Displacement Errors in Antenna Near-Field Measurements and Their Effect on the Far Field

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 for both the on-axis and off-axis directions. The x- y- and z-displacement error in planar scanning are studied and 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. >

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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 onor 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 The fractional error in the spectrum D(_k_) due to position errors is then (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) ,.,^^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. MO) = Iv^xi^,^ĵ^^^B (x) which in the one-dimensional case is (prime denotes differentiation) 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), assuming a « 1, one obtains c = 2. As will be seen below, this is, indeed, a good estimate.
The scale length i represents variations in the near field B(x) either parallel or perpendicular to _k^, the wave vector of interest. Two alternative expressions to eq (4) can be shown to be, using 6 for the unit amplitude near field and denoting derivatives by ', and aD(0) 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 subject to the constraint where L is the interval of interest. We use the Lagrange multiplier n [4] to with respect to 6x. Thus, 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 firstand 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 For a first-order expansion in 6x the condition must hold, and similarly for higher-order terms. This implies where n is determined by eq (11). If we further assume that then from eqs (13) and (11) satisfies eq (11). Consistent with the first-order approximation developed here, the integral in eq (8) is approximated as and similarly for higher-order terms. Using eq (15), eq (12) becomes and eqs (14) and (18) are consistently satisfied by 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) 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 has been sought. We can use the first-order expression (15) to get r,^<3.2> 2,_,2 _].
(21b) 2T he 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^i s the error-contaminated spectrum, one can 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.

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 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 essentially 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 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., 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 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.
Then expanding eq (26) for small 5x(x) and from eqs (11) 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^t o maximize eq (29) subject to constraint (30).
A more detailed treatment of a specific case of eq (26) is given in section 3.4 and Appendix B.

Maximization for K^t
If in the region of interest the near field is real, the off -axis error displacement function 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 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 where, depending on the magnitude of 9, one of the terms is negligible compared 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.
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 gives the error contaminated spectrum in terms of the known exact spectrum and the known displacement error function. Equation (43) can be written as E(l<',l<) =/e""'e" "" d2x. 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.

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, 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 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. -1 X where 9 = sin n->^n d 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 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. However, as we have seen in section 3.2, the maximization of |l| in eq (50) is given by the linear combination 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.

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. 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) 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).

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.
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, This expression should be compared to the expansion above eq (12). To first order then, where the supercript indicates first order, then the constraint (11) gives 5x = a -^^^1,^(B6) <b'2> The second-order maximization of the imaginary part of eq (Bl), using (B12), is To maximize the amplitude of AD(e), we construct the linear combination 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. 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). 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. NBS Interagency Reports (NBSIR)-A special series of interim or final reports on work performed by NBS for outside sponsors (both government and non-government). In general, initial distribution is handled by the sponsor; public distribution is by the National Technical Information Service, Springfield, VA 22161, in paper copy or microfiche form.