COMPUTER IMPLEMENTED METHOD FOR ESTIMATING INTERFERERS OF RADIOFREQUENCY SYSTEM, COMPUTER PROGRAM, AND DEVICE

Abstract

A method comprising: Obtaining observations Z_n, and building an observation vector Z=[Z_1, . . . , Z_n, . . . , Z_N ], Defining a latent variable V_n, to build a vector of latent variables V=[V_1, . . . , V_n, . . . , V_N ], and Implementing a Dirichlet process involving a Gibbs sampling with a Markov chain, the sampling being repeated as follows until convergence: For n?1, . . . , N, if the observation Z_n is associated to a source, remove observation Z_n from a source corresponding to latent variable V_n, and retrieve a position posterior of this source as the observation Z_n is belonging to this source; Draw a new value of latent variable V_n, based on a conditional probability ; Associate the observation Z_n to the source, and update the posterior distribution of the position for the source, and, upon convergence of the algorithm, operating a separation of the interfering sources into K independent measurement sets, and an estimation of each source position.

Claims

1. A computer implemented method for estimating interferers of a radiofrequency system embarked in a moving vehicle having a known trajectory at each time n, with a non overlapping condition between the interferers considered as K independent active sources of interference having respective positions ?=[?.sub.1, . . . , ?.sub.k, . . . , ?.sub.k], the method comprising: Obtaining observations Z.sub.n corresponding to measurement of interference from time instant 1 to N, and building an observation vector Z=[Z.sub.1, . . . , Z.sub.n, . . . , Z.sub.N], Defining a latent variable V.sub.n indicating which source is activated at moment n, to build a vector of latent variables V=[V.sub.1, . . . , V.sub.n, . . . , V.sub.N], and Implementing a Dirichlet process involving a Gibbs sampling with a Markov chain defined by vector V=[V.sub.1, . . . , V.sub.n, . . . , V.sub.N], the sampling as follows, being repeated until convergence: For n=1, . . . , N, If the observation Z.sub.n is already associated to a source, remove observation Z.sub.n from its current associated source corresponding to latent variable V.sub.n, and retrieve a position posterior of this source as the observation Z.sub.n is no longer belonging to this source, Draw a new value of latent variable V.sub.n, based on a conditional probability P(V.sub.n=k|V.sub.?n, Z.sub.?n, Z.sub.n) depending on whether a source k pre-existed or not, Associate the observation Z.sub.n to the source corresponding to latent variable V.sub.n, and update the posterior distribution of the position for the source corresponding to latent variable V.sub.n, and, upon convergence of the algorithm, operating thereby: a separation of the interfering sources into K independent measurement sets related respectively to the K interfering sources, and an estimation of each source position with the sources thus separated.

2. The method of claim 1, wherein a probability is evaluated to identify which source an observation Z.sub.n belongs to, said probability being given by:
P(V.sub.n|V.sub.?n,Z.sub.?n,Z.sub.n)?P(Z.sub.n|V.sub.?n,Z.sub.?n,V.sub.n)P(V.sub.n|V.sub.?n)=P(V.sub.n|V.sub.?n)?p(Z.sub.n.sup.(Vn)|Z.sub.?n.sup.(V.sup.n.sup.),?.sub.V.sub.n).p(?.sub.V.sub.n|Z.sub.?n.sup.(V.sup.n.sup.))d?.sub.V.sub.n where ( ).sub.?n refers to an index other than n, the probability p(Z.sub.n.sup.(V.sup.n.sup.)|Z.sub.?n.sup.(V.sup.n.sup.),?.sub.V.sub.n being a conditional probability of an observation Z.sub.n to be associated to the source corresponding to latent variable V.sub.n, given other measurement Z.sub.?n already associated to this source.

3. The method of claim 2, wherein the posterior distribution of the position is updated progressively to estimate the source's position by implementing: $P (?_{V_{n}} .Math. Z_{- n}^{(V_{n})}, Z_{n}^{(V_{n})}) = \frac{P (Z_{n}^{(V_{n})} .Math. Z_{- n}^{(V_{n})}, ?_{V_{n}}) P (?_{k} .Math. Z_{- n}^{(V_{n})})}{P (Z_{n}^{(V_{n})} .Math. Z_{- n}^{(V_{n})})}$ where ( ).sub.?n refers to an index other than n.

4. The method of claim 3, wherein the probability p(Z.sub.n.sup.(V.sup.n.sup.)|Z.sub.?n.sup.(V.sup.n.sup.), ?.sub.V.sub.n) is expressed as: $p (Z_{n}^{(V_{n})} .Math. Z_{- n}^{(V_{n})}, ?_{V_{n}}) = \frac{1}{\sqrt{2 ? ?_{n .Math. - n}^{{(V_{n})}^{2}}}} e^{\frac{{(Z_{n - ?_{n .Math. - n}^{(V_{n})}})}^{2}}{2 ?_{n .Math. - n}^{{(V_{n})}^{2}}}}$

5. The method of claim 4, wherein the terms ?.sub.n|?n.sup.(V.sup.n.sup.) and ?.sub.n|?n.sup.(V.sup.n.sup.) are calculated as follows: ${\begin{matrix} ?_{n .Math. - n}^{(V_{n})} = ?_{n}^{(V_{n})} + {.Math.}_{n .Math. - n}^{(V_{n})} {.Math.}_{- n}^{{(V_{n})}^{- 1}} (Z_{- n}^{(V_{n})} - ?_{- n}^{(V_{n})}) \\ ?_{n .Math. - n}^{{(V_{n})}^{2}} = ?^{2} - {.Math.}_{n .Math. - n}^{(V_{n})} {.Math.}_{- n}^{{(V_{n})}^{- 1}} {.Math.}_{- n .Math. n}^{(V_{n})} \end{matrix},$ where: ?.sub.n.sup.(V.sup.n.sup.) denotes a mean in a gaussian distribution for the observation Z.sub.n, and expressed as ?.sub.n.sup.(V.sup.n.sup.)=a+b log?T.sub.n??.sub.V.sub.n? ?.sub.n|?n.sup.(V.sup.n.sup.) denotes a correlation matrix between observation n and the other observation than n of source V.sub.n, ?.sub.?n.sup.(V.sup.n.sup.) denotes an auto-correlation matrix of observations other than n of source V.sub.n, ?.sub.?n.sup.(V.sup.n.sup.) denotes the mean at observations other than n of source V.sub.n, ?.sub.?n|n.sup.(V.sup.n.sup.) denotes the correlation matrix between observations other than n and observation n of source V.sub.n.

6. The method according to claim 2, wherein the conditional probability p(V.sub.n=k|V.sub.?n) is given by: $p (V_{n} = k .Math. V_{- n}) = \frac{N_{k} + ? / K}{N - 1 + ?}$ where N.sub.z is a number of observations associated to a source corresponding to V.sub.n, N is the total number of observations, a being a concentration parameter.

7. The method of claim 6, wherein the number of possible interferers K is unknown and: the conditional probability for an observation to belong to a pre-existing source k is given by: $p (V_{n} = k .Math. V_{- n}) = \frac{N_{k}}{N - 1 + a}$ and the conditional probability for an observation to belong to a new source k is given by: $p (V_{n} = k^{} .Math. V_{- n}) = \frac{?}{N - 1 + a}$

8. The method of claim 2, wherein the conditional probability equals to: $b \frac{N_{k}}{N - 1 + ?} ? p (Z_{n}^{(k)} .Math. Z_{- n}^{(k)}, ?_{k}) .Math. p (?_{k} .Math. Z_{- n}^{(k)}) d ?_{k},$ for an existing source k, or to $b \frac{?}{N - 1 + ?} ? p (Z_{n} .Math. ?) .Math. G_{0} (?) d ?,$ for a new source, where b is an appropriate normalizing constant making the above given probabilities sum to one.

9. The method according to claim 1, wherein the Dirichlet process involves a Dirichlet mixture model defined as: ${\begin{matrix} Z_{n} .Math. V_{n}, ? ? N (?_{n}^{(V_{n})}, ?_{n}^{2^{(V_{n})}}) \\ V_{n} ? Discrete (p_{1}, .Math., p_{K}) \\ p ? Dirichlet (? / K) \\ ?_{k} ? G_{0} \end{matrix}$ Where G.sub.0 is a base distribution of position of a source.

10. The method according to claim 1, comprising further an estimation of a likelihood p(Z.sub.n.sup.(Vn)|Z.sub.?n.sup.(V.sup.n.sup.), ?.sub.V.sub.n), as a function of the mobile vehicle position T.sub.n, given by: $p (Z_{n}^{(V n)} .Math. Z_{- n}^{(V_{n})}, ?_{V_{n}}) = A .Math. \exp (- B {(\underset{n}{.Math.} c_{n} \log (? - T_{n}) - h^{})}^{2}) .$

11. The method according to claim 10, wherein position ? is discretized into discrete values in a discrete space ?.sub.?, each value of position ? in said discrete space being associated with a probability.

12. The method according to claim 10, wherein a continuous position determination is performed by partitioning a space around the mobile vehicle position into sub-partitions s=1, . . . , S, each sub-partition being represented by a center C.sub.s, the vehicle position T.sub.n being in sub-partition s, and the likelihood being given by: $p (Z_{n}^{(V n)} .Math. Z_{- n}^{(V_{n})}, ?_{V_{n}}) = A \exp (- B {(\underset{n}{.Math.} c_{n} \log (? - T_{n}) - h^{})}^{2})$ $? A \exp (- B {(\underset{s}{.Math.} w_{s} \log (? - C_{s}^{n}) - h^{})}^{2})$ Where C.sub.s.sup.i is determined by $C_{s}^{n} = \underset{C_{s}}{argmin} .Math. T_{n} - C_{s} .Math.,$ And said base distribution G.sub.0 is given by $G_{0} (?) = A_{0} \exp (- B_{0} {(\underset{s}{.Math.} w_{s}^{0} \log (? - C_{s}) - h_{0}^{})}^{2}) .$

13. Computer program comprising instructions for performing the method according to claim 1 when such instructions are executed by a processing circuit.

14. Device comprising a processing circuit configured to implement the method according to claim 1.

Description

BRIEF DESCRIPTION OF DRAWINGS

[0062] FIG. 1 shows a CBTC radiofrequency system measuring the interference at each position on its trajectory.

[0063] FIG. 2 shows measured power of interference and the true position of interferers.

[0064] FIG. 3 shows two first interferers being detected.

[0065] FIG. 4 shows the appearance of a third source of interference.

[0066] FIG. 5 shows the appearance of a fourth source of interference.

[0067] FIG. 6 shows the estimate in the end of a first iteration.

[0068] FIG. 7 shows the converged estimate.

[0069] FIG. 8 shows an algorithm having steps of the method presented above, according to a possible embodiment.

[0070] FIG. 9 shows a local coordinate system.

[0071] FIG. 10 shows an example of embodiment of a device for performing the method presented above.

DESCRIPTION OF EMBODIMENTS

[0072] Reference is made now to FIG. 1 to present the main concept of geolocation of several waysides interferers from measurements made in a train. The several positions enable for having different geometrical views of the problem and thus for estimating the position of one interferer. When several interferers are present in the system, source separation is to be performed in order to associate each observation with each interferer and then use the previously mentioned geolocation technique from observation at several positions. However, the number of interferers is usually unknown.

[0073] It is proposed here to solve the two aforementioned questions of source separation and source geolocation, by using a machine learning approach relying on Dirichlet processes and Gibbs sampling. In particular, the geo-location problem cannot be expressed in a form which can allow a conventional Gibbs sampling. Thus, it is proposed two approaches to cope with this issue: [0074] A grid of interferers positions, in which it is addressed in particular the mechanism of updating and retrieving the probability when adding and removing (respectively) an observation. These features (adding and removing an observation) enables the Gibbs sampling. [0075] The approximation of geo-location problem, in which the exact problem is approximated by a closed-form, which allows the update and retrieve in a more simple way by adjusting several parameters of this closed-form.

[0076] Taking an example where power measurements of interference on the train are obtained as Z=(Z.sub.1, . . . , Z.sub.n, . . . , Z.sub.N) according to train's position as T=(T.sub.1, . . . , T.sub.n) . . . , T.sub.N), a plot (T, Z) is given in the upper sub-figure of FIG. 2. The objective is to find the true position of interferers plotted in the lower sub-figure of FIG. 2.

[0077] Knowing the measurements are mixed between sources, therefore to efficiently geolocate the sources, firstly it is needed to separate them into independent ones, and it is introduced here a latent variable that indicates which source is activated at moment n:

V=[V.sub.1, . . . , V.sub.n, . . . , V.sub.N].

[0078] Furthermore, in the following, ?=(?.sub.1, ?.sub.2, . . . , ?.sub.K) denotes the vector of position of K interferers. The Dirichlet mixture model can be stated, according to the problem to solve, as follows:

[00016] ${\begin{matrix} Z_{n} .Math. V_{n}, ? ? ? (?_{n}^{(V_{n})}, ?_{n}^{2 (V_{n})}) \\ V_{n} ? Discrete (p_{1}, .Math., p_{K}) \\ p ? Dirichlet (? / K) \\ ?_{k} ? G_{0} \end{matrix}$ [0079] where custom-character (?.sub.n.sup.(V.sup.n.sup.), ?.sup.2.sup.(V.sup.n.sup.).sub.n) denotes the normal distribution, ? being the concentration parameter, K is the number of sources, and G.sub.0 is the base distribution of position of a source (?.sub.k denoting a source position parameter). The number of sources is supposed to be unknown, then K=?.

[0080] The algorithm can mainly be structured as follows: [0081] For very first observation Z.sub.1, create a very first source and associate Z.sub.1 to this source (V.sub.1=1), then update the posterior distribution of its position by Bayes rule as follows

p(?.sub.1|Z.sub.1,V.sub.1=1)?p(Z.sub.1.sup.(1)|?.sub.1).G.sub.0 [0082] From the second observation, for any n: [0083] Draw a sample for new value of V.sub.n, based on the conditional probability as follows (Gibbs sampling): [0084] For an existing source k:

[00017] $b \frac{N_{k}}{N - 1 + ?} ? p (Z_{n}^{(k)} .Math. Z_{- n}^{(k)}, ?_{k}) .Math. p (?_{V_{n}} .Math. Z_{- n}^{(k)}) d ?_{k}$ [0085] For a new source:

[00018] $b \frac{?}{N - 1 + ?} ? p (Z_{n} .Math. ?) .Math. G_{0} (?)$ [0086] Here, b is the appropriate normalizing constant that makes the above probabilities sum to one, N.sub.k is the number of observations in the source k. [0087] Associate Z.sub.n to the source V.sub.n, and update the posterior distribution of its position by Bayes rule.

[0088] In this example, a plot of the result after three first observations is given in FIG. 3. As it can be seen, two first observations are associated to the first source and the third is associated to the second source. The posterior distribution of position of these sources are plotted as contours in the lower sub-figure. Since, the first source has two observations, the distribution of its position is denser compared the second one which has only one observation associated.

[0089] FIG. 4 shows the appearance of the third source, plotted B in the upper sub-figure.

[0090] An algorithm which is presented in details below with reference to FIG. 8 continues the estimation and decides itself whether there is another source or not until the end of observation. The estimate is plotted in FIG. 5. Bearing in mind that there are only three sources here, this estimate is still not accurate.

[0091] The estimation can be continued until the end of observation, meaning n=N. The estimate at this moment is shown in FIG. 6. The algorithm estimates there are at least four sources and the probability for their position is as in the lower sub-figure of FIG. 6.

[0092] The algorithm is not stopping here, and comes back to the first observation and: [0093] Dissociate the observation from current associated source [0094] Retrieve the posterior distribution of the related source as the observation is no longer in this source [0095] Draw a sample for new value of V.sub.n, based on the conditional probability as follows (Gibbs sampling): [0096] For an existing source k:

[00019] $b \frac{N_{k}}{N - 1 + ?} ? p (Z_{n}^{(k)} .Math. Z_{- n}^{(k)}, ?_{k}) .Math. p (?_{V_{n}} .Math. Z_{- n}^{(k)}) d ?_{k}$ [0097] For a new source:

[00020] $b \frac{?}{N - 1 + ?} ? p (Z_{n} .Math. ?) . G_{O} (?)$

[0098] The process is repeated iteratively until the estimate converge as the result shown in FIG. 7 showing finally only three sources which are well defined spatially.

[0099] To make Gibbs sampling feasible, it should be possible to update or retrieve the posterior distribution whenever an observation is deemed to be associated to or dissociated from a source. Regarding the specific problem, there does not exist a closed form for the distribution to be updated or retrieved and to cope with this issue. Hence, two approaches are proposed here: [0100] Discretization: To simplify the problem, the parameter position ? is discretized into discrete values defining thus a discrete space ?.sub.?. For example, considering a 2D space of interferer's position, ? is discretized uniformly in both X and Y axis. Each value of ? in this grid is associated with a probability. The update or retrieve for each source is done by adjusting the probability for every value of ? in ?.sub.?, [0101] Approximation: the exact problem is approximated by partitioning into several forms, which allows the update and retrieve in a more simple way by adjusting several parameters of the sub-form.

[0102] A corresponding flowchart reflecting the algorithm is presented in FIG. 8. The algorithm is presented below with definitions and conditions given hereafter: [0103] Non-overlapping condition: it is supposed that there is no collision between interferers such that, at a time moment, only one interference source emits signal. This condition can be satisfied by using CSMA/CA protocol or CSMA/CD for example. [0104] Known train trajectory (in step S1): data of the train positions T.sub.1, T.sub.2, . . . , T.sub.n, . . . , T.sub.N are received successively and the algorithm can know its position as well as its velocity and direction at any moment. [0105] The notations of the local coordinate systems are given as shown in FIG. 9. [0106] Distance between train and interferer k, at time n, is given by:

d.sub.n.sup.(k)=?T.sub.n??.sub.k?(1) [0107] Interference received power Z.sub.n on the train, if the interferer k is active, is measured in step S1 at time n (the path loss model being considered as known)

Z.sub.na+b log d.sub.n.sup.(k)+w.sub.n.sup.(k),(2) where a and b are two coefficients of path-loss model and w.sub.n.sup.(k) denotes the shadow fading on the train with respect to interferer k, at time n. [0108] The shadow fading follows the multivariate Gaussian distribution with the correlation between two train positions T.sub.n and T.sub.m expressed as

[00021] $\begin{matrix} ? = ?_{0} e^{- \frac{.Math. T_{n} - T_{m} .Math.}{d_{c}}}, & (3) \end{matrix}$ [0109] where ?.sub.0, d.sub.c are coefficients of shadowing model (also known). [0110] The path-loss and shadowing model are known. [0111] Assuming that there are K sources of interference, therefore the parameters to be estimated are their positions:

?=[?.sub.1, . . . , ?.sub.k, . . . ,?.sub.K].(4) [0112] On the train, the radio module measures the power level of interference from time instant 1 to N, the observation vector can be written as stated in step S1 of FIG. 8:

Z=[Z.sub.1, . . . ,Z.sub.n, . . . ,Z.sub.N]. (5)

[0113] However, the information of which one among K interferers is active at any time instant, is unknown. In this sense, the observation is mixed among sources. Combining with non-overlapping condition, at a time instant there is only one emitting source.

[0114] Moreover, a latent variable that indicates which source is activated at moment n, is introduced:

V=[V.sub.1, . . . ,V.sub.n, . . . ,V.sub.N], (6)

[0115] In order to geolocate the interferers, two problem need to be solved: [0116] Source separation: the power measurement on the train is a mixed signal, needing to separate them into K independent sets which are related to K interferers. [0117] Source geolocation: once the sources are separated, their positions can be estimated.

[0118] The two above-mentioned sub-problems are correlated: the performance of one affects the other. The Dirichlet mixture model (described for example in reference [1], the details of which are given at the end of the description, below) can be stated, according to the main problem to solve, as follows:

[00022] ${\begin{matrix} Z_{n} .Math. V_{n}, ? ~ ? (?_{n}^{(V_{n})}, ?_{n}^{2^{(V_{n})}}) \\ V_{n} ~ Discrete (p_{1}, .Math., p_{K}) \\ p ~ Dirichlet (? / K) \\ ?_{k} ~ G_{0} (?_{k}) \end{matrix}$

[0119] Based on this model, a Markov Chain Monte Carlo algorithm is implemented, involving a Gibbs sampling. The principle is to separate the measurements into independent sources and then update the distribution of sources' position in order to be able to geolocate the sources.

[0120] A Markov Chain Monte Carlo simulation is performed by repeating the following operations until convergence:

[0121] After building in step S1 a global set custom-character of observations given by couples (Z.sub.n, T.sub.n) of radiofrequency power measurements Z.sub.n and train positions T.sub.n, [0122] for n=1, . . . , N (and then for an observation at time n given by a couple (Z.sub.n, T.sub.n) as stated in step S21 of FIG. 8): [0123] in step S2, the observation (Z.sub.n, T.sub.n) is removed from the global set custom-character of observations; [0124] If the observation (Z.sub.n, T.sub.n) is already associated to a source V.sub.n (test S3: Yes), dissociate observation (Z.sub.n, T.sub.n) from its current associated source V.sub.n (in step S4), and retrieve the position posterior of this source as the observation at time n is no longer belonging to this source; [0125] In step S5, for the Gibbs sampling implementation, draw a sample for a new value of V.sub.n given with probabilities as follows:

[00023] $b \frac{N_{k}}{N - 1 + ?} ? p (Z_{n}^{(k)} .Math. Z_{- n}^{(k)}, ?_{k}) . p (?_{k} .Math. Z_{- n}^{(k)}) d ?_{k},$

for an existing source k already detected,

[00024] $b \frac{?}{N - 1 + ?} ? p (Z_{n} .Math. ?) . G_{0} (?) d ?,$

for a new source, where b is an appropriate normalizing constant making the above given probabilities sum to one; [0126] If V.sub.n is new (test S6), create a new source (S7); [0127] Associate the observation (Z.sub.n, T.sub.n) to the source V.sub.n, and update its posterior (S8).

[0128] For each iteration, after all measurements are considered (test S9 and loop on S11), a condition (test S10) is added to check whether the convergence is met or not. The convergence condition can be diverse: the convergence of source position, or the convergence on the repartition of measurements, or the maximum number of iterations, etc.

[0129] In step S5 of FIG. 8, the Gibbs sampling is more particularly performed by evaluating the following probability so as to identify which source the observation n belongs to:

P(V.sub.n|V.sub.?n,Z.sub.?nZ.sub.n)?P(Z.sub.n|V.sub.?n,Z.sub.?n,V.sub.n)P(V.sub.n|V.sub.?n)=P(V.sub.n|V.sub.?n)?p(Z.sub.n.sup.(Vn)|Z.sub.?n.sup.(V.sup.n.sup.),?.sub.V.sub.n).p(?.sub.V.sub.n|Z.sub.?n.sup.(V.sup.n.sup.))d?.sub.V.sub.n [0130] where ( ).sub.?n denotes the other than n. The probability p(Z.sub.n.sup.(V.sup.n.sup.)|Z.sub.?n.sup.(V.sup.n.sup.), ?.sub.V.sub.n) is the conditional probability of observation n in the source (V.sub.n), given the other observations already associated to this source. This conditional probability is Gaussian with following mean and variance:

[00025] ${\begin{matrix} ?_{n .Math. - n}^{(V_{n})} = ?_{n}^{(V_{n})} + ?_{n .Math. - n}^{(V_{n})} ?_{- n}^{{(V_{n})}^{- 1}} (Z_{- n}^{(V_{n})} - ?_{- n}^{(V_{n})}) \\ ?_{n .Math. - n}^{{(V_{n})}^{2}} = ?^{2} - ?_{n .Math. - n}^{(V_{n})} ?_{- n}^{{(V_{n})}^{- 1}} ?_{- n .Math. n}^{(V_{n})} \end{matrix},$ [0131] where ?.sub.n.sup.(V.sup.n.sup.) denotes the mean at time n of source V.sub.n, [0132] ?.sub.n|?n.sup.(V.sup.n.sup.) denotes the correlation matrix between observation n and the other observation than n of source V.sub.n, [0133] ?.sub.?n.sup.(V.sup.n.sup.) denotes the auto-correlation matrix of observations other than n of source V.sub.n, [0134] ?.sub.?n.sup.(V.sup.n.sup.) denotes the mean at observations other than n of source V.sub.n, [0135] ?.sub.?n|n.sup.(V.sup.n.sup.) denotes the correlation matrix between observations other than n and observation n of source V.sub.n.

[0136] Hence, the probability p(Z.sub.n.sup.(V.sup.n.sup.)|Z.sub.?n.sup.(V.sup.n.sup.), ?.sub.V.sub.n) is expressed as:

[00026] $p (Z_{n}^{(V_{n})} .Math. Z_{- n}^{(V_{n})}, ?_{V_{n}}) = \frac{1}{\sqrt{2 {??}_{n .Math. - n}^{{(V_{n})}^{2}}}} e^{- \frac{{(Z_{n} - ?_{n .Math. - n}^{(V_{n})})}^{2}}{2 ?_{n .Math. - n}^{{(V_{n})}^{2}}}}$

[0137] Since ?.sub.n.sup.(V.sup.n.sup.)=a+b log?T.sub.n??.sub.V.sub.n?, then ?.sub.n|?n.sup.(V.sup.n.sup.)?.sub.?n.sup.(V.sup.n.sup.).sup.?1?.sub.n|?n.sup.(V.sup.n.sup.) can be rewritten as ?.sub.l?nc.sub.l log(?.sub.V.sub.l?T.sub.l)+h.

[0138] The conditional probability p(V.sub.n=k|V.sub.?n) can be expressed as

[00027] $p (V_{n} = k .Math. V_{- n}) = \frac{N_{k} + ? / K}{N - 1 + ?}$ [0139] where N.sub.k is the number of observations associated to source V.sub.n, and N is the total number of observations.

[0140] The number K can be such that K=? to define a non-parametric model, and then: [0141] The conditional probability for an observation to belong to an existing source k is

[00028] $p (V_{n} = k .Math. V_{- n}) = \frac{N_{k}}{N - 1 + ?}$ [0142] The conditional probability for an observation to belong to a new source k is

[00029] $p (V_{n} = k^{} .Math. V_{- n}) = \frac{?}{N - 1 + ?}$ and the posterior is updated progressively to estimate the source's position:

[00030] $P (?_{V_{n}} .Math. Z_{- n}^{(V_{n})}, Z_{n}^{(V_{n})}) = \frac{P (Z_{n}^{(V_{n})} .Math. Z_{- n}^{(V_{n})}, ?_{V_{n}}) P (?_{k} .Math. Z_{- n}^{(V_{n})})}{P (Z_{n}^{(V_{n})} .Math. Z_{- n}^{(V_{n})})} .$

[0143] To implement the above-mentioned algorithm, an appropriate base distribution G.sub.0 should be able to update and retrieve the posterior in each step of Gibbs sampling. Regarding the likelihood p(Z.sub.n.sup.(Vn)|Z.sub.?n.sup.(V.sup.n.sup.), ?.sub.V.sub.n), this is a function of the train position T.sub.n. The likelihood exists in this form

p(Z.sub.n.sup.(Vn)|Z.sub.?n.sup.(V.sup.n.sup.),?.sub.V.sub.n)=A.exp(?B(?.sub.nc.sub.n log(??T.sub.n)?h).sup.2).

[0144] In order to theoretically update or retrieve the posterior with the above likelihood, proposing a G.sub.0 to be conjugate with the above likelihood is not simple.

[0145] To simplify the problem, a first embodiment can propose to discretize the position ? into discrete values and call this discrete space as ?.sub.?. For example, considering a 2D space of interferer's position, ? is discretized uniformly in both X and Y axis. Each value of ? in this grid is associated with a probability.

[0146] The update or retrieve for each source is done by adjusting the probability for every value of ? in ?.sub.? as follows: [0147] Update the posterior if adding an observation

[00031] $P (?_{V_{n}} .Math. Z_{- n}^{(V_{n})}, Z_{n}^{(V_{n})}) = \frac{P (Z_{n}^{(V_{n})} .Math. Z_{- n}^{(V_{n})}, ?_{V_{n}}) P (?_{k} .Math. Z_{- n}^{(V_{n})})}{P (Z_{n}^{(V_{n})} .Math. Z_{- n}^{(V_{n})})}$ [0148] Retrieve the posterior if removing an observation P(?.sub.k|Z.sub.?n.sup.(V.sup.n.sup.)=q.P(?.sub.V.sub.n|Z.sub.?n.sup.(V.sup.n.sup.),Z.sub.n.sup.(V.sup.n.sup.))/P(Z.sub.n.sup.(V.sup.n.sup.)|Z.sub.?n.sup.(V.sup.n.sup.),?.sub.V.sub.n), where q is a normalizing constant that makes the posterior sum to one.

[0149] In a second embodiment considering a continuous position determination rather than a discrete one, it is possible to partition the space of train's position into sub-partitions s=1, . . . , S. Each sub-partition is represented by a center C.sub.s. To make the update and retrieve of posterior possible, it can be stated that if the train position In is in sub-partition s, the likelihood p(Z.sub.n.sup.(Vn)|Z.sub.?n.sup.(V.sup.n.sup.),?.sub.V.sub.n) can be approximated by replacing T.sub.n by C.sub.s. The likelihood then becomes

[00032] $\begin{matrix} P (Z_{n}^{(Vn)} .Math. Z_{- n}^{(V_{n})}, ?_{V_{n}}) = A \exp (- {B (\underset{n}{.Math.} c_{n} \log (? - T_{n}) - h^{})}^{2}) \\ ? A \exp (- {B (\underset{n}{.Math.} c_{n} \log (? - C_{s}^{n}) - h^{})}^{2}) \end{matrix}$

C.sub.s.sup.i is determined by

[00033] $C_{s}^{n} = \underset{C_{s}}{\arg \min} .Math. T_{n} - C_{s} .Math. .$

[0150] Since the centres of sub-partitions are deterministic, the approximated likelihood can be expressed as

[00034] $p (Z_{n}^{(Vn)} .Math. Z_{- n}^{(V_{n})}, ?_{V_{n}}) ? A \exp (- {B (\underset{s}{.Math.} w_{s} \log (? - C_{s}) - h^{})}^{2}) .$

[0151] The base distribution G.sub.0 in this case can be proposed in the form of:

[00035] $G_{0} = A_{0} \exp (- B_{0} {(\underset{s}{.Math.} w_{s}^{0} \log (? - C_{s}) - h_{0}^{})}^{2}) .$

[0152] In the simplest case, G.sub.0 can be uniform, therefore w.sub.s.sup.0=0 for any s, h.sub.0=0, ? B.sub.0, and A.sub.0 is the normalizing constant that makes G.sub.0 sum to one. The term (?.sub.sw.sub.s log(??C.sub.s)?h).sup.2 can be developed as follows

[00036] ${(\underset{s}{.Math.} w_{s} \log (? - C_{s}) - h^{})}^{2} = d^{T} W_{1} d + w_{2} d + h^{2}$ [0153] where d=(log???C.sub.1?, . . . , log???C.sub.S?), w.sub.1=(w.sub.1, . . . , w.sub.S).sup.T* (w.sub.1, . . . , w.sub.S),

w.sub.2=2*(w.sub.1, . . . , w.sub.S)*h

[0154] In this logic, the posterior takes the same form as the approximated (Z.sub.n.sup.(Vn)|Z.sub.?n.sup.(V.sup.n.sup.), ?.sub.V.sub.n). The update and retrieve of posterior can be done as follows: [0155] Update: w.sub.1.sup.new =w.sub.1.sup.old+w.sub.1, w.sub.2.sup.new=w.sub.2.sup.old+w.sub.2, h.sup.(new)=h.sup.(old)+h [0156] Retrieve: w.sub.1.sup.new=w.sub.1.sup.old?w.sub.1, w.sub.2.sup.new=w.sub.2.sup.old?w.sub.2, h.sup.(new)=h.sup.(old)?h

[0157] FIG. 10 shows a device for implementing the method presented above, and comprising typically a processing circuit including a memory MEM storing data including data of computer program instructions for performing the method presented above when executed by a processor, as well as such a processor PROC, and an interface communication COM for receiving notably radiofrequency measurements.

[0158] The device can be embarked in the train and be responsible to identify the interferers, then feed the information back to a distant server. Besides, radiofrequency measurements can be performed by a module embarked in the train, while such measurement data are transmitted along with the train geolocation data to a distant server operating the steps of the method presented for example in the embodiment of FIG. 8. Therefore, the device to perform such a method can be embarked in the train or can be alternatively a remote server.

[0159] The device does not need to feed back the information for every step of the algorithm, i.e. each time the data of sources change. Instead, a metric can be defined such that the device decides when performing any feedback of the interfering sources. Indeed, the posterior probability of each source's position is updated progressively during the algorithm. The decision of doing the feedback should involve the evolution of posterior. One can state that once the posterior stays stable, the feedback may be proceeded. The feedback decision is taken once the evolution reaches a value which is small enough. That is to say: [0160] When the estimation of considered source is converged (step S10), actions to take can be: [0161] Feeding the posterior for example back to a server; [0162] Keeping the source in consideration for source separation. [0163] When the source is out of vision of the train, there is no longer any observation that is to be added to the source and the action to take can be: [0164] Feeding the posterior back to the server; [0165] Withdraw the source from the consideration of source separation.

[0166] In case of discretization, the posterior can actually be a grid of probability for discrete. By defining ?.sub.k.sup.new, and ?.sub.k.sup.old the grids of this probability after and before each update (if any), respectively, the evolution of posterior are then defined as follows

?=??.sub.k.sup.new??.sub.k.sup.old?

[0167] In the continuous case, the posterior is approximated by a continuous function. By defining H.sub.k.sup.new(?.sub.k) and H.sub.k.sup.old(?.sub.k) are the approximate posterior of source k after and before each update (if any), respectively, the evolution of posterior can be defined as:

?=?|H.sub.k.sup.new(?.sub.k)?H.sub.k.sup.old(?.sub.k)|d?.sub.k

COMPUTER IMPLEMENTED METHOD FOR ESTIMATING INTERFERERS OF RADIOFREQUENCY SYSTEM, COMPUTER PROGRAM, AND DEVICE

Assignee

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Cpc classification

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G01S5/02695

PHYSICS

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H04W4/42

ELECTRICITY

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H04B17/345

ELECTRICITY

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H04W4/029

ELECTRICITY

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G01S5/011

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Classification Explorer

G01S5/0278

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International classification

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G01S5/02

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G01S5/00

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Abstract

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