This is the author’s version of an article that has been published in this
journal. Changes were made to this version by the publisher prior to
publication.
The final version of record is available at
http://dx.doi.org/10.1109/TAP.2020.3008648

# Multi-Frequency Phase Retrieval
for Antenna Measurements

Josef Knapp, Student Member, IEEE, Alexander Paulus, Student Member, IEEE,
Jonas Kornprobst,
Student Member, IEEE, Uwe Siart, Member, IEEE, and Thomas F. Eibert, Senior
Member, IEEE Manuscript received December 27, 2019; revised May 25, 2020;
accepted June 13, 2020; date of this version June 18, 2020. This work was
supported by the German Federal Ministry for Economic Affairs and Energy under
Grant 50RK1923. _(Corresponding author: Josef Knapp.)_ The authors are with
the Chair of High-Frequency Engineering, Department of Electrical and Computer
Engineering, Technical University of Munich, 80290 Munich, Germany (e-mail:
<EMAIL_ADDRESS>hft.ei@tum.de).Color versions of one or more of the
figures in this letter are available online at
http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAP.2020.3008648

###### Abstract

Phase retrieval problems in antenna measurements arise when a reference phase
cannot be provided to all measurement locations. Phase retrieval algorithms
require sufficiently many independent measurement samples of the radiated
fields to be successful. Larger amounts of independent data may improve the
reconstruction of the phase information from magnitude-only measurements. We
show how the knowledge of relative phases among the spectral components of a
modulated signal at the individual measurement locations may be employed to
reconstruct the relative phases between different measurement locations at all
frequencies. Projection matrices map the estimated phases onto the space of
fields possibly generated by equivalent antenna under test (AUT) sources at
all frequencies. In this way, the phase of the reconstructed solution is not
only restricted by the measurement samples at one frequency, but by the
samples at all frequencies simultaneously. The proposed method can increase
the amount of independent phase information even if all probes are located in
the far field (FF) of the AUT.

###### Index Terms:

Phase retrieval, phaseless antenna measurements, broadband receiver.

##  I Introduction

Time-harmonic near field (NF) antenna pattern measurements classically
comprise magnitude and phase measurements of the fields radiated by the
antenna under test (AUT) [1, 2, 3], where the phase information is important
for the calculation of desired far field (FF) quantities (gain, radiation
pattern, etc.) from the measured data. The phase measurement requires
synchronized transmit and receive signals of probe and AUT, e.g., by a
reference phase signal. Providing a stable phase reference to all different
probe locations may call for elaborate measurement setups at high frequencies
or for large AUTs [4], where unavoidable movement of the reference cable may
introduce severe phase errors. Certain measurement setups — e.g., with a probe
fixed to an unmanned aerial vehicle — may cause severe complications if the
phase reference has to be provided at all probe locations.

In order to avoid intricacies associated with phase measurements, and to
render some measurement setups feasible in the first place, it is desirable to
retrieve the phase information from magnitude-only field measurements, which
is a non-linear and non-convex problem in general — and hard to solve [5, 6,
7, 8, 9, 10, 11, 12]. Since it is attractive in a variety of applications, a
large number of algorithms and methods have been proposed to tackle this
problem of phase retrieval in magnitude-only antenna measurements [9, 5, 13,
14, 15, 16, 17, 18, 19, 20] or other cases [21, 22, 23, 24, 25, 26, 27, 12,
28, 29, 30, 31, 11, 10, 8, 7, 32, 33, 34].

In any case, it is understood that one needs to increase the number of
measurement samples to enable phase retrieval as compared to the number of
samples for conventional measurements with magnitude and phase [34, 35, 36,
37, 38, 39, 7, 9, 17, 18, 40]. Phase retrieval algorithms usually rely on
measurement data with sufficiently many independent measurement samples to
allow for a stable reconstruction process. With an increasing number of
independent measurements, the problem becomes more similar to a convex problem
such that even local minimization techniques can be used to find the true
solution [41, 40, 5, 42]. Once the number of independent measurement samples
reaches the square of the effective number of unknowns, the problem even can
be formulated in a linear manner [43, 18, 22, 21]. A still unsolved problem
is, however, to reliably find a measurement setup which provides sufficiently
many independent measurement samples [44].

Classical attempts employ measurements on two or more surfaces with various
distances to the AUT [28, 45, 29, 27, 19]. The idea is that the field
contributions of the AUT interfere differently at different distances as long
as they are in the NF of the AUT. The magnitudes convey information about
these coherent interferences. Since the interference patterns are strongly
affected by the phases of individual field contributions, the reconstruction
of the relative phases might be possible. It is still hard to predict where
the NF measurement locations have to be chosen in order to produce sufficient
dissimilar information about the field interferences.

Specially designed multi-antenna probes have been studied to obtain the
required data [14, 17, 46]. The different probes perform different coherent
linear combinations of the incident fields to form their output signals. Thus,
the magnitudes of the different probe signals heavily depend on the relative
phases of incident plane waves and encode the phase information of the
incident fields. The success of this method strongly depends on the
possibility of utilizing probes which measure different interference patterns
of the radiated AUT field contributions. To be effective, the probes have to
be either very large or rather close to the AUT in order to be able to produce
a useful weighted mean of field samples. For instance in the FF, the field is
practically constant over the profile of the probe and taking different linear
combinations of this field does not improve the situation. The amount of
different information which can be obtained in this way is also limited by the
number of different probes which can be used in the measurement setup. In the
FF, the task of providing large numbers of independent measurement samples
becomes even more difficult.

In this work, we increase the measurement diversity by introducing
relationships between the measured signals at different frequencies. No
absolute phase reference is required to obtain the relative phases between the
measurement samples at different frequencies with the same probe position.
Phase stability is only required over a short time span in which the
measurement samples are obtained for all the frequencies at the individual
measurement locations. Using the same local oscillator (LO) for all
frequencies at the receiver side (which is not required to be synchronized
with the transmit LO), it is possible to assign magnitudes and phases to all
signal samples at the different frequencies. The remaining inverse problem is
to find the phase differences between the measurement locations (i.e., the
phase of the receive LO at each position).

Phase retrieval for radiated or scattered fields at multiple frequencies was
already investigated occasionally [47, 48, 6] but these approaches retrieved
the phases independently at each frequency or utilized the solution for one
frequency as an initial guess at other frequencies. In contrast to this, we
employ the measurement samples at all frequencies simultaneously to find the
global phase solution. Preliminary investigations about this idea can be found
in [49]. The source reconstruction problem at one selected reference frequency
or a mixture of several reference frequencies is constrained by measured
signals at all frequencies. This constraint is implemented by projection
matrices which are designed to remove unphysical portions of the measured
signals. The measurements at all frequencies possibly provide additional
information about the relative phases of the radiated fields at the reference
frequency.

In Section II, the forward problem in antenna measurements is recapitulated
for the classical and the magnitude-only problems. A possible implementation
of a measurement setup which is able to find the phase differences between the
signals at different frequencies is described in Section III in the form of a
feasibility study. In Section IV, we show how the measurements at different
frequencies can be combined using projection matrices. These are used to check
whether a given phase solution for the reference frequency is also physically
reasonable at the other frequencies. Section V discusses a rule of thumb for a
frequency sampling step size leading to independent data. In Section VI, the
performance of the algorithm is investigated for synthetic and measured data.

##  II Notation and Problem Description

First, we establish the linear problem to clarify the notation. For every
angular frequency $\omega_{k}$, an equivalent current method modeling the AUT
by discretized sources [50, 51] (e.g., a collection of elementary dipoles) is
employed. The unknown source coefficients are contained in the vector
$\mathsfbfit{x}_{k}\in\mathbb{C}^{N\times 1}$. With increasing frequency,
usually more unknowns are utilized to represent the AUT accurately.

The measurement vector corresponding to the frequency $\omega_{k}$ is denoted
by $\mathsfbfit{b}_{k}$ and its $\ell$th entry $[\mathsfbfit{b}_{k}]_{\ell}$
denotes the measurement sample at the $\ell$th measurement location. One can
establish a linear relationship between the sources and the measurement
samples giving rise to the linear equation system [52]

$\displaystyle\mathsfbfit{b}_{k}=\mathsfbfit{A}_{k}\mathsfbfit{x}_{k}\,,$ (1)

with the system matrix $\mathsfbfit{A}_{k}\in\mathbb{C}^{M\times N}$.

In a magnitude-only measurement, the phase vector $\mathsfbfit{\phi}_{k}$ with
entries defined by

$\displaystyle\mathrm{e}^{\,\mathrm{j}[\mathsfbfit{\phi}_{k}]_{\ell}}\,=\dfrac{[\mathsfbfit{b}_{k}]_{\ell}}{\left\lvert[\mathsfbfit{b}_{k}]_{\ell}\right\rvert}$
(2)

is unknown for every frequency $\omega_{k}$ and must be determined in a non-
linear inverse problem. With the elementwise absolute value operator
$\left\lvert\cdot\right\rvert$, the phase retrieval problem at frequency
$\omega_{k}$ can be expressed as

$\displaystyle\text{find $\mathsfbfit{x}_{k}\in\mathbb{C}^{N\times 1}$ such
that}\left\lvert\mathsfbfit{b}_{k}\right\rvert=\left\lvert\mathsfbfit{A}_{k}\mathsfbfit{x}_{k}\right\rvert\,.$
(3)

In previous works, this phase retrieval problem has been tackled independently
for each frequency. The absolute phases had to be found for every frequency
component of the measured signals at all measurement positions separately. In
contrast to this, we assume that the phase differences between signal
components at different frequencies are known at each measurement position.
This corresponds to knowing the difference
$\mathsfbfit{\phi}_{k}-\mathsfbfit{\phi}_{i}$ of the phase vectors at
frequencies $\omega_{k}$ and $\omega_{i}$, respectively. This assumption
simplifies the problem as now only one phase value has to be determined at
every measurement position. This phase value determines the absolute phases
related to all frequencies within the measured signal at this location.

In the following, we show how this relative phase information can be obtained
and exploited for the solution of the phase retrieval problem.

##  III Measurement of Relative Phases of Signals with Different Frequencies

The following discussions are intended as a proof of concept to demonstrate
that multi-frequency antenna measurements are realistic and feasible. The aim
of this paper is to provide a valid algorithm for the corresponding phase-
retrieval problem. Efficient hardware implementations of measurement setups
are a topic left for future work.

For most antenna measurement scenarios, phase stability can be ensured up to
several tens of $\mathrm{G}\mathrm{H}\mathrm{z}$ with common technology. At
even higher frequencies, providing a stable phase reference to all measurement
locations or AUT positions becomes challenging and costly. This section
investigates how phase synchronization at a low frequency can suffice to
measure relative phases of signals at different frequencies consistently for
all measurement locations.

Figure 1: Signal path in a completely asynchronous receiver setup. The
receiver LO has a frequency shift $\Delta\omega$ and a phase shift
$\Delta\phi$ compared to the transmitter LO.

Consider the case of a totally asynchronous receiver first for which the
complete signal path is given in Fig. 1 —later we will use the low frequency
base band signal for synchronization. A certain band limited and periodic
input signal

$a_{\mathrm{BB}}\\!\left(t\right)=\sum\limits_{k=1}^{N_{\mathrm{f}}}\alpha_{k}\mathrm{e}^{\,\mathrm{j}\omega_{k}t}$
(4)

with periodicity $T$ consisting of a comb spectrum with a maximum frequency of
$\omega_{N\mathrm{f}}$ and spectral coefficients $\alpha_{k}\in\mathbb{C}$ is
upconverted to the carrier frequency $\omega_{\mathrm{c}}$. The resulting
radio frequency signal

$a_{\mathrm{RF}}\\!\left(t\right)=\sum\limits_{k=1}^{N_{\mathrm{f}}}\alpha_{k}\mathrm{e}^{\,\mathrm{j}\left(\omega_{k}+\omega_{\mathrm{c}}\right)t}$
(5)

is sent from the AUT to the probe antenna over the measurement channel with
transfer function $H(\omega)$. The receive signal

$b_{\mathrm{RF}}\\!\left(t\right)=\sum\limits_{k=1}^{N_{\mathrm{f}}}\alpha_{k}H\\!\left(\omega_{k}+\omega_{\mathrm{c}}\right)\mathrm{e}^{\,\mathrm{j}\left(\omega_{k}+\omega_{\mathrm{c}}\right)t}$
(6)

is first downconverted with an asynchronous receiver LO (frequency
$\tilde{\omega}_{\mathrm{c}}$) to

$\displaystyle b_{\mathrm{IF}}\\!\left(t\right)$
$\displaystyle=\sum\limits_{k=1}^{N_{\mathrm{f}}}\alpha_{k}H\\!\left(\omega_{k}+\omega_{\mathrm{c}}\right)\mathrm{e}^{\,\mathrm{j}\left(\omega_{k}+\omega_{\mathrm{c}}-\tilde{\omega}_{\mathrm{c}}\right)t}\mathrm{e}^{\,\mathrm{j}\Delta\phi}$
$\displaystyle=\sum\limits_{k=1}^{N_{\mathrm{f}}}\alpha_{k}H\\!\left(\omega_{k}+\omega_{\mathrm{c}}\right)\mathrm{e}^{\,\mathrm{j}\left(\omega_{k}+\Delta\omega\right)t}\mathrm{e}^{\,\mathrm{j}\Delta\phi}$
(7)

and then sampled in time domain.

Compared to the transmitter LO, the receiver LO may run at a slightly
different frequency
$\tilde{\omega}_{\mathrm{c}}=\omega_{\mathrm{c}}-\Delta\omega$ and have an
unknown phase shift $\Delta\phi$. Also, in an asynchronous setup, the exact
value for $t=0$ is not known at the receiver side. Fixing $t=0$ (i.e., the
starting time for the sampling interval) to an arbitrary value results in an
unknown time shift of $\Delta t$. The intermediate frequency (IF) signal is
thus written as

$b_{\mathrm{IF}}\\!\left(t+\Delta
t\right)=\sum\limits_{k=1}^{N_{\mathrm{f}}}\alpha_{k}H\\!\left(\omega_{k}+\omega_{\mathrm{c}}\right)\mathrm{e}^{\,\mathrm{j}\left(\omega_{k}+\Delta\omega\right)t}\mathrm{e}^{\,\mathrm{j}\Delta\phi}\mathrm{e}^{\,\mathrm{j}\left(\omega_{k}+\Delta\omega\right)\Delta
t}$ (8)

and contains all these uncertainties. Note that the highest frequency which
occurs in the signal $b_{\mathrm{IF}}$ is
$\omega_{N_{\mathrm{f}}}+\Delta\omega$. If the frequencies of the LOs for up-
and downconversion are about the same, $\Delta\omega$ is considerably smaller
than $\omega_{\mathrm{c}}$. In order to get rid of the undesired time shift
$\Delta t$, it suffices to synchronize the sampling-interval for
$b_{\mathrm{IF}}$ accurately enough for the frequency
$\omega_{N_{\mathrm{f}}}+\Delta\omega$. The periodic base band input signal
$a_{\mathrm{BB}}$ can be used for synchronization. No technically challenging
synchronization of receiver and transmitter around the carrier frequency
$\omega_{\mathrm{c}}$ is required.

In order to verify the procedure, measurements have been obtained with an
antenna configuration similar as the one in Section IV-C. Different channel
functions $H(\omega)$ were realized by moving the probe antenna to ten
different positions around the AUT as depicted in Fig. 2.

Figure 2: Schematic of the measurement positions for the phase difference
measurement verification.

In the considered measurement setup, the in-phase and quadrature components of
the base band signal $a_{\mathrm{BB}}$ consisting of 21 spectral components
with frequencies from $-9\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ to
$11\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ are generated and upconverted
($f_{\mathrm{c}}=$2.489\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$) internally
with a Rohde & Schwarz SMBV 100A signal generator [53]. The radio frequency
signal $a_{\mathrm{RF}}$ in the range from
$2.480\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ to
$2.500\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ is sent to the transmitting
antenna and downconverted with an external Mini-Circuits 4300 mixer [54] after
reception by the receive antenna. The LO signal for downconversion
($\tilde{f}_{\mathrm{c}}=$2.380\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$) is
generated by a separate Rohde & Schwarz SMR 40 signal generator [55]. No
synchronization signal is given to this device, i.e., it constitutes a freely
running LO. Finally, the downconverted signal $b_{\mathrm{IF}}$ together with
the in-phase and quadrature component of the base band input signal
$a_{\mathrm{BB}}$ are fed to a LeCroy WaveMaster 808Zi-A oscilloscope [56].
For each antenna position, single-shot measurements of the three signals from
different times (called “samples” in the remainder of this section) were taken
with the oscilloscope.

The magnitude of the base band transmit signal $a_{\mathrm{BB}}$ — which is
reconstructed from the measured in-phase and quadrature components — is used
for the synchronization of the measured signals in the post-processing.

The procedure was carried out for each antenna at three different times (one
sample was taken at a different day than the others) after all devices have
been switched off and on again to ensure that no long-term drifting problems
occur. After synchronization, the spectral components of all signals were
obtained with a fast Fourier transform (FFT). The absolute phases of all
reconstructed spectral components for the three different measurements differ
by a phase $\Delta\phi$ corresponding to the random phase of the receiver LO.
However, the relative phases with respect to the reference frequency at
$100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ show a good agreement. Figure 3
shows the reconstructed relative phases for an exemplary measurement at
antenna position 1.

Figure 3: Reconstructed phases dependent on frequency in the IF band at
position 1, relative to the signal phase for the first frequency.

The overall agreement between the measurements at different times is better
than $5^{\circ}$. The deviations between the oscilloscope measurements and the
reference measurements from a vector signal generator (of course taken at the
transmit frequencies $\omega_{\mathrm{c}}+\omega_{k}$) reveal a systematic
deviation due to additional cables which were added to connect the mixer and
so on.

This systematic deviation does not matter for our phase reconstruction
algorithm, as a frequency dependent phase offset which is consistently added
to all measurement samples does not hurt, since it cancels out during the
reconstruction. Important is only that the frequency dependent phase bias is
the same for all measurement samples such that the phases for all frequencies
can be reconstructed up to a global phase bias, if the phase is known at one
of the frequencies.

In order to show that the phase bias is constant for all measurement
positions, Fig. 4 shows the phase errors for every position, after the phase
bias from reference position 6 was subtracted. The blue line denotes the mean
of the phase error over all frequencies and the error bar shows the measured
standard deviation. The maximum/minimum values of the phase offset are
depicted by the solid and dashed orange lines, respectively. For all
measurement positions, except position 5, the standard deviation of the
reconstructed phase is well below $5^{\circ}$. At position 5, the channel
function $H(\omega)$ is at a minimum as it is located at a null of the AUT
pattern.

The systematic offset for every measurement frequency (the noisy samples from
position 5 were removed as they would obscure the data) is shown in Fig. 5.
The continuous increase of the mean deviation suggests that the
synchronization can be further improved.

In conclusion, it was shown that the relative phase difference
$\phi_{k}-\phi_{i}$ of the signal samples at two frequencies $\omega_{k}$ and
$\omega_{i}$ of the transfer function between two antennas at high frequencies
can be measured in principle for each measurement location without providing a
phase reference at the radio frequency.

Important parameters and requirements for the implementation of a reliable
system to measure the relative phases between spectral components of a signal
are:

  * •

the bandwidth of the base band signal $a_{\mathrm{BB}}\\!\left(t\right)$.
Larger bandwidths are preferred for the phase retrieval algorithm described in
the sections below. However, the phase error between the spectral components
due to synchronization errors $\Delta t$ depends linearly on the signal
bandwidth, i.e., $\Delta\phi\propto\Delta\omega\,\Delta t$.

  * •

the shape of the base band signal $a_{\mathrm{BB}}\\!\left(t\right)$. In
principle, any periodic signal with a certain bandwidth can be used, but it
helps to have characteristic features in the time domain representation of the
signal (such as a steep slope) as well as in the frequency domain
representation of the signal (such as discrete frequencies). Such features are
beneficial for diminishing synchronization errors $\Delta t$ and frequency
offsets of the downconverting mixer at the receiver.

  * •

the periodicity $T$ of the base band signal
$a_{\mathrm{BB}}\\!\left(t\right)$. The same signal must be measured at
different measurement locations. Averaging over several periods reduces the
impact of noise. In a frequency comb spectrum, the frequency step between
adjacent frequencies determines the periodicity of the signal.

  * •

the IF frequency range of the downconverted receive signal
$b_{\mathrm{IF}}\\!\left(t\right)$ and the corresponding sampling frequency
should be chosen to support the evaluation of the receive signal with low
noise and interference as well as with good accuracy.

Figure 4: Phase errors of the relative phases at each measurement location.
Figure 5: Phase errors for the relative phases of the measurement frequencies
in the IF-band.

##  IV The Multi-Frequency Phase Retrieval Problem

The problem to solve is to find the unknown phase terms $\Delta\phi$ for each
measurement position, such that a physically reasonable measurement vector
$\mathsfbfit{b}_{k}$ is attained for every frequency $\omega_{k}$, i.e., a
measurement vector which could be generated by the given set of AUT sources.

At a given frequency $\omega_{k}$, one can perform a projection of any
(unphysical) right-hand side vector $\mathsfbfit{b}_{k}$ into the space of
physically reasonable measurement vectors by multiplying it with the
projection matrix
$\mathsfbfit{P}_{k}=\mathsfbfit{A}_{k}\mathsfbfit{A}_{k}^{\dagger}$ generated
with the help of a pseudo-inverse $\mathsfbfit{A}_{k}^{\dagger}$. To be
physically reasonable, the vector $\mathsfbfit{b}_{k}$ must be insensitive to
the projection matrix $\mathsfbfit{P}_{k}$, i.e.,
$\mathsfbfit{b}_{k}=\mathsfbfit{P}_{k}\mathsfbfit{b}_{k}$. A physically
reasonable vector is orthogonal to the null-space of the projection matrix. If
we assign a false phase to the measured magnitudes, with a high chance, there
will be some portion of the reconstructed vector $\mathsfbfit{b}_{k}$ which
lies in the null-space of the projection matrix and is, thus, removed. The
projected vector
$\tilde{\mathsfbfit{b}_{k}}=\mathsfbfit{P}_{k}\mathsfbfit{b}_{k}$ is the
closest vector to $\mathsfbfit{b}_{k}$ in the subspace generated by the
discretized AUT sources in $\mathsfbfit{x}_{k}$. The dimension of the
projection matrix $\mathsfbfit{P}_{k}\in\mathbb{C}^{M\times M}$ is independent
from the number of unknowns $N$ in the AUT model and depends only on the
number of measurement samples $M$. Of course, the projection matrices and in
particular the required pseudo-matrices are not necessarily constructed
explicitly. Instead one may use fast iterative methods to approximate the
matrix vector products involving the pseudo-inverses or the corresponding
projection matrices.

In the following, a phase retrieval formulation is presented first, which is
numerically unstable if measurement values with small magnitudes occur. This
equation allows to present the main idea in its simplest form. Subsequently,
we will show measures to eradicate the instabilities.

###  IV-A Basic Formulation

With the knowledge of the measured magnitudes and the relative phases, the
measurement vector at frequency $\omega_{k}$ can be expressed by the
measurement vector at frequency $\omega_{i}$ by

$\displaystyle\mathsfbfit{b}_{k}=\mathsfbfit{U}_{k,i}\mathsfbfit{b}_{i}\,,$
(9)

where $\mathsfbfit{U}_{k,i}$ is a diagonal matrix with elements

$\displaystyle\left[\mathsfbfit{U}_{k,i}\right]_{\ell\ell}=\dfrac{\left[\mathsfbfit{b}_{k}\right]_{\ell}}{\left[\mathsfbfit{b}_{i}\right]_{\ell}}=\dfrac{\left\lvert\left[\mathsfbfit{b}_{k}\right]_{\ell}\right\rvert}{\left\lvert\left[\mathsfbfit{b}_{i}\right]_{\ell}\right\rvert}\,\mathrm{e}^{\,\mathrm{j}\left(\left[\mathsfbfit{\phi}_{k}\right]_{\ell}-\left[\mathsfbfit{\phi}_{i}\right]_{\ell}\right)}\,.$
(10)

The phase retrieval problems at different frequencies are connected to each
other and can be solved simultaneously. A certain solution
$\mathsfbfit{b}_{i}$ at one frequency $\omega_{i}$ also solves the phase
retrieval problem at all other frequencies if the corresponding measurement
vectors $\mathsfbfit{b}_{k}=\mathsfbfit{U}_{k,i}\mathsfbfit{b}_{i}$ are
physically reasonable for every frequency $\omega_{k}$ — at least if the
inter-frequency mapping $\mathsfbfit{U}_{k,i}$ is perfect.

Choosing an arbitrary reference frequency with index $i$, the multi-frequency
phase retrieval problem is formulated as

$\displaystyle\text{find }\mathsfbfit{x}_{i}\in\mathbb{C}^{N\times 1}\text{
such that }\begin{pmatrix}\left\lvert\mathsfbfit{b}_{1}\right\rvert\\\
\vdots\\\ \left\lvert\mathsfbfit{b}_{i}\right\rvert\\\ \vdots\\\
\left\lvert\mathsfbfit{b}_{N_{\mathrm{f}}}\right\rvert\end{pmatrix}$
$\displaystyle=\left\lvert\,\begin{pmatrix}\mathsfbfit{P}_{1}\,\mathsfbfit{U}_{1,i}\,\mathsfbfit{A}_{i}\\\
\vdots\\\ \mathsfbfit{A}_{i}\\\ \vdots\\\
\mathsfbfit{P}_{N_{\mathrm{f}}}\,\mathsfbfit{U}_{{N_{\mathrm{f}}},i}\,\mathsfbfit{A}_{i}\end{pmatrix}\mathsfbfit{x}_{i}\,\right\rvert$
$\displaystyle=\left\lvert\,\tilde{\mathsfbfit{A}}\,\mathsfbfit{x}_{i}\,\right\rvert\,.$
(11)

If we consider a single measurement frequency only, we impose $M$ (possibly
linearly dependent) restrictions on the source vector $\mathsfbfit{x}_{i}$.
Using all $N_{\mathrm{f}}$ measurement frequencies at once, one can increase
the number of restrictions on the source vector by a factor of
$N_{\mathrm{f}}$ and it is clear that phase retrieval algorithms can be
successful with larger probability.

The influence of the projection matrices $\mathsfbfit{P}_{k}$ depends on the
number of degrees of freedom (DoFs) $N_{\mathrm{dof}}$ of the radiated fields
and the number of measurement samples. The projection matrix is designed to
filter out unphysical contributions in the vector ${\mathsfbfit{b}_{k}}$ which
may be introduced by assigning false phases to the measured magnitudes. Only
if the number of measurement samples is sufficiently large, the projection
matrix will have a null-space which allows to separate unphysical solutions.
With insufficient measurements, any vector
$\mathsfbfit{b}\in\mathbb{C}^{M\times 1}$ can be generated by the sources and
appears to be physical ($\mathsfbfit{P}_{k}$ is the identity matrix in this
case). The number of measurement samples should, therefore, exceed the number
of DoFs, $N_{\mathrm{dof}}$, considerably (e.g., by a factor of 2 or 4) at the
highest measured frequency to ensure that unphysical solutions can be detected
at all frequencies.

###  IV-B Numerically Improved Formulation

The problem of a division by zero in (11) can be treated by scaling the
measurement samples at the frequency $\omega_{k}$ with the magnitudes of the
signal at the reference frequency $\omega_{i}$. The diagonal matrix

$\displaystyle\mathsfbfit{B}=\mathrm{diag}\\!\left(\left\lvert\mathsfbfit{b}_{i}\right\rvert\right)$
(12)

contains the magnitudes of the measurement vector $\mathsfbfit{b}_{i}$ at the
reference frequency $\omega_{i}$. The linear relationship between the sources
$\mathsfbfit{x}_{k}$ and $\mathsfbfit{B}\mathsfbfit{b}_{k}$ (i.e.,the scaled
version of the measurement samples at frequency $\omega_{k}$) is given by the
scaled matrix $\mathsfbfit{B}\mathsfbfit{A}_{k}$. The relationship between the
scaled measurement vector at frequency $\omega_{k}$ and the measurement vector
at the reference frequency $\omega_{i}$ is given by the new diagonal matrix

$\displaystyle\tilde{\mathsfbfit{U}}_{k,i}=\mathsfbfit{B}\mathsfbfit{U}_{k,i}$
(13)

and its elements can be found by

$\displaystyle\left[\tilde{\mathsfbfit{U}}_{k,i}\right]_{\ell\ell}=\left\lvert\left[\mathsfbfit{b}_{k}\right]_{\ell}\right\rvert\mathrm{e}^{\,\mathrm{j}\left(\left[\mathsfbfit{\phi}_{k}\right]_{\ell}-\left[\mathsfbfit{\phi}_{i}\right]_{\ell}\right)}\,.$
(14)

A modified projection matrix
$\tilde{\mathsfbfit{P}}_{k}=\mathsfbfit{A}_{k}\left(\mathsfbfit{B}\mathsfbfit{A}_{k}\right)^{\dagger}$
must now account for the magnitude scaling of the measurement vector entries
which is achieved by taking the pseudo inverse of the scaled forward matrix
$\mathsfbfit{B}\mathsfbfit{A}_{k}$. Finally, the stabilized version of the
multi-frequency phase retrieval problem reads

$\displaystyle\text{find }\mathsfbfit{x}_{i}\in\mathbb{C}^{N\times 1}\text{
such that }\begin{pmatrix}\left\lvert\mathsfbfit{b}_{1}\right\rvert\\\
\vdots\\\ \left\lvert\mathsfbfit{b}_{i}\right\rvert\\\ \vdots\\\
\left\lvert\mathsfbfit{b}_{N_{\mathrm{f}}}\right\rvert\end{pmatrix}$
$\displaystyle=\left\lvert\,\begin{pmatrix}\tilde{\mathsfbfit{P}}_{1}\,\tilde{\mathsfbfit{U}}_{1,i}\,\mathsfbfit{A}_{i}\\\
\vdots\\\ \mathsfbfit{A}_{i}\\\ \vdots\\\
\tilde{\mathsfbfit{P}}_{N_{\mathrm{f}}}\tilde{\mathsfbfit{U}}_{{{N_{\mathrm{f}}}},i}\,\mathsfbfit{A}_{i}\end{pmatrix}\mathsfbfit{x}_{i}\,\right\rvert$
$\displaystyle=\left\lvert\,\tilde{\mathsfbfit{A}}\,\mathsfbfit{x}_{i}\,\right\rvert\,.$
(15)

No direct division by small values occurs in this formulation. The division is
de facto transferred to the modified pseudo inverse which may be able to
retrieve the stability of the problem by regularization properties.

Both problem formulations in (11) and (15) have the form of a standard phase
retrieval problem and can be tackled with established algorithms. In the
following sections, we demonstrate that utilizing measurements at different
frequencies provides a reliable method for generating large numbers of
independent measurement samples for the phase retrieval problem such that a
stable phase retrieval should be possible with any reconstruction method.
Furthermore, the usage of different frequencies allows to generate independent
measurement samples even in the FF, where other methods fail to generate new
independent samples, because no new information is obtained by using different
probes or changing the distance to the AUT.

###  IV-C Formulation Without Specific Reference Frequency

Finally, if the measured signal at the reference frequency $f_{i}$ has a low
magnitude for certain measurement samples, the measured reference phase may
have an arbitrary error. Although we did not see this effect to affect our
measured or simulated data, an alternative formulation of the minimization
problem in terms of measurement sample unknowns instead of AUT source unknowns
shall be given in this section for completeness.

For every measurement location, we can choose one of the signal samples at a
certain frequency as our unknown for this location. The remaining signal
samples at this measurement location are then already determined because the
ratio
$\left[\mathsfbfit{b}_{k}\right]_{\ell}/\left[\mathsfbfit{b}_{i}\right]_{\ell}$
is known for all frequency pairs. If the signal magnitude is very low for all
frequencies, the reconstructed phase is irrelevant, as the measured value is
approximately zero for all frequencies. In all other cases, we can pick the
most convenient frequency sample at each location to form our vector of
unknowns and ensure a physically reasonable measurement vector by using the
projection matrices. We have

$\displaystyle\text{find }\hat{\mathsfbfit{b}}\in\mathbb{C}^{N\times 1}\text{
such that }\begin{pmatrix}\left\lvert\mathsfbfit{b}_{1}\right\rvert\\\
\vdots\\\ \left\lvert\mathsfbfit{b}_{i}\right\rvert\\\ \vdots\\\
\left\lvert\mathsfbfit{b}_{N_{\mathrm{f}}}\right\rvert\end{pmatrix}$
$\displaystyle=\left\lvert\,\begin{pmatrix}\tilde{\mathsfbfit{P}}_{1}\,\hat{\mathsfbfit{U}}_{1}\\\
\vdots\\\ \tilde{\mathsfbfit{P}}_{i}\,\hat{\mathsfbfit{U}}_{i}\\\ \vdots\\\
\tilde{\mathsfbfit{P}}_{N_{\mathrm{f}}}\hat{\mathsfbfit{U}}_{{{N_{\mathrm{f}}}}}\end{pmatrix}\hat{\mathsfbfit{b}}\,\right\rvert$
$\displaystyle=\left\lvert\,\hat{\mathsfbfit{A}}\,\hat{\mathsfbfit{b}}\,\right\rvert\,$
(16)

where $\hat{\mathsfbfit{b}}$ is the vector of unknown measurement samples at
the chosen frequencies for each measurement location and
$\hat{\mathsfbfit{U}}_{i}$ is analogous to (14), but with a variable reference
frequency instead of a fixed reference frequency for every measurement
location.

The magnitudes of all measurement samples in all the considered examples were
well above any values which could be problematic for a numerical evaluation.
Therefore, we did not see differences between the formulations with and
without a fixed reference frequency. Since the newly included formulation
introduces more unknowns in the reconstruction process, the originally
proposed version remains the preferable choice and was used in the paper.

##  V Choosing the Frequency Step Size

Our goal in this section is to find a rule of thumb for a reasonable frequency
sample step size. The received signal at a measurement position varies over
frequency due to two different possible reasons for the change. Either the
current distribution on the AUT or the operator $\mathsfbfit{A}_{k}$ changes.
A prediction of the frequency behavior of the AUT current distribution is
difficult, hence we investigate the frequency behavior of the forward operator
$\mathsfbfit{A}$111When the current distribution changes, the proposed method
will always generate independent data. Thus, a detailed investigation of
changing currents is superfluous..

To this end, consider a simple example of $N$ $z$-oriented Hertzian dipoles
placed on a ring with radius $d$ lying in the $xy$-plane as depicted in Fig.
6. These Hertzian dipoles represent a generic AUT. The field is sampled at a
distance $r_{0}$ from the center of the dipole ring.

Figure 6: Setup to find a reasonable frequency step width.

For the linear relationship between the sources and the $z$-component of the
electric field at the observation location, we can write

$E_{z}\\!\left(\omega\right)=\mathsfbfit{a}^{\mathrm{H}}\\!\left(\omega\right)\mathsfbfit{z}\\!\left(\omega\right)\,.$
(17)

For each frequency $\omega$, one obtains a row vector
$\mathsfbfit{a}^{\mathrm{H}}\\!\left(\omega\right)$ describing the
relationship between the sources and the observation point.

To provide new information, two row vectors
$\mathsfbfit{a}^{\mathrm{H}}\\!\left(\omega_{1}\right)$ and
$\mathsfbfit{a}^{\mathrm{H}}\\!\left(\omega_{2}\right)$ at different
frequencies should not be too similar. As a similarity metric we can use the
singular value decomposition (SVD) of the matrix

$\mathsfbfit{A}=\begin{pmatrix}\dfrac{1}{\omega_{1}}\mathsfbfit{a}^{\mathrm{H}}\\!\left(\omega_{1}\right)\\\\[10.00002pt]
\dfrac{1}{\omega_{2}}\mathsfbfit{a}^{\mathrm{H}}\\!\left(\omega_{2}\right)\end{pmatrix}\,.$
(18)

The scaling with the inverse of the frequencies ensures, that the elementary
dipole fields maintain the same field magnitude at different frequencies and
that the two vectors $\mathsfbfit{a}^{\mathrm{H}}\\!\left(\omega_{1}\right)$
and $\mathsfbfit{a}^{\mathrm{H}}\\!\left(\omega_{2}\right)$ have their mean
values on the same order of magnitude. If the two row vectors are similar, the
ratio $\left\lvert\sigma_{2}/\sigma_{1}\right\rvert$ of the two singular
values will be small. If the two vectors are independent, then the ratio will
approach unity.

Fig. 7 shows the singular value ratio for two different radii
$d=$0.18\text{\,}\mathrm{m}$$ and $d=$1.38\text{\,}\mathrm{m}$$ of the AUT
circle on which $N=1000$ dipoles were distributed evenly. The observation
distance is $r_{0}=$2.1\text{\,}\mathrm{m}$$ and the frequency $\omega_{1}$ is
fixed at $\omega_{1}=2\uppi\,$1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$.

Figure 7: Singular value ratio for two different AUT sizes dependent on the
frequencies of the forward operators with
$\omega_{1}=2\uppi\,$1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$.

Clearly, for larger antennas a smaller frequency step size is sufficient to
ensure that the observation at the two frequencies will bring new information.
Overall, as soon as the AUT size exceeds the wavelength associated with the
frequency step $\Delta\omega=\omega_{2}-\omega_{1}$ by some amount, we can
expect that the forward operators for the two frequencies will show enough
variation even if the current distribution on the AUT does not vary over the
different frequencies.

Therefore, to maximize the information we can obtain from measurements in a
certain bandwidth, the frequency step $\Delta\omega$ should not be larger than
$\Delta\omega_{\mathrm{max}}=c_{0}/\left(2d\right)$, where $d$ denotes the
radius of the minimum sphere enclosing the AUT and $c_{0}$ is the speed of
light. If the frequency step size becomes much smaller than this value,
however, many of the frequency samples will be redundant.

##  VI Results

###  VI-A Investigation on the Number of Independent Samples

For a first impression of the usefulness of the proposed multi-frequency
formulation, the suitability of the operator $\tilde{\mathsfbfit{A}}$ from
(15) for the task of phase retrieval is evaluated. Based on the technique
described in [18], one can count, how many “independent” magnitude samples are
possible with a given operator $\mathsfbfit{A}$ by applying an SVD of the
matrix

$\mathsfbfit{Q}=\left\lvert\mathsfbfit{A\
\mathsfbfit{A}^{\mathrm{H}}}\right\rvert^{\circ 2}\,,$ (19)

where $\mathsfbfit{A}^{\circ 2}$ squares each element of $\mathsfbfit{A}$. The
number of independent magnitude samples is then equal to the number of
singular values with a magnitude larger than a given threshold [18]. Larger
values of this quantity correspond to an increased chance of successfully
solving (3).

For the purpose of comparison, we generated different linear operators
$\mathsfbfit{A}\in\mathbb{C}^{M\times N}$ for these unknowns. The reference
case consists of a matrix $\mathsfbfit{A}_{\mathrm{Gaussian}}$, with i.i.d.
Gaussian distributed elements. Such a linear operator states the ideal case
for a measurement matrix as each new measurement sample (i.e., each new row in
$\mathsfbfit{A}_{\mathrm{Gaussian}}$) provides independent information. Once
$N^{2}$ independent measurement samples have been reached, the phase retrieval
problem becomes linear and every additional measurement sample will be
(linearly) dependent on the previous samples.

Practical measurement matrices $\mathsfbfit{A}$ are usually not Gaussian
distributed. To investigate the behavior of measurement matrices in more
realistic measurement scenarios, field measurements on either one sphere
(radius $r_{1}=5\lambda$) or two spheres (radii $r_{1}=5\lambda$,
$r_{2}=10\lambda$) are considered. The measurement locations are distributed
on a Fibonacci spiral [57] on the measurement spheres, respectively. This
ensures that the $M$ measurement locations are relatively evenly distributed
on the surfaces for varying values of $M$. Two measurement samples are taken
at every measurement location, corresponding to the two orthogonal
polarizations.

A synthetic spherical measurement setup is considered. The AUT is modeled by
$N=300$ randomly excited Hertzian dipoles distributed according to a Fibonacci
mapping [57] tangentially on the surface of a sphere with diameter of
$2\lambda$. The random excitation of the dipoles are the same for all
frequencies, such that variations in the matrix $\tilde{\mathsfbfit{A}}$
solely come from variations of the forward operator at different frequencies.

On the single measurement sphere, we also assume a probe array similar to
[17], which can perform (complex valued) linear combinations of two separated
field values before the magnitude of the signal is taken. The separation
between the two field samples was set to $2\lambda$. The special probe
antennas yield $8$ measurements, consisting of four linear combinations for
each of the combinations of two polarizations between the two measurement
locations of the array, where the displacement of the array elements can be
either along the $\vartheta$\- or the $\varphi$-direction. From the magnitudes
of these linear combinations, local phase information in the form of phase
differences can be obtained [17, 18, 16, 47].

Finally, the measurement matrix $\tilde{\mathsfbfit{A}}$ is generated
according to (15) from the measurement matrices $\mathsfbfit{A}_{k}$ at
multiple frequencies for measurement samples on the single measurement sphere
with radius $r_{1}$. For the sake of a fair comparison of the measurement
setups, the number of rows, $M$, in the resulting forward operators is the
same for each approach. This means that fewer measurement locations are
considered for the probe array and multi-frequency setups such that the number
$M$ of actual measurement samples is the same for each setup. The results can
be seen in Fig. 8, where the number of frequencies utilized in the multi-
frequency formulation was set to $N_{\mathrm{f}}=2$ and $N_{\mathrm{f}}=8$,
respectively. For each case, two bandwidths were considered: a small bandwidth
of $100\text{\,}\mathrm{MHz}$ between $3.0\text{\,}\mathrm{GHz}$ and
$3.1\text{\,}\mathrm{GHz}$, and a large bandwidth of $4\text{\,}\mathrm{GHz}$
between $3.0\text{\,}\mathrm{GHz}$ and $7.0\text{\,}\mathrm{GHz}$.

Figure 8: Number of “independent” magnitude samples for different linear
operators corresponding to various NF measurement setups.

For different numbers of samples $M$, we count the number of singular values
$\sigma_{i}$ which have a ratio $\sigma_{i}/\sigma_{1}\geq 10^{-5}$ to the
largest singular value. As a matter of fact, the number of independent samples
is limited to $N^{2}=9000$. While the Gaussian matrix shows an ideal behavior,
where the number of independent samples rises linearly with every additional
sample until the upper limit is reached at $M=9000$, the curves for the more
realistic cases saturate on certain levels. The probe array and the multi-
frequency formulation with the larger bandwidth may yield the best results for
large oversampling in the considered scenario. This suggests that more
valuable information can be collected by using specialized probe arrays or the
broadband information on a single sphere, as compared to measurements on two
spheres with the same probe at the same frequency.

The multi-frequency formulation with the small bandwidth stagnates at a level
only slightly larger than the measurements at one frequency. The measurement
matrices at neighboring frequencies do not differ much from each other and the
rows of the combined operator $\tilde{\mathsfbfit{A}}$ are correlated.

If the number of measurement locations is too small, the co-kernel of the
matrix $\mathsfbfit{A}$ vanishes. The projection matrices
$\mathsfbfit{P}_{k}=\mathsfbfit{A}_{k}\mathsfbfit{A}_{k}^{\dagger}$ collapse
to the unit matrix and have no influence. Once the number of measurement
locations exceeds $N_{\mathrm{dof}}$ for the antenna model at the
corresponding frequency, the projection matrices become effective.

The measurement samples at different frequencies only have an influence, if
enough spatial sampling locations are considered. When the spatial sampling
exceeds the critical value, all of a sudden, the filtering properties of the
projection matrices have an effect and the measured signal samples at the
different frequencies convey independent information. Therefore, the number of
independent samples for the $N_{\mathrm{f}}=8$ cases grow very slowly at first
(only about every eighth sample is useful) before we have a drastic increase
of independent samples (slope greater than one) once the critical sampling
rates for the antenna model at the respective frequencies are reached.

Figure 9: Number of “independent” magnitude samples for different linear
operators corresponding to various FF measurement setups.

In Fig. 9, the same analysis is performed in the FF ($r_{1}=$5000$\lambda$,
$r_{2}=$10\,000$\lambda$). It can be seen that measurements at two distances
or with special probes are not able to generate more independent measurements
than a simple probe at a single FF distance. Only the multi-frequency
measurements _are_ indeed able to increase the number of independent samples
to a level which is comparable to the NF measurements.

###  VI-B Field Transformation of a Simulated Biconical Antenna

Figure 10: Simulated biconical dipole antenna.

In this example, we consider a biconical antenna as depicted in Fig. 10. The
biconical antenna with a minimum sphere diameter of around
$0.33\text{\,}\mathrm{m}$ was simulated with Feko [58] in a frequency range
from $2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ to
$4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. The NF was sampled on three
spherical surfaces, with radii $r_{1}=$0.9\text{\,}\mathrm{m}$$,
$r_{2}=$1.2\text{\,}\mathrm{m}$$ and $r_{3}=$1.5\text{\,}\mathrm{m}$$. On
every surface the two tangential field components were obtained at $8000$
locations.

For the FF reconstruction, the equivalent source model consists of $3001$
Hertzian dipoles distributed on a hull surface enclosing the AUT. The
excitation coefficients of the Hertzian dipoles were obtained by solving (15)
in a Wirtinger-Flow-like minimization procedure [17, 34]. After the source
reconstruction, the corresponding FF is obtained from the reconstructed
sources.

Figs. 11 to 14 show the reconstructed FFs at
$2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ and
$4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ using either a single or eleven
different frequencies in $\tilde{\mathsfbfit{A}}$, respectively. All FFs are
normalized to their respective maximum. The error curve is given by

$\epsilon_{\mathrm{dB}}\\!\left(\vartheta,\varphi\right)=20\log_{10}\\!\left(\left\lvert\dfrac{\left\lvert
E_{\vartheta}\\!\left(\vartheta,\varphi\right)\right\rvert}{E_{\vartheta,\mathrm{max}}}-\dfrac{\left\lvert
E_{\vartheta,\mathrm{ref}}\\!\left(\vartheta,\varphi\right)\right\rvert}{E_{\vartheta,\mathrm{max},\mathrm{ref}}}\right\rvert\right)\,.$
(20)

Adding more frequency samples helps to avoid getting stuck in local minima
during the solution of (15). Using only a single frequency at a time is not
enough to reconstruct a satisfying solution. Adding additional restrictions in
form of measurements at additional frequencies leads to an accurate FF
reconstruction. Naturally, since the relative phases between signal samples at
the different frequencies at every measurement location are known, one obtains
the correct solution at all frequencies, as soon as the correct solution has
been found for one frequency.

Figure 11: Reconstructed $\varphi=0^{\circ}$ FF-cut of the
$\vartheta$-component at $2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ using
three measurement distances and a single frequency. Figure 12: Reconstructed
$\varphi=0^{\circ}$ FF-cut of the $\vartheta$-component at
$2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ using three measurement distances
and eleven distinct frequencies. Figure 13: Reconstructed $\varphi=0^{\circ}$
FF-cut of the $\vartheta$-component at
$4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ using three measurement distances
and a single frequency. Figure 14: Reconstructed $\varphi=0^{\circ}$ FF-cut of
the $\vartheta$-component at $4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ using
three measurement distances and eleven frequencies.

###  VI-C Field Transformation of Horn Antenna Measurements

####  VI-C1 The Measurement Setup

Figure 15: Measurement setup with DRH18 probe and the horn AUT.

NF measurements were obtained in the anechoic antenna measurement chamber at
the Technical University of Munich, demonstrating the feasibility of the
approach for realistic noisy data. The radiated field of a horn antenna with
minimum sphere radius of about $0.42\text{\,}\mathrm{m}$ was sampled with a
DRH18 probe [59] on four spherical surfaces with radii
$r_{1}=$2.512\text{\,}\mathrm{m}$$, $r_{2}=$2.642\text{\,}\mathrm{m}$$,
$r_{3}=$2.892\text{\,}\mathrm{m}$$, and $r_{4}=$3.092\text{\,}\mathrm{m}$$ at
a large number of frequencies in $25$ MHz steps. On each of the measurement
surfaces, $130\,682$ measurement samples were obtained coherently with
magnitude and phase on a regular $1\text{\,}\mathrm{\SIUnitSymbolDegree}$-grid
— this represents a strong oversampling of the field, $N_{\mathrm{dof}}$ is on
the order of several thousands for an antenna of this size at around
$3\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. Fig. 15 shows the AUT horn antenna
and the DRH18 probe antenna mounted in the anechoic chamber.

From the large number of measurement samples, $20\,000$ samples were picked to
reduce the computational complexity. A reference FF from the measured NF data
was calculated with a conventional NF to FF transformation from the coherent
data. The phase information was then removed from the NF data and knowledge of
phase differences between the frequency samples was introduced in the form of
(15). $5000$ Hertzian dipoles placed on a conformal hull around the AUT served
as the AUT model for the transformations.

The single frequency phase-retrieval problem is solved for each individual
frequency with a non-convex solver [17], where an initial guess according to a
spectral method [34] for the matrix $\mathsfbfit{A}_{k}$ is employed. The
procedure for the multi-frequency transformations is as follows. First, we
solve the single frequency phaseless problem for the reference frequency as
mentioned above. This solution $\mathsfbfit{x}_{i}$ is taken as an initial
guess for the multi-frequency phase retrieval problem. The solution of the
multi-frequency problem with the non-convex solver gives an estimate of the
phases of the measurement vectors at each frequency. Then, we obtain complex
measurement vectors by combining the estimated phases with the measured
magnitude vectors. We retrieve the equivalent currents at each frequency by
solving a standard complex problem.

####  VI-C2 The Problem of Local Minima

The initial guess has a large impact on the solution of the phase-retrieval
problem. A severe problem of non-convex phase retrieval techniques (i.e., the
minimization of non-linear non-convex cost functions) are local minima. With
ideal data, we can try to judge whether we are stuck in a local minimum — only
the hopefully unique global minimum exhibits a cost function value of zero. In
measurement scenarios, the problem is worse: The measurement error gives a
limit to the reconstruction deviation, and local minima with cost function
values below the error floor become indistinguishable from the true solution.
This is observed in Fig. 16 for Fibonacci spiral sampling and in Fig. 17 for
randomly picked samples from a regular grid with the pole in the main-beam
direction.

Figure 16: Comparison of magnitude-only and complex NF deviations for single-
frequency measurements on a Fibonacci spiral and four distances. The star
marks the initial guess by the spectral method, the other initial guesses are
random. Figure 17: Comparison of magnitude-only and complex NF deviations for
single-frequency measurements on a regular grid and four distances. The star
marks the initial guess by the spectral method, the other initial guesses are
random.

For $100$ random right-hand sides, we investigate the solver behavior at
$2.8$, $3.0$, and $3.2$ GHz and observe that there is no way to know whether a
lower magnitude-only deviation
$\big{\lVert}\lvert\mathsfbfit{b}\rvert-\lvert\tilde{\mathsfbfit{A}}\,\mathsfbfit{x}\rvert\big{\rVert}_{2}$
correlates with a better true complex NF deviation
$\lVert\mathsfbfit{b}-\tilde{\mathsfbfit{A}}\,\mathsfbfit{x}\rVert_{2}$. With
the good initial guess provided by the mentioned spectral method (marked with
a star), the magnitude-only solver is also stuck in a local minimum. Many
(possibly good or bad) solutions with similar magnitude-only NF deviation
exist. It is not possible to decide whether a good magnitude deviation implies
a good solution since many of the local minima are indistinguishable for the
phaseless solver. Apparently, the four measurement surfaces did not provide
enough diversity in the measured data to avoid local minima.

An advantage of the multi-frequency method is found in the fact that the
quality of the solution can be judged to some extent by looking at the
reconstruction deviation of the multi-frequency solution. With the discussed
measurement data, we can also compare the magnitude reconstruction deviation
to the true complex NF deviation. The result is shown in Figs. 18 and 19 (for
Fibonacci and regular sampling) for the multi-frequency case with three
frequencies, where each of them is chosen as reference in (15).

Figure 18: Comparison of magnitude-only and complex NF deviations for three-
frequencies measurements on a Fibonacci spiral and four distances, three
choices for the reference frequency. The star marks the initial guess from the
single-frequency solution, the other initial guesses are random. Figure 19:
Comparison of magnitude-only and complex NF deviations for three-frequencies
measurements on a regular sampling and four distances, three choices for the
reference frequency. The star marks the initial guess from the single-
frequency solution, the other initial guesses are random.

We observe that local minima with rather low magnitude deviation are still
present, but most of the local minima are shifted to much larger NF
reconstruction deviation values.

Analyzing the accuracy of the different variants, we see that regular sampling
performs here better than spiral sampling.

####  VI-C3 Transformed FF Patterns

The FFs in Figs. 20 and 21 were obtained from the magnitudes at
$3.2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ only.

Figure 20: Reconstructed $\varphi=0^{\circ}$ FF-cut of the
$\vartheta$-component at $3.2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ using
four measurement distances and a single frequency. Figure 21: Reconstructed
$\varphi=0^{\circ}$ FF-cut of the $\varphi$-component at
$3.2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ using four measurement distances
and a single frequency. Figure 22: Reconstructed $\varphi=0^{\circ}$ FF-cut of
the $\vartheta$-component at $3.2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$
using four measurement distances and three frequencies. Figure 23:
Reconstructed $\varphi=0^{\circ}$ FF-cut of the $\varphi$-component at
$3.2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ using four measurement distances
and three frequencies.

Using only a single frequency, the phase reconstruction fails despite having
measurements on four distances to the AUT. The magnitude-only NF deviations
and the complex NF deviations are shown in Fig. 17. The maximum FF errors for
the frequencies $2.8$, $3.0$, and $3.2$ GHz are $-23.0$, $-19.6$, and $-10.3$
dB, respectively.

Figs. 22 to 23 show the reconstructed FFs for the $\vartheta$\- and
$\varphi$\- components of the electric field at
$3.2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, where the knowledge of the phase
differences related to the three frequencies is used in the manner of (15)
with $2.8\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ as reference frequency. The
additional information obtained by using the three frequencies helped in this
case to find a much better solution in the minimization process. The NF
reconstruction deviations for the combined problem are given in Fig. 19, the
subsequent solution of the individual problems yield NF magnitude deviations
of $-35.5$, $-31.1$, and $-33.6$ dB, which correspond to NF errors of $-10.4$,
$-9.2$, and $-10.2$ dB. The reconstructed FF is in agreement with the
reference FF up to a maximum deviation of $-24.0$, $-24.9$, and $-26.1$ dB for
the three investigated frequencies — which is an improvement at each
frequency.

The method also works for larger bandwidths. We performed the same procedure
for the range $2.8$ to $3.6$ GHz with $100$ MHz steps. NF and FF error
improvements are similar as in the discussed case with three frequencies. In
particular, the FF patterns of the single-frequency and multi-frequency cases
are shown in Figs. 24 to 27. Note that this time the shown frequency is
$3.6\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ and the shown cut is
$\varphi=90^{\circ}$. The pattern maximum does not appear in this cut at this
frequency.

Figure 24: Reconstructed $\varphi=90^{\circ}$ FF-cut of the
$\varphi$-component at $3.6\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ using four
measurement distances and a single frequency. Figure 25: Reconstructed
$\varphi=90^{\circ}$ FF-cut of the $\vartheta$-component at
$3.6\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ using four measurement distances
and a single frequency. Figure 26: Reconstructed $\varphi=90^{\circ}$ FF-cut
of the $\varphi$-component at $3.6\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$
using four measurement distances and nine frequencies. Figure 27:
Reconstructed $\varphi=90^{\circ}$ FF-cut of the $\vartheta$-component at
$3.6\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ using four measurement distances
and nine frequencies.

###  VI-D Planar Field Transformation of Synthetic Horn Antenna Measurements

In order to verify that the procedure is not restricted to spherical surfaces,
a planar measurement setup has been synthetically mimicked from the source
distribution found for the horn antenna in the previous example. The $x$\- and
$y$-components of the field were computed on two planar surfaces of size
$$3\text{\,}\mathrm{m}$\times$3\text{\,}\mathrm{m}$$ at distances
$1.5\text{\,}\mathrm{m}$ and $1.7\text{\,}\mathrm{m}$, yielding a valid angle
for the near-field far-field transformation (NFFFT) of about $\pm 60^{\circ}$.
The spatial sampling step was $\Delta x=\Delta y=$0.03\text{\,}\mathrm{m}$$.
The calculated field magnitudes and the phase differences between the
frequency components at 9 frequencies from
$2.8\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ to
$3.6\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ were taken as input for the
multi-frequency algorithm.

Figure 28: Reconstructed $\varphi=0^{\circ}$ FF-cut of the $y$-component at
$3.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ using two measurement distances
and a single frequency. Figure 29: Reconstructed $\varphi=0^{\circ}$ FF-cut of
the $y$-component at $3.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ using two
measurement distances and nine frequencies.

The reconstructed co-polar components of the FFs for the single frequency
phase reconstruction algorithm in Fig. 28 and the multi-frequency algorithm in
Fig. 29 show that the reconstruction algorithm can benefit from multi-
frequency data, regardless of the particular sampling geometry.

TABLE I: Mean NF-reconstruction errors | Single-frequency | Multi-frequency
---|---|---
Frequency | $\epsilon_{\mathrm{mag}}$ | $\epsilon_{\mathrm{compl}}$ | $\epsilon_{\mathrm{mag}}$ | $\epsilon_{\mathrm{compl}}$
$2.8\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ | $-52.3\text{\,}\mathrm{d}\mathrm{B}$ | $-11.3\text{\,}\mathrm{d}\mathrm{B}$ | $-50.8\text{\,}\mathrm{d}\mathrm{B}$ | $-10.6\text{\,}\mathrm{d}\mathrm{B}$
$2.9\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ | $-47.7\text{\,}\mathrm{d}\mathrm{B}$ | $-17.9\text{\,}\mathrm{d}\mathrm{B}$ | $-46.6\text{\,}\mathrm{d}\mathrm{B}$ | $-12.9\text{\,}\mathrm{d}\mathrm{B}$
$3.0\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ | $-36.2\text{\,}\mathrm{d}\mathrm{B}$ | $-7.3\text{\,}\mathrm{d}\mathrm{B}$ | $-47.4\text{\,}\mathrm{d}\mathrm{B}$ | $-11.8\text{\,}\mathrm{d}\mathrm{B}$
$3.1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ | $-33.0\text{\,}\mathrm{d}\mathrm{B}$ | $-2.7\text{\,}\mathrm{d}\mathrm{B}$ | $-50.2\text{\,}\mathrm{d}\mathrm{B}$ | $-12.5\text{\,}\mathrm{d}\mathrm{B}$
$3.2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ | $-35.1\text{\,}\mathrm{d}\mathrm{B}$ | $-1.9\text{\,}\mathrm{d}\mathrm{B}$ | $-49.2\text{\,}\mathrm{d}\mathrm{B}$ | $-12.4\text{\,}\mathrm{d}\mathrm{B}$
$3.3\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ | $-35.2\text{\,}\mathrm{d}\mathrm{B}$ | $-2.3\text{\,}\mathrm{d}\mathrm{B}$ | $-50.7\text{\,}\mathrm{d}\mathrm{B}$ | $-12.8\text{\,}\mathrm{d}\mathrm{B}$
$3.4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ | $-33.8\text{\,}\mathrm{d}\mathrm{B}$ | $-1.7\text{\,}\mathrm{d}\mathrm{B}$ | $-49.5\text{\,}\mathrm{d}\mathrm{B}$ | $-13.0\text{\,}\mathrm{d}\mathrm{B}$
$3.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ | $-33.0\text{\,}\mathrm{d}\mathrm{B}$ | $-0.7\text{\,}\mathrm{d}\mathrm{B}$ | $-50.8\text{\,}\mathrm{d}\mathrm{B}$ | $-13.3\text{\,}\mathrm{d}\mathrm{B}$
$3.6\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ | $-47.4\text{\,}\mathrm{d}\mathrm{B}$ | $-13.2\text{\,}\mathrm{d}\mathrm{B}$ | $-51.6\text{\,}\mathrm{d}\mathrm{B}$ | $-11.6\text{\,}\mathrm{d}\mathrm{B}$

Table I shows the mean magnitude reconstruction error
$\epsilon_{\mathrm{mag}}=\left\lVert\left\lvert\bm{b}\right\rvert-\left\lvert\bm{b}_{\mathrm{ref}}\right\rvert\right\rVert/\left\lVert\left\lvert\bm{b}_{\mathrm{ref}}\right\rvert\right\rVert$
and the mean complex reconstruction error
$\epsilon_{\mathrm{compl}}=\left\lVert{\bm{b}}-{\bm{b}_{\mathrm{ref}}}\right\rVert/\left\lVert{\bm{b}_{\mathrm{ref}}}\right\rVert$
for the single-frequency phase and the multi-frequency reconstruction
algorithm. A comparison reveals that the magnitude must be very accurately
matched by the reconstruction algorithms in order to obtain acceptable phase
reconstructions and that using multi-frequency data evidently increases the
probability of a correct reconstruction.

##  VII Conclusion

In general, phase retrieval for antenna measurements is an extremely hard to
solve problem, since it is non-linear and non-convex. Given that only one
correct and unique global minimum exists (which cannot be ensured in practice
due to measurement errors), there are no reliable algorithms available to find
this global minimum with reasonable effort. Any piece of additional
information helps the algorithms to come (statistically) closer to this
minimum — where it has to be clear that this is by no means a guarantee for a
specific set of measurements.

Using the relative phase differences of signal samples at different
frequencies at the same measurement location can help to improve the
reliability of the non-convex phase retrieval algorithms in antenna
measurements. The relative phase information can be measured and the
corresponding measurement setup is less challenging than a fully coherent
measurement setup at very high frequencies. Having in mind that every
available piece of information is useful, we know that measurements on
multiple surfaces are (statistically) superior to measurements on a subset of
these surfaces. The same holds for adding multi-frequency data: It helps to
make existing methods (e.g., with multiple measurement surfaces) more
reliable.

In our investigations, the multi-frequency formulation was found to be able to
achieve better results than a single frequency formulation for otherwise
unchanged measurement configurations, whereas we never faced the case in which
the single-frequency solution was better.

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|  Josef Knapp (S’18) received the M.Sc. degree in electrical engineering and
information technology from the Technical University of Munich, Munich,
Germany, in 2016. Since 2016, he has been a Research Assistant at the Chair of
High-Frequency Engineering, Department of Electrical and Computer Engineering,
Technical University of Munich. His research interests include inverse
electromagnetic problems, computational electromagnetics, antenna measurement
techniques in unusual environments, and field transformation techniques.
---|---
|  Alexander Paulus (S’18) received the M.Sc. degree in electrical engineering
and information technology from the Technical University of Munich, Munich,
Germany, in 2015. Since 2015, he has been a Research Assistant at the Chair of
High-Frequency Engineering, Department of Electrical and Computer Engineering,
Technical University of Munich. His research interests include inverse
electromagnetic problems, computational electromagnetics and antenna
measurement techniques.
---|---
|  Jonas Kornprobst (S’17) received the B.Eng. degree in electrical
engineering and information technology from the University of Applied Sciences
Rosenheim, Rosenheim, Germany, in 2014, and the M.Sc. degree in electrical
engineering and information technology from the Technical University of
Munich, Munich, Germany, in 2016. Since 2016, he has been a Research Assistant
with the Chair of High-Frequency Engineering, Department of Electrical and
Computer Engineering, Technical University of Munich. His current research
interests include numerical electromagnetics, in particular integral equation
methods, antenna measurement techniques, antenna and antenna array design, as
well as microwave circuits.
---|---
|  Uwe Siart (M’08) received the Dipl.-Ing. degree from the University of
Erlangen-Nürnberg, Erlangen, Germany, in 1996 and the Dr.-Ing. degree from the
Technical University of Munich, Munich, Germany, in 2005. He has been with the
Chair of High-Frequency Engineering, Department of Electrical and Computer
Engineering, Technical University of Munich, since 1996. In 2005, he became a
Senior Research Associate. His research interests are in the fields of signal
processing and model-based parameter estimation for millimeter-wave radar
signal processing and high-frequency measurements. He is working on stochastic
electromagnetic wave propagation, remote sensing the atmosphere, low-power
radar sensors, microwave reflectometry and passive millimeter-wave components.
---|---
|  Thomas F. Eibert (S’93$-$M’97$-$SM’09) received the Dipl.-Ing. (FH) degree
from Fachhochschule Nürnberg, Nuremberg, Germany, the Dipl.-Ing. degree from
Ruhr-Universität Bochum, Bochum, Germany, and the Dr.-Ing. degree from
Bergische Universität Wuppertal, Wuppertal, Germany, in 1989, 1992, and 1997,
all in electrical engineering. He is currently a Full Professor of high-
frequency engineering at the Technical University of Munich, Munich, Germany.
His current research interests include numerical electromagnetics, wave
propagation, measurement and field transformation techniques for antennas and
scattering as well as all kinds of antenna and microwave circuit technologies
for sensors and communications.
---|---