Near-Field Warping Sampling Scheme for Broad-Side Antenna Characterization

In this paper the problem of sampling the field radiated by a planar source observed over a finite planar aperture located in the near-field is addressed. The problem is cast as the determination of the spatial measurement positions which allow us to discretize the radiation problem so that the singular values of the radiation operator are well-approximated. More in detail, thanks to a suitably warping transformation of the observation variables, the kernel function of the relevant operator is approximated by a band-limited function and hence the sampling theorem applied to achieved discretization. It results in the sampling points having to be non-linearity arranged across the measurement aperture and their number can be considerably lowered as compared to more standard sampling approach. It is shown that the proposed sampling scheme works well for measurement apertures that are not too large as compared to the source’s size. As a consequence, the method appears better suited for broad-side large antenna whose radiated field is mainly concentrated in front of the antenna. A numerical analysis is included to check the theoretical findings and to study the trade-off between the field accuracy representation (over the measurement aperture) and the truncation error in the estimated far-field radiation pattern.


Introduction
Antenna testing is a fundamental and necessary step in the manufacturing process of any transmission system. The most advanced testing procedures rely on near-field measurement techniques that consist of measuring the field radiated by the antenna under test at a relatively short range within an anechoic environment [1][2][3] and then to compute the far-field pattern from such measurements. More in detail, near-field measurements are usually collected by mechanically scanning a measurement surface [4] and then the measured data are processed by the so-called "near-field to far-field transformations" [3,[5][6][7], or related approaches [8,9], to obtain the antenna radiation pattern. For large antennas, the number of required measurements may become extremely high. Therefore, in order to control the acquisition time, it is crucial to reduce the number of measurements without compromising the accuracy of the results [10][11][12][13][14][15].
The aim of this contribution is to address this question for the case of a planar source distribution whose radiated field is measured over a planar aperture. For such a case, according to classical plane-wave spectrum reasoning, the probe usually scans the measurement aperture with a sampling step of half the free-space wavelength. The resulting sampling point number is herein assumed as the benchmark against which to achieve data reduction.
From a general perspective, the task of reducing the spatial measurements can be cast as a sensor selection problem [16], where one selects a finite number of positions among the ones available over a generally very dense grid. As is well-known, this type of problem presents a combinatorial complexity x-axis, i.e., J = J(x, y)x, and collect the corresponding tangential y-component of the radiated field (see Figure 1 for a pictorial view of the configuration). If we omit an unessential scalar factor, the radiation problem is described in the frequency domain by the following radiation operator A : J(r) ∈ X = L 2 (SD) → (AJ)(r o ) = E(r o ) = SD K(r o , r)e −jΦ(r o ,r) J(r)dr ∈ Y = L 2 (OD), (1) with L 2 (SD) and L 2 (OD) being the set of square integrable functions supported over the source and the observation domains, respectively, r o ∈ OD and r ∈ SD are the field and the source points and k the free-space wavenumber. Moreover, Φ(r o , r) = k|r o − r| and K(r o , r) = 1 |r o −r| 2 [jk + 1 |r o −r| ] ≈ jk |r o −r| 2 , where the last approximation is because |r o − r| ≥ z o ≥ λ, λ being the free-space wavelength.
We are concerned with the design of a sampling scheme for the observation variable r o which allows to dicretize the data space in such a way to approximate the singular values of A up to a certain index. As is well-known, the singular system {u n , σ n , v n } ∞ n=0 of A, with σ n being the singular values and u n and v n the singular functions that span the source and the field spaces, solves the following equations σ n v n = Au n σ n u n = A † v n , (2) where A † is the adjoint of the radiation operator defined as with f and g being two generic functions belonging to Y and X, respectively. However, for our purposes, it is convenient to address the associated eigenvalue problem Since its finite dimensional approximation entails to discretize r o only. Therefore, in the following we focus on AA † whose explicit expression, apart from an unessential constant, is In order to devise the sampling scheme, the main idea it to recast the kernel function of AA † as a Fourier-like transformation. To this end, it is convenient to rewrite the phase term as with ∇ p denotes the gradient with respect to p, such that p(ν) is a curve whose starting and ending points coincide with r o and r o , respectively, that is p(ν 0 ) = r o and p(ν 1 ) = r o . Now, the curve p(ν) can be properly chosen in order to let the phase term resemble a Fourier kernel. This can be achieved, for example, in the following way. Considerr o ≡ (x o , y o ) and then perform integration in (6) along the polygonal line with nodes r o ,r o and r o , i.e., integration is performed along the segment joining r o and r o and followed by the segment joiningr o and r o . Accordingly, we have that where · denotes the scalar product, w ≡ (w x , w y ) and Now, it can be shown that ∀r o , r o the transformation w : r → w(r o , r o , r) is injective and the corresponding Jacobian matrix full rank (the details concerning this point have been omitted for brevity). This allows us to replace in (5) the integration in r with the integration in w, which yields with being the corresponding integration domain in the w variable and is the corresponding amplitude term which includes the Jacobian determinant, i.e., , of the variable transformation from r to w. To proceed further we focus on the kernel function of (10), which is given by In order to slightly simplify the previous expression, we note that, because H(r o , r o , w) is a constant sign function, Equation (13) clearly shows that the leading order contribution occurs for r o − r o = 0 [28]. This allows us to approximate the amplitude factor by its value assumed for r o = r o , that is By observing that the Jacobian transformation yields we finally have H(r o , w) = 1 and the kernel function is eventually approximated as It is interesting to highlight that (16) shows the kernel function as a 2D spatially varying band-limited function [29], which allows us to expect a non-uniform sampling. This indeed has been already reported in previous contributions [25,[30][31][32][33] for one-dimensional currents.

Sampling Scheme
In order to devise the sampling scheme, we look for a sampling expansion approximation of the kernel (16). To this end, we extend the approach in [25]. Here, the matter is much more difficult because, unlike as in [25], both the size and the shape of the band Ω(r o , r o ) change with the observation variable.
To deal with the change in shape of Ω(r o , r o ) as r o and r o range within OD, we content to approximate (16) by considering a rectangular domain . In order to determine Ω R (r o , r o ), we have to compute the extreme points of Ω(r o , r o ) along w x and w y . This is equivalent in determining w max The latter is a tedious but not a complex task and is pursued in Appendix A under the assumption OD ⊆ SD. Accordingly, once these extreme points have been determined, the parameters of Ω R (r o , r o ) follow as At this juncture, by extending the integration in (16) over the estimated rectangular domain Ω R (r o , r o ), the following closed-form approximation of the kernel function is obtained with w m = (w mx , w my ) and sinc(x) = sin(x)/x. The parameters of Ω R (r o , r o ) reported in (17) are spatially varying with the observation variable. This dependence can be removed by introducing a suitable 'warping' transformation [34][35][36]. This task is relatively easy under the assumption OD ⊆ SD (see Appendix A). Indeed, in this case the warping transformation 'factorizes', in the sense that the observation variables x o and y o can be warped independently from each other. In particular, such transformations are (see Appendix A for details) and Accordingly, Equation (18) rewrites as (see Appendix A for further details) Basically, Equation (19) and Equation (20) transform the rectangular region in (x o , y o ), i.e., the actual observation domain OD = [−X 0 , with ξ xmax and ξ xmin being the maximum and the minimum of the function ξ over the allowed values of x o , and ξ ymax and ξ ymin are the analogous for the ξ y function. Now, we can finally rewrite the eigenvalue problem in (10) in the warped domain (ξ x , ξ y ) by changing the integration variable from (x o , y o ) to (ξ x , ξ y ). Accordingly, the singular functions spanning the field space can be expressed as where By employing similar reasoning used for the amplitude term in (16) we made Section 2, and (23) becomes The advantage provided by reformulating the eigenvalue problem as in (26) is evident since we are now allowed to use the standard sampling theorem (with respect to the introduced warped variables) [26] to build the discrete version of AA † for eingenspectrum computation [37]. More in detail, Equation (26) says thatv n are band-limited functions (becausekern is a band-limited function) and hence can be expanded as with ξ xm = mπ and ξ yl = lπ being the sampling points and m and l integer indexes. Of course, since the singular functionsv n span the field space the same expansion holds true for the field. Hence, Equation (27) is the sampling expansion we were looking for. In particular, in order to pass from the sampling points in (ξ x , ξ y ) to the ones in (x o , y o ) (the actual observation variables), Equation (19) must be used. For example, the sampling position along x o , i.e., x om , are obtained by solving for x om the following equation or equivalently A similar equation of course holds true for the sampling points y om along the variable y o . In order to appreciate the goodness of the proposed sampling scheme, we need to obtain the discrete version of the eigenvalue problem (10). This is achieved by inserting (27) into (26), that yields where v n is the vectorized form of the matrix consisting of the samples ofv n and the entries of the matrix, B α,β , are given by Note that the integer indexes m, l and s and t range over the two-dimensional sampling lattice involved by (27) and the matrix entry indexes α and β vary according to the way the vectorization of v n is achieved.
It is worth remarking that B describes an infinite discrete problem. However, since (27) must be used to represent the field over the measurement aperture, we are allowed to retain only the samples falling within [−∆ξ x , ∆ξ x ] × [−∆ξ y , ∆ξ y ], which corresponds to the observation domain OD. Accordingly, in the sequel, we will consider a truncated version of B, i.e., B N of size N × N, which takes into account only the samples falling within the observation domain. More in detail, being the operator that takes the integer part. Indeed, for classical band-limited kernels, N represents the so-called Shannon number (SN) which is known to give a good estimation of the number of degrees of freedom [23,37]. In particular, in these cases, the singular values exhibit a step-like behavior and the SN basically returns the number of singular values preceding the abrupt decay. However, it also known that to properly capture that part of the singular value behavior, and also to go a bit beyond the 'knee', a slightly greater number of samples are required [37]. Therefore, in the following numerical analysis, an oversampling factor of 1.3 is considered, that is to say, that the sampling step (in ξ x and ξ y ) is fixed at π/1.3.

Numerical Analysis
In this section, we check the previous theoretical findings by some numerical examples. We start by first verifying if the proposed sampling scheme works in approximating the singular values of the radiation operator. Note that the singular values of A are the square root of the eigenvalues of AA † . Therefore, in the sequel, we will speak about the singular values or the eigenvalues without distinction.
We consider a source domain SD = [−8λ, 8λ] × [−4λ, 4λ] (with X s = 8λ and Y s = 4λ) and assume to collect the data over two measurement domains both located at z o = 7λ: the first one is The corresponding results are reported in Figure 2. In particular, in panels (a) and (b) the sampling point distributions returned by the proposed non-uniform sampling scheme are sketched for the two considered observation domains. Panels (c) and (d) instead report the comparison between the eigenvalues of B N and AA † . According to the theoretical derivation, we have a strict constraint on the size of the measurement aperture which should not exceed the one of the source. Nonetheless, in both the cases considered in Figure 2, the observation domain violates such a constraint, especially for the example reported in panels (b) and (d). By looking at such a figure the following conclusions can be drawn. First, the proposed sampling scheme is able to very well approximate the eigenvalues even when the observation domain OD slightly exceeds the source domain SD (see panel (c) which refers to OD 1 ). Instead, in panel (d), where OD 2 is much larger than SD, it is evident that the number of degrees of freedom is underestimated since the 'knee' of the eigenvalues starts before the actual one. This means that the sampling points are not enough (and not properly located). However, the initial part of the eigenvalue behavior is very well-approximated. Hence, we conclude that in this case, the proposed non-uniform sampling strategy is able to approximate only a subset of all possible radiated fields, i.e., the ones spanned by the singular functions corresponding to the singular values that are well-estimated. As a consequence, it is expected that the non-uniform sampling can allow for a good radiated field approximation if the field significantly projects on those singular functions, even when the constraint on the size of OD is not strictly verified.
The second important point that must be highlighted is that the number of samples required by the proposed sampling scheme is actually much lower than the ones arising from a λ/2 sampling. Indeed, for the two cases, our method requires N = 462 and N = 840, respectively for OD 1 and OD 2 , whereas the λ/2 sampling requires 1025 and 7381 samples.
More in detail, the first source gives rise to very low side-lobes and hence it has been considered to see if, and to what extent, they can be estimated by using the proposed sampling scheme. The second source is constant and presents an abrupt decay at the edges of the source domain; its radiation pattern is a sinc-like function. Finally, the third current leads to a steered multi-beam radiation pattern. Basically, these examples present a growing level of difficulty, since moving from J 1 to J 3 the currents project over a large number of singular functions.
In Table 1, the relative error RE is given in dB for the different sources under consideration and the two measurement apertures addressed in Figure 2. As can be seen, for OD 1 = [−10λ, 10λ] × [−6λ, 6λ], the error is relatively low for all the sources. This means that the proposed sampling scheme returns a good approximation for the near-field although the number of samples has been greatly reduced as compared to the λ/2 sampling scheme. This was indeed expected since the proposed sampling scheme works well if the observation domain OD is similar in size to the source domain SD. When the measurement aperture is increased (see the third column of Table 1) the error decreases for J 1 (x, y) and J 2 (x, y) and increases for J 3 (x, y). This is because the field radiated by J 1 and J 2 significantly projects on the singular functions corresponding to the singular values that are well-approximated (see panel (d) of Figure 2). Accordingly, the metric error benefits from the higher number of sampling points (that are required since OD 2 is larger than OD 1 ) that can be used to perform the interpolation. On the contrary, for J 3 , the error increases because the radiated field also projects on the singular functions of A which are not well-approximated by our discretization scheme. In other words, the field radiated by J 3 is also relevant for the points of OD 2 which exceed the limit of OD 1 and hence of SD.  We now pass to analyzing the radiation patterns which are obtained by Fourier transforming (by means of a FFT procedure) the near-field data. The radiation patterns are reported as a function of the spectral variables k x = k sin θ cos φ and k y = k sin θ sin φ, with θ and φ being the usual polar angles, and shown only for the so-called 'visible' domain, that is for k 2 x + k 2 y = k 2 . More in detail, after collecting the field data according to the proposed non-uniform sampling scheme, the field is interpolated over a uniform λ/2 grid and finally, Fourier transformed.
The radiation pattern corresponding to J 1 is reported in Figures 3 and 4 for the measurement aperture OD 1 and OD 2 , respectively. For example, by comparing panels (a) and (b) of Figure 3 the radiation patterns computed by using the proposed method and the usual uniform sampling look very similar. This can be even better appreciated by looking at the cut-views shown in panels (c) and (d), where the blue lines refer to the radiation pattern computed by using the uniform λ/2 sampling and the red ones to the radiation pattern obtained by the proposed method. In particular, herein, the actual radiation pattern (in green lines) is also reported for comparison purposes. Since for this case, the non-uniform sampling succeeds in approximating well the near-field (see Table 1) this very good match between the radiation patterns computed by using the two sampling schemes under comparison was indeed expected. In particular, they also exhibit a similar truncation error in the very low side-lobe region along k y (see panel (d)). This, of course, is because the observation domain is shorter along y axis. However, this error is dramatically reduced in Figures 4 where the larger measurement aperture OD 2 was considered. In fact, since in this case, the aperture has been enlarged and the radiated field still projects well on the singular functions that have been well-approximated through the non-uniform sampling, the three lines overlap very well and appear indistinguishable. . Normalized amplitude of the radiation pattern of J 1 (x, y) with X s = 8λ, Y s = 4λ, z o = 7λ, X 0 = 10λ and Y 0 = 6λ. In (a), the radiation pattern is obtained by employing the near-field data according to the proposed non-uniform sampling scheme over the grid shown in (a) of Figure 2 and then interpolated over a λ/2 grid. In (b), the radiation pattern is obtained by directly employing the near-field data over a uniform λ/2 grid. Panels (c,d) have been obtained by fixing k y = k sin (π/20) sin (π/4) and k x = k sin (π/20) cos (π/4), respectively, and compare the radiation pattern cut-views passing through the main-beam maximum. The green lines report the actual radiation pattern, the blue lines the radiation pattern computed by using the uniform λ/2 sampling and the red ones show the radiation pattern obtained by the proposed method. Noiseless case.
In Figure 5, we consider the same case as in Figure 4 but a complex white Gaussian noise is added to the field data. In particular, a signal to noise ratio (SNR), defined as with E the field data and N the noise, of 20 dB is considered. As can be seen, the two sampling schemes still return similar results (in particular look at panels (c) and (d)). Indeed, both succeed in approximating the first side-lobe of the actual radiation pattern whereas the very low side-lobe region is definitely affected by the noise. However, what matters here is that, though much fewer sampling points have been used by our method, the two sampling schemes show a similar effect of the noise. The results concerning J 2 are reported in Figures 6-8. In particular, Figure 6 refers to OD 1 , Figure 7 to OD 2 and Figure 8 to OD 2 with noisy data and SNR = 20 dB. By looking at Figure 6 it can be appreciated that the radiation pattern computed by the two sampling schemes are still very similar and both exhibit a relevant truncation error (see panels (c) and (d)) since the returned radiation patterns (red and blue lines) are considerably different from the actual one (green lines). This error, however, is reduced to a large extent by increasing the measurement aperture as shown in Figure 7 where the three lines are indistinguishable. Moreover, the estimated radiation pattern through the two sampling schemes shows similar stability against the noise as illustrated in Figure 8.  . Normalized amplitude of the radiation pattern of J 1 (x, y) with X s = 8λ, Y s = 4λ, z o = 7λ, X 0 = 30λ and Y 0 = 15λ. In (a), the radiation pattern is obtained by employing the near-field data according to the proposed non-uniform sampling scheme over the grid shown in panel (b) of Figure 2 and then interpolated over a λ/2 grid. In (b), the radiation pattern is obtained by directly employing the near-field data over a uniform λ/2 grid. Panels (c,d) have been obtained by fixing k y = k sin (π/20) sin (π/4) and k x = k sin (π/20) cos (π/4), respectively, and compare the radiation pattern cut-views passing through the main-beam maximum. The green lines refer to the actual radiation pattern, the blue lines to the radiation pattern computed by using the uniform λ/2 sampling and the red ones show the radiation pattern obtained by the proposed method. Noiseless case. . Normalized amplitude of the radiation pattern of J 1 (x, y) with X s = 8λ, Y s = 4λ, z o = 7λ, X 0 = 30λ and Y 0 = 15λ. In (a), the radiation pattern is obtained by employing the near-field data according to the proposed non-uniform sampling scheme over the grid shown in (b) of Figure 2 and then interpolated over a λ/2 grid. In (b), the radiation pattern is obtained by directly employing the near-field data over a uniform λ/2 grid. Panels (c,d) have been obtained by fixing k y = k sin (π/20) sin (π/4) and k x = k sin (π/20) cos (π/4), respectively, and compare the radiation pattern cut-views passing through the main-beam maximum. The green lines refer to the actual radiation pattern, the blue lines to the radiation pattern computed by using the uniform λ/2 sampling and the red ones show the radiation pattern obtained by the proposed method. Noisy case with additive complex white Gaussian noise and SNR = 20 dB. dB dB Figure 6. Normalized amplitude of the radiation pattern of J 2 (x, y) with X s = 8λ, Y s = 4λ, z o = 7λ, X 0 = 10λ and Y 0 = 6λ. In (a), the radiation pattern is obtained by employing the near-field data according to the proposed non-uniform sampling scheme over the grid shown in panel (a) of Figure 2 and then interpolated over a λ/2 grid. In (b), the radiation pattern is obtained by directly employing the near-field data over a uniform λ/2 grid. Panels (c,d) have been obtained by fixing k y = 0 and k x = 0, respectively, and compare the radiation pattern cut-views passing through the main-beam maximum. The green lines refer to the actual radiation pattern, the blue lines to the radiation pattern computed by using the uniform λ/2 sampling and the red ones show the radiation pattern obtained by the proposed method. Noiseless case.
In (a), the radiation pattern is obtained by employing the near-field data according to the proposed non-uniform sampling scheme over the grid shown in (b) of Figure 2 and then interpolated over a λ/2 grid. In (b), the radiation pattern is obtained by directly employing the near-field data over a uniform λ/2 grid. Panels (c,d) have been obtained by fixing k y = k sin (π/20) sin (π/4) and k x = k sin (π/20) cos (π/4), respectively, and compare the radiation pattern cut-views passing through the main-beam maximum. The green lines refer to the actual radiation pattern, the blue lines to the radiation pattern computed by using the uniform λ/2 sampling and the red ones show the radiation pattern obtained by the proposed method. Noiseless case.
Finally, Figures 9 and 10 show the results concerning J 3 . According to what was reported at the beginning of this section, since for the case of OD 1 the proposed sampling strategy allows to obtain a good estimation of the near-field, the radiation patterns obtained by the non-uniform and the uniform sampling schemes are very similar for the case of OD 1 addressed in Figure 9. However, because of the size of OD 1 , there is a relevant truncation error, as highlighted in panels (c) and (d) of such a figure. The measurement aperture is enlarged at OD 2 in Figure 10. Now, though the truncation error is significantly reduced for both the sampling schemes, the actual pattern (green lines) is much better approximated by the one returned by the uniform sampling (blue lines) (see panels (c) and (d) of Figure 10). This is because, differently from J 1 and J 2 , J 3 presents relevant components over the singular functions that are not well-approximated by the non-uniform sampling scheme. This clearly highlights the role of the type of source.
Summarizing, regardless of the type of source, the proposed sampling strategy returns a good estimation for the near-field when the observation domain, OD, is equal or 'slightly' larger than the source domain, SD. When the measurement aperture is much larger than SD, the representation error depends on the type of sources and is relevant if the source significantly projects on the singular functions of A that are not well-approximated by the proposed discretization strategy. In the latter case, the estimated radiation pattern can suffer from a large deviation from the actual one. Therefore, it can be concluded that the method is better suited to broadside antennas and further theoretical work is required to generalize the sampling scheme to the case of beam-steered antennas.
In (a), the radiation pattern is obtained by employing the near-field data according to the proposed non-uniform sampling scheme over the grid shown in (b) of Figure 2 and then interpolated over a λ/2 grid. In (b), the radiation pattern is obtained by directly employing the near-field data over a uniform λ/2 grid. Panels (c,d) have been obtained by fixing k y = k sin (π/20) sin (π/4) and k x = k sin (π/20) cos (π/4), respectively, and compare the radiation pattern cut-views passing through the main-beam maximum. The green lines refer to the actual radiation pattern, the blue lines to the radiation pattern computed by using the uniform λ/2 sampling and the red ones show the radiation pattern obtained by the proposed method. Noisy case with additive complex white Gaussian noise and SNR = 20 dB. . Normalized amplitude of the radiation pattern of J 3 (x, y) with X s = 8λ, Y s = 4λ, z o = 7λ, X 0 = 10λ and Y 0 = 6λ. In panel (a), the radiation pattern is obtained by employing the near-field data according to the proposed non-uniform sampling scheme over the grid shown in (a) of Figure 2 and then interpolated over a λ/2 grid. In (b), the radiation pattern is obtained by directly employing the near-field data over a uniform λ/2 grid. Panels (c,d) have been obtained by fixing k y = k sin (π/4) sin (π/4) and k x = k sin (π/4) cos (π/4), respectively, and compare the radiation pattern cut-views passing through the main-beam maximum. The green lines refer to the actual radiation pattern, the blue lines to the radiation pattern computed by using the uniform λ/2 sampling and the red ones show the radiation pattern obtained by the proposed method. Noiseless case.  Figure 10. Normalized amplitude of the radiation pattern of J 2 (x, y) with X s = 8λ, Y s = 4λ, z o = 7λ, X 0 = 30λ and Y 0 = 15λ. In (a), the radiation pattern is obtained by employing the near-field data according to the proposed non-uniform sampling scheme over the grid shown in (b) of Figure 2 and then interpolated over a λ/2 grid. In (b), the radiation pattern is obtained by directly employing the near-field data over a uniform λ/2 grid. Panels (c,d) have been obtained by fixing k y = k sin (π/4) sin (π/4) and k x = k sin (π/4) cos (π/4), respectively, and compare the radiation pattern cut-views passing through the main-beam maximum. The green lines refer to the actual radiation pattern, the blue lines to the radiation pattern computed by using the uniform λ/2 sampling and the red ones show the radiation pattern obtained by the proposed method. Noiseless case.

Conclusions
In this paper, the problem of sampling the field radiated by a planar source and observed over a finite planar aperture located in the near-field has been addressed. The problem has been cast as the determination of the measurement spatial positions for which the singular values of the radiation operator are well-approximated. Thanks to suitable variable transformations, which 'warp' the spatial observation variables, the kernel function of AA † has been approximated as a band-limited function and hence the standard sampling theorem used to discretize the problem. Basically, the new content conveyed by this paper consists in the introduction of a sampling scheme which allows us to reduce the number of measurements as compared to the most used sampling scheme in the industry for antenna characterization, to avoid to use of numerical iterative procedures for selecting the measurement positions, to extend our previous results which were concerned for the simpler case of strip currents.
The developed theory rigorously works for measurement apertures that are not too large as compared to the source's size. Therefore, it has been concluded that the proposed method is better suited to broadside antennas. In this regard, it must be emphasized that the mentioned limitations arise because in the derivation we assumed SD ⊇ OD. This greatly simplified the problem because it allowed us to find factorized warping transformation and the related sampling scheme. Accordingly, this contribution can be seen as a preliminary contribution that must be generalized in order to deal with larger measurement apertures and general source types. In view of the great reduction in the number of data points, we are stimulated in addressing this question in future developments.
Finally, it worth remarking that besides the radiation pattern estimation, determining how to sample the radiated field is inherently connected to the inverse source problem [38] and also to the computation of the information content that can be 'communicated' from a source to an observation domain. In fact, it is well-known that the information content (quantified by the Shannon or the Kolmogorov metrics) are explicitly dependent on the singular value behavior of the radiation operator [39].
In particular, Equation (A2) allows to highlight that w x and w y range within intervals that depend only on x o , x o and y o , y o , respectively. This is a very important aspect since it leads to the 'factorized' sampling expansion presented in Section 3. More in detail, by using (A2) in (17), we obtain ]dp x ∆w y (y o , y o ) = In particular, by solving the integrals . (A4) Finally, by using the warping transformations presented in (19) and (20), it readily follows that and ∆w y (y o − y o ) = ξ y (y o ) − ξ y (y o ), and hence from (18) the kernel expression in (21) is obtained.