Multi-Frequency Phase Retrieval for Antenna Measurements
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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 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 1 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 Abstract—Phase retrieval problems in antenna measurements non-convex problem in general — and hard to solve [5]–[12]. arise when a reference phase cannot be provided to all mea- Since it is attractive in a variety of applications, a large number surement locations. Phase retrieval algorithms require sufficiently of algorithms and methods have been proposed to tackle many independent measurement samples of the radiated fields to be successful. Larger amounts of independent data may improve this problem of phase retrieval in magnitude-only antenna the reconstruction of the phase information from magnitude-only measurements [5], [9], [13]–[20] or other cases [7], [8], [10]– measurements. We show how the knowledge of relative phases [12], [21]–[34]. among the spectral components of a modulated signal at the in- In any case, it is understood that one needs to increase dividual measurement locations may be employed to reconstruct the number of measurement samples to enable phase retrieval the relative phases between different measurement locations at all frequencies. Projection matrices map the estimated phases as compared to the number of samples for conventional onto the space of fields possibly generated by equivalent antenna measurements with magnitude and phase [7], [9], [17], [18], under test (AUT) sources at all frequencies. In this way, the [34]–[40]. Phase retrieval algorithms usually rely on measure- phase of the reconstructed solution is not only restricted by the ment data with sufficiently many independent measurement measurement samples at one frequency, but by the samples at all samples to allow for a stable reconstruction process. With an frequencies simultaneously. The proposed method can increase the amount of independent phase information even if all probes increasing number of independent measurements, the problem are located in the far field (FF) of the AUT. becomes more similar to a convex problem such that even local minimization techniques can be used to find the true Index Terms—Phase retrieval, phaseless antenna measure- ments, broadband receiver. solution [5], [40]–[42]. Once the number of independent mea- surement samples reaches the square of the effective number of unknowns, the problem even can be formulated in a linear I. INTRODUCTION manner [18], [21], [22], [43]. A still unsolved problem is, IME-HARMONIC near field (NF) antenna pattern mea- however, to reliably find a measurement setup which provides T surements classically comprise magnitude and phase sufficiently many independent measurement samples [44]. measurements of the fields radiated by the antenna under Classical attempts employ measurements on two or more test (AUT) [1]–[3], where the phase information is important surfaces with various distances to the AUT [19], [27]–[29], for the calculation of desired far field (FF) quantities (gain, [45]. The idea is that the field contributions of the AUT radiation pattern, etc.) from the measured data. The phase mea- interfere differently at different distances as long as they are in surement requires synchronized transmit and receive signals of the NF of the AUT. The magnitudes convey information about probe and AUT, e.g., by a reference phase signal. Providing these coherent interferences. Since the interference patterns a stable phase reference to all different probe locations may are strongly affected by the phases of individual field con- call for elaborate measurement setups at high frequencies tributions, the reconstruction of the relative phases might be or for large AUTs [4], where unavoidable movement of the possible. It is still hard to predict where the NF measurement reference cable may introduce severe phase errors. Certain locations have to be chosen in order to produce sufficient arXiv:2105.09928v1 [eess.SP] 20 May 2021 measurement setups — e.g., with a probe fixed to an unmanned dissimilar information about the field interferences. aerial vehicle — may cause severe complications if the phase Specially designed multi-antenna probes have been studied reference has to be provided at all probe locations. to obtain the required data [14], [17], [46]. The different probes In order to avoid intricacies associated with phase measure- perform different coherent linear combinations of the incident ments, and to render some measurement setups feasible in the fields to form their output signals. Thus, the magnitudes of the first place, it is desirable to retrieve the phase information from different probe signals heavily depend on the relative phases of magnitude-only field measurements, which is a non-linear and incident plane waves and encode the phase information of the Manuscript received December 27, 2019; revised May 25, 2020; accepted incident fields. The success of this method strongly depends June 13, 2020; date of this version June 18, 2020. This work was supported by on the possibility of utilizing probes which measure different the German Federal Ministry for Economic Affairs and Energy under Grant interference patterns of the radiated AUT field contributions. 50RK1923. (Corresponding author: Josef Knapp.) The authors are with the Chair of High-Frequency Engineering, Department To be effective, the probes have to be either very large or of Electrical and Computer Engineering, Technical University of Munich, rather close to the AUT in order to be able to produce 80290 Munich, Germany (e-mail: [email protected], [email protected]). a useful weighted mean of field samples. For instance in Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. the FF, the field is practically constant over the profile of Digital Object Identifier 10.1109/TAP.2020.3008648 the probe and taking different linear combinations of this Copyright © 2020 IEEE. Personal use of this material is permitted. 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The amount of different The measurement vector corresponding to the frequency information which can be obtained in this way is also limited l: is denoted by b: and its ℓth entry b: ℓ denotes the by the number of different probes which can be used in the measurement sample at the ℓth measurement» ¼ location. One measurement setup. In the FF, the task of providing large can establish a linear relationship between the sources and numbers of independent measurement samples becomes even the measurement samples giving rise to the linear equation more difficult. system [52] In this work, we increase the measurement diversity by b = A x , (1) introducing relationships between the measured signals at : : : " # different frequencies. No absolute phase reference is required with the system matrix A: C . 2 × to obtain the relative phases between the measurement samples In a magnitude-only measurement, the phase vector φ: with at different frequencies with the same probe position. Phase entries defined by stability is only required over a short time span in which the b j φ: ℓ : ℓ measurement samples are obtained for all the frequencies at e » ¼ = » ¼ (2) b: ℓ the individual measurement locations. Using the same local j» ¼ j oscillator (LO) for all frequencies at the receiver side (which is unknown for every frequency l: and must be determined in is not required to be synchronized with the transmit LO), it is a non-linear inverse problem. With the elementwise absolute possible to assign magnitudes and phases to all signal samples value operator , the phase retrieval problem at frequency l: j·j at the different frequencies. The remaining inverse problem can be expressed as is to find the phase differences between the measurement # 1 find x: C × such that b: = A: x: . (3) locations (i.e., the phase of the receive LO at each position). 2 j j j j Phase retrieval for radiated or scattered fields at multiple In previous works, this phase retrieval problem has been frequencies was already investigated occasionally [6], [47], tackled independently for each frequency. The absolute phases [48] but these approaches retrieved the phases independently at had to be found for every frequency component of the each frequency or utilized the solution for one frequency as an measured signals at all measurement positions separately. initial guess at other frequencies. In contrast to this, we employ In contrast to this, we assume that the phase differences the measurement samples at all frequencies simultaneously between signal components at different frequencies are known to find the global phase solution. Preliminary investigations at each measurement position. This corresponds to knowing about this idea can be found in [49]. The source reconstruction the difference φ: φ8 of the phase vectors at frequencies l: − problem at one selected reference frequency or a mixture and l8, respectively. This assumption simplifies the problem of several reference frequencies is constrained by measured as now only one phase value has to be determined at every signals at all frequencies.