# Planar near-field antenna measurements
with a uniform step larger than half-wavelength

R. Moretta, , F. Pascariello, G. Petraglia, M. Feo, M.A. Maisto Manuscript
received February 11, 2024.
This work was supported in part by the European Union and the Italian Ministry
of University and Research funding through Programma Operativo Nazionale
Ricerca e Innovazione 2019/2020 under Grant B26C18000080005. (Corresponding
author: Raffaele Moretta.)Raffaele Moretta was a postdoctoral researcher at
University of Campania “Luigi Vanvitelli”, 81031 Aversa (NA), Italy. He is now
an Antenna Engineer at MBDA Italy, 80070 Bacoli (NA), Italy (e-mail:
raffaele.moretta@mbda.it).Fabio Pascariello, Giovanni Petraglia, and Maurizio
Feo are with MBDA, 80070 Bacoli (NA), Italy (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>maurizio.feo@mbda.it).Maria Antonia Maisto is with Dipartimento di Ingeneria,
Università della Campania “Luigi Vanvitelli”, 81031 Aversa (NA), Italy
(e-mail: mariaantonia.maisto@unicampania.it).

###### Abstract

In this paper, a new sampling scheme of the near field radiated by a planar
source is proposed and assessed. More in detail, the paper shows a uniform
sampling criterion that allows representing the near field over a plane with a
number of measurements lower than the classical half-wavelength sampling.

At first, a discretization strategy of the near field based on the warping
method is recalled from the literature. The latter requires to collect a non-
redundant number of field measurements that are non-uniformly arranged over
the observation domain.

Despite this, the warping sampling scheme works well only if the measurement
plane does not overcome the source. When the observation domain is larger, it
does not predict the exact positions of the field samples at the edges of the
measurement plane; accordingly, in these regions it is not possible to recover
the near field behavior by the collected samples.

To overcome this drawback, a spatially varying oversampling is exploited. The
latter is chosen in such a way that the resulting sampling becomes uniform.
Such choice also ensures a growth of the sampling rate only at the edges of
the observation domain permitting the retrieval of the near field by its
samples.

Finally, numerical simulations based on experimental data corroborate the
effectiveness of the approach in recovering both the near and the far field.

###### Index Terms:

Near field sampling, antenna measurements, singular values decomposition,
warping.

##  I Introduction

In the framework of antenna testing, the radiation pattern of the antenna
under test (AUT) can be retrieved by far field or near field measurements.

Far field measurements are carried in free space at a distance fulfilling the
far-field condition depending on the frequency and the source size. Since such
distance may be large (especially for large sources), a high transmitting
power could be necessary. Moreover, far field measurements may be negatively
affected by bad weather conditions, obstacle reflections, ground reflections,
and other signals.

All these drawbacks highlight the need to test antennas in an indoor
environment (like an anechoic chamber) under fully controlled conditions. To
achieve this goal, two different strategies can be adopted.

The first one concerns the use of a compact range to produce the far field at
a short distance [1]. In this case, the far field pattern is obtained by
collimating the field radiated by the AUT with a large reflector or a lens.

The second strategy consists of measuring directly the near field [2]. Then,
once the near field data have been acquired, the radiation pattern of the AUT
is recovered by exploiting a suitable near-field far-field transformation
(NFFFT) [3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

However, since near field measurements can be acquired with a simpler
measurement setup, the second strategy is mainly used in antenna testing [13].

In order to reconstruct the radiation pattern by near field data, a proper
measurement scheme must be set. Over the years different measurement schemes
of the near field have been developed for the planar [14] [15], cylindrical
[16] [17], and spherical scanning [18, 19, 20, 21].

In the case of planar scanning, a sampling step of half wavelength is enough
to recover the visible portion of the far field spectrum and the radiation
pattern of the AUT. Accordingly, a step of $\lambda/2$ represents the actual
standard in planar near field antenna measurements.

Despite its simplicity, the half wavelength sampling does not take into
account for the geometrical parameters of the configuration (the size and the
shape of the source, the size of the measurement plane, and the distance
between the source and the measurement plane); for such reason, it requires to
acquire a redundant number of near field samples. Naturally, this negatively
affects the acquisition time of the field samples which, especially at high
frequencies, may become long.

In this framework, two relevant points are:

  1. 1.

to establish the minimum number of measurements that allows to acquire the
near field without loss of information,

  2. 2.

to develop a sampling strategy of the near field that employs a non-redundant
number of measurements.

As concerns the first point, the minimum number of measurements is equal to
the number of degrees of freedom of the near field. By definition, such a
number represents the “essential dimension” of the field space [22] [23]. It
can be evaluated by determining the number of relevant singular values of the
radiation operator $T$ relating the source current $J$ with the near field $E$
[24] [25].

In reference to the second point, over the years different sampling strategies
employing a reduced number of near field measurements have been proposed [26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. Among these,
the sampling scheme described in [42] for a finite planar source is of
particular interest since it exploits a number of near field measurements
equal to the NDF. Such strategy estimates the sampling points of the near
field $E$ by discretizing the continuous model $TJ=E$ in such a way that the
discretized model shows the same singular values of the continuous one. More
in detail, the problem of discretizing the radiation operator $T$ is cast as
the discretization of the composed operator $TT^{\dagger}$ where $T^{\dagger}$
stands for the adjoint operator. To this end, at first, a warping vector
transformation mapping the two starting observation variables into two new
ones is introduced to rewrite the kernel of $TT^{\dagger}$ as a convolution
and bandlimited function. After, the continuous operator $TT^{\dagger}$ is
discretized by exploiting the Shannon sampling theorem [43] and a sampling
series of the near field is derived.

The vector warping transformation derived in [42] works well until the
observation domain is not larger than the source domain. When this condition
is breached, the derived warping transformation does not provide a good
representation of the field outside the source domain and the resulting
sampling series returns an under-sampled version of the field. Accordingly, a
more complicated warping transformation where each new variables depend on
both starting variables should be introduced. To find such a transformation is
a difficult task and it leads to a complicated no-factorized sampling point
distribution.

To overcome such a problem, a sub-optimal sampling strategy is proposed in
[45]. Basically, the arrangement of points provided by the warping sampling
scheme is thickened by employing an oversampling factor. In particular, since
the oversampling is strictly required only outside the source domain, a
spatially varying oversampling is introduced. The resulting sampling scheme
employs a number of field samples slightly higher than the NDF but
significantly lower than half-wavelength sampling. Hence, it is almost optimal
in terms of number of measurements. Despite this, the arrangement of the
sampling points over the measurement plane is non-uniform; this represents an
issue since in many cases the control software of the near field probe is not
conceived to move it with a non uniform step. Moreover, another issue of the
sampling scheme in [45] is represented by the choice of the constants
appearing in the oversampling factor since they depend on the geometrical
parameters of the configuration in an unknown manner.

With the aim of avoiding the drawbacks of non-uniform sampling strategies, a
uniform sampling scheme will be presented. The latter will be obtained with a
proper choice of the oversampling functions whose structure is uniquely
determined once the geometrical parameters of the configuration have been
specified.

The paper is structured as follows. In section II, the radiation operator
linking the planar current to the near field over a plane is introduced. In
section III, the near field sampling scheme based on the warping technique is
recalled from the literature. In section IV-A, the warping sampling scheme is
extended to the case of a measurement plane larger than the source domain. In
section IV-B, starting from the sampling criterion of section IV-A, a novel
uniform sampling criterion is derived. Finally, a section of numerical results
based on experimental data is included to assess the performance of the new
sampling scheme.

##  II Geometry and mathematical preliminaries

Consider the $3D$ geometry depicted in Figure 1 where a planar antenna
supported over a finite plane $SD=[-a,a]\times[-b,b]$ located at $z=0$ is
shown.

Figure 1: Geometry of the problem.

According to [4], the electromagnetic field radiated by a planar antenna is
equivalent to the one generated by a magnetic density current
$\mathbf{J}(x,y)=J_{x}(x,y)\,\mathbf{\hat{i}_{x}}+J_{y}(x,y)\,\mathbf{\hat{i}_{y}}$.

Such density current radiates an electric field $\mathbf{E}$ whose tangential
components $(E_{x},E_{y})$ are collected over a finite plane
$OD=[-X_{o},X_{o}]\times[-Y_{o},Y_{o}]$ located at $z=d$.

For the considered configuration, the components of the density current are
related to the tangential components of the electric field by the scalar
equations

$E_{y}(\textbf{r})=T\,J_{x}(\textbf{r}^{\prime})$ (1)
$E_{x}(\textbf{r})=T\,J_{y}(\textbf{r}^{\prime})$ (2)

where

  * •

$\textbf{r}^{\prime}=(x^{\prime},y^{\prime},0)$ is a point of the source
domain $SD$,

  * •

$\textbf{r}=(x,y,d)$ is a point of the observation domain $OD$,

  * •

$T:J\in L^{2}(SD)\longrightarrow E\in L^{2}(OD)$ is the radiation operator
with $L^{2}(\cdot)$ denoting the class of square integrable functions over the
set $(\cdot)$.

Since equations (1) and (2) are formally identical, in the following the
simplified notation $E=TJ$ will be used to refer to both the equations.

The radiation operator $T$ can be explicitly written as

$T\,J(\textbf{r}^{\prime})=\int_{SD}h(\textbf{r},\textbf{r}^{\prime})\,e^{-jk|\textbf{r}-\textbf{r}^{\prime}|}J(\textbf{r}^{\prime})\,d\textbf{r}^{\prime}$
(3)

where $k$ denotes the wavenumber, and $h(\textbf{r},\textbf{r}^{\prime})$ is
an amplitude term. Assuming $d$ large enough to neglect the evanescent waves
contribution, the function $h(\textbf{r},\textbf{r}^{\prime})$ can be
approximated as

$h(\textbf{r},\textbf{r}^{\prime})\approx-\dfrac{jkd}{4\pi|\textbf{r}-\textbf{r}^{\prime}|^{2}}.$
(4)

As concerns the adjoint operator $T^{\dagger}:E\in L^{2}(OD)\longrightarrow
J\in L^{2}(SD)$, it is given by

$T^{\dagger}\,E(\mathbb{r})=\int_{OD}h^{*}(\textbf{r},\textbf{r}^{\prime})\,e^{jk|\textbf{r}-\textbf{r}^{\prime}|}d\textbf{r}$
(5)

with $h^{*}(\textbf{r},\textbf{r}^{\prime})$ denoting the conjugate of
$h(\textbf{r},\textbf{r}^{\prime})$.

Since the near field sampling proposed here is based on a Shannon sampling
representation of the left singular functions of the radiation operator, the
singular system of $T$ is introduced. The latter is made of the triple
$\\{u_{n},v_{n},\sigma_{n}\\}$ where

  * •

$\\{\sigma_{n}\\}$ are non-decreasing scalars called singular values,

  * •

$\\{u_{n}\\}$ and $\\{v_{n}\\}$ are two sets of orthonormal functions named
respectively right and left singular functions.

By definition, the singular system of $T$ satisfies the equations
$Tu_{n}=\sigma_{n}v_{n}$ and $T^{\dagger}v_{n}=\sigma_{n}u_{n}$ for each
$n\in\mathbb{N}$. Accordingly, the $n-$th right singular function fulfills the
eigenvalue problem $T^{\dagger}Tu_{n}=\sigma_{n}^{2}u_{n}$ while the $n-$th
left singular function is a solution of
$TT^{\dagger}v_{n}=\sigma_{n}^{2}v_{n}$.

##  III Near field sampling when $OD\subseteq SD$

In this section, the near field warping sampling scheme working until
$OD\subseteq SD$ ($X_{o}\leq a$ and $Y_{o}\leq b$) is recalled from [42].

When the smallest size of the source is larger than the wavelength (
$min\\{a,b\\}>>\lambda/2\pi$ ), the singular values of $T$ exhibit an abrupt
decay at a critical index $N$ which identifies the number of degrees of
freedom. In such a case, the range of $T$ is “essentially” finite-dimensional;
accordingly, the near field can be represented by considering only the first
$N$ left singular functions, i.e.

$E(\textbf{r})\approx\sum_{n=1}^{N}c_{n}v_{n}(\textbf{r})$ (6)

From Equation (6) it is evident that a field sampling criterion can be derived
by a proper discretization of the left singular functions
$\\{v_{1}(\textbf{r}),...,v_{N}(\textbf{r})\\}$. To find such discretization,
the eigenvalue problem $TT^{\dagger}v_{n}=\sigma_{n}^{2}v_{n}$ is considered.
The latter can be expressed as

$\int_{OD}H(\textbf{r},\textbf{r}_{o})\,v_{n}(\textbf{r}_{o})\,d\textbf{r}_{o}=\sigma_{n}^{2}\,v_{n}(\textbf{r})$
(7)

where $\textbf{r}_{o}=(x_{o},y_{o},d)$ and

$H(\textbf{r},\textbf{r}_{o})=\int_{SD}h^{*}(\textbf{r}_{o},\mathbb{r}^{\prime})\,h(\textbf{r},\textbf{r}^{\prime})\,e^{jk\big{(}|\textbf{r}_{o}-\textbf{r}^{\prime}|-|\textbf{r}-\textbf{r}^{\prime}|\big{)}}\,d\textbf{r}^{\prime}.$
(8)

As shown in [42], Equation (7) can be approximated as

$\hskip
9.0pt\int_{\zeta(-Y_{o})}^{\zeta(Y_{o})}\int_{\eta(-X_{o})}^{\eta(X_{o})}e^{-j\,ka\,[\gamma(\eta)-\gamma(\eta_{o})]}e^{-j\,kb\,[\xi(\zeta)-\xi(\zeta_{o})]}\
\ \ \ \ \ \ $
$\begin{split}sinc\,(ka\,(\eta-\eta_{o}))\,sinc\,(kb\,(\zeta-\zeta_{o}))v_{n}(\eta_{o},\zeta_{o})\,d\eta_{o}\,d\zeta_{o}\end{split}$
(9) $\hskip 150.0pt=\,\sigma_{n}^{2}v_{n}(\eta,\zeta)$

where the new variables $\eta=\eta(x)$, $\eta_{o}=\eta(x_{o})$,
$\zeta=\zeta(y)$, $\zeta_{o}=\zeta(y_{o})$ are defined as

$\eta(x)=\dfrac{\sqrt{(x+a)^{2}+d^{2}}-\sqrt{(x-a)^{2}+d^{2}}}{2a}$ (10)
$\zeta(y)=\dfrac{\sqrt{(y+b)^{2}+d^{2}}-\sqrt{(y-b)^{2}+d^{2}}}{2b}$ (11)

and

$\gamma(x)=\frac{\sqrt{(x+a)^{2}+d^{2}}+\sqrt{(x-a)^{2}+d^{2}}}{2a}$ (12)
$\xi(y)=\frac{\sqrt{(y+b)^{2}+d^{2}}+\sqrt{(y-b)^{2}+d^{2}}}{2b}$ (13)

At this juncture, if the assignment

$\hat{v}_{n}(\eta,\zeta)=v_{n}(\eta,\zeta)\,e^{jka\,\gamma(\eta)}e^{jkb\,\xi(\zeta)}$
(14)

is performed, the eigenvalue problem (9) can be rewritten as

$\begin{split}&\int_{\zeta(-Y_{o})}^{\zeta(Y_{o})}\int_{\eta(-X_{o})}^{\eta(X_{o})}sinc{((ka\,(\eta-\eta_{o}))}\,sinc{(kb\,(\zeta-\zeta_{o}))}\\\
&\hskip
90.0pt\hat{v}_{n}(\eta_{o},\zeta_{o})\,d\eta_{o}d\zeta_{o}=\sigma_{n}^{2}\,\hat{v}_{n}(\eta,\zeta)\end{split}$
(15)

Differently by (7), the previous integral equation involves a convolution
operator with a bandlimited kernel. Accordingly, also the eigenfunctions
$\\{\hat{v}_{n}\\}$ are bandlimited with respect to $(\eta,\zeta)$ with a
bandwidth equal to $ka$ and $kb$, respectively. For such reason, they can be
discretized without loss of information with the step-lengths
$\Delta\eta={\pi}/(ka)$ and $\Delta\zeta=\pi/(kb)$. Moreover, the first $N$
eigenfunctions can be represented through the following truncated Shannon
sampling expansion

$\hat{v}_{n}(\eta,\zeta)\approx\hskip 280.0pt\vspace{-7pt}$ (16)
$\sum_{p=-N_{1}}^{N_{1}}\sum_{q=-N_{2}}^{N_{2}}\hat{v}_{n}(\eta_{p},\zeta_{q})\,sinc(ka\,\eta-p\pi)\,sinc(kb\,\zeta-q\pi)$

where $\eta_{p}=p\,\pi/(ka)$, $\zeta_{q}=q\,\pi/(kb)$,
$N_{1}=[\frac{ka}{\pi}\,\eta(X_{o})]$, $N_{2}=[\frac{kb}{\pi}\,\eta(Y_{o})]$
with $[\,\cdot\,]$ denoting the integer part. The scalar $N_{1}$ and $N_{2}$
have been chosen in such a way to consider only the samples
$\\{\hat{v}_{n}(\eta_{p},\zeta_{q})\\}$ supported over the set
$[\eta(-X_{o}),\eta(X_{o})]\times[\zeta(-Y_{o}),\zeta(Y_{o})]$. This choice
makes negligible the error due to the truncation of the indexes $p$ and $q$ in
(16) since all the samples $\\{\hat{v}_{n}(\eta_{p},\zeta_{q})\\}$ falling
outside the measurement plane are not relevant for the representation of
$\hat{v}_{n}(\eta,\zeta)$ over the set
$[\eta(-X_{o}),\eta(X_{o})]\times[\zeta(-Y_{o}),\zeta(Y_{o})]$.

Now, if equation (16) is substituted in (14), the following sampling
representation comes out

${v}_{n}(\eta,\zeta)=e^{-jka\,\gamma(\eta)}e^{-jkb\,\xi(\zeta)}\sum_{p=-N_{1}}^{N_{1}}\sum_{q=-N_{2}}^{N_{2}}{v}_{n}(\eta_{p},\zeta_{q})\hskip
20.0pt\vspace{-4pt}$ (17)
$e^{\,jka\,\gamma(\eta_{p})}e^{\,jkb\,\xi(\zeta_{q})}\,sinc(ka\,\eta-p\pi)\,sinc(kb\,\zeta-q\pi)$

By virtue of (6), the near field $E$ admits a sampling representation
perfectly equal to (17). Moreover, since $\eta=\eta(x)$ and $\zeta=\zeta(y)$,
it can be expressed in terms of the variables $(x,y)$ as below

$\begin{split}&E\,(x,y)=e^{-jka\,\gamma(x)}e^{-jkb\,\xi(y)}\sum_{p=-N_{1}}^{N_{1}}\sum_{q=-N_{2}}^{N_{2}}E\,(x_{p},y_{q})\
\ \ \ \ \ \\\
&e^{jka\,\gamma(x_{p})}e^{jkb\,\xi(y_{q})}\,sinc\big{(}ka\,\eta(x)-p\pi\big{)}\,sinc\big{(}kb\,\zeta(y)-q\pi\big{)}\vspace{2pt}\end{split}$
(18)

where $x_{p}$ and $y_{q}$ are solution of $\eta(x_{p})=\eta_{p}$ and
$\zeta(y_{q})=\zeta_{q}$, respectively. Accordingly, it results that

$x_{p}=\eta_{p}\sqrt{a^{2}-\dfrac{d^{2}}{1-\eta_{p}^{2}}}\ \ \ \ \ \ \
y_{q}=\zeta_{q}\sqrt{b^{2}-\dfrac{d^{2}}{1-\zeta_{p}^{2}}}$ (19)

Equation (19) returns the positions where the field samples must be collected.
Since the non-linear dependence of $\eta(x)$ and $\zeta(y)$ respectively on
$x$ and $y$, uniform sampling of $\eta(x)$ and $\zeta(y)$ returns a non-
uniform grid in $x$ and $y$ with a number of sampling points equal to

$N\approx\left(2\left[\dfrac{ka}{\pi}\eta(X_{o})\right]+1\right)\left(2\left[\dfrac{kb}{\pi}\zeta(Y_{o})\right]+1\right)$
(20)

Since $N$ is very close to the number of degrees of freedom of the near field
[44], the interpolation formula (18) exploits a non-redundant number of field
measurements. In this sense, the described near field sampling criterion is
optimal.

##  IV Near field sampling when $OD\supseteq SD$

The sampling scheme shown in Section III works until the observation domain
($OD$) is lower or equal than the source domain ($SD$). Hence, it can be
exploited only for testing antennas with a broadside radiation pattern.

When the main lobe of the field pattern is scanned away from boresight, in
order to control the truncation error, the near field shall be collected over
a measurement plane larger than the source domain. For such reason, this
section aims at showing two measurement schemes of the near field suitable
when $OD\supseteq SD$ (i.e., when $X_{o}>a$ and / or $Y_{o}>b$). In
particular, in section IV-A a recent non uniform sampling scheme based on
warping technique is recalled by the literature. In section IV-B, a novel
uniform sampling criterion that overcomes all the drawbacks of non-uniform
strategies is proposed.

###  IV-A Non-uniform field sampling

When $OD\supseteq SD$, the sampling expansion (17) does not approximate well
the field on the subset of $OD$ exceeding $SD$. In particular, according to
local band considerations addressed in [45], Equation (17) returns an under-
sampled representation outside $SD$. In this case, the approach developed in
[42] can be still applied but a couple of warping variables where each one
depends on both $x$ and $y$ variables shall be used. This implies that the
sampling along $x$ and $y$ is no more factorized since the arrangement of
samples along $x$ will depend also on $y$ and vice versa.

Despite the non-factorized sampling is optimal in terms of number of samples,
it introduces a complicated sampling scheme. This complication can be overcome
by introducing a uniform oversampling factor in the interpolation formula
(17). The latter reduces the representation error on the near field by using a
slightly redundant number of samples (bigger than the number of degrees of
freedom but lower than the number of samples required by the half wavelength
sampling). In particular, if the oversampling factors $\chi_{1}>1$ and
$\chi_{2}>1$ are properly chosen, the left singular functions
$\\{v_{1},...,v_{N}\\}$ are well approximated by

$v_{n}(\eta,\zeta)=e^{-jka\gamma(\eta)}e^{-jkb\xi(\zeta)}\sum_{p=-M_{1}}^{M_{1}}\sum_{q=-M_{2}}^{M_{2}}v_{n}(\eta_{p},\zeta_{q})\
\ \ \ \ \ \ \ \vspace{-1pt}$ (21)
$e^{jka\,\gamma(\eta_{p})}e^{jkb\,\xi(\zeta_{q})}\,sinc\big{(}\chi_{1}ka\,\eta-p\pi\big{)}\,sinc\big{(}\chi_{2}kb\,\zeta-q\pi\big{)}$

where $\eta_{p}=p\pi/(\chi_{1}ka)$, $\zeta_{q}=q\pi/(\chi_{2}kb)$,
$M_{1}=[\frac{\chi_{1}ka}{\pi}\,\eta(X_{o})]$ and
$M_{2}=[\frac{\chi_{2}kb}{\pi}\,\zeta(Y_{o})]$.

It is worth nothing that, for a correct representation of the left singular
functions, the oversampling is not mandatory in the region where
$|\eta|<\eta(a)$ and $|\zeta|<\zeta(b)$ since in this region the
considerations of Section III can be applied. On the contrary, it is strictly
necessary for $|\eta|>\eta(a)$ and $|\zeta|>\zeta(b)$.

Based on these considerations, in [45] a spatially varying oversampling is
proposed with the aim to re-arrange the sampling points by thickening the
sampling step only outside $SD$. The spatially varying oversampling is
realized by choosing two oversampling factors $\chi_{1}(x)$ and $\chi_{2}(y)$
as below

$\chi_{1}(x)=1-(1-\alpha_{1})sin^{p}\left(\dfrac{\pi}{2X_{o}}x\right)$ (22)

$\chi_{2}(y)=1-(1-\alpha_{2})sin^{p}\left(\dfrac{\pi}{2Y_{o}}y\right)$ (23)

Such factors are proximal to 1 for $|x|\leq a$ and $|y|\leq b$, instead, they
rise monotonically for $\,a<|x|\leq X_{o}\,$ and $b<|y|\leq Y_{o}$. In this
way, the sampling rate is kept at the Nyquist limit in the region of the
measurement plane located in front of the source domain while it is gradually
increased at the edges of the measurement plane.

When a spatially varying oversampling is adopted, the first $N$ left singular
functions of the radiation operator can be still expressed trough a truncated
Shannon sampling series. Indeed, after the introduction of the variables

$\hat{\eta}(x)=\chi_{1}(x)\,\eta(x)\ \ \ \ \ \ \ \
\hat{\zeta}(y)=\chi_{2}(y)\,\zeta(y)$ (24)

the left singular functions $\\{v_{n}\\}_{n=1}^{N}$ can be written as

$v_{n}\,(\hat{\eta},\hat{\zeta})=e^{-jka\,\gamma(\hat{\eta})}e^{-jkb\,\xi(\hat{\zeta})}\sum_{p=-M_{1}}^{M_{1}}\sum_{q=-M_{2}}^{M_{2}}v_{n}\,(\hat{\eta}_{p},\hat{\zeta}_{q})\vspace{-8pt}$
(25)
$e^{jka\,\gamma(\hat{\eta}_{p})}e^{jkb\,\xi(\hat{\zeta}_{q})}\,sinc\big{(}ka\,\hat{\eta}-p\pi\big{)}\,sinc\big{(}kb\,\hat{\zeta}-q\pi\big{)}$

with $\hat{\eta}_{p}=p\,\pi/(ka)$, $\hat{\zeta}_{q}=q\,\pi/(kb)$,
$M_{1}=[\frac{ka}{\pi}\,\hat{\eta}(X_{o})]$, and
$M_{2}=[\frac{kb}{\pi}\,\hat{\eta}(Y_{o})]$. As a consequence, the near field
$E(x,y)$ is given by

$\begin{split}E&\,(x,y)=e^{-jka\,\gamma(x)}e^{-jkb\,\xi(y)}\sum_{p=-M_{1}}^{M_{1}}\sum_{q=-M_{2}}^{M_{2}}E\,(x_{p},y_{q})\\\
&e^{jka\,\gamma(x_{p})}e^{jkb\,\xi(y_{q})}\,sinc\big{(}ka\,\hat{\eta}(x)-p\pi\big{)}\,sinc\big{(}kb\,\hat{\zeta}(y)-q\pi\big{)}\end{split}$
(26)

where $x_{p}$ and $y_{q}$ are solution of

$\hat{\eta}(x_{p})=\chi_{1}(x_{p})\,\eta(x_{p})=p\dfrac{\pi}{ka}$ (27)
$\hat{\zeta}(y_{q})=\chi_{2}(y_{q})\,\zeta(y_{q})=q\dfrac{\pi}{kb}.$ (28)

Accordingly, following this approach, the variables to be uniformly sampled
are not $\eta$ and $\zeta$ but $\hat{\eta}$ and $\hat{\zeta}$.

###  IV-B Drawbacks of non-uniform methods and introduction of a novel uniform
sampling

Figure 2: Diagrams of $\eta(x)$, $\hat{\eta}(x)$ and $\zeta(y)$,
$\hat{\zeta}(y)$ when $a=b=10\lambda$ and $d=5\lambda$.

Despite the introduction of the spatially varying oversampling factors
described in section IV-A is a sub-optimal strategy that allows achieving a
better trade-off between the representation error and the number of sampling
points than the one based on uniform oversampling, the proposed strategy still
shows two drawbacks.

The first one concerns the non-uniform arrangement of the near field samples
over the measurement plane which is due to the non-linear dependence of
$\hat{\eta}(x)$ and $\hat{\zeta}(y)$ on $x$ and $y$, respectively. Indeed,
although the non-uniform sampling of the near field is possible in principle,
the control software of many near field scanner systems does not give the
possibility to move the field probe with a non-uniform step length. For such
reason, non-uniform near field measurements are difficult to perform without
updating the scanning system.

The second drawback concerns the choice of the constants $\alpha_{1}$,
$\alpha_{2}$ and $p$ appearing in the oversampling factors (22) and (23). Such
constants depend on the geometrical parameters of the configuration
$a,b,X_{o},Y_{o},d$ in an unknown manner; accordingly, they must be properly
chosen case by case.

With the aim to overcome these issues, a uniform sampling scheme of the near
field is now proposed. As will be more clear later, such a strategy needs a
number of measurements slightly higher than the non-uniform sampling scheme of
section $IVA$ but significantly lower than the classical half wavelength
sampling leading to a good trade-off between the acquisition time of the near
field samples and the simplicity of the scanner system.

It is worth highlighting that the non-uniformity of the sampling schemes
recalled above derives from the non-linear behavior of the warped variables in
terms of the spatial variables $x$ and $y$. Indeed, due to such non-linearity,
the uniform sampling in the warped variables is mapped into a non-uniform
sampling in the original spatial variables.

Moved by this reasoning and inspired by the spatially varying oversampling
approach, our idea is to introduce two spatially varying oversampling factors
that compensate for the non-linear behavior of the original warped variables
$\eta(x)$ and $\zeta(y)$ in such a way that the resulting sampling in the
original spatial variables $x$ and $y$ will be uniform. In particular, in the
uniform sampling scheme proposed here, the oversampling factors $\chi_{1}(x)$
and $\chi_{2}(y)$ will be chosen in such a way that

  1. 1.

the new warped variables $\hat{\eta}(x)=\chi_{1}(x)\,\eta(x)$ and
$\hat{\zeta}(y)=\chi_{2}(y)\zeta(y)$ are linear functions;

  2. 2.

the oversampling functions $\chi_{1}(x)$ and $\chi_{2}(y)$ are greater or
equal than $1$ over all the observation domain $OD$.

The fulfillment of the first condition ensures that a uniform sampling of the
warped variables $\hat{\eta}(x)$ and $\hat{\zeta}(y)$ is mapped into a uniform
sampling in the original spatial variables $x$ and $y$. Instead, the
fulfillment of the second condition allows to keep the representation error of
the near field coming from the proposed uniform sampling comparable with the
representation deriving from the half-wavelength sampling and non-uniform
sampling of section IV A.

Figure 3: Diagrams of $\chi_{1}(x)$ and $\chi_{2}(y)$ when $a=b=10\lambda$ and
$d=5\lambda$.

To obtain the desired linear expression of the warped variables
$\hat{\eta}(x)$ and $\hat{\zeta}(y)$, the Taylor expansion at the first order
of $\eta(x)$ and $\zeta(y)$ around the points $x=0$ and $y=0$ is considered.
From such expansions, it comes out that

$\hat{\eta}\,(x)=\left.\dfrac{d\eta}{dx}\right|_{x\,=\,0\,}x=\dfrac{1}{\sqrt{a^{2}+d^{2}}}\,x$
(29)
$\hat{\zeta}(y)=\left.\dfrac{d\zeta}{dy}\right|_{y\,=\,0\,}y=\dfrac{1}{\sqrt{b^{2}+d^{2}}}\,y$
(30)

Since the spatially varying oversampling factors $\chi_{1}(x)$ and
$\chi_{2}(y)$ shall be in agreement with (29) and (28), it results that

$\chi_{1}(x)=\dfrac{x}{\sqrt{a^{2}+d^{2}}}\dfrac{2a}{\sqrt{(x+a)^{2}+d^{2}}-\sqrt{(x-a)^{2}+d^{2}}}$
(31)
$\chi_{2}(y)=\dfrac{y}{\sqrt{b^{2}+d^{2}}}\dfrac{2b}{\sqrt{(y+b)^{2}+d^{2}}-\sqrt{(y-b)^{2}+d^{2}}}$
(32)

In Figure 2 the diagrams of the warping variables $\eta(x)$, $\hat{\eta}(x)$
and $\zeta(y)$, $\hat{\zeta}(y)$ are sketched. As can be seen from the figure,
it results that $|\hat{\eta}(x)|\geq|\eta(x)|\ \forall x\in[-X_{o},X_{o}]$ and
$|\hat{\zeta}(y)|\geq|\eta(y)|\ \forall y\in[-Y_{o},Y_{o}]$. This implies that
the choice of $\chi_{1}(x)$ and $\chi_{2}(y)$ ensuring a linear behavior of
$\hat{\eta}(x)$ and $\hat{\zeta}(y)$ also guarantees $\chi_{1}(x)\geq 1\
\forall x\in[-X_{o},X_{o}]$ and $\chi_{2}(y)\geq 1\ \forall
y\in[-Y_{o},Y_{o}]$. Hence, the uniform sampling proposed in this section is
really an over-sampled version of the sampling criterion described in Section
III.

The plot of $\chi_{1}(x)$ and $\chi_{2}(y)$ is shown in Figure 3. From such
plots, it is evident that $\chi_{1}(x)\approx 1$ for $|x|\leq a$ while
$\chi_{1}(x)>1$ for $|x|>a$. Similarly, $\chi_{2}(y)\approx 1$ for $|y|\leq b$
while $\chi_{2}(y)>1$ for $|y|>b$. This allows us to state that also the
uniform sampling scheme employs a number of field measurements proximal to the
minimum in the portion of the measurement plane located in front of the
source. Instead, at the edges of the observation domain, the sampling rate is
not decreased but maintained constant.

Figure 4: Diagram of the sampling step along $x$ (or $y$) in terms of the
source length along $x$ (or $y$) for different value of the distance $d$.

At this juncture, if equations (29) and (30) are considered, the general near
field interpolation formula (26) can be rewritten as below

$\begin{split}E&(x,y)=e^{-jka\gamma(x)}e^{-jkb\xi(y)}\sum_{p=-M_{1}}^{M_{1}}\sum_{q=-M_{2}}^{M_{2}}E(x_{p},y_{q})\,e^{jka\gamma(x_{p})}\\\
&e^{jkb\,\xi(y_{q})}\,sinc\left(\dfrac{ka}{\sqrt{a^{2}+d^{2}}}\,x-p\pi\right)\,sinc\left(\dfrac{kb}{\sqrt{b^{2}+d^{2}}}\,y-q\pi\right)\end{split}$
(33)

where

  * •

$M_{1}=[\frac{ka}{\pi}\,\hat{\eta}(X_{o})]=\left[2aX_{o}/(\lambda\sqrt{a^{2}+d^{2}})\right]$,

  * •

$M_{2}=[\frac{kb}{\pi}\,\hat{\eta}(Y_{o})]=\left[2bY_{o}/(\lambda\sqrt{b^{2}+d^{2}})\right]$.

The exact coordinates $\\{x_{p}\\}$ and $\\{y_{q}\\}$ of the sampling points
can be easily derived by substituting the expression of $\hat{\eta}(x)$ and
$\hat{\zeta}(y)$ provided by (29) and (30) in (27) and (28), respectively. By
doing this and expressing the wavenumber in terms of the wavelength
($k=2\pi/\lambda$), it comes out that

$x_{p}=p\dfrac{\lambda}{2}\dfrac{\sqrt{a^{2}+d^{2}}}{a}\ \ \ \ \ \ \ \ \
y_{q}=q\dfrac{\lambda}{2}\dfrac{\sqrt{b^{2}+d^{2}}}{b}.$ (34)

$\forall p\in\\{-M_{1},\ldots,M_{1}\\}$ and $\forall
q\in\\{-M_{2},\ldots,M_{2}\\}$. Hence, the sampling steps $\Delta x$ and
$\Delta y$ are given by

$\Delta x=\dfrac{\lambda}{2}\dfrac{\sqrt{a^{2}+d^{2}}}{a}\ \ \ \ \ \ \ \ \
\Delta y=\dfrac{\lambda}{2}\dfrac{\sqrt{b^{2}+d^{2}}}{b}.$ (35)

From the previous equation, it is evident that the distance between the field
samples is always greater than $\lambda/2$; accordingly, the required number
of field measurements

$M=\left(2\left[\dfrac{2X_{o}}{\lambda}\dfrac{a}{\sqrt{a^{2}+d^{2}}}\right]+1\right)\left(2\left[\dfrac{2Y_{o}}{\lambda}\dfrac{b}{\sqrt{b^{2}+d^{2}}}\right]+1\right)$
(36)

is always lower than the half-wavelength sampling. In particular, the
steplengths $\Delta x$ or $\Delta y$ of the proposed sampling scheme are
significantly larger than $\lambda/2$ especially when the semi-extension of
source domain edges ($a$ or $b$) is little or comparable with respect to the
distance $d$. This is also confirmed by Figure 4 which illustrates the
sampling step $\Delta x$ (or $\Delta y$) in terms of the source extension
along $x$ (or $y$).

Equations (33) and (34) represent the basis of the proposed sampling scheme
and the main results of this article. The main peculiarity of the proposed
strategy for near field measurements is that it employs a number of field
samples $M$ a few larger than the non-uniform sampling of Section IV-A with
the simplicity and the flexibility of the uniform half-wavelength sampling.
Accordingly, it merges the main advantages of the two sampling schemes by
avoiding their drawbacks.

Finally, it is worth computing also the percentual reduction $R$ of field
samples with respect to half-wavelength sampling. The latter is given by

$R=\left(1-\dfrac{M}{M_{\frac{\lambda}{2}}}\right)100\%=\left(1-\dfrac{ab}{\sqrt{a^{2}+d^{2}}\sqrt{b^{2}+d^{2}}}\right)100\%$
(37)

where $M_{\frac{\lambda}{2}}=(2\,[2X_{o}/\lambda]+1)(2\,[2Y_{o}/\lambda]+1)$
is the number of field samples required by the half-wavelength sampling.

##  V Uniform sampling validation by
experimental field data

Figure 5: Near field obtained by the interpolation of measurements collected
with $\Delta x=0.48\lambda$, $\Delta y=0.48\lambda$. Figure 6: Near field
obtained by the interpolation of measurements collected with $\Delta
x=1.26\lambda$, $\Delta y=0.86\lambda$. Figure 7: Far field spectrum obtained
by the near field samples collected with the steplengths $\Delta
x=0.48\lambda$ and $\Delta y=0.48\lambda$ Figure 8: Far field spectrum
obtained by the near field samples collected with the steplengths $\Delta
x=1.26\lambda$ and $\Delta y=0.86\lambda$ Figure 9: Comparison between the
main cuts of the far field spectrum obtained the half wavelength sampling and
the proposed sampling scheme.

In this section, the uniform sampling scheme proposed in section IV B is
corroborated by exploiting some experimental field data provided by MBDA
Italy.

A rectangular array of Vivaldi antennas arranged over the source domain
according to a triangular lattice has been considered and tested. The array
support is a rectangle whose edges have a semi-extension $a=3\lambda$,
$b=5\lambda$.

The tangential component of the near field radiated by such array has been
measured over a plane at a distance $d=7\lambda$. The semi-extension of the
measurement plane along the $x$ and the $y$ axes are $X_{o}=14.5\lambda$ and
$Y_{o}=9.75\lambda$, respectively.

In order to assess the performances of the proposed sampling scheme, the near
field radiated by the array has been measured by employing both the classical
step length of half-wavelength and the sampling steps provided by (35). Then,
the near field retrieved by collecting data according to the standard half-
wavelength sampling is compared with that obtained by exploiting the proposed
uniform measurement strategy. Finally, also a comparison between the far field
reconstructions is performed.

In the first test case, the near field has been collected by employing the
classical half-wavelength sampling ($\Delta x=0.48\lambda$, $\Delta
y=0.48\lambda$). Accordingly, the number of field samples collected along the
$x$-axis are $M_{x}=61$ while those collected along the $y$-axis are
$M_{y}=41$. The total number of field samples is $M_{\lambda/2}=2501$.
To reconstruct the near field by its samples, the field measurements have been
interpolated with the classical interpolation formula

$\begin{split}&E(x,y)=\sum_{p=-30}^{30}\sum_{q=-20}^{20}E\left(p{\Delta
x},q{\Delta y}\right)\,sinc\left(\dfrac{\pi}{\Delta x}x-p\pi\right)\\\
&\,sinc\left(\dfrac{\pi}{\Delta y}\,y-q\pi\right)\end{split}$ (38)

The amplitude in dB and the phase in degrees of the near field reconstructed
by (38) are sketched in Figure 5.

In the second test case, the near field has been measured by employing the
novel uniform sampling introduced in section IV-B. In such a case, the
sampling steps deriving from (35) are $\Delta x=1.26\lambda$ and $\Delta
y=0.86\lambda$ (note that the step length is smaller along the direction where
the array under test is larger). Accordingly, the number of field samples
collected along the $x$ and $y$ axes is respectively $N_{x}=24$ and $N_{y}=23$
while the total number of field measurements $M$ is $552$.
The near field has been recovered by employing the interpolation formula (33).
The result of this interpolation is shown in Figure 6 which sketches the
amplitude in dB and the phase in degrees of the near field reconstruction.

With the aim to evaluate the mismatching of the near field reconstructions
shown in Fig. 5 and 6, the relative root mean squared error ($rRMSE$) has been
evaluated. In particular, it results that

$rRMSE=\dfrac{||E-E_{\lambda/2}||^{2}}{||E_{\lambda/2}||^{2}}=7.80\cdot
10^{-4}$ (39)

where $||\cdot||$ stands for the Euclidean norm. Accordingly, in the
considered test cases, the proposed sampling scheme achieves the same
reconstruction of the near field with a percentual reduction $R$ of field
measurements equal to $77.93\%$.

Since in the framework of antenna testing the interest is in the far field of
the antenna under test, the far field of the considered array has also been
evaluated.

In Figure 7, the far field spectrum obtained by near field samples collected
with $\Delta x=0.48\lambda$ and $\Delta y=0.48\lambda$ is shown. The figure
has been obtained by computing directly the Fast Fourier Transform of the
measured field samples.

In Figure 8, the far field spectrum computed by the near field measurements
collected with the step lengths $\Delta x=1.26\lambda$ and $\Delta
y=0.86\lambda$ is sketched. In such a case, the far field spectrum has been
evaluated in two steps. The first step is the resampling of the near field in
Fig. 6 with a step length of $\lambda/2$. Then, the Fast Fourier Transform of
the synthetic field samples has been computed to obtain the far field
spectrum.

By comparing Figure 7 and 8, it is evident that the reconstructions of far
field spectrum deriving by the two sampling schemes are essentially the same.
The accuracy in the reconstruction of the far field spectrum can also be noted
by Figure 9 which compares the main cuts of the far field spectrum for both
the classical and the novel uniform sampling strategies.

##  VI Conclusions

In this paper, a new sampling scheme of the near field radiated by planar
source has been proposed and assessed. Such sampling scheme allows a reduction
of the testing time for all the planar antennas whose edges have a length in
the range $[3\lambda,25\lambda]$. In particular, for all antennas of such
dimensions, the proposed sampling scheme maintains the same performance of the
half-wavelength sampling with a smaller number of near field measurements.
Moreover, it employs a number of field samples slightly higher than non-
uniform sampling schemes by avoiding the issues of non-uniform measurement
strategies (not all the near field scanners are able to move the probe
according to a non-uniform step).

Finally, it is worth remarking that for antennas greater than $25\lambda$, the
proposed sampling scheme becomes similar to the half-wavelength sampling;
instead, for antennas smaller than $3\lambda$, the proposed uniform sampling
may still be used if a slight oversampling factor is introduced. However, the
quantification of such oversampling constant is out of the scope of this
article and it will be demanded in a future study.

Other future developments concern the extension of the proposed uniform
sampling to the case of amplitude-only measurements.

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|  Raffaele Moretta (S’ 2018 – M’ 2021) was born in Caserta (Italy) in March
1989. He received the Laurea (summa cum laude) in Electronic Engineering and
the Ph.D degree in Electronic and Computer Science engineering from the
University of Campania “Luigi Vanvitelli” in 2018 and 2021, respectively. At
the end of 2021, he joined the Electromagnetic Fields Group of the University
of Campania ”Luigi Vanvitelli” as Postdoctoral Researcher. After a short
experience in Avio as avionic system engineer, he moved to MBDA Italy where he
is now an antenna engineer. His current research interests include inverse
problems in electromagnetics with particular attention to phase retrieval,
near field measurement techniques, antenna diagnostics and synthesis. Dr.
Moretta is a member of the Institute of Electrical and Electronics Engineers
(IEEE) and of the Italian Society of Electromagnetism (SIEM). Moreover, in
February 2021 he was under consideration by the committee of the IEEE Antennas
Propagation Society for the R.W.P. King Award 2020.
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