ARSAdvances in Radio ScienceARSAdv. Radio Sci.1684-9973Copernicus PublicationsGöttingen, Germany10.5194/ars-15-11-2017Comparison of non-convex cost functionals for the consideration of
phase differences in phaseless near-field far-field transformations
of measured antenna fieldsKnappJosefjosef.knapp@tum.dePaulusAlexanderLopezCarlosEibertThomas F.Chair of High-Frequency Engineering, Technical University of Munich,
Arcisstr. 21, 80333 Munich, GermanyJosef Knapp (josef.knapp@tum.de)21September201715111919December201610March201713March2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://ars.copernicus.org/articles/15/11/2017/ars-15-11-2017.htmlThe full text article is available as a PDF file from https://ars.copernicus.org/articles/15/11/2017/ars-15-11-2017.pdf
This work introduces two methods which extend the non-convex minimization
problem arising in phaseless (NF) far-field (FF)
transformations. With the new extensions, knowledge about phase differences
between measurement points can be incorporated into the minimization problem.
The additional information helps to avoid stationary points of the
minimization cost functional which would otherwise compromise the result of
the near-field far-field transformation. The methods are incorporated into
the Fast Irregular Antenna Field Transformation Algorithm (FIAFTA), analyzed and
compared. Their effectiveness is shown by transforming synthetic near-field
data sets with partial knowledge of phase differences to the far-field.
Introduction
With the rapid development in communication technology, also
the demands for antennas and antenna measurement technologies increase. In
antenna measurements, one of the main interests is to determine the antenna
radiation pattern in the far-field (FF) of the antenna under test (AUT). If
straightforward measurements in the AUT FF are not feasible, for example when
the FF distance exceeds the measurement chamber dimensions, then the
measurements can be obtained in the near-field (NF) of the AUT and afterwards
the AUT FF pattern can be determined by NF to FF transformation (NFFFT). In
general, magnitude and phase information are required in the NF measurements
in order to determine the FF. However, phase measurements become complicated
in the very high frequency regime or might also be omitted when utilizing
cheap scalar measurement equipment. Phaseless NFFFTs, which require only
magnitude data, are needed for the development of leading-edge antenna
technologies. The phaseless NFFFT problem is directly related to the more
general problem of phase retrieval, which has numerous applications in
optics, radiology, and various physical disciplines
. Since the
measurements in phase retrieval scenarios (i.e. the radiation from an antenna
for example) arise in a well defined physical environment, there exists a
relationship between the magnitude of the measurements and its phase. Due to
the non linear nature of this relationship, the phase can not easily be
retrieved from magnitude only measurements.
First successful attempts in retrieving the phase of magnitude only data are
found in . The initial algorithm has been
further developed throughout the years (for example in
) and the topic is still under heavy research. In
the last decade, one has often tried to retrieve the phase via non-convex
minimization
or by a
convex relaxation of the non-convex formulation
.
Because of their high numerical complexity, convex relaxations are suitable
for small and medium sized problems only. Non-convex minimizations suffer
from stationary points of the cost functional. Convergence guarantees have
been established for the non-convex minimization in and . However, the
convergence criteria hold only, if enough measurements can be obtained, which
follow certain probability distributions. It remains unclear, how such
measurements can be obtained in a NF antenna measurement scenario.
Interferometric methods have been used to obtain phase differences from
magnitude only measurements in . There, the
measured phase differences have been set up along a chain from one element to
the other, thus being very inflexible in the measurement setup.
This article aims at representing arbitrary known phase differences in terms
of additional goals for the minimization problem. Phase differences can be
obtained from magnitude only measurements of certain linear combinations as
in or by the assumption that a global reference
phase will be stable for at least two successive measurements. Finding a
global minimum to a cost functional involving both, magnitudes and phase
differences, is equivalent to determining the near field up to a global phase
shift. By introducing the phase knowledge in terms of minimization goals and
not setting phase differences to a fixed value, we allow for solutions which
give an overall best approximation for the phase differences and the
magnitudes, thus being less prone to measurement errors.
The article is structured as follows. Section briefly
revisits the phaseless NFFFT presented in ,
which is similar to the analysis of the Wirtinger Flow minimizations in
and . In
Sect. , two extensions for the cost functional are
presented which introduce phase knowledge to the minimization problem. These
implementations are formally analyzed in Sect. for their
behavior. Finally Sect. shows numerical evidence for the
effectiveness of the proposed methods.
Formulation of the phaseless field transformation
The task of an NFFFT is to determine the electromagnetic FFs of an antenna
from a number of NF measurements. It can be solved as an inverse problem by
finding equivalent sources in the AUT volume or on the surface of the AUT
volume. The equivalent sources are chosen such that they reproduce the
measured NF values. The FF is thereafter easily obtained from the found
equivalent currents. Starting from a discrete set of basis functions for the
sources (e.g. RWG basis functions in ), one
obtains a linear equation system
b=Az,
where b∈CM is the vector of M complex NF measurements,
z∈CN contains the N coefficients for the equivalent
source basis functions and A∈CM×N is the system
matrix in which the entry Amn describes the influence of the nth basis
function on the mth measurement. The minimum mean square error solution of
the normal equation system
z=A†b
with the pseudo-inverse A† yields the coefficients for the
discrete set of equivalent source basis functions for the computation of the
corresponding FF. If the dimensions are high, it becomes computationally
expensive to calculate the system matrix A explicitly. Then,
iterative methods can be utilized to determine the solution of
Eq. (), which only require the evaluation of the matrix
vector products Ax and AHx′, where
AH denotes the Hermitian transpose of A. In this
work the matrix vector products are evaluated efficiently with the Fast
Irregular Antenna Field Transformation Algorithm (FIAFTA) described in
, and
. Due to the hierarchical field representation,
the matrix vector products can be evaluated with a computational complexity
of ONlogN.
By combining the phase retrieval algorithm with FIAFTA, the phaseless NFFFT
described here inherits all the positive properties of FIAFTA such as the
possibility to work with arbitrary irregular and regular measurement grids,
full probe correction for arbitrary probes, and a very flexible source
representation with modal field expansions as well as magnetic and/or
electric surface currents on a triangular mesh representing the geometry of
the AUT.
When the phases of the measurements in b are unknown, we want to find
a solution of
|b|=|Az|,
or equivalently
b∘2=Az∘2,
where ⋅ denotes the elementwise absolute value operator
and the exponent ⋅∘2 denotes that the power of
2 is applied on each element of the vector. Since Eq. () is a
non linear equation, we try to solve the corresponding minimization problem
minz∈CNb∘2-Az∘222
or equivalently
minz∈CNb*∘b︸β-Az*∘Az22,
where ∘ denotes the Hadamard product, β=b*∘b is the vector of the squared magnitudes of the elements of
b and
Az*∘Az yields the
squared magnitudes of Az. For the rest of this paper we will
call y=Azvirtual measurements and
β the goal vector. The cost functional f=β-Az*∘Az22 in Eq. () can be identified as a squared sum of
individual differences between the virtual measurements y=Az and the elements of the goal vector β.
Extension for phase differences
The main idea of this work is to extend the goal vector β by
values, which bring information about phase differences into the problem. In
other words, given we know the phase difference ϕij=ϕj-ϕi
between the measurements bi and bj, how can we define a cost
functional, which yields the correct phase differences, after its
minimization?
Magnitude of linear combinations of two measurements
The (squared) magnitude of a linear combination of the two complex numbers
bi and bj carries information about the phase difference between these
two numbers as can be seen from
bi+bj2=bi2+bj2+2bibjcosϕij.
The phase difference ϕij=ϕj-ϕi between the complex
measurements bi and bj can uniquely be determined from four magnitudes,
namely bi, bjt, bi+bj and bi+jbj for example as in
. These four linear combinations are not the
only possibility to specify the phase differences in terms of magnitudes.
Almost any tuple of magnitudes of four different linear combinations of the
complex numbers bi and bj will define the phase difference between
these two numbers uniquely. The rare tuples which are not suitable to
reconstruct the phases can easily be avoided by a careful choice of the
linear combinations to be measured. A natural choice for an additional row
inside the norm of the cost functional is to use the magnitude of a linear
combination of two already existing measurements. The goal value
βN+1′ for this linear combination magnitude can be obtained from
additional measurements with special probes as in
or can easily be computed from known magnitudes and phase differences
analogous to Eq. (). Formally, the cost functional in
Eq. () can be extended for the newly introduced information
by
f′=f+βN+1-yi+yj22=f+βN+1-Azi+Azj22,
with the new goal
βN+1=bi+bj2.
The additional term introduces an additional penalty for any deviations in
the magnitudes of the considered linear combination which in turn can be
interpreted as an additional phase constraint according to
Eq. ().
In general, any linear combination of arbitrary measurements can be attached,
by extending the cost functional in a similar manner as before. One obtains
f′=f+βN+1-α1yi+α2yj22
with the new goal
βN+1=α1bi+α2bj2.
Also an arbitrary number of additional linear combinations can be considered
by simply adding more terms like Eq. ().
Due to the flexibility of the formulation in this chapter, in principle
magnitudes of any probe or linear combination of probes can be incorporated
into the minimization, without the need of determining phase differences
explicitly. However, if the phase differences are known, the magnitudes of
any linear combination can be calculated and incorporated as presented.
Complex conjugated multiplication
The phase of a product of a complex number bi and the complex conjugate of
another complex number bj* is equal to the phase difference ϕji=ϕi-ϕj between them:
bibj*=bibjejϕji.
If the phase difference between two measurements is known, this information
can be incorporated into Eq. () with usage of this
multiplication identity. Consider the extended cost functional
f′=f+βN+1-yiyj*2=f+βN+1-AziAzj*2,
with the new goal
βN+1=bibj*.
The newly introduced additional term in the cost functional adds a penalty
for any deviation of the product yiyj* from its goal and can be
interpreted as an additional phase constraint according to
Eq. ().
Similar to the linear combinations it is possible, to extend the cost
functional for the complex conjugated multiplication by more than one term,
by simply attaching more terms in the presented scheme. It is also possible,
to combine linear combinations and complex conjugated multiplication costs in
a single cost functional by attaching both of the corresponding cost
functional types.
Notice that the extended goal vector β′ contains complex values
for the complex conjugated multiplication terms as opposed to real and
positive numbers only in the case of magnitudes. This makes the evaluation of
the Jacobian, which is needed for many minimization procedures, more
difficult.
Analysis of the extended cost functionals
In this section the behavior of the newly introduced cost functionals will be analyzed.
To this end, consider the four cost functionals
f1=β1-yi+yj22,f2=β2-yi-jyj22,f3=f1+f2,f4=β4-yiyj*2,
with the goal terms
β1=bi+bj2,β2=bi-jbj2,β4=bibj*.
The variables yi and yj denote the ith and jth virtual measurement
and bi and bj for the corresponding (hypothetical) true complex
measurements. In general, the virtual measurements yi deviate from the
goals bi in magnitude and phase, i.e.
yi=cibiejΔϕi,
with ci∈R+ is the magnitude factor by which the virtual
measurement yi deviates from the goal bi and Δϕi∈0,2π is the phase difference between bi and yi, i.e. ∠yi=∠bi+Δϕi. As discussed previously, only the
magnitudes of bi and bj and the linear combinations or the phase
difference between them might be known (i.e. the absolute phase of bi and
bj is unknown). Since only the phase difference between the virtual
measurements yi and yj is relevant, we can identify the term
ϕϵ=Δϕi-Δϕj
as the error term in the phase difference between y1 and y2. Remember
that ϕji=ϕi-ϕj is the phase difference between the goals
bi and bj, while Δϕi and Δϕj are the phase
deviations of the virtual measurements from the goals bi and bj
respectively, i.e. if the virtual measurements have the same global phase
shift to the actual measurements bi and bj (i.e. Δϕi=Δϕj), the error term ϕϵ for the phase difference is
zero. The cost functionals f1 and f2 correspond to linear combinations
considered in Sect. and the cost functional f3 is the
sum of the cost functionals f1 and f2. The cost functional f4
corresponds to a complex conjugated multiplication presented in
Sect. .
Cost functionals for phase error only. For this example the goal
values have been chosen to be bi=ej55∘ and bj=2.3ej0∘.
Cost functionals with phase and magnitude error. For this example,
additionally to the phase deviation, a multiplicative amplitude deviation of
ci=2.3 and cj=1.3 is assumed.
Figure shows the values of f1 to f4 dependent on the
phase deviation ϕϵ=Δϕi-Δϕj in case that
the virtual measurements yi and yj have the same magnitudes as th
correspondent goals bi and bj. The values for bi=ej55∘ and bj=2.3ej0∘ have been
chosen arbitrarily to yield cost functionals which plainly show the different
behaviors. It can be seen that all four cost functionals have a minimum for
ϕϵ=0∘. However, the cost functionals f1 and f2
also show a false minimum for another ϕϵ≠0∘. For
the case of no magnitude deviations, i.e. ci=cj=1, the cost functional
f3 is a scaled version of the cost functional f4.
In the minimization process, while the minimization has not terminated yet,
in general there will be phase deviations ϕϵ≠0 as well as
magnitude deviations ci≠cj≠1. Figure shows the
values of the cost functionals dependent on the phase deviation
ϕϵ for virtual measurements which deviate from the goals in
phase as well as in magnitude, viz. ci=2.3 and cj=1.3. The false minima
for f1 and f2 are more distinct than in Fig. and the
cost functionals f1 and f2 can return zero and thus reach their global
minimum even though neither magnitudes nor the phase difference matches the
goals. The cost functionals f3 and f4 have a minimum for a single value
of ϕϵ only, i.e. they do not suffer from false minima.
For an analysis, the cost functionals f1 to f4 can be expressed in
terms of their magnitude and phase deviation. For the cost functional f1
we have
f1=β1-yi+yj22=bi+bj2-cibiejΔϕi+cjbjejΔϕj22=bi21-ci2+bj21-cj2+2bibjReejϕji-c1c2ejϕji+Δϕi-Δϕj2=bi21-ci2+bj21-cj2+2bibjReejϕji-c1c2ejϕji+Φϵ2.
Accordingly, for the cost functional f2 we have
f2=β2-yi-jyj22=bi+jbj2-cibiejΔϕi-jcjbjejΔϕj22=bi21-ci2+bj21-cj2+2bibjImejϕji-c1c2ejϕji+ϕϵ2.
The cost functionals f1 and f2 deviate from each other only by
evaluating the real and imaginary part of ejϕji-cicjejϕji+ϕϵ respectively.
Thus, they suffer from the same kind of undesired behavior. For any given
magnitude deviation ci and cj, both cost functionals have two minima
for two different values of ϕϵ (there are two numbers on the
unit circle which have the same imaginary or real part respectively). The
desired behavior would be that the cost functionals have a single minimum for
ϕϵ=0 only. The occurring false minimum reflects the fact that
a single linear combination magnitude does not uniquely define the phase
difference between the two complex numbers bi and bj. Note, that two
distinct minima might occur for each of the cost functionals f1 and f2
even, when the magnitudes yi=bi and
yj=bj are equal to the desired
magnitudes, i.e. c1=c2=1.
For the analysis of cost functional f3, we introduce the auxiliary
variable
A=bi21-ci2+bj21-cj2.
By identifying Eq. () as the squared magnitude of the real
part and Eq. () as the squared magnitude of the imaginary
part of the same complex number, f3 can be written as
f3=f1+f2=1+jA+2bibjejϕji-c1c2ejϕji+ϕϵ2=1+jAe-jϕji+2bibj1-c1c2ejϕϵ2=B-2bibjcicjejϕϵ2
with the auxiliary term B=1+jAe-jϕji+2bibj. From the last row
of Eq. () it is clear that the cost functional f3 is
minimal only for the single value of ϕϵ, when it equals the
phase of B shifted by 180∘. The cost functional obtains its minimum
for ci=cj=1 and ϕϵ=0, as intended, however for general
magnitude deviations ci≠1 and cj≠1, the cost functional f3
may obtain its minimum at ϕϵ≠0. Some tuples
ci,cj,ϕϵ≠(1,1,0) can also lead to f3=0.
However, together with the minimization of the magnitudes as described in
Sect. , the cost functional has a unique global minimum
(up to a global phase) for yi and yj.
The cost functional f4 can be rewritten in the form
f4=β4-yiyj*2=bibj*-bibj*cicjejϕϵ2=bi2bj21-cicjejϕϵ2.
The cost functional f4 has a global minimum for ci=cj=1 and
ϕϵ=0. Also independent on the deviation between yi and
bi and yj and bj, the minimum for f3 occurs a
ϕϵ=0. However the cost functional f3 can return zero also
for any combination of magnitude deviations ci=(1/cj)≠1. Thus for a
unique global minimum with ci=cj=1 and ϕϵ=0 the cost
functional f3 has to be minimized together with the magnitude cost
functional from Sect. .
Mesh defining the RWG unknowns for transforming synthetic data of a
horn antenna.
Location of the sample points.
Numerical results
The effectiveness of the presented methods is shown by numerical examples.
The NF data has been generated synthetically according to
at 3 GHz. The behavior of a horn
antenna has been simulated by 2232 dipoles arranged on a surface enclosing a
CAD model of a real horn antenna depicted in Fig. . The horn
antenna model aperture is 227.40mm×151.60mm in
size. The other side of the antenna builds a transition to a rectangular
waveguide of dimensions 72.14mm×34.04mm. The horn
length is 350 mm and the taper is a (1-cos)-taper. The excitations
for the dipoles have been obtained approximately from a simulation in CST
MICROWAVE STUDIO 2016 . For the NFFFT the triangles on
the same mesh has been used for RWG basis functions
, i.e. the vector z in
Eq. () contains the coefficients of each RWG basis function.
As measurement probes serve Hertzian dipoles. Two probes at a time form a
measurement pair which will be evaluated for the phase differences. A pair
consists of two, horizontally separated dipoles spaced two wavelengths apart.
The measurement points, i.e. the centers of the dipole pairs, are located on
spirals on a spherical surface around the AUT. Figure
shows the 1000 locations of the pair centers along with a black cuboid
denoting the location of the AUT. The spiral sampling shown in
Fig. is more uniformly distributed than the usual
spherical sampling which has equidistant angular steps in ϑ and
φ direction. Since in general two independent polarizations are
needed in NF measurements, the same measurements have been obtained with
rotated dipoles. Thus we have 4000 measurements (not counting any linear
combinations) since at each measurement location two measurements – one to
the left and one to the right – have been obtained with two polarizations.
Any linear combination and phase differences can be obtained from the complex
valued synthetic dipole outputs. In the following linear combinations and
phase differences have been considered only for corresponding dipole pairs.
Note, that the locations of dipoles of neighboring pairs do not interfere in
general, which means that the absolute phases of the measurements at the
individual dipoles cannot be determined by iterating through a chain of known
phase differences.
Four scenarios have been considered. In the first scenario, the coefficients
in z are retrieved from magnitude only measurements as in
Sect. . In the second scenario, additionally to the
magnitudes of the single dipole outputs, the magnitudes of the sum of the
outputs of each dipole pair have been considered according to
Sect. and Eq. (). In the third scenario
also the magnitudes of the phase shifted sums have been considered as in
Eq. (). Finally, in the fourth scenario the phase knowledge
has been included in terms of conjugated multiplication as described in
Sect. and in Eq. (). For all scenarios, the
corresponding cost functional has been minimized with a L-BFGS procedure
, a memory limited version of the
Broyden–Fletcher–Goldfarb–Shanno algorithm (named after its inventors) in
the family of quasi-Newton methods . The
algorithm terminates, once an insufficient relative decrease in the cost
functional is observed. The initial conditions have been chosen to be zi=1+1j∀i for each basis function in all scenarios. After the
minimization has finished, the FF is calculated from the retrieved sources in
z. Table shows an overview over the four
scenarios. The different timings are mostly due to the different number of
iterations. All minimizations used roughly the same memory of about
450 MB. For Table , the complete FF has been
considered, not only the cuts which are shown in the following subsections.
Magnitude only measurements via cost functional f
The dashed blue line in Fig. shows the cut of the copolar
E-field component in the E-plane of the retrieved FF from magnitude only
NFFFT. The orange solid line denotes the ideal FF, which has been computed
from the original source dipoles. Both, the reference and the retrieved FF,
are normalized to their maximum respectively. The dotted purple line shows
the difference between both curves. The maximal error in the FF (relative to
the maximum of any FF) is larger than -20 dB.
Figure shows the magnitude of the cost functional at each
iteration, normalized to its initial value. The termination criterion is met
after 1285 iterations and the progression of the curve hints to the cost
functional being stuck in a local stationary point. Especially outside the
main lobe, the retrieved FF pattern does not match the reference. As shown in
Table , the maximum FF error is at -14.9 dB and
the average FF error is -32.4 dB.
Retrieved FF from magnitude only data,
φ= 90∘.
Progress of the minimization of the cost functional for magnitude
only data.
Magnitudes of single measurements and of pair sums via cost functional
f+f1
For the second scenario, also the magnitudes of the complex sum of the
outputs of each dipole pair have been considered. The E-plane cut of the
retrieved copolar FF component can be found in Fig. . The
values in Table suggest that there hardly is any
improvement compared to the magnitude only minimization. Even though the
retrieved FF in Fig. seems to show a better result than
Fig. , especially outside the main lobe, the error level is
still very high and the minimization again seems to be captured at a local
stationary point which is different from the one in the magnitude only
minimization, even though the initial guess has been the same. This agrees
with the analysis from Sect. , in which it was stated that
false minima may occur for the additional cost functional parts in f1.
Nevertheless, the usage of the magnitude of a linear combination has some
effect since it leads to a termination at a different stationary point than
the magnitude only minimization. In some cases this may lead to a situation
in which local stationary points from the magnitude only data are avoided. In
this particular case, the minimization does not progress anymore after 1075
iterations, as shown in Fig. .
Retrieved FF from magnitudes and one linear combination data,
φ= 90∘.
Progress of the minimization of the cost functional for magnitude
data combined with one linear combination magnitude per diploe
pair.
Retrieved FF from magnitudes and two linear combinations data,
φ= 90∘.
Progress of the minimization of the cost functional for magnitude
data combined with two orthogonal linear combination magnitudes per dipole
pair.
Magnitudes of single measurements, of pair sums, and of phase shifted pair sums via cost functional
f+f3
In the third scenario additionally to the magnitude only minimization
objectives also the two (squared) magnitudes of the linear combinations
bi+bj2 and bi-jbj2
have been considered for the dipole pairs. Figure shows the
retrieved copolar FF in the E-plane. With a maximum FF error of
-54 dB it can be concluded that the minimization process reached
the minimum which corresponds to the correct near fields. This observations
supports the conclusions of Sect. . The progression of the
cost functional shown in Fig. also shows that the
minimization did not reach its minimal value before at least ten times the
number of iterations of the first two scenarios but also shows a steady
decrease. This suggests that all local stationary points have been avoided
thanks to the additional objectives for the linear combinations.
Retrieved FF from magnitudes and conjugated multiplication data,
φ=90∘.
Progress of the minimization of the cost functional for magnitudes
combined with conjugated multiplication data per dipole
pair.
Magnitudes of single measurements and conjugated multiplication
between pairs via cost functional f+f4
In the fourth scenario, parallel to the magnitude only goals, the conjugated
multiplications described in Sect. have been minimized for the
dipole pairs. Figure shows the retrieved copolar FF pattern
in the E-plane. Similar to the previous scenario, the retrieved FF coincides
with the reference. As stated in Table , the maximum FF
error is less than -60 dB. The retrieved FF as well as the progression
in Fig. suggest that any local stationary points have been
avoided and the correct NF has been retrieved.
Conclusions
The phaseless near-field (NF) far-field (FF) transformation
(NFFFT) can be extended to incorporate phase knowledge in terms of phase
differences between measurements and magnitudes of linear combinations. The
cost functional for the non linear minimization is formally extended by
additional terms which penalize the difference between the virtual
measurements and the correspondent goals. The exact build-up of these goals
introduces the phase knowledge either in terms of complex conjugated
multiplication or in terms of linear combinations. It has been shown that
both new cost functionals behave differently during the minimization
procedure. When the phase goals are chosen with care, the cost functional
exhibits a unique global minimum (up to a global phase) which corresponds to
the fully restored radiated NF. From this restored NF, the FF can be computed
easily with means of standard NFFFT. For the cost functionals with a global
minimum corresponding to the correct NF, the FF has been retrieved with a
maximum error of up to -62 dB. However, even though an unique
global minimum exists for the radiated fields, the minimization may not
converge to this minimum due to the non linearity of the problem. In such a
case, the minimization runs into a local stationary point. Incorporating
phase knowledge can help to avoid stationary points in this case. This way,
one can retrieve a barely distorted FF without the need of full phase
measurements.
Underlaying data is available from the corresponding author upon request.
The authors declare that they have no conflict of
interest.
The responsibility for the content of this publication is with
the authors.
Acknowledgements
This work has been supported by the German Federal Ministry for Economic
Affairs and Energy under Contract No. 50YB1512.This work was supported by the German Research Foundation (DFG) and the Technische Universität München within the funding programme Open Access Publishing.
Edited by: R. Schuhmann
Reviewed by: two anonymous referees
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