Nonideal measurement locations in planar near-field antenna metrology

We introduce a near-field to far-field transformation method that relaxes the usual restriction that data points be located on a plane-rectangular grid. It is not always practical or desirable to make uniformly spaced measurements; for example, the maintenance of positioning tolerances becomes more difficult as frequency is increased. Our method can (1) extend the frequency ranges of existing scanners, (2) make practical the use of portable scanners for on-site measurements, and (3) support schemes, such as plane-polar scanning, where data are collected on a nonrectangular grid. Although "ideal" locations are not required, we assume that probe positions are known. (In practice, laser interferometry is often used for this purpose.) Our approach is based on a linear model of the form A/spl xi/=b. The conjugate gradient method is used to find the "unknown" /spl xi/ in terms of the "data" b. The operator A must be applied once per conjugate gradient iteration, and this is done efficiently using the recently developed unequally spaced fast Fourier transform and local interpolation. As implemented, each iteration requires O (N log N) operations, where N is the number of measurements. The required number of iterations depends on desired computational accuracy and on conditioning. We present a simulation that is based on actual near-field antenna data.


Introduction
We introduce a near-eld to far-eld transformation method that relaxes the usual restriction that data points be located on a plane-rectangular grid [1]. It is not always practical or desirable to make uniformly spaced measurements; for example, the maintenance of positioning tolerances becomes more di¢cult as frequency is increased. Our method can (1) extend the frequency ranges of existing scanners, (2) make practical the use of portable scanners for on-site measurements, and (3) support schemes, such as plane-polar scanning, where data are collected on a nonrectangular grid.
Although ideal locations are not required, we assume that probe positions are known.
(In practice, laser interferometry is often used for this purpose.) Our approach is based on a linear model of the form A» = b (see section 2). The conjugate gradient method is used to nd the unknown » in terms of the data b (section 3). The operator A must be applied once per conjugate gradient iteration, and this is done e¢ciently using the recently developed unequally spaced fast Fourier transform [2], [3] and local interpolation (section 4). As implemented, each iteration requires O (N log N) operations, where N is the number of measurements. The required number of iterations depends on desired computational accuracy and on conditioning. In section 5, we present a simulation that is based on actual near-eld antenna data.

The Model
Consider a transmitting test antenna (located in the half space z < 0) and a receiving probe (translated without rotation). According to Kernss theory [4], the probe response w (r) may be modeled as w (r) = X º¹ » º¹ exp (ik º¹ ¢ r), z > 0 (1) [exp (¡i!t) time convention], where » º¹ is the (normalized) coupling product and k º¹ = ¼º L xx + ¼¹ L yŷ +°º ¹ẑ ,°º ¹´s k ¡ µ ¼º We assume that the probe response is negligible outside the interval jxj · L x , jyj · L y for z values of interest. [That is, w (r) is a periodic extension.] To improve conditioning (see below), we include only propagating plane waves (°º ¹ real) in the summation in (1).
Evanescent waves (°º ¹ imaginary) are exponentially attenuated and are negligible in the far-eld region. We must also ensure that evanescent waves are not important contributors to the measured probe response; this is usually accomplished by maintaining a probe-totest-antenna separation of several wavelengths.
In matrix form (1) becomes w = Q» (2) where w´fw (r n )g, r n is the location of the nth measurement point, »´©» º¹ ª , and Q´fQ n;º¹ = exp (ik º¹ ¢r n )g. The objective of near-eld to far-eld transformation is to U .S. G overn m ent contrib ution n ot sub ject to copy right in th e U nited States.
determine the coupling product » from measurements w made in a restricted region near the test antenna.
In practical situations, where the number of measurements often exceeds the number of unknowns, the system (2) is overdetermined and will generally not have a solution. We will actually solve the normal equations Rate of convergence can be estimated with°°°» j ¡ »°°°A · 2 µ c ¡ 1 c + 1 ¶ j°°°» ¡ »°°°A .
Here kyk A´y H Ay and the condition number c is the ratio of the largest to smallest eigenvalue of A. (The condition number of Q is c, c¸1.) Thus, the conjugate gradient algorithm will always converge.
When condition numbers are large (poor conditioning), equations (4) and (5) N) operations] at the points x nx +y nẑ + zẑ for several xed values of z. We then use local interpolation in z to reach the actual measurement locations r n . Since we are dealing with bandlimited functions, the numerical precision of the algorithm can be controlled and is specied as an input parameter. Computational time depends on the desired numerical accuracy and on the spatial distribution of data points.
Our technique is most e¢cient when measurement locations lie close to a plane. Details will be presented elsewhere.

Simulation
We began with planar near-eld data for a radiometer antenna with an aperture diameter  For this example, the condition number is c J 21,°°r '°°= kbk < 10 ¡" , and°°r ' ²The algorithm is iterative, with a xed cost per iteration that is O (N log N ). The memory requirement is O (N ) and is independent of the number of iterations. ²Convergence is guaranteed. Bounds [see (5)] on the convergence rate for the conjugate gradient procedure are tighter than for many alternative iterative techniques. ²Computational error (not measurement error) is bounded by the residual [see (4)]. ²Our current implementation is fully three-dimensional. ²The recipe given in this paper is also applicable to cylindrical and spherical scanning geometries. The basic ingredient is an e¢cient procedure for predicting probe response at the measurement locations, based on an estimated modal spectrum.
The software is available from the authors.