Method for nonlinear estimation of a signal mixture

Description

[0001] The present invention relates to a method for estimating radio signals from several sources, the time / frequency representation of which shows an unknown non-zero proportion of zero components, by means of a network composed of P>2 antennas, when the directional vectors U and V of the sources emitting these signals are known or estimated elsewhere.

[0002] It is common to have to carry out the estimation of radio electrical signals (denoising) coming from radars, communications systems, or acoustic signals (audio or sonar), and received by a listening system consisting of an array of antennas.

[0003] The received signal results from a temporal and spectral mixture of 2 sources, whose directional vectors are assumed to be known, since they were estimated beforehand.

[0004] The criterion classically used is maximum likelihood (ML), leading to processing by spatial linear filtering which improves the signal-to-noise ratio by a factor equal to the number of sensors in the single-source case.

[0005] Under the assumptions we make (known directional vectors), this treatment leads to an unbiased linear estimate with minimal variance of signals for which no a priori knowledge is available.

[0006] For example, we know from the MASTER AS document: “Bayesian two source modeling for separation of n sources from stereo signals”, ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, 2004, a signal estimation process by applying an A Posteriori Maximization technique.

[0007] Other methods, which are more complicated to implement, can be used, such as Capon filtering when the directional vectors are imperfectly or not known ("Robust Adaptive Beamforming", eds P. Stoica and J. Li, Wiley, 2006).

[0008] None of these methods exploits any a priori on the signal, and in particular does not allow the temporal and / or spectral supports of the signal to be correctly delimited since linear (MV) or pseudo-linear (Capon) processing always provides a signal at the output even if the measurement at the input is only composed of noise.

[0009] The problem is to access more detailed knowledge of the signal.

[0010] The aim of the invention is to propose a method for determining the components of the signal more precisely.

[0011] For this purpose, the invention relates to an estimation method according to claim 1.

[0012] The method thus makes it possible to answer the following questions: for each of the signals present (their number being assumed to be limited to 2 locally), what are the temporal and spectral supports of the signal assumed to be described by components obtained by means of a time / frequency analysis? And what is the value of each component when it is non-zero? The answer to these questions makes it possible to improve knowledge of the signal.

[0013] According to particular embodiments, the method comprises one or more of the characteristics of claims 2 to 9.

[0014] The invention will be better understood by reading the following description, given with reference to the attached drawings: there figure 1 is a schematic view illustrating signal sources and an installation for estimating radio signals coming from these sources according to the invention, given solely for information purposes and without the intention of representing reality; figure 2 is a flowchart of one of the methods as implemented in the invention in the single-source case; figure 3 is a flowchart of one of the methods as implemented in the invention in the bisource case.

[0015] The device 8 for estimating a mixture of signals coming from several sources 12 according to the invention illustrated in the figure 1 comprises an antenna array composed of a plurality of antenna elements 10 or sensors. Each antenna element is coupled to a reception channel for, in particular, digitizing the analog signal received by each sensor.

[0016] The invention is suitable for both monopolarized antenna arrays and bipolarized antenna arrays.

[0017] The device further comprises calculation modules. In different embodiments of the signal estimation device according to the invention, the calculation modules can be arranged according to different architectures, in particular each step of the method can be implemented by a separate module or, on the contrary, all of the steps can be grouped within a single information processing unit 14.

[0018] The computing unit(s) are connected to the sensors by any means suitable for transmitting the signals received.

[0019] The computing unit(s) include means of storing information, as well as computing means for implementing the algorithms of the figures 2 et 3 , depending on whether we are in the unclaimed single-source or dual-source case.

[0020] Advantageously, reception is done on a space diversity network (interferometric network) and the demodulation of the signal allowing the "downgrade to baseband" is carried out by the same local oscillator for all the antennas in order to ensure coherence. The received signal is sampled in real or complex (double quadrature demodulation or any other process) on each channel.

[0021] The received signal, filtered in a band of typically several hundred MHz, is modeled by: s 0 ( t ) e i 2 πf 0 t < . This model does not make any assumptions about the type of modulation.

[0022] This signal is sampled at a rate Te such that 1 / Te >> 2 x useful signal bandwidth.

[0023] We calculate the weighted and overlapping Discrete Fourier Transforms (DFT) of this signal on N DFT points. The weighting serves to reduce the side lobes. Since this weighting induces a variation in the contribution of the data to the DFT (the data in the center of the DFT temporal support being given a much greater weight than the data on the edges of the support), which can go as far as a loss of short signals, we assume an overlap of the temporal supports.

[0024] The collected measurements are therefore the results of the DFTs. They constitute a time-frequency grid, the boxes of which are called time-frequency boxes. Each box of the grid contains a complex vector which is the result of the Discrete Fourier Transforms for all the channels, for a given time interval and frequency interval.

[0025] For the signal frequencies and distances involved here, the wavefront is considered to be flat. Therefore, the antennas receive the signal with a phase difference that is a function of the two angles between the wave plane and the antenna plane.

[0026] In the unclaimed single-source case, the measurements collected on the complete network are therefore written as: X n = s n U + W n X n represents the measures: X n is a complex column vector of dimension P, where P is the number of channels and n = 1,2, ... N is a double index (I,c) spanning the space of times (Fourier transform index) and frequencies (channel number for a Fourier transform). More precisely, the index n spans the boxes of a rectangular time x frequency window of size N. The indices I and c correspond to the row and column numbers of the boxes of the window. sn is a complex number representing the signal after DFT W n is the thermal noise in the time / frequency box of index n. W n is a column vector of dimension P. W n is Gaussian on its real and imaginary components, independent from one time / frequency box to another and independent from one antenna channel to another. In other words, W n is white spatially, frequently and temporally. The standard deviation of the noise counted on each real or imaginary component, at the level of a time / frequency bin, is equal to σ. To verify the hypothesis of independence of the noise from one bin to another, we limit the overlap of the TFDs to 50%.

[0027] In the case of a monopolarization interferometric network, U is written in the form: U = g u 1 ⋯ u P , where g is a complex scalar depending on the polarization of the incident wave and its direction of arrival, and where the u i are complex numbers of modulo 1 representing the geometric phase shifts associated with the direction of the incident wave. One of the receiver antennas can be chosen as the phase reference.

[0028] In the case of a bipolarization interferometric network, U is written in the form: U = hH + νV , where h and v are complex scalars such that | h | 2< + | v| 2< = 1 which express the polarization of the incident wave, and where H (resp V) is the response of the grating to a horizontally (resp vertically) polarized wave. H and V depend only on the direction and frequency of the incident wave.

[0029] In all cases, U is of dimension P, where P is the number of antennas used.

[0030] We can consider that U is normed, and that s n carries the signal strength and average network gain.

[0031] U is preserved by the TFD which is a linear transformation, and is found on the signal at the output of TFD.

[0032] In the general case there may be a mixture of K signals (K possibly being greater than the resolution of the network). The signal is then written: X n = ∑ k = 1 N s k n U k + W n ; n = 1 , 2 , … N

[0033] The basic assumption is that when the set of N time / frequency bins is restricted to a rectangular area or window of index j, the complexity of the environment is such that in such a window, the mixing of signals is limited to two signals.

[0034] So the model becomes: X jn = s j 1 n U j 1 + s j 2 n U j 2 + W n

[0035] Windows of predetermined size are defined to cover the time-frequency grid. The set of these windows forms a division of the grid. The size of the windows is chosen so that at most two signals from two sources are present in each window.

[0036] The vector U or vectors U and V designating the unit directional vector(s) formed by the incident signal or signals relative to the network are then extracted (or estimated), by any suitable known means.

[0037] A loop is performed to go through all the windows defining the grid division.

[0038] For each window, a step of estimating the signal(s) is implemented using the conditional expectation and the approximations that can be made to it for the specific model of the signals.

[0039] The final treatment is broken down into: a non-linear step of deciding the situation in each time / frequency box: source 1 present and source 2 present, or: source 1 present and source 2 absent, or: source 1 absent and source 2 present, or: source 1 absent and source 2 absent, then a linear filtering step specific to the situation.

[0040] The processing obtained is unbiased, almost optimal in the sense of the mean square error, and delimits in time / frequency the support of the signal to be estimated.

[0041] The signal estimation is carried out over a window where the situation is single-source or dual-source.

[0042] In single source and in an unclaimed manner, the vector signal measured for the index box n is X n = s n U + W n , n = 1 , 2 , …

[0043] In bisources, we have: X n = s n U + c n V + W n , n = 1 , 2 , …

[0044] In the writings of Equation 3 and Equation 4, U and V are unit directional vectors of a monopolarization or bipolarization network, s n And c n are complex signals that must be estimated; W n is the noise which is spatially and n-white, Gaussian, centered and of covariance: E W n W n * = 2 σ 2 I p Or I p is the unit matrix of C p< (P being the number of reception channels). *< denotes the conjugate transpose.

[0045] In the following, we will assume that U or V are estimated by Û And V̂ for example using a MUSIC type method, and that this estimate having been made on a number of boxes N >>1, we can make the approximation U = Û , V = V̂.

[0046] To take into account the fact that s n (Or c n ) can be zero for some n, without knowing in advance the modulation of the signal, we model s n (Or c n ) as independent samples in n of a random variable whose probability density is a mixture (q, 1-q, q<1) of two centered Gaussians of respective variance 2 σ j 2 (j=1 for s n and j=2 for c n ), and 2 τ 2< , with τ 2 < σ 2 < σ j 2 . 2 σ j 2 is the power of the useful signal if it is present when we neglect τ 2< in the expression 1 − q 2 τ 2 + q 2 σ j 2 of average power.

[0047] τ is a regularization parameter of the model which allows the application of the Bayes formula for probability laws admitting probability densities with respect to the Lebesgue measure; however, the physical reality is that there is an absence of signal when modeling it using the centered Gaussian of variance 2 τ 2< (power 2 τ 2< ). This is why, at the end of the calculations, we only keep the limit of the expressions when τ →0.

[0048] s n And c n are considered independent.

[0049] The measured signal is written in both cases, single source or dual source, in the unique form: X n = MS n + W n , n = 1 , 2 , … N Or M = U And S n = s n in single source, and M = (U V ) , S n = ( s n c n ) T< in bisources, with known M and σ 2< known.

[0050] In the rest of the document, to simplify the notations, it is understood that the processing operates on each index box n independently, we omit the index n e t we note X the measurement in a time / frequency box, s the claimed non-T single-source signal and S = ( s c ) T< the bi-source signal in a time / frequency box.

[0051] A priori knowledge about S is given by a probability density. In the unclaimed single-source case: p s = q 2 πσ 1 2 exp − s 2 2 σ 1 2 + 1 − q 2 πτ 2 exp − s 2 2 τ 2

[0052] In the dual-source case: p S = ∑ j q j π 2 detC j exp − S * C j − 1 S où q 1 = q 2< , q 2 = q 3 = q (1 - q ), q 4 = (1 - q ) 2< if the sources are considered independent and equiprobable (in the following we generalize to a distribution q 1 , q 2, q 3, q4 of the 4 situations not linked by the expressions above), And C 1 = 2 σ 1 2 0 0 2 σ 2 2 , C 2 = 2 σ 1 2 0 0 2 τ 2 , C 3 = 2 τ 2 0 0 2 σ 2 2 , C 4 = 2 τ 2 0 0 2 τ 2 are the covariance matrices of S for the four possible cases.

[0053] We estimate S by means of the conditional expectation using the unclaimed single-source and two-source models (Equation 5, Equation 6 and Equation 5, Equation 7).

[0054] Conditional Expectation (CE) is the estimator Ŝ which minimizes the mean square deviation E(∥ S - Ŝ ∥ 2< ). It is also unbiased, and gives an explicit solution for Ŝ . It is constructed as follows: Let X be the measure; its probability density which depends on the parameter to be estimated S is interpreted as the conditional probability density of X knowing S. We therefore have p(X / S) and p(S) coming from a priori knowledge of S.

[0055] S is given by the explicit formula: S ^ = ∫ domS Sp S / X dS p ( S / X), the conditional probability density of S given X, is obtained by Bayes' formula. p S / X = p X / S . p S p X Avec p X = ∫ domS p X / S . p S dS

[0056] In the event that X = MS + V And p ( S ) is Gaussian, centered with covariance C, we can find analytically Ŝ , which is not generally the case.

[0057] It is a linear function of X. Indeed (in dimension 2): p X / S = 1 π 2 4 σ 4 exp − X − MS 2 2 σ 2 p S = 1 π 2 det C exp − S * C − 1 S p X / S p S = 1 π 4 4 σ 4 det C exp − X 2 2 σ 2 + X * MS 2 σ 2 + S * M * X 2 σ 2 − S * M * MS 2 σ 2 − S * C − 1 S Let's ask Σ − 1 = M * M 2 σ 2 + C − 1

[0058] By completing the “square” in S, we have: p X / S p S = K 2 det C exp − S − Σ 2 σ 2 M * X * Σ − 1 S − Σ 2 σ 2 M * X − X 2 2 σ 2 + X * M Σ M * X 4 σ 4 Or K 2 is a constant (=1 / π 4< 4 σ 4< ) in dimension 2. Σ is a positive definite Hermitian matrix. We deduce: ∫ p domS X / S p S dS = K 2 π 2 det Σ det C exp − X 2 2 σ 2 + X * M Σ M * X 4 σ 4 ∫ domS S . p X / S p S dS = K 2 π 2 det Σ det C exp − X 2 2 σ 2 + X * M Σ M * X 4 σ 4 ⋅ Σ 2 σ 2 M * X

[0059] The complete expressions of Equation 12 will be used to find the desired estimator for our problem.

[0060] In the case where S is a Gaussian sample, Equation 9, Equation 10, Equation 11, Equation 12, and Equation 13 give: S ^ = Σ 2 σ 2 M * X which can also be written: S ^ = 2 σ 2 Σ − 1 − 1 M * X = M * M + 2 σ 2 C − 1 − 1 M * X

[0061] We conclude that if 2 σ 2< C -1< << M *< M , Ŝ reduces to the maximum likelihood estimator of S using the model of Equation 5 without a priori knowledge about S : Ŝ MV = ( M *< M ) -1< M *< X .

[0062] The condition 2σ 2< C -1< << M *< M as matrices, is also expressed by C » 2σ 2< ( M *< M ) -1< which means that the a priori on S which is defined by C does not provide any real information on S.

[0063] The signal estimate in the unclaimed single-source case is as shown below.

[0064] In the unclaimed one-dimensional case for S=s (single source), we have M=U and therefore M*M = 1 ;

[0065] The matrix C is reduced to the constant c; p X / S p S = 1 2 πσ 2 exp − X − Us 2 2 σ 2 ; p s = 1 π . c exp − s 2 c

[0066] We deduce from this Σ − 1 = 1 2 σ 2 + 1 c ou Σ = 2 σ 2 c 2 σ 2 + c p X / S p S = K 1 c exp − 2 σ 2 + c 2 σ 2 c 1 − c 2 σ 2 + c U * X 2 − X 2 2 σ 2 + c 2 σ 2 2 σ 2 + c U * X 2 with K 1 = 1 2 πσ 2

[0067] Then he comes: ∫ doms s . p X / S . p S = K 1 c π 2 σ 2 c 2 σ 2 + c exp − X 2 2 σ 2 + c 2 σ 2 2 σ 2 + c U * X 2 . c 2 σ 2 + c U * X ∫ doms s . p X / S p S = K 1 c π 2 σ 2 c 2 σ 2 + c exp − X 2 2 σ 2 + c 2 σ 2 2 σ 2 + c U * X 2

[0068] We obtain the conditional expectation, in the Gaussian case for s, by the quotient of Equation 15 by Equation 16: s ^ = c 2 σ 2 + c U * X

[0069] Si c >> 2 σ 2< , which expresses that we have no a priori information on s, ŝ reduces to the maximum likelihood estimator: Ŝ MV = U *< X

[0070] The signal estimate in the dual-source case is as shown below.

[0071] The conditional expectation estimator is obtained using Equation 15 and Equation 16 for the mixture density of S given by Equation 7.

[0072] After simplification by the common factor K 2 π 2 exp − X 2 2 σ 2 in all terms in the numerator and denominator, we obtain: S ^ = ∑ j q j det Σ j det C j exp 1 4 σ 4 X * M Σ j M * X Σ j 2 σ 2 M * X ∑ j q j det Σ j det C j exp 1 4 σ 4 X * M Σ j M * X

[0073] With ∑ j − 1 = M * M 2 σ 2 + C j − 1 ou ∑ j = 2 σ 2 M * M + 2 σ 2 C j − 1 − 1

[0074] Let's ask Γ j = 2 σ 2 C j − 1 (dimensionless), Q j = M * M + 2 σ 2 C j − 1 − 1 = M * M + Γ j − 1 = ∑ j / 2 σ 2

[0075] We have: S ^ = ∑ j = 1 4 q j det Q j det Γ j exp X * MQ j M * X 2 σ 2 Q j M * X ∑ j = 1 4 q j det Q j det Γ j exp X * MQ j M * X 2 σ 2

[0076] According to Equation 8, M * M = 1 U * V V * U 1 Γ 1 = σ 2 / σ 1 2 0 0 σ 2 / σ 2 2 , Q 1 − 1 = 1 + σ 2 / σ 1 2 U * V V * U 1 + σ 2 / σ 2 2 Γ 2 = σ 2 / σ 1 2 0 0 σ 2 / τ 2 , Q 2 − 1 = 1 + σ 2 / σ 1 2 U * V V * U 1 + σ 2 / τ 2 Γ 3 = σ 2 / τ 2 0 0 σ 2 / σ 2 2 , Q 3 − 1 = 1 + σ 2 / τ 2 U * V V * U 1 + σ 2 / σ 2 2 Γ 4 = σ 2 / τ 2 0 0 σ 2 / τ 2 , Q 4 − 1 = 1 + σ 2 / τ 2 U * V V * U 1 + σ 2 / τ 2

[0077] We deduce from this: Q 1 = 1 1 + σ 2 / σ 1 2 1 + σ 2 / σ 2 2 − U * V 2 1 + σ 2 / σ 2 2 − U * V − V * U 1 + σ 2 / σ 1 2 Q 2 = 1 1 + σ 2 / σ 1 2 1 + σ 2 / τ 2 − U * V 2 1 + σ 2 / τ 2 − U * V − V * U 1 + σ 2 / σ 1 2 Q 3 = 1 1 + σ 2 / τ 2 1 + σ 2 / σ 2 2 − U * V 2 1 + σ 2 / σ 2 2 − U * V − V * U 1 + σ 2 / τ 2 Q 4 = 1 1 + σ 2 / τ 2 − U * V 2 1 + σ 2 / τ 2 − U * V − V * U 1 + σ 2 / τ 2

[0078] The expression of the signal estimate is approximated to allow its estimation as indicated below in the two-source case.

[0079] The products of determinants in Equation 19 are respectively equal to the following expressions which are approximated for a good signal-to-noise ratio ( σ 1 2 / σ 2 ≫ 1 And σ 2 2 / σ 2 ≫ 1 ) and for τ → 0. det Q 1 det Γ 1 = 1 + σ 2 / σ 1 2 1 + σ 2 / σ 2 2 − U * V 2 − 1 σ 4 / σ 1 2 σ 2 2 ≈ 1 − U * V 2 − 1 σ 4 / σ 1 2 σ 2 2 det Q 2 det Γ 2 = 1 + σ 2 / σ 1 2 1 + σ 2 / τ 2 − U * V 2 − 1 σ 4 / σ 1 2 τ 2 ≈ τ 2 / σ 2 1 + σ 2 / σ 1 2 − 1 . σ 4 / σ 1 2 τ 2 ≈ σ 2 / σ 1 2 det Q 3 det Γ 3 = 1 + σ 2 / τ 2 1 + σ 2 / σ 2 2 − U * V 2 − 1 σ 4 / σ 2 2 τ 2 ≈ σ 2 / σ 2 2 det Q 4 det Γ 4 = 1 + σ 2 / τ 2 2 − U * V 2 − 1 σ 4 / τ 4 ≈ 1

[0080] Similarly, we find for Q j when τ → 0: Q 1 is unchanged. Q 1 ≈ 1 1 + σ 2 / σ 1 2 1 + σ 2 / σ 2 2 − U * V 2 1 − U * V − V * U 1 Q 2 ≈ 1 / 1 + σ 2 / σ 1 2 0 0 0 Q 3 ≈ 0 0 0 1 / 1 + σ 2 / σ 2 2 Q 4 ≈ 0 0 0 0

[0081] It is observed that the products of Q i det Γ i have a finite limit in each of the four situations, as do the matrices Q j , which is satisfactory behavior.

[0082] We thus obtained a first expression of the estimator.

[0083] In reality, only one of the terms in Equation 19 is predominant for each box, which leads to a first simplification. From this, we deduce the new estimation treatment of Ŝ . S ^ = Q j 0 M * X où j 0 = Arg Max j q j det Q j det Γ j exp X * MQ j M * X / 2 σ 2

[0084] Which we simplify to: S ^ = Q j 0 M * X où j 0 = Arg Max j F j = Arg Max j ln q j det Q j det Γ j + X * MQ j M * X / 2 σ 2

[0085] M *< X is given by: M * X = U * X V * X

[0086] The det Q j .det Γ j are given by Equation 21.

[0087] THE Q j are given by Equation 22, Les q j sont donnés par exemple par q j = q 2 , j = 1 q 1 − q , j = 2 , 3 , 1 − q 2 , j = 4 if there is independence of the 4 situations and equiprobability for s ≠ 0, c ≠ 0.

[0088] F(j) is given by: F j 1 = ln q 1 det Q 1 det Γ 1 + X * MQ 1 M * X / 2 σ 2 = ln q 2 1 − U * V 2 σ 4 / σ 1 2 σ 2 2 + X * MQ 1 M * X / 2 σ 2 F j 2 = ln q 2 det Q 2 det Γ 2 + X * MQ 2 M * X / 2 σ 2 = ln q 1 − q σ 2 / σ 1 2 + X * UU * X / 2 σ 2 F j 3 = ln q 3 det Q 3 det Γ 3 + X * MQ 3 M * X / 2 σ 2 = ln q 1 − q σ 2 / σ 2 2 + X * VV * X / 2 σ 2 F j 4 = ln q 4 det Q 4 det Γ 4 + X * MQ 4 M * X / 2 σ 2 = ln 1 − q 2

[0089] For j 0 = 1, we find the maximum likelihood estimator. Indeed in this case Γ 1 = σ 2 / σ 1 2 0 0 σ 2 / σ 2 2 ≈ 0 And Q 1 = ( M *< M ) -1< so that Ŝ = ( M *< M ) -1< M *< X . We note that if U *< V = 0, then the estimates of s and c are completely separate, because then the relationship Ŝ = ( M *< M ) -1< M *< X simplifies and decouples into ŝ =U *< X , ĉ = V *< X . For j 0 = 2, Ŝ T< = ( U *< X, 0) (filtering by the directional vector of source 1) For j 0 = 3, Ŝ T< = (0 , V *< X )(filtering by the directional vector of the source 2) For j 0 = 4 , Ŝ T< = (0,0)

[0090] Where the symbol T< denotes the transpose.

[0091] We have thus "linearized" the optimal treatment, since we have obtained four linear filters, controlled by the decision on the type of situation for each box: both sources are present / source 1 is present / source 2 is present / neither source is present. We call the estimator obtained "Conditional Expectation with 4 Linear Filters".

[0092] In this way, the optimal estimator was simplified, by decomposing it into two steps: Situation detection Applying appropriate filtering to the situation

[0093] It is satisfactory to note that the estimator is independent of τ , which is the expected behavior since τ is not a physical parameter, but an artifice allowing the “absence of signal” situation to be modeled by a very pinched Gaussian.

[0094] This estimator requires the calculation of three quadratic forms and a test. The difficult point remains the calculation of the unknown parameters q j,j =1...4, σ 1 2 , σ 2 2 (the power of noise 2 σ 2< is assumed to be known).

[0095] In the particular case where the two sources are independent, and have the same power and the same presence rate, the parameters can be estimated by calculating the empirical moments of order 2 and 4.

[0096] If we call 2σ' 2< the common value of the variance of the Gaussian representing each source, and q the probability common to the 2 sources, everything happens as if we were in a single-source situation, with a single source of variance 2 σ M ′ 2 = 2 σ ′ 2 and probability of presence q M = 2q . σ M ′ 2 And q M are then given by the following equations: 1 N ∑ n X n 2 = 2 q M σ M ′ 2 P + σ 2 + 1 − q M 2 σ 2 1 N ∑ n X n 4 = 8 q M σ M ′ 2 P + σ 2 2 + 8 1 − q M σ 4

[0097] In the general case (independent sources), there are 4 model parameters: q 1 , q 2, σ 1 2 , σ 2 2 . Those skilled in the art know how to generalize the above method of moments to higher orders to obtain estimates of these parameters.

[0098] A variation of the estimation treatment is to simplify the decision step described previously, as follows: The conditional expectation considers the four possible situations: s ≠ 0 , c ≠ 0 ; s ≠ 0 , c = 0 ; s = 0 , c ≠ 0 ; s = 0 , c = 0

[0099] Normally, a four-hypothesis decision problem would be required.

[0100] To simplify, we propose to test s ≠ 0 against s = 0 independently of c on the one hand, and to test c ≠ 0 against c = 0 independently of s on the other hand. We therefore perform two two-hypothesis tests instead of one four-hypothesis test.

[0101] These tests will be carried out from the pre-processed measurements U *< X, V *< X.

[0102] TEST DE s ≠ 0 CONTRE s = 0 H 1 : U * X = s + U * V . c + u V * X = V * U . s + c + ν , s ≠ 0 H 0 : U * X = U * V . c + u V * X = c + ν , s ≠ 0 where the (.) indicates the scalar x scalar product and where u = U *W, v = V *< W : ( u, v ) is therefore Gaussian, centered and of covariance: E U * W V * W W * U W * V = 2 σ 2 1 U * V V * U 1 , and where c is an unknown parameter.

[0103] This is a problem invariant under the group of vector translations ( U *< V 1) T< and a linear hypothesis problem (see "Testing Statistical Hypothesis", 3rd edition, EL Lehmann, JP Romano, Springer, 2005). It can be treated by first performing a projection onto the orthogonal of (U*V 1) T< to eliminate c, then testing for the presence of s by a chi2 test.

[0104] The projection is written: U * X − U * V V * X = 1 − U * V 2 s + u − U * V ν H 1 u − U * V ν H 0

[0105] Le test at effectuer therefore relates to the measure | U *< X - ( U *< V ) V *< X | 2< : U * X − U * V V * X 2 > ou < λ

[0106] Which is the same as doing the test: s ^ MV 2 = U * X − U * V V * X 2 1 − U * V 2 > ou < λ ′ , Or ŝ MV is the maximum likelihood estimate of s. TEST DE c ≠ 0 CONTRE c = 0

[0107] In the same way for c, we project M*X onto the orthogonal of (1 V *< U ) T< in order to eliminate the terms in s and we obtain the new measure to consider: − V * U U * X + V * X = 1 − U * V 2 c + ν − V * U u H ′ 1 ν − V * U u H ′ 0

[0108] We obtain the following test: V * U U * X − V * X 2 > ou < μ that we can write c ^ MV 2 = V * U U * X − V * X 2 1 − U * V 2 > ou < μ ′ Or ĉ MV is the estimate of c in the sense of the maximum likelihood of c.

[0109] In bisources the proposed estimator consists of carrying out the following operations: As illustrated in the figure 3 , from the calculation of the two-source maximum likelihood estimator Ŝ MV = ( ŝ , ĉ ) = ( M *< M ) -1< M *< X performed in step 220, a thresholding of the modulus of each component of Ŝ MV is performed in step 320 on each of the components | ŝ MV | and | ĉ MV |.

[0110] Then, depending on the situation, spatial filtering is applied in steps 331 to 334 under the following conditions, which makes it possible to obtain the so-called Independent Decision Conditional Expectation (IDCE) estimator: ∘ Si s ^ MV > seuil et c ^ MV > seuil : S ^ ECDI = M * M − 1 M * X ∘ Si s ^ MV > seuil et c ^ MV < seuil : s ^ ECDI = U * X , c ^ ECDI = 0 ∘ Si s ^ MV < seuil et c ^ MV > seuil : s ^ ECDI = 0 , c ^ ECDI = V * X ∘ Si s ^ MV < seuil et c ^ MV < seuil : S ^ ECDI = 0

[0111] In the unclaimed single-source case, the proposed estimator consists of performing the following operations, as illustrated in the figure 2 : From the calculation of the maximum likelihood estimator Ŝ MV =U*X performed in step 210, a thresholding of the module of Ŝ MV is performed in step 350, then, depending on the situation, spatial filtering is applied in steps 361 or 362 under the following conditions: ∘ Si s ^ MV > seuil s ^ ECDI = U * X ∘ Si s ^ MV < seuil s ^ ECDI = 0

[0112] Advantageously, the threshold for steps 320, 350 is set as follows: We call Pfa the probability of deciding s ≠ 0 while s = 0 and Pd the probability of deciding s ≠ 0 while s ≠ 0.

[0113] For example, and in a non-limiting manner, it is proposed to use the Neyman-Pearson criterion which consists of fixing the Pfa (for example a few percent) and in return maximizing Pd, which makes it possible to obtain a threshold on λ' And µ' . For example, and in no way limiting, one can also adjust the value of λ' (resp µ ') such that 1 - Pd = Pfa around a fixed RSB.

[0114] A variant close to Maximum Likelihood and called Thresholded Maximum Likelihood (TSM) consists of carrying out the following operations: Calculation of the maximum likelihood estimator Ŝ MV = ( M *< M ) -1< M * X Thresholding of the modulus of the components of Ŝ MV Depending on the situation, application of spatial filtering ∘ Si s ^ MV < seuil : s ^ MVS = 0 , sinon s ^ MVS = s ^ MV ∘ Si c ^ MV < seuil : c ^ MVS = 0 , sinon c ^ MVS = c ^ MV

[0115] Another variation is to use one of the two previous estimators to obtain an initialization of the unknown parameters q j,j =1..4 , σ 1 2 , σ 2 2 , then apply the Conditional Expectation estimator or the Conditional Expectation estimator with 4 Linear Filters.

Claims

1. A method for the non-linear estimation, by calculation means, of two mixed signals from separate sources, the time / frequency representation of which shows an unknown non-zero proportion of zero components, using an array made up of P>2 antennas, when the directional vectors U and V of the sources emitting these signals are additionally known or estimated, including the following steps: a) Calculating the successive discrete Fourier transforms of the signal received by the antennas and sampled to obtain a time-frequency P-vector grid of the signal; each element of the grid being referred to as a box and containing a complex vector X forming a measurement; b) For each box, calculating an estimation of the dual-source signal S = (s,c)T, s being the signal associated to the first source, and c being the signal associated to the second source, and (.)T being the transposed of (.), by an approximation of the conditional expectation estimator of the signal, or of the signals, the probability density of S being modeled as a Gaussian mixture centered and weighted by coefficients q1, q2, q3, q4 of the for possible situations in each case among, bothe sources present,: source 1 present and source 2 absent, source 1 absent and source 2 present, both sources absent, said approximation being S ^ = ∑ j = 1 k q j det Q j det Γ j exp X * MQ j M * X 2 σ 2 Q j M * X ∑ j = 1 4 q j det Q j det Γ j exp X * MQ j M * X 2 σ 2 where (.) * is the conjugated transpose of matrix (.); and where : Q 1 = 1 1 + σ 2 / σ 1 2 1 + σ 2 / σ 2 2 − U * V 2 1 − U * V − V * U 1 Q 2 = 1 / 1 + σ 2 / σ 1 2 0 0 0 Q 3 = 0 0 0 1 / 1 + σ 2 / σ 2 2 Q 4 = 0 0 0 0 and where det Q1det Γ1, det Q2det Γ2, det Q3det Γ3, det Q4det Γ4 are chosen respectively equal to: 1 − U * V 2 − 1 σ 4 / σ 1 2 σ 2 2 , σ 2 / σ 1 2 , σ 2 / σ 2 2 , and 1 ; and where 2 σ 1 2 , 2 σ 2 2 and respectively the power of the signal from source 1, the power of the signal from source 2, and where 2σ2 is the power of the noise supposed Gaussian, spatially, frequentially and temporally white; and where M = (U V).

2. The method according to claim 1, characterized in that the calculation of the conditional expectation estimator is approximated by a Conditional Expectation with 4 Linear Filters obtained by a four-hypothesis decision processing pertaining to four Hermitian forms of the measurement X, followed by linear filtering commanded by the result of the decision: according to equations : S ^ = Q j 0 M * X where j 0 = Arg max j ln q j det Q j det Γ j + X * MQ j M * X / 2 σ 2 , j ∈ 1 2 3 4 3. The method according to any of the preceding claims, characterized in that it includes a step for estimating parameters q j σ 1 2 σ 2 2 necessary to establish the conditional expectation using the method of moments operating on the boxes of a divided window in the time / frequency grid.

4. The method according to claim 2, characterized in that the calculation of the Conditional Expectation estimator with 4 Linear Filters is approximated by a treatment named "Conditional Expectation with Independent Decisions" obtained by two statistics tests with two-hypothesis, followed by linear filtering commanded by the result of the decision.

5. The method according to claim 4, characterized in that the decision is given by: - if |ŝMV| > threshold and |ĉMV| > threshold : both sources are presents, - if |ŝMV| > threshold and |ĉMV| < threshold : source 1 is present and source 2 is absent, - if |ŝMV| < threshold and |ĉMV| > threshold : source 1 is absent and source 2 is present, - if |ŝMV| < threshold and |ĉMV| < threshold : both sources are absents, where S ^ MV = M * M − 1 M * X .

6. The method according to claims 4 or 5, characterized in that, as a function of the result of the decision, the linear filtering processing yielding the "Conditional Expectation estimator with Independent Decisions" is ŜECDI = (ŝECDI ĉECDI)T: - is both sources are presents: the estimator of the dual-source maximum likelihood for the signal s and c: ŜECDI = (M * M) -1M * X, - if source 1 is present and source 2 is absent: the single-source maximum likelihood estimator for s and, 0 for c: ŜECDI = U * X, ĉECDI = 0, - if source 1 is absent and source 2 is present: 0 for s, and the single-source maximum likelihood estimator for c,: ŜECDI = 0, ĉECDI = V * X, - is both sources are absents: 0 for the signals of the two sources : ŜECDI = 0,7. The method according to any of claims 4 to 6, characterized in that the calculation of the Conditional Expectation estimator with 4 Linear Filters is approximated by a treatment named Thresholded Maximum Likelihood, ŜMVS = (ŝMVS ĉMVS)T obtained by estimating the signal(s) by the dual-sources maximum likelihood method followed by the comparison of each estimate to a threshold, according to equations: - if s ^ MV < threshold : s ^ MVS = 0 , else s ^ MVS = s ^ MV - if c ^ MV < threshold : c ^ MVS = 0 , else c ^ MVS = c ^ MV 8. The method according to any of claims 4 to 7, characterized in that the or each decision threshold is chosen to respect a so-called false alarm likelihood consisting of declaring the signal to be non-zero when it is zero.

9. The method according to claim 7, characterized in that it includes: • A first estimate of the signals done using the Conditional Expectation with Independent Decisions or Thresholded Maximum Likelihood method, • An estimate of parameters q j σ 1 2 σ 2 2 done from the components of the signal obtained in the previous step, • A second estimate of the signals done using the "Conditional Expectation method or the Conditional Expectation with 4 Linear Filters" method, informed of the values of the parameters q j σ 1 2 σ 2 2 obtained in the previous step.