The next topic of interest arose out of the first, namely, during the investiga- tion of the effects of the propellers’ material on the scattered field

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B

UDAPEST

U

NIVERSITY OF

T

ECHNOLOGY AND

E

CONOMICS

F

ACULTY OF

E

LECTRICAL

E

NGINEERING AND

I

NFORMATICS DEPARTMENT OFBROADBANDINFOCOMMUNICATIONS AND

ELECTROMAGNETICTHEORY

On scattering and radiation from moving and composite bodies

Author:

Károly MARÁK

Supervisor:

Dr. Sándor BILICZ

S

UMMARY

April 2, 2020

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1 Background

Around the beginning of my doctoral studies, there was great interest in the topic ofUnmanned Aerial Vehicles’(UAVs, drones) detection through radar from the Microwave Remote Sensing Laboratory of the department. The challenge in this task comes from the small size of drones, which makes their observation by radar difficult, as at lower frequencies, theirRadar Cross Section(RCS) is quite small, and at higher frequencies, they are difficult to distinguish from other objects. However, it was observed experimentally that the modulation of the scattered electromagnetic field caused by the rotation is most pronounced near the resonant frequency of the propeller, and this modulation causes a discrete scattered spectrum [1]. My first task was therefore to create a model of backscattering (monostatic radar detection) from a rotating propeller, which can be used to calculate the scattered field as a function of parameters such as the geometry, materials used, the speed of the rotation, etc.

After I constructed this model and verified it through alternative calculation methods, I expanded my area of research to include more general cases, such as a bistatic scattering experiment or a coupled model of two nearby rotating propellers. All of these scenarios were validated experimentally with a setup constructed by fellow PhD student Tamás Pet˝o.

The next topic of interest arose out of the first, namely, during the investigation of the effects of the propellers’ material on the scattered field. Simulations I had performed on scattering from conductive materials implied that by investigating the results of the scattering experiments, some conclusions could be drawn about the material of the propellers. Much of the scattering theory was in place by this point, however, the inverse problem was yet to be formulated, and, as the typical material for propellers is some kind of carbon-fiber composite, I also had to address the question of anisotopy. After all this was done, I designed the experiment and procured some samples. Again with the help of Tamás Pet˝o, the experiment was set up to verify the feasibility of the idea.

Work on third and final thesis point was started in late 2017, during a short- term scientific stay at the TU in Prague. Within the framework of the COST Ac- tion 1301, I were to investigate antenna array pattern synthesis in the context of radiatedWireless Power Transfer(WPT). After implementing the vanilla algorithm based on Dr. Kracek’s idea, I extended the basic formalism, and tested it on different goal patterns.

Since that time, this problem has branched to different subtopics including multiple pattern synthesis, interaction between nearby antennas, interpolation and minimization schemes, which required coming up with and implementing new ideas, however, earlier results concerning the efficient modelling of inter- acting scatterers were also useful, together with the proficiency gained with numerical software.

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2 Results

From the theoretical standpoint, it was important to gain an understanding of, e.g., the formalism describing scattering from moving bodies [2], properties of anisotropic scatterers [3] and radiation pattern synthesis theory [4]. From the numerical point of view, I learned about and tried several computational schemes (such as theIntegral Equation(IE) [5] or theFinite Element Method (FEM) [6]) and numerical software (such as Matlab or Comsol).

Fortunately, with regards to experimental work, I could rely on the hardware assembled by Tamás Pet˝o, in addition to his expertise in the subject, meaning that I could concentrate mainly on theoretical matters. However, I assisted in al- most all of the performed measurements whose data I used, encountering several questions such as noise filtration on the spectrum analyzer, the choice of appropriate samples, unwanted reflection from the environment and working around the frequency-dependent characteristics of certain antennas.

2.1 Investigation of electromagnetic scattering from rotating propellers Recently, Unmanned Aerial Vehicles (UAVs)—drones—have gotten attention on account of their difficulty of detection with conventional radar systems due to their very small radar cross section (RCS). In order to develop more powerful detection methods, it is necessary to more deeply investigate the patterns in the signal reflected from UAVs.

It is well known that illuminating a body in uniform, linear motion causes the scattered wave to suffer a so-called Doppler shift, meaning a shift in frequency. A similar phenomenon can be observed in the case of more complicated (micro-) motion, such as the rotation of a propeller; this phenomenon is called the micro-Doppler effect.

In the literature [7, 8], most of the time, similar problems are solved using a small-wavelength approximation such as Physical Optics (PO) or Ray Tracing.

In the case when such approximations are not appropriate (as in our case, when the size of the propeller is comparable to the wavelength of the illumination), it is necessary to perform afull wave simulationwith the appropriate considera- tions as to the rotation of the observed object.

In other cases [9], a time domain simulation is constructed. Fundamen- tally, it would be possible to use such a method in the simulations, however, it is simpler (we get ride of thetimevariable) and more consistent with the theory (we are dealing exclusively with harmonic time functions) to remain in the frequency domain.

I outline the following approach in the thesis: by using the properties of the transformation to and from rotating frames, I show that the spectrum of the scattered field contains an infinite but countable number of frequency components

Es(r,t)= X∞ m=−∞

Em(r)e^j(^ω+m^Ω^)t, (1)

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Figure 1: While the FEM model requires a 3D discretization of the propeller and the surrounding medium, in the MoM, a 1D discretization of the scatterer with appropriately chosen parameters (a and l ) and a sinusoidal basis of approxi- mately 5-10 functions yields similar results.

whereΩis the angular velocity of the propeller’s rotation andωis the angular frequency of the illuminating wave.

Afterwards, I show that in the case whenωandΩare several orders of magnitude apart, a so-calledquasi-stationaryapproach [10] is applicable. This is analogous tobandpass samplingorundersamplingknown from signal theory [11]. Using this simplification, I then describe the spectrum of the electromagnetic field scattered in any direction by calculating the scattered field for a number of stationary positions of the propeller. Due to the large aspect ratio of a propeller, I perform the calculation using an efficient 1D discretization scheme by means of an integral equation formulation, making use of the propeller’s geometrical properties; I also validate the results by means of a 3D FEM simulation (the schemes are illustrated in Fig. 1).

This procedure can be generalized to an arbitrary scatterer whose dimensions are comparable (or smaller than) the wavelength. Specially, if it performs a time-periodic motion and its position can be described by parameterΩt=ϕ, then the coefficients in (1) can be calculated as the Fourier coefficients of the functionEs(r,ϕ)

Es(r,ϕ)= X∞ m=−∞

Em(r)e^jϕ. (2)

I also investigate the more general case of multiple adjacent scatterers. The re- sulting spectrum is in the form of

E^sec_s (r,t)= X∞ m1=−∞

X∞ m2=−∞

Em1,m2(r)e^j(ω+^m¹^Ω¹⁺^m²^Ω²^)t, (3) whereΩ1andΩ2are the angular velocities of two adjacent propellers.

Using the experimental hardware/setup assembled by fellow Phd student Tamás Pet˝o, him and I have validated the theoretical results in a number of experiments:monostaticandbistaticscattering experiments using a single CFRP propeller, and also performed experiments withtwo adjacent propellerswith adjustable orientation and relative position. I calculate the radar cross section

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Figure 2: Anechoic chamber measurement scenario for two adjacent propellers.

Figure 3: Measured spectrum for two adjacent propellers d=300 mmapart. The red signs denote the source of the spectral components.

(RCS) of a 11 inch CFRP propeller from the measurement results; it is found that it agrees acceptably well with the theoretical expectations, especially when considering the small size of the scatterer (the experiments imply a value of RCS_exp≈0.4m²compared to the theoretical RCS_th≈0.25m²). The sketch and results of the experimental setup for the investigation of scattering from two adjacent rotating propellers can be seen in Figs. 2 and 3, while results of a bistatic scattering measurement and their comparison with the calculated values are shown in Fig. 4.

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-300 -200 -100 0 100 200 300 f-f₀ [Hz]

-140 -120 -100 -80 -60 -40

P[dBm]

a/3

measurement calculations

Figure 4: Comparison of experimental and computational results of the spec- trum measured in a bistatic scenario. Calculations predict the intensity of the side peaks relatively accurately, while the central peak is more pronounced due to the miscellaneous scattering from the environment and the interference between the transmitter and receiver antennas.

Summary

Thesis 1: I devised a computationally efficient, simplified 1D integral equation (IE) based method in order to describe scattering from rotating carbon fiber rein- forced plastic (CFRP) propellers, the results of which I validated by means of 3D finite element method (FEM) simulations describing the exact propeller geome- try.

• The theoretical framework for treating the scattering problem involves a frequency domain calculation method (such as the aforementioned IE or the FEM), coupled with the quasi-stationary formalism for calculating scattering from moving objects.

• Specifically, the simplified IE model was shown to cause a massive improvement in performance, owing to the much lower number of Degrees of Freedom (DoF) when compared to the complex 3D model.

• The framework was applied to the problems of bistatic and monostatic scattering from a single rotating propeller, as well as monostatic scattering from two adjacent propellers. Together with fellow PhD student Tamás Pet˝o, we designed experiments corresponding to these problems.

• An experimental setup for the purpose of validating the model’s results was assembled byTamás Pet˝oand theMicrowave Remote Sensing and Radar Laboratory. It consists of an adjustable frame capable of setting

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the orientation of the rotation axis of the propellers, as well as their position, antenna(s) for the purpose of illumination (excited with a continu- ous wave signal) and reception (connected to a spectrum analyzer) of the scattered field. The movement of the propeller is controlled by a brush- less DC motor which can be regulated through a programmable micro- controller. Using this setup, we performed several anechoic chamber and free-range experiments corresponding to the aforementioned scattering scenarios. I discuss the accuracy of the measurement method, as well as the prospects for future applications.

Related publications: [C1, C2, C3, C4], [J1, J2]

2.2 Micro-Doppler based conductivity measurement

In microwave applications such as radar detection or antenna design, theelec- trical conductivity(σ) is a material parameter of high importance. There are several methods of its measurement, however, many are limited to lower frequencies [12] (due to the errors introduced into the experimental setup by high frequency effects, particularly at the contacts), and it can show a dependence on frequency for certain materials. Another group of techniques is applicable at higher frequencies [13, 14]. Generally, for the aforementioned reasons, contactless methods are preferred, however, these methods generally require either a large sample compared to the wavelength, or a very precise measurement of EM field quantities.

Based on the results of Thesis 1, I make use of the micro-Doppler effect caused by the rotation of an elongated sample to filter out the background noise of miscellaneous scattering from the environment. From the backscattered spectrum, it is then possible to draw conclusions regarding the conductive properties of the sample’s material near the frequency of the investigation :

• By measuring the relative intensity of the backscattered field and comparing it with a reference sample’s field;

• and by investigating the resonant frequency of the sample.

I perform numerical simulations of the scattering to assess the accuracy of the method; as seen in Fig. 5, I show that it performs the best for conductivities ranging from 1 S/m to 10³S/m. I extend the idea to anisotropic samples, with slight modifications proposed to the experimental setup in case of a significant anisotropy of the material inspected.

Lastly, in collaboration with Tamás Pet˝o, we perform experiments to demonstrate the viability of the procedure on samples with dimensions similar to those of a typical drone propeller, with the results and samples being shown in Figs. 6 and 7, respectively.

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10⁰ 10¹ 10² 10³ 10⁴ [S/m]

10^-2 10^-1

|Es r / Ei| [m]

(a) Dependence of the intensity of the backscattered field on the conductivity of an elongated sample. Eⁱ and E^sindicate the intensities of the incident and scattered fields, and r denotes the distance from the sample at which the scattered field is calcu- lated.

10² 10³ 10⁴ [S/m]

620 630 640 650 660 670

fr[MHz]

(b) Dependence of the resonant frequency on the conductivity of an elongated sample.

Figure 5: Scattered intensity and resonance frequency as functions of conduc- tivity; the dotted red lines signify the interval of calculated intensities which cor- respond to a given conductivity value with±10%deviation from the nominal value.

560 580 600 620 640 660

f[MHz]

-114 -112 -110 -108 -106

received power[dB] sample 1

sample 2

Figure 6: Experimental validation of the proposed measurement method: re- ceived power at frequency(f−2F)(where f is the frequency of the illumination and F is the frequency of the sample’s rotation), corresponding to the -1st peak in the scattering spectrum of the rotating sample. The blue marks correspond to sample 1, while the red marks to sample 2 of a lower conductivity.

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Figure 7: Photograph of the samples used during the demonstrative experiments, 3D printed from a conductive filament, next to a carbon fiber reinforced plastic drone propeller. On the top is sample 1 with an infill percentage of 100%, while in the middle is sample 2 with a 10% infill percentage.

Summary

Thesis 2: I propose an experimental arrangement to measure high frequency elec- tromagnetic material parameters of small, elongates samples; I performed nu- merical simulations to analyze the accuracy of the method, and designed a demon- strative experiment.

• Based on the results of Thesis 1, I propose to employ the micro-Doppler effect caused by the rotation of an elongated sample to filter out the background noise of miscellaneous scattering from the environment. From the backscattered spectrum, one can then draw conclusions about the conductive properties of the sample’s material near the investigation frequency:

1. By measuring the relative intensity of the backscattered field and comparing it with a reference sample’s field;

2. And by investigating the resonant frequency of the sample.

• I performed numerical simulations of the scattering to assess the accuracy of the method; I found that it performs the best for conductivities ranging from 1 S/m to 10³S/m. I extended the idea to anisotropic samples, with slight modifications proposed to the experimental setup in case of a significant anisotropy of the material inspected.

• Lastly, I designed and, in cooperation with fellow PhD student Tamás Pet˝o, performed experiments to demonstrate the viability of the procedure.

Related publications: [C5], [J3]

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2.3 Array synthesis using an iterative algorithm

Antenna array pattern synthesis is one of the challenges arising in the develop- ment of 5th generation (5G) communications [15]. Here, the goal is to increase the efficiency of communication between devices, which requires the concen- tration of the radiated power into a certain direction. Usually, arrays with fixed structure are used, however, this does not make use of the possibilities of the proper placement of the array elements.

Another field of interest is Wireless Power Transfer (WPT). In some applications, devices cannot be powered using conventional galvanic contacts or near field WPT (such as some sensors and actuators, or vehicles on the move);

a possible solution could be provided by a radiated WPT system. Naturally, an appropriate design of the energy transmitting and receiving structures is of paramount importance [16].

Since a general problem of antenna array synthesis has a large number of variables [17] and the running time of numerical optimization increases expo- nentially with the number of variables, many synthesis methods have been de- veloped which reduce the number of variables by imposing some kind of con- straint on the array structure, such as linear or planar arrays with uniformly ori- ented and spaced antennas. The method of antenna placement I propose in the thesis allows for the relaxation of the geometric constraints on the antennas’ placement, which allows for more degrees of freedom and potentially bet- ter performance, while keeping the computational complexity reasonably low.

In order to measure the similarity of radiation patterns, I use an inner product of radiation patterns

f₁,f₂®

= Z

Ω h

f₁(ˆk)i_∗

·f₂(ˆk)dΩ, (4)

wheref₁(ˆk) corresponds to the radiation pattern in a direction corresponding to wavevectork, and the integration is performed over the unit sphere.

In the norm generated by this inner product, I define the difference of two patterns as the functional

kf1−f₂k²=

f₁−f₂,f₁−f₂®

. (5)

When applying the method to a given goal patternf_Aand a basic antenna element with patternf0, we are seeking on the set of allowed geometrical oper- ations (e.g., a combination of translations and rotations) and excitation (magnitude and phase) those values which minimize the difference off_Aandf₀. After some algebraic manipulation, I show that the number of optimization variables can be reduced by the two describing the excitation (phase and magnitude). Af- terwards, the method is recursively applied to the results of the previous step, e.g., the difference of the goal pattern and the fields of the previously placed elements.

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Next, I introduce an alternative algorithm (Algorithm 1), which makes it possible to specify the desired number of array elements (I). In itsI+1th step, in- stead of placing a new antenna, we reposition the 1st antenna and recalculate its optimal excitation. This procedure is then continued on all the array elements a specified number of times, or until the desired accuracy is reached.

Algorithm 1:Synthesis of patternf_Afor given numberIof elements which stops ifGgenerations of array are created, or, alternatively, if errorδnde- creases below∆.

(A2.1) Place new elements untili=I (A2.2) g=1;fS=P_I

j=1c_jfE(x_j) whileg<Gdo

(A2.3) i=0;g+ + whilei<I(orδn≥∆)do

(A2.4) i+ +

(A2.5) f_S=f_S−c_if_E(x_i) (A2.6) £

x_i¤

=_x

¯

¯ D

fE(x),fA−fS E¯

¯

(A2.7) c_i= D

fE(x_i),fA−fS E

(A2.8) f_S=f_S+c_if_E(x_i) (A2.9) n=I g+i; δn=°

°f_A−f_S°

°

I extend the algorithm to be able to synthesize multiple target patterns with a single geometrical arrangement; in this case, the sum of the errors’ squares is minimized. This can for example be used to implement steering or changing the beamwidth when necessary.

For illustration, the algorithm is tested on the simultaneous synthesis of 8 circularly polarized patterns using 32 half-wave dipole antennas. The target patterns are given by the relations

f_A,₊(`,ϑ,ϕ)=N_Asin³^`ϑcos¹⁰^`φ

2( ˆϑ+j ˆφ), (6) f_A,₋(`,ϑ,ϕ)=N_Asin³^`ϑcos¹⁰^`φ

2( ˆϑ−j ˆφ), (7) where ˆϑand ˆφdenote unit vectors in the polar and azimuthal directions, and N_Ais the normalization factor. The patterns are characterized by`=2, 3, 4, 5.

In Fig. 8, we outline the algorithm’s speed of convergence, while in In Fig. 9, we illustrate the goal and synthesized field patterns.

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δn2(error)

100 200 300 400 500 600

0.05 0.10 0.50 1

ℓ=2 ℓ=3 ℓ=4 ℓ=5

n(iteration number)

Figure 8: Errorδ²n (with n=I g+i ) of synthesized patternsf_Swith respect to goal patternsf_A,+for`=2, 3, 4, 5. After i=32, the antennas already placed are repositioned once per generation. Generations are marked by white and gray bars in background of graph.

0 π π 2 3π

π

4 3π

2 4

0

0 π π 3π 2

π

4 3π

2 4

0

0 π π 2 3π

π

0

dB] 0

π π 3π 2

π

0

Figure 9: Synthesized radiation patternsfS plotted for the real part of theϑ- component (dashed) and the imaginary part of theϕ-component (dotted) for

`=2(first column) and`=5(second column) in planes corresponding toϑ=π/2 (first row) andϕ=0(second row). The finely dotted thin line indicates the goal patternf_A, which is the same forϑ- andϕ-components. The graph is normalized with respect to f₀– the maximal value of the goal pattern for`=5.

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Summary

Thesis 3: I implemented an iterative algorithm for the purpose of antenna array pattern synthesis and proved its convergence, furthermore, I extended the scope of the algorithm to allow the synthesis of arrays with a specified number of elements by repositioning, and multiple-goal patterns.

• Based on Dr. Kracek’s idea, I lay down the theoretical foundations of a novel iterative method of antenna array radiation pattern synthesis. The method can be applied to an arbitrary type of a basic antenna element, and is applicable to any goal radiation pattern. The algorithm works by iteratively placing antenna array elements through the rotation and trans- lation of a basic element.

• I prove that the synthesized radiation pattern converges to the best approximation of the goal pattern in the sense of the norm defined on a Hilbert space.

• I extend the algorithm for the correction of already placed elements and for multiple goal patterns.

• Finally, in an example, I design an array capable of producing multiple circularly polarized radiation patterns. The algorithm is able to handle this relatively complicated optimization problem with acceptable accuracy.

An improvement of the method’s numerical implementation could be a possible continuation of the work presented here, as I believe that the study of efficient minimization algorithms [18] and interpolation schemes [19] can further improve the results and speed of the algorithm.

Steering can be implemented through making use of the extension to multiple goal patterns on planar arrays (with different goal patterns corresponding to different radiation directions).

Related publications: [C6], [J4]

Publications unrelated to the Theses: [C7],[C8], [J5]

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Journal articles

[J1] K. Marák, T. Pet˝o, S. Bilicz, S. Gyimóthy, and J. Pávó, “Electromagnetic simulation of rotating propeller blades for radar detection purposes,”IEEE Transactions on Magnetics, vol. 54, no. 3, pp. 1–4, 2018.

[J2] ——, “Bistatic RCS calculation for propellers at near-resonant frequencies,”

International Journal of Applied Electromagnetics and Mechanics, vol. 59, no. 1, pp. 19–26, 2019.

[J3] K. Marák, S. Bilicz, and J. Pávó, “Experimental technique for high- frequency conductivity measurement,”COMPEL-The international jour- nal for computation and mathematics in electrical and electronic engineer- ing, vol. 38, no. 5, pp. 1711–1722, 2019.

[J4] K. Marák, J. Kracek, and S. Bilicz, “Antenna array pattern synthesis using an iterative method,”IEEE Transactions on Magnetics, vol. 56, no. 2, pp. 1–4, 2020.

[J5] K. Marák, “Characterization of the inverse problem in critical dimension measurement of diffraction gratings,”Periodica Polytechnica. Electrical En- gineering and Computer Science, vol. 60, no. 3, p. 187, 2016.

Conference articles

[C1] K. Marák, “Calculation of the radar cross section of small propellers by means of an efficient integral equation method,” inProceedings of the Mesterpróba conference, Budapest, 2017, pp. 20–23.

[C2] K. Marák, T. Pet˝o, S. Bilicz, S. Gyimóthy, and J. Pávó, “Electromagnetic simulation of rotating propeller blades for radar detection purposes,” inPro- ceedings of Compumag 2017 Daejeon. International Compumag Society, 2017, pp. 1–2.

[C3] K. Marák, S. Bilicz, S. Gyimóthy, J. Pávó, and T. Pet˝o, “Bistatic RCS calculation for propellers at near-resonant frequencies,” inProceedings of the In- ternational Symposium on Electromagnetics and Mechanics (ISEM) 2017, Chamonix, 2017, pp. 1–4.

[C4] T. Pet˝o, K. Marák, S. Bilicz, and J. Pávó, “Experimental and numerical studies on scattering from multiple propellers of small UAVs,” inProceedings of the 2018 European Conference on Antennas and Propagation (EuCAP).

IEEE, 2018, pp. 1–4.

[C5] K. Marák, S. Bilicz, and J. Pávó, “Experimental technique for high frequency conductivity measurement,” inIGTE Symposium, Graz, 2018, pp.

1–1.

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[C6] K. Marák, S. Bilicz, and J. Kracek, “Antenna array pattern synthesis using a novel iterative method,” inProceedings of Compumag, Paris. Interna- tional Compumag Society, 2019, pp. 1–2.

[C7] K. Marák, “Investigation of microchannel based heat sinks for vlsi circuits,”

inProceedings of the Mesterpróba conference, Budapest, 2016, pp. 1–2.

[C8] S. Bilicz, S. Gyimóthy, J. Pávó, P. Horváth, and K. Marák, “Uncertainty quantification of wireless power transfer systems,” in Wireless Power Transfer Conference (WPTC), 2016 IEEE. IEEE, 2016, pp. 1–3.

Further references

[1] T. Pet˝o, L. Sz ˝ucs, S. Bilicz, S. Gyimóthy, and J. Pávó, “The radar cross section of small propellers on unmanned aerial vehicles,” in2016 10th European Conference on Antennas and Propagation (EuCAP). IEEE, 2016, pp. 1–4.

[2] J. Van Bladel,Relativity and engineering. Springer Science & Business Me- dia, 2012, vol. 15.

[3] A. Mehdipour, T. A. Denidni, A. R. Sebak, C. W. Trueman, I. D. Rosca, and S. V.

Hoa, “Anisotropic carbon fiber nanocomposites for mechanically reconfig- urable antenna applications,” in2013 IEEE Antennas and Propagation Soci- ety International Symposium (APSURSI). IEEE, 2013, pp. 384–385.

[4] S. J. Orfanidis, “Electromagnetic waves and antennas, 2008,”Online Book Available at: http://www. ece. rutgers. edu/ orfanidi/ewa, 2008.

[5] W. C. Gibson,The method of moments in electromagnetics. CRC press, 2008.

[6] J.-M. Jin,Theory and computation of electromagnetic fields. John Wiley &

Sons, 2011.

[7] V. C. Chen, F. Li, S.-S. Ho, and H. Wechsler, “Micro-Doppler effect in radar: phenomenon, model, and simulation study,”IEEE Transactions on Aerospace and Electronic Systems, vol. 42, no. 1, pp. 2–21, 2006.

[8] S. Y. Yang, S. M. Yeh, S. S. Bor, S. R. Huang, and C. C. Hwang, “Electromag- netic backscattering from aircraft propeller blades,”IEEE Transactions on Magnetics, vol. 33, no. 2, pp. 1432–1435, Mar 1997.

[9] T. Ohtani and Y. Kanai and N. V. Kantartzis, “Wide-Angle Elimination of TF/SF-Generated Spurious Waves in the Nonstandard FDTD Technique,”

IEEE Transactions on Magnetics, vol. 53, no. 6, pp. 1–4, June 2017.

[10] J. Cooper and W. Strauss, “Scattering of waves by periodically moving bodies,”Journal of Functional Analysis, vol. 47, no. 2, pp. 180–229, 1982.

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[11] R. G. Vaughan, N. L. Scott, and D. R. White, “The theory of bandpass sam- pling,”IEEE Transactions on signal processing, vol. 39, no. 9, pp. 1973–1984, 1991.

[12] N. Bowler and Y. Huang, “Electrical conductivity measurement of metal plates using broadband eddy-current and four-point methods,”Measure- ment Science and Technology, vol. 16, no. 11, p. 2193, 2005.

[13] Y. Ju, Y. Hirosawa, H. Soyama, and M. Saka, “Contactless measurement of electrical conductivity of si wafers independent of wafer thickness,”Applied Physics Letters, vol. 87, no. 16, p. 162102, 2005.

[14] B. Yuan, L. Yu, L. Sheng, K. An, and X. Zhao, “Comparison of electromagnetic interference shielding properties between single-wall carbon nan- otube and graphene sheet/polyaniline composites,”Journal of Physics D:

Applied Physics, vol. 45, no. 23, p. 235108, 2012.

[15] W. Xiang, K. Zheng, and X. S. Shen,5G mobile communications. Springer, 2016.

[16] L. Poli, G. Oliveri, P. Rocca, M. Salucci, and A. Massa, “Long-distance wpt unconventional arrays synthesis,”Journal of Electromagnetic Waves and Ap- plications, vol. 31, no. 14, pp. 1399–1420, 2017.

[17] D. W. Boeringer and D. H. Werner, “Efficiency-constrained particle swarm optimization of a modified bernstein polynomial for conformal array excitation amplitude synthesis,”IEEE Transactions on Antennas and Propagation, vol. 53, no. 8, pp. 2662–2673, 2005.

[18] T. Bauernfeind, P. Baumgartner, O. Biro, C. A. Magele, K. Preis, and R. Tor- chio, “Peec-based multi-objective synthesis of non-uniformly spaced linear antenna arrays,”IEEE Transactions on Magnetics, vol. 53, no. 6, pp. 1–4, 2017.

[19] S. Bilicz, “Sparse grid surrogate models for electromagnetic problems with many parameters,”IEEE Transactions on Magnetics, vol. 52, no. 3, pp. 1–4, 2016.