The reconstruction of the field radiated by a source from square amplitude only data falls into the realm of phase retrieval. In this paper we tackle phase retrieval with two different approaches. The first one is based on a convex optimization problem called Phaselift. The latter exploits the lifting technique to recast phase retrieval as a linear problem with an increased number of unknowns and then, because the linear problem is highly undetermined, it adds further constraints (based on the mathematical properties of the solution) to estimate it. The second approach formulates phase retrieval as a least squares problem therefore it requires to tackle the minimization of a quartic functional which will be carried out by applying a gradient descent method. In the second part of the paper, in order to corroborate the effectiveness of both approaches, we present numerical results. Afterwards, we provide a comparison between the two methods and finally we emphasize how the ratio between the number of independent data and the number of unknowns impacts on the performance in both approaches.

The paper addresses the question of efficiently sampling the near field and its intensity over an arbitrary curved line. It aims at finding the minimum number of measurements and their position for discretizing the near field and its square amplitude without loss of information and follows a common approach for both the radiation operator and the correspondent lifting operator. A scalar geometry is examined for both electric and magnetic current sources. At first, the singular values of the radiation / lifting operator are evaluated by an asymptotic approach and a change of variables which lead to an operator with a widely investigated spectrum. As the number of relevant singular values is available in closed form, the dimensions of the near field and its intensity over the curve are found. Next, starting from a sampling expansion of the pertinent left singular functions, a non-redundant sampling representation of the near field and its square amplitude is provided. Some numerical results confirm the analysis. The results may be of great interest especially in antenna testing by near field data (both standard and phaseless) with an unconventional field scanning system as, for instance, an UAV mounted probe.

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.

In this paper, the problem of computing the number of degrees of freedom (NDF) of the field radiated by a strip current along all the possible lines orthogonal to the source is addressed. As well known, the NDF is equal to the number of singular values of the radiation operator that are before a critical index at which they abrupt decay. Unfortunately, in the considered case, the solution of the associate eigenvalue problem is not known in closed-form, and this prevents us from directly evaluating the singular values of the radiation operator. To overcome this drawback, a weighted adjoint operator is exploited. The latter allows obtaining an eigenvalue problem whose solution is known in closed-form but, at the same time, it modifies the behavior of the singular values. However, since the change affects only the dynamics of the singular values but not the critical index at which they abrupt decay, the NDF of the radiated field can be analytically estimated by resorting to the weighted adjoint operator.

In this article, the question of how to sample the square amplitude of the radiated field in the framework of phaseless antenna diagnostics is addressed. In particular, the goal of the article is to find a discretization scheme that exploits a non-redundant number of samples and returns a discrete model whose mathematical properties are similar to those of the continuous one. To this end, at first, the lifting technique is used to obtain a linear representation of the square amplitude of the radiated field. Later, a discretization scheme based on the Shannon sampling theorem is exploited to discretize the continuous model. More in detail, the kernel of the related eigenvalue problem is first recast as the Fourier transform of a window function, and after, it is evaluated. Finally, the sampling theory approach is applied to obtain a discrete model whose singular values approximate all the relevant singular values of the continuous linear model. The study refers to a strip source whose square magnitude of the radiated field is observed in the Fresnel zone over a 2D observation domain.