METHOD FOR OPTIMIZING THE POSITION OF IDENTICAL SENSORS IN A
COMMUNICATION NETWORK
The subject of the present invention relates notably to a method for optimally determining the position of sensors in an array comprising a number of sensors used to perform goniometry. The method relates to uniform arrays consisting of identical sensors in which the response of the array to a direction Θ depends solely on the wave vector k(Θ) and on the positions pn of the sensors.
The invention is applicable for defining a goniometry system and an array of sensors in particular, based on a specification of given technical and economic performance levels. It relates to the definition of products used for goniometry in the high frequency HF and very high frequency V/UHF ranges, as well as the IFF products.
It is for use in cellular radio goniometry systems in which the arrays are often patches of a not-inconsiderable size that are to be installed on a plate of finite dimension.
More generally, the field to which the present invention relates is that of antenna processing which processes the signals from a number of transmitting feeds on the basis of the observations from a number of sensors.
It is also applicable to the medical imaging location methods for locating tumors or epileptic sensors, in sounding applications for petroleum and mining surveys in the field of seismology, in the locating of feeds in an urban or submarine context.
A set of sensors is called an array of antennas 1 or array of sensors. In an electromagnetic context, the sensors are antennas and the radioelectric feeds are propagated according to a polarization. Figure 1 shows an example of an antenna processing system 2 in which an array of sensors Ci receives the signals from a number of feeds of incidences Ei, feeds of incidence Θmp. An antenna processing system notably comprises a processing unit comprising one or more processors, this type of system being known to those skilled in the art. The antenna processing techniques executed by a processor P are designed to separate or extract information associated with each of the feeds by exploiting the space diversity dependent on the geometry of the array of antennas and on the incidence Θmp of the feeds. The field is more particularly that of goniometry which consists in
estimating the directions Θmp of each of the feeds.
The individual sensors of the array receive the feeds with a phase and an amplitude that are dependent in particular on the incidences of the paths and the position of the sensors.
According to figure 2, the incidence of a path is either defined by direction parameters Θm= {θm and/or Δm} or by the wave vector k (Θm) = [um vm wm]T which is of unity standard. The angles θm and Δm are respectively the azimuth and the elevation. The aim of a goniometry technique is to estimate the components either of Θm or of k(Θm). A goniometry is called q-D when the number of parameters to be estimated is q. The other parameters are either known a priori or undetermined. According to figure 1, the vector pn= [xn yn zn]T is the position vector of the nth sensor relative to an origin point O. An array of N antennas is said to be d-D when the space engendered by the position vectors pn (1 ≤ n ≤ N) is of dimension d. Consequently, a linear array is of dimension 1 (1-D) and planar array is of dimension 2 (2-D).
One of the technical problems to be resolved is that of optimizing the goniometry of an antenna array. More particularly, one of the objectives is to determine the position pn of the N sensors for (1 ≤n≤N) from performance criteria contained in a specification such as: single-feed or dual-feed accuracy, resolution of two feeds, subject to constraints such as for example: a given number N of antennas, the more or less omnidirectional character of the antenna array within a certain area of space, a given maximum bulk, allowable and non-allowable positions for location of the sensors, a minimum distance between sensors to minimize the mutual coupling between antennas, a given maximum level of ambiguities. The definition of the optimal positions of the sensors in a given array will thus make it possible to obtain a performance-optimized array.
It should be noted that an array is said to be omnidirectional when the single-feed performance levels of the array are independent of the direction of arrival.
There is a great abundance of literature concerning theoretical performance calculations. These performance levels are generally variances or biases of the parameters to be estimated such as Θm or k (Θm). In the performance articles, to the knowledge of the applicant, just one, entitled "Cramer-Rao Bounds
for Antenna Array Design" by H. Gazzah and S. Marcos and published in the IEEE Trans. Signal Processing journal, 54(1):336-345, Jan. 2006, establishes a link between the stochastic bound and the positions pn of the N sensors in a single-feed situation for a 2-D goniometry with a 2-D array. The authors of this article use this link to establish an analytical directivity criterion dependent on the angle formed by the two branches of a V-shaped two-dimensional array, the array comprises N = 2M + 1 sensors and consists of two branches with a uniform linear array having the first sensor in common.
Despite its benefits, the technique developed by the prior art notably presents the following drawbacks:
• of having established the analytical link between the performance levels and the positions of the sensors only from the stochastic bound in single-feed mode,
• of not having established any criterion linking the resolution of two feeds and the geometry of the sensors,
• of processing just one very particular family of 2-D arrays which are the V-shaped arrays consisting of two uniform linear subarrays observing the non-ambiguity criterion.
Hereinafter in the description, the notation Δ with a single index such as Δm is used to designate angels, whereas the notation Δ with two indices such as Δ12
relates to a distance in the space of the wave vector or else the norm of Δk which is the difference between the wave vectors of two feeds.
The subject of the invention relates to a method for determining the optimum position of identical sensors of dimension d-D (1 ≤_d ≤_3) intended to perform goniometry measurements within a communication network minimizing the ambiguities, by taking into account parameters from a specification such as the physical aperture Dph of an array, characterized in that it comprises at least the following steps:
0 - initialization of a set ψ of arrays and of a value i corresponding to the index of an array,
1 - selecting, by a drawing process, an initial ith array d-D (pd,n ini for (1≤n≤N)), said array comprising N position sensors pn
2 - determining the physical aperture of the selected array by executing the
following steps:
Step No. 1 (DPH): initialization of a value for the physical aperture: Dph = 0 for all the
d-uplets pd ,ni in which (1 ≤ i ≤ d +1) bearing in mind that (1 ≤ ni ≤ N)
Step No. 2 (DPH): calculation of:
— the distance Di between the two sensors when d = 1
— the diameter Di of the circle passing through the three sensors when d= 2
— the diameter Di of the sphere passing through the four sensors
when d=3
Step No. 3 (DPH): calculation of the center p0 of the circle or of the sphere when
d > 1, for d = 1 calculation of the mean position of the two sensors, if (Pd ,n -P0)H (Pd ,n -P0) ≤ D/2 for (1 ≤ n ≤ N) then Dph = D
Step No. 4 (DPH): transition to the next d-uplets and return to step No. 2 (DPH)
3 - transformation of the initial array into an array of physical aperture Dph by
performing: pd ,n =(Dph/Dph 0)(pd ,n ini-P0) for (1≤n≤N)
4- calculating the values of the Di from the specific values of
(Equation Removed)

from the determined values of Di, calculating the parameters σθmin and σθmax by
using σd,1 =σθmin|cos(Δm)| and σd,d =σθ,m|cos(Δm|)
in which σθmin and σθmax are the minimum and maximum values of the azimuth
accuracy at the elevation Δm and an accuracy σΔ at Δm = 0.,
5 - testing to see if the value of the ratio σθmin /σθmax satisfies the technical criterion
given by the specification, and if so, then defining an array with the values of the d-uplets, otherwise, executing a loop on a set of / vectors of parameters of a family of arrays, return to step No. 1 to draw another array with i=i+ 1.
The test to check that the criteria are satisfied may be as follows: if σθmin /σθmax is not less than the given value pθ, execute a loop on a set of / vectors of
parameters of a family of arrays, return to step No. 1 to draw another array with i = i + 1.
Other features and advantages of the device according to the invention will become more apparent from reading the following description of an exemplary embodiment given as a nonlimiting illustration, with appended figures which represent:
• figure 1 represents an array of sensors and the signals transmitted by the transmitter and propagated toward an array of sensors,
• figure 2, the representation of an incidence of a feed,
• figure 3, an exemplary simplified diagram of the operation of the method
according to the invention,
• figure 4, a uniform two-dimensional circular array,
• figure 5, an exemplary array in two-dimensional concentric circles
• figure 6, an exemplary V-shaped array with 2 identical branches of sensors
parameterized by the angle 8 between the 2 branches
• figure 7, the representation of a physical aperture of a two-dimensional array.
The following description will be given in the context of the optimization of a uniform array consisting of N identical sensors with a number of feeds. Modeling the signal and formulating the problem
When there are M feeds, the signal at the output of the array of N sensors is expressed as follows:
(Equation Removed)
In which xn(t) is the signal received on the nth sensor, sm(t) is the signal from the mth feed, n(t) is the additive noise, Θm is the direction of arrival of the feed defined (figure 1) and a(Θm) is the observed directing vector. With model error, the vector a(Θm) is expressed
a(Θm) = a(Θm) + em (2).
In which a(Θm) is the theoretical directing vector such that a(Θm)H a(Θm) = N and em is the model error. According to figure 1, and knowing that the sensors are identical (or uniform), the nth component of a(Θ) is expressed
(Equation Removed)
In which pn =[xn yn zn]T is the position vector, X is the wavelength and k(Θ) is the wave vector such that
(Equation Removed)
In which Θ = {θ, Δ} depends on the azimuth θ and on the elevation Δ. The objective of the goniometry is to estimate the parameters ψm that may be equal either to the direction Θm or to the wave vector k(Θm). A goniometry algorithm gives the estimates ψm of the feeds of direction ψm from a criterion dependent on the parameters ψ=Θ or k(Θ). The performance levels can be given in terms
• of accuracy with
• the bias E[ψm]-ψm
• the variance or EQM which is expressed
MSψm=E[ΔψmΔψmT] with Δψm=ψm-ψm
• of resolution between two feeds: presence of two estimates ψ1 and ψ2 not
associated with a secondary lobe of the goniometry criterion. The resolution will
be defined according to the distance Δ12=||k(Θ2)-k(Θ1)|| between 2 feeds
• of ambiguity of order M: an array is mathematically ambiguous when there is at
least one incidence ψambig ≠ ψm whose directing vector a(ψambig) belongs to
the space engendered by the directing vectors a(ψm) for 1 ≤ m ≤ M.
The directionality and accuracy conditions of the specification in single-feed mode will give the form of the uncertainty ellipse. This ellipse depends on second order statistics of the position vectors pn, which then makes it possible to
transform an initial array according to the statistics of the specification by a whitening then coloring process.

The resolution of two feeds depends on the fourth order statistics of the position of the sensors. It is accepted that arrays that have identical performance levels in single-feed mode may have different resolution mode performance levels. Consequently, in a family of arrays with identical performance levels in single-feed mode, the method can rank the arrays according to their resolution mode performance levels.
Also, it is known that, in a family of arrays that have identical performance levels in single-feed and dual-feed modes, all the arrays do not have the same robustness to ambiguities. It will then be necessary to rank the arrays of this family according to a defined ambiguity criterion.
In practice, and in light of the goniometry objectives, the only parameter
which gives the direction of a feed is none other than the wave vector k(Θ). In
these conditions, the array optimization process will be based on the good
estimation statistics of this vector. One hypothesis is to deposit ψ = k(Θ). In the
case of an array d-D, interest is focused solely on the components of the wave vector being projected into the space engendered by the positions of the sensors. Consequently
• for a 2-D planar array: the uncertainty ellipse will be characterized by its long axis, small axis and its angle of orientation.
• For a 1-D linear array: the uncertainty ellipse will be characterized only by the standard deviation and the bias of the accuracy of estimation of the component of the wave vector being projected onto the line.
• For a 3-D linear array: the uncertainty ellipse is in the general case an ellipsoid which contains 3 axes and two angles. It is also possible in this case to characterize the performance levels for the components of the wave vector being projected in the plane closest to the 3D array. In practice, since the wave vector is normed, the 3rd component is deduced from the others.
Basic tools for optimizing an array
Omnidirectionality and accuracy tools deduced from single-feed performance
levels
The correlation matrix of Δψm = ψm-ψm has the following expression in single-feed mode:

(Equation Removed)
with
(Equation Removed)
Knowing that ψm is a vector of dimension dx1. The values of the coefficient α indicated in the table below depend on the type of performance levels considered.
(Table Removed)
Table1 - Table of the parameters to be associated with the single-feed accuracy criteria, knowing E[n(t)n(t)H] = σ2IN
Generally, in the presence of a multidimensional parameter ψm, the performance levels are given according to an ellipsoid of uncertainty which depends on the specific elements of the matrix MSψm which amounts to decomposing the variable
Δψm into d independent variables Δψm(i) such that:
(Equation Removed)
in which
(Equation Removed)
In which (σψm(i))2 and hψm(i) are respectively the specific values and specific vectors of MSψm . When d = 2, an ellipse is obtained which has, for its long axis σψm(imax) = max (σψm(i)), and for its small axis σψm(imin) =max(σψm(i)) and an
orientation relative to the first component of Δψm with the value φ = angle(hψm (i),[1 0]T).
Thereinafter in the section, the link between the matrix MSψm (the matrix whose
specific elements determine the ellipsoid of uncertainty) and the positions pn of the sensors is established in order to obtain a relationship with the parameters of the ellipse. The link is established initially for ψm =k(Θm). The matrix H(k(Θm)) is expressed
(Equation Removed)
with(Equation Removed)

The latter expression shows that the matrix MSk(Θm) . is independent of the value of
the wave vector k(Θm) by depending on the second order statistics (Dpp) of the positions pn of the sensors. An array is d-D when the rank of the matrix Dpp has the
value d. Without changing anything concerning the generality of the problem, an array is said to be one dimensional or 1-D when yn= zn= 0 and two dimensional or 2-D when zn = 0. For an array d-D, interest will be focused only on the projected
version kd(Θm) of the wave vector k(Θm) on the space engendered by the positions of the sensors. In the table below, a few examples are summarized for the
expressions of the wave vectors krf(Θm) and of the positions of the sensors pn
(Table Removed)
Table 2 - Value of kd (Θm) and pn for 1 ≤ d≤ 3
According to table 2, the expression of the directing vector becomes
(Equation Removed)
The correlation matrices of the pd ,n and pn are denoted in the same way, consequently
(Equation Removed)
with(Equation Removed)
The correlation matrix of the expression (5) therefore becomes:
(Equation Removed)
The decomposition into specific elements of MSk (Θm) depends directly on the specific elements ((Di)2 and hi) of Dpp. More specifically, the following is obtained
(Equation Removed)
Knowing that the decomposition into specific elements of Dpp is as follows
(Equation Removed)
in which (Di)2.) and hi are respectively the specific values and the specific vectors of Dpp. The objective is to calculate the values of σd,i and the hi which define the
axes of the uncertainty ellipses from the technical performance levels given in the specification. The following description establishes the link between σd,i and the hi and the performance levels of the specification. It will be said that there is an omnidirectional array when
D1 =••• = Dd=D0 and therefore DPP=(D0)2 Id (13)
In the equation (13), the concept of omnidirectionality is defined from the following two criteria concerning the estimation accuracy of the wave vector
• the components of the estimation error of the wave vector are independent
• the components of the estimation error of the wave vector have the same variance.
It can be shown that this omnidirectionality criterion is equivalent to stating that the
statistical mean of the angle formed between the vectors kd (Θ) and kd (Θ) is independent of the wave vector kd (Θ). More specifically, by denoting k = kd (Θ) and k = kd (Θ), the following criterion C(k)
(Equation Removed)
with
(Equation Removed)
is independent of k = kd (Θ). This is equivalent to stating that the specific values of MSk(Θm) are identical and therefore that D1=•••= Dd. Thereafter in the development, the specific elements of MSkd(Θm), make it possible to fully define the
ellipses (or ellipsoids) of uncertainty which fully define the directionality of an array of sensors.
Knowing that, for an omnidirectional array:
• UCA (Uniform Circular Array) of diameter Dph in which d = 2 that D0 =Dph/√8
• USA (Uniform Spherical Array) of diameter Dph in which d = 3 that
D0=Dph/M
It can be deduced therefrom that the equivalent aperture of the array of dimension d, Dde in which the index e is equivalent to UCA (or USA), is the physical aperture Dph
of the equivalent omnidirectional uniform array d-D. Consequently
(Equation Removed)
with(Equation Removed)

When an array is omnidirectional, this means that each of the components of the wave vector has the same estimation variance and that the estimates of the different components are independent. Consequently, the performance levels of an omnidirectional array satisfy
(Equation Removed)
in which(Equation Removed)

This last expression shows that the mean square error MSE σd of each of the components of the wave vector is inversely proportional to the equivalent aperture Dd ,e of the array. Consequently, to transform an array of sensors of dimension d-D
each having a position pd ,n with a correlation matrix Dpp into an array of sensors of
omnidirectional position pd ,n omni having an equivalent aperture Dd ,e setting the accuracy, it is sufficient to perform the following matrix transformation:
(Equation Removed)
However, according to the requests of the specification, it is not necessarily desirable to design omnidirectional arrays. Thus, it is possible to optimize an array that has a greater or lesser directivity in a direction Θm. The specification may give,
for example, a minimum and maximum standard deviation of the azimuth for a feed for which the arrival elevation is Δm. For this, it is necessary to return to the notion of ellipse of uncertainty in the space of the angles of arrival Θm which depends on the
specific elements of the matrix . The objective is then to calculate the matrix
D which makes it possible to obtain the performance levels matrix MSΘm in the space of the angles of arrival. Knowing that the equations (10)(11) have established the link between the specific elements of Dpp and those of the matrix defining the ellipse of uncertainty in the space of the wave vector, it is sufficient to
establish the relationship between the specific elements of
MSk d(Θm) and MSΘm
to obtain the link between MSΘm and Dpp. More particularly, the method will define the directivity parameters which will condition the specific elements of the matrices
MSk d(Θm) =αH(k d(Θm))-1 and MSΘm =αH(Θm)-1- The matrix H(Θm) is expressed as follows according to the matrix H(kd(Θm)) which is of full rank
H(Θm) = Jd(Θm)H H(k,(Θm))Jd (Θm) (18)
In which Jd(Θm) is the Jacobien. In order for the matrix H(Θm) to be of full rank,
the number of angular parameters of Θm should be less than or equal to d, consequently
• case of 3-D arrays (d = 3): Θm = {θm, Δm}
• case of 2-D arrays (cf = 2): Θm = {θm' Δm}
• case of 1 -D arrays (d = 1): Θm = Θm|Δ0 ={θm} knowing that Δm = Δ0
• case of 1-D arrays (d = 1): Θm= Θm|θ0 = {Δm} knowing that θm = θ0
The case d = 1 is in fact a degenerate case. In the following table, the possible values of the Jacobien are summarized according to the dimension d of the array.
(Table Removed)
Table 3 - Jacobien value according to the dimension of the array
Matrix MSp, can then be expressed
(Equation Removed)
with(Equation Removed)

In the omnidirectional case where σd ,i = σd for 1 ≤_i≤_d (see (16)), the above expression (19) becomes:
(Equation Removed)
with(Equation Removed)

because the matrices Kd(Θm) are unitary. Knowing that the matrices Ad are
diagonal, the estimations of the azimuths and of the elevations are independent (or decoupled) for omnidirectional arrays when d> 1. In these conditions, the axes of
the ellipse of uncertainty in the space of the azimuth θm and elevation Δm, angles Θm ={θm , Δm} are defined by the standard deviations(Equation Removed)
and
(Equation Removed)
respectively of the azimuths and of the elevations. The
following table summarizes the possible different values of σθ and σΔ for an omnidirectional array according to the dimension of the array.
(Table Removed)
Table 4 - Azimuth and elevation accuracy of an omnidirectional array
In the case where the arrays are not omnidirectional, the directionality will be defined from the incidences Θm in which the estimation of the azimuth and of the elevation are decoupled. This decoupling is satisfied when
(Equation Removed)
or(Equation Removed)
When the condition of equation (21) is obtained, then
(Equation Removed)
with(Equation Removed)
In the case of the 2-D array, the condition of equation (21) is satisfied when hi is
orthogonal to(Equation Removed)
or k2(θm,0). By positing (Equation Removed)
and
(Equation Removed)
with(Equation Removed)

and forΘm={θm+π/2,Δm}
(Equation Removed)
with(Equation Removed)

According to (23) (24), by setting the elevation at Δm, it is shown that the standard deviation in azimuth σθ is bounded as follows:
σθmin ≤σθ ≤σθmax with
(Equation Removed)
When σd, 1 ≤ σd ,2. It is then possible to envisage a process of transformation of any array into a directional array. The specification may give the minimum and maximum values σθmin and σθmax of the azimuth accuracy at the elevation Δm. It should be
noted that the ratio σθmax /σθmin is a directivity factor because, when σθmax =σθmin , the
array is omnidirectional. Consequently, the steps of transformation of an initial array of dimension 2-D consisting of N sensors of position p2 ,n into an array of chosen
directivity are then as follows. The two parameters of the specification of the process detailed hereinbelow are the minimum and maximum values σθmin and σθmax of the
azimuth accuracy at the elevation Δm. The specification gives the azimuth
θdirectional=θm+π/2 for which the azimuth accuracy σθmin is minimum. It should be

noted that the process described below includes the case of the omnidirectional arrays with σθmin = σθmax .
Step (T-2D) No. 1: Calculation of the correlation matrix Dppinit of the p2,n according to
equation (9).
Step (T-2D) No. 2: Calculation of σd,1 =σθmin |cos(Δm)| and σd,2 = σθmax |cos(Δm)| by
using (25).
Step (T-2D) No. 3: Calculation of the array sizes Di according to equation (11) by
performing
(Equation Removed)
Step (T-2D) No. 4: Calculation of the correlation matrix Dppdirectional by using (12)
(Equation Removed)
(The directions of the axes of the ellipse of uncertainty are given by the specification with (Equation Removed)
which gives the direction in which the azimuth accuracy is
minimum).
Step (T-2D) No. 5: Calculation of the position of the sensors of the directional array by performing
(Equation Removed)
The array giving the performance levels is determined from the position of the sensors found in step (27).
In the case of the 3-D array, the condition of equation (21) is satisfied when h, is orthogonal to (Equation Removed)
or (Equation Removed)
. By positing(Equation Removed)
(Equation Removed)
and h3 = k(θm,0), the following is obtained for Θm = {θm,Δ.m}
(Equation Removed)
with(Equation Removed)

and for Θm={θm+π/2,Δm}
(Equation Removed)
with(Equation Removed)

According to (28) (29), by setting the elevation at Δm, it can be shown that the standard deviation σθ satisfies
(Equation Removed)
with(Equation Removed)

When σd ,1 ≤ σd ,3. It will also be noted that σΔ1 = σΔ2 = σd, 2 at Δm = 0. A process of
transformation of any array into a directional array can then be envisaged and defined in a specification. The specification may give the minimum and maximum values σθmin and σθmax of the azimuth accuracy at the elevation Δm and an accuracy
σΔ at Δm =0. The specification gives the azimuth ddirectional=θm+ π/2 for which the azimuth accuracy σΘmin is minimum. Consequently, the steps of transforming an
initial array of dimension 3-D consisting of N sensors of position p3 ,n into an array of
chosen directivity are then as follows:
Step (T-3D) No. 1: Calculation of the correlation matrix Dppinit of the p3 ,n according to
equation (9).
Step (T-3D) No. 2: Calculation of σd ,1 =σθmin |cos(Δm)| and σd ,3=σθmax |cos(Δm)| by
using (30).
Step (T-3D) No. 3: Calculation of σd 2 = σΔ for the elevation accuracy at Δm - 0 to
be σΔ.
Step (T-3D) No. 4: Calculation of the array sizes Z) according to equation (11) by
performing Dt =((√a/8N)/σd ,i)λ
Step (T-3D) No. 5: Calculation of the correlation matrix Dppdirectional by using (12)
(Equation Removed)

(The directions of the axes of the ellipse of uncertainty are given by the specification with θdirectuional=θm+π/2 which gives the direction in which the azimuth accuracy is minimum).
Step (T-3D) No. 6: Calculation of the position of the sensors of the directional array by performing
(Equation Removed)

The optimum array is defined from the positions of the sensors found in this step.
Process of transforming a 3-D array into a directional array In the case of the 1-D array, the case is a degenerate case because one of the two
azimuth or elevation parameters has to be set. In this case, two array optimization situations are envisaged:
• case where the elevation is set at Δ0
• case where the azimuth is set at θ0
When the elevation is set at Δ0, a specification gives the azimuth accuracy σθ. Consequently, the steps of transforming an initial array of dimension 1-D consisting of N sensors of position p1,n into an array of chosen accuracy are then as follows:
Step (T-1D-θ) No. 1: Calculation of the correlation matrix Dppinit = Dppinit of the p1,n
according to equation (9).
Step (T-1D-θ) No. 2: Calculation of σd,1 = σθ|cos(Δ0)sin(θm)| by using the data
contained in table 4.
Step (T-1D-θ) No. 3: Calculation of the array sizes D1 according to equation (11) by
performing(Equation Removed)

Step (T-1D-θ) No. 4: Calculation of the correlation matrix Dppdirectional= Dppdirectional by using (12)
Dpp directional =(D1)2 (33) .
Step (T-1D-θ) No. 5: Calculation of the position of the sensors of the directional array by performing
(Equation Removed)
Determining an array comprising sensors defined by these positions (34).
Process of transforming a 1-D array when the elevation is set
When the azimuth is set at θ0, a specification gives the elevation accuracy σΔ. Consequently, the steps of transforming an initial array of dimension 1-D consisting of N sensors of position p1,n into an array of chosen accuracy are then as follows:
Step (T-1D-Δ) No. 1: Calculation of the correlation matrix Dpp = Dpp of the p1,n
according to equation (9).
Step (T-1 D-Δ) No. 2: Calculation of σd,1 = σΔ |sin(Δm)cos(θ0)| by using table 4.
Step (T-1 D-Δ) No. 3: Calculation of the array sizes D1 according to equation (11) by
performing(Equation Removed)

Step (T-1D-Δ) No. 4: Calculation of the correlation matrix D ppdirectional = Dppdirectional by using (12)
D ppdirectional =(D1)2 35). Step (T-1D-Δ) No. 5: Calculation of the position of the sensors of the directional array by
performing(Equation Removed)

Determining an array comprising sensors defined by these positions (36).
Process of transforming a 1-D array when the azimuth is set
Figure 3 represents a functional diagram of the method according to the invention. Thus, the first step consists in randomly drawing an array from a family of sensor arrays, 10, which results in having a matrix of correlations of the initial positions pn of the sensors that is denoted pnini, then the following step 11 consists in performing a linear transformation of the initial array taking into account the technical constraints given in the specification 12 (for example a maximum physical aperture), which generates a matrix consisting of the positions of the sensors pi(k). A subsequent step 13 may consist in checking whether other criteria imposed in the operation of the system, for example the technical criteria of bulk and/or of resolution and/or of accuracy (for example, an omnidirectionality ratio σθmax„ /σθmin
less than a value pθ or even a σθsmax bounded by a maximum value given in the
specification) are satisfied. If these criteria are not satisfied, 14, then the method will proceed to draw another array of the family and will apply to it all the steps of the method. If, on the other hand, the criteria are observed, 15, then the method will perform a calculation of the ambiguity level 16 and rank 17 the arrays according, for example, to the level of ambiguities. The method will perform iterations 19, to find the best arrays of the arrays that have identical characteristics in terms of accuracy, resolution, etc., by having different ambiguity characteristics. The draw depends on the family of array. For example, if we take the V-shaped arrays of figure 6 where
the initial angle is set at 5, the initial array consists of two identical branches in which the distribution of the sensors follows a normal law.
At this point, the method has determined a set of the "nb" best arrays belonging to a family of arrays, that is to say, the position of sensors in an array. The best arrays are the arrays that are the least ambiguous which satisfy the performance levels of the specification. The continuation of the method comprises, for example, the integration of any additional constraints which may be bulk criteria, resolution criteria, or even other criteria relating to the array of sensors. The process that has just been described can also be implemented with a maximum bulk constraint given in the specification. Resolution tool
In this section, a criterion for quantifying the resolution capacity of an array is constructed. This criterion is, for example, established from the resolution performance levels of the known MUSIC algorithm in the presence of two feeds. In the presence of M = 2 feeds, the MUSIC method applied to the observations of
equation (1) is designed to find the M = 2 minima of the criterion J(Θ) such that
(Equation Removed)
with(Equation Removed)

In which πb is the noise projector of the covariance matrix Rxx such that
(Equation Removed)

in which λ1 >---> λN are the specific values with λi the specific value associated with the specific vector ei. Two feeds are considered to be resolved, for example, when
in(Equation Removed)
which (Equation Removed)
knowing that(Equation Removed)

in which (Jmean, σmean) represents the mean and the standard deviation of the MUSIC criterion on the mean incidence and (J12, σ12) represents the mean and the standard deviation of J12 =(J(Θ1) + J(Θ2))/2. It is known that, in the absence of model error for K infinite, J12 = 0 and Jmean> J12 . The distance between two feeds of incidence Θ, and Θ2 is defined by
(Equation Removed)

in which ψorientation is the angle of orientation between the two feeds. For a 2-D array ψ iruebtatuib ={α, 0} in which a is an angle of orientation in the azimuthal plane and for a 3-D array ψ orientation ={α, ß} in which ß is an angle of orientation of elevation type in the direction α . The resolution limit Δ12lim is the minimum value of Δ12 in the direction ψ oientation for which the condition (39) is satisfied. The expression of the resolution limit is as follows
(Equation Removed)

in which the values of σerror summarized in the table below depend on the type of performance levels considered.
(Table Removed)
Table 5 - Value of σ error of the resolution criterion according to the type of
performance levels
The parameters Dpp(ψorientation) and p(ψorientation) are linked as follows to the second and fourth order statistics of the distribution of the sensors
(Equation Removed)
The positions dn (ψ) are the positions of the sensors projected onto an axis of orientation kd (ψorientation) . The expression (41) makes it possible to
• define an omnidirectionality in resolution for an array d-D: the coefficient
p(ψorientation) is independent of the direction ψ. A uniform circular array is
omnidirectional in resolution with p(ψorientation)2 =1.5.
• define a resolving power criterion in which 1/(p(ψorientation)2-1) should be
close to 0. Hereinafter in the section, more accurate criteria are established concerning the resolution omnidirectionality and the resolution capacity. The expressions
Dpp(ψorientation) and p(ψorientation) of tne expression (42) can be expressed:
(Equation Removed)

In which Dpp is the covariance matrix of the positions defined by the equation (9), is the Kronecker product and k2=kk. More specifically, the coefficient p(ψorientation)2 is the following quadratic form ratio:
(Equation Removed)

It is then possible to define omnidirectionality and resolving power criteria from the following specific values of the matrix Qpp of the equation (44).
(Equation Removed)

In which the values Qi are the specific values associated with the specific vectors vi. The resolution omnidirectionality criterion is as follows:
(Equation Removed)

Consequently, the array is omnidirectional when Romni is minimum, that is to say that Romni=1. Knowing that an array will have a strong resolving power when
|p(ψorientation)2 -1 | is great, the resolving power is then defined by
(Equation Removed)
Consequently, the resolution criteria are linked to the fourth order statistics of the distribution of the sensors Qpp. In an array optimization process, the objective will
be to minimize the criterion Romni and to maximize Rpower. The calculations of these
two criteria from the positions pd ,n of the sensors of an array are summarized in the
following steps:
Step No. 1 (Res): from the positions of the sensors pd ,n calculation of Dpp
according to (9).
Step No. 2 (Res): from the pd ,n calculation of Qpp according to (43).
Step No. 3 (Res): from Dpp and Q , calculation of Qpp according to (44).
Step No. 4 (Res): decomposition into specific elements of Qpp , or:
(Equation Removed)

Step No. 5 (Res): calculation of resolution omnidirectionality criterion Romni
according to (46) from Qi.
Step No. 6 (Res): calculation of the resolving power Rpower according to (46') from
Qi.
Process for calculating criteria for optimizing the resolving power and the
resolution omnidirectionality Array ambiguity
The objective is to establish a contrast between the level of the main lobes and that of the secondary lobes of the MUSIC criterion or of another equivalent criterion. The goniometry is ambiguous when the algorithm gives the incidences of one of the secondary lobes instead of one of the main lobes. The ambiguity criteria will be done in the space of the components of the wave vector. First order ambiguities
In the presence of a feed of incidence Θ1, an array is mathematically ambiguous to the first order when there is an incidence Θ for which the vectors a(kd(Θ!)) and a(kd(Θ)) are collinear. In these conditions, the projection of the
directing vector a(Θ) onto the vector a(Θ1) is maximum.
(Equation Removed)

in which 1 -Diagλ (a(k1),k2) is a distance between the vectors a(k1) and a(k2))
in which n(A) is the projector onto the space engendered by the columns of A such that
π(A) = A(AHA)-1AH (48)
and in which aλ (u, v) is the directing vector for the wavelength λ such that
(Equation Removed)
with(Equation Removed)

Since Diagλ (A,kd(Θ)) is between 0 and 1 and there is a mathematical ambiguity when Diagλ (A,kd(Θ))=1 (the distance between a(kd(Θ)) and A is zero), the criterion of robustness to first order ambiguities denoted η1 (λ) is dependent on the wavelength λ as follows:
(Equation Removed)

because kd (Θ)H kd (Θ) ≤ 1. From a practical viewpoint, the first order ambiguities are considered to be low when η1 (λ) < 0.1.
Conventional method for obtaining η1 (λ) according to the prior art
The conventional method according to the prior art for obtaining the robustness to ambiguities consists in randomly drawing the incidences Θk1 and in calculating the following criterion
η1(λ)k = η1 (1,Θk1) for 1≤k≤nb (51)
In which nb is the number of implementations for establishing the distribution function of η1 (λ) . The robustness to first order ambiguities η1(λ) for the wavelength λ must then satisfy
Pr (η1 (λ)k < η1(λ)) = pfa for 1 ≤k≤nb (52)
In which Pr(.) is a probability and in which pfa is typically 5%.
Optimized method for obtaining η1(λ) according to the invention
Since the array optimization consists in searching for the best array in a wide frequency band, the aim is to obtain, at one and the same time, all the values
of λ1(c/f) for fmin≤f≤fmax in which fmin and fmax are the minimum and
maximum frequencies of use of an array. The optimization which will be described later is based on the fact that
(Equation Removed)

The following property No. 1 is first deduced therefrom
(Equation Removed)

in which 0d = [0 • • • 0]T. Knowing that kd (Θ,)H kd (Θ,) ≤ 1 and that kdH kd ≤ 1, the following is deduced therefrom
ΔkHΔk ≤ 2 (55)
or else ΔkHΔk the distance squared between kd and kc/(Θ1) is bounded by (2) (56)
Consequently, η1(λ) is obtained by performing
(Equation Removed)

From the equation (53), the following property No. 2 is deduced therefrom
(Equation Removed)

It is deduced therefrom that
(Equation Removed)

in which λmax=c/fmax knowing that f≤fmax. Consequently, the criterion of robustness to first order ambiguities within a frequency range where f≤fmax is such that
(Equation Removed)

because η1(λmax) ≤ η1 (λ) when fmax ≥ f.
Ambiguities of order P > 1
An array is mathematically ambiguous to the Pth order when the
directing vector a(kd (Θ)) is engendered by a base of directing vectors a(kd (Θ1))
to a(kd (Θp)). Consequently, the following projection of the directing vector
a(kd(Θ)) onto the space engendered by the vectors a (kd(Θp)) for 1≤p≤P (P vectors indexed p) is zero:
(Equation Removed)

in which the projector π(A) is defined by the equation (48) and in which
Aλ(Φp) = [aλ(kd(Θ1)) - and Φp={kd(Θ1) - kd(Θp)} (62) Since Diagλ(A,kd(Θ)) is between 0 and 1 and there is a mathematical ambiguity when Diagλ(A,kd(Θ))=1, the robustness to Pth order ambiguities for the wavelength λ, is defined by the following parameter ηP (λ):
(Equation Removed)

From a practical viewpoint, it is considered that there is no Pth order ambiguity when ΗP (λ,Φp) < 0.1. From a statistical viewpoint, it is often noted that
(Equation Removed)

The optimized estimation of P=1 of ΗP (λ) for all the incidences is not applicable for
P > 1. Thus, the following conventional method is applied to the maximum frequency.
This method consists in randomly drawing P pairs of incidences ΦPk = {kd (Θ1k) • • • kd (ΘPk)} in order to then calculate

(Equation Removed)

in which nb is the number of implementations for estimating the distribution function of ηp(λ)k . The robustness to Pth order ambiguities ηP (λ) for the wavelength λ must then satisfy
(Equation Removed)

in which Pr(.) is a probability and in which pfa is typically 5%. Conclusion
In the array optimization process, the ambiguity criteria ηP(Λ) will be
estimated for each array in which the positions of the sensors are parameterized by pn. When P = 1, there is a single criterion (60) which can be used to obtain the
robustness to ambiguities η1amblg within a frequency range of maximum frequency
fmax and independently of the direction of the feeds.
Array optimization
The optimization process or processes is/are initialized by families of arrays for which the characteristic parameters are varied. These arrays may be
• Random cf-D arrays (linear, planar or 3-D in space). The distribution of the sensors follows a law which may be Gaussian, uniform or, etc.
• 2-D arrays consisting of two 1-D subarrays having an orientation deviation of 8. These are in particular V-shaped arrays as in figure 5. These arrays are not all omnidirectional. The more or less omnidirectional character depends on the relative position of the two linear branches of the V and on the angle 8 between the branches.
• 2-D arrays consisting of a number of concentric circles as in figure 4. The characteristic parameters are, for example, the radii, the number of sensors and the orientation of each circle. These arrays have the characteristic of being omnidirectional in accuracy. In this family of arrays, there are uniform circular arrays as in figure 3.
• 1-d linear arrays may be
— ULAs in which the sensors are evenly spaced (p1,n = d(n-1))
— Homothetic with p1,n = dxpn-1 in which p is the homothetic ratio
— random
A specification may also give bulk constraints such as: physical aperture Dph: diameter of the smallest circle (2-D array) or of the smallest sphere (3-D array) which encompasses all the sensors of the array. For a 1-D array, it is the distance between the two sensors furthest apart. The process for calculating the physical aperture of a d-D array is described below, bearing in mind that it is possible to define a circle from 3 points and that a sphere is constructed with 4 points.
Step No. 1 (DPH): Initialization: Dph =0
For all the d-uplets pd,ni in which (1 ≤ i ≤ d +1) bearing in mind that (1 ≤ ni ≤ N)
Step No. 2 (DPH): Calculation of
— the distance Di between the two sensors when d = 1
— the diameter Di of the circle passing through the three sensors when d = 2
— the diameter Di of the sphere passing through the four sensors when d=3
Step No. 3 (DPH): Calculation of the center p0 of the circle or of the sphere when
d > 1. For d = 1 calculation of the mean position of the two sensors:
(Equation Removed)
then Dph =D
Step No. 4 (DPH): Transition to the next d-uplets and return to step No. 2 Process for calculating a physical aperture of a d-D array
• Minimum distance between sensors which may be due either to the bulk of the individual sensor or to a desire to minimize cross-coupling.
• Area 1-d, 2-d or 3-d in which the sensors can be installed.
In the exemplary embodiment given below, the optimization of the array takes into account the bulk criterion to obtain arrays that have optimum performance levels for the given bulk.
The following process can be used to find the nb best omnidirectional (or directional) d-D arrays making it possible to produce a P-feed goniometry from:
— a physical aperture Dph.
— a maximum omnidirectionality ratio pθ = σθmin /σθmax
— an array family which may be random or parameterized.
Thus, the summarized steps implemented by the method according to the invention
are as follows:
Step No. B.O: Initialization of the set ψ = Ø , i = 0
Loop to a set of / vectors of parameters of a family of arrays
Step No. B.1: Select, by a drawing method, an initial /th array d-D
(Pd ,n ini for (1≤n≤N)) according to the set of parameters of the family of chosen
arrays, said array notably consisting of N sensors of position pd,n
Step No. B.2: Calculation of the physical aperture Dph0 according to the following
steps:
Step No. 1 (DPH): Initialization: Dph = 0
For all the d-uplets pd ,ni in which (1 ≤ i ≤ d +1) knowing that (1 ≤ ni ≤ N)
Step No. 2 (DPH): Calculation of
— the distance Di between the two sensors when d = 1
— the diameter Di of the circle passing through the three sensors when d = 2
— the diameter Di of the sphere passing through the four sensors
when d = 3
Step No. 3 (DPH): Calculation of the center p0 of the circle or of the sphere when
d > 1. For d = 1, calculation of the mean position of the two sensors

If (p^-p0f(p^-p„)^D/2 for(l≤«≤7V)

then Dph=D
Step No. 4 (DPH): Transition to the next d-uplets and return to step No. 2
Step No. B.3: Transformation of the initial array into an array of physical aperture
Dph by performing: pd ,n=(Dph/Dph0)(pd ,n ini -p0) for (1≤n≤N)
Step No. B.4:
Calculate the values of the sizes of array Di from the known value of
(Equation Removed)
(12), from the determined values of Di, calculate the parameters σθmin and σθmax according to the equations (25) or (30) at Δm depending

on the dimension of the array.
Step No. B.5 (Optional): If σθmin /σΘMAX < pθ then return to step No. 1 to draw another
array with i = i+ 1.
Step No. B.6 (Optional): If the array does not satisfy certain bulk constraints like
the minimum distance between 2 sensors, then return to step No. 1 to draw another
array with i = i + 1.
Step No. B.7: Calculation of the resolution parameters Romni (omnidirectionality
factor in resolution mode) and Rpower (resolving power).
Step No. B.8 (Optional): If Romni > Romnimax then return to step No. 1 to draw another
array with i = i+ 1.
Step No. B.9 (Optional): If Rpower>Rpowermax then return to step No. 1 to draw
another array with i = i+ 1.
Step No. B.10 (Optional): If σθmax > σθmax max then return to step No. 1 to draw
another array with i = i+ 1.
Step No. B.11: Calculation of the criterion η1ambig of the robustness to first order
ambiguities of equation (59) from the knowledge of fmax.
Step No. B.12: Storage of the array ψ = {ψ (η1ambig ,pd,n for (1 ≤ n ≤ N))}
Step No. B.13: Ranking of the elements of ψ according to the level of first order ambiguities such that, ultimately
(Equation Removed)
with
(Equation Removed)
Ranking from the array most robust to first order ambiguities to the least robust
array.
Step No. B.14: i = i+ 1 if i < / then return to step No. 1.
Step No. B.15: Storage of the nb best arrays, that is to say a set of arrays each
consisting of N sensors with respect to the first order ambiguities such that
(Equation Removed)
with
(Equation Removed)
Step No. B.16: Calculation of the Pth order ambiguities of the arrays of ψ
according to (66) to obtain ψopl = {(η1 ambig ,ηp (c/fmax )(k) ,pd,n(k)) for (1 ≤ k ≤ nb)}.
Step No. B.17: Ranking of the elements of ψopt according to the level of the Pth
order ambiguities such that, ultimately
ψopt = {(η1 ambig ,ηp(k) (c/fmax ) ,pd,n(k)) for (1 ≤ k ≤ nb) with ηp(1) ≥ ••• ≤ηp(nb)}:
Ranking from the array most robust to the Pth order ambiguities to the least robust array. Possibly, choice of the "best arrays". The array for which the positions of the sensors give the best performance is retained.
The method according to the invention makes it possible, compared to the prior art, to resolve the following problems:
• the generalization of the calculation of the performance levels in single-feed mode to a single expression,
• the calculation of single-feed performance levels for any 2-D, 1-D and 3-D arrays according to the geometry of the array,
• the establishing of a criterion linking the resolution of two feeds and the geometry of the sensors,
• the establishing of a directivity criterion for any d-D arrays,
• the optimization of any array of sensors with a fixed number of sensors from a specification giving the single-feed accuracy, the resolution under the constraints of maximum bulk, omnidirectionality or areas in which the antennas can be installed.

.

CLAIMS
1 - A method for determining the optimum position for identical sensors in an array of dimension d-D (1 ≤ d ≤3) intended to perform goniometry measurements within a communication network minimizing the ambiguities, by taking into account parameters from a specification such as the physical aperture Dph of an array, characterized in that it comprises at least the following steps:
0 - initialization of a set ψ of arrays and of a value i corresponding to the index of an
array,
1 - selecting, by drawing, an initial ith array d-D (pd,n ini for (1≤n≤N)) set of N
positions sensors pd,ni
2 - determining the physical aperture of the selected array by executing the
following steps:
Step No. 1 (DPH): initialization of a value for the physical aperture: Dph = 0 for all the d-uplets pd,ni in which (1 ≤ i ≤ d +1) bearing in mind that (1 ≤ ni ≤ N)
Step No. 2 (DPH): calculation of:
— the distance Di between the two sensors when d = 1
— the diameter Di of the circle passing through the three sensors when d=2
— the diameter D of the sphere passing through the four sensors when d=3
Step No. 3 (DPH): calculation of the center p0 of the circle or of the sphere when
d > 1, for d = 1 calculation of the mean position of the two sensors,
if (pd,n-P0)H (pd,n -P0)≤D/2 for (1≤n≤N), then Dph= D
Step No. 4 (DPH): transition to the next d-uplets and return to step No. 2 (DPH)
3 - transformation of the initial array into an array of physical aperture Dph by
performing: pd,n =(Dph/Dph0)(pd,n'ini -p0) for(1≤n≤N)
4- calculating the values of the sizes of Di from the specific values of
(Equation Removed)
, from the determined values of Dj, calculating the parameters σθmin and σθmax by using σd,1 =σθmin |cos(Δm)| and σd,d =σθ,m|cos(Δm|)

in which σθmin and σθmax are the minimum and maximum values of the azimuth
accuracy at the elevation Δm and an accuracy σΔ at Δm = 0.,
5 - testing whether σθmin/σθmax satisfies the technical criteria given by the
specification, and if so, confirming the position of the sensors pd,n and defining an array by taking into account the value of said positions, otherwise, executing a loop on a set of l vectors of parameters of a family of arrays, return to step No. 1 to draw another array with i = i+ 1.
2 - The method as claimed in claim 1, characterized in that a check is carried out to
see if σθmin /σθmax is less than a given value pθ and, if it is not less than said value,
executing a loop on a set of / vectors of parameters of a family of arrays, return to step No. 1 to draw another array with i = i+ 1.
3 - The method as claimed in claim 1, characterized in that it includes a step during
which criteria of omnidirectionality and resolving power are determined from the
position pd,n of the sensors of an array, whether it is the array drawn initially or else
the directional array defined by taking into account the covariance matrix Dpp of the
positions of the sensors and
(Equation Removed)

from Qpp and Dpp determining the value of Qpp
decomposing the matrix Qpp into specific elements, which leads to
(Equation Removed)

determining the resolution mode omnidirectionality criterion(Equation Removed)

determining the resolving power Rpower = min|Qi -1|
comparing these two values to values given in the specifications and, if the
determined values do not satisfy the criterion of the specifications, choosing a new array and repeating the various steps for calculating the position of the sensors and the omnidirectionality and resolving power criteria, for arrays of dimension 1, 2 or 3.
4 - The method as claimed in claim 1, characterized in that it includes a step during
which the arrays are ranked according to a degree of ambiguity by executing the
following steps:
initializing a set ψ containing arrays and their characteristics,
determining, for an array, the criterion η1ambig of the robustness to first order
ambiguities from the knowledge of the frequency fmax
(Equation Removed)

in which λmax = c/fmax, bearing in mind that f ≤ fmax. with Δk corresponding to a distance
(Equation Removed)

in which 0d=[0 ••• 0]T of, bearing in mind that kd(Θ1)H kd(Θ1)≤1 and that
kdH/kd≤1
ranking the elements contained in ψ according to the level of the first order ambiguities such that ultimately
ψ = {(η1ambing(k) ,Pd,n(k)) for (1 ≤ k ≤ K) with η1ambig(1) ≥ ••• ≥η1ambig(K)}
ranking from the array most robust to the first order ambiguities to the least robust
array by executing the following steps
storage of the nb best arrays with regard to the first order ambiguities such that
ψopt  ψ with ψopt = {(η1ambig(k), Pd,n(k)) for (1 ≤k ≤ nb)}
repeating said steps for all the arrays i = 1,... .1
calculation of the ambiguities of order P of the arrays of ψopt according to
the criterion of robustness to the ambiguities of order P ηp (λ)k for the wavelength λ satisfying
Pr(ηP (λ) < ηP (λ)) = pfa 'for 1≤k≤nb
in which Pr(.) is a probability and in which pfa is typically 5%.
to obtain ψopl = {(η1ambig(k) ,ηP (c/ fmax)(k) ,pd,n(k)) for (1 ≤ k ≤ nb)}.
ranking of the elements of ψopt according to the level of the ambiguities of order P
such that ultimately
ψopt ={(η1ambig(k) ,ηP (c/ fmax)(k) ,pd,n(k)) for (1 ≤ k ≤ nb)}.
ranking from the array most robust to the ambiguities of order P to the least robust
array.
5 - The method as claimed in one of claims 1 to 4, characterized in that the array of sensors is a V-shaped array in which the sensors are arranged on two branches forming an angle 8 between them.
6 - The method as claimed in one of claims 1 to 4, characterized in that the array of sensors is an array in which the sensors are distributed over a double circle.
7 - The method as claimed in one of claims 1 to 4, characterized in that the initial
array is an array of sensors d-D in which the distribution of the sensors follows a
Gaussian law.
8 - The method as claimed in one of claims 1 to 4, characterized in that the initial
array is an array of sensors d-D in which the distribution of the sensors follows a
uniform law.
9 - The use of the method as claimed in one of claims 1 to 8 for arrays and sensors
operating in the UHF or VHF band.