# Linear Phase Retrieval for Near-Field Measurements with Locally Known Phase
Relations

Alexander Paulus1, Jonas Kornprobst1, Josef Knapp1, and Thomas F. Eibert1
1Chair of High-Frequency Engineering, Department of Electrical and Computer
Engineering,
Technical University of Munich, Munich, Germany<EMAIL_ADDRESS>
###### Abstract

A linear and thus convex phase retrieval algorithm for the application in
phaseless near-field far-field transformations is presented. The formulation
exploits locally known phase relations among sets of measurement samples,
which can in practice be acquired with multi-channel receivers. Due to the
linearity of the formulation, a reliable phaseless transformation is achieved,
which completely avoids the problem of local minima — the Achilles heel of
most existing phase retrieval techniques. Furthermore, the necessary number of
measurements are kept close to that of fully-coherent antenna measurements.
Comparisons with an already existing approach exploiting local phase relations
demonstrate the accuracy and reliability for synthetic data.

###### Index Terms:

phase retrieval, phaseless, source reconstruction, near-field far-field,
partial coherence, inverse problem.

##  I Introduction

Over the past decades, near-field (NF) measurements with a successive NF far-
field (FF) transformation (NFFFT) have established as a low-cost alternative
to measurements in FF or compact ranges. In order to further reduce the cost
and to adapt to the challenges of next generation mobile communication
systems, phaseless transformation algorithms are intensively investigated [1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Supported by the comprehensive
research in the field of phase retrieval, e.g., see [14, 15, 16, 17, 18, 19],
the antenna community has proposed various approaches for obtaining the FF
behavior from NF measurements without reliable phase information, as required
by common NFFFTs [20, 21, 22, 23, 24, 25]. Recent phaseless NF antenna
measurements rely on holography [26] with a fixed and known reference antenna
[11, 12, 27, 28, 29], on measurements on multiple surfaces [3, 4, 10, 30, 13,
31, 32], apply specialized probe antennas [33, 34, 35, 1, 2, 12] or utilize
multi-frequency information [36, 37]. The holographic approach is often
limited by the dynamic range of the measurement setup and suffers from
practical issues related to the positioning of the reference antenna without
obstructing the measurement probe or antenna under test (AUT) at any time.
Most of the approaches utilizing special probe antennas aim at forming chains
of locally coherent measurements, such that phase relations between measured
signals at distant locations can explicitly be established [33]. However,
error propagation makes this approach sensitive to measurement inaccuracies.
Instead of explicitly reconstructing the phase, some attempts have been made
to exploit the additional information provided by the special probe antennas
and by measurements on multiple surfaces with iterative phase retrieval
algorithms, e.g., [4, 1]. The resulting transformation results have often been
observed to be superior to the case when relying on measurements with common
probe antennas and on a single surface. However, it should be noted that due
to the rather complicated nature of the phase retrieval problem, no practical
guidelines or convergence promises for real-world applications exist. As such,
the techniques reported in literature may improve the suitability of
measurement setups for phaseless NFFFTs, however, there exists no algorithm
that can reliably perform a transformation with that kind of structured data.

Recently, a possible foundation for reliable phase retrieval has been reported
[38], when partially coherent observations, e.g., in the form of local phase
relations, are available. In the present paper, in contrast to [38], we
present a formulation in terms of mixed unknowns, corresponding to source
coefficients and phase terms. In this way, the problem formulation features
more flexibility while keeping the computational costs low. The presented
method exploits partial coherence in the measured data while preserving
linearity, and thus convexity. The approach perfectly matches the problem
arising from phaseless NF antenna measurements when a multi-channel receiver
with multiple probe antennas is used. No synchronization between the AUT and
the probes is required, and no coherence among sequentially acquired samples
is assumed. However, due to the receiver architecture, local coherence between
the channels of the receiver at each respective measurement position is given.
This can either be achieved with vectorial measurement hardware, or with
adequate scalar equipment, however, requiring a larger number of channels.
Note that the formulation can also be applied to (incomplete) holographic
measurements with a reference antenna. Furthermore, the need for a fixed
reference antenna is removed, diminishing the problem of limited dynamic range
and mutual shadowing of probe, AUT, and reference antenna.

We start by briefly revising the common NFFFT source reconstruction problem
and discuss the model of partially coherent observations. For the purpose of
comparability, we recapitulate an existing formulation for exploiting
partially coherent measurements, followed by the novel linear phase retrieval
formulation. Results for simulation data are discussed.

##  II Phase Retrieval With Partial Coherence

For the sake of simplicity and without loss of generality, we consider NF
antenna measurement setups where the unknown AUT is transmitting and the field
is sampled by one or multiple probe antennas in receive mode. The AUT is
represented by a vector of complex-valued source coefficients
$\bm{z}\in\mathbb{C}^{n}$, the probe antennas acquire the complex-valued
signals $\bm{b}\in\mathbb{C}^{m}$ and the solution of the relation

$\displaystyle\mathbf{A}\bm{z}$ $\displaystyle=\bm{b}$ (1)

represents the main task of an NFFFT with full phase information. In the case
of phaseless measurements, only $\left|\bm{b}\right|$ is known and (1)
deteriorates into a nonlinear system of equations. In both cases, the desired
fields, e.g., the FF, can be computed from $\bm{z}$, once it has been
determined.

Let us now assume that we have acquired three sets of NF measurements,
$\bm{b}_{1}\in\mathbb{C}^{m_{1}}$, $\bm{b}_{2}\in\mathbb{C}^{m_{2}}$, and
$\bm{b}_{3}\in\mathbb{C}^{m_{3}}$. Without loss of generality, assume that
$\bm{b}_{1}$ has been acquired with some random phase, but the second and
third set of samples $\bm{b}_{2/3}$ have been measured simultaneously by a
two-channel receiver equipped with two probe antennas. Consequently, both sets
feature the same number of entries, $m_{2}=m_{3}$, and there is a known phase
relation between the $k$th entry in $\bm{b}_{2}$ and $\bm{b}_{3}$ for
$k\in\left\\{1,...,m_{2}\right\\}$.

###  II-A Interferometry via Linear Combinations

It is well known that the phase difference between two complex scalars $a_{1}$
and $a_{2}$ can be computed analytically from four magnitude measurements of
the linear combinations [33]

$\displaystyle o_{1}$ $\displaystyle=\left|a_{1}\right|^{2},\hskip
14.22636pto_{3}=\left|a_{1}+a_{2}\right|^{2},$ $\displaystyle o_{2}$
$\displaystyle=\left|a_{2}\right|^{2},\hskip
14.22636pto_{4}=\left|a_{1}+\mathrm{j}a_{2}\right|^{2}$ (2)

via

$\displaystyle\angle\left(a_{1}\overline{a_{2}}\right)$
$\displaystyle=\text{atan}\left[\dfrac{o_{4}-o_{1}-o_{2}}{o_{3}-o_{1}-o_{2}}\right],$
(3)

where $\overline{a_{2}}$ denotes the complex conjugate of $a_{2}$. Coming back
to our example from above, we can thus formulate the task of a phaseless NFFFT
with knowledge of the partial coherence between $\bm{b}_{2}$ and $\bm{b}_{3}$
as

$\displaystyle\left|\begin{bmatrix}\mathbf{A}_{1}\\\ \mathbf{A}_{2}\\\
\mathbf{A}_{3}\\\ \mathbf{A}_{2}+\mathbf{A}_{3}\\\
\mathbf{A}_{2}+\mathrm{j}\mathbf{A}_{3}\end{bmatrix}\bm{z}\,\right|$
$\displaystyle=\left|\begin{bmatrix}\bm{b}_{1}\\\ \bm{b}_{2}\\\ \bm{b}_{3}\\\
\bm{b}_{2}+\bm{b}_{3}\\\ \bm{b}_{2}+\mathrm{j}\bm{b}_{3}\end{bmatrix}\right|,$
(4)

with elementwise magnitude operations on both sides of the equation. Note that
when neglecting the added rows of linear combinations in the forward operator
and the right-hand-side, the common, fully-incoherent problem is obtained. In
terms of the notation in (4), the knowledge of the phase differences thus
directly translates into additional measurement rows. The overall structure
remains the same.

A simple approach of exploiting partial coherence is thus to solve (4) with a
phase retrieval algorithm of your choice. As reported in [2, 1], the
additional rows related to the phase differences increase the chance of a
successful transformation. Still, success is not guaranteed and only limited
theoretical insight into the behavior of the formulation is available [39].

###  II-B Linear Formulation for Partial Coherence

Here, we follow an idea presented in [38] and derive a flexible formulation
with low computational cost. In order to illustrate the new approach, we
consider an example with $m_{1}=1$, $m_{2}=2=m_{3}$, and

$\displaystyle\bm{b}_{1}=\left|b_{11}\right|\mathrm{e}^{\,\mathrm{j}\varphi_{11}},\hskip
2.84544pt\bm{b}_{2}=\begin{bmatrix}\left|b_{21}\right|\mathrm{e}^{\,\mathrm{j}\varphi_{21}}\\\
\left|b_{22}\right|\mathrm{e}^{\,\mathrm{j}\varphi_{22}}\end{bmatrix},\hskip
2.84544pt\bm{b}_{3}=\begin{bmatrix}\left|b_{31}\right|\mathrm{e}^{\,\mathrm{j}\varphi_{31}}\\\
\left|b_{32}\right|\mathrm{e}^{\,\mathrm{j}\varphi_{32}}\end{bmatrix}.$ (5)

We can write

$\displaystyle\mathbf{A}\bm{z}=\bm{b}=\begin{bmatrix}\bm{b}_{1}^{\text{T}}&\bm{b}_{2}^{\text{T}}&\bm{b}_{3}^{\text{T}}\end{bmatrix}^{\text{T}}$
$\displaystyle=\begin{bmatrix}\left|b_{11}\right|&0&0\\\
0&\left|b_{21}\right|&0\\\ 0&0&\left|b_{22}\right|\\\
0&\left|b_{31}\right|\mathrm{e}^{\,\mathrm{j}\left(\varphi_{31}-\varphi_{21}\right)}&0\\\
0&0&\left|b_{32}\right|\mathrm{e}^{\,\mathrm{j}\left(\varphi_{32}-\varphi_{22}\right)}\\\
\end{bmatrix}\begin{bmatrix}\mathrm{e}^{\,\mathrm{j}\varphi_{11}}\\\
\mathrm{e}^{\,\mathrm{j}\varphi_{21}}\\\
\mathrm{e}^{\,\mathrm{j}\varphi_{22}}\end{bmatrix}$
$\displaystyle=\text{diag}\left(\left|\bm{b}\right|\right)\begin{bmatrix}1&0&0\\\
0&1&0\\\ 0&0&1\\\
0&\mathrm{e}^{\,\mathrm{j}\left(\varphi_{31}-\varphi_{21}\right)}&0\\\
0&0&\mathrm{e}^{\,\mathrm{j}\left(\varphi_{32}-\varphi_{22}\right)}\\\
\end{bmatrix}\bm{\psi}$ $\displaystyle=\mathbf{B}\mathbf{C}\bm{\psi},$ (6)

leading to the linear formulation

$\displaystyle\left(\mathbf{A}\mathbf{P}_{1}-\mathbf{B}\mathbf{C}\mathbf{P}_{2}\right)\tilde{\bm{z}}$
$\displaystyle=\bm{0},\hskip
14.22636pt\tilde{\bm{z}}=\begin{bmatrix}\bm{z}^{\text{T}}&\bm{\psi}^{\text{T}}\end{bmatrix}^{\text{T}}\in\mathbb{C}^{n+q}$
(7)

with the stacked vector of unknowns $\tilde{\bm{z}}$ consisting of the unknown
AUT representation $\bm{z}$ and a vector $\bm{\psi}$ of all remaining unknown
phase terms. The matrices $\mathbf{P}_{1/2}$ in (7) are defined via unit and
zero matrices $\mathbf{I}$ and $\mathbf{0}$, respectively, as

$\displaystyle\mathbf{P}_{1}\tilde{\bm{z}}$
$\displaystyle=\begin{bmatrix}\mathbf{I}&\mathbf{0}\end{bmatrix}\tilde{\bm{z}}=\bm{z}\in\mathbb{C}^{n}$
(8) $\displaystyle\mathbf{P}_{2}\tilde{\bm{z}}$
$\displaystyle=\begin{bmatrix}\mathbf{0}&\mathbf{I}\end{bmatrix}\tilde{\bm{z}}=\bm{\psi}\in\mathbb{C}^{q},$
(9)

and extract the respective parts from the stacked vector of unknowns. The
diagonal matrix $\mathbf{B}\in\mathbb{R}^{m\times m}$ contains all measured
magnitudes while the actual information about partial coherence is inside the
sparse matrix $\mathbf{C}\in\mathbb{C}^{m\times q}$, containing all known
phase differences. The more phase differences are added, the less independent
phase terms remain unknown and the fewer columns in $\mathbf{C}$ exist. Note
that for (7) to yield the correct solution, we have to enforce that
$\left|\psi_{k}\right|=1$, $\forall k\in\left\\{1,...,q\right\\}$. However,
this condition is nonlinear and leads to a nonconvex problem. Still, we find
an approximate and linear formulation by writing

$\displaystyle\left(\mathbf{A}\mathbf{P}_{1}-\mathbf{B}\mathbf{C}\mathbf{P}_{2}\right)\tilde{\bm{z}}$
$\displaystyle=\bm{0}$ (10) $\displaystyle\text{s.t.}\,\,\,\psi_{s}=1,$

which only enforces the magnitude constraint on the $s$th phase unknown and
fixes its phase to zero. In this way, a linear and thus convex phase retrieval
formulation for the case of partially coherent observations is obtained. Note
that, in contrast to (18) in [38], the here presented notation is more
flexible and allows for an arbitrary combination of incoherent and coherent
measurements. Also, the large computational effort per matrix-vector product
of (21) in [38], requiring a pseudo-inverse, is avoided. As a downside of
(10), more unknowns are introduced.

##  III Transformation Results

###  III-A Randomly Generated Data

As a first indicator for the performance of the formulation in (10), synthetic
data randomly taken from a complex-valued normal distribution was generated
and processed. The reference formulation in (4) was solved via the phase
retrieval algorithm described in [1], which essentially is a nonconvex
optimization-based phase retrieval algorithm similar to the widely known
Wirtinger flow [16]. The algorithm features an iterative line search algorithm
according to [40] and computes descent directions via the L-BFGS method [41].
Due to its nonconvexity, the formulation heavily depends on the initial guess,
which was computed via $40$ power iterations of a spectral method [16]. While
(10) was solved via a direct inversion in Matlab [42], at most $2000$
iterations were allowed for the iterative solution of (4).

The model with three sets of measurements introduced in Section II-A was
utilized, with $n=20$ and $m_{1}=n$. The number of measurements with local
phase relations $m_{2/3}$ was varied within the range of $$0.05$n$ to $1.5n$
to account for a variable contribution of phase information. Figure 1 depicts
the chance of a successful phase retrieval, where success was declared once a
relative complex NF deviation

$\displaystyle\varepsilon_{\text{c}}$
$\displaystyle=20\,\text{log}_{10}\left(\dfrac{\left\lVert\mathbf{A}\bm{z}-\bm{b}\right\rVert}{\left\lVert\bm{b}\right\rVert}\right)$
(11)

below $-90\text{\,}\mathrm{dB}\text{/}$ was achieved. Note that global phase
shifts between $\mathbf{A}\bm{z}$ and $\bm{b}$ are ignored in the evaluation
of (11) since phase retrieval is always only possible up to an arbitrary
global phase shift. For real-world measurements, only the relative magnitude
NF deviation

$\displaystyle\varepsilon_{\text{m}}$
$\displaystyle=20\,\text{log}_{10}\left(\dfrac{\left\lVert\left|\mathbf{A}\bm{z}\right|-\left|\bm{b}\right|\right\rVert}{\left\lVert\bm{b}\right\rVert}\right)$
(12)

can be evaluated. For each ratio of the total number of measurements to
unknowns, $5000$ random realizations of the forward operators, reference
solution vectors and their corresponding measurement vectors were considered.
For reasons of comparability, results for formulation (4) without the addition
of the linear combination terms — thus without any partial coherence — have
also been added. It becomes evident from Fig. 1 that the nonlinear formulation
exploiting the knowledge of all available magnitudes and phase differences
achieves better success rates for fewer measurement samples. This is expected
to some extent, as the proposed formulation is only linear since $q-1$
restrictions on the magnitudes of the $\bm{\psi}$ terms are ignored — a loss
of “information content”. However, once a certain ratio of measurements to
unknowns is available, the linear formulation always returns the unique and
correct solution. This ratio is equivalent to the bound derived in [38] as a
necessary condition for convergence. Due to the differences in the
formulations, this threshold in [38] can, however, not directly be proven to
be valid for the approach in (10). The utilized nonlinear formulation for
exploiting partially coherent observations may return a suboptimal solution
even for large numbers of measurements — a success rate of $$100$\%$ is
actually never achieved. This is due to the nonconvex nature of the
formulation and can cause unreliable behavior in real-world applications. With
neglect of the information about the partial coherence, i.e., ignoring that
$\bm{b}_{2}$ and $\bm{b}_{3}$ have been acquired simultaneously with two
coherent channels, the success rate drops further and stagnates at
$\approx$95$\%$ in Fig. 1.

Figure 1: Comparison of the formulation in (4), the proposed formulation in
(10), formulation (21) in [38], and (4) without the linear combinations
(incoherent) for complex-valued normally distributed data. The problem
dimensions were picked as $n=20$, $m_{1}=n$, and the number of partially
coherent measurements $m_{2/3}$ was swept from $$0.05$n$ to $1.5n$. For each
sampling ratio, $5000$ randomly generated measurement matrices, solution
vectors, and the corresponding measurement vectors were considered.

###  III-B Synthetic Antenna Data

Figure 2: Near-field measurement arrangement of the DRH18 AUT (left) measured
with an open-ended-waveguide probe (right). Three spherical measurements are
performed sequentially, each with a different location of the probe on the
planar positioner (yellow dots). The data is then combined as if the three
measurements would have been performed in parallel, via a three-element
“L”-shaped probe array and a receiver with three coherent channels. Here, we
only consider synthetic data modeling this setup.

As a final verification, spherical antenna NF data of a dual-ridged horn
antenna (DRH18 [43]) at $2.6\text{\,}\mathrm{GHz}\text{/}$ was processed. The
real-world measurement data with full phase information was utilized to
generate an equivalent AUT model via a vector spherical wave expansion [25].
With this model, NF data for a virtual spherical measurement with a probe
antenna array consisting of three elements in an “L”-arrangement was
generated. Within the transversal probe coordinate system, the elements are
placed at the locations
$\left($0\text{\,}\mathrm{m}\text{/}$,$0\text{\,}\mathrm{m}\text{/}$\right)$,
$\left($1\text{\,}\mathrm{m}\text{/}$,$0\text{\,}\mathrm{m}\text{/}$\right)$
and
$\left($0\text{\,}\mathrm{m}\text{/}$,$0.8\text{\,}\mathrm{m}\text{/}$\right)$,
respectively, forming the three corners of an “L”. This virtual setup is
illustrated in Fig. 2, showing a possible realization in an anechoic chamber
with an open-ended-waveguide probe.

For modeling the AUT during the phaseless transformations, $n=$800$$ Hertzian
and Fitzgerald dipoles placed tangentially on an enclosing box were used.
Since different sources for the data generation and for the treatment of the
inverse problem are used, inverse crime is avoided. From the virtual spherical
measurements, $2000$ samples per set were randomly picked, resulting in a
total number of $m=$6000$$ measurements available for the phase retrieval
algorithms — plus the inherent phase differences between the probe antennas.
Noise according to a signal-to-noise ratio (SNR) of
$60\text{\,}\mathrm{dB}\text{/}$ — with respect to the maximum value of the
measurement signals — was added to the stacked measurement vector. The
nonconvex solvers again started from an initial guess computed via the
spectral method based on $40$ power iterations.

In Fig. 3, the obtained FF radiation characteristic of the dominant field
component in the $\varphi=$90\text{\,}\mathrm{\SIUnitSymbolDegree}\text{/}$$
cut plane through the main beam is depicted. The FF computed from the
spherical multipole expansion was taken as reference. The deviation curves
display the absolute value of the complex-valued difference between the
reference and the transformation result, normalized by the maximum value of
the reference. While the graphs only display the magnitude of the electric
field, large deviations can also be caused by a false phase distribution, as
it seems to be the case in Fig. 3(c). In agreement with the relative NF
deviations reported in Tab. I, the proposed linear transformation results in
good FF accuracy, better than that of the nonlinear transformation exploiting
local phase relations in (4). Ignoring the incomplete phase information, we
obtain the inferior transformation result in Fig. 3(c). Essentially, the
transformation result in Fig. 3(a) is limited by the SNR, while the results in
Fig. 3(b) and 3(c) are deteriorated due to local stationary points. For
completeness, the obtainable deviations between the spherical vector wave
function representation and the dipole representation, when full phase
information is available, is given in Tab. I, labeled as “coherent”.

(a)

(b)

(c)

Figure 3: Dominant FF component of the electric field in the $\varphi=$90\text{\,}\mathrm{\SIUnitSymbolDegree}\text{/}$$ cut plane of the DRH18 horn antenna at $2.6\text{\,}\mathrm{GHz}\text{/}$ obtained with three phase retrieval formulations. Data was generated via an equivalent model of the DRH18 based on a spherical multipole expansion. The transformation results are normalized to their respective maximum values, at which their phase is also set to zero. a) Transformation exploiting partial coherence with (10). b) Transformation exploiting partial coherence with (4). c) Transformation ignoring partial coherence in (4).  TABLE I: NF deviations obtained for the DRH18 horn antenna model | (10) | (4) | Incoherent | Coherent
---|---|---|---|---
$\varepsilon_{\text{c}}$ | $-50.9\text{\,}\mathrm{dB}\text{/}$ | $-30.2\text{\,}\mathrm{dB}\text{/}$ | $-3.2\text{\,}\mathrm{dB}\text{/}$ | $-55.6\text{\,}\mathrm{dB}\text{/}$
$\varepsilon_{\text{m}}$ | $-53.0\text{\,}\mathrm{dB}\text{/}$ | $-40.0\text{\,}\mathrm{dB}\text{/}$ | $-33.7\text{\,}\mathrm{dB}\text{/}$ | $-58.6\text{\,}\mathrm{dB}\text{/}$

##  IV Conclusion

A reliable linear formulation for the task of phase retrieval with partially
coherent measurements was presented and investigated. Suitable near-field data
can be obtained by receivers with two or more coherent channels and
appropriate probe arrays. The presented equations are highly flexible with
respect to the form and availability of the knowledge of partial coherence and
allow for a computationally efficient implementation. While effectively
dropping a portion of the available magnitude restrictions, the algorithm
features superior reliability compared to an existing nonconvex approach once
a sufficient number of measurements has been acquired. The required number of
samples is in the order of that needed for a conventional transformation with
full phase information. Only in the regime of few available measurement
samples, existing nonconvex approaches should be preferred as they do not drop
information and may, by chance, return a more accurate result.

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