AlllDE blb3TM

NATL INST OF STANDARDS & TECH R.I.C.

A1 11 0261 6394

iMuth. Lorant A/Dlsplacement errors In an 3Cld0 .U5753 NO.1306 1986 V198 C.I NBS-P %'i

\

•"•tAU Ot */ NBS TECHNICAL NOTE 1306

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards
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NBS

RESEARCH

INFORr.lATIOli CENTER

Displacement Errors in Antenna ^-

Near-Field Measurements and 'V^^

Their Effect on the Far Field

Lorant A. Muth

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

flrjtft

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

NATIONAL BUREAU OF STANDARDS, Ernest Annbler, Director

October 1986
National Bureau of Standards Technical Note 1306

Natl. Bur. Stand. (U.S.), Tech Note 1306, 36 pages (Oct. 1986)

CODEN:NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1986

For sale by the Superintendent of Docunnents, U.S. Government Printing Office, Washington, DC 20402
CONTENTS

Page

1. Introduction 1 2. General Mathematical Statement of the Problem 2 3. The Maximization Procedure 5 3.1 Some Simple Models 8

3.2 Maximization of General Complex Near-Field Error Integral 10

3.3 Maximization for K it 12

3.4 Steered Beams.... 7 13

4. The Error-Contaminated Far Field 14

5. Displacement Errors in Spherical Scanning 15

5.1 A Simple Model in Spherical Geometry 19

6. Summary 19

7. Acknowledgments 20

8. References 20

Appendix A Maximization of General Complex Near-Field Error

Integrals— Some Further Details 21

Appendix B The Maximization Procedure for Steered Beams 23

Appendix C The Far-Field Error Spectrum 26

Appendix D Radial Displacement Errors in Spherical Scanning 29

m

Displacement Errors in Antenna Near-Field Measurements

and Their Effect on the Far Field

by

Lorant A. Muth

The effects of probe displacement errors in the near-field meas- urement procedure on the far-field spectrum are studied. Expressions are derived for the displacement error functions that maximize the fractional error in the spectrum both for the on-axis and

off-axis directions. Planar x-y and z-displacement errors are studied first and, consequently, the results are generalized to errors in spherical scanning. Some simple near-field models are used to obtain order of magnitude estimates for the fractional error as a function of relevant scale lengths of the near field,

defined as the lengths over which significant variations occur.

Key words: error maximization; far-field spectrum; near fields;

off-axis fractional errors; on-axis fractional errors; planar

scanning; probe displacement errors; spherical scanning.

1. Introduction

The unavoidable errors in the probe's position while scanning the near

field of an antenna show up inevitably in the far field of the antenna being

measured. As is well known [1], the far field of an antenna is obtained by

taking the Fourier transform of the antenna's planar near field and performing

additional algebraic manipulations to remove the effect of the receiving characteristics of the probe, a procedure known as probe correction. In evaluating the accuracy of a near-field range

planar, cylindrical or spherical--

obvious and natural questions arise: 1) what systematic position errors will

lead to maximum far-field errors; 2) what is the dependence of this maximum

far field error function on the wave number _k_, whose magnitude is constant;

and, more generally, 3) what is the exact error-contaminated far field of an antenna if its exact near field and an arbitrary probe displacement error

function are known?

Two previous studies raised the first two questions above [2,3] and

treated them in the context of planar scanning. This paper re-examines the

above question in the planar context from a slightly different point of view,

with the intent to achieve enough generality in the mathematical formalism so
that the analysis can be extended to study position errors in cylindrical and

spherical scanning procedures. Some error expressions are derived in spherical geometry which can serve as the basis for computer simulation. Similar

simulations in the planar case have been discussed in [3]. A general expression that answers the third question is also derived.

To accomplish the objectives of the paper, expressions for maximum systematic errors in all geometries have to be derived. First, simple general

mathematical arguments are used to get the structure and the relevant parameters that appear in the fractional error expressions; then a rigorous procedure for maximization of error is outlined for on-axis errors in real near

fields. This simplified special case is considered first for a mathematical

reason: the error expressions for realistic (complex) near fields for on- or off -axis directions in k-space can be obtained using the procedure worked out

for the simplified on-axis case if a straightforward additional procedure is

incorporated. Once this procedure is worked out, all special complicated

cases such as steered beams and all errors in spherical geometry can be

treated.

2. General Mathematical Statement of the Problem

We can derive general expressions for the fractional error in the spectrum of the near field B(_x) due to arbitrary position errors. Here _x is an arbitrary three-dimensional position vector. If the real function 6£(_x) is

the error in position of the probe at jc^, then the near field measured is

B(x_ + 6£(_x)). The fractional error in the spectrum D(_k_) due to position errors is then

-ik»x

AD(k) / {B(x + 6x(x)) - B(x)} e

' ' d2x

D(k)

" ^nTTx

/ B(x) e

" d2x

(1)

where the integration is over the finite scan plane. The use of a finite

plane of integration is an approximation in the denominator, but is exact in

the numerator, since errors occur only at points where measurements are

taken. In the above expression we also have J<_ = (K_, ±y). where J< = (k^* ky).

^z = ±Y and |k| = -^ = constant for lossless media. We now seek that real
function 5_x(_x) that maximizes the fractional error in a given direction in kspace. The numerator in eq (1) will have a maximum for a finite 6)<J,x) only if

6_x(_x) is subject to the constraint

^ / |6x|2 d2x = a2 (2)

where a = constant, A is the total scan area. Expressions (1) and (2) then

define a variational problem wherein the function 6_x(_x) that maximizes eq (1) can be found with the use of the functional derivative [4]. This simple

procedure will be indicated below for the one-dimensional case. Physically,

the constraint in eq (2) merely restricts the error functions to be considered

to a constant RMS value. Strictly speaking, the variational problem has to be

formulated for either the real or imaginary part of eq (1) separately. To

find the maximum of the absolute value of the fractional error, a slightly

modified procedure has to be followed.

One can write down very general expressions for the maximum fractional error in eq (1) without having to specify 6x_(_x_). By the use of the mean value

theorem for real functions [5] and for J< = (on axis), one can write

(assuming that any z-dependence is specified as a function of x and y, and

suppressing the k^ = +y dependence)

AD(0) ^(^ ' «5^^)) - ^(^)

,., ^^^ B(x)

Where x and x are some points on the scan plane and B(x) is the average of the

near-field measurements. We have assumed here that B(x) is essentially a real

function (complex phase is allowed), since the mean value theorem cannot be

applied to complex functions directly [6].* We can approximate eq (3) as

*To reiterate, these simplifying assumptions are made here in order to develop

an understanding of the relevant parameters and the structure of the error expressions sought. It is not true that this simple example is studied

because more complicated cases cannot be treated. As will be seen below, the error analysis of any special case that is more general than the one considered here follows simply from the considerations in this section and the

next and the additional procedure outlined in section 3.2.
^^ = < . {4b)

MO) = Iv^xi ^,^j ^^^^ B(x)

which in the one-dimensional case is (prime denotes differentiation)

aD(0) _ ^^^^ ^^^""^ B'..v«x

B(x) B{x)

As will be seen below, in a first order approximation a properly normalized

displacement-error function has a maximum proportional to a, hence (using a as constant of proportionality),

B

_ aD(0) ^ max ^ a /ca\ ^^^^ B(x) ^

where c is some constant of order unity, and i is scale length of significant

variation in the near field. In eq (5a) and below, the subscript on y indicates the order of approximation. The exact value of c can be obtained only

from either an actual near field or a model of it. An order-of -magnitude

estimate for c can be obtained as follows: B' « B /ji, B(x) =» B /2 and max max' ^ ' max'

assuming a « 1, one obtains c = 2. As will be seen below, this is, indeed, a good estimate. (Equation (5a) agrees in form with eq (63a) in [3]).

The scale length i represents variations in the near field B(x) either

parallel or perpendicular to _k^, the wave vector of interest. In case

6xiik , then 2." » |k I = -=?-, as can be verified from the simple model of a

- -0 I oi X plane wave of constant amplitude propagating in the direction 6_x. Variations

in the amplitude of the near field orthogonal to 6_x_will be reflected in the

constant in eq (5a), in this case denoted by c . For 6xik, i represents the

scale length in the variation of the near-field profile along surfaces of

constant phase. The constant in this case will be denoted by c . Hence, we

can write

yj"^ < c,,ka„ (5b)

and

a ui') < c _L. (5c)

1 i a
In the case of an on-axis beam (_K = 0) eq (5b) represents the upper bound error resulting from z-displacement errors (out of the scan plane), and eq

(5c) represents the upper bound resulting from displacement errors in the scan plane. The ratio

(5d)

in general

Two alternative expressions to eq (4) can be shown to be, using 6 for the

unit amplitude near field and denoting derivatives by ',

aD(0)

D(0)

< a max

S max

<6> <B'2>

1/2'

(6)

and

aD(0)

0(0)

<[S-(x)]2>

" <6(x)>2

' (7)

where the < > implies the average. These expressions will be derived in the

next section. Again, the derivatives in eqs (6) and (7) represent directions

either parallel or orthogonal to _k_.

In eqs (5) through (7) it has been assumed that the near field is real.

If the fractional error in directions other than J£ = is desired or if the near field is complex (i.e., for any real antennas, steered beams or electrically small antennas), the derivations of scale-length expressions are slightly more complicated, but fundamentally present no great difficulties.

3. The Maximization Procedure

To solve the variational problem as stated in eqs (1) and (2) in complete

generality, we have to proceed in steps. First, we solve the simplified problem, where the near field is a real function and the wave vector in the exponential vanishes. Then we seek to maximize (in 1-D) the integral

I =

/ [B(x + fix) - B(x)] dx,

L

(8)
subject to the constraint

I / (6x)2 dx = 1,

*- L (9)

where L is the interval of interest. We use the Lagrange multiplier n [4] to

maximize

I = / [B(x + 5x) - B(x)] dx - n / (6x)2 dx (10a)

with respect to 6x. Thus,

61

6(6x)

or

f [B'(x + 5x) - 2n 6x] dx =

' L

6x = y- B' (x + 6x).

(10b)

(10c)

Equation (10c) is an implicit statement specifying 6x(x) that will maximize

the integral in eq (8). To obtain 5x(x) explicitly, eq (10c) is expanded in a Taylor series and the constant n is obtained from eq (9). The conditions for

first- and higher-order approximation values can thus be worked out. If we want the displacement-error function to satisfy

i / (5x)2 dx = a 2 ^ 1 (11)

instead of condition (9), we merely have to multiply 6x in eq (9) by a^. Thus

there is no loss of generality in using eq (11), as eq (9) is a special case

of eq (11).

The Taylor series expansion of eq (10c) is

6x = y^ {B'(x) + B"(x) 5x + -2^ B"'(x)(6x)2 + ...}.

For a first-order expansion in 6x the condition

_1 B"'(x)

2! B"(x)

(6x) « 1 (12)
must hold, and similarly for higher-order terms. This implies

6x = B'(x)

^ 2n - B"(x)

where n is determined by eq (11). If we further assume that

(13)

B"(x) « 2n (14)

then from eqs (13) and (11)

4n 2^ / [B'(x)]2 dx = a

and

Sx = a B'(x)

X .. 1/2

<B'2>

(15)

satisfies eq (11). Consistent with the first-order approximation developed

here, the integral in eq (8) is approximated as

with

I = f B'(x) 6x dx

L (17)

1 B"(x)

6x « 1

and similarly for higher-order terms. Using eq (15), eq (12) becomes

""x B"'(x) B'(x)

1 B"(x) ^g.2> 1/2

and eqs (14) and (18) are consistently satisfied by

« 1

B"(x)

^' <B'2>

^/^

« 1.

(18)

(19a)

(19b)

It is easy to see that eqs (19a) and (19b) can be satisfied for a small enough

0^. Special attention must be given to points where B"(x) = 0, but we will

not address that here.

The maximum of eq (17) is obtained using eq (15)
I = a L <B'2>

X V2 (20)

and the fractional error in eq (1) can be written as in eq (7)

,2 = aD(0)

D(0)

<B'2>

= a2

<8'2>

X <B{x)>2 X <B(x)>2

(21a)

where 0(x) is the unit-amplitude near field. Higher-order approximation

schemes to solve eqs (8) and (10) consistently can be worked out, but this

will not be done here, since the unwieldy algebraic manipulations lead to no new results. In eqs (1) and (2) a second-order expansion of the integral in

eq (8) has been found useful, i.e., the maximization of

/ {B'(x) 6x +

Ij- B"(x)(6x)2} dx

has been sought. We can use the first-order expression (15) to get

r,^<3.2> 2,_,2 _]. (21b) ^2 ^

The more exact expression for ^2 using a second-order expansion in eq (10)

would result in a much more complicated form.

Since aD e D - Dq, where D^ is the error-contaminated spectrum, one can write D = Dq(1 + y^ ) or

R(dB) E 20 logio (tt) "^ ^-^ ^i- (22)

This result together with eq (21a) can be compared to that given in [3].

Before proceeding to maximize eq (8) for complex near fields B(x) or for

K ;t (off axis), a few elementary near-field models will be used to exhibit

some explicit results for the fractional error p.

3.1 Some Simple Models

Two basic models that incorporate the most essential features of near

fields will be used in expressions (21) and (22). These d,TQ a triangle and
Figure 1. Simple models for the near field to calculate

the effects of probe displacement errors

a cos^ax, where a = ^. These are illustrated in figure 1. Both of these

models have a scale length i. The model independence of the maximization

process can be surmised by deriving results for both of these models. For the

triangle |6'| =y 6" = 0, except at x = 0, and 
 = V2 (2a)(1)/2£ = V2 • So,

VI < 2-^. (23a)

The second-order term in ^2 contributes only at x = 0, which cannot be calcu2 lated by elementary means. If we approximate b" * —. then

a a ^ = 2 — + (—J . 1 ^ I (23b)

For 6 = COS^aX, B' = - asin2ax, B" = - 2a2cos2aX,

and

/2 * 2.22 —

.

(23c)
In this case, there is no contribution from the second-order term, since the

near-field profile is symmetric around x = 0. Only asymmetric near fields

will contribute here. In practice this will arise for steered beams.

Comparing eqs (23a) and (23b) one can see that the results are essen- tially model independent, since the constant coefficients are essentially

equal and the other parameters enter exactly the same way.

On closer examination, it is found that for this example eq (19b) gives

the most stringent condition on ax> "i'S.,

i « \ i\f (2*)

must hold for the analysis presented in section 3 to be valid. If L = 2ji is

the aperture dimension, then a << L/9. Let L = nx, where x is the wavelength

and n is some constant, then a « m(0.1 x), where m > 1. In practice, such a root -mean-square position error is attainable. Using eq (22), one can write

R(dB) = 40 -^

or ^ = inR. (25)

This last relationship gives the RMS displacement error in units of wavelengths in terms of R, the far-field error in dB, and n, the aperture size, in

units of wavelength.

3.2 Maximization of General Complex Near-Field Error Integral

For general complex near fields or for errors in the off-axis direction,

the expression whose amplitude is to be maximized has both real and imaginary

parts; i.e.,

I = / G (x,5x(x)) dx + i / G.(x,5x(x)) dx. (26)

G (Gi) "is the real (imaginary) part of the integrand in the numerator of

eq (1). One can either maximize the real (imaginary) part using the same

10
procedure as for on-axis real near fields, but the maximum of the amplitude of

I in eq (26) will not, in general, be thus attained. Only if

/ g^g^- dx = (27)

where g^ = [G^]', g. = [G-]', [Q]' = y^l^y | ^^ = 0' ^^^ ^ ^^^ maximum of I be

given by the larger of the maximum of the two integrals in eq (26). The proof

of this simple fact will not be detailed here. If condition (27) does not

hold, we look for the function that will maximize eq (26) as a linear combination of functions that maximize each of the integrals separately. Thus,

6x = a 6x (^^ + S 6x^^) (28)

where 6x^^^ (6x^^') is the displacement error function which maximizes the

real (imaginary) part of the (26), and a and a are constants determined by

conditions (31) and (32) below. Adapting the results in section 3, a firstorder approximation scheme to maximize eq (26) is as follows:

Let 5x^^) =

g

r

6x(^) =g.

I = f g2 dx

I.= / g2 dx

\ -

/ g.g^dx.

Then expanding eq (26) for small 5x(x) and from eqs (11) and (28)

|IP = (ct I^ + 6 IJ2 , (a I,^ + B 1^)2 (29)

2la^ = a2 I + 2a8 I + 6^ I. (30)

X r m 1

^ '

and one determines the parameters a and 6(a) from

ULLl = (31)

da

11
da

= 0. (32)

The details of this calculation are presented in Appendix A, where final

expression for a and 6 are derived as well as conditions that must hold

for the special cases a # 0, g = and a = 0, a ^ to maximize eq (29) subject to constraint (30).

A more detailed treatment of a specific case of eq (26) is given in sec- tion 3.4 and Appendix B.

3.3 Maximization for K ^t

If in the region of interest the near field is real, the off -axis error displacement function

6x = 2. - B" 1^;"^ ll]

(33)

will maximize the integral

/B'(x) 6x

1^i"3 ^IJ}

dx (34)

where n is determined by eq (11). If eq (27) holds, i.e.,

/ [B'(x)]2 cos kx sin kx dx = (35)

either integral with sin kx (cos kx) will maximize the corresponding fractional error y. For near fields symmetric about the origin eq (35) will be

satisfied. In practice, most near fields have a small asymmetric component,

so eq (35) is only approximately satisfied. In case the asymmetry is

significant, the procedure outlined in sections 3.2, 3.4 and in Appendix B has

to be followed. The results are for |D(k)| ?t

,2 = AD(k)

< a <B'2 sin2 kx>

x <B(x) cos kx>2 + <B(x) sin kx>2

(36)

12
and

adoo

D(0)

< 0' <B'2 s1n2 kx>

<B(x)>2

(37)

and similarly for cos kx. Similar calculations to obtain second-order corrections could be easily performed.

In Appendix C, eq (37) is evaluated for the simple model B(x) = cos^ ax

in figure 1, and the results are compared to real simulations.

3,4 Steered Beams

In the case of steered beams, displacement errors both in the scan plane

and perpendicular to the scan plane have components along the off-axis beam

direction. One can model such a beam to zeroth order by the near field

B(x) = b(x) e Tex

(38)

where e is some wave number and b(x) is one of the profiles depicted in

figure 1. Mathematically, the problem of maximizing the error integral (26)

(either for J<_ = e_, or k ^^ e) can be simplified if one keeps in mind the ratio

in eq (5d); i.e., if the beam is steered enough off axis so that displacement

errors in the scan plane correspond geometrically to displacement errors

parallel to _k_, with a small additional effect due to errors perpendicular to

k. If the beam angle is e, a first-order approximation is

I I rC0S9i ^ ^ i rSine,

(39)

where, depending on the magnitude of 9, one of the terms is negligible com- pared to the other. The choice of the trigometric function depends on whether

one is examining z or x-y displacement errors. For example, for 9=0, only

the term c a |ej constributes for z errors, and for x-y errors only the term

c a /i contributes. For 9 = v/2 similar reasoning shows that the role of each

term in eq (39) is interchanged. For angles such that any error has a significant projection both parallel and perpendicular to Jc_, eq (39) is essentially

valid, but the full analysis as outlined in section 3.2 and Appendix A has to

be carried out to determine the constants.

13
For k * e, the expressions in section 3,3 can be easily adopted for beams

steered sufficiently off axis. One merely has to put the symbols in that section as

k -». e - k

B(x) > b(x) (40)

B'(x) > e b(x)

and

k = ^ k = e.

The adaption of the maximization procedure for steered beams is detailed in

Appendix B.

4. The Error-Contaminated Far Field

The error-contaminated spectrum of the near field can be calculated

exactly if the near field of the antenna and the exact displacement-error

function are known. Let Ug(J<) be the error-contaminated far field,

-ik.x

D (K) = / B(x + 6x(x)) e

" " d2x (41)

which, for errors in the x-di recti on only, can be expanded as

;^R 1 a2

-'''•^ ^^(K) =

/ [B(x) +1^ (x) 6x + — -^B(x) (6x)2 + ...] e

' " d2x.

The general expansion is a three-dimensional Taylor series. Since

ik'.x

B(x) = / D(K') e

" d2K' (42)

spatial derivatives of B(x) can be obtained from eq (42). Each differentiation will introduce < with the result that

tion will introduce a factor of ik., j = 1,2,3 into the integrand of eq (41)

i(k'-k).x ik.6x(x)

D (K) = / D(K') [ / e

" " " e

" " " d2x] d2K' (43)

gives the error contaminated spectrum in terms of the known exact spectrum and

the known displacement error function. Equation (43) can be written as

14
D (K) = / D(K') E(K',K) d2K'

*^^^^^ i(k'-k).x ik'.6x(x)

E(l<',l<) =/e""'e" "" d2x. (44a)

If 6x_(j<_) E 0, then E(K' ,K) = 6(_k_' -

J<_) and Dg(J<_) = U(J<) ; i.e., there is no error in the spectrum. If 6x_(x_) ~ c_, the result is essentially the same, as ik.c

merely a phase factor e

" ~ is introduced. Another simple special case follows if 6x(x) = (a^x, a2y) as discussed in [3],

For small 6x_{x) such that k.6x « 1, eq (43) immediately yields a firstorder approximation

d^{K) =

j m') Ei(KM<) d2K'

where

i(k' - k).x

Ei(K',K) = 6(K - K') + / e

" " ' ik.5x d2x. (44b)

If we model 6x as a sum of delta functions, i.e., 5x(x) - V e 5(x - x ) where ep is a complex amplitude and x^ are the grid points where measurements

are taken, then eq (44b) yields

AD(K) = D (K) - D(K) = ik .

( V £pB(x^) e -*"]. (44c)

n

Here, e is an unrestricted amplitude of the displacement function, and hence

could be a complex random number. In such an event, eq (44c) gives the

effects of random displacement errors on the far field.

5. Displacement Errors in Spherical Scanning

It has been observed experimentally that for electrically large antennas

the near-field amplitudes obtained in planar and spherical scanning are essentially equal [3]. Thus,

|B^(r)| ^ |B^(x)| (45)

lb
where ^^{rj is the spherical and Bp(x_) is the planar near field andV is the

general three-dimensional position vector. The phases across the main beam,

however, differ, primarily due to the change in the z displacement between the

probe and the antenna. In a planar scan, the phase is essentially constant,

but in the spherical scan the significant phase is given by

kAz(x) » u (f) i^)\ y =

(^) « 1

(46)

where R is the radius of the scan sphere and x^ + z^ = R2 is the intersection

of the scan sphere and the y = plane.

The simple expressions in eqs (45) and (46) can be exploited in spherical error analysis to take advantage of the results obtained in planar error

analysis: one merely has to transform the phase of Q^{r_) according to eq (46)

and approximate Bp(x_) with Bg(_r). The additional effect due to the variation

of the orientation of the probe in spherical scanning as a function of position relative to the constant orientation of the probe in planar scanning is

only significant at extreme angles and is neglected in this section. Accordingly, errors in spherical scanning will, in general, be a linear combination

of z and x-y errors in planar scanning. The linear coefficients will depend

on an averaged geometric relationship between the sphere and the plane, as will now be shown.

Consider a displacement error function 69^(9) along an arc that projects

onto the x- and z-axis at y = 0. The x- and z-components of 6_0_ are

^ ^ A, AAA

69(e) = 69(9) [(9»x)x + (9»z)z] or (47)

A A R69(e) = 69(9) [Rcos9x - Rsin9z].

In planar notation, eq (47) is

6x(x) = 69(x) RC0S9 = Z 69(x)

(48)

6z(x) = - 69(x) Rsin9 = - x 69(x)

16
-1 X where 9 = sin n-> ^nd R is the scan radius. Both displacement error functions in eq (48) must be taken into account in the maximization procedure, and

section 3 must be altered accordingly.

The constraint (corresponding to eq (11)) is now written as

where de/dx = 1/z. The expression to be maximized is now

(49)

i =

/ {B (x + R6e(x)} - B(x)| dx - XR / -^-^^^ dx (50)

L L

^

which to first order is

' = ^ / iff ^ - f ''I C***' dx - XR /

-t^ dx. (51)

The functional derivative of I in eq (51) with respect to (R69(x)) will give

the maximizing function similar to eq (15). Since -r— = iyB, the integrand in

Z eq (51) is complex, in general, and the procedure outlined in section 3.2 must

be followed. Each term is maximized separately by

and (52)

„ , X 1 .z. .X, SB

«"'^) - n (r^ (r) ilHence,

R6e(x) = 0^ r-^ (53)

9 1/2

•R^ ' X

'x " 8x °"" """ "-"--^ -^^e - RAe

where B^ = -^ and one defines <Q>^ e p^ / Q dx, where Q is any quantity,

Similarly, for z-displacements

17
R«9(x) = a^

Yfz-

(5'')

The integrands in eq (51) become

3 1/2

i = a^RAe <(|) (Bp2>^ (55)

and

2 1/2

I = '^ RA9 <tF)(^)'(B )2> (56) .r;vr; v-^.

and the corresponding on-axis fractional errors are

"?(0) ' i <B(x)>2 (")

or

<(f)(^)^B^)^>

^?(°) ^ ^e <B(x)>2

(58)

Expressions (55)-(58) can be compared to eqs (20)-(21). The presence of the

geometric factors (-d) and (4) in these new expressions merely reflect the fact

that the magnitude of 6^(x) is being optimized rather than the displacements

in the x-y plane.

Equations (52) and (58) individually maximize the respective terms in eq

(51). However, as we have seen in section 3.2, the maximization of |l| in eq

(50) is given by the linear combination

R69(X) = . (|)' f * B (f)(^) Ifll (59)

where a and 6 are determined using the method outlined in section 3.2 and

Appendix A.

The corresponding treatment for radial displacement errors is outlined in

Appendix D.
5.1 A Simple Model in Spherical Geometry

In general, the on-axis fractional errors are essentially (from eqs (57)-

(58))

^?(0) < c2 i^f (|)^ . c2 o2 k2 (|) i^{ . c2^ al [|) (|)'(|) (60)

A /\ where z, z, z, x and x are some intermediate values in the range (0,R). For

narrow-beam antennas ii»X, the second term predominates, but for very narrow

beams, all the terms might be equally important. The simple models in figure 1 can be used to estimate in somewhat more detail the terms in eqs (57),

(58) and (60). These estimates are, to leading order,

3 a 2 max — —1 —r-5 for x-component (61a)

2 ^R^ sin3 e max

, e

1 „. - max ^ / ^1L \ a2k2 —T—r for z-component (61b)

4 sin3 9 ^ and "lax

4 k

-^ a2 -T- sin3 9 for the mixed component. (61c)

These expressions are comparable to eqs (5b) and (5c) and to (23) and (25).

6. Summary

The effects of probe displacement errors on the far-field spectrum have

been examined both for planar and spherical scanning. Expressions for the

displacement errors that maximize the error in the far-field have been derived

using a method well known in the calculus of variations. The treatment of the

planar case is straightforward, but the spherical problem is complicated by

the fact that an error in a spherical coordinate corresponds to both x-y and z errors in planar geometry. Hence, a more complicated maximization procedure

had to be adopted after the spherical data were transformed both in amplitude

and phase onto the plane. To first order all fractional errors can be

expressed as functions of c —. where c is some constant of order unity, i is

the relevant length scale either parallel or orthogonal to the direction in kspace under observation, and a is an integral measure (constraint) of the

total mean-square error of the measuring system.

19
7. Acknowledgments

Numerous discussions with Allen C. Newell on the previous approaches to

the subject area treated in this study are gratefully acknowledged. Useful

interaction with Dr. David Hill and Dr. James Randa is also acknowledged.

8. References

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

antenna-antenna interactions. Nat. Bur. Stand. (U.S.) Monogr. 162; 1981.

[2] Yaghjian, A. D. Upper-bound errors in far-field antenna parameters

determined from planar near-field measurements. Part 1: Analysis. Nat.

Bur. Stand. (U.S.) Tech. Note 567; 1975.

[3] Newell, A. C. Upper-bound errors in far-field antenna parameters determined from planar near-field measurements. Part 2: Analysis and computer

simulation. Lecture notes for National Bureau of Standards Short Course,

Antenna Parameter Measurements by Near-Field Techniques: Boulder,

Colorado; 1975. Private communication.

[4] Mathews, J.; Walker, R. L. Mathematical methods of physics, second edition. Massachusetts: W. A. Benjamin, Inc.; 1970. 501 p.

[5] Taylor, A. E. Advanced calculus. New York: Ginn and Company; 1955.

786 p.

[6] Conway, J. B. Functions of one complex variable, second edition. New

York: Springer-Verlag; 1984. 320 p.

20
Appendix A

Maxiinization of General Complex Near-Field Error Integrals

Some Further Details

In section 3.2, the maximization of general complex near-field error integrals has been outlined. Here we present some important details. All

quantitites used in this appendix have been previously defined in section 3.2.

Condition (32) results in

d6

al,, + Bl

r m

da al + gl. m 1

(Ala)

and conditions (31) and (Ala) give

- I +^ I dg ^ m da r da

J d6 T

'

i ' da m

(Alb)

Equation (Ala) leads to

m da 1

I .^i '

r da m

(A2a)

and eq (Alb) leads to

•dB^2 , d8 rV ^i m

and since the discriminant b^ - 4ac > one obtains real solutions, with

I - I. 1 . ^

' 2 I m [% = . I i /I2 ^ 1. (A3)

da^±

An

The choice of the sign in eq (A3) is arbitrary, g' = (-p) can then be substituted into eq (A2a) to obtain a in terms of 6, i.e.,

21
r ± m

which can be used to obtain 8 from eq (30)

I + Si L

-V2 ! = a /7l (f2 I - 2f I + I.)' ^2. (A5)

Equations (A4) and (A5) provide the coefficients in the displacement error function (28) that maximizes the general error integral (26).

Treating the special case a =!t 0, g = 0, that corresponds to maximizing

only the real part of eq (26), one obtains from (A2a)

d^ T" ^'^^^ m

and from (Alb)

[ / g,gTClx]2 - [/ g2dx]2= (A7)

which is not, in general, satisfied. For example, gp = g-j would satisfy eq

(A7). Similar results hold for a = 0, e # 0, i.e., one must have

[ / g^g. dxj2 - / g2 dx / g? dx = 0. (A8)

This shows that these special cases do not maximize, in general, the amplitude

of the integral in eq (26).

22
Appendix B

The Maximization Procedure for Steered Beams

In this appendix the qualitative physical treatment presented in section

3.4 is made more precise. Only the essential details are presented.

For steered beams, from section 3.4,

B(x) = b(x) e^^^. (38)

The error spectrum is then given by

AD(k) = / [b(x+6x) e^^^""""^^^ - b(x) e^"^^] e'""^^ dx (Bl)

where 6x{x) is subject to constraint (11). The method presented in section 3 can be adapted to yield a condition for 6x(x) that will maximize the real

(imaginary) part of eq (Bl). For k = e, one obtains for the real part

5x = ^r [b'(x+6x) cos(e5x) - eb(x+5x) sin(e6x)] (B2)

corresponding to eq (10c). For small e6x, the zeroth-order approximation immediately follows; i.e., let cos(e5x) » 1, sin(e5x) « 0, and eq (10c) is

recovered. Higher order approximations are obtained by expanding in eq (B2)

the near-field quantities and the trigonometric functions in Taylor and infinite series, respectively, and collecting terms in increasing powers of 5x.

Thus,

6x = 2I[b'(x) + [b"(x) - e2b(x)] fix + ^

[b"'(x) - e2b'] (5x)2 + ...]. (B3)

This expression should be compared to the expansion above eq (12). To first

order then,

5x = ^^^-^ (B4)

*^ 2x - [b"(x) - e2b(x)]

23
If

b"(x) - £2b(x) « Z\^^^ (B5)

where the supercript indicates first order, then the constraint (11) gives

5x = a -^^^1, ^ (B6)

^ <b'2> /2

or

2x(^) =-I <b'2>V2. (B7)

Again, for small enough o^ condition (B5) can be satisfied, as well as conditions corresponding to eqs (19a) and (19b), The second-order maximization of

the real part of eq (Bl), using (B6), is

1 Vo 1 <(b" - e2b^ b'2>

This expression corresponds to eq (22).

The imaginary part of eq (Bl) can be maximized similarly to the above

procedure. One obtains

6x = "2^ [b'(x+6x) sin(e5x) + eb(x+5x) cos(e6x)] (B9)

and

5x =

e b

2x - c b'(x)

where, if 2x » eb'(x), then from the constraint (11),

= -^ <b2> V:

and

^4Ut (BIO)

2x = -^ <b2> /2 (Bll)

a

X

6X = a .y^. (B12)

' <b2>

''2

The second-order maximization of the imaginary part of eq (Bl), using

(B12), is

24
To maximize the amplitude of AD(e), we construct the linear combination

6x = a b(x) + B b'(x) (B14)

and use the method outlined in section 3.2 and Appendix A to solve for a and

6. This will not be presented here. Finally, the case k ^t e could be fully

developed exactly along the lines presented in this appendix.

I

25
Appendix C

The Far-Field Error Spectrum

In this appendix the theoretical result in section 3.3 is evaluated for

the model near field B(x) = cos^ ax, where 2a = k'= y. Equation (37) gives

the imaginary part of the fractional error, whereas the real part is given by

AD(k)

0(0)

< o' <B'2 cos2kx>

<B(x)>2

(CI)

When the averages are evaluated one obtains

AD(k)

DToy

< a^ a^ (1 ± sincq + sinc(q' - q) + sinc(q' + q)) (C2)

where sinc(q) = sin(q)/q, q = 2kji and q' = 2k'z = 2Tr, and the upper (lower)

set of signs give the real (imaginary) part of the fractional error, respectively. These functions of k have been plotted in figure C-1.

CSI

esj

—•fit>

0-

-.25

-25 -20 -15 - 10 -5 5 10

2 1k ( r ad i an s ) 15 20 25

Figure C-1. The amplitude squared of the normalized maximum fractional error as a function of k for a near field B(x) = cos2ax2a = k' = tt/£.

26
Using the error displacement function that gives a maximum error on-axis, the

fractional error spectrum as a function of k is given by

vD(k) ^ ^ . yx .

[ / B'2 coskx dx + i / B'2 sinkx dxl.

"o <B><B

When this is evaluated for B(x) = cos^ax, one obtains

(C3)

vD(k'

I

2 = 23 a2 a2 [\ sine f

- ^ sinc(q' + f)- j sine (q' - TTO f]]

where q(q') have been defined previously.

(C4)

In figure C-2a the effects of experimentally induced near field errors on the far field are shown. The maximum error occurs on axis. In figure C-2b

the expression (C4) is plotted. The qualitative agreement between the experimental and theoretical curves is apparent. For a general (realistic) near

field, the error spectrum will be given by a sum over k' of functions given in

eq (C2) wherein each term is weighted by the spectral component of the square

of the derivative of the near field, as can be seen in eqs (37) and (CI).

0.50

0.40

;E 0.30

0)

0.20

0.10

c 0.00

i.o

s- -0.10

+j -0.20

c:<uo

J- -0.30

<u a.

-0.40

-0.50

• —

M\ A A l\ Li k\ ^ A/ 'V '\\ ^u^ 1 W ^

r1

-90 -72 -54 -36 -18 18 36 Angle in Degrees

54 72 90

Figure C-2a. Error in percent of maximum amplitude in far-field pattern

due to x-position error (same as figure 17 in [3]).

27
CO

CVJ

. 15 -

. 05 -

Figure C-2b. Theoretical fractional error in the spectrum

when the on-axis error is the maximum.

28
Appendix D

Radial Displacement Errors in Spherical Scanning

In this appendix the equations analyzing e-displacement errors in section

5 are adapted to displacement errors in the radial direction, denoted by

6r(9). The error displacement function is

6:(e) = 6r(e) [(|) x + [|) i], (Dl)

corresponding to eq (48), and the equation of constraint is

RijM«r)2^dx. (02)

corresponding to eq (49). The expression to be maximized is

i =

/ {B(x + 6r) - B(x)} dx - XR /

^^^

dx (D3)

L L

^

or

Each term is maximized individually by

I = / tf^ (^) * f (^)} "-'-"I "^ - ^^ /^ z "" t"")

and

a (f)'B (X)

5n(x) =_L^ 1

, (D6)

<U (B )^>.

corresponding to eqs (53) and (54). The on-axis fractional errors are given

by

29
<B(x)>2

and

R^ ' z <(f)'(B )2>

^?(^^^^?-<B00>2-

(D8)

The amplitude of I is maximized by a linear combination of eqs (D5) and (D6),

and an expression similar to eqs (60) and (61) can be written immediately.

30
NBS-n4A IREV. 2-8C) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS/TN-1306

2. Performing Organ. Report No, 3. Publication Date October 1986

4. TITLE AND SUBTITLE

Displacement Errors in Antenna Near-Field Measurements and Their

Effect on the Far Field

5. AUTHOR(S)

Lorant A. Muth

6. PERFORMING ORGANIZATION (If joint or other thon NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

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

10. SUPPLEMENTARY NOTES

\2^ Document describes a computer program; SF-185, FlPS Software Summary, is attached.

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

bibliography or literature survey, mention it herej

The effects of probe displacement errors in the near-field measurement procedure

on the far-field spectrum are studied. Expressions are derived for the displacement

error functions that maximize the fractional error in the spectrum both for the on-axis

and off-axis directions. Planar x-y and z-displacement errors are studied first and,

consequently, the results are generalized to errors in spherical scanning. Some simple

near-field models are used to obtain order of magnitude estimates for the fractional error as a function of relevant scale lengths of the near field, defined as the lengths over which significant variations occur.

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

error maximization; far-field spectrum; near fields; off-axis fractional errors; on-axis

fractional errors; planar scanning; probe displacement errors; spherical scanning

13. AVAILABILITY

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Q For Official Distribution. Do Not Release to NTIS

[X] Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C

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[ I Order From National Technical Information Service (NTIS), Springfield, VA. 22161

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