SOLUTION FOR OPTIMIZING AN ANTENNA / 3D NETWORK PAIRING CONFORMING TO A SURFACE.

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 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 maximum lamination constraint at the frequency fmax, and determining a frequency band [fmin, fmax] meeting a gain constraint, - 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). Figure for the abbreviation: Fig. 6,

Description

Title of the invention: Solution for optimizing a 3D antenna / network pair conforming to a surface Technical field

[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 transmission channels, such as radiogoniometry processing whose objective is to estimate the direction of arrival (0 m) of electromagnetic waves coming from several transmitters in far fields (plane wavefront) from a network of sensors which can be arranged 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 the antenna arrays depend mainly on: - characteristics of the radiating element (antenna) which makes up the network, mainly characterized by its radiation pattern in amplitude, phase and polarization, and its variation as a function of frequency, - the geometry of the network, i.e. the position and orientation of the radiating elements, - the supporting structure of the radiating element network, which has an influence on its behavior and limits its maximum size.

[0003] Goniometry 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 ability improves with the number of antennas in the network and degrades when the network congestion increases for a fixed number of antennas, - the accuracy of estimating the direction of one or more sources, which improves when the dimensions of the network and the number of elements increase, - the resolution between the directions of two sources, which improves as the dimensions of the network increase.

[0004] More particularly, the objective is to design a goniometry 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 at transmission 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 the diversity of polarization to increase the transmission rate. 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 with the same orientation, to perform a 2D goniometry in bearing and elevation (0, A), so as to avoid performing a 1D goniometry in bearing which must make an assumption about the elevation angle of the sources. In many solutions, it is assumed that the goniometry system and the transmitters are located in the same plane by assuming that the elevation angle A is zero. The need for 2D goniometry in (0, A) instead of 1D goniometry in bearing has the effect of reducing robustness to ambiguities: this implies dimensioning radiating elements whose elevation diagram is reproducible and measurable, using the same network in the widest possible frequency domain, which implies using compatible radiating elements in this domain and integrable within a compact structure, being able to integrate on 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 the mutual coupling between antennas in the network, coupling with a platform has the effect of significantly attenuating the gain of all the antennas in the network in certain directions and periodically in the space of the directions of arrival. This coupling between the platform and the antenna network is minimized by placing a metal surface or a ground plane between the antennas and the platform, . • network congestion and weight. This means that a compromise must be determined between performance, size and number of radiating elements of the network taking into account the surface area available for installation.

[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 “Petals” 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 a high-performance antenna array and its limited size compared to a targeted radioelectric performance. The invention can also be applied in an identical manner 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 developed antenna network depends on the ability of the antennas to achieve an advantageous compromise between size, low thickness in the presence of a metallic structure and radioelectric performance over a wide frequency band.

[0006] [Fig. 1a] schematically shows 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 under the expression “capacitive roofs”, advantageously make it possible to improve the radioelectric performance of the antenna at the bottom of the band. The antenna has a height H, a width W and a depth D. [Fig. 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 R2, the latter fixing the total width of the antenna.Then, the Ri / R3 ratio makes it possible to optimize the transitions at the level of the feeding and the terminal folding of the strand.

[0007] [Fig. 1e] represents a side 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 Fi and F2. 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] Let the Cartesian coordinates of points A, B and C be such that: / XA \ , / XB and / Xc . A = 0 B = 0 C = 0 \ZAI \ ZB / \ ZC /

[0009] The equations of the curves are given by two additional curvature parameters: Ci and c2. The equation of Fi is such that: -Yr +(ry \ wyf= 1 - - 1 Likewise' t - t the equation of F2 is such that: , rxv , / xe^-e^2 uq The second strand (^2):X = xB + (xc-xB) (ZBC2, Vze [zB, zc]. € - € of the radiating element is then generated by x-axis symmetry.

[0010] 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 major radius of the large ellipse (Ri), - the minor radius of the large ellipse / major radius of the small ellipse (R2), - the small radius of the small ellipse (R3), - 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 (cO, - the curvature parameter of the second curve (c2).

[0011] Antenna optimization firstly consists of obtaining a radiating element whose radioelectric characteristics such as radiation or impedance matching are stable and vary monotonically with frequency, within a defined footprint relative to the type of array geometry envisaged, while taking into account the supporting structure. This can be done for example by fixing the width and height of the antenna, then refining the other parameters according to their impact on the radiation pattern or impedance matching over the entire target frequency band, for example from simulations. During antenna array optimization, homothety on all the parameters can be performed on the obtained radiating element in order to adapt its dimensions to the desired array geometry without returning to this initial step.

[0012] 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 that is both: - ultra wide band: its impedance matching and its gain make it possible to cover a frequency domain greater than a decade, providing the performance radioelectric training required for this type of system; - compact: the folded shape makes it possible to make maximum use of the occupied volume compared to planar solutions or those 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. It also makes it possible to have network geometries with radiating elements closer to each other to improve protection against ambiguities without degrading the interception range of the targeted transmitters; - sectorial: the shape of the antenna, reminiscent of a "Vivaldi" antenna (in English Tapered Slot Antenna) resting on a ground plane, makes it possible to obtain sectorial radiation with an opening of the radiation lobe at half-power greater than 90° in both planes. This also makes it possible to rest the radiating element directly on the ground plane without degrading the response of the network.

[0013] Processes for optimizing the position of antennas in an antenna array are also known to those skilled in the art. Patent EP 2,462,459 B1 describes an optimization process in the case of an array consisting of identical antennas all having the same orientation. The optimization then consists of determining the position of the antennas making it possible to meet specifications in terms of single-source precision, directivity and / or dual-source resolution. A search is then made in a family of arrays having the same performance for those with the best robustness to ambiguities.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 Optimisation - The mono-source case”, EUS1PCO, Aug 2009, Glasgow, United Kingdom, allowing the positions of the antennas of an array to be linked to a performance coming from a specification.

[0014] Patent application EP 2,458,398 A2 is a generalization of the optimization of a homogeneous network in monopolarization 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 couples (position, orientation) of the antennas was established thanks to the modeling of [Fig.2] in electric component E = (Ex, Ey, Ez) and magnetic H = (Mx, My, Mz) of the elementary antenna. It is thus possible, under a constraint of omnidirectionality in direction of arrival and in polarization, to find the networks most robust to ambiguities. The method was developed more particularly in the case of a cylindrical type carrier structure. These heterogeneous networks in orientation have the advantage of being free 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.

[0015] The main disadvantage of the state of the art is that antenna optimization is carried out separately from network optimization. In particular, the greater the gain specified in a specification for a particular band, the more this will result in producing large antennas having a size much greater than X / 2 (half wavelength), and consequently a gain with significant lamination in the main lobe. On the other hand, the optimization processes require that certain antennas in the network be as close as possible to remove direction finding ambiguities. In particular, in the worst case of regular networks, this spacing must not exceed X / 2. Knowing that network optimization processes take into account the size of the antennas, a large antenna with high gain will result in, at best, networks that are not very robust to ambiguities.

[0016] Another disadvantage 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 arranged between the array and the platform, as shown in [Fig.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 many 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 antennas of the array.

[0017] Arranging an antenna array on a ground plane, however, has drawbacks for 2D direction finding applications. Indeed, the total gain of the antennas is greatly reduced when the sources come from a grazing direction relative to the ground plane. Similarly, 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 from 2D direction finding, particularly in an airborne context. In other words, a planar array greatly limits the angular sector covered by the direction finding system: the problem is both in gain and accuracy for 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 metal surface, for example a . portion of a metal sphere. [Fig.3b] represents the antenna array of [Fig.3a] in the vertical plane. The antennas 301 are arranged on a metal spherical cap 302 having an angle of curvature [3.

[0018] As regards the arrangement and position of the antennas, knowing that one objective is to establish a network allowing 2D goniometry to be carried out in polarization diversity, the method for optimizing a heterogeneous network in orientation described in patent application EP 2,458,398 A2 seems to be a good solution. However, this method 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 metal surface) this gain has the effect of being deformed 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.

[0019] 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 mismatching of the polarization of the antennas of the array to the polarization state of the incident waves. In other words, a homogeneous array reduces the polarization range of the incident electromagnetic waves covered by the direction finding system.

[0020] 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 (or on a ground plane). The model does not take into account the deformation 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 be significantly weakened compared to the same antenna in free space. All of this therefore does not make it possible to determine an optimal network solution with antennas conforming to a surface, such as a spherical cap or a cylinder.

[0021] An aim of the invention is therefore to describe a method for determining the best elementary antenna / antenna array pair optimizing the compromise between the gain of an antenna and the robustness to ambiguities of the array.

[0022] 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

[0023] 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 array of antennas from its support, with N greater than 1. The antenna array aims to be substantially omni-directional in the direction of arrival and in polarization in a frequency band having the maximum frequency fmax. 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 maximum lamination constraint at the frequency fmax, and for each antenna configuration, determining a frequency band [fmin, fmax] meeting a gain constraint, - a second step of calculating, for each of the K antenna configurations, at least one antenna array configuration, the orientations of the N antennas of the antenna array being chosen so as to promote the omnidirectionality of the antenna array in polarization, the arrangement of the N antennas of the antenna array being chosen so as to promote the omnidirectionality of the antenna array in the direction of arrival, - a third stage of selection of the best antenna configuration / antenna array configuration pair(s).

[0024] Advantageously, the antennas are “petal” type antennas, comprising two strands folded towards a ground plane at the center of the antenna.

[0025] In one embodiment, the method for 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).

[0026] When the antennas are “petal” type antennas, said optimization of the configuration may comprise 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.

[0027] According to one embodiment of the method according to the invention, the N antennas are identical.

[0028] According to one embodiment of the method according to the invention, the first step and the second step are implemented from an electromagnetic simulation of a unitary antenna arranged on said metal surface.

[0029] Advantageously, the metal surface is a portion of a metal sphere.

[0030] According to an 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 maximum lamination constraint at the frequency fmax, then for each antenna configuration, the determination of a frequency band [fmin, fmax] meeting a gain constraint, then the selection of K antenna configurations from said K' antenna configurations, considering the length of each antenna and the associated minimum frequency fmin.

[0031] According to one embodiment of the method according to the invention, the second step comprises: - obtaining complex gains G / 0,f) and G / / (0,f) of responses of the unit antenna to the polarizations Eo and Ev following directions of arrival 0 = {0, A] for a plurality of frequencies included in the frequency band [fmin, f max] 5 - for a plurality of frequencies / between fmin and fmax, the calculation of antenna gain modeling parameters, by estimating electromagnetic components em( / ) and interpolation coefficients w( / ) of the total gain of the antenna from the complex gains G / 0,f) and G / 0,0; then for a given number of iterations: - the determination of directions dn of the antennas favoring the omnidirectionality of the antenna network in polarization; - the determination of positions (pb..,pN) of the antennas favoring the omnidirectionality of the antenna network in the direction of arrival; - rejection of the antenna network when its dimension is greater than a maximum dimension; and wherein the third step comprises, for each antenna configuration / antenna array configuration pair: - for a plurality of frequencies included in the frequency band [fmin, f max], the calculation of the robustness to ambiguities of the antenna configuration / antenna array configuration pair by carrying out: • from the parameters {w( / ), em( / )} of the antenna, the wavelength X = c / / of the orientations of the antennas { dn} and their positions { pn}, the calculation of the responses a„(0, Pv) and a„(0, PH) of the N antennas for the polarization Pv = [1 0]T and PH = [0 1]T; • for each direction 0, the orthonormalization of the basis of vectors a(0, Pv) and a(0, PH) to obtain the columns of the matrix Û(©) = î ai5(0) ' • the calculation of the robustness to ambiguities of the confi couple antenna configuration / antenna array configuration from said matrix Ü ( 0 ); - the calculation of the robustness to ambiguities of the antenna configuration / antenna array configuration pair n-network mm f . mm the best antenna configuration / antenna array configuration pair(s) being the one(s) with the highest robustness to array ambiguities.

[0032] According to one embodiment of the method according to the invention, the choice of orientation of the N antennas of the second stage includes: - the random drawing of Nl values ​​x2 to xn, with Ài=l, - the construction of a vector = = ... = , with b ((p ) = c( <p) + js(<p). COS ((p ) and cos fe). the calculation of an angle amm minimizing an orthogonality criterion - the calculation of direction phases î ï ,,, = ; - the calculation of the orientations dn from said direction phases.

[0033] 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 n° = [xy P in a horizontal plane, - calculation of an equivalent aperture matrix j^o of the antenna array, with -pr < 'and p=z' - decomposition of the matrix into proper elements, with ' pp where E is a matrix of eigenvectors of ^eo and A a diagonal matrix pp eigenvalues ​​of Q§eo, 1 pp - calculation of a whitening matrix W, with W=EA1 / 2 - calculation of a set of antenna positions $ ; _yy2 ' -p) ' - calculation of network congestion n1, "n - resizing of network n1 by homothety to respect a constraint of clutter, and calculation of the associated n2 positions.

[0034] In an 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 onto the metal surface. Brief description of the drawings

[0035] The invention will be better understood and other characteristics, details and advantages will appear more clearly on reading the following description, given without limitation, and thanks to the appended figures, given by way of example.

[0036] [Fig.1a] [Fig.1a] schematizes a three-dimensional life of a petal antenna, used to illustrate the implementation of the method according to the invention;

[0037] [Fig.lb] [Fig.lb] represents one of the petals of the petal antenna of [Fig.la] in front view;

[0038] [Fig. 11] [Fig. 11] represents a profile view of the petal antenna of [Fig. 1a];

[0039] [Fig.2] [Fig.2] is a representation with an equivalent diagram of the model of a antenna, in a general case;

[0040] [Fig.3a] [Fig.3a] represents a view of an antenna array in a horizontal plane, for illustration purposes;

[0041] [Fig.3b] [Fig.3b] represents a view of the antenna array of [Fig.3a] in a vertical plane, for illustration purposes;

[0042] [Fig.4a] [Fig.4a] is a representation of the polarization Po of an incident wave to a network of antennas;

[0043] [Fig.4b] [Fig.4b] is a representation of the propagation of a wave vector in V polarization;

[0044] [Fig.4c] [Fig.4c] is a representation of the propagation of a wave vector in H polarization;

[0045] [Fig.5] [Fig.5] represents a practical application of a petal antenna array;

[0046] [Fig.6] [Fig.6] schematically represents the steps of a method for designing an array of N antennas according to one embodiment of the invention;

[0047] [Fig.7a] [Fig.7a] is an illustration of a petal antenna positioned in a orthonormal reference frame (x', y', z');

[0048] [Fig.7b] [Fig.7b] is an illustration of a change of reference linked to a variation from the orientation of the petal antenna of [Fig.7a];

[0049] [Fig.8a] [Fig.8a] is an illustration of the gain of a perfect dipole as a function of its orientation;

[0050] [Fig.8b] [Fig.8b] is an illustration of the gain of a perfect loop as a function of its orientation;

[0051] [Fig.9] [Fig.9] is an illustration of a heterogeneous array of antennas positioned on a plane.

[0052] Identical references may be used in different figures when they designate identical or comparable elements. Description of the embodiments

[0053] An objective of the invention described below is to design an omnidirectional antenna array in the direction of arrival (bearing 6 and elevation A) and in polarization. The proposed array design method jointly optimizes the (antenna, array) pair.

[0054] [Fig. 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 array according to the invention makes it possible to implement goniometric functions in order to determine the direction of arrival in two dimensions of electromagnetic waves transmitted by other equipment 505 and 506.

[0055] 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 an array of identical antennas, conforming to a 3D surface of the metallic spherical cap type. However, the invention applies in an identical manner when the antennas are different, or when the array of antennas is arranged on any metallic surface of equation z = / (x, y).

[0056] 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 a deformation axis of a horizontal ground plane. In the case of a spherical cap, the deformation function is very simple, with y ( xy ) — -^2 _ x2 - y2-

[0057] The method described therefore generalizes the optimization of the antenna network to any surface, unlike 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) = ^R2 - y2-

[0058] The remainder of the description firstly describes the principles of goniometry necessary to understand the implementation of the method according to the invention and to obtain the desired performance. In terms of mathematical notation, 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 conjugate transpose, the operator H represents the conjugate transpose, the operator designates an average.

[0059] [Fig.4a] is a representation of the polarization Po of a wave incident on an antenna array. Reference 401 designates a heterogeneous antenna array. An antenna array is said to be homogeneous when all the sensors in the array are identical radiating elements having the same orientation in space, and heterogeneous otherwise. It is therefore possible to have a heterogeneous array comprising N different sensors, or N identical sensors having different orientations.

[0060] From a general point of view, any wave propagates with a given polarization P (projection of the electric field vector E in the wave plane 402), which is the linear combination of the polarization V where E=kv and the polarization H where 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(0,A), with 0 the azimuth and A the elevation in the xyz plane of the antenna array. We denote by bv and bH the magnetic components of the polarizations V and H.

[0061] Figures 4b and 4c represent the propagation of a wave polarized respectively in V polarization and in H polarization, in the wave plane defined by the orthonormal vectors kv (3, A ) and kH ( 8, A ). These figures show the position of the incident electric field Eo and the incident magnetic field Ho as a function of the polarization of the wave.

[0062] In the presence of M sources, the output signal of a network of N sensors is written: r = | : i = y.â(Q..sP.,.).s.„(,*) L-'aCOJ (1) where is the signal received on the n-th sensor, s (ÿ) is the signal from the m-th source, afÿ is the additive noise, Q is the direction of arrival of the source, p is the polarization as defined in Figures 4b and 4c, and . p ) is the observed direction vector. In the presence of model error the vector à® _ p. 1 is written: at(0,KP = a(® P,.} t e. (2) °ù ...... P. ■ is the theoretical direction vector such that

[0063] a ( ®m, Pm ) Ha ( ®m, Pw ) = N and is the model error. Assuming no mutual coupling, and according to Figure 4a, the n-th component of a ( Pm ) is written as: a„(0,P) = G„(0,P)exp(j^k(0)TpJ (3) where n = Lr vzl^is the position vector, z is the wavelength and k ( 0 ) l ” ■ nn,J is the wave vector such that: ruk(0) = v with vvJ ' u = cos ( 0 ) cos ( A ) v = sin ( 6 ) cos ( A ) , w = sin ( A ) (4) where 0 = { 0, A} depends on the azimuth 6 and the elevation A . A network is said to be heterogeneous when the antenna gains Gm ( 0, P ) are not identical. Without harming any generality the direction vector a ( 0, P ) is written as follows: a(0, P) =U(0)P (5)

[0064] Knowing that the matrix U ( 0 ) of dimension Nx2 depends on the incidence 0 of the source, as well as the positions and orientations of the radiating elements composing the network, the first column of U ( 0 ) can be associated with the response of the network to the polarization P( 1 ) = Eo and the second column to the polarization P(2) = Eç. The algebraic properties of this matrix completely condition the performances of the network.

[0065] The antenna network optimization will be done on the basis of single-source performance (M = 1) with: - the criterion of robustness to ambiguities, - the precision of goniometry.

[0066] A mathematical ambiguity is present when, for a source of direction 0i and polarization Pb there exists another direction / polarization pair (02, P2) such that the vectors a(0i,Pi) and a(02,P2) are collinear. Under these conditions the following criterion is zero: J(©!, ©o, Pn Po) = 1 --JafWZihfï^PzlL—_ || a"(0„ P,) || - || a«(02, P2) || “ (6)

[0067] Thus, the robustness to ambiguities in single source is the following value: / / = min / (©j, 02, Pb P2). 1 (PbP2* (7)

[0068] Considering the following canonical decomposition of the matrix U(0): u ( 0 ) = [a ™ ( ® ) pco a — ] (8) where the vectors a ( ® ) and a( ® ) are the co-polarization and cross-polarization responses of the network forming an orthonormal basis, we can show that: li - H ~ \ \ 2 min .7(0,.02. P,, P,) = 1-U,JU (0011(0,))) p2 ) with Ü ( 0 ) = [a ™ ( 0 ) a ( © ) ] (9) where Xmax(M) is the maximum singular value of M. It should also be noted that a monopolarization network is such that Pcross is zero, and Pco 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 Pcross and Pco vectors are equal, we are then in the case of an omnidirectional network in polarization. According to the last expression, the robustness to ambiguities in single-source is: ,,( / )=1- min (4,OA(ü"(0,)Ü(02)))2 1 * “2) (10) where / = c! At, is the carrier frequency and c is the speed of light. The robustness to ambiguities 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 array.

[0069] Knowing that we are in the presence of model error em or additive noise n (t), the goniometry algorithm estimates the incidence-polarization pair pp = @ -oukf® { P i of the source with an error. We are particularly interested in the variance of the estimate: MSW =£[ AW,flAV] with = * W (11) At 2-1 m where (wp) is an estimate of Pm), and \ T nv * m / JM T m P L m

[0070] The root of MSw [ 1 ] [11 is the estimation precision of the bearing 0m and “m MS(p [2] [2] is the estimation accuracy of the elevation Am. In the reference " High Resolution direction finding: from performance toward antenna array opti-mization - The mono-source case », it is shown that in the presence of a source: MS^ = e[ ATjT] =aH(^I)“1 (12) H(^J ) =2ÀHn( T1)À with A PW1,P) tWi, P) A - n / q / , =I aQV P) 1 N a(Tb P^aCWp P)

[0071]

[0072]

[0073] According to this article and the following table, which gives the parameters to be associated with the single-source precision criteria knowing n ) ïl( / )^ ] = <72Iy' 'cs values ​​of the coefficient ® depend on the type of performances considered. [Tables 1] Type of Performance Value of coefficient a Cramer bound RAO From x(4) for i < k < K Stochastic case i + r ,, « = liCA j \(7 ! Deterministic case a = A c with rss = AE 11 sM | \(j / Performance of MUS1C At finite integration time with for 1 < k < K 1 (Z — / FX \(T / In the presence of model error £ F 8 ^8 1 a~ N This shows that the performance depends only on the matrix H(), which is directly related to the lobe width of the goniometry criterion. According to the following expressions, the matrix contains on its diagonal the variance of estimation of the bearing and the elevation and the matrix MS the variance of estimation of the com- Kl V7 dd posers (uB vb Wi) of the wave vector: MSn = < B Cl << 1 < <0-7 faq >-----------------------1 and (13)

[0074] 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 for respecting omnidirectionality in the direction of arrival and in polarization. In particular, it is shown in the case of a monopolarization 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 matrix is ​​diagonal, and when the matrix is ​​proportional tional to identity.

[0075] [Fig.6] schematically represents the steps of a method for designing an array of N antennas according to an embodiment of the invention, for the design of an antenna array aiming to be omnidirectional in direction of arrival and in polarization.

[0076] Knowing that the maximum working frequency of the goniometer is fmax, 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 step of optimizing the antenna network with a set of K elementary antennas whose impedance matching and gain are compatible by their level for use in the widest possible working frequency band [fmin, fmax].

[0077] This step is advantageously carried out by optimizations using electromagnetic simulations, the simulations being able to use measured radiation data, in order to characterize the response of the collection of K antennas operating up to the frequency fmax. The K antennas selected notably satisfy a constraint of maximum lamination of the total gain of the antenna in the main lobe. The frequency fmin is determined with respect to the gain differential of the lobe peak of the antenna at fmin and fmax. 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 lamination. These variations (decreases) create blind zones of the antenna, and must therefore be avoided. For this, a lamination rate is measured, which corresponds to the difference between the minima and the maximums of the antenna gain zone.

[0078] 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 fmax verifying the maximum admissible lamination rate, then by resizing the antenna with the homothetic ratio fmax0 / fmax. The frequency fmin is determined according to a constraint of maximum peak gain difference between the frequencies fmin and fmax. The optimization will consist of minimizing the pair (antenna size, minimum frequency fmin). This homothety makes it possible to converge quickly towards a solution. It can be followed by a step of fine adjustment of the antenna parameters. For this, the use of the “Petals” antenna of European patent EP 3,335,277 B1 is particularly advantageous given all the optimization parameters that it offers.

[0079] A possible embodiment is as follows: - for a given antenna length L, adjustment of the other antenna parameters to provide an antenna configuration with good radio characteristics, in particular in terms of gain stability and impedance matching over a frequency band increased by fmax; - calculation of the maximum frequency fmax0 up to which the antenna satisfies a maximum lamination criterion, with fmax0 < fmax; - calculation of the minimum frequency fmin0 as a function of a maximum gain variation criterion over the frequency band [fmin0, fmax0], so that the gain peak at the frequency fmax0 and the gain peak at the frequency fmin0 have a difference less than a threshold AG; - resizing of the antenna by homothety of a factor fn / and 1 maxO / / f : ' max calculation of the frequency fmin, with / f 1 mln' f — f * J »m.aO / ' min ' minO 1 / t \ / max allows to obtain an antenna of length L' satisfying the lamination criterion and the minimum gain criterion on the band [fmin, fmax].

[0080] 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, i.e. the K antennas of the smallest dimensions and having the widest operating band, and therefore which minimize the torque (length L', frequency fmin0), are selected to implement the rest of the method.

[0081] Resizing the antennas by homothety makes it possible to converge towards an operation of the antenna at the limit of lamination at the frequency fmax. It is therefore the best possible compromise at this frequency between the geometry of the network and the size of the antenna. The performances of the network can still be This step 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.

[0082] The method for 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.

[0083] The input parameters for this step are as follows: - an electromagnetic simulation of each antenna among the K antennas selected during 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; - the maximum size D of the antenna network in the horizontal plane; - the number N of antennas in the antenna network; - the equation Z — / (x, y) of the surface on which the network antennas are installed.

[0084] 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 metal surface (for example a portion of a sphere) on which it is arranged must be carried out, and the parameters of the model (electromagnetic components and interpolation coefficients of the total gain) are estimated from this data. A random selection of the positions and orientations of the antennas on the metal surface is then carried out. The positions of the antennas are adjusted to obtain an omnidirectional network in the direction of arrival and the orientations of the antennas are adjusted to obtain a network maximizing the polarization diversity (the sum of the orientations of the antennas 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.

[0085] According to one embodiment of the invention, step 602 is implemented by performing the following calculations for each of the K antennas selected during step 601: - Network Step.l: Obtaining the complex gains G / 0,1) and G / / (0,f) of the responses of the unit antenna to the polarizations Eo and Ev following all the directions of arrival 0 ={0, A}, by measurements or by electromagnetic simulation, for a plurality of frequencies between fmin and presence of an insulating metal surface; - Network Step.2: For a plurality of frequencies / between fmin and fmax, calculation of the antenna gain modeling parameters, by estimating electromagnetic components em( / ) and interpolation coefficients w( / ) of the total gain of the antenna from the complex gains G / 0,f) and G / / (0,f), for example according to the process described later as Step A; Then for a given number of iterations: - Stage Network.3: Drawing on a random basis the orientations (q>i,...q>N) of the antennas in the horizontal plane, the orientations being adjusted in order to obtain an omnidirectional antenna network in polarization, for example according to the process described later as Stage D. For each orientation <pn, en déduire les directions dn des antennes selon l’équation (29), à partir de la connaissance de l’équation z=f(x,y) de la surface métallique sur laquelle le réseau est disposé ; - Network Step 4: Drawing on a random basis the positions (pi,...pN) of the antennas in the horizontal plane giving an omnidirectional network in the direction of arrival on the metallic surface of equation z = flx,y) for example, according to the process described later as Step E; - Network step.5: Test to determine if the parameters (pn, dn) are compatible with the antenna footprint. If not, return to Network step.3.

[0086] The characteristics of the antenna / network pair are then stored.

[0087] Variations can be easily implemented on the steps mentioned above, for example by reversing certain steps such as drawing orientations and drawing positions.

[0088] The method for 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).

[0089] This step consists of evaluating the robustness to ambiguities of each antenna array in the band [fmin, fmax], then selecting the antenna / antenna array pair(s) having the best robustness to ambiguities.

[0090] 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 / between / „,„ and fmax, calculation of the robustness to ambiguities of the antenna / antenna network pair by performing: • Network Step.6.1: From the parameters {w( / ), em( / )} of the radiating element, the wavelength "k=clf, the orientations of the antennas {dn} and their positions {pn}, calculation of the responses an (0,PV) and a„(0, PH) of the N antennas for the polarization Pv= [1 0]T and PH= [0 1]T, for example according to the process described later as Step C; • Network Step.6.2: For each direction 0, orthonormalization of the vector basis a(0,Pv) and a(0,PH) to obtain the columns of the matrix £^0\. _j ' • Network Step.6.3: Calculation of the robustness to ambiguities njy) of the antenna / network pair from said matrix Ü ( 0 ); - Network Step.7: Deduction of the robustness to ambiguities of the antenna / antenna network pair by calculating ?7, = min; * network f < f < f * ' • J min J ■' max - Network Step 8: Selection of the antenna / antenna network pair(s) maximizing the network criterion-

[0091] Advantageously, the method for designing an array 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 the 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.

[0092] 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 desired omnidirectional performance for the entire array. However, the 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 from the observed response of the antenna array (for example by modifying the surface of the antennas if they are too close, etc.).Alternatively, an additional step 604 of optimizing the antennas 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 relative size. effective 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 imposed.

[0093] 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 size, these antennas offer a large number of degrees of freedom allowing fine optimization of 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 R2 and R3 to increase the low-frequency radiation efficiency without introducing overlapping of the antennas in the network, or to modify the radii of curvature Ci and c2, to reinforce the directivity of the diagrams to attenuate the lamination, and this without necessarily calling into question the arrangements and orientations calculated in steps 602 and 603.

[0094] Conversely, 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 of compromise between the compactness of the solution and its performance.

[0095] 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.

[0096] The optimization of the antenna array is based on a parametric model of the response G„(0,P) of an antenna given in equation (3). This response depends on the orientation of the antenna in space, and on the electric field vectors E and magnetic field H which characterize it, as shown in [Fig.2].

[0097] The electromagnetic components {E, H] of the antenna are measured or estimated from an electromagnetic simulation in a certain frame (x',y',z'), as illustrated in Figure 7a, where the petal antenna 701 has a direction d0. We can then deduce, by change of frame techniques, the gain of this antenna when it has another orientation dn in the frame (x,y,z) of the network, as illustrated in [Fig.7b]. This is how it is possible to control the algebraic properties of the matrix U( 0) of equation (5), and to give conditions on the orientations of the antennas of the network so that the network is omnidirectional in polarization.

[0098] According to figures 4a, 4b and 4c, the electric and magnetic fields {E, H] of the transmitting antenna are projected into the wave plane defined by the orthonormal vectors kv(0,A) and kH(0,A): i -cosi 4' isin i Ai î -sin {4 (14) k. ty =: -ai ni 4}sb { A i et 1^01 = 1 i eoM AJ L o

[0099] The components (Pv, PH) of the incident electric field Eo projected into the wave plane are the components of the polarization vector along the components EO and Eq>. The incident magnetic field Ho projected into the wave plane is orthogonal to Eo. The wave vector k(0,A) is orthogonal to the wave plane.

[0100] The polarization Po of an incident wave is defined by the components of the electric field in the wave plane. According to [Fig.4a]: œ;j©.B) = —^xkjw^xiu©) j . , . , £ • ' s - - ■ । ; with B = ii H " ! P i ' ' EL ~ ' 7 y J ' ' (15) where Po is a normalized vector and where (Eo, Ho) are respectively the amplitude of the electric field and the magnetic field. According to Figures 8a and 8b, the gain of a perfect dipole or a perfect loop depends on the orientation d of the radiating element, as well as on the total gain GT(0) of the antenna, i.e.: (a P) = Gt (©)

[0101] Indeed, the gain of a loop depends only on its electric field EDipoie, 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 [Fig.4a] by an electric field / magnetic field couple (E, H), which will allow us to give the following expression for the gain: G ( ©, B ) = Gt (©) x [ E% (©, B ) - H(Q,B ) ) (17)

[0102] Thus in polarization V where Pv=l and PH=0, we have the following gain: ( ® ) = j ® ) x ( kr ( ® )+HJ' k ( ® ) ) = ( ® ) xa / ( ® ) x «a with .. a© pei ' i and em=it (18) where em is the vector of electromagnetic components that we wish to estimate from data from measurements or from an electromagnetic simulation of the antenna. In H polarization, where Pv=0 and PH=1, the gain is: Gy (O) = G. (©)x(Erky {©)- H1 kr L with . FeI “ ' ' [HJ (19)

[0103] The expression for the gain of an antenna then verifies ; ii... . ... Gj0:,ïL j= g* (© fPs with < ] K1 O i - iuy î 0 ! ud GN (20)

[0104] The total gain ^(0) of the antenna, which takes into account the influence of the surface me 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. So as not to make any particular assumptions about the physics of the impact of a metal surface where ® is the Kronecker product, 0 is the bearing in radians and A is the elevation.

[0105] Consequently, an antenna can be modeled by the vector em of the electromagnetic components and the vector w containing the interpolation coefficients of the total gain, i.e.: .. ,y _ ... . ,, fi 1 G[0.P-0' IL i ' ' v LHJ (22)

[0106] In the method for designing an antenna array according to one embodiment of the invention, the torque (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 0. We can then deduce the gain of this same antenna for an orientation dn different from that of the initial simulation as illustrated in figures 7a and 7b.

[0107] Whether by measurements or by electromagnetic simulation, it is possible to recover the gains Gv(0i) and 0 / / (0) for a set of incidences {0} covering the entire angular space. To estimate the vector em, we then construct the following criterion J, based on the fact that according to equation (20), the vector g(0;) is collinear with the vector Kr(0;) em, i.e.: g* (OJK' (0,. )em J { em ! = ? 1—_————-—L = eur Qem *7* IT with |K(0. |g|0. jg" (0 )Kr(©. î|; (23)

[0108] The vector em must maximize the criterion J(em). Consequently, the vector em is proportional to the eigenvector associated with the largest eigenvalue Xmax(Q) of the matrix Q. We then obtain the normalized vector em such that: em =• arg m ax J ( hb} with em = 1 «If (24)

[0109] According to the models of equations (20) and (21), the vector g(0) is written: g 0 — Ti axechi© -'K" 0 hü 1 (25)

[0110] The interpolation vector w is estimated in the least squares sense by minimizing the following criterion: w =: sis iras gi® .1- M € h |xa»' (26) — II* II

[0111] The explicit solution of w is known to those skilled in the art.

[0112] From the complex gains Gv(0) and G / / (0;) for a set of incidence {0} of the antenna on a metallic surface, the estimation of the electromagnetic components em and the interpolation coefficients w of the total gain can be obtained by implementing the process of step A which follows: - Step Al: Construction of the vectors g(0) according to equation (20) for all incidences 0 belonging to the set {0} of measurements present; - Step A.2: Construction of the K(0) matrix according to equations (14), (18), (19), and (20) for all incidences 0 belonging to the set {0} of 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 principal eigenvalue of Q; - Step A.5: Search for the interpolation vector w according to equation (26).

[0113] [Fig.9] represents a heterogeneous network comprising 5 antennas 901 902 arranged on a plane 903.

[0114] The network is made up of antennas with positions pn= [xn yn zn]T according to Figure 9, and orientations dn according to Figures 7a and 7b. The objective is to deduce its gain Gn(0= {0, A}) in the network frame, knowing that the coefficients (w, em) have been estimated in the simulation frame such that Gn(0= {0, A}) = G(0'={O', A'}). There is therefore a change of basis to be made to deduce the incidence 0 from the incidence 0'. There therefore exists a rotation matrix Tn such that: k:0)=r„xL(0' s (27) L (t 3'j et dmc { IL = LH ■' .li xy ï 04 K =rA where (En, Hn) are the electric field and magnetic field vectors of the antenna of direction dn in the network frame. We note d0 the direction of the antenna in the simulation frame where, according to figures 7a and 7b, d0 =[0 0 1]'. The vectors^j Q ), , Q'i and Q' j are respectively the wave vector and the wave plane vectors in the simulation frame. The following aims to determine Tn knowing that the antennas are installed on a surface of equation zn = f (xn, y^.

[0115] According to figures 7a and 7b, the characterization or simulation of the antenna is carried out in the orthonormal frame (qn= -r|nAdn, -r|n, dn), of origin O = (0, 0, 0), where r|n is the normal to the plane of the antenna corresponding to the direction of maximum radiation. In the network, the antenna is located in the orthonormal frame of axes (x,y,z) having as origin its position pn.

[0116] Here we construct a network where the normal vector r|n of the antennas is also the normal of the surface of equation zn = / y Consequently this vector has the following expression:

[0117] In the case of a spherical cap with equation _ y 2 _ y 2Y we have ar|n = pn. On the other hand, the direction dn of the antenna will be chosen so that the projection of the direction vector dn in the horizontal plane verifies d(q>n) = [cos(q>n) sin(q>n)]T. The vector dn is then written as follows:

[0118] The last vector qn of the trihedron of the orthonormal reference frame of the electromagnetic simulation is therefore the following vector product according to figures 7a and 7b: = -q,• d (30) The rotation matrix of equation (27) then has the following expression:

[0119] 11 iMO? 1 43 (31) The Step B process for constructing the rotation matrix p of a

[0120] orientation antenna <pn dans le plan horizontal et de position p =[xn yn zn]T sur une surface of equation zn = / is then the following: Step Bl: Calculation of the vector r|n normal to the equation surface Zn = / y ) at the position point p =[xn yn zn]T according to equation (28); Step B.2: Calculation of the orientation vector dn from the angle <pn selon l’équation (29) ; Step B.3: Calculation of the vector qn by the vector product q = ; Step B.4: Construction of the rotation matrix r by performing — q, d,j ■

[0121] For a direction 0, it is then possible to calculate the wave vector ; q \ in the simulation frame, and according to the equation (4), to deduce in the following manner the incidence 0'= {0',A'} in the simulation frame:

[0122] It is then possible to calculate the total gain of the antenna of direction dn and position pn according to equation (21) by carrying out the following calculation: Gr: ' ( 01 ~ wJ x é ( @ | 1, à ' | ) with r exp(- / u) 1 , . ; exp( — / (Zl)c)î €(0)=6^(0)^6^(^) etejÿ) = * " exp(Mi) J (33)

[0123] According to equations (18), (19) and (20), the expression of the matrix K(0) is as follows: 1 M®) k.7(Ç (34) (6 ) 5)j

[0124] According to equation (27), we then know that: (35) where ® is the Kronecker product and I2 the identity matrix of dimension 2. According to equation (22), the gain of the antenna of direction dn and position pn is then written: G.. ( 0. P-, ] = G.7 ( ®} k( g* K ] ôH with g,, = [ FGF, / ] em (36)

[0125] According to equation (3), the response of the antenna of direction dn and position pn is then written as follows: ï U(0, P;J = g / ©. P. ] ( ©, Pa. ) = ^ ( O ) x U( 0; P-.) with ( u ( ©, P< j = K ( 0 ) R iAjO) = G7 (0)xexp^ | (37)

[0126] The Step C process of constructing the response of an antenna to a direction 0 and a polarization P is as follows, knowing that the antenna is modeled by the vector em, the interpolation vector w , its position pn and the calculated rotation matrix r for example by following the steps of step B described above from an orientation <pn de l’antenne dans le plan horizontal : - Step Cl: calculation of the wave vector in the reference frame of the simulation in ef- performing w'f=r / kie); - Step C.2: calculation of the incidence 0' = {0',A'} in the simulation frame by performing and ^=^y4x!^^; - Step C.3: calculation of the total gain of the antenna by performing £ " 10 = ।, knowing that the function is defined at equation (33); - Step C.4: calculation of the matrix K(0) according to equations (14) and (34) then calculation of lt ) = K( q jF; ; - Step C.5: calculation of the vector gn by performing T rr ; gs XX II «ai Step C.6: calculation of fqj») _ g / u (Q^p); - Step C.7: calculation of . f 2,- T \ ; . < 0 ï = c ' Us t exp ' —k 0 fp ( y v'v - Step C.8: Calculation of the antenna response by performing à5 (®5 P ) = cy 0] xl(j0,Pj •

[0127] The remainder of the description describes an embodiment making it possible to determine the positions and orientations of the antennas, in order to best approach the desired conditions of omnidirectionality in the direction of arrival and polarization.

[0128] The direction vector is then written as follows: a (¢, P) = 0I OHG zq (0S P) (38) Or : (Ê » û Ô | g / i (39) ¢)(0)= 0 | 0 0 U ; ït G = _g / d

[0129] The algebraic structure of G conditions the conditions of polarization diversity. We can thus establish conditions on the angles (q>i,... <pN) d’orientation des antennes dans le plan horizontal, en établissant une condition sur le n-uplet (q> i,...q>N) for the network to be omnidirectional in polarization. For this, the columns of G must form an orthonormal basis. To simplify, we consider the case of a planar network verifying: ; $ ! CC-SÎ ¢7 5 J (40) d =d 1 1 and q., = ; Ol a • -J ; vec d { ¢7 ! = | Sïll ï . | ] L 0 1

[0130] We deduce from equations (30) and (31) that ; —sin i J, Ü cos ii JJ = F[Ç7j with T. zoz w 0 smijj i 0 -î 0 i (41)

[0131] According to equations (22), (27) and (36), we know that _ i £ ; LM J IzE JEJ 7 H s J = 1 1 =; " 1 with E = i £. i and H =3 M. 1 r . îH HJ i -J -n L ' - ■ s E i î V | (42)

[0132] Consequently, the matrix G of equation (39) is: G = [G,. xE G, xAJ] -j G, x E, G x AJ ]z [G: xE? G: xM] c($) COjçJN i ul with (G, =[0 0 1] wherec(9)= : ] $(^)= | 1J d |G. =|c(0) s($i 0] )J Jj (43)

[0133] The condition for the network to be polarization diverse is that the matrix [c(<|) ) s( <j>) 1] is of full rank greater than 6. We then see that this requires that there is at least one antenna whose orientation is different from that of the other antennas, so that the vectors c( <j>) or s( <j>) are not collinear with the unit vector 1. On the other hand, we see that to approach a condition of omnidirectionality at the level of polarization, we must find a set of phases (q>i,...q>N) such that the vectors c( <j>) and s( <j>) are orthogonal. Below we propose a method to obtain such a condition, based on the following property, by setting b ( (p ) = c ( <p ) + js( <p) ; the vectors c( <j>) and s( <j>) are orthogonal if and only b ( <p ) et |} ( j sont orthogonaux. (44) On construit alors le vecteur structuré suivant bM] = b{ <p= m (45)

[0135]

[0136]

[0137]

[0138] After a random draw of the n-tuple {x2,... , xN} such that xmin < x; < xmax and Xi = 1, we seek the value amin minimizing the following criterion ( a ): =aigmax (a) with (a) = bJ' (46) The phase n-tuple (q>i,...q>N) is such that q>;= amin Xj. The Step D process of calculating the directions (q>i,...q>N) of the antennas in the horizontal plane can then be as follows: - Step Dl: Drawing of the n-tuples {x2,... , xN] such that xmin< x;< xmax and Xi = 1; - Step D.2: Construction of the vector b ( a ) = b (' <p = { = X}a ■ ■ ■ (pN = xNa} ) pour 0 < a < 2ir sachant that ) •'< esp(.JWi f ' - Step D.3: Search for the angle amin minimizing the criterion Cs(a)=b' for 0< a<2ir; - Step D.4: Calculation of the n-tuples of the direction phases of the antennas in the horizontal plane by performing { qf = • ■ • <PN = •^Jvctmin} ; - Step D.5. Deduction of the orientations dn from the n-tuples <pn selon l’équation (29), à partir de l’équation = / :■,■ v j de la surface métallique. Regarding the determination of a position set pn allowing to obtain omnidirectionality in the direction of arrival, this property is verified when the matrix ^*5^ of equation (13) is diagonal, and the matrix MS^ is proportional to the identity. This condition is true when the matrix g,} of equation (12) is diagonal. According to the article "High Resolution direction finding: from performance toward antenna array optimization - The mono-source case", p^(s)=É^(g (u| = -^7— j " ' z \ g"g G"g ; g = —and g' = Gu HP;;-p(g))(PK-p(g)y | § 8. with jp(g) = £ps»'5(g) -gg" j ' , sjg(?0 l ' g'g (48) with g(n) the n-th component of g. The matrices and j (©y are the Jacobians respective of k(0) and u(0,P) with: - A) XX ■ ; i / .. L-XAf-m (49) 7(®H CÔS(^)€ÔS| A) -siffrisia(A) î and -GO cos(A) i 1 1 1 and where the matrices j । and j { q) are the respective Jacobians of kH(0) and kv(0) with : sia 04) sia (a) — c Os(t?}cOS:(A} § | cos(^) 0 ; (50) ( © ) = -£os(£)sm( A) — s iïi(4)eos(A > s et 0 | 0 -sia (A) | ! 0 0 J

[0139] 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 (q>i,...q>N), with dn = d(<|>). In this case the matrix G has the structure of equation (43). Consequently, we can say according to equation (47) that cüsfçyi j 1 sia(^) ) (51) ,gs=Gxu(ey,P1 =| c(ç) s (ç) |xâ with c(«p) = CO C: Si q» ) s(wH 1 sic (W i where à is a vector of dimension 2 depending on the incidence 0m, the polarization P and the electromagnetic components of the antenna. In this particular context, we can then say that:

[0140] According to equation (47), a necessary condition for approaching omnidirectionality in the arrival direction is that the matrix ys® is propor tional to the identity, exactly as in the case of geometric networks where the condition is also sufficient. The matrixn । ki 0., ) i, Which is a function of the incidence 0m, is then for a large angular sector proportional to the identity.

[0141] 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 {pn0}, then transforming it in the following manner to obtain a set of positions {pn'} associated with a network that is almost omnidirectional in bearing and elevation: i - 1 A ■■ _ v , ,, (FC -p)(K -?) p ! = -p) with and (53)

[0142] The set of positions {pn*} is then modified by a homothetic factor so that the network respects a size D given by the specifications. This can be done in the following way: f D 1 < h - <|| p =i--Cp,: with D. ^max pZ-p? " 1D 1 " v H ' J 11 (54) where ||pi _ p 11| is the distance between the ith and the jth antenna. Knowing that the coordinates p = [xn yn zn]T are constrained to a surface of equation Zn = / (x;î, the calculation of the positions is initially done in the horizontal plane with pn°= [xn yn 0]T.

[0143] The process Step E of calculating the positions of the antennas of an antenna network under a constraint of omnidirectionality in the direction of arrival and congestion D can be the following: Step El: random drawing of antenna appositions in the horizontal plane with {Pn°=[xn yn]T for l <n<n] ; Step E.2: Calculation of the equivalent aperture matrix of the network with Step E.3: decomposition into eigenelements of jj, with D = EAE^ ' °where E is the matrix of eigenvectors and A is the matrix diagonal of eigenvalues; Step E.4: Calculation of the matrix W for whitening with W=EA1 / 2. Step E.5: Calculation of a set of antenna positions giving omnidirectionality by performing i _ opj for 1 < n <n ; Step E.6: Calculation of the network congestion {pn 1} by performing =max^ ' - Step E.7: resizing the network by performing py « F = ( / Art )x: P « ' - Step E.8: calculation of the pn positions conforming to the surface by carrying out the following operations for l <n<n :      r — p« =| y» vJ i

[0144] The method for designing an antenna array according to the invention performs a 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 parameters of the array such as the position and orientation of each of the elementary antennas. It aims to determine the best set of parameters giving an omnidirectional array in the direction of arrival and in polarization, and having good robustness to ambiguities. All this is done from a specification characterized by a frequency band (fmin .. .fmax), a gain constraint and characteristics of the secondary lobes (lamination) in this band and a footprint available on the platform to integrate the elementary antennas.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 volume of the payload).

[0145] The network antenna design method according to the invention is perfectly suited for a network conforming to a 3D metal surface. For this, it comprises 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 is here the complex gain of an antenna when it is adapted to its polarization. This diagram is deformed in the presence of a surface. This modeling is done by interpolating the complex response of the total gain of the elementary antenna measured or simulated in the entire 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 network frame of reference and that in the frame of reference of the reoriented antenna. The same method is used for the modeling of the gain in polarization, characterized by the electromagnetic components of the antenna array.

[0146] The proposed solution makes it possible to work with 3D conformal networks having the advantage of having gain at the horizon (A=0°), and better precision in elevation at the horizon. This has the advantage of improving the performance of instantaneous geolocation techniques for distant sources from 2D goniometry in bearing and elevation. The method described makes it possible to design heterogeneous antenna networks 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. < / n> < / j> < / j> < / j> < / j> < / j> < / j> < / j>

Claims

Claims

1. Method for designing an array of N antennas arranged on a metal surface intended to isolate the antenna array from its support, the antenna array being substantially omnidirectional in the direction of arrival and in polarization in a frequency band having the maximum frequency fmax, with N greater than 1, the design method 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 maximum lamination constraint at the frequency fmax, and for each antenna configuration, determining a frequency band [fmin, fmax] meeting a gain constraint, - a second step (602) of calculating, for each of the K antenna configurations, at least one antenna array configuration,the orientations of the N antennas of the antenna array being chosen so as to promote the omnidirectionality of the antenna array in polarization, the arrangement of the N antennas of the antenna array being chosen so as to promote the omnidirectionality of the antenna array in the direction of arrival, - a third step (603) of selecting the best antenna configuration / antenna array configuration pair(s).,

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

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

4. A 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 a configuration parameter of the antennas among: a width (W), a shape of the strands (Rb R2, R3) and a radius of curvature of the strands (cb c2).

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

6. Method for designing an antenna array according to one of the preceding claims, in which the first step and the second step are implemented from an electromagnetic simulation of a unit antenna arranged on said metal surface.

7. A method of designing an antenna array according to claim 6, wherein said metal surface is a portion of a metal sphere.

8. Method for designing an antenna array according to one of the preceding claims, in which the first step (601) comprises the determination of K' antenna configurations having different geometric characteristics, with K' greater than K, adapted to satisfy a maximum lamination constraint at the frequency fmax, then for each antenna configuration, the determination of a frequency band [fmin, fmax] meeting a gain constraint, then the selection of K antenna configurations from among said K' antenna configurations, considering the length of each antenna and the associated minimum frequency fmin.

9. Method for designing an antenna array according to one of the preceding claims, wherein the second step (602) comprises: - obtaining complex gains Gy(0,f) and G / / (0,f) of responses of the unit antenna to the polarizations Eo and Ev along directions of arrival 0 = {0, A} for a plurality of frequencies included in the frequency band [fmin, fmax]; - for a plurality of frequencies f included between fmin and fmax, calculating modeling parameters of the antenna gain, by estimating electromagnetic components em( / ) and interpolation coefficients w( / ) of the total gain of the antenna from the complex gains Gy(0,f) and G / / (0,f); then for a given number of iterations: the determination of directions dn of the antennas favoring the omnidirectionality of the antenna network in polarization; the determination of positions (pb..., pN) of the antennas favoring the omnidirectionality of the antenna network in the direction of arrival; rejection of the antenna array when its dimension is greater than a maximum dimension; and wherein the third step (603) comprises, for each antenna configuration / antenna array configuration pair: - for a plurality of frequencies included in the frequency band [fmin, fmax], the calculation of robustness to ambiguities of the antenna configuration / antenna array configuration pair by carrying out: - from the parameters { w( / ), em( / )} of the antenna, the wavelength X = c / f, the orientations of the antennas { dn} and their positions { pn}, the calculation of the responses an(@, Pv) and an (0, PH) of the N antennas for the polarization Pv = [1 0]T and PH = [0 1]T; • for each direction 0, the orthonormalization of the basis of vectors a(0, Pv) and a(0, PH) to obtain the columns of the matrix = ' • the calculation of robustness to ambiguities (j of the antenna configuration / antenna array configuration pair from said matrix Ü ( 0 ); • the calculation of the robustness to ambiguities of the antenna configuration / antenna array configuration pair n . = min n.if); 'network f . < f < f 1 i \ / 1mm - 1 1 max the best antenna configuration / antenna array configuration pair(s) being the one(s) with the highest robustness to array ambiguities.

10. Method for designing an antenna array according to one of the preceding claims, in which the choice of the orientation of the N antennas of the second step (602) comprises: the random drawing of Nl values ​​x2 to xN, with Xi=l, the construction of a vector --- ^=xv <M ,avec b (9) = c (9 ) +js(9), c(9) = cos (9) And cos (9n) s(9) = sin (9^ , sin CM. the calculation of an angle amin minimizing an orthogonality criterion the calculation of steering phases such as the calculation of orientations dn of the antennas from said direction phases.

11. Method for designing an antenna array according to one of the preceding claims, in which the choice of the position of the N antennas of the second step (602) comprises the steps of: - random drawing of N antenna positions n° = [xy 1T in rn L n J ni a horizontal plane, - calculation of an equivalent aperture matrix of the pp network of antennas, with s 5 - v , and = LjP* -p.Hp» -p.) - decomposition of the matrix into proper elements, with pp D sso = EAES ' °where E is a matrix of eigenvectors of ggéo and A a diagonal matrix of the eigenvalues ​​of pp pp - calculation of a whitening matrix W, with W=EA1 / 2, - calculation of a set of antenna positions pi _ -ie - p ) ' - calculation of network congestion n1, rn - resizing of network n1 by homothety for respect a space constraint, and calculation of the associated n2 positions. 1 n

12. Method for designing an antenna array according to one of claims 10 and 11, 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 (602) further comprises a step of projecting the positions and / or orientations onto the metal surface.

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