In this paper, we address the problem of discretizing the singular system of the radiation operator concerning the case of a magnetic strip current whose radiated field is observed in near-zone on a bounded line parallel to the source. This question has been already addressed in previous articles with the limitation that the extension of the observation domain does not overcome the source size. In this article, we remove such limitation, hence, we provide a discrete model that well approximates the singular values of the radiation operator in the case where the observation domain is larger than the source

Maisto, Maria Antonia

Moretta, Raffaele

Pierri, Rocco

English

Archivio Istituzionale della Ricerca - Università degli Studi della Campania "Luigi Vanvitelli"

1      A SAMPLING STRATEGY OF THE RADIATION OPERATOR IN  NEAR-ZONE BASED ON AN ASYMPTOTIC KERNEL            Raffaele Moretta(1), Maria Antonia Maisto(1), Rocco Pierri(1) (1) Dipartimento di Ingegneria, Università della Campania “L. Vanvitelli”, Aversa 81031, Caserta, Italia raffaele.moretta@unicampania.it Abstract  In this paper, we address the problem of discretizing the singular system of the radiation operator concerning the case of a magnetic strip current whose radiated field is observed in near zone on a bounded line parallel to the source. This question has been already addressed in previous articles with the limitation that the extension of the observation domain does not overcome the source size. In this article, we remove such limitation, hence, we provide a discrete model that well approximates the singular values of the radiation operator in the case where the observation domain is larger than the source.  Index Terms − inverse source, radiation operator, asymptotic approach, discretization. I. INTRODUCTION  The question of finding an efficient discretization of the field radiated by a source is a relevant task in the framework of inverse source problem. The latter consists in the inversion of the integral equation                                                            𝐸 = 𝐴 𝐽                                                            (1) where 𝐴 stands for the radiation operator, 𝐽 represents the density current of the source, and 𝐸 represents the radiated field. In the case of planar scanning, the most common sampling strategy is based on a uniform sampling with a step length of half-wavelength [1]. Despite its simplicity, such a strategy is not efficient since it requires to acquire a number of samples that may be significantly higher than the dimension of the unknown space. For this reason, the aims of the paper are those of 1) establishing the minimum number of sampling points, 2) finding their optimal position.  As concerns the first point, the minimum number of sampling points is equal to the number of degrees of freedom (NDF) of the radiated field. The latter can be evaluated by counting the number of the most relevant singular values of the radiation operator. As regards the optimal positions of the sampling points, this issue can be recast in how to collect the radiated field in such a way to obtain a discrete model whose singular values well approximate the most relevant singular values of the radiation operator.              For several far-field configurations, the kernel of the related eigenvalue problem involves a bandlimited kernel of difference type. In these cases, the sampling theory approach developed in [2] can be exploited to discretize the continuous model.                           In this paper, we consider the case of a strip current whose radiated field is observed in near zone over a truncated line parallel to the source. Unfortunately, for such a configuration, the integral equation for the computation of the singular values involves a space-variant kernel; hence, the literature mentioned above cannot be directly applied to discretize the model. Despite this, as shown in [3] and [4], a change of the integration variable allows recasting the operator involved in the related eigenvalue problem, as a convolution operator with a bandlimited kernel, at least when the observation domain is smaller than the observation. In this paper, thanks to the use of a weighted adjoint, we remove such limitation; hence, we provide a discretization strategy of the radiation operator that works when the observation domain is larger than the source domain.  II. GEOMETRY OF THE PROBLEM A 1D magnetic current 𝐽 ̅(𝑥) = 𝐽(𝑥) 𝑖̂𝑦 directed along the y-axis and supported on the set 𝑆𝐷 = [−𝑎, 𝑎] of the 𝑥-axis radiates within a homogeneous medium with wavenumber 𝛽. The 𝑥 component of the radiated electric field is observed in near non-reactive zone over 2  a bounded observation domain 𝑂𝐷 = [−𝑋𝑜 , 𝑋𝑜] that is parallel to the source and located along the axis 𝑧 = 𝑧𝑜 (see Fig. 1).                                          FIG. 1 − Geometry of the problem. For the geometry at hand, the radiation operator is given by                                         𝛢 𝐽(𝑥′) = ∫𝑧𝑜𝑅3/2(𝑥′,𝑥𝑜)𝑒−𝑗 𝛽 𝑅 (𝑥′,   𝑥𝑜) 𝐽(𝑥′) 𝑑𝑥′+ 𝑎− 𝑎                             (2) with 𝑅 (𝑥′, 𝑥𝑜) = √(𝑥 − 𝑥′)2 + 𝑧𝑜2. Consequently, a weighted adjoint operator can be expressed in the form                                      𝐴𝑤†  𝐸 = ∫ 𝑤(𝑥𝑜) 𝑧𝑜𝑅3/2(𝑥′,𝑥𝑜)𝑒𝑗 𝛽 𝑅 (𝑥′,   𝑥𝑜) 𝐸(𝑥𝑜)𝑑𝑥𝑜+𝑋𝑜−𝑋𝑜                                (3) where 𝑤(𝑥𝑜) is the weight function that prefilters the data. Further considerations on the effects of the weight function will be made in the next sections.    III. STUDY OF THE RADIATION OPERATOR   In this section, we first evaluate the kernel of 𝐴𝐴𝑤† by exploiting an asymptotic approach. Later, by introducing a suitable change of variables, we show that it is possible to approximate such kernel with a bandlimited function of difference type.                   The operator 𝐴𝐴𝑤† is defined as below                               𝐴𝐴𝑤† 𝐸 = ∫ (𝑧𝑜2 𝑤(𝑥𝑜) ∫𝑒−𝑗 𝛽 (𝑅 (𝑥′,𝑥) −𝑅 (𝑥′,𝑥𝑜)) 𝑅32(𝑥′, 𝑥) 𝑅32(𝑥′, 𝑥𝑜)  𝑑𝑥′+𝑎−𝑎) 𝐸(𝑥𝑜) 𝑑𝑥𝑜+𝑋𝑜−𝑋𝑜                       (4) By setting 𝑓(𝑥′, 𝑥, 𝑥𝑜) = 𝑅−3/2(𝑥′, 𝑥) 𝑅−3/2(𝑥′, 𝑥𝑜) and 𝜙(𝑥′, 𝑥, 𝑥𝑜) = 𝑅 (𝑥′,𝑥) − 𝑅 (𝑥′, 𝑥𝑜)𝑎, the kernel can be expressed by the following integral                                                     𝐻(𝑥, 𝑥𝑜) = 𝑧𝑜2 𝑤(𝑥𝑜) ∫ 𝑓(𝑥′, 𝑥, 𝑥𝑜) 𝑒−𝑗 𝛽𝑎 𝜙(𝑥′,𝑥,𝑥𝑜)  𝑑𝑥′+𝑎−𝑎                           (5) For 𝛽𝑎 ≫ 1, and ∀𝑥 ≠ 𝑥𝑜 the kernel 𝐻(𝑥, 𝑥𝑜) can be approximated through the asymptotic form    𝐻(𝑥, 𝑥𝑜) ≈ −𝑧𝑜2𝑗𝛽𝑎𝑤(𝑥𝑜) 𝑒−𝑗 𝛽𝑎2 (𝜙−𝑎(𝑥,𝑥𝑜)+𝜙𝑎(𝑥,𝑥𝑜)) ⋅                          ( 𝑓𝑎(𝑥, 𝑥𝑜)𝜙𝑎′ (𝑥, 𝑥𝑜)𝑒𝑗 𝛽𝑎2(𝜙−𝑎(𝑥,𝑥𝑜)−𝜙𝑎(𝑥,𝑥𝑜)) −𝑓−𝑎(𝑥, 𝑥𝑜)𝜙−𝑎′ (𝑥, 𝑥𝑜)𝑒−𝑗 𝛽𝑎2 (𝜙−𝑎(𝑥,𝑥𝑜)− 𝜙𝑎(𝑥,𝑥𝑜)))                    (6) which corresponds to the first term of the integration by parts method [5].                                The subscripts − 𝑎 or 𝑎 in Eq. (6) denote that the correspondent function has been particularized in the point 𝑥′ = − 𝑎 or 𝑥′ =  𝑎.                                                        As can be seen from Eq. (6), the operator 𝐴𝐴𝑤† is space-variant in the variables (𝑥, 𝑥𝑜). With the aim to recast it in a form more similar to a convolution operator, the last expression of  𝐻(𝑥, 𝑥𝑜) suggests introducing the following variables  3                      𝜂(𝑥𝑜) =√(𝑥𝑜+𝑎)2+𝑧𝑜2 −√(𝑥𝑜−𝑎)2+𝑧𝑜2 2𝑎,            𝛾(𝑥𝑜) =√(𝑥𝑜+𝑎)2+𝑧𝑜2+√(𝑥𝑜−𝑎)2+𝑧𝑜2 2𝑎                             (7)  The latter allows expressing the operator 𝐴𝐴𝑤† in the following form  𝐴𝐴𝑤† 𝐸 = ∫𝑧𝑜2𝑗𝛽𝑎𝑤(𝜂𝑜)𝑑𝑥𝑑𝜂𝑜𝑒𝑗𝛽𝑎( 𝛾(𝜂𝑜)−𝛾(𝜂)) (𝑓− 𝑎(𝜂, 𝜂𝑜)𝜙− 𝑎′ (𝜂, 𝜂𝑜)𝑒−𝑗𝛽𝑎(𝜂−𝜂𝑜) −𝑓𝑎(𝜂, 𝜂𝑜)𝜙𝑎′ (𝜂, 𝜂𝑜)𝑒𝑗𝛽𝑎 (𝜂−𝜂𝑜)) 𝐸(𝜂𝑜)𝑑𝜂𝑜𝜂(𝑋𝑜)𝜂(−𝑋𝑜)                                                                                                                                        (8) where 𝜂𝑜 = 𝜂(𝑥𝑜), 𝜂 = 𝜂(𝑥), and 𝑑𝑥𝑑𝜂𝑜=1 √𝑎2 (1−𝜂𝑜2)+𝑧𝑜2  𝑎2(1−𝜂𝑜2)2+𝑧02(1−𝜂𝑜2)3/2 .                    Note that the change of the integration variable introduces in the kernel the Jacobian term 𝑑𝑥/𝑑𝜂𝑜 which is singular for 𝜂𝑜 =  ± 1. Such singularities affect also on the kernel 𝐻(𝜂, 𝜂𝑜) if they are not compensated by 𝑤(𝜂𝑜). Since              𝑑𝑥𝑑𝜂𝑜= 𝑂 (1(𝜂𝑜∓1)3/2) ,        𝑓𝑎(𝜂,𝜂𝑜)𝜙𝑎′ (𝜂,𝜂𝑜)= 𝑂 ((𝜂𝑜 ∓ 1)3/4),        𝑓−𝑎(𝜂,𝜂𝑜)𝜙−𝑎′ (𝜂,𝜂𝑜)= 𝑂 ((𝜂𝑜 ∓ 1)3/4)        (9) as 𝜂𝑜 →  ± 1, it results that for 𝜂𝑜 → ± 1 the terms 𝑑𝑥𝑑𝜂𝑜  𝑓𝑎(𝜂,𝜂𝑜)𝜙𝑎′ (𝜂,𝜂𝑜) and 𝑑𝑥𝑑𝜂𝑜 𝑓− 𝑎(𝜂,𝜂𝑜)𝜙− 𝑎′ (𝜂,𝜂𝑜) go to infinity as (𝜂𝑜 ∓ 1)− 3/4. Hence, in order to compensate the singularities, we choose                                                                                                       𝑤(𝜂𝑜) = (𝑑𝑥𝑑𝜂𝑜)− 1/2                                                     (10) In Fig. 2 the diagram of 𝑤(𝜂𝑜) for different values of  𝑧𝑜 is shown.   FIG. 2 − Diagram of 𝑤(𝜂𝑜) normalized with respect to its maximum for different values of 𝑧𝑜 when the size source 𝑎 = 10𝜆.    Such choice of the weight function makes the kernel free from singularities, and it allows rewriting  𝐻(𝜂, 𝜂𝑜) in the form below   𝐻(𝜂, 𝜂𝑜) ≈𝑧𝑜2𝑗𝛽𝑎(𝑑𝑥𝑑𝜂𝑜)1/2𝑒𝑗 𝛽𝑎 ( 𝛾(𝜂𝑜)−𝛾(𝜂)) (𝑓−𝑎(𝜂, 𝜂𝑜)𝜙−𝑎′ (𝜂, 𝜂𝑜)𝑒−𝑗 𝛽𝑎 (𝜂−𝜂𝑜) −𝑓𝑎(𝜂, 𝜂𝑜)𝜙𝑎′ (𝜂, 𝜂𝑜)𝑒𝑗 𝛽𝑎 (𝜂−𝜂𝑜))         (11) At this stage, the 𝐻(𝜂, 𝜂𝑜) is free from singularities but it does not appear space-invariant since the amplitude terms do not seem to depend on the difference (𝜂 − 𝜂𝑜). However, as shown in [4], ∀(𝜂, 𝜂𝑜) ∈ 𝐷1 = [𝜂(−𝑎), 𝜂(𝑎)] × [𝜂(−𝑎), 𝜂(𝑎)] the terms  𝑓𝑎/𝜙𝑎′  and 𝑓−𝑎/𝜙−𝑎′  can be approximated as below                                               𝑓𝑎(𝜂,𝜂𝑜)𝜙𝑎′ (𝜂,𝜂𝑜)≈𝑓−𝑎(𝜂,𝜂𝑜)𝜙−𝑎′ (𝜂,𝜂𝑜)≈ −𝑎𝑧𝑜2 𝑑𝑥𝑑𝜂𝑜 (𝜂−𝜂𝑜)                                    (12) Hence, it results that ∀(𝜂, 𝜂𝑜) ∈ 𝐷1 the kernel can be rewritten apart for an unessential factor in the simple form                                                                             𝐻(𝜂, 𝜂𝑜) ≈  𝑒𝑗 𝛽𝑎 (𝛾(𝜂𝑜) − 𝛾(𝜂))  sin  (𝛽𝑎(𝜂−𝜂𝑜))𝜋 (𝜂−𝜂𝑜)                                 (13)      where we have taken into account that for 𝜂𝑜 ∈ [𝜂(−𝑎), 𝜂(𝑎)] the term (𝑑𝑥/𝑑𝜂𝑜)−1/2 is almost constant and well approximated by the value in 𝜂𝑜 = 0. According to approximation in Eq. (12), for 𝜂𝑜 →  ± 1 the terms 𝑓𝑎/𝜙𝑎′  and 𝑓−𝑎/𝜙−𝑎′  are not an 𝑂((𝜂𝑜 ∓ 1)3/4) as shown in Eq. (9). For this reason, such approximation does not work 4  for (𝜂, 𝜂𝑜) ∈ 𝐷 − 𝐷1 where 𝐷 = [𝜂(−𝑋𝑜), 𝜂(𝑋𝑜)] × [𝜂(−𝑋𝑜), 𝜂(𝑋𝑜)]. In such region, a reasonable approximation of the terms 𝑓𝑎/𝜙𝑎′  and 𝑓−𝑎/𝜙−𝑎′  is given by                                           𝑓𝑎(𝜂,𝜂𝑜)𝜙𝑎′ (𝜂,𝜂𝑜)≈𝑓−𝑎(𝜂,𝜂𝑜)𝜙−𝑎′ (𝜂,𝜂𝑜)≈ −𝑐𝑜𝑛𝑠𝑡(𝑑𝑥𝑑𝜂𝑜)1/2(𝜂−𝜂𝑜)                      (14)                            By substituting Eq. (14) in Eq. (11), it follows that also for (𝜂, 𝜂𝑜) ∈ 𝐷 − 𝐷1 the kernel 𝐻(𝜂, 𝜂𝑜) can be expressed as in Eq. (13). Hence, we can state that ∀ (𝜂, 𝜂𝑜) ∈ 𝐷 the kernel of 𝐴𝐴𝑤† can be approximated as in Eq. (13). In Fig. 3 the actual kernel of 𝐴𝐴𝑤†  is compared with the sinc kernel given by Eq. (13). As it can be seen from the figure, the sinc kernel represents a good approximation of the actual kernel.   FIG. 3 − Amplitude of 𝐾(𝜂, 𝜂𝑜) (numerically computed), and amplitude of the sinc kernel given by Eq. (13).  The diagrams are in dB, and they refer to the configuration 𝑎 = 10𝜆, 𝑧𝑜 = 5𝜆, 𝑋𝑜 = 10𝑎.  IV. NDF AND DISCRETIZATION  At this stage let us discuss about the NDF, and the discretization of the radiation operator in the case where 𝑋𝑜 > 𝑎. Naturally, the best thing would be that of solving such issues when the adjoint operator is not modified by the weight function 𝑤(𝜂𝑜). In the remaining part of the manuscript, we denote the usual adjoint operator by 𝐴†.                As regards the NDF, let us note that the eigenvalues of the operators 𝐴𝐴† and 𝐴𝐴𝑤† exhibit the same behavior. To proof this, as first thing note that by definition the operators 𝐴†𝐴 and  𝐴𝐴† have the same eigenvalues. Later, observe that the operator 𝐴𝐴𝑤† involves a convolution kernel of sinc type with a spatial bandwidth product 𝑐 = 𝛽𝑎 𝜂(𝑋𝑜) exactly like the kernel of 𝐴†𝐴 [6]. This implies that, unless of a constant factor, 𝐴𝐴𝑤†  and 𝐴†𝐴 have the same eigenvalues. Consequently, we can conclude that the weight function does not change the behavior of the eigenvalues, hence, the eigenvalues of 𝐴𝐴† and 𝐴𝐴𝑤† have the same shape. Thanks to this, it is possible to compute the eigenvalues of 𝐴𝐴† by referring to the operator 𝐴𝐴𝑤†. By exploiting the results in [7], we can state that the eigenvalues of 𝐴𝐴† exhibit a step-like behavior with the knee occurring at the index                                                          𝑁 =2𝛽𝑎𝜋𝜂(𝑋𝑜)                                                     (15) As regards the discretization, let us start by considering the eigenvalue problem 𝐴𝐴𝑤† 𝑣𝑛𝑤 = 𝜆𝑛𝑣𝑛𝑤. The latter can be easily recast in the form                                       ∫ 𝑠𝑖𝑛𝑐(𝛽𝑎(𝜂 − 𝜂𝑜))?̅?𝑛 𝑤(𝜂𝑜) 𝑑𝜂𝑜 = 𝜆𝑛 ?̅?𝑛 𝑤(𝜂) 𝜂(𝑋𝑜)𝜂(−𝑋𝑜)                               (16) where ?̅?𝑛(𝜂𝑜) = 𝑣𝑛(𝜂𝑜)𝑒𝑗𝛽𝑎 𝛾(𝜂𝑜). The integral equation expressed by Eq. (16) involves a convolution operator with a bandlimited kernel whose maximum frequency with respect to 𝜂 is equal to 𝛽𝑎/(2𝜋). Consequently, it can be discretized by applying the sampling theory approach shown in [2] with a sampling step Δ𝜂 = 𝜋/(𝛽𝑎). The application of such sampling method provides an eigenvalue problem for a matrix which apart for the truncation error approximates very well the eigenvalues of the operator 𝐴𝐴𝑤†.                   5  However, as specified before, our aim is that of finding a discretization of 𝐴𝐴†. Since if 𝑋𝑜 > 𝑎 the sampling theory approach cannot be used to discretize the operator 𝐴𝐴†, we check if the discretization introduced for 𝐴𝐴𝑤† works also for 𝐴𝐴†. Figure (4) shows the eigenvalues of 𝐴𝐴†, and those of the discrete model for different values of the sampling step Δ𝜂 = 𝜋/(𝜒𝛽𝑎) (at Nyquist rate 𝜒 = 1, with an oversampling factor 𝜒 =1.1 and  𝜒 =1.1) when 𝑋𝑜 = 10𝑎. As can be seen from the figure, if the observation domain is significantly larger that the source domain, a sampling step exactly equal to the Nyquist step Δ𝜂 =𝜋/(𝛽𝑎) is not sufficient to approximate the eigenvalues of the continuous model. However, a sampling step Δ𝜂 a little bit smaller than 𝜋/(𝛽𝑎) already suffices to approximate the eigenvalues of the continuous operator 𝐴𝐴†.   FIG. 4 – Eigenvalues of 𝐴𝐴† and those of the discretized model for different values of the sampling step Δ𝜂.  The figure refers to the case  𝑎 = 10 𝜆, 𝑧𝑜 = 5 𝜆, 𝑋𝑜 = 10 𝑎. Note that the non-linear relation between 𝜂 and 𝑥 implies that the uniform sampling step Δ𝜂 maps into a spatially varying sampling step Δ𝑥. The position of the sampling points in 𝑥 domain can be found by exploiting the equation                 𝑥𝑚 = 𝜂𝑚 √𝑎2 + 𝑧02/(1 − 𝜂𝑚2 )   where  𝜂𝑚 = 𝜂(−𝑋𝑜) + (𝑚 − 1)Δ𝜂     ∀𝑚 = 1, … , 𝑁𝑠𝑎𝑚𝑝𝑙𝑒𝑠    (17)  V. CONCLUSION In this paper, with reference to a strip current observed in near zone on a finite line parallel to the source, a strategy to collect the radiated field in the case 𝑋𝑜 > 𝑎 has been proposed. In particular, it has been shown that to obtain a good approximation of the singular values of the radiation operator the field must be collect according to Eq. (17). REFERENCES [1] E. Joy, D. Paris, "Spatial sampling and filtering in near-field measurements," IEEE Transactions on Antennas and Propagation, vol. 20, no. 3, pp. 253-261, May 1972. [2] K. Khare, "Sampling theorem, bandlimited integral kernels and inverse problems," Inverse problems, vol. 23, no. 4, pp. 1395, 2007. [3] M.A. Maisto, R. Pierri, R. Solimene, “Near-Field Warping Sampling Scheme for Broad-Side Antenna Characterization,” Electronics, vol. 9, no.7, pp. 1047, 2020. [4] R. Pierri, R. Moretta, “Asymptotic Study of the Radiation Operator for the Strip Current in Near Zone,” Electronics, vol. 9, no. 6, pp. 911, 2020.  [5] N. Bleistein, R.A. Handelsman. Asymptotic Expansions of Integrals; Dover Publications: New York, NY, USA, 1986. [6] M.A. Maisto, R. Solimene, R. Pierri, "Resolution limits in inverse source problem for strip currents not in Fresnel zone." JOSA A, vol. 36, no. 5, pp. 826-833, 2019. [7] D. Slepian, H.O. Pollack, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty-I,” The Bell System Technical Journal, vol. 40, no. 1, pp. 43, 1961. 

A sampling strategy of the radiation operator in near-zone based on an asymptotic kernel

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A sampling strategy of the radiation operator in near-zone based on an asymptotic kernel

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