Search

The linear sampling method (LSM) is a simple and effective approach to image the shape of unknown targets via the solution of a linear inverse problem. In this paper, we show that the LSM can also be exploited to devise a novel effective approximation of the scattering phenomenon, that leads to a new noniterative linear inversion method for the estimation of the target's electric contrast. Since the introduced approximation relies on the broad applicability of the LSM, the proposed inversion method is suitable to tackle inverse scattering problems involving nonweak scatterers. As such, it represents an innovative, yet effective, way to tackle quantitative imaging. Examples with numerical and experimental data are given to show the performance of the approach. In particular, results obtained with Fresnel data-sets show that the proposed method is capable of successfully imaging targets which have been so far processed using nonlinear iterative schemes and taking advantage of frequency diversity.

We describe the implementation of a *MATLAB experimental framework for the data conversion of magnetic-resonance (MRI) images to contrast maps. This is done for use in microwave breast cancer detection via the contrast source (CS) model (although the underlying logic can be adapted quite easily to different contexts and applications). The framework is conceived to relieve interested developers from most of the programming burden, and to provide final users with a friendly, consistent approach to the numerical simulations.

The homogeneity of the amplitude of one of the polarizations of the RF fleld B 1 is a crucial issue in Magnetic Resonance Imaging (MRI), and several methods have been proposed for enhancing this uniformity (\Shimming"). The existing approaches aim at controlling the homogeneity of B + 1 and limiting the Speciflc Absorption Rate (SAR) of the RF fleld by independently controlling magnitude and phase of individual excitation currents in MRI scanners, either birdcage or TEM coil system. A novel approach is presented here which allows a joint control of B + 1 uniformity, SAR, and purity of polarization of the total RF B 1 fleld. We propose a convex optimization procedure with convex constraints, and special attention has been devoted to the issue of convexity of the proposed functional. The method is applied to MRI brain imaging; numerical tests have been performed on a realistic head model at low, medium, and high RF fleld in order to assess the efiectiveness of the proposed method. We found that maintaining a speciflc polarization plays an important role also in maintaining the homogeneity of B + 1 amplitude.

Uniform amplitude sparse arrays have recently gained a renewed interest and a number of synthesis techniques, mainly based on global optimization algorithms, have been presented. In this paper, after a discussion about the expected characteristics of such arrays, a simple deterministic approach for the case of pencil beams patterns is proposed and discussed. The approach, which outperforms previous synthesis techniques, takes inspiration from existing (not well-known) density taper procedures to develop a two stages synthesis where the first step is solved in a new closed analytical form, while the second one just requires local refinements or fast 1-D optimizations. As a consequence, the proposed approach avoids the need of multidimensional global optimization procedures and the inherent possibly prohibitive computational costs. Notably the first step, which also exhibits an improved flexibility with respect to other analytical techniques, already outperforms previous approaches in a number of cases of interest. Several numerical results confirm the effectiveness and usefulness of the proposed tools.

In many diagnostics and imaging applications intensity only measurements of the diffracted field are available. Hence, it is of interest to develop procedures able to solve the underlying inverse scattering problem under these conditions. In this paper, we introduce an approach in which the geometrical and electromagnetic properties of unknown complex targets are reconstructed from the squared amplitudes of the diffracted fields. The approach relies on some preliminary considerations on the spatial bandwidth of the diffracted intensities which allow one to fix rules to effectively measure the data and on an alternative formulation of the electromagnetic scattering that lowers the ‘degree of non-linearity’ of the relationship among the electric features of the targets and the fields. The proposed procedure is tested against benchmark experimental data in the microwave frequency range and the obtained results confirm the possibility of achieving faithful quantitative images of unknown targets from phaseless...

Quantitative microwave tomography is an imaging method based on the solution of a non-linear inverse scattering problem, which is able to provide an objective assessment of a region under test. Such a capability is relevant in the framework of non-destructive structural monitoring, where it allows to process ground-penetrating radar data so to achieve accurate information on the inner status of the probed structure. In this paper, we propose a two-step quantitative microwave tomography approach in which the morphological characterization of the targets is first pursued and then their electric contrast is determined by exploiting the information previously gained. A feasibility assessment of the proposed strategy is given against synthetic data concerning the imaging of a leaking pipeline embedded in a wall.

With reference to inverse scattering problems, two different integral equation models belonging to the class of source-type integral equations are reviewed, discussed, and properly compared. In particular, starting from the concept of degree of nonlinearity of the scattering problem, universal plots and useful convenience maps are derived and exploited to foresee which is the most convenient integral equation model to adopt for modeling the scattering phenomena at hand. Numerical examples, dealing with both synthetic and experimental data, are reported and discussed. The results fully confirm the theoretical analysis.

Microwave hyperthermia is a non-invasive treatment for cancer which exploits a selective heating of tissues induced through focused electromagnetic fields. In order to improve the treatment's efficiency, while minimizing side effects, it is necessary to achieve a constrained focusing of the field radiated by the sources. To address this issue, in this paper we present an innovative and computationally effective approach to the field focusing for hyperthermia. The proposed method, after establishing the number of sources to be used, determines the excitations of the given set of sources such to produce a maximum field in a given region of space subject to a completely arbitrary mask for the field amplitude in all other regions. As the approach relies on a formulation of the problem in terms of convex programming, it is able to achieve the globally optimal solution without the adoption of computationally intensive global optimization procedures. A preliminary assessment of the feasibility is given on hyperthermia therapy of breast cancer by means of numerical examples run on realistic 2D phantoms of female breast.

A suitable deterministic partitioning of large planar arrays in two kinds of properly shaped subarrays is shown to improve performances in terms of side-lobes peaks, directivity and aperture efficiency with respect to other existing solutions. The application of this simple idea to the case of large planar arrays with a reduced number of control points, useful in satellite applications, is demonstrated.

The problem of the optimal synthesis of circularly symmetric aperture distributions able to radiate pencil beams fulfilling well-defined requirements on the far-field and/or the directivity is addressed and solved. In particular, constraints concerning both the sidelobe level (SLL) and the directivity within a coverage zone are dealt with while being careful not to incur into the so called super-directive sources or into aperture fields actually difficult to be realized. The case wherein a constraint is enforced on the dynamics of the excitations is also considered. Numerical examples concerning cases of practical relevance are also shown in support to the given theory.

An analytical approach for controlling the zeros of a scattering event (i.e., cloaking) is given, substituting the concept of positive-negative constitutive parameters with a positive-negative contrast formulation. Starting from the zero scattered field condition, a proper manipulation is performed and two different equations are derived: the Contrast-Field-Green Cloaking Equation (CFGCE) and the Contrast Cloaking Equation (CCE). In the present paper, the CCE is deeply analyzed and compared with the existing cloaking approaches in literature. It is shown how Transformation Optics and Scattering Cancellation methods corresponds to the Strong and Weak solutions of the CCE, respectively. This unique equation turns out to be independent from the involved physical events, if the main contrast parameter is properly chosen. Under its main assumptions, the CCE predicts a cloaking effect for arbitrary backgrounds, arbitrary shape objects and arbitrary physics of fields (e.g., magnetic, thermal, acoustic, etc.).

We briefly review the rationale of our 'virtual experiments' approach to inverse scattering problems. The different resulting solution procedures, including very recent developments, are also introduced and summarized.

The Electron Cyclotron Resonance Ion Sources (ECRISs) development is strictly related to the availability of new diagnostic tools, as the existing ones are not adequate to such compact machines and to their plasma characteristics. Microwave interferometry is a non-invasive method for plasma diagnostics and represents the best candidate for plasma density measurement in hostile environment. Interferometry in ECRISs is a challenging task mainly due to their compact size. The typical density of ECR plasmas is in the range 10(11)-10(13) cm(-3) and it needs a probing beam wavelength of the order of few centimetres, comparable to the chamber radius. The paper describes the design of a microwave interferometer developed at the LNS-INFN laboratories based on the so-called "frequency sweep" method to filter out the multipath contribution in the detected signals. The measurement technique and the preliminary results (calibration) obtained during the experimental tests will be presented.

Two different strategies are reviewed in order to counteract the possible occurrence of false solutions in non linear inverse scattering, namely the 'Contrast Source Extended Born' formulation, and the 'Virtual experiments' framework. Article not available.

Time reversal is a well-known method to focus a wave-field into complex unknown/known media. In this letter, we show that when applying time reversal to vector fields, a proper choice of the test source's polarization allows definite advantages as compared to the usual approach of using an a priori fixed polarization. As a matter of fact, by so doing, one can not only focus the target point, but also obtain a field distribution that better accommodates some other requirements (e.g., minimizes sidelobe level). To exploit this possibility, we propose a simple and effective optimization strategy and assess it in case of focusing transverse electric (TE) fields into 2-D scenarios.

The solution of inverse scattering problems is tackled by means of a novel iterative approach that relies on the emerging virtual scattering experiments' paradigm. In the virtual experiments framework, inverse scattering problems can be solved within a linear framework even for non-weak targets, thanks to the behavior of the fields enforced through proper rearrangements of the scattered field data. Such a concept is herein exploited within an iterative scheme, in which the virtual experiments are recast on the basis of intermediate results and the relevant Green's function updated at each iteration step. Examples dealing with both numerical and experimental data are provided to assess the effectiveness of the proposed method and to show that it overcomes the well-known Distorted Born iterative method, while preserving its flexibility and simplicity. As such, these results demonstrate that the virtual experiment framework can promote the exploitation of microwave imaging in real world applications.

Starting from the concept of "degree of non-linearity" of the relationship amongst unknown parameters and scattered fields, we discuss how knowledge of the geometrical features of the targets favorably affects the solution of inverse scattering problems. The same concept is also exploited to optimize the contrast source-extended Born (CS-EB) model, resulting in the optimized CS-EB (O-CS-EB) inversion approach, which remarkably reduces false solutions occurrence. Then, a two-steps approach is proposed as a reliable and effective inversion strategy. The first step consists in retrieving the shape of the targets using the linear sampling method, while in the second step, the estimated shapes and the O-CS-EB are exploited to reconstruct the permittivity profile. The approach has been tested against experimental data, showing that very accurate reconstructions are achieved from monochromatic data.

Efficient and reliable reconstruction of location and shape of dielectric targets via microwave imaging is relevant in many applications. In this respect, the linear sampling method is an effective candidate to pursue this task. However, despite its simplicity and computational effectiveness, still its use is restricted to the mathematical community wherein it has been originally developed. Starting from this observation, in this paper we propose and test a simple and original "physical" interpretation of the linear sampling methods, which shows its relationship with electromagnetic focusing problems. Taking advantage of this result we discuss merits and limitations of the method and suggest new guidelines for a successful application. The analysis is supported with results against experimental data

The problem of synthesizing "optimal" sum and difference patterns subject to arbitrary sidelobe bounds by means of a simple feeding network is dealt with. It is shown how a hybrid optimization procedure properly acting on both clustering into subarrays and excitations of these latter greatly improves performances of known feeding architectures with respect to existing procedures. Results of the proposed technique, which significantly outperform existing methods, renew the interest of these simple feeding architectures, and emphasize the effectiveness of this kind of hybrid procedures exploiting the convexity of the problem with respect to a part of the unknowns

Most global optimization based synthesis procedures act in essentially the same way on all the involved variables. In this paper, it is theoretically discussed and numerically emphasized how explicit exploitation of the convexity of the problem with respect to a part of the degrees of freedom (when available) allows to achieve increased performances with respect to previous results, and to develop new interesting design solutions. Examples are presented for the synthesis of both sum and difference patterns by using nonuniformly spaced arrays and for the so called "optimal compromise amongst sum and difference patterns." Finally, the proposed approach is exploited in developing effective design solutions (based on the concept of interleaved arrays) to the problem of achieving very flexible antennas while using simple feeding networks. As an example, array antennas capable to radiate two different beams with individually steerable patterns, or to perform jammer rejection at the physical layer or even to realize a "flat-top" beam pattern are synthesized and presented

This paper deals with a novel approach to determine the far field of a discrete source starting from the knowledge of far-field intensity data sets only. In particular, by taking advantage from a body of knowledge from antenna synthesis, as well as from the exploitation of suitable norms, we are able to formulate the problem as the superposition of (many) convex problems. Such a circumstance allows (at least in principle) to avoid false solutions.

The linear sampling method (LSM) is a model based inverse scattering approach to image the shape of an unknown target, which does not require approximations and it is effective from a computational point of view. In particular, it provides quasi real-time images of even large regions, thus being very attractive for applications. On the downside, the LSM cannot be applied to traditional GPR multi-monostatic, or multi-bistatic configurations. This contribution considers the application of the LSM to mine clearance, taking advantage of an international research effort aimed at developing an integrated demining system, which will include a multi-view multi-static GPR device. However, the peculiar configuration adopted by the system is different from the canonical ones so far considered to validate LSM in subsurface imaging scenarios. As such, the feasibility of LSM in these working conditions has to be preliminary proved before tackling the experiments. To address this issue, we present the outcomes of a numerical study that demonstrates that LSM can be successfully applied to the forthcoming on-site demonstration.

In this communication we review the rationale and the outcomes of a new way of thinking we have recently introduced for solving inverse scattering problems, i.e., the ‘virtual experiments’ framework.

A proper extension of the contrast source extended Born model, recently introduced in the two-dimensional scalar case, is proposed to establish a novel full-wave inversion method for nonlinear vectorial three-dimensional scattering problems. In particular, a suitable auxiliary function, embedding the parameters of the unknown targets, is introduced as fundamental unknown of the inverse problem instead of the widely used contrast function. Numerical examples confirm effectiveness and reliability of the proposed inversion approach, which also favorably compares with other methods previously known in the literature

Development of fast, flexible, and accurate methods for the analysis (and design) of photonic crystals devices is a relevant topic in order to optimize existing devices and/or developing new design solutions. Within this framework, in this article, we present an improved version of the scattering matrix method [1] for the evaluation of the electromagnetic behavior of two-dimensional finite-extent photonic crystals made of a finite set of parallel dielectric rods. In particular, by relying on features of the fields to be computed within the device and a suitable aggregation into “macrocells”, a considerable reduction of the number of unknowns of the linear system to be solved is achieved. A numerical analysis confirms the accuracy of the proposed method and the remarkable computational benefit. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 2564–2570, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21998

With reference to the case of TM data, in this paper we test the reconstruction capabilities of a recently proposed inverse scattering approach: the contrast source extended Born (CS-EB) inversion method. By relying on a suitable rewriting of the scattering equations, the CS-EB is claimed to be particularly effective whenever losses are present in the background medium and/or in the unknown scatterers. Here, by taking advantage of the Marseille data set concerning lossless inhomogeneous targets in free space, we show the effectiveness of the CS-EB inversion method also in the lossless case. In order to take advantage in a simple fashion of the available multifrequency information, data are processed within a frequency hopping scheme and suitable regularization schemes are exploited at each frequency. Moreover, in order to reduce the computational burden as well as the overall difficulty, the limited bandwidth properties of scattered fields are exploited by means of some simple processing to achieve a preliminary estimation of the region wherein the scatterers are located.

A convenient rewriting of the pertinent integral equation is exploited to introduce a new model for two-dimensional electromagnetic scattering by dielectric objects in lossy media. Exploiting this latter, a new series expansion is introduced to solve the forward problem accurately and effectively. The first term of such a series coincides, in particular situations, with the well-known extended Born approximation. Theoretical tools and results are given on the range of applicability and rate of convergence of the series, which favorably compares with the traditional Born one. These tools allow noticing that the new model exhibits a lower "degree of nonlinearity" with respect to parameters embedding dielectric characteristics as compared to the traditional model, thus suggesting its exploitation in the solution of the inverse problem. Numerical examples assessing effectiveness and convenience of the proposed models, tools and inversion methods are presented.

In microwave imaging, the linearity of the relationship between the incident and the scattered field offers the possibility of a posteriori recombining the performed scattering experiments and then to cast the underlying inverse problem with respect to the resulting, virtual , ones. The interest of such a circumstance is that properly designed virtual experiments can enforce particular and convenient conditions. In this paper, we present an application of this paradigm to the popular contrast source inversion (CSI) method. In particular, we first design a set of virtual experiments capable to induce contrast sources exhibiting circular symmetries (with respect to some pivot points). Then, we devise an original and effective regularized CSI scheme, in which a penalty term is added to the usual cost functional, in order to account for the symmetry of the auxiliary unknowns. Notably, the approach does not require any a priori assumption on the unknown contrast, as it relies on the particular nature of the virtual contrast sources. Results with single frequency Fresnel experimental data are given to assess the capabilities of the proposed approach.

This paper introduces and discusses an innovative approach to focus the electromagnetic power carried by a vector field into a target point, while keeping the field level bounded elsewhere. To overcome the complexity of the underlying NP-hard problem, we introduce a multi-objective optimization framework based on convex programming sub-problems. Notably, such a strategy has the unique capability of achieving the globally optimal solution by means of local search algorithms. A numerical analysis is reported to assess the methods’ performance, including a comparison with previous contributions. The proposed approach is herein presented in the case of 2D vector fields (transverse electric polarization) but it can be easily extended to the general 3D case.

Compressive sensing (CS)-based techniques can represent a very attractive approach to inverse scattering problems. In fact, if the unknown has a sparse representation and the measurements are properly organized, CS allows to considerably reduce the number of measurements and offers the possibility to achieve optimal (or nearly optimal) reconstruction performance. Unfortunately, the inverse scattering problem is nonlinear, while CS theory is well established only for linear recovery problems. As a contribution to overcome this issue, in this letter, we introduce two different CS-inspired approaches that exploit the “virtual experiments” framework, wherein it is possible to cast the inverse scattering problems in a linear form even in the case of nonweak targets.

By using properly designed synthetic (or “virtual”) experiments and an original approximation of the contrast sources, we are able to recast the inverse scattering problem in an algebraic form (in a subset of points of the imaged domain) and, hence, to solve it by means of closed form formulas. The new approximation relies on the assumption that the contrast sources induced by the different virtual experiments are focused in given points belonging to the scatterer. As such, the method involves a preprocessing step in which the outcome of the original scattering experiments is recombined into the new, virtual, ones capable of enforcing the expected contrast sources behavior. Examples with numerical and experimental data are provided to assess the actual possibility of setting such a virtual experiments framework, and show the effectiveness of the proposed method.

An innovative approach exploiting the Compressive Sensing theory in the solution of the non linear inverse scattering problem is presented. In fact, the proposed approach tackles the inverse problem in its full non linearity by adopting a Contrast-Source Inversion method. In particular, the problem is formulated as a minimization of a proper functional into a convex set defined by the i 1 -norm of the unknown.

The paper investigates the feasibility of an approach to design effective cloaking devices by means of a joint exploitation of available degree of freedom in synthesis strategies and the physical mechanism underlying cloaking. A different point of view in the design of dielectric covers that minimize the scattering cross section of dielectric objects is introduced and new opportunities are outlined to perform an effective synthesis of cloaking profiles via inverse scattering techniques.

Synthesizing antenna arrays with a minimum number of elements is a problem of high importance in different applications. A relevant contribution in solving such a problem is currently given by a number of synthesis techniques based on the Compressive Sensing (CS). For this reason, we present herein an innovative approach wherein fast pre-processing and post-processing techniques are proposed in order to further improve performance of CS in the design of linear arrays with shaped patterns. Numerical examples in support of the proposed processing procedure are also provided.

The potentialities of Compressive sensing (CS) in inverse scattering problems are considerable, because of the intrinsic nature of the targets that one has to reconstruct in actual applications. However, the CS theory is well established only for linear recovery problems, whereas inverse scattering problems are nonlinear. Hence, an actual exploitation of CS in inverse scattering based applications is a still open issue. As a step in this direction, in this paper we introduce a solution procedure based on the joint exploitation of CS and a recently introduced inversion approach that, by means of proper preprocessing techniques, allows a linearization of the problem, which holds true for a set of cases largely exceeding the validity range of the Born approximation.

In this paper, two new solution procedures for inverse scattering problems, based on a suitable pre-processing of the original scattered field data, are introduced. The two approaches allow to obtain satisfactory results.

With reference to dielectric scatterers, in this paper we briefly report the performance of a nonlinear inversion approach as tested on the 'Marseille data'. Moreover, a simple, yet very effective, regularization scheme exploiting the homogeneous nature of the scatterers at hand is proposed and tested.

A new approach to the optimal focusing of array fields subject to arbitrary upper bounds is presented. The approach formulates the problem as the minimization of a linear function in a convex set. Unlike other approaches, this one guarantees the achievement of the global optimum by using local optimization techniques and can, moreover, deal with any convex constraint on the unknowns, such as near field constraints. Optimization is performed by two ad hoc developed solution algorithms, which exploit the geometrical characteristics of the problem at hand, thus leading to extremely effective and computationally efficient numerical codes. An extensive numerical analysis has been performed in all cases of linear, planar, and circular arc arrays. The enhanced performance of the proposed technique with respect to the solution algorithms available in the literature fully confirms the effectiveness of the approach.

This paper describes the modeling and the full wave numerical simulation of electromagnetic waves propagation and absorption in an anisotropic magnetized plasma filling the resonant cavity of an electron cyclotron resonance ion source (ECRIS). The model assumes inhomogeneous, dispersive and tensorial constitutive relations. Maxwell's equations are solved by the finite element method (FEM), using the COMSOL Multiphysics® suite. All the relevant details have been considered in the model, including the non uniform external magnetostatic field used for plasma confinement, the local electron density profile resulting in the full-3D non uniform magnetized plasma complex dielectric tensor. The more accurate plasma simulations clearly show the importance of cavity effect on wave propagation and the effects of a resonant surface. These studies are the pillars for an improved ECRIS plasma modeling, that is mandatory to optimize the ion source output (beam intensity distribution and charge state, especially). Any new project concerning the advanced ECRIS design will take benefit by an adequate modeling of self-consistent wave absorption simulations.

Dedicated short-range communications (DSRCs) is a short-to-medium range wireless protocol designed for automotive systems. This article describes the development of a one-layer compact left-hand circularly polarized antenna with a footprint and characteristics suitable for on-board-unit antenna of a DSRC system. The size of the developed prototype is 40 × 40 × 1.55 mm3 and exhibits a circularly polarized gain of about 4.68 dBc with a cross polarization discrimination of about 22.5 dB. © 2014 Wiley Periodicals, Inc. Microwave Opt Technol Lett 56:1843–1846, 2014

Signal reconstruction from phaseless data occurs in electromagnetics both in the diagnostics of large reflector antennas and in the near-field testing of microwave and millimeter wave antennas. The main difficulty in solving such problems arises from its inherent nonlinearity. Because of the latter, in fact, the solution procedure can converge to a function which has nothing to deal with the actual solution. In this paper we review and summarize the properties of a recently proposed approach to phase retrieval problems which allows to understand and possibly avoid the false solution problem. To this end, ill-posedness of the problem is briefly recalled, and a convenient generalized solution is introduced as the global minimum of a proper functional whose local minima are indeed the false solutions of the problem. Then the geometrical meaning of such a functional is considered, and properties of the solution are discussed according to the formulation of the problem as a quadratic inverse one. In particular, the relevance of using nonredundant representations for the unknown of the problem and as many independent data as possible are discussed in full detail from both a theoretical and a numerical point of view. Finally, it is shown how the proposed solution procedure, descending from the adopted formulation based on a weaker nonlinearity with respect to the other ones, has indeed a better behavior as far as local minima problem is concerned.

The retrieving of the phase of a field radiated by a planar source over a plane in near zone is investigated by formulating it as the inversion of a quadratic operator. To solve the problem, a representation of the near field is adopted which takes into account both the dimension and the location of the source. This makes it possible to derive a new uniqueness result which allows one of the main Limitations of previous methods to be removed. The conditions to be met in order to obtain a completely reliable solution procedure are then briefly illustrated by exploiting the quadraticity of the operator to be inverted. Finally, the iterative solution algorithm is described and examples are shown.

A rigorous deterministic approach is proposed for the mask-constrained power pattern synthesis of shaped beams through one-dimensional sparse arrays having uniform-amplitude excitations (and variable phases). The approach includes the well known density taper procedure as a special case. The corresponding technique, which exploits an analytical formulation of the problem and convex programming routines, results in fast computation and is able to achieve optimal solutions without recurring to any global optimization. Numerical results assessing the capabilities of the proposed design method are provided, opening the way to the development of an effective approach for the synthesis of circularly symmetric shaped beams via uniformly-excited circular ring arrays.

In this contribution we discuss the feasibility of a full-Earth coverage from GEO satellite by means of isophoric (that is, equi-amplitude) Direct Radiating Arrays (DRAs). By focusing on the ring-array architecture we show that a large beam scanning requirement imposes some severe geometrical constraints on the final array layout. Furthermore, we introduce a synthesis procedure capable to generate arrays satisfying the desired specifications with a reasonably low number of control points.

We would like to correct a statement that we gave in our comments to [1, eq. (9)] . While such a change does not affect in any way the validity of the inversion approach presented in [1] and the results therein obtained, it is nevertheless important to discuss it, in order to provide the correct physical interpretation of (9).

The problem of determining radiated electromagnetic fields from phaseless distributions on one or more surfaces surrounding the source is considered. We first examine the theoretical aspects and basic points of an appropriate formulation and show the advantage of tackling the problem as the inversion of the quadratic operator, which, by acting on the real and imaginary parts of the field, provides square amplitude distributions. Next, useful properties and representations of both fields and square amplitude distributions are introduced, thus making it possible to come to a convenient finite-dimensional model of the problem, to recognize its ill-posed nature and, finally, to define an appropriate generalized solution. Novel uniqueness conditions for the solution of the problem and questions regarding the attainment of the generalized solution are discussed. The geometrical properties of the functional set corresponding to the range of the quadratic operator relating the unknowns to the data are examined. The question of avoiding local minima problems in the search for the generalized solution is carefully discussed and the crucial role of the ratio between the dimension of the data representation space and that of the unknowns is emphasized.

An algorithm for the determination of the radiation pattern of a radiating source from phaseless measurements over two planes is developed in detail. A rather extensive assessment of its performances with respect to positioning errors and noise uncertainties is included. Moreover, comparisons with experimental data for a low sidelobe antenna demonstrate the wide applicability of the proposed technique, which is now mature for use both at frequencies where phase information cannot be reliably detected and at lower frequencies. The full exploitation of this new concept in near field techniques will allow a drastic reduction of the cost of the whole facility, since many specification parameters can be relaxed without affecting the precision of the measurement set-up.

Due to the continuous growth of satellite communications and the nearly saturation of the Ku band, Ka band has started to be used for broadband communications. For the new generation of spaceborne antennas operating in Ka band, satellite communication operators require some in-orbit features as the flexibility and the re-configurability. The most suitable antenna configuration candidate to provide these characteristics is the Active Direct Radiating Array. This paper describes innovative solutions that, by the use of sparse/thinned lattices, allows to minimize the radiating elements number reducing the complexity for the cooling system integration (e.g. heat-pipes) and beam former implementation.