and -0.6 dB, respectively, while the actual differences for the fifth sidelobes are respectively -1.8 dB and -0.7 dB.

The patterns meet the requirements imposed by the equation (3.IBa-d).

The normalized aperture fields for 0 < r < I are somewhat higher than the aperture field for the 'reference' pattern.

For equal sidelobes we have given some examples in figs. 40-44. In some of these cases even a second approximation does not allow us to obtain differences between required and computed levels less than I dB. The diagrams shown have two or more equally prescribed sidelobes. The efficiencies are higher and the 3 dB beam widths are lower than in the case of the 'reference' pattern. With two -30 dB or -40 dB lobes the required levels are adequately approxi-mated and the aperture fields remain rather smooth. With three or more -30 dB sidelobes the required levels are not reached in two approximations and the aperture fields are not very smooth. Apparently the changes in the levels relative to the 'reference' pattern are too great, and although the method is capable of handling more than two approximations, no more than two have been carried out.

In figs. 45-48 we show some computed patterns, aperture distri-butions and relevant data, starting at different levels for the first sidelobe. The decay rate of the first three extrema is

prescribed. Now the deviations between required and computed levels are less than dB with one or two approximations. The efficiencies and 3 dB beam widths differ little from the 'reference' values

(fig. 37) 0.5 and 4.B due to the chosen sidelobe levels.

The differences 6g according to eq. (3.30), computed for instance for the patterns of figs. 47 and 48, are respectively 2.3 dB and 1.2 dB. The actual differences at the fifth sidelobe extrema are 2.4 dB and 1.2 dB.

The normalized aperture fields of figs. 45-48 are smooth and have changed little compared to the aperture distribution of fig. 37.

In fact many more cases could be handled but the results and observations would be similar to those of section 3.2. The dif-ferences are found in the magnitude of some characteristics. If patterns with the same prescribed levels are synthesized with the source functions of section 3.1. and section 3.3., then with the last-named we find the following:

- the zeros of the radiation pattern are further away from the origin, so that the 3 dB beam width is greater;

- the efficiency is lower;

- the sidelobes not prescribed are lower and decay faster.

This is clearly demonstrated in the patterns of figs. 45 and 31 for sidelobes prescribed at -36, -41 and -46 dB and in the patterns of figs. 47 and 33 for sidelobes prescribed at -48, -52 and -56 dB.

In conclusion, we can say also that the method described in section 3.3., using source functions with both the field and the first derivative of the aperture field equal to zero at r

=

1 enables us

to prescribe several types of sidelobe envelopes. These source functions ensure an even more rapid decay of the unprescribed side-lobes than do the source functions with only the field values equal to zero at r=l.

References

[lJ Borgiotti, G.

DESIGN OF CIRCULAR 1."ERTURES FOR HIGH BEAM EFFICIENCY AND

LOW SIDELOBES. Alta Frequenza, Vol. 40 (1971), No.8, p. 652-657.

[2J C.C.I.R. Recommendation 465-1. REFERENCE EAEFaH STATION

RADIATION PATTERN FOR USE IN COORDINATION AND INTERFERENCE ASSESSMENT IN THE FREQUENCY RANGE FROM 2 TO ABOUT 10 GHz.

In: C.C.I.R. XlIIth Plenary Assembly, Geneva, 1974. P. 155-156. Geneva: International Telecommunication Union, 1975.

[3J FINAL ACTS OF THE WORLD ADMINISTRATIVE RADIO CONFERENCE FOR THE PLANNING OF THE BROADCASTING-SATELLITE SERVICE IN FREQUENCY BANDS 11.7-12.2 GHz (IN REGIONS 2 AND 3) AND 11.7-12.5 GHz (IN REGION 1). Geneva: International Telecommunication Union, 1977. P. 102.

[4] Hansen, R.C.

TABLES OF TAYLOR DISTRIBUTIONS FOR CIRCULAR APERTURE ANTENNAS. IRE Trans. Antennas and Propagation, Vol. AP-8 (1960), No.1, p. 23-26.

[5] Ishimaru, A. and G. ~

ANALYSIS AND SYNTHESIS OF RADIATION PATTERNS FROM CIRCULAR APERTURE. Canadian Journal of Physics, Vol. 38 (1960), No.1, p. 78-99.

[6J 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 Electrical Engineers, 1978. P. 422-427.

[7J Kritskiy, S.V. and M.T. Novosartov

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.

[8J 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.

[9J Rudduck, R.C., D.C.F. Wu and R.F. Hyneman

DIRECTIVE GAIN OF CIRCULAR TAYLOR PATTERNS. Radio Science, Vol. 6 (1971), No. 12, p. 1117-1121.

[101 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.

Appendix A

Derivation of eq. (2.7).

Start with eq. (2.5) for the far field g(u). Use J (u )=J (0) = I.

o 0 0

then g(u) becomes

N -a uJ (u ) g(u) = J 1 (u)

L

n o n = 2 2 n=O u -u _n J 1 (u) [ao-2 _

~U2Jo(UN)]

a1u Jo(u l ) = ₂ 2

...

2 2 u ul-u uN-u

t

o

-al~

_{Jo(u l )} 1 J 1 (u) u

.. , '0(""']

1 uN = 2

...

u 2 I-~ u 2 1-

2

u 1 uN

----J 1 (u) = u 2

The nominator of the form between brackets is a polynomial in u of order N. the zeros of which can be expressed by (u +t )2 with

n n n=I.2 •....• N. so that g(u) can be written as

2J 1 (u) N g(u) = - ' - - - II u n=1 2 1 - _--,u=---,c (u +t ) 2 n n 2 1- ~ 2 u n

if at the same time g(O) is normalized to I.

0 b

8

10 11 I~

,b

18

.to

.u.

l~ -n

-

₀

,

I

,

~ 0 -

j)A

~ U.11

G

(UlitFORM) ~ ~pz.

d&

G

(TAYLOR) >

36.'15 de.

~ - t -10

,

EHI(',II:.NCY' 0.8311 :>- \

...

_{-< \}

'PRt:.sc.IUBED L£.VEl : -30 da

--\ _{r- \} :x: C> _\ ;;JO _\

ii :

_.3

~ \

9

3da

~

!.qs

(\)EGR) 0 _\

...

..

""

-20 \

ItA

1>1

it 11 0 N

'PATT'erUI

...

0 \ c::

WARe

n

.-""

\

- - - .

:0-

...

\ ;;JO

.->- :» --I

...,

_...

I

,...

""

-30 ;J> ;;>0 _rn

--< _N c:: -d

---" .

'"

0 ;£.

--

--

---.."

...

""

_""

...

--t r-- _-'10

"'

t3

""

~

-43

-44.S

:z ~

'"

""

I

-so

...

₀ 0...

...

<II

....

'"

-60

<TO

,....

0 0> IT' V'

0

8

10

1b

1.

10 .22

..,.,

0 ~ !" -..J

'PI>. "

_lUB

G(UrUfORIYI) "~lH d~

~ --i

,..

-10 6(TIIYI.OR) "

_{E HI(lEKC. Y}

₌ 3b.ll.t

_{b. 'i!'181}

d

&

-

-<

....

r-LEVEL"

-?>o

dB

:c

,..

0 _'PP.~s(fIl&E

_D

~

_...

... _\

n:'!

\I

""

\ e~

_{d& '" 1.q&}

(l>E('~)

.r r>

'"

-.to

c::

"'

RAI1IATtoN 'PATIER N

...

,..

.-

\ > \

WARC

':\1-...

- I

_....

_\

--

->- . " \

...,

"'

\

"'

""

-.0 --1 _{0_30 I} c:

---

-

---'"

~

--

?>

'"

_"'

_""

--

'"

-~~

---""

....

...

---I

---rn 6--38.~ ;P _cP z ~-'IO

..,..

-1111 <:> ;10 I

....

_{<:> -50} ~ ~ ~

'"

'"

r--60

0 gO m

'"

-!"

...,

...

:E:

>-...

-<

- I

.-

₀ :t:

'"'

~I ...

...

_,..

"

...

V1 c:: r-

>-,.

>

....

rn

...

- I c::

,..

'"

...

,..

....

....

'"

; 0 :z

...

0-,.

,

...

0 0-00

'"

...

<:I " , r-C>

.,.

".. oJ' o -10

,..-10

....

.-""

....

I--' <: rn ~ -)0 o

:t:

,." ?O

r-CL._IfO ~ -50 -(,o

,

\ \ \

,---Jo.~ - 31

--~~ ,. tUb

G(ULUFO~r1) ~

11.01 dB

6 (TAY LOR) " 3b.!>l d ~

E F FInE-Key : 0.8

bU

"PREStUaEl1 LEVEl :. -

~ll

dB

ii~5

&"

elY> "

l · n ('PEGR) - - U)lATI6N l'AHERN

- - WARC

H

-ii ....

----1>1-:;'---

---1/

\

I

>

0

....

....

!"

...

-10 ~ > -4 H

_....

_-<. r-:r:

""

,.,

:II _r- -10

•

....

0-

,....

'"

C r-:J>

.,.

>-""

" ,

""

..; C " .

'"

...

:P -< -< rn :;0 Z

..,..,

co

-so

:;0 I ~ 0-00

_-1.0

~ 'OJ

'"

...

<>

...

_rn

""

I I \ \ I I \ \ \ \

---

... -lo.~ _'ft"l - _ ... ,

-31.~-••

PI). ~

1.1.18

G(UNrfO~I"1) : 3~.go d~ G(TAYLOIt) = 16.31

de

t:HIClE~[¥ = o.Btl5

n=6

03d! '" 1.

98

(~E6R) - - RA»Io\Tl6M l'ATrER rl

- - - WAlen

--

- - -

---

- -

----I ;J>

0.1 o

m.10

0.5 fIIOIlMlIllHO RAl/iUS

'"

---1

C.lRCULAR AtHTURE

)!STuaUTlOll~ f~ 1AYLOIt llATTflN5 'WllH

1-1

51 DE LOU

B TREMA OF

-lO

di.

0.'

r--__ ~

...

...

-" ' U

....

_co: ::J

:;;:

..u

...

'< •. , 0.1

o

0.5 N()RMALU:~\1 RAPlU& fi~b " . .

---

_n.

-

₅

,

fl6.11:

C.IRCULAR

AfHTUR.E nST"UUTION5

FO~

TAHOR

'PATTERK~

WITH

11.-1

51DE.L08f

EXTREMA OF

-30

dlo.

-64

~

'"

~ -JO ~ ~

'"

0-w

-t,

>-...

I-..:

_....

,

...

0< -SO

- 60

\ \ \ \

\

\ \ \

,

\ I I I \ I

I,

-Jl.,

\

1

I 1 I

fIG. 12. RAvunOI'l l'~nE.«'KS WITI1 ·HI~E.E. -%5 cl~ 511>Hoi!>~~ Accoll>tNG To

EQ,.Ct-15)

AND

Eq.(.z-16 ) - - - .

·15.3

-30.1

-n.1

FIG.

1!1 :

M»rATION

1'.~iT£R~

WITt! TI-\Rf.E

-%'5 d~ SIDELO!>E.,>

olD

...

>

-so

...

~

..

_...

- -60

1 o

6

8

-n.s

10 -36.'1 11. fit f6 18

-'q.l

nn

zuo~

:

U, ' 5.5101 u.:

8.&n1

11.\'

Iqq 1 5

!.l~'

1'i.

gso,

u's' t8.o~t1

"l.'

21.111

b

EfFl<.IEl'K Y: o.o~

11-6AINREDu(.1l0N:

1.b dll

113c1& • 4.15 ~~~~-~~-~~-~~~---'''-o o.~ 1 NOlMAll!E.D ~I'.Ptu!> 't

1'16.111

_{APERTURt H~1",nUTlON ].(1.110'"18 '1,) All:!> RA1>lATION}

lATTElN

10 LOG

I

Jo(~)

I

10 (1/.:\1

t~r.)

_i.

₁

_{~ ~}

-0"

--t

=

100

-11 _U.

₌

_"If

_-}'i>lIt

_(e)

-31

x

I

...

"'"

<!l _It ~ -~D :II: z: _.lll"

'"

I

....

:>

..

""

l!1OTROl'I(.

'"

-~o 1 ~,·31.-1·)l LOG (e) e~ 1"

-60

[ G(.,ll - !>O • LOH&l ,. ]I(

G.:31-

~s

•

L06(&) " ]I[ <1,'15 ~ 1S II LO& (9) "

-1°

FIG. 15 :

~

....

-0 D _fO -30 ...; -'10 ~

...

:z

4 po. -SO

'"

"'"

H ~ oC ~-OO

o

FIG. 1

b :

N

_'0

- 1/ .• 1-It

''I

16

I~

5li)HO'~ lEVEL : l'R£:S(RUfD -11-~

dB

1).1+\:1 ~ lUroS llz. ~

8.6,> 31

E~('ITA nON co E FFIClEt( g .

a

_{o '} ~.q

881t'{

1l 1= -2.~s916 EHlctENC.y

=

D.S20~ GAIN KEDUCTION=

0.86

J~ Ll~dg

=

3.Q

20 ...

u..

COiYIPUTED

-Ilo dS

Aff.RTlJl\~

'l>It"u.HuTIoW

L

dlft.}olLl.",'t)

AlW

RI'tPI,.nON

'PATTERN

o

o

-/0 -.II .

..

s

...

...

..

_~

....

'"

w -JO

1 1 - - - _

;i 0.5 \;

~

(,

a

10 -33.8 1. It Iv

/6

18 .10 -+

u.

- '10. J -fS .

.t

61l>HO~J: llO VEL 'PRE.~C.1&c!) (a<tPUrEp -12.5 dB -1.U dB i.I.,-t

t,

= 4.q55~ IA.l = a.b5~"t nc.n"nOH C:OEFFIC.UMi"!>

a..:

~.~poB <11 = -o'QI311 EFfIW:NC'l':

a·l

bn

['AlM RE.l>UCTION·1.1 & d~

m.q:

A?EHUlE

1>ISTRnUTIo~

I

o.",Jo(,tA.,,:'t)

A~D RAOIHIO~

1'A1TERI'i.

-11

-JI ..

...

-~O ~ o

C>-....

:;;

....

OAf tIC _ z 1 o

..

_;:;

'"

~ l -on

...

A

""

...

'"

...

_....

oC lE ..:

_.,

'"'

-Jo

.

..,

1

IIOl"ALIU) _Anus If,

t

,,,

16 ,8

-u.S"

-'<'.J. .SlIlHO&E Lnn HRO!> ~~C.lTATION

COEFfICIEHTS:

a.o~ ~·l15't& Q.1' 0.~1'11 UFIC.lE ItC Y :

o.

6b

~ 0

GAUl "DLlCTION : 1.1&

de>

u

~.r.

=

It.

.t3

.10 _ I L

-Sol.

C.OI'lPUTE.D

-3D

ds

FlG.

18 :

MUJURE. 111':> TRa

unoI'{

L

11. ... ). (",,,,)

"Ml>

1"l)IAllON 'I'ATTEllM.

-10 w

... -to

~ C>

....

..

0

...

....

::;I

...

...

..

....

'"

....

PI

...

1 ~G.5 ~

_...

...

< A

...

...

:1

I

_..

0

o

0 6 0.5

NO~"'AlIH1) I\Al>tu~ 'II

8 11)

-JI·1

1

u

18 20 _ _

u-!\

-s2.8 !>l1>~L01!>~ L~~h '. ?1\~S(RtaED ('Ol'll'UTE~ -U d.. -.U.'f ciS IE.P.O!o: U, + ~, : b.O-U'i Ul "8.&$11 H(ITATlON. COf.HL(.1EI(T!>:

a..:

~.668~8

a

_{1 ,}

o.

6 09~J HfI(lHIC,(: O. b381 GAlli REJ)uCTIOII: 1.QS d~ U!.di : 4.31

fIG.

1~

:

AP~R1Ul\t. 1)I~

_{n.l&UTID N}

~

(1. ..

1.

(IA. •• :',) AriD R.At>IATlON nTH It I'L

-/0

...

_{,.. -SO}

..

_~ -' wi

•

_-60

1 'Z: CI

...

_...

"

-

:a

_~

:a

po

...

~,.S

j

Q

...

!::l

i

_D 0 O.S

"o"MAU~ ~ 1) II.A Otu:. '"

-35.1

1

1

-s~.~

51VHO&e LEVU: ~REs(R1aE]) CO,",PUTn'

-36 dB

-3s.1

dS

ZERO:': U,+t" b.BlS

Ul.

• 8.6Hl

EtC.l1ATION COHF1qm~;

a" :

U

".'.f

a./ :

D.ll""

HFI'HIlC.Y ;

o.

bl, 1 GAtN II.E1:>UCiIO!'t: %.01.

dB'

L(!> d~

=

1.(.

?,t

FIG. 10 :

AP£'. TUKE

1)1

S

1\\1

~UTICIj

Z

/1.",10

(11".'1.)

AN

~

RA1)lHIOli PATTHIII. 1'/1 ••

o

,

l 8 I. I~ 16 Ii .tp -+ u. '/0

.J'

/--

-"

\ / \ -JD \ (

\

_I I _\ \

_\

,

~

_I

_I

,

_\ -0 _\ -..J .~,

I

I \ ~

\

\ /-~

l

\ I \ I

....

\'

I. .... SO

...

...

...

_...

...

...

-.6D

,. RUN _{~. RUN}

~11>tLO'E LeVU l'~S(U&c]) _{tOM l>UTEP}

-Z'5 d~ _{-Z""1 rJS -.%5".5" dB} -~~ -30. 3 -2~.B

-~5 -~,. 8 -2S.1

-31- 6

-3q

-ltq

_-~0'1

IEtos

IL, t

t,

~.q~eb

4·l.l\8

u'Lt il _l~aBO

_V~

ILl t t l I .H'11 I .'S00S'

11~ _{I~. ~lO9} I~.~lo~

-

\ ~~CITATION (OHFltIEIt T!.

.-11.0

~.lr1

i.~lO

"

\

_It,

_-0.6

_I~ I

-o. \qs

\ _{III 0.81~ol 1. S iO} \

a.l

-1. 51

1

13 -1.80'54 EFFLCIEl(,"Y _{0. 80}

1B

_{D. 8~bo}

IiAl~ RE)IICTtoll _{o.~ 3 dB}

_o.pdB

u.~

dl

_3·~r

3.bs

o

•.

S

HORMMUlP \l.A~tUS. 't

3

FIG. Z 1

A1'EI\.TUR~

1)ISTRUUTlON

L

1N",,]olu. ...

'L)

A!,\1) R.AHHION nnEp.N

-10 -JD ~ ... -SO

;

....

-o

as

1 HORl'IA LL<H R~1>t U ~

t

:I.

SIDHOF>f. lE.Vel "PR.E.~c.RI J£1>

-lO d~ -10

H-R05

E)(C.ITA nON (O~Fflc.rEN T('

40'" ~.SOS65

Il,= -O.II't15 d

_{z: o.}

'4~

bl

E HIC.IEI'IC(

=

0·12

·p

~All\ ~£DUC.TI{)N= '1.3&

dB

U~d;: L!.D4

C.O~fUTEP

-29· 6

.1F> -~H

-},

~ . " - ' :; .. JD I ;J: I 0 I

...

_..,

~-y,

!i:

-...

..

-SO

-60

-1'

1'RUN lO RUN

t.1 "lDELO&~ L~~ELS rREsc.U~E) (OM PUTEP

-30 d' -lB.1d? -30.0

dr.

-10

-

~1.s

_-

t~.8

-;0

=1\\

-

-It"

~o. 0

6

- ~ .S

"It

..

_{ZElOS IA,H, 5.1151 ... t405}

..

llt t t l l6~lO 1·~~~1f

..

~ lll+ '1 10.S 5~ 10.b\,,5

,.

~ U~ 1~.q)O _1~'1

09

...

_.,.

H(ITATIOti COE.fFICIENTS

...

_~

_,

11. ~.~')51~ ~.!.~tD!

...

_-O.'~~

₆₈

~I.i tI., _-0.t~4i4

~ /Jol

-MB\bD

o.ho

...

a.~

-!.In

~ I -O'Q'i213

...

<

..

...

EHIC!EIIICY O. fb85

o·Hil

;

_{G ... IN UI>UnION 1.1"1}

_dJ>

_{1.1.3 dj)} c

~

_{iJ.!>dt 3.~ I}

_;.9

0

'"

₀ 0 1.$ 1 KOlMALUcD MUU5 'l. ~

Fl6 . .t~

:

APc"TU~E 1>1SnHUTI~N

2..

(J,1\.

]oLIJ. ... "-)

A~l)

1I.A11fAHOH l'A1THN.

III, p

D

,

ID I~ IV 16 16 ,(IJ _ ! L -/0

-21

,.-JO

~ ,

-

/ / . - -....

,

-0 \ ~ \ \

""

~ .... -~D ~ ~

-,

0

...

...

-.~

....

I 0-\ c oJ

...

DtI.t,

-7

D I' !tUN l'P-UN

Sll1f.LO&E LE.VH 1'RE.SCI'.I8EV C;O M I' UTE 1>

-)0 dB _{-28.% d8}

-.2%.8

_ela

1 _{-36 -l~.o -3 .1}

'"

-10 -JJ.'1 -)0.0 Q - )0 -U.1 - 3(). 0

...

~ _{"l - ~I.~}

_-19·

'I

=

....

,

U,ii, 1).I'5b1

5.()\1~

....

_,---

_{... lERO~}

""

....

_"

_UZ+~Z

_lVb~

.,.,

_p.

_S

...

_po \ _{Il) t tJ 10.}

_lb

_I

16·2.1"1-....

..

\ 1.1.'1 t f~ 13 .1S~1 _13.

_f6ff

\ _{Us 18. O~II 1&.6111} ~O.6 \

'"

.u \

cXCIT ... noN

confICIENTS

4.~1 1

b1

...

_,

_{11., ~.i.llTD} < £, - o.'Ild6 -0. 'I 1 12.1 ~ III t'~r~1

_0.'1:1.»0

!!!

/I.,

-0·1

~ ~ -1.1<;"ob5 oJ

tl'l

O.q~8, 1.11

ss8

-c

~

:a:

,

Hn'IE~(Y

_o·r

9lb ().8oZIj

,

I.S 1

_GAlH

REIH.CTroM I.b

rdS

o.q

6

dB

NHMALlUl1 kA1)lU~ 'L

U~d"

Ui9

3.8,

~

fIG.

~~: APtRTU~E

lllSTIU6UnOli

fI.~O

4'1\]0(11. ... )

AIID RAPIATIotl 1'ATIERN.