Near-field far-field transformations (NFFFTs) are commonly performed for time-harmonic fields. Considering arbitrary in-situ measurement scenarios with given transmission signals, time-varying aspects of modulated signals have to be taken into consideration. We investigate and characterize two methods for the measurement of modulated fields, which work with a time-domain representation of the radiated fields and, at the same time, allow to employ the standard time-harmonic NFFFT. One method is based on the fact that the modulation signal can be assumed to be constant in a short enough measurement interval under the condition that the modulation and carrier frequencies are several decades apart. The second method performs long-time measurements in order to obtain the complete frequency spectrum in every single measurement. Both methods are verified by the NFFFT of synthetic field data.

With the rise of wireless communication technologies, also the demand for
antenna characterization increases. One of the most important characteristics
of an antenna is its far-field (FF) radiation pattern. A common method to
obtain the radiation pattern involves near-field (NF) measurements of the
antenna from which the FF can be calculated in the post-processing. This is
known as the NF to FF transformation (NFFFT)

In this work, we tackle the problem of handling modulated fields arising in
in-situ measurements. To do so, NFFFT algorithms are reviewed first. For
time-harmonic fields, the NFFFT can be formulated as a linear inverse
problem, where equivalent sources, e.g., equivalent surface current densities
on a surface enclosing the AUT, plane wave spectra or spherical multipoles,
are to be retrieved. Solving this inverse problem, an equivalent source model
is attained that reconstructs the measured NF and from which the FF is
calculated. State-of-the-art algorithms for time-harmonic signals work in the
spectral domain, i.e., for single-frequency signals. For modulated signals,
the single-frequency assumption does not hold any more. Even if there are
NFFFT algorithms that work in the time domain and can deal with modulated
fields

The paper is structured as follows. In Sect.

To cope with continuously modulated (i.e. continuously time-varying with a
certain periodicity and without abrupt phase and magnitude changes) fields
within the time-harmonic NFFFT, the field signals must be available in the
frequency domain. In this work, we concentrate on the case of continuously
amplitude modulated fields. This is only for the reason of simplicity and
demonstration as all investigations hold true for the generic case. A
continuously modulated field

One approach towards the NFFFT of modulated fields is to measure the field
signal in the time-domain for such a long time that all relevant frequency
components can be resolved and distinguished, followed by a Fourier
transform. This approach is called

The long-time measurement approach consists of four steps

The LTM technique relies in particular on the fact that the amplitude and phase relations between the single frequency components are retained throughout the complete processing chain, from the measurement, no matter whether in time domain or in frequency domain, over the NFFFT, until the final signal composition in the FF.

A drawback of the method is the measurement time

In contrast to the LTM, the width of the window function can also be chosen
much shorter. This basically avoids the issue of long measurement times but
brings along other effects which have to be treated carefully. If

Within the short-time measurement approach, the field signal

Assuming a constant modulation signal

Principle of the short-time measurement approach. The envelope of
the time-varying field signal is sampled by each measurement. All samples
sharing the same modulation state are transformed as a data set to the
FF, which is equivalent to the transformation of the single frequency
components within the LTM in Fig.

Since this technique requires a shorter measurement time and allows for a
faster measurement, it is called

Further, both approaches work only if the time-varying field signal is
periodic since one transformable field data set consists of components from
multiple measurements which have to be taken one after the other. Even if the
measurement time can be short for single measurements, the reconstruction of
the modulation signal requires that the sampling theorem is fulfilled for
consecutive measurements. This is due to the fact that the STM is a sampling
of the signal envelope. Therefore, the sampling theorem is given
by

Sometimes it may be assumed that the far-field radiation pattern for the
carrier frequency and for the modulation sidebands does not change
significantly, since

The applicability of the STM and LTM approaches is demonstrated by numerical
results. The NF data is generated synthetically from a horn antenna that is
represented by Hertzian dipoles. To create a realistic antenna model, the
surface currents of a time-harmonic full-wave simulation of the horn antenna
were approximated by 2232 Hertzian dipoles on the PEC surface. The full-wave
simulation was performed in CST Microwave Studio

Arrangement of the 2232 Hertzian dipoles that represent a horn antenna. The excitation of the Hertzian dipoles has been found from the discretization of the horn antenna's surface currents.

The antenna has an aperture of

Time-varying far-field signal at the center of the AUT main beam. The transformed fields of the STM (blue) and LTM (orange) match with the reference (red) signal. Both approaches do not introduce any additional error to the field measurement, the resulting error is due to the NFFFT.

Far-field main cuts at the carrier frequency in comparison with a
reference pattern obtained from a time-harmonic simulation. The phi-cut

Figure

The FF main cuts at the carrier frequency are depicted in
Fig.

A key difference between the STM and the LTM is the actual measurement time.
Lacking a real-world measurement scenario, this can be only estimated based
on the presented example. For the LTM, the field signal has been measured at
each position for a time span of

Two different techniques for the measurement of modulated fields have been presented, which both extend the time-harmonic NFFFT to the case of modulated electromagnetic fields. The validity of the measurement approaches was shown by NFFFTs of synthetically generated NF data of a horn antenna model. It has been demonstrated that the resulting deviation of the LTM and STM are on a very similar accuracy level and that the introduced error does not significantly deviate from the error of the single-frequency-time-harmonic NFFFT. Comparing the two methods, the advantage of the STM over the LTM is the faster measurement time, which enables the use of this technique for UAV-based antenna measurements and other applications where short measurement times are mandatory.

The underlying research data can be requested from the authors.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Kleinheubacher Berichte 2018”. It is a result of the Kleinheubacher Tagung 2018, Miltenberg, Germany, 24–26 September 2018.

This work was supported by the German Federal Ministry for Economic Affairs and Energy (grant no. 20E1711A), as well as by the German Research Foundation (DFG) and the Technical University of Munich (TUM) in the framework of the Open Access Publishing Program.

This paper was edited by Romanus Dyczij-Edlinger and reviewed by Adrian Amor-Martin and Heyno Garbe.