Determining phase profile for beamforming antenna

Abstract

A system and method are provided for determining a phase profile over an aperture of a beamforming antenna to shape a beam of the beamforming antenna. As input, a target beam shape to be established and a characterization of the aperture of the beamforming antenna are received. The stationary phase approximation is used to determine a transport map T(r) which maps intensities in a preferably measure-preserving manner from an antenna domain of the beamforming antenna to a beam domain of the beamforming antenna through a gradient of the phase profile. The transport map is integrated over its irrotational part to obtain the phase profile ϕ(r). The phase profile, or a sampled version thereof, is then output, for example to a transmitter of the beamforming antenna. The above measures may allow the phase profile to be computed in a computationally efficient manner and in real-time or near-real time.

Description

[0001]DETERMINING PHASE PROFILE FOR BEAMFORMING ANTENNA TECHNICAL FIELD The presently disclosed subject matter relates to a system and method for determining a phase profile for a beamforming antenna. The presently disclosed subject matter further relates to an antenna system comprising the beamforming antenna and the system for determining the phase profile, and to a computer-readable medium comprising instructions for causing a processor system to perform the method. BACKGROUND Beamforming is a well-established signal processing technique used in antenna systems for directional signal transmission or reception. The primary objective of a beamforming antenna is typically to focus the radiated power of an antenna in a particular direction or spatial sector, or in other words, to ‘form’ or ‘shape’ the beam of the antenna. Such beamforming may be achieved by employing multiple antenna elements that collaboratively produce a combined signal with a specific directional pattern. Beamforming offers many advantages, such as enhanced signal quality, reduced interference, and increased capacity in wireless communication systems. While traditional beamforming techniques manipulate both amplitude and phase of the signals across the antenna elements to achieve the desired radiation pattern, phase-only beamforming simplifies this process by adjusting only the phase of the signals while maintaining a constant amplitude. By forgoing the amplitude adjustments, phase-only beamforming antennas offer a streamlined design, potentially resulting in reduced hardware complexity, power consumption, and overall system cost. Despite the advantages offered by phase-only beamforming, computing phase profiles for these antennas presents significant challenges. Accurate phase control is pivotal in determining the directionality and shape of the radiated beam. However, without the flexibility of amplitude adjustment, it is challenging to determine a phase profile with which the beamforming antenna establishes a target beam shape. SUMMARY It would be advantageous to obtain an improved way of determining a phase profile for a beamforming antenna, in particular for a phase-only beamforming antenna. In accordance with a first aspect of the invention, a system is provided for determining a phase profile for a beamforming antenna, the system comprising: an output interface; a processor subsystem configured to determine a phase profile over an aperture of the beamforming antenna to shape a beam of the beamforming antenna by: - receiving a target beam shape to be established and a characterization of the aperture of the beamforming antenna; - determining, using stationary phase approximation, a transport map which maps intensities in a preferably measure-preserving manner from an antenna domain of the beamforming antenna to a beam domain of the beamforming antenna through a gradient of the phase profile; - integrating over an irrotational part of the transport map to obtain the phase profile; and - via the output interface, outputting the phase profile or a sampled version of the phase profile. In accordance with a further aspect of the invention, a method is provided for determining a phase profile over an aperture of a beamforming antenna to shape a beam of the beamforming antenna, the method comprising: - receiving a target beam shape to be established and a characterization of the aperture of the beamforming antenna; - determining, using stationary phase approximation, a transport map which maps intensities in a preferably measure-preserving manner from an antenna domain of the beamforming antenna to a beam domain of the beamforming antenna through a gradient of the phase profile; - integrating over an irrotational part of the transport map to obtain the phase profile; and - outputting the phase profile or a sampled version of the phase profile. In accordance with a further aspect of the invention, a transitory or non- transitory computer-readable medium is provided, the computer-readable medium comprising data representing a computer program, the computer program comprising instructions for causing a processor system to perform the method. The above measures involve computing a phase profile for a beamforming antenna, and preferably, for a phase-only beamforming antenna. For that purpose, a target beam shape which is to be established is received as input. The target beam shape may represent the desired shape of the beam in terms of, for example, directionality and strength. The target beam shape may be defined in the spatial domain and may for example be described using radiation patterns, being a graphical representation of the radiation properties of the antenna as a function of space coordinates. Furthermore, a characterization of the aperture of the beamforming antenna is received as input. Here, the term ‘aperture’ may refer to the physical configuration and properties of the antenna array. This may include the number of antenna elements, their arrangement (e.g., linear, planar, or circular), the spacing between them, the individual element radiation patterns, etc. In particular, the aperture characterization may provide details on how each individual antenna element contributes to the overall antenna pattern. The aperture characterization may also indicate the degrees of freedom available for beamforming. For instance, an array with many elements offers more flexibility in shaping the beam than a smaller array. The above measures involve calculating a phase profile to be used in controlling the beamforming antenna. This may involve using a stationary phase approximation to generate a transport map which maps intensities in a preferably measure-preserving manner from an antenna domain of the beamforming antenna to a beam domain of the beamforming antenna through a gradient of the phase profile. This may be explained as follows. The stationary phase approximation (SPA) is a mathematical technique which uses the principle that when integrating a rapidly oscillating function, the predominant contributions to the integration are derived from points where the function's phase remains stationary, meaning its derivative is zero. The above measures leverage the SPA to determine how intensities are transitioned from the antenna domain (e.g., the spatial locations of the antenna elements) to the beam domain (which delineates the spatial radiation pattern or beam shape). This mapping may also be referred to as a transport map. This way, an understanding of the primary determinants of the beam's shape may be obtained, and in particular, it may be identified where the majority of the energy from the antenna domain will be positioned in the beam domain, given a particular phase profile. The transport map may be specifically determined to be measure-preserving. Thereby, the cumulative energy present in the antenna domain may be conserved, at least substantially, when mapped to the beam domain. The phase profile, meaning the distribution of phase values applied to each element within the beamforming antenna to steer or shape the emitted beam, may then be obtained by integrating over an irrotational part of the transport map. Having determined the phase profile, the phase profile is then output, or sampled and then output, to be used to control the beamforming antenna. For example, the phase profile may be output to a transmitter of the beamforming antenna. The sampling may serve to determine individual phases for individual elements of the antenna. While several phase-only beamforming optimization methods are known in the literature, such as gradient methods, alternate projections, global optimization methods (e.g., genetic, particle swarm, invasive weed, etc.), semidefinite programming relaxation, etc., these optimizations suffer from one or more of the following problems: the reduced space of the degrees of freedom in phase-only beamforming, the high non-linearity of the optimization problem which may trap the solution in largely sub- optimal local minima, the large number of elements, the limited efficiency of the these optimization methods, the incapability of obtaining results in real-time, etc. The above measures have the effect of allowing the phase profile to be computed in a computationally efficient and rapid manner. This is highly relevant in beamforming antennas, as computationally resource intensive calculations may prohibit use in various application areas, such as space vehicles in which available resources are limited, while time consuming calculations may prevent real-time or near- real time use. Conversely, the above measures may enable phase profiles to be calculated in real-time or near-real time using only limited computational resources. The following embodiments may relate to the system but may also denote corresponding limitations, e.g., in form of steps, of the method or computer program. In an embodiment, the processor subsystem is further configured to: - receive a first intensity profile ^(^) over the aperture domain and a second intensity profile ^^(^)over the beam domain; - determine the transport map ^(^) from the antenna domain to the beam domain as a solution of a Jacobian-determinant partial differential equation ^det J^^(^)^^ =^(^)^^^^(^)^ ; - determine the phase profile ^(^) over the antenna domain as a potential function satisfying a gradient partial differential equation ^^(^) = ^(^). The above measures compute the phase profile by considering the computation as an irrotational transport map problem in which, given an intensity distribution in the aperture domain (e.g., a power distribution over the aperture of the antenna) and a desired far-field intensity distribution in the beam domain (e.g., the power distribution over the beam), the goal is to determine a transport map which is at least substantially measure-preserving. This transport mapping can be formulated as a Jacobian-determinant partial differential equation. Advantageously, there exist several techniques to find an irrotational transport map which solves the Jacobian-determinant partial differential equation, for example in an exact or approximated manner. In an embodiment, the processor subsystem is configured to analytically solve the Jacobian-determinant partial differential equation and the gradient partial differential equation to obtain the transport map and the phase profile. The inventors have found that in a number of situations, an analytical solution is feasible, in that it may have acceptable computational resource requirements and / or may still be executable in an acceptable timeframe. For example, analytical solutions can be determined when the beam shape corresponds to an affine transformation of the antenna aperture, or when both the antenna and the beam domains and intensities exhibit a rotational symmetry. Advantageously, an analytical solution may exactly solve the Jacobian-determinant partial differential equation and / or the gradient partial differential equation and thereby provide a phase profile which most accurately establishes the target beam shape. In an embodiment, the processor subsystem is configured to search for an approximated solution to the Jacobian-determinant partial differential equation to obtain an approximated transport map. The inventors have found that various approximation techniques may be used to find an approximated transport map which approximately solves the Jacobian-determinant partial differential equation. Such a transport map may not be a fully optimal transport map but may still satisfy a relevant part of the desired constraints and may therefore still provide a suitable phase profile while being significantly less computationally resource intensive and time consuming. In an embodiment, the processor subsystem is configured to search for the approximated solution using a domain mapping method. In an embodiment, the processor subsystem is configured to search for the approximated solution by defining a first set of points in the aperture domain and a second set of points in the beam domain and by using a thin plate spline mapping method to find the approximated transport map by finding a mapping between the first set of points and the second set of points. In an embodiment, the processor subsystem is configured to search for the approximate solution using a Knothe-Rosenblatt transformation method or a conformal mapping method. In an embodiment, the processor subsystem is configured to determine the phase profile as a scalar potential ^(^) of an irrotational component of the approximated transport map ^(^) using a Helmholtz decomposition of ^(^) = ^^(^) + ^ × ^(^)into an irrotational component and a solenoidal component. In an embodiment, the processor subsystem is further configured to determine the phase profile ^(^) by solving the Jacobian-determinant partial differential equation and the gradient partial differential equation together as a Monge-Ampère ) ^(^) partial differential equation ^det J^^^(^ ^^ = ^^^^^(^)^ . In an embodiment, the processor subsystem is configured to determine the transport map or the phase profile by determining a plurality of radial functions for respective angular segments of the antenna domain and which radial functions together represent either the transport map or the phase profile, wherein the plurality of radial functions is determined using adiabatic approximation in which a radial dependency of a respective radial function is assumed larger than an angular dependency of the radial function. The inventors have observed and found that that on antenna and beam domains which are not too irregular, the variability of the transport map with respect to the radius is higher than the variability with respect to the angle. As such, the transport map may be determined by employing a radial reduction of the problem of finding the transport map and finding rotationally symmetric solutions to this problem. Advantageously, such solutions may be determined in closed form. In an embodiment, the output interface is to a transmitter of the beamforming antenna, and wherein the processor subsystem is configured to determine the phase profile in real-time or near-real time. In a further aspect of the invention, an antenna system is provided comprising the beamforming antenna and the system for determining a phase profile for the beamforming antenna. In an embodiment, the beamforming antenna is an antenna array or a shaped-aperture antenna. The measures described in this specification both apply to beamforming antenna arrays as well as beamforming shaped-aperture antennas. In a further aspect of the invention, a space vehicle or aerial vehicle or ground station or radar station is provided comprising the antenna system. It will be appreciated by those skilled in the art that two or more of the above-mentioned embodiments, implementations, and / or aspects of the invention may be combined in any way deemed useful. Modifications and variations of any one of the system, method, computer program, beamforming antenna, vehicle, and station, which correspond to the described modifications and variations of another one of these entities, may be carried out by a person skilled in the art on the basis of the present description. BRIEF DESCRIPTION OF THE DRAWINGS Further details, aspects, and embodiments will be described, by way of example, with reference to the drawings. Elements in the figures are illustrated for simplicity and clarity and have not necessarily been drawn to scale. In the Figures, elements which correspond to elements already described may have the same reference numerals. In the drawings, Fig.1 illustrates the problem of determining a phase profile of an aperture of a beamforming antenna to shape a beam of the beamforming antenna; Fig.2 illustrates phase synthesis as a transport map problem; Fig.3 illustrates use of thin plate splines mapping; Fig.4 illustrates a center-symmetric problem; Fig.5A illustrates adiabatic approximation; Fig.5B illustrates adiabatic approximation for uniform intensities; Fig.6 shows a circular antenna aperture; Fig.7 shows a thin plate splines mapping of the circular antenna aperture to a triangular beam shape; Fig.8 shows a phase profile for the antenna aperture; Fig.9 shows a radiation pattern of the antenna beam; Fig.10 shows a hexagonal antenna aperture; Fig.11 shows a thin plate splines mapping of the hexagonal antenna aperture to a square beam shape; Fig.12 shows a phase profile for the antenna aperture; Fig.13 shows a radiation pattern of the antenna beam; Fig.14 shows a system for determining a phase profile for a beamforming antenna, which system includes an output interface to the beamforming antenna; Fig.15 shows a method for determining a phase profile for a beamforming antenna; and Fig.16 shows a computer-readable medium comprising data; Reference signs list The following list of references and abbreviations is provided for facilitating the interpretation of the drawings and shall not be construed as limiting the claims. 100 system for determining phase profile for beamforming antenna 110 output interface 120 processor subsystem 130 data storage 150 transmitter 160 beamforming antenna (array) 200 method of determining phase profile for beamforming antenna 210 receiving target beam shape and aperture characterization 220 determining transport map 230 integrating over irrotational part of transport map 240 outputting phase profile 300 computer-readable medium (memory card) 310 stored data DETAILED DESCRIPTION OF EMBODIMENTS The presently disclosed subject matter is susceptible of embodiment in many different forms. In the following, some embodiments are illustrated in the drawings and detailed in this description. However, these embodiments, and the mathematical concepts used in the embodiments, are merely examples. In the following, for the sake of understanding, elements of embodiments are described in operation. However, it will be apparent that the respective elements are arranged to perform the functions being described as performed by them. Further, the presently disclosed subject matter is not limited to the embodiments, as features described herein or recited in mutually different dependent claims may be combined. The following measures concern determining a phase profile for a beamforming antenna. For that purpose, as input, a target beam shape to be established and a characterization of the aperture of the beamforming antenna may be received. A stationary phase approximation may then be used to determine a transport map ^(^) which maps intensities in a preferably measure-preserving manner from an antenna domain of the beamforming antenna to a beam domain of the beamforming antenna through a gradient of the phase profile. The transport map may be integrated over its irrotational part to obtain the phase profile ^(^). The phase profile, or a sampled version thereof, may then be output, for example to a transmitter of the beamforming antenna. The following describes with reference to Figs.1-5B further details and implementation options for the abovementioned determining of the phase profile, with reference to Figs.6-13 exemplary results, and with reference to Figs.14-16 respectively the system and a corresponding method and computer-readable medium. Phase Synthesis and Transport Maps Fig.1 illustrates the problem of determining a phase profile of an aperture of a beamforming antenna to shape a beam of the beamforming antenna. Mathematically, the problem may be (re)formulated as determining the phase distribution ^(^) over the planar aperture domain ^, with assigned real amplitude distribution ^(^), so as to obtain a Fourier transform distribution (far-field) ^(^) = with desired magnitude ^^(^) over the target beam domain ^. The variables of the bidimensional spaces may be defined as, with ^^= 2^ / ^ being the free-space wavenumber. The phase synthesis problem may be formulated as assigning real positive magnitudes ^(^), ^ ∈ ^, and ^^(^), ^ ∈ ^, and finding the phase distribution ^(^) such that:|^(^)|≈ ^^(^). Applying the stationary phase method to the asymptotic approximation of ^(^), and introducing the intensities ^(^) =|^(^)|^of the aperture field, and ^(^) = ^ |^(^)|^of the far-field, the phase synthesis problem may be reduced to a partial differential equation of Monge-Ampère type, complemented by the Parseval– Plancherel energy conservation condition. An asymptotic synthesis problem may be formulated as, given an aperture distribution with intensity ^(^) over a planar aperture domain ^, and assigned a desired pattern intensity ^^(^) in the main beam domain ^, both normalized such that, where ^ and ^ are bounded sets in ℝ^, finding the phase function ^(^) over ^ such that ∇^: ^ → ^ ,(3)and the following Monge-Ampère PDE is satisfied, where ^ is the Hessian operator. The Monge–Ampère PDE was first introduced by Monge and then by Ampère for determining the surface with prescribed Gaussian curvature. The link between the Monge–Ampère PDE and the “Optimal Mass Transport” problem, first formulated by Monge, has been explored from several perspectives. The so-called “second boundary value condition” (3) may require that the domain ^ ⊆ ℝ^is fully mapped through the gradient of ^(^) to the domain ^ ⊆ ℝ^. This condition is often replaced by the boundary condition ∇^(^^)= ^^,(^) which only involves the boundaries of the domains, as it is easier to be dealt with in numerical methods. Expressing the Hessian matrix as the Jacobian of the gradient, ^^(^)= , an equivalent problem may be obtained on ^(^) = ∇^(^), a bijective vectorial function ^: ^ → ^ which acts as a mapping function from the antenna domain ^ to the beam domain ^. An equivalent irrotational transport map problem may be formulated as, assigned the positive intensities ^(^), ^ ∈ ^, and ^^(^), ^ ∈ ^, normalized according to (2), finding the bijective transport map ^: ^ → ^ satisfying the Jacobian-determinant partial differential equation: such that the transport map ^(^) is irrotational, ^(^) = ∇^(^), (7) To fulfill the Parseval–Plancherel energy conservation condition, the transport map ^ = ^(^) may need to preserve the transported energy in the transport ^ → ^. A pictorial representation of the link between the asymptotic phase synthesis and the transport map problem is exemplified in Fig.2. The Monge-Ampère PDE, Equation (4), and the equivalent combination the of Jacobian-determinant PDE (6), together with the gradient PDE (7) are intimately linked to the Monge-Kantorovich Optimal Transport Problem of finding a push-forward map ^(^), ^#^ = ^^, that maps the intensity ^(^) into  ^^(^) in a measure-preserving manner and minimizes the total ^(^)  = argmin^^ . 8# ^( )^ ^^^ Approximate Solution Methods The asymptotic phase synthesis problem may be cast in a Monge-Ampère partial differential equation and then addressed, for example as the search for a proper irrotational transport map solving a Jacobian-determinant partial differential equation, or a variational optimal transport problem solving a Monge-Kantorovich Optimal Transport Problem. While the irrotational transport maps exactly solve the Monge- Ampère partial differential equation, their evaluation may require iterative numerical methods which may in some cases, e.g., for some types of antennas, be computational resource intensive and time consuming. The availability of rapidly evaluable approximate transport maps between the source aperture and the target beam may be highly beneficial and may enable real- time applications. A non exhaustive list of possible approximations include: • Knothe-Rosenblatt transformation may be used to generate a proper push-forward map ^#^ = ^^. By construction, the approximate transport map may fulfil the measure-preserving condition (6), but not the irrotational condition (7) or the optimal transport condition (8). • Conformal mapping, Thin Plate Splines or other domain mapping methods may be applied to map the source antenna domain ^ ∈ ℝ^onto the target beam domain ^ ∈ ℝ^. These type of maps may not fully fulfil the measure-preserving condition (6), nor fulfil the irrotational condition (7) or the optimal transport condition (8). • Angular adiabatic approximation: from a careful experimental analysis it was observed that, on not too irregular domains, the variability of the transport map ^(^)with respect to the radius may be higher than the variability with respect to the angle. Exploiting the remarkable property that the asymptotic phase synthesis problem for rotationally symmetric antenna and beam intensities has a radial solution which can be obtained in a closed form, an adiabatic approximation to the solution of the general problem can be reduced to the composition of a plurality of radial functions for respective angular segments of the antenna domain, wherein each radial function is determined as solution of the rotationally symmetric problem for the radial dependence of the antenna and beam intensities along the respective angular direction. Once one of these methods is applied, a non-optimal transport map may be obtained which may satisfy only part of the desired constraints, nevertheless, the approximate transport map may still serve as a basis for deriving the phase profile. Given an approximate transport map, by exploiting the Helmholtz decomposition (e.g., refer to Chapter 1.16 in G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th Edition, San Diego, Academic Press, 2005), ^ = ∇^ + ∇ × ^,(9)the scalar potential of the irrotational component of the map provides an approximate solution to the phase synthesis problem. An approximate map problem may be formulated as, given positive intensities ^(^), ^ ∈ ^, and ^^(^), ^ ∈ ^, identifying a map ^(^) such that: ^: ^ → ^.(10)If needed, ^(^) may be evaluated as the potential of the irrotational component of ^(^),^(^)= ∇^(^)+ ∇ × ^(^).(11)Alternatively, using the angular adiabatic approximation, the phase profile may be obtained by direct integration of approximate transport map along the radial directions. Fast Approximation based on Thin Plate Splines A fast numerical procedure may comprise using Thin-Plate-Splines (TPS) for aperture mapping and integration of the phase as a solution of a Poisson Partial Differential Equation (PDE). The steps of the procedure may be as follows: Finding ^(^) An approximate map may be determined applying the Thin Plate Splines approach which is described hereafter. Thin Plate Splines Problem - Given: • ^^= (^^, ^^), ^ = 1…^ points in the domain ^, • ^^= (^^^, ^^^), ^ = 1,,^ landmark points in the target space ^, find a function ^: ^ → ^ such that 1. ^(^^) = ^^, and 2. each component ^^,^minimizes the functional Fig.3 illustrates use of thin plate splines mapping. The minimization condition may be interpreted as the minimization of the bending energy of two thin plates each represented by one of the components of ^. A solution to the above constrained interpolation problem may be given as follows. Consider matrices Having set have (∥ ^^− ^ ∥), (13) The procedure may be repeated for ^^with ^^= [^^| 0 0 0]^ setting obtaining It can be proved that the above maps ^^and ^^are interpolating, and that they are minimizing the functional (12) within the class of linear combinations of linear functions of the coordinates and functions of the form ^(∥ ^^− ^ ∥). Finding ^(^) The transport map ^(^) may be obtained by applying the TPS method. One may now calculate ^: ^ → ℝ such that ∇^ = ^. To this end, applying the nabla scalar operator (divergence) to the Helmholtz decomposition (11), one may obtain the phase profile ^(^) solving the Neumann problem for the Poisson partial differential equation: for example by a finite-difference method. Note that by virtue of the identity ∇ ∙ ∇ × ^(^)= 0, only the irrotational component of the Helmholtz decomposition contributes to the above Neumann problem (15). Since the solution is unique up to a constant, one may choose a point (^^, ^^) on the grid and require the condition ^(^^, ^^) = 0. The system (15a)-(15b) may be solved by second-order centered or side discretization schemes. Once ^(^) is determined, its irrotational component ^(^) may be determined by solving the system (15a)-(15b) on a discrete grid. A Solution through Adiabatic Approximation The basic principle of the adiabatic approximation is that the degrees of freedom of a system can be separated into fast and slow variables and the solution of the governing equations can be found by freezing the slowly varying variables. In this present case, if the antenna domain ^, and the beam domain ^ of the problem are not too irregular, centroids ^ ∈ ^ and ^′ ∈ ^ can be identified. Then, from a thorough analysis of the obtained numerical solutions, it can be recognized that, superimposing ^′ by translation to ^, and removing the associated planar shift, with respect to a radial line centered at ^, the angular dependence of ^(^) is much slower than the radial dependence. Exploiting the different rate of variation (radial vs angular) one may decompose the mapping problem in a finite set (angular sampling) of radial problems with a parametric dependence on the angle. The existence of a closed form solution for the rotationally symmetric problem allows a fast solution of the individual radial problems and integration of the radial phase dependency. For uniformly illuminated apertures and uniform amplitude shaped beams an analytical approximation may be readily obtained. Analytical Solution of the Center-Symmetric Problem Fig.4 illustrates a center-symmetric problem. In the center-symmetric hypothesis, ^(^)= ^(^, ^)= ^(^), ^^(^)= ^^(^, ^)= ^^(^).(16)The conservation of the intensity yields the equality between the densities: ^(^)^ ^^ ^^ = ^^(^)^ ^^ ^^.(17)By means of the radially cumulative distribution the integral reformulation of the conservation of intensity (17) becomes which may be rewritten as The phase may then be reconstructed as Azimuthal Adiabatic Approximation Fig.5A illustrates adiabatic approximation. In the general case, the intensities in the near-field and in the far-field can be expressed in polar coordinates centered in ^ and ^′, as, respectively ^(^)= ^(^, ^), ^^(^)= ^^(^, ^) (22) Under the hypothesis of slowly azimuthal variation, one may consider the transport map substantially radial, and the azimuthal dependency as a local parameter: ^ = ^^(^). Introducing the normalized near-field and far-field normalized radial intensities, respectively: one may obtain the local radial map ^^(^) applying the center-symmetric problem closed form (20), The local phase may then be reconstructed as Azimuthal Adiabatic Approximation for Uniform Intensities Fig.5B illustrates adiabatic approximation for uniform intensities. Namely, in the particular case when the near-field and the far-field intensities are uniform, one may provide the solution in closed form: that may lead to and finally to the analytical phase profile: As an alternative derivation of the above approximation, one may write the Monge–Ampère equation (4) in polar coordinates, and then assume that the (unknown) phase function ^ depends on ^ only. That leads to an ordinary differential equation that may be exactly solved, with the same results as above. Numerical Results In the following, results of the phase synthesis method based on, by way of example, the Thin Plate Spline mapping and derivation of the irrotational component by Poisson Partial Differential Equation are described. The examples refer to two different source aperture cases, circular and hexagonal, applied to square and triangular beams shaping, respectively. When grayscale-plots of the far-field intensities are shown, they are normalized to the average intensity ^(^) within the main beam domain ^ (contour line). In Figs.7 and 9, the circle represents the visibility contour (√^^+ ^^= 1). Circular Aperture Fig.6 shows a circular antenna aperture with a diameter of 50^. The target beam shape is a triangular beam of ^^= 0.5 encircling radius. The thin plate splines mapping of the circular antenna aperture to a triangular beam shape is shown in Fig.7, the resulting phase profile in Fig.8, and the far-field, referring to the radiation pattern of the antenna beam, in Fig.9. This result demonstrates the robustness of the proposed approach. Hexagonal Aperture Fig.10 shows a hexagonal aperture with an encircling diameter of 50^. The target beam shape is again a triangular beam of ^^= 0.5 encircling radius. The thin plate splines mapping of the hexagonal antenna aperture to a triangular beam shape is shown in Fig.11, the resulting phase profile in Fig.12, and the far-field, referring to the radiation pattern of the antenna beam, in Fig.13. Also in this case, the results show a satisfactory phase-only beam shaping. It is noted that known numerical methods aimed at finding the transport map and its potential are typically based on finite differences discretization of the derivatives and involve numerical solving of non-linear algebraic equations at some stage. Such methods need many iterations to have a reasonable convergence, while for some geometries, convergence remains very difficult. In addition, known numerical methods are generally not concerned with realistic engineering applications, as typically the domain of the transport map is a square and the target has a simple geometric shape. The measures described in this specification improve upon these known numerical methods. For example, the radial approximation may be calculated in real- time due to the closed form solution of the problem. In general, the source and target geometries, e.g., the aperture and the beam shape, may be arbitrary non-regular polygons, yielding a substantial generalization of what is available in the literature. Another example is that the TPS approximation may be calculated very rapidly, as a non-iterative formula directly yields the solution and requiring only, as the most computationally complex part, the inversion of one matrix which depends both on the antenna shape and beam shape. Similarly, the evaluation of the phase profile depends on the inversion of one matrix associated to the discretized Poisson PDE, but which may be calculated only once, as it depends on the antenna shape which is not expected to vary, and which computing time depends on the number of points of the discretization. The estimated improvement of the calculation time with respect to traditional numerical methods is two orders of magnitude in the case of TPS, while a much larger improvement may be obtained in case of the radial approximation. Fig.14 shows a system 100 for determining a phase profile for a beamforming antenna. The system 100 is shown to comprise an output interface 110, which in the example of Fig.14 is shown to be an output interface to a transmitter 150 of the beamforming antenna 160, which is by way of example shown to be an array antenna. Using the output interface 110, the system 100 may control the transmitter, for example on the basis of the determined phase profile. In other examples, the output interface may take other forms. For example, in case the system 100 comprises the transmitter 150, the output interface 110 may be an output interface of the transmitter 150. In another example, the output interface may allow the system 100 to output the phase profile in the form of a file or other types of data. In such cases, the output interface may for example be a network interface or a bus-type interface. The system 100 is further shown to comprise a processor subsystem 120 which may be configured, e.g., by hardware and / or software, to perform the steps described in this specification in conjunction with the determining of the phase profile, for example as described with reference to Figs.1-13. For example, the processor subsystem 120 may be configured to receive a target beam shape to be established and a characterization of the aperture of the beamforming antenna, determine, using stationary phase approximation, a transport map which maps intensities in a preferably measure-preserving manner from an antenna domain of the beamforming antenna to a beam domain of the beamforming antenna through a gradient of the phase profile, integrate over an irrotational part of the transport map to obtain the phase profile, and via the output interface 110, output the phase profile or a sampled version thereof. In some examples, the system 100 may further comprise a data storage 130, for example to store input data such as the target beam shape data, the aperture characterization data, etc., output data such as the phase profile or sampled version thereof, or temporary data for the computation of the phase profile. The data storage 130 may for example be a volatile or non-volatile memory, or a solid-state disk, etc. In general, any system described in this specification, including but not limited to the system 100 of Fig.14, may be embodied as, or in, a device or apparatus. The device may be an embedded device. The device or apparatus may comprise one or more microprocessors which execute appropriate software. For example, the processor subsystem of the system may be embodied by a single Central Processing Unit (CPU), but also by a combination or system of such CPUs and / or other types of processing units. The software may have been downloaded and / or stored in a corresponding memory, e.g., a volatile memory such as RAM or a non-volatile memory such as Flash. Alternatively, the processor subsystem of the respective system may be implemented in the device or apparatus in the form of hardware, e.g., as programmable logic, e.g., as a Field-Programmable Gate Array (FPGA), or as an Application Specific Integrated Circuit (ASIC). In general, each functional unit of the system may be implemented in the form of a circuit. The system may also be implemented on a workstation or a server or in a distributed manner, e.g., involving cloud-based servers. Fig.15 shows a method 200 for determining a phase profile for a beamforming antenna. The method 200 may for example be a computer-implemented method, e.g., implemented in software, but may alternatively also be implemented in hardware or as a combination of hardware and software. The method 200 is shown to comprise, in a step titled “RECEIVING TARGET BEAM SHAPE AND APERTURE CHARACTERIZATION”, receiving 210 a target beam shape to be established and a characterization of the aperture of the beamforming antenna, in a step titled “DETERMINING TRANSPORT MAP”, determining 220, using stationary phase approximation, a transport map which maps intensities in a preferably measure- preserving manner from an antenna domain of the beamforming antenna to a beam domain of the beamforming antenna through a gradient of the phase profile, in a step titled “INTEGRATING OVER IRROTATIONAL PART OF TRANSPORT MAP”, integrating 230 over an irrotational part of the transport map to obtain the phase profile, and in a step titled "OUTPUTTING PHASE PROFILE”, outputting 240 the phase profile or a sampled version of the phase profile. It will be appreciated that, in general, the operations or steps of the method 200 of Fig.15 may be performed in any suitable order, e.g., consecutively, simultaneously, or a combination thereof, subject to, where applicable, a particular order being necessitated, e.g., by input / output relations or for technical reasons. Each method, algorithm or pseudo-code described in this specification may be implemented on a computer as a computer implemented method, as dedicated hardware, or as a combination of both. As also illustrated in Fig.16, instructions for the computer, e.g., executable code, may be stored on a computer-readable medium 300, e.g., in the form of a series 310 of machine-readable physical marks and / or as a series of elements having different electrical, e.g., magnetic, or optical properties or values. The executable code may be stored in a transitory or non-transitory manner. Examples of computer-readable mediums include memory devices, optical storage devices, integrated circuits, servers, online software, etc. Fig.16 shows a memory card. Examples, embodiments or optional features, whether indicated as non- limiting or not, are not to be understood as limiting the invention as claimed. Mathematical symbols and notations are provided for facilitating the interpretation of the invention and shall not be construed as limiting the claims. It should be noted that the above-mentioned embodiments illustrate rather than limit the invention, and that those skilled in the art will be able to design many alternative embodiments without departing from the scope of the appended claims. In the claims, any reference signs placed between parentheses shall not be construed as limiting the claim. Use of the verb "comprise" and its conjugations does not exclude the presence of elements or stages other than those stated in a claim. The article "a" or "an" preceding an element does not exclude the presence of a plurality of such elements. Expressions such as “at least one of” when preceding a list or group of elements represent a selection of all or of any subset of elements from the list or group. For example, the expression, “at least one of A, B, and C” should be understood as including only A, only B, only C, both A and B, both A and C, both B and C, or all of A, B, and C. The invention may be implemented by means of hardware comprising several distinct elements, and by means of a suitably programmed computer. In the device claim enumerating several means, several of these means may be embodied by one and the same item of hardware. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage.

Claims

CLAIMS Claim 1. A system (100) for determining a phase profile for a beamforming antenna, the system comprising: - an output interface (110); - a processor subsystem (120) configured to determine a phase profile over an aperture of the beamforming antenna to shape a beam of the beamforming antenna by: - receiving a target beam shape to be established and a characterization of the aperture of the beamforming antenna; - determining, using stationary phase approximation, a transport map which maps intensities in a preferably measure-preserving manner from an antenna domain of the beamforming antenna to a beam domain of the beamforming antenna through a gradient of the phase profile; - integrating over an irrotational part of the transport map to obtain the phase profile; and - via the output interface (110), outputting the phase profile or a sampled version of the phase profile. Claim 2. The system (100) according to claim 1, wherein the processor subsystem (120) is further configured to: - receive a first intensity profile ^(^) over the aperture domain and a second intensity profile ^^(^) over the beam domain; - determine the transport map ^(^) from the antenna domain to the beam domain as a solution of a Jacobian-determinant partial differential equation ^det J^^(^)^^ =^(^)^^^^(^)^ ; - determine the phase profile ^(^) over the antenna domain as a potential function satisfying a gradient partial differential equation ^^(^)= ^(^). Claim 3. The system (100) according to claim 2, wherein the processor subsystem (120) is configured to analytically solve the Jacobian-determinant partial differential equation and the gradient partial differential equation to obtain the transport map and the phase profile.Claim 4. The system (100) according to claim 2, wherein the processor subsystem (120) is configured to search for an approximated solution to the Jacobian- determinant partial differential equation to obtain an approximated transport map. Claim 5. The system (100) according to claim 4, wherein the processor subsystem (120) is configured to search for the approximated solution using a domain mapping method. Claim 6. The system (100) according to claim 4 or 5, wherein the processor subsystem (120) is configured to search for the approximated solution by defining a first set of points in the aperture domain and a second set of points in the beam domain and by using a thin plate spline mapping method to find the approximated transport map by finding a mapping between the first set of points and the second set of points. Claim 7. The system (100) according to claim 4 or 5, wherein the processor subsystem (120) is configured to search for the approximate solution using a Knothe- Rosenblatt transformation method or a conformal mapping method. Claim 8. The system (100) according to any one of claims 4 to 7, wherein the processor subsystem (120) is configured to determine the phase profile as a scalar potential ^(^) of an irrotational component of the approximated transport map ^(^) using a Helmholtz decomposition of ^(^) = ^^(^) + ^ × ^(^) into an irrotational component and a solenoidal component. Claim 9. The system (100) according to claim 2, wherein the processor subsystem (120) is further configured to determine the phase profile ^(^) by solving the Jacobian-determinant partial differential equation and the gradient partial differential equation together as a Monge-Ampère partial differential equation=Claim 10. The system (100) according to any one of claims 1 to 5 or 8 to 9, wherein the processor subsystem (120) is configured to determine the transport map or the phase profile by determining a plurality of radial functions for respective angular segments of the antenna domain and which radial functions together represent either the transport map or the phase profile, wherein the plurality of radial functions isdetermined using adiabatic approximation in which a radial dependency of a respective radial function is assumed larger than an angular dependency of the radial function. Claim 11. The system (100) according to any one of claims 1 to 10, wherein the output interface (110) is to a transmitter (150) of the beamforming antenna (160), and wherein the processor subsystem (120) is configured to determine the phase profile in real-time or near-real time. Claim 12. An antenna system comprising the beamforming antenna (160) and the system (100) according to any one of claims 1 to 11. Claim 13. The antenna system according to claim 12, wherein the beamforming antenna is an antenna array (160) or a shaped-aperture antenna. Claim 14. A space vehicle or aerial vehicle or ground station or radar station comprising the antenna system according to claim 12 or 13. Claim 15. A method (200) of determining a phase profile over an aperture of a beamforming antenna to shape a beam of the beamforming antenna, the method comprising: - receiving (210) a target beam shape to be established and a characterization of the aperture of the beamforming antenna; - determining (220), using stationary phase approximation, a transport map which maps intensities in a preferably measure-preserving manner from an antenna domain of the beamforming antenna to a beam domain of the beamforming antenna through a gradient of the phase profile; - integrating (230) over an irrotational part of the transport map to obtain the phase profile; and - outputting (240) the phase profile or a sampled version of the phase profile. Claim 16. A transitory or non-transitory computer-readable medium (300) comprising data (310) representing a computer program, the computer program comprising instructions for causing a processor system to perform the method according to claim 15.