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Muth, Lorant A/An Iterative technique to QC100 .U5753 N0.1323 1988 V198 C.I NIST- NIST TECHNICAL NOTE 1323

U.S. DEPARTMENT OF COMMERCE / National Institute of Standards and Technology

NXSTC ^C. fOO .0 5^7^3 rio, /3Z3 /^SB

An Iterative Technique to

Correct Probe Position Errors in

Planar Near-Field to

Far-Field Transformations

Lorant A. Muth

Richard L. Lewis

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Institute of Standards and Technology

(fornnerly National Bureau of Standards)

Boulder, Colorado 80303-3328

K^^'°"c,

Stimulating America s Progfess

1913-1988

U.S. DEPARTMENT OF COMMERCE, C. William Verity, Secretary

NATIONAL INSTITUTE OF STANDARDS & TECHNOLOGY, Ernest Ambler, Director

<i^.^^

^^:<c^s^^'• ,^^Z^ , #^'

Issued October 1988 ^ rf.'^^A*"''v -^ ..A'^'^'^
National Institute of Standards and Technology

Technical Note 1323, 20 pages (Oct. 1988)

CODEN:NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1988

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washiington, DC 20402-9325
CONTENTS

Page

Abstract 1

1. Introduction 1 2. Analytical Error Expressions 2

3. Computer Simulations 5

4. Conclusions and Suggestions for Further Study 6

5. References 7

Appendix 11

in

An Iterative Technique to Correct Probe Position Errors

in Planar Near-Field to Far-Field Transformations

by

Lorant A. Muth and Richard L. Lewis

We have developed a general theoretical procedure to take into ac- count probe position errors when planar near-field data are transformed

to the far field. If the probe position errors are known, we can represent

the measured data as a Taylor series, whose terms contain the error function and the ideal spectrum of the antenna. Then we can solve for the

ideal spectrum in terms of the measured data and the measured position

errors by inverting the Taylor series. This is complicated by the fact that

the derivatives of the ideal data are unknown; that is, they can only be

approximated by the derivatives of the measured data. This introduces

additional computational errors, which must be properly taken into ac- count. We have shown that the first few terms of the inversion can be

easily obtained by simple approximation techniques, where the order of

the approximation is easily specified. A more general solution can also

be written by formulating the problem £ls an integral equation and using

the method of successive approximations to obtain a general solution.

An important criterion that emerges from the condition of convergence

of the solution to the integral equation is that the total averaged position

error must be less than some fraction of the sampling criterion for the

antenna under test. Key words: error-contaminated spectrum; ideal spectrum; integral equation; inversion of Taylor series; method of successive approximations;

probe position errors; Taylor series expansion

1. Introduction

In planar near-field scanning a probe antenna scans the field radiated by

the antenna-under-test in a plane that is located a distance zq away from the test antenna. Ideally, measurements are made on a regularly space grid along the x and

y directions, and at a fixed distance of separation zq between the antennas along

the 2-direction. Naturally, the ideal measurement grid can only be approximated

in practice, that is, position errors in all the coordinate directions axe inevitable, z position errors have the most significant effect on the far field when the main beam

is along the z axis [1,2]. More generally, position errors in the direction of the main

beam are most significant, because they introduce errors that are proportional to

kSz, whercELS displacements in the orthogonal directions introduce errors that are proportional to 1//, where / is the characteristic scale of the near field of the
antenna. Usually / ^ A. Consequently, first-order corrections of phase errors

in near-field data is the most significant step we can take to improve accuracy

of far-field features, such as gain, sidelobe levels, cross polarization, boresight

direction and position of nulls. For example, the relevant phase error parameter

for millimeter wave antennas is as follows: the number of near-field points needed

is very large, since the sampling theorem demands data points at A/2 spacing

for complete characterization of the far field. At 60 GHz the data spacing is 2.5 mm, and the maximum position error on our near-field range, for example,

is approximately 0.2 mm, (which can be reduced to 0.025 mm, if the range is carefully realigned). Such a position error represents a maximum phase error of

Att/X = 14.4° in the main beam direction. Since the near field is a superposition

of an infinite number of plane waves, the actual phase error at any point could be

significantly different from the main beam contribution. We would like to be able

to correct such phase errors in the near field. Furthermore, second- order phase

and amplitude corrections might also be necessary to achieve high accuracy in the

far field.

In the rest of this report we look at the theoretical error expressions that

will be used in the computer simulation study (Section 2); then we discuss the

computer simulation techniques and some preliminary results that show that the

major error correction technique of inverting the Taylor series approximation of

the ideal data is feasible (Section 3), and give some further suggestions for studies

to improve performance assessments of millimeter wave antennas (Section 4).

2. Analytical Error Expressions

The field radiated by an antenna can be described as the superposition of an

infinite number of plane waves whose wavenumbers k are constant [3]. We can

write that k = (K^-y), where k A;=constant, K = (fcx,^j/), and j"^ — k'^ — K^

gives the z component of the propagation vector. The received near-field signal 6q

measured by a probe whose receiving coefficients are 5o2(-^^) is

b',{x, y, z) = F'ao ff T.oiK) • S'^^{K) e'^^ e'^'^dk, dky (1)

where F' = 1/(1 — F/Fp), F/ and Tp are reflection coefficients for the load and

probe, respectively, Tio(K) are the transmission coefficients of the antenna under

test, ao is the amplitude of the incident wave produced by the generator at the

terminal surface 5o, and z is the distance of the near-field scan plane from Si, a plane situated in front of the antenna defining ^ = 0, and the position vector

P = (xix,X2y), where x and y are unit vectors. Equation (1) assumes that

multiple reflections are negligible; the presence of multiple reflections in a real

measurement range is minimized by judiciously choosing the position of the plane

of measurement and the size and design of the probe.

Since eq (1) is a Fourier transform, the quantity

D{K) = 4n\oF'fio{K) S'.^iK) (2)
can be immediately written in terms of the near field. Thus,

D{K) = JJ

b',{P, z)e-''-'dx dy, (3)

— where x = (xi,X2,X2) = P + x^z.

Since the z dependence of the near-field quantity 60 in eq (1) appears only in

the exponential, we can immediately write that

dz^

jj D{K) 7 e'^^e'^-^ dk, dky (4)

and

Similarly, the partial derivatives with respect to Xj for j = 1,2 are given by

dx"!

(6)

If these expressions can be evaluated, then first- order corrections can be introduced into the data. We assume that the probe's position is known accurately and

is given by

X + Sx{x), (7)

where x is the position of the probe on an ideal near-field range, where measure- ments are made on a regularly spaced (a^j, X2) grid, and Sx{x) is the deviation in

the probe's position from the ideal grid. A thorough discussion of the effects of

such displacement errors on the far-field pattern has been presented in [2]. Some

of the basic considerations relevant to the current subject are included here.

The near-field quantity b'Q{x-\-Sx(x)) is measured at the locations given by eq

(7). However, this function is assumed to be defined on the regular grid x when

the spectrum is obtained numerically using Fourier techniques. We can write the

Taylor expansion at x,

h',ix + 6xix)) ^ fe'o(x) + ^^Sx, + ^^^6x,Sx, + •, (8)

which defines the measured data on the left in terms of the unknown field quantities

on the right. If we write 6q = aexp(z0), where a is the amplitude and is the

phase of ig , then

6b' = e"^^6xj -f iae"^^8xj (9) OXj OXj

holds. If we assume that a~^da/dxj <C d(l)/dxj, then eq (9) can be immediately

integrated to give

b'o{x + 6x) = b'^{x)e'I''^''\ (10)
where kj = d(f>/dxj is the j component of the local wave-vector. In realistic

near-field data of antennas the inequality above is satisfied for variations in the z direction. The wavenumber in the z direction can be expanded for small K as

A:, ^7 = Ml-^(f)' + •••] (11)

Also, for a plane wave d(^jdz = k^. Substituting this into eq (10), we obtain, to

zeroth order in (K/k), that

b'oix + Sx) = b{x)e'^^' (12)

which is a zeroth order correction for probe displacement errors in the near field.

This plane-wave model correction technique has been applied to real data for some

time now [4]. To evaluate eq (10) more exactly, we would need to know kj along

the path of integration, or at least at the end points of the path, since kj = d(f)/dxj

is an exact differential. By the mean value theorem, the integral in eq (10) can be written as kjAxj, where kj is some value of kj in the interval of integration.

Thus, the accuracy of 6o(£) will depend on the accuracy with which kj is specified.

For higher order corrections we need to develop a more thorough analysis.

We can write, as in eq (3), the error-contaminated spectrum D^{K) in terms of

the measured data as

De{K)= f f b'Q(x-\-6x{x))e-'^-^dxdy. (13)

Using expansion (8) and the transform relation (3), we can write that

DeiK) = D{K) -h ffi^^^Sxj -{-higher order teTms]e-''^-^ dx dy. (14)

Here j = 1,2,3 and the Einstein summation convention on repeated indexes is understood. In eq (14) D(K) and db'^/Oxj are unknown. We do, however, know

56o(£-|- Sx(x))/dxj through the relationships (4)-(6). We can differentiate eq (8)

with respect to a:^, ^ = 1,2,3, so that

which shows that eq (14) can also be written as

D,{K) = D{K) + ff[?M^^-±M^Sxj + 0{8xi6xj)]e-^^Uxdy. (16)

We now have a first-order expression that gives the true spectrum in terms of the

error-contaminated spectrum and the measured near-field data. A second-order
correction can be obtained in a similar manner. Equation (15) can be differentiated

again, which immediately yields that

X{x + 6x{x)) = ^^

+ 0{6x). (17)

dxidxj dxidxj

The second-order term in eq (8) can now be written in terms of the known quantity

6o(x + (5x(;r)). However, the second-order terms neglected in the first-order approximation must now be included. These are the second and third terms appearing

in eq (15). After replacing the quantities 6o(x) with their first-order approximations, the second-order approximation of the known error-contaminated spectrum

in terms of the true spectrum and the known displacement error function can be

written as

D,{K) = D{K) +

II

dh',{x + 8x) l dH',{x+8x) d dh',{x + 8x)

0{8xi6xj6xe)]e-"'-^ dxdy. (18)

There is a general procedure for writing the nth-order approximation to D{K).

Equation (8) can be rewritten as [2]

D,{K) = fD(K') f

e'ik'-k).x^ik'.6x(x) ^2^ ^j^,^ ^^/^^ (^9^

This expression can be recast into the form of an inhomogeneous integral equation,

which, under certain conditions, has an iterative solution [5]. Some aspects of this are detailed in Appendix A.

3. Computer Simulations

Simulation Techniques

To study the effects of probe errors using computer simulations an error-free

data set is assumed in a plane of measurement, denoted by zq. An error-free spectrum can then be obtained using standard Fourier transform techniques. An in- verse Fourier transform then can yield error-free near-field data at any 2,=constant

plane. If a large set of near-field data is obtained at many different 2,-, then a set of error-contaminated data can be constructed on the mathematical plane zq, ac- cording to an arbitrary error displacement function in the z direction. The error- contaminated data then yield an error-contaminated far-field pattern, which can

then be compared to the original error-free spectrum. The same technique can be

used to introduce displacement errors in the xy plane, which then can be studied

similarly.
A number of error correction schemes can be studied with this technique

since both error-free and error-contaminated data can be generated in a prescribed

manner. The following error correction schemes are worthy of consideration:

1. First order phase-error corrections in the main beam direction only.

2. First order phase-error corrections taking into account the contributions of

plane waves in the off- axis directions.

3. Second order phase-error correction.

4. General second order complex error corrections for x, y and z displacement

errors.

5. General higher order complex error corrections for x^y and z displacement

errors.

Numerical Feasibility Study

In order to execute the numerical studies suggested above, the feasibilty of the

general error correction techniques has to be ascertained; that is, the convergence

of the Taylor series has to be demonstrated in a simple test case. For this purpose,

an actual near field of an array antenna radiating at 3.3 GHz was transformed

from 2o, the original plane of measiirement, to zi = zq -\- 0.02A. This separation

distance was chosen to scale the limits of displacement errors in a well aligned

near-field range at 60 GHz, where these error correction techniques are essential

to obtain acceptable far-field patterns from near-field data. The data set chosen

for this study has no special significance other than it was easily available; data

sets at higher frequencies were not available at the time of this study.

In Table 1, the near-field values at two different z positions separated by

6z = 0.02A are listed as a function of the vertical y-direction in the center of

the near-field scan plane [x = 0). In Table 2 the differences in amplitudes of

the first, second and third order Taylor series expansions of the near-field data

around Zq (which approximate the function at zi ) and the near-field data at Zi are shown. We can easily ascertain form these amplitude differences that the Taylor

series converges to the correct value to an improved order of magnitude with each

successive approximation. In Table 3 the phase differences between the phases

obtained in the first, second and third order approximations and the exact phases

are shown. Again, convergence in easily observed.

4. Conclusions and Suggestions for Further Study

We have developed a theoretical procedure which, in principle, can correct

for probe position errors present in near-field data. The question of numerical

convergence has been addressed on a simple but basic level. The result of this

small study is significant in that it establishes the feasibility of obtaining correc- tions to error-contaminated near-field data, where the error is due to the faulty

positioning of the probe. Since the feasibility of our techniques has been established a full implementation of the techniques is recommended as the next phase

of this study. After successful completion of this phase, a full implementation of
the study at higher frequencies is recommended. At such higher frequencies the

additional problem of an increased number of data points will have to be handled

computationally; this, however, will not present fundamentally new difficulties.

5. References

[1] Newell, A. C. Error analysis techniques for planar near-field measurements.

IEEE Trans. Antennas and Propagat., 36, 754-768; 1988 June.

[2] Muth, L. A. Displacement errors in antenna near-field measurements and their

effect on the far field. Nat. Bur. Stand. (U.S.) Tech.

[3] Kerns, D. M. Plane wave scattering-matrix theory of antenncis and antennaantenna interactions. Nat. Bur. Stand. (U.S.) Monogr. 162; 1981.

[4] Newell, A. C. private communication; 1988.

[5] Stakgold, I. Boundary value problems of mathematical physics, v. 1, MacMillan Series in Advanced Mathematical and Theoretical Physics, London, The MacMillan Company, Collar-MacMillan Limited; 1967. 340 p.

[6] Riesz, F.; Sz.-Nagy, B. Functiomal Analysis, New York, Frederick Ungar PubHshing Co.; 1955. 468 p.

[7] Landu, H. J. Sampling, data transmission, and the Nyquist rate, IEEE Trans.

Antennas and Propagat., 55, 1701-1706; 1967 October.
Table 1. The amplitude and phase of the near-field data at zq (the first two columns) and

at Zq + 0.02A (the third and fourth columns), zq = 380 cm and the frequency is 3.3 GHz

0.70862614E-01

0.68881877E-01

0.66208757E-01

0.63052207E-01

0.59651922E-01

0.56263678E-01

0.53143855E-01

0.50533973E-01

0.48644681E-01

0.47639515E-01

0.47619592E-01

0.48612926E-01

0.50571788E-01

0.53379204E-01

0.56861863E-01

0.60805749E-01

0.64972103E-01

0.69113575E-01

0.72990544E-01

0.76387547E-01

0.79129122E-01

0.81094168E-01

0.82226910E-01

0.82544021E-01

0.82135588E-01

0.81159256E-01

0.79826355E-01

0.78379579E-01

0.77063955E-01

0.76095171E-01

0.75632460E-01

0.75764105E-01

0.76509200E-01

0.77832721E-01

0.79664603E-01

0.81913017E-01

0.84467262E-01

0.87192379E-01

0.89921571E-01

0.92453308E-01

0.94557285E-01

-0.77406448E+02

-0.77066040E+02

-0.76955162E+02

-0.77103645E+02

-0.77551033E+02

-0.78342728E+02

-0.79519890E+02

-0.81101540E-(-02

-0.83061073E+02

-0.85307442E+02

-0.87686134E+02

-0.90009270E+02

-0.92103424E-f02

-0.93848976E+02

-0.95192131E+02

-0.96131607E+02

-0.96695152E+02

-0.96918556E+02

-0.96832283E+02

-0.96455200E+02

-0.95793037E+02

-0.94839607E+02

-0.93579697E-f-02

-0.91993576E+02

-0.90063461E+02

-0.87782776E+02

-0.85168304E+02

-0.82274063E+02

-0.79202957E+02

-0.76109612E+02

-0.73187904E+02

-0.70642082E+02

-0.68648933E+02

-0.67324738E+02

-0.66707481E+02

-0.66757774E+02

-0.67374077E+02

-0.68415756E+02

-0.69726944E+02

-0.71157341E+02

-0.72575905E+02

0.71044020E-01

0.69043353E-01

0.66348597E-01

0.63171484E-01

0.59753377E-01

0.56350138E-01

0.53216860E-01

0.50592951E-01

0.48687033E-01

0.47661930E-01

0.47619943E-01

0.48591554E-01

0.50531447E-01

0.53323638E-01

0.56793861E-01

0.60725752E-01

0.64877853E-01

0.69000728E-01

0.72853938E-01

0.76222844E-01

0.78934476E-01

0.80871642E-01

0.81983767E-01

0.82293473E-01

0.81896998E-01

0.80957592E-01

0.79690382E-01

0.78338675E-01

0.77143587E-01

0.76311313E-01

0.75986728E-01

0.76240897E-01

0.77076137E-01

0.78444391E-01

0.80268413E-01

0.82456067E-01

0.84902853E-01

0.87485127E-01

0.90050764E-01

0.92413932E-01

0.94359033E-01

-0.71065971E+02

-0.70638588E+02

-0.70416214E+02

-0.70429436E+02

-0.70720047E+02

-0.71338348E+02

-0.72334320E+02

-0.73740471E+02

-0.75548035E+02

-0.77684982E+02

-0.80011765E+02

-0.82346024E+02

-0.84508301E+02

-0.86364479E+02

-0.87842964E+02

-0.88926064E+02

-0.89628761E+02

-0.89977943E+02

-0.89998314E+02

-0.89705132E+02

-0.89101845E+02

-0.88180779E+02

-0.86925827E+02

-0.85316986E+02

-0.83337311E-f-02

-0.80982773E+02

-0.78275276E+02

-0.75277008E+02

-0.72101707E+02

-0.68915894E+02

-0.65923607E+02

-0.63335045E+02

-0.61328175E+02

-0.60016930E+02

-0.59435833E+02

-0.59542557E+02

-0.60233273E+02

-0.61364899E+02

-0.62778072E+02

-0.64317627E+02

-0.65846565E+02
Table 2. The differences between the amplitude of the near-field data at zi and the amplitudes of the first, second and third-order Taylor series approximations of these near-field

values

-0.45118481E-03

-0.44924021E-03

-0.44508278E-03

-0.43936074E-03

-0.43277442E-03

-0.42614713E-03

-0.42039528E-03

-0.41628256E-03

-0.41457266E-03

-0.41580573E-03

-0.42020157E-03

-0.42781979E-03

-0.43842569E-03

-0.45155734E-03

-0.46665967E-03

-0.48295408E-03

-0.49973279E-03

-0.51622093E-03

-0.53173304E-03

-0.54571778E-03

-0.55778027E-03

-0.56769699E-03

-0.57550520E-03

-0.58130920E-03

-0.58547407E-03

-0.58839470E-03

-0.59053302E-03

-0.59235841E-03

-0.59417635E-03

-0.59643388E-03

-0.59945136E-03

-0.60362369E-03

-0.60918182E-03

-0.61618537E-03

-0.62443316E-03

-0.63331425E-03

-0.64200163E-03

-0.64958632E-03

-0.65503269E-03

-0.65743923E-03

-0.65617263E-03

-0.30323863E-05

-0.31217933E-05

-0.31739473E-05

-0.32261014E-05

-0.32335520E-05

-0.32037497E-05

-0.31515956E-05

-0.30025840E-05

-0.28051436E-05

-0.25555491E-05

-0.22388995E-05

-0.19259751E-05

-0.16354024E-05

-0.13858080E-05

-0.11883676E-05

-0.10319054E-05

-0.91642141E-06

-0.82701445E-06

-0.75995922E-06

-0.70780516E-06

-0.66310167E-06

-0.67055225E-06

-0.73015690E-06

-0.86426735E-06

-0.10877848E-05

-0.14156103E-05

-0.18551946E-05

-0.24512410E-05

-0.30845404E-05

-0.37848949E-05

-0.44405460E-05

-0.50142407E-05

-0.54091215E-05

-0.55581331E-05

-0.54985285E-05

-0.51930547E-05

-0.46640635E-05

-0.40382147E-05

-0.33155084E-05

-0.25480986E-05

-0.18253922E-05

0.54389238E-06

0.54389238E-06

0.55134296E-06

0.54389238E-06

0.54761767E-06

0.55879354E-06

0.54389238E-06

0.56251884E-06

0.56996942E-06

0.56251884E-06

0.58859587E-06

0.59977174E-06

0.60349703E-06

0.61839819E-06

0.62584877E-06

0.64447522E-06

0.65565109E-06

0.66310167E-06

0.67800283E-06

0.68545341E-06

0.70035458E-06

0.71525574E-06

0.71525574E-06

0.72270632E-06

0.73015690E-06

0.74505806E-06

0.76740980E-06

0.74505806E-06

0.78231096E-06

0.78231096E-06

0.80466270E-06

0.79721212E-06

0.80466270E-06

0.81956387E-06

0.81956387E-06

0.81956387E-06

0.84191561E-06

0.81211329E-06

0.78976154E-06

0.79721212E-06

0.7S231096E-06
Table 3. The differences between the near-field phase at zi and the phases of the first,

second and third-order Taylor series approximations to these near-field values

0.47615051E-01

0.49217224E-01

0.51124573E-01

0.53321838E-01

0.55770874E-01

0.58174133E-01

0.60035706E-01

0.60745239E-01

0.59631348E-01

0.56427002E-01

0.51422119E-01

0.45364380E-01

0.39352417E-01

0.33981323E-01

0.29586792E-01

0.26054382E-01

0.23193359E-01

0.20782471E-01

0.18684387E-01

0.16906738E-01

0.15472412E-01

0.14572144E-01

0.14411926E-01

0.15319824E-01

0.17639160E-01

0.21736145E-01

0.27877808E-01

0.36117554E-01

0.46089172E-01

0.56945801E-01

0.67359924E-01

0.75893402E-01

0.81283569E-01

0.82736969E-01

0.80207825E-01

0.74066162E-01

0.65208435E-01

0.54622650E-01

0.43506622E-01

0.32730103E-01

0.23017883E-01

-0.14038086E-01

-0.14480591E-01

-0.15060425E-01

-0.15800476E-01

-0.16654968E-01

-0.17593384E-01

-0.18615723E-01

-0.19592285E-01

-0.20484924E-01

-0.21118164E-01

-0.21423340E-01

-0.21400452E-01

-0.20996094E-01

-0.20370483E-01

-0.19615173E-01

-0.18836975E-01

-0.18096924E-01

-0.17448425E-01

-0.16914368E-01

-0.16502380E-01

-0.16242981E-01

-0.16105652E-01

-0.16113281E-01

-0.16242981E-01

-0.16479492E-01

-0.16799927E-01

-0.17196655E-01

-0.17593384E-01

-0.17951965E-01

-0.18226624E-01

-0.18394470E-01

-0.18409729E-01

-0.18318176E-01

-0.18154144E-01

-0.17887115E-01

-0.17581940E-01

-0.17219543E-01

-0.16838074E-01

-0.16395569E-01

-0.15945435E-01

-0.15502930E-01

0.22888184E-03

0.22888184E-03

0.23651123E-03

0.22125244E-03

0.22125244E-03

0.23651123E-03

0.22888184E-03

0.25939941E-03

0.25939941E-03

0.27465820E-03

0.30517578E-03

0.28991699E-03

0.32806396E-03

0.33569336E-03

0.34332275E-03

0.33569336E-03

0.33569336E-03

0.32806396E-03

0.32806396E-03

0.33569336E-03

0.32806396E-03

0.33569336E-03

0.32043457E-03

0.32043457E-03

0.31280518E-03

0.31280518E-03

0.29754639E-03

0.28991699E-03

0.27465820E-03

0.26702881E-03

0.24414063E-03

0.23651123E-03

0.23651123E-03

0.21743774E-03

0.23651123E-03

0.24032593E-03

0.25939941E-03

0.24795532E-03

0.28228760E-03

0.29754639E-03

0.30517578E-03

10
Appendix

The True Spectrum as a Solution to an Inhomogeneous Integral Equation

Equation (19) of the main text can rewritten as

D{K) = De{K) +

^

fjD{K')fJ{l-e''^'-''^^^)e'^''-'^^^^ (al)

Here we have used the definition of the ^-function

6{k' - fc) = -i^ //e'^^'-^^-^cfxcfy. (a2)

Equation (al) is of the form of the Fredholm integral equation [5], which in two

dimensions, is written as

f{x^y) = 9{x,y)+ M{x,y;x',y')f{x',y')dx' dy'. (a3)

J a

Symbolically this is / = ^f + Mf. If we compare eq (al) and eq (a3) a straightforward identification of terms is easily done.

A solution of the general integral equation (a3) can be obtained by the method

of successive approximation (Neumann series) [5]. Symbolically, the nth-order

solution is given by

fn = g + Mfn-i ={l+M + M^ + ... + M"-^ )g. (a4)

This solution is unique if the series in eq (a4) converges uniformly [6]. The condition for convergence is that the integral operator M be bounded so that its least

upper bound or norm, ||M||, is less than 1. An alternate condition is that the

product of the range of integration and of the maximum value of the kernel is less

than 1 [6] . An estimate of the norm is given by

\\Mf <

JJ

\M{x, y; x' , y')|' dx dy dx' dy\ (a5)

and the kernel is, in our example,

M{K,K') = -^ f f{l-e'^'-^^^^^)e''^^'-^^-^dxdy. («6)

The maximum value of the kernel occurs at A; = A;', where its first-order approximation can be written as

\M^'\KJ{)\ = ^\ffk-6xdxdy\. {al)

11
Thus,

- -* h f f JcA \M^'\KJ<)\ < —\JJ 6xdxdy\ = -^[Sx] (aS)

where [Sx] is the average of the error displacement function in the scan plane.

Since the range of integration is Ak = 2ki, where k = 27r/A, the second condition

of convergence stated above can be written, to first order, as

or

If the near field is sampled at A/2 intervals, then the ratio on the right side

above is essentially the inverse of the number of measurement points A'^. Then the

inequality (a9) becomes

N[6x\ < -(A/2) (alO)

TT which states that the total averaged displacement error has to be less than a fraction of the grid spacing A/2. This is a rather stringent condition on the size of

displacement errors one can accept in a near-field measurement range, if we want

to recover the true spectrum from error-contaminated near-field data. The condition essentially means that the displacement errors must be small enough so that

the sampling criterion, according to the sampling theorem [7], is not violated in

an average sense. Thus, only displacement-error functions that average to close to

0, or of very small magnitude, will satisfy this criterion.

If [Sx\ = then the second order approximation of the kernel must be examined. This is \M^'\K J<)\ = ^\jj{k 8xf dx dyl (all)

which yields an upper bound of

|M(2)(/?,7?)| < ^1 f f Sx-Sxdxdy\. (al2)

In deriving expression (al2) we have assumed that the components of Sx are uncorrelated, so that mixed terms average to 0. Under these conditions, the smallness

of the second order kernel depends entirely on the smallness of the amplitude of

Sx. Now the condition of convergence of the method of successive approximations

is —^A[Sx-Sx]<l (al3)

ZTT or N[Sx-Sx\ < ^{X/2f. (al4)

12
The interpretation of this expression is similar to the one given expression (alO)

above. In general, both conditions (alO) and (al4) have to be satisfied to guarantee

convergence.

To facilitate evaluation of the integrals in eq (al), we expand the exponential

term containing 6x to second order, and write the second-order iterative solution

of eq (al) as (using the abbreviations dx = dx dy and dK = dkx dky)

D{K) = D,{K) - -^ ffsxje-''^ f[krD,{K')e''' -'^dK'} dx

+ -i^ ffsxjSxee-'''^ ffk'jk'fDe{K')e'^'-^dK'}dx

-^ffdxlSxje-'^-^ffdK' k'je'^'-^ffdx' fsxee-'^'-^'fflk'lD,{K")e'^"-^'dK"

(al5)

Higher-order iterations can be readily obtained, but we will not do so here. Each

iterated integral above is a Fourier transform; hence, these integrals can be evaluated with FFT codes.

13
NBS-n4A iREv. 2-8C) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NIST/TN-1323

2. Performing Organ. Report No 3. Publication Date October 1988

4. TITLE AND SUBTITLE

An Iterative Technique to Correct Probe Position Errors in Planar Near-Field to

Far-Field Transformations

5. AUTHOR(S)

Lorant A. Muth and Richard L. Lewis

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

National Institute of Standards and Technology

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

[^^ 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 si gnificant

bibliography or literature survey, mention it here) We have developed a general theoretical procedure to take into account probe position

errors when planar near-field data are transformed to the far field. If the probe

position errors are known, we can represent the measured data as a Taylor series,

whose terms contain the error function and the ideal spectrum of the antenna. Then

we can solve for the ideal spectrum in terms of the measured data and the measured

position errors by inverting the Taylor series. This is complicated by the fact that

the derivatives of the ideal data are unknown; that is, they can only be approximated

by the derivatives of the measured data. This introduces additional computational

errors, which must be properly taken into account. We have shown that the first few

terms of the inversion can be easily obtained by simple approximation techniques,

where the order of the approximation is easily specified. A more general solution

can also be written by formulating the problem as an integral equation and using

the method of successive approximations to obtain a general solution. An important

criterion that emerges from the condition of convergence of the solution to the

integral equation is that the total averaged position error must be less than some fraction of the sampling criterion for the antenna under test.

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

error-contaminated spectrum; ideal spectrum; integral equation; inversion of Taylor

series; method of successive approximations; probe position errors; Taylor series

expansion

13. A yAlLABILITY

[X] Unlimited

03 ^°' Official Distribution. Do Not Release to NTIS

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

20402.

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

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PRINTED PAGES

20

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