Multi-taper-based automatic correction of non-anechoic antenna measurements

Prototype measurements belong to the key steps in the development of antenna structures. Although accurate validation of their far-field performance can be realized in dedicated facilities, such as anechoic chambers, the high cost of their construction and maintenance might not be justified if the main goal of measurements is to support teaching or low-budget research. Instead, they can be performed in non-anechoic conditions and then refined using appropriate correction algorithms. Unfortunately, the existing post-processing methods suffer from multiple challenges that include manual setup of parameters as well as validation of performance in idealized conditions. In this communication, a multi-taper-based framework for correction of antenna characteristics obtained in non-anechoic environments has been proposed. The algorithm augments one-shot measurements of the structure under test in order to extract the line-of-sight responses while attenuating the interferences pertinent to multi-path propagation and noise from external sources of radiation. The performance of the proposed correction routine has been demonstrated in two test sites using a geometrically small Vivaldi radiator and validated against state-of-the-art techniques from the literature. The uncertainty budget for the measurements performed using the approach amounts to 0.26 dB, which is low given challenging propagation conditions considered for experiments.


Introduction
Experimental validation of prototypes is an inherent step in the development of novel antenna geometries.It is normally performed in professional, dedicated laboratories that include Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.anechoic chambers (ACs), compact-range, near-field, or opentest sites [1][2][3][4][5][6][7][8].Despite high accuracy-resulting from strict control of the propagation environment-high construction cost of the mentioned facilities might be unjustified from the perspective of applications such as training of students, or budget-tight research.From this perspective, neglecting the control of radiation environment in favor of installing the equipment at sites deemed unsuitable for the validation of antenna far-field properties might substantially reduce measurement-related expenses [9].Example locations include office cubicles, hallways, courtyards, or parks [9][10][11].Unfortunately, the responses directly obtained in such non-anechoic conditions are useless for rendering meaningful conclusions on the far-field performance of the antenna under test (AUT) [9][10][11][12].
The problem concerning insufficient fidelity of nonanechoic measurements can be mitigated using appropriate post-processing techniques.Popular methods are based on the concepts of: (i) signal decomposition and (ii) characterization of the propagation environment .The first group involves extracting the useful part of the responsepertinent to line-of-sight transmission between the reference antenna (RA) and AUT-using time-, or frequencydomain methods [9][10][11][12][13][14][15][16][17][18][19][20].The former exploits window functions centered around the signal pulse pertinent to direct transmission [9][10][11][12][13][14].The frequency-based routines focus on approximating the relevant fraction of the response using truncated compositions of basis functions.The latter are in the form of, e.g.complex exponentials or spherical-wave coefficients [15][16][17][18][19][20][21][22][23].Depending on the implementation, the correction process might involve the analysis of either single-, or multi-point data [19,22].The second class of methods is oriented towards extracting the effects of the propagation environment on the RA-AUT transmission.This is predominantly realized based on a series of independent measurements performed in the laboratory [12,20,24].Practical approaches to the problem involve analysis of the AUT in multiple locations within the test-site [24], extraction of the noise floor in the environment [12], or estimation of the equivalent currents on a hull that encloses the radiator [28].
The above-mentioned methods suffer from multiple challenges that make them of limited use for measurements in uncontrolled environments.These include cognition-based (i.e.experience-driven) setup of correction parameters or validation of performance in conditions that represent a much less demanding environment compared to standard office rooms (e.g.ACs, or semi-ACs with installed reflected surfaces) [15][16][17][18][19][20]32].Furthermore, in the literature, the postprocessing parameters are fine-tuned to the given test conditions, which hinders transferring the setup between test sites [18,20,24].From this perspective, the problem pertinent to correcting antenna measurements performed in non-anechoic environments remains open.
In this communication, a framework for automatic correction of measurements performed in uncontrolled environments based on multi-taper functions has been proposed.The approach involves analysis of the RA-AUT impulse response using a series of discrete prolate spheroidal functions so as to augment the relevant part of the signal while attenuating the noise (in the form of electromagnetic radiation from external systems and/or multi-path interferences).Due to multi-modal character of the functional landscape, the post-processing results are obtained as a convex combination of responses refined using locally optimal multi-taper-based functions.The performance of the algorithm has been demonstrated based on a total of 20 experiments performed at two non-anechoic test sites.The tests involved measurements of far-field performance characteristics (radiation patterns) of a geometrically small Vivaldi antenna at a total of 10 unique frequencies.Evaluation of the method in terms of uncertainty budget yields 0.26 dB, which is low given the challenging propagation conditions.From the standpoint of radiology, practical applications of the considered antenna structure might include incorporation as a part of the microwave imaging system for breast cancer detection [33][34][35].A comparison of the method against state-of-the-art techniques from the literature has also been provided.

Methodology
In this section, the multi-taper-based correction procedure is discussed.In order to make the work self-consistent, we provide a formulation of the problem pertinent to the refinement of non-anechoic measurements.Next, the correction procedure that involves transformation of the measurements as well as construction and optimization of the multi-taper kernel functions is explained.The section is summarized by a discussion of the proposed algorithm.The numerical results are provided in section 3, whereas benchmarks of the method and discussions are provided in section 4.

Problem formulation
Let R u (ω, θ) be the family of uncorrected S 21 responses of the RA-AUT system [36].
T the angle of AUT rotation w.r.t.RA.The response R u (ω, θ) is distorted by multi-path interferences and electromagnetic noise (i.e. from external systems that share the same frequency spectrum) which makes it of limited use for evaluating the real-world AUT performance.Next, let R c * = R c * ( f 0 , θ) be the refined response of the radiator (as a function of θ angle) at the frequency of interest.The proposed post-processing involves spectral analysis of R u (ω, θ) using a multi-taper approach in order to obtain R c * ( f 0 , θ), which is an approximation of the measurements conducted in a controlled setting, such as an anechoic chamber [1,37].

Multi-taper-based response correction
A multi-taper approach enables spectral analysis of the signal.Compared to conventional Fourier-based methods, it maintains a trade-off between the frequency resolution and variance of the corrected responses [38][39][40].The method generates a set of orthogonal taper functions-also referred to as discrete prolate spheroidal sequences (DPSSs)-that enable constructing low-bias, statistically consistent estimators capable of reducing spectral leakage of the data [38][39][40][41][42].The multitaper-based correction of the time series responses extracted from R u (ω, θ) is as follows. Let , where n = 1, 2, …, N-1 and s = 1, 2, …, n refer to the segment and step lengths, respectively (both defined in points).Note that n = 1 (and thus s = 1) will produce N segments of T u , each being of unit length (i.e. a single discrete time-instance), while for n = N-1 there will be ⌊(1 + s)/s⌋ segments of N-1 points in length (i.e. for s = 1 yielding N-2 points of overlap between consecutive T u fractions, or a single non-overlapping segment for s ̸ = 1).Additionally, for s = n there is no overlap between the subsequent T u segments.In practice, overlapping of segments results in temporal interpolation of data that could increase the precision of spectral events identification (here, separation of the noise from the relevant part of the RA-AUT transmission) [38].Here, 1 ⩽ s ⩽ n ⩽ N/2 is assumed to ensure a reasonable T u segmentation.
To perform multiple tapering of segments from T u , one has to define appropriate DPSS kernels.The problem corresponds to finding the n-point finite energy sequences that maximize the spectral concentration ratio for the selected 2 W bandwidth, where W < 0.5∂t -1 .The sequences and their concentrations can be represented using a composition of eigenvectors and their corresponding eigenvalues (bounded by 1) of the n-by-n self-adjoint positive semidefinite operator [44][45][46][47].Considering the ith segment of T u , the process of finding all DPSS kernels T κ (t s , w) can be formulated as follows: where w = 0, 1, …, n-1 denotes the order of the identified sequence, γ = 0, 1, …, n−1, whereas λ w is the eigenvalue of the corresponding eigenvector T κ (t s , w) of order w.Note that the 2 W bandwidth defines the spectral resolution of the multitaper method, i.e. the main lobe width in the spectral estimate, which controls the minimum distance between peaks that can be resolved [38,42].The optimal number of DPSS tapers is given as [38,42].The example DPSS sequences-with normalized amplitude and ∂t = 1 ns (for visualization)-obtained using (1) are shown in figure 1(a).Upon identification of kernels, the post-processing of nonanechoic measurements is performed as follows.Each predetermined segment of the data time series is modified using w opt tapers.Then, the n-point Fourier transforms of individual events are calculated, and their convex combination (weighted using the eigenvalues λ w ) is calculated to get the frequencydomain response for each segment.The process for the ith segment T s can be depicted as: where the symbol • denotes the component-wise multiplication, R cs (Ω s , θ a ) is the frequency-domain response for the ith segment T s , and Concatenating the frequency-domain responses for every segment and applying arithmetic mean to the overlapping points (when s < n), will yield the frequency-domain response The corrected nonanechoic response of the RA-AUT system R c ( f 0 , θ a ) is extracted at f 0 ∈ Ω (note that ω ∈ Ω) and θ a angle.To find the R c ( f 0 , θ), the above-outlined procedure is executed for all θ a angles.

Identification of optimal DPSS parameters
The DPSS-based correction performance heavily depends on the determination of appropriate segment lengths n and steps s, whereas the effects of W and t HB on the behavior of kernel functions are less pronounced (here, t HB = 4 is used).The optimal number of segments and steps required to refine the measurements performed at the given frequency of interest are identified using the scalar objective function of the form: where x = [x 1 x 2 ] T = [n s] T is the vector of setup parameters, R c (x) is the corrected R c ( f 0 , θ) response obtained using the DPSS kernels resulting from x, and is the reference performance figure (here radiation pattern) obtained from EM simulations.The multiplicative factor represents an analytical solution to the curve fitting problem that minimizes the discrepancy between R c (x) and R f through appropriate (single-value) scaling of the latter.The goal of scaling is to ensure that minimization of (3) is oriented towards identification of DPSS components that maintain the appropriate shape of R c (x) rather than match its amplitude to the one resulting from simulations.In other words, it aids in rejecting the responses that do not resemble the expected antenna response.As shown in figure 1(b), the example DPSS-based functional landscape resulting from (3) is highly multimodal, which makes identification of optimum parameters difficult.Here, the problem is mitigated using the exhaustive search procedure where the objective function ( 3) is evaluated for a set of designs X = {x p } 1⩽p⩽P .Next, the domination-based ranking is performed in order to identify a set of X opt ⊂ X designs that correspond to local minima.The corrected responses R c (X opt ) are then used to extract the final result R c * .The latter is obtained as an average of solutions R c (x β ), β = 1, 2, …, and x β ∈ X opt , that represent the σ-quantile of all locally optimal responses (here, σ = 0.1).

Summary of the approach
The presented framework for correction of non-anechoic measurements using a multi-taper-based approach can be summarized as follows: 1. Set f 0 , K, B, N, and measure R u (ω, θ) data; 2. Set t HB and define a set of X = {x p } 1⩽p⩽P DPSS design parameters for exhaustive search; set p = 1; 3. Identify DPSS tapers from (1) using x = x p and setup (2) for all segments of the time series; 4. Determine R c (Ω, θ a ) and extract the ath element R c ( f 0 , θ a ) of the R c ( f 0 , θ) vector; 5.If a = A, go to Step 6; otherwise, set a = a + 1 and go to Step 4. 6. Evaluate the function U(x p ) and store the R c (x p ); 7. If p < P, set p = p + 1 and go to Step 3; otherwise, go to Step 8; 8. Rank the obtained U(X) responses to identify R c (X opt ), and extract R c * as the average of R c (x β ), x β ∈ X opt , that correspond to σ-quantile locally optimal solutions; END.
The resulting vector R c * ( f 0 , θ) represents the corrected AUT responses.Note that the final post-processing step can involve normalization of R c * followed by its conversion to decibels.It should be emphasized that the presented approach executes correction based on one-shot measurements and that the DPSS taper parameters are adjusted in an automated manner (i.e.without engineering inference).To the best knowledge of the authors, this is the first implementation of the multitaper approach for the correction of non-anechoic antenna measurements.

Correction results
The proposed correction framework has been validated based on a series of 20 experiments performed in two non-anechoic test-sites (i.e. office rooms) that are not tailored to farfield experiments in any way except for the installation of the mobile positioning towers, the vector network analyzer (VNA), and the required cables and adapters [9].The dimensions of the considered sites are 8.4 m 3 × 4.5 m 3 × 3.1 m 3 for room A and 5.5 m 3 × 4.5 m 3 × 3.1 m 3 for room B, respectively (cf figure 2).The AUT considered for the experiments is a spline-parameterized antipodal Vivaldi antenna of figure 3 [9,48].Owing to its far-field properties and broadband operation, the structure could be utilized as a part of the microwavebased imaging system dedicated to evaluation of living tissues, e.g. for the purpose of breast cancer detection [33][34][35].Note that, for the considered experiments, the structure is also used

Case study-test site A
Our first case study encompasses the measurements performed at the site A. The post-processing, associated with the multitaper method (see section 2), is executed independently at each f 0 frequency of interest.Upon data acquisition, an optimal setup of DPSS tapers is identified based on the exhaustive search procedure outlined in section 2.3.The candidate DPSS kernels have been defined according to the set of test designs X.The lower-and upper-bounds for the latter are specified as:  the R c (X) responses are ranked according to their corresponding U(X) values.The final solution is then identified as the average of R c (x β ) responses featuring the highest rank among the corrected measurements.Here, the number of considered designs amount to five which corresponds to the σ-quantile of the locally optimal solutions (with σ = 0.1).The procedure is repeated for all of the considered f 0 frequencies.The antenna performance characteristics before and after correction are summarized in table 1.The refinement results indicate that the average improvement of the non-anechoic measurements fidelity due to the application of the proposed framework-expressed in terms of e R -amounts to over 7.3 dB (i.e.decrease of the e R factor from-16 dB to-23.3dB) with maximum and minimum change of the nonanechoic response quality of 11.2 dB at 4 GHz and 4.9 dB at 8 GHz, respectively.Note that the difference between the responses fidelity before and after the correction process is expressed in terms of 4 shows a comparison of the radiation pattern characteristics obtained for the antenna at the selected frequencies of interest before and after the post-processing.As already indicated by the results of table 1, the fidelity improvement of the real-world antenna performance evaluated in a non-anechoic environment due to post-processing is substantial.

Case study-test site B
For our second case study, the usefulness of the proposed correction framework has been demonstrated based on a total of ten experiments performed at the second test site.The latter (due to the smaller volume as well as the increased amount of more tightly packed furniture) represents a more challenging propagation environment compared to room A.

Discussion and comparisons
The proposed correction framework has been subjected to a comparative analysis (in both test sites) against the state-ofthe-art post-processing algorithms that involve time-domain analysis of the signal.The setup of benchmark methods includes the determination of intervals (pertinent to the relevant part of the RA-AUT transmission) based on: (i) manual estimation the RA-AUT distance (with rectangular window function), (ii) cognition-based analysis of the impulse response (with Hann function), and (iii) determination of intervals using a composite window [12][13][14]32].To ensure an equitable comparison, the configuration parameters, i.e. f 0 , K, B, N remain the same for all of the methods.The results presented in table 3 demonstrate that, for the considered test sites and the AUT at hand, the proposed multi-taperbased framework yields the most effective correction performance in terms of the averaged e R factor.The improvement of the refined radiation patterns compared to the benchmark approaches ranges from 1.6 dB to 1.8 dB for site A and from 0.4 dB to 1 dB for site B, respectively.Furthermore, it should be noted that the average correction results obtained using the proposed framework are consistent across the test sites, whereas for the benchmark methods, the errors range from 0.6 dB to 1.2 dB, respectively.Another important remark is that, contrary to (i) and (ii), the presented approach does not rely on engineering insight.At the same time, compared to (iii)-where composite function parameters are based on a set of pre-defined threshold values-the exhaustive search of parameters implemented in the proposed algorithm ensures an increased number of degrees of freedom for tuning the kernel functions, which might result in improved flexibility in terms of the range of radiators and test-sites that can be supported.Evaluation of the method in terms of uncertainty has also been considered.Due to challenging and time-consuming setup, the experiments have been performed only in the test site B and at a single frequency of interest, i.e. 4 GHz.The uncertainty budget has been estimated as follows [6]: where each contributor δ p , p = 1, …, P (here, P = 5) has been evaluated based on a series of measurements tailored to extract the factors such as: (i) dynamics of the environment (here understood as temporal changes of propagation conditions due to external factors)-δ 1 , (ii) mutual alignment of the RA-AUT system components-δ 2 , (iii) VNA calibrationδ 3 , (iv) repeatability of interconnections realized through microwave adapters-δ 4 , and (v) cable bends-δ 5 [6].It should be noted that uncertainty of the propagation environment is subject to (slight) variations of temperature, but also repeatability of angular alignment maintained by stepper motors in the rotary towers, noise introduced by the rotary adapters, and/or aliasing effects introduced in the course of the post-processing using the proposed algorithm.Consequently, the mentioned factors cannot be easily isolated to account for their individual contribution to (i) in the course of the postprocessing.The extracted values of individual factors amount to 0.11 dB, 0.03 dB, 0.01 dB, 0.01 dB, and 0.23 dB-for (i)-(v), respectively.The resulting uncertainty budget derived from (4) is 0.26 dB, which indicates a high performance of the proposed correction approach.It is worth noting that, for (v), a relatively large variations are due to unfavorable testing conditions through measurements of the scattering parameters through a cable coiled around the 30 mm diameter cylinder (single turn), as well as upon its straightening (which is exaggerated compared to routine experiments).

Conclusion
In this work, a framework based on multi-tapered functions has been proposed for correcting the antenna measurements performed in non-anechoic environments.The approach involves augmenting the RA-AUT transmission obtained as a result of a series of one-shot measurements (each for a unique angular position of the RA-AUT system) using a series of spheroidal functions so as to separate the useful fraction of the signal from the interferences and noise.The performance of the algorithm has been demonstrated based on a series of 20 measurements of a geometrically small antipodal Vivaldi structure performed at two non-anechoic test sites.For the considered structure and experiments, the average improvement of the measured responses resulting from the application of the method amounts to around 8 dB w.r.t.uncorrected signals.The approach has been favorably compared against the state-of-the-art techniques from the literature that involve time-domain-based processing of the far-field responses.The uncertainty budget for correction using the proposed method amounts to 0.26 dB, which is low given the challenging experimental conditions provided by the considered non-anechoic measurement system.Future work will focus on the enhancement of the proposed algorithm so as to enable automatic determination of all parameters that are relevant from the perspective of correction performance.Combination of the algorithm with alternative techniques oriented towards augmenting the amount of information that can be extracted from one-shot measurements will also be considered.

Figure 2 .
Figure 2. Non-anechoic test-sites considered for experiments with highlight on location of the positioning towers: (a) room A and (b) room B. Light-and dark-shade colors represent short and tall furniture, respectively.The antennas mounted on towers are denoted using red dots.

Figure 3 .
Figure 3.A photograph of the spline-parameterized antipodal Vivaldi antenna considered for experiments: (a) top and (b) bottom views.

Table 1 .
Test Site A: Correction Performance.

Table 2 .
Test Site B: Correction Performance.

Table 3 .
Benchmark of the Proposed Method.
a Averaged over the frequencies of interest.