Specification
METHOD FOR MEASURING INCOMING ANGLES OF COHERENT SOURCES USING SPACE SMOOTHING ON ANY SENSOR NETWORK
The invention relates notably to a method making it possible to interpolate steering
vectors of a network of any sensors by using omnidirectional modal functions.
It also relates to a method and a system making it possible, notably, to estimate
arrival angles of coherent sources via a smoothing technique on a network of
nonuniform sensors.
It is used, for example, in all the location systems in an urban context where the
propagation channel is disrupted by a large number of obstacles such as buildings.
In a general manner, it may be used to locate transmitters in a difficult propagation
context, urban, semi-urban (airport), inside buildings, etc.
It may also be used in medical imaging methods for locating tumors or epileptic
focal spots.
It applies in sounding systems for mining and oil research in the seismic field. These
applications require estimates of arrival angles with multipaths in the complex
propagation medium of the earth's crust.
The technical field is that of the processing of antennae which process the signals of several transmitting sources based on a multisensor receiving system. In an electromagnetic context, the sensors are antennae and the radioelectric sources are propagated according to one polarization. In an acoustic context, the sensors are microphones and the sources are sound sources. Figure 1 shows that an antenna processing system consists of a network of sensors receiving sources with different incoming angles mp. The field is, for example, that of goniometry which consists in estimating the incoming angles of the sources.
The elementary sensors of the network receive the sources with a phase and an amplitude that is dependent in particular on their angles of incidence and on the position of the sensors. The angles of incidence are in parametric representation in 1D by the azimuth m and in 2D by the azimuth m and the elevation m. According to figure 2, a 1D goniometry is defined by techniques which estimate only the
azimuth supposing that the source waves are propagated in the plane of the sensor
network. When the goniometry technique jointly estimates the azimuth and the
elevation of a source, it is a question of 2D goniometry.
The objective of antenna processing techniques is to make use of spatial diversity
which consists in using the position of the antennae of the network to make better use
of the differences in incidence and distance of the sources.
Figure 3 illustrates an application to goniometry in the presence of multipaths. The
mth source is propagated on P paths of incidences mp (1≤p≤P) which are caused by
P-\ obstacles in the radioelectric environment. The problem treated in the method
according to the invention is notably the situation of coherent paths where the
propagation time difference between the direct path and a secondary path is much
less than the inverse of the band of the signal.
The technical problem to be solved is also that of the goniometry of coherent paths
with a reduced calculation cost and a network of sensors with a nonuniform
geometry.
Knowing that the goniometry techniques with a reduced calculation cost are suitable
for networks of equally-spaced linear sensors, one of the objects of the method
according to the invention is to use these techniques on networks of nonuniform
sensors.
The algorithms making it possible to process the case of coherent sources are, for
example, the algorithms of Maximum Likelihood [2] [3] which can be applied to
sensor networks with nonuniform geometry. However, these algorithms need
multiparameter estimates which induce an application with a high calculation cost.
The maximum likelihood technique is adaptable for the cases of equally-spaced
linear sensor networks via the IQML or MODE [7] [8] methods. Another alternative
is that of the spatial smoothing techniques [4] [5] which have the advantage of
processing the coherent sources with a low calculation cost. The goniometry
techniques with a low calculation cost adapted for linear networks are either the
ESPRIT method [9] [10] or techniques of the Root type [11] [12] amounting to
searching for the roots of a polynomial.
The techniques making it possible to transform networks with nonuniform geometry
into linear networks are described, for example, in documents [6] [5] [11]. These methods consist in interpolating on an angular sector the response of the sensor network to a source: Calibration Table.
The document of B. Friedlander and A. J. Weiss entitled "Direction Finding using spatial smoothing with interpolated arrays", IEEE Transactions on Aerospace and Electronic Systems, Vol. 28, No. 2, pp. 574-587, 1992, discloses a method which consists in:
• interpolating the sensor network via a linear network in a determined angular sector with an interpolation function that is not omnidirectional in azimuth,
• decorrelating the paths by a spatial smoothing technique. This technique, although powerful, has the disadvantages:
• of processing the case of coherent sources present in the same angular sector, hence of processing a single angular sector;
• of interpolating with a function that is not omnidirectional in azimuth.
The invention relates to a method for determining the angles of arrival of coherent sources in a system comprising several nonuniform sensors, the signals being propagated along coherent or substantially coherent paths between a source and said receiving sensors of the network. It is characterized in that use is made of at least one
modal interpolation function z()k that is omnidirectional in azimuth where z() = exp(j) with 9 corresponding to an angle sector on which the interpolation
of the steering vectors a () of the sensor network is carried out in order to process the signals transmitted by the sources and received on the sensor network and a spatial smoothing technique is applied in order to decorrelate the coherent sources, the interpolation function W e() is expressed in the following manner:
(Equation Removed)
The matrix W of dimension Nx(2L + l) is obtained by minimizing in the sense of
the least squares the deviation ||a() - We()||2 for azimuths verifying 0 ≤ < 360°, the length of the interpolation 2L+1 depends on the aperture of the network.
The method according to the invention notably offers the following advantages:
• It interpolates the sensor network with omnidirectional functions in azimuth.
• It processes the case of coherent sources on different angular sectors.
• It uses the algorithm from 0 to 360° in azimuth.
• It applies a spatial smoothing technique in order to decorrelate the coherent sources.
Other features and advantages of the present invention will appear more clearly on reading the following description of an exemplary embodiment given as an illustration and being in no way limiting with the addition of figures which represent:
• figure 1, an example of signals transmitted by a transmitter and being propagated to a sensor network,
• figure 2, the presentation of the incidence of a source on a sensor plane,
• figure 3, the propagation of multipara signals,
• figure 4, an example of position sensor networks (xn,yn),
• figure 5, a network of sensors consisting of two subnetworks that do not vary by translation,
• figure 6, the length of interpolation with modal functions according to the ratio R/, of the network,
• figure 7, the amplitude error for R/=0.5 where 89=50°,
• figure 8, the interpolation according to the invention on two angular sectors,
• figure 9, a zone of interpolation on two sectors,
• figure 10, the complete meshing of the space for the calculation of the matrices Wjk.
Before giving details of an exemplary embodiment of the method according to the invention, a few notes on modeling the output signal of a sensor network are given.
Modeling the output signal from a sensor network
In the presence of M sources with Pm multipaths for the mth source, the output signal, after receipt on all the sensors of the network:
(Equation Removed)
where xn(t) is the output signal of the nth sensor, A=[ A1... AM], AM=[ a(m1)... a(mPm)], s(t)=[ s1(r)T ... SM(t)T]T, sm(t)=[ sm{t-m1)... sm(t-mPm)]T, n(t) is the additional noise, a() is the response of the sensor network to a source of direction 6 and mp, mp, mp are respectively the attenuation, the direction and the delay of the pth paths of the mth source. The vector a() which is also called the steering vector depends on the positions (xn,yn) of the sensors (see figure 4) and is written:
(Equation Removed)
where X is the wavelength and R the radius of the network. In the case of an equally spaced linear network, the vector a() is written:
(Equation Removed)
where d is the distance between sensors.
In the presence of coherent paths, the delays verify m1 =...=mpm- In these
conditions, the signal model of the equation (1) becomes:
(Equation Removed)
where a(m,pm) is the response of the sensor network to the mth source, m=[ m1 • • mpm ]T and m = [m1 ... mpm]T • The steering vector of the source is no longer a(ml) but a composite steering vector a(M,m) which is different and
which depends on a number of more important parameters.
A problem with the algorithms of the prior art in the presence of coherent
sources
The algorithm MUSIC [1] is a high-resolution method based on the breaking down into elements specific to the matrix of covariance Rx=E[x(t) x(t)H ] of the multisensor signal x(t) (E[.] is the mathematical hope). According to the model of the equation (1), the expression of the covariance matrix Rx is as follows:
(Equation Removed)
The alternative to MUSIC for coherent sources is the algorithm of Maximum Likelihood [2] [3] which requires the optimization of a multidimensional criterion depending on the incoming directions mp of each of the paths. The latter estimate mpfor (1 ≤ m ≤ M) and (1≤ p ≤ Pm) of a criterion with K= Σm=1 Pm dimensions requires a high calculation cost.
Spatial smoothing techniques
The object of spatial smoothing techniques is notably to apply a preprocess to the covariance matrix Rx of the multisensor signal which increases the rank of the covariance matrix Rs of the sources in order to be able to apply algorithms of the MUSIC type or any other algorithm having equivalent functionalities in the presence of coherent sources without needing to apply an algorithm of the maximum likelihood type.
When a sensor network contains invariant subnetworks by translation as in figure 5, the spatial smoothing techniques [4][5] can then be envisaged. In this case, the signal received on the ith subnetwork is written:
(Equation Removed)
where ai() is the steering vector of this subnetwork which has the particular feature of verifying:
(Equation Removed)
The mixed matrix A' of the equation (6) is then written
(Equation Removed)
Knowing that Ai=[ A1i... AMI] and Ami'=[ a' (m1)... a' (m pm)]. In the case of the linear network of the equation (3) this gives
(Equation Removed)
The smoothing technique is based on the structure of the covariance matrix Rxi=E[ X(T) IX(T) iH] which, according to (6)(8), is written as follows:
(Equation Removed)
The spatial smoothing technique therefore makes it possible to apply a goniometry algorithm like the MUSIC algorithm on the following covariance matrix:
(Equation Removed)
where I is the number of subnetworks. Specifically this technique makes it possible to decorrelate to the maximum I coherent paths because
(Equation Removed)
and thus rank{ Rs }A_dB_ref then 86=86/2 and return to step
A.2
Step No. A.5: Calculation of K = 180 /(Pδ)
Step No. A.6: For all P-uplets (iv..iP) verifying 0 ≤ i1, ≤... ≤ iP < K:
Step No. A.6.1: Calculation of the ip =2δ x ip for 1 ≤ p ≤ P
Step No. A.6.2: Calculation of the matrix Wi1...ip by minimizing in the
sense of the mean squares ||a()-Wi1...ip e()||2 for |-ip| <δ and
1≤p≤P.
Step No. A.6.3: Return to step A.6.1 if all the P-uplets (i1...iP) verifying
1 ≤, i1, ≤, ...≤ip≤ K are not explored.
The steps for carrying out the goniometry with an interpolation on P sectors use the
interpolation matrices calculated during the steps A. The steps of the goniometry are
then as follows:
Step No. B.0: Initialization of the assembly Θ at O
Step No. B: For all P-uplets (i1...iP) verifying 0 ≤i1,≤...≤ip< K:
Step No. B.1: Calculation of yi1..ip (t) = Wi1...ip-1x(t)
Step No. B.2: Calculation of ip =2δxip for1≤p≤P
Step No. B3: Application of a spatial and/or Forward-Backward smoothing technique to the observation yi1... ip(t) then application of a goniometry of the
ESPRIT type in order to obtain the incidences k for 1 ≤, k ≤ Ki1...ip .
Step No. B.4: Selection of the estimated incidences k εΘi1...ip, where
(Equation Removed)
according to the following MUSIC[1] criterion in which
(Equation Removed)
where II b is the noise projector extracted from the covariance matrix Rx (the
equation (7) forms part of the passage in orange that has been deleted). Hence the proposition; according to a known equation of the methods of goniometry of the MUSIC type. (The threshold is chosen typically at 0.1.)
Step No. B.5: 0=Θ Θi1...ip assemblies of the angles of incidence verifying the step B.4 for all the sectors associated with the P-uplets (i1...iP) processed by the algorithm. Step No. B.6: Return to step No. B.l so long as all the P-uplets (i1,..ip)
verifying 0≤i1≤...≤ip A_dB_ref, then do δ=δ2 and return to
step A.2
Step No. A.5: Calculation of K = 180/{PS0)
Step No. A.6: For all P-uplets (i1...iP) verifying0 ≤ i1 ≤...≤ iP < K with K being the number of sectors on which the interpolation is carried out:
Step No.A.6.1: Calculation of ip =2S0xipfor 1 ≤ p ≤ P
Step No.A.6.2: Calculation of the interpolation matrix Wi1.. ip by minimizing in the sense of the mean squares ||a()-Wi1..ip e()||2 for |-ip|, <δ and 1≤p≤P.
Step No. A.6.3: Return to step A.6.1 if all the P-uplets (i1..iP) verifying
1 ≤i1,≤...< ip ≤ K are not explored.
2 - The method as claimed in claim 1, characterized in that the value of L is
determined in the following manner:
(Formula Removed)
where L is the minimal value verifying AdB less than 0.1 dB. Specifically, AdB is zero when the interpolation is perfect and therefore when a() = We().
3 - The method as claimed in claim 2, characterized in that, for networks in
which the length of the interpolation 2L+1 is greater than N, the network is
interpolated by K sectors of width SO =180/K with square interpolation matrices
Wk where
(Formula Removed)
where the K matrices Wk are squared with N = 2L0 +1,
the matrices Wk are obtained by minimizing the deviation ||a()-Wke()||2 in the
sense of the mean squares the deviation for | - k | < δ,
the width of the interpolation cone δis determined from the following amplitude error criterion:
(Formula Removed)
where δ is the minimal value verifying that AdB is less than 0. ldB because AdB is zero when a() = Wke().
4 - The method as claimed in one of claims 1 to 3, characterized in that it
comprises a goniometry step comprising at least the following steps:
Step No. B.0: Initialization of an assembly 0 at 0
Step No. B: For all P-uplets (i1,...ip) verifying 0 ≤/i1≤...≤ip< K:
Step No. B.1: Calculate yi1..ip (t) = Wi1...ip-lx(t)
Step No. B.2: Calculate the ip =2δxip for1≤p≤P
Step No. B.3: Apply a spatial and/or Forward-Backward smoothing technique to the observation yi1..ip'(t) and then apply a goniometry algorithm
A
in order to obtain the incidences k for 1 ≤ k ≤ Ki1..ip .
Step No. B.4: Select estimated incidences k ε Θi1...ip where Θi1...ip ={
(Formula Removed)
Step No. B.5: Θ =Θ Θi1...ip assemblies of the angles of incidence verifying
the step B.4 for all the sectors associated with the P-uplets (i1,...ip) processed
by the algorithm. Step No. B.6: Return to step No. B.1 so long as all the P-uplets (i1..iP)
verifying 0 ≤ i1 ≤...≤ip, < K are not explored.
5 - The method as claimed in claim 1, characterized in that, for coherent sources present in different sectors, the steering vector a() is interpolated jointly on several sectors.