Introduction
Modeling a source or a scatterer using an equivalent radiator is of interest in many applications as antenna synthesis [Reference Bucci, Franceschetti and D’Elia1], electromagnetic compatibility [Reference Regué, Ribó, Garrell and Martín2], the design of complex waveform generators [Reference Capozzoli, Curcio and Liseno3], computational electromagnetics and inverse scattering [Reference Yaghjian, Hansen and Devaney4, Reference Zaridze, Bit-Babik, Tavzarashvili, Economou and Uzunoglu5].
The problem can be framed as an extension of one of the classical equivalence theorems. It can be formulated as that of determining the shape and dimensions of a radiating surface capable to produce, in a targeted region of space Ω, an electromagnetic field as close as possible to that generated by any “primary” radiator/scatterer, contained in a prefixed region of space Γ, according to a prefixed tolerance. In [Reference Capozzoli, Curcio and Liseno6], the dimensioning problem has been dealt with as the determination of effective subspaces associated to the two operators linking the radiator/scatterer or the equivalent radiating panel Γ to their respective fields radiated over the region of interest. A solution has been provided by aid of the singular value decomposition (SVD) of the two mentioned operators. In particular, the singular functions of such operators associated to the significant singular values identify the linear subspaces to which the fields radiated by the radiator/scatterer and by the panel belong. The dimensions of the equivalent radiator are determined to reduce, as much as possible, the error by which the field radiated by the equivalent panel approximates the primary one, independently from any allowed primary field itself. In [Reference Capozzoli, Curcio and Liseno6], the case of a flat and rectangular Γ and of a flat and rectangular Ω parallel to Γ has been considered. For the sake of simplicity, a scalar problem has been dealt with.
Once dimensioned the radiating panel, a practical implementation is in order. A possibility is to consider a discretization of the equivalent panel based on the use of a non-uniformly spaced array of radiators, a problem already explored in [Reference Capozzoli, Curcio and Liseno7]. The element positions should be properly chosen so that the array is capable to approximate, with an adequate degree of accuracy, any field radiated by the equivalent radiator. Following the idea in [Reference Capozzoli, Curcio and Liseno7], in this paper, the array element positioning is worked out by exploiting a quadrature rule [Reference Barclay and Marinos8–Reference Marchaud, de Villiers and Pike10] transforming an integration into a summation. In other words, taking into account that the singular functions supported on the region of interest associated to the significant singular values of the radiation operator are related to those supported on the equivalent panel by a radiation integral, the quadrature rule replaces the integral representation of the singular functions into an approximation by a weighted summation. The quadrature rule defines also a set of weights which are essential in the definition of the element excitation coefficients from the knowledge of the source distribution on Γ.
For the sake of simplicity, in this paper, we consider a one-dimensional problem and a Legendre quadrature rule. The approach is numerically assessed by checking the capability of the array to radiate, within a prefixed tolerance, all the singular functions associated to the region of interest.
The paper is organized as follows. In section “The Radiation Operator and Its SVD,” the radiation operator from the equivalent panel to the region of interest is briefly introduced along with its SVD. In “Legendre Quadrature” section, the guidelines for the Legendre quadrature rule are recalled. Section “Panel Discretization” is devoted to the panel discretization by the quadrature-defined array. The performance of the approach is assessed in the following section. Finally, conclusions are drawn and future developments are foreseen in last section.
The radiation operator and its SVD
The geometry of the problem is depicted in Fig. 1. We consider the two-dimensional case of a strip source with current 
$\underline{J} (x^{\prime})=J(x^{\prime}){\skew7\widehat{i}}_y$, radiating in free-space and laying on a radiating panel of size 
$2a^\prime$ whose field is of interest over a portion Ω of a quiet plane of size 2a. The only (y) component of the radiated field is denoted by 
$E(x,z)$ and is expressed, apart from the unessential factor 
$-\omega \mu_0/4$, as:
\begin{equation}
\begin{aligned}
E(x,z) ={\mathcal A}(J)= &\int_{-a^\prime}^{a^\prime}J(x^\prime)H_0^{(2)}(\beta R)dx^\prime,\\
&\qquad\qquad z=d, \; x\in(-a,a),
\end{aligned}
\end{equation}
Figure 1. Geometry of the radiating panel.
where ω is the angular frequency, µ 0 is the free-space magnetic permeability of the embedding medium, 
$H_0^{(2)}$ is the Hankel function of zero-th order and second kind, 
$R=\sqrt{(x-x^\prime)^2+d^2}$ and 
${\mathcal A}$ is the radiation operator mapping J into the field on 
$(-a,a)$.
We denote by 
$\lbrace \sigma_l, u_l(x^\prime), v_l(x)\rbrace_{l=0}^{+\infty}$ the singular system of 
${\mathcal A}$, where the σl’s are the singular values, the ul’s are the right singular functions expanding the radiating current J and the vl’s are the left singular functions expanding the radiated field E. From a practical point of view, the spectral representation of the radiation operator can be limited to the only part of the singular system corresponding to the singular values deemed to be significant, namely, 
$\lbrace \sigma_l, u_l(x^\prime), v_l(x)\rbrace_{l=0}^{L-1}$. In other words,
\begin{equation}
E(x,d)={\mathcal A}(J)=\sum_{l=0}^{L-1} \sigma_l \lt J,u_l \gt _{(-a^\prime,a^\prime)}v_l(x),
\end{equation} where 
$ \lt \cdot,\cdot \gt _{(-a^\prime,a^\prime)}$ is the scalar product in 
${\mathcal L}^2(-a^{\prime},a^{\prime})$. As a consequence, the field radiated by J belongs essentially to the finite dimensional space spanned by the vl’s, 
$l=0,\ldots,L-1$.
Legendre quadrature
In this section, we shortly review Legendre quadrature which will be used in section “Panel Discretization”.
To this end, let us firstly state that, following a change of variables, any one-dimensional integral over a domain (b, c) can be set up as an integral over 
$(-1, 1)$, namely:
\begin{equation}
\int_{b}^{c} f(x)dx =\frac{c-b}{2} \int_{-1}^1 f\left(\frac{c-b}{2}\xi+\frac{b+c}{2}\right)d\xi,
\end{equation}where f is a generic complex-valued function of a real variable.
Following the application of an N points Gaussian quadrature, the integral in (3) can be expressed as:
\begin{equation}
\int_{b}^{c} f(x)dx =\frac{c-b}{2} \sum_{n=0}^{N-1}w_n\, f\left(\frac{c-b}{2}\xi_n+\frac{b+c}{2}\right),
\end{equation} where the 
$\lbrace \xi_n \rbrace_{n=0}^{N-1}$ are the quadrature nodes and the 
$\lbrace w_n \rbrace_{n=0}^{N-1}$ are the quadrature weights. The quadrature nodes are expressed as the zeros of the Nth degree orthogonal polynomials 
$p_n(\xi)$, while the quadrature weights, which are all real and positive, are also expressible in terms of the same orthogonal polynomials [Reference Gautschi11]. The polynomials 
$p_n(\xi)$ are defined by the following recursive relation:
\begin{equation}p_{n+1}(\xi)=(\xi-\alpha_n)p_n(\xi)-\beta_np_{n-1}(\xi),\;\;\;n=0,1,\dots,\end{equation} with 
$p_0(\xi)=1$ and 
$p_{-1}(\xi)=0$ and the αn’s and the βn’s specifically define the relevant polynomials. The Legendre quadrature theory states that, once defined the tridiagonal symmetric Jacobi matrix:
\begin{equation}
\underline{\underline{M}}_N= \begin{bmatrix}
\alpha_0 & \sqrt{\beta_1} & & & 0\\
\sqrt{\beta_1} & \alpha_1 & \sqrt{\beta_2} & & \\
& \sqrt{\beta_2} & & & \ddots \\
& & \ddots & \ddots & \sqrt{\beta_{N-1}} \\
0 & & & \sqrt{\beta_{N-1}} & \alpha_{N-1}
\end{bmatrix},
\end{equation}the quadrature nodes can be determined as its eigenvalues, while the quadrature weights can be retrieved according to its eigenvectors [Reference Golub and Welsch12]. In particular, the weights are expressed as:
\begin{equation}
w_n=\beta_0 \gamma_{n1}^2, \;\;\; n=0, \ldots, N-1,
\end{equation} where 
$\gamma_{n1}$ is the first component of the corresponding eigenvector 
$\underline{\gamma}_n$.
For our purposes, Legendre quadrature is employed for which 
$\alpha_n=0$, 
$n=0,\ldots,N-1$, and 
$\beta_0=2$ and 
$\beta_n=(4-n^{-2})^{-1}$, 
$n=1,\ldots,N-1$.
Panel discretization
In this section, we introduce a discrete version of the panel (array) capable to synthesize any field radiated by J over Ω. The array will generally be non-uniform and since, according to eq. (2), any field radiated over Ω is represented as a sum of functions vl’s, 
$l=0,\ldots,L-1$, then the array must be capable to radiate any individual vl over Ω within the prefixed accuracy.
Here, the radiating panel is replaced by a linear, non-uniform array made of N elements located on the 
$x^{\prime}$ axis, having unique positions 
$\underline{x}^\prime = (x_0^\prime,x_1^\prime,\ldots,x^\prime_{N-1})$ and complex excitation coefficients 
$\underline{b}^l=(b_0^l,b_1^l,\ldots,b_{N-1}^l)$, 
$l=0,\ldots,L-1$, which depend on the field vl to be represented (see Fig. 2). The element positions and excitation coefficients are determined according to quadrature rules [Reference Barclay and Marinos8–Reference Marchaud, de Villiers and Pike10], in this paper dealt with as the above recalled Legendre quadrature.
Figure 2. Array geometry.
To reach the targeted goal, the field radiated by the array over Ω is represented as:
\begin{equation}
E^l(x,d)=\sum_{n=0}^{N-1}b_n^l H_0^{(2)}(\beta R_n), \;\;\; l=0,\ldots,L-1,
\end{equation} where 
$R_n=\sqrt{(x-x^\prime_n)^2+d^2}$. Synthesizing an array discretizing the radiating panel amounts to finding 
$\underline{x}^\prime$ and 
$\underline{b}^l$, 
$l=1,\ldots,L$, so that:
\begin{equation}
E^l(x,d)= v_l(x), \;\;\; l=1,\ldots,L, \;\;\; |x|\leq a.
\end{equation}Due to the link between left and right singular functions:
\begin{equation}
v_l(x)=\frac{1}{\sigma_l}\int_{-a^\prime}^{a^\prime} H_0^{(2)}(\beta R) u_l(x^\prime)dx^\prime,
\end{equation}the quadrature enables representing approximately the vl’s by summations as:
\begin{equation}
v_l(x)\simeq \widetilde{v}_l(x)=\frac{1}{\sigma_l} \sum_{n=0}^{N-1}w_n H_0^{(2)}(\beta R_n) u_l(x^\prime_n),
\end{equation} where 
$\underline{w}=(w_0,w_1,\ldots,w_{N-1})$ is a unique set of quadrature weights. On comparing eqs. (8), (9) and (11), then the array excitation coefficients are:
\begin{equation}
b_n^l = w_n \frac{u_l(x^\prime_n)}{\sigma_l}, \;\;\; l=0,\ldots,L-1.
\end{equation} We remark that the developed approach does not introduce limitations on the class of fields that can be equivalently radiated by the considered array since 
$E(x,d)$ belongs to the space spanned by 
$v_l(x)$, 
$l=0,\ldots,L-1$, and the addressed quadrature rule, as it will be clearer in the next section, allows to adequately represent all the relevant singular functions 
$v_l(x)$’s.
Numerical results
In this section, we provide results to validate the quadrature technique for the discretization of the radiating panel.
To this end, we consider the same results of the panel dimensioning problem in [Reference Capozzoli, Curcio and Liseno6] and consider a domain Γ with 
$a^\prime=14\lambda$, a domain Ω with 
$a=5\lambda$ and a reciprocal distance of 
$d=10\lambda$. With this setup, L = 18.
The first step toward the array definition is the choice of the number of elements N. The order of the quadrature depends on the order of the polynomials one wants to integrate exactly. An N-nodes quadrature integrates exactly Legendre polynomials of degree up to 
$2N-1$. From this point of view, when increasing N, we expect that the overall performance of the array improve. In order to assess the array performance, the percentage mean square error (PMSE) is adopted:
\begin{equation}
\Phi(\underline{x}^\prime, \underline{w})=100 \cdot \frac{\sum_{l=0}^{L-1} \| \widetilde{v}_l(x) - v_l(x)\|^2}{L}.
\end{equation} For the considered test case, the PMSE against N is reported in Fig. 3. Once assigned the maximum tolerable PMSE, the number N of nodes to be employed can be determined. In particular, from Fig. 3, to reach a maximum PMSE of 
$1\%$, we should have 
$N\ge 39$. As a consequence, henceforth a Legendre quadrature rule with N = 39 is considered for which 
$PMSE=0.96\%$.
Figure 3. Percentage mean square error.
Figs. 4 and 5 display the array element positions and weights wn’s corresponding to a Legendre quadrature rule with N = 39. On the other side, Table 1 reports the interelement spacings 
$d_n=x_{n+1}-x_n$. As it can be seen, the spacing between outermost two elements on the left and on the right is very small and thus unpractical to realize. In order to avoid too close interelement spacings, we change the quadrature rule by replacing elements electrically too close each other with a single element having coordinate and weight equal to the average of their respective coordinates and weights. Obviously, such a procedure introduces further errors. Nevertheless, as it will be shortly clear, the performance of the array is not impaired. Indeed, Figs. 6, 7, 8 and 9 enable the comparison between the singular functions vl’s radiated by the panel and the 
$\widetilde{v}_l$’s approximated by the array for 
$l=0, 4, 10, 13$. A very good match between the twos can be appreciated. Analogous results have been observed for all the other relevant singular functions.
Figure 4. Quadrature nodes.
Figure 5. Quadrature weights.
Figure 6. Radiated singular function 
$v_0(x)$.
Figure 7. Radiated singular function 
$v_4(x)$.
Figure 8. Radiated singular function 
$v_{10}(x)$.
Figure 9. Radiated singular function 
$v_{13}(x)$.
Table 1. Element spacings
Conclusions and future developments
A practical implementation of a continuous equivalent radiating panel synthesized according to [Reference Capozzoli, Curcio and Liseno6] has been considered and provided by a non-uniform array. The element positions have been chosen by a quadrature rule so that the array is capable to approximate, with an adequate degree of accuracy, the fields radiated by the equivalent radiator. The quadrature rule has enabled also to choose a set of weights which are essential in the definition of the element excitation coefficients from the knowledge of the source distribution on the equivalent panel. Indeed, the capability of switching from a radiated field to another is based on the reconfigurability of the element excitation coefficients.
For the sake of simplicity, a one-dimensional problem with a Legendre quadrature rule has been considered. The approach has been numerically assessed by checking the capability of the array to radiate, with a satisfactory degree of accuracy, the singular functions associated to the region of interest.
Future developments will involve the optimization of the element position and weights [Reference Capozzoli, Curcio and Liseno7] by taking also into account constraints concerning the minimum allowed interelement spacing and the maximum allowed size [Reference Capozzoli, Curcio, D’Elia, Liseno and Vinetti13, Reference Capozzoli, Curcio, D’Elia, Liseno and Vinetti14] and the extension to the two-dimensional case using two-dimensional quadrature rules [Reference Sukri, Hoe and Khairuddin15].
Funding statement
This research received no specific grant from any funding agency, commercial or not-for-profit sectors
Competing interests
The authors report no conflict of interest.
Amedeo Capozzoli received the Laurea degree (summa cum laude) in electronic engineering and the Ph.D. degree in electronic engineering and computer science from Università di Napoli Federico II, Italy. He is a Full Professor of electromagnetic fields with the same university. His honors and distinctions include: the Telecom Italia Prize for the best thesis in Electronic Engineering at Università di Napoli Federico II, the Barzilai Prize for young scientists in 2002, the Italian Society of Electromagnetism, the Best Technical Paper Award, the Antenna Measurement Technique Association (AMTA) in 2009 and 2010, the Honorable Mention at EuCAP2011, the Nomination for the Best Paper Award at EuCAP2014 and at EuCAP2018, the 2016 Best Italian EMC Poster Prize at the IEEE EMC Young Professional Paolo Corona Day. Papers he co-authored received the MiniCircuits Harvey Kaylie Best Student Paper Award at IEEE-COMCAS in 2021 and the Best Student Paper Award from AMTA in 2021.
Claudio Curcio is currently Associate Professor at Università di Napoli Federico II. His main fields of interest are antenna measurements, standard and phaseless effective near-field/far-field transformation techniques, optical beamforming techniques for arrays, array and reflectarray synthesis. In 2009 and 2010 he received the Best Technical Paper Award at the AMTA Symposium. In the 2011 he was recipient of the Honorable Mention for the Best Antenna Measurement Paper at EUCAP, while in 2013 and 2018 he was finalist for the Antenna Measurements Best Paper Award at the EUCAP. In the 2016 he received the IEEE Best Italian EMC Poster Prize from the IEEE EMC Society, Italy Chapter. Since May 2020 he is member of the IEEE Working Group for the revision of the Standard IEEESTD1720 “Recommended Practice for Near-Field Antenna Measurements”. Since January 2022 he is Associate Editor for IEEE Access.
Francesco D’Agostino received his Laurea degree in electronic engineering from the University of Salerno. At the same university, he also received the Ph.D. degree in information engineering, where he was appointed Associate Professor of electromagnetics and is currently working. His research activity includes application of sampling techniques to electromagnetics and to innovative NF-FF transformations, diffraction problems, radar cross section evaluations, and electromagnetic compatibility. In this area, Dr. D’Agostino has coauthored 5 books and over 260 scientific papers, published in peer-reviewed international journals and conference proceedings. He serves as a regular Reviewer for several journals and conferences and has been invited speaker and chair for international events and conferences. Dr. D’Agostino is a senior member and fellow of Antenna Measurement Techniques Association (AMTA), senior member of the Institute of Electrical and Electronics Engineers (IEEE), member of the European Association on Antennas and Propagation (EurAAP) and of the Italian Electromagnetic Society (SIEM).
Angelo Liseno received the “Laurea” degree (summa cum laude) and a Ph.D. in 1998 and 2001, respectively, both in Electrical Engineering, from Seconda Università di Napoli, Italy. In 2003–2004, he was a research scientist with the Institut für Hochfrequenztechnik und Radarsysteme of the Deutsches Zentrum für Luft- und Raumfahrt (DLR), Oberpfaffenhofen, Germany. From 2005 to 2015, he was a Researcher with the Università di Napoli Federico II, Dipartimento di Ingegneria Elettrica e delle Tecnologie dell’Informazione, Napoli, Italy. Since 2015, he has been an Associate Professor with the same department. Angelo Liseno was awarded the AMTA Best Technical Paper Award for two consecutive years in 2009 and 2010, received the Honorable Mention for the Best Antenna Measurement Paper at EuCAP2011, the Nomination for the Best Paper Award at EuCAP2014 and at EuCAP2018 and the Best Italian EMC Poster Prize from the IEEE EMC Society, Italy Chapter, in 2016. Angelo Liseno is Associate Editor of Antennas and Propagation, Advanced Electromagnetics and IET Microwaves, Antennas and Propagation.
Luigi Pascarella was born in Caserta, Italy, in 1999. He received the Bachelor degree and the Master degree (summa cum laude), both in Electronic Engineering, from Università di Napoli Federico II, Naples, Italy, in 2020 and 2022, respectively. Since November 2022, he is a Ph.D. candidate in Industrial Engineering at the University of Salerno, Fisciano, Italy, with the Research Group in Applied Electromagnetics. His main interests include antenna measurements, array synthesis and echo generators design.
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