# Patent

## Title
Method of obtaining an antenna gain

## Abstract

A method for obtaining a gain function by using an array of antennae and a weighting of the signals received or to be transmitted by vectors of complex coefficients. The complex coefficients are referred to as weighing vectors, according to which, a reference gain function being given, the reference gain function is projected orthogonally onto the sub-space of a gain function. The sub-space of the gain function is generated by the weighting vectors of the space of the gain functions. The weighting vectors of the space of the gain functions are provided in advance with a norm, and a weighing vector generating the reference gain function thus projected is chosen as the optimum weighting vector.

## Background
BACKGROUND OF THE INVENTION
(a) Field of the Invention
The present invention concerns in general terms a method of obtaining an antenna gain function. More particularly, the present invention relates to a method of obtaining an antenna gain for a base station in a mobile telecommunication system. It makes it possible to obtain an antenna gain function, in transmission or reception mode, which is invariant by change of frequency.
(b) Description of the Related Art
The formation of channels or the elimination of interfering signals is well known in the field of narrow-band antenna processing. Both of these use an array of antennae, generally linear and uniform (that is to say with a constant pitch) and a signal weighting module. More precisely, if it is wished to form a channel in reception mode, the signals received by the different antennae are weighted by means of a set of complex coefficients before being added. Conversely, if it is wished to form a channel in transmission mode, the signal to be transmitted is weighted by a set of complex coefficients and the signals thus weighted are transmitted by the different antennae.
FIG. 1 illustrates a known device for obtaining antenna gain in transmission and reception mode. The device comprises an array of antennae (100),(101), . . . ,(10N−1), a transmission weighting module (11) and a reception weighting module (15). The signals received by the different antennae, (xi), i=0 . . . N−1 are weighted at (130),(131) . . . ,(13N−1) by a set of complex coefficients (bui), i=0, . . . ,N−1 before being added at (14) in order to give a signal Ru. Conversely, a signal to be transmitted Sd is weighted at (120),(121) . . . ,(12N−1) by means of a set of complex coefficients (bdi), i=0, . . ,N−1, before being transmitted by the different antennae.
If respectively the vector of the signals received and the vector of the weighting coefficients is denoted {overscore (x)}=(x0,x1, . . . ,xN−1)T and {overscore (bu)}=(bu0,bu1, . . . ,buN−1)T, it is possible to write:

Ru={overscore (b)}u{overscore (x)}  (1)

The complex gain (or the complex gain function of the antenna) in reception mode can be written:




  
    
      
        
          G
          ⁡
          
            (
            
              
                
                  b
                  u
                
                _
              
              ,
              θ
            
            )
          
        
        =
        
          
            
              
                
                  b
                  u
                
                _
              
              T
            
            ·
            
              
                
                  e
                  u
                
                ⁢
                θ
              
              _
            
          
          =
          
            
              ∑
              
                i
                =
                0
              
              
                N
                -
                1
              
            
            ⁢
            
              
            
            ⁢
            
              
                b
                ui
              
              ·
              
                exp
                ⁡
                
                  (
                  
                    -
                    
                      jφ
                      i
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




where euθ represents the vector {overscore (x)} corresponding to a flat wave arriving at an angle of incidence θ, and

φi=(2πd/λ).i.sin(θ)=(2πdf/c).i.sin(θ)  (3)

is the difference in operation between consecutive antennae for a uniform linear array of pitch d, λ and f being respectively the wavelength and the frequency of the flat wave in question;

φi=2πRΔθ/λ.sin(θ−θi)=2πRfΔθ/c.sin(θ−θi)  (4)

for a circular array where θi is the angle between a reference axis and the normal to the antenna of index i, R the radius of curvature of the array, Δθ is the angular difference between two consecutive antennae in the array.
Likewise the complex gain (or the complex gain function) in transmission mode can be written:




  
    
      
        
          G
          ⁡
          
            (
            
              
                
                  b
                  d
                
                _
              
              ,
              θ
            
            )
          
        
        =
        
          
            
              
                
                  b
                  d
                
                _
              
              T
            
            ·
            
              
                
                  e
                  d
                
                ⁢
                θ
              
              _
            
          
          =
          
            
              ∑
              
                i
                =
                0
              
              
                N
                -
                1
              
            
            ⁢
            
              
            
            ⁢
            
              
                b
                di
              
              ·
              
                exp
                ⁡
                
                  (
                  
                    jφ
                    i
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        5
        )
      
    
  




with the same conventions as those adopted above and where {overscore (edθ)} designates the vector {overscore (x)} corresponding to a flat wave transmitted in the direction θ. The weighting vectors in reception and transmission mode respectively will be called {overscore (bu)} and {overscore (bd)}.

Clearly, the antenna gain in transmission or reception mode depends on the frequency of the signal in question. There are however many situations in which the antenna gain must remain unchanged whatever the frequency of the signal. For example, in so-called FDD (Frequency Division Duplex) mobile telecommunication systems, where the frequency used on the downlink, that is to say from the base station to the mobile station, differs from that used on the uplink. Similarly, in frequency-hopping radar systems, it is necessary to ensure the invariance of the gain function, notably in order to aim a transmission or reception beam in a given direction or to eliminate the interference coming from a given direction, whatever the frequency used.
In more general terms, it is desirable to be able to obtain, for a given signal frequency, an antenna gain function which is as close as possible, in the sense of a certain metric, to a reference gain function. The reference gain function can notably be a gain function obtained at a given frequency which it is sought to approximate to the greatest possible extent during transmission or reception at another frequency.
SUMMARY OF THE INVENTION
The aim of the invention is to propose a method of obtaining a gain function making it possible, for a given signal frequency, to approach a reference gain function as closely as possible.
A subsidiary aim of the invention is to propose a method for best approaching an antenna gain function obtained at a given frequency when the network is transmitting or receiving at another frequency.
To this end, the invention is defined by a method of obtaining a gain function by means of an array of antennae and a weighting of the signals received or to be transmitted by vectors ({overscore (b)}) of N complex coefficients, referred to as weighting vectors, N being the number of antennae in the array, according to which, a reference gain function being given, the said reference gain function is projected orthogonally onto the sub-space of the gain functions generated by the said weighting vectors of the space of the gain functions, provided in advance with a norm, and a weighting vector generating the reference gain function thus projected is chosen as the optimum weighting vector.
The gain functions are preferably represented by vectors ({overscore (G)}), referred to as gain vectors, of M complex samples taken at M distinct angles, defining sampling directions and belonging to the angular range covered by the array, the space of the gain functions then being the vector space CM provided with the Euclidian norm and that, for a given frequency (f), the reference gain vector is projected on the vector sub-space (Imf) of the gain vectors generated by the array operating at the said frequency in order to obtain the said optimum weighting vector.
Advantageously, M is chosen such that M>πN.
According to one example embodiment, the sampling angles are distributed uniformly in an angular range covered by the array.
The reference gain vector can be obtained by sampling the reference gain function after anti-aliasing filtering.
The gain vectors ({overscore (G)}) being the transforms by a linear application (hsf) of CN in CM weighting vectors of the array and Hf being the matrix, of size M×N, of the said linear application of a starting base of CN in an arrival base CM,, the said optimum weighting vector, for a given frequency f, is preferably obtained from the reference gain vector {overscore (G)} as {overscore (b)}=Hf+.{overscore (G)} where Hf+=(H*fT.Hf)−1.H*fT is the pseudo-inverse matrix of the matrix Hf and where H*fT the conjugate transpose of the matrix Hf.
The said starting base being that of the vectors ēk, k=0, . . . ,N−1, such that ēk=(ek,0,ek,1, . . . ,ek,N−1)T with




  
    e
    
      k
      ,
      i
    
  
  =
  
    exp
    ⁡
    
      (
      
        
          j
          ·
          
            
              2
              ⁢
              π
              ⁢
              
                
              
              ⁢
              fd
            
            c
          
          ·
          i
          ·
          sin
        
        ⁢
        
          
        
        ⁢
        
          θ
          k
        
      
      )
    
  




and θk=kπ/N k=−(N−1)/2, . . . ,0, . . . ,(N−1)/2 and the arrival base being the canonical base, the matrix Hf then has the components:





  
    H
    pq
  
  =
  
    
      exp
      ⁡
      
        (
        
          
            j
            ⁡
            
              (
              
                N
                -
                1
              
              )
            
          
          ⁢
          
            
              Ψ
              pq
            
            /
            2
          
        
        )
      
    
    ·
    
      
        sin
        ⁢
        
          (
          
            N
            ⁢
            
              
            
            ⁢
            
              
                Ψ
                pq
              
              /
              2
            
          
          )
        
      
      
        sin
        ⁢
        
          (
          
            
              Ψ
              pq
            
            /
            2
          
          )
        
      
    
  




with Ψpq=πη(sin(ρπ/N)−sin(qπ/M)) and η=f/f0 with f0=c/2d, d being the pitch of the array.

If the reference gain vector is obtained by sampling the gain function generated at a first operating frequency f1 of the array by a first weighting vector {overscore (b1)}, the optimum weighting gain vector for a second frequency f2 is obtained by {overscore (b2)}=H+f2.Hf1{overscore (b1)}.
The frequency f1 of operation of the array is for example the frequency of an uplink between a mobile terminal and a base station in a mobile telecommunication system and the frequency f2 of operation of the array is for example the frequency of a downlink between the said base station and the said mobile terminal.



BRIEF DESCRIPTION OF THE DRAWINGS
The characteristics of the invention mentioned above, as well as others, will emerge more clearly from a reading of the following description given in relation to the accompanying figures, amongst which:
FIG. 1 depicts schematically a known device for obtaining an antenna gain function;
FIG. 2 depicts schematically a device for obtaining an antenna gain function according to one example embodiment of the invention.



DETAILED DESCRIPTION OF THE INVENTION
A first general idea at the basis of the invention is to best approximate a reference gain function by virtue of a linear combination of base functions.
A second general idea at the basis of the invention is to sample the reference gain function and to best approximate the series of samples obtained by means of a linear combination of base vectors.
The first embodiment of the invention consists of approximating the reference gain function by means of a linear combination of base functions.
Let h be the linear application of CN in the vector space F of the complex functions defined on [−π/2,π/2] (or [−π,π]) which associates with any vector {overscore (b)} of complex numbers the function h({overscore (b)}) such that h({overscore (b)})(θ)=G({overscore (b)},θ) where G is a complex gain function in transmission or reception mode as defined at (2) or (5). CN being a vector space of dimension N on C, the image of CN by h is a vector sub-space of F of dimension at most equal to N, which will be denoted Imf to emphasise that the image depends on the frequency f in question in expression (2) or (5).
Let G be a reference complex gain function, the problem is to find the weighting vector {overscore (b)} such that h({overscore (b)}) is as close as possible to G in the sense of a certain metric. For a uniform linear array, the metric corresponding to the scalar product on F




  
    
      w
      1
    
    ·
    
      w
      2
    
  
  =
  
    
      ∫
      
        
          -
          π
        
        /
        2
      
      
        π
        /
        2
      
    
    ⁢
    
      
        
          
            w
            1
          
          ⁢
          
            (
            θ
            )
          
        
        ·
        
          
            w
            2
            ′
          
          ⁢
          
            (
            θ
            )
          
        
      
      ⁢
      
        cos
        ⁢
        
          (
          θ
          )
        
      
      ⁢
      
        
      
      ⁢
      
        d
        θ
      
    
  




and therefore to the norm





  
    
      
      w
      
    
    2
  
  =
  
    
      ∫
      
        
          -
          π
        
        /
        2
      
      
        π
        /
        2
      
    
    ⁢
    
      
        
          
          
            w
            ⁡
            
              (
              θ
              )
            
          
          
        
        2
      
      ⁢
      cos
      ⁢
      
        
      
      ⁢
      
        θ
        ·
        
          d
          θ
        
      
    
  




is chosen. The case of the circular array can b a similar manner (the chosen norm does not then include the term cos(θ)). The space F2 of the functions of F of bounded norm is itself a vector space normed by the above norm. If G is an element of F2, the element of the sub-space Imf closest to G is then the projection of G onto this sub-space.

If the vector sub-space corresponding to the inherent frequency of the array is considered to be Imf0, it is possible to demonstrate that the functions ek(θ), k=0, . . . ,N−1 defined by:
ek(θ)=h({overscore (bk)})(θ)=G({overscore (bk)},θ), where {overscore (bk)} is the vector of components bki=exp(j.2πki/N), are orthogonal. Being N in number, they therefore form a base of Imf0. In more general terms, it can be shown that, if two vectors {overscore (b)} and {overscore (b′)} are orthogonal, that is to say are such that {overscore (bb′)}={overscore (b)}T{overscore (b′)}=0, the functions h({overscore (b)}) and h({overscore (b′)}) of Imf0 are orthogonal.

This is because:





  
    
      
        
          
            
              
                
                  h
                  ⁡
                  
                    (
                    
                      b
                      _
                    
                    )
                  
                
                ·
                
                  h
                  ⁡
                  
                    (
                    
                      
                        b
                        ′
                      
                      _
                    
                    )
                  
                
              
              =
              
                
                  ∑
                  
                    i
                    =
                    0
                  
                  
                    N
                    -
                    1
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    ∑
                    
                      
                        i
                        ′
                      
                      =
                      0
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      
                        b
                        i
                      
                      ·
                      
                        b
                        
                          i
                          ′
                        
                        ′
                      
                    
                    ⁢
                    
                      
                        ∫
                        
                          
                            -
                            π
                          
                          /
                          2
                        
                        
                          π
                          /
                          2
                        
                      
                      ⁢
                      
                        
                          exp
                          ⁡
                          
                            (
                            
                              
                                j
                                ⁡
                                
                                  (
                                  
                                    i
                                    -
                                    
                                      i
                                      ′
                                    
                                  
                                  )
                                
                              
                              ⁢
                              
                                φ
                                ⁡
                                
                                  (
                                  θ
                                  )
                                
                              
                            
                            )
                          
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            θ
                            )
                          
                        
                        ⁢
                        
                          d
                          θ
                        
                      
                    
                  
                
              
            
          
        
        
          
            
              =
              
                
                  ∑
                  
                    i
                    =
                    0
                  
                  
                    N
                    -
                    1
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    ∑
                    
                      
                        i
                        ′
                      
                      =
                      0
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      
                        b
                        i
                      
                      ·
                      
                        b
                        
                          i
                          ′
                        
                        ′
                      
                      ·
                      sin
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      c
                      ⁡
                      
                        (
                        
                          πη
                          ⁡
                          
                            (
                            
                              i
                              -
                              
                                i
                                ′
                              
                            
                            )
                          
                        
                        )
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        6
        )
      
    
  




with φ(θ)=2πfd/c.sin θ=πηsin θ where η=f/f0≦1, is the ratio of the frequency used at the maximum frequency f0=c/2d, which can resolve the array without ambiguity, which will be referred to as the natural frequency of the array, and where sinc. is the cardinal sine function. For η=1, the terms below the sum signs of the second member of equation (6) are zero if i≠i′ and therefore the second member is equal to zero if the vectors {overscore (b)} and {overscore (b′)} are orthogonal.

Consider now the general case of a frequency f≦f0. Let ek(θ), k=0, . . . ,N−1 be an orthogonal base of Imf. By definition, ek(θ)=h({overscore (bk)})(θ)=G({overscore (bk)},θ) where {overscore (bk )}is a vector of CN. Consider now a gain function G of F2. It can be projected onto the vectors ek(θ). If λk=G.ek is written, then the vector of CN,




  
    
      b
      G
    
    _
  
  =
  
    
      ∑
      
        k
        =
        0
      
      
        P
        -
        1
      
    
    ⁢
    
      
    
    ⁢
    
      
        λ
        k
      
      ⁢
      
        
          b
          k
        
        _
      
    
  




is such that h({overscore (bG)}) best approximates the function G.

The second embodiment of the invention consists of approximating a vector of samples of the reference gain function by means of a linear combination of base vectors.
Let G0(θ) be the antenna gain function obtained without weighting for a linear uniform array, it is easily shown that:




  
    
      
        
          
          
            
              G
              0
            
            ⁡
            
              (
              θ
              )
            
          
          
        
        =
        
          
            
              
                sin
                ⁡
                
                  (
                  
                    N
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      /
                      2
                    
                  
                  )
                
              
              
                sin
                ⁡
                
                  (
                  
                    φ
                    /
                    2
                  
                  )
                
              
            
            ⁢
            
              
            
            ⁢
            with
            ⁢
            
              
            
            ⁢
            φ
          
          =
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              
                fd
                /
                c
              
              ·
              sin
            
            ⁢
            
              
            
            ⁢
            θ
          
        
      
    
    
      
        (
        7
        )
      
    
  




This function has zeros for the values φk=2kπ/N, k integer non-zero such that φkε[−π,π[ that is to say in the directions for which sin θk=k.c/Nfd, when this expression has a direction. The phase difference between two consecutive zeros of the gain diagram is constant and is equal to Δφ=2π/N. The angular difference between two consecutive zeros of the diagram varies in terms of Arcsin., a function whose derivative is increasing on [−1,1] and is therefore at a minimum for the angular difference between the first and second zeros. It is therefore bounded by Δθmin=c/Nfd if N is sufficiently great. It will be assumed that the frequencies used are less than f0 where f0 is the natural frequency of the array. It can be concluded from this that the spectrum of the function G0(θ) has a support bounded by 1/Δ θmin=N/2.
In more general terms, let G(θ) be the antenna gain function obtained by means of a weighting vector {overscore (b)}. G can be expressed as the Fourier transform (FT) (in reception mode) or the inverse Fourier transform (in transmission mode) of the complex weighting distribution of the antenna, namely:




  
    b
    ⁢
    
      (
      x
      )
    
  
  =
  
    
      ∑
      
        i
        =
        0
      
      
        N
        -
        1
      
    
    ⁢
    
      
    
    ⁢
    
      
        b
        i
      
      ·
      
        δ
        ⁢
        
          (
          
            x
            -
            
              x
              i
            
          
          )
        
      
    
  




with xi=i.d; this gives: Gb1(θ)=B(sin θ) with





  
    B
    ⁢
    
      (
      u
      )
    
  
  =
  
    
      ∫
      
        -
        ∞
      
      
        +
        ∞
      
    
    ⁢
    
      
        b
        ⁢
        
          (
          x
          )
        
      
      ⁢
      
        exp
        ⁡
        
          (
          
            
              -
              j
            
            ⁢
            
              
            
            ⁢
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              ux
              /
              λ
            
          
          )
        
      
      ⁢
      
        
      
      ⁢
      
        d
        x
      
    
  




and likewise Gd(θ)=B′(sin θ) with





  
    
      B
      ′
    
    ⁢
    
      (
      u
      )
    
  
  =
  
    
      ∫
      
        -
        ∞
      
      
        +
        ∞
      
    
    ⁢
    
      
        b
        ⁢
        
          (
          x
          )
        
      
      ⁢
      
        exp
        ⁡
        
          (
          
            j2π
            ⁢
            
              
            
            ⁢
            u
            ⁢
            
              
            
            ⁢
            
              x
              /
              λ
            
          
          )
        
      
      ⁢
      
        
      
      ⁢
      
        
          d
          x
        
        .
      
    
  




The function b(x) being delimited by N.d, the difference between two zeros of the function B or B′ is at least λ/N.d and therefore even more so 2/N. Given the increase in the derivative of the function Arcsin. the minimum difference between two zeros of the function G is 2/N. The function G therefore has a spectrum delimited by N/2.

According to the Shannon sampling theorem, it is concluded that it is possible to reconstitute the function G(θ) if sampling is carried out at a frequency greater than the Nyquist frequency, i.e. N. In other words, for an angular range [−π/2,π/2], at a minimum M>π.N samples are necessary, where M is integer. In practice K.N samples can be taken with K integer, K≧4.
For a circular array, it can be shown that 1/Δθmin=N and the angular range being [−π,π], M (M>π.N and M integer) angularly equidistributed samples also suffice to reconstitute the function G(θ).
In the general case of the sampling of any gain function G(θ), it is necessary to previously filter G(θ) by means of an anti-aliasing filter before sampling it. It then suffices to take M samples of the filtered diagram over the whole of the angular range in order to reconstitute the filtered diagram.
Let (gk), k=0, . . ,M−1 be the samples of the complex diagram, possibly filtered by an anti-aliasing filtering if necessary, that is to say gk=G′(θk) where the θk are M angles equidistributed over [−π/2,π/2] or [−π,π] and where it is assumed that G′ was the filtered version of the reference complex diagram.
It is now possible to define a linear application, hfs of CN in CM which makes the vector hsf({overscore (b)})={overscore (G)}=(g0,g1, . . . ,gM−1)T, where gk=G({overscore (b)},θk), correspond to any vector {overscore (b)}. The image of CN by hfs, is a vector sub-space of CM of dimension at most equal to N, which will be noted Imf. If a base of CN is chosen, for example the canonical base, and a base of CM, the linear application hfs can be expressed by a matrix Hf of size M×N which is at most of rank N.
Let {overscore (G)} be any gain vector corresponding to a sampled gain function. The problem is to find a vector {overscore (b)} such that hsf({overscore (b)}) is the closest to {overscore (G)} in the sense of a certain metric. The Euclidian norm on CM, namely




  
    
      
        
        
          G
          _
        
        
      
      2
    
    =
    
      
        ∑
        
          k
          =
          0
        
        
          M
          -
          1
        
      
      ⁢
      
        
      
      ⁢
      
        
          
          
            g
            k
          
          
        
        2
      
    
  
  ,




will be taken as the norm. If it exists, the sought-for vector {overscore (b)} is then such that hsf(b)={overscore (G)}p where {overscore (G)}p the orthogonal projection of the vector {overscore (G)} onto Imf. If the matrix Hf is of rank N, the sought-for vector {overscore (b)} exists and can be written:

{overscore (b)}=Hf+.{overscore (g)}  (8)

where Hf+=(H*fT.Hf)−1.H*fT is the pseudo-inverse matrix of the matrix Hf with transposed H*fT the conjugate of the matrix Hf.
In the discrete case as in the continuous case, the reference gain function (sampled in the discrete case) is projected onto the sub-space generated by the functions (continuous case) or the vectors (discrete case) associated with the array weighting vectors.
In order to express the matrix Hf, it is necessary to agree a base of the starting space and a base of the arrival space. It is possible to choose as a base of CM the canonical base and as a base of CN a base adapted to the description of the flat waves of frequency f. Consider the distinct vectors ēk, k=0, . . ,N−1, such that ēk=(ek,0ek,1, . . ,ek,N−1)T with




  
    e
    
      k
      ,
      i
    
  
  =
  
    
      exp
      ⁡
      
        (
        
          
            j
            ·
            
              
                2
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                fd
              
              c
            
            ·
            i
            ·
            sin
          
          ⁢
          
            
          
          ⁢
          
            θ
            k
          
        
        )
      
    
    =
    
      exp
      ⁡
      
        (
        
          j
          ⁢
          
            
          
          ⁢
          
            π
            ·
            η
            ·
            i
            ·
            sin
          
          ⁢
          
            
          
          ⁢
          
            θ
            k
          
        
        )
      
    
  




with η=f/f0 and where the θk belong to the interval [−π/2,π/2]. The vectors ek are the weighting vectors of the array making it possible to form beams in the directions θk. The vectors ēk form a base if the determinant of the coordinates of the ēk in the canonical base of CN is non-zero. This determinant is a Vandermonde determinant which is equal to





  
    
      ∏
      
        p
        ≠
        q
      
    
    ⁢
    
      
        (
        
          
            exp
            ⁡
            
              (
              
                j
                ⁢
                
                  
                
                ⁢
                
                  φ
                  p
                
              
              )
            
          
          -
          
            exp
            ⁡
            
              (
              
                j
                ⁢
                
                  
                
                ⁢
                
                  φ
                  q
                
              
              )
            
          
        
        )
      
      ⁢
      
        
      
      ⁢
      with
      ⁢
      
        
      
      ⁢
      
        φ
        k
      
    
  
  =
  
    π
    ⁢
    
      
    
    ⁢
    η
    ⁢
    
      
    
    ⁢
    sin
    ⁢
    
      
    
    ⁢
    
      
        θ
        k
      
      .
    
  




This determinant is cancelled out if and only if there are two angles θp and θq such that sin θp−sin θq=2/η. In other words, for η<1 the N vectors ēk always form a base, and for η=1 only the case θp=−θq=π/2 is excluded. The directions can, for example, be chosen so as to be equidistributed, that is to say such that θk=kπ/N with k=−(N−1)/2, . . . ,0, . . ,(N−1)/2. In this case, the matrix Hf has as its components:





  
    
      
        
          
            
              H
              pq
            
            =
            
              
                ∑
                
                  i
                  =
                  0
                
                
                  N
                  -
                  1
                
              
              ⁢
              
                
              
              ⁢
              
                
                  exp
                  ⁡
                  
                    (
                    
                      jπη
                      ·
                      i
                      ·
                      
                        sin
                        ⁡
                        
                          (
                          
                            p
                            ⁢
                            
                              
                            
                            ⁢
                            
                              π
                              /
                              N
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                ⁢
                
                  exp
                  ⁡
                  
                    (
                    
                      
                        -
                        jπη
                      
                      ·
                      i
                      ·
                      
                        sin
                        ⁡
                        
                          (
                          
                            q
                            ⁢
                            
                              
                            
                            ⁢
                            
                              π
                              /
                              M
                            
                          
                          )
                        
                      
                    
                    )
                  
                
              
            
          
          ⁢
          
            
          
          ⁢
          
            

          
          ⁢
          
            or:
          
          ⁢
          
            

          
          ⁢
          
            
              
                
                  
                    H
                    pq
                  
                  =
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      
                        N
                        -
                        1
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      exp
                      ⁡
                      
                        (
                        
                          jπη
                          ·
                          i
                          ·
                          
                            [
                            
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    p
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      π
                                      /
                                      N
                                    
                                  
                                  )
                                
                              
                              -
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    q
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      π
                                      /
                                      M
                                    
                                  
                                  )
                                
                              
                            
                            ]
                          
                        
                        )
                      
                    
                  
                
              
            
            
              
                
                  =
                  
                    
                      exp
                      ⁡
                      
                        (
                        
                          
                            j
                            ⁡
                            
                              (
                              
                                N
                                -
                                1
                              
                              )
                            
                          
                          ⁢
                          
                            
                              Ψ
                              pq
                            
                            /
                            2
                          
                        
                        )
                      
                    
                    ·
                    
                      
                        sin
                        ⁡
                        
                          (
                          
                            N
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                Ψ
                                pq
                              
                              /
                              2
                            
                          
                          )
                        
                      
                      
                        sin
                        (
                        
                          
                        
                        ⁢
                        
                          
                            Ψ
                            pq
                          
                          /
                          2
                        
                        )
                      
                    
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
              with
              ⁢
              
                
              
              ⁢
              
                Ψ
                pq
              
            
            =
            
              πη
              ⁡
              
                (
                
                  
                    sin
                    ⁡
                    
                      (
                      
                        p
                        ⁢
                        
                          
                        
                        ⁢
                        
                          π
                          /
                          N
                        
                      
                      )
                    
                  
                  -
                  
                    sin
                    ⁡
                    
                      (
                      
                        q
                        ⁢
                        
                          
                        
                        ⁢
                        
                          π
                          /
                          M
                        
                      
                      )
                    
                  
                
                )
              
            
          
        
        ⁢
        
          
        
      
    
    
      
        (
        9
        )
      
    
  




Alternatively, it is possible to choose as a starting base another base adapted to the frequency f, the one formed by the vectors {overscore (e′)}k, such that e′k,i=exp(j π.η.i.sin θk) with sin θk=2k/ηN and k=−(N−1)/2, . . . ,0, . . ,(N−1)/2. These vectors exist if |sin θk|≦1, ∀k, that is to say for η>1−1/N and in this case the vectors {overscore (e′)}k form a base which has the advantage of being orthogonal.
Alternatively, it is possible to choose as a starting base the canonical base of CN, which has the advantage of not depending on the frequency. In this case, the matrix H′f expressed in this base is written:

H′f=Hf.T−1   (10)

where T is the matrix of the coordinates of ēk in the canonical base, that is to say Tpp′=exp(jπp sin(p′/N)). It was seen above that this matrix had a non-zero Vandermonde determinant and was consequently reversible.
Whatever the chosen base, consider now a gain function obtained at a first frequency f1,f1≦f0 and {overscore (G1)}=hsf1({overscore (b1)}) the vector of the samples associated with this gain function. Let there be a second working frequency f2, f2≦f0. {overscore (G1)} belonging to CM, if the matrix Hf2 is of rank N, it is possible to find a vector {overscore (b2)} such that hsf2({overscore (b2)}) is the projection of hsf1({overscore (b1)}) onto Imf2. The vector {overscore (b2)} is obtained by means of the matrix equation:

{overscore (b2)}=Hf2+.Hf1 {overscore (b1)}  (11)

This equation makes it possible in particular to obtain, at a second working frequency, a sampled gain diagram which is as close to possible to the one, referred to as the reference one, obtained at a first working frequency.
Equation (11) advantageously applies to the array of a base station in a mobile telecommunication system operating in FDD. Equation (10) makes it possible to directly obtain the weighting vector to be applied for the “downlink” transmission at a frequency fd on the weighting vector relating to the “uplink” transmission at a frequency fu. For the paired frequencies fd and fu, it is then possible to write:

{overscore (bd)}=Hd+.Hu{overscore (bu)}  (12)

The base station can thus direct transmission beams to the mobile terminals using a gain function optimised for the reception of the signals transmitted by these terminals.
FIG. 2 depicts an example of an embodiment implementing the second embodiment. The device comprises a transmission weighting module (31) and a reception weighting module (35) with a structure identical to that of the modules (11) and (15) respectively. The module (35) is associated with a module (36) supplying the complex coefficients for the formation of reception channels and/or the elimination of signals in the interference directions. The module (36) determines, in a manner known per se, a weighting vector {overscore (bu)} which maximises the signal received in the useful direction or directions and minimises it in the interference directions. Advantageously bu is calculated adaptively from the signals received by the different antennae. The vector is on the one hand used by the reception weighting module (35) and on the other hand transmitted to a projection and inversion module (32) determining the vector {overscore (bd)} from equation (12). The vector {overscore (bd)} is used for weighting the signals to be transmitted in the module (31). As seen above, the transmission gain diagram at frequency fu will minimise the difference, in the sense of the Euclidian distance, between the transmission gain vector {overscore (Gd)} and the reception gain vector {overscore (Gu)}.
Although the invention has been essentially described, for reasons of simplicity of presentation, in the context of a uniform linear array, it can apply to any type of antenna array and notably to a circular array.

## Claims

1. A method of obtaining a gain function from an array of antennae and a weighting of signals received or to be transmitted by vectors ({overscore (b)}) of N complex coefficients, referred to as weighting vectors, N being the number of antennae in the array, comprising the steps of:
generating a sub-space which is normed and orthogonal with respect to a space of gain functions, the gain functions being generated by the weighting vectors;
projecting a desired reference function onto the sub-space; and
choosing a weighting vector which generates a gain function approximate to the projection of the desired reference gain function in the sub-space, as an optimum weighting vector.

2. The method of obtaining the gain function according to claim 1, wherein the gain functions are represented by vectors ({overscore (G)}) referred to as gain vectors, of M complex samples taken at M distinct angles, defining sampling directions and belonging to the angular range covered by the array, further comprising:
providing the space of the gain functions being the vector space CM with an Euclidian norm; and
projecting the reference gain function for a given frequency (f) onto the vector sub-space (Imf) of the gain vectors generated by the array operating at the frequency in order to obtain the optimum weighting vector.

3. The method of obtaining the gain function according to claim 2, wherein M is chosen such that M>πN.

4. The method of obtaining the gain function according to claim 2 or 3, wherein the M distinct angles are uniformly distributed in the angular range covered by the array.

5. The method of obtaining the gain function according to claim 2, wherein the reference gain function is obtained by sampling the reference gain function after an anti-aliasing filtering.

6. The method of obtaining the gain function according to claim 2, further comprising:
transforming the gain vectors ({overscore (G)}) by a linear application (hsf) of CN in CM of the weighting vectors of the array and Hf being the matrix of size M×N of the said linear application of a starting base of CN in an arrival base CM, the optimum weighting vector for a given frequency is obtained from the reference gain vector {overscore (G)} as {overscore (b)}=Hf+·{overscore (G)} wherein Hf+=(Hf*T·Hf)−1·Hf*T is the pseudo-inverse matrix of the matrix Hf and where Hf*T is the conjugate transpose of the matrix Hf.

7. The method of obtaining the gain function according to claim 6, wherein said starting base being that of the vectors ēk, k=0, . . . ,N−1, such that ēk=(ek, 0,ek, 1, . . . ,ek, N−1)T with




  
    e
    
      k
      ,
      i
    
  
  =
  
    exp
    ⁡
    
      (
      
        
          j
          ·
          
            
              2
              ⁢
              π
              ⁢
              
                
              
              ⁢
              fd
            
            c
          
          ·
          i
          ·
          sin
        
        ⁢
        
          
        
        ⁢
        
          θ
          k
        
      
      )
    
  




and θk=kπ/N, k=−(N−1)/2, . . . ,0, . . . ,(N−1)/2 and the arrival base being a canonical base, the matrix Hf having the components:




  
    H
    pq
  
  =
  
    
      exp
      ⁡
      
        (
        
          
            j
            ⁡
            
              (
              
                N
                -
                1
              
              )
            
          
          ⁢
          
            
              Ψ
              pq
            
            /
            2
          
        
        )
      
    
    ·
    
      
        sin
        ⁢
        
          (
          
            N
            ⁢
            
              
            
            ⁢
            
              
                Ψ
                pq
              
              /
              2
            
          
          )
        
      
      
        sin
        ⁢
        
          (
          
            
          
          ⁢
          
            
              Ψ
              pq
            
            /
            2
          
          )
        
      
    
  




with Ψpq=πη(sin(pπ/N)−sin(pπ/M) and η=f/fo with fo=c/2d, d being the pitch of the array.

8. The method of obtaining the gain function according to claim 6, wherein the reference gain vector is obtained by sampling the gain function generated at a first operating frequency f1 of the array by using a first weighting vector {overscore (b)}1 and wherein the optimum weighting gain vector for a second frequency f2 is obtained by {overscore (b)}2=Hf2+·Hf1·{overscore (b)}1.

9. The method of obtaining the gain function according to claim 8, wherein the operating frequency f1 of the array is the frequency of an uplink between a mobile terminal and a base station in a mobile telecommunication system and in that the operating frequency f2 of the array is the frequency of a downlink between the base station and the mobile terminal.

10. The method of obtaining the gain function according to claim 7, wherein the reference gain vector is obtained by sampling the gain function generated at a first operating frequency f1 of the array using a first weighting vector {overscore (b)}1 and wherein the optimum weighting gain vector for a second frequency f2 is obtained by {overscore (b)}2=Hf2+l ·Hf1·{overscore (b)}1.

11. The method of obtaining the gain function according to claim 3, further comprising:
transforming the gain vectors ({overscore (G)}) by a linear application (hsf) of CN in CM of the weighting vectors of the array and Hf being a matrix of size M×N of the linear application of a starting base of CN in an arrival base CM, the optimum weighting vector for a given frequency f is obtained from the reference gain vector {overscore (G)} as {overscore (b)}=Hf+·{overscore (G)} wherein Hf+=(Hf*T·Hf)−1·Hf*T is the pseudo-inverse matrix of the matrix Hf and where Hf*T is the conjugate transpose of the matrix Hf.

12. The method of obtaining the gain function according to claim 1, wherein the norm provided to the vector space is an Euclidian norm.

13. The method of obtaining the gain function according to claim 2, further comprising:
approximating a vector of samples of the reference gain function by using a linear combination of base vectors.

14. The method of obtaining the gain function according to claim 1, wherein the array of antennae is a circular array.