NBS

" Reference Pubiications

I^BSIR 80-1630

AlllCH—75^51-

DETERMINATION OF MUTUAL COUPLING

BETWEEN CO-SITED MICROWAVE ANTENNAS

AND CALCULATION OF NEAR-ZONE ELECTRIC FIELD

NAT;l INST. OF STAND & TECH R.I.C.

A111D5 03bTfl3

C.F. Stubenrauch

A.D. Yaghjian

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

IGO

. U56

80-1630

1981

V )

June 1981

NBSIR 80-1630 JL^Tiuj^al hi.'hf:*'' or BTandard.'

IOBKAKT

AUG 7 1981

DETERMINATION OF MUTUAL COUPLING 't%o

BETWEEN CO-SITED MICROWAVE ANTENNAS ^ AND CALCULATION OF NEAR-ZONE ELECTRIC FIELD

C.F. Stubenrauch

A.D. Yaghjian

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

June 1981

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director

TABLE OF CONTENTS

INTRODUCTION

Page

...1

1 . FORMULATION OF THE MUTUAL COUPLING BETWEEN TWO ANTENNAS

1.1 The Basic Coupling Formula (Transmission Integral)

1.1.1 The Plane-Wave Scattering Matrix Approach

1.1.2 The Coupling Quotient in Terms of Far Field of Each Antenna....

1.1.3 Coupling Quotient When the Roles of Transmitting and Receiving

Are Exchanged

1.2 Eulerian Angle Transformations Describing the Arbitrary Orientation

of the Antennas

1.2.1 Rotational Transformations from (k^, k^) to the Far-Field

A J Direction in the Fixed Coordinate System of Each Antenna

1.2.2 Vector Component Transformations Required to Compute the

Coupling Dot Product

1.3 The Sampling Theorem, Limits of Integration, and Fast Fourier Transform

1.3.1 The Point Spacing of k^ and ky Required by the Sampling Theorem

1.3.2 The Limits of Integration and'^Number of Points Required

1.3.3 Application of the Fast Fourier Transform

1.4 Preliminary Numerical Results

.3.4

.4

.6

.8

.9

.9

1215

151619

20

2. TRANSFORMATION FROM FAR FIELD TO NEAR FIELD 25

2.1 Relationship of Near-Field Intensities to Power Input and

Antenna Gain or Efficiency 26

3. PHYSICAL OPTICS MODEL FUR REFLECTOR ANTENNAS 28

3.1 Physical Optics Subroutines Employed by USC 30

3.2 Test of Near-Field Program 31

4. COMPARISON OF PHYSICAL OPTICS AND MEASURED FAR FIELDS 34

5. COMPARISON OF PREDICTED AND MEASURED NEAR-FIELD COUPLING 50

6. CONCLUSIONS AND RECOMMENDATIONS 53

ACKNOWLEDGMENT 57

REFERENCES 58

APPENDIX A. POMODL - PHYSICAL OPTICS ANTENNA MODEL 59

A.l GENERAL OVERVIEW OF COMPUTER PROGRAM 59

A. 1.1 PROGRAM POMODL 61

A. 1.2 SUBROUTINE FAR2D(EPL ,HPL ,EY ,NTHETA,NPHI ,DATAX, IR2X2 , IC2T0N) 69

A. 1.3 SUBROUTINE FFKXY(0ATAY ,NTHX2 ,NPHI ,DATAX, IR2X2 , IC2T0N) 72

A. 1.4 SUBROUTINE NFKXY(DATA, IR2X2, IC2T0N) 77 A. 1.5 SUBROUTINE ETIOGAM(DATA( 1 ,C0L ) ,NR0W,NC0L , ICOL , ISGN,FLMDA,DELX

,

DELY,DIST) 82

A. 1.6 SUBROUTINE PHSC0R2(DATA,NRX2,NC0L) 85

A. 1.7 SUBROUTINE SWAP(NRX2,NC0L,DATA) 88

A. 1.8 SUBROUTINE ARAYPTR(DATA,NRX2,NC0L) 91 A. 1.9 SUBROUTINE FF0UT(DATA,NRX2,NC0L,LU0UT) 94

A. 1.10 SUBROUTINE F0URT(DATA,NN,N0IM, ISIGN, IF0RM,W0RK) 97 A.l. 11 SUBROUTINE PARAB (F0D,D0L .BLOCK, OFOCUS ,AC0SE ,AC0SH , THETA, ETHETA,

EPHI) 105

A.l. 12 SUBROUTINE PLT120R{X, Y ,XMAX ,XMIN,YMAX, YMIN .LAST , I SYMBOL , NO .MOST) 113

A. 2 SAMPLE PROGRAM INPUT AND OUTPUT 115

Appendix 8. CUPLNF - CALCULATION OF COUPLING BETWEEN ANTENNAS 126

B. l GENERAL OVERVIEW OF COMPUTER PROGRAM 126

-i ii-
B.1.1 PROGRAM CUPLNF{ INPUT, OUTPUT, TAPE 1,TAPE 3,..., TAPE 8) 128

B.1.2 SUBROUTINE ANGLGEN(PKXOXK, PKYOXK, PHI , THETA, PSI ,PHIP ,THETAP ,PSIP

,

PHIT,THETAT,PHIR,THETAR) 143

B.1.3 SUBROUTINE FINDFF { IDAYHR ,LUIN ,LUA,LUOY ,LUOZ ,DATA,NRX2 ,NCOL ,FFY

FFZ,STOR) 147

B.1.4 SUBROUTINE VECTGEN(FOX,FOY,FOZ,PH,THET,PS,FX,FY,FZ) 153

B.1.5 SUBROUTINE MINMAX(Z,ZMIN,ZMAX,LEX,LEY) 156

B.2 SAMPLE PROGRAM INPUT AND OUTPUT 158

LIST OF FIGURES

Page

Figure 1. Coupling Schematic for two antennas (0 and O' will be chosen at roughly the center of the radiating part of their respective antenna) 5

Figure 2. Definition of coordinates for the left antenna of figure 1 10

Figure 3. Definition of coordinate systems for the right antenna of figure 1 13

Figure 4. Physical i nterpretation for limits of integration. To a good approximation,

only that portion of the spectrum within a is required to compute the

coupling quotient h^/a^ for the two antennas 18

Figure 5. Hypothetical circular antennas directly facing each other

in the near field 22

Figure 6. Coupling of circular antennas computed first using FFT integration,

and then directly from far field along direction of separation 23

Figure 7. Typical coupling curve for antennas skewed in their near field 24

Figure 8. Geometry of vectors for surface integral 29

Figure 9a. Field strength in a uniformly illuminated aperture calculated using

physical optics far fields. Dashed line indicates theoretical

distribution 32

Figure 9b. Phase of field in a uniformly illuminated aperture calculated using

physical optics for fields 33

Figure 10a. Comparison of measured and calculated far-field patterns for antenna

No. 1. E-plane cut, solid line - measured pattern, dashed line - physical optics 35

Figure 10b. Comparison of measured and calculated far-field patterns for antenna

No. 1. H-plane cut, solid line - measured pattern, dashed line - physical optics 36

Figure 11a. Comparison of measured and calculated far-field patterns for antenna

No. 2. E-plane cut, solid line - measured pattern, dashed line - physical optics 37

Figure 11b. Comparison of measured and calculated far-field patterns for antenna

No. 2. H-plane cut, solid line - measured pattern, dashed line - physical optics 38
Figure 12a. Comparison of measured and calculated far-field patterns for antenna

No. 3. E-plane cut, solid line - measured pattern, dashed line - physical optics 39

Figure 12b. Comparison of measured and calculated far-field patterns for antenna

No. 3. H-plane cut, solid line - measured pattern, dashed line - physical optics 40

Figure 13. Comparison of measured and calculated far-field patterns for antenna

No. 4. H-plane cut, solid line - measured pattern, dashed line - physical optics 41

Figure 14. Comparison of effective current distribution used in physical optics

and geometrical theory of diffraction calculations. (Uniform

distribution assumed) 43

Figure 15. Diagram of multiple reflections involving feed structure 43

Figure 16a. Comparison of measured and calculated far-field patterns for antenna

No. 1 with feed region covered with absorber. E-plane cut, solid curve - measured pattern, dashed curve - physical optics 44

Figure 16b. Comparison of measured and calculated far-field patterns for antenna

No. 1 with feed region covered with absorber. H-plane cut, solid curve - measured pattern, dashed curve - physical optics 45

Figure 17a. Feed region of antenna with absorber collar 46

Figure 17b. Feed support struts with absorber attached 46

Figure 18a. Comparison of measured and calculated far-field patterns for antenna

No. 1 with feed region covered with absorber. E-plane cut, solid curve - measured pattern, dashed curve - physical optics 47

Figure 18a. Comparison of measured and calculated far-field patterns for antenna

No. 1 with feed region covered with absorber. H-plane cut, solid curve - measured pattern, dashed curve - physical optics 48

Figure 19. Photograph of experimental set up for measuring coupling between

two reflector antennas 51

Figure 20. Schematic showing relative orientations of antennas for the

three test cases 52

Figure 21. Mutual coupling between 1.2 meter reflector antennas.

Case 1: 6p=0°, 9^=0°. Solid lines indicate envelope of measured

mutual coupling 54

Figure 22. Mutual coupling between 1.2 meter reflector antennas.

Case 2: 0^=15°, 0-(-=O°. Solid lines indicate envelope of measured

mutual coupling 55

Figure 23. Mutual coupling between 1.2 meter reflector antennas.

Case 3: 0^=15°, 0^=20°. Solid lines indicate envelope of measured

mutual coupling 56

-V-
. .V. .'*

.

:

. M"

-**^4iil*p "'MEi---~^ ,V ,,i-t.i, >v* '•^>'<1..

' -^NiiSSl . ^.' .

. ' S' .

^

' <

„ . , ,v '“f ‘„<ifnw|v«i, s' "'

:|»*’.- vA’ v>'-^

,<?!it' jf;-r U'-«. \r -

• - - •"...•

.i',vS .- ,_^_

1.

' - ;.t . „ .IT

• •.

'

.V* *

, „ ,. ,, ^ ,»V • 4

• '• -••.-,

'^ -wsffl^'

'

»(. !((jS»U"

.•« • ‘)'i' 1

‘'

. ...

^ „ .. . .**. t

><; .«! .. 'J. ".'.w^e .?.';:,f

J.' ..; ikai'

;

V^: ' vv*'*-'! '• . '.

...4Si '

f

:' ti, -!. >>

.'v' t^1hR<ir.i3<yi» r,^. V

;Sft.i!

. ..

»';r ^>«4, :' ,..:.^;TMrTf^

-4-.

' ' *N)L» Stp^

|JV,^

Sw; , ./iSi»i'S •

'• '

' ^ •— • ’r.

...‘•.Sr VW', .'SVA ,^'».i'.f-, ^>;»^il|^

- > »

. . , «., W. t .. • bV. • Ml.-* * f • ,

1 « . c • » * *'^ ' :*•

... " ' --^ ' '-:^

..,,^.a,^4f.; ,Mr . ^ .

.. ;. "".'<5/. . '..;

* i

V' . . . V/ %:
DETERMINATION OF MUTUAL COUPLING BETWEEN CO-SITED MICROWAVE

ANTENNAS AND CALCULATION OF 'NEAR-ZONE ELECTRIC FIELD

By

C. F. Stubenrauch and A. D. Yaghjian

The theory and computer programs which allow the efficient computation of

coupling between co-sited antennas given their far-field patterns are developed.

Coupling between two paraboloidal reflector antennas is computed using both

measured far-field patterns and far-field patterns which were obtained from a physical optics (PO) model. These computed results are then compared to the

coupling measured directly on an outdoor antenna range. Far fields calculated

using the PO model are compared to those obtained from transformed near-field

measurements for several reflector antennas. Theory and algorithms are also

developed for calculating near-field patterns from far fields obtained from the

PO model. Documentation of the near-field and coupling computer programs is presented in the appendices. Conclusions and recommendations for future work are

i ncl uded.

Key words: Co-sited antennas; coupling; far fields; near fields; physical

optics; plane-wave spectrum; reflector antennas.

INTRODUCTION

This report discusses work done at the National Bureau of Standards (NBS) concerning

problems related to the prediction of mutual coupling between antennas and the prediction

of antenna near fields. In addition, comparisons for several paraboloidal reflector

antennas were made between far-field patterns obtained from near-field measurements and

those which were predicted using a physical optics (PO) model for the antennas.

A consequence of the scattering matrix theory of antennas and antenna-antenna

interactions developed at NBS over the past 20 years [1] is that mutual coupling and near fields can be calculated provided the plane-wave spectra for the antenna or antennas are known. The essential, propagating part of a spectrum is related by a simple expression to the antenna's far-field pattern which may be determined, e.g., through model computation,

direct far-field measurements, or transformed near-field measurements. For engineering

studies of co-sited coupling or antenna near fields, expressing the quantities of interest

in terms of the far fields proves especially convenient. In many cases the measured

patterns are unavailable. Because it is possible to predict these patterns by employing a suitable model, part of the work described herein discusses the capability of a particularly convenient and efficient model: the physical optics computer program obtained

from the University of Southern California (USC).
Formulations of the mutual coupling problem in terms of antenna far fields are well

known [7]; however, calculations using previous theories have been deficient because of

the large amounts of computation time and data required. In this work, it is shown that

the functions to be integrated can be made band-limited; and thus the sampling theorem

can be employed to determine the required point spacing, rather than the more usual

tri al -and-error method of testing convergence. Further, it is shown that the evaluation of

mutual coupling requires only the far fields lying within the mutually subtended angles of

the antennas. As a result of these improvements in the theory, an efficient program for

calculating mutual coupling was written.

Section 1 of this report details the theory which allows rapid calculation of the

mutual coupling between two antennas without restrictions on the separation distances.

Section 2 discusses the specific problem of obtaining the near fields of an antenna given

the far-field pattern. The PO model for reflector antennas is briefly discussed in

section 3 as is the particular model employed. Far-field patterns which were predicted by

the PO model and far-field patterns obtained from near-field measurements of actual

antennas are compared in section 4. In section 5 coupling values measured directly in the

laboratory are compared to those predicted from the theory of section 1 employing both

modeled and measured far-fields. Conclusions and recommendations are given in section 6.

The appendices describe the computer programs which perform the coupling and

near-field calculations. Appendix A discusses and documents POMODL, a program which uses a PO model to calculate the far-field pattern for a reflector antenna and which calculates

from this pattern the near-field distribution on a specified plane. The predicted far

field also provides output for use as input by the program CUPLNF (described in sec. 1 and

documented in Appendix B) which calculates the mutual coupling between two arbitrarily

located and oriented antennas from their far-field patterns.

2
1. FORMULATION OF THE MUTUAL COUPLING BETWEEN TWO ANTENNAS

The plane-wave scattering matrix (PWSM) description of antennas, introduced by Kerns

at the NBS, forms an ideal theoretical framework on which to base the determination of

mutual coupling between two collocated antennas. In fact, the basic PWSM formula required

for the determination of the coupling between two antennas has existed for nearly twenty

years [1]. However, before the existing formulas could be translated into a convenient

program which computed coupling efficiently, three important tasks needed to be accomplished

1) The Kerns coupling formula or transmission integral, as he calls it, was originally

written in terms of the appropriate plane-wave spectrum for each antenna. For our purposes,

we wanted to express the near-field mutual coupling in terms of the far field of each

antenna (assuming reciprocal antennas) because usually the far field most conveniently

characterizes an antenna and is most efficiently computed from, e.g., a PO-GTD (physical

optics and/or geometrical theory of diffraction) program or from near-field measurements.

This task, although straightforward, requires careful attention to the details of definition of the far field, the plane-wave spectrum, and the reciprocity for each antenna.

2) The far fields of each antenna are usually expressed in a Cartesian coordinate system

fixed in each antenna. To compute coupling for an arbitrary separation and orientation of

two antennas, the coupling formula requires an integration of the dot product of the two vector far-field patterns in reoriented coordinate systems. Thus, task two consisted of

expressing the reoriented coordinates of each antenna in terms of the Eulerian angles from

the preferred or fixed coordinates in which the far field of the antenna was given. In addition, a similar transformation had to be applied in order to compute the dot product of

the two vector far-field patterns. Again this task was fairly straightforward, yet rather

tedious.

3) Finally, even though tasks (1) and (2) above recast the coupling or transmission

integral in terms of the far fields of each antenna expressed in the preferred coordinate

system fixed in each antenna, repeated evaluation of the double integrals (actually a double Fourier transform) would require a prohibitive amount of computer time for electrically large microwave antennas unless the sampling theorem and FFT (fast Fourier transform)

algorithm could be applied effectively. However, the application of the sampling theorem

to these double Fourier transforms requires a sample spacing which, in general, is so small

that repeated evaluation even by means of the FFT still becomes prohibitive. Moreover, the

required sample spacing becomes smaller with increasing separation distance between antennas

Thus, the third major task was to discover a way to reduce drastically the computer time

needed to evaluate the final form of the double integrals expressing the mutual coupling

between two antennas.

The details of these three tasks and their accomplishment are described in the following three major sections (1.1, 1.2, 1.3).

3
1.1. The Basic Coupling Formula (Transmission Integral)

This section begins with the transmission integral derived by Kerns [1] for the coupling

of two antennas (when multiple reflections are neglected) in terms of the transmitting and

receiving spectra of the respective antennas. The receiving antenna is assumed reciprocal,

and its receiving spectrum is written in terms of its transmitting spectrum through the

reciprocity relations. The transmitting spectrum of each antenna is then expressed in terms of the antenna's far electric field, which in turn yields a transmission integral or coupling formula in terms of the dot product of the vector far fields of each antenna.

Finally, reciprocity is invoked for both antennas to prove that the mutual coupling is essentially the same when the roles of transmission and reception are exchanged.

1.1.1. The Plane-Wave Scattering Matrix Approach

Consider an arbitrary antenna transmitting with time dependence to the left of

an arbitrary receiving antenna, as shown in figure 1. The antennas may have arbitrary

separation and orientation. Assume that only one mode propagates in the waveguide feed to each antenna.^ The incident waveguide mode coefficients for the left antenna are labeled

a^ and b^ respectively, and for the right antenna, a^ and b^ respectively . The reflection

coefficients of the right (receiving) antenna and its passive termination are denoted by

and respectively.

The quantity b^/a^, which we shall call the coupling quotient , is a measure of how

much signal couples into the receiving antenna per unit input into the transmitting antenna.

If the same type of waveguide feeds each antenna and the receiving waveguide is terminated

in a perfectly matched load, |bya^|2 equals the amount of power coupled to the receiving

antenna per uni t power i ncident to the transmitting antenna . (This power ratio expressed

in decibels is commonly referred to as the insertion loss ratio. ) Thus, b^/a^ is indeed

the major parameter of interest in determining mutual interference between antennas.

The transmission integral which gives the coupling quotient in terms of transmitting

and receiving plane-wave spectra of the respective antennas can be found directly from

Kerns [lb]:

b 0

a 0 1-^L^o

^2 * -10 ( 1 )

where ^ "complete" transmitting and receiving spectra defined with

respect to plane waves traveling in the common J< direction but with phase reference to the

^If more than one mode propagates in one or both of the feeds, this analysis can be applied

for each possible transmit-receive pair of modes; and thus the analysis can be applied to "out-of-band" coupling.

4
5

03 CP O

C C rH C -H (/It 0) -IJ 4J -p e

U) p c

M-l 03 0)

(U M Oj

.J p tn

p

+Jo

Figure 1. Coupling Schematic for two antennas (0 and O' will be chosen at roughly the center of the radiating part of their respective antenna).

5
origins 0 and O' of the left (transmitting) and right (receiving) antennas respectively.

The z axis is chosen to run from 0 to O', with the distance d = 00' and the x-y axes perpendicular to the z axis at 0 (see fig. 1). J< = k g +k § is the transverse part of the

2tt X X y y 2 2 1/2 propagation vector k = K+y§ (k = where X is the wavelength), and y = (k -K ) is

taken positive real for K<k and positive imaginary for K>k. dK is shorthand notation for

the double differential dk^dk^.

Equation (1) is an exact result from Maxwell's equation for two linear antennas

operating with time dependence in free space, when multiple reflections between the

antennas are neglected. (In other words, the b^/a^ computed from eq (1) neglects power

which enters the receiver after having been reflected from receivinq antenna to transmitting

antenna and back one or more times.) No other restrictive assumptions are involved. For

example, the antennas may be lossy or even nonreciprocal.

Of course, eq (1) cannot be used to evaluate b^/a^ unless the spectra ^nd are determined explicitly in terms of commonly measured or computed characteristics of the

antenna. Toward this end, both spectra and eq (1) are recast in the next subsection in terms of the far electric fields of the antennas.

1.1.2. The Coupling Quotient in Terms of Far Field of Each Antenna

As a preliminary to expressing eq (1) in terms of the far fields of the antennas, assume that the receiving antenna contains no nonreciprocal devices or material so that its

receiving functions related to its transmitting functions simple reciprocity formula [lb].

'o ^2 (K) l20 ( 2 )

All quantities in eq (2) have been defined in the previous section except the impedance of

free space and n^, which is the characteristic admittance of the propagated mode in the

feed waveguide of the right (receiving) antenna of figure 1.

Substitution of f’^om eq (2) into eq (1) giveSi

.-1 i-r,T'

;

0 0 ys’g(-K) • s^f^{K)e

10 ' iyd

dK.

K<k

( 3 )

Note that the integration limits in eq (3) have been made finite by eliminating the

integration over the evanescent part of the spectra (included in the original infinite

limits of eq (1)), thereby leaving only the radiating part of the spectra. This is permissible for all nonsuper-reactive antennas which are separated by a distance greater than a wavelength or so, i.e., if the antennas are outside each other's reactive field zone [2];

6
and if the contribution from the integration in eq (3) near the critical point K = k is negligible, as is usually the case.

A major advantage of the PWSM techniques is that the radiating part of the spectrum of

an antenna is proportional to the vector far field £(r)^^ of the antenna. Specifically,

if £(_r) refers to the normalized, complex far-electric-field pattern of the left (transmitting) antenna of figure 1, i.e..

f(r) E

_ re

-i kr (4)

then the radiating spectrum, related to the complex far-field pattern by

the disarmingly simple proportionality [lb].

iiofy = 7 ity (5)

Although f is shown as a function of _r in eq (4), we know that the complex far-field pattern

is a function only of the direction of £; and thus f(J<) in eq (5) is also only a function

of the direction of j< which is determined solely by the relative size of k^ and k^, the

integration variables of eq (3).

Similarly, the radiating spectrum, ^20 right (receiving) antenna in

figure 1 can be written in terms of the normalized, complex, far-electric-field pattern f' of that antenna:

where, as in eq (4), f' is defined in terms of the far-electric-field £'(n)j,_^ of the right

antenna when it is radiating:

f' (r)

-i kr

lie F'(rl

a

' — — r-^ (7)

Substitution of the spectra from eqs (6) and (7) into eq (3) produces the coupling

quotient for two antennas as a double integral over the dot product of the complex farelectric-field patterns of the antennas:

0 _ p

I

K<k

e^^° dK . Y

( 8 )

In eq (8), C is a consolidated notation for the "mismatch factor" (I-TlI^o^

7
1.1.3. Coupling Quotient When the Roles of Transmitting and Receiving are Exchanged

The coupling quotient b^/a^ in eq (8) is a measure of the signal which is received by

the passively terminated antenna on the right side of figure 1 when an input mode of unit

amplitude is applied to the transmitting antenna on the left. A natural and important

question is what will be the coupling to the left antenna when the right antenna transmits

at the same frequency and the left antenna is terminated in a passive load. Specifically,

what is the expression for b^/a^ and how is it related to b^/a^ of eq (8).

The answer to this question can be obtained immediately by retracing the steps in the

derivation of eq (8) but with the left antenna in figure 1 receiving and the right antenna

transmitting. So doing, yields an expression for b^/a^ very similar to eq (8).

= -C f(-k') • f'(k') ^iy'd

K'<k Y

dK' (9)

where the "mismatch factor" C is defined as before.

C = 1-r, r„;

L O'

( 10 )

Tq and are now the reflection coefficients to the antenna on the left and its passive

termination, respectively. And is now the characteristic admittance of the propagated

mode in the waveguide feed to the left antenna. Because g ,= -g , we can choose g , = g

y z' z’ y' y

and g , XX= -g . Then changing the dummy integration variables in eq (9) from k' and k' to

X y

k^ and -k^ shows that the integration in eq (9) is identical to eq (8), i.e..

- -C

f(k) • f(-k)

iyd dK

K<k

( 11 )

Comparing eqs (8) and (11), we see that the two coupling quotients, b^/a^ and b^/a^,

are related merely through a constant factor, i.e.

This means that if the coupling between two antennas i s measured or computed wi th one of

the antennas transmitting and the other receiving , the coupl ing , when the roles of trans - mitting and receiving are reversed , i s al so known (through eq (12) ) . A separate measurement

or computation need not be done. Use of eq (12), of course, requires knowledge of the

reflection coefficients and input admittances of each antenna contained in the definitions

of C and C

.

8
As a check, eq (12) was also derived directly from the "system two-port" equations

describing the two antennas, by applying the Lorentz reciprocity theorem [lb] and knowing

that multiple reflections between the antennas are being neglected. It can be further

proven that if scattered fields are also negligibly received by the transmitting antenna,

then the available power at the receiving antenna per unit input power to the transmitting

antenna is the same when the rules of receiving and transmitting are reversed.

1.2. Eulerian Angle Transformations Describing the Arbitrary Orientation of the Antennas

From a quick look at eq (8), it might be concluded that the analysis required to compute the coupling between two antennas is essentially finished. All we need to do is compute or measure the vector far-field patterns of each antenna, take their dot product,

and perform the double integration on a computer.

Unfortunately, a major problem, ignored so far, is the fact that the far-field pattern of

an antenna is given with respect to a Cartesian coordinate system which is fixed in the

antenna and which is not, in general, aligned with the Cartesian system shown in figure 1 to which the far-field patterns f(j<) and £'(-]<) in eq (8) are referenced. Thus, to use

eq (8), it is mandatory that the far-field direction in the coordinate system fixed in each

antenna corresponding to a given (k , k.,) in eq (8) be determined explicitly. Moreover, to

A y

evaluate the dot product f ' •£, the rectangular components of f and f' in the x-y-z system

of figure 1 must be expressed in terms of the rectangular components of the coordinate

systems fixed in the antennas.

Fortunately, all these necessary transformations can be accomplished by specifying the

Eulerian angles required to align the axes fixed in each antenna with the (x, y, z) axes chosen in figure 1, as the following two subsections explain.

1.2.1. Rotational Transformations from (k^, k^) to the Far-Field Direction

in the Fixed Coordinate System of Each Antenna

Assume the left antenna in figure 1 has a fixed coordinate system with rectangular

axes (x^, y^, z^ centered at 0) in which the normalized far-electric-field pattern is given in terms of the spherical angles (|)^ and e^, as shown in figure 2a. That is, we have

at our disposal, obtained from either measurement or computation, the vector far-field

pattern as a function of (fiy^.and e^.

Let ((p, 0, i|;) be the Eulerian angles needed to rotate the (x^, y^, z^) axes in

line with the (x, y, z) coupling axes of figure 1. Specifically, as shown in figure 2b,

rotate an angle 4>( 0£4i< 2Tr) about the positive z^ axis, thereby changing the direction of x^

and y^ but not z^. Then rotate an angle ©(O^e^ir) about the new positive y^ axis, thereby

changing the direction of z^ (to z) and again x^ but not y^. (4> and e are the usual spherical

angles.) Finally, rotate an angle ij;(0<ij;<2Tr) about the positive z axis to align the new x^

and y^ axes with x and y. These are fairly common definitions of Eulerian angle rotations

found in a number of textbooks such as reference [3].

9
Figure 2. Definition of coordinates for the left antenna of figure 1.

10
To understand the transformation needed to evaluate eq (8), note in eq (8) that £ and

£' are written as functions of § +k § +y@ or, in other words, as functions of k and

^ ^ y y ^ ^

k because y is determined from k and k . However, we are given as known (measured or y X y

^ computed) f as a function of and 6^, not k^ and k^. Consequently, to evaluate eq (8)

numerically, a transformation is needed which will convert (kj^,k^) to under the

given Eulerian angles ((p,e,:p) defining the x^-y^-z^ system with respect to the x-y-z

system. This Eulerian transformation, which is a straightforward, rather lengthy, linear

transformation found in a number of textbooks [3], will not be derived here but simply

stated in the form useful for our purposes of evaluating eq (8).

Before actually writing the required expression for cp^ and 0^, the antenna on the

right side of figure 1 should also be discussed because it will require a similar transformation to convert k^ and k^ to the spherical angles of its preferred system. That is,

if the far-field pattern f_' of this right antenna is known (measured or computed) in

terms of spherical angles cj)p and

6p with respect to (Xp, yp, Zp) axes fixed to the antenna

(and centered at O'), then (<}>p»0p) needed as functions of (k^,k^) in order to evaluate

f_'(-J^) in eq (8) (see fig. 3). (An important point to remember is that |)'(-_k) denotes

the value of the far-field pattern in the -J< direction. ) Also, as shown in figure 3, let

(f)', 0', and ijj' denote the Eulerian angles which rotate the (Xp, yp, Zp) axes fixed in the

right antenna parallel to the ((-x), y, (-z)) coupling axes of figure 1.

Both transformations, from (k^, k^) to (cj)^, 0^) and (cj)p, 0p), are similar and can be

written explicitly as:

cos cos

'P sin

k + cos 1

k (13a)

tan

(sin|^4> 1 cos :m

cos

I :m

;sin|r,J; 1

1 sin|:m

jcos|

CD

CD sin

1 |-cos|:m

jcos 1 li-'JII

k 1 /+sin sin

I

CD

CDI

k

j^cos Ik (M

cos 1^!

|cos| f<t>

'

U'j

|sin'IM1’ [cos|[-)>

1

1 cos IM

sin|

i;'! +sin|fi' 1

I cos [::t: ^ +COS C''

|sin|IM

The top signs in eqs (13) go with (cj)^,0^), the bottom with (cj)p,0p). Equations (13) look

rather cumbersome at first sight, yet computationally they are quite manageable because they

involve only sines and cosines of the Eulerian angles and linear dependence upon k^, k^,

and Y (which equals /k^-(k^+k^) ). The computer program merely contains a subroutine which

yields (<Pn>Qa^ (<}) ,0 ) from eqs (13) when given the Eulerian angles ((j),0,ijj), (cj)' ,0' ),

and (k^,ky) as input.

With the transformations of eqs (13), eq (8) can now be expressed in terms of (0^5<|)^)

and (<l)p,ep):

11
= -c dK. (14)

K<k

1.2.2. Vector Component Transformations Required to Compute the Coupling Dot Product

In the previous subsection a transformation was written that yielded f and £' in eq (14) as functions of the spherical angles ((f)^ 50 ^) and (<l>p 50 p) In which the far-field

patterns were measured or computed. Still, a method is needed to compute the dot product

£' •£, because the components of £ and f^' are given in terms of unit vectors of the

(x^,Ya»Za) and (^p’Yp’^p) coordinate systems fixed respectively in the left and right

antennas of figure 1. And these two sets of unit vectors have relative directions which

depend also on the Eulerian angles ((J),0,iJ;) and (cj)' ,0 ' )

.

A convenient way to evaluate £' *f is to first write £ and f' in the (x,y,z) and

(x',y',z‘) rectangular components respectively shown in figures 2 and 3,

f-fg + fg + fg - XX y y z z

—f =f',g , +f',g I +f',g I X X y y z z

(15a)

(15b)

Because by definition,

= -e. and g. -e

z’ (16)

the dot product becomes

f

' -f -f ,f + f ,f - f ,f

X X y' y z z (17)

Next, we express the rectangular components of eq (17) in the rectangular components

with respect to the fixed axes (x^,y^,z^) and (^p’yp’Zp), again through the appropriate

Eulerian transformation. In matrix notation

> f

X

fy

f

z

(cos (j) cos 0 cos 4> -sin (p sin i|j)(sin (|) cos 0 cos ip + cos c[) sin ijj)(-sin 0 cos \p)

(-cos (j) cos 0 sin ip -sin c}) cosiI))(-sin cos 0 sin + cos cj) cos i|j)(sin 6 sin ip)

(cos (j)Sine) (sincj)Sin0) (cos 0)

xA

yA

^A

( 18 )

12
Coordinate systeir fixed to the right antenna in which the far field f is known as a function of i , £' ~ P P

Eulerian angles ((}' , 8 ' , ip' ) needed to rotate the fixed axes x ,y ,z to the coupling axes x ' , y

' , z

' ,^which^are in the direction of the (-x) , (y) , (-z) axes of Fig. 1.

Figure 3. Definition of coordinate systems for the right antenna of figure 1.

13
The counterpart equation for fyi> f

^

^

(<}) ' >0 ' >4^'

)

and (fxp’^yp’'*'zp^ replacing (ct),e,4^) and (^xA’^yA’^zA^ ’ >"espectively . It should also be

noted that the x, y, and z components of the far field are not independent because there is no radial component of far field. Using f^, for an example, the rectangular components are related by cos cf)^ sin + sin c|)^ sin + cos " 0-

If the far-field components

(^xA’^yA’^zA^

antenna and (^xp’^yp’^zp^

the right antenna of figure 1 are known, eq (18) and its counterpart equation yield

(f ,f ,f ) and (f ' . ,f ' I ,f ' I ) in terms of the given Eulerian angles. In turn, eq (17)

X y 2 X y z yields the dot product jf ' • f . Again, the computer program which computes the double integral

(14) need only contain a simple subroutine to evaluate eq (18), and the dot product £' •£

is immediately computable from eq (17).

One other set of transformations often proves useful, however. Usually, the far field

of an antenna is given not in terms of rectangular components but in terms of spherical

components. If the far-electric-field pattern of the left and right antennas of figure 1 are known in terms of respectively, then the rectangular components

are related to these spherical components by the spherical angles. Specifically,

'^xA^ -sin (|)^ cos 0^ cos (f

^(t>A

^yA

= cos (f)^ cos 6^ sin cj)^

^zA 0 -sin ^6A

(19)

The counterpart equation giving

(^xp’^yp’^zp^ functions of (^^p’^Qp) is formed from eq

(19) merely by replacing (4>^,9^) in the matrix with (^l>p>9p)-

In summary, if ^nd (f^Jjp’f^p) ^re the known far-electric-field patterns in

the fixed coordinate systems of the left and right antennas of figure 1, respectively,

eq (19) and its counterpart transform these spherical components to rectangular components.

Equation (18) and its counterpart transform these rectangular components in the fixed

systems to rectangular components in the coupling (x,y,z) or (x',y‘,z') coordinates.

Finally, eq (17) yields the required dot product from the transformed components.

These transformations must be done for each (k^,k^) within the limits of integration

needed to evaluate eq (14). Moreover, eqs (13) must be evaluated for each (k^,k^).

Fortunately, the nature of the integrals in eq (14) allows the application of the sampling

theorem and fast Fourier transform, as well as the limits of integration to be reduced

inversely proportional to d. These topics, which enable the efficient computer evaluation

of the mutual coupling quotient, are covered in the following section.

14
1.3. The Sampling Theorem, Limits of Integration, and Fast Fourier Transform

This section shows how the sampling theorem converts the double integration in eq (14)

to a double summation which can be summed using the fast Fourier transform (FFT) algorithm.

In addition, the effective limits of integration are shown to reduce inversely proportional to d, the separation distance 00' between the two antennas.

1.3.1. The Point Spacing of and Required by the Sampling Theorem

Equation (14) represents the coupli-ng quotient for the two antennas positioned in

figure 1. If the antenna on the right side of figure 1 is displaced by a vector ^ perpendicular to the z axis, the integrand in eq (14) changes only by the phase factor

exp(ij<*^) = exp(ik^x+ik v) . That is, eq (14) can be written more generally as

b;(R,d)

= -C j'yd

e''- - dK

K<k

( 20 )

The sampling theorem [4] could be applied to convert the double Fourier transform in

eq (20) to a double Fourier series, if b^(^,d) were zero outside a finite |^| = R^. Now

b^(_R,d) behaves as l//R^+d^ as R ^ «>, and thus, strictly speaking, will never vanish for

finite R . However, if we choose R »d, b' is small and the "aliasing" error introduced

by using the sampling theorem should be small, especially near ^=0, even though b^ is not strictly "band limited" (i.e., zero outside a finite range).

In view of the decay of b^ with R, choose

^^0 = Bd ( 21 )

where B is a number much greater than 1. (Computations show that in practice, a B no larger than 1 or 2 is often sufficient for the accurate calculation of b^(^,d) near ^

= 0 from eq (23) below. For larger greater B is generally required. Also, R^ should never

be smaller than about the sum of the diameters of the two antennas.) The sampling theorem

applied to eq (20) then requires a sample spacing no larger than

Ak Ak , X y ^ A

k

’ k 2Bd ’ ( 22 ) .

in order to convert eq (20) to the double summation.

(R,d)

= -C Ak^Ak^

M L

I I m=-M £=-L ^£m

£m ,£m^

'’'lira'*

where

1/ —£m _ £A * , mA *

k " 2Bd ^x 2Bd ®y ’

(23)

( 24 )

15
and £,iTi are integers which range to cover the limits of integration |J<j |<k (i.e..

The beauty of eg (23) is not only that the integrals have been converted to summations,

which can be performed on a computer, but also that the summation is ideally suited for

evaluation by means of the FFT algorithm, which decreases the running time considerably

when the coupling quotient over a range of ^ is desired.

1.3.2 The Limits of Integration and Number of Points Required

The number of points required to compute the double summation of eq (23) is approximately (2Bd/A) for each separation (^,d) and orientation of the antennas. For d/A of

appreciable size, the number of points can become so large that the computer time required

to evaluate eq (23) over a range of even using the FFT, can become exorbitant. For example, if d = 10 meters and A = 3cm, choosing a typical value of B = 2 yields (2Bd/A) =

1.8x10^ terms to be summed for each separation and orientation of the antennas. Fortunately,

however, it can be shown that the effective limits of integration, i.e., M and L in eq (23), can be reduced inversely proportional to the separation distance d to keep the total

number of summation points bounded to a manageable number regardless of the value of the

separation d between antennas.

Consider eq (20) and rewrite the phase factor e^—*— in the plane-wave form e^—

,

where r^ = For r much larger than the dimension of either antenna, the function

e^—*— oscillates more rapidly than the oscillations of the far-field pattern dot product

f

' *f

,

except when k is in the directions approximately parallel to _r. This means that the

integration in eq (20) will essentially cancel to zero except for the contribution near

equal to provided the contribution from near the critical point K = k is negligible, as

is usually the case. In particular, a more thorough analysis of the integration in eq (20)

reveals that in order to compute the coupling quotient for values of |^| between 0 and R, only the part of the spectrum defined by

K R , (D+D')

k r r (r > R + D + D'

)

(25)

contributes significantly to the integration (under the assumed provision of negligible

contribution from the end critical point). The quantities D and O' in the inequality

(25) refer to the overall dimension of each of the antennas except when D and/or D' is

less than 2A, in which case D and/or D' is set equal to 2A. For example, if each antenna

were an electrically large, circular aperture type of radiator, D and D' would be their

respective diameters; but if one or the other of the antennas were a short dipole, its

effective diameter would be set equal to 2A. Of course, nearly all microwave antennas have

dimensions much greater than 2A.

2 Equation (25) assumes implicitly that the origins 0 and O' for the two antennas by which r is defined (r = 00') are chosen near the physical centers of their respective antennas.

16
For R<<(D+D' ) , i.e.

,

reduces to simply K/k<

coupling along the z axis as shown in figure 1, the criterion

I —, and the limits of integration in eq (20) become

(25)

lj(. D+D')

^ d (d>D+D'>>R). (26)

As d gets much larger than the sum of the overall dimensions of the two antennas, eq (26)

shows that the effective limits of integration become much less than the original K<k. This means that the summation limits L and M of eq (23) reduce to

L,M ^ 2B(D+D'

)

X (27)

The result (27), which holds for all separation distances for fixed B, is

significant. It implies that the number of terms in the summation which evaluates the

coupling quotient depends only on the electrical size of the antennas and not on the

separation distance of the antennas. We will now show as a result of this reduction in effective limits of integration that the Ak^, Ak., sample spacing can be increased beyond

A y that of eq (22) to an interval independent of the separation distance d until d reaches the

mutual Rayleigh distance; and thus the summation limits L and M can be decreased with

increasing d below the values given by eq (27).

Physically, eq (26) has a very simple interpretation. Referring to figure 4, it says

that to a good approximation, for ordinary antennas larger than a couple of wavelengths

across, only that portion of the plane-wave spectrum within the sheaf of angles mutually

subtended by the smallest spheres circumscribing the radiating part of both antennas

(including feeds, struts, edges and all other parts of the antenna which radiate

significantly) is required to compute the coupling quotient. Thus, if the coupling quotient

is desired only near R = 0, i.e..

R«(D + D'), (28)

the integration limits in eq (20) need extend only over K given by criterion (26). In other

words, the spectrum can be set equal to zero outside the mutually subtended angle of figure

4. This means that the coupling quotient b^(R,d) computed from the limited integrations

will no longer be equal, even approximately, to the actual coupling quotient for R greater

than about (D+D‘), but will in fact become zero more rapidly beyond (D+D'). Specifically, a more detailed analysis shows that limiting the range of integration to K<k(D+D')/d also

artifically band-limits the coupling quotient to

R^ = larger of

B(D + D')

Bxd

(D + O'

)

From eq (22), the sampling theorem spacing is then

(29)

17
18

Figure

4.

Physical

interpretation

for

limits

of

integration.

To

a

good

approximation,

only

that

portion

of

the

spectrum

within

a

is

required

to

compute

the

coupling

quotient

b'/a

for

the

two

antennas.
smaller of (30)

Ak Ak

X y _

k ’ k 2B(D + D'

'

(D + D' )' 2Bd

and from this equation and eq (26), the summation limits become

L,M - larger of

2B(D + D' )^

Ad

2B

(31)

2 Note that when the separation d becomes larger than the "mutual Rayleigh distance," (D+D‘) /A,

only a few (2B) points of integration are required, as one might expect from physical

intuition because only the near-axis plane waves contribute to the coupling as the far

field is approached.

1.3.3. Application of the Fast Fourier Transform

As mentioned above, eq (23) is amenable to computation by means of the efficient

algorithm often referred to as the fast Fourier transform (FFT) [5]. The particular FFT algorithm we use is called FOURT and was written by Norman Brenner of MIT Lincoln Laboratories.

FOURT, like all FFT algorithms, requires the summation in eq (23) to be written in a specific

form, namely

b'(R,d) -ik(a,x+b,y) (a,+a.,) (b.+b.,) ^1 ^2 2Tri C e

- - / N I I A[j ,J2] e ^ '^2 j^=l J2=l

^

'(jl-l)(m^-l) (j2-l)(m2-l

'7

:32)

The definition of the various parameters in eq (32) in terms of quantities defined

previously can probably be best understood by referring back to eq (20). As usual, C is the mismatch factor (defined after eq (8)), and (x,y) are the components of the transverse

vector The real numbers (a^,a2) and (b^,b2) define the limits of integration on k^ and

ky‘, specifically.

-a^ £ 5 a2 (33a)

k -b^ ^ ^

b2 • (33b)

N^ and N2 are the number of terms in the k^ and k^ summations respectively, and are equal

to (2M+1) and (2L+1) defined under eq (23). (In light of the discussion leading to eqs (26)

and (31), for ^ near zero, apa2,bp and 62 will all lie within a circle of radius

k(D+D')/d (d>D+D') in the k^k^ plane; and N^ and N2 need be no larger than about twice the

L,M given in eq (31).) The exponential immediately following C in eq (32) arises from

making the summation indices range only over positive integers.

In eq (32) the FFT will compute the double summation for the following values of x and y:

19
X (34a)

(-N^/2+m^-l)A

[a^+a^T

Tbj+bp

(34b)

where

I5 2 , ••9

^2 ^ ^

(35a)

(35b)

Finally, the matrix A(j^,j

2

) in eq (32) needs defining:

A(ji,J2) = :^r(4>p,0p)-I(cl)A.0A)

J

1 +J 2 (36)

where (4> ,0^) and ((J)„,0fl) are determined from the transformations (13) for given Eulerian

P P MM

angles and (k^,k ), which are defined in terms of ( 02902 )

(a,+a^)

(37a)

k-Jt =

k N

b,+bj

(02-1) -bi (37b)

01+02

The (-1) factor in eq (36) arises from requiring the algorithm FOURT to yield the

coupling quotient directly for every value of x and y without the need of "rearranging."

The z component y of the propagation vector is 9 of course 9 determined from k^ and k^ through a simple relation, which for completeness will be repeated here:

y

= A^-k^-kJ . (38)

The dot product £' A is also computed as explained in section 1.2.2.

In short, eq (32) for the coupling quotient between two antennas is ready for efficient evaluation on the computer using the FFT algorithm FOURT.

1.4. Preliminary Numerical Results

In order to build confidence in the computer program which was written to evaluate

coupling products from eq (32), the far fields of two hypothetical antennas were inserted

into the program. The hypothetical antennas were linearly polarized (in x direction),

uniform, circular aperture antennas for which the complex far-field patterns are well known

in terms of simple analytic expressions involving the first-order Bessel function [6]. The

radius and operating frequency of the antennas could be chosen arbitrarily along with their

mutual orientation and separation.

20
One check performed on the program is displayed graphically in figure 5, which shows

the coupling quotient for two identical antennas facing each other in their very near

field. Here the coupling should be very high, actually approaching unity when the antennas

are directly aligned, as figure 5 confirms. (It should be mentioned that the curve in fig 5 and those in figs 6 and 7 took no more than a few seconds to compute.)

A second check of the computer program involves computing the coupling when the

antennas are separated by a large enough distance for coupling to take place mainly between

the far fields along the direction between the antennas. As mentioned in section 1.3.2.,

this critical distance which we call the "mutual Rayleigh distance" can be shown to be

approximately (D+D') /X. In figure 6 the coupling between the antennas is computed at this

mutual Rayleigh distance for the antennas by two methods--first, by the FFT integration of

eq (32), and then directly from the far-field coupling along the direction of separation.

The close agreement between the two results again imbues confidence in the correctness of

the coupling computer program.

Finally, figure 7 shows a typical coupling curve for two antennas skewed in the near

field of each other. Note that a small lateral displacement appreciably less than an antenna diameter can make a 20 dB or more change in coupling.

In summary, the results of these and numerous other sample computations with hypothetical circular antennas yielded reasonable curves in every case; thus, we entered the

experimental stage of the program, confident of the reliability of the computer program.

21
22

X

(WAVELENGTHS)
Figure 6 Coupling of circular antennas computed first using FFT integration,

and then directly from far field along direction of separation.

23
40

dB

o

t °e/?q I lN3Ii0nt) ONHdnOO

(sson Noiia3SNi)

1CO

•OoVO

24

Figure

7.

Typical

coupling

curve

for

antennas

skewed

in

their

near

field.
2. TRANSFORMATION FROM FAR FIELD TO NEAR FIELD

This section details the theory which underlies the transformation from far field to near field. As in the case of coupling between antennas, the techniques are based on the

scattering matrix theory of antennas developed at NBS. A brief review of the points

applicable to the calculation of near fields is presented here. For a more thorough

discussion, see Kerns [lb].

We consider a finite antenna system which is located between the planes z

= and

z = z^\ zi<Z 2

* The fields to the right -of plane Z 2 can be expressed by a superposition of plane waves in the following form

i(r) = ^ Jj [bCjDe'l ^ K a(K)e'''l h e-*- dK , (39)

- 00

where

^(j<) is the spectral density function for plane waves travelling to the right (outgoing);

^(J<) is the spectral density function for plane waves travelling to the left (incoming);

J< = kxex + kyCy is the transverse propagation vector;

Y

= (k^ - k^ - k|)^/^ = (k^ - is positive real or imaginary,

k^ = pe; and

dj< = dkxdky.

Each plane wave is specified by its propagation vector

k— = ke + ke +ye = K+yc. - XX yy-z z

Further, each component satisfies the transversal ity relation

^= 0 ; ^= 0 .

We note that eq (1) indicates a Fourier transform relation exists between the electric

field and the spectrum.

25
A surprisingly simple relationship exists between the far-field radiation from a finite antenna and its spectrum, as noted in section 1.1.2, and is given by

Hence, knowledge of the far-field pattern immediately permits calculation of the

spectrum, from which we can calculate the near-field pattern at any point using eq (39).

For our purposes here, we consider an antenna radiating into free space; hence,

there are no waves travelling left for z>Z2

* Thus, a^ (J<) = 0 and eq (39) becomes

has been introduced as a constant which normalizes the magnitude of the far field. It

will be evaluated in the following section.

The constant will be determined oy the power input to the antenna and the

intrinsic properties of the antenna itself. We will let the property be the antenna gain

as it is the one most often measured or specified. In the case of a reflector antenna,

with lE(_r) determined by a mathematical model, we use the physical size and efficiency to provide the appropriate normalization.

Recall that, for a single antenna radiating into free space

E.'^(jl) = -iyk (B.k/r) e^**'"^/r. ( 40 )

r

(41)

2.1 Relationship of Near-Field Intensities

to Power Input and Antenna Gain or Efficiency

( 42 )

- 00

Further, as shown by Kerns, the gain of an antenna is given by

26
'1 (1-1 r

0 ' 0

where, as in section 1, Yq = l/Zg is the admittance of free space, Hq the

characteristic admittance of the feed mode, and Tq is the antenna input reflection

coefficient.

Now we are interested in normalizing our calculation to the gain in a single

direction. This is usually the boresight or "on axis" direction (though in the case of a monopulse difference pattern we may need to specify the gain in a different direction.)

For the antennas and models considered in this study, however, the boresight direction

corresponds to the peak of the main lobe and thus makes a convenient normalization point.

Solving for ^^q(J<= 0) in terms of the boresight gain and substituting into eq (42)

gives

where

a ,Ai (1- r

0 V 0 0 G(0)

4tt Y k 0

J Y F

iK»R

e dK

(K)

(44)

Now, for an antenna connected to a source which delivers an average power input Pq, we have

P 0 = i \ (I

but because bg = r gag

P 0 (1

Substituting this into eq (44) gives

E(r) 1_

2tt

P G(0)

_o

2tt Y k^

0

(K) e

I Y z iK*R

' e dK (45)

27
For the case of an antenna pattern determined from a model, we may estimate the gain

of the antenna from its physical size and assumed efficiency. The receiving cross section

0 , can be related to its physical area by the expression

0 = T A ,

where

n = aperture efficiency

A = physical area of the antenna,

Further, for a reciprocal antenna, gain and receiving cross section are related by

G

=

4tto 2

*

Finally, for a circular antenna we have

r

2 2 G = n TT d ^

,

where d^ = y ''s the diameter expressed in wavelengths.

A

3. PHYSICAL OPTICS MODEL FOR REFLECTOR ANTENNAS

In order to calculate the radiated fields of a reflector antenna, it is necessary to employ some sort of approximate theory because an exact solution is essentially impossible

to complete. Of several approximate theories, the one most appropriate for prediction of

the antenna is main beam and near sidelobes is physical optics (PO). For farther out sidelobes, better results can usually be obtained from asymptotic theories such as the

geometrical theory of diffraction (GTD).

The model employed in this work was physical optics and the basic theory will be

discussed here. Several good references are available on the subject of physical optics.

Here, we follow the development of Rusch [8,9].

As is well known, the fields in space can be calculated if all currents are known. A general expression for these fields can be written in terms of the free-space dyadic

Green's function [10]. This expression is quite complicated if we want to calculate fields

at any point. However, if we desire only "far-field" expressions, considerable

simplification can be made.

28
Figure 8. Geometry of vectors for surface integral.

Here, 0 is the origin of the reference coordinate system, P is the field point, R is a A ~

vector which locates P in the reference system, and af^ is a unit vector parallel to The integration point is located by the vector £, while the vector _r designates the

A location of P with respect to the integration point and a^, is a parallel unit vector.

Now, under the usual far-field assumptions r >> X and

write the electric field at P as

p « R or r, we can

E(R) dS . (46)

This expression can be evaluated relatively easily using numerical techniques, provided

that ^ is known. The crux of the problem, then, is the evaluation of

A useful approximate theory for obtaining ^ is PO. Simply stated, PO approximates

the surface currents with those that are obtained by the assumption of a local plane-wave

reflection field, i.e..

J“S

= 2[n X H . ] -1 nc (47)

where n is the unit normal to the surface and is the incident magnetic field.

Numerical evaluation of the two-dimensional integral in eq (46) can be time consuming

for many cases. The size of the cell required to obtain a given accuracy with the

numerical integration scheme decreases as the observation point moves off axis, and may

29
approach a small fraction of a wavelength. Thus, we see that calculation of the fields off

axis for a large aperture antenna requires a large number of points. Further, the

near-field calculations which are to be performed using the far-field patterns require a large number of individual far-field calculations.

In order to arrive at a practical model, some simplifications must be employed. The

model, which is employed by the USC programs, assumes that the reflector is axially

symmetric. This assumption allows the performance of the azimuthal integration in eq (46)

analytically, thus reducing drastically the number of points required in the integration.

Details of this simplification may be found in Rusch [8].

Another consequence of the assumption of axial symmetry is that a complete

far-field pattern (i.e., specification for all values of (?) requires that the field be

calculated only in the E- and H-planes, i.e., 0

= tt/ 2 and 0, respectively. The field at any point (R,0,0) is given by

ikR

i(R,e,0) = [F (0) sin0a + E (e) cos0a ]. (48)

K L 6 n p

For the purposes of this study, we require the rectangular components of the antenna

pattern, which are given by

ikR

E“RE = — [E^(0) cose - E,,(e)] COS0 sin0a

H X

2 2 + [E (e) cose sin 0 + E (e) cos 0] a - E (e) sine sin0^ . t H y h z

3.1 Physical Optics Subroutines Employed by USC

The subroutines used to compute the PO fields of the paraboloidal reflector antennas were written by Prof. W. V. T Rusch, of the University of Southern California and obtained

at a short course. Reflector Antenna Theory and Design , given in June 1976.

The subroutine package will calculate far-field patterns for an axially symmetric

reflector antenna which has a circular blockage on axis caused by the feed. Further, it

allows the feed pattern to be specified in the E- and H-planes independently to control the

reflector illumination function.

Three options are available for the feed pattern. These are: uniform illumination,

dipole illumination, and cos'^0' illumination where 0' is the angle measured from the feed

axis. For this case, the feed patterns in the E- and H-planes are given by

30
E . r

n = cos e

H

r- n I E = cos 6 . H

other parameters of the antenna which are required as input include focal length to diameter ratio, fractional diameter blockage, diameter in units of wavelength, and axial

position of the feed relative to the focal point of the reflector.

The subroutines use a Romberg type of algorithm to perform the necessary integrations.

This is an adaptive algorithm in the sense that it selects the necessary interval size

based on a required accuracy. The result is a rapidly executing program, because advantage

can be taken of the fact that rather large increments can be used near the main beam, thus

reducing time to compute the far fields for these points.

If the integration routine is unable to achieve the required accuracy, either because

of accumulated round-off error or because the integration range cannot be sufficiently

subdivided, an appropriate error flag is set. This condition is noted in the program

output, so that this data may be deleted in further calculations. Further discussion of

these errors occurs in the program description.

3.2 Test of Near-Field Program

In order to check the operation of the near-field transformation in conjunction with

the far-field PO model, a test case consisting of a 52-wavelength, uniformly illuminated

aperture was run. Near fields were calculated in the aperture plane from the far fields

calculated using PO, and were compared with the original uniform distribution. Results are shown in figure 9. As can be seen, the calculated results agree well with the uniform

distribution. Note that the scale is electric field in volts/meter, not relative field in dB. Total variation from the original distribution is +1.1 dB, -0.55 dB.

The ripple can be attributed to several causes. Since the PO program encounters

round-off error problems for angles which lie too far off boresight, the far field must be

truncated beyond a critical angle. For this example, the truncation occurred at an angle

of 10.2 degrees, which was also chosen because it was a null position. Even so, eight

sidelobes were included in the far-field pattern, the last one having an amplitude of about

-40 dB relative to the main beam. The spacing of far-field points also affects the ripple

to some extent. Here, there were about 10 points per sidelobe. Finally, evanescent modes were neglected because of the point spacing chosen in k-space. The results do indicate

that useful near fields can be calculated from the model for this case.

31
ELECTRIC

FIELD

-

VOLTS/METER

50

45

40

35

30

O' ^ » * — i * .00 .14 .29 .43 .57 .72 .86 1.00 1.14 1.29 1.43

X-POSITION - METERS

Figure 9a. Field strength in a uniformly illuminated aperture calculated using

physical optics far fields. Dashed line indicates theoretical distribution.

32
PHASE

-

DEGREES

Figure 9b. Phase of field in a uniformly illuminated aperture calculated using

physical optics for fields.

33
4. COMPARISON OF PHYSICAL OPTICS AND MEASURED FAR FIELDS

As noted in section 3, the PO model represents an approximation to the true fields

generated by the reflector antenna. Because of the approximations involved, it was considered desirable to compare the results obtained using a PO model to actual measured

far-field patterns. Four cases were considered, and some additional experimental work was done in one case to attempt to determine the cause of observed discrepancies. The four

cases are listed in table 4.1.

TABLE 4.1

Antenna Frequency

GHz

Di ameter

m(x)

Fractional

Aperture

Blockage

E

n H

n MeasuredGain

dB

1 4.0 1.22(16.25) .164 1.57 1.72 29.66

2 4.0 1.22(16.25) .164 1.02 1.07 28.34

3 12.73 1.22(51.8) .143 1.09 1.09 40.70

4 57.5 .45(87.5) .120 1.10 46.3

Each antenna had an essentially circular blockage at the feed, and each had three

support struts. Antennas 1, 2, and 3 were essentially identical, being built by the same manufacturer, the only difference being in the feed. The feeds of antennas 1 and 2 were

adjusted in the NBS near-field facility to obtain optimum focus and coincidence of

electrical and mechanical axes.

The adjustment procedure consisted of moving the feed axially and laterally in order

to obtain a minimum near-field phase curvature (focus adjustment) and a near-field phase

with no linear component (boresight adjustment). It should be noted that for antennas 1 and 2, at least, it was not possible to obtain a flat phase front in both E- and H-planes.

A compromise adjustment was made. Thus, either the E- or H-plane pattern can be somewhat

improved, but only at the expense of a worse pattern in the other plane. It is not known

whether the problem exists in the case of antennas 3 and 4, as these antennas had been

previously measured at NBS and were not available for further experimentation.

In order to determine the parameters n^^ and n^ for antenna 3, the dimensions of

the feed were obtained and the patterns estimated using standard horn theory. For antenna 4, a Cassegrain antenna, the near-field data obtained were used to estimate the

parameters when the antenna was calibrated at NBS. For antennas 1 and 2, the feed patterns were measured on a far-field range before the feeds were installed on the reflector.

The far-field patterns for these antennas are shown in figures 10 to 13, with the

PO predicted patterns superimposed. We note that, in general, the agreement between the

34
Figure 10a. Comparison of measured and calculated far-field patterns for antenna

No. 1. E-plane cut, solid line - measured pattern, dashed line - physical optics.

35
AZIMUTH ANGLE - DEGREES

Figure 10b. Comparison of measured and calculated far-field patterns for antenna

No. 1. H-plane cut, solid line - measured pattern, dashed line - physical optics.

36
ELEVATION ANGLE - DEGREES

Figure 11a. Comparison of measured and calculated far-field patterns for antenna

No. 2. E-plane cut, solid line - measured pattern, dashed line - physical optics.

37
AZIMUTH ANGLE - DEGREES

Figure 11b. Comparison of measured and calculated far-field patterns for antenna

No. 2. H-plane cut, solid line - measured pattern, dashed line - physical optics.

38
Figure 12a. Comparison of measured and calculated far-field patterns for antenna

No. 3. E-plane cut, solid line - measured pattern, dashed line - physical optics.

39
Figure 12b. Comparison of measured and calculated far-field patterns for antenna

No. 3. H-plane cut, solid line - measured pattern, dashed line - physical optics.

40
FAR-FIELD

AMPLITUDE

I

AZIMUTH ANGLE - DEGREES

Figure 13. Comparison of measured and calculated far-field patterns for antenna

No. 4. H-plane cut, solid line - measured pattern, dashed line - physical optics.

41
PO computations and measurements improves as the diameter to wavelength ratio

increases; and further, by comparing 1 and 2, we note that a higher value of edge

illumination seems to allow a better prediction.

Several possible explanations for the discrepancies exist. These can be grouped into

five categories: edge effects, diffraction by struts, aperture blockage effects, back and

sidelobe radiation from the feed, and violation of the assumed circular symmetry.

The first of these arises because of the sharp discontinuity in current which occurs

at the edge of the reflector surface. The effect of this discontinuity is imperfectly

accounted for by the PO model. In order to better describe edge effects, it is necessary

to employ the geometrical theory of diffraction (GTD) or similar asymptotic theories to predict more accurately the sidelobes generated by these edge effects. To clearly see the

difference between the edge as described by PO and GTD, it is useful to consider the

"effective" currents which are used. These are illustrated in figure 14. We note that, in

both cases, there is a sharp discontinuity in current density at the edge of the reflector

surface. The GTD model includes the effect of the singularity in the current at an edge.

GTD models usually assume a sharp edge. However, the antennas used in this study were made

with a rolled edge as is common; and thus the normal GTD theory will not apply. The

effect of the edge singularity manifests itself more as the angle off boresight increases.

It is thus assumed that the use of PO rather than GTD is not significant in explaining the

observed discrepancies.

The remaining processes are more likely candidates for the observed discrepancies.

While blockage is taken into account, diffraction from the feed structure is not. In addition, because of the structure of the particular antennas used, multiple reflections

between the feed structure and the reflector surface are likely to occur. An approximate

cross section is shown in figure 15.

In order to test the multiple reflection hypothesis, the feed support plate was lined

with rf-absorbing material, and near-field scans were again taken. The resulting far

fields are shown in figure 16. Note that the agreement between the PO model and measured

far fields is better. This suggests that at least part of the problem is in neglecting

multiple reflections between the feed housing and the reflector.

The struts were now covered as shown in figure 17 to try to minimize diffraction by

them. Results of this test showed an increase in the discrepancy between experiment and

theory as shown in figure 18. However, this should not be taken to mean that strut reflection is negligible because, as can be noted in the photograph, there is significantly

more blockage for rays travelling off axis than in the uncovered strut case. A better

method for determining the strut diffraction effect experimentally would be to support the

feed with dielectric material and measure patterns in this configuration. The asynmetry

observed in the E-plane pattern is an indication of significant strut effects.

42
CURRENT

DENSITY

} REFLECTOR \

I SURFACE ,

Figure 14. Comparison of effective current distribution used in physical optics

and geometrical theory of diffraction calculations. (Uniform

distribution assumed).

Figure 15. Diagram of multiple reflections involving feed structure.

43
ELEVATION ANGLE - DEGREES

Figure 16a. Comparison of measured and calculated far-field patterns for antenna

No. 1 with feed region covered with absorber. E-plane cut, solid curve - measured pattern, dashed curve - physical optics.

44
0

m

a

-5

-10

-15

-20

2 -25

<

Q

-I

UJ^ -30

cc<

ll -35

-40

-45

-50

f\

L

- u

/r 1

I \!

1

/ /

‘V

1 ll 1; '1

ll 1

ll \

ll \

\

1 1 \ \\

\ \

[

1 J\

/

/ / /

ii

i|

1

ll ^

ll

1

\\

\ \V X

1

1

i

i

1

f

1

1

\\

\\

1\ M \

/

^

/

/ / z' 'A:

t1

1

1

1

1 ' m/

> '

\ \ \

\ \

>

l\

-/

' ^ ll

' M

1\n

\

\

\ -60 -48 -36 -24 -12 0 12 24 36 48 60

AZIMUTH ANGLE - DEGREES

Figure 16b. Comparison of measured and calculated far-field patterns for antenna

No. 1 with feed region covered with absorber. H-plane cut, solid curve - measured pattern, dashed curve - physical optics.

45
Figure 17a. Feed region of antenna with absorber collar.

Figure 17b. Feed support struts with absorber attached.

46
I

!

Figure 18a. Comparison of measured and calculated far-field patterns for antenna

No. 1 with feed region covered with absorber. E-plane cut, solid curve measured pattern, dashed curve - physical optics.

47
AZIMUTH ANGLE - DEGREES

Figure 18a. Comparison of measured and calculated far-field patterns for antenna

No. 1 with feed region covered with absorber. H-plane cut, solid curve - measured pattern, dashed curve - physical optics.

48
Backlobe radiation from the feed antenna is not considered. It is difficult to estimate the magnitude of this effect. While patterns were taken for the feeds of antennas

number 1 and 2, it is in a completely different mounting structure when in place in the

antenna; and, as a result, the pattern in the rear hemisphere for the feed will not give

any valid data about its back lobes.

The following general conclusions can be stated concerning the usefulness of this

particul ar PO model

.

1. The model appears to give better results for larger D/X

ratios.

2. Sidelobe positions are fairly accurately predicted for

the first few sidelobes.

3. The magnitudes of the predicted sidelobes can be as much

as several dB off for small (<50X) antennas.

4. A contributor to the observed differences in the case of

antenna 1 (and also 2 because its construction was the same)

is multiple reflections between the reflector surface and

feed structure.

5. For far sidelobe regions (beyond 4 or 5 lobes) it appears

that a better model such as a PO-GTD combination should be

empl oyed.

6. A model which takes struts into account would be useful.

Because the theoretical model is used to predict near-zone fields and coupling, it is

useful to consider the effect of discrepancies between the modeled and actual fields on the

prediction of near-fields and coupling.

For determination of the near-field radiation in front of the antenna, it is expected

that the sidelobe discrepancies will have a negligible effect. The major source of error

will occur because the true gain is not known and must be estimated. The current PO model

will not give results in the region far off boresight or in the back direction.

The coupling results will be affected by the sidelobe region, however. Calculation of

the coupling depends on that portion of the far field of each antenna which is subtended by

the other; hence, the sidelobe structure is important. Because the locations of the

sidelobes are accurately predicted, the basic structure of the coupling as a function of

49
relative position of the two antennas will be retained. Any errors in the magnitude of the

far field predicted by PO will be carried over into the coupling ratio.

5. COMPARISON OF PREDICTED AND MEASURED NEAR-FIELD COUPLING

In order to utilize the near-field coupling program (CUPLNF) to predict actual

near-field coupling, the two C-Band reflector antennas (numbers 1 and 2 of table 4.1) which

were modeled using PO, were set up to measure the near-field coupling directly for various

relative orientations and separations. The frequency of operation was 4.0 GHz which gives

a diameter of 16.25 wavelengths and a combined or mutual Rayleigh distance (D|+D2)^./X

of about 80 meters.

The antennas were mounted on movable wooden towers at a height of about 7 meters above

the ground. The coupling was measured as a function of separation distance for separations

ranging from 1 to 8.5 meters and for three relative orientations of the antennas. This

procedure also gives a measure of the level of multiple reflections between the antennas

(which are neglected in the calculations).

A photograph of the experimental setup is shown in figure 19, and figure 20

illustrates schematically the three relative orientations employed.

For cases two and three, the angle of the receiving antenna was varied over approximately a +4° range at a fixed separation of 3.5 meters to test the coupling as a function of angle.

Small angles were deliberately chosen for two reasons. Because the PO model used here

does not perform well in the si delobe region, the measurements must be restricted to small

angles so the model can successfully predict coupling from boresight. Further, the planar

scan data yields far fields which are valid only to about 45° to 50°, and, this too,

limits the angles. For wider angle coverage, nonplanar scanning techniqes such as cylindrical or spherical would prove useful.

It should be noted that in case 1, the primary source of coupling is the interaction

of the main lobes of the two antennas. Case 2 corresponds to the main lobe of the

transmitting antenna interacting with the sidelobes of the receiving antenna. In case 3,

the sidelobes of each antenna interact with each other.

Calculation of the coupling between the antennas was carried out for five separations

in the range 1.5 to 7.5 meters for each case measured. Far fields used as input were from

two sources. The experimentally determined far fields obtained from transformation of

near-field data were used in one set of calculations, and the far fields obtained from the

model using the adapted USC PO subroutines were used in the other calculations.

50
51I

two

reflector

antennas.
CASE 3

= 0*. = 180*

9^. * 20*. 9^ = 15*

4/ = 0*. S' = 180*

't ' r

Figure 20. Schematic showing relative orientations of antennas for the

three test cases.

52
The results of the three cases are shown in figures 21 to 23. In each case, the

envelope of the measured data is shown, rather than the actual data, which consists of

approximately sinusoidal oscillations of period \/2 superimposed on the data which arise

because of multiple reflections between the two antennas.

We note fairly good agreement between the measured data and that predicted using

measured far fields except in the case of the (0°, 15°) data. This disagreement will

be discussed shortly. In the (QO, 0°) case, the prediction using actual far-field data

is approximately 0.5 dB low, and follows the shape of the average of the measured data very

well. In the (20°, 15°) case, we again ob-serve fairly good agreement between the shape

of the predicted and measured curves with an average error of about 2 dB. As in the case

of the measurement of low si delobes, this error is not unacceptable. It might be expected

that a greater error would occur when the sidelobes are interacting because of their

complicated structure and resultant sensitivity to orientation. While every effort was made in the experimental procedure to ensure accurate positioning, the accuracy was probably no better than 1/2° about all three axes.

We now discuss the (0°, 15°) case where agreement is not good. Here, we suggest

that slight misalignment may be the primary cause. In the rotation performed at a separation of 3.5 meters, a peak of -25.2 dB occurred at about 12.0°. Calculations show

that a peak in the predicted coupling occurs at an angle of 12.4° with a magnitude of

-26.4 dB. Predicted and observed nulls also occur at about 20° to 22°, though the

magnitude comparison of the null depth is not so good. Because of multiple reflections

and multipath and because the cross-polarized component is not included in the

calculations, null comparisons cannot be expected to be so good as that observed at relative maxima. It would thus appear that the discrepancy at (0°, 15°) can be

explained by a small error in orientation.

6. CONCLUSIONS AND RECOMMENDATIONS

Programs and subroutines were written to calculate near fields of reflector antennas

and to calculate mutual coupling between antennas whose separation and orientation are arbitrary. The basic data required for these calculations are the twif-dimensional complex

far-field patterns of the antennas involved.

Documentation for the programs including listings and sample input and output are given

in Appendices A and B.

It was seen that the coupling program provides good results if proper far fields are used as input data. When a model such as the physical optics discussed here is employed,

the coupling program fails to adequately predict the coupling for off-axis directions.

53
INSERTION

LOSS

Figure 21. Mutual coupling between 1.2 meter reflector antennas.

Case 1: 0^=0°, 0-(-=O°. Solid lines indicate envelope of measured

mutual coupling.

54
Figure 22. Mutual coupling between 1.2 meter reflector antennas.

Case 2: e =15°, 9^=0°. Solid lines indicate envelope of measured

r V mutual coupling.

55
Figure 23. Mutual coupling between 1.2 meter reflector antennas.

Case 3: e^=15°, e^=20°. Solid lines indicate envelope of measured

mutual coupling.

56
Several areas would appear to be worth pursuing. Certainly better models can be

obtained. For the types of data required (complete two-dimensional, far-field patterns), a two-dimensional integration PO model is probably not practical. For this type of approach

each far-field point would require a two-dimensional rather than a one-dimensional

numerical integration. Further, because no symmetry is assumed, all needed far-field

points must be computed rather than only the E- and H-plane cuts. Because of these

considerations, the computation of the complete pattern by a model which requires

two-dimensional integration appears to be impractical. An alternative would be to calculate the main beam and first few sidelobes with PO, and use a GTD analysis for points

farther off axis. Such a combination would use the best features of each technique.

A second alternative would be to reformulate the PO model in terms of aperture fields

rather than surface currents. This approach would allow efficient computations using the

FFT.

Contrasted with the above is the question of whether application might permit the use of less sophisticated models which would give upper-bound values for the desired

quantities. Note that regardless of the sophistication of the model employed, certain

antennas of a given type may fail to perform as predicted because of unit-to-unit

variations. These variations have been observed to be as large as the discrepancies

observed between measured and modeled fields for certain types of antennas.

With this in mind, we suggest two alternatives to the use of a sophisicated model.

First, a catalog of measured far fields for antenna types in use could be compiled and

these data used in coupling or near-field calculation. It would probably be necessary to measure several samples in order to determine expected unit-to-unit variations. An alternate approach would be to employ an envelope type of far-field pattern, such as the

amplitude pattern that the CCIR recommended (32-25 log 0 function), if a reasonable phase

function is also included.

It is recommended that these approaches be studied to determine if, in fact, they can give useful results.

ACKNOWLEDGMENT

The physical optics computer program was supplied by Prof. W. V. T. Rusch of the

University of Southern California. Near-field measurements were performed by

Mr. D. P. Kremer, who also assisted in the mutual coupling measurements. Helpful

discussions with Dr. R. C. Baird, Mr. A. C. Newell, and Prof. Rusch are also acknowledged.

57
REFERENCES

[la] Kerns, D. M., and Dayhoff, E. S., Theory of diffraction in microwave interferometry,

J. Res. Nat. Bur. Stand., 64B , 1-13 (Jan. -March 1960).

[lb] Kerns, D. M. , Plane-wave scattering-matrix theory of antennas and antenna-antenna

interactions: Formulation and applications, J. Res. Nat. Bur. Stand, SOB , 5-51

(Jan. -March 1976).

[2] Yaghjian, A. D., The reactive and far-field boundaries for arbitrary antennas derived

from their quality factor. National Radio Science Meeting (USNC/URSI), University of

Colorado at Boulder (Jan. 9-13, 1978).

[3] Ames, J. S., and Murnaghan, F. D. , Theoretical Mechanics, Ch. II (Ginn, Boston,

Massachusetts, 1929).

[4] See, e.g., Cathey, W.T., Optical Information Processing and Holography, Ch. 2 (John

Wiley & Sons, Inc., New York, N.Y.. 1974).

[5] See, e.g., the June 1967 issue of the IEEE Trans, on Audio and Electroacoustics.

[6] See, e.g., Johnson, C. C. , Field and Wave Electrodynamics, section 10.5 (McGraw-Hill,

New York, N.Y., 1965).

[7] Hu, Ming-Kuei, Near zone power transmission formulas, IRE Convention Record, 6, Pt. 8, 128-138 (1958).

[8] Rusch, W. V. T., Reflector antennas, in Numerical and Asymptotic Techniques in

Electromagnetics, R. Mittra, Ed. (Springer-Verlag, New York, N.Y., 1975).

[9] Rusch, W. V. T., Course notes for short course. Reflector Antenna Theory and Design;

University of Southern California, Los Angeles, California (July 1976).

[10] Tai, C-T, Dyadic Green's Functions in Electromagnetic Theory (Intext, New York, N.Y.,

1972).

58
APPENDIX A. POMODL - PHYSICAL OPTICS ANTENNA MODEL

This appendix includes detailed documentation of the program which models reflector

antennas using physical optics and, at the user's option, calculates a two-dimensional

far-field pattern for use by CUPLNF and also calculates near-field patterns on a specified

plane. Each subroutine is documented individually, except for those which were obtained

from other institutions and used unaltered, in which case only a brief description and

listing is included. The final section of the appendix includes a sample input deck and a sample program output.

A.l GENERAL OVERVIEW OF COMPUTER PROGRAM

The program POMODL and its associated subroutines are described in detail in the following

subsections. The flow chart below is presented in order to give the reader an overview of

the operation of the program package.

59
60
A.l .1 PROGRAM POMODL

PURPOSE :

To control input, output and flow of far-field calculation and transformation to near

fiel d.

GENERAL DISCUSSION :

This subroutine is a modified and extended version of SUBROUTINE PDRIVE written by

Professor W. V. T. Rusch of the University of Southern California (USC). This

subroutine reads data cards which specify the physical parameters of a paraboloidal

reflector antenna and the parameters of the desired near-field patterns. It is basically a driver program for the USC PO subroutine PARAB and the subroutines which

perform the far- to near-zone transformation.

The program produces plots and tables for far field in the E- and H-planes and near- field cuts on a plane or planes perpendicul ar to the axis of symmetry of the

reflector. In addition, the program calculates the near field on the complete plane

and stores it in an array. This data may be obtained by a minor program modification.

The far field presented in a table of values at equally spaced increments in (kx,ky) space is also available at the user's option.

Because the techniques used require a substantial amount of computer core, it is

recommended that the DIMENSION and COMMON statements specifying the size of arrays EY and DATAX be changed to suit the problem considered. Minimum size for EY is

2 X (number of points to be calculated in 6-direction) x (number of points in 0-

direction). For DATAX, the size must be at least 2 x (number of near-field points in

x-direction) x (number of near-field in y-di rection) . Because arrays EY and DATAX are

not directly used by the main program but are dimensioned only for storage allocation

purposes, they may be dimensioned as single dimensioned arrays whose sizes are greater

than or equal to the values specified above.

INPUT CARDS

The input card deck consists of two groups of cards. The first five cards must be

included in every run and specify the parameters of the antenna being modeled and the

ranges and increments for the far field.

The second group of cards specifies the desired parameters of the near field to be

calculated. If no cards of this group are present (i.e. only five input cards), only

the E- and H-plane far-field patterns will be calculated and plotted. The near-field

61
parameters are specified by a single card. Near fields for planes lying at different

z-distances can be calculated by including multiple cards.

In addition, it is possible to specify that the far-field array which is calculated at evenly spaced points in (k^, ky), space may be written out to logical unit 20 for

use as input data for the mutual coupling program CUPLNF.

The following is a list and description of the data cards.

Group I

Card 1 Col. 1-40 This card contains alphanumeric information, usually the name and telephone extension of the person submitting

the job.

Card 2 Col. 1-80 An alphanumeric identifier which is used to identify the case being studied. It appears as headings of tables

and plots and on identification records of output

fi 1 es.

Card 3 This card specifies antenna parameters. All numbers on this card must have the decimal point explicitly

specified.

Col. 1-10 FOD - the F/D ratio for the reflector.

Col. 11-20 FOL - the diameter in wavelengths of the reflector.

Col. 21-30 BLOCK - the feed blockage as a fraction of the reflector

diameter.

Col. 31-40 DFOCUS - amount of axial defocussing in wavelengths,

positive defocussing if the feed is beyond the focal

point.

Col. 41-50 ACOSE-E-pl ane illumination factor.

If ACOSE < -100. aperture is uniformly illuminated.

-100. ^ACOSE < 0. feed is a y-directed electric dipole.

ACOSE >_ 0. E-plane feed pattern is cos^COSE

62
,1

Card 4

\

Card 5

Group II

Card 6

Col. 51-60 ACOSH - H-plane illumination factor.

If ACOSH 0. H-plane feed pattern is

cos^COSH^^.q)

Col. 61-70 FREQ - frequency in GHz.

This card specifies parameters related to the far-field

pattern calculated from PO. Except as noted, decimal

pointy must be explicitly specified.

Col. 1-10 THETHF - initial value of theta - degrees.

Col. 11-20 DTHETA - theta increment - degrees.

Col. 21-30 PHIF - initial value of phi - degrees.

Col. 31-40 DLPH - phi increment - degrees.

Col. 41-45 NTHETA - number of theta points desired, no decimal

point, right justified in field.

This card gives data which allow calculation of

magnitude of near electric field.

Col. 1-10 PIN - power input to antenna, a blank in field gives

default value of 1.0 watt.

Col. 11-20 EFF - assumed aperture efficiency of antenna in percent,

a blank in field gives default value of 100 percent.

This card specifies the parameters of the near field

which is to be calculated. This card may be repeated to calculate near fields on different planes. If card 6 is

omitted, only a far field will be computed and plotted.

Col. 1-10 DELX - near field x-increment in meters.

Col. 11-20 DELY - near-field y-increment in meters,

63
Col. 21-30 DIST - distance from focal point of antenna reflector

to near-field plane in meters.

Col . 31-40 Blank - field not used.

Col. 41-45 IR2T0N - number of y points desired in near field, no decimal point specified, right justified in field.

Col. 46-50 IC2T0N - number of x points desired in near field, no decimal point specified, right justified in field.

OUTPUT

A copy of typical output for the program is included in section A. 2. A table of input

parameters is given first followed by the E- and H-plane far-field patterns for the

antenna. Page printer plots for the E- and H-plane are then included.

The near-field parameters are then printed in a table giving the x- and y- near-field

centerline cuts. Finally, the amplitude and phase of the near-field centerline cuts are plotted.

SYMBOL DICTIONARY:

ACOSE = E-plane aperture illumination factor

ACOSH = H-plane aperture illumination factor

BLOCK = Fractional diameter blocking

CASEID = Alphanumeric identifier

CEE = Speed of light x lO"^

DATAX(I,J) = Array reserved for far field versus k^ and ky DELX = Near-field x-increment

DELY = Near-field y-increment

DFOCUS = Amount of axial defocussing beyond focus in wavelengths

DIST = Distance between near-field plane and focal plane in meters

DLPH = Far-field phi increment in degrees

DOL = Reflector diameter in wavelengths

DTHETA = Far-field increment in degrees

EFF = Assumed antenna efficiency

EPFAZE = Phase of EPHI in degrees

EPHDB(I) = Normalized phi component magnitude expressed in dB EPHI = Phi component of far field

EPLANE(I) = y-component of s^g

EPMAG = Intermediate variable - magnitude of EPHI EPREF = Magnitude of EPHI(l) used for normalization purposes

64
ETFAZE

ETHDB(I)

ETHETA

ETMAG

ETREF

EY(I,J)

FKAY

FOD

FREQ

GAIN

GDB

HPLANE(I)

IC2T0N

IDIR2T0N

JTH2M1

JTHETA

JTHX2

NPHI NTHETA PARAB

PHIF

PIPNRM

PIN PNRM

RTD THETA(I)

THETAF

= Phase of ETHETA in degrees

= Normalized theta component magnitude expressed in dB

= Theta component of far field

= Intermediate variable - magnitude of ETHETA

= Magnitude of ETHETA(l) used for normalization purposes

= Array reserved for far field versus 9 and 0

= Propagation constant = 2Tr/wavelength

= Reflector focal length/diameter

= Frequency in GHz.

= Theoretical gain of antenna

= Gain of antenna expressed in dB

= x-component of s^q

= Number of near-field points in x-direction

= Alphanumeric identifier, usually programmer's name

= Number of near-field points in y-direction

= 2 X JTHETA - 1 used for array indexing

= Theta loop index

= 2 X JTHETA used for array indexing

= Number of phi points to be calculated

= Number of theta points to be calculated PARAB

= Main subroutine to calculate far field of paraboloidal reflector antenna

= Initial value of phi in degrees

= 7T = 3.14159

= Power normalization factor

= Input power to antenna

= Power normalization factor

= Radians to degrees conversion factor = tt/180

= Polar angle measured from boresight axis

= Initial value of theta in degrees

COMMON BLOCKS :

The labeled common used in POMODL is described below with a list of routines in which

it is used. The variables are defined in the symbol dictionary.

COMMON /CNTRL/ DTHETA, DLPH, DELX, DELY, FREQ, DIST, PNRM

Routines using /CNTRL/: POMODL, FAR2D, FFKXY, NFKXY

65
T ppn^p^“ PHMnnL (Tf^PUT, "1IITPUT, T A P P 5 = I N PUT , TA P F6 = GUT PUT . TAPE20) POMODL 1 C PCMODL 2

r OPIVFP PPOr-PAI POP, PFYSTCAL OPTICS SUPPOUTTNF PAPAB. WPITTFN BY POMODL 3

r ppQFPSPqP W. V. T. PUFCH np THE UNIVFPSTTY OF SOUTHEPN POMODL A

K C CALIF^RMTA, WHTCF INCLUDES CAPABILITY OF CALCULATING POMODL 5 c NEAP FIELDS ON A SPPCIFIED PLANF. POMODL 6

r POMODL 7

r 7-AVTP IS AYIS OF SYYFETPY POINTING AWAY FPQM PAPARPLOID. X IS P(nMOOL 0

C POLAP ANGLE THETA-PPINE NEASUPED FROM THE PQSITIVE-I AXIS. POMODL 9

: n c X = °I IS THE DIPFCTION OF THE REFLECTOR VFPTEX. POMODL 10

c POMODL 11

r XP IS THP POLAP ANGLE THFT a-DOU B L E- P R I NE MEASURED FROM THE POMODL 12

c POPTTTVF ?-AVtS WITH THE DFFGCUSFD FFFO AS ORIGIN. POMODL 13

r POMODL lA

1 r TPP FTFins OP THE FPPC APP THF FIELDS OF A CIRCULAR APERTURE POMODL 15

C. PXriTFn IN THP Mil A7IMIJTHAL MDDF. THF F-PLANE IS THE POMODL 16

r Y7-PLANF and THF H-PLANE IS THF X7-PLANE. THE COMPLEX POLAR POMODL 17

C pATTFpnS AKTP) and DKTP) are SUCH THAT MOST OF THE POWER POMODL IB

c TP PADIATED T^WAPP THF PEFLECTOP AND VFPY LITTLE POWER IS POMODL 19 ?0 r paptaTFD INTO THF HALP-SPACF XP.LT.PI/?. F U RTH E P M OP f

.

Tn POMODL 20

c ASSURE CONTINUITY OF THE FIELD WHEN XP » PI, IT IS NECESSARY POMODL 21

r THAT Dl(PI) = -Al(PI). POMODL 22

c POMODL 23

c POMODL 2A

c pro = pppLFCTnp E/n POMODL 25

r ncL = PFFLFCTOP niAMFTFP IN WAVELENGTHS POMODL 26

r Plori< = FRACTIONAL DIAMETER BLOCKING POMODL 27

r PFOrUS = AMOUNT OF AYTAL DFFOCUSING BEYOND THE FOCUS IN W A VE L E NG TH P 0 MDDL 28

c TF(ArOSF.LT. (-1CC.0) ) THE APFPTUPF IS UNIFORMLY ILLUMINATED POMODL 29

’0 c I F ( ACOS F . r,E .

( -100 . 0 ) . AND. LT . 0. 0 ) THF FEFD IS Y-DIRECTED ELECTRIC POMODL 30

r PTOPL F POMODL 31

r I F ( ArnsF . GF .0 .0 ) A1 I (COS ( PI-XP )

) TT ACOSF , xp.GE.PI/2 PQMODL 32

r = C, XP.LT.PI/2 POMODL 33

c D1 =- (COS ( PI-CP n ** ACOSH, xp.C-E.PI/2 POMODL 3A

?5 r = 0. XP.LT.PI/2 POMODL 35

r FPPC = FPPOUFNCY POMODL 36

r POMODL 37

r THFTAF = INITIAL VALUE OF THETA, DEGREES POMODL 38

c DTHFTA = DIFPERENTIAL VALUE OF THETA POMODL 39

^•0 r PH IF = INITIAL VALUE FF PHI POMODL AO

r DLPH = DTEFFPENTIAL VALUE dE PHI POMODL A1

C NTHFTA = NU’^REP Qc THETA VALUES PQMODL A2

r pomqdl A3

c PIN = PDWFP INPUT TO ANTENNA FOP NFAP-70NE FIELD STRENGTH POMODL AA

4

c r FEE = APPPTHRF EFFICIENCY OF ANTENNA POMODL A5

r POMODL A6

c POMODL A7 COMPLFY ETHFTA,EPHI POMODL A8

r POMODL A9 DIMCNSIDN FTHDP(200), FPHDR(2C0), THETA(2C0) PQMODL 50 DIMpnipion FRLANE(AOO), HPLANE(ACO) POMODL 51 DIYENPinN ID(A) POMODL 52

r POMODL 53 CCMMPN EY(6C00), DATAX(81<?2) POMODL 5A

1=; CCMmpn /id/ CASFID(P) POMODL 55 Cpmmpm /cntpl/ dtmfta, dlpw, dflx, dely, frfo, dist, pnrm POMODL 56

c POMQDL 57

r INPUT POMODL 58 READ (5, 5000) ID POMODL 59

^•0 PRINT EOOl, ID POMODL 60

P F AD ( " , 5 0DD ) C AS F ID POMODL 61

WP ITP ( 6,6000 ) CASF ID POMODL 62 PFAD(5,5C20)FnD,nCL,BLnrK,nFDCUS,ACOSE,AC0SH, FRFO PCMODL 63 RFAD(5, 5DA0) THFTAF, OTHFTA, PHIF, DLPH, NTHFTA POMODL 6A

A ^ PFAD(5. 50R0) PIN, FFF POMODL 65

IF (PIN ,FC. 0.) PIN = l.C POMODL 66

IF (FFF .FO. 0.) FEE = 1. PQMODL 67 wPTTF(6,Ao?o)Fcn,nrL,PLorK,oEocus, fpfq POMODL 68 TF(AC0SE.LT.(-1CD.C))WPITE(6,6C05) POMODL 69

70 TF(ArnSE.GF.(-10O,0).AND.ACnSF.lT.0.0)URITE(6,6C10) POMODL 70 IE(ACOSF.GP.O.O)WPITF(6.6015)ArOSE,ACCSH POMODL 71

r POMODL 72

c 'MISCELLANEOUS POMODL 73

C POMODL 7A 75 PI = A.OTATAN(l.O) POMODL 75 PTD = 180.0/PI POMODL 76 TEF = .?Qg7Q?5 POMODL 77

66
FKA Y = 2.YPTYFPFC/CFF POMOOL 78 NPHT = 90./0LPH + l.CCOCCl POMODL 79

eo GAIN = FFF*PH'PIYDnLYDrL PDNGDL 80 GOB = 10n4L0G10(GATM PGMGDL 81 pNRY = SQP T ( 1 5 .* PINYG AI N /FK A Y/FK A Y ) / PI POMODL 82 WRITF( 6 , 6002) FFFYICC.. GOB. PIN PGMGDL 83 WPITF(6,6030) PPMPDL 84

PS r POMODL 85

r ENTFP THE THETA LCCP POMODL 86

c POMODL 87 nc 100 JTHETA = l.NTFFTA POMODL 88 THETA ( JTHETA ) = THETAF + ( J T HE T A-1 ) DTH ET

A

POMODL 89 QO GALL PAPAR(Fnn,DOL.RLCCK.OFOCUS.ACOSE,APOSH,THFTA!JTHETA), cTHETA,POYODL 90

IF PHI ) POMODL 91 ETMAG = CABStETHETA) POMODL 92

IF ( JTHETA . FO . 1 ) FTREF = ETHAG POMODL 93 BTHOP (JTHETA ) = 20.OYAL0G10 ( ETMAG/ETPEF

)

POMODL 94

05 FTFA7E = -AT AN2( AIMAG (ETHETA ) . PEAL ( ETHETA )

) PTD POMODL 95 FPMAG=0APS(FPHI) POMODL 96 TF(JThpTA.FO.I) FPPEF = FPYAG POMODL 97 EPHDB ( JTHETA ) = 20.0 + ALQG10( EPYAG/EPPEF) POMOOL 98 EPFA2F = -ATAH2 ( A IMA G ( EOMT ) . PF AL ( EPHI ) )*PTn POMOOL 99 100 PPINT 6040, THFTA(JTFFTA), FTMAG, ETHDB(JTHETA), ETFAZF, FPMAG, POMOOL 100 lEPHOR ( JTHETA ) , EPFAZE POMODL 101 JTHY? = JTMETA*2 POMODL 102 JTHPMl = JTHXZ - 1 POMODL 103

c POMODL 104

IPS r NPPMalIZF E to l.C AT THETA = 0. AND CALCULATE SIO. POMODL 105

C POMODL 106

EPL ANE ( JTH2M1) = C A B S ( FT HE T A ) / FT R E F / COS ( T HFT A ( J T HF T A ) / R T 0

)

POMODL 107

F PL ANF ( J THY? ) = FTFAZE/RTO POMODL 108

HPL ANF ( JTH?Ml ) = PAPS (FPHI) /FTPFF/CCS(THFTA( JTHETA) /RTO) POMODL 109 no HPLAME(JTHX2) = EPFAZE/RTD POMODL no

100 CONTINUE POMODL 111

r POMODL 112

c PLOT E AND H-PLANE AMPLITUDES POMODL 113

c POMODL 114

115 CALI. PI T120P ( ThfTA, ETHPP, 60., -60 ., 0., -50., NTHETA, IHY, 1, 1 ) POMOOL 115 PRINT 6000, CASEID, ICH E-PLANE POMODL 116 CALL PLT120R ( THFTA , FPHDR, 60., -60., 0., -50., NTHETA, 1H+, 1, 1 ) POMOOL 117 PPINT 6080, CASEID, ICH H-PLANE POMODL 118

c POMOOL 119

1?0 0 POMODL 120

c READ data for near FIELD POMOOL 121

c POMODL 122

c DFLY = NEAP FIELD Y-RPACING POMODL 123

r deLY = NFAR FTFIO Y-RPACING POMODL 124

125 c CIST = Z-PQSITICN CP NEAR FIFLD PLANE. (DIST = 0. IS FOCAL POMODL 125

c PLANE qP PAPABDLA) POMOOL 126

r nijM>^Y - not CURPFNTLY used POMODL 127

r 1;p7T0N = NUMBER OF POINTS IN Y-APRAY POMODL 128

c TC2TPN = number of points IN Y-APPAY POMODL 129

130 c POMODL 1 30

1 RFAD 5D40, DELY, DELY, DIST, DUMmy, IP2TGN, IC2T0N POMODL 131

IP ( PPP ( 5 ) ) 200,

?

POMODL 132

2 PRINT 6070, DELYAIOO., TCBTON, DELYAIOO., IR2TCN, 0IST*10O. POMOOL 133 CALL FARPO (EPLANE, HPLANE, EY, NTHETA, NPHI, DATAY, IR2T0N72, POMODL 134

135 Hr2T0N) POMODL 135

GP TO 1 POMODL 136

20 ^ NR ITF ( 6, 6060

)

POMODL 137 rpco FTP'' AT (RAID) POMODL 138

5020 FPPM4T(PF10.0) POMODL 139

140 5040 format ( 4F10. C, 21 5

)

POMODL 140 MOOO FnRMATIlHl.TT.+PADIATinN PATTERN DF AN AYIALLY OFF OC USED PARAROLOIPOMODL 141 *P+,//,T7,8A10) POMODL 142 AOOl FOPMATdH , PAID) POMODL 143 6CC? FORMAT (T16, YARRUMFC RFFICIFNCY = 7. F10.2, 4 PERCENT^./, POMODL 144

145 1 T16, YNOMXNAL GAIN = 4, F10.2, 4 DB4,/, T16, 4P0WFP INPUT = 4, POMODL 145

2 F10.2, 4 WATTS4,//) POMODL 146

6005 P OR M AT ( 1 HO , T7 . * A P FRTL’P F IS UNIFORMLY ILLUMINATED WHEN FEED IS FOCUPOMODL 1 47

*S F04, / ) POMODL 148 8010 F OP AT ( 1 HO , T7, 4THR PEED IS AN FLFCTPIC DIPCLF ALONG THE Y-AYIR*, / ) POMODL 149

1 50 6.015 FOR M A T ( IHQ, T7, YF pro F-P| ANE PATTERN = ( C 0 S ( Y ) ) 4 4 ( < , F 5 . 2 , ,! ) 4 , / , POMODL 150

4 T ] 2 , <H-P L AN E pattern = - ( C OS ( Y ) ) 4 4 ( y , F 5 . 2 , 4 ) 4 , /

)

POMODL 151 6020 FnPMAT(lH ,T7.4RFFLFCT0R RARAMFTFRR - 4 , / / , T 1 6

, 4 F / 0 = 4,P5.3,/» POMODL 152

4 T1 6 . 40 I A MF T F P = 4,F6.2,4 W A U F L EN GT HR 4 , / , T 1 6 , 4 F P A C T I ON A L OTAMFTFR 8P0M0DL 153 4LrrPTNG = *,f5.3,/,tt6,4ayial defoc using ' 4,f6.3,4 wavelengths PFPOMODL 154

67
l‘=5 *YPNn FnCU'!*t/. T 1 6 . * FUEODFNC Y = *,FP.^.* GH7. + ./) PCPODL 60?0 FrRNAT(]Hn,T?0,’t>F-PL/lNF*,T6?,>l'H-PL4NE*./» PGMOOL

>>Tn,>l'THFTA=('.T22.<<M4G<'.T3?.'!'MAG+,TAl> + PHASE*» POMOOL *T5A,YYAG*,T6A,+rAG*>T73.«PHASFY,/, PO^DDL

* Til. + ( OFG ), T20. * ( VCLTS ) , T31> ( DP ) *> TA1> + ( DFG ) » PPMOOL

leO +T52. * ( VDLTS ) *. Tfc3. * ( DR )* . T73 . t OEG ) + > / ) POMODL

60A0 FrPRATdH , T9, F7. P , T2C, F6. 3 . T2<5, F7 . 2 .T3Q. F7. 2,T52, E6 .3 . TM , E7. 2 . POA'PDL *T71,F7.2) PQRCOL

6C60 EQRA<AT( /// ,* EGE ECUKC G^' LU5+) PCRDDL 6D70 FORRATC///, T7, +NEAP-ETELD P A P A y f TF p S-* , / / , T 1 6 , yt-SPACING =+, pOMGOL

1F6.2, CY + , TRO, 16,* PGIMTS*, /,T16, *Y-SPACING = . F6.2, Y CM*, PQMDDL

2T50,ie. * POINTS*./, T16, *DISTANCE FROM REFLECTOR FOCAL POINT =*,PGMGDL

3FP.2, * CM AWAY FFTM RFFLECTOP SURFACE*,//) POMGDL

AORQ FGPMAT(/, RX, 8A1C, EX, AlC) POMGDL

END POMODL

155 156

157

158

159 160

161 162

163 164

165 166

1 67 168 169

68
A. 1.2 SUBROUTINE FAR2D(EPL,HPL,EY,NTHETA,NPHI ,DATAX,IR2X2,IC2T0N)

PURPOSE :

To produce a two dimensional array of far-field data given the E-and H-plane cuts for

an axially symmetric antenna.

ARGUMENTS :

EPL is a complex vector containing the far-field. E-plane pattern of the antenna

stored in amplitude phase form. HPL is a complex vector containing the far-field, H-plane pattern of the antenna

stored in amplitude-phase form.

EY is a complex array which contains a two-dimensional, far-field array, arranged as a function of theta and phi.

NTHETA is the number of points in theta direction.

NPHI is the number of points in phi direction.

DATAX is not used in this subroutine, see FFKXY.

IR2X2 is twice the number of rows of data produced as a function of and ky.

IC2T0N is the number of columns of data in far-field array produced as a function of

kx and ky.

METHODS :

Circular symmetry is assumed in the antenna, hence, it is necessary to calculate the

far-field pattern over one quadrant only. The subroutine calculates the main

rectangular component of the far field which is given by

2 2 Ey(0,0) = ^ ^

where

EyE(e) = Electric field in E-plane as a function of theta.

E^^(e) = Electric field in H-plane as a function of theta.

Under the assumption of circular symmetry, this subroutine calculates the far-field ycomponent as a function of 9 and 0 given the E- and H-plane patterns (0=0, tt/2) as a function of 6.

This subroutine uses library functions: ATAN, COS, and SIN.

69
SYMBOL DICTIONARY:

COSTH

DATAX(I,J)

DTR

EPL(I)

EY(I,J)

HPL(I)

IIC2TON

ICOL

IR2T0N

IR2X2

I ROW

NPHI NPHIMl

NTHETA

NTHMl

NTHX2

PHI

PIRTD SINPH

TEMI TEMR THETA

= cos(THETA)

= Array of angular spectrum data as a function of and ky

= Degree to radian conversion factor = tt/ 180

= y-component of

= Array of angular spectrum data as a function of 6 and 0.

= x-component of ^]^g

= DO loop index

= Number of columns of data in DATAX

= Column loop index

= Number of rows of data in DATAX array

= 2 X IR2T0N

= Row loop index

= Number of data output points in phi direction

= NPHI - 1

= Number of data output points in theta direction

= NTHETA - 1 = 2 X NTHETA

= Azimuth angle - far-field pattern coordinate

= TT = 3.14159....

= Radian to degree conversion factor = 180 /tt = sin(PHI)

= Intermediate variable

= Intermediate variable

= Polar angle from boresight - far-field pattern coordinate

70
1 SUBROUTINF F&R2D(EPL, HPL. EY, NTHETA, NPHI, OATAX, IR2X2> IC2TGN)FAR2D 1 C- FAR2D 2 C- THIS SUPROUTINF takes p-PLANF AMD H-PLANE DATA GENERATED AS A FAR2D 3 c- FUNCTION OF ANGLE FROM BORESIGHT, AND GENERATES A TWO-DIMENSIONAL FAR2D A

5 c- iPRAY OF DATA AS A FUNCTION OF THETA AND PHI WHERE THETA AND PHI FAR20 5 c- ARF THE USUAL SPHERICAL ANGLES DEFINED IN A COORDINATE SYSTEM F AR2D 6 c- WHOSE POLAR AXIS COINCIDES WITH BORESIGHT. FAR2D 7 c- FAR2D 8 r- TNOTF* THIS SUBROUTINE ONLY PRODUCES VALID RESULTS FOR ANTENNAS F AR2D 9 10 c- WHICH M4VF SEPARABLE FAR-FIELD PATTERNS. FAR2D 10 c- F AR2D 11 DIMENSION ERL(D* HPL(l). F Y ( N THE T A , N PHI ) , DATAX(IR2X2» IC2T0N) FAR20 12 COMMON /CNTRL/ DLTH> DLPH. DFLX. DFLY, FRF0> DIST, PNRM FAR2D 13 CDHPLFX FY FAR2D lA

15 PI = A. + ATAN(1.) T RTD = 180. /PI I DTP = PI/ 180. F AR2D 15 NTHX? = ?*NTHETA F AR2D 16 IP2T0N = IR2X2/2 FAR2D 17

or 10 I = 1, NTHETA FAR2D 18 TFMP C FPL(2YI - 1 ) *COS ( FPL ( 2* I ) ) *PNR

M

FAR2D 19

70 TFNI * FRL(2*I - 1 ) *S IN( FPL ( 2*1 )

) YPNRM FAR2D 20

EY( I , 1) = CMPLX (TFMR. ’’EMI) FAP2D 21 TFMP = HPL(2*I - 1 ) +COS { HPL ( 2 + 1 )) PNRM FAR2D 22 TFMI = HPL{2+I - 1)+SIN(HPL (2+1) )+PNRM FAR2D 23 FY(I, NRHI) = CNPL X (TEMP , TEM

n

FAR20 2 A 25 10 CONTINUE FAR2D 25 NTHMI = NTHFTA - 1 t NPHMl = NPHI - 1 FAR2D 26 no 20 IPOW * 1, NTHFTA FAR20 27 THETA = (IROW - n+OLTH + DTR F AP2D 28 COSTH = CPS(THFT.A) F AR2D 29

30 DO 30 ICOL * 2. NPHMl FAR2D 30

PHI * (ICOL - D+DLPH+DTR FARED 31 SIMPH = SIN(PHI) FARED 32 EY(TRQW, ICOL) = (FY(IPOW» D+COSTH - EYdROW. NPH I ) ) +S I NPH + FARED 33

1 SINPH + FY(IPOW» NPHI) FARED 3A 35 30 CONTINUE FARED 35 20 CONTINUE FARED 36 CALL FFKXY (EY, NTHX2. NPHI, DATAX, IR2T0N+2, IC2T0N) FARED 37

R FTUR N FARED 38 FND FARED 39

71
A. 1.3 SUBROUTINE FFKXY(DATAY,NTHX2,NPHI ,DATAX,IR2X2,IC2T0N)

PURPOSE :

To produce an array of two-dimensional, far-field data which is equally spaced in the

coordinates and ky, given an array which is equally spaced in the coordinates 6 and 0.

ARGUMENTS :

DATAY is a two-dimensional array of far-field values, expressed as a function of

equally spaced 0 and 0 coordinates in the quadrant 0 _< 0 ^ tt/2. Complex far-field

values are expressed with real and imaginary parts adjacent in storage, such as FORTRAN IV stores them. Note, after execution, DATAY is expressed in polar form

because of a call to ARAYRTP.

NTHX2 is twice the number of points in 0 direction.

NPHI is the number of 0 points in one quadrant.

DATAX is the output array of far-field points which are equally spaced in k^ and

ky. Complex far-field values are expressed in polar form with amplitudes and phases

stored in adjacent locations. This array contains far-field values of an entire

hemisphere rather than a single quadrant as is the case for DATAY.

IR2X2 is twice the number of rows (ky values) in the DATAX array.

IC2T0N is the number of columns (k^ values) in the DATAX array.

METHODS :

FFKXY is basically an interpolation routine which fills each point in the DATAX array,

by calculating the corresponding values of 0 and 0 locating the four nearest points

corresponding to these values in the DATAY array. The value stored in DATAX is then a weighted average of these four points. The program assumes that the far-field input

array is from a single quadrant such as produced by FAR2D, and produces a far-field

output array over the entire hemisphere by reflecting about the lines k^ = 0 and

ky = 0.

Because the FFT is used to calculate the near-field distribution, it is necessary to have a far field which is sampled on equally spaced points in k^ gnd ky. Further,

we chose the spacing so that the near-field spacing will satisfy the sampling theorem

criteria. Thus, the far-field increments k^ and ky are fundamentally related to

72
the near-field spacing which is specified and transmitted into the subroutine via common CNTRL. Relationship between k^, ky, the far-field increment, and 6x>

6y, the near-field spacings, are.

A k = X

2tt 6 N X X

2ti 6 N y y

Beginning at the center (kx=ky=0) of the DATAX array, the value of 0 and 0 corresponding to k^ and ky are calculated. These are given by

-1 / 2 2 2 2

6 = cos 1 - R /k - k /k

X y

= tan ^(k /k ) y X

The indices corresponding to the four elements in the DATAY array that lie closest to the value of 0 and 0 are computed. A linear two-dimensional interpolation is then

performed using these four points in order to compute the value desired. The

interpolation is performed on the amplitude and phase, not on the real and imaginary

parts of the DATAY array.

Care must be exercised in interpolating the phase, because the phase is only given

modulo 360°. This causes errors in performing the interpolation when the phase

function makes a jump between two points in question unless a correction is applied to one of the phases. In this subroutine, three of the four phases are reset to lie on the same cycle as the reference phase by testing to see that the absolute value of the

phase difference between the point in question and the reference is less than 180°.

This procedure is valid provided that the far-field data points are spaced closely

enough. A reasonable requirement would be to have at least 4 or 5 far-field points in

an angular range of a sidelobe, a requirement which is met anyway if a sufficiently

smooth pattern is produced.

The interpolation is performed by taking a weighted average of the amplitude or adjusted phases of the form surrounding points, the weighting of an individual point

being inversely proportional to its distance from the point in question.

SYMBOL DICTIONARY :

C(I) = Coefficients used to calculate k^ and ky from near-field spacing

CEE = Speed of light x lO"^

D33J1 = Intermediate variable used in phase test

D43J = Intermediate variable used in phase test

D43J1 = Intermediate variable used in phase test

73
DATAX(I,J)

DATAY(I,J)

= Far-field data array as a function of and ky

= Far-field data array as a function of 6 and 0 DFI = Fractional part of FI DFJ = Fractional part of FJ DLPHI = 0 increment in radians

DLTHTA = 9 increment in radians

DTEMPl = Intermediate variable

DTEMP2 = Intermediate variable

DTEMP3 = Intermediate variable

DTR = Degree to radian conversion factor = tt/180.

FI = Reference theta position for interpolation

FJ = Reference phi position for interpolation

FKAY = k = Propagation constant

FKAYSQ = k2 FKX = kx = x-component of propagation vector

FKXSQ = k2 FKY = ky = y-component of propagation vector

FKYSQ

FLMDA = Wavelength

I = Integer part of FI

11 = Interpolation point index

12 = Interpolation point index

13 = Interpolation point index

14 = Interpolation point index

IC = Column interpolation loop index

IC2D2 = IC2T0N/2 = Center column of far-field array DATAX

IC2T0N = Number of points in k^ direction in DATAX array

ICN = Row counter for filling remaining three quadrants of DATAX

IR = Row interpolation loop index

IR2 = Intermediate index

IR2D2 = IR2T0N/2

IR2T0N = Number of rows in DATAX array

IR2X2 = 2 X IR2T0N

IRN = Row counter for filling remaining three quadrants of DATAX

IRX = Index for center row of far-field array

J = Integer part of FJ NPHI = Number of points in 0 direction in DATAY array

NTHX2 = 2 X Number of points in 9 direction in DATAY

PHI = 0

= Azimuth angle in far-field

PHIO = Initial value of 0 PI = 7T = 3.14159

PIX2 = 2tt THETAO = Initial value of 9 THMAX = Maximum value of 9 in radians

TST = Test variable to determine if z-component of propagation vector is real

74
1 3PPPni|TINF FF'tXYfDAT/'Y, NTHY?, KOHT> DAT4X. IR?Y7, IC2T0N) F FK X Y 1 r - FFR YY 2 r- THIS SUORnUTINE TNTERPCLATFS AN ARRAY CF FAS-FIFLO DATA WHICH FFKYY 3 r- TS EOUALIY SPAOF'^ IN THFTA ANO IN PHI JC PRODUCE AN ARRAY WHICH FFKYY A

5 r- IS FCIIALLY SRACFO IN RY AND KY, FFKYY 5

i: - FFKYY 6 rCMYCN /CNTRL/ DLTH, CLPH> DFLY, HFLY, fDpQ, 0IST> P N P M FFKYY 7 niMFNSinN OATAYINTHY? , NPHI), '(P), nATAY(IP?Y?> IC2TGN ) FFKYY 8 FFKYY q

: 0 LlinilT = 20 FFKYY 10 IRPTON = IR2Y2/2 FFKYY 11

PI = A. + ATAN(1.) S PIY? = 2. Y PI FFKYY 1 2 CFF = .29?7925 t FLYDA = FpF/ppfQ FFKYY 13 OTP = OI/IRO. FFKYY IF

15 FRAY = PIY2/FLNDA 1 FKAYSO = FKAY+FKAY FFKYY 15 THFTO > 0. % PHIO = 0. FFKYY 16 OLTHTA = OLTH+DTR t OLPHI =: niPH + DTR FFKYY 17 THMax = (NTHY2/2 - 2)YDLTHTA FFKYY 18 T(l) = PIYR/(nP(. yyicPTON) FFKYY iq

?0 C(?) = pIY2/(DFLY+IP2T0N) ' FFKYY 20 IC2D2 = IC2T0N/2 1 IR2D2 = IR2TCN/2 FFKYY 21 c- FFKYY 22

c- CHANGE OATAY ARRAY FRON RECTANGULAR TO POLAR FCRN. FFKYY 23 c- FFKYY 2A

CALI. ARAYRTP ( DATAY, NTHY2. NPHI) FFKYY 25 ICN = 0 FFKYY 26 no 61 IC = TC2D2. IC2TCN FFKYY 27 FRY = C(1)*(TC - IC2C2) $ FKYSO = FKYYFRY FFKYY 28 IPN = 0 FFKYY pq

30 DC 62 IR = IP2D2. IR2T0N FFKYY 30 IP2 = IRYR - 1 FFKYY 31 FRY = C(2)Y(IP - IR2D2) « FRYSQ = FRY+FRY FFKYY 32 TST = FRAYSQ - F.KYSQ - FRYSO FFKYY 33

IF (TFT .LT. 0. ) GC TO ao FFKYY 3^

3 THFTA = AC OS ( ( SORT! FK A YSO - FRYSO - F K Y S 0 ) ) / F R A Y

)

FFKYY 35 IF (THETA .GT. THMAY) GO TO 00 FFKYY 36

IF (FRY .LT. 0.) THFTA = -THFTA FFKYY 37

IF (fky ,fo. C. .ano. fry .FO. C.) GO TC f-3 FFKYY 38 PHI ^ ATAN2IFRY, FRY) FFKYY 39

i.0 GO TO HA FFKYY AO

6R PHI = 0. FFKYY 61 6A IF (PHI ,LT. 0.) PEI = PHI + PI FFKYY 62

r- FFKYY 63

r - XMTFPPnLATF OATAY ARRAY TO PRODUCF OATAY ARRAY WHICH IS EOUALLY FFKYY 66

C- SPACEC IN RY and RY. FFKYY 65 r- FFKYY 66

c PINO THF INDICES FTP THE INPlJT DATA WHICH IDENTIFY THE COGROINATES FFKYY 67

r CLOSEST TO the DESIRED THETA AND PHI VALUES. INTERPOLATE TO FIND THF FFKYY 68

C PROOF PATTFPN AT THE DFSIPFD POINT. FFKYY 6Q

50 C FFKYY 50 FI=((TheTA-THET0)/DLTHTA)+1.0 FFKYY 51

Fj = ((PHi-PHio)/rLPHT) + .qp<5Qqq'3q FFKYY 52 IFIPHI .FQ. 0.) FJ=1. FFKYY 53

I = F I FFKYY 56 =5 J = FJ FFKYY 55 DFI=FI-I FFKYY 56 DFJ=f j-j FFKYY 57

1 1 = 2YI-1 FFKYY 58 12= Il + P FFKYY 5q

‘'0 13=2*1 FFKYY 60

I A = I 3 + 2 FFKYY 61

IRY = IP2D?*? - 1 FFKYY 62

C FFKYY 63

r DFTFRNINF AMP AT (THETA, PHI) BY WFIGHTED AVERAGE OF ALL 6 POINTS FFKYY 66

65 r ARDLIND theta, phi. FFKYY 65

r FFKYY 66 DATAX(IP?,IC)=(DFI*DATAY(I2,J)+(1. C-DF I )*DATAY(I1,J) )*(1.0-DFJ) FFKYY 67 1+(DFI*DATAY(T2,J+1)+(1.0-DFI)*DATAY(I1,J+1))*0FJ FFKYY 68

r FFKYY &q

TO r RESET phases at THREF CORNERS TO BF ON SAME CYCLE AS REFFRENCE AT FFKYY 70

r ( 13, J ) FFKYY 71

C FFKYY 72

c FFKYY 73

c RESET PHASE AT (16, J) IF NECESSARY SC! THAT THF ABSOLUTE VALUE OF 063J FFKYY 76

75 c TS LESS THAN 180.0 DEGREES. FFKYY 75

r FFKYY 76 063J=nATAY(IF,J)-nATAY(IP,J) FFKYY 77

75
TF{083J.GT. 180.0) FFKXY lpTFrDl=OATAY(Ti.,J)-36C.C FFKX Y °0 IF(n83J.LF. 180.0. AND. 083J.GF. -180.0) FFKXY ]DTF|xD^=nAT4Y(I8.J) FFKXY lF(n8^J.LT. -180.0) FFKXY inTF8Pl = DATAY( I8,J)+3 8 0.0 FFKXY

r FFKXY

8 5 r PFGFT PHAFF 4T (T3.J+1) IF NFCFSSAPY SO THAT THF ABSOLUTF VALUE OF FFKXY

C D33J1 TS LFSS THAN 180.0 FFKXY

r FFKXY n33Jl=OATAY( T3.J+1 )-rATAY( 13. J) FFKXY IF(n33Jl.GT. 180.0) FFKXY

OP •1PT8yp? = P4TAY(I3.J + ] )-360.0 FFKXY IF(m^ J 1. LF. 180.0. A Nn.O33J1.0,F. -180.0) FFKXY ir)TFNPF=nATAY(I3»J + l) FFKXY IF(n33Jl.LT.-18C.O) FFKXY 10Tfyc3=0ATAY(13.J+1)+360.0 FFKXY

95 r FFKXY

r OFSFT PHAFF AT (T8.J+1) TF NECFSSARY SO THAT THF ABSOLUTF VALUE OF FFKXY

C P83J1 IF 1. FFF than IRC.O OFC'^FFS. FFKXY

c FFKXY n83Jl=nATAY(I8,J+l)-CATAY(I3,J) FFKXY

ion IF ( D83 J 1 . GT . 1 80 .0 ) FFKXY 10TFMP3=DATAY(I8,J+l)-3e0.0 FFKXY TF(n83Jl.LF.l«0.0. AND. 083J1.GF. -180.0) FFKXY

inTF''P8 = DATAY( I8,,)+l) FFKXY IF(083J1.LT. -180.0) FFKXY

1 P5 inTENP3=DATAY(T8,J+l)4360.0 FFKXY

r FFKXY

c OFTERNINF “HASP AT (THETA, PHI) BY WEIGHTED AVERAGE CE ALL 8 POINTS FFKXY

r APnilND THFTA.RHI. FFKXY

r FFKXY

110 DATAX(IP241,IC) = (DFI*CTEMP1 + (1.0-DFI)«DATAY(I3,J))>!‘(1. O-OF J ) FFKXY l+(nFI*nTEMP3+{l.n-DFI)*0TFMP2)TDEJ FFKXY IF(FkY.LT.O.O) nAT4Y(IP?+l,IC)=n4TAX(IR2+l,lC)-180.0 FFKXY DATAX(IR? + 1,IC)=AN0D(CATAX(IR2+1,IC). 360.0 FFKXY TE(nATAX(TP2+l,IC).LT.0.0) DATAX(TR2+l,TC)=nATAX(IR2+l ,10+360.0 FFKXY

: 1 = r FFKXY

GO TO 90 FFKXY

so 0ATAX(IP3. TO = 0. FFKXY CATAX(IP2 + 1, IC) = C. FFKXY

9 0 CONTI NiJF FFKXY

120 FFKXY

IF (TCPD2 - ICN .LF. C) GO TO 102 FFKXY IFdPX-IPN.LE. C) GO TO 101 FFKXY DATAX(TRX - TRN.IO » DATAX(IP2,IC2D? - ICN) = FFKXY 1DATAX(IPX - IPN,IC2D2 - ION) = 0ATAX(IR2, IC ) FFKXY

125 DATA.XdPX - IPN + 1,IC) = DATAX(IR2 + 1.IC2D2 - ICN) = FFKXY lOATAXdPX - IPN + l,ir?D? - ICN) = DATAX(IS2 + 1, IC) FFKXY

G F TO 100 FFKXY

ICl DATAX(IP2, IC2D2 - ICN) = DATAX(IR2, ID FFKXY OATAYdP? + 1, IC2D2 - ICN) = DATAX(IR2 + 1, IC) FFKXY

130 GO TO 100 FFKXY

102 IF (IRX - IPN .LE. 0) GO TO 100 FFKXY nATAY(TPX - IPN, ID = 0ATAXdR2,TC) FFKXY OATAVdPX - IPN + 1,10 = DATAXdPp + 1,IC) FFKXY

ICO CrNTI-'HJE FFKXY

1 3 5 IPN! = IPN + ? FFKXY

F 2 CONTINUE FFKXY

ICN = ICM + 1 FFKXY

61 CONTIN'IF FFKXY

C FFKXY

1 4 n C WRITF FAP-FIFLD OUT ON UNIT LUOUT FFKXY

r F FK XY CALI P FOOT ( DAT AX , IP2X2, IC2T0N, LUOUT) FFKXY

r FFKXY

C CALCULATE MEAR-FIELD FFKXY

1 85 r FFKXY

CALL NFKXY(OATAy, IP2X?, K2T0N) FFKXY

C FFKXY

2POO FOPNAT (d, (T6, 10F12.8)

)

FFKXY

2001 format (IHO) FFKXY dn P F T U

R N FFKXY

FNO FFKXY

7879

80

818?

83

888586

87

8889

90

9192

93

9895

96

97

9899 100

101 102

103

108

105

106 107

108 109

110

111

112 113

118

115 116

117

118 119

120

121 122 123

128 125 126

127

128 129

130

131 132

133

138

135 136

137

138

139 180

181 182

183

188

185 186

187

188

189 150

151

76
A. 1.4 SUBROUTINE NFKXY(DATA,IR2X2,IC2TON)

PURPOSE :

To calculate an array of near-field electric field values for an antenna given an array expressing the far-field radiation pattern of the antenna.

ARGUMENTS :

DATA - A two-dimensional array whicTi on entry contains one component of the far-field

radiation pattern of an antenna expressed in polar form as a function of equally

spaced kx and ky coordinates. The amplitudes and phases are stored in adjacent

locations in memory. On exit, this array contains the near-field pattern in polar

form as a function x and y.

IR2X2 is twice the number of rows (ky or y values) in the array DATA.

IC2T0N is the number of columns in the array DATA.

METHODS:

The expression evaluated in this subroutine is basically eq (45) repeated below.

E (r) y “ (45)

The quantity Siq(I<) is the normalized transmitting coefficient and is given in terms of the far field by

s

E(9,0)

This quantity is multiplied by the power normalization factor in front of the integral

and stored in the input array on entry to the program. The integral is converted into

a discrete Fourier transform (DFT) and evaluated using the FFT algorithm. The

resulting summation is.

E 1_-

2tt

'P^G(O)

2 Y k^

0

AK Ak

X y

E (0) y

NE

i=-N

ME

j=-M

E(i»j:

IJ

ly. -z e

iK. . R.

.

ij iJ e

77
The subroutine ETOIGAM is called for each column in order to multiply the input data

by

This array DATA is then converted from polar to rectangular form using subroutine

ARAYPTR. The Fourier transform is then performed using FFT routine FOURT and the

results converted to polar form using subroutine ARAYRTP.

The results of the FFT must be corrected in two ways because of the nature of the FFT algorithm and the indexing system used in FORTRAN. First, the summation indices must

be changed to 1 to 2N(M)+1, rather than -N to N, as in eq (1). Second, the output is

in a range 0 to 2tt rather than -it to tt. The first is equivalent to a shift in origin,

and, by the shifting theorem of Fourier analysis, produces a linear phase shift after

transformation to the near field. The second effect causes the center of the near field to be located at the point (1,1) in the output array and the negative and x- and

y-positions are in the outer portions of the array. The output data are rearranged in order to place the center of the near field at the center of the array. This is

accomplished using subroutine SWAP. The phase shift is corrected in PHSC0R2.

The data in corrected form now reside in array DATA. Printer plots are produced using

subroutine PLT120R. This subroutine uses library functions, ATAN and subroutines

ARAYPTR, ETOIGAM, FOURT, SWAP, and PHSC0R2.

SYMBOL DICTIONARY;

CEE

DATA(I,J)

E(I)

FACTOR

FLMDA

I

ICIC2T0N

ICOL

IKIR2T0N

IR2X2

I ROW

ISIGN

NN(I)

P(I)

= Speed of light x 10"^

= Angular spectrum which is transformed to near-electric field

= Near-field magnitude array for plot (single cut)

= Scale factor to give near-field units of volts/meter

= Wavelength

= Index for plotting array

= Column loop index

= Number of columns in array DATA

= Column loop index

= Row loop index

= Number of rows in array DATA

= 2 X IR2T0N

= Row loop index

= +1 for forward Fourier transform: -1 for inverse Fourier transform

= Array specifying the dimensions of the FFT to be processed in each

di rection

= Near-field phase array for plot (single cut)

78
PIPIX2

RTD

X(I) XMAX

XMIN

= TT = 3.14159....

= 2tt = Radian to degree conversion factor = ISO/ti

= x-coordinate array used in near-field plots

= Maximum value of x for plots

= Minimum value of x for plots

79
1

5

1 0

1

20

P'5

3 0

35

45

50

55

frO

70

75

12

3A

56

7

89 10

11

12

1314

151617

1319

20

21

22

2324

2526

27

28

29

30

3132

33

3435

36

37

38

39

40

414?

43

44

4546

47

43

49

50

515253

54

555657

58

5960

61

626364

6566

67

6869

70

7172

73

74

7576

77

S

U

30 nu tint MFKXY( 0 ATA, TP?X 2 . irZTON)

C-C- CALCULATF'; THF N'FAO-FIFLD OISTPIBUTIGN IN A PLANE GIVEN THE

r- fao-ftflo angular SpfCTPUH.

cniMFNSIoiN P( 128 ). P( 178 ), X( 12 P), Y(l? 8 ) OIMFNSION nATA(TP2X2, ICPTHM), MN{?)

r CO^YriN /lO/ CASFI 0 ( 8 ) COY'-TN /GNTPL/ DLTH, OLPH, OFLX. OELY, FPEO, OIST, PNRY

PI = 4 .*ATAN( 1 .) % PIX 2

» 2 .X<PI PTD = 180 . /PI

NN( 1 ) = TO? ton = TP 2 XP /7 $ NN(?) = IC 2 TGN

CFF = .2997925 I FLMOA = CFE/FRFQ

ISir-N = 1 r _ r- apply F to IYGAMMAAO PHAFF COORFOTinN COLUMN BY COLUMN

CPO 50 TC = 1 » ICPTON

CALL FTQ IG AM ( OATA {

1

. IC ) , IRpTHN, IC 2 T 0 N, IC. + 1 > FLMOA, DELX,

1 OFLY. OIST)

50 CONTINUE

r -

NFKXY NFKXY

NFKXY

NFKXY NFKXY

NFKXY NFKXY

NFKXY NFKXY

NFKXY NFKXY

NFKXY NFKXY

NFKXY

NFKXY

NFKXY NFKXY

NFKXY

NFKXY NFKXY

NFKXY

NFKXY NFKXY

c- CHANGE OATA AOOAY FROM POLAO TO RFCTANGULAO FORM. NFKXY

C- NFKXY

CALI ASAYPTR (DATA, IR2X7, IC2T0N) NFKXY

C- NFKXY

r_ pepfopm fouptFP transform Qf DATA ARRAY TO PRODUCE NEAR-FIFLO. NFKXY

C~

r-CC55

r

ICO

c

110

c

NFKXY

CALL FOlIRTfDATA, NN, 2, ISIGN, +1, 0) NFKXY NFKXY

C-MANGF NFAR-FIFLO OATA FROM RECTANGULAR TO POLAR FORM. NFKXY

NFKXY

CALL ARAYRTP ( DATA, IP2X2, IC2T0N) NFKXY

factor = PI X ? / ( FLO A T (I ORTON ) * FLOAT ( IC 2T0N ) *0 FLX + DPL Y ) NFKXY

on 55 TOOL = 1, IC2T0N NFKXY

on 5M lonw = 1, IR2X2, 2 NFKXY OATAdPOW, ICOL) = OATA(TROW, ICGL)*FACTOR NFKXY CONTINUE NFKXY

CALL SW AO { IR 2X2, I C2T0N , D AT A ) NFKXY

CALI PHSC 0R2 ( n A TA , IR2 X2 , TC 2T ON ) NFKXY

NFKXY

DO ICC IX = 1, lopTON NFKXY

X(IX) = nFLX*(IX - IP2TON/2) NFKXY

NFKXY

DO no TY = 1, IC2T0N NFKXY

Y(IY) = OFLYAdY - IC2TON/21 NPKXY

NFKXY

PRINT 3001 NFKXY

PRINT 2002, (X((IK + 1)/2),0ATA(IK, IC2TON/2), nATA(IK + l, IC2T0N/2), NFKXY

1 Y((TK + l)/2), 0 ATA (IP2T0N-1 , (IK+D/2), DATAdRpTON, (IK + D/2), NFKXY

2 IK = I, IR2X2, 2) NFKXY

XMIN = X(I) S YMAX * X(IR2TPN) NFKXY

OP 60 IK = 1, IR2X2, 2 NFKXY I=(TK+1)/2 NFKXY

Fd)- = DATA (IK, IC2T0N/?) NFKXY

P(I) » OATA (IK + 1, IC2TON/2) NFKXY rPNTTNUF NFKXY NFKXY

PLOT t--PLANF amplitude and PHASE. NFKXY

NFKXY

FALL olt120R(X, E, XMAX, XMIN, 10., 0., IC2TQN, IH*, 1, 1) NFKXY

PRINT 2003, CASFIO, lOHY-7 PLANF , lOHAMOLITUDF NFKXY

CALI PLTIROP(X, P, XMAX, XMIN, R60., C., IC2T0N, 1H4, 1, 1) NFKXY

PRINT ?003, CASEIO, lOHY-2 PLANE , lOHPHASF NEKXY

YMIN = Yd) S YMAX = Y(IC2T0N) NFKXY

DO 61 I = 1, IC2T0N NFKXY

F(I) = OATAdRRTGN - 1, I) NFKXY

P(T) = OATAdORTON, T) NFKXY

51 CONTTNUF NFKXY

0- NFKXY

C- PLOT H-PLANF AMPLITUDF and PHASF. NFKXY

NFKXY

CALL PLT12QR(Y, E, YMAX, ymin, iq,, 0., IC2T0N, IHY, 1, 1) NFKXY

POINT 2003, CASFIO, ICHX-Z PLANF , 1 OH A M P L I TL'O F NFKXY

CALL olt120P(Y, 0, YMAY, ymin, 360., 0., IC2TGN, 1H+. 1, 1) NFKXY

PRINT 3003, CASFIO, lOWY-7 PLANF , lOMPHASF NFKXY

80
0 E T 1 1 R M NFK X Y 78 -CR''4T( / // ,T6^»*CFNTEPLINE DATA*,//>T37»’i'X-Z PLANE** T97, *Y-7 PLANFKXY 79 iMF»./,T??,4<x+,T40.*AMO*.T5q.+PHASE*»T82.*Y + . T100.*AMP+,T119,*PMASENFKXY 80 ?*) NFKXY 81 ?no? P0PM4T(T^t 6E70.A) NFKXY 82 ZOC? FOP*^ AT ( / , 5X , 8A10> 8V, ?A1C) NFKXY 83

FN n NFKXY 8A

81
A. 1.5 SUBROUTINE

ETOIGAM(DATA(l,ICOL) ,NROW,NCOL,ICOL,ISGN,FLMDA,DELX,DELY,DIST)

PURPOSE :

To multiply each element of complex array DATA by the factor exp(+iYd).

ARGUMENTS :

DATA is a two-dimensional complex array in polar form whose magnitude and phase are adjacent in storage.

NROW is the number of rows in array DATA. NCOL is the number of columns in array DATA.

ICOL is column number of the data to be operated on. ISGN = +1 depending on whether DATA is to be multiplied by exp(+iyd).

FLMDA operating wavelength.

DELX x-increment of desired near-field data. DELY y-increment of desired near-field data.

DIST spacing between antenna reference point and desired near-field plane.

METHODS :

The subroutine does not employ complex arithmetic. It is assumed that the numbers in array DATA are the magnitude and phase stored in adjacent locations. If DATA contains

complex data in real and imaginary form, a call to ARAYRTP must be made prior to the

call to ETOIGAM. The pertinent relationships are

DATA = DATA e^^^

Y

= \/k2 _ 1<2 _

k = to yjic

kx = k sine cos0

ky = k sine sin0.

Because DATA is assumed to be in magnitude, argument form, we calculate

ARG(DATA) = ARG(DATA) + Yd

for Y real , and

MAG(DATA) = MAG(DATA) Exp(-Yd)

for Y imaginary

Y is computed from the row and column positions of the data elements.

82
ky =

2tt (IROW-NROW/2)

NROW • Ay

II X

2tt (ICOL-NCOL/2)

NCOL . Ax

Array DATA is assumed to correspond to points equally spaced in k^, ky

with kx=ky=0 being the center point of the array.

SYMBOL DICTIONARY:

DATA = Input data array

DELX = Near-field x-increment

DELY = Near-field y-increment

DIST = Distance from antenna reference point to desired near-field plane

DTOR = it/180 = degree to radian conversion

FACTOR = Amplitude correction factor for imaginary y

FKAY = k = 2tt/X = free space wave number

FKAYSQ = k2 FKX = kx = x-component of propagation vector

FKXSQ = k2 FKY = ky = y-component of propagation vector

FKYSQ = k2

FLMDA = Operating wavelength

ICOL = Running index for column number

I ROW = Running index for row number

ISGN = +1 = desired sign for exponential phase factor

NCOL = Number of columns in far-field array

NROW = Number of rows in far-field array

PHACORR = Phase correction factor added to data array phases for real

PI = TT = 3.14159....

PIX2 = 2tt RTOD = 180/tt = radian to degree conversion

SUMSQ = k2 + k2 TEMP = Intermediate variable

83
1 3|iRi?nUTINe FTGIGAM (CAT&» NRGW» NCDL. ICOL . ISGN. FLMOA, OELX. OELETOIGAM 1 lY, nisi) ETOIGAM 2

r ETOI GAM 3 nyMPMSTON oata (1) FTCIGAM 4

r ETOIGAM 5 PT = 3. 1415926535 ETOIGAM 6 PTX? = ?. * PI ETOIGAM 7 PkAY = PTX2 / FI. MOA ETOIGAM 8 FK A VFO =; Fk AY * * 2 ETOIGAM 9

1 0 PTnri = 1«0. / PI ETOIGAM 10 pthr = 1, / k rOO ETOI GAM 11 FKX = PIX? * (ICHL - (NFOL / ?)) / DFLX / NCOL ETOIGAM 12 FKXSO = FKX * * Z ETOIGAM 13

C ETOIGAM 14

IF (MRTW ,LT. 1) GO TO 130 FTOIGAM 15 pr l^o TROW = 1, NROU ETOIGAM 16 fkY = PIX? (IRGW - (NRQW / 2)) / DELY / NROW ETOIGAM 17 FKYSO = FKY « ? FTOIGAM 18

SIIY30 = FKXSO + FKYSC ETOIGAM 1 9 ^0 PHACGPp = 0.0 ETOIGAM 20 FACTPR = 1.0 ETOI GAM 21

IF (FI.MFO ,gT. fkaYSC) GG to IOC ETOIGAM 2?

C ETOIGAM 23 dmaCGPR = ISGM * '^QPT (FKAYSO - SUMSO) + DIST >! RTOO ETOI GAM 24

P’i GO TO 110 ETOIGAM 25

r ETOIGAM 26

1 00 FACTHP = (SORT (SUYSO - FKAYSO)) + CIST ETOIGAM 27

IF (FACTOR .GT. 100.)FArTGR = 100. FTOIGAM 28 factor "= EXP (ISGN * FACTOR) E lUl GAM 29

30 r ETOIGAM 30 110 CGNT IM'JF ETOIGAM 31

r F rOIGAM 32 data (7 + TRPW - 1) = CATA (2 + IROW - 1) * FACTOR FTOIGAM 33 TPHASF = DATA (? Y IprW) ETOIGAM 34

35 Ti^MP = data (2 Y loGW) + PHACDRR ETOIGAM 35 TFMP = TEYP - INT (TEMP / 36C.) * 360. ETOIGAM 36

IF (TEMP .LT. 0.0)TEMP = TEMP + 360.0 ETOIGAM 37 DATA (7 + IPGW) = TFMP ETOIGAM 38 ETOIGAM 39 1?0 C ON T IM'JF ETOIGAM 4 0 1?0 COMTINUE ETOIGAM 41 1?00 format nx, 4110. 5F12.3.//) ETOIGAM 42

0 FORMAT (IX. PI^. 8F12.3) ETOIGAM 43 PFTURN ETOI GAM 44

‘>3 FMD ETOIGAM 45

84
A. 1.6 SUBROUTINE PHSC0R2(DATA,NRX2,NC0L)

PURPOSE :

j To correct the phase of the near-field data which arises because the reference point

' of the EFT algorithm is the point (1,1) rather than the center of the far-field

array.

ARGUMENTS :

!

I DATA is a two-dimensional array containing the complex near-field data in polar form.

Amplitude and phase in degrees are located adjacent in storage.

NRX2 is twice the number of rows in the array DATA.

I NCOL is the number of columns.

METHODS :

As shown by the shifting theorem, a shift in coordinates in one domain introduces a linear phase shift in the transformed domain. This subroutine corrects for the phase

shift which occurs as a result of the different reference points of far-field pattern

and the FFT algorithm. The shift added because of this change of origin is

^i (ax+by)

where a and b are the shifts in far-field origin in the and ky directions

respectively and x and y are the coordinates of the specific near-field point.

The subroutine adds a phase shift equal to

to the phase of each complex number in the array in order to compensate for the

above shift. It is assumed in this factor that the center of the far-field pattern

lies at (NR0W/2,NC0L/2).

An additional phase of 90° is added to each element in order to allow the near-field

phase to be conveniently plotted in the range 0O-360°.

This subroutine uses inline functions FLOAT and INT.

85
SYMBOL DICTIONARY:

Cl = Phase correction for column ICOL

C2 = Phase correction for row IROW

CONSTl = Column phase increment

C0NST2 = Row phase increment

DATA = Input data array

ICOL = Column loop index

102 = IROW/2

I ROW = Row loop index

NCOL = Number of columns in array DATA

NRX2 = Twice the number of rows in array DATA

TEMP = Intermediate variable, the corrected phase at point (102, ICOL)

86
1 ^:U3PnuTINF PHSC 0P2 ( data, NPX2. NCOU PHSC0R2 1 c P HSCOR 2 2 PT^FNSinM nATA(Npy?> nopl) PHSCDR2 3 c PHSCPR2 A

c NPFIW >= MPX? / 2 PHSC0R2 5 c PHSC0R2 6

c PHSCCR2 7 CPNFTl . -130. FLOAT (NCPL - 2 ) / p L 0 AT ( NC OL

)

PHSCDR2 8 CC1NST2 = -IBO .FLOAT (NPnw - 2 ) / F L 0 A T ( NP OW ) PHSCOP 2 9 10 IF (NJOHL .LT. 1) on TO 130 PHSC0R2 10 on 100 Tcni = 1 , MCCL PHSC0R2 11

Cl = CnMSTl ( TCTL - 1

)

PHSC0R2 12

IF (NPX2 .LT. 2) GO TO 110 PHSCOP 2 13 OP 100 TROW = 2. NPX2, 2 PHSCQP2 lA

] IP2 = TPPW / R PHSC0R2 15

C2 - CPNST3 (IPR - 1) + Cl PHSC0R2 16 TEMP = OATAdPOW, ICCL) + C2 + PO. PHSC0R2 17 TFMP ' TFMP - INKTFMP / 36.0.) 360. PHSC0R2 18

TF (TFMP .LT. 0.) TEMP = TEMP + 360. PHSC0R2 19

20 DATA ( IPPW. T CPI ) * TEMP PHSCPR2 20 ICO CONTIN'IJF PHSC0R2 21

1 10 CPNTINDF PHSC0P2 22 130 CPNTTNUF PHSCGR2 23

c PHSC0R2 2 A

P FTPRK' PHSCQR2 25

FN 0 PHSC0P2 26

87
A. 1.7 SUBROUTINE SWAP(NRX2,NC0L,DATA)

PURPOSE:

To perform the rearrangement of data necessary to place center of near field at center

of near-field data array.

NRX2 is twice the number of rows in the array DATA. NCOL is the number of columns in the array DATA.

DATA is an array containing the near-field pattern of an antenna which is to be

rearranged.

The FFT algorithm fundamentally takes data over a range of 0-2tt and transforms them

into a domain of 0-2tt. Suitable scaling is employed to fit the far-field (angular

spectrum) and near field (x-y position) into these ranges. The negative portion of

the x-y range occurs from tt to 2tt. Thus, to have a continuous near field at x,y=0,

the data are rearranged.

The rearrangement is done in place, the rearranged array replacing the original one in core, requiring only three temporary storage locations. The rearrangement takes place

in two steps. First, the edges of the array are moved to the center and the center to the edges by columns. The process is then repeated by rows.

The array DATA contains complex numbers which may be in either polar or rectangular

form. This routine does not use complex arithmetic.

ARGUMENTS:

METHODS:

SYMBOL DICTIONARY:

DATA

ICOL

ICPNC

I ROW

IRPNR

NCMl

NCOL

NC02

NROW

NRX2

NR2M2

= Complex array to be rearranged

= Column loop index

= Intermediate subscript

= Row loop index

= Intermediate subscript

= NCOL -1

= Number of columns in DATA

= NCOL/2

= Number of rows of complex data

= 2 • NROW = dimension of DATA in row direction

= NRX2-2

88
TEMP = Intermediate variable

TEMPI = Intermediate variable

TEMP2 = Intermediate variable
1 SURROLITINF SWAP(NRX3, MCOL. DATA) S WAP 1 DImeNFTPN 0ATA(NRX2, NCHL) SWAP 2 SWAP S

c SWAP 4

c NPPW = NPX2 / 2 SWAP 5 NCO? NCOL / 2 SWAP 6

r SWAP 7 ^-^'nvT^'n fooes of array Tr center and vice VERSA RY COLUMNS SWAP 8

c SWAP -9 in IF (HPX7 ,LT. 1) GO TO 220 SWAP 10 on 700 IRON = 1, NPX2 SWAP 11 IF (NCn? ,LT. 1) GO TO 210 SWAP 12

nn 700 TCPL = 1, NC02 SWAP 13 IPPNC = TCPL + NC02 SWAP 14

15 TFNP = OATAURPW, ICPNC) SWAP 15 OATAdPOU, ICPNC) = DATAdRPU, ICOL) SWAP 16 ?oo DATAdPnw. TCPL) = TEMP SWAP 17

?] 0 CPNTINIIF SWAP 18 RRO c n N T T N 1 1

F

SWAP 19

r SWAP 20 NCNl = NCPL - 1 SWAP 21

IF (NRX 7 . lT . 1 ) GO TO 31

0

SWAP 22 OP 3P0 IPPW = 1. NPX2 SWAP 23 TFMPl = DATAdROW, 1) SWAP- 24. ?5 IF (MpNI ,LT. 1) GP TC 7R: SWAP 25 np ?30 TC3L = 1. NCMl SWAP 26 230 OATAdPpw, TCPL) = DATA( IPP'.d TCPL + 1 ) SWAP 2-7 ?P0 C3NTTNIJE SWAP 28 POO OATA( TROW. NCOL ) = TFNPl SWAP 29 ’0 310 CP.NTTNIIE S-WAP- SC- SWA P SI C-wnVTNG OE ARRAY TP CENTER AMO VICE VERSA P Y ROWS S WAP 32

r SWAP 33 IF (NCPL .LT. 1) CP TO 340 SWAP 34

0 c no 3?n ICOL = 1. NCPL SWAP 35 IF (NPPW .LT. 1) GO TP 330 SWAP 36- nr 370 lonw = i, nrow SWAP 37 d p\ip = TO Ow + NPPW SWAP 38 TF«P = DATAIIRPNR, ICOL) SWAP- 3 9- 40 PATAtlRONR, ICOL) = DATAdROW, ICOL) SWAP 40

3 70 DATAdROW. ICOL) = TEMP SWAP 41 330 CONTIN'JE SWAB- A2

3 iO C3NTTMUE SWAP 43

c SWAP 44 45 MR3M7 = NRX2 - 2 SWAP 45

IF (NCPL .LT. 1) GO TO 3P0 SWAP 46

OP 37C TCPL = 1. NCOL SWAP 47 TFYPl = OATAd, TCDL ) SWAB -48 TR^-RP = PATA( 7, TCCL) SWAP 49

50 IF (NR7M7 .lT. 1) GO TO 360 SWAP 50 DO 350 IRQW . 1, NR2.M2 S WAP 51

3 r)ATA(TRnw, ICOL) = DATAdROW + 2. ICOL) SWAB 52 350 CPNTTNIJF SWAP 53 OAT A (NR2M2 . 1, ICOL ) = TEMPI SWAP. 54

55 370 oata(mr2M2 + 2, ICOL) = TFMP2 S WAP 55 3P0 CPNTIN'IF SWAP 56

r SWAP 5.7

R FTUR N SWAP 58

FNO SWAP 59

90
A. 1.8 SUBROUTINE ARAYPTR(DATA,NRX2,NC0L)

PURPOSE:

To convert a two-dimensional complex array from polar form to rectangular form or from

rectangular form to polar form (ENTRY ARAYRTP).

ARGUMENTS :

DATA is a two-dimensional complex array whose real and imaginary parts are adjacent in storage, such as FORTRAN IV places them. On exit, DATA contains adjacent amplitudes

and phases.

NRX2 is twice the number of rows in DATA.

NCOL is the number of columns in DATA.

ENTRY POINT :

ARAYRTP performs rectangular to polar conversion.

METHODS:

This subroutine does not use complex arithemtic. However, array DATA is stored in the same fashion as is required by FORTRAN IV for complex numbers. Thus, while the

subroutine operates on a complex array, complex FORTRAN functions are not used.

1. ARAYPTR

DATA( IR0W,IC0L) contains magnitude of complex number.

DATA( IROW+1 ,IC0L) contains phase of complex number, expressed in degrees.

Re(DATA) =

| DATA | cos ( ANGLE(DATA)) . Im(DATA) = I DATA | sin ( ANGLE (DATA)) .

2. ARAYRTP

DATA(IR0W,IC0L) contains real part of the complex number.

DATA( IROW+1 ,IC0L) contains imaginary part of the complex number.

DATA I = \/[Re(DATA)]2 + [Im(DATA)]2.

ARG(DATA) = tan

-1 Im(DATA)

X 1800/tt.

Re(DATA)

This subroutine uses library functions, SIN, COS, ATAN2, and SQRT

,

91
SYMBOL DICTIONARY

ANGLE

DATA

DTOR

FIMAG

FREAL

I COL

I ROW

IRPl

NCOL

NROW

NRX2

PIRTOD

TAMP

Intermediate variable, phase angle of compl ex ijgumber

Input data array

tt/180 = degree to radian conversion factor

Imaginary part of complex number

Real part of complex number

Column loop running index

Row loop running index

IROW + 1 Number of columns in input array DATA

Number of rows in input array DATA

2 • NROW

= 3.14159

180/it = radian to degree conversion factor

Intermediate variable, amplitude of complex number

92
1 <:iJPDnUTTNF 4 YDTP ( DATA , NPX?, MCnt) APA YPTP 1 r 4PA YP TP 2 0T»'FMSTOm 0AT4(A'3X?, Nro^) AP AY PTR 3 r APAYPTR 4 PI = 3.1415026535 APAYPTP 5 pyP'O = l”o. / pj APAYPTR 6 OTPD = 1 . / p TOn APAYPTO 7

C APAYPTR 8

f F (M C^I . LT . 1 ) r,n in 130 APAYOTR 0

1 9 on 170 loni - 1, NCHL APAYPTR 10

IP ( YP X7 . IT . 1 ) GO TO 1 10 ARA YPTR 11 pp 100 jpnw = 1, NOX?, ? APAYPTR 12 TPPl = IPPV + 1 APAYPTR 13 T4WD = OATA(Ipnij, lOCL) AR AYPTP 14

1

=; A VOLF = OATAfipol, ICCL) * nrnp APAYPTR 15 0ATA{T7nw, IPHI) = tamp a GPS(4 5'P'LF) APAYPTR 16 nAT4(IPPl, KHL) = TAMP A SIN(AN0,LF) 4RAYDTP 17 10 0 Cn^Tl^!')F ARAYPTR 18 no rnNT TMMF APAYPTR 19

7') 1 ?o c ONTl M'lr APAYPTR 20

1 ?o ''POTT 'IMF APAYPTR 21

D F T l.i? M APAYPTR 2?

r APAYPTR 23 PMTPY AP6YRTP ARAYPTR 24

2'v r ARAYPTR 25

DT = 7.1415026536 APAYPTR 26 PTPn = IPP, ! PI APAYPTR 27 OTPP = 1 . / PTor APAYPTR 2 8 r APAYPTR 29

30 IF (NCL .LT. 1) GO TO 1 op APAYPTR 30

pp IPO TC OL = 1 , VOni APAYPTR 31

IF (\ip X 7 . L T . 1 ) GO TP 170 APAYPTR 32 nn 160 IPnu = 1 . vpx?, 7 APAYPTR 33 TPPj = TPnv) + 1 ARAYPTR 34 FOCAL = p4TA(I0P'J, ICOL) APAYPTR 3 5 CT''AG = OATA(I?oi, TP on APAYPTR 36 OATA(Tppv. TCC'L) = SpRT(CRC4L * FPFAL + FIMAG * FIMAP,

)

APAYPTR 37

TF (FFCaL .FO. n.o .AKT. FTMaG .FO. 0.0) GP TFI 160 APAYPTR 38 PAPA(IP°1. IPPIL) = ATAA'7(FIM4r-, FPCa|_) a PTOO ARAYPTR 39

40 1 ^0 r c"! t I

c

APAYPTR 40

] 70 PPA'TIAJUC 4 p A YP TP 41

1 PO roNi T T 'J 1 IP APAYPTR 42 IQO C PM T IM MF ARAYPTR 43

r ARAYPTR 44

45 P c T M P '! APAYPTR 45

r V P APAYPTR 46

93
A. 1.9 SUBROUTINE FF0UT(DATA,NRX2 ,NCOL , LUOUT)

PURPOSE:

This subroutine writes the array DATA out to logical unit LUOUT. A header record is written as the first record of the file.

ARGUMENTS:

DATA is the array to be written out. NRX2 is the number of floating point numbers in a row. NCOL is the number of columns.

LUOUT is the logical unit on which file is to be written.

METHODS:

A file consisting of NCOL + 1 records is written on unit LUOUT. The first record is an identification (ID) record, and each of the NCOL rows in the array DATA is a record. The records are written using unformatted WRITE statements.

The ID record consists of ten ten-character words, these are listed below

WORD 1 WORD 2 WORD 3-7

WORD 8 WORD 9 WORD 10

PHYSICAL 0 PTICS SIML

Alphanumeric information from the first 5 words in common block

CASE ID MMDDYYHHNN Month, Day, Year, Hour, Minute

Number of columns in DATA = NCOL

Twice the number of rows in DATA = NRX2

Words 1 through 8 are written in Hollerith format, and words 9 and 10 are written in

integer (I) format. The ID record may be read with a (8A10,2I10) format. The date

and time for word 8 are generated by calls to DATE and TIME and are thus the date and

time when the output file was created.

The subroutine uses library functions DATE and TIME.

SYMBOL DICTIONARY :

CASEID = Hollerith identification supplied from calling program

DATA = Array to be written to unit LUOUT

DT = Date information obtained from function DATE

DY = Day of month

94
HR = Hour

I = DO loop index

IC = Column loop index

ID = Identification array

IR = Row loop index

LUOUT = Output logical unit number

MN = Mi nute

MON = Month

NCOL = Number of columns in DATA

NRX2 = Number of rows in DATA

SC = Second

TM = Time information obtained

YR = Year

function TIME

95
1 SURROUTINF FFnUKnsTA, KPX2. NCOLf LUOUT) FFOUT 1 C FFOUT 2

r THIS SURPTUTINP WPITFS 4PRAY DATA TO FILF LUCUT FFOUT 3

c FFOUT A 5 OIMFNSinH DATA(NPy2. NrOL)» 10(10) FFOUT 5 CGHHON / 10/ rASFIO(P) FFOUT 6 INTFGFR CASEID FFOUT 7

c FFOUT 8 10(1) = lOHPHYSICAL C FFOUT 9 in IC(2) => lOMPTICS SIML FFOUT 10 DP 10 T » 3. 7 FFOUT 11

10 ID(T) = CASFTOd - ?) FFOUT 12 CALL OATF( OT) FFOUT 13 CALI TTMF(TY) FFOUT lA

16 DFCnOP(10, 1500, OT) YP, MON, OY FFOUT 15 OFCPOedO, 1 500, T^*) FP, MN , SC FFOUT 16 FN^'HOPdO, 1510, ID(8)) MON, OY, YR, HP, MN FFOUT 17 10(0) = NCQL FFOUT 18 10(10) = NPX2 FFOUT 19

?o WPITF (LUniJT) (lOd). I

>= 1,10) FFOUT 20 PRINT 1520, IP FFOUT 21 DP PO ir = 1, NCPL FFOUT 22 WRTTF (LUOUT) (DATA(IP, IC), IR = 1, NRX2) FFOUT 23

20 CPNT INIJF FFOUT 2A FNDFILF LUOUT FFOUT 25

P FT URN FFOUT 26

1 >^00 FPPYAT (IX, ?(A2, IX)) FFOUT 27

1 510 FPPNAT (5AP) FFOUT 28 15?0 FORMAT (+OQUTPUT FILF ID — Y, lOX , 0A1O, 2110) FFOUT 29

30 FNO FFOUT 30

96
A. 1.10 SUBROUTINE FOURT(DATA,NN,NDIM,ISIGN,IFORM,WORK)

PURPOSE:

To compute the discrete Fourier transform of the array DATA using the fast Fourier

transform algorithm.

DATA is a multidimensional complex array whose real and imaginary parts are adjacent

in storage, such as FORTRAN IV places them.

NN is an array giving the lengths of the array in each dimension.

NDIM is the number of dimensions of the array DATA, hence the number of elements in

array NN.

I SIGN is +1 for a forward transform -1 for a reverse transform.

IFORM If all imaginary parts of the input array are zero (input array is real), set IFORM = 0 to reduce running time by approximately 40 percent, otherwise set IFORM = +1.

WORK if all dimensions of DATA are not integral powers of 2, specify array WORK in

calling routine with dimension greater than largest non 2^^ dimension, otherwise set WORK = 0.

Using the Fast Fourier transform algorithm, FOURT replaces the array DATA with its

discrete Fourier transform given by

01=1 02=1

For a more complete description of the subroutine and its usage, see the comments

included at the beginning of its listing or the supplementary comments by the

programmer, Norman Brenner of MIT.

Uses external library functions COS, SIN, FLOAT, and MAXO.

Note: Brenner, Norman, "F0UR2 and FOURT program description," private communication.

ARGUMENTS:

METHODS:

TRANSF0RM(K1,K2,...)

NN(1) NN(2)

E E DATA(01,02) e

1968

97
1

s

1 0

1

70

7<^

30

35

i, ''

50

“^5

5C

65

70

75

12

34

56

7

89 10

111213

14151617

18

19

20

212223

2425

2627

2829

30

31

3233

34

3536

3738

39

40

41

424344

454647

48

49

50

51

52

53

545556

57

5859

60

61

6263

6 4 656667

68

6970

7172

7374

7576

77

r

rr

r

r

r

r

r(

rc

c

r

r

r

rr

rC

C

c

r

rr

c

c

rc

r

r

cr

rr

cr

r

rr

0

rr

c

r

C

r

C

r

r

c

C

c

r

r

c

c.

snarriiTTMr coopt (osts. mn. noi?', ttign, ifopm, woro foupt

FOURT

THF noiFY-TUKFY fast fouRIFP TRANSFORM' IN USASI BASIC FORTRAN FOURT

FOUPT TpaNSFOPM(ki,k?, ...) >= SIIN(OATA(J1,J2,...)*FXP(ISION*2TPI + SORT(-1 ) FOUPT

* ( ( Jl-1 ) *( Kl-1 ) /NN ( 1 ) + ( J2-1 ) 7 ( K2-1 ) /NN ( 2) + . . . ) ) ) . SUynFO FOR ALL FOURT

Jl. RI nCTwFFN 1 ANO NN ( 1 ) , J2, K7 8FTWEFN 1 AND NN(P), ETC. FOURT

TMFPC IS NO LIMIT TO Tfif number OF SUBSCRIPTS. DATA IS A FOURT

m'JLTTOTMCNS TONAL COMPLFX ARRAY WHOSE PEAL AMO IMAGINARY FOUPT

PARTS APC adjacent IN STOPAGF, SUCH AS FORTRAN IV PLACES THEM. FOURT

IF all imaginary parts ARF ZFRO (DATA APF OISGUISFD PEAL). SET FOURT

TPOPm th 7C0Q TO CUT THC punning TI“F BY L'P TO FORTY PERCENT. FOUPT OTHFPUTSC. IFOPM = +1. THE LENGTHS OF ALL DIMENSIONS ARE FOURT STORFD IN ARRAY NN, OF LENGTH NDIM, THEY MAY BE ANY POSITIVE FOURT

IMTFGPPS. THO THE PPOGPAM PIJNS FASTcr ON COMPOSITE INTEGERS. AND FOURT FSPFCIAILY fast on NUMBERS RICH IN FACTORS OF TWO. I S I GN IS +1 FOURT

OP -1. IF A -1 TPANSFOPM IS FOLLOWED BY A +1 ONE (OR A +1 FOURT

BY A -1) TFIF original data reappear, MULTIPLIED BY NTOT (=NN(1)* FOURT NN(?)T...). transform values are always complex, and are returnedfourt

IN ARRAY DATA. REPIACINC- THE INPUT. IN ADDITION, IF ALL FOURT DTmfNSTPNS APF NOT POWFPF pp TWO, ARRAY WORK MUST BE SUPPLIFD, FOURT rpKPLcy OF length FOLAL TO THE LARGEST NON 2+TK DIMENSION. FOURT OTMFPWTSF, pfplaCF WQpk by ?fR0 in the CALLING SEQUENCE. FOURT MPPMAL FnPTPAN DATA OPPFPING IS EXPECTED, FIPST SUBSCRIPT VARYING FOUPT FASTCST. all subscripts RFGIN aT ONF. FOURT FOURT PUNNIMG TIME IS MUCH SHOPTPP THAN THE NAIVE NT0T+*2. BEING FOURT

GIVBN BY THE FOLLOWING FORMULA. DFCOMPOSE NTOT INTO FOURT

* 5**^5 * .... LET SUMP ’ 2*K2, SUMF * 3*K3 + 5*K5 FOURT

+ ... AND NF = K3 + K5 + .... THF time TAKEN BY A mijlTI- FOURT OImfmftpmaL TPANBFQRM on THFSF NTOT DATA IS T = TO + NT0T*(T1-f FOURT

TP('SIimp + T3*SUMF + T4*NF ) . ON THE CDC 3300 (FLOATING POINT ADD TIME FOURT

OF SIX MJCROSFCONDS ) . T = 3000 + N T OT * ( 50 0 + 4 3 S UM 2 +6 B * S U M F + FOURT

320TNF) MTCPOSECONDS ON COMPLEX DATA. IN ADDITION, THE FOURT AOniPACY IS GPFATLY TMPPOVFO, AS THF RMS RELATIVE ERROR IS FOURT

pnijMpFn PY 3 *PTA ( -B ) * SUM ( factor ( J ) 7X^1 . 5 ) , WHERE B IS THE NUMBER FOURT

OF BITS TN the floating POINT FRACTION AND FACTOR(J) ARE THE FOURT

PPTMF factors of NTOT. FOURT

FOURT PDPGPAM by mopmaN BRFNNfp from the basic program by CHARLES FOURT

RAOFR. RALPH ALTFR SUGGESTED THE IDEA FOR THE DIGIT REVERSAL. FOURT

MIT LINCOLN LAPOPATOPY, AUGUST 1967. THIS IS THE FASTEST AND MOSTFOURT

VfpfatilE VERSION OF IHE ppT KNOWN TO THF AUTHOR. SHORTER PRO- FOURT

GPAMS FOIJRI and four? RfSTRICT DIMENSION LENGTHS TO POWERS OF TWO. FOURT

SEE — ICFF audio transactions (JUNE 1967), SPECIAL ISSUE ON FFT. FOURT

FOUPT

THF niSCPETF FOURIFP transform places THRFF RESTRICTIONS UPON THE FOURT

data. fqi_|pt 1. THF number of input data AND THE NUMBER OF TRANSFORM VALUES FOURT

MUST BF the same. FOURT

2. both the input data ANO THE TRANSFORM VALUES MUST REPRESENT FOUPT FQUTSPACFD POINTS IN THFip RfSPFCTIVF DOMAINS OF TIME AND FOURT

frequency. calling these SPACINGS DELTAT ANO DELTAF, IT MUST BE FOUPT

TRUE Tmat D flt a F= p *P I / ( nn( I

) yd ELTAT ) . OF COURSE, DFLTAT NEED NOT FOURT bfTmfSAmffoREVERYDIMENSION. FOURT

3. CONCEPTUALLY AT LEAST, THE INPUT DATA AND THE TRANSFORM OUTPUTFOURT

rFPpfsfmT single cycles of PERIODIC FUNCTIONS. FOURT

FOURT example 1. THRFF-DIMFNSIONAL FORWARD FOURIER TRANSFORM OF A FOURT CPmplfy appay DIMcnsICNPD 32 BY 25 BY 13 IN FORTRAN IV. FOURT

DIMFNSiriN DATA ( 32. 25, 13) , WORK ( 50 ) , NN( 3) FOURT

r complex data

C data NN/3P.P5.13/

C DO 1 1=1, br

C DO 1 J = 1 , 25

P DO 1 K = 1 ,13

C 1 D AT A ( T , J , K ) = COMPLF X VALUE

0 CAIL BOUPT ( DATA ,NN, 3 ,-l, 1, WORK

)

r

G Fy^wDLP P. one-dimensional forward

r lbngth 64 IN fopTPAN II.

r dimension DATA(2,64)

C OP P I =1 , 64

C nATA(l.I)=PFAL PAOT

C P 0ATA(P.I)=0.

r OJLL four T( data

,

64, 1 ,-l ,0,0

)

c

FOURT

FOUR T FOURT FOURT

FOURT FOUPT

FOURT FOURT TRANSFORM OF A REAL ARRAY OF FQURT

FOURT FOURT

FOURT FOURT

FOURT FOUPT FOURT

98
0TMPM2T0M OATA (1), NN (1). IFAPT (32)> WORK ( 1) FOURT 78 WP = 0. FQURT 79

PO WI = 0. FOURT 80

W 2 T P P = 0 . FOURT 81

w 9 T p j = n

,

FOURT 82 TwnPT = 9.PP318P707 FOURT 83

IF (NOTM _ 1 ) 12P0. 100, 100 FOURT 89

R*; 100 NIPT = 2 FOURT 85 nr no ioim = i, ndim FOURT 86

IF (MM (iniA'))1280, 12B0, no FOURT 87 no k)TOT = MTOT * ^N (TOIM) FOURT 88

c FOURT 89 90 r MAIM LPnP FPP FACH DTMeMSIGM FOURT 90 FOURT 91

M PI = ? FOURT 92 on 1270 miM = I, NDTM FOURT 93

M = MM ( T D IM ) FOURT 99

05 MP? = NPl + N FOURT 95

IP (N - 1 ) 1?P0, 12^,0. 120 FOURT 96

c FQURT 97

c FACTOP M FOURT 98

r FOURT 99 lOO no M = M FOURT 100 MTwr = MPl FOURT 101

IF = 1 FOURT 102 miv = 2 FOURT 103

1 ?o lOIJPT = M / IDIV FOURT 109

10‘5 IPFM = M - ipiv * lOUCT FOURT 105

IF (IQllPT - T0TV)710, ]40, 190 FOURT 106 190 T F (TPP'^ ) no, 150, IFO FOURT 107

1 “^O NTwn = NTWn + NTWO FOURT 108

M = ICMPT FOURT 109

no GO TO 170 FOURT no

160 rniv = 3 FOURT in

1 70 toupt = m / niv FOURT 112 IPpM '= _ iniv 7 TOUPT FQURT 113

IF (TOUPT - TOTV)270, IPO, 180 FOURT 119

11 ^ IPO TF (IPFM)200, 190, ?C0 FOURT 115 190 IF ACT (IF) = lOTV FOURT 116

IP = TP + 1 FOURT 117

M = TOUPT FOURT 118 GO TP 170 FOURT 119

1 ?0 ?00 inn/ = T n I V + 2 FOURT 120

GP TP 170 FOURT 121

710 IF (IRpm)770, 220, 230 FOURT 122 270 NTWr = MTWP + MTWP FOURT 123

G P TP 7 9 0 FOURT 129

1 pp 270 T fact ( TF ) = M FOURT 125

c FOURT 126

r SFPAPATF four CA2FS-- FQURT 127

c 1. CPMPIFy TRAN2FCPM pp PfAL TRANSFORM FOR THE 9TH, 5TH,ETC. FOURT 128

r 01 MP MS TPN7 . FOURT 129 IPO r 7, PFAL TPAMPFPPM FPP TMF 2ND np 7PP nrMFNSION. MFTHOP

FOURT 130

r TPANSFPPM half Tur DATA, SUPPLYING TMF pthfr HALF RY CON- FOURT 131

c JUG AT F S YMM F TR Y

.

F OUR T 132

r 7. RFAL transform FOP THE 1ST OIMFNSTON, N ODD. MFTHOO

FOURT 133

r PPAMSFORM MALF THF OATA AT FACH stage. SUPPLYING THE OTHER FOURT 139

r HALF 7Y CONJUGATF SYM«FTPY . FOURT 135

r 9. PFAL TPANSFPPM FOR THF 1ST PTMENSIPN, n FVFN . METHOD

F (^URT 136

C TPANSFPPM A rpMPLPy ARRAY OF LPNGTH N/2 WHOSE PFAL PARTS FOURT 137

r APF THF FVFN NUMBFPfd RFAL VALUES ANO WHOSE IMAGINARY PARTS FOURT 138

r APF THF GPO NUMOPPEO PFAL VALUES. SFPAPATF AND SUPPLY FOUPT 139

1 r THF ^fconp half by conjugatp symmftpy. FOURT 190

r FOUPT 191

Mr\i9 - mol 7 (NP2 / A TWO) FOURT 192 TPASr = 1 FOURT 193

IF (iniM _ 9)250, 30C, 300 FOURT 199

IF (TFPPM)7A,0, 290, PCC FOURT 195

? ^0 T r A S F = 2 FOURT 196

IF ( IP TM - 1) ?70 , 270 , 700 FOURT 197

?70 n AS F = 7 FOURT 198

IF (MTWP - NP1)300, 300, 280 FOURT 199

150 ?P0 TC A sc = it FOURT 150 NTWP = NTWP / 2 FOURT 151

N = N / 2 FOURT 152

KIP? = NP2 / 7 FOURT 153 MTPT = NTPT / 2 FOURT 159

99
1 T = 3 FOURT 155

on ?90 J = 2. NTOT FOURT 156

n/STA (J) > DATA (I) FOURT 157

290 1 = 1+2 FOURT 158

300 I IP Nip = NOl FOURT 159 1^0 TF ( lOASF - 2 )320 . 310, ’20 FOURT 160 310 JIPMO = NPO ( 1 + NPRPV / 2) FOURT 161

r FOURT 162

c SHIJFFLP ON THF FACTOPS OF TWO IN N. AS THE SHUFFLING FOURT 163

r CAN 8F OONF RY STMPLF INTFPCHANGF, NO WORKING APPAY IS NgEDFO FOURT 169

c FOURT 165 320 IF (NTWG - NP1)700, 700. 330 FOURT 166

330 NPPHP = MP2 / ? FOURT 167

J = 1 FOURT 168

DO 390 I’ = 1, NP2, N0N2 FOURT 169 170 TF (J - I2)’90, 360, 360 FOURT 170

3^0 IlMAY = T2 + N0N2 - 2 FOURT 171 no 350 1 1 = 1 2 , UMAX, ’ FOURT 172

no 350 13 = 11, NTOT, NP2 FOURT 173

J3 = J + I 3 - 12 FOURT 179

17'5 TFNPP = DATA (13) FOURT 175 tempi . data (13 + 1) FOURT 176 DATA (13) = data (J3) FOURT 177 DATA (13 + 1 ) = DATA ( J3 + 1

)

FOURT 1 78 data ( j 3 ) = TF NPR FOURT 179

1 fiO 350 DATA (J’ + 1) = TEMPI FOURT 160

350 M = NP2HF FOURT 181 370 TF (J - H)390, 390, 3R0 FOURT 182

3P0 J = J - M FOURT 183

M = M / 7 FOURT 189

1 TF (M - N0M2)390, 370, 370 FOURT 185 390 J = J + M FOURT 186

C FOURT 187

C matn loop for factors of two. P F R F QPM FOURIER TRANSFORMS OF FOURT 188

r lfngth foijp, with one of length two if NEEDED. THE TWIDDLE EACTOREOURT 1 89

1 oo c W = EXP( TSTGNY2YPIYS0PT(-T )=!>M/(9*MMAX) ) . CHFCK FOR W= IS IGN + SQRT( -DFOURT 190

c AND RFPFAT for W= TS T GN+SORT (-1 ) YCON jugate ( W)

.

FOURT 191

c FOURT 192 N0N2( = N0N2 + N0N2 FOURT 193 TRAP = NT WO / NDl FOURT 199

19 5 ^00 IF (TPAR - 2)990. 92C, 910 folipt 195

ifio IPAP = TRAP / 9 FOURT 196

GO TO 900 FOURT 197 920 DO 930 11 = 1, IIRNG, 2 FOURT 198

DO 930 J3 = 11, N0N2, NPl FOURT 199 200 DO 930 K1 = J3, NTOT, N0N2T FOURT ’00

K2 = K1 + N0N2 FOURT 201 TFMPP = data (K2) FOURT 202 TFWPI = DATA (K2 + 1) FOURT 203 DATA (K2) = DATA (Kl) - TEMPP FOURT 209

205 DATA (K2 + 1 ) = DATA (Kl + 1 ) - TEMPI FOURT 205 DATA (Kl) = DATA (Kl) + TEMPP FOURT 206

30 DATA (Kl + 1 ) = DATA (Kl + 1 ) + TEMPT FOURT 207

^<40 MMAX = NON 2 FOURT 208

9 50 IF (MYAX - NP2HF)960, 700, 700 FOURT 209

210 950 LMAX = MAXO (N0N2T, MMAX / 7) FOURT 210

IF (MMAX - NPN2)500, 500, 970 FOURT 211 970 THETA = - TWOPT + FLOAT (N0N2) / FLOAT (9 * MMAX) FOUR T 212

TF (ISTGN)990, 980, 980 FOURT 213 980 theta = - THFTA FOURT 219

’15 990 iJP = CDS (THFTA) FOURT 215

WI = SIN (THFTA) FOURT 216 U^TPP = - p, WI Y WT FOURT 217 WSTPT = 7. Y WP * WI FOURT 218

5D0 on 693 L = N0N2, LMAX , NHN?! FOURT 219

’20 M = L FOURT 220

IF ("MAX - NQN2 ) 520

,

520, 510 FOURT 221

510 W2R=WP*WP-WI* WI FOURT 222

W’l = 2. Y UP Y WI FOURT 223

W3P = V2P Y WP - W2I * WI FOURT 229

22 5 W 3 I = W 2

P * W T + W 2 I Y WP FOURT 225

5’0 on 690 T1 = 1, IIRMG, 2 FOURT 226

DD 690 J3 = 11, NCN2, NPl FOURT 227 KMIN = J3 + IPAR Y M FOURT 228

IF (MMAX - NON2)53n, 530, 590 FOURT 229

230 5 30 K M I M = J 3 FOURT 230

590 KDTF = IPAP Y MMAX FOURT 231

100
550 K STF° = 4 5 K D IF FnUPT 237

on 6^0 K1 = KMIM, NTHT, INSTEP FQUPT 733

K? = K1 + KOIF FOUPT 234 735 K 3 = KP + KDIF FOUPT 235

K 4 = K 3 + K 0 I p FHUPT 236

IF ( « MAX - NiPN7 ) 560 , 560 , 5q0 FOUPT 737

550 U 1

P = oat A (Kl) + DATA (K7) FOUPT 733 Ull = data (K1 + 1) + DATA (K? + 1

)

FOUP T 23 9 ?40 IJ70 = DATA (K3) + DATA ( K4) F G U P T 240

U7I = DATA (K3 + 1) + data (K4 + 1) FOUPT 741

II3P = DATA (Kl) - DATA (K7) FOUPT 242

U31 = data {Kl + 1) - DATA (K? + 1) FOUPT 243

IF (T5T0M)k70. ‘^60» '=60 FOUPT 244

74? 570 U4R = DATA (K3 + 1) - data (K4 + 1) FOUPT 245

1)4 1 = DATA ( K4 ) - DATA (K3

)

FOUPT 246

GO TO 670 FOUPT 247

560 'J4P = DATA (KA + 1) - data (K? + 1) FOUPT 249

1)41 = DATA (K3) - DATA (K4). FOUPT 249

7 50 GO TO 6

7 o FOUPT 250

5Q0 T 7 P = U7P * DATA (K?) - W3I # data (K7 + 1) FDIJPT 251

T7I = VZP A DATA (K7 + 1) + W?I 7 DATA (K7) FOUPT 252

T3P = WP 7 data (K3) - WT 7 DATA (K3 + 1) FOUPT 253

T3T = WP 7 data (K3 + 1) + wI 7 DATA (K3) FOUPT 254

755 T4 P = W'3P * data (K4) - W3I 7 data (K4 + 1) FOUPT 255

T4I = W3P 7 data (K4 + 1) + W3I 7 DATA (K4) FOUPT 256

DIP = DATA (Kl) + T?P FOUPT 257

HIT = data (Kl + 1) + T’l FOUPT 258

1)7 P = T3P + T4P FOUPT 259

7h0 1J7I = T3I 7 T4I FOUPT 260

U3P = data (K 1 ) - ( FOUPT 261

1)3 T = DATA (Kl + 1) - T7T FOUPT 262

IF ( I S IGN ) 600, 610, 610 FQUPT 263

600 U4P = T3T - T4I FOUPT 264

765 U4I = T4P - T3R FOUPT 265

GO TO 670 FOUPT 266 610 'I4P = T4I - T3I FOUPT 267

U4I = T3P - T4P FOUPT 268 62C 0AT4 (Kl) = DIP + U2P FOUPT 269

770 DATA (Kl + 1) = IJII + U7I FOUPT 270 data (K?) = IJ3P + IJ4P FOUPT 271 DATA (K7 + 1) = U3I + U4I FOUPT 272

data (K3) = UIP - U?P FOUPT 273

DATA ( K 3 + 1 ) = Mil - LI ? I FOUPT 274 775 DATA (K4) = IJ3P - LI4P FOUPT 275

630 DATA (K4 + 1) = U3I - IJ4I FOUPT 276 KMIN =47 (KMTN - J3) + J3 FOUPT 277

KDIF = KGTEP FOUPT 273

IF (KOTF - N'P?)!550 » 640 FOUPT 279

7°0 640 r PNJT TM'IF FOUPT 280

M = MM&X - M FOUPT 281

TF (TST^-N)660f 660, 660 FOUPT 282 650 T FM P9 = WP FOUPT 283

WR * - WI FQUPT 284

7B5 = - T F D P P FOUPT 285

GH TO 6,70 FOUPT 286

660 T F M P R = WP FOUPT 287

WP = WT FOUPT 2 88 WT = TFMPR FQUPT 289

7<?0 67a IF (M - L^IAX ) 680, 680, 510 FQUPT 290

6SC T FMPP = WP FQUPT 291

WP = WP 7 W5TPP - WT 7 W^TPI + WP FOUPT 292

690 WI = WI * WSTOP + TFMPP * WSTPI + WI FOUPT 293 TPAR = 3 - IPAP FOUPT 294

795 N|MAX = MMAX + 3MAX FOUPT 295

GD TP 4'=0 FQUPT 296

C FOUPT 297

C MAIN LOOP FOP FACFQPP NOT EQUAL TO TWO. AOPLY THE TWIDDLE FACTOR FOUPT 298

r W = FXP(ISIGN>l<2*PI*SQPT(-l)^(J?-l)-l'(Jl-J2)/(NP2*IFPl)), THEN FOUPT 299

300 c ’FPFOPM A FOIJPIFR TPANSFQPM OF LFNGTH IFACT(IF), MAKING USE OF FOUPT 300

c rPNJUGATF PYMMFTPTF':. FOUPT 301

c FOUPT 302 700 IF (NTWO - NP2)710, 990, 990 FOUR T 303 710 IFPl = non? FQUPT 304

3C5 IF 1 FQUPT 305

NP] HF = NPl / 7 FOUPT 306

770 IFP? = IFPl / IFACT ( IF) FOUPT 307

J ] png = N P7 FOUPT 308

101
IF (TAft7F _ 7)740, 7?C, 740 FOUPT 30<?

^ 1 0 770 JIPMG = (NP? + IF°1) / ? FQURT 310 J7ST0 = M07 / IF4CT (IF) FOUPT 311 J1P07 = (J7STP + IFP2) / 7 FOUPT 31? 7^0 J7MTN = 1 + TCP? FOUPT 313

IF (TFOl - MP?)750, POO, 500 FOUPT 314

7'=. 0 nr 700 J7 = J7*^IN, IFPl, IFP? FOUPT 315 THFT4 = - TWOPI « FLOAT (J? - 1) / FLOAT (NP?) FOUPT 316

IF (ISt0N)770, 760, 760 FOUR T 317 760 THFTA = - TMFTA FOUPT 318 770 S

1 VTH = FIN (THFTA / 7 .

)

FOUPT 31Q

^ 7 A WFTPP = - 7. * SINTH 7 5TNTH FOUPT 3?0 WFTPT = SIM (THFTA) FOUPT 3?1

UP = WFTPP + 1. FOUPT 322

WT = WSTPT FOUPT 323 JU'IN = J7 + IFPl FOUPT 324 nr 700 J1 = JIF'IN, JIPNO, IFPl FOUPT 325 ir-AV = J1 + I1PN3 - 7 FOUPT 326

on 770 11 ’ Jl, IIMAT, 7 FOUPT 327

nr. 7P0 13 = n, NTOT, NP7 FOUPT 328

,|7MAX = 13 + IFP? - NPl FOUPT 329 770 nn 7°0 J3 = 13, J?max, MPI FOUR! 330 TFHPP = RATA ( J3

)

FOUPT 331 OATA (J3) = DATA (J3) W9 - DATA (J3 + 1) * WI FOUPT 332

7i=C OATA (J3 + 1) = TFHPP 7 WI + OATA (J3 + 1) WP FOUPT 333

TFA'Pp = WO FOUPT 334

7 7 «; UP I UP 7 WFTPR - WI 7 WSTPI + UP FOUPT 335

7Q0 Ul = tpmpp WSTPT + WT 7 WSTPP + WI FOUPT 336

5 0 0 THFTA = - TWODI / FLOAT (IFATT (IF)) FOUPT 337

IF (IFIGM)7?0, 810, 810 FOUPT 338

510 THFTA = - THFTA FOUPT 339

570 SINTH = SIN (THFTA / 7.) FOUPT 340 WSTPP =-2.7 SINTH 7 SINTH FOUPT 341 WSTPI = SIN (THFTA) FOUPT 342 KSTFP = 7 7 N / IFACT (TF) FOUPT 343

FPANf- ^ RSTFP 7 (IFAOT (TF) / 7) + l FOUR T 344

7 i •; nr ORA XI = 1, TIPN'G, 7 FOUPT 345

or PRO 13 = 11 , mtqt, np? FOUPT 346

pn QIP WHIN = 1, KPAK'G, KSTFP FOUPT 347 jIHAY = T? + JIPNG - IFPl FOUPT 348

no PRO Jl = 13, JIMAX, IFPl FOUPT 349 350 J3HAY = Jl + ICP? - NPl FOUPT 350

nr P80 J7 = Jl. J3MAX, Noi FOUPT 351 J7HAX = J3 + IFPl - IFP2 FOUPT 352

K = K«IN + (J3 - Jl + (Jl - 13) / IFACT (TF)) / NPIHF FOUPT 353

IF (KNIN - 1)530, 53C, 550 FOUPT 354

755 R70 F|!MP = 0. FOUPT 355

SUM I = 0

.

FOUPT 356

nn 740 J? = J3, J7NAX, IFP? FOUPT 357

SUM? = SlIHP + OATA (J?) FOUPT 358

5 40 FIJMJ = SIIHI + OATA (J2 + 1) FOUPT 359 760 wrPK (K) = SUMP FOUPT 360 WrOK (K + 1) = SDMI FOUPT 361

G r TO 5 p

0

FOUPT 362

5 50 WCnNJ = K + 2 * (N - KMTN + 1) FOUPT 363

J7 = J ?M AX FOUPT 364

75‘= Sljyp = OATA (j?) FOUPT 365

SlIMT = OATA (J7 + 1) FOUR T 366

n L n F p = 0

.

FOUPT 367 niOFI = 0. FOUPT 368

J7 = J? - 1FP2 FOUPT 369

7 0 8F0 TFMPO = SIJ^^P FOUPT 370 TFMPj = S'lMT FOUPT 371

SIIMP = TUOWP 7 SIJMP - OLDSP + DATA (J2) FQURT 372 FU^r = Twnwp 7 SU^I _ OunST + DATA (J2 + 1) FOUPT 373

ni n 5P = TF MR p FOUPT 374

n|_nsi = TFMPl FQURT 375

J? = J7 - IFP? FQURT 376

IF (J7 - J3)R70, 570, 860 FOUPT 377

« 70 TFMPP = WP 7 SIJMP - 0[OSP + DATA ( J 2 ) FOUPT 378 TFMPJ = uj + FIJMI FOUPT 379

FO WORK (K) = TEMPP - TFMPl FOUPT 380 wnpw (KTONJ) = TFMPP + TFMPl FOUPT 38 1 TFMPP = WP 7 SIIMI - nunsi + DATA (J2 + 1) FOUPT 382 TFMPl = WI 7 SUMP FOUPT 383 WnpK (W + 1) = TFMPk icMPI FOUPT 384

Q R WroK (KCONJ + 1) = TFMPP - TFMPl FOUPT 385

102
P°0 0

ON! TIN IIP FGUPT 386

IP (RMTN - 1)800. 80C. OOO FOUPT 387

POO WP = WSTPR + 1. FOUPT 388

W T = WOT P

I

FOUPT 3 81 100 on Tn 010 FOUPT 390 000 TPMpp = yp FOUPT 39 1 WR = UP + WSTOP _ wi WSTPI + UP FOUPT 392

UI = tpmdp * WOTPI + WI * WSTPP + WI FOUPT 303

010 TWOWP = UR + WP FOUPT 394

3'J5 IF ( T”* lOE - 3 )030, 0?C, 030 FOUPT 395 070 IF (TfPl - NP?)O50. 010, 030 FOUPT 396

030 R = 1 FOUPT 397 I7N4V = 13 + NPl - MDI FOUR T 398

on Q40 17 = 13. 1 7MAX , NPl FOUPT 399 400 oat A ( 17 ) = WHPK (K ) FQL'P T 400 DATA ( T2 + 1 ) = WnPK (K + 1) FOUPT 401 040 K = K + 7 FOUPT 402

r, p TP 0 8 0 FOUPT 403

r FOUPT 404 405 c CPNPLPTF a PFAL TPANSFCIPM in the 1ST DIMENSION, M ODD, RY CON- FOUPT 405

r JUGATF symmfjpIES at each stagf. FOUPT 406

c FOUPT 407

050 JIM AX = 11 + IFP? - NPl FOUPT 408

no 070 J3 = 13, JINAX, NPl FOUP T 409

410 J7NAX = J3 + NP7 - J7PTP FOUPT 410

00 070 J7 = Jl, J7MAX, J2STP FOUPT 411 JIMAX = J7 + J1P07 - IFP? FOUPT 412 JICNJ = J3 + J2MAX + J20TP - J2 FOUPT 413 on 070 Jl = J7, JIM AX, IFPl FOUPT 414

4 15 K = 1 + J 1 - 11 FOUPT 415 data (Jl) =. wnpk (R) FOUPT 416 data (Jl + 1) = WORK (K + 1) FOUPT 417

TF (Jl - J7)O70, 070, °60 FOUPT 418

060 data (JICMJ) = WQPR (K) FOUPT 4ig

4 ?0 DATA (JIONJ + 1) = - WPPK (K + 1) FOUPT 420

070 JICNJ = JICNJ - TFP? FOUPT 421

0 PO CPNTINHF FOUPT 422

IF = IF + 1 FOUPT 423

IFPl = TFP? FOUPT 424

475 IF (IFPl - NPDOOO, 000, 7?0 FOUPT 425

r- FOUPT 426

r CnwPLFTF A PFAL TPANSFOPM IN THF 1ST dimension. N EVEN, BY CON- FOIJPT 427

r JIIC-ATF SYMMFTPIFO. FOUP T 428

c FGUPT 429

‘tin 000 CP TO (1160, IIRO, 1260, 1000), ICASF FGUPT 430

1000 N H A L F = A' FOUP T 431

N = N 4- N FOUPT 432 XHFTA = - TWGPl / FLi;aT (N) FOUR T 433

TF (I0TGN)102O, 1010, ICIO FOUPT 434 4?5 1010 thfta = - TMFTA FOUPT 435 1070 SINTH = OIM (THFTA / ?.) FOUPT 436 WSTPP = - 7. + SINTH * PINTH FOUPT 437 WSTPI = SIN (THFTA) FOUPT 438

UP = w P T P P + 1 . FOUPT 439

440 W I = V S T 0 I FOUPT 440

I MIN = 3 FOUPT 441 JMIN = 2 * NHALF - 1 FOUPT 442

GO TP 1050 FOUPT 443

1 030 J = JMIN FOUPT 444

445 DC 104P I = Imin, NTOT, NP’ FOUPT 445

SIJMP = (DATA (T) + DATA (J)) / 7, FOUPT 446

SIIMI = (DATA (I + 1) + PATA (J + 1 ) ) / 2. FOUPT 447 OTFP = (OATA (T) - DATA (J)) / 2. FOUPT 448

DIPT = (PATA (I + 1) - DATA (J + 1 ) ) / 2. FOUPT 449

4 5 0 TFMOp = WP + SLIMI + WI 7 OIFP FOUPT 450

TF-MPT = WI * SDMI - WP * niFP FOUPT 451 data (I) = SUMP + TFYPP FOUPT 452 data (I + 1) = PIFI + TFMPI FOUPT 453 data (J) = SUMP - TFMPP FOUPT 454 455 DATA (J+1)=-0IFI+ TEMPI FOUPT 455 1740 J = J F NP7 FOUPT 456 IMIN = TMIN + 7 FOUPT 457 JMIN = JMIN - 2 FOUPT 458 TFMPP = WP FOUPT 459

46 0 WP = VR A WSTPP - WI * WSTPI + WP FOUPT 460

WI = tcmPP 7 WSTPI + WI * WSTPP + W I FOUPT 461

1 C5C IF (IMIN - JMIN)1010, 105Q, 1000 FOUPT 462

103
1060 IP ( I ST G^J ) 1070, 1070, 1070 FQURT 463 107C no 1080 T = IMTN, NTOl, NP? FOURT 464 465 1080 DATA (I + 1) = - DATA (I + 1) FOURT 465

10<50 NP? = NP7 + NP? FOURT 466 NTOT = NTQT + NTHT FOURT 467

J = ntot + 1 FOURT 468

INAX . ntot / 7 + 1 FOURT 469

47r> 1100 ININ = INAX - 2 * NHALF FOURT 470

I = ININ FOURT 471

GO TO 1180 FOURT 477 1110 OATA ( J ) = OATA ( I

)

FOURT 473 data U + 1) * - DATA (I + 1) FOURT 474 475 1170 1 = 1 + 8 FQURT 475

J = J - 2 FOURT 4 76 IF (I - TNAxnilO, 1130, 1130 FOURT 477 mo DATA (J) = DATA (ININ) - DATA (ININ + 1) FOURT 478 DATA (J + 1) = 0. FOURT 479

IF (I - J)1150, 1170, 1170 FOURT 480

1 1 40 DATA ( J ) = DATA ( I ) FOURT 481 DATA (J + 11 = DATA (I + 1) FOURT 482

1160 1 = 1-2 FOURT 483

J = J - 7 FOURT 484

4R 5 TP (T _ TNTN)1160, 1160, 1140 FOURT 485 1160 data (J) = DATA (ININ) + DATA (ININ + 1) FOURT 43 6 data ( j + 1 ) = 0 . FOURT 487

TMAX = I"TN FOURT 488

on Tn 1100 FOURT 489

60Q 1 170 DATA (1) = DATA (1) + DATA (2) FOURT 490

data (7) = 0. PQURT 491

GP TO 1760 FOUR T 492

C FOURT 493

C CPNPLCTF A PFAL TPANSFORH FOR THF 2ND OR 3RD DINENSION BY FOURT 494

40 6 C CON JUGATE SYNNFTRIFS, FOURT 495

r FOURT 496 1180 IF (IIRNO - NPDllQC, 1760, 1260 FOURT 497

1 ICO nn 1?60 T3 = 1, NTOT, NP7 FQURT 493 T?wax = 13 + NP2 - NPl FOURT 499 50C on 1750 12 = 13, I2NAX, Npi FOURT 500 I«IN = 17 + IIRNG FOURT 501 TNAX =12+ NPl- 2 FOURT 502

INAX = 7 * T3 + NPl - I-NIN FOURT 503

IF (17 - 13)1210, 1710, 1200 FOURT 504 505 1700 J'^AX = JMAX + NP? FOURT 505 1210 IF ( iniN - 2 ) 1240 , 1740, 1720 FOURT 506 1770 J = JNAX + NPO FOUR T 507

DP 1730 T = ININ, INAX, 7 FOURT 508 DATA (I) = DATA (J) FOURT 509

610 OATA (I + 1) = - OATA (J + 1) FOURT 510 1730 J = J - 8 FOURT 511 1740 J = jnax FOURT 512

DP 1250 T = ININ, INAX, NPO FOURT 513 PATA (I) = DATA (J) FOURT 514 515 OATA (I + 1) = - DATA (J + 1) FOURT 615 I860 J = J - NPO FOURT 516

C FOURT 517

C END OF L-nOP ON FACH niNFNSION FOURT 518

0 FOURT 519

5?0 1260 NPO = NPl FOURT 570

NPl = NP7 FQURT 521 1270 N P p F V = N FOURT 522 1?°0 R PTHD N FQURT 523

FNO FOURT 524

104
A. 1.11 SUBROUTINE

PARAB(FOD,DOL, BLOCK, DFOCUS,ACOSE,ACOSH, THETA, ETHETA.EPHI)

PURPOSE :

This subroutine calculates the E- and H-plane far electric field for an axially

defocused, circularly symmetric, paraboloidal reflector antenna at a specified angle

from the axis.

ARGUMENTS :

FOD is the focal length to diameter ratio for the reflector.

DOL is the diameter of reflector in wavelengths.

BLOCK is the fractional diameter blockage.

DFOCUS is the amount of axial defocusing in wavelengths (positive direction

corresponds to feed beyond focal point).

ACOSE is the E-plane aperture illumination factor.

ACOSH is the H-plane aperture illumination factor. (NOTE: See discussion of POMODL

for a more complete discussion of ACOSE and ACOSH.)

THETA is the angle from axis at which field values are desired in degrees.

ETHETA is the electric field in E-plane.

EPHI is the electric field in H-plane.

DISCUSSION :

This and associated subroutines EPINT, ETINT, QATRC, and BESFUN were written by

Professor W. V. T. Rusch of the University of Southern California. This discussion is

intended to indicate the computations performed and is not a detailed description of

the operation of the subroutines.

The subroutine uses PO as discussed in section 3 of the report. It is assumed that

the antenna is rotationally symmetric, thus allowing very rapid execution.

Aperture illumination may be of three types: uniform, dipole, or cosPe', where 0'

is the angle from the axis of the feed. These are selected with parameters ACOSE and

ACOSH, and the E- and H-plane tapers are independently specified.

The integration is performed by subroutine QATRC. This subroutine has error flags

which are set when the desired accuracy is not achieved either because of accumulated

round-off errors or because the integration range could not be sufficiently

subdivided. PARAB prints an error message indicating the type of error. These errors occur at larger values of THETA. Care should be taken to delete any far-field points

known to be in error.

105
This subroutine requires that functions ETINT, EPINT, and subroutines QATRC and

BESFUN be supplied. In addition, library functions ATAN, COS, SIN, ATAN2, CEXP, SQRT,

CABS, and inline functions CMPLX and ABS are employed.

106
1 SUQRHUTTNP P 4P# B( FCD. D CL. BLOCK, OFOCUS./SCOSE, AC CSH» THETA, E THE TAjEPHP ARAB 1 *I ) PARAB 2

r RADIATION patterns FROM A DEFnCMSFD PARABOLOID PAR AB 3

r programmer - w.v.T. rusch PARAB A

C 16 maJ 1°7A PARAB 5 C MOOIFTED 1? MAY iq?6 PARAB 6 COMPLEY AUX(11),R0MB.CMPLX,A1,D1,ETHFTA,FPHI PARAB 7 COMMON/DAT A/FQL, P I. SINT.COST.DFOCSS, ACGSEE,ACOSHH PARAB 8 EXTERNAL FTTNT.EPINT PARAB 9 10 OFOC SS=OF OCUS PARAB 10 ACOSFFxACOSE PARAB 11 AC0RHM*ACOSH PARAB 12

PI *A .0 + AT AN ( 1 .0) PARAB 13 dtp=pi/i«o.o PARAB lA

IB RTD=180.0/PI PARAB 15 FOL =• FODYOOL PARAB 16

A = 2.07ATAN(A.0+F0D) PARAB 17 TFtBLOCK.LT. 0.0001) P » PI' PARAB 18

I F ( BLOCK. GE . C. CCOl ) B = 2 . 0+ AT AN ( A . 0* F 00/ BLOCK

)

PARAB 19

?0 COS T * cost THET AtDTR ) PARAB 20 SINT = SI Nt THFTAYDTR ) PARAB 21 CALL 0ATRCtA,B,1.0R-C3,ll.FTINT,R0MR,IER,AUX) PAPAB 22

IF tIER .EQ. 1) PRINT 1000 PARAB 23

IF tIFR .>^0. 2) PRINT 1010 PARAB 2 A

P‘5 ETHETA • CMPLXtO.0,1 .0)Y2.0*PIYF0L*R0MB PARAB 25 CALL OATRCtA,R,1.0F-C3,ll,EPTNT,ROMB,IER,AUX) PARAB 26

IF tlFR ,P0. 2) PRINT 201^ PAPAB 27

IF t IFR . FO. 1 ) PRINT 2000 PARAB 28 EPHI = CMPLX t 0. 0, 1 .0) *2 .0+PI*F0L*R0MB PARAB 29

BO RETURN PARAB 30

C PARAB 31 1000 FORMATt* REOUIRFD ACCURACY MOT ACHIEVED IN E-PLANF DUE TO R OUNOINPARAB 32 IG frror S .* ) PARAB 33 1010 FORMATt* PFOUIRED ACCURACY MOT ACHIEVED U' E-PLANE DUE TO INSUFFIPARAB 3A

3B ICIFNT NUMBRR OF INTEGRATION STEPS.*) PARAB 35 2000 FOPMATt* PFOUIRED ACCURACY NOT ACHIEVED IN H-PLANE DUE TO ROUNOINPARAB 36

IG ERRORS.*) PARAB 37 2010 FORMATt* PFOUIRED ACCURACY NOT ACHIEVED IN H-PLANF DUE TO I NSUFFIPARAB 38 ICIRNT NUMBER OF INTEGRATION STEPS.*) PARAB 39 AC END PARAB AO
1 c SL'OPOUTTNF 04TP0('(L.Kl).'^PS.NniM,FCT,Y,IFP,AUX) 0 ATRC •QATRC

1

2 r QATRC 3

c SMRPniJTINP 04TRC QATRC 4

5 r rnMPLFY VERSION op SSp-PGUTINF CATp, sept. 72. MS-J QATRC 5

r PUP POS F QATRC 6

0 xn cnMpijxg 4 KJ APPPOX 1 M AT ION FOP INTFGPAL OF COfAPLEX QATRC 7

c function FCT(X) with peal BOUNDARIES XL AND XU. QATRC B

r QATRC Q 10 c 1 1 S A G P QATRC 10

c CALL QATPCIXL.XU.FPS.NDIY.FCT.Y.IEP.AUX) QATRC 11

c DAPA''FTPP FCT PFOUIPFS AN FXTEPNAL STATFYFNT. QATRC 12

r QATRC 13

c OFSCPIPTION OF PARAYETFRS QATRC 14

15 c XL - THF LOWFP 30UNO OF THE INTERVAL. QATRC 15

r xi| - THE UPPER ROUNO OF THE INTERVAL. QATRC 16

r FPS - THF UPPFP ROUND OF THE ARSOLUTF ERROR. QATRC 17

r NOIY - THF DIMFNSION OF THE AUXILIARY STOP AGP ARRAY AUX. QATRC 18

r NOIY-i IS THF maximal NUMRFP OR BISECTIONS op QATRC 19 20 r THE interval (XL. XU). QATRC 20

r FfT - THE NAME OF THE EXTERNAL FUNCTION SUBPROGRAM USED. QATRC 21

r Y - THE RESULTING APPROXIMATION FOP THE INTEGRAL VALUE .QATRC 22

c lER - A resulting error PARAMETER. QATRC 23

c AUX - AN AUXILIARY STQPAGP ARRAY WITH DIMENSION NOIM. QATR C 24

25 r QATRC 25

c P F ^ P K S QATRC 26

c CPDQP papameteR Iep is coded IN THE PQLLOWING E(0RM QATRC 27

c IFR=o - IT WAS POSSIBLE TO REACH THE PEOUIPED ACCURACY. QATRC 28

c NO ERROR. QATRC 29

00 r Ier=1 - IT IS IMPOSSIBLE TO PEACH THE REOUIRED ACCURACY QATRC 30

c PFCAUSF OF ROUNDING ERRORS. QATRC 31

c IFR=2 - IT WAS IMPOSSIBLE TO CHECK ACCURACY BECAUSE NDIM QATRC 32

r IS LESS THAN 5. OR THE REOUIPED ACCURACY COULD NOT QATRC 33

c BE PEACHED within NDIM-1 STEPS. NDIM SHOULD BE QATRC 34

35 c increased. QATRC 35

c QATRC 36

r SUBROUTINES and function SUBPROGRAMS PEOUIPED QATRC 37

r THE EXTERNAL FUNCTION SUBPROGRAM FCT(X) MUST BE COOED BY QATRC 38

c THE USER. ITS ARGUMENT X SHOULD NOT BE DESTROYED. QATRC 39 40 r QATRC 40

c ME THOn QATRC 41

r cvaLUaTTON np Y IS DONE BY means OF TPAPEIOIOAL RULE IN QATRC 42

c connection with phmbfrgS principle, on return y contains QATRC 43

r THF REST POSSIBLE APPROXIMATION OF THE INTEGRAL VALUE AND QATRC 44

45 c vector iUX THF UPWARD DIAGONAL OF ROMBERG SCHEME. QATRC 45

r '•OMPONENTE AUX(I) IEND, with IEND LESS THAN OR QATRC 46

r FOUAL TO NDIM) BECOME APPROXIMATIONS TO INTEGRAL VALUE WITH QATRC 47

r OPruFASING ACCURACY BY MULTIPLICATION WITH (XU-XL). OA TR C 48

r pnp PFPFPE'^ICFj QATRC 49

50 c (1) FILIPPI, DAS VFRFAHRFN VON R 0 MB E R G- S T I E E E

L - R A U E R ALS QATRC 50

r SPFZIALEALL DFS ALLGFMETNFN PRIN7IPS VON RICHARDSON, QATRC 51

0 maTHEMA TIK-TECHNIK-W IRTSC HAFT, VOL. 11, ISS.2 (1954), QATRC 52

0 PP . 49-54 , QATRC 53

c (?) BAUER, algorithm f-0, CACM, VOL. 4, ISS.6 (1961), PR. 2 55 . QATRC 54

r QATRC 55

c •QATRC 56

r QATRC 57

c QATRC 58

r QATRC 59

f-n COMPLEX FCT, Y, SM , AUX ( NDIM) QATRC 60

c QATRC 61

C PREPARATIONS OF ROMBFRG-LOOP QATRC 62 AUX(1)=.5+(FCT(XL)+FCT(XU)) QATRC 63 H=XU-VL QATRC 64

A =

1 F ( ND TM_i ) s, P ,

1

QATRC 65

1 I E ( M) ?, 10 ,

2

(DATRC 66

C QATRC 67

c N.niM IS GREATER THAN 1 AND H IS NOT EQUAL TQ 0. QATRC 68

2 HH=H QATRC 69

70 E=FPS/ ABS (H) QATRC 70 DELT?=0. QATRC 71

P = 1 . QATRC 72

J J = 1 QATRC 73 DO 7 1=2, NOIM QATRC 74

75 Y = AIIX( 1 ) QATRC 75 nFLTl=DELT? QATRC 76 Hn= HIM QATRC 77

108
H H = . 5 H H OATRC 78

P = . P* P OATPC 7P

PO V =VL + MH OATRC 80 S^”=(0.,0. ) OATRC 81 OP 3 J= 1 . J

J

OATRC 82

S>^ = <;y + FCT (X ) OATRC 83

3 X = X 4-HP OATRC 8A

°5 AtlX(I) = .1 + AUX(I-l) + P*SM OATRC 85

r A MFU APPRPX IM4TIPN OF INTFGPAL VALI.IF IS CnPPUTEO BY MEANS OF OATRC 86

C TPAPFZDTOAL PULE. OATRC 87

r OATPC 88

C ST4PT OF P3WRCPGC c XTP A POL A T I PN ME THOn

.

OATRC 80

QO 0=1, OATRC 90

J I = T-1 OATRC 91 pn A J = 1 . J I OATRC 92

I T=I-J OATPC 93 0=0 + 0 OATRC 9A

Q R 0=0 + 0 OATRC 95 AllX(TI) = Al)X(II + l) + (4IIX(n + l)-Aliy(n))/(0-l.) OATPC 96

r FNO OF R TAR FP G-ST FP OATRC 97

C OATPC 98 0FLT?=CARS(Y-AUX(1)) OATPC 99 ion IF ( T-5 ) 7. 5, 1 OATPC 100

5 I F ( DCLT7-P ) 1 0, 10.

1

OATRC 101

f I F ( OF LT7-0PLT1 ) 7, 11 , 11 OATPC 102

7 JJ=JJ+JJ OATPC 103

P T FPr? OATPC lOA 101 9 Y = M + AI)X ( 1 ) OATPC 105 PFTUOM OATRC 106

10 I FP =0 OATPC 107

GO X n 7 OATPC 108

11 I FR = 1 OATRC 109 110 Y =M A Y OATPC 110

R F T U P N OATPC 111 FNO OATPC 112
1 COMPLEX FUNCTION ETINT(X) ETINT 1 CnNMON/nAT4/FCL.PT.SINT,COST,nFaCUS»ACOSF,4COSM ET INT 2 OTMFNSION BJ(IOOO) ETINT 3

C NOTF THAT nS(PI-XP) = ( 0 F OC U

S -R HO TC OS X ) / P HQ P R I M F » R2/PHP0VL ETINT 4

5 complex cex p, c mplx ,a 1 ,ni.Hx E TINT 5 SINX = SIN(X) ETINT 6 COSX = cos

(

X ) FTINT 7 RHOOVL • 2.0+F0L/ ( 1 .O-COSX

)

ETINT 8 RHPOVL = SOPT( PHOOVL+RHOOVL+PFOCUS+OFCCUS-2 .*DF0CUS*RH00VL*C0SX) ETINT 9 10 RL = 9HQ0VL*STNX ETINT 10 P? = OFOCUS - PHCOVL+COSX ETINT 11 XP * PI - ATAN2(P1,R? ) ETINT 12 CSPMXP = R2/RHO0VL ETINT 13 F47F = 2 .0» PI * ( PHOOVL’>COSX*COST-R HPOVL ) ETINT 14

15 BFTA = 2.0TPT*PH00VL+SINX+SINT ETINT 15 IF(BETA.f,T.O.O) GO TO 2 ETINT 16 PFSSO • 1.0 ETINT 17 PESSl * 0.0 ETINT 18 3FSS2 » 0.0 ETINT 19 20 60 TO 3 ETINT 20

? CALL PES FUN ( PFT A, BJ , 4) ETINT 21 PES30 = PJ ( 1) ETINT 22 PFSSl * BJ(2) ETINT 23 BFSS2 * PJ(3) ETINT 24

35 3 continue ETINT 25

IF ( ACOSF . GF .

( -100. 0) ) GO TO 20 ETINT 26

A1 = P . 0/ ( 1. 0 + CSPMXP ) ETINT 27

01 « -41 ETINT 28 GO TO 50 ETINT 29

30 20 I F ( 4C0SE . GF . 0. C )G0 TO 40 ETINT 30

41 = CSPMXP ETINT 31

01 ' -1.0 ETINT 32 GO TO 50 FTINT 33

40 41 = rspMXP+4AC0SP ETINT 34 35 01 = -CSPMXP++ACOSH ETINT 35 50 CONTINUE ETINT 36

HX » (RHOOVL/ RHPOVL )*{A1*C0ST*(PESS0-BESS2)-D1+C0ST+(BESS0+3ESS2)*ETINT 37 SIN(XP-(X/2.0))/SIN(X/2.0)-2.0X-CMPLX(0.0,1.0)'I-4 1*PESS1*COS(X/2.0)/ETINT 38 *SIN ( X/2.0 ) +SI NT) ETINT 39 40 HX = HX M 1 .O-COSX

)

ETINT 40 FTINT . HX + SINX’»CEXP(CMPLX (0.0. FAZE) ) ETINT 41

P ETIJPN ETINT 42 END ETINT 43

110
1 CnMPLFV FUNCTION FPINT(X) EPINT 1 rOMNnN/DATA/FCLjDi.siM.CriST.DFnFUS.iCOSFjiCOSH EPINT 2 OIN’FNSIPN BJ(IOOO) EPINT 3

r NOTF TH4T cnS(PI-XP) = (PFOCUS-PHO+COSX )/RHOPPIMF = P3/RHP0VL EPINT A

5 CCf-PLFX CF XP , CNPLX, 41,01 ,HX EPINT 5

S INX * S IN ( X ) EPINT 6 cosx = cns(x) EPINT 7 PHOrvL = P .0 *FOL /( 1 . O-COSX

)

EPINT 8 PHPOVL = SOPT(PHn0VL+PHnnVL + DF0CUS*DF0CUS-?.*CF0CU'5’*‘PH0GVL*COSX) EPINT <5 10 PI = RHOnvL*SINX EPINT 10 P? = DFOCUS - PHOOVL*CnSX EPINT 11

XP = PI - 4TAN2 (P1,R2) EPINT 12 CSPMXP = P2/PHPrVL EPINT 13 FA 2 F > 2, 0»P I* ( RHCOVL»CnSX*COST-PHPOVL

)

EPINT lA

1’5 BFTA = 2.0*PI*PH00VL+STNX*SINT EPINT 15

IF ( 0 FT4 ,GT .0 .0 ) GO TO 2 EPINT It RESFO = 1.0 EPINT 17 BESSl = 0.0 EPINT 18 RFSF2 « 0.0 EPINT 19

?0 GO TO 3 EPI N1 20

? CALL BFSFUN ( PET 4 , R J, A

)

EPINT 21 PFSFO = BJ ( 1) EPINT 22 PESSl = P J ( 2 ) EPINT 23 PFSS? - BJ(3) EPINT 2A

?’> 3 continue EPINT 25

I F ( ACOS F . GE . (-100. 0) ) GO TO 20 EPINT 26

41 = 2.0/ ( 1. 0+CSPMXP) EPINT 27

D1 = -41 EPINT 28 GC TO 50 EPINT 29

30 ?0 IF ( ACOSE . GE .0 .0) GO TO AO EPINT 30

A1 = CSPMXP EPINT 31 01 = -1.0 EPINT 32 GO TO 50 EPINT 33

'.O A1 = CSPMX D*+ ACOSR EPINT 3A

01 « -CSPIXP-^aaCOSH EPINT 35 50 CONTINUF EPINT 36 MX = (RHGOVL/RMPOVL ) t A 1 + { R E S S 0 + P F S S 2

) -01 ( RESS0-BESS2)*SIN(XP-(X/EPINT 37 2.0))/SIN(X/2.)) EPINT 38 HX = MX/ ( 1 ,0-COSX

)

EPINT 39

‘•O print >= MX+S I NX*C EXP ( CMPLX( 0 .0, F AZE ) ) EPINT AO RETURN EPINT A1 ENO EPINT A2
1 5UPR0UTINE BESFUN(X,BJ>NM4X) BESEUN 1 niMFNS TON 5 J ( 1

)

PESFUN 2 C NOTF BJ(1)=>J0 5J(?) = J1 ... BJ( 200)-Jiqq P ESFUN 3 10 IF( X.GT.l.OF-03 ) GO TO IF PESFUN 4

5 B J( 1 ) =1 .0 BES FUN 5 DO 15 JRJ=>2.Nr<AX B ESFUN 6 15 3 J ( J3 J ) =0 .0 PESFUN 7

» FTUPN PESFUN 8 15 TF(X.0,T.NM4X) M.? + X + 7 BESFUN 9 10 IF ( X. L T .N*! 4X ) N'=2*NMAX + 7 BFSFUN 10

IF ( F .L T. qqo) GO TO 19 BFSFUN 11 WRITF(6.2000) BESFUN 12 2000 FOpmaT ( lOX , 33H M EXCEEDS 190. EXFCUTICF' APOBTFO) PESFUN 13 STOP PESFUN 14

1 5 iq FM1=1 0. F-2R BESFUN 15

EF = 0 .0 BESFUN 16

AL PHA-O. BESFUN 17 IF(M-(M/2)+7 ) 20,30.20 BESFUN 18

20 JT = 1 BESFUN 19

?0 GO TO 40 BESFUN 20

30 JT»-1 BESFUN 21 AC M 1 BESFUN 22 DO 160 K=1,M2 BESFUN 23

MK =M-K BESFUN 24 ?5 XFK=^K BESFUN 25 RMK=2.TXMK*EM1/ X-FM BESFUN 26 FM=FM1 BESFUN 27 Fm = RM'< BESFUN 28

8 J ( MK ) «=BMK BESFUN 29

30 JT=-JT BESFUN 30

S>=1 + JT BFSFUN 31 ALPHA= ALPHA+BMK + S BFSFUN 32 160 C CNTI N'lE BESFUN 33 PMK .2 .07 FM 1/X-FM BESFUN 34 35 BJ ( 1 ) «PMK BESFUN 35 4LPHA = 4L PH4 + R MK BESEUN 36 00 200 IN=1,NMAX BFSFUN 37

PJ( IN)=BJ( IN) /ALPHA BESFUN 38 200 C PNTI NUF BESFUN 39

1,0 PETUPN BFSFUN 40 END BESFUN 41

112
A. 1.12 SUBROUTINE

PLT120R(X, Y,XMAX,XMIN,YMAX,YMIN, LAST, ISYMBOL, NO, MOST)

PURPOSE :

To make a page plot of array Y versus array X.

ARGUMENTS:

X

YXMIN

Array containing abscissa values of the function to be plotted.

Array containing ordinate values of the function to be plotted.

Minimum abscissa value. XMAX Maximum abscissa value. YMIN Minimum ordinate value.

YMAX Maximum ordinate value.

LAST

ISYMBOL =

Number of points to be plotted.

A Hollerith variable containing the plotting symbol, e.g., to plot with the

symbol "X" ISYMBOL = IHX.

NO

MOST

Number of plot on page.

Total number of plots to be made on one page.

DISCUSSION:

This subroutine produces a "quick and dirty" plot of Y versus X on the page printer.

The size of the plotting area is 50 x 120 units. Multiple plots may be made on a single page. A page eject is performed before the first plot of a series is begun,

but no eject is performed after completion of a series. This allows a title to be

printed at the bottom of the plot. The subroutine uses inline function FLOAT.
1 <:[ippniiT INF PLT1?07(X. Y, Xy4Y, YMIN, ymax , YM IN , LAST, ISYMRQL, NQPLT170R 1 1, MOST) PLT170B 7

o viQn jcT FO n /4 /6P PL T170R 3 OTMFMptOM y(]), Y(1), 7Y(13), GPAPH(121» 51) PLT170R 4

s TWTcr^cs <;pAPMt CHLU-MN^t ftLANK, RHPOcp PT.T170R 5 P6TA (LTNP^ = '51). {COLUMNS = 1?1) PLT170R 6 KMAY = rntUMMS / 10 + 1 PLT170P 7 TC (MP ,NC. 1) GH to IQC PL T170R 8 YLAP = VM&Y PLT170R 9

s VMJM PLT170R 10 y^AD = YMAY Pt T170R 11

= YMJN' PLT170R 1? nnpppD = -[mt PLT170P 13 BLAMK = IM PLT170P 14

1 c MATPiy = roi UMMS * LINFS PLT170P 15

TF fMATPT^ .LT. 1) GO TO 120 PLT170R 16 on inn t = i, matrtv P1.T-120P 17

1 r>G r p ^ D H ( T ) = p l_ A N K PLT170P 18

1 70 mNTTNi'F PLT120R 19

70 TF (LIMPS .LT. 1 ) GO TO 140 PLT170R 70

DO 130 T = 1, LTMFS PLTl? OP 71 170 GPADi-'d, n = GPiPH( niDMNS. I) = PPBOFP PLT170R 77 140 c r M T r N II c 2T.Tl?flP 73

TF (rOLiiYMS ,LT. 1) (?n TO 160 PLTl 70R 74

7 5 nr 150 T = 1. mLl.iYMF PLT17CP 75

1 50 OIAPHiT, 7 f.) - IH. DL T170R 26 150 rnM TTM i|F PLT170R 27 YSC4LF = (XLAP - XSi'A) / (COLUMNS - 1 . ) PLT170P 28 YFOALF = (YLAS - YSMA’ / (LINFS - 1.) PLT170R 79

7 0 TF (i/«Ay ,lt. 1) GO TO irq PLT17CP 30

-in 17P V =: 1, KYXX PLT170P 31 170 7X(K) = 10. * FL0AT(B - 1) * XSCALF + XSMA PL T170R 3? IPO P

ni.' T T N 1 IF PLTl 20P 33 loo TF (LA7T .LT. 1) GO TO 750 PLT170R 34

7 S HQ ?40 T = 1. LAST PLTlr2GP 35

T'7 (X(T) .ST. XLAP .np, x(I) .LT. XSMA) GO TO 740 PLT170R 36

TF (Y(T) .C-T. y|_ap .op. Y(I) .LT. YSMA) GO TO 740 PLT170R 37

IX = (X(I) - XSHA) / XSGALF + 1.5 PLT120R 38

lY = (Y(T) - YSMA) / YSGALF + .5 PLT170R 39

U 0 TY = L TNFS - T

Y

PLT170R 40 SDADL|(TV. TY) =: TS YMROL PLT1-70R 41

7 A 0 rnMTTV'Ic PLT170R 42

7 = C .-ONTTMiir PLT170R 43

IF (NO .NF. Y.nPT) pFTUPN PLTl 20R 44

4"; DPI NT 1500 PLT170R 45 YFS = YLAP + YSOALP PLT170R 46

TF (LTMFS .LT. 11 GO TO ?70 OLT17-0P 47

rn f = x, LI'''PS PLT170 R 48 YEP = YFS - yscalf PLT170R 49 PPTMT 1510, YFS, (GPAPH(J, T), J = 1, COL LIMNS ) PLTl 70R 50 740 mNTTNUF PLT170R 51

7 70 r n M T I N ' J F PLTl 70R 52 PPJNT 1570 0LT170R 53 npiMT 16S0. 7X PLT120R 54

55 DF Y',1 P\i PLT170R 55

] 50C FOP.MAT (1H1,CX,74(5(-I....)1HI) OLT170R 56 1510 fodmat (lu , F R . 7 ,

]

X , 1 ? lA 1 ) PLT170R 57 1570 FFipmat (1H ,QX, 74 ( 5HI . . . . ) 1 hi ) PLT170P 58

1 5 7.0. FTTP-YAT : 1-H ,-2X, 13T-lX,FQdO ) Ptri-TDR 5^9

f-.o F N. n PLT170R 60
APPENDIX A. 2 SAMPLE PROGRAM INPUT AND OUTPUT

Illustrated below is a typical input card deck for program POMODL. The output

produced using this deck is reproduced on the following pages.

0. A4 64

'

1 1 "1 1 1 L J 1 1 I 1 1_ ! 1 1 1 1 1 . 1.

1 1 . 1 1 : 1 1 1 1 1 1 1 1 l 1 1 0. i

.

t 1 1

!’) , S . A i F ^ !

1 -- !

1 ~

1 1 1- ! L_

1

^ 1

1 S iO.

1 ( 1 .i 0. I. i. 4.1 ] 1 1 1 1 1 1 1 _ 1 1 i I >114 iJLATinN TEST rill.

i_ . 1 - 1

i

1

' 1 I 1 1 1 1 ! 1

1

1 1 1 STL C. F.

_ -I —- 1- -i_ -

- "<-39£?

_I ! I I 1 ! L__ l_ 1. . _ 1

1

1 1 t

1 1 2 Bri'6 in 9(Hfci 17 iiiin'isHH 18 ;0l2l 22 23 24 25’ 26 27 16 29 J'j|3l 33 3! aj'OS 37 13 .’O 40|4I 13 43 41 l;'46 47 43 49 OOlSi 52 53 T'J 55*CC 5/ 58 59 tolsi 62 63 64 SO'SC 07 a8 69 -'0|7I 72 73 74 75’ 76 77 78 79 80 {

1 1 1 ~ 1 1 1 1 1 1 1 1 3

4"^ 6 ^ C 3 1 ol 1 1 |'2 l'3 l'4 15 16 ( I8l920|2l 22232425262728293 01 31 3233343536373S39 10|4 4243444546-57*94960 51 '.'2 53 54 65 56 57 c'h 59 5'ol

1 1 , 1 1 1 1 1

' ' '

n n n S „ n nin n n n n'

I 2 3 4 5 |8 7 8 d 10 II n 1 1

1

11

22 , 2^2 2 2 2 2

I 2 3 4 5 6 7 6 9 10

3 3 3 3|3 3 3 3

44 44444,4

1 2 3 4 5 6 7 B S 10 5355 I 5555

1 2 3 4 5 C 7 B 9 10

I

11 12 13 14 15'IS 11 ia 19 20

1 1 11 1|1 11 1 1 2222222222

M 12 13 14 Is'lS 17 18 19 20

3 3

'3 3 3 3 44444^4444

11 12 13 14 15 16 17 19 19 20

5 5 5 5 5l5 5 5 5 5

6 6 6 6 6|6 6 6 6 6 6 6 6 6 6,6 6 6 6

1 2 3 4 5 6 7 8 9 10 7777777777

© I ©

Oit)

11 12 13 H 15 (C 17 IB 19 TO 7777777777

_ @ I ©

11 12 13 14 15 16 1/ 18 19 20

I 9 9 9 9 9'9 9 9 9 9 9 9 9 9'9 9 9 9 9

21 22 23 24 25 26 27 29 29 30 1111111111

22222 I 2 2 2 2

21 22 21 :i 2: 29 27 29 29 19

3 3 3 3|3 3 3 3 3

4 4 4 4 4,4 4 4 4

:i 23 23 24 25 26 27 28 29 30

5 5 5 5 s's 5 5 5 5

6 6 6 8 6^6 6 6 6 6

2' r: 23 24 25 26 27 29 29

7 777 I 7 777 ©

31 32 33 34 35'36 37 38 39 40 11 1 1 1,1 n n

2222222222

31 3 : 33 2 : -j La 37 33 39 40

3 3 3 3 3'3 3 3 3 3

41 42 43 44 45 46 47 48 49 SO

1 1 1 1 1,1 1 11 1 2222222222

41 43 44 15 4G 47 43 49 50

3 3 3 3 3'3 3 3 3

11 17 33 31 35 35 37 18 31 13

5 5 5 5 5,5 5 5 5 5

444414444

11 11 13 41 43 4C ,17 43 49 51

6686666666

31 12 33 34 35 38 37 li 30 30

7 7 7 7 7

I 7 7 7 7 7 7 0 1 ©

31 32 33 34 ' ;• ,7 ?3 L9 40

9 9 9 9 i9 9 9 9 9|9 9 9 9 9 I 9 9 9 9 9 | 9 9 9 9 9.9 9 9 9 9

2222222222

.. 5! 5 : 53 54 55 5' 57 58 55 60

3 3 3 3 3 3

I 3 3 3 3

5 5 5 5 515 5 5 5 5 5

6 8 6 6 6,6 6 6 6 6

41 4: 43 :i 4i 4b 4/ ;j 49 -.3 777777777

0 I 0

51 52 53 54 5S'5S 57 58 59 6C(61 62 63 64 6SlS6 67 68 69 7C 11 1

1

i.n 1 11 2222222222

- 61 6: 53 64 65 06 67 68 63 70 .

.

333333I333333

52 63 54 55 56 57 53 59 tO

5 5 5 515 5 5 5 5

6 B 6 6 8 6 6 6 6 6

I 51 :? 53 54 55 56 57 53 59 60

7 7 7 7 7'7 7 7 7 7 0 i_ _©

1111111111

4 .1 4 4|4 4 4 4 4 4

91 97 63 64 95 66 67 63 68 70

5 5 5 5 5,5 5 5 5 5

6 6 6 6 8,6 6 6 6 6

61 57 63 61 65 'eS 67 63 63 70 mil'll m

0 I 0

51 52 53 54 55 56 57 58 59 bO 61 S2 63 64 65 66 67 68 89 70 9 9 9 9 9'9 9 9 9 919 9 9 9 919 9 9 9 9

71 72 73 74 75'75 7? 7B 79 W

11 11 1.1 n 1

1

2222222222

71 72 73 74 75 76 77 78 79 SC 3333I 33333

4 4 4 4|4 4 4 4 4

71 72 73 74 75 76 77 73 79 80

5 5 5 5 5|5 5 5 5 5 6666666666

II 72 13 II li'ii 1) 18 79 a.'

7 7 7 7 7'7 7 7 7 7 0 I 0

71 72 73 74 75 76 77 78 79 80

9 9 9 9 9'9 9 9 9 9

GlOBc 501674
P/D

=

.5CC

DIAMETER

=

10. OC

WAVELENGTHS

FRACTIONAL

diameter

BLOCKING

•

.100

AXIAL

OFFOCUSING

=

0.000

WAVELENGTHS

BEYOND

FOCUS

FREQUENCY

»

A,

0000

GHZ.

OfVJO fMOtnOi-HOh'tT(<SJfnOO<VOsO«-*'J*OC yCcCOOO'T^COf^-Of^COCD^DGcnCi^O'T

oo'C7'0'cro'000"r*“»£>'t'«r\mcD(\im>r>^

I i I I I I

if-H»of^or^'^f^r^h-roCT'or^ooDr^r*-f-i<^>

I I I I «— «•— •»—*rH«-n

( III

<I CD

_j c cci Q. Z. d.

ocT-0'tr»Or~i'4’n''<'or'-c?‘roO'J''»xr'~-ir\h-rHO'rom'fi<)’0<\'’^r'“<M«—<\jO'Oir>r-ioL-o c ^D ocg«-Hf''t-4inNOOao‘no'rHr->4-n-, ^rc^oOf^<Mf-fo^o^-o•4:»^“«-^^^lOO'^''J•r'-^f^JO-ao O I «~*<M»r\ccmfnorHOf-»<4-<\jo»-»0'f\jOi-HfM(7'ooO‘»-''Oor'“'0'OC^fsjf-ir“m>i'ir\'C;cD

I I t t I I I t I I I I I I I I I I I I I 1 I I I I I

I I 1 I I I

NOf^'0'OcosJ’Of»'/tnsj-r'«-(\jflOC\jiri('uOc<^0*mcC'rooor'-cnrn»rc\j»--st'mcumr'.cDCMQ'f'~iri

mo'iriCco‘i--*Gcr'-»-^r-'a''C<^sOOcy-'^r'“0(?‘'CXi-Th-ar'*r-'si'r'->CfMr^O‘sra:)0<t'm(vi

O'lriinCT'Oi—irjr^f00'OO>DfvJr-tc\;mf\j»-io<vf»>>rr^-cjf-HrHi-Hi-*rHrHOOf-it-Hrvj,-ii-»f-x ooor'--ON»'<v *-H

O' o

• o

O'

</'> ir. D D

CJ C->

o»—'f'- '^cr fr'0'(\.'0coir<0'or'' o- 'J’Oc-rH«s.-r*Dip,<\-^o'irx'T<v->j'cr>if'Lr'Of^-‘>rooac oc mr'^

0'OOOoC^OO^C^-•£>'£)'OtAfcO'lfO<t^^-Hf^u tiilltll I I I I I

»comccoDmir\ir.

I

I I I f—< t—< f—t r-< i-H *—< i-H I f—I r—t (—i

i J

<I CD —

^ <I CT COO^<v«^aco'Oir\r'~'4"irsm*3-a for'-r-iT'^NriT'. rr, cc lpic-cOsJ-nJ r'-rrfviO'0'<t i-hcviit- C<c>»-<f'-r-<tn»co<\jr*-»-<foo'OmO'trrT'Oro‘r^r'“»-HOJOir\,—(CC'a.'4'ir>QC(\jiriir>-£,oaCf^r o I rHroiriacmrT!rH,-HO»H’^<vmf\jO(Mcro<voC'Of<'a‘'OOh-r'-ccOol^'C'Occou\

I I I I ,-HrNjrYOC\jfvjfvjcvjcoN3'r''rn<'Or'‘. cYDrn(^J^^.fnr'-f^^J-^r^«Dc^fn^of<. rrinDfr)m>3--4^

1 I I I I I I I I I i I I I I I I i I I

I I I I I

I I t I I t i I

^o^-^Ol^^cDoooDO'Clnoo^''-^T'^^JONrccIO'0^-(^iajmO''J‘00^-o<^>ou•<^'•~c^oo

triOir'h-r^O'O^OOh“ro»-*’4-f'^f'-rr;>D(M*-«'C'ir'afO'J-co‘^''0*«J'sj-<\i»-*cv'3''4.sOfnCiu^ omU'OOr-<fsjr^n'iooCO-.Cf^orv<^'C\jr-^*—

000'f^*0»J'<\J I-H

- ZD -I o s. u.

</0 51 3: to O O

<r z Q. OOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOO OOOOOOOOOCOOCOOOOOOCOOOOOOOOOOCOOOOOOOO

Cr-'Cvcn^f-ir^NCr'' o'. OOi cvf^<rir'<Df'' cnoc’

116
3P.00

.

01^9

-A6.98

37.98

.119

-39.30

95.86

\

I

^ f\j rsjr^s3-oocoif'»-H<rf^O'

ir. cpr><7^<\jt-40cr>ri—'i—itT'roirvh-cDODfn'rf'^fsj •4'inO'00cDsj-fO<VN£)<^»-t'0‘^Guin<v«^oc>t\,r*»

j I roif'*

I I I I I

a cninh-r~r-otr u‘if--»fOOOoo"'<\.r-o<\j<\jr“

*0 LTl 10. -O CC (\i fOfTjf<7<rifnro%3'

I t I I I 1 I

-tj OD O'

I i I I I i

lo- C7> iTi sj- sji I 1 t

<\J O O' CC <t <t cn till

f-< r- <\j O' o

nO oc h- sC C\J ^^£>>cr-<vmc:Nroccfnr^>c

oc>3't— I—tm.o««'0 ooc ooooooooo«-»

O f-Co

OfT-. Q^Osj-cju'3voacr' srccfT, fo.£;(\jQ.^ajOcv'

.-•rvjc iniT\Nr>oor''>rc:>c«-oor''mr>-N£>ocv tr OrnO'T»£>Omr'^mi-4>ij'^oa.>T o co (\, c- imiri'XC^cDOO'OOoof^'*' | |fvcnu''s£'

I I I I I I I I » t I I I I I I

cjvC^CMoa'J‘Lr\c')i-Hroh-i-HC“r'‘0*4'or'-f^^“ 0(\jO h-mio >j-r^co'i. oomoiOsa- cno-3-n''oc' mocc>oooO(\j«J’r^OfV)U''cr. 'i/CoOr^-OioiTiio

I I I I I t I I I I I I I I I I I I I I I

C- ^rCf«-or<3r«''rocc %or^OC'< 'Or^O>>rGL O >L/ r>-oco<\j»-<ocro'>3'mf\j.-'»-<«-HfvjfO'Tmif\ir'ir Of-ii-H,-i»-l,-«OOOOOOOOC ooc ooo

oocoooooooooooocooooc: OOCOOOOOOOOOOOOOOOOOO

o fv o'

-j- >r >j- >r

.j- IT ^ r- <r

sj- >3- nt >r •j- o o f-*

'j' m lo <N<^sj-ir. >oh-<r o c. ir'ioioirir'irir'iOsO

117
OCOCOOOC-OOOOOOOOCOOOOOOOOOOOOOOOOOCCCCOOOOOOOOOCOO + + ^ + + + + + + + + -f UjUjU>li-LLLl-U_UJtL»UJUJU U_UJUjU_'UJU'U-UJU-'LL)LL-LLU-'ULl-Ll-U,U-'U.UJU-U-.UJU_UJLLiU-LL/UjUJLL'LULL LL'U_'U.LlJli- c/Oooooocoot-<<vjmvru^'Or«-oco'OrHfv'r^»'^io*C'r^coo'Oi-(cofn*j“m>c^ xc3'OfHtvr0'j'u>»cr^cLa'O

O I I I I I I 1 I I I I I I I I I I t I I t I I I I t I I I I i I 1 I I I i I i

118

STMULATTHN

TFST

MH.

1

E-PLANE
I •-« t-H l-» O

1/1 (/^ o

I— -

z: 2 o

s: z t-' O O Ul^^ IT LU

r- Lu

• • Ol CO n“.

COOOOCOOOOOC'OOCOOOOCOOOOOCCOOOOOOOOOOOOCOOCCOOOOOO •' + + *»- + ‘f + -f-f-f + + + -*- + 4-+ + *f + -»*-»'4'-f+4- + + *«'‘«- + + + *«->f + + -f + + *«' + + + + + ++ + +

Ll_LLU-'IJ-LLlLLLLU-tLLLUU-UjU.Lt-LL-'U. UL'U’O.iUJU'UJLLLL.LL’U. Ll-li-'UJU U'LLlLljUJlJjLLU.U. U-lJjU.'Li-iUJU-UJJ-l-'LUliJLL'UJ COOCoC/OCOO^-lf^.m.^Jlx^^^-ccO‘O^-HfvJrl^^l^^^^"000•o*~'^J^^''3lf>'<^^^ccO'CI—i(vjrnvrtr''i-'r^a:'CT-c o I I ) I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

119

NFAP-FIELD

PARAMFTFPS-
OUTPUT

FIL'^

ID

—

PHYSICAL

OPTICS

S

I

^L

S

I

MUL

ATI

ON

TFST

NO.

1

0501791229

64

128

(

UJon

<rX

CL

h- (M O

O' lO O'

CO (\J O' O O vO r-4

O rH O' O' rH fO 00 '4’ GO CO O CO tH cn ^

^ o m r- m

in f-t >o C\J O O' tn in

O'

>0 «-H 00 >0

(\j CO o r*“ IT* »H <VJ

rH >0 fO (T) m fvj CO ^ O f''* O CO cn m >.> (O ^C\i (\J O CO O OJ m mO o

CO CO

(\J O'

r-i O'

fvj ^4

sT>o

CD

r- o

CD o

O O'

on >-4

oDotnocoor-^o

0<0f"'>00'0ao0 mo'«\jO'moo-4' mo <oom«-H<D,-«

nJ- C\J O'

sO rH CO n- ^ O' GO nj O'

O cvj CM m

00 m O' r~

(\» CM O 00 rH m m rH

cn HD

'J- 00 O f"- cn >r

f-H sT m r- O' <M m >D 00 «-H O' >1- r- o m

in m CM <M rH

HDCMm

CMom

rH rH O CO >0 rHinO'CO>4-

cn CM CM CM CM

ro rH in CD CO in in in

CO CM CO CM

CO O' o O' CM CM o nT O' o o o

CM vO f'- O' h- rH CM CM rH (O

cn fs- ^ fO CD CM CM CO £) rH

CM rH (O

rH CD C^> O CO CO <r m ^ CO

CO ro CO <o fO

00 in CM in CM

in m m in

CO CO ro CO CO m CO CO ^ CO or m vT CO sT in '* CO ro CO ro CO CO

rH CM CD CO rH r- CO

CO >r rH >0 CO CM

ro CO rH CM O O O O '5" O

rH f'“ ^ f'~ •>0 CM

CO rH CM CM

O'

O'

CM

inf-HCO'.J-«DOD'OCDCMf'-'Tr'-CMOCMmcOOmr^CMO'CMCM>Df-HrOCMOO'<7'00'OOCMCOrHHDCMCMO'<Mr''“mOrOmCM'OCMr'''«r 00'r'-COO'J'^rHCOOOOOCO,OrH^flOrHCOCMin'OCCrHcOOOO'r'~CDrHOrHCOr''*0'OOCOrHOO«OinCMCOrHCO'^rH'OCOOCO|

rHsO'0'<J'CO»DCMONOOCDinrHaOrHCMi-lrH<MCMinCOO'r''“0«J'CMf'~'^'COrHsOrHCOsrr'-CM'«»-Of'“0-COinCMCMfHrHCMrHCDrHinOOl rHr^Or^rHr^CMr^(MCMrHmrH^O'3CMmODOst^rcOCO'#>OCDO'«i^CM^-OrHO^''fMsrO'GO^D'J^OrO'^•'JOCDmCM-4COr^COrHj

rHrHrHCM'i'm,D00r'-r'-a0f''-O'rHO'f'''0Dr'~-r'-C0N0in’4’CMiHrHrH |

inomoinomomoinotnomomoincmomoinoinoinomomoinoinomcr'moinomoinoinoir'oin

cMinr'-0CMinr'-0cvmf''-0fMinC'-0CMinr^0CMmr'-0cMmf'~0cvinr'^0r'-incM0r'-incM0t''-incM0r'^mcM0r'^incvi0r'-

>OCMODinrHf''-rooocMCDinrHr-roO'OCMCC'inrHf'~c<'0^fMcrinr-*h“COOror^iHinoon.''Oocof-rHinoocMxf)OrOf'~rHmcD _ Hh-rOOOCMCDinrHr-rOO'OCMCC'in

>“rHrHoooo'0'0'ODccr'-r''-r'-'DHDsDmin>f«j'

rHrHrHrHrHilllllllllllllllllllllllllO I I I I I <o

Q.tu

Z"UJo

LU

onX

a.

'O CM rH O' 'O O

n- C f'- m m rH rH rH >3- in CM CO >0 O'

CM HD CO O l''“ m CM

HDHDOmHDrHr^

Hj CM CO -J" O' O '*» nT ro CO rH o CO O' tn in M- h- CO CM

ro O CO CO oc CM CM COmrHinHiO'sJ- roHDaD'J-HJorHT fO m IT. r~- ^ CO

o f" r- '4- O' o o

OT' O CO ^ OC sT rH m CM O O O CD rH

OC O o r'“ '4’ CM nT

a o h- m r- o CD r'~cD«4-r'''4’ccr'-

ro HJ m CM in o CO

ro CO ^ in CO ro

O'0 0'-4*r'-

rH *4- 00 >0 CO rH 00 O' O' O

^ CM '4' h- C

r- O CM o CD '4

' cM in r"- c a. CO

CM or CO oo o

O' iTi If'' m

O HD 4" a; i£) HD r" m fo

Cf)

CO

r-

•4- O'CM

>o

O'o

m4-

hrH CM ^0 rH O'

HD >0 m in -4 CM sj- m ^CM CO CM ro

CO <\|

cv in O cr> CM CMi

cot'“inf'^aoccrH 0‘O'Ccr cc cv nT m»cm,-(r^inrHCMHDrHincoco-4*

rH rH COCMrH rCiCOrOCO

CM m m r'- m ro nT ro CO ro to

'J" ro tn CO m in m m

CO o". CO to ^ r" tn tn m s3- CO >3"in’' re rci CO

CM >4 cv 00 m 4“ CO CO

CO CO CO CO

CO o o- in rH HD CM CO rH

rH cr CD r- in CD rH if ' rH in sC m

CM CO rH rH m

roCM

.L

'(

z

<3-J a.

a. z:

<i

I'M

J X

O'HDCOrrH CP cr vC rH

rH in f'^ OCj o

CO CM CM rH rH

m CO m rH ^

OCMrHO-Cf"“f'- o CO O' in o 4' m

rH O O O i-H O O

O O' m in o m O'

O' CO rH ,—i sf> '“I CO ^ 00 O' O' CM o H. rH O rH CM CM '4- rO' • CO o '4" O' O

H.I cr r- vj- rH r- O' O CO CM <v cc rH fo ro rH in 0O4-

•4- •4- HD CO m o CO

O' O C rH CO O CO

oc rH O' O' in o in

oc m rH CD I-H rH rH

<0'D4-'4 0 0''4-

rH O O' O O' r- rH

O' o rH CL' CC nT CM' tt rH in cc >r CL m

OcomO'O'inomif', 0' 0 •4•4

'

•4•^“ac>HD^or-HDrHrH(ocT^'-^'- CMfn>DO>OOfMO' 0'a>H^lf'’4

' rHCOmrHf"->3'CMCMrHOrHOO

r^rHCMM•m^ccn^~^'-ar. C'-O' O'r'-ccr^r'-crHDin'prvfrHrH

m o in o m

CM m n- o CM ^ CM oc m rH X rH I-H C O o

o in c m

m f'- o CM

1"“ CO O HD

O' O' O' GO

c in o m

in r- o

CM a* in I-H cc r- f'- f'-

o in. o m r- o

O' c

HD >C HD

m. o m O m

CM m t"- O CM

sCi CM CO If' I-H

If'I in '4' 4- >3

o in o m o

in. r- o CM m

f''- cr. CD/ HD Cs. <Oi m CO CM fMI

m o m o m

f'^ O CM m F"- a. in I-H n

rH rH rH O C

o m o m o

o h- m CM o

o CO n- I-H IT. O O C rH

m o IT' O ITf'- m CM onoo n. -hj o cTi rH CM’ CM CO roO in. o in o

m CM o N m

r- rH m 00 iM

to 4- s3- -4

- m

m o m O m

CM o 3- m CM

HD C CO n- f-H in HD sD HD f"-

c in O pu> CO

r- n- i-HrHrHrHrHlIllllllllllllllllllllllllO I I I I I

120

1
8250

.2007

203.5065

cD'roD<^f^cr triOiPt mr-^coo(7'<^J'^oOf-i r«-ro>0'^*^'OOC)®

QDir>*—•rO'OCOO'—t^irN

CMCD»OCD'C>J'0^»-^>^CO

-OIT'

Csjt—«C0i-^rHr^O»^^*^

iTi O iTv O iT' CM o lT« fvJ *0 o cn rCO (7- O' O' Go m o m o o h- IT' <M O

ir> CD <\j >0 O O *-• t-» <M

O 'O m fv: m ro

nJ- m

sO G >o OsC «-• f-t m

O' 'O cn m osj ix\ f^

,-1 sT <VJ nO 'O

ipi r- c r*

CVJ in sj- rH rH o ro Co CO

cv>f in'3'C\j(7‘tHODrv *— or~n.rcro>rmin'C'£'>r‘

(V CJ

.j-,-Hmroinh-00'CO‘»-<

O'T'-'Cs.Or-^'i-CCnf-HCC

oir>cr ^OOOr-i»-tf-H(MroC'OJ

o o IT' o

m fM o ic> Cvj'^Of<'C“

cc cr o a cr

m. O m o

CM o

»—I m CO CM O O o *-H

m o

CM O

'O c

f-H CM

121
I -fr »-H <\J

-'O'

OOOOCOOOOOCOOOOOOOCOOOOOOOOOCOC/OCOOOOOOOOOOOOcCOOOf-i + + + + + + + + + + + + + + + +

liia.'U^U_UjLLU.'UJU LL.UjLLiUJLL LL’UJU-LLajLUU.Ll_LJJU-U-LLU-UJajLLU.U-LUU.tU-ajU ULUujUJUjIJ-'U.IJJUjUJ11:LLLL.'U_ occ^^(sjOcc'4;>^fNaooc>^j’4-rooou«c->4‘<vocc.o<rfv.caiNO>3- c\uOa sC'»c\jooc^>j-<voa:j^ ^t^o000oc,0'

»-'0‘0'0 0‘0‘ccoooDODOC’h-r^r-r-h->C'^'C't>»oifNir«iru'‘m-4 {^rnrnm(\jrMC\jr\.{\JrH«-t»-Hi-ti-'CC«4)*^<vjX

I

122

SIMULATTHN

test

Nn,

1

Y-Z

PLANE

AMPLITUDE
o

o

(M

j

I

-H-

•tt-

•»

•«-

ft-

**

Jfita-

*•

*

•»

*

«•

•»*

*

•N-

«

cn

K-. >j-

(7O

I

-«•

-»

*

*

c^cnfnrnrnr^^rrirofr, <T, cof»i(^frfnrnfTi(T', (No<\.fvjcvjCV)CM(\jC\JOiJ t-n OOCOOOOOOOOCOOCOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOf- I + + + + 4-4-«*- + + -f4-*f++-»-+-f + -*'+++4-+4- + + + + '*' + + + -f++ + + + + -»''+- + + + «*-++-*'‘«“l

IX IX U- U- tt- U_' U- U- Ui U-' U U- U- LU U- LL- LU LL. LU LL IX' LL‘ IX' ix: U. U. U-l UJ U-' U.J LL' UJ U.’ UJ Ui U. LL' U-' U' LU U- U-- LL' li U. IX’ LU UJ U. X-'iT. insTrrtVfvjt—'OcO‘Ctf^r^^C'U^»^s^fnc\J(^Jt-^OC^C^cc^^f^'£'U^'^•sr(^C<J(^J^-^o^^'OC^^ol^^(Xo<^'00fV'•Jc\JO'

123

i

SIMULATION

TFST

no,

1

Y-Z

PLANE

PHASE
o

o

«\l

-H-

*

*

*

»

*

*

a-

•fr

*

*

*

C.OOOOCOOOOCOOOCOOOOOOOOCOOC OOOOOOOOCCOOOOOCCOC'0 000<-H I 4--*- + -*--*- + -f-t- + + + -*- + + + 4- + + 4'+-*' + -*-+ + + -»-4-*»- + + + + + '»- + + **--4- + ++ + + + -*-'+-f + -f| LLlXaiXUJUJLLU_U.UJlijUJLLU_UjU.LULiJU-U LL'LLiLl-'UjU:LJ.ll^U-LJJLJjlJ-'U_lJ>U.U-LL’LLU-LLU'U'U-Ll-’U-t-l-'UJU.U-UjLrL^ ocLON3-(\;O<x«£.>jc>JCcC'‘f>3*rjoa:->c«j-rv-O0c>£j'j-<V'O<X''£)>4-<vj<->cL>>£?««3’(\joa.Nt<rcNjOa:'<j'j00OOOOO0‘

SIMULATTON

TEST

NO.

1

X-Z

PLANE

AMPLITUDE
• *

C’ r~ n cn c^i H" r'j rr. rn ro fo O' fTi rr. rr, rr, f»-, m ri n~i m m rr* rf) n COOOOOOOOOOOOOOOOOOOOOOOCO + + + + + + + + + +

LL LL U UJ UJ U- LU LL U LU QJ tl- IX' U- U IXJ IXJ LIJ !Xi U. U-' LX’ U. Lil U' LX' '^?ir\iT'sj-on<\^<\.f-HCO C^aJ^-^“ NCir\sj''j cncv<\i-»ocr o <Xi O' m m 0*1 O'' m o*j O' fo oI c\j cvj oo Cu Oo fM CO <\j <\i rvj <\i I—' rH T-i

o^

>C o^o^o^f^o o^o^fTio^o^o". ojojrooocvj(vrvc\.(\j(\j<\j(\j»--o i-h COOOCCCOOOOOCOOOOOOOCOOCt-H I 4--l- + + -f4--f4-4- + + ++ 4- + *X'*- + 4-*f-»--«- + + J U' U LX ix; IX LL' U' U- LX’ U LL) U IXJ IX U' U' IXJ LL IX LX' IX IX LX' IX' LL >j-o'i<v(\jrHO'3'«oC'<\,ma c.fOsDC7'<vj'sr(v<7-

LT)^ O^ fVI Oj f-H f'- OJ

SIHULATTON

test

no.

1

X-Z

PLANE

PHASE
APPENDIX B. CUPLNF - CALCULATION OF COUPLING BETWEEN ANTENNAS

This appendix includes detailed documentation of the program which calculates coupling

between two antennas given their far-field patterns. This program, as presented here, uses only a single component of the far field for each antenna, and is thus applicable only for

linearly polarized antennas oriented with the major components of their polarization

vectors lying in a common plane. The inclusion of the cross component in the calculation

is not a difficult extension to the program. Each subroutine is individually documented

except for those which are also used in PROGRAM POMODL and are discussed in Appendix A. The final section of the appendix includes a sample input deck and a sample program

output.

B.l GENERAL OVERVIEW OF COMPUTER PROGRAM

The program CUPLNF and its associated subroutines are described in detail in the following

subsections. The flow chart below is presented in order to give the reader a general

overview of the program package.

126
127
B.1.1 PROGRAM CUPLNF (INPUT, OUTPUT, TAPE 1, TAPE 3, ..., TAPE 8)

PURPOSE :

To compute and plot the mutual coupling between a transmitting and receiving antenna

of arbitrary orientations and separation from the given complex far-electric-field

pattern of each antenna.

METHOD :

Evaluate eg (32) of the main text along x and y perpendicular lines or cuts, using the

fast Fourier transformation.

GENERAL DISCUSSION:

The main program divides conveniently into six subsections which list sequentially as

fol 1 ows:

1) General information about program,

2) Specification statements,

3) Definition and reading of input data,

4) Limits of integration and number of integration points,

5) Filling of the input matrices (AX and AY) to the FFT FOURT, and

6) Printout and plotting.

General Information about the Program

This subsection is a self-explanatory aid providing the program user with specific

definitions of the main input parameters required by the program, as well as with a general feel for what the program does.

Specification Statements

This subsection merely dimensions, equivalences, and comments the appropriate arrays, and

declares the necessary complex and integer variables.

Definition and Reading of Input Data

This subsection defines and reads from data cards the input variable parameters to the

program. A list of the required data cards follows:

Card 1 Col. 1-40 An alphanumeric identifier, usually the name and telephone

extension of the person submitting the job.

128
Card 2 Col. 1-80 An alphanumeric identifier specifying the particular case being

studied.

Except where specifically noted all data on the following cards must have the decimal

point explicitly specified.

Card 3 Col. 1-10 Frequency of operation in GHz.

Col. 11-20 Distance between origins of the reference coordinates of the

two antennas in meters.

Col. 21-30 x-spacing corresponding to the near-field spacing for the

transmitting" antenna.

Col. 31-40 y-spacing corresponding to the near-field spacing for the

transmitting antenna.

Col. 41-50 x-spacing correspondi ng to the near-field spacing for the

receiving antenna.

Col. 51-60 y-spacing correspondi ng to the near-field spacing for the

receiving antenna.

Col., 61-70 Ratio of transmitting to receiving antenna feed mode

admittances.

Col.. 71 Set equal to 1 if spectrum rather than far-field pattern is

gi ven

Card 4 Col., 1-10 Maximum value for plot. If this field is left blank, the scale

is chosen to fill the plot page.

The remaining data on this card are integer data, and must be right

justified in the field provided.

Col.. 11-15 Lower index in the increment loop.

(Set equal to 1 if field is blank).

Col

,

. 16-20 Upper index in the increment loop.

(Set equal to 1 if field is blank).

Col.. 21-25 Lower index in the integration limits loop.

(Set equal to 1 if field is blank).

Col. 26-30 Upper index in the integration limits loop.

(Set equal to 1 if field is blank).

129
Card 5

Card 6

Card 7

Col. 1-10 Transmitting antenna gain in dB.

Col. 11-20 Magnitude of transmitting antenna far-field pattern on boresight. (Allows for normalization of far-field pattern).

Col. 21-30 Radius of transmitting antenna in meters.

Col. 31-40 PHI

I Col. 41-50 THETA | - Eulerian angles of reoriented transmit antenna.

Col. 51-60 PSI

I

Col. 61-65 NROWT - Number of rows of data in transmit antenna pattern.

(Integer data right justified in field).

Col. 66-70 NCOLT - Number of columns of data in transmit antenna

pattern. (Integer data right justified in field).

Col. 71-80 Transmit antenna pattern file identifier.

Col. 1-10 Receive antenna gain in dB.

Col. 11-20 Magnitude of receive antenna far-field pattern on (Allows for normalization of far-field pattern).

boresi ght

.

Col. 21-30 Radius of receive antenna in meters.

Col. 31-40

Col. 41-50

Col. 51-60

PHIP

1 THETHP

1 - Eulerian angles of reoriented receive

PSIP

1

antenna

.

Col. 61-65 NROWR - Number of rows of data in receive antenna

(Integer data right justified in field).

pattern.

Col. 66-70 NCOLT - Number of columns of data in receive antenna pattern

(Integer data right justified in field.)

Col. 71-80 Receive antenna pattern file identifier.

Col. 1-20 GAMT - Transmit antenna reflection coefficient.

(Real part 1-10, imaginary part 11-20).

Col. 21-40 GAMR - Receive antenna reflection coefficient.

(Real part 21-30, imaginary part 31-40).

130
Col. 41-60 GAML - Receiving load reflection coefficient.

(Real part 41-50, imaginary part 51-60).

Limits of Integration and Number of Integration Points

In the analysis of the main text, it is shown that only the far-field pattern within the

sheaf of angles mutually subtended by the two antennas is necessary to accurately compute

the coupling between the antennas. These reduced limits of integration artifically

bandlimit the coupling and thus increase the integration increments required by the

sampling theorem. In all, the number of integration points is drastically reduced. This

subsection of CUPLNF computes a maximum solid angle mutually subtended by the antenna and

translates this information into specific limits of integration for and ky. In addition, the integration increments and subsequently the number of integration points in

the X and y directions are also obtained in this subsection.

Filling of the Input Matrices (AX and AY) to the FFT FOURT

Now that the previous subsection has computed the points and limits of integration, the

far-field patterns of each antenna must be retrieved from input files at the specified

points of integration. These far-field arrays are inserted as input into the FFT FOURT in

order to compute the coupling quotient from eq (32) of the main text. The subroutine

FINDFF, documented separately, takes the required array of far-field integration points

(directions) searches the input files containing the far field for the value of far field

in the required directions, and outputs the array of far-field values to be used eventually

by FOURT.

Before calling FINDFF, the program must calculate the far-field directions corresponding to the integration variables k^, ky in the integral (summation) of eq (32). This is accomplished through the subroutine ANGLGEN, which is documented separately.

After the far-fields are obtained from FINDFF, their dot product must be determined as eq (36) demonstrates explicitly. This scalar product is accomplished through the short

subroutine VECTGEN, which has been documented separately from CUPLNF.

The dot products of the far-fields found from VECTGEN are appropriately placed in two

arrays, AX and AY, from which the FFT subroutine FOURT computes the near-field coupling

quotient along two mutually perpendicular x and y cuts.

131
Printout and Plotting

This subsection simply prints and plots the magnitude of the coupling quotient (i.e.,

coupling loss ratio) along the mutually orthogonal x and y cuts which lie normal to the

line of separation. Because of the possible lack of a common plotting system, curves are made using a general purpose page printer subroutine (PLT120R). The coupling is plotted in the X and y directions over a distance approximately equal to twice the sum of the

diameters of the two antennas.

SYMBOL DICTIONARY :

Variables (in alphabetical order)

ABL

ACLCUT

AX, AY

(A1,A2), (B1,B2)=

BFAC

CEE

COEF

CUPLDB

C1,C2

DATA

DIAMR, (DIAMT)

DIAMSUM

DKOK

DLX,DLY

DLXR,DLXT

DLYR,DLYT

Intermediate variable for defining the range of k^/k and ky/k. The

range of ABL beyond XKLIM is zero filled.

A real array used to store the magnitude of the coupling quotient along

XO and YO perpendicular axes or cutter.

Complex arrays used to store first the coupling far-field product then

the coupling quotient along XO and YO cuts, respectively.

The limits of integration of k^/k and ky/k, respectively.

Variable which adjusts the integration increments, and should be

approximately 1 or 2; making BFAC larger tests whether convergence has

been reached.

Speed of light in gigameters per second = .2997925.

The coefficient of the summation in eq (32) of the main text with the

exponential factor omitted . Coupling quotient for two antennas expressed in dB.

The k^/k and ky/k increments, respectively, i.e., (ai+a 2 )/Ni

and (b]^+b 2 )/N2 in eq (32).

Array containing far-field pattern of transmitting or receiving antenna,

used in SUBROUTINE FINDFF and included here for storage allocation

purposes only.

Twice the larger of RADR (RADT) or WAVELGTH of the receiving

(transmitting) antenna.

DIAMR plus DIAMT.

The approximate kj^/k and ky/k integration increments, i.e.,

N1 == (A1+A2)/DK0K and N2 = (B1+B2)/DK0K.

Subsequent labels for (DLXR and DLXT), (DLYR and DLYT).

X - increment which corresponds to the k^ increment of the receiving and

transmitting antenna, respectively.

y

- increment which corresponds to the k^ increment of the receiving and

transmitting antenna, respectively.

132
DPMI, DPHIP, DPSI

DTHETA, DTHETAP

DTR DX,DY

ETOER

FDOTFP

FFRMX,FFTMX

FMM

FREQ

FX,FY,FZ

FXP,FYP,FZP

FXR,FYR,FZR

FXT,FYT,FZT

GAINR,GAINT

GAML,GAMR,GAMT =

HEAD

IBFAC

ID(I)

IDAYHRR.IDAYHRT =

ISCL

ISPECT

IXLIM J1,J2

LMl, M2

NBF1,NBF2

NCOLR,NCOLT

, DPSIP,

The Eulerian angles PHI, PHIP DTHETAP expressed in degrees rather

than radians.

Degree to radian conversion factor.

The increments in XO and YO, respectively, over which the coupling

quotient is computed by the FFT. Ratio of characteristic admittance of the transmitting antenna feed mode

to the characteri Stic admittance of the receiving antenna feed mode.

The dot product of the complex far-electric-field pattern of the two

antennas

.

Magnitude of unnormalized far-field pattern at THETA = 0 for the

receiving and transmitting antennas, respectively.

The mismatch factor, 1/(1-GAMR»GAML) in the right, receiving antenna.

Frequency in Hz. The complex rectangular components of far electric field in the preferred

coordinate system fixed in the left, transmitting antenna. (FX and FY are also used later in the program as intermediate complex variables.)

The complex rectangular components of far electric field in the preferred

coordinate system fixed in the right, receiving antenna.

The complex rectangular components of the far electric field of the right

receiving antenna in its mutual coupling coordinate system.

The complex recangular components of the far electric field of the left,

transmitting antenna in its mutual coupling coordinate system.

Gain in dB of receiving and transmitting antennas, respectively.

Reflection coefficient of receiving load, receiving antenna, and

transmitting antenna, respectively.

Integer array identifier for case under study.

Loop index for varying BFAC.

Integer array (with index I) identifier for programmer's name and one extension.

File identifier for receiving and transmitting antenna data,

respecti vely

.

Integer indexer for conditional statements.

Spectrum flag. Set equal to 1 if spectrum rather than far-field

patterns specified.

Loop index for varying XLIM.

Dummy loop indices used in the filling of the AX, AY coupling product

arrays, and later in the printout statements.

Dummy index for write and read statements.

Dummy loop indices used in the multiplication of the sum in eq (32) by

the preceeding factors.

Begin and end index for range of BFAC.

Number of columns of data in receive and transmit patterns,

respectively.

133
NN1.NN2

NROWR.NROWT

NRX2R.NRX2T

NX, NY NXL1.NXL2

N1,N2

N1MAX,N2MAX

N1MIN,N2MIN

N10,N20

N11,N22

PHI, THETA, PSI PHIP,THETAP,

PSIP

PHIR,THETAR

PHIT,THETAT

PIR.T

RADR,RADT

RCUT

RG,TG

SUM2

TKOKSQ

TSUM21

WAVLGTH

WORK

X

= Integer arrays of dimension (1) used in call to EFT subroutine FOURT and

equal to N1 and N2, respectively.

= Number of rows of data in receive and transmit pattern, respectively.

= NROWR and NROWT x 2.

= Four times N1 and N2, respectively.

= Begin and end index for range of XLIM.

= Integers equal to the number of and ky integration points,

respectively.

= Integers determining maximum of the x and y range, respectively, over which the coupling quotient is plotted.

= Integers determining the minimum of the x and y range over which the

coupling quotient is plotted.

= Intermediate integers used to define (N1MIN,N1MAX) and (N2MIN,N2MAX)

,

respecti vely.

= Number of points in the x and y range, respectively, over which the

coupling quotient is plotted.

= Eulerian angles of the left transmitting antenna as shown in figure 2.

= Eulerian angles of the right, receiving antenna as shown in figure 3.

= Spherical angles in the preferred coordinate system fixed in the right,

receiving antenna, corresponding to the direction k^/k, ky/k

(0p,6p of eqs (13) and shown in figure 3).

= Spherical angles in the preferred coordinate system fixed in the left,

transmitting antenna, corresponding to the direction k^/k, ky/k

(0/\j 0A of eqs (13) and shown in figure 2).

= TT = 3.14159...

= Complex array containing the spherical angle coordinates for the

coordinate system fixed in the receiving and transmitting antenna,

respecti vely.

= Radii of the smallest sphere circumscribing the right receiving and left

transmitting antenna, from their respective origins.

= Maximum ordinate value for plots. If RCUT equals 0, plot is

sel f-scal ed.

= Input reflection mismatch factor for receiving and transmitting antenna,

respecti vely.

= Dummy summation variables used in the filling of the AX, AY matrices.

= Magnitude squared of the transverse part of the propagation vector.

= Summation variable used to compute the coupling quotient at XO = 0,

XO = 0 by summing directly without the use of the FFT (as a check).

= Wavelength in meters.

= Complex array required only by FFT subroutine FOURT.

= Array containing the abscissa values for plots.

134
XDUM

XKXKLIM

XKMAX

XKMIN

XKXOK,XKYOK

XLIM

XMAX.XMIN

XNX,XNY,XNZ

XO,YO,ZO

File Names

= Dummy variable used in MINMAX when this subroutine is used with a one dimensional array.

= 2tt/X.

= Real variable which limits the range of and ky/k integration

when its value is less than XKMAX.

= An upper bound (less than 1.0) on XKLIM; except for very close antennas

XKLIM will usually be less than XKMAX anyway.

= Sum of the diameters of the two antennas divided by their separation

distance; this variable is approximately proportional to the mutual

angle subtended by the two antennas; XKLIM = XKMIN times XLIM when this

product is less than XKMAX.

= k^/k and ky/k, respectively.

= Intermediate variable used for adjusting XKLIM; making XLIM larger

tests whether a wide enough spectrum has been included.

= Maximum and minimum abscissa values for plots.

= Variable used for incrementing k^/k, ky/k, and y/k, respectively.

= X, Y, Z coordinates of the origin of the right receiving antenna in the

mutual coupling coordinate system of the left transmitting antenna;

specifically ZO is the separation distance (d in eq (32)) between

antennas

.

INPUT, OUTPUT, TAPE 1, TAPE 3, ..., TAPE 8

Subroutines Not Within FORTRAN Library (in alphabetical order)

ANGLGEN (Documented below)

FINDFF (Documented below)

FOURT (Standrd FFT subroutine with documentation within its own comment cards)

MINMAX (Documented below)

PLT120R (Page printer subroutine)

VECTGEN (Documented below)

135
Functions Inline or within Computer Library (in alphabetical order)

AMAX1(X,Y)

AMIN1(X,Y)

ATAN(X)

CABS(C)

CEXP(C)

CMPLX(X,Y)

EOF EXIT

SQRT(X)

= Maximum of X and Y.

= Minimum of X and Y.

= Angle between -tt/2 and tt/2 whose tangent is X.

= Absolute value of complex number C. = Complex exponential of complex number C.

= Complex number X + iY.

= E0F(End of File).

= (Terminates execution and returns control to operating system.)

= Square root of X.

List of Complex Quantities

AX, AY, COEF, ETA, FDOTFP, FMM, FX, FXP, FXR, FXT, FY, FYP, FYR, FYT, FZ, FZP, FZR, FZT,

R, SUM2, T, TSUM21, WORK, CEXP, CMPLX.

COMMON BLOCKS:

The labeled common in CUPLNF is described below with a list of routines in which it is used. The variables are defined in the symbol dictionary

COMMON /FAR/ Nl, N2, NX, NY, DLX, DLY, XK, ISPECT

Routines using /FAR/: CUPLNF, FINDFF.

136
1

5

10

1 5

?0

??

30

35

A5

50

55

60

^-5

70

75

12

3

56

7

89 10

1112

1314

1516

17

18

1920

21

22

2324

252627

2829

30

31

32

33

343536

37

3839

40

41

42

4344

4546

47

48

49

5051

5253

545556

57

585960

61

6 2 63

64

6566

676869

70

7172

7374

75

7677

PP00P4"! CUPLNF (INPUT, OUTPUT, TAPPl, TAPF3, TAPE4, T4PF5,

1 T4Pe6, TAPF7, TAPfB, T A P F 6 0 = I N P UT

)

GFNCPAL TNFOPMATION APOUT PPGGPAH

TMIS pphgpa'i computes the coupling ouotifnt between a TPANSMITTING ANTENNA ON THE LEFT AND A RECEIVING ANTENNA ON THE RIGHT OF ARBITRARY RELATIVE ORIENTATION ANO SEPARATION,

from thf given cqnplex far-field pattern he each antenna.

THE COUPLING OLIUTTENT IS COMPUTED ALONG XO AND YO PERPENDICULAR

1 INFS OP CUTS

.

AX, AY, AND WORK SHOULD BE DIMENSIONED ,GE. THE LARGER QF (N1,N2).

ACLCUT AND X SHOULD RE DIMENSTONEO AT LEAST 2 GREATER THAN THE

(_ aR GFP OF ( N 1 , N? ) . FYT, FYP, EXT, FZR, P, AND T SHOULD BE DIMENSIONED .GE. N2. DATA SHOULD BE LARGE ENOUGH TO CONTAIN ALL OF THE INPUT FAR-FIELD

data for either ANTENNA'IE. .GF. 2+NROWT+NCOLT OR 2YNRDWR+NC0LR.

PHI, theta, PSI APE THE EULEPIAN ANGLES OF THE RFORIENTED

TRANSMITTING AXES WITH PESPFCT TO THE AXES FIXED IN THE TRANSMITTING antenna. phIO.THFTAP, PSIP APE THE EULEPIAN ANGLES OF THE REORIENTED

RFCFIVING AXES WITH RESPECT TO THE AXES FIXED IN THE RECEIVING ANTENNA.

(XO,Y0,70) ARE THE CCOPDINATFS GF THE ORIGIN QF THE RECEIVING

ANTFNNA IN THE PEOPIENTED RECTANGULAR SYSTEM OF THE TRANSMITTING

ANTFNNA . Tmf pfopifntED CORDINATF systems of faCH ANTFNNA ARE THE COMMON

MUTUAL COUPLING coordinate SYSTEMS OF THE ANTENNAS.

THF coordinate SYSTEM FIXED IN FACH ANTENNA IS THE xpreFFRREDm

SYSTEM IN WHICH THE FAP-FIFLOS OF EACH ANTENNA ARE GIVEN.

ZC must be specif IFD, BUT THE RANGE OF XO AND YO ARE DETERMINED

tmplicttly by the reouirements of the algorithm FOUPT. paDT=RADIUS of smallest sphere which CIRCUMSCPIBFS thf transmitting ANTFNNA FROM ITS ORIGIN,

RA0R=RAPIUS of smallest sphere which circumscribes THE RECEIVING ANTFNNA FROM ITS ORIGIN,

niAMT = TWICE -THE LARGER OF R A DT OP WAVLGTH

DIAMP = twice THR LAPGER of RADR or WAVLGTH

BFAC ADJUSTS THE INTEGRATION INCREMENTS, ANO SHOULD BE approximately 1 OR- 2. MAKING BE4C LARGER TESTS WHETHER CONVERGENCE HAS BFEN RRACHEO.

XLIM ADJUSTS THF NONZERO-FILL PORTION OF THE INTEGRATION RANGE,

AND- SHOULD BE- APPROX, 1 OR 2, MAKING XLIM LARGER TESTS WHETHER

A WTDF ENOUGH INTEGRA! ION RANGE HAS BEEN INCLUDED. INCREASE NG XLIM ALSO OFCREASES the INTEGRATION INCREMENTS PROPORTIONATELY

T-0 -PPE-VEN-T ALIASING.

A1,A2,B1,82 DEFINE THE TOTAKWITH Z E R C- F I L L ) I NT EGR AT I ON RANGES (KX/K-FROM -A1 TO approx ,A2 ) AND (KY/K FROM -PI TO APPROX. B2),

IN INCRFmfntS of (A1+A2)/N1 or (R1+B2)/N2 APPOX. equal TO OKOK.

DKOK = WAVLGTH/ ( 2+ ( DI AMT + 0 I A MR ) *B F AC * XL I M ) .

IF SOPT( (KX/K )+ + 2+(KY/K)5Y2) IS .GE. XKLIM THE SPECTRUM

TS SET equal TO ZERC. ( A P PR E C I AL B LE ZERO FILLING IS AN OPTION

DE-SIGNED -TO -ALLOW FINER INCREMFNTS DX AND OY AT WHICH THE COUPLING QUOTIENT IS COMPUTED BY THF EFT.) XKLIM must be equal to or less THAN 1 BECAUSE

CUPLNF

CUPLNF

CUPLN F CUPL NF CUPLNF CUPLNF

CUPLNF CUPLNF

CUPLNF CUPLNF

CUPLNF CUPLNF

CUPLNF

CUPLNF

CUPLNF

CUPLNF

CUPLNF

CUPLNF CUPLNF

CUPLNF

CUPLNF

CUPLNF

CUPLNF CUPLNF

CUPLNF

CUPLNF

CUPLNF CUPLNF

CUPLNF CUPLNF

CUPLNF

CUPLNF

CUPLNF CUPLNF CUPLNF

CUPLNF CUPLNF

CUPL NF CUPLNF

CUPLNF

CUPLNF CUPLNF

CUPLNF

CUPLNF

CUPLNF

CUPLNF

CUPLNF

CUPLNF CUPLNF

CUPLNF

CUPLNF CUPLNF CUPLNF

CUPLNF

CUPLNF

CUPLNF

CUPLNF CUPLNF

CUPLNF

CUPLNF CUPLNF

rG

CC

C

C

CG

C

rc

c

c

c

cc

thf program NEGLECTS THE EVANESCENT MODES. IN ORDER NOT TO GET CUPLNF

TOXT CLOSE -TO THG -1//SAMMA SINGULARITY, IT IS SAFER TO CHOOSE XKLIM CUPLNF

NO lAPGFR than XKMAX= ABOUT .P. CUPLNF

XLIM adjusts XKLIM. IF AN ACCURATE COUPLING QUOTIENT IS CUPLNF

J3.F0UIRED only FO-R SMALL VALUES OF XO/((DIAMT + 0 I AMR ) B F AC *X L I M ) CUPLNF

AND YO/((OIAMT + 0 T A M R ) t B F A C* X L I M ) , XLIM NEED NOT BE MORE THAN CUPLNF

1 OR 2. IF ACCURACY IS DFSIRED FOR LARGER XO AND YO AS WELL, CUPLNF

XLIM 8HDULD BE MADE CORRESPONDINGLY LARGER, AS MENTIONED ABOVE, CUPLNF

MAKING XLIM LARGER TESTS WHETHER A WIDE ENOUGH SPECTRUM CUPLNF

HAS BFBN INCLUDED. CUPLNF

CUPLNF

THE XO AND YO INCREMENTS ARE D X * W A V L GTH / ( A 1 + A 2 ) AND CUPLNF

OY = WAVLGTH/ ( B1+B2 ) . CUPLNF

THE RAJJGF OF BOTH XO AND YO 18 GIVFN A P P R 0 X IM A T F L Y BY CUPLNF -(DIAMT ni AMR) *BFAC*XLIM TO -KDIAMT + D I A M P ) + B F A C * X L I M , BUT ONLYCUPLNF

-(OIAMT * DIAMP) TO +(DIAMT + DIAMR) APPROXIMATELY IS PRINTED AND CUPLNF

137
r OLOTTcn (WMFN vlIM*pfaC TF GOF&TCP THAN OP FOUAL TO 1). CUPLNF 78

r OFF IF THF SPFFD OF LIGHT IN GIGAMFTFPS PFR SECOND. CUPLNF 79

PO r PMM IS THF NIS"ATCH FACTOR FOP THE RECEIVING ANTENNA, CUPLNF 80

C CUPLNF 81

1* CUPLNF 82

r- SPETIFICATION STATFMFNTS CUPL NF 83

r CUPLNF 8^

p •) rOMPLFT AX(IOOO), AYdCOO). T(IOOO), R(IOOO), WOPK(IOOO) CUPLNF 85 CPHPLFX FYT(IOOO). FYP(inOO)» F7T(1CC0)» FZP(ICOO) CUPLNF 86 gonplfy fn'', chef, gamt, gamp, gaml CUPLNF 87

CCIMPLFY FXT, FXR CUPLNF 88 rpMPLPX FX.FY.FT CUPLNF 89 QC POMFLE^ FXP,FYP,E7P CUPLNF 90 complex footep CUPLNF 91 PPM P| F X TSUMp I , SUM2 CUPLNF 92

c CUPLNF 93 niMFNSION NNl(l), NNP(l). HEAD(8). ID(^) CUPLNF 9^r 05 DIMFNSIHN ACLCUK 1010) . X(IOIO), DATA(8192) CUPLNF 95

r CUPLNF 96 Intfgfp head CUPLNF 97

C CUPLNF 98 EOUIVALENCF (T,EZT), (P,FZR)» ( AC L CD T , F YT ) . (WOPK>FYR) CUPLNF 99 ion FaUIVALFNCF (AX, DATAd)), (AY, 0ATA(25OO)), (X, T) CUPLNF 100

r CUPLNF 101 CpmmON /far/ Ml, N2, NX, NY, OLX. DLY, XK, ISPFCT CUPLNF 102

C CUPLNF 103

c CUPLNF 10^

105 r - offinition and rfading of input data CUPLNF 105

r CUPLNF 106

c- ID AND HFAD ARF AL°HANUMFPIC IDENTIFIERS CUPLNF 107

C- ID IS PPOGRAMMfPS NAMF and phone EXTENSION CUPLNF 108

c- HEAD IS THF IDENTIFIER FOR THE CASF UNDER STUDY CUPLNF 109

iin 0 CUPLNF 110

f- FRFO = FPEOUFNCY OF OPERATION IN GHZ. CUPLNF 111

ZO = SEPARATION RETUEFN ANTFNNA REFERENCE POINTS (SEE COMMENTS CUPLNF 112

r APOVE

)

CUPLNF 113

r - DLXT = X-INCRFMfmt which CORRESPONDS TO KX INCREMENT XMIT CUPLNF 118

115 c- DLYt = Y-INCRFMENT WHICH CORRESPONDS TO KY INCREMENT XMIT CUPLNF 115 c- DLXR = X-INCRFMENT WHICH CORRESPONDS TO KX INCREMENT RECV CUPLNF 116

r - DLYR = Y-INPPFMFNT which CORRESPONDS TO KY INCREMENT RECV CUPLNF 117

C- FTPFR = RATIO OF C H A R AC T F R I ST I C ADMITTANCE OF TRANSMITTING CUPLNF 118

r ANTFNNA FFED MODE TO C HA P A C T EP I S T I C ADMITTANCE OE THE CUPLNF 119

1 ?o c PPCFIVINP, ANTFNNA FFED MODE CUPLNF 120

c- ISPECT = SPFCTPUM FLAG - SET EQUAL TO 1 IF INPUT DATA IS SPECTRUM CUPLNF 121

r - RATHFR THAN FAR FIELD CUPLNF 122

r CUPLNF 123

C- rCLIT = MAXIMUM ORDINATE VALUE FOR PLOTS. IF PCUT .FO. 0, PLOT CUPLNF 124

1?5 r- IS SFLF-SCALFD CUPLNF 125 C- NEF1,NRF? = RFGIN AMD END INDEX FOR RANGE OF BFAC CUPLNF 126 C- NXLI.NYL? = PEGIN AND END INDEX FOR RANGE OF XLIM CUPLNF 127

C CUPLNF 128 C- GAIMT =r GAIN np XMIT ANTFNNA IN OR. CUPLNF 129

1 ^0 r- FFTMX = magnitude or FAR-cTELD pattern (UNNORMALIZED) AT CUPLNF 130

r- THFTA = 0, XMIT CUPL NF 131

r - PADT = RADIUS OR TRANSMIT ANTENNA (SEE COMMENTS ABOVE) CUPLNF 132

c- PHI, THPTA, PSI = EULER ANGLES IN DEGREES FOR XMIT ANTFNNA (SEE CUPLNF 133 C- COMMENTS ABOVE) CUPLNF 134 1?5 c- NPOWT = NIJMBEP OF ROWS OF DATA IN TRANSMIT PATTERN CUPLNF 135 c- NCCLY = number of columns of data in TRANSMIT PATTERN CUPLNF 136

r _ IDAYHRT = file identifier for xmtt data CUPLNF 137

c CUPLNF 138

C- GA TNP = GAIN OF RECV ANTENNA IN DR CUPLNF 139

1 40 r- FP PM y = MAGNITUDR OF FAR -field pattern (UNNORMALIZFD) AT CUPLNF 140

r- THFTA = 0, RECV CUPLNF 141 C- P 40P = RADIUS OF RECEIVE antenna (SFE COMMFNTS ABOVE) CUPLNF 142

c- PHTP, THFTAP, PSIP = ELLER ANGLFS IN DEGREES FOR RECV ANTENNA ( SEE CUPLNF 143

c- comments ABOVE) CUPLNF 144 145 c- M on wp = number of ROWS OF DATA IN RECEIVE PATTERN CUPLNF 145

r- HC.niQ = NUMBER OF COLUMNS OF DATA IN PECFIVF PATTERN CUPLNF 146

C- lOAYHPP = FILE identifier FOP XMIT DATA CUPLNF 147

C CUPLNF 148

c- CAMT = reflection COEFFICIENT OF TRANSMITTING ANTENNA CUPLNF 149 150 c- GAMP = reflection COEFFICIENT OF RFCEIVIMG ANTENNA CUPLNF 150

r- game = REFLFCTION COEFFICIENT OF RECEIVING LOAD CUPLNF 151

C CUPLNF 152 print 5005 CUPLNF 153 READ 5000, (10(1), 1*1,4) CUPLNF 154

138
1

= 5 PRINT 5001 . ( ID( I ) , I = 1. 4) CUPLNF 155

20 OfiO 5000, HE40 CUPLNF 156

IP (EPP(60n 1001.21 CUPLNF 157

21 POINT 5001. HEAD CUPLNF 158

C CUPLNF 156 160 PP40 5022. FPFO, 70. DL7T. OLYT. DLYP, OLYP, ETOER, ISPECT CUPLNF 160 PRINT 5023. FREO. 70, OLYT, DLYT, OLYP, DLYP, ETOER, ISPEOT CUPLNF 161 READ 5030, RCUT, NBFl, NBF2, NXLl, NXL2 CUPLNF 162 PRINT 5031, POUT, NBFl, NRF2, NXLl, NXL2 CUPLNF 163 READ 5010, GAINT, FETYX, RADT, PHI, THFTA, PSI, NROWT, NCGLT, CUPLNF 16A 165 i IDAYHRT CUPLNF 165 PRINT 5011, GAINT, FFTNX, RADT, PHI, THETA, PSI, NROWT, NOOLT, CUPLNF 166

1 IDAYHRT CUPLNF 167 READ 5010, GAINR, FFRMX, RADR, PHIP, THFTAP, PSIP, NPQWR, NCOLR, CUPLNF 168

1 IDAYHRP CUPLNF 166

170 PRINT 5011, GAINR, FFRNX, RADR, PHIP , THETAP, PSIP, NRDWR, NCOLR, CUPLNF 170

I IDAYHRP CUPLNF 171 READ 5020, GANT, GAMP, GANL CUPLNF 172 PRINT 5021, GAMT. GAMP, g*AML CUPLNF 173

G CUPLNF 17A 175 r CUPLNF 175 PI=A. AATANt 1.0) CUPLNF 176

CEE = .2667625 CUPLNF 177

DTP = oi/lPO. CUPLNF 178 WAVLGTH = C EF/FRFO CUPLNF 176

1 PO XK=2 .API AWA VLGTH CUPLNF 180

FMN = 1 . / ( 1 . - GAMR*G A«L

)

CUPLNF 1 81 PHIP = PHIP+DTP CUPLNF 182

PHI = phiaotp CUPLNF 183 THFTAP = THFTAP+DTP CUPLNF 18A

1 P5 TMFT4 THFTAYDTR CUPLNF 185 PSIP = PSIPADTR CUPLNF 186

PSI = PSI*DTP CUPLNF 187 GAINT = lO.+A (GAlNT/10.

)

CUPLNF 188 GAINR = 10.A+(GAINR/10.) CUPLNF 186

160 diamt = 2.TAMAX1 (WAVLGTH, RADT) CUPLNF 160 niAMP = 2 . +A MAX 1 ( WA VL GTH, RADR) CUPLNF 161 DIAM5UM = DIAMT + DIAMP CUPLNF 162 PRINT 7, 0 I AMSHMAO I AMSUM/W AVLGTH CUPLNF 163

C CUPLNF 16A

165 I SOL = 5 CUPLNF 165

IF(PGIJT .FO. 0.) ISOL = 1 CUPLNF 166 IF(NPF1 .FO. 0) NBFl = 1 CUPLNF 167 IE(NBF2 .FO. 0) N8F2 = 1 CUPLNF 168 IF(NXL1 .pQ. 0) NXLl = 1 CUPLNF 166

200 IF(NXL2 .60. 0) NXL2 = 1 CUPLNF 200 IF(FFTMX ,E0. 0.) FFTMX = 1. CURL NF 201 IFtFFRMX .FO. 0.) FFPMy » 1. CUPLNF 202

r CUPLNF 203

C CUPLNF 204 ’05 C- LIMITS DF INTEGRATinw AND NUMBER OF INTEGRATION POINTS CUPLNF 205

C CUPLNF 206

DO 1000 IXLTM = NXLl, NXL2 CUPLNF 207 XLTM = IXLIM CUPLNF 208

nr 1000 ibeac = nbfi, nbf2 CUPLNF 206

210- BFAr=IBFAO CUPLNF 210 XKMAX= .6 CUPLNF 211 XKMIN = diamsum/zo CUPLNF 212 xklIM=xlImaxkmin CUPLNF 213

XKl

I

m=amIN1(XKLIM,XKMAX) CUPLNF 214

21'^ C xmlIm DFFINFS THF NONZPRO-FILL LIMITS OF INTEGRATION. CUPLNF 215

ABL = XRLIM CUPLNF 216 A1=4BL tA2«ABL 5R1=ABL tB2*A0L CUPLNF 217

c FROM XKLIm to ABL THERE IS ZERO FILLING. NEXT WE COMPUTE C UPLNF 218

c- i^HMBpp OR INTEGRATION pnisjTS. CUPLNF 216

220 DKOR = WAVLGTH/2 . /DIAMSLIM CUPLNF 220 OKOK = 4“IN1(0K0R, 0 IAMSUM / ZO/2 . ) CUPLNF 221

OKflK = DKOK/ (BFAOAXLIM) CUPLNF 222

NNl (1 )

=> ( A1 + A2 ) /DRCK CUPLNF 223

NNl ( 1 ) =2A ( ( NN 1 ( 1 ) +1 ) /2

)

CUPLNF 224

225 NN2 ( 1 ) =

( P 1 + B 2 ) / OKOK CUPLNF 22 5 nn?(1)=2a( (nn2(1)+1)/2) CUPLNF 226 N1*NN1 ( 1 ) CUPLNF 227

N 2 = N N 2 ( 1 ) CUPLNF 228

r CUPLNF 226

2 30 ri=(Al+A2) /N1 CUPLNF 230 C2=(Bl+02)/N2 CUPLNF 2 31

139
rOFF = -C14C2*FMM»S0PT( GAINT+GAINP ) / ( 4.*P I*FFTNX+FFRMX

)

CUPLNF 232

PG = S0PT(1. - C APS ( GAMP )*1. 2 ) CUPLNF 233

TG = 50PT(1. - C APS ( GAMT ) 2

)

CUPLNF 234

p 3 F COFF = COFF+PG*TG*SQPT( FTOFR

)

CUPLNF 235

C CUPLNF 236

r CUPLNF 237

c- FILLING PF THF input MATRICFS (AX AND AY) TO THE FFT FOURT. CUPLNF 238

c CUPLNF 239 7A0 TSUM71= ( 0. . 0 . ) CUPLNF 240 NY = 4+N2 CUPLNF 241 NX = 4*NI CUPLNF 242

OP 10 J2 = 1, NY CUPLNF 243 AX(J2) = (0.,0.) CUPLNF 244

245 AY(J2)=(0.,0.) CUPLNF 245

10 CONTIN'IF CUPLNF 246 PRINT 15. Nl, N2 CUPLNF 247

nn 200 J1 . 1, Nl CUPLNF 248

XNX = P1*(J1 - 1.) CUPLNF 249

250 XKYPK = XNX - A1 CUPLNF 250

on 100 J2 = 1, N2 CUPLNF 251

XNY =. C2F(J2 - 1.) CUPLNF 252 xkYOK = XNY - R1 CUPLNF 253 TKPKSO = XKXOR*XKXnK + XKYDK4XKY0K CUPLNF 254

?

c t; IF (TROKso .r,F. xklim^xklim) go TP no CUPLNF 255 CALL AMGLGFN( XRXCK, XKYOK, PHI, THFTA, PSI, PHIP). THETAP, CUPLNF 256

1 PSIP, PHIT, thftat, phir, TMETAR) CUPL NF 257

T(J2) = CMPLX(PHIT, THFTAT) CUPLNF 258

P ( J2 ) = CMPL X( PHIR, THETAR ) CUPLNF 259 PYO C-n TO 100 CUPLNF 260

110 T( J2) = R( J2) = (C., -1 .

)

CUPLNF 261 100 C PNTI Nl|F CUPLNF 262

v'PITF (3) (T(L), L = 1, N2) CUPLNF 263 WRITE (4) (R(L), L = T» N2) CUPLNF 264

265 700 CPNTIN IJF CUPLNF 265

N RY2T I NPOWT + 2 CUPLNF 266

NPX ?P = NPPWP *2 CUPLNF 267

OLX = OLXT T ply = OLYT CUPLNF 268 CALL FTNOFF (TOAYHBT, 1. 3> 5. 1, DATA, N'PY2T, NCOLT, FYT, FZT, CUPLNF

?70 1 WPDK) CUPLNF 270

OLV = riLYP t CLY = DLYP CUPLNF 271

CALI FTNDFF (lOAYHRR, 1, 4 , 6, 8, DATA, NRX2R, NCOLR. FYR, FZR, CUPLNF 272

1 WORK) CUPLNF 273

r CUPLNF 274

P75 DC 7P0 J1 = 1, Nl CUPLNF 275

XNX = ri+(ji - 1.) CUPLNF 276

XKXOK = XNX - A1 CUPLNF 277 5UM2 = (0.. C.) CUPLNF 278 RFAD ( 5 ) ( FYT ( L ) , L= 1

.

N 2) CUPLNF 279

PPO RPAD (7) (FZT(L), L= 1, N7 ) CUPLNF 280 RPAD (6) (FYP(L), L= 1, N2 ) CUPLNF 281 °FAD (8) (FZR(L), L= 1, N2 ) C UPLNF 282

FXT = ( 0. , 0 . ) CUPLNF 283

FXR = ( 0. , 0. ) CUPLNF 284

7 F5 op 740 J? = 1 , N2 CUPLNF 285

xmy = 07 *( J? - 1. ) CUPLNF 286 XKYOK = XNY - 81 CUPLNF 287 TKOKSO = xkxok+xkxok + XKYOK*XKYOK CUPLNF 288

IF (TKOKSO .GE. XKLIM*XKLIM) GO TP 240 CUPLNF 289 790 XNZ = S0PT(1. - TKOKSO) CUPLNF 290 CALL VECTGFN(FXT, FYT(J2>, FZT ( J2 ) , PHI, THETA, PSI, FX, FY, CUPLNF 291

1 FZ ) CUPLNF 292

CALL VFCTGFN(FXR, FY0(J7), FZR(J2), PHIP, THETAP, PSIP, FXP, CUPLNF 293

1 fyb, FZP) CUPLNF 294

'>Q5 FDOTFO * -FX + FXP 4 FYX<FYP - fZ*FZP CUPLNF 295 FOOTFO = FD0TFP4CFXP(CMPLX ( 0. , X K 4 XN Z + Z 0 ) ) / X N

Z

CUPLNF 296 SUM? = FOOTFP 4 SUM2 CUPLNF 297

AY(J2 4 3*N2/7) = FDOTFP 4 AY(J2 4 34N2/2) CUPLNF 298

?<*() CONTINUE CUPLNF 299 300 AX(J1 4 7*N1/?) = -SUM2*(-1) 4*( J1 4 3*Nl/7) CUPLNF 300

TS'|M?1 = SMM2 4 TSU«21 CUPLNF 301 770 CCNTI NUF CUPLNF 302

c CUPLNF 303 DO 1? J 7 = 1 , NY CUPLNF 304

AY( J7)=-AY( J?)*(-l )*4J? CUPLNF 305

1 7 C PNTI NUF CUPLNF 306

r CUPLNF 307

M N 1 ( 1 ) = NX CUPLNF 308

140
NN ? ( 1 ) = NY CUPLNF 307 ’10 C CUPLNF 310

CALL <=nU7TfAX.NNl.l,+l,+l,wn‘?K) CUPLNF 311 CALL FniJO T ( 4 Y, NN2 , 1 , + 1 , 1 , W OP K ) CUPLNF 312 nn 40<i . 1 , NX CUPLNF 313

yO = (-NX^l. + Ml _ 1 ,

) * WAVL CT H/ ( 4 1 + A2 ) CUPLNF 314 ’15 FX = CPXO(CMPLX(0.»-XK441»Y0) ) CUPLNF 315 4X(M1)=fx«C0FF*AX(M1) CLIPL NF 316 400 r HNT I N'lP CUPLNF 317

on 300 Ml = 1 , ny CUPLNF 313

YO = (-NY/7. + M7 - 1 .

) *U4VLGTM/ ( B1 + B2) CUPLNF 310

J 7 0 Fv = 0'^XP(rMPLX(O.»-XKAPl*YO) ) CUPLNF 320

4Y( M? ) = FY + ''nFF>l-4Y (M? ) CUPLNF 321 ’00 CONTTNUF CUPLNF 322

C CUPLNF 323

r CUPLNF 324

17 5 C- PPINinilT AMO plotting. CUPLNF 325

c CUPLNF 326 PPTNT 5. XLTY, RFAO CUPLNF 327 POINT 15, NX , NY CUPLNF 328 POINT 15>W4VLO,TH,OAnT.P4 00,ZO CUPLNF 327 ’’0 0PHI = dMI*150. /PT 5 D THFTA = TH FT 4 + 1 RO . / PI $ 0 P S I = P S I * 1 3 0 . / P

I

CUPLNF 330 0PMIP = OMlP*lRO./oi sothf-TAP=THFTAP* 1RO./PI tOPSIP ,P5IP41P0./PI CUPLNF 331 POINT 4 5, n PHI , DTHF TA , OPS I , D PH I P , 0 TH F TA 0 , D P S I P CUPLNF 332

OX =: W4VLGTH/(A1 + A71/4. 1 OY = WAVLGTH/( B1 + B2) /4. CUPLNF 333 POINT 55, -DX+NX/?., DX*(NX/2. - 1. ) , Dx CUPLNF 334 3’5 POINT 65, -DY+NY/7., 0Y+(NY/2, - 1. ) , DY CUPLNF 335 POINT 75, -Al+4, 42+4 - Cl, Cl C UPLNF 336 PPINT R5, -Rl+4, 92*4 - C2, C2 CUPLNF 337 POINT R'^.XKLI'' CUPLNF 338

CLIOLDP = 20 , + ALOGIC ( C APO ( TSHM21 +COFF )

)

CUPLNF 337 140 PPINT 05 , TSUM71*cnFF, CUPLDB CUPLNF 340

c CUPLNF 341

c ppINToijt of XO and yC CFNTFPi inf C'JTS OESPFCTIVELY CUPLNF 342

c CUPLNF 343 POINT 17 CUPLNF 344 345 POINT 75, ( 4X( Jl) , J1 = 1, NX) CUPLNF 345 PRINT 20 CUPLNF 346 POINT 25, (AY( J2) , J2 = 1, NY) CUPLNF 347

c CUPLNF 34 8

c PLOT IF MAGNITUDF of XO AND YO CFNTEPLINE CUTS CUPLNF 347 150 c CUPLNF 350 POINT 510 CUPLNF 351

r CUPLNF 352 NIO = NX /

(

XL I Y + BF 4C ) + . 000001 CUPLNF 353 NIMTN = NX/7 + 1 - NlO/2. CUPLNF 354 155 N1M4X » NX/2 + 1 NlO/7. CUPLNF 355

OP 501 Jl = NIMIN, N1M4X CUPLNF 356 ACLCnUJl - NIMIN + 1) = CABS(4X(JD) CUPLNF 357

X(J1 - NIMIN + 1) = {-MX/2. + Jl - 1 . )*W4VLGTH/{ 41 + 42 ) /4 . CUPLNF 353 POINT 515, ACLC'JKJI - NIMIN + 1. ), X ( Jl - NIMIN 1 ) CUPLNF 357

360 501 CPNTINUF CUPLNF 360 Nil = NlMAX - NIMTN + 1 CUPLNF 361

r CUPLNF 362 XMIN = X ( 1 ) 5 XM4X = X( Nil ) CUPLNF 363

IF ( TSCL .NF. 1 ) GO TP 511 CUPLNF 364 365 CALL mINmaX ( 4CLCUT, XOLIM, PCUT, Nil, 1 ) CUPLNF 365 511 CALI PLT1200(X, aCLCUT, XMAX, Xmin, RCUT, 0., Nil, 1H+, 1, 1) CUPLNF 366 PRINT 5041, HPAD, lOHMAGNITlIPF , lOH VS xn CUPLNF 367 POINT 1 0 CUPLNF 363 N?C = NY / ( XL I M + P F AC ) + . 000001 CUPLNF 367

370 N2MIN = NY/7 + 1 - N20/2. CUPLNF 370 N2M4X = NY/2 + 1 + N2C/2, CUPLNF 371 on 601 J? = N2MIN, N2MAX CUPLNF 372 ACL0'JT(J2 - N?min + 1 ) = CARS(4Y(J2)) CUPLNF 373 X(J2 - N7MIN * 1) = (-NY/2, + J2 - 1. )*WAVLGTH/ (31 + B2)/4. CUPLNF 374

375 POINT 615, 4CLCUT(J? - N2MIN + 1,), X(J2 - N2MIN + 1) CUPLNF 375

601 CONTINUF CUPLNF 376 N12 = N7M4X - N2MIN + 1 CUPLNF 377

r CUPLNF 378

XMT4' = Xfl) T XMAX = X(K'22) CUPLNF 377

’3 0 IF (ISCL .NF. 1) GH TO 611 CUPLNF 380

CAl L minmax ( ACLCUT, XDUM, PCUT, N22, 1) CUPLNF 381 611 CALL PLT1200(X, ACLCUT, XMAX, XMIN, PCUT, 0., N22, 1H+, 1, 1) CUPLNF 382 POINT 5041, HEAD, lOHMAGNITUDE , lOH VS Yp CUPLNF 383

r CUPLNF 384 385 C CUPLNF 385

141
7 QO

•>Q5

«tOO

^05

^IC

415

4?0

4?5

430

4^5

4 40

PFWTNO FJLF5 CUPLNF 386 CUPLNF 387

P'^WINO 1 CUPLNF 388 RFWTNO 3 CUPLNF 389 PrwINO 4 CUPLNF 390 PFWTNO 5 CUPLNF 391 RFWINO 6 CUPLNF 392 RFWIMD 7 CUPLNF 393 RFWINO 5 CUPLNF 394 ICOO 0 ONT T Nijc CUPLNF 395

GO TO 70 CUPLNF 396

1001 C M. 1 F X T T CUPLNF 397 CUPLNF 398 CUPLNF 399

- FORMATS CUPLNF 400 CUPLNF 401

5 format (* XLIM := *, FI?. 5. 5X. *nFAr = 7, F1 2. 5/ /

)

CUPLNF 40?

7 format {*1MUTUAL RAYLFIGH DISTANOF = ( DIAMSUM )SOUARFD/WAVLGTH= 7,. CUPLNF 403

1 F17.5. 7 yFTFRS*//) CUPLNF 404

15 format ( 1 X , *N1=* I6,5X,*N2=* I6,5X,*THFY BOTH SHOULD B F FVFN7//) CUPLNF 405

75 FORMATdx, (10F12.51) CUPLNF 406

77 FORM AT ( 1 X , 7 XO-CUT* /

)

CUPLNF 407

7q format; 1 X ,/ /7 yO-CUT*/) CUPLNF 408

3 5 rrPMAT(lX,7WAVLGTH,RA0T.RADR*AH0 7Q =*4F12.57 meters RESPECTIVELYCUPLNF 409

17// ) CUPLNF 410

45 FORM AT ( IX . 7FUL FR A M GL F S ( P H , TH , PS ) OF TR. AND PE . ANTS. PESP . ARE7CUPLNF 411

1 3F10.47 AN073F10.47 oFGRFFS*//) CUPLNF 412

55 format; 1 X, 7X0 RANGES FR0M7F12.57 T07F12.57 IN INCREMENTS 0E7F12 .CUPLNF 413

1 57 MFTFPS*//

)

CUPLNF 414

65 FORMAT;1X,7YO RAMGFS FR0MYF1?.57 T07E12.67 IN INCREMENTS 0F7F12 .CUPLNF 415

157 MFTFPS7//) CUPLNF 416

75 FDPM AT ; IX , 7THF INTEGRATION VARIABLE KX /K P ANGFS FROM* F12 . 57 TO+E CUPLNF 417

1 17.57 IN INCRFMFNTS GF7F12.5//) CUPLNF 418

fl5 fopmatiix.ythf integration variaplf KY/R RANGFS from+fip.s* TO + F C UPL NF 419

1 12.57 IN INCREMFNTS OF7F12.5//1 CUPLNF 420

P7 FORMAT ; IX , 7THE SPFCTRIM IS ZERO FILLED BEYOND SOP T ; K X 2+K Y 2

)

K TIMECUPLNF 421 1S7C12. 5// ) CUPLNF 422

q5 format ; IX ,/ /7 THE COUPLING OUOTIFNT AT X0=0 AND Y0=0, SUMMED DIP ECTCUPLNF 423

ILY WITHOUT THF FFT, EQUALS*. 2E12.5, 7 OP 7, FIO .2, 7 DR7/ / ) CUPLNF 424 510 FOR MA T ; 1 X , / /* maGNITL'DF (XO-CUT)*/) CUPLNF 425

515 FORM AT ; IX , FI 2 . 5* X0 = *F12.5) CUPLNF 426

610 format; IX, //* mao,nitude (yo-cut)*/) CUPLNF 427

515 fppmaT ; 1 X , F 12 . 57 YC = 7F12.5) CUPL NF 428 5000 format ;baio) CUPLNF 429

5001 FORMAT ; 1 HD, RAID) CUPLNF 430

5005 FORMAT ;iHl) CUPLNF 431 5010 FORMAT (6F10.4, 215, AID) CUPLNF 432

5511 FORMAT ;iX, 6F10.4, 2T5, AlO) CUPLNF 433 5070 format ; RF 10 .4

)

CUPLNF 4 34 5021 format (IX, 8F10,4) CUPLNF 435 5027 format (7F10.4, ID CUPLNF 436

5023 FORMAT ;ix, 7F10.A, 15) CUPLNF 437

5030 format ;f10.4, 415) CUPLNF 438 5031 format ;ix, Flo. 4, 415) CUPLNF 439

5041 format ; / . 5X , R aid , 5X , 2A10) CUPLNF 440

FND CUPLNF 441

142
B.1.2 SUBROUTINE ANGLGEN

(PKXOXK, PKYOXK, PHI, THETA, PSI,PHIP,THETAP,PSIP,PHIT,THETAT,PHIR,THETAR)

PURPOSE :

To compute in the coordinate system fixed in each antenna, the far-field angles

corresponding to a given direction of the propagation vector in the mutually common

xyz coordinate system (see fig. 1).

METHOD :

Evaluate eqs (13a) and (13b) of the main text.

ARGUMENTS:

Input Parameters (in order of appearance)

PKXOXK, PKYOXK

PHI, THETA, PSI

PHIP,THETAP,

PSIP

= X and y components of the normalized propagation vector (kx/k,ky/k of

eqs (13)).

= Eulerian angles of the antenna on the left as drawn in figure 2 (0,9, of

eqs (13)).

= Eulerian Angles of the antenna on the right as drawn in figure 3

(01,01,4^1 of eq (13)).

Output Parameters (in order of appearance)

PHIT,THETAT

PHIR,THETAR

= Spherical angles in the coordinate system fixed in antenna, corresponding to the direction k^/k, ky/k

eqs (13) and shown in figure 2).

= Spherical angles in the coordinate system fixed in

antenna, corresponding to the direction k^/k, ky/k

eqs (13) and shown in figure 3).

the left transmitting

(0A>9a

the right, receiving

(0p>6p of

SYMBOL DICTIONARY :

Variables (in alphabetical order)

CSPH,CSPHP,

CSPS,CSPSP,

CSTH.CSTHP = Cosine of PHI, PHIP, PSI, PSIP, THETA, and THETAP, respectively.

GAMOXK = Normalized z-component of propagation vector (y/k).

PI = TT = 3.14159

143
RD,RN

R1,R2,R3,R4,

R41,R42,R5,

R51,R52,R6,

R7,R71,R72,

R8,R81,R82,

R9

SNPH.SNPHP,

SNPS.SNPSP,

SNTH.SNTHP

TD,TN

T1,T2,T3,T4,

T41,T42,T5,

T51,T52,T6,

T7,T71,T72,

T8,T81,T82,

T9

XCOSR,XCOST

= Denominator and numerator respectively of eq (13b) when computing 0p.

= Intermediate variables used in computing 0p and 6p from eqs (13).

= Sine of PHI, PHIP, PSI, PSIP, THETA, and THETAP, respectively.

= Denominator and numerator respectively of eq (13b) when computing 0;\.

= Intermediate variables used in computing 0/\ and Qjs^ from eqs (13).

= Right side of eq (13a) for 0p and 0^, respectively.

Functions Inline or within FORTRAN Library (in alphabetical order)

ABS(X)

ACOS(X)

ATAN(X)

ATAN2(X,Y)

COS(X)

SIN(X)

SQRT(X)

= Absolute value of X.

= Angle between 0 and tt whose cosine is X.

= Angle between -it/2 and tt/2 whose tangent is X.

= Angle between -tt and tt whose tangent is X/Y.

= Cosine of X. = Sine of X.

= Square root of X.

List of Complex Quantities

(None)

144
1 SlIPPnilTTM'^ AN'GLGPN(PKVnYk,PKYnXK,PHI,THFTA,PSI,PHIP,THFT4P.PSIPf ANGLGEN 1

] P HT T, THFT AT, PH TP , THFT AR ) ANGLGEN 2

r ANGLGEN 3

r PKXnvK and pKYOYK ape the X AND Y COMPONENTS OF THE NOPMALIZEO ANGLGEN 4

i r po PP4 ^,^T I ON vector . ANGL GEN 5

r PHT,THFTA,ANn PST APF THF EMLFPIAN ANGLES OF THE PQTATEO SYSTEM ANGLGEN 6

r np THE left, TPANS'^ITTTNG antenna T WTTH PFSPECT to the axes ANGLGEN 7

r FTXFO TN THE TRANSMITTING ANTENNA. ANGLGEN 8

c PHIP.THETAP. AND PSIp APE THE EULFPIAN ANGLES OF THE ROTATED ANGL GEN 9 10 r system OF THF right, PFCFIVING ANTENNA R WTTH RESPECT TO THE ANGL GEN 10

r AXES FTXFD TN THF PECETVTNG ANTENNA, ANGLGEN 11

r TmfTAT AMO phIT are THF ANGLES IN THE FIXED CCCRDINATE SYSTEM OF ANGLGEN 12

r T noo F Fpp>NO I NG TO THF DIRFCTTON ° K X PX K , P K Y 0 X K . ANGLGEN 13

c THFTAR and PHIR are the angles in the FIXED CnORDINATF SYSTEM OF ANGLGEN 14

1 5 r Q corresponding to THF DIRECTION P K X 0 X K , P K YO X K . ANGL GEN 15

C THFTAT and THETAR range from from 0 TP PI, ANGLGEN 16

r PHTT AND PHIR range from 0 TO ?PI, ANGLGEN 17

r ANGLGEN 18

c ANGLGEN 19 ?0 p 1 = 4 , * AT AN n . n

;

ANGLGEN 20 GAMnxKiSOPTtl.-PWXOXKY+E-PKYPXKTY?) ANGLGEN 21

r ANGLGEN 22 CSTH = COS(THFTA) ANGLGEN 23 FMTH = FTN(THFTA) ANGLGEN 24

7 5 CFTHO = COS( THFTAR) ANGLGEN 25 SMTHP = SIN(THFTAP) ANGL GEN 26

r sps = c OF ( PE I ) ANGLGEN 27 FNP s = S TN ( PS I ) ANGLGEN 28 CSOSR = Cns(PSIP) ANGLGEN 29

00 SNPSP = SIN(PSIP) ANGL GEN 30 CSPH = '-PG(PHI) ANGL GEN 31 SNPH = STN(PHI) ANGLGEN 32 CSPHP = COS( RHIP) ANGLGEN 33 SNPHP = SIN(PHIR) ANGL GEN 34

35 r ANGLGEN 35

r rpMPlJTATinN OF THFTAT AND THFTAR. ANGLGEN 36

r ANGLGEN 37

T1 = S NTHYCSP S* PW xrx

K

ANGL GEN 38

R1 = <;MjHP*rSPSPYP'< XCXK ANGLGEN 39 40 T? = SNTH-XSNPS + PNYOXK ANGLGEN 40

R? = SNTHP+SNPSPtPWYrXK ANGLGEN 41

T3 = CSTH+G AMPXK ANGLGEN 42 RR = rSTHP+GAMOXK ANGLGEN 43 xrnsT=-Ti+T?+T3 ANGLGEN 44

45 xCnSR=-Rl-R2+R? ANGLGEN 45 THFTATiACnS(XCPST) ANGLGEN 46 THFTAR =ACnS ( XCnSR ) ANGLGEN 47

C ANGLGEN 48

r rnMoijTATinN of phtt and phtr. ANGLGEN 49

=0 r ANGLGEN 50

I&l = CSPH+CSTHYCSPS ANGLGEN 51 T4? = S NPH^SNPS ANGL GEN 52 T4 = (T41-T4? )A:PKXnXK ANGLGEN 53

r ANGLGEN 54 55 PAl = rSPHP*CSTHP*CS RSP ANGL GEN 55 R42 = SN PHP-XS NPS P ANGLGEN 56

R 4= (R41-P4?. ) +PKXOXK ANGL GEN 57

c ANGL GEN 58 T51 = CSPH+CSTHYSNPS ANGLGEN 59

4-0 = <5k;ph*CSPS ANGLGEN 60

T5 = ( T5 1 + T5 7 ) PK YPX K ANGLGEN 61

r ANGLGEN 62 R51 = c SPHP’ACS THP + SN P SP ANGLGEN 63

p = SNDHP*CSPS P ANGLGEN 64

4- 5 R5=(R5lFR57)5PKYnX'< ANGL GEN 65

C ANGLGEN 66

Tp = CEPH>!>SNTH*GAMuXK ANGLGEN 67 Rt = cephp+snthp+gamoxk ANGLGEN 68 TD=T4-T5+T6 ANGLGEN 69

70 rD=R4+R5eR6 ANGLGEN 70

r ANGLGEN 71 ANGLGEN 72

T?l = SNPHX<cSTHX-CSPS ANGLGEN 73 T72 * CSPH+SNPS ANGLGEN 7 4 75 T7= (T71+T72 ) YPKXOXK ANGL GEN 7 5 ANGL GEN 7 6 P7I = fkjp'to+cSTHdX'CSPSP ANGLGEN 77

145
P77 = rpp4P>)c$MP$p ANGLGFN 7P R7=(P7]+P7?)4PKX0XK ANGLGEN 70

eo c ANGLGFN 80

TB 1 = OS PN7CS PS ANGLGEN 81 TP? = pmdh*CSTH*SNPS ANGLGFN 8? TBn( jp]_trP)+PKYOXK ANGLGFN 83

r ANGLGFN 84

R5 Pfil = OSPNP«CSPS= ANGLGPN 85 RPP = '5MPHP+r$THP*SMPSP ANGLGEN 86 pp=(pPl_pgp)*ot<YnXK ANGLGEN 87

c ANGLGFN 88 TC = pmpm*SNTH7GA‘^0XK ANGLGEN 80 on pq = BMPHP*SNTHP»r, JwrXK ANGLGEN 00

TN = T7*-TB+Tq ANGLGEN 01 PN=P7-PB+pq amc-lgfn 02

r ANGL GEN 03

C CHAMOF nP P4NGF OF PHTT FPOMf-PI.PI) TO (0. ?PI ) . ANGLGEN 04

Q <= IP ( ( 4P F (TN ) +AHF ( TO) ) . FO .0 . ) GO TO 10 ANGLGFN 05 PHIT=AT4M7(TN,Tn) ANGLGEN 06

IF ( PHI T.L T .0 . ) PHIT=? ,*PT+PHIT ANGL GFN 07

GO TO ?0 ANGLGFN 08

10 CDNTINUF ANGLGEN 00 in PHITiO. ANGLGFN 100

?0 r PNTT K"IF ANGLGEN 101

C PHiMPF PP PANGF of pHIO FPnM(-PI,PI) TO ( 0. ?P I ) . ANGLGEN 102

I F ( ( APS ( PM ) + ASS ( PO ) ) .FO . 0.) GO TO 30 ANGLGFN 103 PHIP=ATAMP(PN,PP) ANGLGPN 104 106 IFIPHTP.LT.O. ) PH IR = ? .*°I + PH IP ANGLGFN 105

GP TP 40 ANGL GEN 106

30 C P M T 1 M 1 J n ANGLGEN 107 DHIP=0. ANGLGEN 108

40 CONTI Ml IF ANGLGFN 100 no R PTLIPN ANGLGFN no

FN D ANGL GEN 111

146
B.1.3 SUBROUTINE FINDFF( IDAYHR ,LUIN ,LUA,LU02 ,LUOE , DATA, NRX2,NC0L ,FFY, FEZ ,STOR)

PURPOSE :

To read from an input file, spectrum or far-field data whose coordinates are and

ky referred to the antenna's preferred coordinate system and from this produces a file containing far fields whose coordinates are specified by the angles specified on

a second input file.

ARGUMENTS:

IDAYHR

LUIN

LUA

LUOY

LUOZ

DATA

NRX2

NCOL

FFY

FFZ

STOR

= File identifier for file on which far field resides.

= Logical unit on which far field or spectrum resides.

= Logical unit on which angle information resides.

= Logical unit on which y-component of far field is to be written.

= Logical unit on which z-component of far field is to be written.

= Two-dimensional array in which this input far field is stored, included

in argument list for dimensioning purposes only.

= Twice the number of rows in DATA.

= Number of columns in DATA.

= y-component of far field, included in argument list for storage

allocation purposes only.

= z-component of far field, included in argument list for storage

allocation purposes only.

= Intermediate array, included for storage allocation purposes only.

METHODS : The subroutine reads the first record of the file containing the far-field or spectrum

data from unit LUIN and compares the eighth word of the record with IDAYHR in order to make sure the proper data file is used. If LUIN contains the incorrect file,

execution terminates. After correct file verification, the entire file is read in and

stored in array DATA. Because the input data exist in polar form, a conversion to rectangular form is also performed in the operation of filling DATA.

All data transfers use FORTRAN unformatted READ and WRITE operations.

The desired far-field angles are assumed to be stored on unit LUA. These are read one record at a time into complex vector FFY with the real part containing the

0-coordinate and the imaginary part, the 0-coordinate. For each element of the vector

FFY, the angles stored are used to locate the nearest far-field point in the array

DATA. The z-component is then calculated and stored in FFZ. When all angles in FFY have been changed to the corresponding far-field values, the vectors FFY and FFZ are written out as a record on units LUOY and LUOZ, respectively.

147
The correct point in the far-field array is found by the following procedure.

Calculate the reference index for the x and y directions by

ky

I = integer part of -f—

c

^ Ak

X

k y

= integer part of

y

where

k^ = k sine COS0

ky = k sine sin0

and AKx and Aky are the far-field k^ and ky increments. These increments are given by

Ak

X

2tt N 6 X X

Ak

y

2tt N S y y

where N^, Ny are the number of x or y points and 6x> 5y are the x or y

near-field spacing.

The far-field increments are given in terms of near-field spacings because it is assumed that the far field is obtained either by a near-field scan or the PO model

program POMODL (see appendix A), which calculates its far-field array based on a desired near-field spacing.

The row and column indices and specify "lower left-hand corner" of the

square in (k^jky) space which contains the point specified by the angles 0 and 0. The fractional part of

k k X or X Ak Ak

X y

is used to determine which corner of the square lies closest to 6 and 0.

148
The z-component is found from the relationship

E = E tan0 sin0

z y

Because the far-field array DATA may not contain values for all angles, a test is performed to determine if DATA does, in fact, contain a far-field value at the

requested 0 and 0. If it does not, the y- and z-components are set to zero.

SYMBOL DICTIONARY:

ANGLE

DATA

DCOL

DLKX DLKY

DROW

DTR FCOL

FFY

FKSQ

FKX FKXMAX

FKY FKYMAX

FROW

IICOL

IDIDAYHR

IFC

IFR

I ROW

IR2X2

ISPECT

J

LNROW

PHI

PIPIX2

= Intermediate variable, phase angle of far-field input data.

= Far-field array as a function of antenna's k^, ky system.

= Fractional part of FCOL.

= Input far-field point spacing in k^ direction.

= Input far-field point spacing in ky direction.

= Fractional part of FROW.

= tt/ 180 = degree to radian conversion factor.

= k^/Ak^.

= y-component of far field, also used as temporary storage for the

far-field angles.

=; 1(2 -L 1(2

= kx = x-component of propagation vector.

= Maximum value of k^ for which there are far-field data.

= ky = y-component of propagation vector.

= Maximum value of ky for which there are far-field data.

= ky/Aky.

= DO loop index.

= Input DO loop column index, also column index for far field.

= Identification array for far-field data.

= Far-field file identification = ID{8) for correct file.

= Integer part of FCOL.

= Integer part of FROW.

= Input DO loop row index, also intermediate variable.

= 2 X IROW.

= 1 if DATA contains spectrum rather than far field.

= Search loop index.

= Input or output DO loop index.

= Number of rows of input far-field data.

= 0

= angle in far field.

= 7T = 3.14159

= 2it.

149
TAMP THETA

XNZ

= Intermediate variable, amplitude of input far-field data.

= e = angle in far field.

= Cose.

150
1

5

: 0

1 5

?0

?0

35

6 0

4'"

50

5 5

AO

A5

70

75

12

34

567

P9 10

1112

13

14

151617

18

1920

21

2223

24

2526

27

2829

30

3132

33

3435

36

3738

3940

41

4243

44

4546

47

4849

50

5152

5354

55

5657

5359

60

61

62

63

64

65

66

6768

69

7071

72

73

7475

76

77

SURPTITIA'E FIMOFF (TOAYHP,

1 NCni, FFY. FF7, STOP)

LI!TN, LU4» LIIDY, LU07, DAT4, NPX2i

THIS Sl'PPnUTTNF RE40S FAP-FIELO OR SPECTPUH 04T4 FROM LUIN 4M0 STORES IT TM 4PR4Y D4T4. ANGLES C 0 R R F SO PN 0 I N G TO FAR-FIELO

niRFCTIPNS IN THF 4NTFNN4S CnOPOINATE SYSTEM APE READ IN FROM LUA. DATA IF SEAPCHFO FOP THE CLOSEST POINT AND THF Y-CGMPONENT OF THE FIFIO AT THE GIVEN ANGLE IS USED TO CCNPUTE THE Z-COMPONEMT.

THFSF FIFLO components APE WRITTEN ON LUCY AND LUGZ. COMPLEX FFY(l). FFZ(l)

COMMON /FAR/ Nl, N2, NX, NY, DLX, DLY, XK, ISPFCT dimension DATA(NPX?, NC0L)» STOP(l), ID(IO)

cr~ MISCFLLANFOUS

r - PRINT 1020, LOIN LUA, LIIOY, LLinz, NPX2, NCOL

1020 format (61701

TSF = 0 PI = 4.TATAN(1.)

PTX2 = 2,YPI

OTP = “I /IPC. NPPU = NRX7/7

ni Y Y = PTX7/NP0W/nLY

niKX = RTX7/NC0L/nLX

FKXM.AX = OLYX + CNCOL - 1)/?

FKYM4X = 0LYY4(NR0W - IW?

PPTNT 1000, OLKX, OLPY, FKXMAX,

1000 FORMATdX, 5G20.5) FKYMAX. XK

F INDFF

F INOFF FINDFF

FINDFF FINDFF

FINDFF FINDFF

F INOFF FINDFF

FINDFF

FINDFF FINDFF

FINDFF

FINDFF FINDFF FINDFF FINDFF

FINDFF

FINDFF FINDFF

FINDFF FINDFF

FINDFF

FINDFF FINDFF

FINDFF

FINDFF

FINDFF FINDFF

c- FINDFF

C - FIND fnpoFCT FAP-FIFLD FTLF ON LOGICAL UNIT 1 IIIN. FINDFF

r - FINDFF

120 RFAD( 1 HIN ) ( TDd ), I = 1, 10) FINDFF PRINT 1510 , ID FINDFF

I F ( ID ( 8

)

,F0. IDAYHR) GO TO 130 FINDFF

I F (FnF(LUIM)) 125, 130 FINDFF ]piSPr^MT 1 530 FINDFF

CALL FXIT FINDFF

130 CCNTINIJF FINDFF

C- FINDFF

C- READ FAP-FIFLD INTO ARRAY DATA. FINDFF

C- FINDFF

DO 140 TCHL = 1, NCOL FINDFF REAndUIN) (STOP(I)> I = 1* NRX2) FINDFF

Dp ISO IROW » 1, NPCW FINDFF

JPY7 = IP0VJ42 FINDFF

TAMP s STCPdRYZ - 1) FINDFF

ANGI.F = 5TPP ( IRX2)4DTP FINDFF 0ATA(IPX? - 1, ICCL) = TAMR4CnS( ANGLE) FINDFF DATAdRX?, ICOL) = TAMP4SI N( AN3L F) FINDFF

150 CONTINUE FINDFF

I 40 GONTINIIF FINDFF

FINDFF

r- PPPLATF valuer of angles WITH C 0 P P F S P ON C I N G FAR-FIELD. FINDFF FINDFF REWIND LUA FINDFF

READ (LUIN) FINDFF

IF (FnF(LUINl) 500, 600 FINDFF

600 continue FINDFF PACKS^APF LUIN FINDFF

'dO CQNTTNUF FINDFF

DO 200 T = 1, Nl FINDFF RFAD(LUA) (FFY(L), L = 1, N2) FINDFF nn300J=l,N2 FINDFF PHI=PEAL(FFY(jn ^FINDFF THcTA = 4 I MA G ( F FY { J ) ) FINDFF

IF (THFTA ,LT. 0.) GO TO 310 FINDFF

^ FINDFF

r- FIND the indices fop the array data which CDRRFFPDND to THF FINDFF

r_ rrnpoiNATPS CLOSEST TO THF DFSIPFD THFTA AND PHI VALUES. FINDFE

f._

“ FINDFF FYX=xK4SIN(THETA)*C0S(PHI) FINDFF FPY=XY*STN(THFTA)4SIN(PHI) FINDFF FXR0=FKX4FKX+FKY4FKY FINDFF

IF (FKSO ,GT. XK*XK) GF) to 310 FINDFF jF(4nc;(FKX).GF. FKXMAX) CO TO 310 FINDFF

YF (A8S(FKY) ,GF. FYYMAX) GO TO 310 FINDFF

151
pony = ckY/OLKY + MPTW/1 F INOFF 7P IF? = F P 0 W FINDFF 79

PO OPny = FPOU - IFP FINOFF 80

I p nw = IFP + 1 FINDFF 81

IF (OROW .LT. .5) TBQW = IFP FINDFF 82 FCOL = FKV/nLKX + ^'COL/2 FINDFF 83 IFO = FCOL FINDFF 8«i 85 OCOL = FCGt - IFC FINDFF 85

lent = IFC + 1 FINDFF 86

IF (OCf^L .LT. .5) TCOL = IFC FINDFF 07 IP2X2 = IPnwY? FINDFF 80 FFY(J) = CYPLXIOATi (IP2X? - 1, ICCL). PATA(IP2X2, ICOD) FINDFF go DO PP7(J) = -T AN ( THF T A ) Y SIN ( PHI ) *PF Y ( J

)

FINDFF 90

IF ( I5PFCT .NF. 1 ) GO TO 3C0 FINDFF 91 XN7 = SOPTfXKYXK - FK$Q)/XK F INDFF 92 FFY( J ) * FFY( J)YXNZ FINOFF 93 FF7U) = FFZ(J)YXNZ FINDFF 9 ^ Q5 GO TO 100 FINDFF 95 110 ffy ( J ) = ( 0. . 0.

)

PINDFF 96 FF7(J) = (0.. 0.) FINDFF 97

ino CHNT INIIE FINDFF 98 WBITP(LUQY) (FFY(L). L = 1> N2) FINDFF 9g 100 wPTTF(LUrZ) (FFZ(L). L = 1. N2) FINDFF 100

200 CnNTINlJF FINDFF 101 PFWINO L"GY FINDFF 102 RFWINO LUOZ FINDFF 103

0 ftupn FINDFF 108 105 1510 FOPMATCIX, PAIC, 215) FINDFF 105 1520 FDR.NATIY file *• T5» * SKIPPED FIN LU Y. 15) FINDFF 106

1 '^?0 PnPMAT(+ PILE NOT EOUNC. EXFCUTinN ABCRTF OY) FINDFF 107

E'''n FINDFF 108

152
B.1.4 SUBROUTINE VECTGEN (FOX,FOY,FOZ,PH,THET,PS,FX,FY, FZ)

PURPOSE

Given the components (FOX,FOY,FOZ) of a complex vector in a right-hand rectangular

coordinate system, find the transformed components (FX,FY,FZ) of that vector in a second coordinate system formed by rotation of the first through the Eulerian angles

(PH,THET,PS).

METHOD :

Use the transformation given by eq (18) of the main text.

ARGUMENTS :

Input Parameters (in order of appearance)

F0X,F0Y,

FOZ = X, y, z rectangular components of the given complex vector in the

unrotated coordinate system.

PH,THET,PS = Eulerian angles of the rotated coordinate system.

Output Parameters (in order of appearance)

FX,FY,FZ = Transformed x, y, z rectangular components of the given complex vector in the rotated coordinate system.

SYMBOL DICTIONARY:

Vari abl es

A11,A12,

A13,A21,

A22,A23,

A31.A32,

A33

CSPH.CSPS,

CSTH

SNPH.SNPS,

SNTH

= The nine elements of the 3x3 matrix on the right side of eq (18).

= Cosine of PH, PS, and THET, respectively.

= Sine of PH, PS, and THET, respectively.

153
Functions Inline within FORTRAN Library

COS(X) = Cosine of X. SIN(X) = Sine of X.

List of Complex Quantities

FX, FY, FZ, FOX, FOY FOZ

154
1 SUBROUTTNf VFCTGFN(F0XtF0Y.F07.PH,THFT »PSa FX.FYaFZ) VECTGEN 1 r TF THF COMPONENTS (FOX. FOv, FOZl 0 F A. CC.MPLFX VECTOR ARE GIVEN IN -VECTGEN 2

C 4 p jqMT-hANPFD RECTANCOLi® COGPOINATF SYSTEM, AND A SECOND VECTGEN 3 C OnOPOINATF SYSTFM 15 cppMFD PY POTATION THROUGH EULERIAN ANGLES VECTGEN A

5 C (dh,thet, PS ), Thfn (FX,fy,FZ) are thf COMPONENTS OF TR4I VECTOR VECTGEN 5

r IN THI'; SECOND ROTATED SYSTEM. VECTGEN 6

r VECTGEN 7 COMRLRX FQM.FOY^FOZ VECTGEN 8 rOMPLRx PX.cY.FZ VECTGEN 9 10 C VECTGEN 10

r VECTGEN 11

C COMPUTATION OF THF NINE ELEMENTS OF THE ROTATIONAL VECTGEN 12

C transformation matrix. VECTGEN 13

c VECTGEN lA

1 5 E5RM = (~r)5(D|J) VECTGEN 15 SNPM = SIN (PH) VECTGEN 16

pcp c = C(^S (R.S ) . VECTGEN 17

SNPR = SIN(P$) VECTGEN 18 CSTH = COS ( THFT

)

VECTGEN 19 SNTH = SIN(THET) .VECTGEN 20

c VFCTGEN 21 All = CSPH*CSTH+CSPS - SMDHYSNPS VECTGEN 22

Al? = SN°H*CS THTCSP? + CSRH’ASNPS VECTGEN 23 A13 = - SNTHYCSPS VECTGEN 2A ’5 A21 = -(CSPHTCSTH^SNRS + SNPHTCSPS) VECTGEN 25 A2? = -SNRH’XCSTHTSnpS + CSRH*CSPS VECTGE.N 26

art = 5a!TH*SN°F VECTGEN 27 A31 = f'S°H'XSNTH VECTGEN 28 AT? = SNPHHSNTH .VECEGEN 29 30 A?T = CSTH VECTGEN 30

c VECTGEN 31

F X = A 1 lY* F 0 X + A 1 ? + P 0 Y + 1 07 VECTGEN 32 FY=A21*FOX + A2?X=FOY + A?3*F07 VECTGEN 33 FZ=A31-PnX+A3?#F0Y+A33YFnZ VECTGEN 3A

3 5 P F T 1, 1 R N J/ECTGEN. 35 END VECTGEN 36

155
B.1.5 SUBROUTINE MINMAX(Z,ZMIN,ZMAX,LEX,LEY)

PURPOSE:

To determine the maximum and minimum values stored in the array Z.

ARGUMENTS:

Z is a two-dimensional array which is to be searched for its maximum and minimum

val ues. ZMIN contains the minimum value in the array Z on exit. ZMAX contains the maximum value in the array Z on exit.

LEX is the number of rows in Z. LEY is the number of columns in Z.

Array Z has dimensions (LEX, LEY). Initially ZMIN and ZMAX are set equal to Z(l,l).

Each value of Z is tested to determine if it is less than ZMIN or greater than ZMAX.

If either condition is satisfied, ZMIN or ZMAX is appropriately changed.

SYMBOL DICTIONARY:

METHODS:

JTZ

= Row DO 1 oop index.

= Column DO loop index.

= Temporary variable, Z(I,J).

156
c 1^1 D n 1 ! T T M F '*T''1N’AV(7, 7"^TM. 7_max, LPy. LFY) fAINMiX 1

( L T X . L ^Y) y I

N' M 4 X 2 ZW|M=7 (1,1

)

t 7 A Y = 7 n , 1 ) M I N y 4 X 3

1 ? 0 T = 1 . LTV M IMM 4 X 4

n n 170 J = It LTV MN^AX 5 T7 = 7( T. 1 ) yiN'y4X 6 TP ( T7 . T . 7 M I M ) z V T N’ = T 7 MINW4X 7 TP f T7 . GT , T V A y ) 7M ft X = TZ ,v| I N M A X 3 Cr\T TM" r 1 1NM 4 X Q

7 F T| 17 N) VI NVA X 10 FMO MIMyAX 11

167
APPENDIX B.2 SAMPLE PROGRAM INPUT AND OUTPUT

Illustrated below is a typical input card deck for program CUPLNF. Far-field data for

the two antennas were generated using POMODL. The output obtained for one of these runs is illustrated in Appendix A. 2. The output produced by CUPLNF is reproduced on the following

pages.

‘I

,

_ L

K 37 :i 33 j) 55 b'Te: 6'< 6-1 6:-'66 C7 65 JOl'! 72 73 .'-i 75* 77 78 7;- &: I 4^«-t -H

I 9 i9 iO 2 < 22 23 24 29 25 2 7 32 33 34 35 3S 37 ? 5 3? 4 '-a 43 isojit ‘‘i ‘i ff 55 57 56 55

OO'OOO 0

1 2 3 4 5|e 7 8 9 10 n 1 n 1

1

' n

22 111111

1 : . 4 5 6 : 6 5 'C 3' 3 3 3l3 3 3 3

ll

44 44444 4

I 2 3 4 5 S 7 8 9 iO 5555 i 5555

6 6 B 5 C |6 6 6 6 6

1 : 3 < 5 6 ; e 9 10 mmiin

0 I ©

8888883688

’ 2 3 4 5 S 7 8 5 10

9 9 9 9 9'3 9 9 91

0 '00000009

11 12 13 14 IS'lS 17 13 19 20

1 n I i|i 1 1 11 2222222222

II 12 13 14 ;;'ii i’ 19 19

3 3

'3 3 3 3

4 4 4 4 4^4 4 4 4 i!

11 12 13 14 15 16 1* ij ID 2'3

5 5 5 5 5|5 5 5 5 5

6 6 6 6 6|6 6 6 6

II 12 13 14 13 lb 17 18 13 20 7777777777

© I ©

3803 88

II 12 r. 14 15 15 17 19 13 20

9 3 9 8 9'0 9 9 9 9

0 0 0 OlO 0 0 0 9 0 0 0 0 0|9 0 0 D o'

21 22 23 24 2sl:s 27 23 25 30 21 32 33 34 35 35 37 38 29 40 n n n 11 11 1 11 11,11 1 11

2

2 2 2 ,'2 2 2 2 2

3 3 3 313 3 3 3 3

4 4 4 4 4.4 4 4 4

21 22 23 24 25 27 23 25 20

5 5 5 5 5 5 5 5 5 5

4 4 4 4 4 4|4 4 4 4 4 4 4 4 4 4!4 4 4 4 4 4 4 4 4 4]4 4 4 4

: 53 54 55 56 57 5c 13 63

6

6 6 6 6^6 G B 6 6

21 :: :3 r: :6 ;3

7

7 7 7i.' 7 7 7 7 © ©

8 8 8 8 8|8 8 8 G 3

9 9 9 9i£ 9 9 9 9

2222222222

3 3 3 3 3'3 3 3 3 3

5 5 5 5 5|5 5 5 5 5

6 6 6 G S 6 6 6 6 6

0 0 0 0 op 0 0 0 0

41 42 43 44 45 48 4 7 48 ‘3 53

1 1 1 1 1|1 1 1 1 I 2222222222

3 3 0 3 3'3 3 3 3 3

©

3

8 G 8

31 32 . :i 3' 38 -

9 5 9 9 °!s 9 9 9 3

5 5 5 5 5|5 5 5 5 5

6 6 6 6 GG 6 3 6 6

7 7 7 7 7

I 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 © I ©

888888388

9 9 9 9 5 9 9 9 9 9

0 0 0 0 G.O 0 0 C C

1 1 1 11,11 1 1 1 22222^2212

57 5 r .. .

3 3 3 3 3

I 3 3 3 3 3

5 5 5 5 515 5 5 5 5

6 6 6 6 6p 6 6 G 6

51 12 53 5J 55 5C 57 5S

7 7 7 7 7'7 7 7 7 7 0 I 0

888888688888

0 0 0 0 Op 0 0 0 0

51 i-2 63 64 65i.'-‘. 57 53 65 70 1111111111

! 2222222222

62 63 64 65 .06 57 68 69 .0

3 3 3 3 3^ 3 3 3 3

4 4 4 4 4 4l4 4 4 4 4

61 S; -i 61 65 55 57 64 -9 70

5 5 5 5 5,5 5 5 5 5

8 G 8 8 6 8 6 £ 3 G

51 62 57 : I 65 56 57 66 6 5 76 nm'mn

© i ©

8 8 8 £-3 8 3 8 3

?• 62 5,4 65 0-- 67 68 69 70

9 9 9 9 9'3 9 9 9 9|9 9 9 9 9l9 9 9 9 9

0 0 C 0 CjO 0 0 0 0

71 72 73 74 75'76 77 78 79 60

1 1 1 1 1,1 1 1 1 1

i 2222222222

3 3 3 3 3:3 3 3 3 3

4 4 4 4 4|4 4 4 4 4,

7. .1 15 15 n 73 79 6!

5 5 5 5 5|5 5 5 5 5 6668666666

71 72 .J 74 75'7S 17 78 79 !" nm'mn

© I ©

8 8 3 8 e'a 8 8 8 8

71 72 73 74 75 76 77 78 79 69

9 9 9 9 9'9 S 9 9 9

0 !

FOUR

158
miITI'AI.

HT'^TAN'i'F =(TSY

SHY 1s

OIJAP

FO/W 4

VI

^TH=

67. 5A

67?

mftfRS

O >o

LUUJ

OL

<J>

ki.< oooooo

CDcoc

c-

<D CU OJ fvj

>J- -t >D O

O' xC

r'- CM <x> •-i

CV) O' ^ o

O' r- O' .r ("- nO r^ • o • o

(\j

m c\j IT CD CJ 0^0

IT\ >o 0c1

(/• Tca

c:, cT

i/- T

(T- a

enQm

r**•

sOI

ocOm0

Lu

C\J cn

>o

(\J1

r»' 0« LL- nOm

ro

nT

sj-1

<iZ>o

r.'

'j- 0 m cn 1 o o

UJ I I fVJ UJ UJ

O' sD >c

O' ir» CO o ® >c

-*DO

c^m ro 0 o o 1 I t U- UJ UJ 00 CO cc r- m oo

rH iT» sj" O sj- GO

CO ,H 1-1

CO CO CO O O o t I I UJ UJ UJ

CO o o in CO ^

CO sO rH o m sj- CC> CO o

CNJ CM CO 0 o o 1 I I UI UJ UJ

00 >r o O cc CO

CO CO

<r ^ f"- I-H I-H CO

CM CM o o o I I I UJ UJ u.' ^ CD a CO ^ ^ o

CD nJ- CO

r-< CO CM

CO (\J CO 0 o o 1 } I UJ UJ UJ ^ o r* O' CO o

r- 'O >j- o 'O O' O .-I '4- I CO0 CO ^ 1 o o

u- I I CO UJ UJ

f-om ^

OD

O o

a: •4- ro C'- oo

CO ,-H I I

ro ro CO 0 o o 1 I I UJ UJ UJ m >0 "O' i-t lA C'- O' f-0 CO

^0 'J’ o

rH 1-1 "T I i

><r CO CO 0 o o 1 I I UJ UJ UJ CO O CO

*-l OC' rH m >c lA CO C- rH

O' c- I-H I

fV CO CO 0 o o 1 I i UI Ul LU

UJ in CO

I—* 'J' ro

cr, c- O'

»~l .-H >0

f—^ CNJ I cj ro CO 0 o o 1 I I UJ LU UJ cj O '£J 0^0 CO r- o

f'“

m 'O CO CJ *-4 I I I

CJ CT' CJ 0 o o 1 t I LU LU UJ o r- o m rH o

sT r- c- »M CO CO

ro CO i-i I I I I I O'4’>J’r0'i'CrtC0»J- CoCOCJfvroCJCJsr CjCO I OOOCOOOOOOOOOOOOO UJ I I I I i i i I I I i I I j I i t ^lU.'U UJLULUUJUJUUjLUUJUUULUUjLU

CJ-'J'rHCvjr-40 CDOC«’‘COCOOinaDr*“ 0'OaO

i-( 0 «-' 0 *~‘ 0-cof'“a:/'C<a>a-r''->ocor'-r'-in •4'inf''cocMs3-r'^mc\jir'inr'->0 'J'inO'Oco

>Dcjirv^r-t-4',oomr'“f'-ino3 o • O'OC'COcvjcO C“ ro rU O' I-ICVJO rH(T' I r* I i I I I I I OCOsj-coroco, rorococviCMCv CMco J ocoooooooocroo u t t I I < i I I I I I I I Cvi LU UJ LL’ Ul LU U U) U U- U_ U U> ir«'j'cosj-i-<oincjfrr'~srr'-Ofo

UAfOor O'f'^'J* >rrcC'CDNj-ir*r\jO ooo‘*-*r'~'C oh-sj-O' iTiCMir'-in fOiTfriAininOO-'O-oJ- mcoCT rH • >TcOr-lr-<CjCOCDr''-i—li—tCMr-lco

CO CO 0 o1 I

om ^ <c OD Cv

u^O -O' t o

Ul I C" u

CM Co C C"

O' ir> 1 I I i I I

CO CM o o t I U LU

"O in

sT in

CM' C\l CM fCI o CO I c U I O U

CO -o- cf. o o o o I 1 i I U u LU U

f-(

O cc

nC CM CM CM

cc o .-t W'C i-iC- I m fo Of ro I I ro CO ro o o c o I I t I U LU u u •4" c I-I <r

fOi ^0

U . CO

a rr <t

c« I CO CO

O CM CM' CO ro CM ro CM CM' ooooooooooocr I I i I ( I I I I I I I UUiU'U U U LL'LUUU-'UUJ incMOO'0's3->rocfom'£>o

OJ'4-*a' 0 'Oroccoo-'4’co, ,-t CMO'vr<MCO%rinCMrHC'C<-'

m, CTr-tOOCMOr-HOO- mr-H

IT. COf-lsT.—..-HCOO'fOsJ-i—«i-i I i I I

CT'cr ro>fMfOcvfM(MsCCMlC^CM OCOC OOOOOOOO I I i i I I I i I I I I LUUllU LUUiULUU'UULU

fMCM^C:«—lrHCOain'4- Cr-A

'Oi-tsCrH'J’Cr. r-HC-rooDr'-CL

rr, o-ip'^-r'-OG ^ r'-xCC'-OO

f—I'T o oi^cMiC-C 4-fMin <r c\ -O'er C.'r-tOCCVrHfncOfV'Cf—I I i ( I i i I O M- I J a. u I cr< (*• ( m u . I m O’ - ' m o I

CO fO. c: oI I

fO. ro CO C01 fO' ro CM Cl CM CM CM CMI OCOC CC'OMjOOOOC I I I I I I I ( I I i I I U U LU U U. u U-' U- u LU U u lU

cr <rccr co;^ or\ifTO»-'crC

r"-co^C- rocMin>cir P''Cor'-C or-cr, CT Cro, P-OM" CM >£. 1—110 r'-M-erc-afo, cru • CO CM O' 1 fO) c CM ro

ororo<*-ro, o-r-fT>frr ..rcMcvfOfvfvcN I c c c c c o o c C' o c c o C-. c c r U I I I I I I I I I I I I I I I I I

c- IT' Cl O' C'

in c c m

a

,-1 cr ru 4.1 c o- cr c u ' 'U cr Lf 1 oc ro j- m. 4. rU'

f" cC

a - c\ . in

r-1

r

n vCf^ni—'Ocf'-

o- m. r-H 4~ a a. c r>-n>tu r-iM-rHC acvo-C cr<\i—>i—I

III I C -4 cr C Cl c*. M- I r c c c C' c

U’ I I I I I I C

CO

C C' c ' c c I I I I I I f I ' cr a c' c o c c r c I I f I U U LL U c C Cl c C C c c c c c I I I i I I t I I I t I

CM IT vT C' e CC '£'CC

C •4*4

O

O Ci-ii-.

U xCst CM

C l O mM-lTr-'lT. rMff CO,—.'

Ib9
-)pQ

7

F_r;?

_.l?q77F-n?

.f,f,o^0E-04

-,l?l?6E-0?

.?q5q5F-03

-.11148F-02

.^85qqF-03

-.

10X53E-02

.63727E-03

m m rn CO o o o o till 4X LU UJ LiJ O' O' q:- O'

sC> CO r- r^ o m (\j <c

O' rH >C o

f*~ CO CO

I I

fO ro CO CO 0 o o o

1 I I I U.I UJ LU LU

>r cc <\j

,-4 ^

ir> h- o CO

IT> ifN CSJ nJ" m r\j ir»

CO >.» CO m CO 0 o o o o

1 I I I i UJ UJ UJ U. UJ

>0 »•} O *0 nT C?' r-4 O' 'O 'O ir^ m -4- <\l CO CO

>» O CO ix>

I I I

CO *j- CO ro cci 0 o o o o

1 I I I I UJ UJ UJ UJ LU

>T G <t >-t

iTv O O rH sT

•—I ro »-< O

,-H ^ lO r-t ro (J' CO O'

fO CO fO CO o o o o o

} I I I I Uj UJ UJ UJ UJ

CO O nO r-4 O

CO CO >r CD o

^ OD <\j CO u> >3- CO r> <D ro ro o

I « I CO nJ- CO CO CO O O o o o 111(1 UJ UJ UJ UJ LLI t-tf-Hi-nr-r-

•<3' O I-H vO CM

'J r-» O *-• I-* O O 4D ro CO Co oc ro nT nO

CO fO CO fO o o O o o (lilt UJ UJ UJ UJ UJ m o (7“ CO

to 0“ ro m ttj i-« •£> vj- C

OJ n3- ^0 O 00

r- rj sO CO

I I

CO CO CO CO CO O O O O O

I I I I I u.' U' UJ UJ UJ CO <—I O'

>r sO T-i CO 'O

ir* o o >r "T O GO o O' «-« OD nJ- CO CO »—

I

CO CO ro ro 'J' 0 o O o o

1 I I I I U! UJ Uj UJ UJ

CM CO r* rH o

i-t O rH >r o O r-u oD ro ic\oO‘Cor'-

rH r-4 I-H CO

I I I CO ^ ro sj- >} 0 o o o o

1 I I I I UJ UJ UJ UJ UJ m ir> sO o fM rH m O O CO

cr h- ro CO

O' CO 00 fO 'J'

sj- O' CM CT- O'

I 1 I I I

CO CO CO CO CO CO 0 o o o o o

1 I I I I I Uf UJ U,' UJ U UJ ^ C'- CM >u r- cc

CM cc) o 'J' cr i3->riOi-(Nj'

CO CM CM r-4 o cr

cc «J* ro nC m rH

III I

CO cr, CO <o ^ CO o o o o o o

I I

I I I I Uj UJ Uj U' U-' U

CM >r O CD vC »-» -H o O 00 OD G0 0'0'>£)'-T

CO 'J' o iTN CO C' 'J" cc> O CM 'J'

I

I

CO CO CO >3* CO CO 0 O O O o o

1 I I I I I U- U' U.' LU U Uj O CL' OC i—t iTi CO

CM r-* r- sD r-j 'T CO m GO (V o O

O IC> CM in CD CO

nr CO sD rr-) ^

I I I

CO CO m ro »3- >? o O o o o o

I I I ) ) I LU U- U-' Uj u ' LU o GO in CM r- OD i-iO'iUO-'J'f^

00 Oj GO vO in

C? O -O- cr CO O' CM cv rH m cv

v3- 0

1 Uj

xT

r-0

>oco

1 COO

I

inCM

'3' GO

I

COO

I

OLO

>D

>T

'J’'3’'3"'^r'^in>3‘co>3’ ro>rcocoin,j' coco OOOOOOOOOOOOOOOOO I I I I I I I I I I I I I I I I I UjLULUUJUJLUU-'UJUjUJUJUJUJUJUJUJUJ

0'f''inc0O0'inc0'j-ccc00'0'inin's3-f''“ |-^GOc^J<rcoOln*3oDOCM^'“fMroln^''-oo OCOCJ'fOCMinr'-CMvOCsJOCMO'f^GOCOO' OcocMcoinO'J-'J'cocM^sO'J'CT'CMmcD

COCMf'“ Crs0Lnr''-r-trHrHc0rH»-Cin'Of-»O'

I I I I till

cocoroCMcMCM-j-cocococo ro>^fo >3-in OooOoOOOoOOOOOOO I I I I I I I I I I I I I I I I LUU.'UJUJUJUiUJUJUJUjUJUJLUUJULU'

rjtJ'^rHr-.roO'CyCMrOrHCMCMCO'J'O

f-H <J- coo sO'3"'£>iris3‘'y OOO'.HCCfO

0^-0‘CM>3^O^n^/^OrOs3*HcOln^o*-'

O'in»o»-crooo's0f-io0‘ cocomO'O

r0'OcOCMrHCMO'inCMrH^fHsOrHi-»»3- II I I I I I I

fO CO -d" •4' ro 0 o o o o

1 i I I t UJ UJ LU U' LU

in CO v3" O' CO f'- O o> crOcD^f'-

ro CO lf^ ^£> O

rH .-H CO CO .-H

CO CO >3' -J- fO CO 0 O o o o o

1 I I I I I LU UJ LU LU LU LU

C'- O- '3’ C^ ^ »0 CO CD in o »r CO

CM rH O' CO O CO

t-H r- in rH CM in rH 1-H CM ro ro «-H

s3- ro cc' CO ro CO 0 o o o o o

1 I I I I I LU LU LU LU LU LU

r- f—< o f-H O' CM

vO f"- f"'. rH ,0 CM

CM O 00 sO «-H

>3' CM f-H ,J- >3" CO

CO rH rH CO CM in

ro CO CO CO ro 0 o o o o

1 I I I I LL< UJ LU LU UJ o 00 m i-H r- o s3- ro CM O *-H o in i-H m o o O' rH ^ ^ rH

COCOcOCOcr^J- 0 O O O o o

1 I I I I I LU LU Uj LU LL LU

O' '3‘ ro 00 CO 'O r- CD O' co tn CO

«£) CO sj- ir\ 'C rH >0 in r-f CO »0

O' CM CM I-H O’)

'4-CO»3's3-^3' 0 o o o o

1 I I t I LU LU UJ LU LU

rH O' C^ ro o CO r- -O' sO

I-H >o O' CO CO

in rH f- rr, rH CO rH CD O m

I I

s3- >4^ '3 CO o o o o o

I I I I I LULULUUU.

<r o

CM rH -r r- O' CO o O' ^

ro CO nO "J- m

s3- D rn. rH CO

I I I

>3- 4- -sT ro sT 0 o o o o o

1 I I I I I LULULL'U'LULU o 'O o

m CO ir^ CO CO O’

O' CM rH CM GO a

CM r" r"- CO in 00 CM «-H »-H rH rH rH

I I

C^r <M 0 o o o o o

1 I I I I I LU LU LU LU- LU LU o O' rH VT CO O U ‘ ro f"- O rH O GO iH c in>r

iH 'O 4- r- o a-> f-* '3- rH rH O rH

I I I rOCV<MCM4- 0 o C o o

1 I I I I LULULULULU

Go sT in CM in

CO f'- oo m

IT\ CD r- O' CM

O' rH r- in CM

CO ro rH CM CM

III (

CO CM CO m

0 o o o o

1 I I I 1 LU Uj LU Uj LU

OC CD CO rH CO o m CM c^' sO CM O 4- CM O CM OC CD 4" CM CO ^ ^ CV

I

I I

4" (O CO 4" CO CO 0 O O O O o

1 I I I I I LU' LU' UJ U’ U' LL, mcr,>ooLnr- OCOCMO>U4'

O CM rH in nO CM ^ CO CM rH 4" rH

CD iH fH r- <v

I I CO CO m CO o o o o till LU LL' LU Uj

f''- CO >D 4" rH m CO nC

O' m cc> (M 4 ^ 'C n-

^0 CM '4 in till

fOi fO, 4 CO o o o o

I I I I LU LU U U'

r-. o O' cc' C in o >3 o r- cr rH

ir> CM' O' o

(X in r"- nO

I I

ro 4 CO CO 0 o o o

1 I I I U LU' UJ LU

in O' 4 in CO CO a <r OD ^ or. CO

O' O O f" CM CD rH a:

I I I CO CO ro f<j 0 o o o

1

I I I LL LU Ui LU

>C COi %C' 4- CL f"- in 4

in >0 o o

CO. H-' CM rH

U'3 CM CM in

I I

CO 4 fO CD 0 o o o

1 I I I U Lu LL UJ

r"- O* 4 CM

CM in CM CO

CO, f'- ro o

in iH O' rCC' rH CM

I I

ro n rri fr* O O O o till LU LU LL LU

CD r- m if\

CM' O' CO 4

a 4 O' t-H in o CM rr, rH O' CM in

I t

CO CO rr . ro O O o o

I I I I LU U- LU' U

rr", rH CM CM

<\i rH 4 CO CD O iH f.- CO 4 O' ro

CM r'- o~'

i I

<Oi (Tj CO ro 0 o O o

1 I I I LL' U LU U-

»-^ O *0 4-

<v O iTs nD

CO o', oc CM

f". cn 4 CD in >c 'O CM

I I CO CO m CO 0 O o o

1 I I I Uj UI LU' LL <0 0-0 4

ro in O CM r- r- 'D c

4^ O' fM ro rH 4 rH 4

I ro ro, rr o, o o o o

I I I I LU LU- U u.

r- CO o cn oG cc in f'- rH 4) IX'. f"- 4 C O OH 4 CO m r4 CO 4 CO 0 o o o

1 I I I LU LL' LU Lu' m. GO ro in

I— nC o CM

'O a m o O CM in rH

n- fO CO rH

I

ro ^ M- 4 0 O o o

1 I I I U U U- LU

4j in 4, in 4 rH 4) 4 O If CC r- 4r CM O' rH 4 4 O'

I I

CO CO fr cr, <3000 11)1 LL U LL LU

IT rH 4 oH C 4 O' CD c 'D> •£) 4 o a OC'

r«- CO cr. in

I I I I

ro cr, 4 CO C <3 O O

I I I I LL U LU LU

r^i C <H cv; CD O' CO CO c m 45 'H-. CM CM CO m

a 4" .D ifI I

cr 4 ^ ro

c. o o o

I I I I LU LL' LL LU

<3 CO CO CM

cr, rH 4 rH

O' 4 o

f"- CM IX'' f"- CO ro l"-

I I t m ri CO rr O <-J C_2 o 1)1) LU u- u. LL,

r-j CM CO <f in o n. 4T 4 O O' -Dj 4' 4 O D

If fo I—' in

I I I

CO ro fo ro o c o o 111) LU LU LU U.

CM' O 4 4^ O 4 rH 4

O CM n- rH 4 CM C"- CO

CT- oH CM' CM

I ) I 4 CO' CO cr, o o o o

1 ) 1 ) LUU.IU.UJ O fG- O If' 4 CM o O

4 Cj f"- O'

If D, r- CM

CM CC 4 If'

) ) 4 ro CO, rr c o c o 111) LLI u^ LU U

r- CM 4) cr rH CM a CO, D- a CO CO

rH 4 c\ 4

4!' f"- rH r--

) I

CO; r I ft-, Ct) <3000 11)1 a u u- LU c in O cr> r-HCMnCT

cr, fT'i CM o

O 4 cr OL' CM a '3 4

) I 4 CO 4 fT o o c <r III) LL' UJ Lu U

rH m O' cr m CM r*" CM

oc 4> 4> in •H CO 4 O' rH in in o

I

CO. r I ro cr c o o c

I I I I U. LU Lu LU

n- o o a.^ 4 cv. o a 4 n- o.' rI-H O CM CM 4 iH 4 IT

) I I

ft' CO 4 CO o o o o III) LU LL Lu LU

r- 4 in 4 <0404

cr . ^ m GO 4 4 c CM

CM CO 4 »H

I I O 4 CO ft-) O o o o 1)11 Uj U. Lu» UJ 4 cv 4 CM h- r- 4 4

4 r-- CM <3 O O CO u'» CO in rH I—I

in0

1 UJ

in

t"- in

CM

) 40

1 u-4

CM

CM

CO

i 4o

I oo

in

4-

)

U'. o

I LL- crCD41

O'

4 m 4 CO 4 0 o o o o

1 I I I I Lu LU LU U- LU

in iH 4> OC' (Ml O CC <G> fO (O 4 CMJ 4 CO CC Cm rH CC rH

CO I-H rH cv , CD

4- fO <0 CO 4

0 o o o o

1 I I I I UJ Uj LU LU LL' in rH 4, in O’ 4- CM O' a 4

4 rH 4 cs 4

CC fG) lo cc O' 45 rH rH rH hin CO fo 0 o cr

1 I I LU LU UJ

(0 4 4

fO rH ,-H IT 4 ro in CO o

CM' CM rH

CO fo ro 0 o o

1 I I LU UI u_

<r‘, CM O' 4 O’ n

CO o 4 o 4 in rH cr, fO

CM fO CO 0 o o

1 I I U' u; UJ

CO CM. O' o o a

O' rH O

CMI o CO

I-H CM 4

CM CM’ D' 0 o o

1 I I LU 111 III 4 CM rr. O' 4 CO

n- o I-H CO rH O' rH 4 rH

CM) CO CO fO fY> O C O O CO

I I I I I LL LU U' LU’ LU

rH o r'- 4 rH

(O CD c o fO O 4 cr in ro 4inOf-Hr^ 4 rH in CM rH

4 4 CO 4 4 CO 0 o o o o o

1 I I I I I UJ LU' LU u LU U

Of m O' If'i GD iH 4 h- m o CM ' CO r- C CM' 4 O' 4 o 4 0 CV 4 CV

CMI in fH rH CD iH

I I I

O’, 4 0 c

1 I LL LL-

<M, r- cr IX

'

in nu o CM rH a

fO, .4- 0 o

1 I U- Lu

CT- Hj in rrr CM rH CT| fH CM

4 D> 0 o

1 I u u

rH O

CM 4

CM IH

CO a

CM |H

I ) ) UA CO ro Nj 4 4 0 o o o o c

1 I I I I

I LU U . U U U LL r’“4cor^cD'U

CT cv. in 4 rH CM

fOi CM CT CT m I-H in 4 rH fH n- rH a fH I-H 4 4 4

I I ) III

4in4fO<^iCOCMCO, CM OOOOOOOOO 1111)1111 U U LU u U' U U' UI LL f. ("-OQu IHIX-..J-4 ro r-CTOcr'iO'CT’fMf04 00’CMOC4400or

P'-0D4P'-rHCT4 4'CT

in. fOOrHCCiHr-r-CMi

I I ill II m CO ro cTi fo 4 4 r in cri OOCOOOOOOOO I I I I I I I I I I I LU LU LL U U U U u U LU U

C m ^ in f-H ro 4 If . oc rH <r» rH4CT 40CO 44C0Om fOf-H4CMf-Hr-iHCCCMlCT’rH 4 roa r-CMO' CMrHCM4C

rHin4 4CMiCTinir i-hsOi-h

I I 4 4 in 4 4 4

c o o o o o

I I I I I I LU U- u LL LU U

rH 4 CD fO. C\J rH

(T CT C». CD rH cr

r-- 4 CT c: 4 ro a r- 4 D- 4 CM

CMI 4 r- rH CD 4

I

I I P'..,X4rOCr' 0 o c o o

1 I I I I Uj u LU LU U

CD CM 4 4 in C^’CJ'CM4 0' cMCM.if'r'-r- 4 4 CM fO cv

I-H .4- r- (M rH

I 4fOfrco>j- 00000 till) LU LU LU U LU

CO cr CO ft-; 4; O m CM rH 4 4 O' CM CL r^

rH U (T. CT fO tn fH r- rH rH

II) I

CO fO CO CM' <M CM o o c o c o III) ))

UI LU- U LU U U 4 CM* O 4 C 4

in CM h- ro 4- a

CM 4 4 CO 4 4

cv; 4 f"'- in o fO

n- 4 CD CM rH <0

I CO CM.’ CO CO 0 O o o o

1 I i I I UULUULU

r'- 4 ro in 4 o roi u-- r- rCO CD 4 CMIC'- O- 4 O' rH in

<VI cv I—I Cv I-H

I I 4 in CO ^ 4 4 0 o O c o (3

1 I I I I I U U-' U’ u LL U

(MrHiH4CMr''“

in CO. CM' in CO 4

CM O ro rH 4 r-- D- O' CM D GO 4 4 rH O' CO O'

I 4 4 4 O <3 O

I I I U u LU' C- CMI CM

CO (3 O' 4 O O' a 4 r'j (o cv 4

4 4 4 000

I I I U LU u

cr 4 o

r-- CT CO 4 4 ^’- 4 O D- CV 4 f“

4 4 CO O O C-.

I I I UJ U' u 4 su fO 4 v1- CM

CMJ U . CM. CT o n- rH cv rH

4 CO. 4- O O <3

I I I U u u

CM 4 4

CM' >C- fO, 4 00 f-H 4 '3’ r- ro I-H rI I 4 44-f04CMn O. D.CM ‘-'0<3;o 0000 c o 11(111(111 U ' U U U' U U LU u U U

D~CVf\ i-H4COrH<Varr'- iHr-ocra irco^ cr o <300004 CG] 4- ff, 4

CO CL 4 (O r- CM H' o 4 IT' 4 f"- 4 CM 4 CV. r- rH ^ 4

I I I I II

CV, D CV 4 fo ft, r«i cti ro ft-, 4

^ o O C o o o O O’ O C' o

I I I I I I I I I I I ULUUUU;LU. U U U UU

Or~.r-Cr. CTC04r-CV'40

f--- cv: U rH C\ (\ CMj U CM' CT 4 I 0'C'Cri4f-'r'“4CC4'<M|

iHOOr-'-u <r4u 4 0 4 I rHrHrH<o,-if\.rHf-HrH,-«Cr»

I

I I fo ro fO fo. f- 4-4 O C C C O o O

I I i I I I I U 1- U' LU LU U U

IT o a c N r- 4

c D- <0, o f' 4 0

c 4 CM r*-' If r- a

c. r- ro li 4 r- <1 a ft c*'' in 4 <0 ^

I i t I I I

II II

ft ct-iroroff. ..j-ft-Nfr-ft c O C C C c c c c

I 1 I I I I I I t UU UUUUUUU 4 4'OClTOCMr'-4 cvr'-c\u-ocr"C4

ar-r'-ft'c: irrHco.

ft O’ cv i-H IT, o. r- .X a 40Ctrt|-HC I'-'r-r- til II I

I III

fo, ro 4 fo ft ft) ft <r. OOCC'C COO I I I I I I I I U. U U. U U U’ U LLIf'. fOi r- C If' C O

IX CV r-nOCV ft.C 4

cvcrrt, <3CMaif f''- Lf <3 1— r< o c cv. f^ r- If- a r-H ft. ff. r-'

I I I

111 I I 4O ft' (o <0 ft. 4- ft-, fO ro ft 4 4 fO 4 Ct' ft ft, ft-)

I ocooooocococcrrccoc'

u I I I

I I I I I I I I I I I I 1 I 4U:U u. UUU U UUU. U’UUuU U'U

4rt Cft nl Lno<vjr'-f“4inc'cr ccr'-oX ftU'. ccr'-inOinU'cr O coCTrH,—<4CTCV

CTO 4nX u.in, if4r''si.u 4 4 <n. r--c 4DCTsii— o 0CCO4U ftiu*-r-^4cMctr'.-r'.-

• i— I— rHCMCVi-Hr-ir-i-HCs. CLft f rHr-.fOf\-

I

I I i I III I I I

II III t I t I I CM<MCM'inCMrOft4ctm4 44Ct4ro. OCC'C OCOC OC'CC CCOcCT

I I I I I I I I I I I I I

I I I LUU U-UlLU'UU UU U Uu Ulf. U

4, (MCMr-^f04sjf'--CMa: cv >X. CVOfV Ok rHf\4i— CTCTct. 4tXrHO 4 or r-0|

4fvr'ftr'-0'44ft. on-oor n 4 ! r"C4a r--f'--rH4 ft.co’r''CMctcrr

,-t|-HlX.4 |-Hin(0, 4 CM 4 r,i-HCT I— rHr-'

I I I I I I II

ft ft ff, fO, ft • ft", Q-. ft ft COCCC. Or oc

I I I I I I I I I ULiUUU ULl-UU

1—.r-r-cvn cjcr'-rt-. 4 4C, 0'44'U'C4

C 'L vL 4CT4ff '£'4 4»-'^r-a CTO' r r- rH 4 4 ..j- ft, sf .

ft. o, m If, fr, ft- ft'. Oi 0 c c o c c c o

1 I I I I I I I uuuu uuuu

4 c- 4- fi O’ vt ft- vO CTCMrHG r.'f-lHCD oacv si 0rH40'

C CC ft 4 4 4 ft r

D- 4 4 G Or sf. ro

4for o fo. rrf0 4

<_ C C C o c c. o

I I I I I I I I UUUUUU'UU

G sr4CT r'’“r''-n'C' 4 CC r- rH 4 4 ,£ 4. G (Ml 4 G cv- s£. r UI xf C r— O' 4' C

u st D -U. ,_.rHrH^

I I I ftftrtGfor’

r r c <3 c C

I I I

I I I UUU UUU

o. I— r. c G_ cv O rH (V. rH cr- cv rH 4 a- -.L C.' CM c 4 CM 4 . 4 fH

I 1 I I I I I

f I 4 fO fo ft. ft c c c c o c

I I I I I I U U U u u u

X u G o f" n

r O H. C 4 LL

ft- rH G c c. G

r- O CT ct' CT rH

CT- 4 O in CM, 4 III I

(I II

G ft- G G G G G

C <3 C c C c C

I I 1 I I I I U U L-- u U u u

r 4 G a G CT CT 4 rH r- 'L 4 CT 4

a G O (CT cv CT 4

O a CM O , fv D 4

I-H 4* 4 f'- G. 4 Cm

I I I

II I I 4 r r G G st c c c c c c

I I I I I I U. u u u u u

c ix- a I-' c ^ CTO 4' r n C 4 4 4 c CT 4

C CL CM rn CM CM CM 4 CM CM. rH O

I I I

Iu

CT- 44

4

I

G 4 4

C_ O CI I I a u -u

vl 4 If' G a o

CT O O

C. rH 4

O' 444 CtG444in44GiG

C' O C O O C <3 C C C. C r> C

I I I I I I I I I I I I I I uau uuuuLuiuu uuuu 4rOGU'04CTCM4r r. 4 4CV

4>GnOCMO'CT OCTrH4CT'QC 4G

CM 4- in O' a 4’ C 4. r_ IP G 4 4 r-H

ix - o n 4 - G G O ' 4 xX G c IX 4

r-«4rHC'rGr-'Grt'G CT 4 nL g4

r. f\GirMriCMfMGG4 Ct'frG44 r o c C C C O c c r r o o C' c C c.

I I I I t 1 i I I I 1 I I I I I U'Uuu UUU. uu'UULauuau

cvCT 4 t—* 4 G 4 G rvr c ir'40’r r p.4CT-OCMiGr'-Cv CTGCCOG444

CTIX r-CT'GCT-rrG'4f— U 440 CG, GvtCTru Gxf. rGrHCTrG-CT4G

4'G r--4rH4rHirrC f- rHr-O G r- ill I 4 4 st 4 st 4 4 4 G G G

C C C <3 c c c C C C C

I I

I I I I I I I I I U uu UUUUUuUu.

r.O 1- 4 CCTr-xi rH<3r

c. CO CTaCT-GCT4f-HrH 4 4 0 4 4- CT 4. 4' cv f G 4fVU OaoCr-OOrH4

xUCT-<M ft. sUCT-G^^'CT'r rH

I 111 I I

I I I t I I I 4G r4GGrt s.1-CVGG-r G 0 cr c c c c o c c r c G c

1 I I I I I I

I I I I I

I uu au uu uUuuu^uu

a Gix c?’ OstmCT r Of— sX ^ 4. I—‘GO oOln^ CTrGCif—

'

srC'CCv4COi-H4 4 CO'<M

CM 4 I-H G, G CM 4 »— O; C 4 rH CM

(— CVCTf—'O. 4 <Mi-'G a 4 «—

'

I I I I I I I

I I I I I

i G G 4 vt sT 4 G r st r cccccc c. c

I I I I I I I I I u U U U U. U U. U U CTGar CM. a''i4r

CTGG rH4aCT4'r- 4440 rHCTrH4-4 44. C LTGC<-CT4

G 4 a in cMj <x CM G. 4

I

I

160

i
OOOOOOOOO I t I I

I I i 1 I LULULiJLi. LULULLiU-'LU r~Of-*oc\Jir>cciO>y

CJ‘<VJir'<\j«-4 NC«—CrHfSj OOOOOOOOO I I 1 I I I i I I LUtJ-LljLL'LL/UJUJUJlU

o->0 >3-'-HO'OOfnm

fNJOC CS-i-H-J-Om^CT

ro-J^ ,-hQC .-ho>£)Oco

I I I

<r>j- OOOOOOOOO I t I I I i I I i

.-H ^ vf ir> m o (V- iT\ o OL r-. iTi O O IT* m r«- o

>j- sT

c\j cr CO o m

m <r ^ c r-t cc o o nT ir> fv r. f-H CM nJfT. ITv ^

0 o o

1 I I LU U. UJ

cn CM ^

r- >c r- O f-i ^

CD »-H f\j

1-4 nCi cr

I I I

•J- sj- -J- 0 o o

1 (

I LU UJ U. CD

Lf> cvt C 'J- vp CO CD

CD a

•J- nJ- nJ- O O O

i I I OJ LU LU 01104 0 m CO ri/'' I cc nJ- sT>

tf’ vjr». cT. >o coo I I I UJ u U

r-1 CM sC m u . O'

OL. r- (\ CD CM O’ f— CM fTi

m c^i COO < I I LL U LL- I—I nC> f-» cr CD

OL o O nJ- 4J

»-« I—t ID

rr: »J nJ- 0 o O

1 i I LL LL LL ^' UJ O' CM vf r- cfi

i-i r- !- if CO

) I

•4

- sJ- sT o <r o

iL ‘

' o nC o

CM D- sC CM O') .-H ID a a

CM >r cv

t

•4 nJ- CD 0 C O

1 i 1 CU LU LL >0 0-^ O »-H n4

ID CD o ID sT CD ID f-*

I I

>4 re : cr C- O O

i I I LU Uf LL u: o 0- O' I—

'

00’ ID ID

CC ID ID

«D t ,—

I

I I I

>4 »J- C') ^4 O c^ o O (III Lu- U U. LL Dl CM C\ IT r- t-* cr D

cr CD ^ o

«4- o o o

rH D- nT

>T '4 sj- sT r»"i o o o c_; o

I I I I I LLi LU U- U- U

ID r- cr O' CD

O’. 0- cr r-J D

ID I CU o D

«-• O' f-i •— CM

cv O- vO sL •—

I

vr Nr <r ID 0 o c o

1 I I I LL u LL. u

a. s4 u r- u. o C O- cDaOiTCM c—' oi

CM D- D

>4-lDN3-<4-vr o o c C c till! U U' LL- LL- U

ir o ic u f'w CC a C-rO'Do- o c f-' u: o r- «j D X

a CM tv nU D eVr-tOr^U O r— (7 su cr (J r—'O-iT r-'-cr D>TiDUi>Jn'D«—

'

0^00 Ni >jf\ era

cv CMi— ir— ir->0

D- iT ^4 CM C- CC ,D C CV >4 u. a O' f-H re, r-i o c a r-- vT, -c-

' vC >4 O D- ID Ce •-* COC-OO O OCT

D-C CCf'-li C^,-.o^

u r"crocvNjsCD-o

-O rr 0> CM 1—• C- O Ci. D

o r- tDO..-* 0‘ sT^rD

a. CL o. a a D r- O'- t"-

ir re r-< c a rC '4 ^ re ir- >£.' CT CD

0- >C u • >4 r»- o. CM C a u- <r cv c o

o- uj >u UT U nC

DOCLU.'4 c\^0 'J>TD'n’'—'CD idU:

I— c cr G . or . r- >f u vC-4'T-«OD'iro"i—

I

U'- ITi If >4 nT nJ- sjDlDO’,—)Of'-ir '4CM crC'D ij uvD-o c-ho. M'vjC'TCv.. iOCP-0 CD idCD,—O' vT'JCVi

rro". o re. Cv.fsiDCMrv.

CD f»- s4 >4 O I O C C <- o

I I I I I U. Li LU LI U

P" o ID LD «—

'4-Cvrror'4

' cr cr '-D ID cv ^ o cr O

CM rv D-

>4 U'' M >4 >j c e c c_ c

I t i I I LL LL u. u LL C U .<4 LDcvr<"i rr cv ID cr a D- c xT o

C' O re ,-1 xT. O cv vt D If

i I I I i

>4 o xx-! x4 O' re f< xj p O C C' c C r c- C c

I I I I I I I I I I U. LLUU LLU-UU L.-U OCD O'DxJi-^xL C' x4 MxTDciC lDaOOxT

O I* • D LT- 4 D O f" ID xD xCU O q xffe ,—,p U),-i

O' I—' o LD cr. I r-i a »-i

I I I t I

DDCMO eVDO. rv DiCMCV D

I t I I i > I I I I I I t I I I I t I I I I I I I I I I i I I I I 11(111

iS‘ <I

s:

II II II It II II II II II It II II II II II It II II c. croccror-oc^c c c c-crcrcoc

rep'rep'O'rep p' 0 re 0 ' 0''4 '--eDp-r're

c c c r' C' c' r c c c crcoc-r-cc

I > I t I I I I I I I I I I I I I I UIJ.UUUUULI Lj-LLLLU LluLLUUU

u D c f'' f'' ID p c CC-. r- f- U o u.' o. f— O. I—

U M cvev «4i—c n't—ix4x4r-^xJ-0 t— O O O f—D iDC'C-t—o t—lOcvaoD. -or-erco

x4 xC D ,-H o xT o O' nT. cr t-i I—• O'* a o o ic’- f' I—It— CVO‘0 >4 x4'4 '4 xJP d c cl cvee*. r

II II II II II II II II II II II II II II II II r~. r <- c r c c- r cr cr c? r c cr c“' c- >>x>>>xx>vx>>x>>

o'ppre. pprppppppvcTp-p cr-c~cccDC c cT'C<"C r cr c

I I I I I I I I I I I I I I I I U U U LL: U U U U U L> U U Li U Li u r~U t-O pD-rvO x4i— CC xt ClT'0''4 i-'0«4r'-ira c.-rea xcorc a c^vo

lfu“a OCMOreor'-OD's4x4re-C_r<

D vrOre'QLircr-s40i-ix4CCrOCL

p- o^ CM tv r-) f—< D. cv re, p xj M Lf' ID nU IX

H II II II II II n n It II II II II II II II c r c C' cr c ' o o c ' c cr c o o c r

> > > > X > XXXXXX>XXV

p ppr^rep-fTppp p evo POO

o<rc'C<rrc<rr<~c"Ccrrc'<r

! I ( I ( I I ( I I I I I I I I LLULi.aLLLj.LLu auaauL. llll

u. c sTxLvCxLO c;c«—‘csxcr-xfxv^p

v+r'x'JC ua DO P‘cvr'“ircraDx4 ax4 U.i—<ffx4 '4 O 0Lifcra U't—Dcv

sf c'eUIif Oir crC'i—'fex4 lDxDDr'~a CLOOC CL'CCG r-lt—It-Hi—It-Ir-Ii—

161
a CO r-iT

sTcr <2 <\ sTu r-o-

>r u' r". rv; r-’ c) cf.Nf»TjvCa^v£<r

r-i r-t «—I ^ I- C_ O O

I i i I I I I I

O' I—• C to r- C a 'Ll i-nn.Nj^. a Otv r lOO- Ooar-Ni'i-i/ sjocv

c\ crf^iT o r-^fTo-ior".

C o 0

- 0 u-ou,cr. ao' ^ I » I I I t I I 1

I

•Jo Oa M >j-r',f-JCf^ C I—’ rr >r CL C Cv C IT' r-n-<cocrr-r-'Cu vj »-'(J‘Ovr(\C'Ci sL>T<v

o'h-r- r^r'r-sO'L.'L sT

i I I I i I I I I I

LO r , I— o h- U >3 (\ £- (rcKrvf-.c.occar' Oovivtocr^tf*^

>c- u" lo IT- tr. nt o vf -sT

i I I I I I I I I

a 'Lvjo. ca>fir'r‘

o‘ 1/ o- C t' <r vO n" ' 'L U ^ C. O'. C\ c O

'C?-r^iof^i— a

>4-r‘p^, ft,o.rr<\ <\ ix

I I I I I t I I I

r-ah-ti O. f-(0 CT't

rt a r->L. ir ^ >jO". (x ojccC'i 'J'ooa nl

C\ <\ir— (r-*i— lOO

I I I I I 1 I I I

>jCvCfX>jsL:C O ^raOasf^Jf^C.

O O r— fx o. <

>r fx o <v nJ- nT' O' C' r o o c c. C' c- »-

I I o

II II II II II M II It II II Cf~Cf“COCr c~ c

> V> >' > > >^ > > >r <r c o c c c

> > > V

c rcc<rccc-oc<ocrc“ccc<-r. r-c >NVV>vx>>>'>^>>v>>>'>.>x r c c CL ' c o r r r- o c c

r c c trccrcc. rcr

i I I t I i I I < I I i I U l- IL U U- Ll U U 11. Li. U U U

C cnr-u'ca rvvfaC'— c r'l^CO NTn'r-^OLO ff O ccr

r-CL/r'if-o-ou

Cr-'f' <t -L o: O ,—-f\o f\ r\. r.

r-H c\ ro t\ <x- cv cx o O' c". r<* r” o'

rooocvr'f'rvooro rrcrcTC^rrcTC-rr

I I I I I I I I I I I i U. U Ll U U U U Ij U IL U. I-' ^L^xL<,^_cxv3^^.^ r o, vi>/

r or-irovcrr'Ou aa COi— UCL.O a o r-^a csL CNs+iOLT'^iro'r-ca ao 0.0 oi O' O'. O' o 0-0 c o r

r r r or r r o o o ooo

roc <^rf~C('^C'r^rr~

> I I I I I i I

I I I I I UlLUUUd Uli L' UUUU ci T' o u ' a c o .— a o O' ir u c Osj-ODirO^o 'LO'Os.tc

cxo looa o CLo-CTf— Oo <f 04 <3000>x4.i'Cr'>f'4'lf'

o O <XiO ,—.1—

o or oor r oor o*'^ c'c'rc'rr f- ccoorI i I I i t I i I I I I u u u u u - u u u u LI- u l: vJi— v3 Or-^O-u T- r-C\ O sj 4 c-oa4 a4 >0001—0

u oa<fr—iOfr>ro4.4.o

vJo-'O OO 4LlT

C C r ccr <r~rcr-f-c^c.

I t I

I 1 I I I I I I I I U U U Ll tl u U U Ll i: i; L u

f— rciou oc^o oc^-o

4 0 >c 0 vtO 4-4^^iro c c ccr'-oa >jLi >jsC'4 Ci—

O I-^C OvTO c*—‘O'OC >3 sf 04.if>J-or'.syu 4Toaaa

162
rfu r'Q^ru kTCLC

r-u r i—iOaa sjfv,-. m>L.r^a c o Cf—

'j-'A.a c<^'0’r“ CTf— r*"' r— «-Hr— fvCVC\CVf' O.

cvo-stooc .j-u r cr

Cr r IT' r cv. o Q- '>3' r r'<jiT'ir'aa.oc.-<

irr Qi— r'irr^(7<Vv.r

rt« r" n* sT vj ir ir'

cr, hCr-f\.>j->nao

^cF^iT-r^fvc^a vC'J'f^'

c\ - o - ^ IT- >i- r- r- a c o

vOa Cr. •«j'*.a

ir'ifN'i/^. vC'i-'i.

(\ xT'i-crar-ir'. irr^ o

.— cjr^tT'r c\ <z a yc 'i c—‘t— sj LTN'l-'Lr^CL C>^n-ir r-c .— r if>r- r-d-OLa O.CLO <J O' O'

I—lo If r-G c.fv-4-Nf:cx.c

(^ »— O u ‘ nT C.' C cr >4.1^, cr <-• c; r-H cv r . >3- IT u

CT <\ .f GV O C <T yC a c

O COC OT-i^,-.,-Hf— fv

(\>3U r-C7i-'r'lfh-0»—

fOr-fc r^ ott 'Cu'

cc. o a c t—t f\ o ,j- >j- u' si, c\ 'j ^1

' O' I ir- r- o 1— 0 . fV C\j f\, <\ r, m O'. C*'. cr. <T <

c cc c~c-rc<rc coccc<" >>><>>>>'>>->X>><>><

r^r’o r G r'rrr«'r"r» rr<jr*r r*~ C r c cr-occ r'CCO<-C'

I I I I I I I I I I I I I I I UULiL:.ULlUUUL>UUL!.UU »-CVa<>fU<\r-Lrr*U OlTr-G

f— iT'ar'r-vti^cr'-'n r'OG.- <t ^ cr a IT ft U’ ft, r*-. o. . U' fr c H' U i i-'r-.— LfG C'<^'T-j:c'rr'-,-«r-cr

crr'~r ^:u u >j-rr<\<\i—ir'-t-trvrt'

II II n II II II 11 II II II II II II II II II rc (Tcccc. cc'c'cccrcoc'

ftft r"frft fs <\ rrfrn-ffffrrr ffC"

C 0 <-CCr CC'C'CCC'C Cc c

I I I I I I I I i i I > I i I 1 UULIUU UUUUUUU U Q u u

>Jsjr'-rt,4.c. <ioa(s-t— -X.Q i~r'~Nj- Ci-><\s3-r ff CvONT

a r^OjU-f— t—.G ft'CVC’C ff, r-«f— Nyor <\-crr'CO'>rc\ar-c-,-ir-''f

u «rao-Of-‘r-iacOffa'aacra:

II II II II II II M II II II II II II II II II II C r- C OC~COf OOOOr'CC'OC >>>XX>X>>>>X>> > > >

r'o fto'c n'o rtf Gr-. c (^ oc f-c<rc'rc<^crcrGcr c<L<r

I I I I I I I I I 1 I I I I I i I u liu Liu UUdU-'Li-IJl-^U UUUU' OCu •—••4‘C-sja f'-u a c o iT't-'.-t'iOcr'*Circ irscr"

vi c. 'Ja^cCt-'Cai-if->r rvf-iO Njr'N^O'f r r-itrC'f-ia ffOt-ir-i

f'-lTfr I—>{\.n3 iru.u'r r-i crI U a *f-i

II II M II II II II II i; II II II II II II II OOC CGOCCGGCCOC OC

r ft-fftr rn'Cr r'nrrr-co

crcccccc. ococc ccc

r I I I I II I 1 I I I i i ) I u U till U-llUU U u LLLLU. L1 'U U- <rci— ^l.^~U.^3

*-‘ 0 ^ Gl^' 0-^-U ft co<NNTo<vf^or.>3-r*',fr'-v3^r C' sL'Oo^G cr^*-''TOU'ccr''f—fv

O-C'^fVQ.C'C'Cv'Jht— C'JCvTnJ'J

LTCL't->J-lf'>LCC0‘O0 r163
vL-crocvxf'X.a.crr-'O i/'r> O r^sT sj-fvccr aac .-'<^f' ^ sfu.scr-rsC or

.—I r- ir> r* cr c

-4- c\- o a h- IT O C‘ f-H 0^ or >j- ir> r- o-

• O Q.J si >t

•j -Jir uvir'ir'>£. r^r'r^i'‘t^coaa>c£,CLaao‘a^

ICvfN.C^CVCOC f\)<\<\.'

II II II (I M II II II II II II II II II II II <r c c <r <r (O' c c- c C' o cr c o o o >X^>>>'xV>X>«XX>>->>'

II II II II II II II II c-occrcc er oof~c

X X^ X X

r <0 c cr (o XXXXXXXX

r~ rf c' O' r^f^o'r'r'r^ 0 c C O C' c c c <- c

1 I I t I I I I I I U LLU UU’I-ILUUU U CO »-i r ir u c If o

r cr>ya.vfa r-iNsrO'

a C'r'*— I—it\ra c^j lcr^^^ n-( CT to rf

CC'

rr, re cr O' c f^r'C'r ccccoccor I I I I I ( I I i U ' U. U U u. Ill U Li U

CL vj si <N. C LTsO'C. 'O Cvirsf r-f' r’'r'sj-<

O.fT'r^sl If. -sTsisCO

rHsC s^r^c cr*^i^rT

r-i<\iif r^u o r^U'Cv

«T O’ r" r" r~ fT' CCCCCCfT'O*"

I I I I i I ) I I UUUU-LIUUU. u c u fvc«jr-r». cc

irr'O ifr^fT-f^sj-a scr^c (\stf^c<v(\

o.

r-- f— c\- o . r o c. rr r rr re. r* coccr cT'C'cr

I I I I I I I I I U U U.LLU LiU U U

sC C f- CV C “ C C >cr-s£u cr a <t C

r- CK o sC (T <\i <r r-t CC sj- ifs c. C t-H C Cs. lf‘'i.r-i<VO .—ii—<*— r-i

164
T

3

T

:^7

F-0

?

^

+

tt- * *

-a- «• <«

K-

•W- -K- *!• *•

•K- •«• *

fr *

M- -B- •a- -B-

-B- »

*

B-

•H-

•fr *

• O

• oo

• I

* -M- -a •«-

•a

a- *

B- *

fr

-» •«-

• «

• -v- *

• ^

• -fr

•a-

« rt—• m

• I

i—i

• o- •

• I

«• -K- a- * *

-aa- *

-K-

•a- -a

•a -a-

«• -a a -a •a

fVj

I

a

a

<r (\ I— O’

^3

- si' cx/ a

a r- IT

sj- c a

Csj c\ I

fN-o».(\,C‘.<\‘(\,(\cvrs^o.cvfvcf. o'rr-rf. r"rr^f<i<^r«:r -Ju

<> <r c c oocc c>oococcoc:c-cc-.r cc-cr-ccooc-occoccccc’ooc-oc .-i

I I ( I I I I I I I I I I I II

I

I I I I I I ( I I I I i I I I t I I 1 I t t I I I I I I UU U LLUIXLXLL.UU LLa U-U.Li U-U'Ll-U LLUU_LLU.[j.ULi.U-U. U u.LlU-LL.U.U_U- LiLi LLU-U-Ll U UJ <N.i-HOoaoLr-r' oc\;f\,HCO o ar-h><. sfcrcvi^^,<iOr- ^-c fv-«o<v i-< C rOfnt^)<\;CV,tVt\ t\. CV<<\,(\ Cv(v.— li—irnt-Hf-iCya. U U rrojfN-«~ir'<v

I

165

MAGMITiirF

(YO-rviT)
Y0=

-’.3651Y sjcvOa.Nrsyfvc.o r-

•sj fv c" cc.

r- u r-, ^ rr c 'i r f~^ rr. \jr\ sL <X’ cr CK' r- < m <r

T-{ O' ^ <T ^ Oa7s£)*«J a.r^h r--r^r-<- sO'irvi c or 4: n3 c\. fOr-LT ^<^-C:xsCv3 : c a r- u r

r? cc -1 c

I c c CL c r- sO If' I c, a c o r- >c a 'T 'T <v c oc

O' . c\! r fv Ov fv .—

(

.CO c c\ ^ IT. r^

s3 n fv r-<

(N. c cc 'L

r >r vt; cr. c tvj c. ir c c - c a r- xf sCo a CL o.

I I I 1 { I I ( I I I I I I I I I I II I I I 1 I I I I i I I I I I I I > I I I I I t I

orr-ccccrrr c c c <z r O' c <~ r c ccrc >>>->-> >>>> > >>> >>>->-> > cccf-'^ccc'crf rcc<rrcc

>>>>>>>>>->>-•>->> > > > >

c-c rc rc c«r cccc <rococrrc <- (T c~

I I I I I I ( I I I I I I I I I I I

I I > I I I I I I t I I i I c C.' c o c

U. Li U U Ll.

C r . C r-< 0 P'- U cp-,-<<rr-<VnH(N

^ p

• cxj-r-

<\.C'p.O-

• 'T nL sj

U U LL LI 'L 0 ’«T'i C.C\j

r- If xL si o a rII Li U l' P ; If' P xL

O' O li p c

I I i > I I I r r r C- c cc.r'ccrc'cc'i. rccc'c-c.cc

t I I I I i I II I t I t I i i I I I i i 1

i<\ifx£sicc. f'-iff'-rx.

a o xj >r p cr o xj-rp-f >fxj-Opr-ar-xjO 'r iPP'-'il'- O', r p OLr(vc

U L. LL u. U’ Ll U LL U PCsLC CUxixfP.lf'

Cfo p’o-p'scif i'' p p'Px'xjchLP P- x1- P CP

^ t-w r-

»-i p- sj O' f-t OL sj xT P IT p C a xC xi xf p

( P P sj- P.) P P P- Px.

166
n

1

YT=

P

??76

'4 fVCa'l-'7 <\i—iC-f^ ^ 1-r xjvi.'ac'<vr’U'i

r-ii— c:cr cc»'-r--

r-’ a r- >j cv- o cr >T' ''T cv arr-r-r-r-sfst'txf

I I I I I I I I I I

II II 1 ) II II II II It II II c c c c c <r <- c. c c V> >>>->->>'>>

c< r r f' cr r*- r' r- n r c <~ r <r c

I I I I I I I I I

I U li U It U u l' u LL C i-^a t— C’ ^ C O <r IT. sC r a c 0- ^

iT\ O’ 1 O" C- a r’^a<~'cv:>j-xra co o'

o". cv c o c a r- sC' 0 'XsC <rf\ C r-U'O

>£jia“ u'ir'xj-'4'sr,jNj

I I I I I I I I I I

II tl M H II II II II II II c o r c O' r o c c~ rO 0"O'v1O O~‘0 0'0 o r c oc c r

I I I I I I I I I I UUUUULlLLUlLU

sj a o. e o c u ^

r-(\c.c s^^^ ^c\.oa

o“tr. <No

ir r* >T lo sL' o c or -j o O^lfV-CCO'O r-ifVir~- 0v

't. >! <\j c rr' >t LO O' i-i o rU" r-ltV'sT^rO- Or-'o'i

u >x o ro (V f— Cl CJ (j a h rrr'iro’f-io’r^'J^rvoa

o. O’ o"> o^. I fv o. o<. C\i Cv. ,-4

I t I I I I I I I I I

H It II II II It II II II II II cr'c'crccc'cc'cc

o O’O.O'O'O'O O O' f^O’

r- c c- c c" f c- r- o c r

t

I I I

I I ( Ii I I LiUULtl-U Uti LiULa

>- u If L lA O' c u . f\ ir O' If o'a i/'C '£ o- uiQrCN o vf h* o^ u' ir. o. vj c

CC o~'J O -JO (>o r

t—.U<\C'JO"Or-HaO'f^

iro'. i-iO’cr Nt <3 fvc^^j ir'r-c-oo-^Ni. ccc,a

^ru >3 ^ 0', cv. I— oc<-

vTxjojocr ^ vjfvioo. ^,-,r-i,-iOOOC OO

I I I I I I I I <-

II II II II II II II IT II II rcccrcr'f^r cr c >>-> >->->>> >>

(NOO^OvOO OfVO O C'’~c’<~-<-rc'c<rc'

I > I ( I I I I I t UUUULlUUU. LlU coaao-o a Sj-ULU-

.--o if'cr-a a Osjo

a r-aif f\.U'oc o’r—

cio>cf (vr~u-<rc<->^rv ^U (\ rs.c’ •'*- r-.f-HNi.Ov

'O’ >f a c »—I O' m O’ r-H O'

'T nJ- On C o lA O' I—' o

rHC' O.NT'OlT'.C. r-U OO

'Jvj.'crooN'NO'vOaCo C c:- O • r-l I—I f—

1

On O <V

II II II II II II II II II II II C' c* c c c. r- c f~ r o o >>>>->>>>>>>

0 O' f\ o., o o • c\ r o (\ <\' cc r'C'Oc T c c c <r

1 I I I I t I I I I I aubUQUuuu. uu

I— r- u’ . u’ sL >c f-i -O’ c Nf •— irr'-f-Ho o r'-C'*— ^ c o

O’' o c a f\ O' >1. c u- Nt r-r- LToo 'f O'O *jo >r i-if-sTf— OvOr-i'* <r Ov r-H

U' vL a O Ow 'T sC GC C. On nJ- vC’^Jr t—*o r'lro'. <v-Oa

I— f\. o. o. 'J IT. Nf; r- cc a o-Of—•o'Li r'-O'i—loirrOj<\ p 0.0. O’JfrivJ- >r>3 vjII II II II It II II II II II II ccc ooc c c ooc >->>>>>-' ’>->>>-

O 0.0 0 0. 0 o o OO O” c r- c r c C' c o c c o

t I I I I I I I I I < U-Ub-llULlUt- ULiU

O’ O O. r— O vL. f— nC O , U . >i. COO oa Ncio-j-r r'-h

u oa o.O'f-or'-r-'—'si. ClTli'Q.LOC'O.. a O r-.,-. f-HO.O O. Oif-tU'i-.r-U (V

167
YO=

.^9965

r- a r-< n-' U' r- o- sT fv f-» c* r- If' r- O •—I cv cvi f<i in

fv sj- ^ a c <N <r

Li' If m LT\ >f- nO sCrHf\-J,CCCO0v ^^

n coixC'J^f^'t-'Oh

sCf'-r''Cco-Oi-i.-HC\ xDO Oojsrr- rr ^ r<

a Oi-'r u f'-O’i-H

IT o~. (\ O o vi nT rf 'j in >c 'f' f" a. O

IT. r^a^ninr o- ixcija.oaac7C

ru n-aorv.'jxf o o fi 'j m lo

<Nj <r si rr O fv sj v£- OCC Ci-Hi— r-lt-'

o OcVvtli'f'- O r-<

sA. u^ O' f-J O' If sj^ ^Lr“ac7•pCf-JC^u

rrCr.si-xLc>,_jr'

I—* Os (V f\- a> Oi

O'Lrr'-O.—

Os o a sT L» 0 < 'J sj m 'C

u ' r- a I—• m

01 fo py 'J "J"

cv sj sf a c. O. O' '4- CP a c o

m r O' I—> <r < s3 >j- in Ifo, sT 'L-' a- O'

nJ o O cc Os 0< Oj .y

sf. a. c_) Os' v'l- iA in >4- >L sC

^ o m O'! m

sT cr. sC '4J

li. r-

»-' O' n r- C o

f'- rO «-H A

CC o

-j" »i; nOi in

-J Os c>C t-' O' a

II II M II ,. II II II c c r o o o c c >>>>>->->->II II II II II II II II II II II II II M II c o c c c c o C' o c e c r c <r >->->->>-> >->>>>->>- > >II II II It II II II II c c c o c o o r. >>->>>>>>

II II II It II c c o c r >>>->->II II II II II II o o c c c c >>->>->>II II II It II II C O C O c r

> > > > > >

II It II II II II II II II II II c oor'C'C' cr'C c c

o o

C.' c

I I U 11. U >1

cr n

r* O' Oi o

r o c C

I I I I U Ll' U LL- o in <r r-i 0- o If • o

r—I nJ- I-' f'* o ir» r- o

1

-

1 ,-t Os O

r .-oo o 'sj o c e c o

I I I I I UL UliU-'

c\ o n c- in. c C' A irv in

B. IT Oi C in

OL o^ Os oc. r\ f-* o i-i cc

fO o r c~

I I u u

Oi a

O' <r o O'- c: c' r cr till U U L' U O If C\: I"-* O- IT >J' a o c 'J

sL' o O a

r r~ c*- r- r ' o or

c- cr c cr cr c ro- ll l I t l i i U-UULlU- UUUo Nf r- •'f ^ 'f -4- c

o. o sT a O' c 0^ ii‘ 0- A O', a o c Os o.

IT a O' li'' O' Os o If I

1-

l^r-Hi-.t^CVt-lr-'

•sj O c c

I I U I u

r- o

1^ r

u a

o. 0 0

x^-O.vfO vTOO'OO'JOsJOO' ooc'c rcr-crc r roo I I I I I I I I I I I I I I UUUaLlU lJ LLUU UUULis

O', i—o oa ao.sno-cc na o <. sj-sT»C’in.iro c<\. i-10-o.iO inr- a.r'-ajr-iOo kjootf't-Hn--oif

sj-o-a 'ro''J'C<i'sf»— O'. r-iKj-O' o t-H a <\ U r-4 (\. r-l »-i U' o o • I-I r-i

or sj O' O' si- O' sj O' vt o c r c c C' C‘ r c r- c' r c c- r

I I I I I I I

I I I I I I I ULiUUU-UUUUUULLU-li.

a-tvO-r-Ovriru o si o'r-nr'-o irh-sfu oinr-U'r'-if'U'M-Of'- ooinn-sAinc\'f.'»-'<rrCO»j

in. O o o o o OL o a i vf •J' c o- r-ir-»0 OsO.Os CTr- C\ .-.If'C -4 t-*

O-sTO-JO <*-oo c c c c r cr c o

I I I I I I I I UUULIUULLU

a. A r o o t—'J sj occ-sjr'-c. uo 0<\OOOlf o'lf

C' 'i O' ' in 0 ' in

r-. O' rH O r-t a f~-

168
7Q?7AF-ni,

vn=

1

.

B??07

crC-tv •sJ^L aocv<ru r^Cr—'n irf^O O' a r^iomi-iGcc^-o-fvcar^u r t— ca^co

a o Oi-Hi-Hcvc<’'0'irt*''r>f'- <x- a r-O'i-nr^i^'r CT't--<r^ay<^(\.<r-ocr<-J^ crcLccu o o a GC_c.oOf-<r-(r-»-‘r-'Cvj<N'r

( r-< . I—1 r-H I— » r-i f\. O' fs <V C\t C\ O, C\J Cv. O' f\ fV

II II II II II II II II II II II II II II II II II II II II 0 00000000 OOC'OOCC'CC'<~C

>v> >->>> >-x>->->>->v>-v>>-> ooor'ooor' oo-o vt-oorr n oof'

cor occc or ooc c cc oo>oo

I

I

I I t t 1 1 I I I I I I I I

i

I t u LLLiLlU-U-UU UU. U U U U U LLLLU<-i- 'ca r- o-ci''r--r' O’ ^-a-C7 r-^^ gO i— f\.r.o- rr'c^ao o-'Cr-Lfr-'oirCo.airr'-ccxC

r-r'-uc r«ii—'IT. C/0'r"-iriroiO'0 crifa.O’

Lf o Of—i'£.r*~,—'iro-f^vCoi^®-<^ f-vTi—

r-lCL r- f—iNl r-i>rc-if—'CLf-HO'-^U-i-lf—l<-CL»—

I

169
7

<ir_o

7

7

‘jf-o? 7?c_n? 7

PF

-07

70

F-0?

*-H «. I-I <

«• «

«• •«

-B-

•fr

CN. (\ Tv c <r c

I I i

Cw <\ C\ f\ f\ <- 00000 I I I

I I I

>£' s£' < >C

(\ rv Cv cv r f\ (\ fv.o.Cv f'-o cvjrvitv- c c c: o c c o c o o c o c c c o

I I I

i I I I t I I I I I I t I LiL:ULi-Ul1. U.'U.LI-UUJI-L’L LiUU’l-»I.U OsCifp*. cvoc7h->i iCaf^iT'>rc\j

fvc\ rs.'(\j^ c* 0 c o o c c o c r o c o c o

1 I I I I I I I I i ) I I I Ll U LL Li U U u Li UJ 11 LL U' Li u c/r-^»j-Of-ioc.r'u-Nj(\i— <s (\ C\ fv (\ <vf^J{\ •—I,—1|— iO

n c m rf, fT 0 c o t o

1 I I I I U- LL Li u ’ U

170

ST'll'i.MTnM

tfST

magnitude
NBS-114A IREV. 2-8C) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

PUBLICATION OR

REPORT NO.

NBSIR 80-1630

2. Performing Organ. Report No. 3.

4. TITLE AND SUBTITLE

Publication Date

June 1981

DETERMINATION OF MUTUAL COUPLING BETWEEN CO-SITED MICROWAVE ANTENNAS AND

CALCULATION OF NEAR-ZONE ELECTRIC FIELD

5. AUTHOR(S)

C. F. Stubenrauch and A. D. Yaghjian

6. PERFORMING ORGANIZATION (If joint or other than NBS, see in struction s) 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

DAEA-76-F-D760

8. Type of Report & Period Covered

NBSIR

July 1976 - June 1978

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

US Army Communications Electronics Engineering

Installation Agency

Fort Huachuca, Arizona 85613

10. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document Includes a significant

bibliography or literature survey, mention it here) The theory and computer programs which allow the efficient computation of

coupling between co-sited antennas given their far-field patterns are developed.

Coupling between two paraboloidal reflector antennas is computed using both

measured far-field patterns and far-field patterns which were obtained from a physical optics (PO) model. These computed results are then compared to the

coupling measured directly on an outdoor antenna range. Far fields calculated

using the PO model are compared to those obtained from transformed near-field

measurements for several reflector antennas. Theory and algorithms are also

developed for calculating near-field patterns from far fields obtained from the

PO model. Documentation of the near-field and coupling computer programs is

presented in the appendices. Conclusions and recommendations for future work

are included.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) Co-sited antennas; coupling; far fields; near fields; physical optics; planewave spectrum; reflector antennas.

13. AVAILABILITY 14. NO. OF

[X^] Unlimited

PRINTED PAGES

1 1 For Official Distribution. Do Not Release to NTIS

Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

176

15. Price

\ 3 Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $12.00

USCOMM-DC 6043-P80
• A *

1 ^. 1

' » -h'L'L

m~ ^ - -la.j.- iiiJti

’ .’ttif'tVt g.'ii'T )i

.

t <1 wjSji

VSJ'' Hasps' Sl'iT'^a Is'

...

8^

M

iI

}'* ' ' ^ '

I

• _/®\ 1

'.vA^pSr^ in I4<,'i' ?fo ii

I-

»l V ‘Sf-3ffM<||.

"?^f. '' I.’ "'^n. )('>.•(•*.*' ), _.*•. ;. iPf-ff

f —’•'

aL II iiyi g* »|

' 5.'f

.- -ii'- ' „s

•: . : ^f')^jd'j*r( >tiW

?.'>

** ' ’ t-’>'^‘*‘.•*' iv /.. ' "sSr/j^SB >.k«V

[jm^z ...i..'\

_j^i* i

1^ 'l 9

^

1^^g0-508/4|S);|lei^^


At-J