COMPUTATION OF ANTENNA SIDE-LOBE

COUPLING IN THE NEAR FIELD USING

APPROXIMATE FAR-FIELD DATA

Michael H. Francis

Arthur D. Yaghjian

National Bureau of Standards

U.S. Department of Commerce

Boulder, CO 80303

August 1982

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COMPUTATION OF ANTENNA SIDE-LOBE

COUPLING IN THE NEAR FIELD USING

APPROXIMATE FAR-FIELD DATA

Michael H. Francis

Arthur D. Yaghjian

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

U.S. Department of Commerce

Boulder, CO 80303

August 1982

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
I

I
CONTENTS

Page

Glossary of Symbols for the Text..... iv

Abstract. 1

1. Introduction. 2

2. Side-Lobe Coupling of Two Antennas Using Their Exact

and Approximate Far Fields................................... 4

2.1 Statement and Formulation of the Problem.... 4

2.2 The Hypothetical Circular Antenna. 5 2.2.1 The Exact Far Field..... 5 2.2.2 The Approximate Far Field for Approximation-1...... 6

2.2.3 The Approximate Far Field for Approximation-2............ 6

2.2.4 Limitations on the Approximations......... 7 2.3 Results and Comparisons... 8

2.4 Conclusions..... 10

3. Mathematical Upper Bounds Derived from Equation (1 11

3.1 Assumptions and Integration. 11

3.2 Comparison of Upper Bound from Formulas to the

Exact Upper Bound............ 15

3.3 Conclusion.............. 15

4. Summary and Concluding Remarks 16

5 . References 18

6. List of Figures 19

Appendix A. Derivation of AyMAy and A^y in Terms of

Antenna Gains and Rel ati ve

A Side-Lobe Levels....... 40

Appendix B. Documentation of ENVLP, the Computer Program to

Estimate Coupling Between Two Antennas... 42

B.l General Overview of Computer Program ENVLP. 42

B.2 Computer Code and Sample Output........................ 48

Appendix C. Subroutine FOURT...... 70

Appendix D. Subroutine PLT120R..... 78
Glossary of Symbols for The Text

a

:

a T,a R: a°: aMAX :

ATMAX’ ARMAX :

C'

:

d: D t

,D r:

E_:

G:

Gt

,G r

: Vk_: k:

k o:

K_:

K:

radius of the antenna aperture.

radius of the transmitting and receiving antennas, respectively,

incident wave-amplitude in the transmitting antenna feed,

the maximum amplitude of the far field in the direction of the

axis of separation.

the maximum amplitude of the far field in the direction of the

axis of separation for the transmitting and receiving antennas,

respectively.

the emergent wave-amplitude in the receiving antenna feed.

1 n Z (1 - r r )

* oo o L the separation distance between antennas.

the diameter of the smallest sphere which circumscri bes the

transmitting and receiving antennas, respectively,

the electric field.

the far field of the receiving and transmitting antennas,

respectively, with components f

x

, f', f'z and f x

, f , f

z

. the antenna gain. y y

the gain of the transmitting and receiving antennas,

respecti vely

.

the Bessel function of first order.

the propagation vector with components k x

, k

y

, y.

/k « k = 2tt/A . MDt + D R )/2d.

k e + k e . xx y y

/irnr .

^i nput

L

rR

S

ST

,S

R

Z

oY

r

rn

Lo0

A

^R’ e R’^R

the input power to the antenna,

position vector with components x, y, z. magnitude of _r« A A xe + ye . x

J y

the relative side-lobe level in the direction of the separation

axis.

the value of S for the transmitting and receiving antennas,

respectively.

the wave impedance of free space,

the z-component of k_. the reflection coefficient of the receiving load.

the reflection coefficient of the receiving antenna.

the characteristic admittance for the propagated mode in the

waveguide feed of the receiving antenna.

the wavelength.

the Euleri an angles of the transmitting antenna,

the Euleri an angles of the receiving antenna.
COMPUTATION OF ANTENNA SIDE-LOBE COUPLING IN

THE NEAR FIELD USING APPROXIMATE FAR-FIELD DATA

M. H. Francis and A. D. Yaghjian

National Bureau of Standards

Boulder, Colorado 80303

Computer programs, in particular CUPLNF and CUPLZ, are presently in

existence to calculate the coupling loss between two antennas provided

that the amplitude and phase of the far field are available. However,

for many antennas the complex far field is not known accurately. In

such cases it is nevertheless possible to specify approximate far fields

from a knowledge of the side-lobe level of each antenna along the axis

of separation, and the electrical size of each antenna. To determine

the effectiveness of using approximate side-lobe level data instead of

the detailed far fields, we chose as our test antennas two hypothetical,

linearly polarized, uniformly illuminated circular antennas for which

the exact far fields are given by a simple analytic expression. The

exact far fields are supplied to the program CUPLNF to compute the exact

near-field coupling loss. Approximate fields are supplied to a new program ENVLP developed for the purpose of computing the approximate

near-field coupling loss. The comparison of the results from ENVLP to

those of CUPLNF indicates that the use of approximate far fields gives

an estimate of the coupling loss which is good to about +5 dB. In addition, the plane-wave transmission formula for coupling between two

antennas is used to estimate upper-bound values of coupling loss. These

upper bounds are compared with the maximum coupling losses obtained from

programs CUPLNF and ENVLP.

Key words: antenna coupling; antenna theory; coupling loss; near-field

measurements.

1
1. Introduction

Three years ago, the Antenna Systems Metrology Group, Electromagnetic Fields

Division of the National Bureau of Standards, under the sponsorship of the

Electromagnetic Compatibility Analysis Center, completed the development of a highly

efficient computer program, CUPINF, for calculating the coupling loss between two

antennas given the far fields of each antenna [1,2]. For antennas arbitrarily

oriented and separated in free space, CUPLNF computes the coupling loss at a single

frequency as a function of antenna displacement in a plane transverse to the axis of

separation of the antennas. Multiple reflections between antennas are neglected but

no other restrictive assumptions are involved. CUPLNF automatically computes electric

fields from the transmitting antenna when a "virtual antenna" of uniform far field

replaces the receiving antenna. Coupling loss for antennas hundreds of wavelengths in

diameter is computed in a few minutes of CPU time within the central memory core,

e.g. , of a CDC 6600. 1

The major limitation of CUPLNF is its requirement for the magnitude and phase of

the electric far-field components of each antenna within the solid angle mutually

subtended by the antennas. For example, if two antennas are oriented so that coupling

occurs mainly through their side-lobe region, the complex vector far field of each

antenna covering this angular side-lobe region must be supplied to the program CUPLNF.

In practice, one may not have such detailed information on the side-lobe far

fields to estimate the coupling loss between two antennas co-sited in their near

field. Often, one has some knowledge of the side-lobe levels of one's antennas, even

if detailed phase and amplitude information of the field components is not available. Thus a natural and important question to answer is whether this limited

information, specifically side-lobe level of each antenna near their axis of

separation, can be used to estimate the antenna coupling in the near field using a new program, ENVLP. Of course, as the separation distance between the antennas approaches

the mutual Rayleigh distance, i.e., the antennas lie in each others far field, the

ordinary far-field formula for coupling between two antennas can be used to estimate

coupling loss. For separation distances much less than the mutual Rayleigh distance,

the far-field formula would not be expected to give realistic estimates of coupling.

It is in this near-field region that we set out to decide if ENVLP will give reasonable values of side-lobe coupling (±6 dB accuracy) by utilizing only side-lobe levels

i The specific computer is identified in this paper to adequately describe the

computer program. Such identification does not imply recommendation or en- dorsement by the National Bureau of Standards nor does it imply that the

computer identified is necessarily the best available for the purpose.

2.
along the axis of separation of the antennas (in addition to their size, separation

distance, and feed characteristics)

.

To determine the efficacy of ENVLP using approximate side-lobe level information

instead of the detailed far fields, we chose as our test antennas two hypothetical,

linearly polarized, uniformly illuminated circular aperture antennas for which the

exact far fields are given by a fairly simple analytic expression involving functions

no more complicated than the first order Bessel function* With these hypothetical

antennas, the exact far fields could be supplied to the program CUPLNF to compute

near-field coupling loss without introducing the errors incumbent with measured farfield data. An approximate side-lobe far field can be substituted for the exact far

field and a comparison of the near-field coupling loss computed with the exact and

approximate far fields can be made simply and straightforwardly.

At first thought it may seem overly optimistic to want to obtain reasonable

estimates of near-field coupling having only side-lobe level information, i.e., having

only the envelope of the magnitude of the far fields in the direction of separation.

Fortunately, however, we can also estimate the phase variation of the side lobes of

most microwave antennas merely from the frequency of operation and the radii of the

antennas, more precisely, from the radius in wavelengths of the smallest sphere which

circumscribes all significantly radiating parts of each antenna.

Also, even though polarization characteristics of the side-lobe far fields may be

unknown, we can assume polarization match of the far-fields of the antennas over the

relevant range of integration. This assumption of polarization match tends to produce

an upper-bound estimate of coupling loss, which may be of appreciable value to the user. We also assume the user has a knowledge of (1) the separation distance between

the antennas, and (2) the reflection coefficients in the waveguides feeding the

antennas--! n all a rather minimal amount of information about one's antennas.

In Appendix B, we document program ENVLP, which is a modification of the program

CUPLNF requiring as input this minimal amount of input data, i.e., the antennas

1

radii, separation distance, reflection coefficients, and relative side-lobe levels in

the direction of separation.

Finally, we work directly with the plane-wave transmission formula for coupling

between two antennas to estimate upper-bound values of coupling loss, comparing these

upper bounds with the maximum coupling losses obtained from program CUPLNF and program

ENVLP using exact and approximate far-field input. We anticipate these simple upperbound formulas being especially valuable when the major requirement for co-sited

antennas is that the coupling interference or fields lie below a certain threshold.

3
2. Side-Lobe Coupling of Two Antennas Using Their Exact

and Approximate Far Fields

2.1 Statement and Formulation of the Problem

Yaghjian [1] showed that the coupling between two antennas in free space,

neglecting multiple reflections, is given by

b'o

a o II

- iyd i K *R

f(k)e e

K<k

dK , ( 1 )

where C = =

n

=~ r

-, and 1/(1 - r r. )is the mismatch factor of the receiving

antenna, n 0 is the characteristic admittance of the propagated mode in the waveguide

feed of the receiving antenna, Z Q

is the wave impedance of free space, k is the

magnitude of the propagation vector, K_ = k x e

x + k

y

eyis the transverse part of the

propagation vector, y = (k 2 - K

2

) 1 ^ 2

, dK_ is an abbreviation for dk

x dk

y

, and the

coordinates (R_, z = d) give the position of the origin O' fixed in the receiving

antenna with respect to the (x,y,z) coordinate system fixed at 0 in the transmitting

antenna (see fig. 1). The functions _f and _f‘ are the electric far fields of the

transmitting and receiving antennas, respectively, without the presence of the other,

r Q

and are the reflection coefficients of the receiving antenna and receiving load

respectively, aQis the amplitude of the input to the transmitting antenna, and is

the amplitude of the output of the receiving antenna. Formula (1) assumes the

receiving antenna is reciprocal, although the formula easily generalizes to include

non-reci procal antennas. Yaghjian [1] also showed that for most practical purposes

the integration range in (1) could be limited to K/k < (Dy + D^)/2d for R = 0 provided

( D-j- + D^)/2 < d < (Dj + Dr)^/A. Dy is the diameter of the smallest sphere which

circumscribes the transmitting antenna and D

R is the diameter of the smallest sphere

ci rcumscribi ng the receiving antenna. (If Dy or D^ is less than twice the wavelength,

y, Dy, or is replaced by 2A.) For coupling in the transverse plane over the range

- (Dy + D^) < R < Dy + D

r the integration range should be doubled [1].

There are some situations where detailed information of the far fields (_f

1 , f) is

lacking and it is desirable to get an estimate of the coupling between two antennas.

For example, both amplitude and phase information may not be available. We would

like, therefore, to consider some possible methods of approximating the coupling and

in particular of finding a good estimate for the maximum coupling between two antennas

in the general direction of the separation axis (the separation axis is drawn from a

4
point centrally located on the transmitting antenna to a point centrally located on the receiving antenna for the antennas in their initial position, R_= 0).

To find the maximum coupling in the general direction of the separation axis we

shall calculate the coupling over a range of values of R_ (R_ is perpendicul ar to the

separation axis). Specifically, we shall move the receiving antenna in the x and y

directions, over a range from - (Dy + D

R

) to + (Dy + D R

) for both the x and y directions. In order to save computer time and cost we will hold y

= 0 while varying x (the XO cut) and hold x = 0 while varying y (the YO cut).

If we are to determine how accurate our approximate results for coupling loss

are, we need to compare them to the results that we would have obtained if we knew and

used the exact far fields of the antennas for finding the coupling quotients. For

this purpose we will use two hypothetical circular antennas and compare results

obtained using the exact far fields to those obtained using approximate far fields.

These results will be compared for different values of the separation distance,

diameters of the antennas, frequency, and orientation of the antennas.

2.2 The Hypothetical Circular Antenna

2.2.1 The Exact Far Field

We consider the case of a uni formally illuminated circular aperture with the

transverse electric field polarized in the x-di recti on. Then the far field for a point in the (9,<j>) direction is given by [3]:

f

x — cose

/tt

Ji(ka sine ) "ka sine

(2a)

(2b)

f

z ka

/tt

J ! ( ka sine)

s1ne “* "

Tea We- • (2c)

For this formula, the circular aperture lies in the xy plane with the center of

the aperture at the origin. The symbols, e,<|> denote the usual angles in spherical

coordinates, k is the magnitude of the propagation vector, a is the radius of the

aperture, and J

L is the Bessel function of first order. An example of this pattern

can be seen in figure 2, where ka = 20.94.

D
2.2.2 The Approximate Far Held for Approximation-1

Approximation-1 is specifically geared to the uniformly illuminated circular

aperture antenna. For this approximation we replace

J , (ka sine)

ka sine

by Hka sine

o

cos(ka(e - e ))

ka sine

o

where e Qis the direction of the separation axis relative to the preferred antenna

coordinate system. 0 Qequals 0^ for the transmitting antenna and e R for the receiving

antenna (see the sketches on figures 6 through 20). This electric field pattern can be

found for e o

= 60° in figure 3. The values of the field are given relative to the main

beam at e o

= 0° in figure 2, i.e. the field is normalized to a main beam equal to 1.

The above approximation may be justified as follows. J^ka sine)/ka sine may be

replaced using the asymptotic expansion for J . This yields:

J -j ( ka sine)

ka sine

/ZlZ

/irka sine

cos(ka sine - 3 tt/ 4

)

ka sine

The cosine term represents the' variation of the sidelobe while the rest of the term is

the amplitude of the envelope. We replace the terms involving the amplitude of the

envelope by their values at 0 Oand expand the cosine term about e Q

. Thus,

cos(ka sine - 3u/4) ~ cos (ka sine_ - 3 tt/ 4 + ka(0 - eHcoseH. The ka sine,, can be

lumped with -3tt/4 into a phase factor which we will ignore since we are finding only

an average coupling over a solid angle. The cos0

Q

term has been replaced by 1 because

actual antennas do not display this cose

o dependence of sidelobe width peculiar to the

hypothetical finite planar aperture distribution.

2.2.3 The Approximate Far Field for Approximation-2

The approximation-2 far field is a general approximation that can be made for a large class of coupling cases. For this approximation we replace JF

1 ( -k ) *f

( JO by

A

TMAX

cosk

x

a Tt o sk

y

a T

A

RMAX

cosk

x

a R cosk

y

a R

.

ATMAX anc* Armax are rnax1

' mum ma§ n i Guides of the far fields in the general direction

of the separation axis (i.e., the side-lobe level) for the transmitting and receiving

antennas, respectively, k x

= ksin0 cos<j>, and ky = ksin0 sin<j>. Hence, Aj^^ anc! a rmax

are the magnitudes of the approximate electric far fields in the direction of the

6
separation axis and are given in absolute SI (mksA) units. In Appendix A we derive

aTMAX and aRMAX in terms the side-lobe levels and the gains of the transmitting and

receiving antennas, ay and a^ are the radii of the smallest spheres circumscribing

the transmitting antenna and the receiving antenna, respecti vely

/vWe have chosen the above approximation because for many antennas the components of

the far-field patterns vary roughly as cosk

x

a coskxa, where a is the radius of the

smallest sphere circumscribing all the significantly radiating parts of the

antenna.^ Note that for approximation-2 we assume that the polarization of the

receiving and transmitting antennas are matched and thus approximation-2 tends to

yield an upper bound. The approximation-2 pattern for <f> = 0° or 90° and the separation axis in the direction of 9 = 60° can be found in figure 4. The values of the

field are given relative to the main beam at e = 0° in figure 2.

2.2.4 Limitations on the Approximations

Both approximation-1 and approximation-2 are valid over a relatively narrow range

of angles. In particular, the approximations should be valid (under the above stated

limitations) if the amplitude of the envelope of the far field of each antenna varies

by not more than about 3 dB over the range of angles mutually subtended by the

transmitting and receiving antennas. Thus, for any case in which the integration of

equation (1) must be performed over a large range of angles, approximation-1 and

approximation-2 may be poor. This will, in general, include those cases involving

coupling with the main beam. It will also include electrically small broad beam

antennas and the unusual cases where both the receiving and transmitting antennas are identical and also have identical Eulerian angles [1] (see figure 5 for a definition

of these angles). In addition, for 9 q = 0 the approximation of the far field for

p‘•The far side-lobe region of the far-field pattern is predominantly caused by

diffraction from edge points of the antenna. These edge points are usually

separated by a distance of approximately an antenna diameter and this leads to

a side-lobe pattern which has a null approximately every x/2a radians, as numerous hypothetical and measured far-field patterns confirm (see for example

Johnson et al . [4], and Newell and Crawford [5]). We assume a smooth variation in the form of a cosine, and that the k and k variation is separable.

This leads to the cosk

x

a cosk

y

a dependence aoout the axis of separation - a fairly reasonable approximation provided the antenna is not highly elongated

(i.e., the length in one direction is not much greater than the length in the

other direction). In the case of a highly elongated antenna the value of a is

approximately half the longer side so that we obtain a variation which is good

in the direction of the long side but poor in the direction of the short side.

For such elongated antennas, a better approximation can be obtained by using a variation of the form cosk

x

a

x cosk a y

where a

x is half the length of the long

side and a is half the length of the short side, assuming the x and y axes are aligned with the long and short sides of the antenna, respectively.

7
approximation-1 goes to infinity. Thus, approximation-1 is never good for

Dr + D (D + D

r

) 2

the main beam. Finally, d must be in the range ^ < d < ^ .

2.3 Results and Comparisons

The results using the exact far fields and the approximation-1 and approximation-2

far fields in equation (1) were obtained using computer programs based on CUPLNF [2].

The program for approximation-2 (titled ENVLP) was written for general use and can be

found documented in Appendix B. Results were obtained for various values of antenna orientation, antenna

diameter, separation distance and frequency, and are summarized in table 2.1 and in

figures 6 through 20.

Column 1 of table 2.1 is an identifier which corresponds to the identifier in

the caption of figures 6 through 20. Columns 2 to 4 give the Eulerian angles of the

transmitting antenna; columns 5 through 7 give the Eulerian angles of the receiving

antenna; columns 8 and 9 give the diameters of the transmitting and receiving antennas, respectively, in units of wavelength. Columns 10 and 11 give the side-lobe

levels relative to the main beam and within the integration range (as specified above)

for the transmitting and receiving antennas, respectively. Column 12 gives the

separation distance in terms of the mutual Rayleigh distance; column 13 gives the

frequency in hertz. Columns 14 through 16 give the maximum coupling amplitude as given by the use of the exact far field, the approximation-1 far field, and the

approximation-2 far field, respectively. The maximum coupling magnitude implied by

the far-field formula is given in column 17.

The far-field formula which is used to determine the maximum coupling is

b' A A

0 TMAX RMAX X a n z (1 - r r ) d

0 oo o L

Column 18 gives the maximum coupling amplitude as implied by the upper-bound

equations (9), which will be discussed in section 3. Finally, columns 19, 20, and 21

contain the RMS mean coupling for the exact, approximation-1, and approximation-2 far

fields, respectively.

Figures 6 through 20 are plots of the magnitude of the coupling quotients in the

x-di recti on at y

= 0 (the X0 cut) for the exact far fields (_), approximation-1 (•...),

and approximation-2 ( ). The Y0 cut is not shown because it is similar to the X0

cut. Each figure corresponds to one of the cases in table 2.1.

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An inspection of table 2.1 will show that the maximum coupling predicted by

approximation-1 compares favorably with the result given by the exact far fields.

With the exception of case Omega, the difference in the maximum coupling loss between

the exact far-field result and the result from approximation-1 is less than 5 dB while

for case Omega, the difference is about 10 dB. The explanation for the large difference for case Omega is that the wavelength for this case is so large that the electric

field pattern lies near the first null (see, e.g., figure 2); thus, approximation-1

which is based on the envelope of the Bessel function is very poor in this case. We find that with the exception of two cases (Beta-3 and Delta-1) the maximum

coupling loss predicted by approximation-2 is within 10 dB of that given by the use of

the exact far fields. For Beta-3, Dy = 0.2m, D^ = 0.8m, and d = 1.0 m; thus, for this case the coupling integration covers a broad angular region and the polarization does

not match at the wider angles as it does at the center. This means that approximation-2, which assumes perfect polarization match over the entire range of integration,

will appreciably overestimate the coupling for this case. In the case of Delta-1,

even though the approximation-2 estimate of the maximum coupling is about 13 d3 low,

this is a relatively good estimate of coupling since the exact coupling lies below

-116 dB.

If we now examine figures 6 through 20 and columns 19 through 21, we find that,

in general, the RMS mean level of coupling generally agrees to within about ±5 dB of

the exact for approximation-2 and considerably closer for approximation-1. On the

other hand, approximate and exact coupling loss at individual points can differ by a substantially greater amount (occasionally more than 20 dB).

It is of interest to compare our results to those obtained from the far-field

formula, equation (3). It can be seen that, in general, if the separation distance

between the antennas is greater than about one-quarter of a mutual Rayleigh distance,

(D-j. + D

R

) 2 /A, the far-field formula gives results closer to the exact results than do

approximation-1 and approximation-2. On the other hand, if the separation distance is

less than one-tenth of a mutual Rayleigh distance, the results from approximation-1

and approximation-2 are in general substantially better than those given by the farfield formula. For separation distances between one-tenth and one-quarter of a mutual

Rayleigh distance both the far-field formula and approximation-1 or approximation-2

yield values of coupling loss of about the same degree of accuracy.

2.4 Conclusions

We conclude that approximation-1 and approximation-2 can be used in the computer

programs to obtain a reasonable estimate of the maximum coupling in the general

direction of the separation axis. It should be emphasized that while we can use

10
approximation-1 and approximation-2 to find the maximum coupling or the mean coupling

over a narrow range of directions in the general direction of the separation axis we

do not necessarily get a good estimate in the exact direction of the separation

axis. Approximation-1 is limited to uniformly illuminated circular aperture antennas

while approximation-2 can be used for general antennas. In using either

approximation-1 or approximation-2 care must be used so as to stay within the

limitations stated in section 2.2.4.

In the event that the separation distance is greater than about one-quarter of a Rayleigh distance the far-field formula (equation (3)) gives better results than

either approximation-1 or approximation-2 and the far-field formula should be used in

those cases to estimate the maximum coupling.

The real advantage of the computer program for approximation-2 (Program ENVLP) is

that it estimates the coupling loss between arbitrary antennas arbitrarily oriented in

the near field of each other from a mere knowledge of (1) the separation distance

between antennas, (2) the side-lobe level of each antenna in the direction of the axis

of separation, and (3) the radius of each antenna.

3. Mathematical Upper Bounds Derived from Equation (1)

3.1 Assumptions and Integration

It is possible to derive an upper bound to the magnitude of b'Q /a Q if we make some simplifying assumptions that will allow us to integrate equation (1); initially, let

us make the following extremely crude assumptions which will allow us to immediately

derive an upper bound for coupling quotient by performing the integration in equation

(1). Afterwards a smaller, more realistic upper bound will be derived.

-'(“-* * -- A

TMAX

A

RMAX

(d

t + d

r )2\1/2 Y Y min M ^ 4d2

iyd i K*R ,

? = e = 1 .

(4a)

(4b)

(4c)

aTMAX and aRMAX are tlie max1

’ muP1 values of _f and f^_, respectively, near the axis of

separation.

With the assumptions of equations (4) we find after integrating equation

-k(D + D ) +k(D + D ) (1) from vyr to «-r—— that

11
( 5 )

b*

0

<

a 0

a tmaxa rmax

( D t+Dr ) 2

n Z (1 - r r ) oo o L

1 -

(d

t

+ d

r

)2 d

2

By using this integration range we limit the validity of equation (5) to

(D

t + Dr )/2 < d < (D T

+ D R

) 2 /A.

Equation (5) gives an absolute upper bound to the magnitude of the coupling

quotient, neglecting multiple reflections, provided that coupling does not occur

through the main beam of either antenna, that (Dy + D

R )/2 < d < (Dy + D

R

) 2 /A and that

the integration range we have used [i.e., -k ( Dy + D

R )/2d to +k(Dy + D R /2d)] is

adequate (as explained in section 2.2.4).

As d approaches the mutual Rayleigh distance we expect equation (5) to approach

the far-field formula, equation (3). We can see that this is indeed the case if we

al low \ 1 -

(d

t DR ry/2

to approach 1 (note that (Dy + D

R

) 2 will be much less

than 4d2 at the mutual Rayleigh distance) and replace one of the d's in d

2

by (Dy + D

R

) 2

/ A . We thus obtain

b' A A

0

< TMAX RMAX A

a n Z (1 - r r ) d

0 oo o L

which is just the far-field formula. If we compare equation (5) to the far-field

formula for separation distances less than a mutual Rayleigh distance, equation (5)

always gives a larger value.

Instead of the assumptions of equations (4), we can derive more realistic upper

bounds by making the more realistic approximations

( y

Dy Dy D

R f'(k) • f(k) = AyMAX

cos (k

x-y + <h)cos(k -y + <J) 3) A

RMA x cos(k

x~2 + ^

°R

X COSUy-yy + (f) 4 )

Y = Y . = Ml - mi n

( 0

T + DR )^/2

4d2 /

(6a)

!6b)

e iYd = e ia

, (6c)

12
where a is a constant and the <j>'s are arbitrary phase shifts. The last assumption

(6c), is made because e

lycl varies more slowly than the cosine terms or the e

1 — *— term

for the integration range being considered. Under assumptions (6), equation (1)

becomes

A

TMAX

A

RMAX

e

1 a k k o o

n Z (1 - r r )k 2 /I - (d

oo t o L I T R

roml /

cos(k

xT

+ *i>cos(k- +

V 4<F

-k -k

o o

Df Dn

X cos(k^-^-+ ) cos (

y—

+ <j) 4 )(cosk

x + isink^Hcosk^y + i sink

y

y)dk

x dk

y

(7)

where k = k(DT + D D )/2d.

0 I K The integration of equation (7) is rather lengthy, but strai ghtforward, and can

be written after some rearrangement, as

i a

b'o A

TMAX

A

RMAX

G

n Z (1 r r )k 2/ (D_ + D_) Z\V? oo oL T R

' ~

1 " 4d^~

( 8 )

We wish to find the maximum value of equation (8). First, we notice that

each term in the two pairs of brackets, [ ], is of the form e”^i ~ ^ ^(sinBK

Q

)/B.

We might be tempted to say that the maximum value of a term of this form is

13
(1/B) but this would be true only if Bk > 1. For Bk. < 1 this term has

a maximum value of L at B = 0.

U We note that there are four values of x and

four values of y for which B+0. They are

x, y

= - (°T

0

R

) (d

t

- D

r

)- +

{d t

- V and +

(0

T

+ V 2

s 2 2 2

only one term of the x bracket and one term of the y bracket at a time

In general

,

have Bk

o | < 1. However, if either Dy or D R or both are sufficiently small more than one term in the x bracket and more than one term in the y bracket

can have Bk. < 1. The maximum value of equation (8) as a function of phase

will occur when each of the phases is an integral number of 2-rr radians.

Keeping this in mind and substituting for k

Q

, we have the following five

expressions for the maximum coupling:

b'

0 < a tmax

a rmax

"k(D

i

+ v

a 0 n Z (1 - r r, )( oo o L 1 - (d

t

+

^ /i a 2 )

i h- 4k 2 2d

1 1 1 +

67

+ 6T ivy (9a)

for D

2 U T °R

4d

D

2 > — i k D2 U > — R k

lb*

1 0 < atmax

a rmax [2k(D

T

+ D

r’ 1 + _ _ i

. T

i 3

l o n Z ( 1 - r r

)

oo o L 2d

| D r

- °R

Max(D

T

,D R

) (9b)

for |D 2 - D|| max (D 2 ,D 2 )>

4d

k min (D 2 ,D 2

) ~ ^

b'

0

<

atmaxa rmax 2k(D

T

+ V 1 1 1 + -f

2

a 0

n Z (1 - r rV? (DT+ D

d

) 2\V2

(•- ,/ )

«*

2d

°T °R

for D

2 - n2

l J

R D

2 > 4d

T» D| > 4d~T

(9c)

14
b'

0

/ atmax

a rmax

3k( D rD

) x ]

2

a 0 n Z (1 - r r)/ (DT

+ D) 2\V2

°°

(l - 4 / )4k2

2d Max(D

T

,D

p

) (9d)

for |D| - D

2

| < ijjj-, min ( ,Dg) < max (D 2 ,D2

) > ^

b'o atmaxa rmax(d

t + y

2

Z(l-rr)/ (D_ + D ) AV2d

2 oo o L|. T R 1 1 - 4d2

(9e)

for IDf-D|l <« D|<« D|<«

Of course, in equation (9e), the latter two conditions imply the first. Notice

that equation (9e) gives the same result as does equation (5).

3.2 Comparison of Upper Bound from Formulas to the Exact Upper Bound

The results obtained from equations (9) for each case in section 2 is given in

the last column of table 2.1. The small letter in parentheses in column 18 of table

2.1 indicates which set of conditions of equations (9) is applicable for each case. There is a minimum of one case for each set of conditions in equations (9). We notice

that equations (9) give results which differ from the exact results by more than 20 dB

in some cases. However, we further notice that equations (9) always give results

which are greater than the results found using the exact far fields.

3.3 Conclusion

We conclude that equations (9) can be used to obtain an upper bound to the

coupling between two antennas neglecting multiple reflections, provided that coupling

does not take place through the main beam of either antenna and provided that the

integration range we have chosen is valid, i.e., it is adequate to integrate k x and ky

only over the range -k ( D j

- D

p )/2d to +k(Dy + Dp ) /2d and we limit d to the range

(Dj + D p )/2 < d < (Dy + Dp

)

2 / A . It has been shown that this range of integration is

adequate for determining side-lobe coupling for nearly all realistic microwave

antennas [1].

15
4. Summary and Concluding Remarks

In this report we have performed a feasibility study to see if it is possible to

get an estimate of the maximum coupling between two antennas when the details of the

far-field amplitude and phase are unknown. We conclude from a study using

hypothetical uniformly illuminated circular antennas that approximation-1 and

approximation-2 are especially useful methods of estimating the maximum coupling of

two antennas when the separation distance is less than one- tenth of a Rayleigh distance. Approximation-1 is limited to uniformly illuminated circular aperture

antennas. Approximation-2 replaces the dot product of the fields in equation (1) by

the product of the maximum field magnitudes in the general direction of the separation

axis multiplied by cosine functions of k

x and k^ and is an approximation applicable to

general antennas. Approximation-2 is used in computer program ENVLP, which is

documented in Appendix B. For distances greater than about one-quarter of a mutual Rayleigh distance, the

far-field formula [equation (3)] used with Ay^ and ARMAX gives an increasingly

accurate estimate of the maximum coupling between two antennas that is generally more accurate than the estimates provided by approximation-1 and approximation-2. For

distances between one-tenth and one-quarter of a mutual Rayleigh distance

approximation-2 and the far-field formula give equally reasonable estimates. For

separation distances less than about one-tenth of a mutual Rayleigh distance

approximation-2, i.e., program ENVLP, gives substantially more accurate values of

coup! i ng.

In section 3 we derived a set of expressions [equations (9)] that allow one to

determine an upper bound to the coupling between two antennas at arbitrary separation

distance. For many applications the upper-bound expression may be adequate,

especially when the requirement is simply that coupling lies below a given value.

The appeal of approximation-2 and the correspond!’ ng computer program ENVLP, is

also their simplicity and the few input parameters that they require. In particular,

the near-field coupling is computed between arbitrary antennas from a mere knowledge

of (1) the separation distance of the antennas, (2) the side-lobe level of each

antenna along the axis of separation, and (3) the radius of each antenna.

Having obtained encouraging results for approximation-2 and the upper-bound

equations (9) for hypothetical antennas we suggest that these results be tested

experimentally using real antennas, and that a similar approximation be formulated and

tested for the computer program CUPLZ.

16
The authors wish to thank Carl Stubenrauch of MBS for the original suggestion of

using approximate far fields as well as for helpful discussions and suggestions

throughout the work. Helpful discussions were also held with Allen Newell and

Andrew Repjar of MBS. In addition, we wish to gratefully acknowledge the

Electromagnetic Compatabil ity Analysis Center of the Defense Department for its

financial support of this work.

17
5. References

[1] Yaghjian, A. D. Efficient computation of antenna coupling and fields within the

near-field region, IEEE Trans. Antennas Rropag. AP-30, pp. 113-128; January 1982.

[2] Stubenrauch, C. F.; Yaghjian, A. D. Determination of mutual coupling between co- sited microwave antennas and calculation of near-zone electric field, Nat. Bur.

Stand. (U.S.) NBSIR 80-1620, pp. 97-104, 113-114, 126-150; June 1981.

[3] Johnson, C. C. Field and wave electrodynamics. New York: McGraw-Hill, Sec. 10.5;

1965.

[4] Johnson, R. C. ; Ecker, H. A.; Hollis, J. S. Determination of far-field patterns

from near-field measurements, Proc. IEEE 61(2), pp. 1668-1694; December 1973.

[5] Newell, A. C.; Crawford, M. L. Planar near-field measurements on high performance

array antennas, Nat. Bur. Stand. (U.S.) NBSIR 74-380; July 1974.

[6] Kerns, D. M. Plane-wave scatteri ng-matrix theory of antennas and antenna-antenna

interactions: Formulation and applications, J. Res. Nat. Bur. Stand. (U.S.) 80B,

p. 24; January-March 1976.

18
6. List of Figures

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Figure 11.

Figure 12.

Figure 13.

Figure 14.

Figure 15.

Figure 16.

Figure 17.

Figure 18.

Figure 19.

Figure 20.

Schematic of two antennas arbitrarily oriented and separated in free

space.

The exact far-field pattern of the x-component of electric field as a function of 0 with X = 3 cm and a = 10 cm for a uniformly illuminated

circular antenna.

The approximation-1 far-field pattern of

as a function of 0 with A = 3 cm and a

= :ircular antenna.

Ihe approximation-2 far-field pattern as 30°, X = 3 cm, a = 10 cm for a At (f> = 0° k y = ksine, k v

= 0;

uni formly

at <J) Xy - INITIO, k - u, at = 90' <p - ulerian angles (<j>,0,ijj) 'needed to rotate

oupling axes x,y,z of figure 1.

he coupling quotient for the X0 cut using the

pproximation-2 far fields: Case Alpha-1,

he coupling quotient for the X0 cut using

pproximation-2 far fields: Case Alpha-4,

he coupling quotient for the X0 cut using

pproximation-2 far fields: Case Alpha-7,

he coupling quotient for the X0 cut using

pproximation-2 far fields: Case Beta-1,

he coupling quotient for the X0 cut using

pproximation-2 far fields: Case Beta-2,

he coupling quotient for the X0 cut using

pproximation-2 far fields: Case Beta-3,

he coupling quotient for the X0 cut using

pproximation-2 far r''v1J

the x-component of electric field

10 cm for a uni formly illuminated

a function of 0, with <j) = 0° or illuminated circular antenna.

*, k = ksin0, k x

= 0)

.

they fixed axes x A

,y A

,z A

to the

the

the

the

the

the

the

uAMiiouuii-t fields: Case Beta-4.

2 coupling quotient for the X0 cut using

Droximation-2 far fields: Case Gamma-1.

2 coupling quotient for the X0 cut using

Droximation-2 far fields: Case Gamma-2.

2 coupling quotient for the X0 cut using

Droximation-2 far fields: Case Delta-1.

? coupling quotient for the X0 cut using

Droximation-2 far fields: Case Delta-2.

d coupling quotient for the X0 cut using . Droximation-2 far fields: Case Epsilon-1.

d coupling quotient for the X0 cut using the

Droximation-2 far fields: Case Epsilon-2.

2 coupling quotient for the X0 cut using the

Droximation-2 far fields: Case Epsilon-3.

5 coupling quotient for the X0 cut using the

Droximation-2 far fields: Case Omega.

the

the

the

the

the

exact. approximati on-1 and

exact, approximation-1 and

exact. approximati on-1 and

exact, approximati on-1 and

exact, approximati on-1 and

exact, approximation-1 and

exact, approximation-1 and

exact, approximation-1 and

exact, approximati on-1 and

exact. approximation-1 and

exact, approximation-1 and

exact. approximation-1 and

exact. approximati on-1 and

exact, approximation-1 and

exact, approximation-1 and

19
20

Figure

1.

Schematic

of

two

antennas

arbitrarily

oriented

and

separated

in

free

space.
Figure 2. The exact far-field pattern of the x-component of electric field as a function of 0 with X = 3 cm and a = 10 cm for a uniformly illuminated

circular antenna.

21
kass 20.94

Figure 3. The approximation-1 far-field pattern of the x-component of electric field

as a function of 6 with X = 3 cm and a = 10 cm for a uniformly illuminated

circular antenna.

22
ka = 20.94

Figure 4. The approximation-2 far-field pattern as a function of 6, with <j> = 0° or 90°, X = 3 cm, a = 10 cm for a uniformly illuminated circular antenna.

(At <p = 0° k x = ksine, k - 0; at <p = 90°, k = ksine, k x

= 0).

23
Figure 5. Eulerian angles (<j>,0,\{>) needed to rotate the fixed axes xA

,yA

,z A

to the

coupling axes x,y,z of figure 1.

24
FREQ—

10

GH;

d=

33.3A

N j

(E a o

oo

co

CMOd

CMrr

COr

COi©

<r-I

OC OoX . o o m m

25

Figure

6.

The

coupling

quotient

for

the

XO

cut

using

the

exact,

approximation-1

and

approximation-2

far

fields:

Case

Alpha-1.
FREQ

-

10

QH;

d

=

33.3A

oc o

„ ©i ©

n j«

26

Figure

7.

The

coupling

quotient

for

the

XO

cut

using

the

exact,

approximation-1

and

approximation-2

far

fields:

Case

Alpha-4.
FREQ

=

10

GHz

d

=

33.3 X

=

0.67A

D

r

=

6.67 A • o

• O CO © © V

II II II esc

®

&o

© N- ©

II II IS

' H K H

© © © © © © © © q7 d 80 © in d TO o TO ©

to CO K ® © CM TO TO

8 I 8 8 8

CD

! ! 5 l

C

_ O

XI o&

27

Figure

8.

The

coupling

quotient

for

the

XO

cut

using

the

exact,

approximation-1

and

approximation-2

far

fields:

Case

Alpha-7.
FREQ

=

10

QM;

oc o

is n si si c tr s oc O ^

® ® 9 °

« *o S © ° 5

si a ii si V

t- H- K »- © ©* «> •*•

© O © © ©

© m 6 ad 6

** c© I*

3*® ®

8 i 5 I i

ffl “DC

_ o ©

jQ 0

V- cm o

8 1 T~ S e © o

<3=3 — cq « 0

E E

K X <0 *5 © o u& a a

x a a

UJ < <

1 : i

’ CM

9 • B tr Q

© + © KQ

CM W

o

1 X —

r

CO

i‘

CO

1°

1

1 q

© © © ©

«> 6 id © © CM CO m

‘J®-

8 1 1 i

28

Figure

9.

The

coupling

quotient

for

the

XO

cut

using

the

exact,

approximation-1

and

approximation-2

far

fields:

Case

Beta-1.
FREQ

=

10

GHz

d

=

33.3A

=

6.67A

O

n

=

20.0A

X3c

«T3

-4s

e

"4

• ©

• © ©

© ®

II II II IEEE -©» «»

. © • o ® o

II II II r- H- » I- O •$. <©. <*.

V •

<_ .

<*» £M

i I II« ©

E E

X X

*s © o

a a a

x a a

m < <

i

oo

<0

CM ^BE Q

© +

d “

CMr

r©

r©

r

o

X

©

d

© © © © © © © © o ©7

d d d o d o’ d o’ d o 'F* CO © K © CM CO tf)

i e i i s 1

i T»

I F3*

1 ”a

i co

•r— +>

fDE

xo

s-a.

Cl

OS

4-3 U

fOX

a>

O)

4-> CDe

to3

M3

U

oX

<D £4->

i.O

CM

4-> c

a> +j

o

3

cr

Cn

CL3

o

<_>

a;

CD

. o

JO oa

<u

£-3

CD

29

approximation-2

far

fields:

Case

Beta-
oc

fO

aQ

£E o+©x i co

+j

fOeXo

i~CL

CL

to

+>u

<T5 X

OI

<L>

cnc • •f- oo

tn3

4-> 3ooX

<D £+-> o

4-

+->

<D +jO

cnc

CL3ou

<D * O © n a

O)

l.3

cn

30

approximation-2

far

fields:

Case

Beta-
FREQ

=

10

Ql

-oem

31

approximation-2

far

fields:

Case

Beta-
FREQ =5

GHz

d=

18. 7X

.= 3. 33 A

Do=

3.33 A

*0c

ra

tz

C

_

© om ! Co

4-> raEXou

ClCL

(Ou

fOX

O)

a> .c

4-> CD •

i/)

+J3OoX

a; G+J

1.O

4-

+Jc

a> +jo3cr

o~> c

0.3ou

aj .c

ro

cu3

CD

32

approximation-2

far

fields:

Case

Gamma-
FREQ =

20

GHz

d=

66.7A oc

<13

<r O

•oc

oc o+oX - o o

.O CO I co

-t-> rexo

t.ClQ.

re

4-> O

rex

QJ

Q>

CD • C CM

*-> 3OoX

QJ C+J1.o

4-> c

QJ r—

+Jo3

CrCDC

Cl3Oa

Q>

Q)

i~3

CD

33

approximation-2

far

fields:

Case

Gamma-
FREQ =

10

GHz

d~

1688./ A oc

fO

OX . © ©

83 coM

«3E

•r-Xo

s-CLCL

<13

-MO

fCX

O)

QJ

er> c .

to3

+J3oX

o> JC

+-> o

+Je

o>

•r

o3

CTcnc

CL3oo

QJ

tr>

O)

s-3

I3>

34

approximation-2

far

fields:

Case

Delta-
FREQ =

10

GHz

d=

233.3 A =

0.87A Db=

13.33 A oc

<T3

cc oc

ac Q4- i co

4->

<T3 Xo

i-D.CL

reu

rex

OJ

0) M

CT> • sr c\j

to3

+-> =3UoX

<u +j

s-oc

0>

•s—

4-> o3

ercn

c:

D.3O

a;

CD . ©

jQ o

03 a>

s~3

CD

35

approximation-2

far

fields:

Case

Delta-
FREQ =

10

0Hz

d16.

7A S

®J?A

Do=

®.6

?A

TJ

c:to

a.

•a £QOX „ © ©

jq

a i co

+-> TOXo

s_CL

Cl

TOO

TOX

0>

<D JC

+J t CDe

*/>

+>3aoX

a> sz4->

S_o

•Mc

QJO3

crc

CL3Oo

0J Jc

a>

i-3

cn

36

approximation-2

far

fields:

Case

Epsilon-
FREQ =

10

GHz

d=

66.7A =

6.87 AD0=

6.67A

TTE

fO

cc o c

<E O+oX l co

+J

fDEXo

a.

Cl

to

4-> O

raX

QJ

QJ JE

+-> . CM

en

CO

-M3OoX

o> .E

4-> o

+-> c

<v

4-> o

cr

cn

D.3oo

a>

co

_ o

JS o

CD

<v

i-3

CD

lZ

37

approximation-2

far

fields:

Case

Epsilon-
FREQ

=

10

GHz

d

ss

133.3 A

.=

®.@?A

0o

=

®.®/A

JE Q

T3

- © o M &

38

Figure

19.

The

coupling

quotient

for

the

XO

cut

using

the

exact,

approximation-1

and

approximation-2

far

fields:

Case

Epsilon-3.
GOo

£E o

_ o

JO o

CO

39

Figure

20.

The

coupling

quotient

for

the

X0

cut

using

the

exact,

approximation-1

and

approximation-2

far

fields:

Case

Omega.
Appendix A. Derivation of AyMAX

Side-Lobe Levels

and A in Terms of Antenna Gains and Relative RMAX

In this section we derive the amplitudes, AyMAX and ARMAX , of the approximate

pattern for the transmitting and receiving antennas, respectively (see section 2.2.3)

in terms of the antenna gains, Gy and G

R

, the side-lobe levels in dB, Sy and SR

, the

reflection coefficients of the antennas, r _ and r n , the characteristic admittances

oT oR

for the propagated mode in the waveguide feeds of the antennas, nQyand nQR , and the wave impedance of free space, ZQ

. The magnitude of the vector _f (which is just A^) in equation (1) of section 2.1

in the direction r_ is given by [1]:

|E(r)|r

I fill I * 1y^ * (AD

where E_(_r) is the electric field in the direction _r, and aQ

is the amplitude of the

incident mode of the waveguide feed to the antenna. AMAX is either AyMAX or ARMAX

depending on whether the transmitting or receiving antenna is being considered. The

relative side-lobe level S in the direction £ (Sy for the transmitting antenna and SR for the receiving direction), is:

S

= -20 log

where £0 is the direction of the main beam. It is well known that the gain for an antenna, G (Gy for the transmitting antenna and G

R for the receiving antenna), is

given in dB by

G = 10 log

4tt I—(^o^ |

2 rZ

2 7 p.

* 0 input

(A3)

Assuming a single propagating mode in the waveguide feeding the antenna, the input

power to an antenna, P-j npu y, can be expressed as [6]:

input

17

( A4)

where n

Q

is n

Qy

for the transmitting antenna and n 0R

for the receiving antenna and

r Q

is r oTfor the transmitting antenna and t qR

for the receiving antenna.

Substituting (A4) into (A3) for Pjp

t we find that the gain is

G = 10 log

4tt E (r ) — -0

2 r2

Z n 0 0a o

2 (1 r 0

2

)

( A5)

40
( A6)

|E_( r ) We solve for -

r-p r in (A5) to get

l a

ol

Substitution of |£(l0 )j

from (A2) into (A6) gives

£(r^)jr

-p^j- 10 amax

10s,2° I>.w)

V2

10 G/20

Thus, A^ax turns out to be simply

amax

= |r 0 |2)V

/2

iqC<G - S)/20)

and in particular.

atmax

=

(

\V2

l r ol|

2

>)

t—O

CD

— 1 I GO —1

ro 0

A

RMAX

= ^ \V2

1 r oR

1

2

7

((G

r

- S

r )/20)

10

K K

( A7

)

( A8)

(A9a)

( A9b)

In summary, equations (A9) give the magnitude of the approximate electric far- field

pattern of the transmitting and receiving antenna along their axis of separation in

terms of the gain (in dB), side-lobe level (in dB) along the axis of separation, the

antenna input reflection coefficient, the characteristic admittance of the waveguide

feeding the antenna, and, of course, the impedance of free space.

41
Appendix B. Documentation of ENVLP, the Computer Program to Estimate

Coupling Between Two Antennas

This appendix documents the program ENVLP which estimates the coupling loss

between two antennas given their radii, separation distance, and side- lobe levels

along their axis of separation. Subroutines used by this program have also been used

by CUPLNF (except for CRTPLT3) and are documented in NBSIR 80-1630[2]

.

B.l General Overview of Computer Program ENVLP

The techniques used by ENVLP for evaluating the coupling loss are basically the same as those used by the computer program CUPLNF [2]. The flow chart for ENVLP is

presented below to give the reader a general understanding of the program package.

Purpose : To calculate an estimate of the coupling loss between two antennas given the

radii of the antennas, their separation distance, and their side-lobe levels along

their separation axis.

Method : Evaluate equation (1) of the main text using approximation-2 (see section

2.2.3) to estimate the dot product of the far fields.

General Discussion : The main program ENVLP divides into the following sections:

1. Genera] information,

2. Specification statements,

3. Definition and reading of input data,

4. Far-field coupling computation,

5. Limits of integration and number of integration points,

6. Filling the input matrices to the FFT subroutine FOURT, and

7. Calculation of maximum, minimum, mean coupling; printout and plotting of the

XO and YO cuts.

General Information : This section provides a general description to the program

user of what the program does and also defines the more important parameters of

the program. A reading of this section will be sufficient for most users to

begin using the program.

Specification Statements : This section dimensions arrays, places arrays in

correnon, and declares complex variables.

42
FLOW CHART FOR PROGRAM ENVLP
Definition and Reading of Input Data : This section defines basic constants such

as the speed of light and reads from data cards the antenna parameters. The

required data cards are:

Card 1

Card 2

Card 3

Card 4

Card 5

Card 6

Col. 1-10

Col. 5

Col. 10

Col. 1- 10

Col

.

11 -20

Col. 1- 10

Col. 11 -20

Col

.

21 -30

Col 31 -40

Col 41 -50

Col 1- 10

Col 11 1 CM o Col. 21 -30

Col 31- 40

Col. 41 -50

Col. 51 -60

Col. 61 -70

Col. 1- 10

An alphanumeric identifier of the user's choice used to

identify the case being computed; the identifier appears at

the top of the printout, (HEAD (2)).

The maximum value that the user wishes XLIM to assume; it

should be 1 or 2. XLIM adjusts the integration range (see

below), ( ILIMAX)

.

The maximum value that the user wishes BFAC to assume; it

should be 1 or 2. BFAC adjusts the integration step size

(see below) , (IBFMAX).

The separation distance in meters, (ZO).

The frequency in Hz.

The radius in meters of the smallest sphere circumscribing

the effective transmitting antenna, (RADT).

The gain of the transmitting antenna in dB, ( GAINT )

.

The relative side-lobe level in dB of the transmitting

antenna along the axis of separation (ST).

The real part of the input reflection coefficient, GAMMAOT,

for the transmitting antenna in free space.

The imaginary part of the input reflection coefficient,

GAMMAOT, for the transmitting antenna in free space.

The radius in meters of the smallest sphere circumscribing

the effective receiving antenna, (RADR).

The gain of the receiving antenna in dB, (GAINR).

The relative side-lobe level in dB of the receiving antenna

along the separation axis (SR).

The real part of the input reflection coefficient, GAMMAOR,

for the receiving antenna in free space.

The imaginary part of the input reflection coefficient,

GAMMAOR, for the receiving antenna in free space.

The real part of the reflection coefficient, GAMMALR, for

the passive termination on the receiving antenna.

The imaginary part of the reflection coefficient, GAMMALR,

for the passive termination on the receiving antenna.

The characteri stic admittance of the propagating mode in the

waveguide feed of the transmitting antenna, (ETAT). If

44
ETAT=0, the program will set ETAT=1/CAPZ0.

Col. 11-20 The characteristic admittance of the propagating mode in the

waveguide feed of the receiving antenna, (ETAR). If ETAR=0,

the program will set ETAR=1/CAPZ0.

Far-Field Coupling Computation : This section computes the coupling between two

antennas directly from their far fields if their separation distance is greater

than a mutual Rayleigh distance (equal to the square of the sum of the effective

antenna diameters, divided by the wavelength; see [1]).

Limits of Integration and Number of Integration Points : For XLIM=1, the limits

of integration are computed using the specification of section 2.1 of the main

text. These limits can be doubled to see if a wide enough integration range has

been included by setting XLIM=2. Strictly speaking, approximation-2 (see

section 2.2.3) is only good for the XLXM=1 integration range. However, if the

results of the XLIM=2 integration range for the mean coupling differs by a huge

amount (e.g., 10 dB) from the mean coupling result for XLIM=1, it is questionable whether ENVLP gives a good estimate of the coupling loss between the

two antennas being considered.

This section also chooses the integration increment size small enough to

prevent aliasing. The increment size can be reduced by increasing BFAC.

Normally, BFAC is set equal to 1 and convergence is tested by halving the

integration increment size, i.e., letting BFAC=2.

Filling the Input Matrices to the FFT Subroutine FOURT : This section calculates

the dot product of the far fields in a square array using approximation-2. The

dot products are then summed in the k (k ) direction for each value of k (k ) y x x y

and placed in the array AX(AY). FOURT then performs a one dimensional FFT on the array AX (AY) to obtain the values of the coupling quotient along the XO(YO)

cut.

Calculation of Maximum, Minimum, Mean Coupling; Printout and Plotting of the XO

and YO Cuts : This section computes the maximum, minimum, mean, and RMS mean coupling for the XO and YO cuts. It further computes the standard deviation in

the coupling for each cut. It prints out the values of the coupling quotient

for each cut and then plots the values of the coupling quotient for each cut

from -(DIAMR+DIAMT) to +(DIAMR+DIAMT)

.

45
Symbol Dictionary (in alphabetical order):

ABL

ACL CUT

AMAX

AMEAN

AM!NX,(AMINY)

ARMAX, (ATMAX)

AX, (AY)

(A1,A2) ,(B1,B2) -

BFAC

C

CAPZO

CEE

CLCUTXA( CLCUTYA) =

COEF

Cl s C2

DIAMR, (DIAMT)

DIAMSUM

DKOK

DX,(DY)

etat,(etar)

FDOTFP

FMM

FREQ

Intermediate value for defining the range of k /k and k /k. The

x y

range of ABL beyond XKLIM is zero filled.

A real array used to store the magnitude of the coupling quotient

along XO and YO cuts.

The maximum value of the coupling quotient for either the XO or YO

cut.

The mean value of the coupling quotient for either the XO or YO

cut.

The minimum value of the coupling quotient for the XO(YO) cut.

The side-lobe level for the receiving (transmitting) antenna.

Complex arrays used to store first the far-field product then the

coupling quotient along the XO(YO) cut.

The limits of integration of k /k and k /k respectively.

x y

Variable which adjusts the integration increments and should be

about 1 or 2; making BFAC larger tests for convergence by making

the integration increments proportionally smaller.

The increment size for the XO and YO cuts when the far-field

formula is used.

The wave impedance of free space.

The speed of light in a vacuum in nmeters/second.

The values of the coupling quotients in the XO(YO) cuts stored for

plotting.

2.*PI*FMM/ETA/CAPZ0/XK for the far-field formula,

-FMM*C1*C2/ETA/ CAPZO otherwise.

The k

x /k and k^/k increments respectively.

Twice the larger of RADR (RADT) or WAVLGTH.

DIAMR plus DIAMT.

The approximate k /k and k /k increments,

x y

The increments in XO(YO) over which the coupling quotient is

computed by the FFT.

The characteristic admittance for the propagated mode for the

waveguide feed of the transmitting (receiving) antenna.

The dot product of the electric far-field pattern of the two

antennas.

The mismatch factor, l/(l-r

0

r£, for the receiving antenna.

Frequency in Hz.

46
FX,FY

GAINT, (GAINR)

Intermediate complex variables.

The gain in dB of the transmitting (receiving) antenna.

GAMMALR,GAMMAOR,GAMMAOT = The reflection coefficients of the receiving load.

HEAD

receiving antenna, and transmitting antenna, respectively.

Integer array identifier.

IBFAC Loop index for varying BFAC.

IBFMAX The maximum value assumed by BFAC; it should equal to 1 or 2.

ILIMAX The maximum value assumed by XLIM; it should be 1 or 2.

I XL I

M

Loop index for varying XLIM.

J1,J2 Loop indices used in the filling of AX (AY) before the FFT.

Ml, M2 Loop indices used in completing the coupling quotient computation

after the FFT.

N The number of points in an XO(YO) cut when the far-field formula

is used.

NN1,NN2 Integer arrays of dimensions one used in the call to the FFT

subroutine FOURT.

NSX , ( NSY

)

The number of values of the coupling quotients to be plotted in

the XO(YO) cut.

N1,N2

N1MAX , (N2MAX)

The number of k and k integration points respectively,

x y

Integers determining the maximum of the XO(YO) range over which

the coupling is printed.

M1MIM, (M2MIN) = Integers determining the minimum of the XO(YO) range over which

the coupling is printed.

N10,N20 Intermediate integers used to determine ( N1MIN ,N1MAX) and

( N2MI N , N2MAX )

.

PI it = 3.14159...

RADR, (RADT) The radius of the smallest sphere circumscribing the receiving

(transmitting) antenna in meters.

RMS The RMS mean coupling quotient for the XO or YO cut.

RO The total distance between the two antennas = (X 2 + Y

2 + l

z fa.

SDEV The standard deviation in the coupling quotient for the XO or YO

cut.

SR, ST The relative side-lobe levels, in dB, of the receiving and

transmitting antennas along their axis of separation.

SUM , SUMRMS , SUMSQ = Intermediate variables used to calculate AMEAN, RMS and SDEV,

respectively.

SUM2 Summation variable used in the filling of the AX and AY matrices.

TEMP An intermediate variable used to calculate ATMAX and ARMAX.

47
TSUM21 Summation variable used to compute the coupling quotient at XO = 0

by summing directly without the use of the FFT.

WAVLGTH = Wavelength in meters.

WORK = Complex array required by the FFT subroutine FOURT.

X ,

( Y

)

= Array containing the abscissa value for the plots of the XO(YO)

cuts.

XK = 2tt / X

.

XKLIM - Variable which limits the range of k /k and k /k integration when

x y

its value is less than XKMAX.

XKMAX — An upper limit (less than 1.0 and usually chosen at .9) on XKLIM;

except for close antennas XKLIM will usually be less than XKMAX.

XKMIN = Sum of the diameters of the two antennas divided by their

separation distance.

XKOK - The square root of the sum of the squares of XKXOK and XKYOK

.

XKXQK ,XKY0K k /k and k /k, respectively,

x y

XLIM Variable used for adjusting XKLIM; making XLIM larger tests to see

if a large enough integration range has been included; XLIM should

equal 1 or 2.

XNX ,XNY ,XNZ = Variables used for incrementing k /k, k /k, and y/k, respectively.

” y

XYO = Intermediate variable used for calculating RO.

XO, Y0 S ZQ X S Y,Z coordinates of the origin of the receiving antenna in the

mutual coupling coordinate system of the transmitting antenna;

specifically ZO is the separation distance.

ID, IFL, IU, NOFRAME = Variables used by the CRT plotting routine CRTPLT3, which is

a specialized routine for the NOAA/NBS CYBER 170/750.

Subroutines Not Available in the FORTRAN Library :

FOURT (a standard FFT subroutine documented with program CUPLNF in NBSIR 80-1630C2].)

PLT120R (printout plotting routine documented in NBSIR 8Q-1630[2].)

CRTPLT3 (a CRT plotting routine specifically written for the NOAA/NBS Cyber 170/750;

we suggest you substitute your own subroutine).

Note : If the electric far field, including phase as well as amplitude, is available

use the program CUPLNF documented in NBSIR 80-1630[2].

B.2 Computer Code and Sample Output

A copy of the computer code for program ENVLP and a sample output for the case

BETA-1 are found below.

48
PPCGPAP

FKVtP(

TNPUT»PUTPUT

I

o

flu

O'©

a.

CNJ

© o © 1 1 mJ 1

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x b > z © X o —

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Appendix C.* Subroutine FOURT ( DATA, NN,NDIM, I SIGN, IFORM, WORK)

Purpose : To compute the discrete Fourier transform of the array DATA using the fast

Fourier transform algorithm.

Arguments :

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.

ISIGN 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.

Methods : Using the Fast Fourier transform algorithm, FOURT replaces the array DATA

with its discrete Fourier transform given by

TRANSFORM^!, K2,...) =

NN ( 1

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NN ( 2

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l

J2=l

i 2 tt I S I GN { DATA ( J 1 , J 2 ) e

(J1-1MK1-1)

NN( 1 ) + (J2-1MK2-1) j NN ( 2 ) +

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, "FQUR2 and FOURT program description," private communication,

1968.

*This appendix is taken from appendix A. 1.11 of Stubenrauch and Yaghjian [2].

70
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69

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Subroutine EDURT (DATA, NO If* 1? ISM* If 0RM, WORK I FOURT l F0U*T 2 THF mifT-TUKFT FAS? FOURIER TRANSFORM IN USASI SASIC FORTRAN FOURT 1 FOURT 6 TR *N^fO®f IK1.K?, ...1 « SUN<DATA( Jl. J • I »F I P t I S I GN*P »R I *S0R T

(

-1 } F 0U» T 9 • mji-iib(ki-ii/nniiimj2-iib<k2-ii/nn<2i*...iii. summed for an fourt 0 Jl. «1 BETWEEN 1 AND NN(11, JP, K? BETWEEN 1 AND NNIP1, ETC. F0U»T 7 There I< NO 11*1* TO IMF NU«BER Of SU8SC»I*TS. DATA IS A FOURT 9 MULTIDIMENSIONAL COfPtE* AMAY WHOSE REAL AND If A6 IN A* Y FOUST 9 PARTS ARf AOJACFNT IN STORAGE. SUCH AS FORTRAN IV PLACES THEN. FOURT 10

IF All IMAGINARY PARTS ARE IFRP (DATA ARE OIS6UISFO REAL ) . SET FOURT 11 TFO»* TO Tfoo TO CUT T«e RUNNING Tiff BY L'P TO FORTY PERCENT. FOURT 12 "TmFPWTSF. I FOR f » !. THE LFNGTHS OF ALL DIMENSIONS ARE FOURT 13

STORED IN ARRAY NN, OF LENGTH NDI*. THEY NAY IE ANY POSITIVE FOURT 16 INTEGERS, Tho the RROGRAf RUNS faster ON COMPOSITE INTEGERS, AND FOURT 19 FSPFCIAUY FAST on NUMBERS RICH IN FACTORS OF TVO. I SIGN IS *1 FOURT 16

OR -1. If A -1 TRANSFORM IS f 01 LOWED BY A ! QN| «0R A *1 FOURT 17

BY A -11 THE ORIGINAL DATA RFAPPEAR. fULTIPLIEO BY NTOT I «NN 111* FOUR T 18

NN(?)*...t. TRANSFORM VALUFS ARE ALWAYS COMPLES. ANO At* RETURNEOFOURT 19

IN assay DATA. SERIATING THE INPUT . IN ADDITION. IF ALL FOURT 20 DIMENSIONS ARF NOT POWERS OF TWO. ARRAY WORK MUST 9E SUPPLIED. FOURT 21 PO“Pl f> «F LENGTH FOLAL TO THE LARGEST MON 2**K DIMENSION. FOURT 22 OTHERWISE. replace WO®k *Y f FR 0 IN THE CALLTN6 SEQUENCE. FOURT 23 normal FORTRAN DATA 0* PF* ING IS E*PFCTEO» FIRST SUBSCRIPT VARYING FOUPT 26 FASTEST. ALL SUBSCRIPTS BEGIN AT ONF. FOURT 29

FOURT 26

RUNNING TIME IS MUCH SHORTER THAN THE NAIVE NT0TPP2. BEING FOU*T 27 GIVEN BY THF FOLLOWING FORMULA. DECOMPOSE NTOT INTO F0U*T 28

7 ppk P • 3RBK3 • 9**«9 • .... LET SUM? • 2*K2, SUMF • 3*K3 9«K9 FOURT 29

... AND NF K3 4 K9 .... THE TIME TAKEN BT A MULTI- FOURT 30

P I ME ME t DN

A

i T® AM* FQ® f ON TwsSF NTOT DATA IS T a TO NTOT® I T ! FOUR T 31 T?*SUfP* T3*SUMFaT6RNF I . ON THE COC 3366 I FLOATING POINT ADO TIME FOURT 32

OF SI* MICROSECONDS! . T * 3006 NT0TB(906y63*SUM2y6B®SUMFy FOUPT 33 120*NPJ MICROSECONDS ON CAMPLE* DATA. IN ADDITION,, THE FOURT 36 ACCURACY is greatly IMPROVED. AS THF RMS RELATIVE ERROR IS FOURT 39 bounded by 3*?**(-B)»SI'M(FACTCR(J 1*61.91. WHERE B IS THE NUM8EP FOURT 36

OF BITS IN THE FLOATING POINT FRACTION AND FACT0»«J! A»f THE FOURT 37 PRIME factors of NTOT. FOURT 39 FOURT 39

PROGRAM BY N09PAN BRENNER FROM THE BASIC PROGRAM BY CHARLES FOURT 60 RADFR. RALPH ALTFR SUGGESTED THE IDEA FOR THE DIGIT REVERSAL. FOURT 61 MIT LINCOLN LABORATORY, AUGUST 1967. THIS IS THE FASTEST AND M0STFOURT 62 VERSATILE VERSION OF THE EFT known TO The AUTHOR. SHORTER PRO- FOURT 63

GRAMS F0IJR1 ANO FOUR 2 RFSTRICT DIMENSION LENGTHS TO ROWERS OF TWO. FOURT 66 SFF— I f F F AUDIO TRANSACTIONS (JUNE 19678. SPFCIAL ISSUE ON FFT. FOURT 69

FOUR? 66

THF DISCRETE FOURIER TRANSFORM PLACES THREE RESTRICTIONS UPON THE FOURT 67

DATA. FOURT 60

1. THF N'J"BFR OF INPUT DATA AND THE «U**BE0 0F TRANSFORM VALUFS FOURT 69 must be THF SAHF. FOURT 90

2. BOTH THE INPUT DATA ANO THF TRANSFORM VALUFS MUST REPRESENT FOURT 91 eeUTSRACFD RPINTS IN THEIR »fSPFCTIVf DOMAINS OF TIME ANO FOURT 92 FREQUENCY. CALLING THFSE BRACINGS DELTA! ANO DEITAF, IT MUST BE FOURT 91

TRIJF That DFLTIf.?rpi/inn( I1RDFLTAT1 . OF COURSE. DELTA! NEE 0 NOT FOURT 96 bf IMF SAMF FOR EVERY DIMENSION. FOURT 99

3. CONCEPTUALLY AT LEAST, THE INRUT DATA AND THE TRANSFORM OUTRUTFOURT 96 RFRRF$fnT SINGLE CYCLES OF PERIODIC FUNCTIONS. FOURT 97 FOURT 98 FYAMRIF 1, THRfF-g IMF NS TONAL FORWARD FOUR IF R TPANSFORN OF A FOUPT 99

COMRIF* ARRAY DIMFNSTCHF9 12 BY 29 BY 13 IN FORTRAN IV. FOURT 60

DIMFNFION 9ATA(32.?9,13»»W0RK(901.NN(38 FOURT 61

CPM»LF* DATA FOURT 62 DATA NNS32.Pf.l3/ FOUR? 63 DO 1 I-I.m? FOURT 66

OO l J • 1 e> ?9 FOURT 69 or l K.1,11 FOUR? 60

1 DATA( T.J.KI.COMBIFS VAU'F FOURT 6T

CALL FOU9T(OATA,MN,?,-l,l,Wt3RKl FOURT 68

FOURT 69

FVAfRLF ?. ONE-DIMENSIONAL FORWARO TRANSFORM OF A REAL ARRAY OF FOURT 70

LENGTH 66 IN FORTRAN II. FOURT 71 DIMENSION DATA! 2,661 FOURT 72 DP P T *1 » 66 FOURT 73

PATAU. I8»RF»l PART FOURT T6

P OATACP.IlsO. FOURT T9 CALL FOUPTIDATA, 66.1,-1,0.01 FOURT 76

FOURT 77

71
muflMIN 04T* (11) NN (1). tf*?T <921# WOBK (11 F0U9T TB M» • 0. F0U9T 79

*0 MI « 0. F0U9T 80 Wjvpp • 0. F0U9T 81 V<*PI • 0. * F0U9T 82 two*t • *.>93189707 *OU*T 83

s ? «Kn?f - nme. ioo e ion F0U9T 8* M 109 NTOT a J FOUPT 89 nr no ioi« » i» ndi«* FOUNT 86

i* inn iniMmio. i 28o. no F0U9T 87 no KTOT • NTOT • NN (!0!«t FOU*T 98

c FOU*T 89 90 r 94JN loop FOP F4f M DIMENSION FOUPT 90

* FOUNT 91 PCI • ? FOU»T 92 nn i??o TO

I

n

8 ® 1. ND!« FOUNT 93

Ni i idini FOUPT 9*

«9 •DPP N?1 * N FOUNT 93

1* <N - 1112*0. 1?*C. 120 FOUPT 96

c FOUPT 97

c F4CT0* *1 FOU»T 99

f FOUPT 99

inp 120 » m H FOUPT 100

»tryr n»i FOU»T 101

TF • 1 FOU»T 102 TOTV • 2 FOU*T 103

110 lotjnr « m t ioiv FOUPT 10*

109 in" • - IOIV * 1QU0T FOU»T 109

?F noi.int - T0TVI210, 1*0. 1*0 FOUPT 106

1*0 IF (IP*® 1 1*0. 190. 1*0 FOUPT 107

1*0 NTw f' « MTWO NTWO FOUPT 108

n ® lounr FOUPT 109

no SP YO no FOUPT no

no TDIV • 9 FOUPT in

no TOlinT a * / mv FOUPT n?

JPF" ' * «* TOTV IOUOT FOU»T 113

1* CTO'JOT - I91VI290. 190. 180 FOU»T 11* ns i«o i* riPFwnoo. ieo. ?ce FOU»T 119 190 IF»CT f I*| « IOIV FOUPT 116

1* - 1* 1 FOUPT 117

« a touot FOUPT 119 or t" no FOUPT 119

l?r . ?oo I n t V * T"*!V t FOU*T 120 on rn i?o FOUPT 121 no IF II****>*30. 220. 298 FOUPT 122 2?0 STWP • *»TVO NTwn FOUPT 123 gP TO Jfc(5 FOUPT 12*

] ’« no met ( t* i • “ FOU»T 129

c FOUPT 126

r SFPAP#?* FOUP C**FS — FOUNT 127

c l. rn*PtF» TPiH9FC»" p* •to. TP4NSF0*" F0« THE *YH» 9TH.ETC. FOU»T 128

r OT"*NSIONS. FOUPT 129 l»o r >. *F4t TP4N*F0PN FO® ?*F *N0 n« i»p PINENSION. NETHO0 — FOUPT 130

r T94NSF0P" M*1 F Tnr 0»T4, SUPPLYING TNF PTMfO HALF »Y CON- FOUPT 131

c JUfUTF SVNNFTPV. F0U9T 132

f 9. #F*L TP 4N$F0P N FPP T«C 1ST OINENSiriN, N 000. NEYHO0— FOUPT 133

r *P4NSF0*P H4LF TH* 04T4 4T E4CH S7*GF. supplying THE 07HE9 FOUPT 13*

n* r HALF 9 Y C0NJU64TF SVN*ETPY « FOUPT 139

r *. •* 4 L Y84NSF0B* FOB YHF 1ST OTNFMSION, N EVFN. «FTM0O— FOUPT 136

c TRANSFORM 4 f(V»Pl*» 4944V OF l«N6TH N/2 whose PFit, 0 4BTS FOUPT 137

p 4PF T*F f VFN NUN®F8E0 9f4L V4LUFS *N0 WHOSE INAGIN49Y P4BTS FOUR T 138

r **F Tmp <jpo NUM*F»EO »F4L V*LU*S. SFP*«4TE *N

0

SUPPLY FOUPT 139 1*0 f YMF BFfnwO *41 F |Y CPN J8JS4 YF S YN*f T* Y« FOUPT 1*0

p FOUPT 1*1 ?*o N*“N f a (VP> } FYVOI FOUNT 1*2 K4U a 1 FOUPT 1*3

I* »TM» - *1290. 30C. 900 FOUPT 1**

i*« »*o IF (IF 0 *®

|

2 * 0 . 2 * 0 . ice FOUPT 1*9

»*c YP 4 S F • 9 FOUPT 1*6

IF CIOT* - 1 1 >70 » 270, 100 FOUPT 1*7

>70 I r 4? F a 9 FOUPT 1*8

IF (NtwO - *>11300, 300, 280 FOUPT 1*9

1*0 2P0 IC49F c * FOUNT 190 NTwn S WYWH t 2 FOU»T 131

N • N / 2 FOUPT 192 NP? ® N®» t 7 FOUPT 193

MYP1Y • NTOT / J FOUNT 19*

72
1M f • 3 FOURT 199 On ?9P J • 2. NTOT FOURT 196 PATA < J > • DAT* (It FOUR? 197

>00 t • I > FOUBT 19#

SCO I1»N0 ® *»>1 FOUBT 19«

160 I* IIC»9f - 2)320. Ilf. 320 FOUBT 160

110 11*96 NPO NMtV / 2) FOURT 161

F0U9T 162

c SHUFFLE ON THF FACTORS OF TWO IN N. AS the shuffling FOURT 163

c CAN 9F DONC AY STNFIF INTFRCHANCF, NO W0RKTN6 A8PAT IS NCEOFO FOURT 16*

169 c FOURT 169

3>Q I* INTWO - n»i )700* TOO. 330 FOUR? 166

130 NP?UF HP 9 / 2 FOUPT 167

J • 1 FOUPT 169

OC 300 I? • 1. N»2, N0N2 FOURT 169

170 IF (j » 12)3*0. 360. ICO FOURT 170

310 H"A* • 12 NON? . ? FOURT 171 no nr ii • if, i in a i» 2 COURT 172

00 390 13 • n. NTOT. N>2 FOUBT 173

43 « J « 13 • 12 FOURT 17*

179 ten®p > data i id FOURT 179 TEN® I • DATA III 1) FOURT 176

DAT* (ID a PATA U3> FOURT 177

OAT* 113 1) • OATA M3 1) FOURT 178 DATA M3) TFNP8 FOURT 179

180 390 -OAT A ( 43 1) a TENPI FOURT ieo

3*0 N a NP2HC FOURT 161

370 TF M - N)3O0. 300. 330 FOURT 182

3*0 J a J - N FOURT 183

N a » f 9 FOUPT 18*

189 IF ( N - NON2I390, 370, 370 FOURT 189

300 J a J N FOURT 186

C FOURT 187

C NATN LIP* fo* factors of two. ®e*fprn FPUPIEP tbansfpbns of FOUST 188

c LFNCTm four, with one OF length TWO IF NEFOFO. THE TWI09LE FACTORFOUPT 189

loo c WaF*P( IDGN*2*®I®30PT|-1)*N/I*®NNA») ) . CHFC* FOR W» IS IGH®SO*TC -DFOURT 190

c ANP B c® e 4 T FOB WaT?IGN®30BTI-l)®CnNJU6ATF|W). FOUPT 191

c FOURT 192

N0N2T a N0N2 N0N2 COURT 193

T ®AB a NTWO / N®

1

FOURT 19*

191 *00 IF (TRAP ~ ? )**0. *2C» AID FPURT 199

*10 TPA® a TPAB / * FOURT 196

00 TO *00 FOURT 197

*20 or *30 Ii a l, 1 1

8

N 5 » 2 FOURT 198

OP *30 J3 a 11, NON?, NP 1 FOU»T 199

200 DO *30 K 1 • J3, NTOT, N0N2T FOURT >00

*> I Kl N0N2 FOURT 201

TF N PP a OATA 1*7 ) FOURT 202 TF-PI a OATA 1*2 1) FOURT 203

DATA |K2) a OATA INI) - TENP® FOURT 20*

>09 DATA (*? * \) <* OATA 1*1 1) - TENPI FPURT 209 OATA 1*1) a DATA 1*1) TENPB FOURT 206

*30 DATA 1*1 1) • PATA 1*1 l) TENPI FOURT 207

**0 ««AV a Nns> FOU»T >08

*90 IF (nna* - NP>HF ) *60 , TOO, 700 FOUBT 209

210 **o LNA* a NATO | NON? T , NNA* / 2) FPURT 210

IF {nna* - NPN2 )100. ICO, *70 FOU*T 211 *>0 theta a - Twn»T 6 FLOAT (N0N2 ) / FLOAT (4 * CHAD FOURT 212

IF I 13 TON ) *00. *30, *00 FOURT 213

*•0 THE T A a - TMTTA FPURT 21*

>19 *70 WB a COS (THETA) FOURT 219

WI a StN ITMFTA) FOUPT 216

WSTPP a . MI MI FOUPT 21? WSTP I • 2, MM U! FOURT 218

9P0 PP 697 l a N0N2, LNA*. NON?T FOURT 219

>20 N a L FOURT 220

IF (-NA* - N0W2)920, 92°. 910 FOURT 221 910 W2* a W® • WR - WI • WI FOURT 222

w> I a ?. • WP • WT FOURT 223

W3P a V7® * W® - W2I • WI FOURT 22*

>29 W3I a W>® • WT W2I • W» FOU*T 229

9>9 on 6*0 Tl • l. IltNG. 2 FOURT 226

»>n 9*0 J 3 • II, NCN2, N®1 FOURT 22? WNJN a J3 IPAP ® N FflURT 228

ic (n«a* - N0N2 ) 930, 530, 9*0 FOURT 229

>30 930 * N I H a J3 FOUPT 239

9*0 *PIF a IPAt • N*A* FOU*T 231

73
990

M5

2*0

2*9

»90

25 9

?60

?65

770

779

»*©

?85

?©0

?©9

300

309

N$Tfte a * » eoTF

90 *30 K1 a K*IW® N?OT » K$TEF

K? • K1 K9JF

«3 • K? «D1F «* ® «3 «<0T E 1? |««**S • N"N?556e» 990, 990

999 »Jie » 9*7* mj| *3*74 ««? 8 U1J ® 9*7* «K1 « 1) « ©ST A <*2 4 IS U?» a 0*74 j«|| 4 0*74 («*)

U2I • DATA |K3 4 l) 0474 |K* 4 il U38 * 0*7* |Kl| - 0*7* |K»}

U3I • 0*7* («1 4 II® DAT* lit? 4 J| IF I T 7 fiM J * TO* 990, 996

970 UA® » 9*7* IKS II • 0*7* |K* 4 11 U*I • 0*»A l«*5 - ©AT* IK3)

§0 70 670

980 U*B * 0*74 |K* 4 u . e»*T4 IKS 4 II U*1 a 0*7* |«?| - 0*7* | K A

|

f,C TP 6?0

999 ??B a MSB * 0*T A |«?| - M?J * ©»TA |K?

7? I c Wge ft OAT* IK? 4 || 4 tf? I 9 0*7*

73® • W© ft «*7* IK35 » Ml * OfiT* IK3 4

T ST 3 M9 • OAT* ««s 4 l» 4 Ml * n»T* I TAB a M3© ft 0*7 * JK*J « M3I 9 0*74 I 7*| * M3 6 *> 0*74 IK* 4 11 4 MSI • 0*

111® a 94T& lesll 4 T?»

111? » 0*7* |«1 4 || 4 T?1 U?® » 738 4 7* ft im 7si * t*i

U3» * 9*7* IKK -

US I » 0*7* IKl 4 11 - T?T

if miswifeos, «io» 0 i© AGO «J*ft e 731 «* 7*1 UAI • T*t -> TSft g0 TO 6?0

*10 it*e ® T*J » T3f

u*1 » 73© - T *8 62C 9*7* 1811 a 01» 4 y?B DAT* I«l 4 II a uil * U?I C*7* |K?| a US® 4 U*»

0474 |K? 4 II e im 4 88* I 0*7* |»3J a Ul» - U?e oat* iks 4 ii • un - u?i

0*7* | K * 1 s use - UAB 630 0*7* IK* 4JS® US! - U*

1

K*XN e A ft IKH1N « J2I 4 JS KPIF a KSTfft

If IKP1F - M»?SS90, 6*@, 6*9

6*0 POMTTMUft

« ® »P4® v. 9

IF nST«M56W» 66Q, 666

6*0 TENS® a M® m * • t»i

9? s s - fSKft* §0 TO 9.T9 660 TFKftB a Mil M* * V?

Ml a 67© Jf l» - 19ASJ68©, Aft©, 91©

#>®0 TF**©® * «»

W» a M© » WSTPS ® tel ft «KT®1 4 Uft

FOUBT 23? FOUST 233 FOUBT ?3* FOUST 235 FOUST 236 FOUBT 237

FOUST 238 FOUST 23© FOUST 2*0 FOUST ?41 FOUBT 2*2 FOUST 2*3 FOUBT 2** FOUST 2*5 F0U©? 2*6

FOUST 2*7

FOUST 2*9 FOUST 2*9 FOUBT 250

4 1J FOUBT 251

IK? 1 FOUST 232

15 FftU»T 253

35 FOUST 25*

4 11 FOUBT 255

SK*S F0UB7 256

FOUBT 257

FOUBT 238 FOUBT 25©

FOUBT 260

FOUST ?61 FOUBT 26? FOUST 263

FOUST 26* FOUBT 265

FOU©T 266

FOUBT 267

FOUBT 26? FOUBT 26©

FOUST 270

FOUST 271

FOUBT 272 FOU©T 273 FOUBT 27* FOUBT 275

FOUST 276

FOUBT 277

FOUBT 27ft FOUST 27© FOU®T 280

F0U9T 281 FOUBT 28 2 FOUBT 283 FOUST 28* FOU®T 285

FOUBT 286 FOUBT 287 FOUBT 288

FOUBT 28©

FOUBT 2®Q FOUBT 2© 1 FOUST 2©2

690 MJ ® MI © MST®B ft MSTFI 4 MI FOUST

fftflO a 9 - ISA® F0UBT

NP** e n*»&t 4 NHA7 FOUB?

©9 ?n fouet

6 FOUST

c »*w inoft fob wnr seust ?o tmo, a®ftiv the tmiobip ?*c?oe eou»t

f ««F8Busi9H©2»8ies©8Ti-i)ftu?-i)4ui-j?8/cH»?ftifftin, them ffiust

t ftfeBOBM * Ff8U#I?9 fe*^SF9»H 0ft SFaCTIIFI, ^ *k IMS USE OF FOUST

£ £0MJU«®7f KV)«Wf 7»lf K, F0U®T

£ POUBT

TO0 Sf «K?S0 - ^9©, @1© fOUET

Tie Jf PI 9 MON? EQUBT

If • 1 FOUST

HP1MF s NS1 / ? FOUST

??© IF©? o 1**1 i SFS£? ||FJ F9U87

JS?N6 « N»? FOU&?

?9S

I©*

?©6

?©7

298

300

301

102 303

30*

305 306

30 T 308

74
19 (TfftlF ® 3 ) 7*0* 73C» 7*0 FOU»T 309

no 7 ?

0

J19NC- 6 (NF? 19011 / t F 0U8 T 31®

J71T# • MO* / JMCT (IF) FOUFT 311

Jl»f» a (J?STP ft IFPfl / » FOUST 312

7*0 J?"TN e 1 !*P? FOU*T 313

IF (IF®! » NP?)7?0, »00, 000 FOUST 31* M5 7*0 Of 770 J7 ® J»M|N, IFF!, IFF? FOU»T 313

THFT4 • - TWO®! • FlOftT U2 - 1» / FLO&T (NP2) FOU®? 316

IF ( t 11 FN) 770» 7fc0. 760 FOUST 317

760 TMFTft 9 - TMFTft FOUFT 319

770 3I*7M ® SIM (TMfTft ) S.) FOU#f 319

WlTFS . - SINTM • * IMTM FOUST 320

WlTPT 9 STM (TMFTftl FOUST 321

we * wst ®8 l. FOUST 322

WT ® WST®I fous? 111 JX®IN • J> ft SFP| FOUST 32*

!?5 or 790 J 1 9 J1**IM» 4 IFNS, IFF! FOU®T 325

I 1*4* JX ft I1FMS - 2 FOUST 326

Of 7«0 11 « Jl, l 1*47, 9 FOUST 327

nr. 7*0 T3 9 n, MT07, NF» FOUST 329

J3*ft* « I! IFF? - M®1 FOUST 329

•10 fn 7*0 J7 * 13. J1M4*. N®1 FOUST 130 TFMFB a OftTft ( J3 ) F0UB? 331

OftTft (J1) 9 OftTft (J1» • MB ~ QftTft (J3 ft J j ® MJ FOUST 332

7«C Oft?* (J3 X* • TFmff * wi ft OftTft U3 ft 1) • MR FOUST 333

TE*F® a MS FQU8T 33*

W® * M* 4 M«T»I - VI • «S?®I ft US FOUFT 333

790 ft I a Tf:«ft» • wSTFf ft MI ft MS T*t ft U| FOUST 336

«C0 TMFTft 0 » TM0»I 1 F104T (JSftCT «SP)t FOUST 337

IF IIMGN>*?0. 810, 810 FOUST 330

•10 T NO T ft c «, TMFTft FOUFT 33®

?*e fl?0 SIMTm 9 SIN (THFT* / 2„» F0U®T 3*0

MSTOO 9 - 2 . ft SINTH • SINTM FOUST 3*1 MSTFI 9 SIM (TMFTft) FOUST 3*2

« STF8 a ? © M s IFftCT (IF) FOUST 3*3

sr«4Mf. 0 <STFP ft (IFftfT (IF) i f) ft 1 F0U«T 3** Or o«n U « 1, Il®*'6» ? FOUST 3*9

OP <5*0 T3 9 11, MTOT, N»? FOU® T 3*6

on q?p ««IM a 1, FPftNG. NSTfP FOUST 3*7

Jl*4* 9 13 ft JISW6 - IP®! FOUST 3*®

on P80 Jl 9 13. JXHftS. !FP1 FOUST 3*4

150 J3*ft* 9 ji fee? „ NP1 FOUST 390 on F30 Jft 9 Jl, J3»6», M F

1

FOUST 391 J7PAS 9 JIT ft IFPl - IFF? FOUST 352

K 9 M •! M ft (J3 - Jl (Jl - 11) t IFftCT (IF)) $ MPIHF FOUST 393

if (KMTN - 1)330, P3C» 980 FOUST 35*

810 SIMP 9 P e FOU»T 399

S("*I ® 0. FOUST 196 nn 9*0 J? • J3, J *84 * » IFF? FOUST 397

SUMS S SU*9 ft Oft?* (J?) FOUS? 158 s&e sir* ? 9 shni OftTft tJS * I) FOU»T 390

160 WfFF (M) « SUMS FOUST 360

are* |< , n < S»i*I FOUST 361

or to spo FOUST 362

9*0 < fnq j 0 K ft 2 * (M - KM IM ft 1) FOUST 363

J? a J?PftT FOUST 36*

V* SUTF a OftTft JJ?| FOUST 369

SU*>T a OftTft (J? ft 1) FOUST 366 otr«e * 0 , FOUST 367

nosi • n. FOUST 369

J* • J* - IFF? FOU»T 369

i7n iff) TFMFP 9 SU«» FOUST 370 TF"»t 9 SdMf ®eu»? 371

St)®® * tmowp * SU"F » OIOS# OftTft (J?) FOUST 172

<!?•*! e Twnwo a su**l - OtnSI OftTft (J2 11 FOUS? 373

n(0?9 a TFMFS FOUST 37*

179 niOSl a TF«FT FOUST 373

J2 • J* “ IFF? FOUST 176

IF (J* ~ J 3 ) 870* *7®, *60 FOUST 377

9?0 tfmfb 9 wf a SUP® -• OlOSF OftTft (J?| FOUST 378

TF “F 1 e Ml * SUFI FOUST 178

IPO WO** ( K ) 9 Tf«P® “ TFHFf FOUST 380

«n#w (wrONj) 9 TFMFF T*SFf FOUST 381

?£•* • m# • SUFI - OftOSI ft OftTft (j? 4 II FOUST 382

TFNPf 9 MI 4 S«l*» FOUST 383

wnpw (« 4 U » TFMFb <» i«MFI FOUST 38*

•»9 Mr®* (KCfMJ 4 1) a TFPP# - TFPFJ FQU*? 389

75
• •6 CONTINUE FOUPT 366

IP <K»TN - 1 ) 690. ••©» *00 FOU®T 36?

• 90 «» • K1TM 1. FOUPT 369 Ml • WSTP1 FOUPT 369

• O'! (SO TO 01

9

FOUPT 390

• CO Tr*P« • UP FOU®T 391 VP a v® 4 VST#® . VI • WSTPI * VP FOUPT 39?

vi • TP*®# • V$T P f VI • VSTPP MI FOUPT 3®3

•19 TvOW® a VP VP FOUPT 39* !9» IF IKftSE - 11030. 9?C, ^30 FOU»T 395

• >0 I F «tf*l - NP21990, 010. 930 FOU»T 39 6 • 30 « » 1 FOUPT 397

I*P*» all* N®» - N *1 FOUPT 396

00 9*0 I? • 13. 12***. MP1 FOUPT 399

*eo OPT * I I?) » W9®« 1 * J FOU»T *00

OPT* 1 12 4 11 • WOP* l« 11 FOUPT *01 •*© K a K * ? FOUPT *02

(SO TO OPO FOUPT *03

c FOUPT * 0 * *05 e Cr«PLFT? * P F PI TPPWSFPO* IN THE 1ST OINENSIQN, N 900. «T CON- FOUPT *05

r JUGPTE SY«NFT®IfS *T E#CH STP6F. FOUPT *09

c FOUPT *07

•90 J3N** a 13 * ISP? - "PI FO'JRT *06 no 97n J 3 . 13. n?i FOU»T *09

*10 J?NP* a J 3 N*> - J*STP FOU»T *19

0(1 9*0 J? a Jft, J?»p*. J2STP FOUPT *11

J1NPX a J? J1P/J7 - IF©? FOUPT *12

JKNi • J 3 J2PPB 4 J2STP - J2 FOUPT *13 no ®70 J1 ® J 2. 11** 1 . I*Pf FOUPT * 1 * *15 K • 1 4 J1 - I* FOUPT *15

OPT* nil • WO©* (M$ FOUPT *16

opt* ni ii a woe* i k ii F0U»T *17

IF Ul - J*|970, 070. 0(SO FOUPT *16

• 90 OPT 4 U1CNJ1 • VD*K |K 1 FOUPT *19

*20 OPT* IJ19NJ * 11 a . VOPK |K * 11 FOUP* *20

e ?0 jicn • JKNJ - if*? FOUPT *21

eoo rrvTiNUF FOUPT *22

IF a IP 1 FOUPT *23

I F *1 a JF P* FOUPT * 2 * *75 IF UFP1 - MP11009, *00. 7*0 F9UPT *25

0 FOU*T *26

C CPP®LFTF ft P F ft l TPPVSF99" IN THF 1ST DINFNSIPN, N EVFN, tr CON- FOUPT *27

r JUGATE STKPFTPIfT. FOUPT *26

C FOUPT *29

©39 • QO fP TO ( 1*99. ll«e» 1260. 100C1. ICPSE FOUPT *30

190© VHPIF a ¥ FOU*T *31

M a M N FOUPT *32 TMFT» a - TV9* | ! *li:PT m FOU»T *33

IF II* T S ** 1 10?P# 1919, 1C10 FOUPT *3*

*?9 1010 TMFT* a » TMFTft FOUPT *35

10*0 SINTH a SIN ITHFT* / *.} FOU#T *36

WSTPP a - ?. • SI**TH • KIVTH FOUPT *37

VSTPJ a SIN ITMFTftl FOUPT *35

UP a W«?P® 1, FO(J07 *39

**n VI a WST»I FOUPT **0

1*11 N a 3 FOUPT **1

J N f N a * • NMftl F - 1 FOUPT **?

09 To 1090 FnUPT **3

1099 J e J * IN FOUPT *** **5 CC 10*0 I a ININ, NT9T, NP> FOU»T **5

SiJPP a 1 9»Tft HI OPT* mi / P. FOUPT ** 9

SUP! a IO»Tft II 11 4 OftTft U 4 11) / 2 . FOUPT ** 7 OIF® a 10»T* III - OftTft Ull / t . FOUPT ** 6 Pin a IPAT4 II 4 11 - 9 ATP (J 4 111 f 2. FOUPT **9

*50 TF»P® a V* • SU«I 4 VI 4 OIF® FOUPT *50 TFNPT a vl p SU«T - V® • OIF® FOUPT *51

8 PTP III a SUMP 4 TEPPP FOUR! *52

OAT* II 4 1» a 01 FI 4 TFNPJ FOUPT *53

OPT* m a SUMP - TFNPP FOU*T *5* *55 OPT* 1 J 4 11 a - ni*I 4 TEN*! FOUPT *55

104© J a J 4 NP? FOUPT *56 ININ a 1NJN 4 f FOUPT *57 JPIN a J«*IN - ? FOU*T *59 TFWPP a V® FOU»T *59

*to VP a VP • WSTPP - VI • VST*! 4 VP FOU*T *60 — VI B TFNPP 4 WS?®1 a vl • WSTPP 4 Ml FOUPT *61 ie*o IF IINTN - JNIM1030. 10*0. 1090 FOU»T *62

76
1000 18 C I ST ON ) 1070» 1090. 1000 EOU»T *63

1070 00 1080 1 • 1"IH, NTOT . NR2 EOURT *6*

*#5 1030 Oil* Cl * 1) • - OAT * II 1) F0U8T *65

i-joo NP? • 822 88? EOURT *66

8T0T NT OT 4 8T07 EOUPT *67

J a NTOT l EOU«T *68

l^AS • ntqt / ? 4.1 80URT *69

*7C 1100 ININ a I*** - 2 * NNAIE EOURT *70

I 1 81

N

EOURT *71

SO TO 11*0 EOU*T *7?

1110 OAT* (J) • 0*T A (II EOURT *73

OAT* CJ 1) • - OAT* II 4 11 8 QirB T *7*

*75 1120 I l 4 2 80URT *75

J J - 2 80URT *76

18 II - T8*»)1H0. 1130. 1130 80URT *77

1110 d*t* m • o*t* c i**T2>*» - o*t* iibin n 80URT *70 n«T« (4*11* 0. 80U*T *79

*R0 18 (I - J11150. 1170# 1170 EOURT *80

11*0 OAT* m OtT* II) 80U#T *81

0*T* 14 11 • OAT* If n 80U»T *62

1180 I • I - 2 80UBT *83

4 • J - 2 80U»T *8*

*8 5 18 II - |8 in ) 11*0# 11*0# 11*C 80U»T *85

11*0 0*T» (J| » 0*T A ( TNJ M 4 0*T* (ININ 4 11 EOURT *56

0*T« (J 4 l) • 0. EOU® T *87

T"AY a I8TN FOURT *88

or *0 1100 FOURT *8«

40Q 1170 OAT* (11 • PAT* III 4 DATA (?) 80URT *90

OAT* (?t 0. e OU*T *91 CO TO 1**0 80UBT *92

C EOURT *93

c CrNP|.8T8 A t e Al TRANSEORB 208 THE 2ND OR 3R0 DIMENSION BY EOURT *9*

408 c CONJUGATE 5Y88ETRIES. 8 0U3 T *95

c EQUBT *96

1180 TP (I1BNR - N8D11QC# 12*0. 1260 EOURT *9 7 11«0 nr 1**0 13 • 1# NTOT # N8? 80UBT *96

T2***» • 13 4 NP? - 8PJ EOUBT *99

500 On 1250 i? - 13. I2«ai# NP1 EOURT 500

I-IN » I* 4 I1BNG EOURT 501

l 8 AT a 12 4 NP 1 - ? EOU#T 502

IBS'S • ? B T3 4 NP 1 - JMJN EOURT 503

18 (I* > 13)1210. 1210# 1200 EOURT 50*

80? 1?P0 4“A* • J * A * 4 NP? EOU*T 505

1210 18 (IOIB - 2)12*0# 1?*C# 1220 EOURT 506

122C 4 J 8 A V 4 N®

0

EOURT 507

on 1*30 I • 1 8Jn # IB**, ? EOURT 508 DATA (T> • DATA (4) EOURT 509

810 OAT* U 4 1) • - OATA (4 4 1) EOURT 510

1230 4*4-7 EOURT 511

12*0 4 • 4* AY EOUBT 512

or 1250 I • I" IN # IB*t, n»9 FOURT 513 OATA II) * OAT* (J) EOURT 51*

515 OATA (I 4 1) - OATA 14 1) EOURT 515

1>80 J • J - NPQ FOURT 516

C EOURT 517

C ENn 08 LOOP ON EACH 0 1 B 8 N S I ON EOURT 518

C EOURT 519

??0 12*0 NPO N*1 EnuRT 520

NP 1 • 8*2 EOUBT 521

1270 NPB8V a N FOURT 522

12*0 • 8TI'P N EQURT 523

8N0 EOU*T 92*

77
Appendix D.* Subroutine PLT120R(X,Y,XMAX,XMIN,YMAX,YMIN,LAST, I SYMBOL, MO, MOST)

Purpose: To make a page plot of array Y versus array X.

Arguments

:

X

YXMI

N

XMAX

YM1M

YMAX

LAST

ISYMBOL =

MO

MOST

Array containing abscissa values of the function to be plotted.

Array containing ordinate values of the function to be plotted.

Minimum abscissa value.

Maximum abscissa value.

Minimum ordinate value.

Maximum ordinate value.

Number of points to be plotted.

A Hollerith variable containing the plotting symbol, e.g., to plot with the

symbol "X" ISYMBOL = 1HX.

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.

’This appendix is taken from Appendix A. 1.12 of Stubenrauch and Yaghjian [2].

78
1 Sl!8®ftMTINF n?l?e®( X. V, X«**S» ««!•»» VN4S» VI IN. LAST « ISY*4«H» NQR1T120® 1 1. »rsT» RfeTlSO* 2

c •O-'IETfi u /* /fc* •IT120* s ftTNFNMOn Ml)* VllS, 2YI13S, 8*49)4(121, 511 RLT120B 4

9- tuTCfts* es*»a» CPlt>*M, «1 *nk, IW«0M •1T120* %

I5*t* UINFS 91). icislunns • 1?1) HT129R 6

s«nai a cniliPNS / 10 ® 1 •IT120* T

ye |NP .nf. 1» «n TO 19C HTHM 8 Vl*» • VN*t •IT120R 9 1« Vtt| a V«S|N RIT120R 19

*1*9 a *«** PHim 11

««•»* a f«TN RIT120R 12

NftBfF® a 1HJ RIT120R 13

!R4N« a IK HT120® 14 19 ««T8I V a fOttJNN? • 1INFS RLT120R 15

TP |n*Tn* .IT. IS CO TO 129 •171209 16

Qfl Iftft T a 1. BATBTW •47120*- **-

1 f'O R8%0MJTS ® ®l»NK •171209 IS

1»ft r»*NTT'ii)x PIT120R 19

>0 IF HINES .IT. IS SO TO 140 S»iT12©K 10

CO 110 T a 1, tines RIT1209 21

19ft G9*pufl, IS a e-9**HCC0LU«MS» IS RPRDFR RIT120R 22 140 CrtiTTMliC *iTl 209- 23

IF 1 CHU««NS .IT. is 60 TO 160 9111209 24

• 5

Oft 150 T 8 1, CI?U'“KS •11120® ?§

150 CR4PMII. >f> • 1H. ®l 7120R 26

160 Oft*|TT*<l|F •171299 21 *SC it® « HU* - *$»&) t ICOtynns - l.S PIT120P 29 VSCAlF a IT149 - VS* 4 ! f (UNE9 •> l.S nmw 20

3ft IF l*«M» ,l T . 15 GO TC 190 •IT120R 30

ft ft 1?C V a 1. H»*S •LT12Q# 11

l?fi Trim a 10. » F tn&T(tf - is e XSC41F XSRA •IT1209 32 1*0 CftFTIN'ff •IT120B 33

1®0 IF U49T .IT. is 00 TO 299 •LT120® 34 PO 2*0 T o 1, |*5? R4T1209- H

J e (»ITS .ST. I L 4 R .09. SIU .IT. SSNAJ CP Y|J 2*0 •LT1209 36

t

f

mri .ct. v i s® . 0 ®. tin .it. ys*»*s eo to ?ao •IT1208 1?

is 3 («in - »S")4) t SSfALf * 1.5 •into® 38 n a inn - YSnS / YSCAIE .9 PIT 1208 39

*9 IT a LINES - If RLT120R 40

S04©U(T», f » 5 a fSVNDftt •1-I1-20R- 41

TArt ft fturt vie PIT120R 42

’*.0 ftftNTTNiJp 91T1209 43

IF lift . NF , «ft«T| 9§TU»N •LT129R 44

45 ®9 I NT 1900 •LT120® 45

VFS a VIA® Y<C*l® •IT120R 46

IF HINES .IT. is CP tn ??© •t*n©#- 4*

ftft ?*ft T a 1, L !“®s •LT120 R 46 YE« . VPS ~ YSCUF •LT 120 ® 49

90 point mo. VIS. IS849HIJ. n» J « 1* m«WSS R1T120® 50

»A0 fftNTTNUF •LT120® 31

?-»ft rftNTn'JF •111209 52

99 1 NT 559ft •1U20® 93

®® JNT 1900. *S •1 Y120R 54

95 BE Ti t 8 Y •LT1209 55 hoc F ft§ 9* ? I1N1.9X.E4I5FI....UHIS •IT120R 56

1510 8ftB0*T HU ,121419 •1T120R 57

l«i*0 Fft®*4T m e 9»,?4l5HI....SlW|| •1T1209 58 V5aa FOBT 4.T UW iau*4 -W

Aft SNft •1T120R 6ft

79
NBS-114A (REV. 2-8C) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publ ication Date

SHEET (See instructions) NBSIR 82-1674 Auqust 1982

4. TITLE AND SUBTITLE

Computation of Antenna Side-Lobe Coupling in the Near Field Using Approximate

Far-Field Data

5. AUTHOR(S)

M. H. Francis and A. D. Yaghjian

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) 7 . Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP)

Dept, of Defense

I ITRI/ECAC

North Severn

Annapolis, MD 21402

10. SUPPLEMENTARY NOTES

Q2] Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT ( A 200-word or less factual summary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here) Computer programs, in particular CUPLNF and CUPLZ, are presently in existence to

calculate the coupling loss between two antennas provided that the amplitude and phase

of the far field are available. However, for many antennas the complex far field is

not known accurately. In such cases it is nevertheless possible to specify approximate

far fields from a knowledge of the side-lobe level of each antenna along the axis of

separation, and the electrical size of each antenna. To determine the effectiveness of

using approximate side-lobe level data instead of the detailed far fields, we chose as our test antennas two hypothetical, linearly polarized, uniformly illuminated circular

antennas for which the exact far fields are given by a simple analytic expression. The

exact far fields are supplied to the program CUPLNF to compute the exact near-field

coupling loss. Approximate fields are supplied to a new program ENVLP developed for the

purpose of computing the approximate near-field coupling loss. The comparison of the

results from ENVLP to those of CUPLNF indicates that the use of approximate far fields

gives an estimate of the coupling loss which is good to about ±5 dB. In addition, the

plane-wave transmission formula for coupling between two antennas is used to estimate

upper-bound values of coupling loss. These upper bounds are compared with the maximum

coupling losses obtained from programs CUPLNF and ENVLP.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

antenna coupling; antenna theory, coupling loss; near-field measurements.

13. AVAILABILITY 14. NO. OF

Unl imited PRINTED PAGES

| | For Official Distribution. Do Not Release to NTIS 84

n Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

15. Price 20402.

[X] Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $10.50

&U.S. GOVERNMENT PRINTING OFFICE:1981 -580-327 / 383 USCOMM-DC 6043-P80