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:

(1)\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 σ_l’s are the singular values, the u_l’s are the right singular functions expanding the radiating current J and the v_l’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,

(2)\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 v_l’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:

(3)\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:

(4)\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:

(5)\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 α_n’s and the β_n’s specifically define the relevant polynomials. The Legendre quadrature theory states that, once defined the tridiagonal symmetric Jacobi matrix:

(6)\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:

(7)\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 v_l’s, $l=0,\ldots,L-1$, then the array must be capable to radiate any individual v_l 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 v_l 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:

(8)\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:

(9)\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:

(10)\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 v_l’s by summations as:

(11)\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:

(12)\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:

(13)\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 w_n’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 v_l’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].