# Patent

## Title
Antenna processing method for potentially non-centered cyclostationary signals

## Abstract

An antenna processing method for centered or potentially non-centered cyclostationary signals, comprises at least one step in which one or more nth order estimators are obtained from r-order statistics, with r=1 to n−1, and for one or more values of r, it comprises a step for the correction of the estimator by means of r-order detected cyclic frequencies. The method can be applied to the separation of the emitter sources of the signals received by using the estimator or estimators.

## Background
BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention relates to an antenna processing method for potentially non-centered or centered cyclostationary sources.
It can be applied for example to CPFSK sources with integer modulation index.
The invention relates, for example, to a method for the separation of potentially non-centered, cyclostationary signals received by a receiver of a communications system comprising several sources or emitters. The term “cyclostationary” also designates the particular case of stationary signals.
It can also be applied to the angular localization or goniometry of potentially non-centered cyclostationary sources.
The invention can be applied especially in radiocommunications, space telecommunications or passive listening to these links, in frequencies ranging from the VLF (Very Low Frequency) to the EHF (Extremely High Frequency).
In the present description, the term “blind separation” designates the separation of emitters with no knowledge whatsoever of the signals sent, the term “centered signal” refers to a signal without any continuous component that verifies E[x(t)]=0, and the term “non-stationary signal” refers to a signal whose statistics are time-dependent.
2. Description of the Prior Art
In many contexts of application, the reception of signals of interest for the receiver is very often disturbed by the presence of other signals (or sources) known as parasites, which may correspond either to delayed versions of the signals of interest (through multiple-path propagation), or to interfering sources which may be either deliberate or involuntary (in the case of co-channel transmissions). This is especially the case with radiocommunications in urban areas, subject to the phenomenon of multiple paths resulting from the reflections of the signal on surrounding fixed or moving obstacles potentially disturbed by the co-channel transmissions coming from the neighboring cells that re-use the same frequencies (in the case of F/TDMA or Frequency/Time Division Multiple Access networks). This is also the case with the HF (High Frequency) ionospherical links disturbed by the presence, at reception, of the multiple paths of propagation resulting from the reflections on the different ionospherical layers and of parasitic emitters due to high spectral congestion in the HF range.
For all these applications, whether it is for purposes of radiocommunications or for listening and technical analysis of the sources received, the sources need to be separated before other processing operations specific to the application considered are implemented. Furthermore, for certain applications such as passive listening, the sources received are totally unknown to the receiver (there are no available learning sequences, the waveforms are unknown, etc.) and their angular localization or goniometry may prove to be difficult (because of coupling between sensors) or costly (because of the calibration of the aerials) to implement. This is why it may prove to be highly advantageous to implement a source separation technique in a totally autodidactic or self-learning way, that is, by making use of no a priori information on the sources, apart from the assumption of the statistical independence of these sources.
The first studies on the separation of sources by self-learning appeared in the mid-1980s in the work of Jutten and Herault [1]. Since then, these studies have been constantly developing for mixtures of sources, both convolutive (time-spread multiple-path propagation channels) and instantaneous (distortion-free channels). A conspectus of these studies is presented in the article [2] by P. Comon and P. Chevalier. A certain number of techniques developed are called second-order techniques because they use only the information contained in the second-order statistics of the observations, as described in reference [3] for example. By contrast, other techniques, known as higher-order techniques, described for example in the reference [4], generally use not only second-order information but also information contained in statistics above the second order. These include the techniques known as cumulant-based, fourth-order techniques which have received special attention owing to their performance potential (reference [2]) and the relative simplicity of their implementation.
However, almost all the techniques of self-learned source separation available to date have been designed to separate sources assumed to be stationary, centered and ergodic, on the basis of estimators of statistics of observations qualified as being empirical, asymptotically unbiased and consistent on the basis of the above assumptions.
Two families of second-order separators are presently available. Those of the first family (F1) (reference [3], using the SOBI method shown schematically in FIG. 1) are aimed at separating statistically independent sources assumed to be stationary, centered and ergodic whereas those of the second family (F2) (reference [6], using the cyclic SOBI method) are designed to separate statistically independent sources assumed to be cyclostationary, centered and cycloergodic.
Two families of fourth-order separators are presently available. For example, those of the first family (F3) (reference [4] by J. F. Cardoso and A. Souloumiac, using the JADE method shown schematically in FIG. 2) are aimed at separating statistically independent sources assumed to be stationary, centered and ergodic while those of the second family (F4) (reference A. Ferreol and Chevalier [8] using the cyclic JADE technique) are designed to separate statistically independent sources assumed to be cyclostationary, centered and cycloergodic.
However, most of the sources encountered in practice are non-stationary and, more particularly, cyclostationary (with digitally modulated sources) and in certain cases deterministic (pure carriers). Furthermore, some of these sources are not centered. This is especially the case for deterministic sources and for certain digitally and non-linearly modulated sources as in the case of CPFSK sources with integer modulation index. This means that the empirical estimators of statistics classically used to implement the current techniques of self-learned source separation no longer have any reason to remain unbiased and consistent but are liable to become asymptotically biased. This may prevent the separation of the sources as shown in the document by A. Ferreol and P. Chevalier [5] for centered cyclostationary sources (linear digital modulations).
SUMMARY OF THE INVENTION
The invention relates to an antenna processing method for centered or potentially non-centered cyclostationary signals, comprising at least one step in which an nth order statistics estimator is obtained from r-order statistics, with r=1 to n−1, and for one or more values of r, a step for the correction of the empirical estimators, by means of r-order detected cyclic frequencies, exploiting the potentially non-centered character of the observations.
It comprises for example a step for the separation of the emitter sources of the signals received by using one of the second-order estimators or fourth-order estimators proposed.
It is also used for the angular localization or goniometry of the signals received.



BRIEF DESCRIPTION OF THE DRAWINGS
Other features and advantages of the inventions shall appear more clearly from the following description of a non-exhaustive exemplary embodiment and from the appended figures, of which:
FIGS. 1 and 2 show prior art separation techniques, respectively known as the SOBI and JADE techniques,
FIG. 3 exemplifies a receiver according to the invention,
FIG. 4 is a diagram of the steps according to the invention applied to the SOBI technique of separation,
FIG. 5 is a diagram of the steps of the method according to the invention applied to the JADE technique of separation,
FIGS. 6A and 6B show results of separation using a classic estimator or an estimator according to the invention,
FIG. 7 shows the convergence of the estimator,
FIG. 8 is a drawing of the steps implemented according to the invention for the cyclic JADE method, and
FIG. 9 is a high level drawing of the steps implemented according to the invention.



MORE DETAILED DESCRIPTION
The method according to the invention enables the processing especially of potentially non-centered or centered cyclostationary sources. It can be applied, for example, to the separation of the sources or to their angular localization or goniometry. FIG. 9 illustrates wherein the method comprises at least the steps of: (802) receiving a mixture of cyclostationary and cycloergodic signals from independent sources; (804) generating an asymptotically unbiased and consistent estimator of a cyclic correlation matrix for centered observations of cyclostationary and cycloergodic signals; (806) generating an asymptotically unbiased and consistent estimator of a cyclic covariance matrix for non-centered observations of cyclostationary and cycloergodic signals; and (808) performing angular localization on the received signals using the at least one nth order estimator.
FIG. 3 gives a diagrammatic view of an exemplary receiver comprising, for example, N reception sensors (11, 12, 13, 14) each connected to several inputs of a processing device 2 such as a processor adapted to executing the steps of the method described here below.
Before describing various alternative modes of implementation of the method according to the invention, we shall describe the way to obtain the estimators, for example second-order or fourth-order estimators, as well as the detection of the cyclic frequencies.
Modelling of the Signal
It is assumed that an antenna with N sensors receives a noise-ridden mixture from P (P≦N) narrow-band (NB) and statistically independent sources. On the basis of these assumptions, the vector, x(t) of the complex envelopes of the output signals from the sensors is written, at the instant t, as follows:




  
    
      
        
          x
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                ∑
                
                  p
                  =
                  1
                
                P
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    m
                    p
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ⁢
                
                  e
                  
                    j
                    ⁡
                    
                      (
                      
                        
                          2
                          ⁢
                          π
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            f
                            p
                          
                          ⁢
                          t
                        
                        +
                        
                          φ
                          p
                        
                      
                      )
                    
                  
                
                ⁢
                
                  a
                  p
                
              
            
            +
            
              b
              ⁡
              
                (
                t
                )
              
            
          
          ⁢
          
            =
            Δ
          
          ⁢
          
            
              A
              ⁢
              
                
              
              ⁢
              
                
                  m
                  c
                
                ⁡
                
                  (
                  t
                  )
                
              
            
            +
            
              b
              ⁡
              
                (
                t
                )
              
            
          
        
      
    
    
      
        (
        1
        )
      
    
  




where b(t) is the noise vector, assumed to be centered, stationary, circular and spatially white, mp(t), Δfp, φp and ap correspond respectively to the complex, narrow-band (NB), cyclostationary and potentially non-centered envelope (deterministic as the case may be), to the residue of the carrier, to the phase and to the directional vector of the source p, mc(t) is the vector whose components are the signals mpc(t)Δmp(t) exp[j(2πΔfpt+φp)] and A is the matrix (N×P) whose columns are the vectors ap.

Statistics of the Observations

First Order Statistics

In the general case of cyclostationary and non-centered sources, the first-order statistics of the vector x(t), given by (1), are written as follows:




  
    
      
        
          
            e
            x
          
          ⁡
          
            (
            t
            )
          
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          
            E
            ⁡
            
              [
              
                x
                ⁡
                
                  (
                  t
                  )
                
              
              ]
            
          
          =
          
            
              
                ∑
                
                  p
                  =
                  1
                
                P
              
              ⁢
              
                
                  
                    e
                    p
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ⁢
                
                  e
                  
                    j
                    ⁡
                    
                      (
                      
                        
                          2
                          ⁢
                          π
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            f
                            p
                          
                          ⁢
                          t
                        
                        +
                        
                          φ
                          p
                        
                      
                      )
                    
                  
                
                ⁢
                
                  a
                  p
                
              
            
            ⁢
            
              =
              Δ
            
            ⁢
            
              
                
                  ∑
                  
                    p
                    =
                    1
                  
                  P
                
                ⁢
                
                  
                    
                      e
                      pc
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ⁢
                  
                    a
                    p
                  
                
              
              ⁢
              
                =
                Δ
              
              ⁢
              
                A
                ⁢
                
                  
                
                ⁢
                
                  
                    e
                    
                      m
                      ⁢
                      
                        
                      
                      ⁢
                      c
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




where ep(t), epc(t) and emc(t) are the mathematical expectation values respectively of mp(t), mpc(t) and mc(t). The vectors ep(t) and epc(t) accept a Fourier series decomposition, and we obtain:





  
    
      
        
          
            e
            pc
          
          ⁡
          
            (
            t
            )
          
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          
            E
            ⁡
            
              [
              
                
                  m
                  pc
                
                ⁡
                
                  (
                  t
                  )
                
              
              ]
            
          
          =
          
            
              
                ∑
                
                  
                    γ
                    pc
                  
                  ∈
                  
                    Γ
                    pc
                    1
                  
                
              
              ⁢
              
                
                  e
                  pc
                  
                    γ
                    pc
                  
                
                ⁢
                
                  e
                  
                    
                      j2πγ
                      pc
                    
                    ⁢
                    t
                  
                
              
            
            =
            
              
                ∑
                
                  
                    γ
                    p
                  
                  ∈
                  
                    Γ
                    p
                    1
                  
                
              
              ⁢
              
                
                  e
                  p
                  
                    γ
                    p
                  
                
                ⁢
                
                  e
                  
                    j
                    ⁡
                    
                      [
                      
                        
                          2
                          ⁢
                          
                            π
                            (
                            
                              
                            
                            ⁢
                            
                              
                                Δ
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  f
                                  p
                                
                              
                              +
                              
                                γ
                                p
                              
                            
                            )
                          
                          ⁢
                          t
                        
                        +
                        
                          φ
                          p
                        
                      
                      ]
                    
                  
                
              
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




where





  
    
      Γ
      p
      1
    
    =
    
      
        
          {
          
            γ
            p
          
          }
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        
          Γ
          pc
          1
        
      
      =
      
        {
        
          
            γ
            pc
          
          =
          
            
              γ
              p
            
            +
            
              Δ
              ⁢
              
                
              
              ⁢
              
                f
                p
              
            
          
        
        }
      
    
  
  ⁢
  
    
  




are the sets of the cyclic frequencies γp and γpc respectively of ep(t) and epc(t),





  
    e
    p
    
      γ
      p
    
  
  ⁢
  
    
  
  ⁢
  and
  ⁢
  
    
  
  ⁢
  
    e
    pc
    
      γ
      pc
    
  




are the cyclic mean values respectively mp(t) and mpc(t), defined by





  
    
      
        
          e
          p
          
            γ
            p
          
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          <
          
            
              
                e
                p
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              e
              
                
                  -
                  
                    j2πγ
                    p
                  
                
                ⁢
                t
              
            
          
          ⁢
          
            >
            c
          
        
      
    
    
      
        (
        4
        )
      
    
  
  
    
      
        
          e
          pc
          
            γ
            pc
          
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          
            <
            
              
                
                  e
                  pc
                
                ⁡
                
                  (
                  t
                  )
                
              
              ⁢
              
                e
                
                  
                    -
                    
                      j2πγ
                      pc
                    
                  
                  ⁢
                  t
                
              
            
            ⁢
            
              >
              c
            
          
          =
          
            
              e
              p
              
                γ
                
                  pc
                  -
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    fp
                  
                
              
            
            ⁢
            
              e
              
                jφ
                p
              
            
          
        
      
    
    
      
        (
        5
        )
      
    
  




where the symbol





  
    <
    
      f
      ⁡
      
        (
        t
        )
      
    
    ⁢
    
      >
      c
    
  
  ⁢
  
    =
    Δ
  
  ⁢
  
    
  
  ⁢
  
    
      lim
      
        T
        →
        ∞
      
    
    ⁢
    
      
        (
        
          1
          /
          T
        
        )
      
      ⁢
      
        
          ∫
          
            
              -
              T
            
            /
            2
          
          
            T
            /
            2
          
        
        ⁢
        
          
            f
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            
          
          ⁢
          
            d
            t
          
        
      
    
  




corresponds to the operation of taking the temporal mean in continuous time f(t) on an infinite horizon of observation. Consequently, the vectors emc(t) and ex(t) also accept a Fourier series decomposition and, by using (2) and (3), we get:





  
    
      
        
          
            e
            
              m
              ⁢
              
                
              
              ⁢
              c
            
          
          ⁡
          
            (
            t
            )
          
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          
            ∑
            
              γ
              ∈
              
                Γ
                x
                1
              
            
          
          ⁢
          
            
          
          ⁢
          
            
              e
              
                m
                ⁢
                
                  
                
                ⁢
                c
              
              γ
            
            ⁢
            
              e
              
                j2πγ
                ⁢
                
                  
                
                ⁢
                t
              
            
          
        
      
    
    
      
        (
        6
        )
      
    
  
  
    
      
        
          
            e
            x
          
          ⁡
          
            (
            t
            )
          
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          
            
              ∑
              
                γ
                ∈
                
                  Γ
                  x
                  1
                
              
            
            ⁢
            
              
                e
                x
                γ
              
              ⁢
              
                e
                
                  j2πγ
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
          
          =
          
            
              
                ∑
                
                  γ
                  ∈
                  
                    Γ
                    x
                    1
                  
                
              
              ⁢
              
                A
                ⁢
                
                  
                
                ⁢
                
                  e
                  
                    m
                    ⁢
                    
                      
                    
                    ⁢
                    c
                  
                  γ
                
                ⁢
                
                  e
                  
                    j2πγ
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                
              
            
            =
            
              
                ∑
                
                  p
                  =
                  1
                
                P
              
              ⁢
              
                
                  ∑
                  
                    
                      γ
                      pc
                    
                    ∈
                    
                      Γ
                      pc
                      1
                    
                  
                
                ⁢
                
                  
                    e
                    pc
                    
                      γ
                      pc
                    
                  
                  ⁢
                  
                    e
                    
                      
                        j2πγ
                        pc
                      
                      ⁢
                      t
                    
                  
                  ⁢
                  
                    a
                    p
                  
                
              
            
          
        
      
    
    
      
        (
        7
        )
      
    
  




where





  
    Γ
    x
    1
  
  =
  
    
      ⋃
      
        1
        ≤
        p
        ≤
        p
      
    
    ⁢
    
      {
      
        Γ
        pc
        1
      
      }
    
  




is the set of the cyclic frequencies γ of emc(t) and ex(t),





  e
  
    m
    ⁢
    
      
    
    ⁢
    c
  
  γ




and





  
    e
    x
    γ
  
  ⁢
  
    
  
  ⁢
  and
  ⁢
  
    
  
  ⁢
  
    e
    x
    γ
  




are respectively the cyclic means of mc(t) and x(t), defined by:





  
    
      
        
          e
          
            m
            ⁢
            
              
            
            ⁢
            c
          
          γ
        
        =
        
          <
          
            
              
                e
                
                  m
                  ⁢
                  
                    
                  
                  ⁢
                  c
                
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              e
              
                -
                j2πγt
              
            
          
          ⁢
          
            >
            c
          
        
      
    
    
      
        (
        8
        )
      
    
  
  
    
      
        
          e
          x
          γ
        
        =
        
          <
          
            
              
                e
                x
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              e
              
                -
                j2πγt
              
            
          
          ⁢
          
            >
            c
          
        
      
    
    
      
        (
        9
        )
      
    
  




Assuming these conditions, the (quasi)-cyclostationary vector x(t) can be decomposed into the sum of a deterministic and (quasi)-periodic part ex(t) and a (quasi)-cyclostationary, centered, random part Δx(t) such that:

Δx(t)Δx(t)−ex(t)=AΔmc(t)+b(t)  (10)

where Δmc(t)Δmc(t)−emc(t) is the centered vector of the source signals, with components Δmpc(t)Δmpc(t)−epc(t)=Δmp(t) ej(2πΔfpt+φp) where Δmp(t)Δmp(t)−ep(t).

Special Cases

By way of an indication, ep(t)=0 for a digitally and linearly modulated source p, which is centered. However, ep(t)≠0 for a deterministic (carrier) source p as well as for certain digitally and non-linearly modulated sources such as CPFSKs with integer index, whose complex envelope is written as follows:




  
    
      
        
          
            m
            p
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            π
            p
            
              1
              /
              2
            
          
          ⁢
          
            
              ∑
              n
            
            ⁢
            
              
            
            ⁢
            
              exp
              ⁢
              
                {
                
                  j
                  ⁡
                  
                    [
                    
                      
                        θ
                        pn
                      
                      +
                      
                        2
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        
                          f
                          dp
                        
                        ⁢
                        
                          
                            a
                            n
                            p
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  T
                                  p
                                
                              
                            
                            )
                          
                        
                      
                    
                    ]
                  
                
                }
              
              ⁢
              
                
              
              ⁢
              
                
                  Rect
                  p
                
                ⁡
                
                  (
                  
                    t
                    -
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      
                        T
                        p
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




where Tp corresponds to the symbol duration of the source, πpΔ<E[|mp(t)|2]>c is the mean power of the source p received by an omnidirectional sensor, the





  a
  n
  p




are the transmitted Mp-ary symbols, assumed to be i.i.d and taking their values in the alphabet ±1, ±3, . . . , ±(Mp−1), where Mp is generally a power of 2, Rectp(t) is the rectangular pulse with an amplitude 1 and a duration Tp, fdpΔhp/2Tp is the frequency deviation, hp is the index of modulation of the source and θpn, which corresponds to the accumulation of all the symbols up to the instant (n−1)Tp, is defined by:





  
    
      
        
          θ
          pn
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          2
          ⁢
          π
          ⁢
          
            
          
          ⁢
          
            f
            dp
          
          ⁢
          
            T
            p
          
          ⁢
          
            
              ∑
              
                k
                =
                
                  -
                  ∞
                
              
              
                n
                -
                1
              
            
            ⁢
            
              
            
            ⁢
            
              a
              k
              p
            
          
        
      
    
    
      
        (
        12
        )
      
    
  




For Mp-ary symbols, the associated CPFSK source is called an Mp-CPFSK source. For this type of source, the set Γp1 of the first-order cyclic frequencies of the source p is written as





  
    Γ
    p
    1
  
  =
  
    {
    
      
        
          γ
          p
        
        =
        
          
            ±
            
              (
              
                
                  2
                  ⁢
                  k
                
                +
                1
              
              )
            
          
          ⁢
          
            f
            dp
          
        
      
      ,
    
  




0≦k≦(Mp−2)/2}. Thus we get:





  
    
      
        
          e
          p
          γ
        
        =
        
          
            
              ±
              
                π
                p
                
                  1
                  /
                  2
                
              
            
            ⁢
            
              1
              
                M
                p
              
            
            ⁢
            
              
            
            ⁢
            for
            ⁢
            
              
            
            ⁢
            γ
          
          ∈
          
            Γ
            p
            1
          
        
      
    
    
      
        (
        13
        )
      
    
  




Second-Order Statistics

Based on the above assumptions (non-centered, cyclostationary sources), the second-order statistics of the observations are characterized by the two correlation matrices Rxε(t, τ) for ε=1 and ε=−1, dependent on the current time t and defined by:

Rxε(t,τ)ΔE[x(t)x(t−τ)εT]=ARmcε(t,τ)AεT+η2(τ)δ(1+ε)I  (14)

where ε=±1, with the convention x1Δx and x−1Δx*, * is the complex conjugation operation, δ(.) is the Kronecker symbol, T signifies transposed, η2(τ) is the correlation function of the noise on each sensor, I is the identity matrix, the matrix Rmcε(t, τ)ΔE[mc(t) mc(t−τ)εT] introduces the first and second correlation matrices of the vector mc(t).

In the general case of non-centered cyclostationary sources using (10) in (14), the matrix Rxε(t, τ) takes the following form:

Rxε(t,τ)=RΔxε(t,τ)+ex(t)ex(t−τ)εT  (15)

where RΔxε(t, τ) introduces the first and second matrices of covariance of x(t) or of correlation of Δx(t) , defined by

RΔxε(t,τ)ΔE[Δx(t)Δx(t−τ)εT]=ARΔmcε(t, τ)AεT+η2(τ)δ(1+ε)I  (16)

where RΔmcε(t, τ)Δ E[Δmc(t) Δmc(t−τ)εT] defines the first and second matrices of covariance of mc(t) such that:

Rmcε(t,τ)=RΔmcε(t,τ)+emc(t)emc(t−τ)εT  (17)

Using (1), (10) and the assumption of statistical independence of the sources in (16), we get:





  
    
      
        
          
            
              
                
                  R
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    xɛ
                  
                
                ⁡
                
                  (
                  
                    t
                    ,
                    τ
                  
                  )
                
              
              =
              
                
                  
                    ∑
                    
                      p
                      =
                      1
                    
                    P
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      
                        r
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          p
                          ⁢
                          
                            
                          
                          ⁢
                          ɛ
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          ,
                          τ
                        
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        j
                        ⁡
                        
                          [
                          
                            
                              
                                (
                                
                                  1
                                  +
                                  ɛ
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    πΔ
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      f
                                      p
                                    
                                    ⁢
                                    t
                                  
                                  +
                                  
                                    φ
                                    p
                                  
                                
                                )
                              
                            
                            -
                            
                              2
                              ⁢
                              πΔ
                              ⁢
                              
                                
                              
                              ⁢
                              
                                f
                                p
                              
                              ⁢
                              τɛ
                            
                          
                          ]
                        
                      
                    
                    ⁢
                    
                      a
                      p
                    
                    ⁢
                    
                      a
                      p
                      
                        ɛ
                        ⁢
                        
                          
                        
                        ⁢
                        T
                      
                    
                  
                
                +
                
                  
                    
                      η
                      2
                    
                    ⁡
                    
                      (
                      τ
                      )
                    
                  
                  ⁢
                  I
                
              
            
          
        
        
          
            
              =
              
                
                  
                    ∑
                    
                      p
                      =
                      1
                    
                    P
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      
                        r
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          p
                          ⁢
                          
                            
                          
                          ⁢
                          cɛ
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          ,
                          τ
                        
                        )
                      
                    
                    ⁢
                    
                      a
                      p
                    
                    ⁢
                    
                      a
                      p
                      
                        ɛ
                        ⁢
                        
                          
                        
                        ⁢
                        T
                      
                    
                  
                
                +
                
                  
                    
                      η
                      2
                    
                    ⁡
                    
                      (
                      τ
                      )
                    
                  
                  ⁢
                  
                    δ
                    ⁡
                    
                      (
                      
                        1
                        +
                        ɛ
                      
                      )
                    
                  
                  ⁢
                  I
                
              
            
          
        
      
    
    
      
        (
        18
        )
      
    
  




where rΔpε(t, τ)ΔE[Δmp(t) Δmp(t−τ)ε], rΔpcε(t, τ)ΔE[Δmpcε(t) Δmpc(t−τ)ε]. From the expressions (2) and (18) a new writing of the expression (15) is finally deduced and is given by:





  
    
      
        
          
            R
            xɛ
          
          ⁡
          
            (
            
              t
              ,
              τ
            
            )
          
        
        =
        
          
            
              ∑
              
                p
                =
                1
              
              P
            
            ⁢
            
              
                
                  r
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    p
                    ⁢
                    
                      
                    
                    ⁢
                    cɛ
                  
                
                ⁡
                
                  (
                  
                    t
                    ,
                    τ
                  
                  )
                
              
              ⁢
              
                a
                p
              
              ⁢
              
                a
                p
                
                  ɛ
                  ⁢
                  
                    
                  
                  ⁢
                  T
                
              
            
          
          +
          
            
              ∑
              
                p
                =
                1
              
              P
            
            ⁢
            
              
                ∑
                
                  q
                  =
                  1
                
                P
              
              ⁢
              
                
                  
                    e
                    pc
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ⁢
                
                  
                    
                      e
                      qc
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        τ
                      
                      )
                    
                  
                  ɛ
                
                ⁢
                
                  a
                  p
                
                ⁢
                
                  a
                  p
                  
                    ɛ
                    ⁢
                    
                      
                    
                    ⁢
                    T
                  
                
              
            
          
          +
          
            
              
                η
                2
              
              ⁡
              
                (
                τ
                )
              
            
            ⁢
            
              δ
              ⁡
              
                (
                
                  1
                  +
                  ɛ
                
                )
              
            
            ⁢
            I
          
        
      
    
    
      
        (
        19
        )
      
    
  




For cyclostationary sources, the functions of covariance rΔpcε(t, τ), 1≦p≦P, and hence the matrices RΔmcε(t, τ) and RΔxε(t, τ) are (quasi-) or polyperiodic functions of the time t accepting a Fourier series decomposition, and we get:




  
    
      
        
          
            r
            
              Δ
              ⁢
              
                
              
              ⁢
              p
              ⁢
              
                
              
              ⁢
              c
              ⁢
              
                
              
              ⁢
              ɛ
            
          
          ⁡
          
            (
            
              t
              ,
              τ
            
            )
          
        
        =
        
          
            ∑
            
              α
              
                
                  
                
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    p
                    ⁢
                    
                      
                    
                    ⁢
                    c
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                  
                  ∈
                  
                    ΓΔ
                    pc
                    
                      [
                      
                        1
                        ,
                        ɛ
                      
                      ]
                    
                  
                
              
            
          
          ⁢
          
            
              
                r
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  pc
                  ⁢
                  
                    
                  
                  ⁢
                  ɛ
                
                
                  α
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    p
                    ⁢
                    
                      
                    
                    ⁢
                    c
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                  
                
              
              ⁡
              
                (
                τ
                )
              
            
            ⁢
            
              e
              
                j2π
                ⁢
                
                  
                
                ⁢
                
                  α
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    pc
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                
              
            
          
        
      
    
    
      
        (
        20
        )
      
    
  
  
    
      
        
          
            R
            
              Δ
              ⁢
              
                
              
              ⁢
              x
              ⁢
              
                
              
              ⁢
              ɛ
            
          
          ⁡
          
            (
            
              t
              ,
              τ
            
            )
          
        
        =
        
          
            ∑
            
              α
              
                
                  
                
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    x
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                  
                  ∈
                  
                    ΓΔ
                    x
                    
                      [
                      
                        1
                        ,
                        ɛ
                      
                      ]
                    
                  
                
              
            
          
          ⁢
          
            
              
                R
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  x
                  ⁢
                  
                    
                  
                  ⁢
                  ɛ
                
                
                  α
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    x
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                  
                
              
              ⁡
              
                (
                τ
                )
              
            
            ⁢
            
              e
              
                j2π
                ⁢
                
                  
                
                ⁢
                
                  α
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    x
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                
              
            
          
        
      
    
    
      
        (
        21
        )
      
    
  




where





  
    Γ
    
      Δ
      ⁢
      
        
      
      ⁢
      pc
    
    
      [
      
        1
        ,
        ɛ
      
      ]
    
  
  =
  
    {
    
      α
      
        Δ
        ⁢
        
          
        
        ⁢
        pc
        ⁢
        
          
        
        ⁢
        ɛ
      
    
    }
  




is the set of the cyclic frequencies αΔpcεof





  
    
      r
      
        Δ
        ⁢
        
          
        
        ⁢
        pc
        ⁢
        
          
        
        ⁢
        ɛ
      
    
    ⁡
    
      (
      
        t
        ,
        τ
      
      )
    
  
  ,
  
    
      Γ
      
        Δ
        ⁢
        
          
        
        ⁢
        x
      
      
        [
        
          1
          ,
          ɛ
        
        ]
      
    
    =
    
      
        {
        
          α
          
            Δ
            ⁢
            
              
            
            ⁢
            x
            ⁢
            
              
            
            ⁢
            ɛ
          
        
        }
      
      =
      
        
          ⋃
          
            1
            ≤
            p
            ≤
            P
          
        
        ⁢
        
          {
          
            Γ
            
              Δ
              ⁢
              
                
              
              ⁢
              pc
            
            
              [
              
                1
                ,
                ɛ
              
              ]
            
          
          }
        
      
    
  




is the set of cyclic frequencies αΔxεof RΔxε(t, τ), the quantities





  
    r
    
      Δ
      ⁢
      
        
      
      ⁢
      pc
      ⁢
      
        
      
      ⁢
      ɛ
    
    
      α
      
        Δp
        ⁢
        
          
        
        ⁢
        c
        ⁢
        
          
        
        ⁢
        ɛ
      
    
  
  ⁡
  
    (
    τ
    )
  




define the first (ε=−1) and second (ε=1) functions of cyclic covariance of mpc(t) while the matrices





  
    R
    
      Δ
      ⁢
      
        
      
      ⁢
      x
      ⁢
      
        
      
      ⁢
      ɛ
    
    
      α
      
        Δ
        ⁢
        
          
        
        ⁢
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
    
  
  ⁡
  
    (
    τ
    )
  




define the first and second matrices of cyclic covariance of x(t), respectively defined by





  
    
      
        
          
            r
            
              Δ
              ⁢
              
                
              
              ⁢
              pc
              ⁢
              
                
              
              ⁢
              ɛ
            
          
          ⁢
          
            ,
            
              α
              
                Δ
                ⁢
                
                  
                
                ⁢
                pc
                ⁢
                
                  
                
                ⁢
                ɛ
              
            
          
          ⁢
          
            
              (
              τ
              )
            
            =
            
              <
              
                
                  
                    r
                    
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      pc
                      ⁢
                      
                        
                      
                      ⁢
                      ɛ
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      ,
                      τ
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      j
                    
                    ⁢
                    
                      
                    
                    ⁢
                    2
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    
                      α
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        pc
                        ⁢
                        
                          
                        
                        ⁢
                        ɛ
                      
                    
                    ⁢
                    t
                  
                
              
              ⁢
              
                >
                c
              
            
          
        
        ⁢
        
          
        
      
    
    
      
        (
        22
        )
      
    
  
  
    
      
        
          R
          
            Δ
            ⁢
            
              
            
            ⁢
            x
            ⁢
            
              
            
            ⁢
            ɛ
          
        
        ⁢
        
          ,
          
            α
            
              Δ
              ⁢
              
                
              
              ⁢
              x
              ⁢
              
                
              
              ⁢
              ɛ
            
          
        
        ⁢
        
          
            (
            τ
            )
          
          =
          
            
              <
              
                
                  
                    R
                    
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      ɛ
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      ,
                      τ
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      j
                    
                    ⁢
                    
                      
                    
                    ⁢
                    2
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    
                      α
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        x
                        ⁢
                        
                          
                        
                        ⁢
                        ɛ
                      
                    
                    ⁢
                    t
                  
                
              
              ⁢
              
                >
                c
              
            
            =
            
              
                A
                ⁢
                
                  
                
                ⁢
                
                  
                    R
                    
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      m
                      ⁢
                      
                        
                      
                      ⁢
                      c
                      ⁢
                      
                        
                      
                      ⁢
                      ɛ
                    
                    
                      α
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        x
                        ⁢
                        
                          
                        
                        ⁢
                        ɛ
                      
                    
                  
                  ⁡
                  
                    (
                    τ
                    )
                  
                
                ⁢
                
                  A
                  
                    ɛ
                    ⁢
                    
                      
                    
                    ⁢
                    T
                  
                
              
              +
              
                
                  
                    η
                    2
                  
                  ⁡
                  
                    (
                    τ
                    )
                  
                
                ⁢
                
                  δ
                  ⁡
                  
                    (
                    
                      α
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        x
                        ⁢
                        
                          
                        
                        ⁢
                        ɛ
                      
                    
                    )
                  
                
                ⁢
                
                  δ
                  ⁡
                  
                    (
                    
                      1
                      +
                      ɛ
                    
                    )
                  
                
                ⁢
                I
              
            
          
        
      
    
    
      
        (
        23
        )
      
    
  




where





  
    R
    
      Δ
      ⁢
      
        
      
      ⁢
      m
      ⁢
      
        
      
      ⁢
      c
      ⁢
      
        
      
      ⁢
      ɛ
    
    
      α
      
        Δ
        ⁢
        
          
        
        ⁢
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
    
  
  ⁡
  
    (
    τ
    )
  




defines the first and second matrices of cyclic covariance of mc(t). Using the expressions (7) and (21) in (15), we get





  
    
      
        
          
            R
            
              x
              ⁢
              
                
              
              ⁢
              ɛ
            
          
          ⁡
          
            (
            
              t
              ,
              τ
            
            )
          
        
        =
        
          
            
              ∑
              
                
                  α
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    x
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                  
                
                ∈
                
                  
                
                ⁢
                
                  Γ
                  
                    Δ
                    x
                  
                  
                    [
                    
                      1
                      ,
                      ɛ
                    
                    ]
                  
                
              
            
            ⁢
            
              
                
                  R
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    x
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                  
                  
                    α
                    
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      ɛ
                    
                  
                
                ⁡
                
                  (
                  τ
                  )
                
              
              ⁢
              
                e
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    πα
                    
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      ɛ
                    
                  
                  ⁢
                  t
                
              
            
          
          +
          
            
              ∑
              
                γ
                ⁢
                
                  
                
                ∈
                
                  
                
                ⁢
                
                  Γ
                  x
                  1
                
              
            
            ⁢
            
              
                ∑
                
                  ω
                  ⁢
                  
                    
                  
                  ∈
                  
                    
                  
                  ⁢
                  
                    Γ
                    x
                    1
                  
                
              
              ⁢
              
                
                  e
                  x
                  γ
                
                ⁢
                
                  e
                  x
                  
                    ω
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                    ⁢
                    
                      
                    
                    ⁢
                    T
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      j
                    
                    ⁢
                    
                      
                    
                    ⁢
                    2
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    ω
                    ⁢
                    
                      
                    
                    ⁢
                    τ
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                  
                
                ⁢
                
                  e
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    2
                    ⁢
                    
                      π
                      ⁡
                      
                        (
                        
                          γ
                          +
                          
                            ω
                            ⁢
                            
                              
                            
                            ⁢
                            ɛ
                          
                        
                        )
                      
                    
                    ⁢
                    t
                  
                
              
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




which shows that the matrices Rxε(t, τ) accept a Fourier series decomposition





  
    
      
        
          
            R
            
              x
              ⁢
              
                
              
              ⁢
              ɛ
            
          
          ⁡
          
            (
            
              t
              ,
              τ
            
            )
          
        
        =
        
          
            ∑
            
              
                α
                
                  x
                  ⁢
                  
                    
                  
                  ⁢
                  ɛ
                
              
              ∈
              
                
              
              ⁢
              
                Γ
                
                  
                    
                  
                  x
                
                
                  [
                  
                    1
                    ,
                    ɛ
                  
                  ]
                
              
            
          
          ⁢
          
            
              
                R
                
                  x
                  ⁢
                  
                    
                  
                  ⁢
                  ɛ
                
                
                  α
                  
                    x
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                  
                
              
              ⁡
              
                (
                τ
                )
              
            
            ⁢
            
              e
              
                j
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  
                
                ⁢
                
                  πα
                  
                    x
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                  
                
                ⁢
                t
              
            
          
        
      
    
    
      
        (
        25
        )
      
    
  




where





  
    Γ
    x
    
      [
      
        1
        ,
        ɛ
      
      ]
    
  
  =
  
    
      {
      
        α
        
          x
          ⁢
          
            
          
          ⁢
          ɛ
        
      
      }
    
    =
    
      
        Γ
        
          Δ
          ⁢
          
            
          
          ⁢
          x
        
        
          [
          
            1
            ,
            ɛ
          
          ]
        
      
      ⋃
      
        {
        
          
            Γ
            x
            1
          
          ⁢
          
            o
            ɛ
          
          ⁢
          
            Γ
            x
            1
          
        
        }
      
    
  




is the set of the cyclic frequencies αxε of





  
    
      R
      
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
    
    ⁡
    
      (
      
        t
        ,
        τ
      
      )
    
  
  ,
  
    
      
        Γ
        x
        1
      
      ⁢
      
        o
        ɛ
      
      ⁢
      
        Γ
        x
        1
      
    
    =
    
      {
      
        
          α
          =
          
            
              γ
              +
              
                ɛω
                ⁢
                
                  
                
                ⁢
                where
                ⁢
                
                  
                
                ⁢
                γ
              
            
            ∈
            
              Γ
              x
              1
            
          
        
        ,
        
          ω
          ∈
          
            Γ
            x
            1
          
        
      
      }
    
  
  ,




the matrices





  
    R
    
      x
      ⁢
      
        
      
      ⁢
      ɛ
    
    
      α
      
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
    
  
  ⁡
  
    (
    τ
    )
  




are the first (ε=−1) and second (ε=+1) matrices of cyclic correlation of x(t), defined by





  
    
      
        
          
            R
            
              x
              ⁢
              
                
              
              ⁢
              ɛ
            
            
              α
              
                x
                ⁢
                
                  
                
                ⁢
                ɛ
              
            
          
          ⁡
          
            (
            τ
            )
          
        
        =
        
          
            <
            
              
                
                  R
                  
                    x
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                  
                
                ⁡
                
                  (
                  
                    t
                    ,
                    τ
                  
                  )
                
              
              ⁢
              
                e
                
                  
                    -
                    j
                  
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  
                    α
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      ɛ
                    
                  
                  ⁢
                  t
                
              
            
            ⁢
            
              >
              c
            
          
          =
          
            
              A
              ⁢
              
                
              
              ⁢
              
                
                  R
                  
                    m
                    ⁢
                    
                      
                    
                    ⁢
                    c
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                  
                  
                    α
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      ɛ
                    
                  
                
                ⁡
                
                  (
                  τ
                  )
                
              
              ⁢
              
                A
                
                  ɛ
                  ⁢
                  
                    
                  
                  ⁢
                  T
                
              
            
            +
            
              
                
                  η
                  2
                
                ⁡
                
                  (
                  τ
                  )
                
              
              ⁢
              
                
              
              ⁢
              
                δ
                ⁡
                
                  (
                  
                    α
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      ɛ
                    
                  
                  )
                
              
              ⁢
              
                
              
              ⁢
              
                δ
                ⁡
                
                  (
                  
                    1
                    +
                    ɛ
                  
                  )
                
              
              ⁢
              
                
              
              ⁢
              I
            
          
        
      
    
    
      
        (
        26
        )
      
    
  




where





  
    R
    
      mc
      ⁢
      
        
      
      ⁢
      ɛ
    
    
      α
      
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
    
  
  ⁡
  
    (
    τ
    )
  




defines the first and second matrices of cyclic correlation of mc(t). In particular, for the zero cyclic frequency, the matrix





  
    R
    
      x
      ⁢
      
        
      
      ⁢
      ɛ
    
    
      α
      
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
    
  
  ⁡
  
    (
    τ
    )
  




corresponds to the temporal mean in t, Rxε(τ), de Rxε(t, τ) which is written, using (14),

Rxε(τ)Δ<Rxε(t,τ)>c=ARmcε(τ)AεT+η2(τ)δ(1+ε)I  (27)

where, using (17), the matrix Rmcε(τ)Δ<Rmcε(t, τ)>c is written

Rmcε(τ)Δ<Rmcε(t,τ)>c=RΔmcε(τ)+<emc(t)emc(t−τ)εT>c=R Δmcε(τ)+Emcε(τ)  (28)

where RΔmcε(τ)Δ<RΔmcε(t, τ)>c and Emcε(τ)Δ<emc(t) emc(t−τ)εT>c.

Empirical Estimator of the Second-Order Statistics

In practice, the second-order statistics of the observations are unknown in principle and must be estimated by the taking the temporal mean on a finite period of observation, on the basis of a number K of samples x(k) (1≦k≦K), of the observation vector x(t), using the property of ergodicity of these samples in the stationary case and of cycloergodicity of these samples in the cyclostationary case. If Te denotes the sampling period, the estimation of the cyclic correlation matrix




  
    R
    
      x
      ⁢
      
        
      
      ⁢
      ɛ
    
    α
  
  ⁡
  
    (
    τ
    )
  




for τ=lTe, which corresponds to a matrix of cumulants only for centered observations, is done only by means of the estimator





  
    
      
        R
        ^
      
      
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
      α
    
    ⁡
    
      (
      
        lT
        e
      
      )
    
  
  ⁢
  
    (
    K
    )
  




qualified as empirical and defined by





  
    
      
        
          
            
              R
              ^
            
            
              x
              ⁢
              
                
              
              ⁢
              ɛ
            
            α
          
          ⁡
          
            (
            
              lT
              e
            
            )
          
        
        ⁢
        
          (
          K
          )
        
        ⁢
        
          
        
        ⁢
        
          Δ
          
            _
            _
          
        
        ⁢
        
          1
          K
        
        ⁢
        
          
            ∑
            
              k
              =
              1
            
            K
          
          ⁢
          
            
              x
              ⁡
              
                (
                k
                )
              
            
            ⁢
            
              
            
            ⁢
            
              
                x
                ⁡
                
                  (
                  
                    k
                    -
                    l
                  
                  )
                
              
              
                ɛ
                ⁢
                
                  
                
                ⁢
                T
              
            
            ⁢
            
              e
              
                
                  -
                  j
                
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                α
                ⁢
                
                  
                
                ⁢
                
                  kT
                  e
                
              
            
          
        
      
    
    
      
        (
        29
        )
      
    
  




The method according to the invention comprises, for example, a novel step to determine a second-order estimator.
Novel Estimator of the Second-Order Cyclic Cumulants of the Observations
In the presence of non-centered observations, the matrices of correlation of the observations Rxε(t, τ) no longer correspond to the matrices of covariance or of second-order cumulants of the observations, as indicated by the expression (15). This is also the case with the matrices of cyclic correlation,




  
    
      R
      
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
      
        α
        
          x
          ⁢
          
            
          
          ⁢
          ɛ
        
      
    
    ⁡
    
      (
      τ
      )
    
  
  ,




defined by (26) which, for non-centered observations, no longer correspond to the matrices of covariance or matrices of second-order cyclic cumulants





  
    
      R
      
        Δ
        ⁢
        
          
        
        ⁢
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
      
        α
        
          x
          ⁢
          
            
          
          ⁢
          ɛ
        
      
    
    ⁡
    
      (
      τ
      )
    
  
  ,




defined by (22) for αΔxε=αxε. Thus, an efficient operation of the second-order separators F1 and F2 with respect to the potentially non-centered cyclostationary sources can be obtained only by making use of the information contained in the matrices





  
    R
    
      Δ
      ⁢
      
        
      
      ⁢
      x
      ⁢
      
        
      
      ⁢
      ɛ
    
    
      α
      
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
    
  
  ⁡
  
    (
    τ
    )
  




rather than in the matrices





  
    
      R
      
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
      
        α
        
          x
          ⁢
          
            
          
          ⁢
          ɛ
        
      
    
    ⁡
    
      (
      τ
      )
    
  
  .




Thus, in as much as, for cyclostationary and cycloergodic sources, the empirical estimator (29) is an asymptotically unbiased and consistent estimator of the matrix of cyclic correlation





  
    
      R
      
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
      α
    
    ⁡
    
      (
      
        lT
        e
      
      )
    
  
  ,




another estimator is used for non-centered observations. This other estimator is aimed at making an asymptotically unbiased and consistent estimation of the matrix of cyclic covariance





  
    
      R
      
        Δ
        ⁢
        
          
        
        ⁢
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
      α
    
    ⁡
    
      (
      
        lT
        e
      
      )
    
  
  .




From the expressions (24) and (25), we deduce the expression of the matrix of cyclic covariance




  
    R
    
      Δ
      ⁢
      
        
      
      ⁢
      x
      ⁢
      
        
      
      ⁢
      ɛ
    
    α
  
  ⁡
  
    (
    
      lT
      e
    
    )
  




as a function of that of





  
    
      R
      
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
      α
    
    ⁡
    
      (
      
        lT
        e
      
      )
    
  
  ,




given by





  
    
      
        
          
            R
            
              Δ
              ⁢
              
                
              
              ⁢
              x
              ⁢
              
                
              
              ⁢
              ɛ
            
            α
          
          ⁡
          
            (
            
              lT
              e
            
            )
          
        
        =
        
          
            
              R
              
                x
                ⁢
                
                  
                
                ⁢
                ɛ
              
              α
            
            ⁡
            
              (
              
                lT
                e
              
              )
            
          
          -
          
            
              ∑
              
                ω
                ∈
                
                  Γ
                  x
                  1
                
              
            
            ⁢
            
              
                e
                x
                
                  α
                  -
                  
                    ω
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                  
                
              
              ⁢
              
                e
                x
                
                  ω
                  ⁢
                  
                    
                  
                  ⁢
                  ɛ
                  ⁢
                  
                    
                  
                  ⁢
                  T
                
              
              ⁢
              
                e
                
                  
                    -
                    j
                  
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  ω
                  ⁢
                  
                    
                  
                  ⁢
                  ɛ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    lT
                    e
                  
                
              
            
          
        
      
    
    
      
        (
        30
        )
      
    
  




Thus, the estimation of the matrix of cyclic covariance




  
    R
    
      Δ
      ⁢
      
        
      
      ⁢
      x
      ⁢
      
        
      
      ⁢
      ɛ
    
    α
  
  ⁡
  
    (
    
      lT
      e
    
    )
  




is made from the estimator





  
    
      
        R
        ^
      
      
        Δ
        ⁢
        
          
        
        ⁢
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
      α
    
    ⁡
    
      (
      
        lT
        e
      
      )
    
  
  ⁢
  
    (
    K
    )
  




defined by





  
    
      
        
          
            
              
                R
                ^
              
              
                Δ
                ⁢
                
                  
                
                ⁢
                x
                ⁢
                
                  
                
                ⁢
                ɛ
              
              α
            
            ⁡
            
              (
              
                lT
                e
              
              )
            
          
          ⁢
          
            (
            K
            )
          
          ⁢
          
            
          
          ⁢
          
            Δ
            
              _
              _
            
          
          ⁢
          
            
          
          ⁢
          
            
              
                R
                ^
              
              
                x
                ⁢
                
                  
                
                ⁢
                ɛ
              
              α
            
            ⁡
            
              (
              
                lT
                e
              
              )
            
          
          ⁢
          
            (
            K
            )
          
        
        -
        
          
            ∑
            
              ω
              ∈
              
                Γ
                x
                1
              
            
          
          ⁢
          
            
              
                
                  e
                  ^
                
                x
                
                  α
                  -
                  
                    ω
                    ⁢
                    
                      
                    
                    ⁢
                    ɛ
                  
                
              
              ⁡
              
                (
                K
                )
              
            
            ⁢
            
              
                
                  
                    e
                    ^
                  
                  x
                  
                    ω
                    ⁢
                    
                      
                    
                  
                
                ⁡
                
                  (
                  K
                  )
                
              
              
                ɛ
                ⁢
                
                  
                
                ⁢
                T
              
            
            ⁢
            
              e
              
                
                  -
                  j
                
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                ω
                ⁢
                
                  
                
                ⁢
                ɛ
                ⁢
                
                  
                
                ⁢
                
                  lT
                  e
                
              
            
          
        
      
    
    
      
        (
        31
        )
      
    
  




where





  
    
      
        R
        ^
      
      
        Δ
        ⁢
        
          
        
        ⁢
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
      α
    
    ⁡
    
      (
      
        lT
        e
      
      )
    
  
  ⁢
  
    (
    K
    )
  




is defined by (29) and where





  
    
      e
      ^
    
    x
    ω
  
  ⁡
  
    (
    K
    )
  




is defined by





  
    
      
        
          
            
              e
              ^
            
            x
            ω
          
          ⁡
          
            (
            K
            )
          
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          
            1
            K
          
          ⁢
          
            
              ∑
              
                k
                =
                1
              
              K
            
            ⁢
            
              
            
            ⁢
            
              
                x
                ⁡
                
                  (
                  k
                  )
                
              
              ⁢
              
                e
                
                  
                    -
                    j
                  
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  ω
                  ⁢
                  
                    
                  
                  ⁢
                  
                    kT
                    e
                  
                
              
            
          
        
      
    
    
      
        (
        32
        )
      
    
  




Thus, assuming cyclostationary and cycloergodic observations, whether centered or not, the estimator (31) is an asymptotically unbiased and consistent estimator of the matrix of cyclic covariance or matrix of second-order cyclic cumulants




  
    
      R
      
        Δ
        ⁢
        
          
        
        ⁢
        x
        ⁢
        
          
        
        ⁢
        ɛ
      
      α
    
    ⁡
    
      (
      
        l
        ⁢
        
          
        
        ⁢
        
          T
          e
        
      
      )
    
  
  .




In particular, the separators F1 must exploit the estimator (31) for α=0 and ε=−1, written as





  
    
      
        
          R
          ^
        
        
          Δ
          ⁢
          
            
          
          ⁢
          x
        
      
      ⁡
      
        (
        
          l
          ⁢
          
            
          
          ⁢
          
            T
            e
          
        
        )
      
    
    ⁢
    
      (
      K
      )
    
  
  ,




and defined by





  
    
      
        
          
            
              
                R
                ^
              
              
                Δ
                ⁢
                
                  
                
                ⁢
                x
              
            
            ⁡
            
              (
              
                l
                ⁢
                
                  
                
                ⁢
                
                  T
                  e
                
              
              )
            
          
          ⁢
          
            (
            K
            )
          
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          
            
              
                
                  R
                  ^
                
                
                  
                    
                  
                  ⁢
                  x
                
              
              ⁡
              
                (
                
                  l
                  ⁢
                  
                    
                  
                  ⁢
                  
                    T
                    e
                  
                
                )
              
            
            ⁢
            
              (
              K
              )
            
          
          -
          
            
              ∑
              
                ω
                ∈
                
                  Γ
                  x
                  1
                
              
              
                
              
            
            ⁢
            
              
            
            ⁢
            
              
                
                  
                    e
                    ^
                  
                  x
                  ω
                
                ⁡
                
                  (
                  K
                  )
                
              
              ⁢
              
                
                  
                    
                      e
                      ^
                    
                    x
                    ω
                  
                  ⁡
                  
                    (
                    K
                    )
                  
                
                †
              
              ⁢
              
                e
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  ω
                  ⁢
                  
                    
                  
                  ⁢
                  l
                  ⁢
                  
                    
                  
                  ⁢
                  
                    T
                    e
                  
                
              
            
          
        
      
    
    
      
        (
        33
        )
      
    
  




where {circumflex over (R)}x(lTe)(K) is defined by (29) with α=0 and ε=−1.

According to another alternative embodiment, the method includes a step for determining a new fourth-order estimator.
Third-Order Statistics
Based on the above assumptions, the third-order statistics (moments) of the observations are defined by (34)

Txε(t,τ1,τ2)[i,j,k]ΔE[xi(t)xj(t−τ1)εxk(−τ2)]=E[Δxi(t)Δxj(t−τ1)εΔxk(t−τ2)]+ex[i](t)E[Δxj(t−τ1)εΔxk(t−τ2)]+ex[k](t−τ2)E[Δxi(t)Δxj(t−τ1)ε]+ex[j](t−τ1)εE[Δxi(t)Δxk(t−τ2)]+4ex[i](t)ex[j](t−τ1)εex[k](t−τ2)

where exx[i](t)=exi(t) is the component i of ex(t). The third-order cumulants are the quantities E[Δxi(t)Δxj(t−τ1)εΔxk(t−τ2)]. Assuming that the term Txε(t, τ1, τ2)[i, j, k] is the element [i, N(j−1)+k] of the matrix Txε(t, τ1, τ2), with a dimension (N×N2), we get an expression of this matrix given by:





  
    
      
        
          
            T
            xɛ
          
          ⁡
          
            (
            
              t
              ,
              
                τ
                1
              
              ,
              
                τ
                2
              
            
            )
          
        
        =
        
          
            A
            ⁢
            
              
            
            ⁢
            
              
                T
                mce
              
              ⁡
              
                (
                
                  t
                  ,
                  
                    τ
                    1
                  
                  ,
                  
                    τ
                    2
                  
                
                )
              
            
            ⁢
            
              
                (
                
                  A
                  ⊗
                  
                    A
                    ɛ
                  
                
                )
              
              
                ɛ
                ⁢
                
                  
                
                ⁢
                T
              
            
          
          =
          
            
              ∑
              
                i
                ,
                j
                ,
                
                  k
                  =
                  1
                
              
              P
            
            ⁢
            
              
            
            ⁢
            
              
                
                  
                    T
                    mce
                  
                  ⁡
                  
                    (
                    
                      t
                      ,
                      
                        τ
                        1
                      
                      ,
                      
                        τ
                        2
                      
                    
                    )
                  
                
                ⁡
                
                  [
                  
                    i
                    ,
                    j
                    ,
                    k
                  
                  ]
                
              
              ⁢
              
                
                  
                    a
                    i
                  
                  ⁡
                  
                    [
                    
                      
                        a
                        j
                      
                      ⊗
                      
                        a
                        k
                        ɛ
                      
                    
                    ]
                  
                
                ɛT
              
            
          
        
      
    
    
      
        (
        35
        )
      
    
  




where Tmcε(t, τ1, τ2) is the matrix (P×P2) whose coefficients are the quantities Tmcε(t, τ1, τ2)[i, j, k] defined by





  
    
      
        
          
            
              
                
                  
                    T
                    
                      mc
                      ⁢
                      ɛ
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      ,
                      
                        τ
                        1
                      
                      ,
                      
                        τ
                        2
                      
                    
                    )
                  
                
                ⁡
                
                  [
                  
                    i
                    ,
                    j
                    ,
                    k
                  
                  ]
                
              
              =
              
                
                  E
                  ⁡
                  
                    [
                    
                      
                        
                          m
                          ic
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          
                            m
                            jc
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              
                                τ
                                1
                              
                            
                            )
                          
                        
                        ɛ
                      
                      ⁢
                      
                        
                          m
                          
                            kc
                            ⁢
                            
                              
                            
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              τ
                              2
                            
                          
                          )
                        
                      
                    
                    ]
                  
                
                =
                
                  
                    
                      E
                      ⁡
                      
                        [
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              m
                              ic
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          ⁢
                          
                            
                              
                                m
                                ic
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    1
                                  
                                
                                )
                              
                            
                            ɛ
                          
                          ⁢
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              m
                              ic
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  2
                                
                              
                              )
                            
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      δ
                      ⁡
                      
                        (
                        
                          i
                          -
                          j
                        
                        )
                      
                    
                    ⁢
                    
                      δ
                      ⁡
                      
                        (
                        
                          i
                          -
                          k
                        
                        )
                      
                    
                  
                  +
                  
                    
                      
                        e
                        ic
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    ⁢
                    
                      E
                      ⁡
                      
                        [
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                m
                                jc
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    1
                                  
                                
                                )
                              
                            
                            ɛ
                          
                          ⁢
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              m
                              jc
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  2
                                
                              
                              )
                            
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      δ
                      ⁡
                      
                        (
                        
                          j
                          -
                          k
                        
                        )
                      
                    
                  
                  +
                  
                    
                      
                        e
                        kc
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            2
                          
                        
                        )
                      
                    
                    ⁢
                    
                      E
                      ⁡
                      
                        [
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              m
                              ic
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          ⁢
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                m
                                ic
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    1
                                  
                                
                                )
                              
                            
                            ɛ
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      δ
                      ⁡
                      
                        (
                        
                          i
                          -
                          j
                        
                        )
                      
                    
                  
                  +
                
              
            
             
          
          ⁢
          
            
              
                e
                jc
              
              ⁡
              
                (
                
                  t
                  -
                  
                    τ
                    1
                  
                
                )
              
            
            ɛ
          
          ⁢
          
            E
            ⁡
            
              [
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  
                    m
                    ic
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ⁢
                
                  
                
                ⁢
                Δ
                ⁢
                
                  
                
                ⁢
                
                  
                    m
                    ic
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        2
                      
                    
                    )
                  
                
              
              ]
            
          
          ⁢
          
            δ
            ⁡
            
              (
              
                i
                -
                k
              
              )
            
          
        
        +
        
          4
          ⁢
          
            
              e
              ic
            
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            
              
                e
                jc
              
              ⁡
              
                (
                
                  t
                  -
                  
                    τ
                    1
                  
                
                )
              
            
            ɛ
          
          ⁢
          
            
              e
              kc
            
            ⁡
            
              (
              
                t
                -
                
                  τ
                  2
                
              
              )
            
          
        
      
    
    
      
        (
        36
        )
      
    
  




For cyclostationary sources, the matrices of third-order moments, Tmcε(t, τ1, τ2) and Txε(t, τ1, τ2), are (quasi-) or poly-periodic functions of the time t accepting a Fourier series decomposition and we get:




  
    
      
        
          
            T
            xɛ
          
          ⁡
          
            (
            
              t
              ,
              
                τ
                1
              
              ,
              
                τ
                2
              
            
            )
          
        
        =
        
          
            ∑
            
              
                v
                xɛ
              
              ∈
              
                Γ
                x
                
                  [
                  
                    1
                    ,
                    ɛ1
                  
                  ]
                
              
            
          
          ⁢
          
            
          
          ⁢
          
            
              
                T
                xɛ
                
                  v
                  xɛ
                
              
              ⁡
              
                (
                
                  
                    τ
                    1
                  
                  ,
                  
                    τ
                    2
                  
                
                )
              
            
            ⁢
            
              e
              
                j2π
                ⁢
                
                  
                
                ⁢
                
                  v
                  xɛ
                
                ⁢
                t
              
            
          
        
      
    
    
      
        (
        37
        )
      
    
  




where





  
    Γ
    x
    
      [
      
        1
        ,
        ɛ1
      
      ]
    
  
  =
  
    {
    
      v
      xɛ
    
    }
  




is the set of the cyclic frequencies vxεof Txε(t, τ1, τ2) and





  
    T
    xɛ
    
      v
      xɛ
    
  
  ⁡
  
    (
    
      
        τ
        1
      
      ,
      
        τ
        2
      
    
    )
  




a matrix of cyclic third-order moments of x(t), defined respectively by





  
    
      
        
          
            T
            xɛ
            
              v
              xɛ
            
          
          ⁡
          
            (
            
              
                τ
                1
              
              ,
              
                τ
                2
              
            
            )
          
        
        =
        
          
            <
            
              
                
                  T
                  xɛ
                
                ⁡
                
                  (
                  
                    t
                    ,
                    
                      τ
                      1
                    
                    ,
                    
                      τ
                      2
                    
                  
                  )
                
              
              ⁢
              
                e
                
                  
                    -
                    j2π
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    v
                    xɛ
                  
                  ⁢
                  t
                
              
            
            ⁢
            
              >
              c
            
          
          =
          
            A
            ⁢
            
              
            
            ⁢
            
              
                T
                mcɛ
                
                  v
                  xɛ
                
              
              ⁡
              
                (
                
                  
                    τ
                    1
                  
                  ,
                  
                    τ
                    2
                  
                
                )
              
            
            ⁢
            
              
                (
                
                  A
                  ⊗
                  
                    A
                    ɛ
                  
                
                )
              
              
                ɛ
                ⁢
                
                  
                
                ⁢
                T
              
            
          
        
      
    
    
      
        (
        38
        )
      
    
  




where {circle around (x)} corresponds to the Kronecker product.

Fourth-Order Statistics

Based on the above assumptions (relating to non-centered and cyclostationary signals) the fourth-order statistics of the observations are the fourth-order cumulants defined by:




  
    
      
        
          
            Q
            ⁢
            
              
            
            ⁢
            x
            ⁢
            
              
            
            ⁢
            
              
                ζ
                ⁡
                
                  (
                  
                    t
                    ,
                    
                      τ
                      1
                    
                    ,
                    
                      τ
                      2
                    
                    ,
                    
                      τ
                      3
                    
                  
                  )
                
              
              ⁡
              
                [
                
                  i
                  ,
                  j
                  ,
                  k
                  ,
                  l
                
                ]
              
            
          
          ⁢
          
            =
            Δ
          
          ⁢
          
            
              Cum
              ⁡
              
                (
                
                  
                    
                      x
                      i
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ,
                  
                    
                      
                        x
                        j
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            1
                          
                        
                        )
                      
                    
                    ζ1
                  
                  ,
                  
                    
                      
                        x
                        k
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            2
                          
                        
                        )
                      
                    
                    ζ2
                  
                  ,
                  
                    
                      x
                      l
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          3
                        
                      
                      )
                    
                  
                
                )
              
            
            =
            
              

            
            ⁢
            
              
                Cum
                (
                
                  
                    
                      
                        Δ
                        ⁢
                        x
                      
                      i
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ,
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        
                          x
                          j
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              τ
                              1
                            
                          
                          )
                        
                      
                      ζ1
                    
                  
                  ,
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        
                          x
                          k
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              τ
                              2
                            
                          
                          )
                        
                      
                      ζ2
                    
                  
                  ,
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        x
                        l
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            3
                          
                        
                        )
                      
                    
                  
                
                )
              
              =
              
                

              
              ⁢
              
                
                  
                    E
                    ⁡
                    
                      [
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            x
                            i
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        ⁢
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            
                              x
                              j
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  1
                                
                              
                              )
                            
                          
                          ζ1
                        
                        ⁢
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            
                              x
                              k
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  2
                                
                              
                              )
                            
                          
                          ζ2
                        
                        ⁢
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            x
                            l
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              
                                τ
                                3
                              
                            
                            )
                          
                        
                      
                      ]
                    
                  
                  -
                  
                    

                  
                  ⁢
                  
                    
                      E
                      ⁡
                      
                        [
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              x
                              i
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          ⁢
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                x
                                j
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    1
                                  
                                
                                )
                              
                            
                            ζ1
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      E
                      ⁡
                      
                        [
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                x
                                k
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    2
                                  
                                
                                )
                              
                            
                            ζ2
                          
                          ⁢
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              x
                              l
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  3
                                
                              
                              )
                            
                          
                        
                        ]
                      
                    
                  
                  -
                  
                    

                  
                  ⁢
                  
                    
                      E
                      ⁡
                      
                        [
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              x
                              i
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          ⁢
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                x
                                k
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    2
                                  
                                
                                )
                              
                            
                            ζ2
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      E
                      ⁡
                      
                        [
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                x
                                j
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    1
                                  
                                
                                )
                              
                            
                            ζ1
                          
                          ⁢
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              x
                              l
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  3
                                
                              
                              )
                            
                          
                        
                        ]
                      
                    
                  
                  -
                  
                    

                  
                  ⁢
                  
                    
                      E
                      ⁡
                      
                        [
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              x
                              i
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          ⁢
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              x
                              l
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  3
                                
                              
                              )
                            
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      E
                      ⁡
                      
                        [
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                x
                                j
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    1
                                  
                                
                                )
                              
                            
                            ζ1
                          
                          ⁢
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                x
                                k
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    2
                                  
                                
                                )
                              
                            
                            ζ2
                          
                        
                        ]
                      
                    
                  
                
                =
                
                  

                
                ⁢
                
                  
                    E
                    ⁡
                    
                      [
                      
                        
                          
                            x
                            i
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        ⁢
                        
                          
                            
                              x
                              j
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  1
                                
                              
                              )
                            
                          
                          ζ1
                        
                        ⁢
                        
                          
                            
                              x
                              k
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  2
                                
                              
                              )
                            
                          
                          ζ2
                        
                        ⁢
                        
                          
                            x
                            l
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              
                                τ
                                3
                              
                            
                            )
                          
                        
                      
                      ]
                    
                  
                  -
                  
                    

                  
                  ⁢
                  
                    
                      
                        e
                        xi
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    ⁢
                    
                      E
                      ⁡
                      
                        [
                        
                          
                            
                              
                                x
                                j
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    1
                                  
                                
                                )
                              
                            
                            ζ1
                          
                          ⁢
                          
                            
                              
                                x
                                k
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    2
                                  
                                
                                )
                              
                            
                            ζ2
                          
                          ⁢
                          
                            
                              x
                              l
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  3
                                
                              
                              )
                            
                          
                        
                        ]
                      
                    
                  
                  -
                  
                    

                  
                  ⁢
                  
                    
                      
                        
                          e
                          xj
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              τ
                              1
                            
                          
                          )
                        
                      
                      ζ1
                    
                    ⁢
                    
                      E
                      ⁡
                      
                        [
                        
                          
                            
                              x
                              i
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          ⁢
                          
                            
                              
                                x
                                k
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    2
                                  
                                
                                )
                              
                            
                            ζ2
                          
                          ⁢
                          
                            
                              x
                              l
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  3
                                
                              
                              )
                            
                          
                        
                        ]
                      
                    
                  
                  -
                  
                    

                  
                  ⁢
                  
                    
                      
                        
                          e
                          xk
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              τ
                              2
                            
                          
                          )
                        
                      
                      ζ2
                    
                    ⁢
                    
                      E
                      ⁡
                      
                        [
                        
                          
                            
                              x
                              i
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          ⁢
                          
                            
                              
                                x
                                j
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    1
                                  
                                
                                )
                              
                            
                            ζ1
                          
                          ⁢
                          
                            
                              x
                              l
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  3
                                
                              
                              )
                            
                          
                        
                        ]
                      
                    
                  
                  -
                  
                    

                  
                  ⁢
                  
                    
                      
                        e
                        xl
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            3
                          
                        
                        )
                      
                    
                    ⁢
                    
                      E
                      ⁡
                      
                        [
                        
                          
                            
                              x
                              i
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          ⁢
                          
                            
                              
                                x
                                j
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    1
                                  
                                
                                )
                              
                            
                            ζ1
                          
                          ⁢
                          
                            
                              
                                x
                                k
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    2
                                  
                                
                                )
                              
                            
                            ζ2
                          
                        
                        ]
                      
                    
                  
                  -
                  
                    

                  
                  ⁢
                  
                    
                      E
                      ⁡
                      
                        [
                        
                          
                            
                              x
                              i
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          ⁢
                          
                            
                              
                                x
                                j
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    1
                                  
                                
                                )
                              
                            
                            ζ1
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      E
                      ⁡
                      
                        [
                        
                          
                            
                              
                                x
                                k
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    2
                                  
                                
                                )
                              
                            
                            ζ2
                          
                          ⁢
                          
                            
                              x
                              l
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  3
                                
                              
                              )
                            
                          
                        
                        ]
                      
                    
                  
                  -
                  
                    

                  
                  ⁢
                  
                    
                      E
                      ⁡
                      
                        [
                        
                          
                            
                              x
                              i
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          ⁢
                          
                            
                              
                                x
                                k
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    2
                                  
                                
                                )
                              
                            
                            ζ2
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      E
                      ⁡
                      
                        [
                        
                          
                            
                              
                                x
                                j
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    1
                                  
                                
                                )
                              
                            
                            ζ1
                          
                          ⁢
                          
                            
                              x
                              l
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  3
                                
                              
                              )
                            
                          
                        
                        ]
                      
                    
                  
                  -
                  
                    

                  
                  ⁢
                  
                    
                      E
                      ⁡
                      
                        [
                        
                          
                            
                              x
                              i
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          ⁢
                          
                            
                              x
                              l
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  3
                                
                              
                              )
                            
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      E
                      ⁡
                      
                        [
                        
                          
                            
                              
                                x
                                j
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    1
                                  
                                
                                )
                              
                            
                            ζ1
                          
                          ⁢
                          
                            
                              
                                x
                                k
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    2
                                  
                                
                                )
                              
                            
                            ζ2
                          
                        
                        ]
                      
                    
                  
                  +
                  
                    2
                    ⁢
                    
                      
                        e
                        xi
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                e
                xj
              
              ⁡
              
                (
                
                  t
                  -
                  
                    τ
                    1
                  
                
                )
              
            
            ζ1
          
          ⁢
          
            E
            ⁡
            
              [
              
                
                  
                    
                      x
                      k
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          2
                        
                      
                      )
                    
                  
                  ζ2
                
                ⁢
                
                  
                    x
                    l
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        3
                      
                    
                    )
                  
                
              
              ]
            
          
        
        +
        
          

        
        ⁢
        
          2
          ⁢
          
            
              e
              xi
            
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            
              
                e
                xk
              
              ⁡
              
                (
                
                  t
                  -
                  
                    τ
                    2
                  
                
                )
              
            
            ζ2
          
          ⁢
          
            E
            ⁡
            
              [
              
                
                  
                    
                      x
                      j
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          1
                        
                      
                      )
                    
                  
                  ζ1
                
                ⁢
                
                  
                    x
                    l
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        3
                      
                    
                    )
                  
                
              
              ]
            
          
        
        +
        
          

        
        ⁢
        
          
        
        ⁢
        
          2
          ⁢
          
            
              e
              xi
            
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            
              e
              xl
            
            (
            
              t
              -
              
                
                  τ
                  3
                
                ⁢
                
                  E
                  ⁡
                  
                    [
                    
                      
                        
                          
                            
                              x
                              j
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  1
                                
                              
                              )
                            
                          
                          ζ1
                        
                        ⁢
                        
                          x
                          k
                        
                      
                      +
                      
                        
                          (
                          
                            t
                            -
                            
                              τ
                              2
                            
                          
                          )
                        
                        ζ2
                      
                    
                    ]
                  
                
              
              +
              
                

              
              ⁢
              
                2
                ⁢
                
                  
                    
                      e
                      xk
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          2
                        
                      
                      )
                    
                  
                  ζ2
                
                ⁢
                
                  
                    
                      e
                      xj
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          1
                        
                      
                      )
                    
                  
                  ζ1
                
                ⁢
                
                  E
                  ⁡
                  
                    [
                    
                      
                        
                          x
                          i
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          x
                          l
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              τ
                              3
                            
                          
                          )
                        
                      
                    
                    ]
                  
                
              
              +
              
                

              
              ⁢
              
                2
                ⁢
                
                  
                    e
                    xl
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        3
                      
                    
                    )
                  
                
                ⁢
                
                  
                    
                      e
                      xj
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          1
                        
                      
                      )
                    
                  
                  ζ1
                
                ⁢
                
                  E
                  ⁡
                  
                    [
                    
                      
                        
                          
                            x
                            k
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              
                                τ
                                2
                              
                            
                            )
                          
                        
                        ζ2
                      
                      ⁢
                      
                        
                          x
                          i
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    ]
                  
                
              
              +
              
                

              
              ⁢
              
                2
                ⁢
                
                  
                    
                      e
                      xk
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          2
                        
                      
                      )
                    
                  
                  ζ2
                
                ⁢
                
                  
                    e
                    xl
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        3
                      
                    
                    )
                  
                
                ⁢
                
                  E
                  ⁡
                  
                    [
                    
                      
                        
                          x
                          i
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          
                            x
                            j
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              
                                τ
                                1
                              
                            
                            )
                          
                        
                        ζ1
                      
                    
                    ]
                  
                
              
              -
              
                

              
              ⁢
              
                6
                ⁢
                
                  
                
                ⁢
                
                  
                    e
                    xi
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ⁢
                
                  
                    
                      e
                      xj
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          1
                        
                      
                      )
                    
                  
                  ζ1
                
                ⁢
                
                  
                    
                      e
                      xk
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          2
                        
                      
                      )
                    
                  
                  ζ2
                
                ⁢
                
                  
                    e
                    xl
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        3
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        39
        )
      
    
  




where ζΔ (ζ1, ζ2) and (ζ1, ζ2)=(1, 1), (−1, 1) or (−1, −1). Assuming that the term Qxζ(t, τ1, τ2, τ3)[i, j, k, l] is the element [N(i−1)+j, N(k−1)+l] of the matrix Qxζ(t, τ1, τ2, τ3), known as the quadricovariance matrix, with a dimension (N2×N2), we obtain an expression of this dimension given by:





  
    
      
        
          
            
              Q
              ⁢
              
                
              
            
            
              x
              ⁢
              
                
              
              ⁢
              ζ
            
          
          ⁢
          
            (
            
              t
              ,
              
                τ
                1
              
              ,
              
                τ
                2
              
              ,
              
                τ
                3
              
            
            )
          
        
        =
        
          
            
              (
              
                A
                ⊗
                
                  A
                  ζ1
                
              
              )
            
            ⁢
            
              
                Q
                
                  m
                  ⁢
                  
                    
                  
                  ⁢
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  ζ
                
              
              ⁡
              
                (
                
                  t
                  ,
                  
                    τ
                    1
                  
                  ,
                  
                    τ
                    2
                  
                  ,
                  
                    τ
                    3
                  
                
                )
              
            
            ⁢
            
              
                (
                
                  
                    A
                    ζ2
                  
                  ⊗
                  A
                
                )
              
              T
            
          
          =
          
            
              ∑
              
                i
                ,
                j
                ,
                
                  k
                  =
                  1
                
              
            
            ⁢
            
              
            
            ⁢
            
              
                
                  Q
                  mcζ
                
                ⁡
                
                  [
                  
                    i
                    ,
                    j
                    ,
                    k
                    ,
                    l
                  
                  ]
                
              
              ⁢
              
                
                  
                    
                      (
                      
                        t
                        ,
                        
                          τ
                          1
                        
                        ,
                        
                          τ
                          2
                        
                        ,
                        
                          τ
                          3
                        
                      
                      )
                    
                    ⁡
                    
                      [
                      
                        
                          a
                          
                            i
                            .
                          
                        
                        ⊗
                        
                          a
                          j
                          ζ1
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        a
                        k
                        ζ2
                      
                      ⊗
                      
                        a
                        l
                      
                    
                    ]
                  
                
                T
              
            
          
        
      
    
    
      
        (
        40
        )
      
    
  




where Qmcζ(t, τ1, τ2, τ3) is the quadricovariance of the vector mc(t) whose elements are Qmcζ[i, j, k, l](t, τ1, τ2, τ3) Δ Cum(mic(t), mjc(t−τ1)ζ1, mkc(t−τ2)ζ2, mlc(t−τ3)).

For cyclostationary sources, the matrices of quadricovariance, Qmcζ(t, τ1, τ2, τ3) and Qxζ(t, τ1, τ2, τ3), are (quasi-) or poly-periodic functions of the time t accepting a Fourier series decomposition, and we obtain




  
    
      
        
          
            
              Q
              ⁢
              
                
              
            
            
              x
              ⁢
              
                
              
              ⁢
              ζ
            
          
          ⁢
          
            (
            
              t
              ,
              
                τ
                1
              
              ,
              
                τ
                2
              
              ,
              
                τ
                3
              
            
            )
          
        
        =
        
          
            ∑
            
              
                β
                xζ
              
              ∈
              
                Γ
                x
                
                  [
                  
                    1
                    ,
                    ζ1
                    ,
                    ζ2
                    ,
                    1
                  
                  ]
                
              
            
          
          ⁢
          
            
              
                Q
                ⁢
                
                  
                
              
              
                x
                ⁢
                
                  
                
                ⁢
                ζ
              
              
                β
                xζ
              
            
            ⁢
            
              (
              
                
                  τ
                  1
                
                ,
                
                  τ
                  2
                
                ,
                
                  τ
                  3
                
              
              )
            
            ⁢
            
              e
              
                j2π
                ⁢
                
                  
                
                ⁢
                
                  β
                  xζ
                
                ⁢
                t
              
            
          
        
      
    
    
      
        (
        41
        )
      
    
  




where





  
    Γ
    x
    
      [
      
        1
        ,
        ζ1
        ,
        ζ2
        ,
        1
      
      ]
    
  
  =
  
    {
    
      β
      xζ
    
    }
  




is the set of cyclic frequencies βxζof Qxζ(t, τ1, τ2, τ3) and





  
    Q
    xζ
    
      β
      xζ
    
  
  ⁡
  
    (
    
      
        
          τ
          ⁢
          
            
          
        
        1
      
      ,
      
        
          τ
          ⁢
          
            
          
        
        2
      
      ,
      
        
          τ
          ⁢
          
            
          
        
        3
      
    
    )
  




is a matrix of cyclic quadricovariance of x(t), defined respectively by





  
    
      
        
          
            Q
            
              x
              ⁢
              
                
              
              ⁢
              ζ
            
            
              β
              
                x
                ⁢
                
                  
                
                ⁢
                ζ
              
            
          
          ⁡
          
            (
            
              
                
                  τ
                  ⁢
                  
                    
                  
                
                1
              
              ,
              
                
                  τ
                  ⁢
                  
                    
                  
                
                2
              
              ,
              
                
                  τ
                  ⁢
                  
                    
                  
                
                3
              
            
            )
          
        
        =
        
          
            <
            
              
                
                  Q
                  
                    x
                    ⁢
                    
                      
                    
                    ⁢
                    ζ
                  
                
                ⁡
                
                  (
                  
                    t
                    ,
                    
                      
                        τ
                        ⁢
                        
                          
                        
                      
                      1
                    
                    ,
                    
                      
                        τ
                        ⁢
                        
                          
                        
                      
                      2
                    
                    ,
                    
                      
                        τ
                        ⁢
                        
                          
                        
                      
                      3
                    
                  
                  )
                
              
              ⁢
              
                e
                
                  
                    -
                    j2π
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    β
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      ζ
                    
                  
                  ⁢
                  t
                
              
            
            ⁢
            
              >
              c
            
          
          =
          
            
              (
              
                A
                ⊗
                
                  A
                  ζ1
                
              
              )
            
            ⁢
            
              
                Q
                
                  m
                  ⁢
                  
                    
                  
                  ⁢
                  c
                
                
                  β
                  x
                
              
              ⁡
              
                (
                
                  
                    
                      τ
                      ⁢
                      
                        
                      
                    
                    1
                  
                  ,
                  
                    
                      τ
                      ⁢
                      
                        
                      
                    
                    2
                  
                  ,
                  
                    
                      τ
                      ⁢
                      
                        
                      
                    
                    3
                  
                
                )
              
            
            ⁢
            
              
                (
                
                  
                    A
                    ζ2
                  
                  ⊗
                  A
                
                )
              
              T
            
          
        
      
    
    
      
        (
        42
        )
      
    
  




In particular, the cyclic quadricovariance for the zero cyclic frequency corresponds to the temporal mean in t, Qxζ(τ1, τ2, τ3), de Qxζ(t, τ1, τ2, τ3), which is written as follows:

Qxζ(τ1, τ2, τ3)Δ<Qxζ(t, τ1, τ2, τ3)>c=(A{circle around (x)}Aζ1)Qmcζ(τ1, τ2, τ3)(Aζ2{circle around (x)}A)T  (42b)

where Qmcζ(τ1, τ2, τ3) is the temporal mean in t of Qmcζ(t, τ1, τ2, τ3).

Novel Estimator of the Fourth-Order Cumulants of the Observations

From (39) and from the Fourier series decomposition of the statistics appearing in this expression, we deduce the expression of the element [i, j, k, l ] of the matrix of cyclic quadricovariance




  
    Q
    xζ
    β
  
  ⁡
  
    (
    
      
        
          τ
          ⁢
          
            
          
        
        1
      
      ,
      
        
          τ
          ⁢
          
            
          
        
        2
      
      ,
      
        
          τ
          ⁢
          
            
          
        
        3
      
    
    )
  




in the general case of non-centered observations, given by:





  
    
      
        
          
            
              
                
                  M
                  xζ
                  β
                
                ⁡
                
                  (
                  
                    
                      
                        τ
                        ⁢
                        
                          
                        
                      
                      1
                    
                    ,
                    
                      
                        τ
                        ⁢
                        
                          
                        
                      
                      2
                    
                    ,
                    
                      
                        τ
                        ⁢
                        
                          
                        
                      
                      3
                    
                  
                  )
                
              
              ⁡
              
                [
                
                  i
                  ,
                  j
                  ,
                  k
                  ,
                  l
                
                ]
              
            
            -
            
              
                ∑
                
                  
                    γ
                    ∈
                    
                      Γ
                      x
                      1
                    
                  
                  ⁢
                  
                    
                  
                
              
              ⁢
              
                
              
              ⁢
              
                {
                
                  
                    
                      
                        e
                        x
                        γ
                      
                      ⁡
                      
                        [
                        i
                        ]
                      
                    
                    ⁢
                    
                      
                        
                          T
                          xζ
                          
                            β
                            -
                            γ
                          
                        
                        ⁡
                        
                          (
                          
                            
                              
                                
                                  τ
                                  ⁢
                                  
                                    
                                  
                                
                                1
                              
                              -
                              
                                
                                  τ
                                  ⁢
                                  
                                    
                                  
                                
                                3
                              
                            
                            ,
                            
                              
                                τ
                                2
                              
                              -
                              
                                
                                  τ
                                  ⁢
                                  
                                    
                                  
                                
                                3
                              
                            
                          
                          )
                        
                      
                      ⁡
                      
                        [
                        
                          l
                          ,
                          j
                          ,
                          k
                        
                        ]
                      
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          j2π
                        
                        ⁢
                        
                          (
                          
                            β
                            -
                            γ
                          
                          )
                        
                        ⁢
                        
                          τ
                          3
                        
                      
                    
                  
                  +
                  
                    

                  
                  ⁢
                  
                    
                      
                        
                          e
                          x
                          γ
                        
                        ⁡
                        
                          [
                          j
                          ]
                        
                      
                      ζ1
                    
                    ⁢
                    
                      
                        
                          T
                          xζ2
                          
                            β
                            -
                            γζ1
                          
                        
                        ⁡
                        
                          (
                          
                            
                              τ
                              2
                            
                            ,
                            
                              τ
                              3
                            
                          
                          )
                        
                      
                      ⁡
                      
                        [
                        
                          i
                          ,
                          k
                          ,
                          l
                        
                        ]
                      
                    
                    ⁢
                    
                      e
                      
                        -
                        
                          j2πγζ1τ
                          1
                        
                      
                    
                  
                  +
                  
                    

                  
                  ⁢
                  
                    

                  
                  ⁢
                  
                    
                      
                        
                          e
                          x
                          γ
                        
                        ⁡
                        
                          [
                          k
                          ]
                        
                      
                      ζ2
                    
                    ⁢
                    
                      
                        
                          T
                          xζ1
                          
                            β
                            -
                            γζ2
                          
                        
                        ⁡
                        
                          (
                          
                            
                              τ
                              1
                            
                            ,
                            
                              τ
                              3
                            
                          
                          )
                        
                      
                      ⁡
                      
                        [
                        
                          i
                          ,
                          j
                          ,
                          l
                        
                        ]
                      
                    
                    ⁢
                    
                      e
                      
                        -
                        
                          j2πγζ2τ
                          2
                        
                      
                    
                  
                  +
                  
                    

                  
                  ⁢
                  
                    

                  
                  ⁢
                  
                    
                      
                        e
                        x
                        γ
                      
                      ⁡
                      
                        [
                        l
                        ]
                      
                    
                    ⁢
                    
                      
                        
                          T
                          xζ
                          
                            β
                            -
                            γ
                          
                        
                        ⁡
                        
                          (
                          
                            
                              τ
                              1
                            
                            ,
                            
                              τ
                              2
                            
                          
                          )
                        
                      
                      ⁡
                      
                        [
                        
                          i
                          ,
                          j
                          ,
                          k
                        
                        ]
                      
                    
                    ⁢
                    
                      e
                      
                        -
                        
                          j2πγτ
                          3
                        
                      
                    
                  
                
                }
              
            
            -
            
              

            
            ⁢
            
              

            
            ⁢
            
              
                ∑
                
                  α
                  ∈
                  
                    Γ
                    x
                    
                      [
                      
                        1
                        ,
                        ɛ1
                      
                      ]
                    
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    
                      R
                      xζ1
                      α
                    
                    ⁡
                    
                      (
                      
                        τ
                        1
                      
                      )
                    
                  
                  ⁡
                  
                    [
                    
                      i
                      ,
                      j
                    
                    ]
                  
                
                ⁢
                
                  
                    
                      
                        R
                        xζ2
                        
                          ζ2
                          ⁡
                          
                            (
                            
                              β
                              -
                              α
                            
                            )
                          
                        
                      
                      ⁡
                      
                        (
                        
                          
                            τ
                            3
                          
                          -
                          
                            τ
                            2
                          
                        
                        )
                      
                    
                    ⁡
                    
                      [
                      
                        k
                        ,
                        l
                      
                      ]
                    
                  
                  ζ2
                
                ⁢
                
                  e
                  
                    
                      j2π
                      ⁡
                      
                        (
                        
                          α
                          -
                          β
                        
                        )
                      
                    
                    ⁢
                    
                      τ
                      2
                    
                  
                
              
            
            -
            
              

            
            ⁢
            
              
                ∑
                
                  γ
                  ∈
                  
                    Γ
                    x
                    
                      [
                      
                        1
                        ,
                        ɛ2
                      
                      ]
                    
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    
                      R
                      xζ2
                      γ
                    
                    ⁡
                    
                      (
                      
                        τ
                        2
                      
                      )
                    
                  
                  ⁡
                  
                    [
                    
                      i
                      ,
                      k
                    
                    ]
                  
                
                ⁢
                
                  
                    
                      
                        R
                        xζ1
                        
                          ζ1
                          ⁡
                          
                            (
                            
                              β
                              -
                              γ
                            
                            )
                          
                        
                      
                      ⁡
                      
                        (
                        
                          
                            τ
                            3
                          
                          -
                          
                            τ
                            1
                          
                        
                        )
                      
                    
                    ⁡
                    
                      [
                      
                        j
                        ,
                        l
                      
                      ]
                    
                  
                  ζ1
                
                ⁢
                
                  e
                  
                    
                      j2π
                      ⁡
                      
                        (
                        
                          γ
                          -
                          β
                        
                        )
                      
                    
                    ⁢
                    
                      τ
                      1
                    
                  
                
              
            
            -
            
              

            
            ⁢
            
              
                ∑
                
                  ω
                  ∈
                  
                    Γ
                    x
                    
                      [
                      
                        1
                        ,
                        1
                      
                      ]
                    
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    
                      R
                      xζ1
                      ω
                    
                    ⁡
                    
                      (
                      
                        τ
                        3
                      
                      )
                    
                  
                  ⁡
                  
                    [
                    
                      i
                      ,
                      l
                    
                    ]
                  
                
                ⁢
                
                  
                    
                      
                        R
                        xζ12
                        
                          ζ1
                          ⁡
                          
                            (
                            
                              β
                              -
                              ω
                            
                            )
                          
                        
                      
                      ⁡
                      
                        (
                        
                          
                            τ
                            2
                          
                          -
                          
                            τ
                            1
                          
                        
                        )
                      
                    
                    ⁡
                    
                      [
                      
                        j
                        ,
                        k
                      
                      ]
                    
                  
                  ζ1
                
                ⁢
                
                  e
                  
                    
                      j2π
                      ⁡
                      
                        (
                        
                          ω
                          -
                          β
                        
                        )
                      
                    
                    ⁢
                    
                      τ
                      1
                    
                  
                
              
            
            +
            
              

            
            ⁢
            
              2
              ⁢
              
                
                  ∑
                  
                    
                      γ
                      ∈
                      
                        Γ
                        x
                        1
                      
                    
                    ⁢
                    
                      
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      
                        ω
                        ∈
                        
                          Γ
                          x
                          1
                        
                      
                      ⁢
                      
                        
                      
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          
                            e
                            x
                            γ
                          
                          ⁡
                          
                            [
                            i
                            ]
                          
                        
                        ⁢
                        
                          
                            
                              e
                              x
                              ω
                            
                            ⁡
                            
                              [
                              j
                              ]
                            
                          
                          ζ1
                        
                        ⁢
                        
                          
                            
                              R
                              xζ2
                              
                                β
                                -
                                γ
                                -
                                ωζ1
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  τ
                                  2
                                
                                -
                                
                                  τ
                                  3
                                
                              
                              )
                            
                          
                          ⁡
                          
                            [
                            
                              l
                              ,
                              k
                            
                            ]
                          
                        
                        ⁢
                        
                          e
                          
                            j2πωζ1
                            ⁡
                            
                              (
                              
                                
                                  τ
                                  3
                                
                                -
                                
                                  τ
                                  1
                                
                              
                              )
                            
                          
                        
                        ⁢
                        
                          e
                          
                            
                              j2π
                              ⁡
                              
                                (
                                
                                  γ
                                  -
                                  β
                                
                                )
                              
                            
                            ⁢
                            
                              τ
                              3
                            
                          
                        
                      
                      +
                      
                        

                      
                      ⁢
                      
                        
                          
                            e
                            x
                            γ
                          
                          ⁡
                          
                            [
                            i
                            ]
                          
                        
                        ⁢
                        
                          
                            
                              e
                              x
                              ω
                            
                            ⁡
                            
                              [
                              k
                              ]
                            
                          
                          ζ2
                        
                        ⁢
                        
                          
                            
                              R
                              xζ1
                              
                                β
                                -
                                γ
                                -
                                ωζ2
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  τ
                                  1
                                
                                -
                                
                                  τ
                                  3
                                
                              
                              )
                            
                          
                          ⁡
                          
                            [
                            
                              l
                              ,
                              j
                            
                            ]
                          
                        
                        ⁢
                        
                          e
                          
                            j2πωζ2
                            ⁡
                            
                              (
                              
                                
                                  τ
                                  3
                                
                                -
                                
                                  τ
                                  2
                                
                              
                              )
                            
                          
                        
                        ⁢
                        
                          e
                          
                            
                              j2π
                              ⁡
                              
                                (
                                
                                  γ
                                  -
                                  β
                                
                                )
                              
                            
                            ⁢
                            
                              τ
                              3
                            
                          
                        
                      
                      +
                      
                        

                      
                      ⁢
                      
                        

                      
                      ⁢
                      
                        
                          
                            e
                            x
                            γ
                          
                          ⁡
                          
                            [
                            i
                            ]
                          
                        
                        ⁢
                        
                          
                            e
                            x
                            ω
                          
                          ⁡
                          
                            [
                            l
                            ]
                          
                        
                        ⁢
                        
                          
                            
                              
                                R
                                xζ12
                                
                                  ζ1
                                  ⁡
                                  
                                    (
                                    
                                      β
                                      -
                                      γ
                                      -
                                      ω
                                    
                                    )
                                  
                                
                              
                              ⁡
                              
                                (
                                
                                  
                                    τ
                                    2
                                  
                                  -
                                  
                                    τ
                                    1
                                  
                                
                                )
                              
                            
                            ⁡
                            
                              [
                              
                                j
                                ,
                                k
                              
                              ]
                            
                          
                          ζ1
                        
                        ⁢
                        
                          e
                          
                            j2πω
                            ⁡
                            
                              (
                              
                                
                                  τ
                                  1
                                
                                -
                                
                                  τ
                                  3
                                
                              
                              )
                            
                          
                        
                        ⁢
                        
                          e
                          
                            -
                            
                              j2πβτ
                              1
                            
                          
                        
                      
                      +
                      
                        

                      
                      ⁢
                      
                        

                      
                      ⁢
                      
                        
                          
                            
                              e
                              x
                              γ
                            
                            ⁡
                            
                              [
                              k
                              ]
                            
                          
                          ζ2
                        
                        ⁢
                        
                          
                            
                              e
                              x
                              ω
                            
                            ⁡
                            
                              [
                              j
                              ]
                            
                          
                          ζ1
                        
                        ⁢
                        
                          
                            
                              R
                              x1
                              
                                β
                                -
                                γζ2
                                -
                                ωζ1
                              
                            
                            ⁡
                            
                              (
                              
                                τ
                                3
                              
                              )
                            
                          
                          ⁡
                          
                            [
                            
                              i
                              ,
                              l
                            
                            ]
                          
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              
                                j2πγτ
                                2
                              
                            
                            ⁢
                            ζ2
                          
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              
                                j2πωτ
                                1
                              
                            
                            ⁢
                            ζ1
                          
                        
                      
                      +
                      
                        

                      
                      ⁢
                      
                        

                      
                      ⁢
                      
                        
                          
                            e
                            x
                            γ
                          
                          ⁡
                          
                            [
                            l
                            ]
                          
                        
                        ⁢
                        
                          
                            
                              e
                              x
                              ω
                            
                            ⁡
                            
                              [
                              j
                              ]
                            
                          
                          ζ1
                        
                        ⁢
                        
                          
                            
                              R
                              
                                x
                                ⁢
                                
                                  
                                
                                ⁢
                                ζ2
                              
                              
                                β
                                -
                                γ
                                -
                                ωζ1
                              
                            
                            ⁡
                            
                              (
                              
                                τ
                                2
                              
                              )
                            
                          
                          ⁡
                          
                            [
                            
                              i
                              ,
                              k
                            
                            ]
                          
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              
                                j2πωτ
                                1
                              
                            
                            ⁢
                            ζ1
                          
                        
                        ⁢
                        
                          e
                          
                            -
                            
                              j2πγτ
                              3
                            
                          
                        
                      
                      +
                      
                        

                      
                      ⁢
                      
                        

                      
                      ⁢
                      
                        
                          
                            e
                            x
                            γ
                          
                          ⁡
                          
                            [
                            l
                            ]
                          
                        
                        ⁢
                        
                          
                            
                              e
                              x
                              ω
                            
                            ⁡
                            
                              [
                              k
                              ]
                            
                          
                          ζ2
                        
                        ⁢
                        
                          
                            
                              R
                              
                                x
                                ⁢
                                
                                  
                                
                                ⁢
                                ζ1
                              
                              
                                β
                                -
                                γ
                                -
                                ωζ2
                              
                            
                            ⁡
                            
                              (
                              
                                τ
                                1
                              
                              )
                            
                          
                          ⁡
                          
                            [
                            
                              i
                              ,
                              j
                            
                            ]
                          
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              
                                j2πωτ
                                2
                              
                            
                            ⁢
                            ζ2
                          
                        
                        ⁢
                        
                          e
                          
                            -
                            
                              j2πγτ
                              3
                            
                          
                        
                      
                    
                    }
                  
                
              
            
            -
            
              

            
            ⁢
            
              

            
            ⁢
            
              6
              ⁢
              
                
                  ∑
                  
                    
                      
                        γ
                        ∈
                        
                          Γ
                          x
                          1
                        
                      
                      
                        e
                        
                          
                            j2πτ
                            3
                          
                          ⁡
                          
                            (
                            
                              γ
                              -
                              β
                            
                            )
                          
                        
                      
                    
                    ⁢
                    
                      
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      
                        ω
                        ∈
                        
                          Γ
                          x
                          1
                        
                      
                      ⁢
                      
                        
                      
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        
                          δ
                          ∈
                          
                            Γ
                            x
                            1
                          
                        
                        ⁢
                        
                          
                        
                      
                    
                    ⁢
                    
                      
                        
                          e
                          x
                          γ
                        
                        ⁡
                        
                          [
                          i
                          ]
                        
                      
                      ⁢
                      
                        
                          
                            e
                            x
                            ω
                          
                          ⁡
                          
                            [
                            j
                            ]
                          
                        
                        ζ1
                      
                      ⁢
                      
                        
                          
                            e
                            x
                            δ
                          
                          ⁡
                          
                            [
                            k
                            ]
                          
                        
                        ζ2
                      
                      ⁢
                      
                        
                          e
                          x
                          
                            β
                            -
                            ωζ1
                            -
                            δζ2
                          
                        
                        ⁡
                        
                          [
                          l
                          ]
                        
                      
                      ⁢
                      
                        e
                        
                          j2πωζ1
                          ⁡
                          
                            (
                            
                              
                                τ
                                3
                              
                              -
                              
                                τ
                                1
                              
                            
                            )
                          
                        
                      
                      ⁢
                      
                        e
                        
                          j2πδζ2
                          ⁡
                          
                            (
                            
                              
                                τ
                                3
                              
                              -
                              
                                τ
                                2
                              
                            
                            )
                          
                        
                      
                    
                  
                
              
            
          
          ⁢
          
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                where
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      T
                      xζ
                    
                    ⁡
                    
                      (
                      
                        t
                        ,
                        
                          τ
                          1
                        
                        ,
                        
                          τ
                          2
                        
                      
                      )
                    
                  
                  ⁡
                  
                    [
                    
                      i
                      ,
                      j
                      ,
                      k
                    
                    ]
                  
                
              
              ⁢
              
                =
                Δ
              
              ⁢
              
                E
                ⁡
                
                  [
                  
                    
                      
                        x
                        i
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    ⁢
                    
                      
                        
                          x
                          j
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              τ
                              1
                            
                          
                          )
                        
                      
                      ζ1
                    
                    ⁢
                    
                      
                        
                          x
                          k
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              τ
                              2
                            
                          
                          )
                        
                      
                      ζ2
                    
                  
                  ]
                
              
            
            ,
            
              

            
            ⁢
            
              
                
                  
                    T
                    xζ
                    β
                  
                  ⁡
                  
                    (
                    
                      
                        τ
                        1
                      
                      ,
                      
                        τ
                        2
                      
                    
                    )
                  
                
                ⁡
                
                  [
                  
                    i
                    ,
                    j
                    ,
                    k
                  
                  ]
                
              
              ⁢
              
                =
                Δ
              
              ⁢
              
                <
                
                  
                    
                      
                        T
                        xζ
                      
                      ⁡
                      
                        (
                        
                          t
                          ,
                          
                            τ
                            1
                          
                          ,
                          
                            τ
                            2
                          
                        
                        )
                      
                    
                    ⁡
                    
                      [
                      
                        i
                        ,
                        j
                        ,
                        k
                      
                      ]
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        j2πβ
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
                ⁢
                
                  >
                  c
                
              
            
            ,
            
              
            
            ⁢
            
              

            
            ⁢
            
              
                
                  
                    T
                    xɛ
                  
                  ⁡
                  
                    (
                    
                      t
                      ,
                      
                        τ
                        1
                      
                      ,
                      
                        τ
                        2
                      
                    
                    )
                  
                
                ⁡
                
                  [
                  
                    i
                    ,
                    j
                    ,
                    k
                  
                  ]
                
              
              ⁢
              
                =
                Δ
              
              ⁢
              
                E
                ⁡
                
                  [
                  
                    
                      
                        x
                        i
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    ⁢
                    
                      
                        
                          x
                          j
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              τ
                              1
                            
                          
                          )
                        
                      
                      ɛ
                    
                    ⁢
                    
                      
                        x
                        k
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            2
                          
                        
                        )
                      
                    
                  
                  ]
                
              
            
            ,
            
              
            
            ⁢
            
              

            
            ⁢
            
              
                
                  
                    T
                    xɛ
                    β
                  
                  ⁡
                  
                    (
                    
                      
                        τ
                        1
                      
                      ,
                      
                        τ
                        2
                      
                    
                    )
                  
                
                ⁡
                
                  [
                  
                    i
                    ,
                    j
                    ,
                    k
                  
                  ]
                
              
              =
              
                <
                
                  
                    
                      
                        T
                        xɛ
                      
                      ⁡
                      
                        (
                        
                          t
                          ,
                          
                            τ
                            1
                          
                          ,
                          
                            τ
                            2
                          
                        
                        )
                      
                    
                    ⁡
                    
                      [
                      
                        i
                        ,
                        j
                        ,
                        k
                      
                      ]
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        j2πβ
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
                ⁢
                
                  >
                  c
                
              
            
          
        
        ⁢
        
          
        
      
    
    
      
        (
        43
        )
      
    
  




In particular, the matrix of cyclic quadricovariance exploited by the separators of the family F3 corresponds to the matrix Qxζβ(τ1, τ2, τ3) for β=0ζ=(−1, −1) and (τ1, τ2, τ3)=(0, 0, 0) and its element Qxζβ(τ1, τ2, τ3)[i, j, k, l], denoted as Qx[i, j, k, l], is written:




  
    
      
        
          
            
              Q
              x
            
            ⁡
            
              [
              
                i
                ,
                j
                ,
                k
                ,
                l
              
              ]
            
          
          ⁢
          
            =
            Δ
          
          ⁢
          
            
              <
              
                
                  
                    Q
                    x
                  
                  ⁡
                  
                    (
                    
                      t
                      ,
                      0
                      ,
                      0
                      ,
                      0
                    
                    )
                  
                
                ⁡
                
                  [
                  
                    i
                    ,
                    j
                    ,
                    k
                    ,
                    l
                  
                  ]
                
              
              ⁢
              
                >
                c
              
            
            =
            
              
                
                  M
                  x
                  0
                
                ⁡
                
                  [
                  
                    i
                    ,
                    j
                    ,
                    k
                    ,
                    l
                  
                  ]
                
              
              -
              
                

              
              ⁢
              
                

              
              ⁢
              
                
                  ∑
                  
                    
                      γ
                      ∈
                      
                        Γ
                        x
                        1
                      
                    
                    ⁢
                    
                      
                    
                  
                
                ⁢
                
                  
                
                ⁢
                
                  {
                  
                    
                      
                        
                          e
                          x
                          γ
                        
                        ⁡
                        
                          [
                          i
                          ]
                        
                      
                      ⁢
                      
                        
                          
                            T
                            x
                            γ
                          
                          ⁡
                          
                            [
                            
                              j
                              ,
                              l
                              ,
                              k
                            
                            ]
                          
                        
                        *
                      
                    
                    +
                    
                      
                        
                          e
                          x
                          γ
                        
                        ⁡
                        
                          [
                          l
                          ]
                        
                      
                      ⁢
                      
                        
                          
                            T
                            x
                            γ
                          
                          ⁡
                          
                            [
                            
                              j
                              ,
                              i
                              ,
                              k
                            
                            ]
                          
                        
                        *
                      
                    
                    +
                    
                      
                        
                          
                            e
                            x
                            γ
                          
                          ⁡
                          
                            [
                            j
                            ]
                          
                        
                        *
                      
                      ⁢
                      
                        
                          
                            T
                            x
                            γ
                          
                          ⁡
                          
                            [
                            
                              i
                              ,
                              k
                              ,
                              l
                            
                            ]
                          
                        
                        *
                      
                    
                    +
                    
                      
                        
                          
                            e
                            x
                            γ
                          
                          ⁡
                          
                            [
                            k
                            ]
                          
                        
                        *
                      
                      ⁢
                      
                        
                          T
                          x
                          γ
                        
                        ⁡
                        
                          [
                          
                            i
                            ,
                            j
                            ,
                            l
                          
                          ]
                        
                      
                    
                  
                  }
                
              
              -
              
                

              
              ⁢
              
                
                  ∑
                  
                    α
                    ∈
                    
                      Γ
                      x
                      
                        [
                        
                          1
                          ,
                          
                            -
                            1
                          
                        
                        ]
                      
                    
                  
                
                ⁢
                
                  {
                  
                    
                      
                        
                          R
                          x
                          α
                        
                        ⁡
                        
                          [
                          
                            i
                            ,
                            j
                          
                          ]
                        
                      
                      ⁢
                      
                        
                          R
                          x
                          
                            -
                            α
                          
                        
                        ⁡
                        
                          [
                          
                            l
                            ,
                            k
                          
                          ]
                        
                      
                    
                    +
                    
                      
                        
                          R
                          x
                          α
                        
                        ⁡
                        
                          [
                          
                            i
                            ,
                            k
                          
                          ]
                        
                      
                      ⁢
                      
                        
                          R
                          x
                          
                            -
                            α
                          
                        
                        ⁡
                        
                          [
                          
                            l
                            ,
                            j
                          
                          ]
                        
                      
                    
                  
                  }
                
              
              -
              
                
                  ∑
                  
                    ω
                    ∈
                    
                      Γ
                      x
                      
                        [
                        
                          1
                          ,
                          1
                        
                        ]
                      
                    
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      C
                      x
                      ω
                    
                    ⁡
                    
                      [
                      
                        i
                        ,
                        l
                      
                      ]
                    
                  
                  ⁢
                  
                    
                      
                        C
                        x
                        ω
                      
                      ⁡
                      
                        [
                        
                          j
                          ,
                          k
                        
                        ]
                      
                    
                    *
                  
                
              
              +
              
                

              
              ⁢
              
                2
                ⁢
                
                  
                    ∑
                    
                      
                        γ
                        ∈
                        
                          Γ
                          x
                          1
                        
                      
                      ⁢
                      
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        ∑
                        {
                      
                      
                        
                          ω
                          ∈
                          
                            Γ
                            x
                            1
                          
                        
                        ⁢
                        
                          
                        
                      
                    
                    ⁢
                    
                      
                        e
                        x
                        γ
                      
                      ⁡
                      
                        [
                        i
                        ]
                      
                    
                    ⁢
                    
                      
                        
                          e
                          x
                          ω
                        
                        ⁡
                        
                          [
                          j
                          ]
                        
                      
                      *
                    
                    ⁢
                    
                      
                        R
                        x
                        
                          ω
                          -
                          γ
                        
                      
                      ⁡
                      
                        [
                        
                          l
                          ,
                          k
                        
                        ]
                      
                    
                  
                
              
              +
              
                
                  
                    e
                    x
                    γ
                  
                  ⁡
                  
                    [
                    i
                    ]
                  
                
                ⁢
                
                  
                    
                      e
                      x
                      ω
                    
                    ⁡
                    
                      [
                      k
                      ]
                    
                  
                  *
                
                ⁢
                
                  
                    R
                    x
                    
                      ω
                      -
                      γ
                    
                  
                  ⁡
                  
                    [
                    
                      l
                      ,
                      j
                    
                    ]
                  
                
              
              +
              
                
                  
                    e
                    x
                    γ
                  
                  ⁡
                  
                    [
                    i
                    ]
                  
                
                ⁢
                
                  
                    e
                    x
                    ω
                  
                  ⁡
                  
                    [
                    l
                    ]
                  
                
                ⁢
                
                  
                    
                      C
                      x
                      
                        ω
                        +
                        γ
                      
                    
                    ⁡
                    
                      [
                      
                        k
                        ,
                        j
                      
                      ]
                    
                  
                  *
                
              
              +
              
                

              
              ⁢
              
                
                  
                    
                      e
                      x
                      γ
                    
                    ⁡
                    
                      [
                      k
                      ]
                    
                  
                  *
                
                ⁢
                
                  
                    
                      e
                      x
                      ω
                    
                    ⁡
                    
                      [
                      j
                      ]
                    
                  
                  *
                
                ⁢
                
                  
                    C
                    x
                    
                      ω
                      +
                      γ
                    
                  
                  ⁡
                  
                    [
                    
                      i
                      ,
                      l
                    
                    ]
                  
                
              
              +
              
                
                  
                    e
                    x
                    γ
                  
                  ⁡
                  
                    [
                    l
                    ]
                  
                
                ⁢
                
                  
                    
                      e
                      x
                      ω
                    
                    ⁡
                    
                      [
                      j
                      ]
                    
                  
                  *
                
                ⁢
                
                  
                    R
                    x
                    
                      ω
                      -
                      γ
                    
                  
                  ⁡
                  
                    [
                    
                      i
                      ,
                      k
                    
                    ]
                  
                
              
              +
              
                
                  e
                  x
                  γ
                
                [
                
                  l
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  
                    
                      
                        e
                        x
                        ω
                      
                      ⁡
                      
                        [
                        k
                        ]
                      
                    
                    *
                  
                  ⁢
                  
                    
                      R
                      x
                      
                        ω
                        -
                        γ
                      
                    
                    ⁡
                    
                      [
                      
                        i
                        ,
                        j
                      
                      ]
                    
                  
                
                }
              
              -
              
                

              
              ⁢
              
                

              
              ⁢
              
                6
                ⁢
                
                  
                    ∑
                    
                      
                        γ
                        ∈
                        
                          Γ
                          x
                          1
                        
                      
                      ⁢
                      
                        
                      
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        
                          ω
                          ∈
                          
                            Γ
                            x
                            1
                          
                        
                        ⁢
                        
                          
                        
                      
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            δ
                            ∈
                            
                              Γ
                              x
                              1
                            
                          
                          ⁢
                          
                            
                          
                        
                      
                      ⁢
                      
                        
                          
                            e
                            x
                            γ
                          
                          ⁡
                          
                            [
                            i
                            ]
                          
                        
                        ⁢
                        
                          
                            
                              e
                              x
                              ω
                            
                            ⁡
                            
                              [
                              j
                              ]
                            
                          
                          *
                        
                        ⁢
                        
                          
                            
                              e
                              x
                              δ
                            
                            ⁡
                            
                              [
                              k
                              ]
                            
                          
                          *
                        
                        ⁢
                        
                          
                            e
                            x
                            
                              δ
                              +
                              ω
                              -
                              γ
                            
                          
                          ⁡
                          
                            [
                            l
                            ]
                          
                        
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          

        
        ⁢
        
          
            
              where
              ⁢
              
                
              
              ⁢
              
                
                  M
                  x
                  0
                
                ⁡
                
                  [
                  
                    i
                    ,
                    j
                    ,
                    k
                    ,
                    l
                  
                  ]
                
              
            
            ⁢
            
              =
              Δ
            
            ⁢
            
              <
              
                E
                ⁡
                
                  [
                  
                    
                      
                        x
                        i
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    ⁢
                    
                      
                        
                          x
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      *
                    
                    ⁢
                    
                      
                        
                          x
                          k
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      *
                    
                    ⁢
                    
                      
                        x
                        l
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  ]
                
              
              ⁢
              
                >
                c
              
            
          
          ,
          
            

          
          ⁢
          
            
              
                T
                x
                β
              
              ⁡
              
                [
                
                  i
                  ,
                  j
                  ,
                  k
                
                ]
              
            
            ⁢
            
              =
              Δ
            
            ⁢
            
              <
              
                
                  E
                  ⁡
                  
                    [
                    
                      
                        
                          x
                          i
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          
                            x
                            j
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        *
                      
                      ⁢
                      
                        
                          x
                          k
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    ]
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      j2πβ
                    
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                
              
              ⁢
              
                >
                c
              
            
          
          ,
          
            

          
          ⁢
          
            
              
                R
                x
                α
              
              ⁡
              
                [
                
                  i
                  ,
                  j
                
                ]
              
            
            ⁢
            
              =
              Δ
            
            ⁢
            
              <
              
                
                  E
                  ⁡
                  
                    [
                    
                      
                        
                          x
                          i
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          
                            x
                            j
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        *
                      
                    
                    ]
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      j2πα
                    
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                
              
              ⁢
              
                >
                c
              
            
          
          ,
          
            
              
                C
                x
                α
              
              ⁡
              
                [
                
                  i
                  ,
                  j
                
                ]
              
            
            ⁢
            
              =
              Δ
            
            ⁢
            
              <
              
                
                  E
                  ⁡
                  
                    [
                    
                      
                        
                          x
                          i
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          x
                          j
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    ]
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      j2πα
                    
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                
              
              ⁢
              
                >
                c
              
              .
            
          
        
      
    
    
      
        (
        44
        )
      
    
  




Thus, the estimation of the matrix of cyclic quadricovariance




  
    
      Q
      
        x
        ⁢
        
          
        
        ⁢
        ζ
      
      β
    
    ⁡
    
      (
      
        
          
            l
            1
          
          ⁢
          
            T
            e
          
        
        ,
        
          
            l
            2
          
          ⁢
          
            T
            e
          
        
        ,
        
          
            l
            3
          
          ⁢
          
            T
            e
          
        
      
      )
    
  
  ⁡
  
    [
    
      i
      ,
      j
      ,
      k
      ,
      l
    
    ]
  




is made on the basis of the estimator





  
    
      Q
      ^
    
    x
    β
  
  ⁢
  
    
      ζ
      ⁡
      
        (
        
          
            
              l
              1
            
            ⁢
            
              T
              e
            
          
          ,
          
            
              l
              2
            
            ⁢
            
              T
              e
            
          
          ,
          
            
              l
              3
            
            ⁢
            
              T
              e
            
          
        
        )
      
    
    ⁡
    
      [
      
        i
        ,
        j
        ,
        k
        ,
        l
      
      ]
    
  
  ⁢
  
    (
    K
    )
  




defined by:





  
    
      
        
          
            
              
                
                  Q
                  ^
                
                
                  x
                  ζ
                
                β
              
              ⁡
              
                (
                
                  
                    
                      l
                      1
                    
                    ⁢
                    
                      T
                      e
                    
                  
                  ,
                  
                    
                      l
                      2
                    
                    ⁢
                    
                      T
                      e
                    
                  
                  ,
                  
                    
                      l
                      3
                    
                    ⁢
                    
                      T
                      e
                    
                  
                
                )
              
            
            ⁡
            
              [
              
                i
                ,
                j
                ,
                k
                ,
                l
              
              ]
            
          
          ⁢
          
            (
            K
            )
          
        
        =
        
          
            
              
                
                  
                    M
                    ^
                  
                  
                    x
                    ζ
                  
                  β
                
                ⁡
                
                  (
                  
                    
                      
                        l
                        1
                      
                      ⁢
                      
                        T
                        e
                      
                    
                    ,
                    
                      
                        l
                        2
                      
                      ⁢
                      
                        T
                        e
                      
                    
                    ,
                    
                      
                        l
                        3
                      
                      ⁢
                      
                        T
                        e
                      
                    
                  
                  )
                
              
              ⁡
              
                [
                
                  i
                  ,
                  j
                  ,
                  k
                  ,
                  l
                
                ]
              
            
            ⁢
            
              (
              K
              )
            
          
          -
          
            

          
          ⁢
          
            
              ∑
              
                
                  γ
                  ∈
                  
                    Γ
                    x
                    1
                  
                
                ⁢
                
                  
                
              
            
            ⁢
            
              {
              
                
                  
                    
                      
                        e
                        ^
                      
                      x
                      γ
                    
                    ⁡
                    
                      [
                      i
                      ]
                    
                  
                  ⁢
                  
                    (
                    K
                    )
                  
                  ⁢
                  
                    
                      
                        
                          T
                          ^
                        
                        
                          x
                          ζ
                        
                        
                          β
                          -
                          γ
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              (
                              
                                
                                  l
                                  1
                                
                                -
                                
                                  l
                                  3
                                
                              
                              )
                            
                            ⁢
                            
                              T
                              e
                            
                          
                          ,
                          
                            
                              (
                              
                                
                                  l
                                  2
                                
                                -
                                
                                  l
                                  3
                                
                              
                              )
                            
                            ⁢
                            
                              T
                              e
                            
                          
                        
                        )
                      
                    
                    ⁡
                    
                      [
                      
                        l
                        ,
                        j
                        ,
                        k
                      
                      ]
                    
                  
                  ⁢
                  
                    (
                    K
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        
                          j2π
                          ⁡
                          
                            (
                            
                              β
                              ⁢
                              
                                
                              
                              -
                              γ
                            
                            )
                          
                        
                      
                      ⁢
                      
                        l
                        3
                      
                      ⁢
                      
                        T
                        e
                      
                    
                  
                
                +
                
                  

                
                ⁢
                
                  
                    
                      
                        e
                        ^
                      
                      x
                      γ
                    
                    ⁡
                    
                      [
                      j
                      ]
                    
                  
                  ⁢
                  
                    
                      (
                      K
                      )
                    
                    ζ1
                  
                  ⁢
                  
                    
                      
                        
                          T
                          ^
                        
                        
                          x
                          ζ2
                        
                        
                          β
                          -
                          γζ1
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              l
                              2
                            
                            ⁢
                            
                              T
                              e
                            
                          
                          ,
                          
                            
                              l
                              3
                            
                            ⁢
                            
                              T
                              e
                            
                          
                        
                        )
                      
                    
                    ⁡
                    
                      [
                      
                        i
                        ,
                        k
                        ,
                        l
                      
                      ]
                    
                  
                  ⁢
                  
                    (
                    K
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        
                          j2πγζ2l
                          1
                        
                      
                      ⁢
                      
                        T
                        e
                      
                    
                  
                
                +
                
                  

                
                ⁢
                
                  

                
                ⁢
                
                  
                    
                      
                        e
                        ^
                      
                      x
                      γ
                    
                    ⁡
                    
                      [
                      k
                      ]
                    
                  
                  ⁢
                  
                    
                      (
                      K
                      )
                    
                    ζ2
                  
                  ⁢
                  
                    
                      
                        
                          T
                          ^
                        
                        
                          x
                          ζ2
                        
                        
                          β
                          -
                          γζ2
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              l
                              1
                            
                            ⁢
                            
                              T
                              e
                            
                          
                          ,
                          
                            
                              l
                              3
                            
                            ⁢
                            
                              T
                              e
                            
                          
                        
                        )
                      
                    
                    ⁡
                    
                      [
                      
                        i
                        ,
                        j
                        ,
                        l
                      
                      ]
                    
                  
                  ⁢
                  
                    (
                    K
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        
                          j2πγζ
                          2
                        
                      
                      ⁢
                      
                        l
                        2
                      
                      ⁢
                      
                        T
                        e
                      
                    
                  
                
                +
                
                  

                
                ⁢
                
                  

                
                ⁢
                
                  
                    
                      
                        e
                        ^
                      
                      x
                      γ
                    
                    ⁡
                    
                      [
                      l
                      ]
                    
                  
                  ⁢
                  
                    (
                    K
                    )
                  
                  ⁢
                  
                    
                      
                        
                          T
                          ^
                        
                        
                          x
                          ζ
                        
                        
                          β
                          -
                          γ
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              l
                              1
                            
                            ⁢
                            
                              T
                              e
                            
                          
                          ,
                          
                            
                              l
                              2
                            
                            ⁢
                            
                              T
                              e
                            
                          
                        
                        )
                      
                    
                    ⁡
                    
                      [
                      
                        i
                        ,
                        j
                        ,
                        k
                      
                      ]
                    
                  
                  ⁢
                  
                    (
                    K
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        j2πγ
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        l
                        3
                      
                      ⁢
                      
                        T
                        e
                      
                    
                  
                
              
              }
            
          
          -
          
            

          
          ⁢
          
            

          
          ⁢
          
            
              ∑
              
                α
                ∈
                
                  Γ
                  x
                  
                    [
                    
                      1
                      ,
                      ɛ1
                    
                    ]
                  
                
              
            
            ⁢
            
              
                
                  
                    
                      R
                      ^
                    
                    
                      x
                      ζ1
                    
                    α
                  
                  ⁡
                  
                    (
                    
                      
                        l
                        1
                      
                      ⁢
                      
                        T
                        e
                      
                    
                    )
                  
                
                ⁡
                
                  [
                  
                    i
                    ,
                    j
                  
                  ]
                
              
              ⁢
              
                (
                K
                )
              
              ⁢
              
                
                  
                    
                      R
                      ^
                    
                    
                      x
                      ζ2
                    
                    
                      ζ2
                      ⁡
                      
                        (
                        
                          β
                          -
                          α
                        
                        )
                      
                    
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          
                            l
                            3
                          
                          -
                          
                            l
                            2
                          
                        
                        )
                      
                      ⁢
                      
                        T
                        e
                      
                    
                    )
                  
                
                ⁡
                
                  [
                  
                    k
                    ,
                    l
                  
                  ]
                
              
              ⁢
              
                
                  (
                  K
                  )
                
                ζ2
              
              ⁢
              
                e
                
                  
                    j2π
                    ⁡
                    
                      (
                      
                        α
                        -
                        β
                      
                      )
                    
                  
                  ⁢
                  
                    l
                    2
                  
                  ⁢
                  
                    T
                    e
                  
                
              
            
          
          -
          
            

          
          ⁢
          
            
              ∑
              
                γ
                ∈
                
                  Γ
                  x
                  
                    [
                    
                      1
                      ,
                      ɛ2
                    
                    ]
                  
                
              
            
            ⁢
            
              
                
                  
                    
                      R
                      ^
                    
                    
                      x
                      ζ2
                    
                    γ
                  
                  ⁡
                  
                    (
                    
                      
                        l
                        2
                      
                      ⁢
                      
                        T
                        e
                      
                    
                    )
                  
                
                ⁡
                
                  [
                  
                    i
                    ,
                    k
                  
                  ]
                
              
              ⁢
              
                (
                K
                )
              
              ⁢
              
                
                  
                    
                      R
                      ^
                    
                    
                      x
                      ζ1
                    
                    
                      ζ1
                      ⁡
                      
                        (
                        
                          β
                          -
                          γ
                        
                        )
                      
                    
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          
                            l
                            3
                          
                          -
                          
                            l
                            1
                          
                        
                        )
                      
                      ⁢
                      
                        T
                        e
                      
                    
                    )
                  
                
                ⁡
                
                  [
                  
                    j
                    ,
                    l
                  
                  ]
                
              
              ⁢
              
                
                  (
                  K
                  )
                
                ζ1
              
              ⁢
              
                e
                
                  
                    j2π
                    ⁡
                    
                      (
                      
                        γ
                        -
                        β
                      
                      )
                    
                  
                  ⁢
                  
                    l
                    1
                  
                  ⁢
                  
                    T
                    e
                  
                
              
            
          
          -
          
            

          
          ⁢
          
            
              ∑
              
                ω
                ∈
                
                  Γ
                  x
                  
                    [
                    
                      1
                      ,
                      1
                    
                    ]
                  
                
              
            
            ⁢
            
              
                
                  
                    
                      R
                      ^
                    
                    
                      x
                      ζ1
                    
                    ω
                  
                  ⁡
                  
                    (
                    
                      
                        l
                        3
                      
                      ⁢
                      
                        T
                        e
                      
                    
                    )
                  
                
                ⁡
                
                  [
                  
                    i
                    ,
                    l
                  
                  ]
                
              
              ⁢
              
                (
                K
                )
              
              ⁢
              
                
                  
                    
                      R
                      ^
                    
                    
                      x
                      ζ2
                    
                    
                      ζ1
                      ⁡
                      
                        (
                        
                          β
                          -
                          ω
                        
                        )
                      
                    
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          
                            l
                            2
                          
                          -
                          
                            l
                            1
                          
                        
                        )
                      
                      ⁢
                      
                        T
                        e
                      
                    
                    )
                  
                
                ⁡
                
                  [
                  
                    j
                    ,
                    k
                  
                  ]
                
              
              ⁢
              
                
                  (
                  K
                  )
                
                ζ1
              
              ⁢
              
                e
                
                  
                    j2π
                    ⁡
                    
                      (
                      
                        ω
                        -
                        β
                      
                      )
                    
                  
                  ⁢
                  
                    l
                    1
                  
                  ⁢
                  
                    T
                    e
                  
                
              
            
          
          +
          
            

          
          ⁢
          
            2
            ⁢
            
              
                ∑
                
                  
                    γ
                    ∈
                    
                      Γ
                      x
                      1
                    
                  
                  ⁢
                  
                    
                  
                
              
              ⁢
              
                
                  ∑
                  
                    
                      ω
                      ∈
                      
                        Γ
                        x
                        1
                      
                    
                    ⁢
                    
                      
                    
                  
                
                ⁢
                
                  {
                  
                    
                      
                        
                          
                            e
                            ^
                          
                          x
                          γ
                        
                        ⁡
                        
                          [
                          i
                          ]
                        
                      
                      ⁢
                      
                        (
                        K
                        )
                      
                      ⁢
                      
                        
                          
                            e
                            ^
                          
                          x
                          ω
                        
                        ⁡
                        
                          [
                          j
                          ]
                        
                      
                      ⁢
                      
                        
                          (
                          K
                          )
                        
                        ζ1
                      
                      ⁢
                      
                        
                          
                            
                              R
                              ^
                            
                            
                              x
                              ζ2
                            
                            
                              β
                              -
                              γ
                              -
                              
                                ω
                                ζ1
                              
                            
                          
                          ⁡
                          
                            (
                            
                              
                                (
                                
                                  
                                    l
                                    2
                                  
                                  -
                                  
                                    l
                                    3
                                  
                                
                                )
                              
                              ⁢
                              
                                T
                                e
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            l
                            ,
                            k
                          
                          ]
                        
                      
                      ⁢
                      
                        (
                        K
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            
                              j2πωζ
                              1
                            
                            ⁡
                            
                              (
                              
                                
                                  l
                                  3
                                
                                -
                                
                                  l
                                  1
                                
                              
                              )
                            
                          
                          ⁢
                          
                            T
                            e
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            j2π
                            ⁡
                            
                              (
                              
                                γ
                                -
                                β
                              
                              )
                            
                          
                          ⁢
                          
                            l
                            3
                          
                          ⁢
                          
                            T
                            e
                          
                        
                      
                    
                    +
                    
                      

                    
                    ⁢
                    
                      

                    
                    ⁢
                    
                      
                        
                          
                            e
                            ^
                          
                          x
                          γ
                        
                        ⁡
                        
                          [
                          i
                          ]
                        
                      
                      ⁢
                      
                        (
                        K
                        )
                      
                      ⁢
                      
                        
                          
                            e
                            ^
                          
                          x
                          ω
                        
                        ⁡
                        
                          [
                          k
                          ]
                        
                      
                      ⁢
                      
                        
                          (
                          K
                          )
                        
                        ζ2
                      
                      ⁢
                      
                        
                          R
                          ^
                        
                        
                          x
                          ζ1
                        
                        
                          β
                          -
                          γ
                          -
                          
                            ω
                            ζ2
                          
                        
                      
                      ⁢
                      
                        
                          (
                          
                            
                              (
                              
                                
                                  l
                                  1
                                
                                -
                                
                                  l
                                  3
                                
                              
                              )
                            
                            ⁢
                            
                              T
                              e
                            
                          
                          )
                        
                        ⁡
                        
                          [
                          
                            l
                            ,
                            j
                          
                          ]
                        
                      
                      ⁢
                      
                        (
                        K
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            j2πωζ2
                            ⁡
                            
                              (
                              
                                
                                  l
                                  3
                                
                                -
                                
                                  l
                                  2
                                
                              
                              )
                            
                          
                          ⁢
                          
                            T
                            e
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            j2π
                            ⁡
                            
                              (
                              
                                γ
                                -
                                β
                              
                              )
                            
                          
                          ⁢
                          
                            l
                            3
                          
                          ⁢
                          
                            T
                            e
                          
                        
                      
                    
                    +
                    
                      

                    
                    ⁢
                    
                      

                    
                    ⁢
                    
                      
                        
                          
                            e
                            ^
                          
                          x
                          γ
                        
                        ⁡
                        
                          [
                          i
                          ]
                        
                      
                      ⁢
                      
                        (
                        K
                        )
                      
                      ⁢
                      
                        
                          
                            e
                            ^
                          
                          x
                          ω
                        
                        ⁡
                        
                          [
                          l
                          ]
                        
                      
                      ⁢
                      
                        (
                        K
                        )
                      
                      ⁢
                      
                        
                          
                            
                              R
                              ^
                            
                            
                              x
                              ζ2
                            
                            
                              ζ1
                              ⁡
                              
                                (
                                
                                  β
                                  -
                                  ω
                                  -
                                  γ
                                
                                )
                              
                            
                          
                          ⁡
                          
                            (
                            
                              
                                (
                                
                                  
                                    l
                                    2
                                  
                                  -
                                  
                                    l
                                    1
                                  
                                
                                )
                              
                              ⁢
                              
                                T
                                e
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            j
                            ,
                            k
                          
                          ]
                        
                      
                      ⁢
                      
                        
                          (
                          K
                          )
                        
                        ζ1
                      
                      ⁢
                      
                        e
                        
                          
                            j2πω
                            ⁡
                            
                              (
                              
                                
                                  l
                                  1
                                
                                -
                                
                                  l
                                  3
                                
                              
                              )
                            
                          
                          ⁢
                          
                            T
                            e
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            j2πβ
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            l
                            1
                          
                          ⁢
                          
                            T
                            e
                          
                        
                      
                    
                    +
                    
                      

                    
                    ⁢
                    
                      

                    
                    ⁢
                    
                      
                        
                          
                            e
                            ^
                          
                          x
                          γ
                        
                        ⁡
                        
                          [
                          k
                          ]
                        
                      
                      ⁢
                      
                        
                          (
                          K
                          )
                        
                        ζ2
                      
                      ⁢
                      
                        
                          
                            e
                            ^
                          
                          x
                          ω
                        
                        ⁡
                        
                          [
                          j
                          ]
                        
                      
                      ⁢
                      
                        
                          (
                          K
                          )
                        
                        ζ1
                      
                      ⁢
                      
                        
                          
                            
                              R
                              ^
                            
                            
                              x
                              1
                            
                            
                              β
                              -
                              γζ2
                              -
                              ωζ1
                            
                          
                          ⁡
                          
                            (
                            
                              
                                l
                                3
                              
                              ⁢
                              
                                T
                                e
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            i
                            ,
                            l
                          
                          ]
                        
                      
                      ⁢
                      
                        (
                        K
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            
                          
                          
                            
                              -
                              j2πγ
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              l
                              2
                            
                            ⁢
                            
                              T
                              e
                            
                            ⁢
                            ζ2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            j2πω
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            l
                            1
                          
                          ⁢
                          
                            T
                            e
                          
                          ⁢
                          ζ1
                        
                      
                    
                    +
                    
                      

                    
                    ⁢
                    
                      

                    
                    ⁢
                    
                      
                        
                          
                            e
                            ^
                          
                          x
                          γ
                        
                        ⁡
                        
                          [
                          l
                          ]
                        
                      
                      ⁢
                      
                        (
                        K
                        )
                      
                      ⁢
                      
                        
                          
                            e
                            ^
                          
                          x
                          ω
                        
                        ⁡
                        
                          [
                          j
                          ]
                        
                      
                      ⁢
                      
                        
                          (
                          K
                          )
                        
                        ζ1
                      
                      ⁢
                      
                        
                          
                            
                              R
                              ^
                            
                            
                              x
                              ζ2
                            
                            
                              β
                              -
                              γ
                              -
                              ωζ1
                            
                          
                          ⁡
                          
                            (
                            
                              
                                l
                                2
                              
                              ⁢
                              
                                T
                                e
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            i
                            ,
                            k
                          
                          ]
                        
                      
                      ⁢
                      
                        (
                        K
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            
                          
                          
                            
                              -
                              j2πγ
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              l
                              1
                            
                            ⁢
                            
                              T
                              e
                            
                            ⁢
                            ζ1
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            j2πγ
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            l
                            3
                          
                          ⁢
                          
                            T
                            e
                          
                        
                      
                    
                    +
                    
                      

                    
                    ⁢
                    
                      

                    
                    ⁢
                    
                      
                        
                          
                            e
                            ^
                          
                          x
                          γ
                        
                        ⁡
                        
                          [
                          l
                          ]
                        
                      
                      ⁢
                      
                        (
                        K
                        )
                      
                      ⁢
                      
                        
                          
                            e
                            ^
                          
                          x
                          ω
                        
                        ⁡
                        
                          [
                          k
                          ]
                        
                      
                      ⁢
                      
                        
                          (
                          K
                          )
                        
                        ζ2
                      
                      ⁢
                      
                        
                          
                            
                              R
                              ^
                            
                            
                              x
                              ζ1
                            
                            
                              β
                              -
                              γ
                              -
                              ωζ2
                            
                          
                          ⁡
                          
                            (
                            
                              
                                l
                                1
                              
                              ⁢
                              
                                T
                                e
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            i
                            ,
                            j
                          
                          ]
                        
                      
                      ⁢
                      
                        (
                        K
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            
                          
                          
                            
                              -
                              j2πω
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              l
                              2
                            
                            ⁢
                            
                              T
                              e
                            
                            ⁢
                            ζ2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            j2πγ
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            l
                            3
                          
                          ⁢
                          
                            T
                            e
                          
                        
                      
                    
                  
                  }
                
              
            
          
          -
          
            

          
          ⁢
          
            

          
          ⁢
          
            6
            ⁢
            
              
                ∑
                
                  
                    γ
                    ∈
                    
                      Γ
                      x
                      1
                    
                  
                  ⁢
                  
                    
                  
                
              
              ⁢
              
                
                  ∑
                  
                    
                      ω
                      ∈
                      
                        Γ
                        x
                        1
                      
                    
                    ⁢
                    
                      
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      
                        δ
                        ∈
                        
                          Γ
                          x
                          1
                        
                      
                      ⁢
                      
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        
                          e
                          ^
                        
                        x
                        γ
                      
                      ⁡
                      
                        [
                        i
                        ]
                      
                    
                    ⁢
                    
                      (
                      K
                      )
                    
                    ⁢
                    
                      
                        
                          e
                          ^
                        
                        x
                        ω
                      
                      ⁡
                      
                        [
                        j
                        ]
                      
                    
                    ⁢
                    
                      
                        (
                        K
                        )
                      
                      ζ1
                    
                    ⁢
                    
                      
                        
                          e
                          ^
                        
                        x
                        δ
                      
                      ⁡
                      
                        [
                        k
                        ]
                      
                    
                    ⁢
                    
                      
                        (
                        K
                        )
                      
                      ζ2
                    
                    ⁢
                    
                      
                        
                          e
                          ^
                        
                        x
                        
                          β
                          -
                          γ
                          -
                          ωζ1
                          -
                          δζ2
                        
                      
                      ⁡
                      
                        [
                        l
                        ]
                      
                    
                    ⁢
                    
                      (
                      K
                      )
                    
                    ⁢
                    
                      e
                      
                        
                          j2πωζ1
                          ⁡
                          
                            (
                            
                              
                                l
                                3
                              
                              -
                              
                                l
                                1
                              
                            
                            )
                          
                        
                        ⁢
                        
                          T
                          e
                        
                      
                    
                    ⁢
                    
                      e
                      
                        
                          j2πδζ2
                          ⁡
                          
                            (
                            
                              
                                l
                                3
                              
                              -
                              
                                l
                                2
                              
                            
                            )
                          
                        
                        ⁢
                        
                          T
                          e
                        
                      
                    
                    ⁢
                    
                      e
                      
                        j2π
                        ⁢
                        
                          
                        
                        ⁢
                        
                          l
                          3
                        
                        ⁢
                        
                          
                            T
                            e
                          
                          ⁡
                          
                            (
                            
                              γ
                              -
                              β
                            
                            )
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        45
        )
      
    
  




where





  
    
      M
      ^
    
    x
    β
  
  ⁢
  
    
      ζ
      ⁡
      
        (
        
          
            
              l
              1
            
            ⁢
            
              T
              e
            
          
          ,
          
            
              l
              2
            
            ⁢
            
              T
              e
            
          
          ,
          
            
              l
              3
            
            ⁢
            
              T
              e
            
          
        
        )
      
    
    ⁡
    
      [
      
        i
        ,
        j
        ,
        k
        ,
        l
      
      ]
    
  
  ⁢
  
    (
    K
    )
  




is defined by the expression:





  
    
      
        
          
            
              
                
                  M
                  ^
                
                
                  x
                  ⁢
                  
                    
                  
                  ⁢
                  ζ
                
                β
              
              ⁡
              
                (
                
                  
                    
                      l
                      1
                    
                    ⁢
                    
                      T
                      e
                    
                  
                  ,
                  
                    
                      l
                      2
                    
                    ⁢
                    
                      T
                      e
                    
                  
                  ,
                  
                    
                      l
                      3
                    
                    ⁢
                    
                      T
                      e
                    
                  
                
                )
              
            
            ⁡
            
              [
              
                i
                ,
                j
                ,
                k
                ,
                l
              
              ]
            
          
          ⁢
          
            (
            K
            )
          
        
        =
        
          
            1
            K
          
          ⁢
          
            
              ∑
              
                m
                =
                1
              
              K
            
            ⁢
            
              
                
                  x
                  i
                
                ⁡
                
                  (
                  m
                  )
                
              
              ⁢
              
                
                  x
                  j
                
                ⁡
                
                  (
                  
                    m
                    -
                    
                      l
                      1
                    
                  
                  )
                
              
              ⁢
              
                
                  x
                  k
                  
                  
                    
                    
                      ζ
                      1
                    
                
                ⁡
                
                  (
                  
                    m
                    -
                    
                      l
                      2
                    
                  
                  )
                
              
              ⁢
              
                
                  x
                  l
                  
                  
                    
                    
                      ζ
                      2
                    
                
                ⁡
                
                  (
                  
                    m
                    -
                    
                      l
                      3
                    
                  
                  )
                
              
              ⁢
              
                e
                
                  
                    -
                    j
                  
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  β
                  ⁢
                  
                    
                  
                  ⁢
                  m
                  ⁢
                  
                    
                  
                  ⁢
                  
                    T
                    e
                  
                
              
            
          
        
      
    
    
      
        (
        46
        )
      
    
  




given in the reference [5] and where





  
    
      
        
          
            e
            ^
          
          x
          γ
        
        ⁡
        
          [
          i
          ]
        
      
      ⁢
      
        (
        K
        )
      
    
    ,
    
      
        
          
            
              R
              ^
            
            
              x
              ⁢
              
                
              
              ⁢
              ɛ
            
            α
          
          ⁡
          
            (
            
              lT
              e
            
            )
          
        
        ⁡
        
          [
          
            i
            ,
            j
          
          ]
        
      
      ⁢
      
        (
        K
        )
      
    
    ,
    
      
        
          
            
              T
              ^
            
            
              x
              ⁢
              
                
              
              ⁢
              ɛ
            
            α
          
          ⁡
          
            (
            
              
                
                  l
                  1
                
                ⁢
                
                  T
                  e
                
              
              ,
              
                
                  l
                  2
                
                ⁢
                
                  T
                  e
                
              
            
            )
          
        
        ⁡
        
          [
          
            i
            ,
            j
            ,
            l
          
          ]
        
      
      ⁢
      
        (
        K
        )
      
      ⁢
      
        
      
      ⁢
      and
      ⁢
      
        
      
      ⁢
      
        
          
            
              T
              ^
            
            
              x
              ⁢
              
                
              
              ⁢
              ζ
            
            α
          
          ⁡
          
            (
            
              
                
                  l
                  1
                
                ⁢
                
                  T
                  e
                
              
              ,
              
                
                  l
                  2
                
                ⁢
                
                  T
                  e
                
              
            
            )
          
        
        ⁡
        
          [
          
            i
            ,
            j
            ,
            l
          
          ]
        
      
      ⁢
      
        (
        K
        )
      
    
  
  ⁢
  
    
  




are defined respectively by (47), (48), (49) and (50).





  
    
      
        
          
            
              e
              ^
            
            x
            γ
          
          ⁡
          
            [
            i
            ]
          
        
        ⁢
        
          (
          K
          )
        
        ⁢
        
          
        
        ⁢
        
          Δ
          
            _
            _
          
        
        ⁢
        
          
        
        ⁢
        
          1
          K
        
        ⁢
        
          
            ∑
            
              k
              =
              1
            
            K
          
          ⁢
          
            
              
                x
                i
              
              ⁡
              
                (
                k
                )
              
            
            ⁢
            
              e
              
                
                  -
                  j2π
                
                ⁢
                
                  
                
                ⁢
                γ
                ⁢
                
                  
                
                ⁢
                k
                ⁢
                
                  
                
                ⁢
                
                  T
                  e
                
              
            
          
        
      
    
    
      
        (
        47
        )
      
    
  
  
    
      
        
          
            
              
                R
                ^
              
              
                x
                ⁢
                
                  
                
                ⁢
                ɛ
              
              α
            
            ⁡
            
              (
              
                lT
                e
              
              )
            
          
          ⁡
          
            [
            
              i
              ,
              j
            
            ]
          
        
        ⁢
        
          (
          K
          )
        
        ⁢
        
          
        
        ⁢
        
          Δ
          
            _
            _
          
        
        ⁢
        
          
        
        ⁢
        
          1
          K
        
        ⁢
        
          
            ∑
            
              k
              =
              1
            
            K
          
          ⁢
          
            
              
                x
                i
              
              ⁡
              
                (
                k
                )
              
            
            ⁢
            
              
                x
                j
              
              ⁡
              
                (
                
                  k
                  -
                  l
                
                )
              
            
            ⁢
            
              e
              
              
                
                  -
                  j
                
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                α
                ⁢
                
                  
                
                ⁢
                k
                ⁢
                
                  
                
                ⁢
                
                  T
                  e
                
              
              
                
                ɛ
            
          
        
      
    
    
      
        (
        48
        )
      
    
  
  
    
      
        
          
            
              
                T
                ^
              
              
                x
                ⁢
                
                  
                
                ⁢
                ɛ
              
              α
            
            ⁡
            
              (
              
                
                  
                    l
                    1
                  
                  ⁢
                  
                    T
                    e
                  
                
                ,
                
                  
                    l
                    2
                  
                  ⁢
                  
                    T
                    e
                  
                
              
              )
            
          
          ⁡
          
            [
            
              i
              ,
              j
              ,
              l
            
            ]
          
        
        ⁢
        
          (
          K
          )
        
        ⁢
        
          
        
        ⁢
        
          Δ
          
            _
            _
          
        
        ⁢
        
          
        
        ⁢
        
          1
          K
        
        ⁢
        
          
            ∑
            
              k
              =
              1
            
            K
          
          ⁢
          
            
              
                x
                i
              
              ⁡
              
                (
                k
                )
              
            
            ⁢
            
              
                x
                j
              
              ⁡
              
                (
                
                  k
                  -
                  
                    l
                    1
                  
                
                )
              
            
            ⁢
            
              
                
                   
                  ɛ
                
                ⁢
                
                  
                
                ⁢
                
                  x
                  l
                
              
              ⁡
              
                (
                
                  k
                  -
                  
                    l
                    2
                  
                
                )
              
            
            ⁢
            
              e
              
                
                  -
                  j
                
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                α
                ⁢
                
                  
                
                ⁢
                
                  kT
                  e
                
              
            
          
        
      
    
    
      
        (
        49
        )
      
    
  
  
    
      
        
          
            
              
                T
                ^
              
              
                x
                ⁢
                
                  
                
                ⁢
                ζ
              
              α
            
            ⁡
            
              (
              
                
                  
                    l
                    1
                  
                  ⁢
                  
                    T
                    e
                  
                
                ,
                
                  
                    l
                    2
                  
                  ⁢
                  
                    T
                    e
                  
                
              
              )
            
          
          ⁡
          
            [
            
              i
              ,
              j
              ,
              l
            
            ]
          
        
        ⁢
        
          (
          K
          )
        
        ⁢
        
          
        
        ⁢
        
          Δ
          
            _
            _
          
        
        ⁢
        
          
        
        ⁢
        
          1
          K
        
        ⁢
        
          
            ∑
            
              k
              =
              1
            
            K
          
          ⁢
          
            
              
                x
                i
              
              ⁡
              
                (
                k
                )
              
            
            ⁢
            
              
                x
                j
              
              ⁡
              
                (
                
                  k
                  -
                  
                    l
                    1
                  
                
                )
              
            
            ⁢
            
              
                x
                l
                
                
                  
                  
                    ζ
                    1
                  
              
              ⁡
              
                (
                
                  k
                  -
                  
                    l
                    2
                  
                
                )
              
            
            ⁢
            
              e
              
              
                
                  -
                  j
                
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                α
                ⁢
                
                  
                
                ⁢
                k
                ⁢
                
                  
                
                ⁢
                
                  T
                  e
                
              
              
                
                
                  ζ
                  2
                
            
          
        
      
    
    
      
        (
        50
        )
      
    
  




Thus, assuming cyclostationary and cycloergodic observations, centered or non-centered, the estimator (45) is an asymptotically unbiased and consistent estimator of the element




  
    
      
        Q
        
          x
          ⁢
          
            
          
          ⁢
          ζ
        
        β
      
      ⁡
      
        (
        
          
            
              l
              1
            
            ⁢
            
              T
              e
            
          
          ,
          
            
              l
              2
            
            ⁢
            
              T
              e
            
          
          ,
          
            
              l
              3
            
            ⁢
            
              T
              e
            
          
        
        )
      
    
    ⁡
    
      [
      
        i
        ,
        j
        ,
        k
        ,
        l
      
      ]
    
  
  .




In particular, the separators F3 must exploit the estimator (45) for β=0, ζ=(−1, −1) and l1=l2=l3=0, denoted {circumflex over (Q)}x[i, j, k, l](K) and defined by:





  
    
      
        
          
            
              
                Q
                ^
              
              x
            
            ⁡
            
              [
              
                i
                ,
                j
                ,
                k
                ,
                l
              
              ]
            
          
          ⁢
          
            (
            K
            )
          
          ⁢
          
            
          
          ⁢
          
            Δ
            
              _
              _
            
          
          ⁢
          
            
          
          ⁢
          
            
              
                M
                ^
              
              x
              0
            
            ⁡
            
              [
              
                i
                ,
                j
                ,
                k
                ,
                l
              
              ]
            
          
          ⁢
          
            (
            K
            )
          
        
        -
        
          
            ∑
            
              γ
              ⁢
              
                
              
              ∈
              
                
              
              ⁢
              
                Γ
                x
                1
              
            
          
          ⁢
          
            {
            
              
                
                  
                    
                      e
                      ^
                    
                    x
                    γ
                  
                  ⁡
                  
                    [
                    i
                    ]
                  
                
                ⁢
                
                  (
                  K
                  )
                
                ⁢
                
                  
                    
                      T
                      ^
                    
                    x
                    γ
                  
                  ⁡
                  
                    [
                    
                      j
                      ,
                      l
                      ,
                      k
                    
                    ]
                  
                
                ⁢
                
                  
                    (
                    K
                    )
                  
                  *
                
              
              +
              
                
                  
                    
                      e
                      ^
                    
                    x
                    γ
                  
                  ⁡
                  
                    [
                    l
                    ]
                  
                
                ⁢
                
                  (
                  K
                  )
                
                ⁢
                
                  
                    
                      T
                      ^
                    
                    x
                    γ
                  
                  ⁡
                  
                    [
                    
                      j
                      ,
                      i
                      ,
                      k
                    
                    ]
                  
                
                ⁢
                
                  
                    (
                    K
                    )
                  
                  *
                
              
              +
              
                
                  
                    
                      e
                      ^
                    
                    x
                    γ
                  
                  ⁡
                  
                    [
                    j
                    ]
                  
                
                ⁢
                
                  
                    (
                    K
                    )
                  
                  *
                
                ⁢
                
                  
                    
                      T
                      ^
                    
                    x
                    γ
                  
                  ⁡
                  
                    [
                    
                      i
                      ,
                      k
                      ,
                      l
                    
                    ]
                  
                
                ⁢
                
                  (
                  K
                  )
                
              
              +
              
                
                  
                    
                      e
                      ^
                    
                    x
                    γ
                  
                  ⁡
                  
                    [
                    k
                    ]
                  
                
                ⁢
                
                  
                    (
                    K
                    )
                  
                  *
                
                ⁢
                
                  
                    
                      T
                      ^
                    
                    x
                    γ
                  
                  ⁡
                  
                    [
                    
                      i
                      ,
                      j
                      ,
                      l
                    
                    ]
                  
                
                ⁢
                
                  (
                  K
                  )
                
              
            
            }
          
        
        -
        
          
            ∑
            
              ω
              ⁢
              
                
              
              ∈
              
                
              
              ⁢
              
                Γ
                x
                
                  [
                  
                    1
                    ,
                    1
                  
                  ]
                
              
            
          
          ⁢
          
            
              
                
                  C
                  ^
                
                x
                ω
              
              ⁡
              
                [
                
                  i
                  ,
                  l
                
                ]
              
            
            ⁢
            
              (
              K
              )
            
            ⁢
            
              
                
                  C
                  ^
                
                x
                ω
              
              ⁡
              
                [
                
                  j
                  ,
                  k
                
                ]
              
            
            ⁢
            
              
                (
                K
                )
              
              *
            
          
        
        -
        
          
            ∑
            
              α
              ⁢
              
                
              
              ∈
              
                
              
              ⁢
              
                Γ
                x
                
                  [
                  
                    1
                    ,
                    
                      -
                      1
                    
                  
                  ]
                
              
            
          
          ⁢
          
            {
            
              
                
                  
                    
                      R
                      ^
                    
                    x
                    α
                  
                  ⁡
                  
                    [
                    
                      i
                      ,
                      j
                    
                    ]
                  
                
                ⁢
                
                  (
                  K
                  )
                
                ⁢
                
                  
                    
                      R
                      ^
                    
                    x
                    
                      -
                      α
                    
                  
                  ⁡
                  
                    [
                    
                      l
                      ,
                      k
                    
                    ]
                  
                
                ⁢
                
                  (
                  K
                  )
                
              
              +
              
                
                  
                    
                      R
                      ^
                    
                    x
                    α
                  
                  ⁡
                  
                    [
                    
                      i
                      ,
                      k
                    
                    ]
                  
                
                ⁢
                
                  (
                  K
                  )
                
                ⁢
                
                  
                    
                      R
                      ^
                    
                    x
                    
                      -
                      α
                    
                  
                  ⁡
                  
                    [
                    
                      l
                      ,
                      j
                    
                    ]
                  
                
                ⁢
                
                  (
                  K
                  )
                
              
            
            }
          
        
        +
        
          2
          ⁢
          
            
              ∑
              
                γ
                ⁢
                
                  
                
                ∈
                
                  
                
                ⁢
                
                  Γ
                  x
                  1
                
              
            
            ⁢
            
              
                ∑
                
                  ω
                  ⁢
                  
                    
                  
                  ∈
                  
                    
                  
                  ⁢
                  
                    Γ
                    x
                    1
                  
                
              
              ⁢
              
                {
                
                  
                    
                      
                        
                          e
                          ^
                        
                        x
                        γ
                      
                      ⁡
                      
                        [
                        i
                        ]
                      
                    
                    ⁢
                    
                      (
                      K
                      )
                    
                    ⁢
                    
                      
                        
                          e
                          ^
                        
                        x
                        ω
                      
                      ⁡
                      
                        [
                        j
                        ]
                      
                    
                    ⁢
                    
                      
                        (
                        K
                        )
                      
                      *
                    
                    ⁢
                    
                      
                        
                          R
                          ^
                        
                        x
                        
                          ω
                          -
                          γ
                        
                      
                      ⁡
                      
                        [
                        
                          l
                          ,
                          k
                        
                        ]
                      
                    
                    ⁢
                    
                      (
                      K
                      )
                    
                  
                  +
                  
                    
                      
                        
                          e
                          ^
                        
                        x
                        γ
                      
                      ⁡
                      
                        [
                        i
                        ]
                      
                    
                    ⁢
                    
                      (
                      K
                      )
                    
                    ⁢
                    
                      
                        
                          e
                          ^
                        
                        x
                        ω
                      
                      ⁡
                      
                        [
                        k
                        ]
                      
                    
                    ⁢
                    
                      
                        (
                        K
                        )
                      
                      *
                    
                    ⁢
                    
                      
                        
                          R
                          ^
                        
                        x
                        
                          ω
                          -
                          γ
                        
                      
                      ⁡
                      
                        [
                        
                          l
                          ,
                          j
                        
                        ]
                      
                    
                    ⁢
                    
                      (
                      K
                      )
                    
                  
                  +
                  
                    
                      
                        
                          e
                          ^
                        
                        x
                        γ
                      
                      ⁡
                      
                        [
                        i
                        ]
                      
                    
                    ⁢
                    
                      (
                      K
                      )
                    
                    ⁢
                    
                      
                        
                          e
                          ^
                        
                        x
                        ω
                      
                      ⁡
                      
                        [
                        l
                        ]
                      
                    
                    ⁢
                    
                      (
                      K
                      )
                    
                    ⁢
                    
                      
                        
                          C
                          ^
                        
                        x
                        
                          ω
                          +
                          γ
                        
                      
                      ⁡
                      
                        [
                        
                          k
                          ,
                          j
                        
                        ]
                      
                    
                    ⁢
                    
                      
                        (
                        K
                        )
                      
                      *
                    
                  
                  +
                  
                    
                      
                        
                          e
                          ^
                        
                        x
                        γ
                      
                      ⁡
                      
                        [
                        k
                        ]
                      
                    
                    ⁢
                    
                      
                        (
                        K
                        )
                      
                      *
                    
                    ⁢
                    
                      
                        
                          e
                          ^
                        
                        x
                        ω
                      
                      ⁡
                      
                        [
                        j
                        ]
                      
                    
                    ⁢
                    
                      
                        (
                        K
                        )
                      
                      *
                    
                    ⁢
                    
                      
                        
                          C
                          ^
                        
                        x
                        
                          ω
                          +
                          γ
                        
                      
                      ⁡
                      
                        [
                        
                          i
                          ,
                          l
                        
                        ]
                      
                    
                    ⁢
                    
                      (
                      K
                      )
                    
                  
                  +
                  
                    
                      
                        
                          e
                          ^
                        
                        x
                        γ
                      
                      ⁡
                      
                        [
                        l
                        ]
                      
                    
                    ⁢
                    
                      (
                      K
                      )
                    
                    ⁢
                    
                      
                        
                          e
                          ^
                        
                        x
                        ω
                      
                      ⁡
                      
                        [
                        j
                        ]
                      
                    
                    ⁢
                    
                      
                        (
                        K
                        )
                      
                      *
                    
                    ⁢
                    
                      
                        
                          R
                          ^
                        
                        x
                        
                          ω
                          -
                          γ
                        
                      
                      ⁡
                      
                        [
                        
                          i
                          ,
                          k
                        
                        ]
                      
                    
                    ⁢
                    
                      (
                      K
                      )
                    
                  
                  +
                  
                    
                      
                        
                          e
                          ^
                        
                        x
                        γ
                      
                      ⁡
                      
                        [
                        l
                        ]
                      
                    
                    ⁢
                    
                      (
                      K
                      )
                    
                    ⁢
                    
                      
                        
                          e
                          ^
                        
                        x
                        ω
                      
                      ⁡
                      
                        [
                        k
                        ]
                      
                    
                    ⁢
                    
                      
                        (
                        K
                        )
                      
                      *
                    
                    ⁢
                    
                      
                        
                          R
                          ^
                        
                        x
                        
                          ω
                          -
                          γ
                        
                      
                      ⁡
                      
                        [
                        
                          i
                          ,
                          j
                        
                        ]
                      
                    
                    ⁢
                    
                      (
                      K
                      )
                    
                  
                
                }
              
            
          
        
        -
        
          6
          ⁢
          
            
              ∑
              
                γ
                ⁢
                
                  
                
                ∈
                
                  
                
                ⁢
                
                  Γ
                  x
                  1
                
              
            
            ⁢
            
              
                ∑
                
                  ω
                  ⁢
                  
                    
                  
                  ∈
                  
                    
                  
                  ⁢
                  
                    Γ
                    x
                    1
                  
                
              
              ⁢
              
                
                  ∑
                  
                    δ
                    ⁢
                    
                      
                    
                    ∈
                    
                      
                    
                    ⁢
                    
                      Γ
                      x
                      1
                    
                  
                
                ⁢
                
                  
                    
                      
                        e
                        ^
                      
                      x
                      γ
                    
                    ⁡
                    
                      [
                      i
                      ]
                    
                  
                  ⁢
                  
                    (
                    K
                    )
                  
                  ⁢
                  
                    
                      
                        e
                        ^
                      
                      x
                      ω
                    
                    ⁡
                    
                      [
                      j
                      ]
                    
                  
                  ⁢
                  
                    
                      (
                      K
                      )
                    
                    *
                  
                  ⁢
                  
                    
                      
                        e
                        ^
                      
                      x
                      δ
                    
                    ⁡
                    
                      [
                      k
                      ]
                    
                  
                  ⁢
                  
                    
                      (
                      K
                      )
                    
                    *
                  
                  ⁢
                  
                    
                      
                        e
                        ^
                      
                      x
                      
                        δ
                        +
                        ω
                        -
                        γ
                      
                    
                    ⁡
                    
                      [
                      l
                      ]
                    
                  
                  ⁢
                  
                    (
                    K
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        51
        )
      
    
  




where {circumflex over (M)}x0[i, j, k, l](K) is defined by (46) (defined here above) with β=0, l1=l2=l3=0, ζ=(−1, −1), {circumflex over (T)}xγ[i, j, l](K) is defined by (49) with α=γ, l1=l2=0, ε=−1, {circumflex over (R)}xα[i, j] and Ĉxα[i, j] are defined by (29) with l=0 and respectively ε=−1 and ε=+1.

As indicated here above and as can be seen from the previous estimators, the method according the invention also comprises a step in which the second-order estimator is corrected by using cyclic frequencies which have to be detected a priori. The following detailed example is given in the case of a detection of the first-order cyclic frequencies.
Detector of the First-Order Cyclic Frequencies
The detection of the first-order cyclic frequencies of the observations γ by a detector of cyclic frequencies and the constitution of an estimate,




  
    
      Γ
      ^
    
    x
    1
  
  ,




of the set,





  
    Γ
    x
    1
  
  ,




of the cyclic frequencies γ are done, for example, by computing the following standardized criterion:





  
    
      
        
          
            V
            ⁡
            
              (
              α
              )
            
          
          =
          
            
              
                1
                N
              
              ⁢
              
                
                  ∑
                  
                    n
                    =
                    1
                  
                  N
                
                ⁢
                
                  
                    
                    
                      
                        
                          
                            e
                            ^
                          
                          x
                          α
                        
                        ⁡
                        
                          [
                          n
                          ]
                        
                      
                      ⁡
                      
                        [
                        K
                        ]
                      
                    
                    
                  
                  2
                
              
            
            
              
                γ
                ^
              
              x
            
          
        
        ⁢
        
          
        
        ⁢
        
          

        
        ⁢
        
          With
          ⁢
          
            
          
          ⁢
          
            
              γ
              ^
            
            x
          
        
        =
        
          
            
              1
              N
            
            ⁢
            
              1
              K
            
            ⁢
            
              
                ∑
                
                  k
                  =
                  1
                
                K
              
              ⁢
              
                
                  
                    
                    
                      
                        x
                        n
                      
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                    
                  
                  2
                
                ⁢
                
                  
                
                ⁢
                and
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      e
                      ^
                    
                    x
                    γ
                  
                  ⁡
                  
                    [
                    n
                    ]
                  
                
                ⁢
                
                  (
                  K
                  )
                
              
            
          
          =
          
            
              1
              K
            
            ⁢
            
              
                ∑
                
                  k
                  =
                  1
                
                K
              
              ⁢
              
                
                  
                    x
                    n
                  
                  ⁡
                  
                    (
                    k
                    )
                  
                
                ⁢
                
                  
                
                ⁢
                
                  exp
                  ⁡
                  
                    (
                    
                      
                        -
                        j
                      
                      ⁢
                      
                        
                      
                      ⁢
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      γ
                      ⁢
                      
                        
                      
                      ⁢
                      k
                      ⁢
                      
                        
                      
                      ⁢
                      
                        T
                        e
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        52
        )
      
    
  




It may be recalled that xn(t) is the signal received at the nth sensor. The estimator êxγ[n](K) may be computed in an optimized way for K cyclic frequencies αk=k/(K Te) (0≦k≦K−1) by an FFT (a Fast Fourier Transformation) on the temporal samples xn((k+k0)Te) such that 0≦k≦K−1. The criterion V(α) is standardized between 0 and 1 because γx represents the temporal mean of the mean power of the signal on the set of sensors. It being known that, on the assumption that xn(kTe) is a Gaussian noise, the criterion V(α) approximately follows a chi-2 relationship, the following detection test is deduced therefrom:

    
    
        The frequency α is a cyclic frequency γn of E[x(t)]: V(α)>=μ(pfα)
        The frequency α is not a cyclic frequency: V(α)<μ(pfα)

where μ(pfα) is a threshold as a function of the probability of false alarm pfα

    
    


The rest of the description gives several alternative modes of implementation of the method for the separation of statistically independent, stationary or cyclostationary sources. The associated separators are respectively called F′1, F′2, F′3 and F′4.
Proposed Second-Order Separators
Separators F1′
The separators of the family F1′ are self-learning second-order separators implementing the following operations:
Whitening Step


    
    
        The detection of the first-order cyclic frequencies of the observations γ by any detector of cyclic frequencies and the constitution of an estimate,
    
    






  
    
      Γ
      ^
    
    x
    1
  
  ,




of the set,





  
    Γ
    x
    1
  
  ,




of the cyclic frequencies γ.

    
    
        The estimation of the matrix RΔx(0) by {circumflex over (R)}Δx(0)(K) defined by (33) and (32) for l=0 on the basis of a given number K of samples.
        The detection of the number of sources P from the decomposition of {circumflex over (R)}Δx(0)(K) into eigen elements. (All the non-deterministic sources are detected).
        The computation of the whitening matrix of the observations, {circumflex over (T)}, where {circumflex over (T)}Δ{circumflex over (Λ)}s−1/2 Ûs†, with a dimension (P×N), where {circumflex over (Λ)}s is the diagonal matrix (P×P) of the P greatest eigen values of {circumflex over (R)}Δx(0)(K)−λmin I, λmin is the minimum eigen value of {circumflex over (R)}Δx(0)(K) and Ûs is the matrix of associated eigen vectors. We write z(t) Δ{circumflex over (T)}x(t). (The directional vectors of the non-deterministic sources are orthonormalized).

Identification Step

        Choice of Q values of non-zero delays, lq, (1≦q≦Q).
        For each value, lq, of the delay, estimation of the matrix of averaged second-order cumulants of the observations, RΔx(lqTe), by {circumflex over (R)}Δx(lqTe)(K) defined by (33), (29) and (32)
        Computation of the matrices {circumflex over (R)}Δz(lqTe)(K) Δ{circumflex over (T)}{circumflex over (R)}Δx(lqTe)(K){circumflex over (T)}†and self-learned identification of the directional vectors of the whitened sources by maximization, with respect to UΔ(u1, u2, . . . up), of the criterion
    
    






  
    C1
    ⁡
    
      (
      U
      )
    
  
  ⁢
  
    
  
  ⁢
  
    Δ
    
      _
      _
    
  
  ⁢
  
    
  
  ⁢
  
    
      ∑
      
        q
        =
        1
      
      Q
    
    ⁢
    
      
        ∑
        
          l
          =
          1
        
        P
      
      ⁢
      
        
          
          
            
              u
              l
              †
            
            ⁢
            
              
                R
                z
              
              ⁡
              
                (
                
                  τ
                  q
                
                )
              
            
            ⁢
            
              u
              l
            
          
          
        
        2
      
    
  




given in the reference [7] where Rz(τq) is replaced by {circumflex over (R)}Δz(lqTe)(K). The solution matrix U is denoted Ând′ and contains an estimate of the whitened directional vectors of the non-deterministic sources.

Filter

    
    
        The computation of an estimate of the matrix of the directional vectors of the non-deterministic source Ând=Ûs{circumflex over (Λ)}s1/2Ând′
        The extraction of the non-deterministic sources by any spatial filtering of the observations constructed on the basis of Ând.

Processing of the Deterministic Sources

        The construction of the orthogonal projector on the space orthogonal to the columns of Ând: Proj=I−Ând[Ând†Ând]−1Ând†
        The implementation of the SOBI algorithm [3] on the basis of the observations v(t)Δ Proj x(t) to identify the directional vectors of the deterministic sources and extract them.

Separators F2′

    
    


The separators of the family F2′ are second-order self-learning separators implementing the following operations:
Whitening Step


    
    
        The detection of the first order cyclic frequencies of the observations γ by any unspecified detector of cyclic frequencies and the constitution of an estimate,
    
    






  
    
      Γ
      ^
    
    x
    1
  
  ,




of the set,





  
    Γ
    x
    1
  
  ,




of the cyclic frequencies γ.

    
    
        The estimation of the matrix RΔx(0) by {circumflex over (R)}Δx(0)(K) defined by (33) and (32) for l=0 from a given number of samples K
        The detection of the number of sources P from the decomposition into eigen elements of {circumflex over (R)}Δx(0)(K). (all the non-deterministic sources are detected).

Identification Step

        The computation of the whitening matrix of the observations, {circumflex over (T)}, where {circumflex over (T)}Δ{circumflex over (Λ)}s−1/2 Ûs†, with a dimension (P×N), where {circumflex over (Λ)}s is the diagonal matrix (P×P) of the P biggest eigen values of {circumflex over (R)}Δx(0)(K)−λmin I, λmin is the minimal eigen value of {circumflex over (R)}Δx(0)(K) and Ûs is the matrix of the associated eigen vectors. We write z(t)Δ{circumflex over (T)}x(t). (The directional vectors of the non-deterministic sources are orthonormalized).
        The detection of the second-order cyclic frequencies of the observations [9], αε, for ε=−1 and ε=+1 by any detector of cyclic frequencies and constitution of the estimates,
    
    






  
    
      
        Γ
        ^
      
      x
      
        [
        
          1
          ,
          
            -
            1
          
        
        ]
      
    
    ⁢
    
      
    
    ⁢
    and
    ⁢
    
      
    
    ⁢
    
      
        Γ
        ^
      
      x
      
        [
        
          1
          ,
          1
        
        ]
      
    
  
  ,




the sets respectively, Γx[1, −1] and Γx[1, 1], the cyclic frequencies respectively of the first and second matrices of correlation of the observations.

    
    
        The choice of an arbitrary number of pairs (αm, εm) (1≦m≦M) such that, for each of these pairs, at least one source possesses the second-order cyclic frequency αm for a matrix of correlation indexed by εm.
        For each pair (αm, εm), (1≦m≦M):
        
            The choice of an arbitrary number Qm of delays, lmq, (1≦q≦Qm)
            
                For each value of the delay, lmq (1≦q≦Qm),
                The estimation of the matrix of second-order cyclic cumulants of the whitened observations,
            
            
        
        
    
    






  
    
      R
      
        Δ
        ⁢
        
          
        
        ⁢
        z
        ⁢
        
          
        
        ⁢
        
          ɛ
          m
        
      
      
        α
        m
      
    
    ⁡
    
      (
      
        
          l
          mq
        
        ⁢
        
          T
          e
        
      
      )
    
  
  ,
  
    by
    ⁢
    
      
    
    ⁢
    
      
        
          R
          ^
        
        
          Δ
          ⁢
          
            
          
          ⁢
          z
          ⁢
          
            
          
          ⁢
          
            ɛ
            m
          
        
        
          α
          m
        
      
      ⁡
      
        (
        
          
            l
            mq
          
          ⁢
          
            T
            e
          
        
        )
      
    
    ⁢
    
      (
      K
      )
    
  




defined by (31), (29), (32) with the index z instead of x.

    
    
        
        
            The detection of the number of non-deterministic sources P(αm, εm) having the second-order cyclic frequency αm, for the matrix of correlation indexed by εm. This detection test may consist of a search for the maximum rank of the signal space of the matrices
            {circumflex over (R)}Δzε, αmm(lmqTe)(K). Û(αm, εm) denotes the unit matrix (P×P(αm, εm)), obtained by SVD of the previous matrices, whose columns generate the space generated by the whitened directional vectors associated with the sources having the second-order cyclic frequency, αm, for the matrix of correlation indexed by εm.
            Reduction of dimension: We write v(t)ΔÛ(αm, εm)†z(t), with a dimension (P(αm, εm)×1) and carry out a computation, for each delay lmq (1≦q≦Qm), of the matrices
        
        
    
    






  
    
      
        
          R
          ^
        
        
          Δ
          ⁢
          
            
          
          ⁢
          v
          ⁢
          
            
          
          ⁢
          
            ɛ
            m
          
        
        
          α
          m
        
      
      ⁡
      
        (
        
          
            l
            mq
          
          ⁢
          
            T
            e
          
        
        )
      
    
    ⁢
    
      (
      K
      )
    
  
  =
  
    
      
        U
        ^
      
      
        (
        
          
            α
            ⁢
            
              
            
            ⁢
            m
          
          ,
          
            ɛ
            ⁢
            
              
            
            ⁢
            m
          
        
        )
      
      †
    
    ⁢
    
      
        
          R
          ^
        
        
          Δ
          ⁢
          
            
          
          ⁢
          z
          ⁢
          
            
          
          ⁢
          
            ɛ
            m
          
        
        
          α
          m
        
      
      ⁡
      
        (
        
          
            l
            mq
          
          ⁢
          
            T
            e
          
        
        )
      
    
    ⁢
    
      (
      K
      )
    
    ⁢
    
      
        U
        ^
      
      
        (
        
          
            α
            ⁢
            
              
            
            ⁢
            m
          
          ,
          
            ɛ
            ⁢
            
              
            
            ⁢
            m
          
        
        )
      
      
        *
        
          ɛ
          m
        
      
    
  




    
    
        
        
            The self-learned identification of the doubly whitened directional vectors associated with the pair (αm, εm) by maximization in relation to UΔ(u1, u2, . . . , up(αm, εm)), of the criterion
        
        
    
    






  
    C1
    ⁡
    
      (
      U
      )
    
  
  ⁢
  
    
  
  ⁢
  
    Δ
    
      _
      _
    
  
  ⁢
  
    
  
  ⁢
  
    
      ∑
      
        q
        =
        1
      
      Q
    
    ⁢
    
      
        ∑
        
          l
          =
          1
        
        P
      
      ⁢
      
        
          
          
            
              u
              l
              †
            
            ⁢
            
              
                R
                z
              
              ⁡
              
                (
                
                  τ
                  q
                
                )
              
            
            ⁢
            
              u
              l
            
          
          
        
        2
      
    
  




given in the reference [7] where Rz(τq) is replaced by





  
    
      
        R
        ^
      
      
        Δ
        ⁢
        
          
        
        ⁢
        v
        ⁢
        
          
        
        ⁢
        
          ɛ
          m
        
      
      
        α
        m
      
    
    ⁡
    
      (
      
        
          l
          mq
        
        ⁢
        
          T
          e
        
      
      )
    
  
  ⁢
  
    (
    K
    )
  
  ⁢
  
    
      
        R
        ^
      
      
        Δ
        ⁢
        
          
        
        ⁢
        v
        ⁢
        
          
        
        ⁢
        
          ɛ
          m
        
      
      
        α
        m
      
    
    ⁡
    
      (
      
        
          l
          mq
        
        ⁢
        
          T
          e
        
      
      )
    
  
  ⁢
  
    
      
        (
        K
        )
      
      †
    
    .
  




The solution matrix U is written as Ând(αm, εm)′ and contains an estimate of the doubly whitened directional vectors of the non-deterministic sources associated with the pair (αm, εm).

Filter

    
    
        The computation of an estimate of the matrix of the directional vectors of the non-deterministic sources associated with the pair (αm, εm): Ând(αm, εm)=Ûs{circumflex over (Λ)}s1/2Û(αm, εm)Ând(αm, εm)′
        The concatenation of the matrices Ând(αm, εm) for all the pairs (αm, εm), (1≦m≦M). We obtain the matrix (N×P) of Ând of the directional vectors of the non-deterministic sources.
        The extraction of the non-deterministic sources by any spatial filtering of the observations constructed from Ând.

Processing of the Deterministic Sources

        The construction of the orthogonal projector on the space orthogonal to the columns of Ând: Proj=I−Ând[Ând†Ând]−1Ând†
        The implementation of the SOBI algorithm [3] from the observations w(t)Δ Proj x(t) to identify the directional vectors of the deterministic sources and extract them.

Proposed Fourth-Order Separators

Separators F3′

    
    


The separators of the family F3′ are fourth-order self-learning separators implementing the following operations:
Whitening Step


    
    
        The detection of the first-order cyclic frequencies of the observations γ by any detector of cyclic frequencies and the constitution of an estimate,
    
    






  
    
      Γ
      ^
    
    x
    1
  
  ,




of the set,





  
    Γ
    x
    1
  
  ,




of the cyclic frequencies γ.

    
    
        The detection of the second-order cyclic frequencies of the observations, αε, for ε=−1 and ε=+1 by any detector of cyclic frequencies and the constitution of the estimates,
    
    






  
    
      
        Γ
        ^
      
      x
      
        [
        
          1
          ,
          
            -
            1
          
        
        ]
      
    
    ⁢
    
      
    
    ⁢
    and
    ⁢
    
      
    
    ⁢
    
      
        Γ
        ^
      
      x
      
        [
        
          1
          ,
          1
        
        ]
      
    
  
  ,




of the sets respectively,

    
    
        Γx[1, −1]and
    
    






  
    
      Γ
      ^
    
    x
    
      [
      
        1
        ,
        1
      
      ]
    
  
  ,




of the cyclic frequencies respectively of the first and second matrices of correlation of the observations.

    
    
        The estimation of the matrix RΔx(0) by {circumflex over (R)}Δx(0)(K) defined by (53) and (52) for l=0 from a given number of samples K
        The detection of the number of sources P from the decomposition of {circumflex over (R)}Δx(0)(K) into eigen elements. (All the non-deterministic sources are detected).
        The computation of the whitening matrix of the observations, {circumflex over (T)}, where {circumflex over (T)}Δ{circumflex over (Λ)}s−1/2 Ûs†, with a dimension (P×N), where {circumflex over (Λ)}s is the diagonal matrix (P×P) of the P greatest eigen values of {circumflex over (R)}Δx(0)(K)−λmin I, λmin is the minimum eigen value of {circumflex over (R)}Δx(0)(K) and Ûs is the matrix of associated eigen vectors. We write z(t)Δ{circumflex over (T)}x(t). (The directional vectors of the non-deterministic sources are orthonormalized).

Identification Step

        The estimation of the quadricovariance, Qz, of the vector z(t) by the expressions (51), (29), (49), (2) and (32) with the index z instead of x.
        The decomposition into eigen elements of {circumflex over (Q)}z and the estimation of the P eigen matrices Mzi (1≦i≦P) associated with the P eigen values of higher-value moduli.
        The joint diagonalization of the P eigen matrices Mzi weighted by the associated eigen values and the obtaining of the matrix of the whitened directional vectors of the non-deterministic sources Ând′.

Filter

        The computation of an estimate of the matrix of the directional vectors of the non-deterministic sources Ând=Ûs{circumflex over (Λ)}s1/2Ând′
        The extraction of the non-deterministic sources by any spatial filtering of the observations constructed from Ând.

Processing of the Deterministic Sources

        The construction of the orthogonal projector on the space orthogonal to the columns of Ând: Proj=I−Ând[Ând†Ând]−1Ând†
        The implementation of the algorithm JADE [4] from the observations v(t)ΔProj x(t) to identify the directional vectors of the deterministic sources and extract them.

Separators F4′

    
    


The separators of the family F4′ are fourth-order self-learning separators associated with the reference [8] implementing the following operations:
Whitening Step


    
    
        The detection of the first-order cyclic frequencies of the observations γ by any detector of cyclic frequencies and the constitution of an estimate
    
    






  
    
      Γ
      ^
    
    x
    1
  
  ,




of the set,





  
    Γ
    x
    1
  
  ,




of the cyclic frequencies γ.

    
    
        The detection of the second-order cyclic frequencies of the observations, αε, for ε=−1 and ε=+1 by any detector of cyclic frequencies and the constitution of the estimates,
    
    






  
    
      
        Γ
        ^
      
      x
      
        [
        
          1
          ,
          
            -
            1
          
        
        ]
      
    
    ⁢
    
      
    
    ⁢
    and
    ⁢
    
      
    
    ⁢
    
      
        Γ
        ^
      
      x
      
        [
        
          1
          ,
          1
        
        ]
      
    
  
  ,




of the sets respectively,





  
    
      Γ
      x
      
        [
        
          1
          ,
          
            -
            1
          
        
        ]
      
    
    ⁢
    
      
    
    ⁢
    and
    ⁢
    
      
    
    ⁢
    
      Γ
      x
      
        [
        
          1
          ,
          1
        
        ]
      
    
  
  ,




of the cyclic frequencies respectively of the first and second matrices of correlation of the observations.

    
    
        The estimation of the matrix RΔx(0) by {circumflex over (R)}Δx(0)(K) defined by (33) and (32) for l=0 from a given number of samples K.
        The detection of the number of sources P from the decomposition of {circumflex over (R)}Δx(0)(K) into eigen elements. (All the non-deterministic sources are detected).
        The computation of the whitening matrix of the observations, {circumflex over (T)}, where {circumflex over (T)}Δ{circumflex over (Λ)}s−1/2 Ûs\, with a size (P×N), where {circumflex over (Λ)}s is the diagonal matrix (P×P) of the P greatest eigen values of {circumflex over (R)}Δx(0)(K)−λmin I, λmin is the minimum eigen value of {circumflex over (R)}Δx(0)(K) and Ûs is the matrix of the associated eigen vectors. We write z(t) Δ {circumflex over (T)} x(t). (The directional vectors of the non-deterministic sources are orthonormalized).

Identification Step

        The choice of a triplet (αm, εm, lmq)
        The estimation of the matrix of second-order cyclic cumulants of the whitened observations,
    
    






  
    
      
        R
        
          
            Δ
            ⁢
            
              
            
            ⁢
            z
            ⁢
            
              
            
            ⁢
            
              ɛ
              m
            
          
          ⁢
          
            
          
        
        
          α
          m
        
      
      ⁡
      
        (
        
          
            l
            mq
          
          ⁢
          
            T
            e
          
        
        )
      
    
    ,
    
      by
      ⁢
      
        
      
      ⁢
      
        
          
            R
            ^
          
          
            Δ
            ⁢
            
              
            
            ⁢
            z
            ⁢
            
              
            
            ⁢
            
              ɛ
              m
            
          
          
            α
            m
          
        
        ⁡
        
          (
          
            
              l
              mq
            
            ⁢
            
              T
              e
            
          
          )
        
      
      ⁢
      
        (
        K
        )
      
    
  
  ⁢
  
    
  




defined by (31), (29), (32).

    
    
        The computation of a unit matrix Û(αm,εm)=[e1. . . ep,] where (P(αm, εm)≦P) corresponds to the number of non-zero eigen values of
    
    






  
    
      
        R
        ^
      
      
        Δ
        ⁢
        
          
        
        ⁢
        z
        ⁢
        
          
        
        ⁢
        
          ɛ
          m
        
      
      
        α
        m
      
    
    ⁡
    
      (
      
        
          l
          mq
        
        ⁢
        
          T
          e
        
      
      )
    
  
  ⁢
  
    (
    K
    )
  




and ek (1≦k≦P′) are the eigen vectors of





  
    
      
        R
        ^
      
      
        Δ
        ⁢
        
          
        
        ⁢
        z
        ⁢
        
          
        
        ⁢
        
          ɛ
          m
        
      
      
        α
        m
      
    
    ⁡
    
      (
      
        
          l
          mq
        
        ⁢
        
          T
          e
        
      
      )
    
  
  ⁢
  
    (
    K
    )
  




associated with the P(αm, εm) highest eigen values.

    
    
        The choice of the set (αm, ζm, lm1Te, lm2Te, lm3Te).
        The reduction of dimension: v(t)ΔÛ(αm, εm)†z(t) is written, with a dimension (P(αm, εm)×1) and a computation is made of the estimate of the cyclic quadricovariance
    
    






  
    
      
        Q
        ^
      
      
        v
        ⁢
        
          
        
        ⁢
        ζ
        ⁢
        
          
        
        ⁢
        m
      
      
        α
        m
      
    
    ⁡
    
      (
      
        
          
            l
            
              m
              ⁢
              1
            
          
          ⁢
          
            T
            e
          
        
        ,
        
          
            l
            
              m
              ⁢
              2
            
          
          ⁢
          
            T
            e
          
        
        ,
        
          
            l
            
              m
              ⁢
              3
            
          
          ⁢
          
            T
            e
          
        
      
      )
    
  
  ⁢
  
    (
    K
    )
  




of v(t) from





  
    
      
        Q
        ^
      
      
        v
        ⁢
        
          
        
        ⁢
        ζ
        ⁢
        
          
        
        ⁢
        m
      
      
        α
        m
      
    
    ⁡
    
      (
      
        
          
            l
            
              m
              ⁢
              1
            
          
          ⁢
          
            T
            e
          
        
        ,
        
          
            l
            
              m
              ⁢
              2
            
          
          ⁢
          
            T
            e
          
        
        ,
        
          
            l
            
              m
              ⁢
              3
            
          
          ⁢
          
            T
            e
          
        
      
      )
    
  
  ⁢
  
    (
    K
    )
  




and from Û(αm, ζm).

    
    
        The decomposition into eigen elements of
    
    






  
    
      
        Q
        ^
      
      
        v
        ⁢
        
          
        
        ⁢
        ζ
        ⁢
        
          
        
        ⁢
        m
      
      
        α
        m
      
    
    ⁡
    
      (
      
        
          
            l
            
              m
              ⁢
              1
            
          
          ⁢
          
            T
            e
          
        
        ,
        
          
            l
            
              m
              ⁢
              2
            
          
          ⁢
          
            T
            e
          
        
        ,
        
          
            l
            
              m
              ⁢
              3
            
          
          ⁢
          
            T
            e
          
        
      
      )
    
  
  ⁢
  
    (
    K
    )
  




and the estimation of the P(αm, εm) eigen matrices Mvi (1≦i≦P(αm, εm)) associated with the P(αm, εm) eigen values with higher-value moduli.

    
    
        The joint diagonalization of the P(αm, εm) eigen matrices MviMvi†weighted by the associated eigen values and the obtaining of the matrix of the directional vectors of the doubly whitened non-deterministic sources associated with the set (αm, ζm, lm1, lm2, l3): Ând(αm, ζm)′

Filter

        The computation of an estimate of the matrix of the directional vectors of the non-deterministic sources associated with the pair (αm, ζm): Ând(αm, ζm)=Ûs{circumflex over (Λ)}s1/2Û(αm, εm)Ând(αm, εm)′
        The concatenation of the matrices Ând(αm, ζm) for all the pairs (αm, ζm), (1≦m≦M). The matrix (N×P) Ând of the directional vectors of the non-deterministic sources is obtained.
        The extraction of the non-deterministic sources by any spatial filtering whatsoever of the observations constructed on the basis of Ând.

Processing of the Deterministic Sources

        The construction of the orthogonal projector on the space orthogonal to the columns of Ând:
    
    






  
    
      
        A
        ^
      
      nd
    
    :
    Proj
  
  =
  
    I
    -
    
      
        
          
            
              A
              ^
            
            nd
          
          ⁡
          
            [
            
              
                
                  
                    A
                    ^
                  
                  nd
                
                †
              
              ⁢
              
                
                  A
                  ^
                
                nd
              
            
            ]
          
        
        
          -
          1
        
      
      ⁢
      
        
          
            A
            ^
          
          nd
        
        †
      
    
  




    
    
        The implementation of the algorithm JADE [4] from the observations w(t) Δ Proj x(t) to identify the directional vectors of the deterministic sources and extract these sources.

Exemplary Simulation of the Method According to the Invention with the Separator F3′

    
    


FIGS. 6A and 6B respectively give a view, in a level graph expressed in dB, for two non-centered cyclostationary sources with a direction (↓1=50° and ↓2=60°), of the spectrum level after separation of the sources (in using the method JADE bringing into play the set of first-order cyclic frequencies and the two sets of second-order cyclic frequencies) and of the channel formation, after separation of the sources by using the classic estimator (FIG. 6A) and by using the proposed estimator (FIG. 6B).
It can be seen that, with the empirical estimator, the separation works badly because a source is localized at ↓=55° while the sources have angles of incidence of ↓1=50° and ↓2=60°. However, with the proposed estimator, the two sources are localized at 50° and 60° because the channel formation method on the first identified vector finds a maximum at 49.8° and, on the second vector, it finds a maximum at 60.1°. The signal-to-noise+interference ratios at output of filtering of the two sources are summarized in the following table:














Source at θ1 = 50°
Source at θ2 = 60°




















Empirical estimators
SNIR1 = 15.3 dB
SNIR2 = 7.35 dB



Rxa and Qxa



Proposed estimators
SNIR1 = 22 dB
SNIR2 = 22 dB



Rx and Qx










In the following simulation, the proposed estimator is known as an exhaustive estimator. This example keeps the same signal configuration as earlier when the two sources nevertheless have a signal-to-noise ratio of 10 dB. The classic empirical estimator is therefore compared with the exhaustive estimator. The SNIR in dB of the first source is therefore plotted as a function of the number of samples K.
FIG. 7 shows that the exhaustive estimator converges on the asymptote of the optimum interference canceling filter when K→+∞ and that the classic estimator is biased in converging on an SNIR (signal-to-noise+interference ratio) of 5 dB at output of filtering.
Separators F4′
The separators of the family F4′ are fourth-order self-learning separators implementing the following operations:

    
    
        The detection of the first-order cyclic frequencies of the observations γ by any detector whatsoever of cyclic frequencies and the constitution of an estimate
    
    






  
    
      Γ
      ^
    
    x
    1
  
  ,




of the set,





  
    Γ
    x
    1
  
  ,




of the cyclic frequencies γ. The estimation of





  
    
      Γ
      ^
    
    x
    1
  
  ,




can be done as in F1′.

    
    
        The detection of the second-order cyclic frequencies of the observations, αε, for ε=−1 and ε=+1 by any detector whatsoever of cyclic frequencies and the constitution of the estimates,
    
    






  
    
      
        Γ
        ^
      
      x
      
        [
        
          1
          ,
          
            -
            1
          
        
        ]
      
    
    ⁢
    
      
    
    ⁢
    and
    ⁢
    
      
    
    ⁢
    
      
        Γ
        ^
      
      x
      
        [
        
          1
          ,
          1
        
        ]
      
    
  
  ,




of the sets respectively,





  
    
      Γ
      x
      
        [
        
          1
          ,
          
            -
            1
          
        
        ]
      
    
    ⁢
    
      
    
    ⁢
    and
    ⁢
    
      
    
    ⁢
    
      
        Γ
        ^
      
      x
      
        [
        
          1
          ,
          1
        
        ]
      
    
  
  ,




of the cyclic frequencies respectively of the first and second matrices of correlation of the observations.

    
    
        The estimation of the matrix RΔx(0) by {circumflex over (R)}Δx(0)(K) defined by (53) and (52) for l=0 from a given number of samples K.
        The detection of the number of sources P from the decomposition of {circumflex over (R)}Δx(0)(K) into eigen elements. (All the non-deterministic sources are detected).
        The computation of the whitening matrix of the observations, {circumflex over (T)}, where {circumflex over (T)}Δ{circumflex over (Λ)}s−1/2 Ûs†, with a size (P×N), where {circumflex over (Λ)}s is the diagonal matrix (P×P) of the P greatest eigen values of {circumflex over (R)}Δx(0)(K)−λmin I, λmin is the minimum eigen value of {circumflex over (R)}Δx(0)(K) and Ûs is the matrix of the associated eigen vectors. We write z(t)Δ{circumflex over (T)}x(t). (The directional vectors of the non-deterministic sources are orthonormalized).
        The choice of a triplet (αm, εm, lmq)
        The estimation of the matrix of second-order cyclic cumulants of the whitened observations,
    
    






  
    
      R
      
        Δ
        ⁢
        
          
        
        ⁢
        z
        ⁢
        
          
        
        ⁢
        
          ɛ
          m
        
      
      
        α
        m
      
    
    ⁡
    
      (
      
        
          l
          mq
        
        ⁢
        
          T
          e
        
      
      )
    
  
  ,
  
    by
    ⁢
    
      
    
    ⁢
    
      
        
          R
          ^
        
        
          Δ
          ⁢
          
            
          
          ⁢
          z
          ⁢
          
            
          
          ⁢
          
            ɛ
            m
          
        
        
          α
          m
        
      
      ⁡
      
        (
        
          
            l
            mq
          
          ⁢
          
            T
            e
          
        
        )
      
    
    ⁢
    
      (
      K
      )
    
  




defined by (31), (29), (32).

    
    
        The computation of a unit matrix
    
    






  
    Û
    
      (
      
        
          α
          ⁢
          
            
          
          ⁢
          m
        
        ,
        
          ɛ
          ⁢
          
            
          
          ⁢
          m
        
      
      )
    
  
  =
  
    
      [
      
        
          e
          
            1
            ⁢
            
              
            
          
        
        ⁢
        ⋯
        ⁢
        
          
        
        ⁢
        
          e
          
            p
            ′
          
        
      
      ]
    
    ⁢
    
      
    
    ⁢
    where
    ⁢
    
      
    
    ⁢
    
      (
      
        
          P
          
            (
            
              
                α
                ⁢
                
                  
                
                ⁢
                m
              
              ⁢
              
                
              
              ,
              
                ɛ
                ⁢
                
                  
                
                ⁢
                m
              
            
            )
          
        
        ≤
        P
      
      )
    
  




corresponds to the number of non-zero eigen values of





  
    
  
  ⁢
  
    
      
        R
        ^
      
      
        Δ
        ⁢
        
          
        
        ⁢
        z
        ⁢
        
          
        
        ⁢
        ɛ
      
      
        α
        m
      
    
    ,
    
      
        
           
          m
        
        ⁢
        
          (
          
            
              l
              mq
            
            ⁢
            
              T
              e
            
          
          )
        
      
      ⁢
      
        (
        K
        )
      
    
  




and ek (1≦k≦P(αm, εm)) are the eigen vectors of





  
    
      R
      ^
    
    
      Δ
      ⁢
      
        
      
      ⁢
      z
      ⁢
      
        
      
      ⁢
      ɛ
    
    
      α
      m
    
  
  ,
  
    
      
         
        m
      
      ⁢
      
        (
        
          
            l
            mq
          
          ⁢
          
            T
            e
          
        
        )
      
    
    ⁢
    
      (
      K
      )
    
  




associated with the P(αm, εm)) higher eigen values.

    
    
        The reduction of dimension: v(t)ΔÛ(αm, εm)†z(t) is written with a dimension (P(αm, εm)×1) and a computation is made of the estimate of the cyclic quadricovariance
    
    






  
    
      
        Q
        ^
      
      
        v
        ⁢
        
          
        
        ⁢
        ζ
        ⁢
        
          
        
        ⁢
        m
      
      
        α
        m
      
    
    ⁡
    
      (
      
        
          
            l
            
              m
              ⁢
              1
            
          
          ⁢
          
            T
            e
          
        
        ,
        
          
            l
            
              m
              ⁢
              2
            
          
          ⁢
          
            T
            e
          
        
        ,
        
          
            l
            
              m
              ⁢
              3
            
          
          ⁢
          
            T
            e
          
        
      
      )
    
  
  ⁢
  
    (
    K
    )
  
  ⁢
  
    
  
  ⁢
  of
  ⁢
  
    
  
  ⁢
  
    v
    ⁡
    
      (
      t
      )
    
  




from





  
    
      
        
          Q
          ^
        
        
          
            v
            ⁢
            
              
            
            ⁢
            ζ
            ⁢
            
              
            
            ⁢
            m
          
          ⁢
          
            
          
        
        
          α
          m
        
      
      ⁡
      
        (
        
          
            
              l
              
                m
                ⁢
                1
              
            
            ⁢
            
              T
              e
            
          
          ,
          
            
              l
              
                m
                ⁢
                2
              
            
            ⁢
            
              T
              e
            
          
          ,
          
            
              l
              
                m
                ⁢
                3
              
            
            ⁢
            
              T
              e
            
          
        
        )
      
    
    ⁢
    
      (
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    ⁢
    
      
    
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    ⁢
    
      
    
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  ,




    
    
        The choice of a set (αm, ζm, lm1, lm2, lm3) (1≦m≦M) such that, for each of these pairs, at least one source possesses the fourth-order cyclic frequency αm for a quadricovariance indexed by ζm and (lm1Te, lm2Te, lm3Te),
        The set of these values is chosen, for example, so that there is compatibility with the parameters of the second-order cumulants (αm, εm, lmq)
        For each set (αm, ζm, lm1, lm2, lm3) (1≦m≦M)
        The estimation of the cyclic quadricovariance,
    
    






  
    
      
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  ,




of the vector v(t) . We obtain





  
    
      
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  ⁢
  
    
      (
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    .
  




    
    
        The decomposition into eigen elements of
    
    






  
    
      
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and the estimation of the P(αm, εm) eigen matrices Mvi(1≦i≦P(αm, εm)) associated with the P(αm, εm) eigen values with higher-value moduli.

    
    
        The joint diagonalization of the P(αm, εm) eigen matrices MviMvi†weighted by the associated eigen values and the obtaining of the matrix of the directional vectors of the doubly whitened non-deterministic sources associated with the set (αm, ζm, lm1, lm2, lm3): Ând(αm, ζm)′
        The computation of an estimate of the matrix of the directional vectors of the non-deterministic sources associated with the pair (αm, ζm): Ând(αm, ζm)=Ûs{circumflex over (Λ)}s1/2Û(αm, εm)Ând(αm, εm)′
        The concatenation of the matrices Ând(αm, ζm) for all the pairs (αm, ζm), (1≦m≦M). The matrix (N×P) Ând of the directional vectors of the non-deterministic sources is obtained.
        The extraction of the non-deterministic sources by any spatial filtering whatsoever of the observations constructed on the basis of Ând.
        The construction of the orthogonal projector on the space orthogonal to the columns of Ând: Proj=I−Ând[Ând†Ând]−1Ând†
        The implementation of the algorithm JADE [4] from the observations w(t)Δ Proj x(t) to identify the directional vectors of the deterministic sources and extract them.
    
    


The implementation is summarized in FIG. 8 for M=1 (only one set (αm, ζm, lm1, lm2, lm3) is used).
The steps of the method according to the invention described here above are applied especially in the above-mentioned separation techniques, namely the SOBI, cyclic SOBI, JADE and cyclic JADE techniques.
Without departing from the scope of the invention, the steps of the method described here above are used, for example, to carry out the angular localization or goniometry of signals received at the level of a receiver.
For classic angular localization, the method uses for example:

    
    
        MUSIC type methods described in the reference [10] of R. O Schmidt with the matrix of covariance: {circumflex over (R)}Δx(0)(K)
        Cyclic goniometry type methods given in the reference [11] by W. A. Gardner with the matrix of cyclic covariance: RΔxε, αmm(lmqTe)
        Fourth-order goniometry methods described in the reference [12] by P. Chevalier, A. Ferreol, J P. Denis with the quadricovariance: {circumflex over (Q)}xζ, 0m(0, 0, 0)(K)
        Methods of fourth-order cyclic goniometry with the cyclic quadricovariance
    
    






  
    
      
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          ⁢
          
            
          
          ⁢
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          ⁢
          
            
          
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      )
    
  
  ⁢
  
    (
    K
    )
  




Bibliography


    [1] C. JUTTEN, J. HERAULT, <<Blind separation of sources, Part I: An adaptive algorithm based on neuromimetic architecture>>, Signal Processing, Elsevier, Vol 24, pp 1-10, 1991.
    [2] P. COMON, P. CHEVALIER, <<Blind source separation: Models, Concepts, Algorithms and Performance>>, Chapter 5 of the book Unsupervised adaptive filtering—Tome 1—Blind Source Separation, pp. 191-235, Dir. S. Haykins, Wiley, 445 p, 2000.
    [3] A. BELOUCHRANI, K. ABED-MERAIM, J. F. CARDOSO, E. MOULINES, <<A blind source separation technique using second-order statistics>>, IEEE Tran. Signal Processing, Vol. 45, No. 2, pp 434-444, February 1997.
    [4] J. F. CARDOSO, A. SOULOUMIAC, <<Blind beamforming for non-Gaussian signals>>, IEEE Proceedings-F, Vol. 140, No6, pp 362-370, December 1993.
    [5] A. FERREOL, P. CHEVALIER, <<On the behavior of current second and higher order blind source separation methods for cyclostationary sources>>, IEEE Trans. Signal Processing, Vol 48, No6, pp. 1712-1725, June 2000.
    [6] K. ABED-MERAIM, Y. XIANG, J. H. MANTON, Y. HUA, <<Blind source separation using second-order cyclostationary statistics>>, IEEE Trans. Signal Processing, Vol. 49, No4, pp 694-701, April 2001.
    [7] P. CHEVALIER, <<Optimal separation of independent narrow-band sources—Concept and Performance>>, Signal Processing, Elsevier, Special issue on Blind Source Separation and Multichannel Deconvolution, Vol 73, No1-2, pp. 27-47, February 1999.
    [8] A. FERREOL, P. CHEVALIER, <<Higher Order Blind Source Separation using the Cyclostationarity Property of the Signals>>, ICASSP, Munich (Germany), pp 4061-4064, April 1997.
    [9] SV SCHELL and W. GARDNER, “Detection of the number of cyclostationnary signals in unknown interference and noise”, Proc of Asilonan conf on signal, systems and computers 5-7 November 90.
    [10] R. O Schmidt, A signal subspace approach to multiple emitters location and spectral estimation, PhD Thesis, Stanford University, CA, November 1981.
    [11] W. A. Gardner, “Simplification of MUSIC and ESPRIT by exploitation cyclostationarity”, Proc. IEEE, Vol. 76 No. 7, July 1988.
    [12] P. Chevalier, A. Ferreol, J P. Denis, “New geometrical results about 4-th order direction finding methods performances”, EUSIPCO, Trieste, pp. 923-926, 1996.

## Claims

1. An antenna processing method that enables separation or angular localization of cyclostationary signals, comprising:
receiving at an antenna with N sensors, a mixture of cyclostationary and cycloergodic signals from independent sources;
making at least one nth order estimator including:
generating an asymrtotically unbiased and consistent estimator of a cyclic correlation matrix for centered observations of cyclostationary and cycloergodic signals; and
generating an asymrtotically unbiased and consistent estimator of a cyclic covariance matrix for non-centered observations of cyclostationary and cycloergodic signals;
thereby allowing separation, or angular localization of the received signals using the at least one nth order estimator.

2. The method according to claim 1, comprising a step for separating emitter sources of the signals received by using the at least one estimator.

3. The method according to claim 2 wherein the at least one estimator is a second-order estimator.

4. The method according to claim 2 wherein the at least one estimator is a fourth-order estimator.

5. The method according to claim 2 wherein the cyclic frequencies are detected first-order or second-order frequencies.

6. The method of separating non-centered cyclostationary sources, according to claim 1, further comprising
separating the signals using at least one of SSOBI, cyclic SOBI, JADE, and cyclic JADE separation techniques.