# Patent

## Title
Method for the higher-order blind demodulation of a linear waveform transmitter

## Abstract

Process for the blind demodulation of a linear waveform source or transmitter in a system comprising one or more sources and an array of sensors and a propagation channel. The process comprises at least the following steps:

    
    
        the symbol period T is determined and sampled to Te such that T=ITe (I being an integer);
        a spatio-temporal observation z(t), the mixed sources of which are symbol trains from the transmitter, is constructed from the observations x(kTe);
        an ICA-type method is applied to the observation vector z(t) in order to estimate the Lc symbol trains {am-i} that are associated with the channel vectors ĥz,j=ĥz(kj);
        the Lc outputs (âm,j, ĥz,j) are arranged in the same order as the inputs (am-i,hz(i)) so as to obtain the propagation channel vectors ĥz,j=ĥz(kj); and
        the phase αimax associated with the outputs is determined.

## Background
The object of the invention relates to a process for the blind demodulation of signals output by several transmitters and received by an array made up of at least one sensor.
For example, it applies to an array of antennas in an electromagnetic context.
The subject of the invention is in particular the demodulation of signals, that is to say the extraction of the symbols {ak} transmitted by a linearly modulated transmitter.
FIG. 1 shows an antenna processing system comprising several transmitters Ei and an antenna processing system T comprising several antennas Ri receiving from radio sources at different angles of incidence. The angles of incidence of the sources or transmitters may be parameterized either in 1D with the azimuth θm, or in 2D with the azimuth angle θm and the elevation angle Δm.
FIG. 3 shows schematically a modulation/demodulation principle for the symbols {ak} output by a transmitter. The signal propagates via a multipath channel. The transmitter outputs the symbol ak at the instant k.T, where T is the symbol period. The demodulation consists in estimating and in detecting the symbols in order to obtain, at the output of the demodulator, the estimated symbols âk. In this figure, the train of symbols {ak} is linearly filtered upon transmission by a transmission filter H, also called a wave-shaping filter h0(t).
In the rest of the description, the expression “blind demodulation” is understood to mean techniques that basically use no information on the signal output, examples being a wave-shaping filter, a learning sequence, etc.
The last ten years have seen the development of SIMO (single-input, multiple-output) blind demodulation techniques called subspace techniques using 2nd-order statistics, as described in reference [7]. However, these algorithms have the drawback of not being robust to either underestimation or overestimation of the order of the propagation channel, resulting in temporal spreading dependent on the multipaths and on the wave-shaping filter. A linear prediction technique, described in reference [11], has been proposed for overcoming this problem, but this has the drawback of being less effective when the length of the channel is known. To improve the subspace techniques, the method described in [16] proposes a parametric technique, but unfortunately this requires knowledge of the wave-shaping filter.
In reference [13], the authors propose a technique based on covariance matching, but this has in particular the drawback of being very difficult to implement. An easier but suboptimal technique, described in reference [12], was therefore developed by minimizing a likelihood criterion and assuming the symbols to be Gaussian in character. This assumption is not verified for the widely used linear modulations such as PSK (Phase Shift Keying) or QAM (Quadrature Amplitude Modulation).
It is also known, in CMA (Constant Modulus Algorithm) methods, to use a spatio-temporal approach described for example in reference [6]. However, this family of methods has the drawback of being suitable only for one particular class of modulations, such as PSK, which are constant-modulus modulations. This method is iterative and therefore has the drawback of having to be correctly initialized. Finally, the CMA methods have the disadvantage of converging more slowly than the abovementioned subspace method. Moreover, reference [20] describes a subspace method making use of higher-order statistics for non-minimum-phase FIR (finite impulse response) channel identification.
The subject of the present invention is a process based in particular on blind source separation techniques known to those skilled in the art and described for example in references [4], [5], [15] and [19] assuming that the symbols transmitted are statistically independent. To do this, the process constructs a spatio-temporal observation whose mixed sources are symbol trains from the transmitter. Each symbol train is for example the same symbol train but shifted by an integral number of symbol periods T.
The invention relates to a process for the blind demodulation of a linear-waveform source or transmitter in a system comprising one or more sources and an array of sensors and a propagation channel, said process being characterized in that it comprises at least the following steps:

    
    
        the symbol period T is determined and samples are taken at Te such that T=ITe (I being an integer);
        a spatio-temporal observation z(t), the mixed sources of which are symbol trains from the transmitter, is constructed from the observations x(kTe);
        an ICA-type method is applied to the observation vector z(t) in order to estimate the Lc symbol trains {am-i} that are associated with the channel vectors ĥz,j=ĥz(kj);
        the Lc outputs (âm,j, ĥz,j) are arranged in the same order as the inputs (am-i, hz(i)) so as to obtain the propagation channel vectors ĥz,j=ĥz(kj); and
        the phase αimax associated with the outputs is determined.
    
    


The process according to the invention offers in particular the following advantages:

    
    
        it makes no assumption about the symbol constellations, unlike the methods described in the prior art;
        it requires no knowledge of the wave-shaping filter;
        the modulus of the symbols is not assumed to be constant;
        it is robust to channel length overestimation;
        it can handle propagation channels with correlated paths; and
        it is direct and simple to implement, with no correlated-path crosscheck step.
    
    





Other features and advantages of the subject of the present invention will become more clearly apparent on reading the following description, given by way of illustration but implying no limitation, and on examining the appended figures, which show:
FIG. 1, an example of an architecture;
FIG. 2, the angles of incidence of the sources;
FIG. 3, the linear modulation and demodulation process for a symbol train;
FIG. 4, the diagram of a linear-modulation transmitter;
FIG. 5, a summary of the general principle employed in the invention;
FIG. 6, the representation of a constellation;
FIG. 7, a first example of the implementation of the method, in which the signal is received baseband;
FIG. 8, a second example in which the signal is received in baseband and the multipaths are decorrelated; and
FIG. 9, a third example in which the signal is received in baseband and the multipaths are groupwise decorrelated.



To explain the process according to the invention more clearly, the following description relates to a process for the higher-order blind demodulation of a linear-waveform transmitter in an array having a structure as described in FIG. 1 for example.
Before explaining the steps for implementing the process, the model of the signal used will be described.
Model of the Signal Output by a Source or Transmitter Linear Modulation
FIGS. 3 and 4 show the process for the linear modulation of a symbol train {ak} at the rate T by a wave-shaping filter h0(t).
The comb of symbols c(t) is firstly filtered by the wave-shaping filter h0(t) and then transposed to the carrier frequency f0. The NRZ filter, which is a time window of length T, very often defined by h0(t)=ΠT(t−T/2), is one particular nonlimiting example of a transmission filter. In radio communication, it is also possible to use a Nyquist filter, the Fourier transform of which, h0(f)≈ΠB(f−B/2), approaches a band window B when the roll-off is zero, therefore h0(f)=ΠB(f−B/2) (the roll-off defines the slope of the filter away from the band B).
The modulates signal s0(t), output by the transmitter, may be expressed at time tk=kTe (Te being the sampling period) as a function of the comb of symbols c(t):




  
    
      
        
          
            s
            0
          
          ⁡
          
            (
            
              kT
              e
            
            )
          
        
        =
        
          
            ∑
            i
            
              
            
          
          ⁢
          
            
              
                h
                0
              
              ⁡
              
                (
                
                  iT
                  e
                
                )
              
            
            ⁢
            
              c
              ⁡
              
                (
                
                  
                    (
                    
                      k
                      -
                      i
                    
                    )
                  
                  ⁢
                  
                    T
                    e
                  
                
                )
              
            
          
        
      
    
    
      
        (
        1
        )
      
    
  




Let the symbol time T be equal to an integer number of times the sampling period, i.e. T=ITe, and let k=mI+j where 0≦j<I. Since c(t)=Σrarδ(t−rITe), in other words c(t)=au for t=uITe and c(t)=0 for t≠uITe, the only values for i for which c((k−i)Te) is non zero satisfy the equation k−i=uI, that is to say such that i=mI+j−ul=nI+j, where n=m−u. Finally, equation (1) becomes:




  
    
      
        
          
            s
            0
          
          ⁡
          
            (
            
              
                mIT
                e
              
              +
              
                jT
                e
              
            
            )
          
        
        =
        
          
            
              ∑
              
                n
                =
                
                  -
                  
                    L
                    0
                  
                
              
              
                L
                0
              
            
            ⁢
            
              
                
                  h
                  0
                
                ⁡
                
                  (
                  
                    
                      nIT
                      e
                    
                    +
                    
                      jT
                      e
                    
                  
                  )
                
              
              ⁢
              
                a
                
                  m
                  -
                  n
                
              
              ⁢
              
                
              
              ⁢
              for
              ⁢
              
                
              
              ⁢
              0
            
          
          ≤
          j
          <
          I
        
      
    
    
      
        (
        2
        )
      
    
  




The parameter L0 is the half-length of the transmission filter which is spread over a duration of (2L0+1)ITe. In the particular case of an NRZ transmission filter, L0=0. As regards the transmitted signal s(t), this satisfies the equation s(t)=s0(t)exp(j2Πf0t) as it is equal to the signal s0(t) transposed to the frequency f0. Under these conditions, the expression s(mITe+jTe) is, from (2):




  
    
      
        
          
            
              
                
                  s
                  ⁡
                  
                    (
                    
                      
                        mIT
                        e
                      
                      +
                      
                        jT
                        e
                      
                    
                    )
                  
                
                =
                
                ⁢
                
                  
                    ∑
                    
                      n
                      =
                      
                        -
                        
                          L
                          0
                        
                      
                    
                    
                      L
                      0
                    
                  
                  ⁢
                  
                    
                      
                        h
                        0
                      
                      ⁡
                      
                        (
                        
                          
                            nIT
                            e
                          
                          +
                          
                            jT
                            e
                          
                        
                        )
                      
                    
                    ⁢
                    
                      exp
                      ⁡
                      
                        (
                        
                          j2π
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              f
                              0
                            
                            ⁡
                            
                              (
                              
                                nI
                                +
                                j
                              
                              )
                            
                          
                          ⁢
                          
                            T
                            
                              e
                              _
                            
                          
                        
                        )
                      
                    
                  
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    a
                    
                      m
                      -
                      n
                    
                  
                  ⁢
                  
                    exp
                    ⁡
                    
                      (
                      
                        j2π
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            f
                            0
                          
                          ⁡
                          
                            (
                            
                              m
                              -
                              n
                            
                            )
                          
                        
                        ⁢
                        
                          IT
                          e
                        
                      
                      )
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      ∑
                      
                        n
                        =
                        
                          -
                          
                            L
                            0
                          
                        
                      
                      
                        L
                        0
                      
                    
                    ⁢
                    
                      
                        
                          h
                          
                            F
                            ⁢
                            
                              
                            
                            ⁢
                            0
                          
                        
                        ⁡
                        
                          (
                          
                            
                              nIT
                              e
                            
                            +
                            
                              jT
                              e
                            
                          
                          )
                        
                      
                      ⁢
                      
                        b
                        
                          m
                          -
                          n
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      such
                      ⁢
                      
                        
                      
                      ⁢
                      that
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                  ≤
                  j
                  <
                  I
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          where
          ⁢
          
            
          
          ⁢
          
            
              
                
                  
                    
                      h
                      
                        F
                        ⁢
                        
                          
                        
                        ⁢
                        0
                      
                    
                    ⁡
                    
                      (
                      
                        iT
                        e
                      
                      )
                    
                  
                  =
                  
                  ⁢
                  
                    
                      h
                      ⁡
                      
                        (
                        
                          iT
                          e
                        
                        )
                      
                    
                    ⁢
                    
                      exp
                      ⁡
                      
                        (
                        
                          j2
                          ⁢
                          
                            
                          
                          ⁢
                          π
                          ⁢
                          
                            
                          
                          ⁢
                          
                            f
                            0
                          
                          ⁢
                          
                            iT
                            e
                          
                        
                        )
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    and
                  
                
              
            
            
              
                
                  
                    b
                    i
                  
                  =
                  
                  ⁢
                  
                    
                      a
                      i
                    
                    ⁢
                    
                      exp
                      ⁡
                      
                        (
                        
                          j2π
                          ⁢
                          
                            
                          
                          ⁢
                          
                            f
                            0
                          
                          ⁢
                          
                            iIT
                            e
                          
                        
                        )
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




Reception of the Signals by the Sensors
The transmitted signal s(t) (FIG. 3) passes through a propagation channel before being received on an array made up of N antennas. The propagation channel may be modelled by P multipaths of angle of incidence θp, delay τp and amplitude ρp (1≦p≦P). At the output of the antennas is the vector x(t), which corresponds to the sum of a linear mixture of P multipaths and noise, assumed to be white and Gaussian. This vector of dimension N×1 is given by the following expression:




  
    
      
        
          
            
              
                x
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
                
                  
                    ∑
                    
                      p
                      =
                      1
                    
                    P
                  
                  ⁢
                  
                    
                      ρ
                      p
                    
                    ⁢
                    
                      a
                      ⁡
                      
                        (
                        
                          θ
                          p
                        
                        )
                      
                    
                    ⁢
                    
                      s
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            p
                          
                        
                        )
                      
                    
                  
                
                +
                
                  b
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
          
        
        
          
            
              
                
              
              ⁢
              
                =
                
                  
                    A
                    ⁢
                    
                      
                    
                    ⁢
                    
                      s
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  +
                  
                    b
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        4
        )
      
    
  




where ρp is the amplitude of the pth path, b(t) is the noise vector, assumed to be Gaussian, a(θ) is the response of the array of sensors to a source with angle of incidence θ, and A=[a(θ1) . . . a(θp)] and s(t)=[s(t−τ1) . . . s(t−τp)]T. Noting that τp=rpT+Δτp (where 0≦Δτp<T=ITe and rp is an integer), and inserting equation (3) into equation (4), the vector received by the antennas is given by:





  
    
      
        
          x
          ⁡
          
            (
            
              
                mIT
                e
              
              +
              
                jT
                e
              
            
            )
          
        
        =
        
          
            
              ∑
              
                p
                =
                1
              
              P
            
            ⁢
            
              
                ∑
                
                  
                    
                  
                  ⁢
                  
                    n
                    =
                    
                      -
                      
                        L
                        0
                      
                    
                  
                
                
                  L
                  0
                
              
              ⁢
              
                
                  ρ
                  p
                
                ⁢
                
                  a
                  ⁡
                  
                    (
                    
                      θ
                      p
                    
                    )
                  
                
                ⁢
                
                  
                    h
                    
                      F
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                  ⁡
                  
                    (
                    
                      
                        nIT
                        e
                      
                      +
                      
                        jT
                        e
                      
                      -
                      
                        Δτ
                        p
                      
                    
                    )
                  
                
                ⁢
                
                  
                    b
                    
                      m
                      -
                      n
                      -
                      r
                    
                  
                  p
                
              
            
          
          +
          
            b
            ⁡
            
              (
              
                
                  mIT
                  
                    e
                    ⁢
                    
                      
                    
                  
                
                +
                
                  jT
                  e
                
              
              )
            
          
        
      
    
    
      
        (
        5
        )
      
    
  




By making the change of variable according to up=n+rp, the vector received by the antennas is given by:




  
    
      
        
          x
          ⁡
          
            (
            
              
                mIT
                e
              
              +
              
                jT
                e
              
            
            )
          
        
        =
        
          
            
              ∑
              
                p
                =
                1
              
              P
            
            ⁢
            
              
                ∑
                
                  
                    
                      u
                      p
                    
                    =
                    
                      
                        r
                        p
                      
                      -
                      
                        L
                        0
                      
                    
                  
                  ⁢
                  
                    
                  
                
                
                  
                    r
                    p
                  
                  +
                  
                    L
                    0
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  ρ
                  p
                
                ⁢
                
                  a
                  ⁡
                  
                    (
                    
                      θ
                      p
                    
                    )
                  
                
                ⁢
                
                  
                    h
                    
                      F
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          (
                          
                            
                              u
                              p
                            
                            -
                            
                              r
                              p
                            
                          
                          )
                        
                        ⁢
                        
                          IT
                          e
                        
                      
                      +
                      
                        jT
                        
                          e
                          ⁢
                          
                            
                          
                        
                      
                      -
                      
                        Δτ
                        p
                      
                    
                    )
                  
                
                ⁢
                
                  b
                  
                    m
                    -
                    
                      u
                      p
                    
                  
                
              
            
          
          +
          
            b
            ⁡
            
              (
              
                
                  mIT
                  e
                
                +
                
                  jT
                  e
                
              
              )
            
          
        
      
    
    
      
        (
        6
        )
      
    
  




Notating rmin=min{rp} and rmax=max{rp}, equation (6) may be rewritten as follows:




  
    
      
        
          x
          ⁡
          
            (
            
              
                mIT
                e
              
              +
              
                jT
                e
              
            
            )
          
        
        =
        
          
            
              ∑
              
                p
                =
                1
              
              P
            
            ⁢
            
              
                ∑
                
                  u
                  =
                  
                    
                      r
                      min
                    
                    -
                    
                      L
                      0
                    
                  
                
                
                  
                    r
                    min
                  
                  +
                  
                    L
                    0
                  
                
              
              ⁢
              
                
                  ρ
                  p
                
                ⁢
                
                  a
                  ⁡
                  
                    (
                    
                      θ
                      p
                    
                    )
                  
                
                ⁢
                
                  
                    h
                    
                      F
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          (
                          
                            u
                            -
                            
                              r
                              p
                            
                          
                          )
                        
                        ⁢
                        
                          IT
                          e
                        
                      
                      +
                      
                        jT
                        
                          e
                          ⁢
                          
                            
                          
                        
                      
                      -
                      
                        Δτ
                        p
                      
                    
                    )
                  
                
                ⁢
                
                  
                    Ind
                    
                      [
                      
                        
                          rp
                          -
                          
                            L
                            ⁢
                            
                              
                            
                            ⁢
                            0
                          
                        
                        ,
                        
                          rp
                          +
                          
                            L
                            ⁢
                            
                              
                            
                            ⁢
                            0
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    (
                    u
                    )
                  
                
                ⁢
                
                  b
                  
                    m
                    -
                    u
                  
                
              
            
          
          +
          
            b
            ⁡
            
              (
              
                
                  mIT
                  e
                
                +
                
                  jT
                  e
                
              
              )
            
          
        
      
    
    
      
        (
        7
        )
      
    
  




where Ind[r,q](u) is the usual indicatrix function (Ind[r,q](u)=1 for r≦u≦p and Ind[r,q](u)=0 otherwise) defined over the set of integers relating to the value in the binary set {0,1}, characterized by Ind[r,q](u)=1 if u belongs to the interval [r,q] and Ind[r,q](u)=0 otherwise. Thus, denoting the channel vector by v(t):





  
    
      
        
          v
          ⁡
          
            (
            
              
                uIT
                e
              
              +
              
                jT
                e
              
            
            )
          
        
        =
        
          
            ∑
            
              p
              =
              1
            
            P
          
          ⁢
          
            
              ρ
              p
            
            ⁢
            
              a
              ⁡
              
                (
                
                  θ
                  p
                
                )
              
            
            ⁢
            
              
                h
                
                  F
                  ⁢
                  
                    
                  
                  ⁢
                  0
                
              
              ⁡
              
                (
                
                  
                    
                      (
                      
                        u
                        -
                        
                          r
                          p
                        
                      
                      )
                    
                    ⁢
                    
                      IT
                      e
                    
                  
                  +
                  
                    jT
                    
                      e
                      ⁢
                      
                        
                      
                    
                  
                  -
                  
                    Δτ
                    p
                  
                
                )
              
            
            ⁢
            
              
                Ind
                
                  [
                  
                    
                      rp
                      -
                      
                        L
                        ⁢
                        
                          
                        
                        ⁢
                        0
                      
                    
                    ,
                    
                      rp
                      +
                      
                        L
                        ⁢
                        
                          
                        
                        ⁢
                        0
                      
                    
                  
                  ]
                
              
              ⁡
              
                (
                u
                )
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




where t=uITe+jTe and equation (5) becomes:





  
    
      
        
          
            x
            ⁡
            
              (
              
                
                  mIT
                  e
                
                +
                
                  jT
                  e
                
              
              )
            
          
          ⁢
          
            
              ∑
              
                u
                =
                
                  
                    r
                    min
                  
                  -
                  
                    L
                    0
                  
                
              
              
                
                  r
                  min
                
                +
                
                  L
                  0
                
              
            
            ⁢
            
              
                v
                ⁡
                
                  (
                  
                    
                      uIT
                      e
                    
                    +
                    
                      jT
                      e
                    
                  
                  )
                
              
              ⁢
              
                b
                
                  m
                  -
                  u
                
              
            
          
        
        +
        
          b
          ⁡
          
            (
            
              
                mIT
                e
              
              +
              
                jT
                e
              
            
            )
          
        
      
    
    
      
        (
        9
        )
      
    
  




Inter-Symbol Interference
The observation vector x(t) coming from the antenna array at t=mITe+jTe involves, from equation (9), the symbol bm, but also the symbols bm-u where u is a relative integer lying within the [rmin−L0, rmax+L0] interval, which phenomenon is more widely known as ISI (inter-symbol interference). Let Lc be the number of symbols participating in the ISI and let the interval of values taken by the latter be limited. From equation (9), if the intersection of the intervals [rp−L0, rp+L0] is non-empty, then Lc=|rmax−rmin|+2L0+1. Consequently, when rmax=rmin, that is to say when all the multipaths are correlated, the lower bound of Lc is reached and is given by Lc=2L0+1. This case is also mathematically expressed by




  
    
    
      
        
          max
          p
        
        ⁢
        
          {
          
            τ
            p
          
          }
        
      
      -
      
        
          min
          p
        
        ⁢
        
          {
          
            τ
            p
          
          }
        
      
    
    
  
  <
  
    T
    .
  




On the other hand, if the intersection of said intervals is empty and if all the intervals [rp−L0, rp+L0] are disjoint, then Lc=P×(2L0+1), which constitutes an upper bound for the set of values that can be taken by Lc. The latter situation corresponds specifically to the case of multipaths that are all pairwise decorrelated. This may also be mathematically expressed as ∀i≠j, |ri−rj|>2L0, this condition being obtained whenever |τi−τj|>(2L0+1)T. To summarize, the quantity Lc is in general bounded as follows:

2L0+1≦Lc≦P×(2L0+1)  (10)

The equation expressing the vector received by the sensors can then be rewritten in the following manner, but this time only the Lc symbols bm-u of interest appear:




  
    
      
        
          X
          ⁡
          
            (
            
              
                mIT
                e
              
              +
              
                jT
                e
              
            
            )
          
        
        =
        
          
            
              ∑
              
                l
                =
                1
              
              
                L
                0
              
            
            ⁢
            
              
                h
                ⁡
                
                  (
                  
                    
                      
                        n
                        ⁡
                        
                          (
                          I
                          )
                        
                      
                      ⁢
                      
                        IT
                        e
                      
                    
                    +
                    
                      jT
                      e
                    
                  
                  )
                
              
              ⁢
              
                b
                
                  m
                  -
                  
                    n
                    ⁡
                    
                      (
                      I
                      )
                    
                  
                
              
            
          
          +
          
            b
            ⁡
            
              (
              
                
                  mIT
                  e
                
                +
                
                  jT
                  e
                
              
              )
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




where ∀1≦I≦Lc and rmin−L0≦n(I)≦rmin+L0 and where:





  
    
      
        
          h
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            ∑
            
              p
              =
              1
            
            P
          
          ⁢
          
            
              ρ
              p
            
            ⁢
            
              a
              ⁡
              
                (
                
                  θ
                  p
                
                )
              
            
            ⁢
            
              
                h
                F0
              
              ⁡
              
                (
                
                  t
                  -
                  
                    τ
                    p
                  
                
                )
              
            
          
        
      
    
    
      
        (
        12
        )
      
    
  




ICA Techniques
The process uses ICA techniques based on the following model, given by way of entirely nonlimiting illustration:




  
    
      
        
          
            
              
                u
                k
              
              =
              
                
                  
                    ∑
                    
                      i
                      =
                      1
                    
                    
                      
                        
                      
                      ⁢
                      L
                    
                  
                  ⁢
                  
                    
                      g
                      i
                    
                    ⁢
                    
                      s
                      ik
                    
                  
                
                +
                
                  n
                  k
                
              
            
          
        
        
          
            
              =
              
                
                  Gs
                  k
                
                +
                
                  n
                  k
                
              
            
          
        
      
    
    
      
        (
        13
        )
      
    
  




where uk is a vector of dimension M×1 received at time k, sik is the ith component of the signal sk at time k, n is the noise vector and G=[g1 . . . gL]. The objective of the ICA methods is to extract the I=L components sik and to identify their signatures gi (the vectorial response of source i through the observation uk) on the basis of the observation uk. The number I=L of components must not exceed the dimension M of the observation vector. The methods of references [4], [5] and [15] use 2nd- and 4th-order statistics of the observations uk. The first step uses 2nd-order statistics for the observations uk (these observations may be functions of the signals received by the sensors) in order to obtain a new observation zk, such that:





  
    
      
        
          
            
              
                z
                k
              
              =
              
                
                  W
                  1
                
                ⁢
                
                  u
                  k
                
              
            
          
        
        
          
            
              =
              
                
                  
                    ∑
                    
                      i
                      =
                      1
                    
                    
                      
                        
                      
                      ⁢
                      L
                    
                  
                  ⁢
                  
                    
                      
                        g
                         ̆
                      
                      i
                    
                    ⁢
                    
                      s
                      ik
                    
                  
                
                +
                
                  
                    n
                    ~
                  
                  k
                
              
            
          
        
        
          
            
              =
              
                
                  
                    G
                     ̆
                  
                  ⁢
                  
                    s
                    k
                  
                
                +
                
                  
                    n
                    ~
                  
                  k
                
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




where the signatures {hacek over (g)}i (1≦i≦L) are orthogonal, {hacek over (G)}=[{hacek over (g)}1 . . . {hacek over (g)}L] and sk=[s1k . . . sLk]T. The second step consists in identifying the orthogonal base of the {hacek over (G)} values using 4th order statistics of the whitened observations zk. In this way, the signals sk may be extracted by effecting:

ŝk={hacek over (G)}#zk={hacek over (G)}#W1uk  (15)

where Ŝk is the estimate of the signals sk and where # is the pseudo-inversion operator defined by {hacek over (G)}#=({hacek over (G)}H{hacek over (G)})−1{hacek over (G)}H.

The ICAR method [19] uses only 4th-order statistics to identify the matrix G=[g1 . . . gk] of signatures.
To summarize, the idea employed in the process according to the invention is to construct a spatio-temporal observation, the mixed sources of which are symbol trains from the transmitter. Each symbol train is for example the same symbol train but shifted by an integral number of symbol periods T.
Several ways of implementing the method will be described below, some of which are explained by way of non-limiting illustration.
First Way of Implementing the Process.
FIG. 7 shows a first illustrative way of implementing the process in which the signal is received in baseband.
The method comprises a step I.1 of determining the symbol time Te, for example by applying a cyclic detection algorithm, such as that described for example in [1] [10].
The next step I.2 consists in interpolating the observations x(t) with I samples per symbol, such that T=ITe.
Under these conditions where f0=0 and bk=ak, equation (11) for the vector becomes:




  
    
      
        
          x
          ⁡
          
            (
            
              
                mIT
                e
              
              +
              
                jT
                e
              
            
            )
          
        
        =
        
          
            
              
                ∑
                
                  l
                  =
                  1
                
                
                  L
                  0
                
              
              ⁢
              
                
                  h
                  ⁡
                  
                    (
                    
                      
                        
                          n
                          ⁡
                          
                            (
                            I
                            )
                          
                        
                        ⁢
                        
                          IT
                          e
                        
                      
                      +
                      
                        jT
                        e
                      
                    
                    )
                  
                
                ⁢
                
                  a
                  
                    m
                    -
                    
                      n
                      ⁡
                      
                        (
                        I
                        )
                      
                    
                  
                
              
            
            +
            
              
                b
                ⁡
                
                  (
                  
                    
                      mIT
                      e
                    
                    +
                    
                      jT
                      e
                    
                  
                  )
                
              
              ⁢
              
                
              
              ⁢
              for
              ⁢
              
                
              
              ⁢
              0
            
          
          ≤
          j
          <
          I
        
      
    
    
      
        (
        16
        )
      
    
  




Since equation (16) is valid for 0≦j<I, the method constructs the next spatio-temporal observation (step I3) from the observations x(kTe):




  
    
      
        
          
            
              
                
                  z
                  ⁡
                  
                    (
                    
                      mIT
                      e
                    
                    )
                  
                
                =
                
                ⁢
                
                  [
                  
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      x
                                      ⁡
                                      
                                        (
                                        
                                          mIT
                                          e
                                        
                                        )
                                      
                                    
                                  
                                
                                
                                  
                                    
                                      x
                                      ⁡
                                      
                                        (
                                        
                                          
                                            mIT
                                            e
                                          
                                          +
                                          
                                            T
                                            e
                                          
                                        
                                        )
                                      
                                    
                                  
                                
                              
                            
                          
                          
                            
                              ⋮
                            
                          
                        
                      
                    
                    
                      
                        
                          x
                          ⁡
                          
                            (
                            
                              
                                mIT
                                e
                              
                              +
                              
                                
                                  (
                                  
                                    I
                                    -
                                    1
                                  
                                  )
                                
                                ⁢
                                
                                  T
                                  e
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      ∑
                      
                        l
                        =
                        1
                      
                      
                        L
                        c
                      
                    
                    ⁢
                    
                      
                        
                          h
                          z
                        
                        ⁡
                        
                          (
                          
                            n
                            ⁡
                            
                              (
                              I
                              )
                            
                          
                          )
                        
                      
                      ⁢
                      
                        a
                        
                          m
                          -
                          
                            n
                            ⁡
                            
                              (
                              I
                              )
                            
                          
                        
                      
                    
                  
                  +
                  
                    
                      
                        b
                        z
                      
                      ⁡
                      
                        (
                        
                          mIT
                          e
                        
                        )
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    where
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        h
                        z
                      
                      ⁡
                      
                        (
                        n
                        )
                      
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  [
                  
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      h
                                      
                                        n
                                        ,
                                        0
                                      
                                    
                                  
                                
                                
                                  
                                    
                                      h
                                      
                                        n
                                        ,
                                        1
                                      
                                    
                                  
                                
                              
                            
                          
                          
                            
                              ⋮
                            
                          
                        
                      
                    
                    
                      
                        
                          h
                          
                            n
                            ,
                            
                              I
                              -
                              1
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                
                  with
                  ⁢
                  
                    
                  
                  ⁢
                  
                    h
                    
                      n
                      ,
                      j
                    
                  
                
                =
                
                ⁢
                
                  
                    h
                    ⁡
                    
                      (
                      
                        
                          nIT
                          e
                        
                        +
                        
                          jT
                          e
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  and
                
              
            
          
          
            
              
                
                  
                    b
                    z
                  
                  ⁡
                  
                    (
                    
                      mIT
                      e
                    
                    )
                  
                
                =
                
                ⁢
                
                  
                    
                      [
                      
                        
                          
                            b
                            ⁡
                            
                              (
                              
                                mIT
                                e
                              
                              )
                            
                          
                          T
                        
                        ⁢
                        ...
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            b
                            ⁡
                            
                              (
                              
                                
                                  mIT
                                  e
                                
                                +
                                
                                  
                                    (
                                    
                                      I
                                      -
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  
                                    T
                                    e
                                  
                                
                              
                              )
                            
                          
                          T
                        
                      
                      ]
                    
                    T
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




Since it is known that x(t) has the dimensions N×1, the vector z(t) has the dimensions NI×1.
h(k) is a vector whose nth component is the kth component of the filter that linearly filters the symbol train {am} on the nth sensor. The filter for the vector coefficient h(k) depends both on the wave-shaping filter and on the propagation channel.
To extract the Lc symbol trains {am-i} of interest (the number of symbols participating in the ISI), the method samples the received signal with I=(2L0+1), assuming that P≦N.
Since it is known that the NRZ filter satisfies 2L0+1=1 and the Nyquist filter 2L0+1=3 for a roll-off of 0.25, the symbol trains may be extracted for these two wave-shaping filters when P≦NI and 3P≦NI, respectively.
Having determined the observation vector z(t), the process applies an ICA-type method to estimate the Lc symbol trains {am-i} associated with the channel vectors ĥz,j=ĥz(kj).
The jth output of the ICA method gives the symbol train {âm,j} associated with the channel vector ĥz,j. The estimated symbol trains {âm,j} arrive in a different order from that of the {am-i} trains satisfying:

âm,j=ρexp(jαi)am-i and ĥz,j=ĥz(i)  (18)

The symbol trains {âm,j} are estimated with the same amplitude because the symbol trains {am-i} all have the same power, satisfying the equation:

E[|am-n(1)|2]= . . . =E[|am-n(Lc)|2].

The next step I.4 of the process has the objective of ordering the Lc outputs (âm,j,ĥz,j) in the same order as the inputs (am-i, hz(i)) so as to obtain the channel vectors ĥz,j=ĥz(kj). To do this, the process intercorrelates pairwise the outputs âm,i and âm,j, calculating the following criterion ci,j(k):




  
    
      
        
          
            c
            
              i
              ,
              j
            
          
          ⁡
          
            (
            k
            )
          
        
        =
        
          
            E
            ⁡
            
              [
              
                
                  
                    a
                    ^
                  
                  
                    m
                    ,
                    i
                  
                
                ⁢
                
                  
                    a
                    ^
                  
                  
                    
                      m
                      -
                      k
                    
                    ,
                    j
                  
                  *
                
              
              ]
            
          
          
            
              
                E
                ⁡
                
                  [
                  
                    
                      
                        a
                        ^
                      
                      
                        m
                        ,
                        i
                      
                    
                    ⁢
                    
                      
                        a
                        ^
                      
                      
                        m
                        ,
                        i
                      
                      *
                    
                  
                  ]
                
              
              ⁢
              
                E
                ⁡
                
                  [
                  
                    
                      
                        a
                        ^
                      
                      
                        
                          m
                          -
                          k
                        
                        ,
                        j
                      
                    
                    ⁢
                    
                      
                        a
                        ^
                      
                      
                        
                          m
                          -
                          k
                        
                        ,
                        j
                      
                      *
                    
                  
                  ]
                
              
            
          
        
      
    
    
      
        (
        19
        )
      
    
  




When the function |ci,j(k)| is a maximum in k=kmax, the ith and jth outputs satisfy the equation: âm,i=âm-k max,j. The algorithm for classifying the outputs âm,n(1) . . . âm,n(Lc) is for example composed of the following steps:
Step A.1: Determination of the output âm,imax associated with the channel vector of higher-modulus ĥz,jmax.
Step A.2: For all the outputs âm-k,j where j≠imax, determination of the indices k=kj maximizing the |cimax,j(k)| criterion. From this is deduced, for each j, that âm,imax=âm-kj,j. Since it is known that cimax,j(kj)=exp(jαimax−jαj) the jth output is reset to the same phase as the iith imax output by taking âm-kj=cimax,j(kj)âm,j. The channel vectors are also reset in terms of phase by taking: {circumflex over (ĥ)}z(kj)=ĥz,jcimax,j(kj)*.
Step A.3: This step reorders the outputs âm-kj and the channel vectors {circumflex over (ĥ)}z,j=ĥz(kj) in the increasing order of the Kj, since it is known that âm=âm,imax and that {circumflex over (ĥ)}z(0)=ĥz,imax.
After these three steps, the symbol trains {âm-k} associated with the channel vectors ĥz(kj) are obtained. Since it is known that the estimated symbols satisfy the equation âm-k=exp(jαimax)am-k, the last step of the process consists in estimating this phase αimax. To do this, the constellation of symbols ak is firstly identified among a database consisting of the set of possible constellations. This database consists of known constellations such as nPSK, n-QAM. Each time that a new constellation is detected or becomes known, this is added to the database.
FIG. 6 shows an example of an 8-QAM constellation when αimax=0 and αimax≠0. In this implementation example, the process then includes the following steps:
The next step I.5 consists in determining the output phase associated with the channel vector of higher modulus. To identify the constellation and determine the phase, the process performs, for example, the following steps:
Step I.5=Steps B.1, B.2 and B.3
Step°B.1: Estimation of the positions of the states of the constellation (red points in the figure) by seeking the maximum of the 2D histogramme of the points Mk=(real(âk), imag(âk)). For a constellation consisting of M states, M pairs (ûm,{circumflex over (v)}m) for 1≦m≦M are obtained.
Step°B.2: Determination of the type of constellation by comparing the position of the states (ûm,{circumflex over (v)}m) of the constellation of {âk} symbols with a database comprising the set of possible constellations. The closest constellation is made up of the states (um,vm) for 1≦m≦M.
Step B.3: Determination of the phase αimax by minimizing in the sense of the least squares the following system of equations:

ûm=cos(αimax)um−sin(αimax)vm and {circumflex over (v)}m=sin(αimax)um+cos(αimax)vm for 1≦m≦M.

The process may include a step of estimating the propagation channel parameters of angle θp and delay τp, of equation (8) by the algorithm proposed in [8]. The step consists in extracting firstly the vectors h(nITe+jTe) for 0≦j<I from the channel vectors ĥz(nj) defined in equation (17). Followed by construction of the matrix H=[h(n(1)ITe) . . . h(n(Lc)ITe)] from equation (11) with the h(nITe+jTe) values in order to apply the parametric estimation method [8] for the multipaths: (θp, τp) 1≦p≦p.
Second Way of Implementing the Process
FIGS. 8 and 9 show schematically another way of implementing the process, which may include two variants corresponding to the decorrelated multipath case and to the groupwise correlated multipath case, respectively.
Decorrelated Multipath Case.
The signal is received in baseband with {bk}={ak}.
The multipaths, the delays of which satisfy the relationship |τj−τi|>(2L0+1)T, have the advantage of being decorrelated with one another, satisfying the equation: E[s(t−τi)s(t−τj)*]=0. By examining equation (4), it may therefore be seen that it is sufficient to apply an ICA type method when P≦N to the observation x(t) in order to obtain the signals s(t−τp) for each of the multibars. After estimating the signals for the various multipaths, the process determines their powers in order to keep the signal s(t−τpmax) of the multipath of higher amplitude ρpmax. This main path is determined using the fact that the outputs of the ICA methods asymptotically satisfy:




  
    
      
        
          
            
              
                x
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
                
                  ∑
                  
                    p
                    =
                    1
                  
                  P
                
                ⁢
                
                  
                    ρ
                    p
                  
                  ⁢
                  
                    a
                    ⁡
                    
                      (
                      
                        θ
                        p
                      
                      )
                    
                  
                  ⁢
                  
                    s
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          p
                        
                      
                      )
                    
                  
                
              
            
          
        
        
          
            
              =
              
                
                  ∑
                  
                    p
                    =
                    1
                  
                  P
                
                ⁢
                
                  
                    
                      a
                      ^
                    
                    i
                  
                  ⁢
                  
                    
                      
                        s
                        ^
                      
                      i
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  with
                
              
            
          
        
        
          
            
              
                
                  
                    
                      s
                      ^
                    
                    i
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                =
                
                  
                    
                      s
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            p
                          
                        
                        )
                      
                    
                    
                      
                        γ
                        p
                      
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  and
                
              
              ⁢
              
                
              
            
          
        
        
          
            
              
                
                  a
                  ^
                
                i
              
              =
              
                
                  
                    γ
                    p
                  
                
                ⁢
                
                  ρ
                  p
                
                ⁢
                
                  a
                  ⁡
                  
                    (
                    
                      θ
                      p
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        20
        )
      
    
  




where γp=ρp2E[|s(t−τp)|2]. Since the vectors a(θp) are normed, satisfying the equation a(θp)Ha(θp)=N, the path of maximum amplitude will be associated with the imaxth output where αimax=âimaxHâimax is a maximum. From equation (3), the output ŝimax(t)=s(t−τpmax) satisfies the equation:





  
    
      
        
          
            
              s
              ^
            
            
              i
              ⁢
              
                
              
              ⁢
              max
            
          
          ⁡
          
            (
            
              
                mIT
                e
              
              +
              
                jT
                e
              
            
            )
          
        
        =
        
          
            ∑
            
              n
              =
              
                -
                
                  L
                  0
                
              
            
            
              L
              0
            
          
          ⁢
          
            
              
                h
                
                  F
                  ⁢
                  
                    
                  
                  ⁢
                  0
                
              
              ⁡
              
                (
                
                  
                    nIT
                    e
                  
                  +
                  
                    jT
                    e
                  
                  -
                  
                    τ
                    
                      p
                      ⁢
                      
                        
                      
                      ⁢
                      max
                    
                  
                
                )
              
            
            ⁢
            
              a
              
                m
                -
                n
              
            
          
        
      
    
    
      
        (
        21
        )
      
    
  




such that 0≦j<I

and it is possible to constitute the following observation vector:





  
    
      
        
          
            
              
                z
                ⁡
                
                  (
                  
                    mIT
                    e
                  
                  )
                
              
              =
              
                [
                
                  
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      
                                        s
                                        ^
                                      
                                      
                                        i
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        max
                                      
                                    
                                    ⁡
                                    
                                      (
                                      
                                        mIT
                                        e
                                      
                                      )
                                    
                                  
                                
                              
                              
                                
                                  
                                    
                                      
                                        s
                                        ^
                                      
                                      
                                        i
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        max
                                      
                                    
                                    ⁡
                                    
                                      (
                                      
                                        
                                          mIT
                                          e
                                        
                                        +
                                        
                                          T
                                          e
                                        
                                      
                                      )
                                    
                                  
                                
                              
                            
                          
                        
                        
                          
                            ⋮
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            s
                            ^
                          
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            max
                          
                        
                        ⁡
                        
                          (
                          
                            
                              mIT
                              e
                            
                            +
                            
                              
                                (
                                
                                  I
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              
                                T
                                e
                              
                            
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
                
                  ∑
                  
                    n
                    =
                    
                      -
                      
                        L
                        0
                      
                    
                  
                  
                    L
                    0
                  
                
                ⁢
                
                  
                    
                      h
                      z
                    
                    ⁡
                    
                      (
                      n
                      )
                    
                  
                  ⁢
                  
                    a
                    
                      m
                      -
                      n
                    
                  
                
              
            
          
        
        
          
            
              
                where
                ⁢
                
                  
                
                ⁢
                
                  
                    h
                    z
                  
                  ⁡
                  
                    (
                    n
                    )
                  
                
              
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      h
                                      
                                        n
                                        ,
                                        0
                                      
                                    
                                  
                                
                                
                                  
                                    
                                      h
                                      
                                        n
                                        ,
                                        1
                                      
                                    
                                  
                                
                              
                            
                          
                          
                            
                              ⋮
                            
                          
                        
                      
                    
                    
                      
                        
                          h
                          
                            n
                            ,
                            
                              I
                              -
                              1
                            
                          
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  
                
                ⁢
                and
              
            
          
        
      
    
    
      
        (
        22
        )
      
    
  




where hn,j=hF0(nITe+jTe−τpmax). According to the model of equation (22), it is sufficient to apply an ICA method to the observation z(mITe) in order to estimate the 2L0+1 symbol trains {am-n} with −L0≦n≦L0. To extract the angles of incidence θp of the propagation channel, it is sufficient from equation (20) to find, for each signature âi (1≦i≦P), the maximum of criterion c(θ)=|a(θ)Hâi|2. To extract the delays τi−τ1 of the propagation channel, it is sufficient from equation (20) to find, for each signal ŝi(t) (1≦i≦P), the maximum of the c(τ)=|ŝi(t−τ)ŝ1(t)*|2 criterion.

To summarize, this variant comprises, for example, the following steps:
Step II.a.1: Determination of the symbol period T, applying a cyclic detection algorithm as in [1] [10].
Step II.a.2: Sampling of the observations x(t) with I samples per symbol such that T=ITe.
Step II.a.3: Application of an ICA method to the observations x(t) in order to obtain ŝi(t) and âi for 1≦i≦P.
Step II.a.4: Determination of the output i=imax where αi=âiHâi is its maximum.
Step II.a.5: Formation of the observation vector z(t) of equation (22) from the signal ŝimax(t).
Step II.a.6: Application of an ICA method for estimating the symbol trains {am-n} where −L0≦n≦L0. From the symbol trains is chosen that one which is associated with the higher-modulus vector hz(n), namely {âm}.
Step II.a.7: Determination of the phase αimax of the output associated with the higher-modulus vector hz(n) applying steps B.1, B.2 et B.3.
Step II.a.8: Phase-resetting of the symbol train {âm} by taking {circumflex over (â)}m=âmexp(−jαimax). The symbol train {{circumflex over (â)}m} constitutes the output of the demodulator of this subprocess.
Step II.a.9: Estimation of the propagation channel parameters, namely angle θp and delay τp, by maximizing, for 1≦i≦P the |a(θ)Hâi|2 and |ŝi(t−τ)ŝ1(t)*|2 criteria for the angles and delays respectively.
General Case for any or Groupwise-Correlated Multipaths
In this variant, the diagram for which is given in FIG. 9, the process considers that some of the multipaths are correlated. Considering that the transmitter is received according to Q groups of correlated multipaths, the signal vector received by the sensors becomes, from equation (4):




  
    
      
        
          
            
              
                x
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
                
                  
                    ∑
                    
                      q
                      =
                      1
                    
                    Q
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        p
                        =
                        1
                      
                      
                        P
                        q
                      
                    
                    ⁢
                    
                      
                        ρ
                        
                          p
                          ,
                          q
                        
                      
                      ⁢
                      
                        a
                        ⁡
                        
                          (
                          
                            θ
                            
                              p
                              ,
                              q
                            
                          
                          )
                        
                      
                      ⁢
                      
                        s
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              τ
                              
                                p
                                ,
                                q
                              
                            
                          
                          )
                        
                      
                    
                  
                
                +
                
                  b
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
          
        
        
          
            
              =
              
                
                  
                    ∑
                    
                      q
                      =
                      1
                    
                    Q
                  
                  ⁢
                  
                    
                      A
                      q
                    
                    ⁢
                    
                      Ω
                      q
                    
                    ⁢
                    
                      s
                      ⁡
                      
                        (
                        
                          t
                          ,
                          
                            
                              τ
                              _
                            
                            q
                          
                        
                        )
                      
                    
                  
                
                +
                
                  b
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        23
        )
      
    
  




Where Aq=[a(θ1,q) . . . a(θPq,q)], Ωq=diag([ρ1,q . . . ρPq,q]) and s(t, τq)=[s(t−τ1,q) . . . s(t−τPq,q)]T with τq=[τ1,q . . . τPq,q]T. The following signals and signatures are estimated as output of the separator by applying an ICA method:




  
    
      
        
          
            
              
                A
                ^
              
              =
              
                [
                
                  
                    
                      a
                      ^
                    
                    1
                  
                  ⁢
                  
                    
                  
                  ⁢
                  ...
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      a
                      ^
                    
                    
                      PQ
                      ,
                      Q
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
                
                  [
                  
                    
                      A
                      1
                    
                    ⁢
                    
                      U
                      1
                    
                    ⁢
                    
                      
                    
                    ⁢
                    ...
                    ⁢
                    
                      
                    
                    ⁢
                    
                      A
                      Q
                    
                    ⁢
                    
                      U
                      Q
                    
                  
                  ]
                
                ⁢
                Π
                ⁢
                
                  
                
                ⁢
                and
              
            
          
        
        
          
            
              
                
                  s
                  ^
                
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
                Π
                ⁡
                
                  [
                  
                    
                      
                        
                          
                            V
                            
                              1
                              ⁢
                              
                                
                              
                            
                          
                          ⁢
                          
                            s
                            ⁡
                            
                              (
                              
                                t
                                ,
                                
                                  
                                    τ
                                    _
                                  
                                  1
                                
                              
                              )
                            
                          
                        
                      
                    
                    
                      
                        ⋮
                      
                    
                    
                      
                        
                          
                            V
                            Q
                          
                          ⁢
                          
                            s
                            ⁡
                            
                              (
                              
                                t
                                ,
                                
                                  
                                    τ
                                    _
                                  
                                  Q
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
                [
                
                  
                    
                      
                        
                          
                            s
                            ^
                          
                          1
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                  
                    
                      ⋮
                    
                  
                  
                    
                      
                        
                          
                            s
                            ^
                          
                          
                            
                              P
                              Q
                            
                            ⁢
                            
                              x
                              Q
                            
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




where Π is a permutation matrix, UqVq=Ωq and VqE[s(t,τq)s(t,τq)H]VqH=IPq. Thus, the paths decorrelated such that E[s(t−τp,q)s(t−τp′,q′)*]=0 are received on different channels ŝi(t) and ŝj(t). The correlated paths where E[s(t−τp,q)s(t−τp′,q′)*]≠0 are mixed in the same channel ŝi(t) and are present on PQ at the same time. In the 1st step of this subprocess, we use this result to identify the Q group of correlated multipaths. Taking the outputs i and j of the separator, the two following hypotheses may be tested:





  
    
      
        
          H
          0
        
        ⁢
        
          :
        
        ⁢
        
          {
          
            
              
                
                  
                    
                      
                        
                          s
                          ^
                        
                        i
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    =
                    
                      
                        b
                        i
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
              
                
                  
                    
                      
                        
                          s
                          ^
                        
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    =
                    
                      
                        b
                        j
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
            
            ⁢
            
              
            
            ⁢
            and
            ⁢
            
              
            
            ⁢
            
              H
              1
            
            ⁢
            
              :
            
            ⁢
            
              
            
            ⁢
            
              {
              
                
                  
                    
                      
                        
                          
                            s
                            ^
                          
                          i
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      =
                      
                        
                          
                            α
                            i
                          
                          ⁢
                          
                            s
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  p
                                
                              
                              )
                            
                          
                        
                        +
                        
                          
                            b
                            i
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                      
                    
                  
                
                
                  
                    
                      
                        
                          
                            s
                            ^
                          
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      =
                      
                        
                          
                            α
                            j
                          
                          ⁢
                          
                            s
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  p
                                
                              
                              )
                            
                          
                        
                        +
                        
                          
                            b
                            j
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        25
        )
      
    
  




where E[bi(t) bj(t−τ)*]=0 whatever the value of τ. Thus for the H0 hypothesis, no multipaths exist common to the two output i and j, and for the H1 hypothesis there is at least one of them. The test consists in determining whether the outputs ŝi(t) and ŝj(t−τ) are correlated for at least one of the τ values satisfying |τ|<τmax. To do this, the Gardner test [3] is applied, which compares the following likelihood ratio with a threshold:





  
    
      
        
          
            
              
                
                  V
                  ij
                
                ⁡
                
                  (
                  τ
                  )
                
              
              =
              
              ⁢
              
                
                  -
                  2
                
                ⁢
                K
                ⁢
                
                  
                
                ⁢
                
                  ln
                  ⁡
                  
                    (
                    
                      1
                      -
                      
                        
                          
                            
                            
                              
                                
                                  r
                                  ^
                                
                                ij
                              
                              ⁡
                              
                                (
                                τ
                                )
                              
                            
                            
                          
                          2
                        
                        
                          
                            
                              
                                r
                                ^
                              
                              ii
                            
                            ⁡
                            
                              (
                              0
                              )
                            
                          
                          ⁢
                          
                            
                              
                                r
                                ^
                              
                              jj
                            
                            ⁡
                            
                              (
                              0
                              )
                            
                          
                        
                      
                    
                    )
                  
                
                ⁢
                
                  
                
                ⁢
                with
              
            
          
        
        
          
            
              
                
                  
                    r
                    ^
                  
                  ij
                
                ⁡
                
                  (
                  τ
                  )
                
              
              =
              
              ⁢
              
                
                  1
                  K
                
                ⁢
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    K
                  
                  ⁢
                  
                    
                      
                        
                          s
                          ^
                        
                        i
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    ⁢
                    
                      
                        
                          
                            s
                            ^
                          
                          j
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            τ
                          
                          )
                        
                      
                      *
                    
                    ⁢
                    
                      
                    
                    ⁢
                    or
                  
                
              
            
          
        
        
          
            
              
                
                  
                    V
                    ij
                  
                  ⁡
                  
                    (
                    τ
                    )
                  
                
                <
                
                ⁢
                η
              
              ⇒
              
                hypothesis
                ⁢
                
                  
                
                ⁢
                
                  H
                  0
                
              
            
          
        
        
          
            
              
                
                  And
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      V
                      ij
                    
                    ⁡
                    
                      (
                      τ
                      )
                    
                  
                
                ≥
                
                ⁢
                η
              
              ⇒
              
                hypothesis
                ⁢
                
                  
                
                ⁢
                
                  H
                  1
                
              
            
          
        
      
    
    
      
        (
        26
        )
      
    
  




The threshold η is determined in [3] in relation to a chi square law with 2 degrees of freedom. The output associated with the 1st output are firstly sought by starting the test by 2<j≦PQ×Q and i=1. Next, removed from the list of outputs are all those associated with the 1st which will constitute the 1st group with q=1. The same series of tests is then restarted with the other outputs not correlated with the 1st output in order to constitute the 2nd group. This operation will be carried as far as the last group where, in the end, no output channel will remain. After the sorting, what will finally be obtained are:

Âq=AqUq and ŝq(t)=Vqs(t,τq) for (1≦q≦Q)  (27)

The angles of incidence θp,q are determined from the Âq values for (1≦q≦Q) applying the MUSIC [1] algorithm to the ÂqÂqH matrix. The matrices Aq are deduced from these goniometry values. Since is it known that xq(t)=Âqŝq(t)=AqΩqs(t, τq), s(t, τq) is deduced therefrom to within a diagonal matrix by taking ŝ(t,τq)=Aq#Xq(t). Since the elements of ŝ(t,τq) are composed of the signals ŝ(t−τp,q), the delays τp,q−τ1,1 are determined by maximizing the c(τ)=|ŝq,p(t−τ)ŝ1,1(t)*|2 criteria where ŝq,p(t) is the pth component of ŝ(t,τq).
Since it is known that E[ŝq(t)ŝq(t)H]=IPq, that AqHAq=N IPq and that Âqŝq(t)=AqΩqs(t,τq), it is deduced therefrom that the group of multipaths associated with the largest amplitudes Ωq maximizes the following criterion: cri(q)=trace(ÂqHÂq). From this is deduced the best output associated with Âqmax and ŝqmax(t). Since from equation (3) the vector s(t, τqmax) satisfies equation:




  
    
      
        
          
            
              
                s
                ⁡
                
                  (
                  
                    
                      mlT
                      e
                    
                    +
                    
                      jT
                      
                        e
                        ,
                        
                          
                            T
                            _
                          
                          
                            q
                            ⁢
                            
                              
                            
                            ⁢
                            max
                          
                        
                      
                    
                  
                  )
                
              
              =
              
                [
                
                  
                    
                      
                        s
                        ⁡
                        
                          (
                          
                            
                              mlT
                              e
                            
                            +
                            
                              jT
                              e
                            
                            -
                            
                              τ
                              
                                q
                                ⁢
                                
                                  
                                
                                ⁢
                                max
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                          
                          )
                        
                      
                    
                  
                  
                    
                      ⋮
                    
                  
                  
                    
                      
                        s
                        ⁡
                        
                          (
                          
                            
                              mlT
                              e
                            
                            +
                            
                              jT
                              e
                            
                            -
                            
                              τ
                              
                                q
                                ⁢
                                
                                  
                                
                                ⁢
                                max
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  P
                                  
                                    q
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    max
                                  
                                
                              
                            
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
                
                  ∑
                  
                    n
                    =
                    
                      -
                      
                        L
                        0
                      
                    
                  
                  
                    L
                    0
                  
                
                ⁢
                
                  
                    
                      h
                      
                        F
                        ⁢
                        
                          
                        
                        ⁢
                        0
                      
                    
                    ⁡
                    
                      (
                      
                        
                          nlT
                          e
                        
                        +
                        
                          jT
                          
                            e
                            ,
                            
                              
                                T
                                _
                              
                              ⁢
                              q
                              ⁢
                              
                                
                              
                              ⁢
                              max
                            
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    a
                    
                      m
                      -
                      n
                    
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      0
                      ≤
                      j
                      <
                      
                        1
                        ⁢
                        
                          
                            
                          
                          ⁢
                          
                            
                          
                        
                        ⁢
                        and
                        ⁢
                        
                          
                        
                        ⁢
                        with
                        ⁢
                        
                          
                        
                        ⁢
                        
                          h
                          
                            F
                            ⁢
                            
                              
                            
                            ⁢
                            0
                          
                        
                      
                    
                  
                
                
                  
                    
                      (
                      
                        
                          nlT
                          e
                        
                        +
                        
                          jT
                          
                            e
                            ,
                            
                              
                                T
                                _
                              
                              ⁢
                              q
                              ⁢
                              
                                
                              
                              ⁢
                              max
                            
                          
                        
                      
                      )
                    
                  
                
              
              =
              
                [
                
                  
                    
                      
                        
                          h
                          
                            F
                            ⁢
                            
                              
                            
                            ⁢
                            0
                          
                        
                        ⁡
                        
                          (
                          
                            
                              nlT
                              e
                            
                            +
                            
                              jT
                              e
                            
                            -
                            
                              τ
                              
                                
                                  q
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  max
                                
                                ,
                                1
                              
                            
                          
                          )
                        
                      
                    
                  
                  
                    
                      ⋮
                    
                  
                  
                    
                      
                        
                          h
                          
                            F
                            ⁢
                            
                              
                            
                            ⁢
                            0
                          
                        
                        ⁡
                        
                          (
                          
                            
                              nlT
                              e
                            
                            +
                            
                              jT
                              e
                            
                            -
                            
                              τ
                              
                                
                                  q
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  max
                                
                                ,
                                
                                  P
                                  
                                    q
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    max
                                  
                                
                              
                            
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        28
        )
      
    
  




for

it is possible to constitute the following observation vector from equation (27):





  
    
      
        
          
            
              
                z
                ⁡
                
                  (
                  
                    mlT
                    e
                  
                  )
                
              
              =
              
                [
                
                  
                    
                      
                        
                          
                            s
                            ^
                          
                          
                            q
                            ⁢
                            
                              
                            
                            ⁢
                            max
                          
                        
                        ⁡
                        
                          (
                          
                            mlT
                            e
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            s
                            ^
                          
                          
                            q
                            ⁢
                            
                              
                            
                            ⁢
                            max
                          
                        
                        ⁡
                        
                          (
                          
                            
                              mlT
                              e
                            
                            +
                            
                              T
                              e
                            
                          
                          )
                        
                      
                    
                  
                  
                    
                      ⋮
                    
                  
                  
                    
                      
                        
                          
                            s
                            ^
                          
                          
                            q
                            ⁢
                            
                              
                            
                            ⁢
                            max
                          
                        
                        ⁡
                        
                          (
                          
                            
                              mlT
                              e
                            
                            +
                            
                              
                                (
                                
                                  I
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              
                                T
                                e
                              
                            
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
                
                  ∑
                  
                    n
                    =
                    
                      -
                      
                        L
                        o
                      
                    
                  
                  
                    L
                    0
                  
                
                ⁢
                
                  
                    
                      h
                      z
                    
                    ⁡
                    
                      (
                      n
                      )
                    
                  
                  ⁢
                  
                    a
                    
                      m
                      -
                      n
                    
                  
                
              
            
          
        
        
          
            
              
                where
                ⁢
                
                  
                
                ⁢
                
                  
                    h
                    z
                  
                  ⁡
                  
                    (
                    n
                    )
                  
                
              
              =
              
                [
                
                  
                    
                      
                        h
                        
                          n
                          ,
                          0
                        
                      
                    
                  
                  
                    
                      
                        h
                        
                          n
                          ,
                          1
                        
                      
                    
                  
                  
                    
                      ⋮
                    
                  
                  
                    
                      
                        h
                        
                          n
                          ,
                          
                            l
                            -
                            1
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        29
        )
      
    
  




where hn,j=VqmaxhF0(nITe+jTe,τqmax). From the model of equation (29), it is sufficient to apply an ICA method to the observation z(mITe) in order to estimate the 2L0+1 symbol trains {am-n} such that −L0≦n≦L0.

To summarize, this variant comprises the following steps:
Step II.b.1: Determination of the symbol period T by applying a cyclic detection algorithm as in [1] and [10].
Step II.b.2: Sampling of the observations x(t) with I samples per symbol such that T=ITe.
Step II.b.3: Application of an ICA method to the observations x(t) in order to obtain ŝ(t) and Â from equation (24).
Step II.b.4: Sorting of the outputs according to Q groups of correlated multipaths in order to obtain Âq and ŝq(t) for (1≦q≦Q): to do this, a correlation test for all the output pairs i and j with the two-hypothesis test of equation (26). Firstly the outputs associated with the 1st output will be sought by starting the test for 2<j≦PQ×Q and i=1. Next, removed from the list of outputs are all those associated with the 1st that will constitute the 1st group with q=1. The same series of tests is repeated with the other outputs that are not correlated with the 1st output in order to constitute the 2nd group. This operation is continued to the last group where in the end no output channel will remain.
Step II.b.5: Determination of the better group of multipaths where (ÂqHÂq) is a maximum in q=qmax.
Step II.b.6: Constitution of the observation vector z(t) of equation (29) from the signal ŝqmax(t).
Step II.b.7: Application of an ICA method for estimating the symbol trains {am-n} where −L0≦n≦L0. From the symbol trains is chosen that one which is associated with the higher-modulus vector hz(i), namely {âm-i}.
Step II.b.8: Determination of the phase αimax of the output associated with the higher-modulus vector hz(i) applying steps B.1, B.2 and B.3.
Step II.b.9: Phase-resetting of the symbol trains {âm} by taking {circumflex over (â)}m=âmexp(−jαimax). The symbol train {{circumflex over (â)}m} constitutes the output of the demodulator of this subprocess.
Step II.b.10: Estimation of the propagation channel parameters, namely the angle θq,p and the delay τq,p. The angles of incidence θq,p are determined from the Âq values for (1≦q≦Q) applying the MUSIC [1] algorithm to the matrix ÂqÂqH. The matrices Aq are deduced from these goniometry values in order to deduce therefrom an estimate of s(t, τq) taking ŝ(t,τq)=Aq#Xq(t). Since the elements of the ŝ(t,τq) are composed of the signals s(t−τq,p), the delays τp,q−τ1,1 are determined by maximizing the c(τ)=|ŝq,p(t−τ)ŝ1,1(t)*|2 critera where ŝq,p(t) is the pth component of ŝ(t,τq).
Another way of Implementing the Process Estimation of the Carrier Frequency and Deduction of the {am} Symbols.
This technique consists in estimating the carrier frequency f0 of the transmitter or the complex z0=exp(j2πf0Te) in order thereafter to deduce the symbols {am} from the symbols {bm}, taking, from equation (3):

am=bmexp(−j2πf0mITe)=bmz0−mI  (30)

This step is applied after step I.4 of reordering the symbols and the channel vectors. From equations (3), (17), (7) and (8), the following channel vectors are used:




  
    
      
        
          
            
              h
              ^
            
            z
          
          ⁡
          
            (
            n
            )
          
        
        =
        
          
            
              [
              
                
                  
                    
                      
                        z
                        0
                        nl
                      
                      ⁢
                      
                        h
                        ⁡
                        
                          (
                          
                            nlT
                            e
                          
                          )
                        
                      
                    
                  
                
                
                  
                    
                      
                        z
                        0
                        
                          nl
                          +
                          1
                        
                      
                      ⁢
                      
                        h
                        ⁡
                        
                          (
                          
                            
                              nlT
                              e
                            
                            +
                            
                              T
                              e
                            
                          
                          )
                        
                      
                    
                  
                
                
                  
                    ⋮
                  
                
                
                  
                    
                      
                        z
                        0
                        
                          nl
                          +
                          
                            (
                            
                              l
                              -
                              1
                            
                            )
                          
                        
                      
                      ⁢
                      
                        h
                        ⁡
                        
                          (
                          
                            
                              nlT
                              e
                            
                            +
                            
                              
                                (
                                
                                  I
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              
                                T
                                e
                              
                            
                          
                          )
                        
                      
                    
                  
                
              
              ]
            
            ⁢
            
              
            
            ⁢
            for
            ⁢
            
              
            
            ⁢
            n
          
          ∈
          Ω
        
      
    
    
      
        (
        31
        )
      
    
  




where Ω={Ind[rp−L0, rp+L0](n)=1 for a p such that 1≦p≦P}

Since it is known that Ω={n1< . . . <nLc}, a grand vector b is obtained from the vectors ĥz(n), such that:




  
    
      
        w
        =
        
          [
          
            
              
                
                  
                    
                      h
                      ^
                    
                    z
                  
                  ⁡
                  
                    (
                    
                      n
                      1
                    
                    )
                  
                
              
            
            
              
                
                  
                    
                      h
                      ^
                    
                    z
                  
                  ⁡
                  
                    (
                    
                      n
                      2
                    
                    )
                  
                
              
            
            
              
                ⋮
              
            
            
              
                
                  
                    
                      h
                      ^
                    
                    z
                  
                  ⁡
                  
                    (
                    
                      n
                      K
                    
                    )
                  
                
              
            
          
          ]
        
      
    
    
      
        (
        32
        )
      
    
  




The search for f0 consists in maximizing the following criterion:




  
    
      
        
          
            Carrier
            ⁢
            
              
            
            ⁢
            
              (
              
                f
                0
              
              )
            
          
          =
          
            
              
              
                
                  w
                  H
                
                ⁢
                
                  c
                  ⁡
                  
                    (
                    
                      exp
                      ⁡
                      
                        (
                        
                          j2π
                          ⁢
                          
                            
                          
                          ⁢
                          
                            f
                            0
                          
                          ⁢
                          
                            T
                            e
                          
                        
                        )
                      
                    
                    )
                  
                
              
              
            
            2
          
        
        ⁢
        
          

        
        ⁢
        
          
            where
            ⁢
            
              
            
            ⁢
            c
            ⁢
            
              (
              
                z
                0
              
              )
            
          
          =
          
            
              [
              
                
                  
                    
                      c
                      ⁡
                      
                        (
                        
                          
                            n
                            1
                          
                          ,
                          
                            z
                            0
                          
                        
                        )
                      
                    
                  
                
                
                  
                    
                      c
                      ⁡
                      
                        (
                        
                          
                            n
                            2
                          
                          ,
                          
                            z
                            0
                          
                        
                        )
                      
                    
                  
                
                
                  
                    ⋮
                  
                
                
                  
                    
                      c
                      ⁡
                      
                        (
                        
                          
                            n
                            K
                          
                          ,
                          
                            z
                            0
                          
                        
                        )
                      
                    
                  
                
              
              ]
            
            ⁢
            
              
            
            ⁢
            and
          
        
        ⁢
        
          
        
        ⁢
        
          
            where
            ⁢
            
              
            
            ⁢
            
              c
              ⁡
              
                (
                
                  n
                  ,
                  
                    z
                    0
                  
                
                )
              
            
          
          =
          
            [
            
              
                
                  
                    z
                    0
                    nl
                  
                
              
              
                
                  
                    z
                    0
                    
                      nl
                      ⁢
                      
                        
                      
                      +
                      
                        
                      
                      ⁢
                      1
                    
                  
                
              
              
                
                  ⋮
                
              
              
                
                  
                    z
                    0
                    
                      nl
                      ⁢
                      
                        
                      
                      +
                      
                        
                      
                      ⁢
                      
                        (
                        
                          l
                          ⁢
                          
                            
                          
                          +
                          
                            
                          
                          ⁢
                          1
                        
                        )
                      
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        33
        )
      
    
  




The steps of the process suitable for the case of a transmitter with a non-zero frequency are the following:
Step III.a.1: Step I.1 to step I.4 described above in order to obtain the symbol trains {{circumflex over (b)}m-k,} associated with the channel vectors ĥz(kj).
Step III.a.2: Construction of the vector w of equation (32) from the ĥz(kj).
Step III.a.3: Maximization of the carrier (f0) criterion of equation (33) in order to obtain f0.
Step III.a.4: Application of equation (30) in order to deduce the symbols {am} from the symbols {bm}.
Step III.a.5: Step I.5 to step I.7 described above.

    
    
        In the case of a transmitter with non-zero frequency and for decorrelated multipaths, the steps are the following:
    
    


Step III.b.1: Step II.a.1 to Step II.a.4 described above in order to obtain the vector z(t) of equation (22).
Step III.b.2: Application of the ICA methods [4], [5], [15] and [19] in order to estimate Lc symbol trains {{circumflex over (b)}m,j} associated with the channel vectors ĥz,j.
Step III.b.3: Reordering of the symbol trains {{circumflex over (b)}m,j} and of the channel vectors ĥz,j applying steps A.1, A.2 and A.3 in order to obtain the symbol trains {{circumflex over (b)}m-k,} associated with the channel vectors ĥz,j=ĥz(kj).
Step III.b.4: Construction of the vector w of equation (32) from the ĥz(kj).
Step III.b.5: Maximisation of the carrier (f0) criterion of equation (33) in order to obtain f0.
Step III.b.6: Application of equation (30) in order to deduce the symbols {am} from the symbols {bm}.
Step III.b.7: Choice of the symbol train associated with the higher-modulus vector hz(i), namely, {âm-i}.
Step III.b.8: Step II.a.7 to step II.a.9 described above.
In the case of a transmitter with non-zero frequency and for correlated multipaths, the steps are for example the following:
Step III.c.1: Step II.b.1 to step II.b.6 No. 2.2 in order to obtain the vector z(t) of equation (29).
Step III.c.2: Application of ICA methods [4], [5], [15] and [19] in order to estimate the Lc symbol trains {{circumflex over (b)}m,j} associated with the channel vectors ĥz,j.
Step III.c.3: Reordering of the symbol trains {{circumflex over (b)}m,j} and of the channel vectors ĥz,j applying steps A.1, A.2 and A.3 so as to obtain the symbol trains {{circumflex over (b)}m-kj} associated with the channel vectors ĥz,j=ĥz(kj).
Step III.c.4: Construction of the vector w of equation (32) from the ĥz(kj).
Step III.c.5: Maximization of the carrier criterion (f0) of equation (33) in order to obtain f0.
Step III.c.6: Application of equation (30) for deducing the symbols {am} from the symbols {bm}.
Step III.c.7: Choice among the symbol trains of that one which is associated with the higher-modulus vector hz(i), namely {âm-i}.
Step III.c.8: Step II.b.8 to step II.b.10 described above.
REFERENCES
[1] R. O. Schmidt, “A signal subspace approach to multiple emitter location and spectral estimation”, November 1981.
[2] W. A. BROWN, “Computationally efficient algorithms for cyclic spectral analysers”, 4th ASSP Workshop on Spectrum Modelling, August 1988.
[3] S. V. SCHELL and W. GARDNER, “Detection of the number of cyclostationary signals in unknowns interference and noise”, Proc, Asilonan Conference on Signal, Systems and Computers, 5-9 Nov. 1990.
[4] J. F. CARDOSO and A. SOULOUMIAC, “Blind beamforming for non-Gaussian signals”, IEE Proceedings-F, Vol. 140, No. 6, pp. 362-370, December 1993.
[5] P. COMON, “Independent Component Analysis, a new concept?”, Signal Processing, Elsevier, April 1994, Vol 36, No. 3, pp. 287-314.
[6] S. MAYRARGUE, “A blind spatio-temporal equalizer for a radio-mobile channel using the Constant Modulus Algorithm CMA”, ICASSP 94, 1994 IEEE International Conference on Acoustics Speech and Signal Processing, 19-22 Apr. 1994, Adelaide, South Australia, pp. 317-319.
[7] E. MOULINES, P. DUHAMEL, J. F. CARDOSO and S. MAYRARGUE, “Subspace methods for the blind identification of multichannel FIR filters”, IEEE Transactions On Signal Processing, Vol. 43, No. 2, pp. 516-525, February 1995.
[8] V. VANDERVEEN, “Joint Angle and delay Estimation (JADE) for signal in multipath environments”, 30th ASILOMAR Conference in Pacific Grove, IEEE Computer Society, Los Alamitos, Calif., USA, 3-6 Nov. 1996.
[9] P. CHEVALIER, V. CAPDEVIELLE, and P. COMON, “Behavior of HO blind source separation methods in the presence of cyclostationary correlated multipaths”, IEEE SP Workshop on HOS, Alberta (Canada), July 1997.
[10] A. FERREOL, Patent No. 98/00731. “Procédé de détection cyclique en diversité de polarisation” [Cyclic detection method in polarization diversity], 23 Jan. 1998.
[11] C. B. PAPADIAS and D. T. M. SLOCK, “Fractionally spaced equalization of linear polyphase channels and related blind techniques based on multichannel linear prediction”, IEEE Transactions On Signal Processing”, March 1999, Vol. 47, No. 3, pp 641-654.
[12] E. DE CARVALHO and D. T. M. SLOCK, “A fast Gaussian maximum-likelihood method for blind multichannel estimation”, SPAWC 99, Signal Processing Advances in Wireless Communications, 9-12 May 1999, Annapolis, US, pp. 279-282.
[13] H. ZENG et L. TONG, “Blind channel estimation using the second-order statistics: Algorithms”, IEEE Transactions On Signal Processing, August 1999, Vol. 45, No. 8, pp. 1919-1930.
[14] A. FERREOL and P. CHEVALIER, “On the behavior of current second- and higher-order blind source separation methods for cyclostationary sources”, IEEE Trans. Sig. Proc., Vol. 48, No. 6, pp. 1712-1725, June 2000.
[15] P. COMON, “From source separation to blind equalization, contrast-based approaches”, ICISP 01, Int. Conf. on Image and Signal Processing, 3-5 May 2001, Agadir, Morocco, pp. 20-32.
[16] L. PERROS-MEILHAC, E. MOULINES, K. ABED-MERAIM, P. CHEVALIER and P. DUHAMEL, “Blind identification of multipath channels: A parametric subspace approach”, IEEE Transactions On Signal Processing, Vol. 49, No. 7, pp. 1468-1480, July 2001.
[17] I. JANG and S. CHOI, “Why blind source separation for blind equalization of multiple channels?”, SAM 02, Second IEEE Sensor Array and Multichannel Signal Processing Workshop, 4-6 Aug. 2002, Rosslyn, US, pp. 269-272.
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## Claims

1. A process for the blind demodulation of a linear-waveform source or transmitter in a system including one or more sources and an array of sensors and a propagation channel, said process comprising steps of:
determining symbol period T and taking samples at Te, such that T=ITe, wherein I is an integer number and Te is the sampling period;
constructing a spatio-temporal observation vector z(t), the mixed sources of which are symbol trains from the transmitter, from observations x(tk) taken at times tk, where the time tk corresponds to kTe where k is an integer;
applying an Independent Component Analysis (ICA)-type method to the observation vector z(t) in order to estimate for a number of input symbol trains {am-i} corresponding to a number of symbols Lc participating in an inter-symbol interference, the input symbol trains {am-i} corresponding to observations, where m and i are positive integers, the estimate outputs {âm,i} being associated with the channel vectors ĥz,j=ĥz(kj), z corresponding to a sensor in the array and j corresponding to a number of an estimate output;
arranging the Lc outputs (âm,j, ĥz,j) in the same order as the inputs (am-i,hz(i)) so as to obtain the propagation channel vectors ĥz,j=ĥz(kj); and
determining a phase αimax associated with the outputs.

2. The process as claimed in claim 1, further comprising estimating propagation channel parameters in order to determine a carrier frequency so as to compensate for the symbol trains in order to obtain the symbol trains in baseband.

3. The process as claimed in claim 1, further comprising a step of estimating angle θp and delay τp parameters of the propagation channel, where p is a positive integer.