Abstract
In this paper, we address the problem of how to efficiently sample the radiated field in the framework of near-field measurement techniques. In particular, the aim of the article is to find a sampling strategy for which the discretized model exhibits the same singular values of the continuous problem. The study is done with reference to a strip current whose radiated electric field is observed in the near zone over a bounded line parallel to the source. Differently from far zone configurations, the kernel of the related eigenvalue problem is not of convolution type, and not band-limited. Hence, the sampling-theory approach cannot be directly applied to establish how to efficiently collect the data. In order to surmount this drawback, we first use an asymptotic approach to explicit the kernel of the eigenvalue problem. After, by exploiting a warping technique, we recast the original eigenvalue problem in a new one. The latter, if the observation domain is not too large, involves a convolution operator with a band-limited kernel. Hence, in this case the sampling-theory approach can be applied, and the optimal locations of the sampling points can be found. Differently, if the observation domain is very extended, the kernel of the new eigenvalue problem is still not convolution. In this last case, in order to establish how to discretize the continuous model, we perform a numerical analysis.
Highlights
The inverse source problem is a classical problem in electromagnetics with a lot of applications related to the sources diagnostics and the fields synthesis [1,2,3,4,5,6,7]
With reference to the case where the sinc approximation works, we show how to efficiently discretize the eigenvalue problem T T † vn = σn2 vn (η ) in order to obtain a discrete model that well approximate the mathematical properties of the continuous operator T T †
In the Appendix C all the steps required to pass from the continous model (22) to the discrete model (23) are shown. The latter represents the eigenvalue problem for the matrix A, and (apart for the truncation error of the sampling series (A6) and (A7)) it exhibits the same eigenvalues of the Fredholm integral Equation (21) and (22)
Summary
The inverse source problem is a classical problem in electromagnetics with a lot of applications related to the sources diagnostics and the fields synthesis [1,2,3,4,5,6,7]. We address the problem of finding the optimal location of the sampling points with reference to a strip magnetic current whose radiated electric field is observed in the near zone over a bounded line parallel to the source In this case, differently from far zone configurations [27,28], the kernel of the integral operator involved in the eigenvalue problem is not band-limited and it is not of convolution type. Of the paper, by exploiting asymptotic arguments and a suitable change of variables, we show how to obtain a new eigenvalue problem whose kernel in same conditions is well approximated by a band-limited of difference type In this condition, the sampling-theory approach can be exploited to efficiently discretize the continuous model. We will perform a numerical analysis to establish the sampling frequency that allows approximating well the singular values of the radiation operator
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