WO2011148248A2

WO2011148248A2 - Method for determining an estimate of a radiation pattern of a phased array antenna

Info

Publication number: WO2011148248A2
Application number: PCT/IB2011/001114
Authority: WO
Inventors: Francesco Di Maggio; Saverio Alessandro
Original assignee: Selex Communications S.P.A.
Priority date: 2010-05-24
Filing date: 2011-05-24
Publication date: 2011-12-01
Also published as: WO2011148248A3; IT1400171B1; ITTO20100431A1; WO2011148248A8

Abstract

A method for determining an estimate of a radiation pattern of a phased array antenna formed by a number M of radiating elements (8,10), including the steps of: determining a theoretical function (P_T) establishing a first M-tuple of phase shifts (α₁₁,... α_M1) related to the M radiating elements; determining a plurality of theoretic electromagnetic field values on the basis of the theoretical function; controlling the radiating elements so as to introduce phase shifts equal to the phase shifts of the first M-tuple, and subsequently determining a plurality of electromagnetic field measures; determining an error vector (Err₁), as a function of the plurality of theoretical values and of the plurality of measures. The method further includes the steps of determining an approximating function, on the basis of the error vector and of the first M-tuple of phase shifts, and determining, for a generic M-tuple of phase shifts to be characterized (α_1x, α_Mx), an estimate of an electromagnetic field irradiated by the antenna, on the basis of the approximating function and of the theoretical function.

Description

METHOD FOR DETERMINING AN ESTIMATE OF A RADIATION PATTERN OF A PHASED ARRAY ANTENNA

TECHNICAL FIELD

The present invention relates to a method for determining an estimate of a radiation pattern of a phased array antenna.

As shown by way of example in figure 1, a phased array antenna 1, also known as phased antenna network or phase controlled antenna network, comprises a supply terminal 2 connected to a supply network 4, which has an input, electrically^ coinciding with the supply terminal 2, and a number M of outputs 6, with M>2 and equal, in the described example, to eight; furthermore, by supplying to the supply terminal 2 a generic input signal, the supply network 4 sends energizing signals to each output 6, typically equal to one another and having the same pattern as the input signal over time.

The phased array antenna 1 further comprises M phase shifters 8, each connected to a respective output 6 of the supply network 4, and adapted to generate, starting from the respective energizing signal, a corresponding elementary signal, which is shifted, with respect to the corresponding energizing signal, by a phase difference equal to _m, with l≤m<M and 0°≤a_m≤360°. In practice, each phase shifter 8 introduces a respective shift oi_m; furthermore, the phase shifters 8 are electronically controllable, typically by supplying respective control voltages by means of control terminals (not shown) of the phase shifters 8themselves. In other words, each phase shifter 8 introduces a respective shift a_m, which is a function of the control voltage supplied to the phase shifter itself. Furthermore, typically but not necessarily, the elementary signals have a same width.

The phased array antenna 1 further comprises M radiating elements 10, each connected to a respective phase shifter 8. The radiating elements 10 radiate the elementary signals received from the respective phase shifters 8, each generating a respective elementary electromagnetic field. In each instant of time, the sum of the elementary electromagnetic fields generated by the radiating elements 10 defines an overall electromagnetic field, radiating by the phased array antenna 1 as a whole. Furthermore, the radiating elements 10 can be typically arranged aligned (as shown in figure 1) or have a planar arrangement.

As for any antenna, the performance of the phased array antenna 1 may be characterized by means of the so- called radiation pattern.

BACKGROUND ART

As known, considering a generic antenna, the corresponding radiation pattern refers to a specific frequency, and shows the gain trend (or directivity) of the considered antenna, as a function of a depointing angle Θ, and possibly of an azimuth angle φ. In practice, assuming a spherical reference system centered in the geometric^.center of the considered antenna, and thus assuming an imaginary spherical surface with center coinciding with the geometric center of the considered antenna, each point of such imaginary surface may be identified by means of a respective pair of values of the depointing Θ and azimuth angles φ. The depointing angle is functionally equivalent to the so-called elevation angle, and is generally calculated starting from the so-called "broadside" direction.

Hereinafter, without loosing in generality, reference will be made to radiation patterns depending on the depointing angle Θ only, because the determination of radiation patterns typically contemplates the execution of a given number of operations (described below) , after having assigned a given value cpo to the azimuth angle cp, and possible interactions of such operations for further values of the azimuth angle φ. In such conditions, the depointing angle Θ is sufficient to identify the points belonging to the aforesaid imaginary surface and such that φ=φο, and thus to obtain the radiation pattern related to φ=φο·

In greater detail, once selected φ=φο_< the radiation pattern of the considered antenna can be obtained by measuring the intensity of the electromagnetic field irradiated by the antenna itself in points belonging to the aforesaid imaginary surface and such that φ=φ₀. In practice, each of such points, is identified by a corresponding value of the depointing angle Θ, and thus each intensity measure is associated to the corresponding value of the depointing angle Θ. In other words, the radiation pattern shows the intensity trend (possibly normalized) of the electromagnetic field irradiated by the antenna;. as a function of the depointing angle Θ.

In particular, in the case of phased array antennas, the broadside direction (indicated by B in figure 1) is the direction that passes through the geometric center of the antenna (indicated by Q in figure 1), meaning the geometric center of the array of radiating elements, and is perpendicular with respect either to the direction along which the radiating elements are aligned, in the case of phased array antennas with aligned radiating elements, or to the plane on which the radiating elements lay, in the case of phased array antennas with planar arrangement.

Consequently, with reference by way of example to the phased array antenna 1, in which the radiating elements 10 are aligned along an axis L, a generic point O, chosen among the aforesaid points of the imaginary surface such that φ=φο, is univocally identified by a corresponding value of the depointing angle Θ. Such a depointing angle Θ, as previously mentioned, is indeed the angle formed by the broadside direction B and by a line passing through the geometric center Q of the phased array antenna 1 and through the generic point 0.

This said and assuming that the input signal, and thus the energizing signals and the elementary signals, are monochromatic at a given operating frequency f_0/ the radiation pattern of the phased array antenna 1 depends, in addition to the operating frequency ^'-^■ f_D, on the arrangement in space of the radiating elements 10, and on the M-tuple of phase shifts (αι,...,α_Μ) introduced by the M phase shifters 8.

In greater detail, given an operating frequency value f₀ and given the arrangement in space of the radiating elements 10, the radiation pattern of the phased array antenna 1 can be modified by varying the M- tuple of phase shifts ( ι,...,α_Μ) introduced by the M phase shifters 8. In practice, the radiation pattern can be shaped by modifying the control voltages supplied to the M phase shifters 8, and thus the M-tuple of phase shifts (αι,...,α_Μ) . In other words, the operating frequency f₀ and the arrangement of radiating elements being equal, a respective radiation pattern corresponds to each considered M-tuple of phase shift (oti, ..., a_M) ; such a radiation pattern has a respective maximum angle Q_max , which identifies the maximum radiation direction of the phased array antenna 1, when the phase shifters 8 set the considered M-tuple of phase shifts.

As known, the need to design phased array antennas in which the maximum angle 9_max assumes a default value is known. For this purpose, and again with reference by way of example to the phased array antenna 1 shown in figure 1, numerical models are used today which allow to simulate the behavior of the radiating elements 10, so as to allow to determine the elementary electromagnetic fields irradiated thereby by first approximation and thus also the overall electromagnetic field.

In practice, by assuming a target angle of maximum Qmax.t i an operating target frequency f₀i and a given arrangement of the radiating elements, by using the aforesaid numerical models, a corresponding theoretical M-tuple of phase shifts (αι,...,α_Μ) can be determined, which in theory should be such that the maximum angle Qrnax coincides with the maximum target angle _max__t . In particular, in the case of phased array antennas with radiating elements aligned and equally spaced by a distance d, it is customary to apply gradual shifts such that the phase difference Δα between shifts associated to adjacent radiating elements is constant and dependent on the maximum target angle 9_max__t according to the equation: Δα =—^sin^, (1) with λ equal to the wavelength corresponding to the target operating frequency f ₀i · Consequently, a m-th phase ^'shift a_m of the theoretical M-tuple of phase shifts is equal to: a_m=Acc(m-\) (2)

For example, if the maximum target angle 9_max_t is zero, i.e. for maximum radiation in broadside direction, the aforesaid phase difference Δα must be zero, i.e. all phase shifters 8 must introduce the same phase shift.

In actual fact, the numerical models are inevitably approximate, and thus by setting the aforesaid theoretical M-tuple phase shifts, the corresponding radiation pattern has a first approximate maximum angle Θ max which can be determined by executing a first plurality of experimental measures of the depointing angle Θ; in general, the first approximately maximum angle θ^' _max does not coincide with the maximum target angle 6_max_t · Consequently, iterative operations are then executed, which contemplate setting a first modified M- tuple of phase shifts, obtained starting from the theoretical M-tuple of phase shifts, and executing a second plurality of experimental measures as the depointing angle Θ varies, on the basis of which determining a second approximate maximum angle e"_max. On the basis of the difference between the first and second approximate maximum angle θ ' _max , 0^" _max, and considering the difference between the fist modified M-tuple of phase shifts and the theoretical M-tuple of phase shifts, the described operations can be iterated until a hypothetical approximate maximum angle e^x _max is determined. Such a hypothetical approximate maximum angle 9^x _max approximates the maximum target angle 6_max__t by less of a predetermined tolerance, and corresponds to a given modified M-tuple of phase shifts. Such given modified M-tuple of phase shifts represents the M-tuple of phase shifts which, given the operating target frequency f₀i and given the arrangement of the radiating elements, allows to better approximate the maximum target angle 9_max_t ·

Further methods related to determining radiation patterns of antennas are described, for example, in US6218985, in EP1835301, in US2009/201206 and in W093/11581.

Again with reference to the operations described above, they allow to design, within given tolerances, phased array antennas with predetermined maximum target angles 9_max__t · However, the described operations require to carry out numerous plurality of measures, e.g. by means of automated measuring systems; furthermore, the previously described iterative operations may not converge, i.e. may not allow to determine the aforesaid hypothetic approximate maximum angle 6^x _max, or better the corresponding M-tuple of phase shifts. The latter case may occur, for example, if the theoretical' models of the radiating elements are not sufficiently accurate, and implies the impossibility of determining a M-tuple of phase shifts such that, when set by the phase shifters, the radiation pattern has a maximum angle 9_max which differs from the maximum target angle 9_max__t by less than the predetermined tolerance: ^'

Similar drawbacks may occur if instead of designing the phased array antenna as a function of the maximum target angle 9_ma _t_i the phased array antennas is designed so that the respective radiation pattern has predetermined zeros and/or radiation lobes.

DISCLOSURE OF INVENTION

It is the object of the present invention to provide a method for determining an estimate of a radiation pattern of a phased array antenna which solves at least in part the drawbacks of the prior art.

According to the present invention a method for determining an estimate of a radiation pattern, a computer program product and an antenna as disclosed in claims 1, 9 and 10, respectively, are provided.

BRIEF DESCRIPTION OF THE DRAWINGS

For a better understanding of the invention, embodiments thereof will be described hereafter only by way of non-limitative example, and with reference to the accompanying drawings, in which figure 1 diagrammatically shows a phased array antenna.

BEST MODE FOR CARRYING OUT THE INVENTION.

The present method contemplates determining an estimate of a radiation pattern of a phased array antenna starting from a limited number of experimental measures of the overall electromagnetic fields irradiated by the phased array antenna itself, i.e. starting from a low number of experimental .measures of the radiation pattern itself.

Hereinafter, the method will be described with reference, by way of example and without loosing in generality, to a phased array antenna 1 shown in figure 1, in which radiating elements 10 are arranged aligned along axis L and are spaced apart by a same distance d. Furthermore, hereinafter, the determination of the radiation pattern related to operating frequency f₀ is assumed without loosing in generality.

In detail, it is possible to define a first analytic function P_T, which expresses in analytic, approximate form the radiation pattern of the phased array antenna 1, thus it is indicative of the overall electromagnetic field. In detail, using an index m such that l<m<M (with M equal to the number of radiating elements 10) , the first theoretical function P_T is given by the equation: j (( w-1)^-d cosΘ cos<p+ _m )

P_j = cos# cos >^ a_me^" (3) wherein A=l/f₀, and wherein the factors a_m ^' are energizing coefficients of the radiating elements 10, i.e. indicate the widths of the respective irradiated element electromagnetic fields.

In particular, the equation (3) contemplates the possibility that the phase difference Δα between phase shifts associated to adjacent radiating elements is not constant, and thus the phase shifts a_m introduced by the phase shifters 8 have any value.

Hereinafter, for the sake of simplicity, the present invention will be described with particular reference to determining an estimate of the radiation pattern on plane p=0°, without loosing in generality; furthermore, for reasons of simplicity, it is assumed that the aforesaid energizing coefficients a_m are unitary. In this case, the first analytic function P_T assumes a simplified form, given by the equation:

m=\

Subsequently, a plurality of values of the depointing angle Θ are determined, such values being, for example, comprised between -45° and 45° and equally spaced apart, with spacing equal, for example, to 0.9°; in practice, according to such an example, a number C of values of the depointing angle Θ equal to one hundred are obtained, to which will be referred to hereinafter as the samples of the depointing angle Θ. The one hundred samples of the depointing angle Θ may be stored in a sample vector.

Afterwards, it will be possible to establish, even arbitrarily, a first M-tuple of phase shifts (otii, ..., α_Μι ) . Furthermore, it is possible to determine a first plurality of theoretical data, on the basis of equation (4) , calculated for the samples of the depointing angle Θ contained in the sample vector, and setting A=l/f₀, as well as the first M-tuple of phase shifts, i.e. setting ( ι,...,α_Μ) = (αιι,...,α_Μι) . In this manner, one hundred theoretical data are obtained, each of which related to a respective sample of the depointing angle Θ. The one hundred theoretical data may be stored in a first respective theoretical vector Ρ_τχ .

Then, in manner known in itself, a first plurality of measures related to an overall electromagnetic field actually irradiated by the phased array antenna 1 can be experimentally determined, for example by inserting the phased array antenna 1 in an anechoic chamber. In particular, each measure corresponds to a receptive sample of the depointing angle Θ contained in the sample vector. Furthermore, the first plurality of measures may be stored in a corresponding first experimental vector Psi, also formed, in the described example, by one hundred elements. Furthermore, it is noted that the first plurality of measures is obtained at the same operating freque cy f_D and by setting in the phase shifters 8 the same first M-tuple of phase shifts (an, CXMI) used in the step of determining the first plurality of theoretical data.

Subsequently, it is possible to determine a first error vector Erri, obtained by subtracting the first theoretical vector _rP_T1-..from the first experimental vector Psi-

In practice, from an analytic point of view, the first error vector Erri corresponds to the difference between the radiation pattern actually irradiated (and measured) by the phased array antenna 1 when the phase shifter 8 introduce the first M-tuple of phase shifts (a , a_M1) , and the analytic radiation pattern provided by the first analytic function P_T when the first M-tuple of phase shifts (an , a_Mi ) is set. In other words, the first error vector Err_x corresponds to a hypothetical analytic error function Err given by the equation:

where P_M is a second analytic function which ideally defines the overall electromagnetic field actually irradiated by the phased array antenna 1. By way of example only, hereinafter, for the sake of simplicity, further operations will be described below with reference to the particular case in which the number M of radiating elements .10 is equal to two.. Therefore, in this case:

2π

. j——dcosO+a-,

P_T = cos(k^{J 1} +∞s9e ^{λ "} (6) and

j— a cos #+or,

Err = P_M - P_T = P_M - cos fe^ja> - cos fe ^{λ '} _{( 7} )

In practice, from an analytical point view:

Err =f(a_lr a₂) ₍₈₎

Availing of the first error vector Erri, a numeric regression method can be applied to the elements of such vector, and thus to differences between the first plurality of measures and the first plurality of theoretical data. Furthermore, such differences between the first plurality of measures and the first plurality of theoretical data may be approximated by means of an approximating polynomial of g degree, with g equal to a generic value. Hereinafter, by way of example only, it is assumed that g is equal to three.

In practice, from an analytical point of view, the approximating polynomial approximates the aforesaid analytic error function Err, so that in the assumption above (M=2,g=3), it can be assumed:

Err≡ ( , ,Ο,³ + A_2Xa + A₃a + A_4] )+

+ A₂₂a₂ ² + A₃₂a₂ + A₄₂ ) ( 9 ) The eight coefficients An, A₂i, A₃i, A₄i, Ai₂, A₂₂,

A₃₂ and A₄₂ are unknown. In order to determine such unknown coefficients, it is possible to operate as follows :

a) establishing, even arbitrarily, a second M-tuple of phase shifts, which, given the assumptions ^■ above, is reduced to only two phase shifts αι₂ and ₂₂;

b) determining a first plurality of theoretical data, on the basis of equation (4) , which is calculated for the samples of the depointing angle Θ contained in the sample vector, and setting M=2, A=l/f₀, and the second M-tuple of phase shifts, i.e. - imposing

(Oil , Of₂ ) = ( 0(12, 0(22) 1

c) experimentally determining, for each sample of the depointing angle Θ contained in the sample vector, a corresponding experimental measure, so as to obtain a second plurality of measures, related to the operating frequency f₀ and to the same second M-tuple of phase shifts ( oii2, o(22 ) used in the step of determining the second plurality of theoretical data; and

d) determining a second error vector Err₂.

In greater detail, the second plurality of theoretical data and the second plurality of measures may be stored, respectively, in a second theoretical vector P_T2 and in a second experimental vector P_s2, also formed, in the described example, by one hundred elements. Furthermore, the second error vector Err₂ also, has one hundred elements, and may be obtained by subtracting the second theoretical vector P_T2 from the second experimental vector P_s2.

It is thus possible to iterate the operations a)- d) , establishing further M-tuples of phase shifts and determining, for each one thereof, a respective error., vector, obtained as difference between a respective experimental vector and a respective theoretical vector. The experimental vectors, the theoretical vectors and the error vector thus obtained relate to the operating frequency f₀ and are formed, in the considered example, by one hundred elements, which each refer to a corresponding sample between the samples of the depointing angle Θ contained in the aforesaid sample vector .

At the end of the described operations, a number N of error vectors Erri are available, each referred to a corresponding M-tuple of phase shifts (an,a2i) (it is worth reminding that M=2 has been assumed, for greater clarity in presenting the analytical examples) and equal to the difference between a corresponding experimental vector P_Si and the corresponding theoretical vector P_Ti .

In order to determine the eight coefficients An, A21, A₃i, A41, A12, A₂₂, A₃₂ and A₄₂, for example, the Ordinary Least Squares method may be applied, on the basis of the N error vectors Erri and of the corresponding N M-tuples of phase shifts («ii, ₂i) .

In detail, it is possible to determine a remainder

R² given by: r² = ^[Err_l-(A_l] +A_2i f_i+A_{3l u}+A_4])-(A_n l_i+A₂₂ +A_{32 2},+A.

=1

(10)

It is worth noting that each of the eight unknown coefficients An, A₂i, A_3i, A_4i, A₁₂, A₂₂, A₃₂, A42 is not a scalar, but rather a vector formed by a number C of elements, in turn unknown. In other words, according to the described example, each of the aforesaid unknown coefficients is formed by one hundred unknown subelements .

Subsequently it is possible to determine the unknown coefficients An, A₂i, A_3i, A41, Ai₂, A22, A₃₂, A₄₂, by minimizing the remainder R². Thus, the unknown coefficients An, A₂i, A₃i, A₄₁, A_i2, A₂₂, A₃₂, A42 are determined by setting the partial derivates of the remainder R² with respect to the coefficients An, A₂i, A₃i, A41, A12, A₂2, A₃2, A42 themselves to zero. In other words, the following equations are set, to which reference will be made hereinafter as set of equations (11)

to²

-2∑[Er_ri-(A al +A ^+A_{ii li} + A_4i)-(A_{l2 2} ² _i+A_{22 2} ² _i+A_{i2 2i} + A₄₂)] dA 1 I

-2∑[Err_t - (A at +A₂a² + A_3la + A_I)- (A + A₂₂a₂ ² _i + A₃₂a_2i + A₄₂ )] ²,

= -2∑[£rr, - (A +A_2loc² + A₃a_h +A_4l)- (A_{i2 2} ³ _i + A_{22 2i} + A_i2a_2l + A₄₂ )] = A,

dR²

^■2∑[£/ -(A_u +A_2ia² + A_{3l u} + A. ) ~ (Α_ι2 + A₂₂a_2i + A₃₂a_2i + A_4Z )] dA 41 = l

+ A₂₂a_2i + A₃₂a_2i + A₄₂ )]a₂ ³ _i

dR

^■2 T [Err, - (A +A_2Xal + A_{3l u} + A_4l)- (A a₂ ² _i + A_{22 2} ² _i + A_na_2i + A₄₂ )] _² _t dA 22 =l dR

= -2∑[Err_l-(A_liai_i +A_2l ² + A_3la_u + A_4i)-(A_{l2 2}, + A₂₂a_2! + A₃₂a_2i + A₄₂)) ₂ dA 32 i=l

dR^ N

= -2∑[£rr, - (A_na +A_2l ~ + A_3la_lt +A_4])- {A a₂ ³ _i + A_{22 2} ² _i + A₃₂a_2i + A₄₂)] = 0 dA 42

(11)

The set of equations (11) thus determine the following system of equations (12) :

4ΐ +

4 + ΐΣ« + ₁Σ«,/ +^12∑«2,«h + 2∑«2²,«Η +^32∑«2,^ + =1 ==l1 =l =l =l +

+

ΐΣ«ΐ .Σ«²,«2, + ,Σ + ,∑α2² + ₂Σ Σ Σ

Ν Ν

=Σ

ΐΣ«ΐ .Σ + ΐΣ + ,Σ + ₂Σ Σ <*1 +

ΝΝ ΝΝ

4 , Σ ' + . Σ Σ + ΐ # + 4₂ Σ < + ₂ Σ Σ

Ν

+ ₂^ = 2 >-

(12 ) In a per se known manner, the equation system (12) may be expressed in matrix form as:

In contracted form, the equation system (12) may be written as: a - A = Err

(13)

In detail, the matrix a has dimensions [2*(g+l) x 2*(g+l)], i.e. is formed by eight by eight elements of scalar type and has the following shape:

The matrix ^ is thus known, because both the N and the phase shifts which form the N M-tuples of phase shifts previously established are known.

Instead, A is a vector formed by 2*(g+l) elements, which coincide, respectively, with the unknown coefficients An , A₂i , A₃₁ , A₄i , A_{i 2} , A₂₂ , A₃₂ , A4₂. Thus, each element of vector A comprises, in turn, a respective subvector formed by C subelements, also unknown .

Similarly to A , Err is also a vector formed by 2*(g+l) elements, each of which comprises in turn a respective subvector formed by C subelements. In detail, Err has the following form:

In other words, in principle both A and Err respectively have dimensions equal to [2*(g+l) x C] , i.e. in the described example eight by hundred. Hereinafter, reference will be made in all cases for the sake of simplicity to vector A (unknown) and to vector Err, indicating them as vectors of eight elements, being implicit that the respective elements are in turn vectors .

In greater detail, the elements of the vector Err are known, because the previously determined error Erri and the respective M-tuples of phase shifts are known, as well as the respective M-tuple to which the error vectors Erri are referred. Thus the vector Err represents the so-called known term of the equation system (12) .

By multiplying both members of the equation (13) per i.e. for the matrix transported by the matrix is obtained: a ^T■ A -a^T -Err (14)

The equation (14) may be solved numerically or may be inverted, allowing to obtain the equation:

In either case, it is thus possible to determine the elements of the vector A, i.e. the unknown coefficients An, A₂i, A₃₁, A 1, A_X2, A₂₂, A₃₂, A₄₂.

Being known the coefficients An, A₂i , A₃i, A₄i, A_i2, A₂₂, A₃₂, A42, and thus, for each, the respective subelements, the corresponding radiation pattern can be estimated, for any generic M-tuple of phase shifts (θ(ΐχ,α_2χ) , without needing to actually characterize the phased array antenna 1 experimentally for such generic M-tuple of phase shifts. It is indeed possible to determine an estimate vector P_F, which corresponds to a hypothetic analytic estimate function Pp , given by the sum of the approximating polynomial with the first analytic function P_T.

In detail, it is worth remembering that the specific example contemplates M=2 , and by indicating with οίΐχ and _2x the phase shifts of the aforesaid generic M-tuple of phase shifts, each element of the estimate vector P_F corresponds .to. a respective, sample θ_χ of the depointing angle Θ contained in the vector sample, and is equal to the sum of:

- a first contribution, obtained by calculating the first analytic function P_T (as equation 6) by θ=θ_χ, for A=l/f_Q, and with αχ and ₂ respectively equal to ai_x and a_2x; and

a second contribution equal to

wherein Au_x , A₂i_x , A_{3 ix} , A_{4 ix} , A_{i 2x} , A_22x , A_32x and A_42x indicate respectively the subelements (scalar) of the coefficients An , A₂i , A_{3 i} , A_{4 i} , A_i2 , A₂₂ , A₃₂ , A4₂ which correspond to the sample θ_χ of the depointing angle Θ, as previously determined.

In practice, the estimate vector P_F thus contains C elements, in the case in point one hundred elements, which provide a discrete estimate of the radiation pattern which characterizes the phased array antenna 1 when the phase shifters 8 are controlled so as to set the aforesaid generic M-tuple of phase shifts ( _ix,a_2x), and using an input signal with frequency equal to the operating frequency f₀. In this manner, it is not necessary to execute any experimental characterization corresponding to the aforesaid generic M-tuple of phase shifts (θίΐχ, a_2x) .

The estimate of the radiation pattern represented by the estimate vector P_F may be optimized by increasing the number, of determined M- tuples (i.e. by increasing the number N) , and thus the corresponding number of error vectors Erri, and the number C of samples of the depointing angle Θ and the degree g of the interpolating polynomial .

Furthermore, the previously described operations with particular reference to the case in which G=100, M=2 and g=3 , may be generalized for each C and M value, as well as any degree g of the approximating polynomial.

In particular, by generalizing the teachings described hereto and notwithstanding the assumptions according to which the energizing coefficients a_m are unitary and φ=0°, the analytic error function Err assumes the following form:

M j((m-l)—dcose+a

Err = P_M - e

(16)

Also in this case, the Ordinary Least Squares method may be applied, on the basis of the difference between theoretic vectors P_Ti and experimental vectors P_Si , i.e. of the error vectors Erri, as well as of the corresponding M-tuples of phase shifts (an , ..., _Μι ) . In practice, also in this case the analytic error function Err can be approximated by an approximating polynomial, according to the equation:

M -l g

Err = ^_{/ +}i)_j(r+i )^(_r+i) ) _{( 1 7 )} r=0 / =0

This said, it is obtained ^'a remainder R² equal to:

It is worth noting that also in this case, the remainder R² is a vector formed by C elements.

Subsequently, the partial derivates of the remainder R² may be calculated with respect to the coefficients of the approximating polynomial, for each value of r from 0 to M-l and for each value of t from 0 a g. By setting such partial derivates to zero, the remainder R² is minimized.

By developing the equation (18), for each value of r from 0 to M-l and for each value of t from 0 to g, the following equation (19) is thus obtained:

(19

In matrix form, it is thus obtained again the equation : a- A- Err (20) wherein the matrix g is known and has dimensions equal to [M(g+l)x M(g+1)], while the vectors A (unknown) and Err (known) have dimensions equal to [M(g+1) x 1] . Furthermore, each element of the vectors A and Err is in turn formed by a respective subvector comprising C subelements .

In practice, it is possible to determine the coefficients of the approximating polynomial for any values of M, N and C, as well as for any degree g of the approximating polynomial itself.

The advantages that the present method allows to obtain are clearly apparent from the discussion above. In particular, the present method allows to determine the radiation pattern of any phased array antenna starting from a low number (N) of predetermined -tuples of phase shifts, at which the phased array antenna is actually characterized, i.e. subjected to experimental measures. Subsequently, for any subsequent M-tuple of phase shifts, the present method provides a corresponding estimate of the radiation pattern, without any additional measure being necessary. In such a manner, the characterization of such type of antennas is considerably simplified because the M-tuple of phase shifts which must be set to obtain a given maximum target angle 9_max_t (o^r a zero, a relative maximum etc.) can be determined on the basis of a restricted, and possibly predefined, number of experimental measures.

Furthermore, although described, for reasons of simplicity, with reference to the case φ=0° and only for the operating frequency f.. the described operations may be iterated, in manner known in itself, for any value of the angle φ, as well as for a different operating frequency.

It is finally apparent that changes and variations can be made to the present method without departing from the scope of protection- of the present invention as defined in the appended claims.

For example, the samples of the depointing angle Θ may be comprised in a different range from -45° to 45°; for example, such values may be comprised between -90° and 90°. Furthermore, the samples of the depointing angle Θ may not be equally spaced apart from one another .

Finally, a different linear regression method or a non-liner regression method can be applied instead of the Ordinary Least Squares method. Furthermore, a non necessarily linear interpolation method may be applied.

Claims

1. A method for determining an estimate of a radiation pattern of a phased array type antenna, formed by a number M of radiating elements (8, 10), each radiating element being configured for irradiating a respective elementary electromagnetic field with a respective phase shift (a_m) with respect to a common reference phase, the method comprising the steps of:

determining a theoretical function (P_T) indicating an overall electromagnetic field irradiated, in use, by said antenna;

- setting a plurality of samples of a first angle

(Θ) ;

- establishing a first M-tuple of phase shifts (αη,...,α_Μι) ;

- determining, for each sample of the first angle (θ) , a corresponding theoretical field value, on the basis of the theoretical function (P_T) , of said sample, of a first frequency (f₀), and of said first M-tuple of phase shifts (a , a_M_ ) , so as to obtain a first plurality of theoretical field values;

- supplying said antenna in such a way that it irradiates said overall electromagnetic field at said first frequency (f₀);

- controlling said radiating elements (8, 10) in such a way that the respective phase shifts (ai,...,a_m) are respectively equal to the phase shifts of said first M-tuple of phase shifts (a , a_Mi ) , and then experimentally determining, for each sample of the first angle (θ) , a corresponding measure indicating said overall electromagnetic field so as to obtain a .first plurality of measures;

- determining a first error vector (Erri) , as a function of said first plurality of theoretical field values and of said first plurality of measures;

- determining, as a function of said first error vector (Erri) and of -said first M-tuple of phase _:-shifts

(an, _Mi) , an approximating function; and

for a generic M-tuple of phase shifts to be characterized (a_ix, ..., ^) , determining an estimate of the overall electromagnetic field irradiated by said antenna when said radiating elements (8, 10) are controlled in such a way that the respective phase shifts (ai,...,a_m) are respectively equal to the phase shifts of said M- tuple of phase shifts to be characterized (a_ix, a^x) , as a function of said approximating function and of said theoretical function (P_T) .

2. The method according to claim 1, further comprising the steps of establishing one or more additional M-tuples of phase shifts (an, a_Mi) and, for each M-tuple considered from among said additional M- tuples of phase shifts:

- determining, for each sample of the first angle (θ) , a corresponding theoretical field value, as a function of the theoretical function (P_T) , of said sample, of said first frequency (f₀), and of said M- tuple considered, so as to obtain a corresponding additional plurality of theoretical field values;

- controlling said radiating elements (8, 10) in such a way that the respective phase shifts (¾,...,¾) are respectively equal to the phase shifts of said M- tuple considered, and then experimentally determining, for each sample of the first angle (θ) , a corresponding measure indicating said overall electromagnetic field so as to obtain a corresponding additional plurality of measures ;

determining a corresponding additional error vector (Erri) , as a function of said corresponding additional plurality of theoretical field values and of said corresponding additional plurality of measures;

and wherein said approximating function is moreover a function of said additional M-tuples of phase shifts and of the corresponding additional error vectors determined.

3. The method according to claim 2 , further comprising the steps of:

- storing said first plurality of theoretical field values in a first theoretical vector (P_Ti) / and each plurality considered from among said additional pluralities of theoretical field values in a corresponding additional theoretical vector (P_Ti) ; - storing said first plurality of measures in a first experimental vector (Psi) / and each plurality considered from among said additional pluralities of measures in a corresponding additional experimental vector (P_Si) ;

and wherein said first error vector (Erri) is equal to the difference between the first experimental vector (Psi) and the first theoretical vector (P_Ti) , and each of said additional error vectors (Erri) is equal to the difference between the corresponding .additional experimental vector and the corresponding additional theoretical vector.

4. The method according to claim 2 or claim 3, wherein said approximating function is a polynomial function of degree g, having as variables a number M of phase shifts, and having coefficients

(Aii,A₂i, A₃i, A₄i, A₁₂, A₂2,A₃₂, A₄₂) that are function of said first error vector (Erri), of said additional error vectors (Erri) , of said first M-tuple of phase shifts ( , ..., _Μι) , and of said additional -tuples of phase shifts (ai_l, a_Mi ) .

5. The method according to claim 4, further comprising the step of determining the coefficients of said approximating function so as to minimize a summation (R²) of quadratic deviations between:

said first error vector (Erri) and said approximating function, calculated by assigning to the variables of said approximating function the phase shifts of said first M-tuple of phase shifts (αη,...,α„ι) ; and

- between each vector, considered from among said additional error vectors (Erri) and said approximating function, calculated by assigning to the variables of said approximating function the phase shifts of the corresponding additional M-tuple of phase shifts (an, a_Mi) .

_

6. The method according to claim 5, wherein said coefficients of said polynomial function each comprise a number C of subelements equal to the number of samples of said first angle (Θ) .

7. The method according to claim 6, wherein said step of determining an estimate of the overall electromagnetic field irradiated by said antenna comprises, for each sample of the first angle (θ) , the steps of:

determining a first field contribution by calculating the theoretical function (P_T) for said first frequency (f₀), for said sample, and for said generic M- tuple to be characterized (oii_x O(M_X) ;

determining a second field contribution by calculating said approximating function using the phase shifts of said generic M-tuple of phase shifts to be characterized (oii_x, anx) and the subelements (Aiix, A₂ix, A_31x, A₄ix, Ai2x, A₂2x, _32x, A_42x) of the coefficients of the approximating function (An, A₂i, A₃i, A₄i,Ai₂, A₂₂, A₃₂, A₄₂) relating to said sample; and

adding together said first and second field contributions .

8. The method according to any one of the preceding claims, wherein said theoretical function (P_T) is given by

j = cos cosq> _^ a_me where: Θ is said first angle; φ is a second angle; λ is a wavelength relating to said first frequency ( f₀) ; a_m are factors indicating amplitudes of said elementary electromagnetic fields; _m are the phase shifts of said radiating elements (8, 10) ; and d is a distance between two of said radiating elements (8, 10) .

9. A computer program product that can be loaded into a memory of a computer and configured for implementing, when run, the determination method according to any one of claims 1 to 8.

10. A phased array antenna comprising a number M of radiating elements (8, 10) , each radiating element being configured for irradiating a respective elementary electromagnetic field with a respective phase shift (oi_m) determined on the basis of the method according to any one of claims 1 to 8.