Method for estimating a wireless communication channel, device for estimating a wireless communication channel and computer program therefor

Abstract

A method for estimating a wireless communication channel between a transmitter and a receiver, including a plurality of paths for propagation of a wave, at least one of the transmitter and the receiver being formed of a plurality of antennas. The method includes: for at least one path, determining a characteristic matrix, which depends on a first element representative of at least one propagation direction associated with the path, and a second element representative of a propagation distance associated with the path; and estimating the communication channel from the at least one obtained characteristic matrix.

Claims

1. A method performed by a device and comprising: estimating a wireless communication channel between a transmitter and a receiver, comprising a plurality of paths allowing propagation of a wave, at least the transmitter or the receiver being formed of several antennas, wherein the estimating comprises: for a first path, determining a first estimate of a first propagation distance value associated with the first path, and of a first propagation direction associated with the first path, and determining a first characteristic matrix, for a second path, determining a second estimate of a second propagation distance value associated with said second path, and of a second propagation direction associated with said second path, and determining a second characteristic matrix, the first and second characteristic matrices each having a definition depending on a parabolic wave model describing a distance between positions of at least one pair of antennas formed by an antenna array of the transmitter and an antenna array of the receiver, and a distance between centers of gravity of the antenna arrays of the receiver and the transmitter, the parabolic wave model being solely a function of: a quantity representative of the respective first or second estimate of the respective first or second propagation direction associated with the respective first or second path depending on at least one of the following elements: a propagation start direction vector associated with the respective first or second path, or a propagation arrival direction vector associated with the respective first or second path, and a corrective term depending on the respective first or second estimate of the respective first or second propagation distance value associated with the respective first or second path, and wherein at least one of the first estimate of the first propagation distance value and of the first propagation direction or the second estimate of the second propagation distance value and of the second propagation direction is carried out in a first underestimate, for a first quantity and a second quantity, distinct from the first quantity, wherein the first quantity is fixed and the second quantity can assume a plurality of predetermined values, and then in a second underestimate for the second quantity fixed at a value resulting from the first underestimate and the first quantity being able to assume the plurality of predetermined values, the first quantity or the second quantity being one of the following elements: the respective first or second estimate of the respective first or second propagation distance value associated with the respective first or second path, or the quantity representative of the respective first or second estimate of the respective first or second propagation direction associated with the respective first or second path, and generating an estimate of the communication channel from the first and second characteristic matrices obtained.

2. The method of claim 1, wherein each of the first and second characteristic matrices associated with the first and second paths further depends on at least one of the following two elements: at least one vector connecting the center of gravity of the transmitter antenna array and at least one transmitter antenna, at least one vector connecting the center of gravity of the receiver antenna array and at least one receiver antenna.

3. The method of claim 1, further comprising: precoding a signal to be transmitted via the communication channel according to the estimated communication channel.

4. The method of claim 1, further comprising: receiving a signal after propagation via the communication channel, and demodulating the received signal according to the estimate of the communication channel.

5. A device for estimating a wireless communication channel between a transmitter and a receiver, comprising a plurality of paths allowing propagation of a wave, at least the transmitter or the receiver being formed of several antennas, wherein the device comprises: a processor; and a non-transitory computer-readable medium comprising instructions stored thereon which when executed by the processor configure the device to: implement, for at least a first path, a first estimate of a first propagation distance value associated with the first path, and of a first propagation direction associated with the first path, and a determination of a first characteristic matrix, implement, for a second path, a second estimate of a second propagation distance value associated with said second path, and of a second propagation direction associated with said second path, and a determination of a second characteristic matrix, the first and second characteristic matrices each having a definition depending on a parabolic wave model describing a distance between positions of at least one pair of antennas formed by an antenna array of the transmitter and an antenna array of the receiver, and a distance between centers of gravity of the antenna arrays of the receiver and the transmitter, the parabolic wave model being solely a function of: a quantity representative of the respective first or second estimate of the respective first or second propagation direction associated with the respective first or second path depending on at least one of the following elements: a propagation start direction vector associated with the respective first or second path, or a propagation arrival direction vector associated with the respective first or second path, and a corrective term depending on the respective first or second estimate of the respective first or second propagation distance value associated with the respective first or second path, wherein at least one of the first estimate of the first propagation distance value and of the first propagation direction or the second estimate of the second propagation distance value and of the second propagation direction is implemented by performing: a first underestimate, for a fixed quantity and a second quantity, distinct from the first quantity, wherein the first quantity is fixed and the second quantity can take a plurality of predetermined values, then a second underestimate for the second quantity set to a value resulting from the first underestimate and the first quantity can take the plurality of predetermined values, the first quantity or the second quantity being one of the following: the respective first or second estimate of the respective first or second propagation distance value associated with the respective first or second path, or the quantity representative of the respective first or second estimate of the respective first or second propagation direction associated with the respective first or second path, and generate and estimate of the communication channel from the obtained first and second characteristic matrices.

6. The device of claim 5, wherein each of the first and second characteristic matrices associated with the first and second paths further depends on at least one of the following two elements: at least one vector connecting the center of gravity of the transmitter antenna array and at least one transmitter antenna, at least one vector connecting the center of gravity of the antenna array of the receiver and at least one antenna of the receiver.

7. The device of claim 5, wherein the instructions further configure the device to: precode a signal to be transmitted via the communication channel according to the generated estimate of the communication channel.

8. The device of claim 5, wherein the instructions further configure the device to: receive a signal after propagation via the communication channel, and demodulate the received signal according to the estimate of the communication channel.

9. A non-transitory processor-readable recording medium, comprising instructions stored thereon which when executed by a processor of a device configure the device to estimate a wireless communication channel between a transmitter and a receiver, comprising a plurality of paths allowing propagation of a wave, at least the transmitter or the receiver being formed of several antennas, wherein the estimating comprises: for a first path, determining a first estimate of a first propagation distance value associated with the first path, and of a first propagation direction associated with the first path, and determining a first characteristic matrix, for a second path, determining a second estimate of a second propagation distance value associated with said second path, and of a second propagation direction associated with said second path, and determining a second characteristic matrix, the first and second characteristic matrices each having a definition depending on a parabolic wave model describing a distance between positions of at least one pair of antennas formed by an antenna array of the transmitter and an antenna array of the receiver, and a distance between centers of gravity of the antenna arrays of the receiver and the transmitter, the parabolic wave model being solely a function of: a quantity representative of the respective first or second estimate of the respective first or second propagation direction associated with the respective first or second path depending on at least one of the following elements: a propagation start direction vector associated with the respective first or second path, or a propagation arrival direction vector associated with the respective first or second path, and a corrective term depending on the respective first or second estimate of the respective first or second propagation distance value associated with the respective first or second path, and wherein at least one of the first estimate of the first propagation distance value and of the first propagation direction or the second estimate of the second propagation distance value and of the second propagation direction is carried out in a first underestimate, for a first quantity and a second quantity, distinct from the first quantity, wherein the first quantity is fixed and the second quantity can assume a plurality of predetermined values, and then in a second underestimate for the second quantity fixed at a value resulting from the first underestimate and the first quantity being able to assume the plurality of predetermined values, the first quantity or the second quantity being one of the following elements: the respective first or second estimate of the respective first or second propagation distance value associated with the respective first or second path, or the quantity representative of the respective first or second estimate of the respective first or second propagation direction associated with the respective first or second path, and generating an estimate of the communication channel from the first and second characteristic matrices obtained.

Description

4. LIST OF FIGURES

(1) Further advantages and features of the invention will become clearer upon reading the following description of a particular embodiment of the invention, given merely as an illustrative and non-limiting example, and the appended drawings, among which:

(2) FIG. 1 schematically illustrates a system comprising a transmitter and receivers that may incorporate an embodiment of the invention;

(3) FIGS. 2 and 3 illustrate respectively precoding and demodulation methods implementing an embodiment according to the invention;

(4) FIG. 4 illustrates more precisely and schematically a transmitter/receiver system that can implement an embodiment of the invention;

(5) FIG. 5 illustrates, as an example, a general representation model of the communication system shown in FIG. 4;

(6) FIG. 6 shows a an embodiment of a method for estimating a communication channel estimate according to the invention;

(7) FIGS. 7a and 7b represent embodiments of two alternatives of a preliminary step of validation of a wave model, according to the invention;

(8) FIG. 8 illustrates an example of the implementation of the channel estimate step, shown in FIG. 6;

(9) FIG. 9 illustrates another example of implementing the step of performing a channel estimate, shown in FIG. 6; and

(10) FIG. 10 illustrates a particular way of realizing a processing means for a transmitter or a receiver.

5. DESCRIPTION OF A PARTICULAR EMBODIMENT OF THE INVENTION

(11) FIG. 1 illustrates an example of a communication system in which the invention is used. In this example, a transmitter Tx or a transceiver such as a base station, communicates with two receivers Rx.sub.1 and Rx.sub.2 (or transceivers) respectively formed by one or a plurality of antennas. A wireless communication channel comprising one or more paths (also called paths) connects the transmitter Tx to each of the receivers Rx.sub.1 and Rx.sub.2. The communication channel (or transmission channel) allows signals to be transmitted from the transmitter to each of the receivers, the receiver having access to the transmitted signal distorted by its passage through the channel, and to which thermal noise has possibly been added. It is known that each path constituting the channel is associated with a complex gain, defined by a phase shift and an attenuation.

(12) The first receiver Rx.sub.1 of dimension L1 (for example a characteristic length of the first receiver, or the diameter of a circle in which all the antennas of the first receiver are inscribed) is located at a distance D.sub.1 from the transmitter Tx. This distance is large enough for the plane wave hypothesis to be applicable, despite the size of the receiver.

(13) Conversely, the second receiver Rx.sub.2 of dimension L2 is located at a distance D.sub.2 from the transmitter Tx. The dimension L2 corresponds for example to the diameter of a circle in which a system of 256 antennas associated with each other is inscribed if we place ourselves in the case of a “massive MIMO” receiver. The length D.sub.2 is insufficient to allow the application of the hypothesis plane waves. Indeed the curvature of the waves symbolized by the concentric circles, is too important from the point of view of the Rx.sub.2 receiver.

(14) For a signal to be transmitted from a transmitter to a receiver, a precoding of the signal to be transmitted is performed to maximize the data rate on the wireless communication channel. As illustrated in FIG. 2, a first step 21 of estimating the wireless communication channel is performed, in order to know the parameters defining this communication channel. Then a precoding 22 of the signal to be transmitted via this communication channel is performed based on the parameters obtained during the communication channel estimate step.

(15) For a signal received by a receiver from a transmitter, a demodulation of the signal received via the communication channel is performed in order to maximize the data rate on the channel. As shown in FIG. 3, a first step 31 of estimating the wireless communication channel is performed, in order to know the parameters defining this communication channel, from known transmitted signals. Then a demodulation 32 of the received signal of the communication channel is performed from the estimated channel, or in other words from the parameters obtained in the step of estimating the communication channel.

(16) FIG. 4 illustrates very schematically a device implementing an embodiment of the invention. A transmitter Tx is connected to a receiver Rx.sub.i via a wireless communication channel Cn symbolized by a rectangle. The transmitter Tx comprises a processing means MTx configured, among other things, to implement the steps 21 and 22 described with reference to FIG. 2.

(17) The receiver Tx comprises a processing means MRx.sub.i configured, among other things, to implement the steps 31 and 32 described with reference to FIG. 3.

(18) FIG. 5 illustrates, as an example, a general model of representation of the communication system schematized on FIG. 4, in a case of a transmission in line of sight (“LoS”).

(19) A point O.sub.t (respectively O.sub.r) corresponds to the center of gravity of the transmission (respectively reception) antenna array.

(20) Vectors {{right arrow over (a.sub.t,1)}, . . . , a.sub.t,N.sub.t} and {{right arrow over (a.sub.r,1)}, . . . , a.sub.r,N.sub.r} correspond to the positions of the transmission and reception antennas with respect to their respective centers of gravity (different local benchmarks are used for transmission and reception). Moreover a reference distance D corresponds to the distance between the two centers of gravity O.sub.t and O.sub.r.

(21) The notation D.sub.ij refers to the distance between the j-th transmitting antenna and the i-th receiving antenna. In the example of FIG. 5, the distance D.sub.23 between the second transmitter antenna and the third receiver antenna is shown.

(22) {right arrow over (u.sub.t)} corresponds to the direction of wave propagation at the start of the communication channel in the reference frame centered on O.sub.t and {right arrow over (u.sub.r)} to the direction of wave propagation at the end of the communication channel in the reference frame centered on O.sub.r.

(23) Assuming attenuation and phase proportional to the distance O.sub.ij traveled by the waves, the transmission channel at the frequency f between the J-th transmission antenna and the i-th reception antenna can be expressed as follows:

(24) $h_{\mathrm{ij}}=h\frac{D}{D_{\mathrm{ij}}}{e}^{-j\frac{2\pi }{\lambda }\left(D_{\mathrm{ij}}-D\right)},$
where

(25) $\lambda \stackrel{\Delta }{=}\frac{c}{f}$
is the wavelength, c corresponding to the speed of light, and h corresponds to the channel between the points O.sub.t and O.sub.r.
Moreover, the distances D and D.sub.ij being rather close in practice, the hypothesis

(26) $\frac{D}{D_{\mathrm{ij}}}=1$
is made. Finally, the distance D.sub.ij is expressed as follows
D.sub.ij=√{square root over (D.sup.2+2D({right arrow over (a.sub.r,i)}.Math.{right arrow over (u.sub.r)}−{right arrow over (a.sub.t,j)}.Math.{right arrow over (u.sub.t)})+∥R{right arrow over (a.sub.r,i)}−{right arrow over (a.sub.t,j)}∥.sup.2)},
where R is a rotation matrix allowing to match the local markers used at transmission and reception.

(27) Considering this expression for the distance as accurate, the inventors were able to establish a new definition of the spherical wave model (SWM), as expressed below:

(28) ${\Delta }_{\mathrm{SWM},\mathrm{ij}}\stackrel{\Delta }{=}{D}_{\mathrm{ij}}-D=\sqrt{D^2+2D\left(\stackrel{.fwdarw.}{a_{r,i}}.\stackrel{.fwdarw.}{u_r}-\stackrel{.fwdarw.}{a_{t,j}}.\stackrel{.fwdarw.}{u_t}\right)+{.Math.R\stackrel{.fwdarw.}{a_{r,i}}-\stackrel{.fwdarw.}{a_{t,j}}.Math.}_2^2}-D,$
So we get for the channel expression:

(29) $h_{\mathrm{ij}}={\mathrm{he}}^{-j\frac{2\pi }{\lambda }{\Delta }_{\mathrm{SWM},\mathrm{ij}}}.$

(30) Classically, due to the assumption of the plane wave model, the state-of-the-art physical models used for channel estimate approximate the expression Δ.sub.SWM,ij by
Δ.sub.PWM,ij={right arrow over (a.sub.r,i)}.Math.{right arrow over (u.sub.r)}−{right arrow over (a.sub.t,j)}.Math.{right arrow over (u.sub.t)}

(31) This expression is much less complex than the one used in the case of the spherical wave model assumption, but it poses the problem of lack of precision in the case of receivers and/or transmitters with large antennas.

(32) To solve this problem of complexity (and therefore cost in terms of computing time) posed by the spherical wave model and inaccuracy posed by the plane wave model, the inventors have proposed a new intermediate wave model called Parabolic Wave Model (ParWM), as defined below:

(33) ${\Delta }_{\mathrm{ParWM},\mathrm{ij}}=\stackrel{.fwdarw.}{a_{r,i}}.\stackrel{.fwdarw.}{u_r}-\stackrel{.fwdarw.}{a_{t,j}}.\stackrel{.fwdarw.}{u_t}+\frac{1}{2D}\left[{.Math.R\stackrel{.fwdarw.}{a_{r,i}}-\stackrel{.fwdarw.}{a_{t,j}}.Math.}^2-{\left(\stackrel{.fwdarw.}{a_{r,i}}.\stackrel{.fwdarw.}{u_r}-\stackrel{.fwdarw.}{a_{t,j}}.\stackrel{.fwdarw.}{u_t}\right)}^2\right].$
This new model is a compromise between the PWM plane wave model and the SWM spherical wave model: it is more accurate than the PWM, and less complex than the SWM. Indeed, the expression of Δ.sub.ParWM,ij only includes products and additions while that of Δ.sub.SWM,ij includes a square root.

(34) In the remainder of the description, for simplification purposes, the invention will be placed in the particular context known as MISO (Multiple In Single Out) multipath, i.e., it is considered that the receiver comprises a single antenna. Of course, the person skilled in the art will easily be able to extend the invention to the MIMO context by considering the general expression of the distance difference D.sub.ij or one of its approximations proposed above.

(35) In the MISO context, the receiver has only one antenna, N.sub.r=1. As a result, the Rotation matrix R, a factor of the vector representing the direction of arrival of the signal on the receiver antennas, disappears from the equation. This corresponds for example to 5G networks designed for base stations with a large number of antennas and mobile terminals with only one antenna. In this case, the expressions given above allow to take into account not only communication channels consisting of a single path in direct view but also paths resulting from reflections on planes in indirect view. Each path is associated with a complex gain corresponding to the channel between the centers of gravity of the transmission and reception antenna arrays.

(36) The inventors have proposed a general expression for the communication channel comprising paths p. This expression is valid for the three considered models SWM, PWM and ParWM:
h=√{square root over (N.sub.t)}Σ.sub.k=1.sup.ph.sub.ke({right arrow over (u.sub.t,k)},D.sub.k),
where denotes the chosen physical model,
h.sub.k, {right arrow over (u.sub.t,k)} and D.sub.k are respectively the transmission channel between the centers of gravity of the transmission and reception antenna arrays, the direction of departure during the propagation of the waves for the k-th path and the distance traveled by the waves for the k-th path, and e(√{square root over (u.sub.t,k)},D.sub.k)∈.sup.N′ is a so-called characteristic vector of the physical model for this k-th path. In case where the transmitter and the receiver each comprise several antennas, the communication channel is then defined by a characteristic matrix whose dimensions are given by the number of antennas at the transmitter and the receiver. In the example considered where the receiver has only one antenna, the characteristic matrix has only one column. We speak then of characteristic vector.

(37) This characteristic vector is generally expressed as follows:

(38) $e_{ℳ}\left(\stackrel{.fwdarw.}{u_{t,k}},D_k\right)=\frac{1}{\sqrt{N_t}}\left(\begin{array}{c}{e}^{-j\frac{2\pi }{\lambda }{\Delta }_{ℳ,\mathrm{lk}}}\\ \mathrm{.Math.}\\ e^{-j\frac{2\pi }{\lambda }{\Delta }_{ℳ,N_{t}k}}\end{array}\right).$
With for the spherical wave SWM and parabolic wave ParWM models:

(39) ${\Delta }_{\mathrm{SWM},\mathrm{jk}}=\sqrt{D_k^2+2D_k\left(\stackrel{.fwdarw.}{a_{i,j}},\stackrel{.fwdarw.}{u_{t,k}}\right)+{.Math.\stackrel{.fwdarw.}{a_{i,j}}.Math.}_2^2}-D_k,{\Delta }_{\mathrm{ParWM},\mathrm{jk}}=-\stackrel{.fwdarw.}{a_{i,j}},\stackrel{.fwdarw.}{u_{t,k}}+\frac{1}{D_k}\left[{.Math.\stackrel{.fwdarw.}{a_{i,j}}.Math.}^2-{\left(\stackrel{.fwdarw.}{a_{i,j}},\stackrel{.fwdarw.}{u_{t,k}}\right)}^2\right].$

(40) In the case of the PWM plane wave model with Δ.sub.PWM,jk=−{right arrow over (a.sub.t,j)}.Math.{right arrow over (u.sub.t,k)}, the expression of the vector does not depend on the distance but only on the direction of departure, it is simply a directional vector (“steering vector”) that can be noted e.sub.PWM({right arrow over (u.sub.t,k)}). This model does not take into account the curvature of the wavefronts. On the contrary, the characteristic vectors of the spherical SWM and parabolic ParWM models depend on the distance and allow to take into account the curvature of the wavefront. It should be noted that a single general expression for the communication channel allows the use of all three models.

(41) In the case where it is preferable to take into account the radius of curvature of the wavefront, a method for estimating the communication channel is proposed (e.g., for steps 21 or 31 of FIG. 2 or 3), an embodiment of which is illustrated in FIG. 6.

(42) A first step 61 includes, for one or more paths, an estimate of a value representative of the propagation distance associated with the path under consideration (or an estimate of a value allowing to infer it later).

(43) For example, this distance can be equated to the distance between the center (or barycenter) of the transmitter and the center of the receiver.

(44) Alternatively, this distance can be assimilated to the distance between an antenna of the transmitter and the antenna of the receiver.

(45) This first step 61 also includes an estimate of the direction of departure of the wave {right arrow over (u.sub.t,k)} for the path(s) considered (or of a magnitude representative of said direction which can be inferred at a later stage). The vectors {{right arrow over (a.sub.t,1)}, . . . , {right arrow over (a.sub.t,N.sub.t)}} may also be estimated during this step 61. Alternatively, these vectors can be provided by default as input to step 61.

(46) A second step 62 comprises estimating a feature vector per path according to the invention, from the values estimated during step 61.

(47) An embodiment of the communication channel estimate is then performed in a step 63 from the feature vectors determined in step 62.

(48) According to one embodiment of the invention, it is particularly advantageous to perform a prior validation step of the channel model on which the feature vector depends. One purpose of this step is to confirm that the channel model used is indeed valid for the distance D separating the transmitter from the receiver.

(49) A first example of a preliminary step is shown in FIG. 7a. A phase shift deviation of at most

(50) $\frac{\pi }{8}$
from the phase shift of the spherical wave mode

(51) 0$.Math.\frac{2\pi }{f}{\Delta }_{\mathrm{SWM},\mathrm{jk}}$
is allowed to validate the plane wave model. If

(52) $\frac{2\pi }{\lambda }.Math.{\Delta }_{\mathrm{SWM},\mathrm{jk}}-{\Delta }_{\mathrm{PWM},\mathrm{jk}}.Math.\mathrm{and}\frac{2\pi }{\lambda }.Math.{\Delta }_{\mathrm{SWM},\mathrm{jk}}-{\Delta }_{\mathrm{ParWM},\mathrm{jk}}.Math.$
are bound, where Δ.sub.PWM,jk and Δ.sub.ParWM,jk are the phase shifts of the plane and parabolic wave models respectively, using the fact that |{right arrow over (a.sub.t,j)}.Math.{right arrow over (u.sub.t)}|≤R.sub.t, where R.sub.t≙max.sub.i∥{right arrow over (a.sub.t,i)}∥.sub.2, (∥.∥.sub.2 denoting the classical Euclidean norm), we get:

(53) $D\ge \frac{8R_t^2}{\lambda }$
for the plane wave model, this bound being defined as the Fraunhofer distance,

(54) $D\ge \sqrt{\frac{8R_t^3}{\lambda }}$
for the parabolic wave model, this bound can be defined as the Fresnel distance.
R.sub.t is the radius of the smallest circle in which the antenna array forming the transmitter fits.

(55) As illustrated in FIG. 7a, a first step 71a comprises obtaining the distance D between the center of the transmitter and that of the receiver. Then comparing in a step 72a between said distance D and two thresholds S.sub.1 and S.sub.2 with for example

(56) $S_1=\sqrt{\frac{8R_t^3}{\lambda }}\mathrm{and}\frac{8R_t^2}{\lambda }.$

(57) Finally a validation step 73a of the use of the plane wave model if

(58) $D\ge \frac{8R_t^2}{\lambda },$
the parabolic wave model if

(59) $\frac{8R_t^2}{\lambda }>D\ge \sqrt{\frac{8R_t^3}{\lambda }},$
and the spherical wave model if

(60) $D\ge \sqrt{\frac{8R_t^3}{\lambda }}.$

(61) This alternative is simple to implement, but it has the disadvantage of being based on an arbitrary phase difference of

(62) $\frac{\pi }{8}.$
Moreover it is independent or me position (once the distance is obtained) and the relative orientation of the receiver with respect to the transmitter.

(63) Another alternative shown in FIG. 7b addresses these problems. This alternative relies on a metric called relative model approximation error (rMAE), defined below:

(64) $\mathrm{rMAE}=\frac{{.Math.h-{\mathrm{proj}}_{ℳ}\left(h\right).Math.}_2^2}{{.Math.h.Math.}_2^2}$

(65) where proj(u)≙argmin.sub.x∈ ∥u−x∥.sub.2 where is the model to be validated and h is the reference channel, here the channel obtained with the spherical wave model. The relative approximation error of the model quantifies the minimum approximation error implied by the model M considered.

(66) The model M can be considered valid when the rMAE is low (for example if rMAE<0.05, which corresponds to an error of at most 5%). For example if we consider a uniform linear array of 256 antennas for the transmitter at 30 GHz and a receiver facing the broadside of the transmitter, the parabolic wave model is valid (rMAE<0.05) if the distance D between the transmitter and the receiver is greater than 2.5 m.

(67) More precisely, this alternative comprises a first step 71b where an approximation error rMAE is determined for a given model by considering a reference model REF, for example the spherical wave model. In a test 72b, the relative approximation error of the model is compared to a predetermined error margin value VP, for example 5%. If the relative approximation error of the model is less than 5%, then the given model is validated, step 73b. Otherwise, it is discarded as too inaccurate for the considered transmitter/receiver configuration, step 74b. It should be noted that the parabolic wave model described above becomes sufficiently accurate from only a few meters.

(68) Using the linear structure of the relations between the transmitted signals and the channel coefficients, a realization of a channel estimate can be based on observations of the form:
y=Xh+n,
where h is the channel to be estimated, X is the observation matrix (which contains the pilot symbols used for channel estimate) and n represents the noise.
The problem in estimating the channel can be reformulated as follows:

(69) 0$\underset{E,\alpha }{\mathrm{minimiser}}={.Math.y-\mathrm{XE}\alpha .Math.}_2^2,\hat{h}\leftarrow E\alpha ,$
where E≙(e({right arrow over (u.sub.t,1)}, D.sub.1), . . . , e({right arrow over (u.sub.t,p)}, D.sub.p)), and α≙√{square root over (N.sub.t)}(h.sub.1, . . . , h.sub.p).sup.Tĥ is the estimated channel.

(70) According to an embodiment, a channel estimate realization is proposed using the characteristic vector defined above, this characteristic vector allowing a unified description of the three models: plane, parabolic and spherical waves. The algorithms considered here are called gluttonous, because the paths are estimated one by one, based on a residual from the estimate of the previous paths.

(71) A first alternative comprises an estimate of a characteristic vector of the spherical or parabolic wave model by solving the following optimization problem when estimating the k-th path:

(72) $\stackrel{.fwdarw.}{u_{t,k}},D_k\leftarrow \underset{\stackrel{.fwdarw.}{u_t},D}{\mathrm{argmax}}\frac{.Math.r^{\left(k\right)H}{\mathrm{Xe}}_{ℳ}\left(\stackrel{.fwdarw.}{u_t},D\right).Math.}{{.Math.{\mathrm{Xe}}_{ℳ}\left(\stackrel{.fwdarw.}{u_t},D\right).Math.}_2},$
where denotes the chosen model, e({right arrow over (u.sub.t)}, D) the feature vector for the model and r.sup.(k) is a residual resulting from the previous iteration, such that:
r.sup.(1)=y,
r.sup.(k+1)=y−XE.sup.(k)α.sup.(k),
with the optimal vector α.sup.(k)←(E.sup.(k)HX.sup.HXE.sup.(k)).sup.−1E.sup.(k)HX.sup.Hy where E.sup.(k)≙(e({right arrow over (u.sub.t,1)}, D.sub.1), . . . , e({right arrow over (u.sub.t,k)}, D.sub.k)) represents the state of the matrix E at the k-th iteration. It is specified that H is here the well-known symbol for transconjugation.

(73) FIG. 8 illustrates an implementation of step 63 of FIG. 6, for example, in the case where two iterations are performed, respectively for a first and a second path. The extension to an implementation for k iterations can be easily deduced by the man of the art, from this example of FIG. 8.

(74) A step 81 includes a first estimate fora first path from the determined values of the distance D.sub.1 associated with the first path and the direction {right arrow over (u.sub.t,1)} also associated with the first path. A residual r.sup.(2) obtained as described above is output as input to a second step 82. This second step comprises a second estimate for a second path from the determined values of the distance D.sub.2 associated with the second path, the direction {right arrow over (u.sub.t,2)} also associated with the second path and the residue r.sup.(2) obtained in step 81.

(75) The matrix E.sup.(2) is delivered at the end of step 82, allowing to determine the estimated channel ĥ from the previously mentioned equation: ĥ←E.sup.(2)α.sup.(2)

(76) Note that in the case where only one path is considered, only step 81 is performed.

(77) The advantage of this alternative is a high accuracy in the estimated channel obtained. However, the realization of the channel estimate is possible by building a dictionary of

(78) $N_{\stackrel{.fwdarw.}{u_i}}{N}_D$
feature vectors (corresponding to starting

(79) $N_{\stackrel{.fwdarw.}{u_i}}$
directions and N.sub.D distances). The complexity of the approximate solution of this problem is therefore

(80) $O\left(N_{\stackrel{.fwdarw.}{u_i}}{N}_D\right).$
Indeed, this alternative implies to test all the distances for each direction, which induces a high computational complexity.

(81) Another less expensive implementation is possible by estimating the characteristic vector of the chosen model (spherical or parabolic waves) via the solution of the optimization problem

(82) $\stackrel{.fwdarw.}{u_{t,k}}\leftarrow \underset{\stackrel{.fwdarw.}{u_t}}{\mathrm{argmax}}\frac{.Math.r^{\left(k\right)H}{\mathrm{Xe}}_{\mathrm{PWM}}\left(\stackrel{.fwdarw.}{u_t}\right).Math.}{{.Math.{\mathrm{Xe}}_{\mathrm{PWM}}\left(\stackrel{.fwdarw.}{u_t}\right).Math.}_2},D_k\leftarrow \underset{D}{\mathrm{argmax}}\frac{.Math.r^{\left(k\right)H}{\mathrm{Xe}}_{ℳ}\left(\stackrel{.fwdarw.}{u_{t,k}},D\right).Math.}{{.Math.{\mathrm{Xe}}_{ℳ}\left(\stackrel{.fwdarw.}{u_t},D\right).Math.}_2}.$
This alternative implementation corresponds to a sequential estimate of the propagation directions respectively associated with each path, and then the distances also associated with each path. Its complexity is

(83) $O\left(N_{\stackrel{.fwdarw.}{u_t}}+N_D\right).$
Indeed, the strategy amounts to testing several distances only for the best propagation direction. This best propagation direction can be determined for example with plane wave fronts.

(84) FIG. 9 illustrates an example of this alternative implementation, for example for the first estimate 81 in the previous figure. A first sub-estimate step 91 is to set the value of the distance D.sub.1 and determine the best direction (the one that maximizes the cost function). When this best propagation direction is determined, a second underestimate 92 amounts to testing several distances for this best propagation direction. Conversely, in an alternative embodiment, the first underestimate may consist in finding the best possible distance value for a given propagation direction (for example in the case of a very directive propagation), then during the second underestimate step testing the propagation directions for this best distance value obtained during the first underestimate step.

(85) FIG. 10 illustrates one particular way, among several possible ways, to realize a processing means 100 (for a transmitter or receiver) configured to implement an embodiment of a method according to the invention. The device 100 comprises a random access memory 130 (e.g., a RAM memory), a processing unit 110 equipped with, for example, a processor, and driven by a computer program stored in a read-only memory 120 (e.g., a ROM memory or a hard disk). Upon initialization, code instructions of the computer program are for example loaded into the RAM 130 before being executed by the processor of the processing unit 110.

(86) FIG. 10 illustrates only one particular way, among several possible ways, of realizing the processing means 100 so that it performs certain steps of the method according to the invention. Indeed, these steps can be performed indifferently on a reprogrammable computing machine (a PC computer, a DSP processor or a microcontroller) executing a program comprising a sequence of instructions, or on a dedicated computing machine (for example a set of logic gates such as an FPGA or an ASIC, or any other hardware module).

(87) In case where the processing means is realized with a reprogrammable computing machine, the corresponding program (i.e. the sequence of instructions) can be stored in a removable or non-removable storage medium, this storage medium being partially or totally readable by a computer or a processor.

(88) It goes without saying that the implementation and realization methods described above are purely indicative and in no way limiting, and that numerous modifications can easily be made by the man of the art without going beyond the scope of the invention.