SOLUTION FOR SURFACE-CONFORMING OPTIMIZATION OF A 3D ANTENNA NETWORK TORQUE

Abstract

The invention relates to a method for designing an array of N antennas arranged on a metallic surface, substantially omnidirectional in the direction of arrival and in polarization within a frequency band having a minimum frequency fmin and a maximum frequency fmax, with N greater than 1, comprising the following steps: - a step (601) of determining K antenna configurations having different geometric characteristics, satisfying a constraint of differential gain over the frequency band [fmin, fmax] and a constraint of gain variation in the main lobe of the antenna, - a step (602) of calculating, for each antenna configuration, at least one antenna array configuration, the orientations of which are chosen so as to promote omnidirectionality in polarization, and the arrangements are chosen so as to promote omnidirectionality in the direction of arrival,- a step (603) of selecting the best antenna configuration / antenna array configuration pair(s).

Description

Technical area :

[0001] The invention lies in the technical field of radiocommunications, and more particularly in that of antennas, antenna networks, and antenna processing with signal processing techniques exploiting the signals of several reception and / or emission, such as radio direction finding processing whose objective is to estimate the direction of arrival (θ m) of electromagnetic waves coming from several far-field transmitters (plane wavefront) from a network of sensors which can be placed on a fixed support or on a carrier, such as for example a vehicle, a boat, an airplane or a drone. Prior art :

[0002] The invention relates to a method for designing an antenna array, the characteristics of which entirely determine the performance of the direction finding. The properties of antenna arrays mainly depend on: characteristics of the radiating element (antenna) which makes up the network, mainly characterized by its radiation diagram in amplitude, phase and polarization, and its variation as a function of frequency, of the geometry of the network, i.e. the position and orientation of the radiating elements, of the supporting structure of the network of radiating elements, which has an influence on its behavior and constrains its maximum size.

[0003] Direction finding performance is defined by: robustness to ambiguities, which corresponds to the network's ability not to confuse the directions of the sources with other directions. From a general point of view, this capacity improves with the number of antennas in the network and deteriorates when the congestion of the network increases for a fixed number of antennas, the precision of estimation of the direction of a or more sources, which improves when the dimensions of the network as well as the number of elements increase, the resolution between the directions of two sources, which improves when the dimensions of the network increase.

[0004] More particularly, the objective is to design a direction finding system allowing: to deal with an environment dense in transmitters: this requires networks with good robustness characteristics to ambiguities in a multi-source context, to overcome the polarization of the antennas of the targeted transmitters. Indeed, it is increasingly difficult to control the polarization on emission with transmitters that are increasingly mobile in position and orientation, antennas that are less and less pure in polarization due to integration constraints (mobile telephony , drones, etc.) or even MIMO transmission systems (English acronym for Multiple Input Multiple Output, or multiple inputs / multiple outputs) exploiting polarization diversity to increase transmission throughput. This requires the design of heterogeneous networks composed of identical antennas with different orientations and positions. Note that a heterogeneous network is less robust to ambiguities than a homogeneous network composed of identical antennas having the same orientation, to carry out a 2D goniometry in bearing and elevation (θ, Δ), so as to avoid carrying out a 1D goniometry in the deposit having to make a hypothesis on the elevation angle of the sources. In many solutions, it is assumed that the direction finding system and the transmitters are located in the same plane assuming that the elevation angle Δ is zero. If this is not the case, we detect the source without being able to goniometer it. This need for a 2D goniometry in (θ, Δ) instead of a 1D goniometry in bearing, however, has the effect of reducing robustness to ambiguities: this involves sizing radiating elements whose elevation diagram is reproducible and measurable, to use the same network in the widest possible frequency domain with an acceptable gain, which implies using radiating elements compatible with this domain and integrable within a compact structure, to be able to integrate into a platform in the terrestrial (vehicle), naval (boat), airborne (drone, balloon, plane, etc.) domains by limiting: the coupling of the antennas with the platform. Unlike mutual coupling between network antennas, coupling with a platform has the effect of strongly attenuating the gain of all the network antennas in certain directions and periodically in the space of arrival directions. This coupling between the platform and the antenna array is minimized by having a metallic surface or a ground plane between the antennas and the platform, the bulk and the weight of the array. This means that a compromise must be determined between the performance, size and number of radiating elements of the network taking into account the surface area available to install it.

[0005] Those skilled in the art know how to design antennas whose characteristics in terms of gain, polarization and frequency band depend on a certain number of geometric parameters on the antenna. This is the case for example of the “Petal” antenna described in European patent EP 3,335,277 B1, used in the remainder of the description for its high degree of optimization possibilities, its performance particularly suited to the design of an efficient antenna network and its limited size compared to the targeted radio performance. The invention can also be applied in the same way to any family of antennas whose radioelectric and dimensional characteristics can be modified by adjusting its geometric properties, which in practice is the case for all antennas. However, the performance of the antenna array developed depends on the ability of the antennas to achieve an advantageous compromise between bulk, low thickness in the presence of a metallic structure and radio performance over a wide frequency band.

[0006] There figure 1a schematizes a three-dimensional view of a petal antenna, used to illustrate the implementation of the method according to the invention. It comprises a first strand 101 and a second strand 102, folded towards a ground plane 103 at the center O of the antenna and in phase opposition. The two strands 101, 102 have in this example an elliptical shape and are arranged symmetrically with respect to the center of the antenna. However, other strand shapes are possible. Folds 104 and 105 located at the ends of the ground plane 103, better known as "capacitive roofs", advantageously make it possible to improve the radio performance of the antenna at the bottom of the band. The antenna has a height H, a width W and a depth D. The figure 1b represents one of the petals of the antenna in front view. It is defined by the intersection of two ellipses sharing the same transverse radius R 2, the latter fixing the total width of the antenna. Then, the ratio R 1 / R 3 makes it possible to optimize the transitions at the level of feeding and terminal folding of the strand.

[0007] There figure 1c represents a profile view of the petal antenna. The curvature of the strand towards the ground plane is defined by the points A, B, C and by the equations of the curves F 1 and F 2. The lower part of the intersection of the two ellipses is then fixed at A (feed point 106 of the strand) then passes through B and C.

[0008] Consider the Cartesian coordinates of points A, B and C, such that: A = x A 0 z A , B = x B 0 z B et C = x c 0 z C .

[0009] The equations of the curves are given by two additional curvature parameters: c 1 and c 2 . The equation of F 1 is such that: X = x B + x A − x B e zc 1 − e z B c 1 e z A c 1 − e z B c 1 , ∀ z ∈ z A z B

[0010] Likewise, the equation of F 2 is such that: X = x B + x C − x B e zc 2 − e z B c 2 e z C c 2 − e z B c 2 , ∀ z ∈ z B z C . The second strand of the radiating element is then generated by x-axis symmetry.

[0011] The characteristics of the antenna can be modified by adjusting the physical parameters of the radiating element: the width of the antenna (W), the height of the antenna (H), the depth of the antenna (D), the large radius of the large ellipse (R 1), the small radius of the large ellipse / large radius of the small ellipse (R 2 ), the small radius of the small ellipse (R 3 ), the position of the junction of the strand with the power supply (A), the position of the junction between the curves (B), the position of the end of the strand (C), the curvature parameter of the first curve (c 1), the curvature parameter of the second curve (c 2).

[0012] The optimization of the antenna consists first of all in obtaining a radiating element whose radio characteristics such as the radiation characterizable by its peak gain or the impedance adaptation are stable and vary monotonically with the frequency, in a footprint defined in relation to the type of network geometry envisaged, while taking into account the supporting structure. This is characterized by a gain variation in the targeted frequency band. This optimization maximizing the peak gain can be done for example by fixing the width and height of the antenna, then by refining the other parameters according to their impact on the radiation pattern or the impedance matching over the entire band. target frequency, for example from simulations. During the optimization of the antenna array, a scaling of all the parameters can be carried out on the radiating element obtained in order to reach the target gain of the frequency band and thus minimize its dimensions to adapt it to the desired network geometry without going back to this initial step.

[0013] It is thus possible to design an antenna addressing any frequency range, whose useful bandwidth and radiation data are adjustable. This radiating element technology makes it possible to achieve an elementary antenna at the same time: ultra-wideband: its impedance adaptation and its gain make it possible to cover a frequency domain greater than a decade by providing the radio performance required for this type of system; compact: the folded shape makes it possible to make maximum use of the volume occupied compared to planar solutions or printed on substrates while having a low profile (of the order of a fraction of a wavelength). This guarantees a better compromise between the compactness of the network and the sensitivity of the sensor. This also makes it possible to have network geometries with radiating elements closer to others to improve ambiguity protection without degrading the interception range of the targeted transmitters; sectoral: the shape of the antenna, reminiscent of that of a “Vivaldi” antenna (in English Tapered Slot Antenna ) based on a ground plane, makes it possible to obtain sectoral radiation with an opening of the half-power radiation lobe greater than 90° in both planes. This also makes it possible to support the radiating element directly on the ground plane without degrading the response of the network. This finally makes it possible to design antenna networks exploiting amplitude diversity, using sectoral antennas oriented in different ways. We can thus obtain an omnidirectional gain network.

[0014] Also known to those skilled in the art are processes for optimizing the position of the antennas in an antenna network. Patent EP 2,462,459 B1 describes an optimization process in the case of a network made up of identical antennas all having the same orientation. The optimization then consists of determining the position of the antennas making it possible to respect specifications in terms of single-source precision, directivity and / or dual-source resolution. We then look for those with the best robustness to ambiguities in a family of networks with the same performance. This homogeneous network optimization process is based on the theoretical performance tools described in the article by Anne Ferreol and Pascal Chevalier: “High Resolution Direction Finding: From Performance to Antenna Array Optimization - The mono-source case”, EUSIPCO , Aug 2009, Glasgow, United Kingdom, allowing the positions of the antennas of a network to be linked to a performance coming from a specification.

[0015] Patent application EP 2,458,398 A2 is a generalization of the optimization of a homogeneous network in mono-polarization to the case of polarization diversity networks composed of a set of identical antennas having different orientations. The analytical link between the performances and the torques (position, orientation) of the antennas was established thanks to the modeling of the figure 2 in electric component E = (E x, E y, E z) and magnetic H = (M x, M y, M z) of the elementary antenna. It is thus possible, under a constraint of omnidirectionality in direction of arrival and polarization, to find the most robust networks to ambiguities. The process was developed more particularly in the case of a cylindrical type supporting structure. These heterogeneous networks in orientation have the advantage of freeing themselves from the polarization of the transmitters, knowing that in the network there will always be antennas sufficiently adapted to the polarization of the incident electromagnetic waves. This makes it possible to obtain sufficient network gain whatever the polarization.

[0016] The main drawback of the state of the art is that the antenna optimization is performed separately from the network optimization. In particular, the greater the gain specified in a specification for a particular band, the more this will have the effect of producing large antennas having a size much greater than λ / 2 (half wavelength), and consequently a gain presenting significant foliation in the main lobe, the foliation being the rate of deformation of the main radiation lobe, deformations which are accompanied by the appearance of secondary lobes. On the other hand, optimization processes minimizing ambiguities require that certain antennas in the network be as close as possible to remove direction finding ambiguities. In particular, in the most unfavorable case of regular networks, this spacing must not exceed λ / 2. Knowing that network optimization processes take into account the size of the antennas, a large antenna with high gain will result in networks that are at best not very robust to ambiguities.

[0017] Another drawback of the state of the art comes from the isolation of the antenna array with the platform. This isolation is done using a ground plane placed between the network and the platform, as shown in the figure 3a , which represents a view of an antenna array in the horizontal plane, and where we can see the case of an antenna array composed of seven antennas 301 arranged on a circular ground plane 302. Each antenna has its own position and orientation. This principle has been considered in numerous applications, in particular for airborne carriers. In particular in the presence of a rotary wing drone type platform, the presence of the metal surface is important so that the propeller blades do not generate diffraction and do not alter the response of the network antennas.

[0018] Arranging an array of antennas on a ground plane, however, presents drawbacks for direction finding applications where the transmitters have directions in the plane of the array. Indeed, the total gain of the antennas is significantly attenuated when the sources come from a direction grazing with respect to the ground plane. Likewise, independently of the gain, the elevation accuracy of a planar array is very poor for sources arriving in the plane of the antennas. This has the consequence of very significantly degrading the distance accuracy of a transmitter when applying an instantaneous geolocation technique based on 2D direction finding, particularly in an airborne context. In other words, a planar network strongly limits the angular sector covered by the direction finding system: the problem is both gain and elevation precision for the sources arriving in the plane of this ground plane. In order to obtain an antenna array having an omnidirectional radiation pattern in azimuth and elevation, it is therefore advantageous to arrange the antenna array on a non-planar metallic surface, for example a portion of a metallic sphere. There figure 3b represents the antenna network of the figure 3ain the vertical plane. The antennas 301 are arranged on a spherical metal cap 302 having an angle of curvature β.

[0019] With regard to the arrangement and position of the antennas, knowing that one objective is to establish a network making it possible to carry out 2D goniometry in polarization diversity, the method of optimizing a heterogeneous network in orientation describes in patent application EP 2,458,398 A2 seems to be a good solution. However, this process does not take into account the modeling of the total gain, which corresponds to the gain of the antenna when the incident wave is adapted to its polarization. In the presence of a ground plane (or metallic surface) this gain has the effect of being distorted compared to a situation where the antenna is modeled alone in free space. For the design of a network installed on a ground plane (or surface) isolating the network from the platform, it is therefore necessary to model the total gain of the elementary antenna.

[0020] It should also be noted that in the presence of an array with antennas having the same orientation on a small ground plane, there are losses due to mismatch of the polarization of the array antennas to the polarization state of the incident waves. . In other words, a homogeneous network reduces the polarization range of incident electromagnetic waves covered by the direction finding system. For sources not having suitable polarization, this results in a loss of gain and significantly degraded control of the network response.

[0021] Furthermore, the modeling of the antennas given in patent application EP 2,458,398 A2 is limited to the estimation of the electromagnetic components of the antenna (components of the electric and magnetic fields) from a simulation of the antenna in free space. The model does not take into account the distortion of the total gain when the antenna is in the presence of a ground plane. Indeed, in a plane tangent to the metallic structure, the total gain of an antenna generally tends to weaken significantly compared to the same antenna in free space. All this does not make it possible to take into account the effect of the surface on a network which conforms to it, and which has the advantage of accentuating the diversity in amplitude of the network, and therefore of improving the goniometry performance of sources. arriving in the horizontal plane. Without modeling the total gain, we cannot therefore determine under good conditions an optimal network solution with antennas conforming to a surface, such as a spherical cap or a cylinder.

[0022] An aim of the invention is therefore to describe a method making it possible to determine the best elementary antenna / antenna array pair optimizing the compromise between the gain of an antenna in a wide frequency band and the robustness to ambiguities of the network.

[0023] Another aim of the invention is that the method takes into account the modeling of the total gain of the antennas, in particular the influence of a ground plane or metallic surface located under the antenna array. Summary of the invention :

[0024] To this end, the present invention describes a method for designing an array of N antennas arranged on a metal surface intended to isolate the antenna array from its support, with N greater than 1. The antenna array aims to be substantially omnidirectional in the direction of arrival and in polarization in a frequency band having a minimum frequency f min and a maximum frequency f max. The design method according to the invention comprises the following steps: a first step of determining K antenna configurations having different geometric characteristics, with K greater than 1, adapted to satisfy a gain differential constraint on the frequency band [f min, f max] and a variation constraint of gain in the main lobe of the antenna, a second step of calculating, for each of the K antenna configurations, of at least one antenna array configuration, the orientations of the N antennas of each antenna array configuration being chosen so as to promote the omnidirectionality of the antenna array in polarization, the arrangements of the N antennas of each antenna array configuration being chosen so as to promote the omnidirectionality of the antenna array in the direction of arrival, a third step of selecting the best antenna configuration / antenna array configuration pair(s).

[0025] Advantageously, the antennas are “petal” type antennas, comprising two strands folded towards a ground plane at the center of the antenna.

[0026] In one embodiment, the method of designing an antenna array according to the invention further comprises a fourth step of optimizing the configuration of the antenna(s) selected during the third step, so as to optimize the performance of the associated antenna array(s).

[0027] When the antennas are “petal” type antennas, said optimization of the configuration may include the modification of a configuration parameter of the antennas among: a width, a shape of the strands and a radius of curvature of the strands.

[0028] According to one embodiment of the method according to the invention, the N antennas are identical.

[0029] According to one embodiment of the method according to the invention, the first step and the second step are implemented from an electromagnetic simulation or a measurement of the complex gain of a unit antenna placed on said metal surface.

[0030] Advantageously, the metallic surface is a portion of a metallic sphere.

[0031] According to one embodiment of the method according to the invention, the first step comprises the determination of K' antenna configurations having different geometric characteristics, with K' greater than K, adapted to satisfy a constraint of limiting the foliation rate at the frequency f max, then for each antenna configuration, the determination of a frequency band [f min, f max] responding to a constraint of variation of the peak gain in the frequency band, then the selection of K configurations d 'antenna among said K' antenna configurations, considering the length of each antenna and the associated minimum frequency f min.

[0032] According to one embodiment of the method according to the invention, the second step comprises: obtaining complex gains G V (Θ,f) and G H (Θ,f) responses of the unit antenna to polarizations E θ and E ϕ following arrival directions Θ = {θ, Δ} for a regular mesh of frequencies included in the frequency band [f min, f max]; for said regular frequency mesh f, calculating antenna gain modeling parameters, by estimating electromagnetic components em ( f ) and interpolation coefficients w ( f ) of the total gain of the antenna from the complex gains G V (Θ,f) and G H (Θ,f); then for a given number of iterations: the determination of orientations ( d 1,... d N) antennas promoting the omnidirectionality of the polarization antenna array; determining positions ( p 1,... pN) antennas promoting the omnidirectionality of the antenna network in the direction of arrival; rejection of the antenna array when the positions and orientations of the antennas are not compatible with maximum congestion of the antenna array; and in which the third step comprises, for each antenna configuration / antenna array configuration pair: for a regular mesh of frequencies included in the frequency band [f min, f max], the calculation at each frequency f d 'robustness to the ambiguities of the antenna configuration / antenna array configuration pair by performing: o from the parameters { w ( f ), em ( f )} of the antenna, of the wavelength λ = c / f , orientations of the antennas { d n} and their positions { p n}, calculating the answers a n (Θ, P V) and a n (Θ, P H) N antennas for polarization P V = [1 0] T< and P H = [0 1] T< to obtain vectors a (Θ, P V) and a (Θ, P H); o for each direction Θ at frequency f, the orthonormalization of the vector basis a (Θ, P V) and a (Θ, P H) to get the columns of the matrix Ũ (Θ) = [ a co< (Θ) a cross< (Θ)] ; o the calculation of robustness to ambiguities η 1 ( f ) (1) of the antenna configuration / antenna array configuration pair from said matrix Ũ (Θ), the robustness to ambiguities corresponding to a minimum of the projection of two planes formed respectively by columns of Ũ (Θi) and U (Θj) for any pair of different directions (Θ i, Θ j); the calculation of the robustness to ambiguities of the antenna configuration / antenna array configuration pair η network =min f min≤f≤fmax η 1 ( f ) ; the best antenna configuration / antenna array configuration pair(s) being the one(s) whose robustness to ambiguities η network is the highest.

[0033] According to one embodiment of the method according to the invention, the choice of the orientations of the N antennas of the second step comprises: random drawing of N-1 values x 2 to xN, with x 1 =1, construction of a vector b̃ ( α )= b ( ϕ ={ ϕ 1 = x 1 α ··· ϕN = x N α}) , with b ϕ = c ϕ + j s ϕ , c ϕ = cos ϕ 1 … cos ϕ N And s ϕ = sin ϕ 1 … sin ϕ N , the calculation of an angle αmin minimizing an orthogonality criterion C ϕ ( α ) = b̃ T< ( α ) b̃ ( α ) , the calculation of direction phases { ϕ 1 = x 1 α min ··· ϕN = x N α min} of N antennae; the calculation of orientations ( d 1,... d N) from the direction phases ( ϕn of the N antennas.

[0034] According to one embodiment of the method according to the invention, the choice of the position of the N antennas of the second step comprises the steps of: random drawing of N antenna positions p n 0 = x n y n T in a horizontal plane, calculation of a matrix D ˜ pp g é o equivalent aperture of the antenna array, with D ˜ pp geo = ∑ n = 1 N p n 0 − p ‾ p n 0 − p ‾ T / N , And p ‾ = ∑ n = 1 N p n 0 / N , matrix decomposition D ˜ pp g é o in own elements, with D̃ pp geo< = E Λ E H< , Or E is an eigenvector matrix of D ˜ pp g é o And Λ a diagonal matrix of eigenvalues ​​of D ˜ pp g é o , calculation of a matrix W whitening, with W = E Λ 1 / 2<, calculation of a set of antenna positions p n 1 = W − 1 p n 0 − p ‾ , calculation of network congestion p n 1 , resizing of the network p* by the application of a homothetic ratio between a bulk associated with the positions p n 1 , and a specified maximum footprint.

[0035] In one embodiment of the method according to the invention in which the metal surface is non-planar, the choice of the position of the N antennas and / or the orientation of the N antennas of the second step further comprises a step of projecting the positions and / or orientations on the metal surface. Brief description of the figures :

[0036] The invention will be better understood and other characteristics, details and advantages will appear better on reading the description which follows, given on a non-limiting basis, and thanks to the appended figures, given by way of example, among which: there figure 1a schematizes a three-dimensional life of a petal antenna, used to illustrate the implementation of the method according to the invention; there figure 1b represents one of the petals of the antenna petal of the figure 1a in front view; there figure 1c represents a profile view of the petal antenna of the figure 1a ; there figure 2 is a representation with an equivalent diagram of the model of an antenna, in a general case; there figure 3a represents a view of an antenna array in a horizontal plane, for illustration purposes; there figure 3b represents a view of the antenna array of the figure 3a in a vertical plane, for illustration purposes; there figure 4a is a representation of the polarization P 0 of a wave transmitted between a transmitter and an antenna array; there figure 4b is a representation of the wave plane, the wave vector and the polarization associated with the propagation of a wave in polarization V; there figure 4c is a representation of the wave plane, the wave vector and the polarization associated with the propagation of a wave in H polarization; there Figure 5 represents a practical application of a petal antenna array; there Figure 6 schematically represents the steps of a method of designing an array of N antennas according to one embodiment of the invention; there figure 7a is an illustration of a petal antenna positioned in an orthonormal coordinate system (x', y', z'); there figure 7b is an illustration of a change of reference linked to a variation in the orientation of the petal antenna of the figure 7a ; there figure 8a is an illustration of the gain of a perfect dipole as a function of its orientation; there figure 8b is an illustration of the gain of a perfect loop depending on its orientation; there Figure 9 is an illustration of a homogeneous network of antennas positioned on a plane.

[0037] Identical references may be used in different figures when they designate identical or comparable elements. detailed description :

[0038] An objective of the invention described below is to design an omnidirectional antenna network in direction of arrival (bearing θ and elevation Δ) and in polarization. The proposed network design process jointly optimizes the couple (antenna, network).

[0039] There Figure 5 represents a practical application of an antenna array. In this example, the antenna array 501 is composed of 8 petal-type antennas 502. The antenna array is positioned on a metal disc 503 whose purpose is to isolate the antenna array from the carrier 504 on which it is arranged, in the example a drone. The antenna network according to the invention makes it possible to implement direction finding functions in order to determine the direction of arrival in two dimensions of electromagnetic waves transmitted by other equipment 505 and 506.

[0040] In order to obtain a network allowing sufficient gain and precision in a sufficiently wide 2D angular sector, the remainder of the description relates to the optimization of a network of identical antennas with different orientations, conforming to a 3D surface of metallic spherical cap type. However, the invention applies in the same way when the antenna array is arranged on any metallic surface of equation z = f ( x , y ) .

[0041] In order to simplify the optimization of the network, the omnidirectionality properties in the direction of arrival and in polarization will be at least verified for the network projected in the horizontal plane (x, y). The vertical z axis can be seen as an axis of deformation of a horizontal ground plane. In the case of a spherical cap, the deformation function is very simple, with f x y = R 2 − x 2 − y 2 .

[0042] The method described therefore generalizes the optimization of the antenna network to any surface, contrary to the state of the art, for example patent EP 2,462,459 B1, where the optimization is only implemented for a cylindrical surface of equation z = f x y = R 2 − y 2 .

[0043] The rest of the description firstly describes the principles of goniometry necessary to understand the implementation of the method according to the invention and obtain the desired performances. In terms of mathematical notations, a term in bold designates a vector, a term in capital letters designates a matrix, the operator ∼< and the operator designate an estimate, the operator T< represents the conjugated transpose, the operator H< represents the conjugated transpose, the operator designates an average.

[0044] There figure 4a is a representation of the polarization P 0 of a wave transmitted between a transmitter and an antenna array. The reference 401 designates a network of antennas heterogeneous in their orientations. An antenna network is said to be homogeneous when all the sensors in the network are identical radiating elements having the same orientation in space, and heterogeneous otherwise. We can therefore have a heterogeneous network comprising N different sensors, or N identical sensors having different orientations.

[0045] From a general point of view, every wave propagates with a polarization P (projection of the electric field vector E in the wave plane 402) given, which is the linear combination of the polarization V where E = k V and polarization H Or E = kH . The wave propagates with a magnetic component H which is perpendicular to the electric field E. The electric and magnetic components are included in the wave plane 402, perpendicular to the wave vector k (θ, Δ), with θ the azimuth and Δ the elevation in the xyz plane of the antenna array. We note by b v And bH the magnetic components of the V and H polarizations.

[0046] THE figures 4b and 4c represent the wave plane, the wave vector and the polarization associated with the propagation of a wave polarized respectively in polarization V and in polarization H, in the wave plane defined by the orthonormal vectors k V ( θ ,Δ) and k H ( θ ,Δ). These figures show the position of the incident electric field E 0 and the incident magnetic field H 0 as a function of the polarization of the wave.

[0047] In the presence of M sources, the signal output from a network of N sensors is written: x t = x 1 t ⋮ x N t = ∑ m = 1 M a ˜ Θ m P m s m t + n t Or x n ( t ) is the signal received on the nth sensor, s m ( t ) is the signal from the m-th source, n ( t ) is the additive noise, Θ m is the direction of arrival of the source, P m is the polarization as defined in figures 4b and 4c , And has (Θ m , P m ) is the observed direction vector. In the presence of model error the vector has (Θ m , P m ) is written: a ˜ Θ m P m = a Θ m P m + e m Or a (Θ m , P m ) is the theoretical direction vector such that a ( Θ m , P m ) H< a ( Θ m , P m ) = N And e m is the model error.

[0048] Assuming that there is no mutual coupling, and according to the figure 4a , the nth component of a (Θ m , P m ) is written: a n Θ , P = G n Θ , P exp j 2 π λ k Θ T p n Or p n = [ x n y n z n ] T< is the position vector, λ is the wavelength and k (Θ) is the wave vector such that: k Θ = u v w avec { u = cos θ cos Δ v = sin θ cos Δ w = sin Δ where 0 = { θ , Δ} depends on the azimuth θ and elevation Δ. A network is said to be heterogeneous when the antenna gains G m (Θ, P ) are not identical. Without harming any generality the directing vector a (Θ, P ) is written as follows: a Θ , P = U Θ P

[0049] Knowing that the matrix U ( Θ ) of dimension Nx2 depends on the incidence Θ of the source, as well as the positions and orientations of the radiating elements composing the network, the first column of U ( Θ ) can for example be associated with the response of the network to polarization H, so that P (1) = E θ and the second column at the polarization V, so that P (2) = E ϕ . The algebraic properties of this matrix completely condition the performance of the network.

[0050] The antenna network optimization will be done on the basis of single-source performances ( M = 1) with: the criterion of robustness to ambiguities, the precision of goniometry.

[0051] A mathematical ambiguity in the presence of a source is present when, for a source of direction Θ 1 and polarization P 1, there exists another direction / polarization pair ( Θ 2, P 2) such that vectors a ( Θ 1, P 1) and a ( Θ 2, P 2) are collinear. Under these conditions the following criterion is null: J Θ 1 , Θ 2 , P 1 , P 2 = 1 − a H Θ 1 , P 1 a H Θ 2 , P 2 2 ‖ a H Θ 1 , P 1 ‖ 2 ‖ a H Θ 2 , P 2 ‖ 2

[0052] This is how the robustness to ambiguities in single-source is the following value: η 1 = min P 1 P 2 J Θ 1 , Θ 2 , P 1 , P 2 .

[0053] Considering the following canonical decomposition of the matrix U ( Θ ): U Θ = a co Θ P co a cross Θ P cross P assage where the vectors a co< (Θ) and a cross< (Θ) are the orthonormal responses in co-polarization and cross-polarization (cross-polarization) of the network forming an orthonormal basis and P co and P cross are scalars indicating the polarization diversity character of the network, we can show that: min P 1 P 2 J Θ 1 Θ 2 P 1 P 2 = 1 − λ max U ˜ H Θ 1 U ˜ Θ 2 2 avec U ˜ Θ = a co Θ a cross Θ where λ max ( M ) is the maximum singular value of M . It should also be noted that a mono-polarization network is such that P cross is zero. P co is then the polarization of the network. In all other cases, we have a polarization diversity network. In the particular case where the norms of the vectors P cross and P co are equal, we are then in the case of an omnidirectional polarization network. According to the last expression, the robustness to single-source ambiguities is: η 1 f = 1 − min Θ 1 ≠ Θ 2 λ max U ˜ H Θ 1 U ˜ Θ 2 2 Or f = c / λ is the carrier frequency and c the speed of light. Robustness to ambiguities η network of an antenna array will be equal to the lowest robustness in the useful frequency band. This robustness will depend on the positions and orientations of the antennas in the network.

[0054] Knowing that we are in the presence of model error e m or additive noise n (t), the goniometry algorithm estimates the incidence-polarization couple (Ψ m = Θ m Or k ( Θ m ), P m ) from the source with an error. We are particularly interested in the variance of the single-source incidence estimate: MS Ψ m = E Δ Ψ m Δ Ψ m T avec Δ Ψ m = Ψ ^ m − Ψ m where (Ψ̂ m , P̂ m ) is an estimate of (Ψ m , P m ), And Ψ m = Θ m P m = ϑ m Δ m P m .

[0055] The root of MS Ψm [1][1] is the estimation precision of the bearing θ m and MS Ψm [2][2] is the estimation precision of the elevation Δ m . In the reference “High Resolution direction finding: from performance toward antenna array optimization - The single-source case » , it is only mounted in the presence of a source: MS Ψ 1 = E Δ Ψ 1 Δ Ψ 1 T = α H Ψ 1 − 1 H Ψ 1 = 2 A ˙ H ∏ Ψ 1 A ˙ with A ˙ = δ a Ψ 1 P δ Ψ 1 1 ⋯ δ a Ψ 1 P δ Ψ 1 d ∏ Ψ 1 = I N − a Ψ 1 P a H Ψ 1 P a Ψ 1 P H a Ψ 1 P I N being the identity matrix of size NxN.

[0056] According to this article and the following table, which gives the parameters to be associated with the single-source precision criteria knowing [ n ( t ) n ( t ) H< ] = σ 2< I N , the coefficient values α depend on the type of performance considered. [Table 1] Type of Performance Coefficient value α Cramer RAO terminal Stochastic Case α = 1 + Nr ss x 2 − 1 K r ss σ 2 avec r ss = E s m t k 2 From x ( t k )for 1 ≤ k ≤ K Deterministic case α = 1 K r ^ ss σ 2 avec r ^ ss = 1 K ∑ k = 1 K s m t k 2 MUSIC Performance At finite integration time with x ( t k ) for 1 ≤ k ≤ K α = 1 K r ss σ 2 In the presence of model error α = E e m H e m N

[0057] This shows that the performance depends only on the matrix H (), which is directly linked to the lobe width of the goniometry criterion. According to the following expressions, the matrix MS Θ1 contains on its diagonal the variance of estimation of the bearing and the elevation and the matrix MS k(Θ1) the variance of estimation of the components (u 1, v 1, w 1) of the wave vector: MS Θ 1 = E θ 1 − θ ^ 1 2 E Δ 1 − Δ ^ 1 2 et MS k Θ 1 = E u 1 − u ^ 1 2 E v 1 − v ^ 1 2 E w 1 − w ^ 1 2 k Θ 1 = u 1 = cos θ 1 cos Δ 1 v 1 = sin θ 1 cos Δ 1 w 1 = sin Δ 1

[0058] The state of the art shows that this matrix can be written relatively simply. This provides important tools for network optimization, because it makes it possible to obtain conditions to respect omnidirectionality in the direction of arrival and in polarization. In particular, we show in the case of a mono-polarization network that the condition of omnidirectionality in the direction of arrival depends only on the position of the antennas in the network. Omnidirectionality in the direction of arrival is obtained when the MS matrix Θ1 is diagonal, and when the MS matrix k(Θ1) is proportional to the identity.

[0059] There Figure 6 schematically represents the steps of a method for designing an array of N antennas according to one embodiment of the invention, for the design of an antenna array aiming to be omnidirectional in direction of arrival and in polarization.

[0060] Knowing that the maximum working frequency of the goniometer is f max, the method according to the invention comprises a first step 601 which consists of firstly optimizing the geometric parameters of the antenna, in order to provide a subsequent optimization step with of the antenna network a set of K elementary antennas whose impedance adaptation and gain are compatible by their level of use in a working frequency band [f min, f max] as wide as possible.

[0061] This step is advantageously done by optimizations using electromagnetic simulations, the simulations being able to exploit measured radiation data, in order to characterize the response of the collection of K antennas operating up to the frequency f max. The K antennas selected satisfy in particular a constraint of limiting the foliation of the total gain of the antenna in the main lobe. The frequency f min is determined in relation to a gain differential constraint of the antenna lobe peak between the frequencies f min and f max. The gain variations observed in the radiation pattern, characteristic of the appearance of secondary lobes or alterations of the main lobe by coupling or resonance, are called antenna leafing. These variations (decreases) create blind areas of the antenna, and must therefore be avoided. For this, a foliation rate is measured in simulations, which corresponds to the difference between the minimums and the maximums in the area of ​​the main lobe of the antenna gain.

[0062] In an embodiment where the length L of the antenna is an adjustment variable of the antenna, step 601 can be implemented by determining the frequency f max checking the maximum admissible lamination rate, then by resizing the antenna with the homothetic ratio f max0 / f max. The frequency f min is determined according to a constraint of maximum peak gain difference between the frequencies f min and f max. The optimization will consist of minimizing the torque (antenna size, minimum frequency f min). This homothety allows us to quickly converge towards a solution. It can be followed by a step of fine adjustment of the antenna parameters. For this, the use of the “Petal” antenna of European patent EP 3,335,277 B1 is particularly advantageous given all the optimization parameters that it offers.

[0063] A possible embodiment is as follows: for a given antenna length L, adjustment of the other antenna parameters to provide an antenna configuration having good radio characteristics, in particular in terms of gain stability and impedance matching over a frequency band increased by f max; calculation of the maximum frequency f max0 up to which the antenna satisfies a maximum foliation criterion, with f max0 ≤ f max; calculation of the minimum frequency f min0 as a function of a maximum gain variation criterion over the frequency band [f min0, f max0], so that the gain peak at frequency f max0 and the gain peak at frequency f min0 have a deviation less than a threshold ΔG; resizing of the antenna by scaling by a factor f max 0 f max and calculation of the frequency f min, with f min = f min 0 ∗ f max 0 f max . This step makes it possible to obtain an antenna of length L' satisfying the lamination criterion and the minimum gain criterion on the band [f min, f max].

[0064] The preceding steps are repeated a number K' of times, with K' ≥ K. When the K' antenna configurations have been calculated, the K best antennas, that is to say the K antennas of the smallest dimensions and having the widest operating band, and therefore which minimize the torque (length L', frequency f min), are selected to implement the rest of the process.

[0065] Resizing the antennas by scaling makes it possible to converge towards operation of the antenna at the foliation limit at the frequency f max. It is therefore the best possible compromise at this frequency between the geometry of the network and the size of the antenna. The performance of the network can be further improved by a final optimization of the radiating element taking into account the impact on its radiation of the complete network and the effective dimensions of the supporting structure.

[0066] The method of designing an array of N antennas according to the invention then comprises a second step 602 of calculating, for each of the K antenna configurations selected during the first step, at least one antenna array configuration . The orientations of the N antennas of the antenna array are chosen so as to promote the omnidirectionality of the antenna array in polarization. The arrangements of the N antennas of the antenna array are chosen so as to promote the omnidirectionality of the antenna array in the direction of arrival.

[0067] The input parameters for this step are: an electromagnetic simulation of each antenna among the K antennas selected in the previous step, in the presence of a metallic surface. In the case where this surface is a spherical cap, the simulation of the antenna must be carried out on a portion of a sphere whose radius is the radius of curvature of the spherical cap; the 3D footprint of a unit antenna; clutter D maximum of the antenna array in the horizontal plane; the number N antennas of the antenna array; the equation z = f ( x , y ) of the surface on which the network antennas are installed.

[0068] Step 602 aims to determine an optimal heterogeneous network of antennas for each of the K best antennas at the output of antenna optimization step 601. For each selected antenna, a simulation or measurement of the performance of the antenna on the metallic surface (for example a portion of a sphere) on which it is placed must be carried out, and the parameters of the model (electromagnetic components and interpolation coefficients of the total gain) are estimated from these data. A random selection of the positions and orientations of the antennas on the metal surface is then carried out. The antenna positions are adjusted to obtain an omnidirectional array in the direction of arrival and the antenna orientations are adjusted to obtain an array maximizing polarization diversity (the sum of the antenna orientations must be zero). Of course, if the metal surface on which the antennas are arranged is flat, the antenna array will not be completely omnidirectional in elevation since the gain will be greatly weakened in the plane tangent to the ground plane. This is why the use of a non-flat metal surface is particularly advantageous.

[0069] According to one embodiment of the invention, step 602 is implemented by carrying out the following calculations for each of the K antennas selected during step 601: Network Step.1 : Using complex gains G V (Θ,f) and G H (Θ,f) of the responses of the unit antenna to polarizations E θ and E ϕ following all arrival directions Θ ={θ, Δ}, obtained by measurements or by electromagnetic simulation, for a regular mesh of frequencies between f min And fmax , in the presence of an insulating metal surface; Network Step.2 : For a plurality of frequencies f between f min And fmax , calculation of antenna gain modeling parameters, by estimating electromagnetic components em ( f ) and interpolation coefficients w ( f ) of the total gain of the antenna from the complex gains G V (Θ,f) and G H (Θ,f), for example according to the process described below as Step A; Then for a given number of iterations: Network Step.3 : Drawing on a random basis the orientations (ϕ 1,... ϕ N) of the antennas in the horizontal plane, the orientations being adjusted in order to obtain an omnidirectional antenna array in polarization, for example according to the process described below as Step D. For each orientation ϕ n, deduce the directions d n of the antennas according to equation (29), from knowledge of the equation z = f ( x,y ) of the metal surface on which the network is placed; Network Step.4 : Random drawing of positions ( p 1,... p N) antennas in the horizontal plane giving an omnidirectional network in the direction of arrival on the metallic surface of equation z = f ( x,y ) for example, according to the process described below as Step E; Network Step.5 : Test to determine if the parameters ( p not , d n) are compatible with the size of the antenna. If not, return to the Network step.3.

[0070] The characteristics of the antenna / network pair are then memorized.

[0071] Variations can be easily implemented on the steps mentioned above, for example by reversing certain steps like drawing orientations and drawing positions.

[0072] The method of designing an array of N antennas according to the invention finally comprises a third step 603 of selecting the best antenna configuration / antenna array configuration pair(s).

[0073] This step consists of evaluating the robustness to ambiguities of each antenna array in the band [f min, f max], then selecting the antenna / antenna array pair(s) having the best robustness to ambiguities.

[0074] According to one embodiment of the invention, step 603 can be implemented by carrying out, for each of the stored antenna / network pairs: Network Step.6: For a plurality of frequencies f between f min And fmax , calculation of the robustness to ambiguities of the antenna / antenna array pair by performing: ∘ Network Step.6.1 : From the parameters { w ( f ), em ( f )} of the radiating element, of the wavelength λ=c / f , orientations of the antennas { d n} and their positions { p n}, calculation of answers a n (Θ, P V) and a n (Θ, P H) N antennas for polarization P V = [1 0] T< and P H = [0 1] T<, for example according to the process described below as Step C; o Network Step.6.2 : For each direction Θ, orthonormalization of the vector basis a (Θ, P V) and a (Θ, P H) to get the columns of the matrix Ũ (Θ) = [ a co< (Θ) a cross< (Θ)] ; ∘ Network Step.6.3 : according to equation (10), calculation of robustness to ambiguities η 1 ( f ) of the antenna / network couple from said matrix Ũ (Θ); Network Step.7 : Deduction of the robustness to ambiguities of the antenna / antenna array pair by calculating η network =min f min≤ f ≤ f max η 1 ( f ) ; Network Step.8 : Selection of the antenna / antenna array pair(s) maximizing the criterion η network.

[0075] Advantageously, the method of designing a network of N antennas according to the invention comprises an additional step 604 of optimizing the configuration of the antenna(s) of the pair(s) selected during the third step, so as to optimize performance. of the associated antenna array(s) to take into account the impact of the final shape of the supporting metal structure and the coupling phenomena between the elements of the array.

[0076] Indeed, the implementation of the first three steps of the method makes it possible to jointly select an efficient antenna / antenna array pair, so as to obtain the omnidirectionality performances sought for the entire network. However, antennas can sometimes be further optimized on certain points (size, coupling, etc.). This additional optimization can be implemented by carrying out several iterations of steps 601 to 603, by improving at each iteration the configuration of the antennas of the first step based on the observed response of the antenna array (for example by modifying the surface of the antennas). antennas if they are too close, ...). Alternatively, an additional step 604 of antenna optimization can be carried out, comprising the fine characterization of the properties of the antenna array, and the adjustment of the characteristics of the antennas in order to take into account the inter-element couplings and / or the couplings with the effective relative size of the structure which can generate resonances, alter the main lobe or accentuate the secondary lobes of the radiation. This additional adjustment step makes it possible to arrive at a solution with a high level of performance taking into account the size or the number of antennas required.

[0077] For this purpose, the use of so-called “petal” antennas described in European patent EP 3,335,277 B1 is particularly advantageous since in addition to having a reduced bulk, these antennas offer a large number of degrees of freedom making it possible to optimize finely the gain, the impedance adaptation and the radiation properties, compared to an imposed size. It is thus possible to adjust secondary characteristics such as the width of the antenna or the parameters of the curved strands R 1, R 2 and R 3 to increase the low frequency radiation efficiency without introducing overlap of the antennas in the network, or to modify the radii of curvature c 1 and c 2, to reinforce the directivity of the diagrams to reduce lamination, and this without necessarily calling into question the arrangements and orientations calculated in steps 602 and 603.

[0078] Conversely, the families of radiating elements presenting few degrees of freedom for optimizations or offering little radiation efficiency compared to an imposed volume constraint, do not offer the same degree of adjustment, and therefore compromise between the compactness of the solution and its performance.

[0079] The remainder of the description describes in more detail different embodiments making it possible to implement the second (602) and the third (603) step of the method according to one embodiment of the invention.

[0080] The antenna array optimization is based on a parametric model of the response G n ( Θ , P ) of an antenna given in equation (3). This response depends on the orientation of the antenna in space, and the electric field vectors E and magnetic fields H which characterize it, as represented on the figure 2 .

[0081] Electromagnetic components {E, H} of the antenna are measured or estimated from an electromagnetic simulation in a certain reference frame (x',y',z'), as illustrated in the figure 7a , where the petal antenna 701 has a direction d 0. We can then deduce, by reference change techniques, the gain of this antenna when it has another orientation d n in the coordinate system (x,y,z) of the network, as illustrated in the figure 7b . This is how it is possible to master the algebraic properties of the matrix U( Θ ) of equation (5), and to give conditions on the orientations of the antennas of the network so that the network is omnidirectional in polarization.

[0082] According to figures 4a, 4b and 4c , electric and magnetic fields {E, H} of the transmitting antenna are projected into the wave plane defined by the orthonormal vectors k V (θ, Δ) and k H (θ, Δ), orthogonal to the wave vector k (θ, Δ): k V Θ = − cos θ sin Δ − sin θ sin Δ cos Δ et k H Θ = − sin θ cos θ 0

[0083] The components (P V , P H) of the electric field E 0 = P V k V (θ, Δ) + PH k H (θ, Δ) incident projected in the wave plane are the components of the polarization vector following the components Eθ and Eϕ. The magnetic field H 0 incident projected in the wave plane is orthogonal to E 0. The wave vector k (θ, Δ) is orthogonal to the wave plane.

[0084] Polarization P 0 = [ P V , P H ] T< of an incident wave is defined by the components of the electric field in the wave plane. According to figure 4a : { E ˜ 0 Θ P 0 = E 0 E 0 = P V × k V Θ + P H × k H Θ H ˜ 0 Θ P 0 = H ˜ 0 H 0 = − P H × k V Θ + P V × k H Θ avec P 0 = P H P V Or P 0 is a normalized vector and where ( E 0, H 0) are respectively the complex amplitude of the electric field and the magnetic field. According to figures 8a And 8b , the gain of a perfect dipole or perfect loop depends on the orientation d of the radiating element, as well as the total gain G T (Θ) of the antenna, i.e.: G Dipole Θ P 0 = G T Θ × d T E ˜ 0 Θ P 0 = G ˜ T Θ × E Dipole T E ˜ 0 Θ P 0 G Boucle Θ P 0 = G T Θ × d T H ˜ 0 Θ P 0 = G ˜ T Θ × H Boucle T H ˜ 0 Θ P 0

[0085] Indeed, the gain of a loop depends only on its electric field E Dipole, because its magnetic field is zero, and vice versa for the loop which does not emit an electric field. From a general point of view, we can characterize an antenna according to the figure 4a by an electric field / magnetic field couple ( E , H ), which will make it possible to give the following expression of the gain: G Θ P 0 = G ˜ T Θ × E T E ˜ 0 Θ P 0 + H T H ˜ 0 Θ P 0

[0086] Thus in polarization V where P V =1 and P H =0, we have the following gain: G V Θ = G ˜ T Θ × E T k V Θ + H T k H Θ = G ˜ T Θ × u V T Θ × em avec u V Θ = k V Θ k H Θ et em = E H Or em is the vector of the electromagnetic components that we wish to estimate from data from measurements or an electromagnetic simulation of the antenna. In polarization H, where P V =0 and P H =1, the gain is: G H Θ = G ˜ T Θ × E T k H Θ + H T k V Θ = G ˜ T Θ × u H T Θ × em avec u H Θ = k H Θ − k V Θ et em = E H

[0087] The expression for the gain of an antenna then verifies G Θ P 0 = g T Θ P 0 avec { g Θ = G H Θ G V Θ = G ˜ T Θ × K H Θ em K Θ = u H Θ u V Θ

[0088] The total gain G̃T(Θ) of the antenna, which takes into account the influence of the metal surface on which the antenna is installed, depends particularly on this metal surface, as well as on the distance of the antenna from this surface. Generally its value is low (or even zero) in the plane tangent to the surface. In order not to make any particular assumptions about the physics of the impact of a metallic surface on the antenna, we model this gain as follows: G ˜ T Θ = w T × e Θ avec e Θ = e L θ ⊗ e L Δ et e L ς = exp − jLς exp − j L − 1 ς ⋮ exp jLς where ⊗ is the Kronecker product, θ is the bearing in radians and Δ is the elevation.

[0089] Consequently, an antenna can be modeled by the vector em electromagnetic components and the vector w containing the interpolation coefficients of the total gain, i.e.: G Θ P 0 = w T e Θ × em T K Θ P 0 avec em = E H

[0090] In the method of designing an antenna array according to one embodiment of the invention, the couple ( w,em ) is estimated from measurements or an electromagnetic simulation. This solution is different from that of patent EP 2,462,459 B1, where the total gain was assumed to be independent of the direction of arrival Θ. We can then deduce the gain of this same antenna for an orientation d n different from that of the initial simulation as illustrated on the figures 7a And 7b .

[0091] Whether by measurements or by an electromagnetic simulation, it is possible to recover the gains at each frequency G V (Θ i ) and G H (Θ i ) for a set of incidences {Θ i} covering the entire angular space. To estimate the vector em, we then construct the criterion J following, relying on the fact that according to equation (20), the vector g (Θ i ) is collinear with the vector K T< (Θi) em , either : J em = ∑ i g H Θ i K T Θ i em 2 g H Θ i g Θ i = em H Qem avec Q = ∑ i K Θ i g Θ i g H Θ i K T Θ i 2 g H Θ i g Θ i

[0092] The vector em must maximize the criterion J ( em ). Consequently the vector em is proportional to the eigenvector associated with the largest eigenvalue λ max (Q) of the matrix Q . We then obtain the vector em normalized such that: em = arg max em J em avec em H em = 1

[0093] According to the models of equations (20) and (21), the vector g (O) is written: g Θ = h Θ × w avec h Θ = K K Θ × em × e T Θ

[0094] The interpolation vector w is estimated in a least squares sense by minimizing the following criterion: w = arg min w ∑ i ‖ g Θ i − h Θ i × em ‖ 2

[0095] The vector w is determined in a least squares sense.

[0096] From complex gains G V (Θ i ) and G H (Θ i ) for a set of incidence {O i} of the antenna on a metallic surface, the estimation of the electromagnetic components em and interpolation coefficients w of the total gain can be obtained by implementing the process of step A which follows: Step A.1 : Construction of vectors g (O) according to equation (20) for all incidences Θ belonging to the set {Θ i} of present measurements; Step A.2 : Construction of matrices K (Θ) according to equations (14), (18), (19), and (20) for all the incidences Θ belonging to the set {Θ i} of the present measurements; Step A.3 : Calculation of the matrix Q according to equation (23); Step A.4 : Calculation of the eigenvector em associated with the main eigenvalue of Q ; Step A.5 : Search for the interpolation vector w according to equation (26).

[0097] There Figure 9 represents a homogeneous network comprising 5 antennas 901 902 arranged on a plane 903.

[0098] The network is made up of position antennas p n = [ x n y n z n ] T< according to Figure 10, and orientations d n according to figures 7a And 7b . The objective is to deduct your gain G n (Θ = {θ, Δ}) in the network reference frame, knowing that the coefficients (w,em) were estimated in the simulation benchmark such that G n (Θ = {θ, Δ}) = G(Θ'={θ', Δ'}). There is therefore a change of base to be made to deduce the incidence Θ from the incidence Θ'. There therefore exists a rotation matrix Γ n such that: { k Θ = Γ n × k ˜ Θ ′ k V Θ = Γ n × k ˜ V Θ ′ k H Θ = Γ n × k ˜ H Θ ′ et donc { E n = Γ n E H n = Γ n H d n = Γ n d 0 Or (E not , Hn) are the electric field and magnetic field vectors of the steering antenna d n in the network marker. We notice d 0 the direction of the antenna in the simulation reference frame where, according to the figures 7a And 7b , d 0 =[0 0 1]'. Vectors k̃ (Θ'), k̃ H (O') and k̃ V (O') are respectively the wave vector and the wave plane vectors in the simulation reference frame. The following aims to determine Γ n knowing that the antennas are installed on a surface of equation z n = f ( xn,yn ).

[0099] According to figures 7a And 7b , the characterization or simulation of the antenna is carried out in the orthonormal reference frame ( q n = - η n^ d not , - η not , d n), original O = (0, 0, 0), where η n is the normal to the antenna plane corresponding to the direction of the radiation maximum and the operator ^ designates the vector product. In the network, the antenna is located in the orthonormal coordinate system of axes (x,y,z) originating from its position p not .

[0100] We construct here a network where the normal vector η n of the antennas is also the normal of the equation surface z n = f ( xn,yn ) . Consequently this vector has the following expression: η n = 1 η n − ∂ ƒ x n y n ∂ x − ∂ ƒ x n y n ∂ y 1 avec η n = 1 + ∂ ƒ x n y n ∂ x 2 + ∂ ƒ x n y n ∂ y 2

[0101] In the case of a spherical cap with equation z n = R 2 − x n 2 − y n 2 , we have η n = p not . On the other hand, the management d n of the antenna will be chosen so that the projection of the direction vector d n in the horizontal plane checks d (ϕ n ) = [cos(ϕ n ) sin(ϕ n )] T< . The vector d n is then written as follows: d n = 1 1 + ƒ n 2 d ϕ n ƒ n avec { d ϕ = cos ϕ sin ϕ ƒ n = ∂ ƒ x n y n ∂ x cos ϕ n + ∂ ƒ x n y n ∂ y sin ϕ n

[0102] The last vector q n of the triad of the orthonormal reference frame of the electromagnetic simulation is therefore the following vector product according to the figures 7a And 7b : q n = − η n ∧ d n

[0103] The rotation matrix of equation (27) then has the following expression: Γ n = q n − η n d n

[0104] The Step B process for constructing the rotation matrix Γ n of an antenna with orientation ϕ n in the horizontal plane and position p n =[x n y n z n ] T< on an equation surface z n = f ( x n , y n ) is then the following: Step B.1 : Calculation of the vector η n normal to the equation surface z n = f ( xn, y n ) at the position point p n =[x n y n z n ] T< according to equation (28); Step B.2: Calculation of the orientation vector d n from the angle ϕ n according to equation (29); Step B.3 : Calculation of the vector q n by the vector product q n = - η n^ d not ; Step B.4: Construction of the rotation matrix Γ n by carrying out Γ n = [ q n - η n d n ].

[0105] For a direction Θ, it is then possible to calculate the wave vector k̃ (Θ') =[ u ' v'w '] T< = Γ nT< k (Θ) in the simulation reference frame, and according to equation (4), to deduce the incidence Θ'= {θ', Δ'} in the simulation reference frame as follows: θ ′ = angle u ′ + jν ′ Δ ′ = angle u ′ cos θ ′ + ν ′ sin θ ′ + jw ′

[0106] It is then possible to calculate the total gain of the steering antenna d n and position p n according to equation (21) by performing the following calculation: G ˜ n T Θ = w T × e Θ ′ = θ ′ , Δ ′ avec e Θ = e L θ ⊗ e L Δ et e L ς = exp − jLς exp − j L − 1 ς ⋮ exp jLς

[0107] According to equations (18), (19) and (20), the expression of the matrix K (Θ) is as follows: K Θ = k H Θ k V Θ − k V Θ k H Θ

[0108] According to equation (27), we then know that: K Θ ′ = I 2 ⊗ Γ n T × K Θ where ⊗ is the Kronecker product and I 2 the identity matrix of dimension 2. According to equation (22), the gain of the steering antenna d n and position p n is then written: G n Θ P 0 = G ˜ n T Θ × g n T K Θ P 0 avec g n = I 2 ⊗ Γ n T T em

[0109] According to equation (3), the response of the steering antenna d n and position p n is then written as follows: a n Θ P 0 = a n Θ × U n Θ P 0 avec { U n Θ P 0 = g n T u Θ P 0 u Θ P 0 = K Θ P 0 a n Θ = G ˜ n T Θ × exp j 2 π λ k Θ T p n

[0110] The Step C process of constructing the response of an antenna to a direction Θ and a polarization P is the following, knowing that the antenna is modeled by the vector em, the interpolation vector w , his position p n and the rotation matrix Γ n calculated for example by following the steps of step B described above from an orientation ϕ n of the antenna in the horizontal plane: Step C.1 : calculation of the wave vector in the simulation reference frame by performing k̃ (Θ') = [ u ' v ' w '] T< = Γ not T< k (Θ); Step C.2 : calculation of the incidence Θ' = {θ', Δ'} in the simulation reference frame by carrying out θ' = corner ( u' + jv' ) and Δ'= corner ( u 'cos( θ ')+ v 'sin( θ ')+ jw' ) ; Step C.3: calculation of the total gain of the antenna by performing G̃ n T< (Θ) = w T< × e (Θ ' ={ θ ',Δ'}), knowing that the function e(O) is defined in equation (33); Step C.4 : calculation of the matrix K (Θ) according to equations (14) and (34) then calculation of u (Θ, P 0 ) = K (Θ) P 0 ; Step C.5 : calculation of the vector g n by carrying out g n=( I 2 ⊗ Γ nT< ) T< em ; Step C.6: calculation of U n (Θ, P 0 ) = g nT< u (Θ, P 0); Step C.7 : calculation of a n Θ = G ˜ n T Θ × exp j 2 π λ k Θ T p n ; Step C.8: calculation of the antenna response by performing a n (Θ, P 0 )= a n (Θ)× U n (Θ, P 0 ).

[0111] The rest of the description describes an embodiment making it possible to determine the positions and orientations of the antennas, in order to best approach the conditions of omnidirectionality in the direction of arrival and in polarization sought.

[0112] The direction vector is then written as follows: a Θ P = Φ Θ × G × u Θ P Or : Φ Θ = a 1 Θ 0 0 0 ⋱ 0 0 0 a N Θ et G = g 1 T ⋮ g N T

[0113] The algebraic structure of G conditions the conditions of polarization diversity. We can thus establish conditions on the angles (ϕ 1,... ϕ N) of orientation of the antennas in the horizontal plane, by establishing a condition on the n-tuple (ϕ 1,... ϕ N) so that the network is omnidirectional in polarization. For this it is necessary that the columns of G form an orthonormal base. To simplify we consider the case of a planar network verifying: d n = d ϕ n et η n = 0 0 1 avec d ϕ = cos ϕ sin ϕ 0

[0114] We deduce according to equations (30) and (31) that Γ n = Γ ϕ n avec Γ ϕ = − sin ϕ 0 cos ϕ cos ϕ 0 sin ϕ 0 − 1 0

[0115] According to equations (22), (27) and (36), we know that g n = Γ ϕ n E Γ ϕ n H = E n H n avec E = E x E y E z et H = M x M y M z

[0116] Consequently the matrix G of equation (39) is: G = G x × E x G x × M x + G y × E y G y × M y + G z × E z G z × M z avec { G x = − s ϕ c ϕ 0 G y = 0 0 1 G z = c ϕ s ϕ 0 où c ϕ = cos ϕ 1 ⋮ cos ϕ N s ϕ = sin ϕ 1 ⋮ sin ϕ N 1 = 1 ⋮ 1

[0117] The condition for the network to have polarization diversity is that the matrix [ c ( ϕ ) s ( ϕ ) 1 ] is of full rank greater than 6. We then see that this requires that there be at least one antenna whose orientation is different from that of the other antennas, so that the vectors c ( ϕ ) Or s ( ϕ ) are not collinear with the unit vector 1. On the other hand, we see that to get closer to a condition of omnidirectionality at the polarization level, it is necessary to find a set of phases (ϕ 1,... ϕ N) such that the vectors c ( ϕ ) And s ( ϕ ) are orthogonal. We propose below a method to obtain such a condition, based on the following property, by posing b(ϕ) = c(ϕ) + j s ( ϕ ): les vecteurs c φ sont orthogonaux si et seulement b φ et b ' φ sont orthogonaux .

[0118] We then construct the following structured vector b ˜ α = b ϕ = ϕ 1 = x 1 α ⋯ ϕ N = x N α

[0119] After a random drawing of the n-tuple {x 2,..., x N} such that x min < x i < x max and x 1 = 1, we look for the value α min minimizing the criterion C ϕ ( α ) following : α min = arg max 0 ≤ α ≤ 2 π C ϕ α avec C ϕ α = b ˜ T α b ˜ α

[0120] The phase n-tuples (ϕ 1 ,... ϕ N ) is such that ϕ i = α min X i .

[0121] The Step D process of calculating the directions (ϕ 1,... ϕ N) of the antennas in the horizontal plane can then be as follows: Step D.1 : Drawing of the n-tuples {x 2,..., x N} such that x min < x i < x max and x 1 = 1; Step D.2: Construction of the vector b̃ ( α ) = b ( ϕ = { ϕ 1 = x 1 α ... ϕN = x N α}) for 0 ≤ α < 2π knowing that b ( ϕ ) =[exp( jϕ 1 ) ··· exp( jϕ N )] T< ; Step D.3: Finding the angle α min minimizing the criterion C ϕ ( α ) = b T< ( α ) b̃ ( α ) for 0 ≤ α < 2π; Step D.4 : Calculation of the n-tuples of the direction phases of the antennas in the horizontal plane by performing { ϕ 1 = x 1 α min... ϕN = x N α min} ; Step D.5. Deduction of directions d n from the n-tuples ϕ n according to equation (29), from the equation z n = f ( xn,yn ) of the metal surface.

[0122] Regarding determining a positional game p n making it possible to obtain omnidirectionality in the direction of arrival, this property is verified when the matrix MS Θ1 of equation (13) is diagonal, and the matrix M.S. k ( Θ 1) is proportional to identity. This condition is true when the matrix H ( k (Θ 1 )) of equation (12) is diagonal. According to this article " High Resolution direction finding: from performance toward antenna array optimization - The single-source case », the expression of this matrix is ​​as follows: H k Θ m = 2 N 2 π λ 2 D pp g m G λ { g m = G × u Θ m P D pp g G λ = D pp g + J ˜ u Θ T D gg u J ˜ u Θ × 2 π λ − 2 J ˜ u Θ T = J Θ J Θ T J Θ − 1 J u Θ T where the aperture matrices D pp ( g ) Then D gg ( u ) are written: { D ˜ pp g = ∑ n = 1 N w n g p n − p ‾ g p n − p ‾ g T D ˜ gg u = G H G g H g − gg ‾ H avec { g ‾ = G H g g H g et g = Gu p ‾ g = ∑ n = 1 N p n w n g w n g = g n 2 g H g with g (n) the nth component of g. Matrices J (Θ) and J u (Θ) are the respective Jacobians of k ( Θ ) and you( Θ , P ) with : J Θ = − sin θ cos Δ − cos θ sin Δ cos θ cos Δ − sin θ sin Δ 0 cos Δ et J u Θ = P V × J V Θ J H Θ + P H × − J H Θ J V Θ and where the matrices J H (Θ) and J V (Θ) are the respective Jacobians of k H ( Θ ) And k v ( Θ ) with : J V Θ = sin θ sin Δ − cos θ cos Δ − cos θ sin Δ − sin θ cos Δ 0 − sin Δ et J H Θ = cos θ 0 sin θ 0 0 0

[0123] According to equation (40), we consider in the process an array of antennas whose orientations in the horizontal plane depend on the N-tuples of phases (ϕ 1,... ϕ N), with d n = d( ϕ ). In this case the matrix G has the structure of equation (43). Consequently, we can say according to equation (47) that g m = G × u Θ m P = c ϕ s ϕ × u ˜ avec c ϕ = cos ϕ 1 ⋮ cos ϕ N et s ϕ = sin ϕ 1 ⋮ sin ϕ N Or û is a 2-dimensional vector depending on the incidence Θ m, the polarization P and the electromagnetic components of the antenna. In this particular context, we can then say that: D ˜ pp g 1 = c ϕ + j c ϕ 2 = D ˜ pp g 2 = c ϕ − j c ϕ 2 = D ˜ pp geo { D ˜ pp geo = 1 N ∑ n = 1 N p n − p ‾ p n − p ‾ T p ‾ = 1 N ∑ n = 1 N p n car { g 1 n = exp jϕ n g 2 n = exp − jϕ n et done w n g 1 = w n g 2 = 1 N

[0124] According to equation (47), a necessary condition to approach omnidirectionality in the direction of arrival is that the matrix D̃ pp geo< is proportional to the identity, exactly as in the case of geometric networks where the condition is also sufficient. The matrix H ( k (Θ m )), which is a function of the incidence Θ m, is then for a large angular sector proportional to the identity.

[0125] As proposed in patent application EP 2,458,398 A2 and in patent EP 2,462,459 B1, the choice of positions is made by randomly drawing a first set of positions { p n 0<}, then transforming it in the following way to obtain a position set { p n 1<} associated with an almost omnidirectional network in bearing and elevation: p n 1 = W − 1 p n 0 − p ‾ avec D ˜ pp geo = WW H et { D ˜ pp geo = 1 N ∑ n = 1 N p n 0 − p ‾ p n 0 − p ‾ T p ‾ = 1 N ∑ n = 1 N p n 0

[0126] The position set { p n 1<} is then modified by a homothetic factor so that the network respects congestion D given by the specifications. This can be done in the following way: p n = D D ini × p n 1 avec D ini = max i , j ‖ p i 1 − p j 1 ‖ where ∥ p i 1< - p j 1< ∥ is the distance between the i th< and the j th< antenna. Knowing that the coordinates p n = [x n y n z n ] T< are constrained to an equation surface z n = f ( xn,yn ), the calculation of positions is initially done in the horizontal plane with p n 0< = [x n y n 0] T<.

[0127] The Step E process of calculating the positions of the antennas of an antenna array under a constraint of omnidirectionality in the direction of arrival and congestion D can be as follows: Step E.1 : random drawing of N antenna positions in the horizontal plane with { p n 0< =[x n y n ] T< for 1≤ n ≤ N} ; Step E.2: calculation of the equivalent aperture matrix of the network with D ˜ pp geo = ∑ n = 1 N p n 0 − p ‾ p n 0 − p ‾ T / N And p ‾ = ∑ n = 1 N p n 0 / N ; Step E.3: decomposition into proper elements of D̃ pp geo< , with D pp geo< = EΛE H< , Or E is the matrix of eigenvectors and A is the diagonal matrix of eigenvalues; Step E.4 : calculation of the matrix W for whitening with W = E Λ 1 / 2< ; Step E.5: calculation of a set of antenna positions giving omnidirectionality by carrying out p n 1< = W -1< ( p n 0< - p ) for 1 ≤ n ≤ N ; Step E.6: calculation of network congestion { p n 1<} by performing D ini = max i , j ‖ p i 1 − p j 1 ‖ ; Step E.7: resizing the network by performing p n 2< = [ x n y n ] T< =( D / D ini )× p n 1< ; Step E.8 : calculation of positions p n conform to the surface by performing the following operations for 1≤ n ≤ N : p n = [ x n y n f ( x n , y n )] T< .

[0128] The method for designing an antenna array according to the invention carries out joint optimization of the elementary antenna with the antenna array. The optimization is done from the geometric parameters of the antenna (length, width, height, curvatures of the petal, etc.) as well as network parameters such as the position and orientation of each of the elementary antennas. It aims to determine the best set of parameters giving an omnidirectional network in direction of arrival and polarization, and having good robustness to ambiguities. All this takes place based on specifications characterized by a frequency band (f min ...f max) with a gain constraint and a maximum lamination rate in this band, and a footprint available on the platform to integrate the elementary antennas of the entire network. The specifications can also set the maximum number of elementary antennas for the network in order to adapt to an available reception system limited in number of channels (this condition can also be linked to constraints of mass, consumption and payload volume).

[0129] The array antenna design method according to the invention is perfectly suited for an array conforming to a 3D metallic surface. For this, it includes the modeling of the total gain of the elementary antennas not pointing in the same direction in the presence of a 3D surface. The total gain diagram here is the complex gain of an antenna when it is matched to its polarization. This diagram deforms in the presence of a surface. This modeling is done through an interpolation of the complex response of the total gain of the elementary antenna measured or simulated throughout the angular space. This makes it possible to model an elementary antenna in the presence of the 3D surface on which the network will be installed. To obtain the gain for another orientation of the antenna, it is then sufficient to carry out the change of base between the elementary antenna in the reference frame of the network and that in the reference frame of the reoriented antenna. The same method is used for modeling the polarization gain, characterized by the electromagnetic components of the antenna array.

[0130] The proposed solution makes it possible to work with 3D conformal networks having the advantage of having gain at the horizon (Δ=0°), and better precision in elevation at the horizon. This has the advantage of improving the performance of instantaneous geolocation techniques for distant sources based on 2D direction finding in bearing and elevation. The method described makes it possible to design heterogeneous antenna arrays with polarization diversity making it possible to determine the direction of arrival of transmitters with a precision almost independent of their polarization, and following an extended angular sector.

Claims

1. Method for designing an array of N antennas arranged on a metallic surface intended to isolate the antenna array from its support, the antenna array being substantially omnidirectional in the direction of arrival and polarized in a frequency band having a minimum frequency f min and a maximum frequency f max , with N greater than 1, the design process being characterized in that It comprises the following steps: - a first step (601) of determining K antenna configurations having different geometric characteristics, with K greater than 1, adapted to satisfy a gain differential constraint over the frequency band [f min , f max] and a gain variation constraint in the main lobe of the antenna, - a second step (602) of calculation, for each of the K antenna configurations, of at least one antenna array configuration, the orientations of the N antennas of each antenna array configuration being chosen so as to promote the omnidirectionality of the antenna array in polarization, the arrangements of the N antennas of each antenna array configuration being chosen so as to promote the omnidirectionality of the antenna array in the direction of arrival, - a third step (603) of selection of the best antenna configuration / antenna array configuration pairs.

2. Method of designing an antenna array according to claim 1, wherein the antennas are "petal" type antennas, comprising two elements (101, 102) folded towards a ground plane at the center (102) of the antenna.

3. Method of designing an antenna array according to any one of claims 1 and 2, further comprising a fourth step (604) of optimizing the configuration of the antenna(s) selected in the third step (603), so as to optimize the performance of the associated antenna array(s).

4. Method of designing an antenna array according to claim 3, wherein the antennas are of the "petal" type, comprising two strands (101, 102) folded towards a ground plane at the center (102) of the antenna, said optimization of the configuration of the antenna(s) comprising the modification of an antenna configuration parameter among: a width (W), a shape of the strands (R1, R2, R3) and a radius of curvature of the strands (c1, c2).

5. Method of designing an antenna array according to any one of the preceding claims, wherein the N antennas are identical.

6. Method for designing an antenna array according to any one of the preceding claims, wherein the first and second steps are implemented from an electromagnetic simulation or a measurement of the complex gain of a unit antenna disposed on said metallic surface.

7. Method of designing an antenna array according to any one of the preceding claims, wherein said metallic surface is a portion of a metallic sphere.

8. A method for designing an antenna array according to any one of the preceding claims, wherein the first step (601) comprises determining K' antenna configurations having different geometric characteristics, with K' greater than K, adapted to satisfy a constraint limiting the lamination rate at frequency f max , then for each antenna configuration, the determination of a frequency band [f min , f maxresponding to a constraint of peak gain variation in the frequency band, then the selection of K antenna configurations from said K' antenna configurations, considering the length of each antenna and the minimum frequency f min associate.

9. A method for designing an antenna array according to any one of the preceding claims, wherein the second step (602) comprises: - obtaining complex gains G V (Θ,f) and G H (Θ,f) of unit antenna responses to polarizations E θ And E ϕ following arrival directions Θ = {θ, Δ} for a regular mesh of frequencies within the frequency band [f min , f max ] ; - for said regular frequency mesh f, Calculating antenna gain modeling parameters by estimating electromagnetic components em ( f ) and interpolation coefficientsw ( f ) of the total antenna gain from the complex gains G V (Θ,f) and G H (Θ,f); then for a given number of iterations: - the determination of orientations ( d 1,... d N ) antennas promoting omnidirectionality of the polarized antenna array; - the determination of positions ( p 1,... p N ) antennas promoting omnidirectional antenna array coverage in the direction of arrival; - rejection of the antenna array when the positions and orientations of the antennas are not compatible with maximum antenna array footprint; and wherein the third step (603) comprises, for each antenna configuration / antenna array configuration pair: - for a regular mesh of frequencies within the frequency band [f min , f max], the calculation at each frequency f of a robustness to ambiguities of the antenna configuration / antenna array configuration pair by performing: o from the parameters { w ( f ), em ( f )} of the antenna, of the wavelength λ = c / f , antenna orientations { d n} and their positions { p n}, the calculation of the answers a n (Θ, P V ) and a n (Θ, P H ) of the N antennas for polarization P V = [1 0] T And P H = [0 1] T to obtain vectors a (Θ, P V ) And a (Θ, P H ); o for each direction Θ at frequency f, the orthonormalization of the basis of vectors a (Θ, P V ) And a (Θ, P H) to obtain the columns of the matrix Ũ (Θ)=[ a co (Θ) a cross (Θ)] ; o the calculation of robustness to ambiguities η 1( f ) of the antenna configuration / antenna array configuration pair from said matrix Ũ (Θ), the robustness to ambiguities corresponding to a minimum of the projection of two planes formed respectively by columns of Ũ (Θi) and of Ũ (Θj) for any pair of different directions (Θ i , Θ j ) ; - calculation of the robustness to ambiguities of the antenna configuration / antenna array configuration pair or réseau = min fmin≤f≤fmax η 1( f ); the best antenna configuration / antenna array configuration pair(s) being the one(s) whose robustness to ambiguities or réseau is the highest.

10. A method for designing an antenna array according to any one of the preceding claims, wherein the selection of the orientations of the N antennas in the second step (602) comprises: - the random selection of N-1 values x 2 to x N , with x1 =1, - the construction of a vector b̃ ( α ) = b ( ϕ = { ϕ 1 = x 1 α ··· ϕ N = x N α}) , with b ϕ = c ϕ + j s ϕ , c ϕ = cos ϕ 1 … cos ϕ N And s ϕ = sin ϕ 1 … sin ϕ N - calculating an angle α min minimizing an orthogonality criterion C ϕ ( α )= b̃ T ( α ) b̃ ( α ) , - the calculation of direction phases { ϕ 1 = x1α min ... ϕ N = x N α min } of the N antennas; - the calculation of orientations ( d 1,... d N ) antennas, from the direction phases ϕ n of the N antennas.

11. A method for designing an antenna array according to any one of the preceding claims, wherein the selection of the position of the N antennas in the second step (602) comprises the steps of: - random selection of N antenna positions p n 0 = x n y n T in a horizontal plane, - calculation of a matrix D ˜ pp g é o equivalent opening of the antenna network, with D ˜ pp geo = ∑ n = 1 N p n 0 − p ‾ p n 0 − p ‾ T / N , And p ‾ = ∑ n = 1 N p n 0 / N - matrix decomposition D ˜ pp g é o in its own elements, with D̃ pp geo = ELE H , Or E is a matrix of eigenvectors of D ˜ pp g é o And Λ a diagonal matrix of eigenvalues ​​of D ˜ pp g é o - calculating a matrix W of whitening, with W = E Λ 1 / 2 - Calculation of a set of antenna positions p n 1 = W -1 ( p n 0 - p ), - calculation of network congestion p n 1 - network resizing p n 1 by applying a homothetic ratio between a bulk associated with the positions p n 1 , and a specified maximum size.

12. Method of designing an antenna array according to any one of claims 10 and 11, wherein the metallic surface is non-planar, the choice of the position of the N antennas and / or the orientation of the N antennas of the second step (602) further includes a step of projecting the positions and / or orientations onto the metallic surface.