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Fernando
Las-Heras received the M.S. degree in 1987 and the
Ph.D. degree in 1990, both in Telecommunication
Engineering, from the Universidad Politcnica de Madrid
(UPM), Spain. Since 1991 he has held a position of
Associate Professor in the Dept. of Signals, Systems
and Radiocommunications at the UPM. He has participated
in a number of R&D Projects and is the author of
communications and papers regarding research topics in
numerical methods applied to electromagnetics, high
frequency methods for RCS estimation, analysis and
design of antennas, and electromagnetic inverse problem
with application to antenna measurement and synthesis.
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This article addresses the Near-Field to
Far-Field (NF-FF) transformation for antenna measurement, using
an equivalent current representation with a matrix-method
solution. The authors propose efficient techniques to improve
numerical cost efficiency and accuracy, achieving improvement
with equivalent magnetic currents and a specific coordinate
representation. This technique, which the authors developed for
cases of cylindrical and spherical scanning, leads to the
decoupling of integral equations relating field and source
components.
Introduction
Near-field systems are one of the main tools for antenna
measurement and electromagnetic diagnosis. The use of
equivalent currents representing the radiating system has been
presented more recently in 
as an alternative for Near Field to Far Field
(NF-FF) transformation. The intermediate step of reconstruction
of equivalent currents has been also presented in 
and 
from spherical and cylindrical measurements.
These techniques are based in the application of the
equivalence theorem to any radiating structure to obtain
equivalent currents in an unbounded medium.
However, the resultant large matrix equations, inherent to
the resolution of the integral equations that relate current
components and the field components, constitute a computational
drawback in the methods based on equivalent currents. In an
arbitrary acquisition, you cannot de-couple the integral
equations relating equivalent current components and field
components. They are directly de-coupled in the planar
acquisition case, can be de-coupled in the cylindrical
acquisition, but are coupled in the spherical acquisition.
Selecting an appropriate cylindrical system relating only
equivalent magnetic sources and fields, we present a sequential
solution procedure, where each integral equation can be solved
for only one component of the sources. This results in a
computationally efficient source reconstruction algorithm from
the knowledge of electric-field components over a cylindrical
surface. In addition, we present an efficient numerical
procedure to calculate the equivalent currents representing the
radiating structure from a spherical acquisition. The procedure
is based on the use of two de-coupled integral equations"under
consideration of zero value of the radial component of the
electric field"solving each integral equation by model
estimation. In order to overcome the applicability limitations
derived from neglecting the radial electric field, this article
presents an iterative algorithm that retrieves the radial
component of the electric field.
Both algorithms are directly applicable to NF-FF
transformation. This feature allows one to calculate the
far-field pattern of the antenna under test from the tangential
components of its near field, measured over cylindrical or
spherical surfaces, which, in practice, are available in many
near-field antenna-measurement facilities. We also show
analytical and measurement results of far-field pattern
calculation.
Equivalent Problem
The uniqueness theorem
allows the representation, in a source-free
region, of the radiated fields from some sources by a set of
equivalent electric and magnetic currents over a surface S'
enclosing the electromagnetic sources. If we consider the
surface S' to be an infinite plane, you get an external
equivalent problem (after substituting a perfect electric
conductor and applying image theory for the inner medium) with
only equivalent magnetic currents, 
,
over a plane radiating in an unbounded medium.
In a homogeneous medium, the electric field radiated by a
magnetic current density over S' can be obtained outside the
sources by:
where 
is the propagation constant and l
the wavelength.
Sequential Reconstruction for the Cylindrical Case
Consider a y-axis cylindrical-coordinate system with the
z-axis perpendicular to the S'-plane. Figure 1 defines
the proposed equivalent problem:
Figure 1: Equivalent problem with the equivalent
currents on the XY-plane and a y-axis cylindrical
representation of the electric-field components.
Under this configuration, the tangential field components in
the cylindrical surface (Eg, Ey) are related to the
magnetic current components over the S'-plane by:
where 
and Mx = Mx(x', y')
and My = My(x', y') are the equivalent
magnetic current components in the 
directions respectively. Initially, two
integral equations arise and you have to solve the two
components of the equivalent magnetic current distribution.
However, you get a cost-effective solution by the sequential
calculation of each of the equivalent current components in the
following fashion:
- Solve the integral equation that relates Ey
and Mx
- Insert the calculated value of Mx in the
integral equations that relate the field component
Eg and the current
components My, Mx.
With this method, you only need to solve integral equations
relating one field component and one current component.
Iterative Radial Field Retrieval for the Spherical Case
Figure 2 shows a spherical representation of the
field components. Two scalar integral equations appear such
that each angular component of the electric field depends on
both Cartesian components of the equivalent magnetic current
density.
Figure 2: Equivalent problem with the equivalent
currents on the XY-plane and a spherical representation of the
field components.
With this representation, many unknowns representing the
equivalent magnetic currents are involved in the numerical
solution of this system of integral equations.
However, we obtain two independent integral equations if we
neglect the radial component of the acquired near field. Under
this assumption, you can derive the Cartesian components of the
electric field from the acquired angular electric-field
components using the following approximation:
You can now apply integral equations relating Cartesian
equivalent current components and Cartesian field components.
The advantage of this technique is that you get two de-coupled
integral equations, each of them relating one Cartesian
component of the electric field and one set of Cartesian
components of the equivalent magnetic current distribution.
Using spherical coordinates to represent the angular field
components and Cartesian components to represent the components
of the equivalent magnetic current distribution over the S'
plane:
Only half the number of unknowns of the conventional matrix
representation are involved in solving each independent
integral equation.
Sometimes, due to the size of the antenna, problem
frequency, and maximum dimensions of the anechoic chamber, you
cannot neglect the radial component of the electric field when
using the Cartesian component of the electric field in the
proposed formulation. A desired accuracy may also make this
assumption valid or invalid for a particular measurement
configuration. To account for the value of the radial component
of the electric field when obtaining the field's Cartesian
components from its angular components, we propose and
implement an iterative algorithm for radial-field retrieval.
Start with the value of the radial component at zero and
perform an equivalent current reconstruction according with the
formulation previously presented in this paper. Using these
initial values, we calculate the equivalent magnetic current
distribution from the electric field over the hemi-spherical
scanning surface, replacing the zero radial component values by
the calculated values. Now calculate the Cartesian components
of the measured field using all spherical field components,
perform a new source reconstruction step, and continue the
iterative procedure until reaching some ending criterion.
Summarizing the algorithm:
- Make an initial guess for the equivalent magnetic current
components My, Mx.
- Obtain an initial zero value of the radial component of
the electric field. The Cartesian components of the electric
field Ex, Ey are obtained from the
angular components Eq,
Ef, supposing
zero-value of the radial component.
- Reconstruct each source component, solving integral
equations for My, Mx.
- Calculate of the radial component of the electric field
Er from the magnetic current components
My, Mx.
- Calculate the Cartesian components of the electric field
Ex, Ey from the angular components
Eq, Ef and the calculated radial
component Er.
- Go back to Step 3.
Solving the Source Component-Field Component Integral
Equations
Model estimation is used to numerically solve each of the
integral equations, relating one field component and one
magnetic current component. Let the measured data of one of the
electric field components be represented by vector Y, and its
correspondent theoretical evaluation, at all the scanning
points, represented by the vector E = GM, where M is the vector
with the discretized representation of the correspondent
equivalent current component over the surface S'.
Defining a least-squares merit function:
a quadratic form in M is obtained:
where H is related with the Hessian matrix. Then, the
optimization procedure to obtain M reduces to finding the minimum
of the quadratic form. Deriving and setting to zero, the
optimum values satisfying the above equation are given by:
Results
We used a nominal field distribution over a virtual aperture
of 1m x 1m, at 3 GHz, with the following value:
to synthesize the near electric field over a
cylindrical-surface hemi-cylindrical region defined by:
With the near-field data, we used the proposed cylindrical
NF-FF to calculate the far-field data, comparing the results
with the far-field data obtained directly from the nominal
aperture distribution (Figure 3). The pattern on the left in Figure 3 shows the far-field data obtained directly
from the nominal aperture. The pattern on the right in Figure 3 shows the far-field data calculated using the
cylindrical NF-FF from the synthesized near-field data.
A panel of the ASAR antenna was measured out of the designed
frequency band (2.225 GHz). The spherical scanning of this
antenna is done at a distance of 5.5 m with a 2.5-degree
spacing in both spherical coordinates. The results of the
proposed spherical NF-FF transformation are shown in Figure 4
along with the results when using a spherical-wave expansion
method such as the one used with the Sniftd software.
Figure 4: Phi = 0 co-polar pattern (dB) of the ASAR
panel. This array antenna (24 x 16 elements, 4 cm spacing) was
measured at 2.225 GHz, out of the designed frequency band.
Conclusions
To summarize, we implemented an efficient matrix method for
NF-FF transformation, in terms of decoupling integral equations
relating source components over a planar domain and field
components, for the case of cylindrical and spherical scanning.
The source reconstruction techniques inherent in the method
assists in the calculation of aperture fields in some type of
antennas and may help with system diagnostics. Our results have shown an accuracy comparable to standard wave transformation methods.
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This work was partially supported by the DGES of the
Ministerio Educacin y Cultura, Spain, under grant
PB97-0573.
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