Patent Publication Number: US-7912680-B2

Title: Direction-of-arrival estimation apparatus

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is based upon and claims the benefit of priority of the prior Japanese Patent Application No. 2008-088091, filed on Mar. 28, 2008, the entire contents of which are incorporated herein by reference. 
     FIELD 
     The present invention generally relates to a direction-of-arrival estimation apparatus, a direction-of-arrival estimation method, and a direction-of-arrival estimation program, and more particularly to a direction-of-arrival estimation apparatus, a direction-of-arrival estimation method, and a direction-of-arrival estimation program for estimating a direction in which an arriving signal has traveled from a target with use of a plurality of sensors. 
     BACKGROUND 
     A digital beam forming method (DBF), a subspace method (SSM), a maximum likelihood method (ML), and the like are used to estimate a direction of arrival (DOA) of an arriving signal from a target (signal transmitting source or signal reflection object) with use of a plurality of sensors. 
     In order to perform a high-speed DOA estimation with high accuracy, International Patent Publication No. WO2006/067869 (Patent Document 1) discloses a technique of estimating a direction of arrival with use of a spatial average covariance matrix in which correlation vectors of baseband signals are combined. 
     The computation speed and robustness of the DOA estimation disclosed in Patent Document 1 can be improved in some aspects. It is, thus, an object of the present invention to provide a direction-of-arrival estimation apparatus, a direction-of-arrival estimation method, and a direction-of-arrival estimation program capable of high-speed estimation for a direction of arrival with high accuracy. 
     SUMMARY 
     According to an aspect of the present invention, there is provided a direction-of-arrival estimation apparatus capable of high-speed direction-of-arrival estimation with high accuracy. The direction-of-arrival estimation apparatus has a signal vector generation unit operable to generate a signal vector v composed of N baseband signals v 1  to v N  from arriving signals received from a target by N sensors. The direction-of-arrival estimation apparatus includes a Hankel matrix generation unit operable to preferentially set an order of a column of a matrix at a natural number M where 1≦M and M≦(N−1)/2 and to generate at least one of (N−M)×M matrices R f1 , R f2 , R b1 , and R b2  from elements v 1  to v N-1  or v 2  to v N  of the signal vector. The matrix R f1  is an (N−M)×M Hankel matrix including elements v 1  to v N-1  of the signal vector. The matrix R f2  is an (N−M)×M Hankel matrix including elements v 2  to v N  of the signal vector. The matrix R b1  is an (N−M)×M Hankel matrix including complex conjugate elements v 1 * to v N-1 * of the signal vector. The matrix R b1  is defined by R b1 =J N-M R f1 *J M  where J N-M  is an (N−M)×(N−M) anti-diagonal unit matrix and J M  is an M×M anti-diagonal unit matrix. The matrix R b2  is an (N−M)×M Hankel matrix including complex conjugate elements v 2 * to v N * of the signal vector. The matrix R b2  is defined by R b2 =J N-M R f2 *J M  where J N-M  is an (N−M)×(N−M) anti-diagonal unit matrix and J M  is an M×M anti-diagonal unit matrix. The direction-of-arrival estimation apparatus also includes an estimation unit operable to generate a matrix R using at least one of the matrices R f1 , R f2 , R b1 , and R b2 . The estimation unit is operable to divide the matrix R into two submatrices R 1  and R 2  by R=[R 1 |R 2 ] T  and estimate a direction of arrival of the arriving signal based on the submatrices R 1  and R 2 . 
     According to another aspect of the present invention, there is provided a direction-of-arrival estimation method capable of high-speed direction-of-arrival estimation with high accuracy. In this direction-of-arrival estimation method, N baseband signals v 1  to v N  are generated from arriving signals received from a target by N sensors. An order of a column of a matrix is preferentially set at a natural number M where 1≦M and M≦(N−1)/2. At least one of (N−M)×M matrices R f1 , R f2 , R b1 , and R b2  is generated from elements v 1  to v N-1  or v 2  to v N  of the signal vector. A matrix R is generated using of at least one of the matrices R f1 , R f2 , R b1 , and R b2 . The matrix R is divided into two submatrices R 1  and R 2  by R=[R 1 |R 2 ] T . A direction of arrival of the arriving signal is estimated based on the submatrices R 1  and R 2 . 
     According to another aspect of the present invention, there is provided a direction-of-arrival estimation program for executing the above procedure with a computer. 
     According to a direction-of-arrival estimation apparatus, a direction-of-arrival estimation method, and a direction-of-arrival estimation program of the present invention, a direction of arrival can be estimated at a high speed with high accuracy. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram showing receiving sensors; 
         FIG. 2  is a block diagram showing a direction-of-arrival estimation apparatus according to a first embodiment of the present invention; 
         FIG. 3  is a flow chart showing processes performed in the direction-of-arrival estimation apparatus according to the first embodiment of the present invention; 
         FIG. 4  is a block diagram showing a direction-of-arrival estimation apparatus according to a second embodiment of the present invention; 
         FIGS. 5A to 5D  are diagrams showing angular spectra computed by indicated methods; 
         FIG. 6  is a diagram showing an angular spectrum at a constant distance with respect to angles; 
         FIG. 7  is a diagram showing an angular spectrum in a comparative example; 
         FIG. 8  is a diagram showing an angular spectrum in the second embodiment of the present invention; 
         FIG. 9  is a block diagram showing a direction-of-arrival estimation apparatus according to a third embodiment of the present invention; 
         FIG. 10  is a flow chart showing processes performed in the direction-of-arrival estimation apparatus according to the third embodiment of the present invention; 
         FIG. 11  is a block diagram showing a direction-of-arrival estimation apparatus according to a fourth embodiment of the present invention; and 
         FIG. 12  is a flow chart showing processes performed in the direction-of-arrival estimation apparatus according to the fourth embodiment of the present invention. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     First, there will be described some factors to allow DOA estimation using the technique disclosed in Patent Document 1 to be improved in computation speed and robustness. In the following description, a superscript “H” following a matrix or vector represents complex conjugate transpose, a superscript “T” following a matrix or vector represents transpose, and a superscript “*” complex conjugate. A superscript “−1” following a matrix represents an inverse matrix. Furthermore, a matrix J P  is a P×P anti-diagonal unit matrix in which the anti-diagonal elements are equal to one while all elements not on the anti-diagonal are equal to zero. A matrix I P  is a P×P unit matrix. A matrix 0 P×Q  is a P×Q zero matrix, and a matrix 0 P  is a P×P zero square matrix. 
       FIG. 1  is a diagram showing receiving sensors (e.g., antennas  10 ) according to the present invention. In  FIG. 1 , N antennas A 1  to A N  are arranged at equal intervals d on a line in a direction of an X-axis. These antennas  10  form a uniform linear array antenna (ULA). It is assumed that each of M independent targets  12  (signal transmission sources or signal reflection objects) is located at a distance r m  from the antenna  10  and an angle θ m . The angle θ m  is set to be zero in the positive direction of a Y-axis and increases clockwise. For the sake of brevity, only one target is illustrated in  FIG. 1  on behalf of the M independent targets  12 . 
     For convenience of explanation, a radar using a high-frequency signal having a wavelength λ as a carrier signal will be described as a specific apparatus. In this example, the radar transmits a detection signal to targets from a transmitting antenna, which is provided at a location spatially separated from the receiving antennas  10 . The antennas  10  receive echo signals generated by reflection of the detection signal on the targets. Then the radar mixes the echo signals with the transmission signal and demodulates the resultant signals into baseband signals. Furthermore, the radar conducts appropriate signal processing to estimate the position or speed of each target. With the arrangement shown in  FIG. 1 , echo signals x m (t) from the M targets (M≦N−1) arrive at each antenna  10 . Here, m=1, . . . , M, and t represents time. 
     At that time, an input signal from the nth antenna A n  is demodulated to obtain a signal v n (t) given by 
                       v   n     ⁡     (   t   )       =         ∑     m   =   1     M     ⁢         x   m     ⁡     (   t   )       ⁢     exp   ⁡     (     j   ⁢           ⁢     ϕ     n   ,   m         )           +       n   n     ⁡     (   t   )                 (   1   )                 ϕ     n   ,   m       ≡         2   ⁢           ⁢   π     λ     ⁢     (     n   -   1     )     ⁢     ⅆ           ⁢   sin     ⁢           ⁢     θ   m               (   2   )               
where x m (t) is a baseband signal, n n (t) is an additive Gaussian noise, and λ is a wavelength of a carrier signal.
 
     The signals, for n=1 to N, v n (t) can be represented as a baseband signal vector (hereafter refereed as signals, a signal vector, or the like) defined by 
                       v   ⁡     (   t   )       =       [             v   1     ⁡     (   t   )               ⋰               v   N     ⁡     (   t   )             ]     =       [               ∑     m   =   1     M     ⁢           x   .     m     ⁡     (   t   )       ⁢     exp   ⁡     (     j   ⁢           ⁢     φ     1   ,   m         )           +       n   1     ⁡     (   t   )                 ⋰                 ∑     m   =   1     M     ⁢         x   m     ⁡     (   t   )       ⁢     exp   ⁡     (     j   ⁢           ⁢     φ     N   ,   m         )           +       n   N     ⁡     (   t   )               ]     =       Ax   ⁡     (   t   )       +     n   ⁡     (   t   )               ⁢     
     ⁢   where           (   3   )                 A   ≡     [       a   ⁡     (     θ   1     )       ,   …   ⁢           ,     a   ⁡     (     θ   M     )         ]       =     [             exp   ⁡     (     j   ⁢           ⁢     ϕ     1   ,   1         )       ⁢                 …         exp   ⁡     (     j   ⁢           ⁢     ϕ     1   ,   M         )               ⋰                   ⋰             exp   ⁡     (     j   ⁢           ⁢     ϕ     N   ,   1         )           …         exp   ⁡     (     j   ⁢           ⁢     ϕ     N   ,   M         )             ]             (   4   )                 x   ⁡     (   t   )       ≡       [         x   1     ⁡     (   t   )       ⁢           ⁢   …   ⁢           ⁢       x   M     ⁡     (   t   )         ]     ⁢             T               (   5   )                 n   ⁡     (   t   )       ≡       [         n   1     ⁡     (   t   )       ⁢           ⁢   …   ⁢           ⁢       n   N     ⁡     (   t   )         ]     ⁢             T               (   6   )               
Assuming that there is no correlation between x(t) and n(t), a covariance matrix R vv  of v(t) is computed from the formula (3) by
 
 R   vv   ≡E{v ( t ) v   H ( t )}= AR   xx   A   H +σ 2   I   (7)
 
where R xx  is a covariance matrix of the baseband signal vector x(t). R xx  is defined by
 
R xx ≡E[x(t)x H (t)]  (8)
 
     The DOA estimation is performed on R vv . 
     Since radar is used in the present embodiment, the ULA receives signals that have been transmitted from the same signal source and reflected from the individual targets. Thus, those received signals have a high coherence. Therefore, the rank of R vv , which is an N×N matrix, degenerates into one. In such a situation, it is difficult to estimate DOA from R vv . 
     Accordingly, forward spatial smoothing (FSS) is used to recover the rank of R vv . In forward spatial smoothing, (N−L+1) submatrices having the order of L×L (L&lt;N) are extracted along the main diagonal direction from R vv . Those submatrices are summed up and averaged to recover the rank of R vv . 
     Furthermore, backward spatial smoothing (BSS), which inverses a reference point of the ULA and performs operation similar to FSS, may also be used to recover the rank of R vv . Usually, forward backward spatial smoothing (FBSS), which is a combination of FSS and BSS, is used to recover the rank of R vv . 
     For example, DOA estimation using Capon method is performed by using 
                       P   Capon     ⁡     (   θ   )       =     1             a   H     ⁡     (   θ   )       ⁡     [     R   vv   FBSS     ]         -   1       ⁢     a   ⁡     (   θ   )                   (   9   )               
where R vv   FBSS  is a matrix obtained by applying FBSS to R vv , and a(θ) is an L-order mode vector (or an array mode vector) and has a structure similar to that of the vector elements a(θ m ) in the formula (4).
 
     With a multiple signal classification (MUSIC) method, eigenvalue decomposition is performed on R vv   FBSS  by using
 
 R   vv   FBSS   =E   S Λ S   E   S   H +σ 2   E   N   E   N   H   (10)
 
Then a matrix E N  is calculated, and DOA estimation is performed by using
 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       MUSIC 
                     
                     ⁡ 
                     
                       ( 
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                       ) 
                     
                   
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                         ⁡ 
                         
                           ( 
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                           ( 
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                       ⁢ 
                       
                         a 
                         ⁡ 
                         
                           ( 
                           θ 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Specifically, in the formula (9) or (11), a mode vector a(θ) including a parameter θ is defined by a(θ)=[1, exp(j2πα sin θ), . . . , exp(j2πα(L−1)sin θ)] T  where α=d/λ. While the parameter θ is scanned, angle information included in the matrix is examined by computation using the formula (9) or (11). Thus, a direction of arrival is estimated. 
     Usually, a signal covariance matrix inevitably includes a noise component σ 2 I as seen in the formula (7). In order to reduce influence from the noise component for improving estimation accuracy of DOA, the following process is performed in the technique disclosed in Patent Document 1. Specifically, signals received by the respective antennas, which form a ULA, are demodulated to generate baseband signals. Then correlation vectors are computed from the baseband signals to generate a pseudo covariance matrix. Thereafter, a propagator matrix is generated from the pseudo covariance matrix. DOA estimation is performed by using the propagator matrix. This estimation method will be described below in detail. 
     For the sake of brevity, the following description is focused only on FSS. A correlation vector r v1  of the baseband signals is computed by
 
 r   v1   =E[v ( t ) v   N *( t )]  (11)
 
     Hereinafter, M denotes the maximum number of targets of which direction of arrival can be estimated with a ULA formed by N antennas. For example, M is defined as a maximum natural number not more than (N−1)/2. 
     Next, the elements of r v1  are rearranged to generate a matrix R f1  defined by 
     
       
         
           
             
               
                 
                   
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     In this case, r v1 (k) is given by 
                       r     v   ⁢           ⁢   1       ⁡     (   k   )       =     E   ⁡     (               v   k     ⁡     (   t   )       ⁢       v   N   *     ⁡     (   t   )                 ⋰                 v     k   +   M   -   1       ⁡     (   t   )       ⁢       v   N   *     ⁡     (   t   )               )               (   14   )               
where k=1, . . . , N−M. Therefore, simple manipulations reveal that R f1  can be represented in the form of R f1 =AX. Hereafter, symbol “t” representing an instantaneous time is accordingly omitted for simplicity. Specifically, the matrix R defined by
 
R≡[R f1 ]  (15)
 
includes the same phase information as the covariance matrix R vv  to which spatial smoothing is applied.
 
     An example in which N=5 and M=2 will be described below. In this example, the matrix R is computed as 
     
       
         
           
             
               
                 
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     Next, the matrix R is divided into an M×M submatrix R 1  and an (N−2M)×M submatrix R 2  as shown by 
     
       
         
           
             
               
                 
                   
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     Then, using the submatrices R 1  and R 2 , an M×(N−2M) matrix Γ is generated by Γ=(R 1 R 1   H ) −1 R 1 R 2   H . 
     Subsequently, using the matrix Γ and an (N−2M)×(N−2M) unit matrix I N-2M , an (N−M)×(N−2M) propagator matrix Π is generated by Π=[Γ|−I N-2M ] T . 
     For example, using the propagator matrix Π, an (N−M)×(N−M) core matrix Ω is defined by Ω=Π(Π H Π) −1 Π H . For example, using this core matrix Ω and the array mode vector a(θ) defined by a(θ)=[1, exp(j2πα sin θ), . . . , exp(j2παa(N−M−1)sin θ)] T  where α=d/λ and an angular spectrum P(θ) is defined by 
     
       
         
           
             
               
                 
                   
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     Then, the angular spectrum P(θ) is computed while the parameter θ is scanned. An arrival angle θ m  of the arriving signal can be obtained from a position of a peak in the angular spectrum P(θ). Alternatively, an arrival angle θ m  of the arriving signal can be obtained from a solution of an algebraic equation (a(1/z) T Ωa(z)=0) where z=exp(j2πα sin(θ)) and α=d/λ. 
     As described above, according to Patent Document 1, the matrix R is computed as shown by the formulas (15) and (16), and the propagator matrix Π is then computed from the matrix R. Thereafter, the core matrix Ω is computed to compute a direction of arrival. 
     For the sake of brevity, an example in which N=5 and M=2 as shown by the formula (16) will be described below. The matrix Γ, which is a primary part of the propagator matrix Π, can be represented by 
     
       
         
           
             
               
                 
                   
                     
                       
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                                             2 
                                           
                                            
                                         
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           v 
                                           1 
                                         
                                         ⁢ 
                                         
                                           v 
                                           2 
                                           * 
                                         
                                       
                                       + 
                                       
                                         
                                           v 
                                           2 
                                         
                                         ⁢ 
                                         
                                           v 
                                           3 
                                           * 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           v 
                                           2 
                                         
                                         ⁢ 
                                         
                                           v 
                                           1 
                                           * 
                                         
                                       
                                       + 
                                       
                                         
                                           v 
                                           3 
                                         
                                         ⁢ 
                                         
                                           v 
                                           2 
                                           * 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                            
                                           
                                             v 
                                             2 
                                           
                                            
                                         
                                         2 
                                       
                                       + 
                                       
                                         
                                            
                                           
                                             v 
                                             3 
                                           
                                            
                                         
                                         2 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             v 
                                             1 
                                           
                                           ⁢ 
                                           
                                             v 
                                             3 
                                             * 
                                           
                                         
                                       
                                       
                                         
                                           
                                             v 
                                             2 
                                           
                                           ⁢ 
                                           
                                             v 
                                             4 
                                             * 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             v 
                                             2 
                                           
                                           ⁢ 
                                           
                                             v 
                                             3 
                                             * 
                                           
                                         
                                       
                                       
                                         
                                           
                                             v 
                                             3 
                                           
                                           ⁢ 
                                           
                                             v 
                                             4 
                                             * 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             1 
                             
                               
                                 
                                    
                                   
                                     v 
                                     5 
                                   
                                    
                                 
                                 2 
                               
                               ⁢ 
                               
                                 
                                    
                                   
                                     
                                       
                                         ( 
                                         
                                           v 
                                           2 
                                         
                                         ) 
                                       
                                       2 
                                     
                                     - 
                                     
                                       
                                         v 
                                         1 
                                       
                                       ⁢ 
                                       
                                         v 
                                         3 
                                         * 
                                       
                                     
                                   
                                    
                                 
                                 2 
                               
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   … 
                                 
                               
                               
                                 
                                   … 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     To know phase components of vectors forming the matrix Γ is sufficient to perform DOA estimation. Thus, a factor 1/|v 5 | 2  multiplying the components in the formula (19) has no direct effect on the DOA estimation and may inhibit high-speed computation. Additionally, in practical computation, such a factor may cause cancellation of significant digits or rounding errors, thereby making it difficult to increase the accuracy of DOA estimation. 
     In view of the above discussion, a direction-of-arrival estimation apparatus according to the present invention performs DOA estimation with use of a matrix that does not include the above factor. 
     The order of the column of a matrix is preferentially set at a natural number M that meets 1≦M and M≦(N−1)/2, and the order of the row of the matrix is set at N−M. Elements v 1  to v N-1  of the baseband signal vector v composed of N signals v 1  to v N  are arranged as elements of the matrix to thereby generate a Hankel matrix R f1  as defined by 
     
       
         
           
             
               
                 
                   
                     R 
                     
                       f 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             v 
                             1 
                           
                         
                         
                           … 
                         
                         
                           
                             v 
                             M 
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           
                               
                           
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             v 
                             
                               N 
                               - 
                               M 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             v 
                             
                               N 
                               - 
                               1 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     For example, in a case where N=5 and M=2, a Hankel matrix R f1  is defined by 
     
       
         
           
             
               
                 
                   
                     R 
                     
                       f 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             v 
                             1 
                           
                         
                         
                           
                             v 
                             2 
                           
                         
                       
                       
                         
                           
                             v 
                             2 
                           
                         
                         
                           
                             v 
                             3 
                           
                         
                       
                       
                         
                           
                             v 
                             3 
                           
                         
                         
                           
                             v 
                             4 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Then, for example, a matrix R is defined as shown in the formula (22). The matrix R is divided into an M×M submatrix R 1  and an (N−2M)×M submatrix R 2 . 
     
       
         
           
             
               
                 
                   
                     R 
                     ≡ 
                     
                       [ 
                       
                         R 
                         
                           f 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               
                                 v 
                                 1 
                               
                             
                             
                               
                                 v 
                                 2 
                               
                             
                           
                           
                             
                               
                                 v 
                                 2 
                               
                             
                             
                               
                                 v 
                                 3 
                               
                             
                           
                           
                             
                               
                                 v 
                                 3 
                               
                             
                             
                               
                                 v 
                                 4 
                               
                             
                           
                         
                         ) 
                       
                       ≡ 
                       
                         ( 
                         
                           
                             
                               
                                 R 
                                 1 
                               
                             
                           
                           
                             
                               
                                 R 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       v 
                                       1 
                                     
                                   
                                   
                                     
                                       v 
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     
                                       v 
                                       2 
                                     
                                   
                                   
                                     
                                       v 
                                       3 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       v 
                                       3 
                                     
                                   
                                 
                                 
                                   
                                     
                                       v 
                                       4 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     Using those submatrices R 1  and R 2 , an M×(N−2M) matrix Γ is computed by 
     
       
         
           
             
               
                 
                   
                     
                       
                         Γ 
                         ≡ 
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   R 
                                   1 
                                 
                                 ⁢ 
                                 
                                   R 
                                   1 
                                   H 
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             R 
                             1 
                           
                           ⁢ 
                           
                             R 
                             2 
                             H 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               [ 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           v 
                                           1 
                                         
                                       
                                       
                                         
                                           v 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           v 
                                           2 
                                         
                                       
                                       
                                         
                                           v 
                                           3 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           v 
                                           1 
                                           * 
                                         
                                       
                                       
                                         
                                           v 
                                           2 
                                           * 
                                         
                                       
                                     
                                     
                                       
                                         
                                           v 
                                           2 
                                           * 
                                         
                                       
                                       
                                         
                                           v 
                                           3 
                                           * 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     v 
                                     1 
                                   
                                 
                                 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     v 
                                     2 
                                   
                                 
                                 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     v 
                                     3 
                                     * 
                                   
                                 
                               
                               
                                 
                                   
                                     v 
                                     4 
                                     * 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         
                                            
                                           
                                             v 
                                             1 
                                           
                                            
                                         
                                         2 
                                       
                                       + 
                                       
                                         
                                            
                                           
                                             v 
                                             2 
                                           
                                            
                                         
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           v 
                                           1 
                                         
                                         ⁢ 
                                         
                                           v 
                                           2 
                                           * 
                                         
                                       
                                       + 
                                       
                                         
                                           v 
                                           2 
                                         
                                         ⁢ 
                                         
                                           v 
                                           3 
                                           * 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           v 
                                           2 
                                         
                                         ⁢ 
                                         
                                           v 
                                           1 
                                           * 
                                         
                                       
                                       + 
                                       
                                         
                                           v 
                                           3 
                                         
                                         ⁢ 
                                         
                                           v 
                                           2 
                                           * 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                            
                                           
                                             v 
                                             2 
                                           
                                            
                                         
                                         2 
                                       
                                       + 
                                       
                                         
                                            
                                           
                                             v 
                                             3 
                                           
                                            
                                         
                                         2 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       v 
                                       1 
                                     
                                     ⁢ 
                                     
                                       v 
                                       3 
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       v 
                                       2 
                                     
                                     ⁢ 
                                     
                                       v 
                                       4 
                                       * 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       v 
                                       2 
                                     
                                     ⁢ 
                                     
                                       v 
                                       3 
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       v 
                                       3 
                                     
                                     ⁢ 
                                     
                                       v 
                                       4 
                                       * 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             1 
                             
                               
                                  
                                 
                                   
                                     
                                       ( 
                                       
                                         v 
                                         2 
                                       
                                       ) 
                                     
                                     2 
                                   
                                   - 
                                   
                                     
                                       v 
                                       1 
                                     
                                     ⁢ 
                                     
                                       v 
                                       3 
                                       * 
                                     
                                   
                                 
                                  
                               
                               2 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   … 
                                 
                               
                               
                                 
                                   … 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     While the matrix in the formula (23) holds phase information to be used for DOA estimation, it does not include an extra factor 1/|v 5 | 2 , which is included in the formula (19). 
     Thereafter, using the matrix Γ and an (N−2M)×(N−2M) unit matrix I N-2M  as with the formula (17), a propagator matrix Π is generated by Π=[Γ|−I N-2M ] T . 
     For example, using the propagator matrix Π, an (N−M)×(N−M) core matrix Ω is defined as Ω=Π(Π H Π) −1 Π H . For example, using this core matrix Ω and the array mode vector a(θ) defined by a(θ)=[1, exp(j2πα sin θ), . . . , exp(j2πα(N−M−1)sin θ)] T  where α=d/λ, an angular spectrum P(θ) is defined by 
     
       
         
           
             
               
                 
                   
                     P 
                     ⁡ 
                     
                       ( 
                       θ 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           a 
                           H 
                         
                         ⁡ 
                         
                           ( 
                           θ 
                           ) 
                         
                       
                       ⁢ 
                       
                         a 
                         ⁡ 
                         
                           ( 
                           θ 
                           ) 
                         
                       
                     
                     
                       
                         
                           a 
                           H 
                         
                         ⁡ 
                         
                           ( 
                           θ 
                           ) 
                         
                       
                       ⁢ 
                       Ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         a 
                         ⁡ 
                         
                           ( 
                           θ 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     Then, the angular spectrum P(θ) is computed while the parameter θ is scanned. An arrival angle θ m  of the arriving signal can be obtained from a position of a peak in the angular spectrum P(θ). Alternatively, an arrival angle θ m  of the arriving signal can be obtained from a solution of an algebraic equation (a(1/z) T Ωa(z)=0) where z=exp(j2πα sin(θ)) and α=d/λ. 
     According to a direction-of-arrival estimation apparatus of the present invention, a matrix R f1  as shown in the formula (21) (formula (20) in general) is used instead of the matrix R f1  as shown in the formula (13). With the matrix R f1  as shown in the formula (21), a matrix Γ generated by the formula (23) does not include a factor such as 1/|v 5 | 2  in the formula (19). Accordingly, computation of the matrix Γ and a propagator matrix Π can be performed at a high speed with high accuracy. Several embodiments based on the above principle of the present invention will be described below. 
     First Embodiment 
       FIG. 2  is a block diagram showing a direction-of-arrival estimation apparatus according to a first embodiment of the present invention.  FIG. 3  is a flow chart showing processes performed in the direction-of-arrival estimation apparatus. The first embodiment of the present invention will be described with reference to  FIGS. 2 and 3 . 
     In  FIG. 2 , the direction-of-arrival estimation apparatus  100  includes a plurality of antennas  10  (antennas A 1  to A N ), a receiving portion  20 , and a DOA estimation portion  30 . The plurality of antennas  10  receive arriving signals from targets. The receiving portion  20  includes a receiver  22  and a signal vector generation unit  24 . For example, the receiver  22  receives arriving signals with a carrier signal having a frequency of 76 GHz, demodulates them in accordance with a modulation method, and downconverts them into baseband signals having an appropriate band. Furthermore, the receiver  22  performs A/D conversion on those baseband signals into digital signals (hereinafter simply referred to as baseband signals). The signal vector generation unit  24  shapes baseband signals corresponding to each antenna as a column vector. Thus, the signal vector generation unit  24  generates baseband signal vector v composed of N signal elements v 1  to v N  represented by the formula (3) (Step S 10  of  FIG. 3 ). 
     The DOA estimation portion  30  includes an arriving signal count predicting/setting unit  31  for predicting or setting the number of arriving signals, a Hankel matrix generation unit  32 , and an estimation unit  45 . The DOA estimation portion  30  is operable to estimate DOA from the baseband signals. The estimation unit  45  includes a propagator matrix generation unit  33 , a core matrix generation unit  34 , and a DOA computation unit  35 . The DOA estimation portion  30  may be implemented by an arithmetical unit such as a central processing unit (CPU) in a computer, a program executed by such an arithmetical unit, or an entity on a storage device in a computer. Furthermore, the entire structure of the receiving portion  20  and the DOA estimation portion  30  may be implemented by an arithmetical unit in terms of a software radio device. 
     The arriving signal count predicting/setting unit  31  is operable to predict or set the number of arriving signals M (Step S 12  of  FIG. 3 ). Akaike&#39;s information criteria (AIC), minimum description length (MDL), or other appropriate indexes based on a maximum likelihood method can be used to predict the number of arriving signals M. In a case where the arriving signal count predicting/setting unit  31  does not predict the number of arriving signals M, the value M may automatically be determined as [(N−1)/2] from the number of the antennas N. The symbol [x] gives a maximum natural number that is not more than a real number x. 
     The estimation unit  45  first determines whether or not an SN (signal-to-noise) ratio of the signal vector in the formula (3) is at least a predetermined threshold value (Step S 14 ). If the SN ratio is lower than the threshold value, the estimation unit  45  computes a correlation vector conj(v N )×[v 1 , . . . , v N ] T  where conj(v N ) is a conjugate complex number of a signal v N , for example, and redefines the signal vector as the computed correlation vector (Step S 16 ). 
     The Hankel matrix generation unit  32  generates a Hankel matrix R f1  as shown in the formula (20). If the signal vector has been redefined in Step S 16 , the Hankel matrix generation unit  32  generates a matrix having a structure (made up of previously mentioned correlation vectors) as shown in the formula (16), the case of which will not hereinafter be mentioned. The propagator matrix generation unit  33  defines a matrix R=[R f1 ] (Step S 18 ). 
     The propagator matrix generation unit  33  decomposes the matrix R into two submatrices R 1  and R 2  and generates a matrix Γ=(R 1 R 1   H ) −1 R 1 R 2   H  as described above. Furthermore, the propagator matrix generation unit  33  generates a propagator matrix Π=[Γ|−I N-2M ] T  from the matrix Γ and a unit matrix I N-2M  (Step S 22 ). 
     The core matrix generation unit  34  generates a core matrix Ω=Π(Π H Π) −1 Π H  (Step S 24 ). 
     The DOA computation unit  35  generates an angular spectrum P(θ) as shown in the formula (24) with use of an array mode vector a(θ), which includes the parameter θ, defined by a(θ)=[1, exp(j2πα sin θ), . . . , exp(j2πα(N−M−1)sin θ)] T  where α=d/λ, and of the core matrix Ω (Step S 26 ). The DOA computation unit  35  computes an angular spectrum P(θ) while scanning the parameter θ. The DOA computation unit  35  computes an arrival angle θ m  of the arriving signal from the peak of the angular spectrum P(θ). Alternatively, the DOA computation unit  35  computes an arrival angle θ m  from a solution of an algebraic equation (a(1/z) T Ωa(z)=0) where z=exp(j2πα sin(θ)) and α=d/λ. The DOA computation unit  35  outputs the arrival angle θ m  as an estimated direction of arrival (Step S 28 ). 
     Instead of the Hankel matrix R f1 , the Hankel matrix generation unit  32  may generate a Hankel matrix R f2  as defined by 
                     R     f   ⁢           ⁢   2       =     (           v   2         …         v     M   +   1               ⋮                   ⋮             v     N   -   M   +   1           …         v   N           )             (   25   )               
The Hankel matrix R f2  includes other elements v 2  to v N  of the signal vector. Then the propagator matrix generation unit  33  may generate a propagator matrix Π with use of this Hankel matrix R f2 .
 
     Furthermore, instead of the Hankel matrix R f1 , the Hankel matrix generation unit  32  may generate a Hankel matrix R b1  including complex conjugate elements v 1 * to v N-1 * of the signal vector by R b1 =J N-M R* f1 J M  where R* f1  is a matrix including a conjugate complex number of every element in R f1 . Then the propagator matrix generation unit  33  may generate a propagator matrix Π with use of this Hankel matrix R b1 . 
     For example, in a case where N=5 and M=2, the Hankel matrix R b1  is represented by 
     
       
         
           
             
               
                 
                   
                     R 
                     
                       b 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   = 
                   
                     
                       
                         J 
                         
                           N 
                           - 
                           M 
                         
                       
                       ⁢ 
                       
                         R 
                         
                           f 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                         * 
                       
                       ⁢ 
                       
                         J 
                         M 
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     v 
                                     1 
                                   
                                 
                                 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     v 
                                     2 
                                   
                                 
                                 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                               
                                 
                                   
                                     v 
                                     3 
                                   
                                 
                                 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                             
                             ) 
                           
                           * 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                           
                           ) 
                         
                       
                       = 
                       
                         ( 
                         
                           
                             
                               
                                 v 
                                 4 
                                 * 
                               
                             
                             
                               
                                 v 
                                 3 
                                 * 
                               
                             
                           
                           
                             
                               
                                 v 
                                 3 
                                 * 
                               
                             
                             
                               
                                 v 
                                 2 
                                 * 
                               
                             
                           
                           
                             
                               
                                 v 
                                 2 
                                 * 
                               
                             
                             
                               
                                 v 
                                 1 
                                 * 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     Next, a matrix R is defined by 
                     R   ≡     [     R     b   ⁢           ⁢   1       ]       =         (           v   4   *           v   3   *               v   3   *           v   2   *               v   2   *           v   1   *           )     ≡     (           R   1               R   2           )       =     [           (           v   4   *           v   3   *               v   3   *           v   2   *           )                     (     v   2   *               v   1   *     )                 ]               (   27   )               
Furthermore, the matrix R is divided into an (N−2M)×M submatrix R 1  and an (N−2M)×M submatrix R 2 .
 
     If a matrix Γ is defined as Γ=(R 1 R 1   H ) −1 R 1 R 2   H  using the submatrices R 1  and R 2 , then 
     
       
         
           
             
               
                 
                   
                     
                       
                         Γ 
                         ≡ 
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   R 
                                   1 
                                 
                                 ⁢ 
                                 
                                   R 
                                   1 
                                   H 
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             R 
                             1 
                           
                           ⁢ 
                           
                             R 
                             2 
                             H 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               [ 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           v 
                                           4 
                                           * 
                                         
                                       
                                       
                                         
                                           v 
                                           3 
                                           * 
                                         
                                       
                                     
                                     
                                       
                                         
                                           v 
                                           3 
                                           * 
                                         
                                       
                                       
                                         
                                           v 
                                           2 
                                           * 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           v 
                                           4 
                                         
                                       
                                       
                                         
                                           v 
                                           3 
                                         
                                       
                                     
                                     
                                       
                                         
                                           v 
                                           3 
                                         
                                       
                                       
                                         
                                           v 
                                           2 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                             
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                   28 
                   ) 
                 
               
             
           
         
       
     
     The matrix Γ in the formula (28) does not include a factor such as 1/|v 5 | 2  in the formula (19) while it holds phase information to be used for DOA estimation, as with the matrix Γ in the formula (23). Accordingly, computation of the matrix Γ and the propagator matrix Π can be performed at a high speed with high accuracy. 
     Furthermore, instead of the Hankel matrix R f1 , the Hankel matrix generation unit  32  may generate a Hankel matrix R b2  using complex conjugate elements v 2 * to v N * of the signal vector by R b2 =J N-M R* f2 J M  where R* f2  is a matrix including a conjugate complex number of every element in R f2 . Then the propagator matrix generation unit  33  may generate a propagator matrix Π using this Hankel matrix R b2 . 
     The propagator matrix generation unit  33  may combine K Hankel matrices R f1 , R f2 , R b1 , or R b2  having the order of (N−M)×M to form a matrix R having the order of (N−M)×(K×M). For example, in a case in which K=2, the matrix R may be defined as R=[R f1 |R f2 ], R=[R b1 |R b2 ], R=[R f1 |R b2 ], or R=[R f2 |R b1 ]. In a case in which K=4, the matrix R may be defined as R=[R f1 |R f2 |R b1 |R b2 ]. In these cases, the matrix R is divided into an M×(K×M) submatrix R 1  and an (N−2M)×(K×M) submatrix R 2  so that R=[R 1 |R 2 ] T . 
     As described above, the propagator matrix generation unit  33  uses at least one of the matrices R f1 , R f2 , R b1 , and R b2  to generate a matrix R and divides the matrix R into two submatrices R 1  and R 2  having an appropriate order. 
     According to the direction-of-arrival estimation apparatus  100  in the first embodiment, the Hankel matrix generation unit  32  generates at least one of the matrices R f1 , R f2 , R b1 , and R b2 . The estimation unit  45  uses at least one of the matrices R f1 , R f2 , R b1 , and R b2  to generate a matrix R and divides the matrix R into two submatrices R 1  and R 2  so that R=[R 1 |R 2 ] T . The estimation unit  45  uses those submatrices R 1  and R 2  to compute the matrices Γ, Π, and Ω, thereby estimating a direction of the arriving signal. Thus, with use of the matrix Γ (see the formula (23)) that does not include a factor such as 1/|v 5 | 2  in the formula (19), computation of the matrix Γ and the propagator matrix Π can be performed at a high speed with high accuracy. 
     Second Embodiment 
       FIG. 4  is a block diagram showing a direction-of-arrival estimation apparatus according to a second embodiment of the present invention. As shown in  FIG. 4 , a direction-of-arrival estimation apparatus  100   a  differs from the direction-of-arrival estimation apparatus  100  shown in  FIG. 2  of the first embodiment in that a DOA estimation portion  30   a  includes a scaling matrix generation unit  36 . Other components shown in  FIG. 4  are the same as those in  FIG. 2  of the first embodiment and will not be described repetitively. 
     The scaling matrix generation unit  36  uses submatrices R 1  and R 2  of a matrix R which are generated by the propagator matrix generation unit  33  to generate an (N−2M)×(N−2M) scaling matrix Λ defined by Λ=R 2 R 2   H −R 2 R 1   H Γ. The core matrix generation unit  34  generates a core matrix Ω defined by Ω=Π(Λ) −1 Π H  from the propagator matrix Π and the scaling matrix Λ. Alternatively, the core matrix generation unit  34  may generate a core matrix Ω defined by Ω=Π′(Λ) −1 (Π′) H  where Π′=Π(Π H Π) −1/2 . 
     The core matrix generation unit  34  may generate a core matrix Ω defined by Ω=ΠΠ H  instead of Ω=Π(Λ) −1 Π H . In this case, the angular spectrum is defined by 
     
       
         
           
             
               
                 
                   
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                           ( 
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                   ( 
                   29 
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     The algebraic equation is defined by
 
 a (1 /z ) T [ΠΠ H   ]a ( z )=0  (30)
 
a(z)≡(1,z, . . . ,z N-2M-1 ) T   (31)
 
where z=exp(j2πα sin(θ)) and α=d/λ.
 
     The core matrix generation unit  34  may generate a core matrix Ω defined by Ω=Π[α(Π H Π)+(1−α)Λ] −1 Π H  where 0≦α≦1 (of course, this linear combination parameter “α” differs from the constant d/λ). Thus, the core matrix generation unit  34  may combine a plurality of matrices derived from a Hankel matrix to generate a core matrix. 
     Assuming that α≠0 or 1 and that Π H Π and Λ have no singularity, the core matrix Ω is given by 
                         Ω   =         Π   ⁡     [       α   ⁢           ⁢   A     +     β   ⁢           ⁢   B       ]         -   1       ⁢     Π   H                   =         Π   ⁡     [     I   +       β   α     ⁢     A     -   1       ⁢   B       ]         -   1       ⁢       (     α   ⁢           ⁢   A     )       -   1       ⁢     Π   H                   =       Π   ⁡     [     I   -       (       β   α     ⁢     A     -   1       ⁢   B     )       -   1         ]       ⁢       (     α   ⁢           ⁢   A     )       -   1       ⁢     Π   H                   =         1   α     ⁢   Π   ⁢           ⁢     A     -   1       ⁢     Π   H       -       Π   ⁡     (       α   β     ⁢     B     -   1       ⁢   A     )       ⁢     1   α     ⁢     A     -   1       ⁢     Π   H                     =         1   α     ⁢       Π   ⁡     (       Π   H     ⁢   Π     )         -   1       ⁢     Π   H       -       1   β     ⁢       Π   ⁡     (   Λ   )         -   1       ⁢     Π   H                       (   32   )               
where A=Π H Π, B=Λ, and 1−α=β.
 
     Thus, this core matrix Ω is equal to a weighted average of known core matrices Π(Π H Π) −1 Π H  and Π(Λ) −1 Π H . Accordingly, it is possible to perform an angle estimation for a target for which angle would be hard to estimate with sole use of the known core matrices. 
     While the core matrix generation unit  34  in the first embodiment generates a core matrix Ω from the propagator matrix Π, the core matrix generation unit  34  in the second embodiment generates a core matrix Ω from the propagator matrix Π and the scaling matrix Λ. Then, by DOA estimation unit  35 , a direction of arrival may be computed from the angular spectrum or the algebraic equation as with the first embodiment. 
     Furthermore, the core matrix generation unit  34  may generate a core matrix Ω by using the propagator matrix Π and a substitute matrix which properly combines a matrix A defined by A=R 1 R 1   H , a matrix B defined by B=R 1 R 2   H , a matrix C defined by C=R 2 R 1   H , and a matrix D defined by D=R 2 R 2   H , such as D, (B H A −1 B), (B H B), (D−B H B), or Λ(Π H Π), instead of using the scaling matrix Λ. 
     For example, if the angular spectrum using the matrix D is defined by 
     
       
         
           
             
               
                 
                   
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     The algebraic equation using the matrix D is defined by
 
 a (1 /z ) T   [ΠD   −1 Π H   ]a ( z )=0  (34)
 
a(z)≡(1,z, . . . ,z N-M-1 ) T   (35)
 
where z=exp(j2πα sin(θ)) and α=d/λ.
 
     And, the DOA computation unit  35  may compute a direction of arrival by using the the substitute matrix for the scaling matrix Λ. 
       FIGS. 5A to 8  show examples in which the second embodiment was applied to a frequency modulated continuous wave (FMCW) radar to orient a target. In these figures, the unit of distance is bin. 
       FIGS. 5A to 5D  are diagrams showing angular spectra with respect to angles and distances in a case where two targets were located. The targets were set at a distance of 40 m and an angle of 0° and 3°, respectively.  FIG. 5A  shows an angular spectrum computed by using an FFT-DBF method,  FIG. 5B  by using an FBSS-MUSIC method,  FIG. 5C  by using a method disclosed in Patent Document 1 (Comparative Example), and  FIG. 5D  by using a method according to the second embodiment. In these figures, if the angular spectrum is intensive at locations of the two assumed targets, i.e., if clear points are seen near the two targets in an angular spectrum, then the angular spectrum shows that directions of arrival are estimated for those targets with high accuracy. 
       FIG. 6  is a graph superposing the angular spectra shown in  FIGS. 5A ,  5 C, and  5 D that were cut in parallel to the angle axis at a distance of 40 m. It can be seen from  FIGS. 5A to 6  that the method of Comparative Example (Patent Document 1) and the method of the second embodiment each performed angle estimation for the targets with higher accuracy as compared to the FFT-DBF method and the FBSS-MUSIC method. 
       FIGS. 7 and 8  are examples in which the orientation performance for a plurality of targets arranged at spatially proximate locations was evaluated by the angular spectra on the assumption of five targets, which are indicated at the cross-marked points in the figures. The peaks of the angular spectrum are located at areas in which contours are intensively illustrated. Since those figures are illustrated in monochrome, it may slightly be difficult to see those peaks.  FIG. 7  shows the angular spectrum computed by using the method disclosed in Patent Document 1.  FIG. 8  shows the angular spectrum computed by using a core matrix defined by Ω=Π[α(Π)+(1−α)Λ] −1 Π H  where α=0.5 in the second embodiment. In  FIG. 8 , contours are more intensive near the cross-marked points as compared to  FIG. 7 . Thus, the peaks of the angular spectrum are located near the assumed targets. By using a combined core matrix, it is possible to accurately detect many targets which produce echo signals having high coherence and would thus be hard to estimate. 
     Third Embodiment 
     An example of a direction-of-arrival estimation apparatus mounted on a vehicle such as an automobile will be described in a third embodiment of the present invention. In the third embodiment, a direction-of-arrival estimation apparatus  100   b  is configured to output an angle (an angle of arrival) for a target that may threaten a vehicle having millimeter-wave radar with the direction-of-arrival estimation apparatus  100   b . For example, a vehicle such as an automobile traveling ahead of the vehicle on which the direction-of-arrival estimation apparatus  100   b  is mounted may be a target. 
       FIG. 9  is a block diagram showing the direction-of-arrival estimation apparatus  100   b  according to the third embodiment of the present invention. The direction-of-arrival estimation apparatus  100   b  includes a threat evaluation portion  40  provided between a receiving portion  20  and a DOA estimation portion  30 . The threat evaluation portion  40  includes a distance/speed/direction estimation unit  37 , a target selection unit  38 , and a beam formation unit  39 . 
       FIG. 10  is a flow chart showing processes performed in the direction-of-arrival estimation apparatus  100   b  according to the third embodiment. The third embodiment will be described with reference to  FIG. 10 . For example, for each of the receiving antennas, sampling is conducted on echo signals during one cycle of a signal used for performing FM modulation on millimeter-wave signals. The signal vector generation unit  24  extracts data sets of signal vectors corresponding to each antenna (Step S 30 ). Specifically, the data include an N×Q matrix (data set) where N is the number of the antennas and Q is the number of the samples. The distance/speed/direction estimation unit  37  performs fast Fourier transform (FFT) on the data set for each antenna in which Q-order vectors are arranged in N sets along the time domain (Step S 32 ). The distance/speed/direction estimation unit  37  performs FFT on a Q-order vector N times to generate an N×Q matrix in which the data set has been converted into a data set in the frequency domain. The distance/speed/direction estimation unit  37  estimates a relative distance r m  and a relative speed s m  between each target and the present vehicle from a bin position corresponding to a peak that can be seen in the frequency domain as a result of the FFT. The bin position may be considered as a value of a frequency index. Furthermore, the distance/speed/direction estimation unit  37  extracts an N-order data vector at each target (bin) position and performs, for example, FFT-DBF (digital beam forming based on the maximum ratio combining approach) to roughly estimate an angular position of each target (Step S 34 ). Moreover, the distance/speed/direction estimation unit  37  conducts pairing on the estimated relative distance, relative speed, and angular position as identification information specific to each target. 
     The target selection unit  38  evaluates a threat score of each target based on the relative distance r m  and the relative speed s m  of each target (Step S 36 ). The threat score increases when the relative distance r m  has a smaller value and the relative speed s m  is negative and has a larger absolute value. In other words, the threat score increases when the target is approaching the present vehicle from its vicinity at a high speed. For example, the threat score employs an evaluation function defined by 
     
       
         
           
             
               
                 
                   
                     
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     In the above evaluation function, s REF  and r REF  are predetermined in accordance with safety standards of a vehicle manufacturer or the like. For example, s REF  and r REF  may be varied dynamically according to the traffic situation obtained by information from an infrastructure system such as dedicated short range communications (DSRC). 
     The target selection unit  38  determines whether or not the highest threat score of the targets is at least a reference (threshold) value (Step S 38 ). If the maximum threat score of the targets is at least the reference value, then the target selection unit  38  determines whether or not the target having the highest threat score is to intersect a traveling path of the present vehicle, such as a lane of the present vehicle, with reference to the angle of the target which has previously been estimated by the distance/speed/direction estimation unit  37  (Step S 40 ). 
     If it is determined in Step S 38  that the highest threat score of the targets is lower than the reference value, or if it is determined in Step S 40  that the target having the highest threat score is not to intersect the traveling path of the present vehicle, then the threat score estimation unit  40  transfers the data (bin) of the relative distance r m  and the relative speed s m  of each target orientated on the frequency axis to the subsequent DOA estimation portion  30 . The DOA estimation portion  30  applies a standard DOA estimation process having relatively low accuracy but requiring a small amount of computation, such as digital beam forming, to the N-order data vector for each specified bin position, thereby estimating an angle of each target (Step S 42 ). As a matter of course, the DOA estimation portion  30  may use the DOA values estimated by the distance/speed/direction estimation unit  37  or by the method according to the first or second embodiment of the present invention. The DOA estimation portion  30  stores the identification information for each target which includes the relative distance, the relative speed, and the estimated angle, and updates the data previously stored in the last measurement (Step S 44 ). The threat evaluation portion  40  uses the updated data as dynamic reference values to reevaluate a threat score of each target based on data to be obtained in the next measurement. 
     If it is determined in Step S 40  that the target having the highest threat score is to intersect the traveling path of the present vehicle, then the target selection unit  38  extracts a corresponding N-order data vector, based on the data (bin) of the relative distance r m  and the relative speed s m  of the target that has been determined to have the highest threat score, and transfers the N-order data vector to the beam formation unit  39 . The beam formation unit  39  refers to the angle information (roughly estimated by the distance/speed/direction estimation unit  37 ) of the target having the highest threat score and performs beam formation in a direction toward the target (Step S 46 ). Thus, the threat score estimation portion  40  converts a data vector relating to a target having a high threat score into a beam space data and transfers it to the subsequent DOA estimation portion  30 . The DOA estimation portion  30  estimates a direction of arrival by the method according to the first or second embodiment of the present invention, which can achieve detailed estimation of DOA (Step S 48 ). When the apparatus starts new data measurement, then the same processes are repeated from Step S 30 . 
     The threat evaluation portion  40  according to the third embodiment uses signal vectors to evaluate a threat score indicative of a target&#39;s threat to the present vehicle. Based on the threat score, the estimation unit  45  of the DOA estimation portion  30  determines whether to estimate a direction of arrival for the target. Thus, by providing the threat evaluation portion  40  in the third embodiment, the importance of a target as to safety to the present vehicle can be evaluated by an index of a threat score. Depending upon the threat score of each target, an appropriate DOA estimation method can be applied in conjunction with the DOA estimation portion  30 . Accordingly, a response time of the apparatus can be optimized. 
     Fourth Embodiment 
     An apparatus according to a fourth embodiment of the present invention uses an additional index such as signal quality or a relative separation angle between targets, thereby achieving processes suitable for more practical use.  FIG. 11  is a block diagram showing a direction-of-arrival estimation apparatus  100   c  according to the fourth embodiment of the present invention. In  FIG. 11 , the direction-of-arrival estimation apparatus  100   c  includes a threat evaluation portion  50  provided between a receiving portion  20  and a DOA estimation portion  30 . The basic hardware configuration of the direction-of-arrival estimation apparatus  100   c  is substantially the same as that in the third embodiment and will not be described repetitively. 
       FIG. 12  is a flow chart showing processes performed in the direction-of-arrival estimation apparatus  100   c . The threat evaluation portion  50  is operable to perform the same processes from extraction of signal vectors (Step S 50 ) to estimation of a relative distance and a relative speed for each target (Step S 53 ) as those in the third embodiment. In this case, the rough DOA estimation using FFT-DBF or the like in the third embodiment may not necessarily be performed. 
     For functions of the threat evaluation portion  50 , new steps are added to the preceding stage of the evaluation of a threat score (Step S 56 ). The new steps include evaluation of signal quality (Step S 54 ), reference to target specification data estimated in the last measurement (Step S 66 ), and comparison of a relative separation angle referred to in Step S 66  with a reference value (Step S 68 ). 
     Newly added processes will be described. First, it is determined in Step S 54  whether or not an SN ratio of a signal vector is at least a reference (threshold) value. If the SN ratio is at least the reference value, then the process proceeds to Step S 56 . If the SN ratio is lower than the reference value, then the process proceed to Step S 66 . In Step S 66 , data on a direction of arrival are extracted from target specification data estimated in the last measurement. Then, in Step  68 , a relative separation angle of the extracted value is compared with a reference (threshold) value. If it is determined in Step S 54  that the SN ratio is at least the reference value, or if it is determined in Step  68  that the relative separation angle is smaller than the reference value, then the threat evaluation portion  50  evaluates a threat score of each target (Step S 56 ) and determines whether or not the highest threat score of the targets is at least a reference (threshold) value (Step S 58 ). The evaluation of the threat score can be performed in the same manner as in Step S 36  of the third embodiment. 
     If it is determined in Step S 58  that the highest threat score of the targets is at least the reference value, then the control is transferred from the threat evaluation portion  50  to the DOA estimation portion  30 , which performs detailed DOA estimation (Step S 60 ). If it is determined in Step S 58  that the highest threat score of the targets is lower than the reference value, or if it is determined in Step S 68  that the relative separation angle is at least the reference value, then the control is transferred from the threat evaluation portion  50  to the DOA estimation portion  30 , which performs rough DOA estimation (Step S 70 ). Thus, the threat evaluation portion  50  uses signal vectors to evaluate an SN ratio, a relative separation angle, or a threat score indicative of a target&#39;s threat to the present vehicle. Depending upon these indexes of each target, the DOA estimation portion  30  can hire an estimation method having appropriate angle resolution for an individual target. 
     The differences between the process performed in Step S 60  and the process performed in Step S 70  will be described. If the theorem of an inverse matrix is applied to RR H , then 
                             (     RR   H     )       -   1       =       ⁢       [         A       B             B   H         D         ]       -   1                   =       ⁢     [             A     -   1       +       A     -   1       ⁢       B   ⁡     (     D   -       B   H     ⁢     A     -   1       ⁢   B       )         -   1       ⁢       (       A     -   1       ⁢   B     )     H       -             -     A     -   1         ⁢       B   ⁡     (     D   -       B   H     ⁢     A     -   1       ⁢   B       )         -   1                       (     D   -       B   H     ⁢     A     -   1       ⁢   B       )       -   1       ⁢       (       A     -   1       ⁢   B     )     H               (     D   -       B   H     ⁢     A     -   1       ⁢   B       )       -   1             ]                 =       ⁢       [           A     -   1           0           0       0         ]     +                     ⁢     [               A     -   1       ⁢       B   ⁡     (     D   -       B   H     ⁢     A     -   1       ⁢   B       )         -   1       ⁢       (       A     -   1       ⁢   B     )     H       -               A     -   1       ⁢       B   ⁡     (     D   -       B   H     ⁢     A     -   1       ⁢   B       )         -   1       ⁢       (     -   I     )     H       -                   I   ⁡     (     D   -       B   H     ⁢     A     -   1       ⁢   B       )         -   1       ⁢       (       A     -   1       ⁢   B     )     H                 I   ⁡     (     D   -       B   H     ⁢     A     -   1       ⁢   B       )         -   1       ⁢       (     -   I     )     H             ]                 =       ⁢       [           A     -   1           0           0       0         ]     +                     ⁢       [             A     -   1       ⁢   B               -   I           ]     ⁢         (     D   -       B   H     ⁢     A     -   1       ⁢   B       )       -   1       ⁡     [         (       A     -   1       ⁢   B     )     H     ⁢       (     -   I     )     H       ]                     =       ⁢       [           A     -   1           0           0       0         ]     +     Π   ⁢           ⁢     Λ     -   1       ⁢     Π   H                       (   38   )               
where R 1  and R 2  are submatrices of the matrix R, R 1 R 1   H =A, R 1 R 2   H =B, R 2 R 1   H =B H , and D=R 2 R 2   H =D.
 
     From the formula (38), a core matrix Ω is given by 
     
       
         
           
             
               
                 
                   Ω 
                   = 
                   
                     
                       Π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Λ 
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         Π 
                         H 
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             RR 
                             H 
                           
                           ) 
                         
                         
                           - 
                           1 
                         
                       
                       - 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       R 
                                       1 
                                     
                                     ⁢ 
                                     
                                       R 
                                       1 
                                       H 
                                     
                                   
                                   ) 
                                 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     Estimation of a direction of arrival is performed in Step S 60  by using a core matrix Ω defined by Ω=(RR H ) −1 −[(R 1 R 1   H ) −1 |0 M×(N-2M) ; 0 (N-2M)×M |0 N-2M ]. Estimation of a direction of arrival is performed in Step S 70  by using a core matrix Ω defined by Ω=(RR H ) −1 . Since the matrix used in Step S 70  includes components spanning a signal subspace, the estimation accuracy becomes low. However, (R 1 R 1   H ) −1  can be computed relatively readily in the process for computing the matrix of Step S 70 . Therefore, the method according to the fourth embodiment can facilitate switching of the estimation accuracy. 
     As described above, according to the fourth embodiment of the present invention, detailed DOA estimation is performed in Step S 60  if the quality (SN ratio) of an echo signal is at least a reference value and the threat score of the target is at least a reference value. Rough DOA estimation is performed as in Step S 70  if the threat score of the target is so low that the detail of a position of the target is not required to be obtained, or if the SN ratio of the echo signal is so low that accurate estimation cannot be expected even with a high-loaded computation. In other words, the apparatus according to the fourth embodiment can select an appropriate operation—an operation for quick response or an operation for high accuracy—depending upon its practical use. 
     Although the preferred embodiments of the present invention has been shown and described in detail, the present invention is not limited to those embodiments. It would be apparent to those skilled in the art that many modifications and variations may be made within the scope of the appended claims of the present invention.