EP1685686A1

EP1685686A1 - Method for the higher order blind demodulation of a linear wave-shape emitter

Info

Publication number: EP1685686A1
Application number: EP04804508A
Authority: EP
Inventors: Anne Thales Intellectual Property FERREOL; Laurent Thales Intellectual Property ALBERA; Joséphine Thales Intellectual Prop. CASTAING
Original assignee: Thales SA
Current assignee: Thales SA
Priority date: 2003-11-07
Filing date: 2004-10-29
Publication date: 2006-08-02
Also published as: US20070140380A1; FR2862173B1; WO2005046150A1; US7787571B2; FR2862173A1

Abstract

The invention relates to a method for the blind demodulation of a linear wave-shape source or emitter in a system comprising at least one source, a network of sensors, and a propagation channel. Said method comprises at least the following steps: the time symbol T is determined and sampled to Te samples per symbol such as T=ITe (I whole); a spatio-temporal observation z(t) having mixed sources which are symbol trains of the emitter is constructed from the observations x(kTe); an ICA-type method is applied to the observation vector z(t) in order to estimate the LC symbol trains { am-i } associated with the channel vectors Hzj = Hz (kj); the LC outputs (Am,j, Hzj) are classed in the same order as the inputs (am-i, hz(i)) in order to obtain the propagation channel vectors Hz, j =, Hz (kj); and the αimax phase associated with the outputs is determined.

Description

METHOD FOR BLIND DEMODULATION AT HIGHER ORDERS OF A LINEAR WAVEFORM TRANSMITTER

The object of the invention relates to a method for blind demodulation of signals transmitted by several transmitters and received by a network composed of at least one sensor.

It applies for example for an antenna array in an electromagnetic context. The object of the invention relates in particular to the demodulation of signals, that is to say the extraction of the symbols {a ^ emitted by a linearly modulated transmitter.

FIG. 1 shows an antenna processing system comprising several transmitters Ei and an antenna processing system T comprising several antennas Ri receiving radioelectric sources with different angles of incidence. The angles of incidence of the sources or transmitters can be configured either in 1D with the azimuth θ _m or in 2D with the azimuth angle θ _m and the elevation angle Δ _m . Figure 3 shows schematically a principle of modulation and demodulation of the symbols {a _k } emitted by a transmitter. The signal propagates through a multipath channel. The transmitter transmits the symbol a _k at the instant kT, where T is the symbol period. Demodulation consists in estimating and detecting the symbols in order to obtain the symbols estimated â at the output of the demodulator. In this figure, the symbol train {a} is filtered linearly on transmission by an emission filter H also called shaping filter h ₀ (t).

In the following description, we define under the expression "blind demodulation", techniques which do not use a priori information on the transmitted signal: shaping filter, learning sequence, etc. The last ten years have seen the development of SIMO blind demodulation techniques, abbreviated single input multiple outputs, (in abbreviation Single Input Multiple Output) called subspace using second order statistics, as described in reference [7]. These algorithms have the disadvantage of not being robust either to an underestimation or to an overestimation of the order of the propagation channel: temporal spreading dependent on the multipaths and the shaping filter. To circumvent this problem, a linear prediction technique has been proposed, described in reference [11], which has the disadvantage of being less efficient when the length of the channel is known. To improve the techniques with subspace, the method described in [16] proposes a parametric technique which unfortunately requires the knowledge of the shaping filter.

In reference [13], the authors propose a technique based on covariance matching, notably having the disadvantage of being very difficult to implement. This is how a sub-optimal technique described in reference [12] was developed, which is easier to implement by minimizing a likelihood criterion and assuming the Gaussian character of the symbols. This hypothesis is not verified for commonly used linear modulations such as PSK (Phase Shift Keying) or QAM (English abbreviation for Quadrature Amplitude Modulation).

It is also known in CMA methods (abbreviated as English Constant Modulus Algorithm) to use a spatio-temporal approach described for example in reference [6]. This family of methods has the drawback, however, of being suitable only for a particular class of modulations such as PSKs which have a constant modulus. This method is iterative and therefore has the disadvantage of having to be correctly initialized. Finally, the CMA methods have the disadvantage of converging less quickly than the subspace method previously stated. On the other hand, reference [20] describes a method to subspace exploiting higher order statistics for identifying Finite Impulse Response (FIR) and non-minimum phase channels.

The object of the present invention relates to a method based in particular on techniques for separating blind sources known to those skilled in the art and described for example in references [4] [5] [15] [19] assuming that the symbols issued are statistically independent. For this, the method constructs a spatio-temporal observation whose mixed sources are trains of symbols of the transmitter. Each symbol train is for example the same symbol train shifted by an integer of symbol period T.

The invention relates to a method for blind demodulation of a source or transmitter of linear waveform in a system comprising one or more sources and an array of sensors and a propagation channel, characterized in that it comprises at least the steps following:

• determine the symbol time T and we sample at Te such that T = ITe (I integer),

• from the observations x (kTe), construct a spatio-temporal observation z (t) whose mixed sources are symbol symbol trains,

• apply an ICA type method on the observation vector z (t) to estimate the L ₀ symbol trains {a _m .j} associated with the channel vectors h ^ = £. (&, -), • order the L _c outputs (â _m , „h _ZJ .) In the same order as the inputs

(a _m -i _ι h _z (i)) in order to obtain the propagation channel vectors

• determine the phase αimax associated with the outputs.

The method according to the invention notably offers the following advantages: • It makes no assumption on the constellations of symbols unlike the methods described in the prior art,

• It does not require knowledge of the shaping filter,

• The symbol module is not supposed to be constant,

• It is robust to an overestimation of the length of the channel,

• It makes it possible to deal with the case of propagation channels with correlated paths,

• It is direct and simple to implement without a step of cross-checking the correlated paths.

Other characteristics and advantages of the subject of the present invention will appear better on reading the description which follows given by way of illustration and in no way limiting on reading the appended figures which represent:

Figure 1 an example of architecture,

FIG. 2 the angles of incidence of the sources,

FIG. 3 the process of linear modulation and demodulation of a train of symbols,

FIG. 4 the diagram of a linear modulation transmitter,

FIG. 5 a summary of the general principle implemented in the invention,

Figure 6 the representation of a constellation,

FIG. 7 a first example of implementation of the method where the signal is received in baseband,

FIG. 8 a second example where the signal is received in baseband and the multi-paths are decorrelated,

FIG. 9 a third example where the signal is received in baseband and the multipaths are decorrelated by group. In order to better understand the method according to the invention, the following description relates to a method of demodulation blind to higher orders of a linear waveform transmitter in a network having a structure such as that described in FIG. 1, for example. Before explaining the steps implemented by the method, we describe the signal model used Model of the signal emitted by a source or transmitter Linear modulation

Figures 3 and 4 show the process of linear modulation of a symbol train {a at rate T by a formatting filter ho (t).

The comb of symbols c (t) is firstly filtered by the shaping filter ho (t) and then transposed to the carrier frequency f ₀ . The NRZ filter, which is a time window of length T, very often defined by ho (t) = ITτ (tT / 2), is a particular nonlimiting example of an emission filter. In radiocommunications, it is also possible to use the Nyquist filter whose Fourier transform h ₀ (f) ≈IlB (fB / 2) approaches a window of band B, when the roll-off is zero then ho (f) = IlB (fB / 2) (the roll-off defines the slope of the filter outside of band B). The modulated signal s ₀ (t), emitted by the transmitter, is written at time t _k = kT _θ

(T _θ : sampling period) as a function of the comb of symbols c (t):

(1) s ₀ (kT _θ ) = ∑ ho (iTe) c ((ki) T _θ ) i

Let us take a symbol time T equal to an integer number of times the sampling period, T = IT _Θ and set k = ml + j with 0 <j <I. Since c (t) = ∑ _r a _r δ (t- rlT _e ), in other words, as c (t) = a _u for t = ulT _θ and c (t) = 0 for t ≠ ulT _e , the only values of i for which c ((ki) T _θ ) is no none verify k- i = ul, that is to say such that i = ml + j-ul = nl + j where n = mu. Finally, expression (1) becomes: _. (2)

So (mlT _e + jT _e ) = 2 ^ ho (nlT _e + jT _θ ) a _m . _n for 0 ≤ j <I n = -_ _D

The parameter is the half-length of the emission filter which is spread over a period of (2L ₀ +1) IT _e . In the particular case of an NRZ emission filter, we obtain l_o = 0. As for the signal emitted s (t), it checks s (t) = so (t) exp (j2πfot) because it is equal to the signal so (t) transposed to the frequency fo. Under these conditions the expression of s (mlT _θ + jT _θ ) is according to (2):

s (mlT _e + jT _e ) = J ho (nlT _e + jT _e ) exp (j2πf ₀ (nl + j) Te) a _m - _n expQ2πf ₀ (mn) IT _e ) * '

AT,

= ∑ h _F0 (n IT _θ + jT _e ) b _mn such that 0 <j <I

"-Lo =

where hFo (iT _e ) = h (iT _e ) exp (j2πfoiT _e ) and bj≈aj expQ2îrfoilT _e )

Receiving signals on sensors

The signal s (t) (FIG. 3) transmitted passes through a propagation channel before being received on a network composed of N antennas. The propagation channel can be modeled by P multi-path incidence θ _p> delay τ _p and amplitude ρ _p (1 <p <P). At the output of the antennas we have the vector x (t) which corresponds to the sum of a linear mixture of P multi-paths and of a supposed white and Gaussian noise. This vector of dimension Λ6 1 has the following expression: p (A) x (t) = ∑ _Pp a (θ _p ) s (t-τ _p ) + b (t) = A s (t) + b (t ) ^the p = l

where p _p is the amplitude of the p ^ièmθ path, b (t) is the supposed Gaussian noise vector, a (θ) is the response of the sensor network to a source of incidence θ, A = [a (θι). .. a (θ _P )] and By noting that τ _p = r _p T + Δτ (where (0 <Δτ _p <T = IT _θ ) and r _p is an integer) and using expression (3) in equation (4), we obtain for the vector on the antennas : p L * (5) x (mlTe + jT _θ ) = ∑ J pp a (θ _p ) h _F0 (n IT _e + jT _e -Δτ _p ) £ „, _„ - _. „+ B (mlT _e + jT _e ) p = ln = -Io

By performing the following variable change u _p = n + r _p , the vector received by the antennas is expressed: p + Lo x (mlT _e + jTe) = ∑ ∑ p _p a (θ _p ) h _F0 ((u _p - r _p ) IT _e + jT _e -Δτ _p ) b _m _ _u + b (mlT _e + jT _e )

By noting r _m j _n = min {r _p } and r _max = max {r _p }, equation (6) can be rewritten as follows:

x (mlT _θ + jT _a ) = ∑ ∑ p _p a (θ _p ) h _F0 ((u- r _p ) IT _θ + jT _θ -Δτp) lnd _[r p-L0, + L _O ] (U) b _m _ _u

+ b (mlT _θ + jT _θ )

Where lnd [ _r , q] (u) is the usual indicator function for r <u <p and lnd [r, q] (u) = 0 otherwise) defined on the set of integers relative to value in the binary set {0, 1}, characterized by lnd [ _r , _q ] ( u) = 1 if u belongs to the interval [r, q] and lnd [ _r , q] (u) = 0 otherwise. Therefore, by noting v (t) the channel vector:

v (ulT _e + jT _e ) = where t = ulTβ + jTe and expression (5) becomes: ". (9) x (mlTe + jT _θ ) = v (ulT _θ + jT _θ ) b _mu + b (mlTθ + jT _θ )

Interference between symbols

The observation vector x (t) from the antenna array at time t = mlTe + jTβ makes, according to equation (9), the symbol b _m but also the symbols b _m -u where u is a relative integer belonging to the interval [r _m in-L ₀ , r _ma χ + Lo], a phenomenon which is better known as Interference Between Symbols (IES). Let L _c denote the number of symbols participating in the IES and limit the range of values taken by the latter. According to equation (9), if the intersection of the intervals [r _p -L ₀ , r _p + Lo] is not empty, then we have L _c = | r _ma χ - r _min | + 2L ₀ + 1. Therefore, when r _max = r _mi π, that is to say when, all the multipaths are correlated, the lower bound of L _c is reached and is worth L _c = 2L ₀ + 1. This case also mathematically translates to | maχ {τ _p } -min {τ _p } | <T. On the other hand, if the intersection of said intervals is empty, and that if necessary, all the intervals [r _p -Lo, r _p + Lo] are disjoint, then we have L _c = Px (2L ₀ + 1), which constitutes an upper bound on the set of values likely to be taken by L _c . This last case corresponds concretely to the case of multi-paths all decorrelated two by two, which mathematically can also be written Vi ≠ j, | n - |> 2L ₀ , condition obtained as soon as | τι - τ _j | > (2Lo + 1) T. To sum up, the quantity L _c generally checks the following box: 2L ₀ +1 <<Px (2L ₀ +1) y. (10)

The expression translating the vector received by the sensors can then be rewritten as follows, where this time only the L _c symbols b _mu of interest appear:

X (mlTβ + jT _e ) = ∑ h (n (l) IT _e + jT _θ ) b _m . _{n (l)} + b (mlT _θ + jT _θ ) ⁽¹¹⁾

1 = 1 Where V 1 <I <L _C , and r _min -Lo <n (l) <rm _in + Lo and where: p 2. h (t) = ∑ p _P a (θ _p ) h _F0 (t -τ _p ) ^l

ICA techniques

The process uses ICA techniques based on the following model given by way of illustration and in no way limitative: ^L (13) uκ = ∑ & s _ik + n _k = G s _k + n _k

where U _k is a vector of dimension Mx1 received at time k, If _k is the i ^th component of the signal S _k at time k, nk is the noise vector and G = [gi ... gj. The objective of the ICA methods is to extract the l = L components Sik and to identify their signatures Q (The vector response of the source i through the observation u _k ) from the observations Uk. The number I = L of components must be less than or equal to the dimension M of the observation vector. The methods of references [4] [5] and [15] use the statistics of order 2 and 4 of the observations u _k . The first step uses the statistics of order 2 of the observations Uk (these observations can be functions of the signals received on the sensors) to obtain a new observation z _k such that:

^L (14) z _k = Wi u _k = ∑ gι s, k + n _k = Ô s _k + n _k

(= 1 where the signatures gι (1 <i <L) are orthogonal, 6 = [g-ι ... gj and s _k = [s _1k ... SL _K ] ^T. The second step is to identify the base orthogonal of the 6 from the order 4 statistics of the whitened observations Zk. Under these conditions we can extract the signals Sk by performing: ê _k ≈ G ^ Z _k ≈ Ô ^ U _k (15)

Where ê _k is the estimate of the signals s _k and where ^# is the pseudo-inversion operator defined by Ô ^# = (Ô ^H Ô) - ¹ 6 ^H.

The ICAR method [19] uses only 4th order statistics to identify the matrix G = [gi ... gκl of signatures. In summary, the idea implemented in the method according to the invention is to construct a spatio-temporal observation whose mixed sources are trains of symbols of the transmitter. Each train of symbols is for example the same symbol train shifted by an integer of symbol period T.

The method described below comprises several variant embodiments, some of which are explained by way of illustration and in no way limitative.

First variant of the process

FIG. 7 represents a first example of an alternative embodiment of the method where the signal is received in baseband.

The method comprises a step 1.1 of determining the symbol time Te by applying for example a cyclic detection algorithm, such as that described for example in [1] [10].

The next step I.2 consists in interpolating the observations x (t) to / samples by symbol, such that T = Te. Under these conditions where fo = 0 and bk , the expression (11) of the vector becomes:

^L <(16) x (ml T _θ + jT _θ ) = ∑ h (n (l) IT _e + jT _e ) a _{mn (} i) + b (ml T _e + jT _e ) for 0 <j <l

1 = 1 As equation (16) is verified for 0 <j </, the method constructs the following spatio-temporal observation (step I.3) from the observations x (kT _e ):

z (ml with h _nJ = h (nl T _e + jT _θ ) and b _z (ml T _e ) = [b (ml T _e ) ^τ ... b (ml T _e + (l-1) T _θ ) ^τ ] ^τ . Knowing that x (t) is of dimension Nx1, the vector z (t) is then of dimension Nlx1. h (k) is a vector whose n ^ièm8 component is the k ^{iô θ} coefficient of the filter linearly filtering the symbol train {a _m } on the n ^{iè β} sensor. The vector coefficient filter h (k) depends on both the formatting filter and the propagation channel.

In order to extract the L _c symbol trains {a _m -i} of interest (number of symbols participating in the IES), the method samples the signal received at l = (2Lo + 1), assuming that P <N. Knowing that the NRZ filter checks 2l_o + 1 == 1 and the Nyquist filter

2Lo + 1 = 3 for a roll-off of 0.25, the symbol trains can be extracted for these two formatting filters respectively when P <NI and 3P <NI.

The observation vector z (t) being determined, the method applies an ICA type method to estimate the L _c symbol trains

{a _mi } associated with the channel vectors h _zj = h _z (k _j ).

The j ^th output of the ICA methods gives the symbol train {â _m , _j } associated with the channel vector i _zJ . The estimated symbol trains {â _m , j} arrive in a different order than the trains {a _m .i} by checking: p exp (jαi) a _mi and h _z = h _z (i) (18)

The symbol trains {â _m , j} are estimated with the same amplitude because the symbol trains {a _mi } are all of the same power by checking:

The objective of the next step I.4 of the method is to order the L _c outputs (â _m , j, h _zj ) in the same order as the inputs (a _m -j, h ₂ (i)) in order to get the channel vectors h _zj . For this, the method intercorrelates two by two the outputs â _mιi and â _mJ by calculating the following criterion Cy (k):

When the function | cy (k) | is maximum in k = k _ma χ the i ^ιemθ and j ^th outputs check: â _m , i = â _m _km _a χj- The algorithm for classifying outputs â _m , _n (i) ... â _m , _{n ( Lc} ) is for example composed of the following steps: Step n ° A.1: Determination of the output â _m , imax associated with the channel vector h _{z ιmax} of higher modulus.

Step n ° A.2: For all the outputs â _m- kj where j ≠ i _ma χ determination of the indices k = k _j maximizing the criterion | Ci _ma χj (k) |. We deduce for each j that â _m , i _ma χ = â _m - _k j, j. Knowing that exp (jαi _max - jαj) the j ^lθmθ output is ^returned to the same phase as the i _m ax ^lth output by performing â _m - j = Cjma _> (kj) â _m , j. We also re-phase the channel vectors by performing: h _z (k _j ) = h _z] Cjm _aX j (kj) ^* .

Step n ° A.3: This step puts in order the outputs â _m - j and the channel vectors h _z = h _z (k _j ) in ascending order of kj knowing that â _m = â _m , i ax and than

h _z (0) = h _z , _{/ max} .

At the end of these three stages, the symbol trains {â _m . k} associated with channel vectors __. (£,). Knowing that the estimated symbols verify â _m . k = expQ α j _a χ) a _m -k, the last step of the process will consist in estimating this phase αj _max . To do this, we first identify the constellation of the symbols ak from a database made up of all the possible constellations. This base consists of known constellations such as nPSK, n-QAM. Each time that we detect or learn of a new constellation, we will enrich the database.

FIG. 6 represents an example of a constellation of 8-QAM when αi _max = 0 and α ι _max ≠ O. In this example of implementation, the method then comprises the following steps:

The next step I.5 consists in determining the output phase associated with the channel vector with the highest modulus. To identify the constellation and determine the phase the process performs for example the following steps: Step I.5 = Steps B.1, B.2 and B.3

Step n ° B.1: Estimation of the positions of the states of the constellation (red dots on the figure) by looking for the maximums of the 2D histogram of the points Mk = (real (âk), imag (âk)). For a constellation with M states we obtain M couples (û _m , v _m ) for 1 <m <M.

Step n ° B.2: Determination of the type of the constellation by comparing the position of the states (û _m , v _m ) of the constellation of {â _k } with a database composed of all the possible constellations. The nearest constellation is made up of the states (u _m , v _m ) for 1 <m <M. Step n ° B.3: Determination of the phase α _imax by minimizing in the least squares sense the following system of equations: û _m = cos (αi _ma χ) u _m - sin (ci _max ) v _m and v, „ = sin (α _imax ) u _m + cos (α _imax ) v _m for 1 <m <M

The method can include a step of estimating the parameters of the propagation channel at angle θ _p and delay τ _p of equation (8) by the algorithm proposed in [8]. The step consists in first extracting the vectors h (nl T _θ + jT _θ ) for 0 <j <l from the channel vectors of the vectors h _z (ri _j ) defined in equation (17). Then build the matrix H = [h (n (1) l T _θ ) ... h (n (L _c ) IT _θ )] of (11) with the h (nl T _θ + jT _e ) to apply the method [8] for parametric estimation of multi-paths: (θ _p> τ _p ) 1 <p <P.

Second variant embodiment of the method FIGS. 8 and 9 schematically show another variant embodiment which may include two variants corresponding respectively to the case of uncorrelated multipaths and to the case of correlated multipaths by group. Case of uncorrelated multipaths. The signal is received in baseband with {bk} = {a _k } Multi-routes whose delays check | XJ-XJ | > (2l_o + 1) T, have the advantage of being uncorrelated with each other by checking: E [s (t-Xi) s (t-Xj) *] = 0. By observing equation (4), we then observe that it suffices to apply an ICA type method when P <N on the observation x (t) to obtain the signals s (tx) of each of the multi-paths. . After the estimation of the signals of the various multi-paths, the method determines their powers to keep the signal s (tx _pma χ) of the multi-path of greatest amplitude p _P max- To determine this main path we use the fact that the outputs of ICA methods verify asymptotically: s (t-τ „) (20) x ^: Wt == ∑ PP a (θp) s (tx _P ) = ∑ âj s _t (t) with _ ?, (*) = and p = l <= 1 where γ _p = p _p ² E [| s (t-τ _p ) | ² ]. As the vectors a (θ) are norms by checking a (θ _p ) ^H a (θ _p ) = N, the path of maximum amplitude will be associated with the i _my χ ^th output where αim _a x≈ âimaχ ^H âj _max is maximum. As according to equation (3) the output ^ _nax W ≈stf- pmax) checks:

(21)

(m IT _e + jT _θ ) = £ ⁿ Fo (n IT _θ + jT _β -τ _pma χ) a „such that 0 <j <l

we can constitute the following observation vector:

z (ml where IT _e + jT _θ -x _P maχ) - According to the model of equation (22), it suffices to apply an ICA type method on the observation z (ml T _e ) to estimate the 2L ₀ +1 symbol trains {a _m - _n } with - L ₀ <n <Lo. To extract the incidences θ _p from the propagation channel, it suffices after (20) to search for each signature ai (1 <i <P) the maximum of the criterion c (θ) = | a (θ) ^H ai | ² . To extract the delays ti - i from the propagation channel, it suffices from (20) to search for each signal _., (?) (1 <i <P) the maximum of the criterion c (τ) = | S. ( ^f - * H (0 * | ^2. In summary this variant includes for example the following steps:

Step n ° ll.a.1: Determination of the symbol time T by applying a cyclic detection algorithm as in [1] [10]. Step n ° ll.a.2: Sampling of observations x (t) to / samples by symbol such as T = / T _θ .

Step n ° ll.a.3: Application of an ICA method on the observations x (t) to obtain s, (t) and ai for 1 <i <P.

Step n ° ll.a.4: Determination of the output i = imax where à "à, maximum. Step n ° ll.a.5: Constitution of the observation vector z (t) of (22) from the signal S _lm (t).

Step n ° ll.a.6: Application of an ICA method to estimate the symbol trains {a _mn } where -L ₀ <n <L ₀ . One chooses among the symbol trains that associated with the vector h _z (n) of higher modulus: {â _m }. Step n ° ll.a.7: Determination of the phase α _ima χ of the output associated with the vector h _z (n) with the highest modulus by applying steps B.1, B.2 and B.3. Step n ° ll.a.8: Re-phasing of the symbol train {â _m } by performing α _m = â _m exp (-jαj _max ). The symbol train {α _m } constitutes the output of the demodulator of this sub-process.

Step n ° ll.a.9: Estimation of the parameters of the propagation channel at angle θ _p and delay x _p by maximizing for 1 <i <P the criteria | a (θ) ^H âj | ² and \ sι (t-τ) _? _x (* | ² for angles and delays respectively. General case with any multipaths or correlated by group In this variant, the diagram of which is given in FIG. 9, the method considers that part of the multipaths are correlated. Considering that the transmitter is received according to Q correlated multi-path groups, the signal vector received by the sensors of equation (4) becomes:

X (t) =

WHERE Aq = [a (θι, _q ) ... a (θ _P q, _q )], ... p _Pq , q]) and S (t, 1 _q ) = [S (t-Xι, _q ) ... S (t- xpq, _q )] ^τ with x _q = [xι, _q .. .xpq, q] ^τ . By applying an ICA method, the following signals and signatures are estimated at the output of the separator:

where π is a permutation matrix, U _q V _q = Ω _q and V _q E [s (t, x _q ) s (t, x _q ) ^H ] V _q ^H = Ip _q . Thus the decorrelated paths such that E [s (tx _p , _q ) s (tx _P ', _q ') *] = 0 are received on different channels _ ?, (*) and S _j (t). The correlated paths where E [s (tx _p , _q ) s (t- t _p . _Q ) *] ≠ 0 are mixed on the same channel s, (t) and are present on PQ at the same time. In the 1 ^st step of this sub-process we use this result to identify the Q correlated multi-path groups. By taking the separator outputs i and j, the following two hypotheses can be tested: i (t) = a _i s (t-τ _p ) + b _i (t) (25)

[S _j (t) = b _j (t) [s _j (t) = a _{j S} (t-τ _p ) + b _j (t)

where E [bj (t) b _j (tx) *] = 0 whatever the value of x. Thus in the hypothesis H ₀ there is no multi-path common to the two outputs i and j and in the hypothesis Hi there is at least one. The test will consist in determining if the outputs §, (t) and S _j (t-τ) are correlated for at least one of the values of x verifying | x | <x _ma χ. For this we can apply the Gardner test [3] which compare the following likelihood ratio to a threshold:

(26)

Vij (x) = - 2K ln (1- ') with r _B (r) = - ∑_J, ($, (* - *) ^* l ^j ? _U (0) r- _dd 0)' ^a * £ ''

Where V (x) <η => hypothesis H ₀

And Vi _j (x)> η = hypothesis Hi

The threshold η is determined in [3] with respect to a chi-2 law with 2 degrees of freedom. We will first look for the outputs associated with the ^first output by launching the test for 2 <J <PQXQ and i = 1. Then from the list of outputs we will remove all those associated with the 1 ^st which will constitute the 1 ^st group with q = 1. We will repeat the same series of tests with the other outputs not correlated with the 1 ^st output to constitute the 2 ^nd group. This operation will be carried out until the last group where in the end there will no longer be any exit route. We will finally get the sorting output:

Âq = Aq U _q and s _g (f) = V _q s (t, x _q ) for (1 <q <Q) (27)

The incidences θ _p , _q are determined from Aq for (1 <q <Q) by applying the MUSIC [1] algorithm to the matrix A, Âq ^H. From these goniometries we deduce the matrices Aq. Knowing that Aq Ω _q s (t, x _q ) _. we deduce s (t, x _q ) to within a diagonal matrix by performing Aq Xq (t). As the elements of s (t, χ _q ) are composed of the signals s (tx pq), the delays x _{p> q} -χι, -ι are determined by maximizing the criteria c (x) = \ s _gp (t-τ ) S _u (t) * | ² where s _qp (t) is the p ^th component of s (t, x _q ).

Knowing that E [s _g (t) s _g (t) ^H ] = lp _q , that Aq ^H Aq = N l _Pq and that Âq S _g (r) = Aq Ω _q s (t, x _q ), we deduces that the group of multi-paths associated with the largest amplitudes Ω _q maximizes the following criterion: cry (q) = trace (A _q ^H Â _q ). We then deduce the best output associated with Âq _ma χ and s _gffiax (r). As after equation (3) the vector s (t, x _q maχ) satisfies

(28) s (ml T _θ + ^h F ° ( ⁿ ' ^T β ⁺ J e. Ïq ax) _tt

for

0 <j <I and with h _F o (n IT _e + we can constitute the following observation vector from (27):

(29)

where h _F0 (n IT _e + jT _θ , x _qm aχ) - According to the model of equation (29), it suffices to apply an ICA type method on the observation z (ml T _e ) for estimate the 2 +1 symbol trains {a _m . _n } such that - L ₀ <n <L ₀ .

In summary this variant comprises the following stages: Stage n ° ll.b.1: Determination of the symbol time T by applying a cyclic detection algorithm as in [1] [10]. Step n ° ll.b.2: Sampling of observations x (t) to I samples by symbol such that T = l T _θ .

Step n ° ll.b.3: Application of an ICA method on the observations x (t) to obtain $ (t) and Â from equation (24).

Step n ° ll.b.4: Sorting the outputs according to Q correlated multi-path groups to obtain Âq and s _g (t) for (1 <q <Q): For this correlation test of all the pairs of outputs i and j by the test with two hypotheses of equation (26). We will first look for the outputs associated with the 1 ^st output by launching the test for 2 <J <PQXQ and i = 1. Then the list of all trips we remove those associated with the 1 ^era that will be the one ^ιer group with q = 1. We will repeat the same series of tests with the other outputs not correlated with the 1 ^st output to constitute the 2 ^nd group. This operation will be carried out until the last group where, in the end, there will be no way out.

Step n ° ll.b.5: Determination of the best group of multi-paths where trace (Âq ^H Âq) is maximum at q = qmax.

Step n ° ll.b.6: Constitution of the observation vector z (t) of (29) from the signal s _gmax (t).

Step n ° ll.b.7: Application of an ICA method to estimate the symbol trains {a _m - _n } where - <n <Lo. One chooses among the symbol trains that which is associated with the vector h _z (i) of higher modulus: {â _mi }. Step n ° ll.b.8: Determination of the phase αi _max of the output associated with the vector h _z (i) of higher modulus by applying steps B.1, B.2 and B.3. Step n ° ll.b.9: Resetting the symbol train {â _m } by performing at _m - â _m exp (-jαjmaχ) - The symbol train {â _m } constitutes the output of the demodulator of this sub- process.

Step n ^c ll.b.10: Estimation of the parameters of the propagation channel at angle θ _q , _p and delay x _q , _p . The incidences θ _Pιq are determined from Â _q for (1 <q <Q) by applying the algorithm MUSIC [1] on the matrix Âq Âq ^H. From these goniometries we deduce the matrices Aq to deduce an estimate of s (t, x _q ) by performing s (t, x _q ) = Aq x _q (t). As the elements of s (t, x _q ) are composed of the signals s (tx _Pιq ), we determine the delays χ _p , _q -χι, ι by maximizing the criteria c (τ) = \ s _gp (t-τ) s _i (t) * \ ² where s _gp (t) is the p ^th component of s (t, x _q ).

Other variant implementation of the process

Carrier frequency estimation and deduction of {a _m }.

This technique will consist in estimating the carrier frequency fo of the emitter or the complex z ₀ = exp (j2τtfoT _θ ) to then deduce the symbols {a _m } from the symbols {b _m } by performing according to (3): a _m = b _m exp (-j2πfomlT _e ) = b _m z ₀ - ^ml (30)

This step is applied after step # l.4 of reordering the symbols and channel vectors. According to equations (3) (17) and (7) (8), we have the following channel vectors:

(31)

where Ω for a p such that 1 <p <P}

Knowing that Ω = {ni <... <nc}, we constitute from the vectors h _z (n) a large vector b such that:

The search for f ₀ will consist in maximizing the following criterion Carrier (f ₀ ) = | w ^H c (expQ2jtf ₀ T _e )) | ² (33)

The steps of the method adapted to the case of a non-zero frequency transmitter are as follows: Step n ° lll.a.1: Steps 1.1 to I.4 described above to obtain the symbol trains {b _m _ _k } associated with the fi ^ & channel vectors. Step n ° lll.a.2: Construction of the vector w of equation (32) from Step n ° lll.a.3: Maximization of the Carrier criterion (fo) of equation (33) to obtain fo.

Step n ° lll.a.4: Application of equation (30) to deduce the symbols {a _m } from the symbols {b _m }.

Step n ° lll.a.5: Steps I.5 to I.7 previously described. In the case of a non-zero frequency transmitter and for uncorrelated multi-paths the steps are as follows:

Step n ° lll.b.1: Steps ll.a.1 to ll.a.4 described above to obtain the vector z (t) of equation (22). Step n ° lll.b.2: Application of ICA methods [4] [5] [15] [19] to estimate the L _c symbol trains {b _mj } associated with the channel vectors h _zj .

Step n ° lll.b.3: Reordering the symbol trains {b ^} and the channel vectors h. , by applying steps A.1, A.2 and A.3 to obtain the symbol trains {b _m _ _k. } associated with the channel vectors h, _z = h _z (k _j ). Step n ° lll.b.4: Construction of the vector w of equation (32) from h _z (k _j ).

Step n ° lll.b.5: Maximization of the Carrier criterion (fo) of equation (33) to obtain f ₀ .

Step n ° lll.b.6: Application of equation (30) to deduce the symbols {a _m } from the symbols {b}. Step n ° lll.b.7: Choice of the symbol train associated with the vector h _z (i) of higher modulus: {â _m -i}.

Step n ° lll.b.8: Steps ll.a.7 to ll.a.9 described above. In the case of a non-zero frequency transmitter and for multi-paths correlated the steps are for example the following:

Step n ° lll.c.1: Steps ll.b.1 to ll.b.6 n ° 2.2 to obtain the vector z (t) of equation (29).

Step n ° lll.c.2: Application of ICA methods [4] [5] [15] [19] to estimate the L _c symbol trains {b _mj } associated with the channel vectors h _z .

Step n ° lll.c.3: Reordering the symbol trains {b _mJ } and the channel vectors fi _z . by applying steps A.1, A.2 and A.3 in order to obtain the symbol trains {b _m _ _k ,} associated with the channel vectors Ê _z . = h _z (^).

Step n ° lll.c.4: Construction of the vector w of equation (32) from h _z (k _j ).

Step n ° lll.c.5: Maximization of the Carrier criterion (fo) of equation (33) to obtain f ₀ .

Step n ° lll.c.6: Application of equation (30) to deduce the symbols {a _m } from the symbols {b _m } - Step n ° lll.c.7: Choice among the symbol trains which one is associated with the vector h _z (i) of higher modulus: {â _m -i} - Step n ° lll.c.8: Steps ll.b.8 to ll.b.10 previously described.

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Claims

1 - Method for blind demodulation of a source or transmitter of linear waveform in a system comprising one or more sources and a network of sensors and a propagation channel characterized in that it comprises at least the following steps:

• determine the symbol time T and we sample at Te such that T = ITe (integer I), • from the observations x (kTe), construct a spatiotemporal observation z (t) whose mixed sources are symbol trains of l '' transmitter,

• apply an ICA type method on the observation vector z (t) to estimate the L ₀ symbol trains {a _m -i} associated with the channel vectors h = h. (k _j ,

• order the L _c outputs (â _m , j, h _Z] .) In the same order as the inputs (a _m -i, h _z (i)) in order to obtain the propagation channel vectors

• determine the phase αjmax associated with the outputs.

2 - Method according to claim 1 characterized in that the parameters of the propagation channel are estimated to determine the carrier frequency in order to compensate the symbol trains to obtain them in baseband.

3 - Method according to claim 1 characterized in that it comprises a step of estimating the parameters of the propagation channel in angle θ _p and delay x ..