Patent Application: US-200913141968-A

Abstract:
a method for jointly determining the azimuth angle θ and the elevation angle δ of the wave vectors of p waves in a system comprising an array of sensors , a number of waves out of the p waves being propagated along coherent or substantially coherent paths between a source and said sensors , includes at least the following steps : selecting a subset of sensors from said sensors to form a linear subarray of sensors ; applying , to the signals from the chosen subarray , an algorithm according to a single dimension to decorrelate the sources of the p waves ; determining a first component w of said wave vectors by applying , to the signals observed on the sensors of the chosen subarray , a goniometry algorithm according to the single dimension w ; determining a second component u of said wave vectors by applying a goniometry algorithm according to the single dimension u to the signals from the entire array of sensors ; determining θ and δ from w and u .

Description:
before detailing an exemplary implementation of the method according to the invention , some reminders concerning the modeling of the output signal from an array of sensors are given . with m sources in which the m - th source contains p m multiple paths , the output signal from the array of sensors is written as follows : x ⁡ ( t ) = [ x 1 ⁡ ( t ) ⋮ x n ⁡ ( t ) ] = ∑ m = 1 m ⁢ ⁢ ∑ p = 1 p m ⁢ ⁢ ρ mp ⁢ a ⁡ ( θ _ mp ) ⁢ s m ⁡ ( t - τ mp ) + n ⁡ ( t ) . ( 1 ) in which x n ( t ) is the signal at the output of the n - th sensor , n the number of sensors , n ( t ) is the additive noise , a ( θ ) is the response of the array of sensors to a source of direction θ =( θ , δ ), θ is the azimuth , δ the elevation and ρ mp , θ mp , τ mp are respectively the attenuation , the direction and the delay of the p - th paths of the m - th source . the vector a ( θ ) which is also called directing vector depends on the positions ( x n , y n ) of the sensors 401 , 402 , 403 , 404 , 405 ( see fig4 ) and is written a ⁡ ( θ _ ) = z ⁡ [ a 1 ⁡ ( θ _ ) ⋮ a n ⁡ ( θ _ ) ] ⁢ ⁢ with ⁢ ⁢ { a n ⁡ ( θ _ ) = exp ⁡ ( j ⁢ 2 ⁢ π λ ⁢ ( x n ⁢ u + y n ⁢ v ) ) u = cos ⁡ ( θ ) ⁢ cos ⁡ ( δ ) v = sin ⁡ ( θ ) ⁢ cos ⁡ ( δ ) . ( 2 ) in which z is the coupling matrix , λ is the wavelength and ( u , v ) are the coordinates of the wave vector in the plane of the antenna . with coherent paths in which the delays of the paths satisfy τ m1 = . . . = τ mpm , the signal model of the equation ( 1 ) becomes x ⁡ ( t ) = ∑ m = 1 m ⁢ ⁢ a ⁡ ( θ _ m , ρ m , p m ) ⁢ s m ⁡ ( t ) + n ⁡ ( t ) ⁢ ⁢ a ⁡ ( θ _ m , ρ m , p m ) = ∑ p = 1 p m ⁢ ⁢ ρ mp ⁢ a ⁡ ( θ _ mp ) . ( 3 ) in which a ( θ m , ρ m , p m ) is the response of the array of sensors to the m - th source , θ m =[ θ m1 . . . θ mp m ] t and ρ m =[ ρ m1 . . . ρ mp m ] t . the directing vector of the source is no longer a ( θ m1 ) but a composite directing vector a ( θ m , ρ m , p m ) dependant on a greater number of parameters . more generally , with k groups of coherent paths , the signal is written : to enable the reader to better understand the method according to the invention , the processing of the coherent sources in azimuth and elevation in the state of the art is explained hereinbelow . a first coherent music algorithm [ 4 ] is first described . the music algorithm [ 1 ] is a high - resolution method based on the breakdown into specific elements of the covariance matrix r x = e [ x ( t ) x ( t ) h ] of the multiple - sensor signal x ( t ), in which e [.] is the mathematical expectation . the expression of the matrix r x is as follows according to ( 4 ): with k groups of coherent paths , the rank of the matrix r x is k . in these conditions , the k specific vectors e k ( 1 ≦ k ≦ k ) associated with the k highest specific values λ k of r x satisfy the n - k specific vectors e i ( k + 1 ≦ l ≦ n ) associated with the lowest specific values of r x are orthogonal to the vectors e k ( 1 ≦ k ≦ k ) of the expression ( 6 ) and define the noise space . since the vectors e i and e k are orthogonal , the directing vectors a ( θ i , ρ i , k i ) are orthogonal to the noise vectors e i . in these conditions , the k minima ( θ k , ρ k , k k ) of the following music criterion j music ⁡ ( θ _ , ρ ) = a ⁡ ( θ _ , ρ , k max ) h ⁢ π b ⁢ a ⁡ ( θ _ , ρ , k max ) a ⁡ ( θ _ , ρ , k max ) h ⁢ a ⁡ ( θ _ , ρ , k max ) ⁢ ⁢ with ⁢ ⁢ π b = ∑ i = k + 1 n ⁢ ⁢ e i ⁢ e i h . ( 7 ) make it possible to give the directions θ k of each path . however , the cost of calculating the criterion of the equation ( 7 ) is very high because it depends on the incidence of k max coherent paths and their relative amplitudes : ( θ , ρ ). the coherent music method described in [ 4 ] is designed to reduce the number of parameters for searching for the music criterion . for this , the vector a ( θ m , ρ m , p m ) of equation ( 3 ) is rewritten as follows : in these conditions , the criterion of equation ( 7 ) is reduced to the following expression : j coherent ⁢ ⁢ music ⁡ ( θ _ ) = det ⁡ ( d ⁡ ( θ _ ) h ⁢ π b ⁢ d ⁡ ( θ _ ) ) det ⁡ ( d ⁡ ( θ _ ) h ⁢ d ⁡ ( θ _ ) ) . ( 9 ) in which det ( m ) is the determinant of the matrix m . the number of dimensions of the criterion is then reduced to 2k max parameters of θ in which k max is the maximum number of coherent paths . consequently , the k minima of the criterion j music - coherent ( θ ) gives the directions θ k ={ θ k1 . . . θ kk max } of the paths of each group of coherent paths for 1 ≦ k ≦ k . thus , with k max = 2 coherent paths , the coherent music criterion still has four dimensions for an azimuth - elevation goniometry . more generally , the coherent music method entails calculating a criterion ( 9 ) having 2 k max dimensions . however , the method makes no assumption as to the geometry of the array because it entails no constraint on the expression of the directing vector a ( θ ). alternative methods that are also known for processing coherent sources are the spatial smoothing [ 5 ][ 6 ] and forward - backward [ 5 ] techniques . these methods make it possible to decorrelate the sources by performing a simple preprocessing on the covariance matrix of the received signals . it is then possible to apply a goniometry algorithm such as music to the new covariance matrix . these techniques derive from the field of spectral analysis whose objective is to model the frequency spectrum of a signal . the spatial smoothing techniques [ 5 ][ 6 ] are applicable to an array of sensors consisting of subarrays 501 , 502 that are unchanging by translation as illustrated in fig5 . with p paths ( coherent or not ), the expression ( 1 ) of the observation vector can be rewritten x ⁡ ( t ) = ∑ p = 1 p ⁢ ⁢ ρ p ⁢ a ⁡ ( θ _ p ) ⁢ s p ⁡ ( t ) + n ⁡ ( t ) = as ⁡ ( t ) + n ⁡ ( t ) ( 10 ) the expression of the signal received on the i - th subarray is then written : x i ⁡ ( t ) = p i ⁢ x ⁡ ( t ) = ∑ p = 1 p ⁢ ⁢ ρ p ⁢ a i ⁡ ( θ _ p ) ⁢ s p ⁡ ( t ) + n ⁡ ( t ) = a i ⁢ ⁢ s ⁡ ( t ) + n ⁡ ( t ) ( 11 ) in which a i =[ a i ( θ 1 ) . . . a i ( θ p )], p i being a matrix consisting of 0 and 1 making it possible to select the signal of the i - th subarray for which the directing vector a i ( θ ) satisfies the following relationship : a i ( θ )= p i a ( θ )= α i ( θ ) a 1 ( θ ) ( 12 ) remembering that the incidence θ =( θ , δ ) depends on the two parameters θ and δ . according to ( 11 )( 12 ), the mixing matrix a i of the i - th subarray satisfies a i = p i a = a 1 φ i with φ i = diag { α i ( θ 1 ) . . . α i ( θ p )} ( 13 ) according to ( 11 )( 13 ) the covariance matrix r x i = e [ x ( t ) i x ( t ) ih ] has the following expression : r x i = a 1 φ i r s φ i * a 1h + σ 2 i n in which r s = e [ s ( t ) s ( t ) h ] ( 14 ) consequently , an alternative to the spatial smoothing techniques consists in applying a music - type algorithm to the following covariance matrix : r x sm ⁢ ∑ i = 1 i ⁢ ⁢ r x i = ∑ i = 1 i ⁢ ⁢ p i ⁢ r x ⁡ ( p i ) h ( 15 ) in which r x = e [ x ( t ) x ( t ) h ]. the aim of this procedure is to obtain a matrix r x sm that has a rank higher than the r x i without destroying the structure of the signal space generated by a 1 . in practice , this technique makes it possible to decorrelate a maximum of i coherent paths because r x sm = a 1 ⁢ r s sm ⁢ a 1 ⁢ ⁢ h + σ 2 ⁢ i n ′ ⁢ ⁢ in ⁢ ⁢ which ⁢ ⁢ r s sm = ∑ i = 1 i ⁢ φ i ⁢ r s ⁢ φ i * ( 16 ) the forward - backward [ 5 ] smoothing technique requires an array of sensors that has a center of symmetry at o as indicated in fig8 . in these conditions , the directing vector has the following structure a ⁡ ( θ _ ) = β ⁡ ( θ _ ) ⁡ [ b ⁡ ( θ _ ) b ⁡ ( θ _ ) * ] ( 17 ) in which , according to fig8 , b ( θ ) is the directing vector of the subarray of coordinates ( x n − x 0 , y n − y 0 ) and b ( θ )* is the directing vector of the subarray of coordinates (− x n − x 0 ,− y n − y 0 ), bearing in mind that ( x 0 , y 0 ) are the coordinates of the center of symmetry o . consequently , the directing vector of the expression ( 17 ) satisfies the following relationship : in which π is a permutation matrix consisting of 0 and 1 . the forward - backward smoothing technique consists in applying a goniometry algorithm such as music to the following covariance matrix r x fb = ar s fb a h + σ 2 i n in which r s fb = r s + φ fb r s φ fb * ( 20 ) the technique makes it possible to decorrelate two coherent paths because rank ⁢ { r s } ≤ rank ⁢ { r s sm } ≤ min ⁡ ( 2 ⁢ ⁢ rank ⁢ { r s } , ∑ m = 1 m ⁢ p m ) φ fb = diag { β ( θ 1 ) . . . β ( θ p )} ( 21 ) the spatial smoothing and forward - backward techniques can be combined to increase the capacity for decorrelation into number of paths . these smoothing techniques make it possible to process the coherent sources with a computation power that is very similar to the application of a single goniometry algorithm such as music . when the array of sensors is disturbed by mutual coupling in which the directing vector is written and in which the directing vector ( θ ) satisfies one of the properties of the equations ( 12 )( 18 ), the spatial smoothing techniques are applicable [ 7 ]. the mixing matrix a of the equation ( 10 ) is then written a = z with =[ ( θ 1 ) . . . ( θ p )] ( 23 ) consequently , the covariance matrix r x = e [ x ( t ) x ( t ) h ] is written as follows : r x = z ( r s h ) z h + σ 2 i n ( 24 ) bearing in mind that i = p i = 1 φ i ( where that π *= φ fb ), the following steps make it possible to apply a spatial smoothing or forward - backward technique in the presence of mutual coupling : step no . l . 1 : break down the covariance matrix r x = e [ x ( t ) x ( t ) h ] into specific elements such that : r x = e s λ s e s h + e b λ b e b h ( 25 ) in which e s and e b are the matrices of the specific vectors respectively associated with the signal space and the noise space according to music 0 and in which λ s and λ b are diagonal matrices respectively consisting of the specific values of the signal space and of the specific values of the noise space . step no . l . 2 : extract the non - noise - affected covariance matrix z ( r s h ) z h by performing : r y = r x - trace ⁡ ( λ b ) n - k ⁢ i n = z ⁡ ( a ⋒ ⁢ r s ⁢ a ⋒ h ) ⁢ z h in which k is the dimension of the signal space such that k ≦ p . step no . l . 3 ( spatial smoothing ): apply the music algorithm to the following covariance matrix r x sm : r x sm = ∑ i = 1 i ⁢ ⁢ p i ⁡ ( z - 1 ⁢ r y ⁡ ( z - 1 ) h ) ⁢ ( p i ) h step no . l . 3 ( forward - backward ): apply the music algorithm to the following covariance matrix r x fb : r x fb =( z − 1 r y ( z − 1 ) h )+ π ( z − 1 r y ( z − 1 ) h )* π t if the directing vector array ( θ ) permits , the two smoothing techniques of steps no . l . 3 can be combined . the spatial smoothing techniques are applicable with mutual coupling . however , this imposes very strong constraints on the geometry of the individual array which have the drawback of requiring a very large number of sensors . in the following example , we will evaluate the minimum number of sensors to process the case of two sources coherent in azimuth - elevation . for this , it is necessary for : constraint c1 : the number of sensors of each subarray to be at least equal to n i = 4 . in practice , because of ambiguities , an array of n sensors makes it possible at most to estimate the direction of arrival of n / 2 sources . constraint c2 : the number of subarrays to be at least equal to 2 . constraint c3 : the subarrays to be planar ( not linear ) in order to be able to perform an azimuth - elevation goniometry . fig6 shows that an array consisting of two subarrays 601 , 602 of four sensors contains at least seven sensors . this array also has the drawback of being weakly open ( or has little spatial bulk ) because the subarrays with four sensors must be unambiguous . since the subarrays consist of four sensors , this ambiguity constraint requires a spacing between sensors less than λ / 2 . in practice , the more open a array is , the more accurate the estimation of the angles of arrival is with a better robustness to calibration errors . for the case where the desire is to perform an azimuth goniometry only , the constraint c3 no longer applies and the array making it possible to perform a goniometry on two coherent sources consisting of two subarrays of four sensors is an evenly - spaced linear array with five sensors . each subarray is then an evenly - spaced linear array with four sensors . fig7 shows that the linear subarray 701 which allows for the azimuth goniometry of two coherent paths has the following differences compared to the array 702 which makes it possible to do so in azimuth and elevation : on the one hand , it consists of fewer sensors : five instead of seven , and on the other hand , it has a greater bulk : 4 d instead of 3 d , bearing in mind that d is a distance less than λ / 2 , λ being the wavelength of the transmitted signals . for the forward - backward technique requiring an array with a center of symmetry as illustrated in fig8 , it is possible to note that , for the spatial smoothing : the decorrelation of two coherent paths for an azimuth - elevation goniometry requires an array of sensors having more sensors and less aperture than the array making it possible to perform an azimuth goniometry only . for an azimuth goniometry , a linear array , not necessarily evenly - spaced , is sufficient . fig9 shows that the forward - backward technique makes it possible , compared to the spatial smoothing technique , to perform an unambiguous goniometry on two coherent paths with an array 901 having a greater aperture ( 10 d instead of 4 d for an array 902 used for spatial smoothing ). the forward - backward technique has the advantage of not imposing any geometry constraint on half the array . the other half of the array is symmetrical to the 1st half . the method according to the invention described combines the coherent music method with a forward - backward technique and / or a spatial smoothing technique . given the advantages and drawbacks of the smoothing techniques and of the coherent music algorithm described above , the method envisages using an array of sensors containing a linear subarray on which a spatial smoothing and / or forward - backward technique can be envisaged . fig1 shows such an array 1001 with a linear subarray 1002 having an orientation α relative to the x axis . more specifically , the method according to the invention can use the array 1101 of fig1 in which the angle α = 90 ° and the linear subarray 1102 consists of 3 evenly - spaced sensors on which a forward - backward technique can be applied . the coordinates ( x n l , y n l ) of the n - th sensor of the linear subarray then have the following expression : { x n l = ρ n ⁢ cos ⁡ ( α ) y n l = ρ n ⁢ sin ⁡ ( α ) ⁢ ⁢ for ⁢ ⁢ 1 ≤ n ≤ n l ( 26 ) in which n l is the number of sensors of the linear subarray . in the absence of coupling and according to ( 2 ), the directing vector a l ( θ ) associated with the linear subarray is written the vector a l ( θ ) then depends on a single parameter w = cos ( θ − α ) cos ( δ ) as follows : a l ⁡ ( θ _ ) = a l ⁡ ( w ) = [ z ρ 1 ⋮ z ρ n l ] ⁢ ⁢ with ⁢ ⁢ z = exp ⁡ ( j ⁢ 2 ⁢ π λ ⁢ w ) ( 28 ) x l ( t ) is used to denote the signal at the output of the linear subarray and p roj the matrix consisting of 0 and 1 that can be used to extract the signals from the linear subarray such that in which x ( t ) is the signal observed on all the sensors of the array . the relationship between the variable w = cos ( θ − α ) cos ( δ ) and the coordinates of the wave vector ( u , v ) of the equation ( 2 ) is as follows : knowing w , the incidence θ becomes a 1d function dependent on the parameter u such that : θ _ ⁡ ( u ) = ( θ , δ ) = f ⁡ ( u , v ) = f ⁡ ( u , w - u ⁢ ⁢ cos ⁡ ( α ) sin ⁡ ( α ) ) ( 31 ) in which the function f ( u , v ) is given by the expression ( 2 ). when α = 0 , the vector of parameter θ cannot depend on the variable u : in this case , the incidence θ depends on the variable v with the function θ ( v )= f (( w − ν sin ( α ))/ cos ( α ), ν ). in the interests of simplicity in the description of the method and without compromising generality , it will be assumed that it is still possible to write θ as a function of u . consequently , with p paths of which at least one group of k max are coherent , the example described of the method according to the invention contains at least the following steps : step a : application of a spatial smoothing and / or forward - backward technique to the observation vector x l ( t ) of the linear array . after a 1d goniometry according to the variable w , the incidence parameters w p = cos ( θ p − α ) cos ( δ p ) are obtained for ( 1 ≦ p ≦ p ). the 1d music criterion has the following expression : j music ⁡ ( w ) = a l ⁡ ( w ) h ⁢ π b l ⁢ a l ⁡ ( w ) a l ⁡ ( w ) h ⁢ a l ⁡ ( w ) ( 33 ) in which π b l is the noise projector extracted from the smoothed covariance matrix . step b : with k max ≦ p coherent paths , application of the coherent music method described above with the variable θ ={ θ 1 . . . θ k max } which is the following function of the variable u ={ u 1 . . . u k max } θ ={ f 1 ( u 1 ) . . . f k max ( u k max )} ( 34 ) in which the coherent music criterion j coherent music is a function of the variable u having k max dimensions . step c : from k ( k being the rank of the covariance matrix r x ) solutions u k minimizing the function j coherent music ( u ) it is possible to extract the p pairs of incidences ( w p , u p ) for ( 1 ≦ p ≦ p ) and deduce the incidences ( θ p , δ p ) therefrom by performing the preceding steps show that the calculation of a criterion with 2k max dimensions for the coherent music algorithm alone in 2d has been replaced by the calculation of a music criterion with one dimension according to the parameter w and the cost of calculation of the 1d coherent music criterion with the variable u having k max dimensions . the gain in computation power is then equal to gain = nb ( 2 ⁢ ⁢ k max - 1 ) 1 + nb ( k max - 1 ) ≈ nb k max ( 36 ) in which nb is the number of points of the meshes of the criteria ( music or coherent music ) according to the variables u and v of the components of the wave vector . in the general case nb is large while being proportional to the size of the array ( nb & gt ; 50 ). it will be assumed that u k is a solution parameter vector for coherent music when j coherent music ( u k )& lt ; η ( k max ) ( 37 ) in which η ( k max ) is a threshold between 0 and 1 because the criterion j coherent music ( u ) is normalized . when the number k ′ of solutions u k is less than the rank k of the covariance matrix r x , it can be deduced therefrom that the number of coherent paths is greater than k max . for the case where k ′& lt ; k max the coherent music algorithm will be applied with k max = k max + 1 . consequently , the method makes it possible to jointly estimate the incidences of the paths with the number of coherent paths . similarly , it will be assumed that w p is a solution parameter of the 1d goniometry of step a when in which η is a threshold between 0 and 1 because the criterion j music ( w ) is normalized . the method envisages treating the case in which at least two coherent paths satisfy w i = w j with u i ≠ u j . this problem can be detected when : the rank of the smoothed covariance matrix remains equal to that of the covariance matrix rx ; the music method does not work on the non - smoothed covariance matrix rx . by assuming that there are k max coherent paths and that 1d music w gives p ′& lt ; k max coherent path solutions , the method consists in complementing the incomplete list of p ′ elements { w p } with k max , − p ′ estimation of the initial list of { w p }. more specifically , with k max = 2 coherent paths and p ′= 1 parameter w 1 detected , it is essential to apply the coherent music method of step b with w 1 = w 1 and w 2 = w 1 or the set of parameters { w 1 , w 1 }. in the case where k max = 3 coherent paths and p ′= 2 , there are two configurations to which the 1d coherent music step b must be applied : { w 1 , w 2 , w 2 } and { w 1 , w 1 , w 2 }. consequently , when p ′& lt ; k max the step b of the method can be applied several times . there are thus l sets of following incidences w p to which the step b of the method must be applied : the number l and the sets ω i can be determined by a conventional arithmetical process . the following steps of the method make it possible to estimate the direction of arrival of p paths in azimuth - elevation bearing in mind that there is at least one group of k max coherent paths and that the array is disturbed by mutual coupling of known matrix z . step no . 1 : breakdown the covariance matrix r x = e [ x ( t ) x ( t ) h ] into specific elements such that in which e is the matrix with the specific vectors and λ is a diagonal matrix consisting of the specific values . step no . 2 : from the specific values of the matrix λ , determination of the number k of dominant specific values giving the rank of r x . in which e s and e b are the matrices of the specific vectors respectively associated with the signal space bearing in mind that dim ( e s )= n × k and in which λ s and λ b are diagonal matrices respectively consisting of the specific values of the signal space and the specific values of the noise space . step no 4 : extraction of the non - noise - affected and coupling - free covariance matrix by performing in which k is the dimension of the signal space such that k ≦ p . step no . 5 : calculation of the noise projector of the matrix r y in the following way : π b = i n − z − 1 ( e s ( e s h z − 1h z − 1 e s ) − 1 e s h ) z − 1h step no . 6 : application of the 2d music algorithm with the criterion j music ( θ )=( ( θ ) h π b h ( θ ))/( ( θ ) h ( θ )) with the vector ( θ ) of the equation ( 22 ). estimation of p 0 ≦ k incidences θ p ( 1 ≦ p ≦ p 0 ) satisfying j music ( θ p )& lt ; η ( 1 ). formation of the set θ ={ θ 1 . . . θ p0 } of the non - coherent paths . if p 0 & lt ; k , go to step no . 7 . step no . 7 : calculation of the covariance matrix of the linear array by performing r x l = p roj r x p roj h bearing in mind that x l ( t )= p roj x ( t ). step no . 8 : application of one ( or both ) of the smoothing techniques to the matrix r x l of the linear array by performing either r ~ x = ∑ i = 1 i ⁢ p i ⁢ r x l ⁡ ( p i ) h for smoothing or { tilde over ( r )} x = r x l + π ( r x l )* π t for the forward - backward . step no . 9 : from a breakdown into specific elements of the matrix { tilde over ( r )} x , estimation of the rank p of the signal space and of the noise projector π b l = e b e b h ( step of the music algorithm reviewed in steps 1 to 3 of this method for the matrix r x ). step no . 10 : application of the 1d music algorithm with the criterion j music ( w )=( a l ( w ) h π b l a l ( w ))/( a l ( w ) h a l ( w )) with the vector a l ( w ) of the equation ( 28 ). estimation of p incidences w p ( 1 ≦ p ≦ p ) satisfying j music ( w p )& lt ; η . step no . 11 : formation of the set ψ of the incidences w p associated with a coherent path such that ψ ={ w p ≠ cos ( θ i − α ) cos ( δ i ) in which θ i ={ θ i , δ i }∈ θ } step no . 12 : if p & gt ; k , then k max = cardinal ( ψ ) and l = 1 with ω 1 = ψ . go to step no . 14 . step no . 13 : if p ≦ k , then k max = k + 1 and formation of the l sets of parameters ω i of the equation ( 39 ) with p ′= cardinal ( ψ ). step no . 15 : application of the 1d coherent music steps b and c described on page 18 with θ = f ( u )={ f 1 ( u 1 ) . . . f k max ( u k max )}, bearing in mind that f p ( u )= f ( u ,( w p − u cos ( α ))/ sin ( α )) in which w p ∈ ω i . obtaining of k i incidences θ k for ( 1 ≦ k ≦ k i ). step no . 16 : for k ranging from 1 to k i if θ k ∉ θ then θ = θ ∪{ θ k } step no . 17 : i = i + 1 . if i ≦ l then return to step no . 14 . one advantage of the method according to the invention is that the minimum number of sensors for estimating the direction of arrival of k coherent paths in 2d is lower than with the methods of the prior art , which require a number of sensors greater than 2 ( k + 1 ), the method according to the invention requiring only a number of sensors greater than k . another advantage of the method according to the invention is that it makes it possible to estimate directions of arrival of the paths in 2d with larger arrays , which enhances the accuracy of the estimation . r o . schmidt , multiple emitter location and signal parameter estimation , in proc of the radc spectrum estimation workshop , griffiths air force base , new york , 1979 , pp . 243 - 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