Abstract
In this paper, the question of evaluating the dimension of data space in an inverse source problem from near-field phaseless data is addressed. The study is developed for a 2D scalar geometry made up by a magnetic current strip whose square magnitude of the radiated field is observed in near non-reactive zone on multiple lines parallel to the source. With the aim of estimating the dimension of data space, at first, the lifting technique is exploited to recast the quadratic model as a linear one. After, the singular values decomposition of such linear operator is introduced. Finally, the dimension of data space is evaluated by quantifying the number of “relevant” singular values. In the last part of the article, some numerical simulations that corroborate the analytical estimation of data space dimension are shown.
Highlights
Antenna testing is a relevant step in the characterization of radiating systems that consists in the determination of the far-field pattern of the considered antenna under test
After a linear model has been obtained by means of the lifting technique, the Singular Value Decomposition can be exploited to estimate a good upper bound of the data space dimension
Since the distance z1 and z2 are very similar, the operators A1 A1†, A1 A2†, A2 A1†, A2 A2† does not differ significantly each other. Their eigenvalues exhibit the same decay. In such case, the second scanning line does not increase significantly the number of significant singular values, and the dimension of data space remains essentially equal to the case of 1 scanning line
Summary
Antenna testing is a relevant step in the characterization of radiating systems that consists in the determination of the far-field pattern of the considered antenna under test. To surmount the question of traps in phase retrieval, in the last decade new methods like PhaseLift [21] and PhaseCut [22] have been introduced The latter exploits the lifting technique which, through a redefinition of the unknown space, allows recasting the original quadratic problem as a linear one. After a linear model has been obtained by means of the lifting technique, the Singular Value Decomposition can be exploited to estimate a good upper bound of the data space dimension In this paper, such quantity is analytically evaluated with reference to the square magnitude of the field radiated by a magnetic current when it is observed on multiple lines in near non-reactive zone.
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