Predicting the Bistatic Scattering of a Multiport Loaded Structure Under Arbitrary Excitation: The S-Parameters Approach

Various applications, including reconfigurable intelligent surfaces (RISs), radio frequency identification, and ambient backscatter devices, are based on scattering. Predicting the scattering properties of these systems accurately and universally in a computationally efficient manner is crucial. In this article, we propose a model for predicting the scattering properties of an electromagnetic structure controlled by loads terminated to multiple ports. This model is based on: 1) S-parameters describing the coupling between the ports; 2) embedded element radiation patterns associated with each port; and 3) structural scattering under multiple incident wave directions. To construct the model, one set of electromagnetic descriptions (e.g., simulations or measurements) needs to be done for a structure before computing the scattering properties of the structure for arbitrary tunable load values. Unlike many other methods, the proposed method fully takes into account structural scattering in different directions simultaneously and requires no simplifications or approximations to the scattering structure, such as the assumption of local periodicity or element identity. This method facilitates characterizing the scattering ability of the structure in terms of bistatic cross section (BCS), also known as bistatic radar cross section (bRCS), and can be beneficial, for instance, in designing RISs and backscatter systems. Simulations and experiments at different frequencies verified the proposed model.


I. INTRODUCTION
I N RECENT years, scattering systems have been widely used in communication technologies.The applications can be primarily divided into two common ideas: redirecting the signal-carrying wave in the required direction and modifying ambient waves to be reused for communication and sensing purposes.
Smart radio environment [1] is an example of a system redirecting signals, for instance, with the help of reconfigurable The authors are with the Department of Electronics and Nanoengineering, Aalto University, 02150 Espoo, Finland (e-mail: aleksandr.kuznetsov@aalto.fi).
Color versions of one or more figures in this article are available at https://doi.org/10.1109/TAP.2024.3418517.
Digital Object Identifier 10.1109/TAP.2024.3418517Fig. 1.Schematic view of the antenna-based approach of a scattering system consideration.
intelligent surfaces (RISs) [2].RISs can create favorable propagation paths by redirecting the incoming waves.Moreover, RISs are applicable for temporal, spectral, and spatial signal manipulations [2], [3].Temporal and spatial signal modifications of the scattered fields are also used in backscatter communications [4] where new information is added to a propagating wave for communication purposes [5].One subset of backscatter communications systems-ambient backscatter systems-is being researched specifically for the needs of the Internet of Things [5], [6], [7], [8].
The metamaterial-based and antenna-based approaches are commonly used to analyze scattering structures for communication purposes [2].The metamaterial-based approach considers a scattering system a medium with electromagnetic properties conditioned by its unit elements, the size of which is much smaller than the wavelength.This approach is well-investigated and has several applications [2], [18].One of them-the digital metasurface concept-became widely popular.In this concept, the metasurface comprises unit elements whose discrete states can be adjusted to achieve the required operation [19], [20], which limits the number of implementable operation modes of the scattering system.
In the antenna-based approach, the scattering system typically contains antenna-like structures (Fig. 1).In many cases, such as in [14], [16], [21], [22], and [23], the scattering system was considered an antenna array with perfectly uncoupled elements.Under this assumption, the scattering of the array may be computed based on the standard antenna array theory.However, the mutual coupling can significantly affect the scattering properties of an antenna system and should therefore not be neglected [24].Gradoni and Di Renzo [13], Akrout et al. [17], Zhang et al. [25], Williams et al. [26], Sneha et al. [27], Peebles [28], Vuyyuru et al. [29], and Li et al. [30] proposed the coupled arrays models where radiated E-field values also consider the mutual coupling of elements.However, these models are best applied using canonical E-field or current distributions (e.g., that of the Hertzian dipole), which are challenging to realize in practice.
Another limitation with some of the methods operating with a bistatic cross section (BCS), for example, those presented in [22], [42], and [43], is that they assume identical scattering elements or disallow their termination with arbitrary loads beyond the digital case.These factors collectively narrow the applicability of the researched algorithms.
Given the limitations of the existing approaches, this article proposes a model for predicting multistatic scattering properties (any incident and scattered propagation directions) with arbitrary load impedances connected to coupled reradiating elements.To perform this aim, the scattering system is characterized using S-parameters, which facilitate the separating of structural, coupling, and load effects on the system.The novelty of the proposed algorithm, compared with the existing models, lies in the absence of assumptions about minimum scattering, the uniformity of scattering elements, the periodicity of the structure, or restrictions on load impedance values.The model is physically consistent and comprehensively describes the scattering properties of the multiport system using S-parameters for an arbitrary set of loads and incident wave angles.This versatility supports its application in optimizing scattering systems for different purposes.
In addition, the model can operate with theoretical, simulation, or measured data and compute scattering in multiple directions simultaneously, enhancing its practical applicability.A coupled antenna-based scattering system with different load values was created, measured, and compared with standard simulation results to verify the proposed model empirically.
This article is organized as follows.Section II introduces the proposed algorithm and explains how the underlying S-parameter model is constructed from the traditional S-parameters, port-specific far-field patterns, and structural scattering values.Section III addresses the testing of the coupled antenna-based structure and its measurement setups.Section IV describes the computation and measurement results and includes an analysis of the applicability of the algorithm.Finally, Section V presents the conclusion.

II. THEORY
In this section, we present an S-parameter-based model to predict the multistatic scattering of the structure terminated with arbitrary loads.The full system-level scattering matrix comprises several submatrices.One submatrix describes structural scattering, two others account for couplings between the ports of the scattering elements, and two more matrices characterize the relationship between the far fields and the ports.First, we address the relationship between the waves and the multiport scheme.Second, we discuss the formation of each submatrix using the values that computations or measurements can produce.Finally, we explain how systemlevel S-parameter values are related to physical BCS values.

A. Developing a Model Based on S-Parameters Matrices
Fig. 2 is the schematic representation of the developed model.First, following [31], the entire scattering system is modeled as an S-parameter matrix describing the connection between N antenna ports and M radiation ports.Second, since all the antenna ports are terminated to the known load values, the input and output signals between load ports can be connected through S-parameters of the loading network analogically to [33], [35].Finally, to introduce the BCS computation, we refer M radiation ports to M different radiation directions.The received model, therefore, describes the relationship between input (a 1 , . . ., a M ) and output (b 1 , . . ., b M ) power waves [44]   For this research, instead of introducing radiation ports for each polarization, common for the models based on [31], we decided to construct four separate [S sys ] matrices which connect signals between radiation ports for the given load values and each combination of polarizations of input and output signals.The following matrix equations could then describe the entire system [35], [37], [45] where the superscript index means polarization or the characterized transition between polarizations and I is the identity matrix.Important to note is that load ports are not connected with polarizations; for all (1), the same [S dd ] and [S L ] matrices must be used, while all other matrices depend on polarizations.The formation of ten S-parameter matrices is thus required.

B. Scattering Matrices Formation
The following text describes the formation of each submatrix and input values to the entries.In every case, calculations, simulations, or measurements can generate the values.
Matrix [S dd ] connecting loading ports characterizes the couplings and mismatch reflections in the reference case and it is the standard S-parameter matrix of the antenna array.
Matrix [S L ] describes the load network connected to the antenna ports.Its diagonal elements represent the reflection coefficient of the power waves from the load computed using [44] S where Z ref is the reference impedance and Z n is the load impedance at port n.Nondiagonal elements describe the coupling between loads.In practice, the [S L ] matrix is usually considered diagonal neglecting coupling between load elements.All other matrices are connected to propagation directions, which requires characterizing input and output signals from the corresponding ports.The chosen parameter for this aim must take into account phase information to address interference effects, be measurable for practical applicability, and describe radiation in different directions.In this research, the E-field radiation patterns are used to fulfill these criteria. .This parameter, connected with the E-field, meets the requirements for signals on radiation ports.Its connection with the E-field pattern must then be revealed.
If the middle point of the scattering system is located at the origin of the local system of coordinates, all the antennas in the system are loaded by conjugate to reference impedance Z * ref (reference case), and a transmitting antenna forms the signal electric field intensity of which near the origin equals ⃗ E s , then from definitions of antenna realized gain and radiation intensity where ⃗ S s is the power density of the signal near the scatterer and P t is the power fed to the transmitting antenna.For the conjugate-matching condition between load and reference impedance, the input and output power of the system equals [44] Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where P r is the power on the load of the receiving antenna.Since for lossless media where η is the impedance of the medium, the amplitude of the power wave on the transmitter is At the same time, for the scattered signal ⃗ E r near the receiving antenna, using the Friis transmission formula, we set After we apply (4b) and ( 5), (7) gives for power waves on the receiver side Power waves are thus connected with obtainable E-field values.For S-parameter matrices' compatibility, the E-field phase refers to the origin of the system.Forming the remaining matrices is therefore possible.For each direction of receiver ϱ = (ϕ r , θ r ) and transmitter τ = for all the polarization combinations, then where α ϱ stands for the phase radiation pattern in ϱ direction with respect to the origin.According to the definitions in [46], [S ff ] hence describes the structural scattering of the structure for Z ref reference impedance.Matrices [S fd ] and [S df ] are computed from the E-field patterns associated with excitation ports.For each port n of the scatterer and receiver direction ϱ with input power wave a ′ n having power P n with termination of other ports to Z * ref , from ( 4) and ( 8) where ⃗ E ϕ r,ϱ and ⃗ E θ r,ϱ represent the ϕand θ-components of the E-field near the receiving antenna in ϱ direction, while α ϕ ϱ and α θ ϱ denote ϕand θ-components of the phase radiation pattern in ϱ direction with respect to the origin, respectively.Applying the reciprocity principle and keeping the same distance between M antennas and the scattering system, we have From ( 9)-( 11), we can compute the required matrices for the connection of radiation ports.Due to the use of the Friis transmission formula, the derivation is performed for a homogeneous medium, which is characterized by impedance η.Based on values ⃗ E s (for scattering) and P n (for radiation pattern), we can compute the E-field pattern ⃗ E r theoretically or using simulation tools at a distance of s t = s r for each direction.Practical measurement of [S ff ] is a measurement of S-parameters between M measurement antennas with scattering structure, all loads of which are equal to conjugate reference impedance value.Measuring [S df ] can be performed similarly but also by connecting the signal source to the ports of the scattering system.
As a result, the model includes descriptions of all the matrices.From the descriptions, all the matrices can be obtained using theoretical, simulation, or practical approaches.These matrices themselves can describe the scattering of antenna-based systems, including structural scattering, for both polarizations, which may be enough for application in communication engineering (analogically to [13], [39]).For cases of a not-free-space environment, each matrix may be redefined based on the scattering properties of the linearly behaving passive objects in the surrounding and their influence on the radiation properties of the multiport scatterer, which is beyond the scope of this article.At the same time, based on the applicability of the superposition principle for S-parameters due to the absence of nonlinear components in the system, it is possible to use obtained [S sys ] matrices for computation of multistatic radar systems' behavior.Moreover, the separate definition of the scattering matrices based on the parameters of the structure allows applying the algorithm for the generation of datasets required for supervised machine learning optimization tasks (e.g., optimization of load values in [47]).

C. Relationship Between the System S-Parameters and BCS
Radar cross section (RCS) is a convenient concept to assess an object scattering in a certain direction.We thus relate the system S-parameters to RCS.Note that our formulation includes a general BCS for different angles in addition to the commonly used monostatic RCS.Using E-field values, we can compute RCS as [46] Following the same computation process as for (9) for the system with loads under study terminated to the scattering system, BCS components for polarization using inverse proportion between the electric field intensity and the square of the distance from the scatterer can be expressed as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE I STRUCTURE SIZES
Exposition of input signal polarization is possible as a combination of ϕand θ-polarizations using the corresponding E-field coefficients p s ϕ , p s θ , which follows: It provides the general BCS values in the corresponding matrix As a result, (15) forms a matrix [σ ] of BCS values connecting all the directions for any combination of loads and polarizations.By its structure, each column [σ ] :,τ represents BCS of the structure for the plane wave coming from τ direction.Moreover, the main diagonal [σ ] ϱ=τ is an array of monostatic RCS values of the structure.The obtained matrix therefore fully describes the scattering ability of the structure.Although (15) includes s t and s r , [σ ] does not depend on these parameters, as an inverse dependence of S sys ϱ,τ on the distance between the scattering system and the measurement antennas occurs.

III. EXPERIMENT
In this section, we introduce a scattering system based on multiple coupled antennas to confirm the theoretical results obtained presented in Section II.

A. Structure of the Scattering System Under Test
The test structure comprises 15 near half-wave (at f = 2.4 GHz) dipole antennas on the printed circuit board (PCB) (Fig. 4).The structure has three identical layers, each containing five antennas.Since the proposed method does not require periodicity of the scattering structure, we chose to Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.use elements of slightly different sizes in a nonregular array.Separation distances between neighboring antennas less than half of the wavelength at the projected frequency were thus chosen to provide coupling, while the operation of all the antenna elements occurred near one dominant frequency.The structure sizes are listed in Table I.
To align layers of the structure, screws were used; ten M3 screws (five along each highest dimension of the PCBs with equal separation between) provided a proper connection between layers.Besides, dielectric block sleeves around screws (two per each) were used to prevent shifts in any direction in the plane of the board and establish equal distance between the antenna structures along the screws.
Dipoles were produced on 0.76-mm-thick RO4350B substrate (ϵ r = 3.66, tanδ = 0.0031) with 35-µm-thick copper foil.For the supporting structure, polyamide screws and nylon nuts were used, while sleeves were made from Preperm RS260 (ϵ r = 2.58, tanδ = 0.0009) material.These options are a compromise between mechanical stability and a low effect on the electromagnetic properties of the structure.
Five different sets of dipole arrays, representing identical electromagnetic structures with different load values, were selected to demonstrate the effect of the load impedances on scattering according to the developed model.In one case (i.e., the reference case), all the ports were terminated with 50-resistances.In other cases, load values were chosen to maximize scattering in specific predefined directions.Thus, the reference impedance used for S-parameters' computation in this research is Z ref = 50 .Table II reveals the values of the connected to the antennas loads in each case.In this research, the capacitive load is a 0.9-pF capacitor (GJM1555C1HR90WB01D) and the inductive load is a 3.6-nH inductor (Coilcraft 0402DC-3N6) only.Fig. 5 shows examples of practical implementations of the developed structure for cases III and IV described in Table II.

B. Measurements Setup
We measured BCSs in an anechoic chamber with a similar setup to that used in [35].BCS values were measured using the Microwave Anechoic Chamber at Aalto University.
Two ETS-Lindgren's dual-polarized 3164-08 Open Boundary Quad-Ridged Horn antennas were used as measurement antennas to illuminate the structure under test and to read the scattered signal.A four-port VNA (Rohde&Schwarz ZNA67) was used to measure S-parameters.Several samples with different loads were prepared (Fig. 5, Table II) to prevent damage connected with the resoldering of load components to the same structure.For practical reasons, the BCS patterns were sampled on a horizontal cut only (θ = 90 • ).
Fig. 6 reveals the measurement setup.For stable and repeatable sample orientation, the sample under test (SUT) was attached, in two points, to a fishing wire stretched between antenna columns in the anechoic chamber.The measurement antennas for all the angles were placed at distance r h = 1.8 m, limited by the chamber size, and the frequency range was limited by f ∈ [2, 3] GHz.Scattered fields were sampled at ϕ = 10 • intervals: Fig. 6(a) demonstrates the possible measurement antenna locations, which were used for all the measurement cases, while Fig. 6(b) demonstrates the measurement process for one structure and one pair of directions.In addition, to remove the effect of nondesired reflections in the measurement setup, we used a calibration process.For that aim, the response of the empty room (i.e., the room without the sample) for each direction combination was measured.For this research, we chose (ϕ τ = 40 • , θ τ = 90 • ) as the income direction, which permanently established the location of the TX measurement antenna (Fig. 6).

C. Postprocessing Methods
To obtain the BCS values, we processed the measured S-parameters.The measured scattering data were corrected to exclude the influence of the empty room.Assuming the total scattering results are a superposition of contributions from the sample and the empty room and no interaction between them occurs, S sys can be computed as where C is the case mark and E is the empty room condition.Afterward, ( 15) is used to obtain BCS values from the measured S-parameters (using s t = s r = r h = 1.8 m).
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The comparison using relative root-mean-square error (RRMSE) metrics corresponding to the measurement and simulation results was implemented to evaluate the precision of the obtained results

RRMSE
where σ meas means measured, σ sim means simulated in CST Studio Suite, and σ comp means computed using the proposed algorithm BCS values.This metric was chosen due to its balance for mismatch accounting without overweighting mismatches near nulls of the patterns.
In addition, we evaluated the impact of uncertainties in the measurement setup on the results.For this purpose, six measurements of S-parameters for each scattering angle and structure were performed.Then, the result of measurements was defined as a mean value over the measured values with the maximal absolute deviation from this value as the uncertainty in both sides.

IV. RESULTS AND DISCUSSION
This section demonstrates the calculated, measured, and simulated results of different prototypes.Since the measurement was performed for linearly polarized scattering antennas, the comparison of the results here is also provided for the same linear polarization.The computed results were calculated using (1), ( 2), ( 9)-( 11), (15), using S-parameters, port-specific embedded radiation patterns, and scattering properties obtained from the reference case from CST simulations.The important feature of the structure under study is its ability to reconfigure its scattering properties by changing the loads terminated to the scattering antennas that should be predictable by the algorithm.Fig. 7(b)-(d) and 8(a) thus demonstrate that one of the BCS pattern lobes may be redirected to the required angle following load values.For example, the comparison of structural scattering in Fig. 7(a) and case II in Fig. 7(b) reveals the increase in the lobe directed to ϕ = 220 • and the occurrence of the comparablein-magnitude lobe directed to ϕ = 90 • for the not-reference case.
Although the prototype was designed to operate at 2.4 GHz, we performed measurements at other frequencies to determine whether the model also correctly predicts scattering in a broad frequency range.Fig. 9 demonstrates the presence of this feature for the proposed algorithm: at f = 2.0 GHz and f = 2.7 GHz, the patterns do not have the lobe directed to ϕ = 300 • (unlike in Fig. 8), and the computed, simulated and measurement (with a small number of outliers) results coincide between each other.These observations prove the reliability of the algorithm for computation at different frequencies.
As described in Section III, the uncertainties on Fig. 8(b), Fig. 9 were evaluated based on the repetition of the experiment.Their possible sources are connected with the measurement setup specificity: location inaccuracy (including the variation of the phase center location for the measurement antennas), an imperfection in the angular alignment of samples (including mechanical instability), equipment imperfection, and the presence of alternative scattering paths.Moreover, the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.proximity of the measured S-parameter values for different conditions increases the influence of the measurement imperfection on the resulting uncertainty.For example, Table III demonstrates the values of the measured S-parameters for the receiving antenna at ϕ = 300 • direction corresponding to the scattering structure at three different frequencies (2.4,2.0, and 2.7 GHz) displayed on Figs.8(b) and 9.It is possible to see a small difference in values between S-parameters for empty room and sample scattering conditions, especially for low scattering levels by the sample (at f = 2 GHz).Then, the combination of uncertainties of these measurements influences stronger on the BCS values due to the complexity of the measurement setup.Nevertheless, the shape of the measured BCS pattern in the azimuthal plane follows computation and simulation results, which support the proposed algorithm.
Table IV displays the evaluation metric values for comparing measurement and computation results.Furthermore, Fig. 10 illustrates the distribution of RRMSE values for comparing the computed results with the simulation in CST and measurement in anechoic chamber results.The mean value of RRMSE sim is less than 0.022% (for all cases at frequencies 2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, and 3 GHz), which indicates the possibility of relying on the computed results with the same degree of trust as with the simulation ones.Fig. 10  demonstrates that only two outliers had RRMSE sim values higher than 0.06%.Moreover, the mean of RRMSE meas values is 2.1%, which indicates the representativeness of the results for practical use.The measured values reveal similar scattering pattern and are located on both sides of the simulated and calculated values (Fig. 7-9), which suggest that deviations are related to measurement uncertainties rather than systematic errors.The increased relative error at f = 3 GHz (Table IV) may be explained by a low level of the scattering of the structure at this frequency and approaching the limit of far-field distance for the measurement setup.As a result, the performance of the proposed algorithm was proven by simulations and practical measurements.Based on the radiation patterns of the structure and its structural scattering, the computed BCS values represent the scattering of the structure with different load values terminated to antennas and can be applied to modify the structure.

V. CONCLUSION
In this article, we present an algorithm for computing the BCS of multiport loaded structures using S-parameters based on antenna parameters.The proposed algorithm works with different sources of characteristics of the scattering system at different frequencies for the nonperiodical location of different scatterers loaded by various impedance values and facilitates simultaneously computing all possible BCS values of the structure based on the arbitrary angle of the incident wave.The provided separation of parameters connected with load values supported by the computational lightness of the model allows the utilization of the algorithm to optimize load impedance values.The presence of structural scattering in the model expands the optimization possibilities in communication engineering, while an absence of necessity to resimulate the structural scattering of the system for each set of loads may be used to generate large datasets for applying supervised machine learning approaches.We supported the performance of the algorithm by comparing its results with the simulation and measurement values.Mean RRMSE values at levels 0.022% and 2.12% correspondingly demonstrate the matching of the received results at different frequencies.The designed examples illustrate the applicability of the algorithm for all the scattering directions, different types of loads, and ambient intended redirection angles.
The algorithm may be developed in several ways.First, reducing the volume of initial data can be achieved by applying statistical methods or approximations.Second, expanding the method connected with near-field illumination may be applicable in some areas.Third, using theory to compute the structural scattering of antenna arrays presents an opportunity to optimize the structural parameters using the same algorithm.Finally, introducing alternative channels between transmitting antennas is valuable for communication engineering purposes.

Manuscript received 11
January 2024; revised 29 May 2024; accepted 17 June 2024.Date of publication 1 July 2024; date of current version 9 August 2024.This work was supported in part by the WALLPAPER Project of the Academy of Finland under Grant 352913.(Corresponding author: Aleksandr D. Kuznetsov.) Fig.2is the schematic representation of the developed model.First, following[31], the entire scattering system is modeled as an S-parameter matrix describing the connection between N antenna ports and M radiation ports.Second, since all the antenna ports are terminated to the known load values, the input and output signals between load ports can be connected through S-parameters of the loading network analogically to[33],[35].Finally, to introduce the BCS computation, we refer M radiation ports to M different radiation directions.The received model, therefore, describes the relationship between input (a 1 , . . ., a M ) and output (b 1 , . . ., b M ) power waves[44] corresponding to M radiation ports using five S-parameter matrices: the M × M matrix [S ff ] connects radiation ports; the N × N matrix [S dd ] connects discrete load ports; the N × N matrix [S L ] describes the S-parameters of the load network used to terminate the antenna ports; and the N × M [S df ] and M × N [S fd ] matrices connect radiation and

Fig. 2 .
Fig. 2. Schematic view of the S-parameters approach for each combination of polarizations of input and output signals.

Fig. 3 .
Fig. 3. Schematic view of the system with a marked E-field.

Fig. 3
illustrates the general structure of the scattering system.Uncoupled antennas are in M directions at far-field distance s m , where m is the number of directions under consideration.Being characterized by antenna impedance Z a m and realized gain G m for conjugate to the reference impedance Z * ref termination case, these uncoupled antennas support creating the connection between power waves ⃗ a and ⃗ b [44]

Fig. 4 .
Fig. 4. Architecture of the structure under research.

Fig. 6 .
Fig. 6.Measurement setup for the BCS in the anechoic chamber.The position of the transmitting antenna (TX) is stable at ϕ t = 40 • , θ t = 90 • , while the position of the receiving antenna (RX) is changed between measurements.(a) Schematic top view of the measurement setup.(b) Photograph of the measurement using the described setup.

Fig. 7 .
Fig. 7. Azimuthal (x y) plane of BCS in m 2 for the structure under study at f = 2.4 GHz.(a) Reference case, (b) case II, (c) case III, and (d) case V.For (b)-(d), light blue arrows mark the intended scattering direction.

Fig. 8 .
Fig. 8. Azimuthal (x y) plane of BCS in m 2 for case IV of the structure under study at f = 2.4 GHz in (a) polar and (b) Cartesian coordinate systems.

Fig. 7 -
Fig. 7-9 show the calculated, simulated, and measured scattering patterns of different cases.Fig. 7 demonstrates the BCS values for the cases under study listed in Table II.Due to the measurement limitations for each plot, only the azimuthal plane of the BCS of the structure at 2.4 GHz is displayed.Each plot includes marked computed points using the proposed algorithm and the cubic spline interpolation based on them (blue, solid), CST simulation results (red, dashed), and measured values (green, points).In addition, an arrow marks the intended scattering direction in Figs.7(b)-(d) and 8(a).Fig. 8 illustrates plots in the polar and Cartesian coordinate systems for case IV.Fig. 8(b) includes the value of the uncertainty of the BCS values from different sources during the measurement process computed from the series of measurements.Fig. 9 demonstrates BCS for case IV but at nondesigned frequencies: 2.0 and 2.7 GHz. Figs.7-9 reveal a high correlation between the computed and measured results and the matching of measurement patterns and computation results.The important feature of the structure under study is its ability to reconfigure its scattering properties by changing

Fig. 9 .
Fig. 9. Azimuthal (x y) plane of BCS in m 2 for case IV of the structure under study at (a) f = 2.0 GHz and (b) f = 2.7 GHz.

TABLE II CONNECTED
LOADS VALUES FOR DIFFERENT CASES OF THE STRUCTURE UNDER TEST Fig. 5. Practical implementations of the structure under test for cases III and IV (TableII).

TABLE III EXAMPLES
OF VALUES OF S-PARAMETERS FOR MEASUREMENT OF CASE IV OF THE STRUCTURE UNDER STUDY AT ϕ = 300 •

TABLE IV VALUES
OF RRMSE MEAS FOR DIFFERENT CASES OF THE STRUCTURE UNDER TEST