METHOD FOR CHARACTERISING EMITTERS BY THE ASSOCIATION OF PARAMETERS RELATED TO THE SAME RADIO EMITTER
The invention relates notably to a passive method of locating fixed or mobile transmitters on the ground. The objective is notably to determine the position of one or more transmitters on the ground on the basis of a mobile reception system.
It also applies in respect of configurations where the transmitters are at altitude and the reception system on the ground.
The associating of parameters also makes it possible to characterize a transmitter by its signal-to-noise ratio. It is also possible to characterize an FH (Frequency Hopping) by its plateau durations, its frequencies of appearance and its direction of arrival. A modulation can also be characterized by identifying for example its amplitude state and phase state.
Figure 1 illustrates airborne location with a mobile reception system, the transmitter 1 is at the position (x0,yo,zo), the carrier 2 at the instant tk is at the position (xkyk,zk) and sees the transmitter 1 at the angle of incidence (tk,xo,yo,zo),A(tk,xo,yo,Zo))- The angles (t,x0,yo,zo) and -(f,x0lyo,z0) evolve over time and depend on the position of the transmitter and also the trajectory of the reception system. The angles (t ,mx0,y0,zo) and (t,x0,yo,zo) are charted with the aid of an array of N antennas that can be fixed under the carrier as shown by Figure 2.
The antennas Ai of the array receive the sources with a phase and amplitude that depend on the angle of incidence of the sources and also the position of the antennas.
Antenna processing techniques generally utilize the spatial diversity of the sources (or transmitters): use of the spatial position of the antennas of the array to better utilize the differences in incidence and in distance of the sources. Antenna processing breaks down into two main areas of activity: 1- Spatial filtering, illustrated in Figure 3, the objective of which is to extract

either the modulated signals 'sm(t), or the symbols contained in the signal (Demodulation). This filtering consists in combining the signals received on the array of sensors so as to form an optimal reception antenna for one of the sources. Spatial filtering can be blind or cooperative.
It is cooperative when there is a priori knowledge about the transmitted signals (directions of arrival, symbol sequences, etc.) and it is blind in the contrary case. It is used for blind source separation, filtering matched to direction of arrival (beam formation) or to replicas, multi-sensor MODEM (demodulation), etc.
2- The objective of estimating the parameters of the transmitters is to determine various parameters such as: their Doppler frequencies, their bit rates, their modulation indices, their positions (xm,ym), their incidences and their direction vectors a (response of the array of sensors to a source with direction etc.
For example, goniometry and blind identification procedures exist in this area:
The objective of goniometry is to determine the incidences 0(t,xm,ym,zm) in 1D or the pair of incidences (t,xm,ym,zm},A(t,xm,ym,zm}) in 2D. For this purpose, goniometry algorithms use the observations arising from the antennas or sensors. When the waves from all the transmitters propagate in the same plane, it suffices to apply a 1D goniometry, in other cases, a 2D goniometry.
The objective of blind identification procedures (ICA) is notably to determine the direction vectors a of each of the transmitters.
The known location techniques according to the prior art generally use histogram techniques to group the parameters together. However, these techniques have the drawback of requiring a priori knowledge about the standard deviation of the parameters in order to fix the stepsize of the histogram.
The invention relies on a new approach consisting notably in judiciously associating the parameters related to one and the same transmitter.

The invention relates to a method for characterizing one or more transmitters and/or one or more parameters associated with a transmitter by using a reception station comprising a device adapted to measuring over time a set of K parameters dependent on the transmitters associated with vectors % representative of the transmitters for 1≤ k≤ K characterized in that
it comprises at least one step of extracting the parameter or parameters consisting in grouping together by transmitter the parameters which are associated therewith by means of a technique of independent component analysis.
The method according to the invention has notably the following advantages:
• It does not require any parameter of settings and no a priori
knowledge about the statistics of the parameters,
• it makes it possible to count the number of transmitters on the basis of
the number of dominant values of a covariance matrix of the
parameters (Rxx): use of the techniques for detecting the number of
sources of the goniometry,
• it makes it possible to identify as many parameter vectors (ή m) as
desired,
• it makes it possible to use the step of associating the parameters of
different nature such as the incidence of the sources, their signal-to-
noise ratio or else their direction vectors,
• it makes it possible to determine the mean incidence of each incident
transmitter on the basis of the measured incidences,
• to perform the mean location of each incident transmitter on the basis
of the measured direction vectors or the measured incidences,
• to extract the phase states of a modulation on the basis of the real and
imaginary parts of the signal of a linear modulation,
• to separate the Frequency Hopping, FH, signals by measuring the

plateau durations and incidences: an FH is characterized by a single incidence and a single plateau.
Other characteristics and advantages of the present invention will be more apparent on reading the following description of an exemplary embodiment given by way of wholly nonlimiting illustration with appended figures which represent:
• Figure 1 an example of airborne location with a mobile reception
system,
• Figure 2 an array of 5 antennas,
• Figure 3 a diagram of spatial filtering by beam formation in one
direction,
• Figures 4, 5 and 6 a numerical example of the use of the method
according to the invention.
The following example is given in conjunction with Figure 1, comprising a transmitter 1 to be located by using an aircraft 2 equipped with devices allowing the measurement of parameters associated with the transmitters, and a processor suitable for executing the steps according to the invention.
In the presence of M transmitters, a location system measures, for example over time, a set of K parameters (representative of the transmitting sources) characterized by the vectors ή k for k≤K. The vectors are for example composed of the azimuth and of the signal-to-noise ratio SNRk of one of the transmitters at the instant tk:ή k=[0k SNRk]T where (T ) denotes the transpose of a vector.
This vector can also be composed of the direction vector a(θk) of one of the sources and of its signal-to-noise ratio: ήk=More generally the kth measurement is marred by an error and is associated with the mth transmitter in the following manner:

where ek is the noise vector associated with the kth measurement and the parameter vector associated with the mth transmitter. The invention consists notably in extracting the M vectors associated with a transmitter m in the midst of the K measurements Procedure for identifying the parameter vectors of the principal sources
By virtue of the location system with which the aircraft is equipped or more generally a system performing parameter measurements, K measurements are available. The prime objective of the invention is to
identify the M vectors associated with the M incident transmitters.
For this purpose, the method comprises a first step consisting in
transforming the measured vectors into vectors of larger dimension.
For a goniometry system in azimuth where the vector % is
equal to (azimuth and signal-to-noise ratio), the transformation step consists in performing the following bijective transformation:
and in elevation where the vector representative of the set of parameters K measured for the M transmitters the transformation step consists in
performing the following bijective transformation:
For a goniometry system in the presence of FH signals whose incidences 9k and plateau durations Tk have been measured, the vector may be written: The transformation step for consists in performing the following bijective transformation:
For a system seeking to extract the phase states of a transmitter from the signal x(kT) of the BPSK, the vector % may be written:
Where (z) and 3(z) denote the real and imaginary parts of the complex z and T is the symbol rate. In the case of a BPSK transmitter with 2 phase states, M=2 states are present, such that:

The determination of the vectors and will make it possible to deduce the phase rotation of the BPSK. In this case it is possible to construct the following vector
The length of the vector determines the maximum number
of identifiable transmitters. In an application of goniometric type that estimates the incidences , it is possible to say for example that this maximum number will not exceed the number of sensors of the array that enabled the goniometry.
On the basis of the vectors the method
thereafter calculates the following covariance matrix:

where denotes the Kronecker product such that and the conjugate transpose. This matrix may also be written:
1where is a noise matrix and pm is the number of vectors associated with the vector ,
the matrix Rxx is the covariance matrix of the K observations According to equation (9) it reduces to the covariance matrix of the signatures of the M transmitters. Knowing that the signatures are all different since the transmitters are associated with different parameters, the principal components of the matrix Rxx (eigenvectors associated with the M largest eigenvalues) define the same space as the M signatures of the transmitters: Rxx is completely related to the vector space of the M signatures of the transmitters.
The rank of the matrix Rxxs thus equal to the number of
transmitters M. This rank can be determined from the eigenvalues of this
matrix.
Thus in the presence of a vector of dimension N, the matrix Rxx is of
dimension N2x N2 and it is then possible to identify at most N2 transmitters.
The method thereafter comprises a step of identifying the
transformed vectors on the basis of Rxx so as thereafter to deduce
therefrom the parameter vectors of each of the transmitters.
For this purpose the first operation consists in decomposing the
matrix Rxx into eigenelements to obtain its eigenvalues.
On the basis of the eigenvalues of the matrix, it is possible to determine the number of sources M by applying, for example, the procedure described in reference [4] or any other "enumeration" procedure which makes it possible to count the number of principal components of the matrix
Rxx. This number in the given example is related to the number M of
transmitters.
On the basis of the M eigenelements associated with the largest
eigenvalues it is possible to determine the square root of the matrix Rxx: such
here diag{..} is a diagonal matrix composed of the elements of {..}, Es and are composed respectively of the eigenvectors and eigenvalues of associated with the M largest eigenvalues: The columns of the matrix B are composed of the signatureseach of the transmitters.
The matrix U is unitary (UH U=lw where IMis the identity matrix of
dimension NxN). Knowing that the columns of the square root RXX1/2 are in
the same space as the columns. of the matrix B, the matrix U is a change of basis matrix. The matrix U is moreover unitary since its columns are mutually orthogonal vectors. Subsequently in the description the method will use this orthogonality property to identify the matrix U. For the identification of U the method will moreover use the redundant structure of B which is related to the Kronecker product ®.
The determination of U is done for example by utilizing the redundant structure of the matrix B, i.e.:
UThe columns of each matrix rn are in the same vector space as the sought-after signatures of each of the M transmitters. The matrices rn differ by change of basis matrices which are equal to a diagonal matrix to within the sought-after unitary matrix U. These properties of the matrices depend on the redundant structure of the matrix B. Accordingly the following matrices :
have the matrix as matrix of eigenvectors denotes the pseudo inverse
such that he method uses the unitary character of the matrix to identify it: The
columns of U are orthogonal vectors.
Under these conditions, to determine the unitary matrix U, the joint
diagonalization of the JADE procedure described for example in reference [5]
of the following matrices or of any other procedure known to the person
skilled in the art is performed:
Once the matrix U has been estimated, it is possible to deduce
therefrom the matrix B to within an amplitude by performing according to
relation (10):
Knowing that the mth column of the matrix B may be written transformed into the following matrix where the are vectors of dimensions .

Knowing that the 1st component of is equal to is deduced from by taking the singular vector em of Bm associated with the largest singular value and by performing where em (1) is the first component of the vector em. The method performs this normalization since the vectors are always constructed with a first component equal to 1. These steps of constructing and normalizing make it possible to resolve the phase ambiguity of the singular vectors em.
The M principal vectors are deduced from the transformed vectors since is bijective.
Having identified the M principal vectors , the method compiles for example statistics on the components of each of the vectors. For an application of location type this step makes it possible notably to give in addition to the mean position of the transmitter, an error bracket for the estimation of the position.
For example for the vecto the method determines the statistics of the azimuth 9m (bias and standard deviation) so as to give the value of the azimuth within a bracket.
The first step consists in determining the set of vectors
associated with the transmitter having mean vector . Knowing that:
Thus all the components of 8 are zero with the exception of the mth which equals 1. It should be noted that the filtering matrix is a separator of the transmitters: By applying this filter to a signature of the mth transmitter only the component associated with this transmitter is nonzero.
Thus to determine the set to which the vector belongs, the
method uses the property of equation (17) (where the objective of the matrix
is to separate the transmitters) by calculating the vector of equation (14) which ought to be close to 8m when is associated with the
mth source.
The vector % belongs to the set of the mth source if the mth component
ßk(m) of largest modulus satisfies: |ßk(m)|. ,
The threshold a is chosen close to 1 (typically a =0.9).
The steps of the method of constructing the sets in the presence of K
vectors comprise for example the following steps:
Step R.1 k=1 and initialization of the M sets to 0 (empty set),
Step R.2 Calculation of the vector ßk using equation (18).
Step R.3 Search for the component such that: for then Step R.5
Step R.6 If k ≤ K return to step R.2.
Once the M sets have been determined,
the method calculates a statistic of the components of the vector for
example as Mean Square Error or MSE.
The Mean Square Error (MSE) of component of may be written:

where card is the cardinal of the set and meanm(/) is the mean value that has to be close to .
In the example of Figures 4 and 5 the vector and the function satisfy:
The following Figure 4 shows the distribution of the goniometry plots in the space:and Figure 5, the evolution of these plots over time: The method gives for the M=2 sources:
In Figures 4 and 5, the mean values estimated by the method appear as a solid line. In the example of Figure 6 the vector and the function satisfy:
Figure 6 the coefficients are represented dotted
and the coefficients as a solid line, where the direction vector a(sm) has been deduced from the M vectors sm estimated by the method. The vector 1 is composed of 1s. The method has detected M=3 categories of sources.
Figure 6 shows that two of the categories are permanently present while the last is present in a much more sporadic manner.
These examples show that the method is applied in an independent manner to the type of parameters of the sources.
Without departing from the scope of the invention, the method can be applied in respect of directions of arrival direction vectors a or else signal-to-noise ratios SNRm. Bibliography
[1] L.ALBERA, A.FERREOL, P.CHEVALIER and P.COMON. GRETSI 2003, Paris, September 2003, "ICAR, un algorithme de ICA a convergence rapide, robuste au bruit" [ICAR, a fast convergence, noise-robust ICA algorithm].
[2] L.ALBERA, A.FERREOL and P.CHEVALIER. ICA2003, Nara (Japan), April 2003, Sixth order blind identification of undetermined mixtures (SIRBI) of sources.
[3] P. COMON, Signal Processing, Elsevier, April 1994, vol. 36", No.3, pp 287-314, Independent Component Analysis, a new concept~?.
[4] O.MICHEL, P.LARZABAL and H.CLERGEOT Test de detection du nombre de sources correlees pour les methodes HR en traitement d'antenne [Test for detecting the number of correlated sources for HR procedures in antenna processing]. GRETSI 91 in Juans les Pins.
[5] J.F. CARDOSO, A. SOULOUMIAC, IEE Proceedings-F, Vol.140, No.6, pp. 362-370, Dec. 1993. Blind beamforming for non-gaussian signals.

CLAIMS
1 - A method for characterizing one or more transmitters and/or one or more
parameters associated with a transmitter by using a reception station
comprising a device suitable for measuring over time a set of K parameters
dependent on the transmitters associated with vectors representative of
the transmitters for characterized in that it comprises at least one step of extracting the parameter or parameters consisting in grouping together by transmitter the parameters which are associated therewith by means of a technique of independent component analysis.
2 - The method as claimed in claim 1 characterized in that the step of
associating the parameters for each transmitter M comprises at least the
following steps:
Step No.1 Transforming the vectors % representative of the set of the K
parameters for the M transmitters into vectors by using a objective function and where the 1st component of f(%) being equal to 1,
Step No.2 determining a covariance matrix R from the vectors and
decomposing it into eigenelements, so as to obtain its eigenvalues,
Where R(V is a matrix related to the vector space of the M signatures
of the transmitters.
Step No.3 Calculating the rank M of the matrix Rxx from its eigenvalues
found in step 2,
Step No.4 Calculating a square root of R^ with its M dominant
eigenelements: where Es corresponds to the eigenvectors
and As to the M largest eigenvalues,
Step No.5 Deducing from the matrices according to
Where the columns of each matrix define the same space vector as the
signatures of the M transmitters,
Step No.6 The unitary matrix U is identified by joint diagonalization of the
Step No.7 Determining the vectors from the columns bm of the matrix
Rxxl1/2U where U is the unitary matrix; Transformation of column bm into the
matrix Bm according to Bm = [bm1 ... bmN] and extraction of
this matrix from the singular vector em associated with the largest singular value so as to perform (1) the component of the vector
Step No.8 Applying the inverse transform of the function f() so as to deduce therefrom the vectors
3.- The method as claimed in claim 2 characterized in that it comprises at
least one step of evaluating the statistics of the components of the vectors found.
4 - The method as claimed in claim 3 characterized in that it comprises a
step of evaluating the mean and the standard deviation of the incidences of
each of the transmitters.
5. - The method as claimed in claim 3 characterized in that it comprises at
least the following steps:
• determining the M sets of vectors % associated with the principal
vector
• determining the statistics of the components of the vector from the
set by using a Mean Square Error procedure.