FR2955950A1 - Central parameter e.g. middle or average parameters, determining method for e.g. Doppler radar, involves comparing calculated distance with threshold value, and obtaining presence of target if calculated distance exceeds threshold value - Google Patents

Abstract

The method involves generating anchorage points of a complex signal (13) received on a set of sensors of a radar receiver. Distance between one of the anchorage points of the complex signal received on the set of sensors of the radar receiver, and a middle or average point associated to the complex signal is calculated. The calculated distance is compared with a threshold value defined by the need of an autoregressive identification algorithm application. The presence of a target is obtained if the calculated distance exceeds the threshold value. An independent claim is also included for a device for transmitting and receiving a signal from a radar.

Description

METHOD AND DEVICE FOR DETERMINING A

CENTRAL PARAMETER ASSOCIATED WITH COEFFICIENTS

The present invention relates to a method executed in particular within a receiver comprising one or more complex signal reception sensors, said method making it possible to differentiate within said complex signal information that is useful for controlling a complex signal. a signal processing upstream of the information process considered as not useful.

1 o In the case of a pulse radar application, the signal received by the receiver after complex demodulation produces a signal sampled on the sine and cosine channels, which constitute a series of complex samples, each sample corresponds to a small domain of the radar range still called distance box.

In the case of a method of smoothing a signal, the idea will be to eliminate the useful information detected from the majority noise.

The method and the associated system according to the invention apply for example for Doppler radars.

The process according to the invention is applicable more generally

20 to embedded real-time systems due to the lack of computation required. In the case of a radar application, the primary objective of a radar is to be able to detect targets in an extended local range in the radar-target radial distance dimension and in the angular dimension. This beach is

25 cut into distance boxes which each contain a number N of shots on the goal (which can be very limited), that is to say of complex measurements (YI, Y2,

YN) from the receiver. In the rest of the description, these N measurements are called sample.

These representative echo measurements contain both the target information and the environmental noise or clutter information. The boxes containing the targets are classically isolated by a threshold adapted to local clutter (TFAC). Conventional radar signal processing methods are based on techniques known as TFAC for "Constant Alarm Fault Rate".

In general, a TFAC method consists in subtracting from the signal an average of the estimated signal environment on a sliding window. Specifically, the average of the atmosphere is subtracted from the logarithm of the amplitude of the signal output Fourier transform filters or FFT ("Fast Fourier Transform") arranged in a bench, then the difference is compared to a threshold on each channel TFF. A logical OR makes it possible to decide a detection if the threshold is exceeded on one of the FFT channels. Unfortunately, for Doppler waveforms with short bursts of less than 16 pulses, for example, the use of an FFT filterbank suffers from several disadvantages, including a low resolution. First, if the Doppler of a target is between two filters, then its power is equally distributed between the two filters. In addition, an intense clutter like soil clutter pollutes all the filters because of the important side lobes. Thus, with a sample of 8 pulses and in the presence of a powerful ground clutter, which theoretically should disturb only the FFT filter corresponding to zero velocity, all 8 FFT filters can be disturbed and the Doppler resolution of each Bench filter is weak to discriminate two near frequencies. Another objective of a radar, for purposes such as mapping, is to distinguish and separate locally homogeneous areas in a globally inhomogeneous clutter. This is particularly the case in doppler weather radar where it is a question of analyzing atmospheric disturbances and distinguishing within them homogeneous zones of different spectral characteristics. This is also the problem in the case of maritime surveillance applications where it is necessary to separate homogeneous clutter areas in an inhomogeneous set and where weak targets must be detected in the local clutter. The general problem is therefore to determine the average characteristic of a collection of P samples each consisting of N complex measurements, either to isolate a remarkable sample, or to eliminate a disturbing element, or to classify the set in separate batches. This can be applied in other areas than radar processing, in all cases where the measurements are in the form of a complex signal, which can also be a real signal. This problem has been addressed for several years in the world of signal processing, both for radar applications and for all kinds of similar applications. One of the solutions explored so far is the search for a median in the spaces of correlation matrices, ie symmetric definite positive matrix matrices placed within the framework of the geometric information theory. The methods thus proposed make use of the diagonalization, the extraction of square roots, the product and the exponentiation of the square matrices with the problems of conditioning that it poses, iterated on all the elements of the collection of samples of which one look for the median and iterated on the number of steps necessary for the convergence of the algorithm. In addition, the empirical correlation matrices are always tainted by high uncertainty on small measurement samples. These algorithms are difficult to apply to real-time systems embedded in the current state of computer technology.

The invention proposes in particular to avoid some of the aforementioned drawbacks of the TFAC methods, which essentially have their origin in the use of FFT filters. The object of the present invention is therefore to determine the average or median characteristic of a collection of P samples, each consisting of N complex measurements, either to isolate a remarkable sample 4 or to eliminate a disturbing element or to classify the sample. set in separate lots. The present invention proposes a simple method for calculating the median or the average, and more generally a central element representative of the measured samples, by a gradient algorithm from Doppler radar measurement sets of size N, when these data are the subject of a self-regressive identification (AR) by the application in particular of the BURG algorithm (trellis filtering). The method also applies to obtain an algorithm of the type Multiple Signal Classification (clusering). The subject of the present invention relates to a method for determining a central parameter, such as the median or the mean, on a set of identification results associated with a complex signal received on a set of sensors of a receiver, leading to to p collections E1, ... EP of M-tuples of complex numbers which correspond to m measurements of the complex signal, said measurements being taken from the receiver Ek = LXk1, Xk 2, ... Xk, MJ k = 1, .. Said method comprising applying an N-1 lattice auto-regressive identification algorithm to said complex signal, providing P points {W1, W2, ... Wp} or points anchoring which are invariant during the implementation of the steps of the method according to the invention, characterized in that it comprises at least the following steps: Each point Wk is a consisting of a sequence of N coefficients of the form Wk = (Wk, Wk,, Wk,, Wk ') in which Wk = 1og (1 E II X k • j II2) and (Wk, ..., Wk Wk-') t the first N-1 complex reflection coefficients from said complex autoregressive identification by implementing a lattice identification algorithm applied to the set of numbers, (Xk ,,, ..., Xk.j, ..., Xk.M) said method is applied to a general set called H in which said anchor points are located and said general set H is the set 2955950 following the form Wk - (Wk, Wk, ..., Wk, .., WkN- ') of N coefficients whose first coefficient is a real number and the following coefficients of complex numbers of modulus strictly less than 1, said method also comprises the application of a gradient descent algorithm which operates on a function defined on the general set H, said function being obtained, either as the sum of the distances of the anchor points at any point in the general set when the process aims at calculating a median, ie as the sum of the squares distances from the anchor points at any point in the general assembly when the method is for calculating an average, and in that: said method is characterized by providing a loop for minimizing the function herein; said gradient algorithm comprising a finite number of steps in which is calculated at each step of said algorithm a gradient vector pointing to a point of convergence which is the result targeted by the method, and that each The gradient algorithm comprises at least the following steps which transform a point z of the general set H from the preceding step of the gradient algorithm into a new point z of the general set: Phase 0: take the point z of the general set resulting from the algorithm in the previous step j-1, for the first step the initial z point will be initialized to any anchor point, Phase 1, (13): to perform on each of the p Anchor points a generalized direct Mebius rotation of z-center to obtain a set of modified P-points corresponding to step j, W'm, Phase 2, (14): for each of the modified points W'm, determine the vector Vm tangent to H at 0, Phase 3, (15a, 15b): determine for each of the modified points, the unit vector Um tangent to H at 0, 6 2955950 Phase 4, (16): calculate the convex linear combination unit vectors weighted by a coefficient am, Phase 5, (17): multiply the linear combination S by a real number corresponding to the increment of the gradient in step j in order to obtain a vector 5 tangent to H at 0, Phase 6, (16): wrap the vector A on the Riemannian manifold H along the geodesic from Z '= 0 to obtain the Z' point of H. At Do, 13.11 .th (1011), ... , IQN 1l .th (~ oN-lh Or: Z '= [AQ, Al 03. (IAlI), ..., ON l Y'OION_l1)] with 0o (x) = th (x) six 2 n / a and th (e ~ six 5 eo x its 15 Phase 7, (19): perform the rotat inverse ion of Mobius of center z to obtain the new candidate z at step j + 1, Phase 8, (20): compare the value of the norm of S with a threshold value and if this value is lower than the threshold value then stop the iteration and define the central element (middle or middle point) as the last determined point, otherwise, do j = j + 1 and return to phase 0, said method comprising a step in which knowing the median point or the average point according to the object of the method, the distance between a point of the set H and the median point or the average resulting from the preceding steps is calculated and then this distance is compared with a threshold value defined by the need of the application, and if this distance exceeds the threshold it is decreed presence of a target. The threshold value can be defined, for example, experimentally. The invention also relates to a device for transmitting and receiving signals for determining a central parameter, such as the median or the average, on a set of identification results associated with a complex signal received on a set of sensors. a receiving module characterized in that it comprises at least one processor adapted to perform the process steps set out above.

The use of this doppler radar method is of several types:

• perform a TFAC to separate the useful targets clutter assimilated to the median amount evaluated on a locally homogeneous area in overall inhomogeneous clutter; • Separate and compare locally homogeneous clutter areas, estimating a median quantity and calculating the distance between them.

This problem is particularly that of doppler weather radar which must separate the various homogeneous zones of a generally inhomogeneous atmospheric disturbance. The method of the invention provides a fast algorithm for determining a midpoint or, if necessary, averaging a set of auto-regressive identification results. A measure of the distance between the

20 median and each of the samples or between several medians is then available.

The goal being, for example:

to smooth the signal by an average value or more effectively a median value,

25 to discard abnormal elements,

to extract remarkable elements,

to compare and classify subsets of different characteristics. Other features and advantages of the present invention will appear better on reading the following description of examples of different implementations of the invention given by way of illustration and not limited to the appended figures which represent:

FIG. 1A, a block diagram of an exemplary system that can implement the method according to the invention in the case of a radar application, FIG. 1B the temporal representation of the signals transmitted and received by an antenna of the system, the FIG. 1C a diagram of the steps according to the prior art and FIG. 1D, a diagram of the steps according to the invention,

FIG. 2A, a block diagram of the steps of the method implemented for a radar application, and FIG. 2B a block diagram of the steps of the method according to the invention in the one-dimensional case,

FIGS. 3 and 4, the result of the process according to the invention in the one-dimensional case,

FIG. 5, the result of the application of the process according to the invention in the case of Gaussians, and

FIG. 6, a block diagram of the method according to the invention applied for correlation matrices. In order to better understand the object of the present invention, the following description will include reminders of Riemannian metrics and other elements useful for understanding the problem. The explanation given in the context of a radar application initially for illustrative and non-limiting purposes will be generalized to other applications.

In summary, the present invention proposes a simple method of calculating the median by a gradient algorithm from N-doppler radar measurement sets, when these data are the subject of a self-regressive identification (AR). by applying the BURG (trellis filtering) algorithm, for example, or any other algorithm having a similar function.

This median is defined in the framework of the geometric information theory which adapts the Riemannian geometry language to the collection of the reflection coefficients obtained by the BURG algorithm.

The present invention also proposes to calculate an average, defined in the same context, the steps of these two variants of implementation will be described below.

The present invention is of general application in complex value signal processing, that is to say for complex signals measured by sensors some examples of which are described below.

Nature and modeling of the signal in the case of a radar application. The useful sign

At the output of the receiver, a Doppler radar analyzing a signal coming from a set of K targets operating at speeds Vk takes the form of a sum of noisy complex sinusoids sampled at the period T: K Sn = Ak.ei, 2 .e.fdk.nT + Bru ta k = 1 fdk = 22k is the Doppler frequency

n = 1, ..., N (At wavelength of the carrier) The number of N measurements available (number of shots at goal) on a modern radar can be of the order of 8 to 16.

The maioritary sign

The majority signal in most distance boxes is in the form of a correlated noise (clutter). The useful signal is rare, the parasitic signal is the majority, it must be possible to separate them.

The analysis The analysis aims to identify the eigenfrequencies of the signal which are the Doppler frequencies fdk to determine the velocities Vk. The autoregressive identification (AR) is on the one hand adapted to the periodic signals, since in the absence of noise, a finite sum of periodic signals, verifies a recurring equation of the AR type, and is on the other adapted to the correlated noise signals which one also wishes to make the spectral analysis. The estimation of the adapted AR model is done within the framework of the present invention by a lattice analysis according to the BURG algorithm improved by a regularization. This analysis, which is interpreted as a Hilbert-Schmidt orthogonalization method, substitutes for the calculation of the coefficients of the AR model, the computation of N-1 complex coefficients called reflection coefficients, necessarily of modulus strictly smaller than 1, which allow by a calculation known to those skilled in the art to find the spectrum and the eigenfrequencies of the signal. Modeling of the radar signal The majority signal takes the form of a correlated noise. It is therefore natural to consider that: • each of the N-tuples of measurements is the realization of a complex Gaussian random vector of dimension N, the distribution of which is characterized by a mean vector and a correlation matrix (complex and hermitian). ). • the signal is stationary over this horizon of a few measurements. This amounts to saying that the correlation matrix is of Toeplitz 20 form (constant on each of the subdiagals). It will be assumed in the following that the average vector, if it is not zero, is systematically subtracted from the signal. Density of Gaussian complex random vector ZN of correlation matrix RN: P (Z \ T / RA) - (ZN mnRAT-1) (Ziv_mNy.NR ZN = ZOO Z1, ..., Zk; , ZN -1Y Zi ECT MN = MO, M1, ..., Mk; ..., MN -1 m EC 11 2955950 We always assume mN = 0 The correlation matrix RN is hermitian complex of Toeplitz form: CO Cl. CN-1 5 CN = CN-1 CN CN-2 CO

Median and mean of P complex Gaussian random variables. In this section, it is a question of defining the notion of mean or median of a collection of P complex random variables Gaussian complex centered N-dimensional and to establish a simple and efficient algorithmic estimator allowing the effective calculation of this median and this average on an achievement. We are thus confronted with a space of complex Gaussian random variables characterized by a set of complex parameters which makes this space an abstract manifold of dimension N (manifold = curved surface in the general sense). It is natural to place oneself within the framework of the geometric information theory, and for that, to define a relevant Riemannian structure on this manifold (local scalar product defined on each of the tangent vector spaces whose matrix depends in a differentiable way parameters). This framework makes it possible to speak of distance between 2 points which are here each, let us remember it, a Gaussian random distribution. We can then speak of a center point or midpoint of a collection of P points of the variety. Barycenter A center point Mo will be a local minimum of the function defined as the sum of the squares of the distances from the point Mo to the points P of the initial collection called anchor points. This minimum will be unique under reasonable conditions of dispersion of the P points on the variety.

On the other hand, on a circle there is no centroid of N points distributed in a star. However, this type of problem does not arise on the variety considered in the present patent application.

Median domain

A midpoint Me will be a local minimum of the function defined as the sum of the distances from the point Me to the P anchor points. The midpoints are rarely unique, they form a closed domain within the variety. For example, the median domain of 2 real numbers is the segment that joins the 2 numbers.

The gradient of the functional to minimize

Let 8 (a, x) be the geodesic distance between an anchor point a and the point x.

On a Riemannian variety, the function b is differentiable outside the point "a". Its x-gradient is a unit vector û (x) in the sense of the local metric at the point x, tangent to the geodesic joining the 2 points and directed from a to x. We can then reach the average by a gradient descent algorithm aimed at canceling the gradient vector of the global function: PPN d (x) = E 8 (xi, x) 2 Vd (x) = E 8 (xi, x) .i (x) l gk1 (x) .Uik.ull = 1 i = 1 1 = 1 k, 1 = 1

Similarly, a point of the median domain will be obtained by a gradient algorithm aimed at canceling the gradient vector of the function: PPN d (x) = E 8 (xi, x) Vd (x) = E iii (x) I gk1 ( x) .uik.u11 = 1 1 = 1 i = 1 k, 1 = 1 This assumes to compute the geodesics as well as the tangent vectors. Theoretical preamble

Variety definition and setting

The present invention is based on the self-regressive (AR) analysis of each of the random vectors, which analysis is in the form of the "lattice" analysis of which the BURG algorithm provides an estimator. A known variant of the prior art makes it possible to introduce a "regularization" in the BURG algorithm that reduces the noise sensitivity of the coefficients of the model with the highest rank of the model. This lattice analysis is nothing more than a Hilbert-Schmidt orthogonalization method exploiting the stationarity of the N random measurements of the sample. These are the lattice formulas well known to those skilled in the art and recalled in the appendix which result in determining a set of N-1 complex parameters of modulus strictly less than 1, referred to as reflection coefficients, crl,. and at a Nth parameter Po which is the common variance of each of the random variables. = [Po, / 11, ..., f IN -1] = [el, 92, ..., ON] PoE + ù {0} ED = {zllzl <1} Po = common variance, 15 p; = i th reflection coefficient of the AR model. These N parameters make it possible to define a differentiable manifold structure that becomes a Riemannian variety when a metric tensor is defined on the tangent bundle which is the set of tangent vector spaces taken at each of the points. 20 Riemannian structure. The Riemannian structure selected is Kählerian. The entropy t of the complex N-dimensional centered Gaussian variable is a concave function whose opposite constitutes a quite satisfactory Kälher potential. To say that the metric is kaherian is to say that the metric tensor on the tangent space at a point O will be the Hessian (second derived matrix) of this potential at this point. 2 (D (Po, pi, uN -1) = Ek 11 (N ù k) .1n [1- I, uk)] + N.ln (rePo) 2955950 The metric tensor (Hessian of the potential) in the plane tangent to the point O is deduced from the elementary vector length (dPo, dul, ..., d, uN_I) which is written as follows: I2 ds 2 = N. / d P o 1 + E - k) . The description of the method according to the invention is simplified by changing in the following the variable Po in In Po 0 = [q = ln (Po), QUI, ..., UN -1] T = [BI, 82, ..., ON] 10 Bi being complex numbers whose modulus is less than 1 2 ds2 = N.dg2 + 1k11 (N ù k). d4 ('-k i ds2 is the square of the length of the elementary vector: (dq, d1u1, ..., duN I) 15, uk is in this case used to designate a reflection coefficient In what follows H will designate the set in which evolves this vector of parameters which is at once the differentiable manifold and the open which constitutes the map parameterizing this manifold: H is the Cartesian product of the set R of the real numbers by N-1 copies 20 of the set D which is the open disc unit of the set of complex numbers D = {z complex such that lzI <1} In continuation D is called disc of Poincaré OEH = RxDN-I 5 25 The geodesics of the Riemannian variety H.

The rotation of Môbius in D.

The search for geodesics is considerably simplified when we notice that in D with the Riemannian manifold structure 2 defined by the element of length ds2 = IdzI, the Mobius rotation of (1-IzI) center "a" contained in D, which at the point z of D associates the point R [a) (z)

with R {a] (z) = z û a, is a bijection of D in D and that this 1

transformation is also an isometry for the Riemannian structure. That is, it leaves the distances unchanged. It transforms geodesics into geodesics, and sends the center "a" of rotation to the point 0 origin of the complex plane.

The advantage that results from using this transformation is that:

• the geodesic connecting any 2 points a and z of D is

transformed into a geodesic connecting the center of the complex plane to the point R [a] (z) of D. • the geodesic thus transformed is the line segment in the complex plane connecting 0 to the z transform, ie it is carried by a radius of the disc.

The generalized rotation in H.

It is then possible to define the generalized rotation T of center A which sends H in H: TA (0) = [q ù qo, R [a1] (z1), R [a2] (z2) ..., R [ aN - 1] (zN - 1)] 0 = [q, z1, z2, ..., zN - A = [go, a1, ..., aN I] a, ED i = 1, .. ., N-1 with R [a] (z) = zu which transforms component to component the elements of H and sends the center A of the rotation to 0 = element of H. The first component which is a real number is translated. The following components that are in complex numbers of D undergo an elementary Mebius rotation. A vector tangent to the differentiable manifold H at point 0 is sent component to component on a vector tangent to H at TA (0) isometrically on each component. The partial ds2 is conserved, as the global ds2 is a sum weighted by constant coefficients of the partial ds2, the global ds2 is itself conserved. T is therefore an isometric transformation of H. Geodesics from A are transformed into geodesics from 0 to H. However, unlike what happens in D, the geodesics in H are not straight lines. Only the projections on each of the complex components are segments of straight lines in D. The metric of H in 0. Let us specify the notion of unit vector in 0. In the vector space tangent to H in 0, the metric is reduced to: ds2 = N.dq2 + EkNH '(N û k). dzkI2 The tangent plane at 0 to H is then a Euclidean vector space with the scalar product deduced from the element of length above. Indeed, the metric is Euclidian with the weighting close and the vector space is a real vector space of dimension 1 + 2 * (N-1).

Equation of the geodesic in H between 0 and any point W of H. 2 ds2 = ao.dro2k ~ Idzkl2 ao = N ak = Nuk -IZkI k = 1, .. N-1 17 2955950 Move in polar coordinates (r module of z and e its phase) on a path joining the point 0 to the point W = (Wo, W1, .. Wk, .., WN -1) T in H: ds 2 = ao.dro 2 + ~ N-lak drk '+ n2 .d 612 k-1 2 (1ûrk2) 2 = dxk2 + dyke = drk2 + rk2.dOk2 withxk = cos (Bk) yk = sin (Ok) dzk dzk = dxk + i.dyk 5 The integral that must be minimized to obtain the geodesic is written symbolically: 2ù W 2 N-1 drk '+ rk2.d8k2 I ~~ 2 N-1 drk j = ~ ao.dr0 + k-1 ak. 2) 2 ao.dro + k-1 ak. 2 0 0 (1 ù rk o (1 ù rk) The functional is diminished by the value it takes when all 0, are constant, which means that on each Poincaré D disk, the projection of the geodesic is a portion of radius of the disk, so the problem moves in H '= R x [0,1 [N-' ds 2 = ao.dro Z + r the ak.d (arg th (rk)). variable change 4 'H' -> RN [ro, ri, r2, ..., rN -1] I> [ro, arg th (ri), arg th (r2), ..., arg th (rN -1)] = [Ro, Ri, R2, ..., RN -1] In this space the metric defined by the element of length ds becomes: ds2 = ao.dRo2 + ~ kl ~ ak.dRk2 10 20 25 Since the ak are constant, it is about Euclidean geometry, the geodesics are segments of lines of RN of parametric equation in A 2 E [0.11: i (2) = [2.Ro max, .t. Rl max, ..., /i..R (N - 1) max]

Length of the geodesic arc, The length d (0, W) of the geodesic arc between the point 0 of H and the point

W of H, after the transformation P, can be computed in the Euclidean space RN since the arc is in this case a segment of the line: Ni1 IIZ J (1) = ao.Ro .2 + Ek = 1 ak. Rk max = d (0, W) Rkmax = argth (rkmax) = argth (IWk) k = 1, ..., N-1 Ro max = ro max = IWoI = q max since the first component is unchanged in all envisaged transformations. The length of the geodesic arc in RN between 0 and the parameter point À will then be:

2.D (0, W) and this length is unchanged during the different transformations since these transformations are isometric and thus retain the lengths.

Parametric equation of the geodesic resulting from 0 in H:

Let us return to the set H by inverse transformation to obtain the equation parametrized by de of the geodesic: 9% x (il) = [2.Ro max, exp (i 81) .th (2.Rl max) ,. .., exp (i B (N 1). th (2.R (N-max)] 8k is the phase of Wk in D. Indeed, as previously noted, the geodesic in H, in restriction to one of the complex components, is a segment of radius of the disk D. ek simply denotes the phase in D of this radius exp (iO) = Wk IWkI independent of λ H (2) = 2. Wo, WI .th (2.argth (W1I), ..., WN WN II .th (/t.argth (IWN-'I) IWII 5 and ITtH (1) II = (1) 11 =. / Ao .Womax2 + 1% T -1 ak.argth (Wk) 2 = d (O, W) Vector tangent to H at 0. A vector tangent to the parameterized arc in H is obtained by deriving H (2, I) relative to h: V (2) = [Ro max, exp (i ei) .RI max (1 th 2 (2, .R) max)), ..., exp (i B (N - 1) ) .R (N -1) max (1- th 2 (2.R (N -1) max))] In At = 0, that is to say at point 0 of H we get: 15 V (0 ) = [Ro max, exp (l01) .RI max, ..., exp (i9 (N - I)) R (N -1) max)] But: ex p (ia) = Wk IWkI Rk max = argth (Wk1) k = 1, ... N-1. The tangent vector V (0) (which is not yet unitary) is therefore written: V (0) = [romax, IW W) WN -1 ll.argth (IWII), ..., IWN I l • argth (WN-II)] It remains to normalize in the local metric in 0 (see previous paragraph) to obtain the vector U (0): unit: U (0) = v (o) IIV (0) II 20 25 5 With: IIV (O) II = Jao.goX2 + Ek I'ax.argth2 (IWkI) = d (0, W) that is: IIV (0) II = IN.gomax2 + Ek 1 ' (Nûk) .argth2 (WkJ) = d (0, W) Parametric equation of the geodesic as a function of a vector tangent to H in O. The parametric equation of the geodesic is expressed as a function of any tangent vector to H in 0: V = (Vo, V1, .., .VN-1) When the maximum of [2.IVk] is small in front of the unit for k = 0 to N-1, the following approximation is valid : 9ÎH (2) [2.Vo, 2.Vi /Î.VN -1] Expression of the inverse generalized rotation. TA (e) = [q û qo, R [α] (zi), R [a2] (z2) ..., R [aN-1] (ZN -,)] TA '(9) = [q + qo, R- '[aI] (z1), R-' [a2] (z2) ..., R-1 [aN-] (zN-1)] e = [q, ZN-1] A = [ go, a ,, a2, ..., aN -,] R-1 [a] (z) = z + a 1 + a.z with the condition that a and z are in D. 2.Vo, V1. th (2.IVii), ..., VN - 1 th (/1.IVN - 1) IV1I VN - II - RH (2) = 21 2955950 The distance between the points U and V of H is written according to a known formula of the prior art: Uo = log (Puo) Description of the Invention Synopsis Diagram FIG. 1A schematizes an example of a system allowing the implementation of the method according to the invention for the example of the radar application.

In this figure are represented a device 1 for transmitting and receiving signals, comprising a transmitter module 2 of a transmitted wave Which will be reflected on a target 3, a reception module 4 of the signal Sr reflected by the target and a processor 5 connected to the receiving module and adapted to implement the steps of the method. The transmitting-receiving device 1 comprises an output 6 connected for example to a processing device 7 of information from the method. Figure 1B shows an antenna and the emission lobe. The antenna performs an angular sweep. The target is in the lobe of the antenna and reflects a signal. The distance boxes usually used in the field of radar are represented. A timing diagram represents the distribution and sequence of emissions and target echoes that are received at the receiver. The round trip time is equal to At = [2 (d + vt) c]. T corresponds to a repetition period of the pulses.

For a distance k, where there are M hits. After reception on the I and Q channels, let X1,... X p be the complex measurements of the received signal. Before demodulation XA e -12g (f ° + M1 = + Noise 1 l L (U, V) = 1 / N. (VoûUo) 2 + Ik 11 (Nûk) .argth2 (Vk-Uk 1ù Vk.Uk) Vo = log (Pro) After demodulation the measurements are expressed in the form: XI = Ale-'2 '(f "T + Noise Where fd is the Doppler frequency for box k and fo the carrier Figure 1C shows the signal processing After demodulation according to the prior art, the implementation of a set of FFT filters on each of the distance boxes is noted to determine a value of TFAC and a value that will be compared with a threshold value to determine the presence or absence of FIG 1D shows the sequence of steps of the method according to the invention which determines for each sample received by an AR analysis and for a given distance box a value of N reflection coefficients, 0 k for j = 1, N. The reflection coefficients are then transmitted to the calculation module of the median TFAC to determine the value of the reflex coefficients. median ion 0jm The median reflection coefficients are then used to calculate the value of the distance for a given distance k box. If this value is greater than a threshold value, then the method decrees the presence of a target in the distance box, otherwise it does not detect a target. The steps of the diagram of FIG ID relating to the method according to the invention will be detailed below. FIG. 2A represents a block diagram of the steps of the method implemented for a radar application, and FIG. 2B a block diagram of the steps of the method according to the invention in the one-dimensional case. The gradient alqorithm for a midpoint or the mean - FIG. 2A Initialization: The starting data is a set of P samples {E1, ... EP} of M complex measurementsM N, 10 23 2955950 The BURG algorithm (possibly regularized), 11, applied to each of these samples generates P points of the set H which are N-tuples of coefficients, the set being referenced Bp, 11, in FIG. 2A.

The first component of the N-tuplet Bp is the natural logarithm of the estimated average power of the sample which will be treated differently from the other coefficients.

The following components are the N-1 complex reflection coefficients of the AR model. This results in P points of H: {W ~, ..., WP} which are the anchor points of the algorithm which remain fixed throughout the procedure. The initialization (step j = 0), starts with a candidate point Z which can be any point, for example one of the P anchor points or the origin 0 of H.

The loop: The loop on step L: which is broken down into 7 phases.

In step j, of the algorithm, we have a candidate ZEH point from the algorithm in the previous step j-1 Z = (zo, zl, ... zn -1) R1-1 phase 1, 13 Perform on each of the P anchor points {W1, ..., WP} the direct generalized 20 rotation of Mbb of center Z to obtain a set of modified P points in H {W'1, ..., W'P} specific to step j. Wm = (Wm, O, Wm, 1, ... Wm, N -1) m = 1, ..., P is transformed in this phase into: 25 W, m = (-W m, O, W ~ m, 1, ... W 'm, N -1) = (W' m, O - Z0, R [ZI] (Wm, l), ..., R [zn - 1] (Wm, N - 1)) where the first component w'm, o which corresponds to the signal strength is translated from z0, while each of the N-1 other components undergoes a rotation of elementary Môbius 24 2955950 R [z] (ww - z 1ù zw R1-2 phase 2, 14

For each of the modified points W'm, calculate the vector Vm (which must be interpreted as a tangent vector at 0 in H) according to the formulas. W'm, l W m, N-1 Vm = [w ', o, arg th (W m, l), ..., arg th (W m, N-II)] 1W'm, 11 W 'm, N-11 To escape the divisions by 0, a variant consists in carrying out the following calculation: Vm = [W m, o, w' m, 1 .0V (w'71 1), ..., W m, N-1 • 0V (W ,, N_1)] with: w (x) = argth (ev) six e, and argth (x) six ev 6'vx ev is a positive real number small in front of 1 for example 0.1 R1- 3-a phase 3, 15a if the object of the algorithm is the calculation of a midpoint For each of the points Vm, calculate the unit vector Um (which must be interpreted as a tangent vector in 0 in H) according to the formulas. Um = with I vmll ~ Vm =. z +, k '(N ù k). arg th 2 (Vm, k To escape the divisions by 0 a variant consists of carrying out the following computation: Um = lVmll If llVmll ~ SU and Sm if lVm <had eU is a positive real number small in front of 1 for example 0.1 R1-3-b phase 3, 15b if the object of the algorithm is the calculation of the mean Do nothing, and rename the vector Vm in Um Um = Vm ~ o R1-4 phase 4, 16 Calculate the convex linear combination S of the vectors Um for m = 1 to P. PS = Eam • Um m = 1P 1 am=1 m = 1 am> _ 0 am real number For example: am = 1 P

15 R1-5 phase 5, 17

Multiply the vector S by the real number y.,> 0 increment of the gradient in step j, to obtain a vector A which is still a vector tangent to H in O. A = y, .. S 20 Developing according to components of H, 0 is written: A = [Ao, A1, ..., Ak, ..., ON-1] R1-6 phase 6, 18 Calculate the vector Z 'of H which is interpreted as I "(Winding up" of the vector A on the Riemannian manifold H along the geodesic from point 0. 26 2955950 r ° Z '= ° o,' th (I °, I), ... ° N .th ( To escape the divisions by 0 the following variant of calculation will be used: Z '= [° ° 1 ° b ° (I °,), ..., ° N1 ° b ° (I °) N 1) j with çb ° (x) = thzx) six> _ e ° and th (e) six e ° e ° is a positive real number small in front of 1 for example 0.1 We can also take the approximation. Z ', - =, [° 0, ° 1 ° 1] if all the components are small in front of 1.

Let Z '= [z'o, Z'l, ..., Z'k, ..., Z'Nù1] R1-7 phase 7, 19 Perform on Z' the inverse rotation of generalized Môbius of Z center who is the candidate point at step j to obtain the new candidate point Z at step j + 1.

Znouveau = Tzù1 (Z ') = [z' + zo, R 1 [zl] (z'1), Ri '[z2] (z'2) ..., R 1 [Zn 1] (z'N_1) ] i = 1, ... N-1 R1-8 stop criterion, 20 The loop stops when the norm of the vector S obtained in phase 4 falls below a real positive number close to 0 which is the stop threshold of the loop. IISII =. \ JN.So + Lk the (N ù k). arg th 2 (Sk) stop threshold R- '[z1] (z' z '+ Z 1 + z; .z'1 with: 27 2955950 The midpoint or average according to the procedure obiet is the last point Z calculated, 21. R1-9-a, 22 restoration of the linear scale for the median power 5 After stopping the loop, as the first component of the N-tuple considered corresponded to the natural logarithm of the average power of the sample, the median or final average power, (depending on whether the object of the procedure is the calculation of the median and the average), is obtained by restoring the linear scale of the first component: 10 Median Power = exp (Zo) R1-9-b, 23 linear scale recovery for average power 15 Average Power = exp (Zo) A step after determining the midpoint or average can be as follows: Knowing the midpoint or the average point according to the object of the process, we calculate the distance between an anchor point and the point from of the method by the application, for example, of the above formula: then this distance is compared to a threshold value defined by the need of the application, and if this distance exceeds the threshold it is decreed presence of a target.

The following example corresponds to the particular case where the algorithm uses a single coefficient. L (U, V) = N. (Vo-Uo) 2 + ~ k ~ '(Nuk) .argth2 (VkuUk 1 - Vk.Uk) 28 2955950 When the envisaged radar processing uses only the first reflection coefficient which is then the correlation coefficient, the algorithm is simplified. In this case, in particular, the power of the signal is considered equal to 1. The logarithm of the power is therefore equal to 0 and the first component of H is always zero. The space H is then reduced to the disk D. This is for example the case in a disturbance mode of a Doppler weather radar. 10 Algorithm of the gradient for a median or average point in dimension 1 ù FIG.2B The framework of the algorithm is then the disk D of Poincaré D = {zl z EC and Izl <1} The P anchor points of the algorithm which remain fixed throughout the procedure are then reduced to p points of D: {W1, ..., WP}. Initialization. The initialization (step j = 0) starts with a candidate point Z which can be any point, for example one of the P anchor points or the origin 0 of D. The loop. The loop on step j: which is broken down into 6 phases. Indeed 2 phases of the general algorithm merge and we go from 7 phases to 6. In step j of the gradient algorithm, we have a point D candidate D from the algorithm in the previous step j-1 25 R2-1 phase 1 (idem RI-1), 30 Perform on each of the P anchor points {W1, ..., WP} the rotation of elementary direct Môbius of center z to obtain a set of P points modified in DP} specific to step j.

W'm = R [z] (Wm) - Wmz 1z.Wm R2-2-a phase 2, 31a if the object of the algorithm is the calculation of a midpoint (idem R1-2 and R1-3a ) Calculate for each of the modified points the unit complex number Um according to the formulas. W 'm m IW'1 To escape the divisions by 0 we will prefer the following calculation: Ov (x) = 1 six ≤. ev and 1 if xev ev x with Um = W'm • Y7 ('W'm I) ev is a positive real number small in front of 1 for example 0.1

R2-2-b phase 2, 31b if the object of the algorithm is the calculation of the mean (idem R1-2 and R1-3b)

Do nothing, and rename the number W'm in Um. R2-3 phase 3 (same R1-4), 32 Calculate the convex linear combination S of the numbers Um for m = 1 to P. 20 PS = Eam.Um m = lp l year, = 1 m = 1 am> _ 0 25 real and coefficient of "weighting" of the unit numbers Um to effect the convex linear combination 2955950 For example: R2-4 phase 4 (same as R1-5), 33 Multiply S by the real number y ,. Where y, is the increment of the gradient in step j, in order to obtain the vector A S .. A = y; .S R2-5 phase 5 (idem R1-6), 34 "Wrap" the vector A on the Riemannian manifold D along the geodesic 1 o from 0 to get the point z 'of D. z' = IQ .th (AI) To escape the divisions by 0 we will prefer the following calculation: z '= A.Oo (DI) with 0o (x) = th (x) if x> _ eo and. th (e °) six <_ so ns is a positive real number small in front of 1 for example 0.1 R2-5bis We can also take the approximation. z '= A if the modulus of A is small in front of 1.

R2-6 phase 6 (idem R1-7), Perform on z 'the inverse rotation of center z to obtain the new candidate number z in step j + 1. Znouveau = Tz 1 (Z ') = R' [z] (Z 'Z' + Z 1+ zz 'R2-7 stop criterion (idem R1-8), 36 The loop stops when the module the complex number S of phase 4 is less than a positive real number close to 0 which is the stop threshold of the loop: IS 5 stop threshold A step after determining the midpoint or the average can be the following: Knowing the median point or the average point according to the object of the process, the distance between an anchor point and the point resulting from the process is calculated by applying the preceding formula: then this distance is compared with a threshold value defined by the need of the application, and if this distance exceeds the threshold, it is decreed that a target is present.Figures 3 and 4 represent the result of the median algorithm in dimension 1, For example, in FIG. 10 complex points in D schematized by (X) points denoted Wi and corresponding to anchor points or 20 coefficients of correlated lation in this embodiment and several points + noted xi represent the candidate points that will be involved in the algorithm. The convergence of the algorithm is represented in particular by the different points xi and the median value by the square in the figure. By taking the steps described above, after applying the Mobius rotation of center x1 over the whole of the figure comprising the different points Wi and x1, the point x1 is found at the center of the unit circle C, hence at x0 (see the arrow FI on the diagram). The Wi points have become modified Wi 'points (which are not shown in the figure). The geodesic arcs to be determined between the central point and the different points L (U, V) = N. (VoùUo) 2 + k ~ '(Nùk) .argth' (Vk-Uk 1 - Vk.Uk) 32 2955950 modified Wi 'correspond to radius segments of the unit circle. For each of the modified points Wi ', the corresponding unit complex number Um is calculated which can be interpreted as a tangent vector at 0 to the Riemannian manifold D according to the formulas R2-2-a in the case of the median, and then a convex linear combination S of the numbers Um for m = 1 to P the number of anchor points considered. In the next step, we will advance in the direction of the vector S over a length of one step y to obtain a new point x2 of the differentiable variety. x2 is a new complex number. 1 o Then we will perform a rotation of inverse Môbius of center xl on point x2 to obtain the new candidate in step j + 1. The modified points Wi 'do not have to be reversed since this operation would only have the effect of reconstituting the anchoring points Wi that are anyway kept in memory. These steps will be repeated as long as the stopping criterion is not checked, to obtain the median represented by the square in the figure. Gradient Algorithm a Reduced Median Midpoint In the initial explanatory exposition, it was assumed that the N dimension of the H space was equal to that of the base samples on which the regularized BURG algorithm operates. This hypothesis is by no means obligatory. Having samples of size M, the regularized BURG algorithm can be interrupted at a number of stages N less than M. The space H on which the invention operates the median and mean algorithms then takes the dimension N. This is particularly useful in applications where the high-order reflection coefficients are insignificant, their estimation unnecessarily consuming computation time. All the steps that have been described for the calculation in one dimension in several dimensions apply.

Application to grouping or "clusterinq" Doppler data for clutter mapping.

Clustering techniques seek to classify data that is already grouped into classes described by real Gaussian non-centered and unreduced variables N (m, Q). method for recognizing a central entity to a whole group of Gaussian variables.

The geometric theory of information is therefore commonly

~ o invoked to solve this type of problem. It is obvious that an algorithm determining the median of a set of Gaussian is welcome and many works go in this direction.

The present invention is immediately applicable to the problem of determining the median of a set of P real 1-dimensional Gaussian variables.

Indeed in the framework of the geometric information theory a real Gaussian random variable is seen as the element of a real 2-dimensional manifold parameterized by the pair (m, a) with m belongs to R set of real numbers and a belongs to R + * together numbers

20 strictly positive real.

This open set is identified with the higher complex half-plane that we will denote H. This manifold is provided with a Riemannian structure using the metric tensor defined at the point (m, a) by the Fisher information matrix of the calculated Gaussian density. at this point.

25 H + with its Riemannian structure is the '/ 2 Poincaré plane. The Cayley transformation u = z ù l <=> z = ùi. u +1 puts H + in bijection z + i u-1

bi-continuous and bi-differentiable (diffeomorphism) with the Poincaré D disk. This is a parameter change (local map). The metric tensor of H + is transferred on D to obtain the Poincaré tensor in D. The tensor of the geometry of the information on the Gaussian variables is equal to 8 times the tensor in the Poincaré disk. Information geometry tensor Gaussian density: 1 1 xm \ zf (xm, 6) _ ù.exp (-.) F2 -T, o 2 \ 6 log-likelihood: 1 1 1 "xiim m, ff)) = û2.log (21r) -. log (6) - Fisher information matrix: L (x, m, a) = 1og (f (x 1 0 F (m, u) = aE has 2 amati L ds2 = 1E aLt dB; = ûEE a2 a2 - am2 L amati L a2 a62 L 2 6_ a2L of..de; ae; ae; 0 2 if we put: x = y = 6 z = x + iy (m \ z 2 dS2 = I.ds2 = 1 dm2 + 2.do-21 dv.y + d6 -1 dx2 + dy2 ldzlz 8 8 62 4 6 -z 4 y2 4 (Im (z)) 2 Transformation of Cayley and passage of H + in D: z u + 1 u = qz = -a. Z + i uu1 dS2 == 1 ldzlz - ldulz 4 (Im (z)) 2 (1 ul2} 2 2955950 The algorithm for finding a midpoint / average among a set of Gaussians

The algorithm for finding a midpoint or average point among a set of real Gaussian Ps then consists in applying the Cayley T transformation in order to transform the P pairs (mk, ok) into P points in the disk D: (mk , ok) = Zk = - + 1.0k> Uk = T (Zk) = Zk-1k = 1, ..., P Zk + l 10 The application of the one-dimensional median algorithm in D provides a midpoint or average UD to which the inverse Cayley transformation T-1 is applied so as to obtain a pair (mD, QD) which is the parameterization of the median or mean Gaussian. MD. ZDT - '(UD) = UD + I ZD = + 1.OD (mD, Ore) UDu1 2 Figure 5 is the result of 10 Gaussian (mk, ok) crosses "x", the median being represented by the square "^".

Mean / median gradient algorithm for Toeplitz shape correlation matrices. When the I and Q sampled data blocks from the receiver are present in the form of correlation matrices, the method proposed by the invention still makes it possible to obtain a matrix representative of the ambient (TFAC). Indeed formulas known to those skilled in the art allow to pass correlation matrices to the reflection coefficients and vice versa. The purpose of this section is to recall these formulas which allow the extension of the invention and thus deals with the passage of a definite hermitian complex correlation matrix of Toeplitz form towards the reflection coefficients of the equivalent complex AR model and vice versa . The correlation matrix RN is defined by the list of correlation coefficients RN = Co C1 and Co CN_I CN-2 The list of reflection coefficients is: {Poäu1, .., un, ..., UN.} With PO power of the signal: Po = co

The reflection coefficients are obtained from the correlation coefficients cn, step by step, by the following iterative algorithm on the index n. This iteration is deduced from the regularized BURG algorithm and is based on intermediate coefficients ak. The iteration is written in the form of the following loop: ao = 1 For n = 1 at the step from 1 to N do: {n-1 n-1 n-1 (a0, al,, ak, n-1) /, in (a0 n,, akn,, an, an-1 '~ al> an) Passage of the coefficients ak-' of step n-1 to the coefficient of reflection, a: n-1 n- 1 Eak'.an (1 + 1) .Ck- (1 + 1) + E Mn) .ak-1 year kk, 1 = 0 k = 1 ~ n ù n-1 n-1 year-1 year '. ck-1 + E Mn) jar 'l2 k 1 k, 1 = 0 k = 0 In this formula: ~ kn) = 71. (2z) 2. (k û n) 2 Passage of the coefficients ak-' and, a to the coefficients ak of step aô = 1

ak = ak '+, one year = k, k = 1, ..., n-1 n an =, a} We will take for example yi = co.10-4. At this stage the correlation coefficients are transformed into a sequence of coefficients {1og (Po) äu1, .., a, .. yuN.} Which constitutes a point of the set H usable by the gradient or average algorithm. described in previous chapters. The above procedure is obviously applied to all correlation matrices (from 1 to p) of the initial matrix data set. This provides P anchors in the set H to which the gradient algorithm for the median or the N-dimensional mean is applied.

The result of the gradient algorithm is an element of the set H which is the estimate of the midpoint or average point in H.

The point of H thus obtained is then transformed back into a correlation matrix by applying the next iteration which constructs a series of symmetrical square matrices Rn of increasing size.

At each iteration step n, a matrix Rn of dimension (n + 1) x (n + 1) is obtained from the matrix Rn_1 of dimension (nxn) computed at the previous iteration by using complex intermediate coefficients ak and a positive real coefficient Enz: At step 0: matrix (lxl) Ro = Po 602 = Po n: 5 38 2955950 For n = 1 at the step from 1 to N do: {I The result of the algorithm is RN , which, depending on the type of algorithm used, is the median or average matrix sought. ag = 1 with ak ak- '+, a year-k' k = l, ..., n-1 ann = / (n n-1 An = aQ an k Rnu1 ùRnù1.An ù Rn ù] .An Enz + A * .Rn ù I.An_ Rn = 39 2955950 Appendix 1 Example of programming in dimension 1. This example shows the simplicity of the algorithm in dimension 1, 5 z complex; w [m] m = 1 ..., M anchor points whose median is searched all are complex numbers of modulus less than 1. Stop = FALSE; y = 1; z = 0; 10 j = 0; as long as ((j <J) AND (Stop == FALSE)) make {gradient = 0, for (m = 1 step 1 to M) make 15 {w [m] ù z w '= 1 ù _; zw [m] w' if (Iw'I s) then gradient = gradient + I 'I fsi w} end for gradient = gradient M 20 Si (gradient) SeuilArret) then stop = TRUE fsi z = y.gradient + z 1 + zygradient j = j + 1} end as The result is z, the midpoint of the collection of the starting w [m].

Appendix 2 Autoregressive lattice identification and reflection coefficients We have a set of N random variables with complex values of zero mean E = {Yo, ... YN -1} By self-regressive identification, we mean the search for a set of N-1 complex coefficients a, such that the actual quantity: E N-1 YN-1 ûai.Y (N-1) -i) i = 1 is minimal. 2 To solve this problem we place ourselves in the space of complex random variables of integrable square on which (X, Y) = E (X.Y) is a scalar product and the standard deviation is a vector standard. This space is a Hilbert space.

We can then adopt a geometric language and speak of orthogonal projection on a vector subspace.

Let P (X / Y) be the orthogonal projection of the variable X on the linear space Y.

For 0_ <k <_tNû1 We define the front and rear innovations by:

e + k (t) = YP (Yt / Yt-1, ... Yt-k) o (t) = where o (t) = Yi ek (t) = Y-kûP (Ytk / Yt, ... Y-k + 1) By stationarity In + (t) and In (t) have the same statistics for all t. In particular: 41 Ile + k (t) II2 = IIE k (t »II2 = EIe + k (t) I2 = EI E k (t ') I2 for all k, t and t' By orthogonality we have: E + k (t) = Yt ûP (Yt / Yt-1, ... Y -k) = YtûP (Yt / Y 1, ... Yt-k + 1) ûP (Yt / Ek-1 (tû1)) 5 _ E + k -1 (t) +, U1k.E k -1 (t -1) E k (t) = Y-kûP (Yt-kY, ... Yt-k + 1) = Yt-kûP ( Y-kYt-1, ... Yt-k + 1) -P (Yt-kE + k-1 (t)) = .E k -1 (t -1) +, u2k.E + k -1 ( t) (e + k-1 (t) k-1 (t-1)) -1)), u1k (t) _ - 2, IIE k-1 (t-1) II \ (E k-1 (t-1) IE + k-1 (t)) 10, ct2k (t) _ ù 2 t, uk IIE + k -1 (1, u2k =, ulk independent of T. Recurrent lattice formulas: 15 e + o (t) = eo (t) = Yt

for k = 1 to N-1: Jek (t) = E + k -1 (t) +, uk.E k -1 (t -1) E k (t) = e + k - 1 (t) - 1) +, Uk.E + k -1 (0 Coefficients of reflection in their theoretical formulation: 20 E (E + k - 1 (t) .ek - 1 (t -1)), uk = - EIE-k -1 (t 1) I 2 ù E (E k - 1 (t -1) .E + k - 1 (t)), uk = - 2 EI E + k - 1 (t) IE (e + k-1 (t) .e k-1 (t-1)) .JEI £ + k-1 (t) 2 .. \ Ele k-1 (t-1) 2 => t [lk ED Rear innovations form a system orthogonal (Hilbert-Schmidt orthogonalization process), taking t = N-1: Eo (t) = Yt, e-1 (t) = Yt-1-P (Yt-1 / Yt), E 2 ( t) = Yt-2-P (Yt-1 / Yt, Yt) 6k (t) = Yt-kuP (Yt-1 / Yt, Yt-1 ... Y-k + 1) The reflection coefficients are the coefficients of the orthogonal projection of Yt on the orthogonal basis of {£ -k (t-1) / k = 0 ... N1} Y = -1 k = 1 pk. & k-1 (t -1) + e + n (t) N-1 YN-1 = [-k 11, L1k, k-1 ((N -1) -1)] + + + N -1 (N -1) = E ai.Y (N -1) - i + £ + N -1 (N -1) i = 1 The variances of the innovations can be expressed according to the coefficients of reflection and the common variance Po des Yt: 1le 2 = ## EQU1 ## where ## EQU1 ## YII2 = n (1 - / 'I ".Po f-1 J = 1 The transition matrix from the Yt base to the orthogonal base of the rear innovations is semi-diagonal lower: 15 10 k-1 ek (t) = Yt-kuP (Yt-1 / Y, Y-1 ... Y-k + l) = Y-k + E% 3k, JY-i J-0 The family of ek (t) = sk (t) 2 is orthonormal Os k (t> II rit = [Yt, ..., Y - k, ... Y - N + l] T

~ .t = [e ". 0 (0, ..., 60-k (t), ... eN-1 (t)] T = L.IIt with: E (IIN -1T.IIN -1) = RN correlation matrix of the random vector / 30.0 0 0 0 0

0 0 0 L / k, 1 / 3k, k 0 0

0 / 3Nu1,1 / N1, k ... / N-1, N-1 / 3k, k = 1 IIS k (t) II E (EN.EN *) = IN = LN.E (IIN.HN " ) .LN * = LN.RN.LN * Where E is the mathematical expectation and RN is the correlation matrix of Ili, IN is the N-dimensional identity matrix N-1 N-1 = Island k (N-1) IIz = PONn (1-I, uk2) N kk = 1 k = 1 1 2 det (RN) = 1 N-1 det (L) 2 fl / 3k, kk = 0 15 N-1 ln (det (RN)) = N. ln (Po) + (N ù k) .ln (1ù, uk 2) k = 1 44 2955950 Appendix 3 Entropy of the Qaussian variable and Kâhlerian metric in H: 5 The Gaussian density is written:

p (ZN / RN) IRN -I. exp (ùZN.RN '.ZN *) = NN .1 RN -I. exp (ùtrace (RN '.ZN.RN *)) 7z-7r û ln (p (ZN / RN)) = N.ln (ir) + 1nJRNI + ZN.RN .ZN * = N.ln (r) + 1nIRNJ + trace (RN '.ZN.RN *) The entropy (D of the complex centered Gaussian variable of dimension N is written as: (D (RN) = E (ûln (p (ZN / RN))) ù N ln (TC) + 1nIRNI + trace (RN .E (ZN.ZN *)) = N.1n (z) + lnNIRN + N N-1 15 ch (RN) _ (D (Poäu1, ..., uN -1) = N.ln (rz.e) + 1n RNI = N.1n (rz.e.Po) + E (N-k) .ln (1i, ukI2) k = I 19 2Po2 ~ (PO, 11) , ..., uN -1) _ N Poe Nûk (1 _ II2) 2

Partial cross derivatives are null. The Hessian of (D is therefore diagonal.

Hence the expression of the dot product in the vector space tangent to the

variety H at the point (Po, ui, ... pN - 1) .10 Appendix 4

The regularized BURG algorithm The BURG algorithm provides on a sample of N measurements z (k) k = 1, .., N an estimator of the reflection coefficients obtained in their theoretical formulation in appendix 2. This estimator is regularized and stabilized the order coefficients> 1. Initialization:

fo (k) = bo (k) = z (k), k = 1, ..., N (N: number of measurements) N P = ù.E 1z (k) r N k = l

e ') = 1

. Iteration (n): For n = 1 to m 2 N n-1 (k) .bn * (k û 1) + 2.E) a (n 1) a ,,, n: k »n-1 -1 kk - - N _ nk = 1 n-1 E If _1 (k) 2 + (k -1) 12 + 2.E e), kn-1) 12 1 N û nk = n + lnk = OI a ,, ) n) = al ') = akn 1) + pn.an, ï-kle, k = l, ..., n-1 an) n û, a f ,, (k) = (k) + / in .bn_1 (k -1) bn (k) = bn_1 (k -1) + p.fn_1 (k) with jain, = y, (27t) 2. (kn) 2 we will take yl = Po.coef with coef = 10-3 or 10-4.

Claims (2)

REVENDICATIONS1. A method for determining a central parameter, such as the median or the mean, on a set of identification results associated with a complex signal from a radar signal received on a set of sensors of a radar receiver, leading to collections E1, ... Ep of M-tuples of complex numbers that correspond to m measurements of the complex signal, said measurements being taken from the radar receiver Ek = LXk 1, Xk 2, ... Xk MJ k = 1, .., P said method comprising applying an N-1 lattice auto-regressive identification algorithm on said complex signal from a radar signal, providing P points {W1, W2 Wp} or points anchoring which are invariant during the implementation of the steps of the method according to the invention, characterized in that it comprises at least the following steps: each point Wk is a consisting of a sequence of N coefficients of the form Wk = where Wk = 1og (M EII Xk j IIZ) and (Wk, ..., W, .., Wk - ') are the N-1 premi ers j = 1 complex reflection coefficients from said complex autoregressive identification by implementing a trellis identification algorithm applied to the set of numbers, (Xk, l, •••, Xk ,;, •••, Xk, M) said method applies to a general set called H in which said anchor points are located and said general set H is the set of sequences of the form Wk = (W, °, Wk, ..., Wk, .., Wk - ') of N coefficients whose first coefficient is a real number and the following coefficients of the complex numbers of modulus strictly less than 1, said method also comprises the application a gradient descent algorithm which operates on a function defined on the general set H, said function being obtained, either as the sum of the distances of the anchor points at any point in the general set when the method 5 is intended to calculate a median, as the sum of the squares of the distances of the anchor points at any point in the general assembly when the method aims to calculate an average, and in that: said method is characterized in that it carries out a loop aiming at minimizing the function it is named by a gradient algorithm comprising a finite number of steps during which is calculated at each step of said algorithm, a gradient vector pointing towards a point of convergence which is the result targeted by the method, and in that each step of the gradient algorithm comprises at least the following steps which transform a point z of the general set H from the preceding step of the gradient algorithm into a new point z of the general set: Phase 0: take the point z of the general set derived from the algorithm in the previous step j-1, for the first step the initial z-point will be initialized to any anchor point, Phase 1, (13): perform on each of the points ts of anchoring a generalized rotation of direct Môbius of center z to obtain a set of modified points P corresponding to step j, W'm, Phase 2, (14): for each modified point W'm, determine the vector 25 Vm tangent to H at 0, Phase 3, (15a, 15b): determine for each of the modified points, the unit vector Um tangent to H at 0, Phase 4, (16): calculate the convex linear combination of the unit vectors weighted by a coefficient rrr ,, Phase 5, (17): multiply the linear combination S by a real number corresponding to the increment of the gradient in step j in order to obtain a vector tangent to H in 0, Phase 6, (16): wrap the vector A on the Riemannian manifold H along the geodesic from 0 to obtain a point Z 'of H, Phase 7, (19): perform the inverse rotation of Mbb of center z to obtain the new candidate z at step j + 1, Phase 8, (20): compare the value of the norm of S with a threshold value and if this v the value is less than the threshold value, then stop the iteration and define the central element (midpoint or middle point) as the last determined point, otherwise, make j = j + 1 and return to phase 0, said method comprising a step during which knowing the midpoint or the average point according to the object of the method, calculating the distance between a point of the set H and the midpoint or average 15 from the previous steps and then comparing this distance a threshold value defined by the need of the application, and if this distance exceeds the threshold is decreed the presence of a target. 2. Method according to claim 1, characterized in that the signal is a signal derived from a Doppler radar signal, channel I and Q sampled and analyzed by block length MN using the BURG algorithm, the BURG algorithm. generates a set of (N-1) uplets of coefficients which are the first N-1 complex reflection coefficients of the sample, said method requires P blocks from the receiver which lead to p points of the set H by application of the BURG algorithm to each of them; each of the P points of the set H, is an N-tuplet whose first component (index 0) is a real number which is the natural logarithm of the power Po of the signal sample, the following N-1 components of the point of H are the N-1 complex reflection coefficients, the method is characterized in that: Phase 1, (13) generates a set of modified P points in H {W'1, ..., W'P} in step j by applying a generalized Mobius rotation Wm = (Wm, o, Wm, 1, ... Wm, N ù 1) W m = (W m, w, W 'm, N ù 1 ) = (W 'm, where Zo Zo, R [ZI] (Wm, l), ..., R [Zn ù 1] (Wm, N ù 1)) R [z] (w) = Wùz 1- For each of the modified points, said method calculates the vector Vm according to the formulas. W'm, 1 W'm, N-1 Vm = [W'm, 0, argth (1w'm, ll), ..., .arg th (IW'm, Nùll) j 1W m, 11 1 W m, N-11 If the object of the algorithm is the calculation of a median point, (15a) the method calculates for each of the points Vm the unit vector Um according to the formulas. 11r / m11 = 'N.Vm 02 + Ek = 11 (N ù k). arg th 2 (I Vm k 1) 20 Um - II Vmll if the object of the algorithm is the calculation of the average, (15b) the process copies the vector Vm in the Um Um = Vm then the process computes the combination convex linear S of the vectors Um for m = 1 to P. 50 2955950 P SûEam.U, m = 1P Fan, = 1 m = 1 am? For example, the following step (14) is to multiply S by the actual number yi> 0 increment of the gradient in step j, to obtain A vector. 0 = 7i.SA = [Ao, 01, ..., Ok, ..., AN ù 1] The phase 6 is executed as follows: D0, Ol .th (1A11), ..., ANi1. th (I ANii1) IA1 ONlI Or: Z ~ = [AI) Al 0e (Io1 I), ..., ONii1 0e (ON-1 I) ~ with (x) = th (x) six and th ( E °) six x E ° then perform, (19) on Z 'the inverse rotation of center Z to obtain the new candidate point Z in step j + 1. Znouveau = Tzù1 (Z,) = [z '+ zo, R-1 [z1] (z'1), R -' [z2] (z'2) ..., Ri1 [zNu1] (z'Nu1) Z '= i = 1, ... N-1 R-1 [zt] (z' z ') + z. 1 + z; z' 20 with: repeating said steps as long as the norm of the vector S is greater than or equal to a positive real number close to 0, which is the stopping threshold of the loop IISII = JN.So2 + Ek 11 (N ù k) arg the (I Sk) threshold stop and in that it comprises the following steps after stopping the loop, execute in the case of the calculation for the median power (22): Median Power = exp (Zo) or in the case of calculating the average power (23) Average power = exp (Zo) 3 - A method according to claim 2 characterized in that the tangent vector Vm is determined by applying Vm = [W m, 0, W m, 1 -0V (W m, 1 W m, N-1OV (W m, N-1)] with: 0V (x) = arg th (V V) if x V V and arg th (x) six _ V x V and the calculation of the vector Um Um = IIVmII if Vm I% U U and EU sl 4 - A method according to claim 3 characterized in that Z'- [Ao, 01 0] if all the components of the vector A are of small modulus in front of 1. - Process according to claim 1, characterized in that the complex signal is a radar signal and in that the framework of the algorithm is then the D disk of Poincaré D = {zl ze C and lzl <1} It results P points of D: {WI, ..., WP} which are the anchor points of the algorithm that remain fixed throughout the procedure. The initialization (step j = 0), starts with a candidate point Z The loop on step i comprising at least the following phases: In step j, the algorithm, we have a zero point D candidate from of the algorithm in the previous step j-1 Perform on each of the P anchor points {W1,..., WP} the rotation of direct Mobius of center z to obtain a set of modified P points in D { W'i, ..., W'r} specific to the stage j., (30) Wm-W'm = R [z] (Wm) = Z - 1 - Z.Wm if the object of the algorithm is the calculation of a midpoint, (31a) Calculate for each of the modified points the unit number Um according to the formulas: _ W '_ mm lJ'ml the object of the algorithm is the calculation of the average, (31b) Do nothing, and rename the number W'm in Um. Um = W'm Calculate, (32), the convex linear combination S of the numbers Um for m = 1 to P. P = E am .Um m = 1P E an, = 1 m = 1 am! 0 am real and coefficient of "weighting" of the unit numbers Um to carry out the convex linear combination We can take for example: a = m 1 P Multiply, (33), S by the real number y,> 0 where yi is the increment of the gradient in step j, to obtain the number A which corresponds to a new point positioned on the course carried out according to the gradient vector obtained. A = .S wrap A on the Riemannian manifold D along the geodesic from 0 to obtain the number z 'of D. z' = ol.th (IAI) Perform, (35), on the inverse rotation of center z to obtain the new candidate number z in step j + 1. z '+ z Znew = Tz1 (z') = R [z] (Z ') _ 1+ ZZ' 20 stop criterion (36) stop the loop when the module of the complex number S is less than a positive real number close to 0 which is the stopping threshold of the loop. The method according to claim 5, wherein it uses the points z '= (IAI) with O ° (x) = thzx) if xe ° and th (EA). if x <ù e ° 5 7 ù A method according to claim 5 characterized in that one makes the approximation. z '= A if the modulus of A is small in front of 1. 8 - The method according to claim 1, characterized in that it uses as starting data P pairs of real numbers representing the average and the standard deviation of P random variables real Gaussian of dimension 1. E ~, ... EP. Ek = (mk, Ok) k = 1, .., P and in that the algorithm for finding a midpoint among a set of real Gaussian Ps then consists in applying the Cayley transform to the P couples (mk, Ok) to obtain P points in the disk D: 15 mk Zk -1 (mk, ok) = Zk = - + l.6k Uk = Zk + l The application of the one-dimensional median algorithm according to claim 6 in D provides a median or mean UD which is then transformed by the inverse Cayley transform into a pair (mD, ap) which is the parameterization of the median Gaussian. UD + 1 mD ZD = ui. = zD = + i.6I'D = (mD, OD) UDu1 - In order to obtain a midpoint or a mean value from a set of real Gaussian P's N (mk, 6k) 9 - A process according to claim 2, characterized in that the data are transformed Doppler radar data blocks putting them in a form of correlation matrix. Application of the method according to one of claims 1 to 8 for Doppler radars and / or weather radars. 11 ù Device for transmitting and receiving signals derived from a radar signal making it possible to determine a central parameter, such as the median or the mean average, on a set of identification results associated with a complex signal received on a sensor assembly of a radar reception module (4) characterized in that it comprises at least one processor (5) adapted to perform the steps of the method according to one of claims 1 to 12.