Patent Publication Number: US-7586437-B2

Title: Efficient methods for wideband circular and linear array processing

Description:
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
   The present invention is based upon work supported and/or sponsored by the Air Force Research Laboratory (AFRL), Rome, N.Y., under contract No. FA8750-06-C-0117. 

   FIELD OF THE INVENTION 
   This invention relates to improved methods and apparatus concerning wideband circular and linear array processing. 
   BACKGROUND OF THE INVENTION 
   Wideband systems are used for improved range resolution without compromising on Doppler resolution. However wideband arrays introduce distortion due to the wide bandwidth introduced, and for problems involving one-dimensional angular search methods have been developed to address this difficulty. [Y. Yang, C. Sun, and C. Wan, “Theoretical and Experimental Studies on Broadband Constant Beamwidth Beamforming for Circular Array”, Proceedings of OCEANS 2003, Vol. 3. pp. 1647-1653, September 2003]. Circular arrays allow open access around the entire 360 degrees and in addition scanning in the elevation direction become possible. Linear arrays are also often used in radar and communication problems. 
   However, processing the data becomes often difficult because of the wideband nature of the problem. Traditionally the wideband data is partitioned into several narrowband data segments, and then they are processed separately or together by stacking up the various frequency components in a vector format. In this later approach, spatial domain, and with temporal domain data when stacked up together with frequency domain results in three-dimensional (3-D) data that adds severe computational burden for processing. [A. O. Steinhardt and N. B. Pulsone “Subband STAP processing, the Fifth Generation,” Proceedings of the Sensor Array and Multichannel Signal Processing Workshop, Cambridge, Mass., March 2000]. In this context, a new signal processing strategy that separates the frequency and the angular variables into two components is investigated in the joint azimuth-elevation domain, so that efficient algorithms can be designed to process the entire wideband data simultaneously. 
   SUMMARY OF THE INVENTION 
   One or more embodiments of the present invention provide signal processing methods for a wide-band circular electronically scanned array (CESA) and a wideband linear electronically scanned array (LESA) for use in surveillance and communications applications. 
   One or more embodiments of the present invention provide an efficient receiver processing strategy for electronically scanned circular array or linear array that operate in a wideband radio frequency (RF) spectrum. The instantaneous bandwidth, sidelobe levels, and beam agility, etc. for the electronically scanned circular array or linear array or one or more embodiments of the present invention are sufficient for both communications techniques and extremely high-resolution synthetic aperture radar (SAR) and ground moving target indicator (GMTI) radar modes. Control of the electronically scanned circular array or linear array is typically sufficient to form multiple beams (possibly for separate, simultaneous modes) around the entire three hundred sixty degrees. 
   In accordance with a method of an embodiment of the present invention, a circular array is analyzed first in the wideband context for simultaneous beam steering both in the azimuth and elevation directions. New signal processing schemes are formulated to exploit the wideband nature of the problem by reducing the computational burden. A new signal processing strategy that separates the frequency and the angular variables into two separate components is investigated in the joint azimuth-elevation domain, and efficient algorithms are designed to process by frequency compensating or focusing the entire wideband data in the frequency domain simultaneously to a single frequency band. This results in new wideband space-time adaptive processing (STAP) methods for a circular array. The frequency compensating or focusing method is also extended to the wideband linear array case for both frequency focusing and adaptive processing. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  shows a diagram of circular array apparatus in accordance with an embodiment of the present invention; 
       FIG. 2A  shows a top view of non-uniform main beam width for a gain function of a wideband circular array, in accordance with an embodiment of the present invention, with twenty-five sensors; 
       FIG. 2B  shows a side view of the non-uniform main beam width for the gain function of  FIG. 2A ; 
       FIG. 3  shows a diagram of a Bessel function as a function of β for three different order values; 
       FIG. 4  shows a diagram of a focused wideband gain pattern for a circular array in the azimuth direction with elevation angle fixed at ninety degrees; 
       FIG. 5A  shows a diagram of a wideband gain unfocused beam pattern for a circular array in the elevation direction with azimuth angle fixed at zero degrees; 
       FIG. 5B  shows a diagram of a wideband gain focused beam pattern for a circular array in the elevation direction with azimuth angle fixed at zero degrees; 
       FIG. 6  shows a block diagram of a frequency compensating or focusing method on the circular array data; 
       FIG. 7  shows a diagram of an unfocused output signal to interference plus noise ratio output (SINR) as a function of frequency and azimuth angle θ for a wideband circular array with twelve sensors; 
       FIG. 8  shows a diagram of a focused output signal to interference plus noise ratio output (SINR) in accordance with one or more embodiments of the present invention as a function of azimuth angle θ for a wideband circular array such as in  FIG. 1  with twelve sensors; 
       FIG. 9A  shows a diagram of an unfocused signal to interference plus noise ratio output (SINR) in the joint azimuth-Doppler domain at a first frequency subband for a wideband circular array such as in  FIG. 1  with twelve sensors and fourteen pulses. Injected target located at zero azimuth angle and ninety degree elevation angle is moving with velocity of twenty-five meters/second; 
       FIG. 9B  shows a diagram of an unfocused signal to interference plus noise ratio output (SINR) in the joint azimuth-Doppler domain at a second frequency subband for a wideband circular array with twelve sensors and fourteen pulses. Injected target located at zero azimuth angle and ninety degree elevation angle is moving with velocity of twenty-five meters/second; 
       FIG. 10  shows a diagram of the focused output SINR in the joint azimuth-Doppler domain in accordance with one or more embodiments of the present invention for a wideband circular array such as in  FIG. 1  using twelve sensors and fourteen pulses. Injected target located at zero azimuth angle and ninety degree elevation angle is moving with velocity of twenty-five meters/second; 
       FIG. 11  shows a diagram of a linear array apparatus in accordance with an embodiment of the present invention; 
       FIG. 12A  shows a top-view diagram of the focused output SINR in the joint azimuth-Doppler domain in accordance with one or more embodiments of the present invention for a wideband linear array such as in  FIG. 11  using fourteen sensors and sixteen pulses. Injected target located at zero azimuth angle and ninety degrees elevation angle is moving with velocity 40 meters/second. Clutter to noise ratio is 40 decibels; 
       FIG. 12B  shows a side-view diagram of the focused output SINR in the joint azimuth-Doppler domain in accordance with one or more embodiments of the present invention for a wideband linear array such as in  FIG. 11  using fourteen sensors and sixteen pulses. Injected target located at zero azimuth angle and ninety degrees elevation angle is moving with velocity 40 meters/second. Clutter to noise ratio is 40 decibels; 
       FIG. 13  shows a diagram of a flowchart of a method in accordance with an embodiment of the present invention; and 
       FIG. 14  shows a diagram of an apparatus for use in accordance with an embodiment of the present invention. 
   

   DETAILED DESCRIPTION OF THE DRAWINGS 
     FIG. 1  shows a diagram of a circular array apparatus  1  in accordance with an embodiment of the present invention. The apparatus  1 , where  1  includes a circular element,  2  of radius r on which is located a plurality of sensors  4   a - 4 N, each shown as a dot or point on the circular element  2   2 . Sensor  4   a  refers to the first reference sensor,  4   n  to the n th  sensor and  4 N to the last sensor. The circular element  2  has a center, origin, or reference point “O”.  FIG. 1  shows a “z-axis” labeled as “OZ”, and x and y axes labeled as OX and OY.  FIG. 1  shows an angle θ between the reference line x axis OX and the line OB through the plane of the circular element  2 , and an angle θ n  that refers to the n th  sensor,  4   n , also with reference to the x axis OX. Similarly φ refers to the angle between the z-axis OZ and the line OA. 
     FIG. 1  shows a circular array  2  with N sensors (sensors  4   a - 4 N) such as that in a helicopter mounted array. The sensors  4   a - 4 N are uniformly placed around the circumference of the circular element  2  at a radial distance r from the reference point O ( FIG. 1 ). In  FIG. 1 , O denotes the origin, OP the reference direction along the first sensor (also the x-axis or OX), or sensor  4   a , and OA denotes the normal to an arbitrary wavefront of interest that generates the plane wave AC when it passes through the n th  receiver  4   n  located at point C. A perpendicular line AB is drawn from the point A to the plane of the circle of the circular element  2  to meet the plane at point B.  FIG. 1  also shows lines OB, BC and OC joined into a triangle. In the configuration of  FIG. 1 : 
                     ∠   ⁢           ⁢   POB     =   θ     ,           ⁢       ∠   ⁢           ⁢   QOA     =   ϕ     ,           ⁢       ∠   ⁢           ⁢   POC     =       θ   n     =       2   ⁢           ⁢     π   ⁡     (     n   -   1     )         N         ,     
     ⁢     n   =   1     ,   2   ,     …   ⁢           ⁢   N             (   1   )               
so that ∠BOC=θ n −θ.
 
   The unknown distance OA=x is of interest, since x/c represents the time delay of the wavefront AC through the n th  sensor at C with respect to the reference point O. Here c italicize represents the speed of light. From the right angled triangle AOB, we have
 
OB=x sin φ, AB=x cos φ, OC=r,  (2)
 
so that from ΔOBC, the law of cosines gives
 
                         BC   2     =       OB   2     +     OC   2     -     2   ⁢     OB   ·   OC   ·     cos   ⁡     (     θ   -     θ   n       )                         =         x   2     ⁢     sin   2     ⁢   ϕ     +     r   2     -     2   ⁢   rx   ⁢           ⁢   sin   ⁢           ⁢   ϕ   ⁢           ⁢       cos   ⁡     (     θ   -     θ   n       )       .                       (   3   )               
Also from the right angled triangle ABC, we get
 
                         A   ⁢           ⁢     C   2       =       AB   2     +     BC   2                   =         x   2     ⁢     cos   2     ⁢   ϕ     +     BC   2                   =       x   2     +     r   2     -     2   ⁢   rx   ⁢           ⁢   sin   ⁢           ⁢   ϕ   ⁢           ⁢       cos   ⁡     (     θ   -     θ   n       )       .                       (   4   )               
Finally from the right angled triangle OAC, we get
   OC   2   =OA   2   +AC   2   (5) or   r   2   =x   2 +( x   2   +r   2 −2 r x  sin φ cos(θ−θ n ))  (6) 
or we get the desired expression
   x=r  sin φ cos(θ−θ n ),  n= 1, 2 . . .  N.   (7) 
   This gives the signal x n (t) at the n th  sensor with respect to the reference signal s(t) to be 
                         x   n     ⁡     (   t   )       =     s   ⁡     (     t   -     τ   n       )         ,           ⁢     n   =   1     ,   2   ,     …   ⁢           ⁢   N       ⁢     
     ⁢   where           (   8   )                 τ   n     =       x   c     =       r   c     ⁢   sin   ⁢           ⁢   ϕ   ⁢           ⁢     cos   ⁡     (     θ   -     θ   n       )                   (   9   )               
and s(t) represents the wideband transmit signal at the origin. Notice that equations (8) and (9) account for both the azimuth angle θ and the elevation direction φ simultaneously. From equations (8)-(9), the wideband clutter data received from the entire field of view will be of the form
 
                       x   n     ⁡     (   t   )       =       ∑   i     ⁢       ∑   k     ⁢       α     i   ,   k       ⁢     s   ⁡     (     t   -       τ   n     ⁡     (     i   ,   k     )         )               ⁢     
     ⁢   where           (   10   )                   τ   n     ⁡     (     i   ,   k     )       =       r   c     ⁢   sin   ⁢           ⁢     ϕ   k     ⁢     cos   ⁡     (       θ   i     -     θ   n       )                 (   11   )               
and α i,k  represents the scatter return from azimuth location θ i  and elevation direction φ k . In addition, when a target is also present in some unknown direction (θ o , φ o ), the data has the form
 
   
     
       
         
           
             
               
                 
                   
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   Clearly from equation (14), any beamforming application must address the frequency dependent nature of the spatial steering vector  a (ω, φ, θ). The steering vector in equation (14) gives rise to the following gain function 
                           G   ⁡     (     ω   ,   ϕ   ,   θ     )       =              1   N     ⁢       w   _     ω   *     ⁢       a   _     ⁡     (     ω   ,   ϕ   ,   θ     )              2                 =              1   N     ⁢       ∑     n   =   1     N     ⁢       w   n   *     ⁢     ⅇ       -   j     ⁢           ⁢   ω   ⁢     r   c     ⁢   sin   ⁢           ⁢   ϕ   ⁢           ⁢     cos   ⁡     (     θ   -     θ   n       )                      2             ⁢     
     ⁢   where           (   15   )                     w   _     ω     =       [       w   1     ,     w   2     ,     …   ⁢           ⁢     w   N         ]     T       ,           ⁢       w   n     =     ⅇ       -   j     ⁢           ⁢   ω   ⁢     r   c     ⁢   sin   ⁢           ⁢     ϕ   o     ⁢     cos   ⁡     (       θ   o     -     θ   n       )                     (   16   )               
represents the weight vector to focus the array pattern to the azimuth angle θ o  and elevation angle φ o . Here onwards, for vectors and matrices such as A, the symbols A T  and A* represent the transpose and the complex conjugate transpose of A respectively.
 
     FIG. 2A  shows a diagram  100  of a top view of non-uniform main beam width for a gain function of a wideband circular array similar to the circular array in  FIG. 1 , in accordance with an embodiment of the present invention, with twenty-five sensors. In this example, the elevation angle φ of the wavefront projected by the target is fixed at ninety degrees and the frequency range is between 235 MHz (Megahertz) and 635 MHz (Megahertz). The diagram  100  includes sections  102 ,  104  and  106 . Section  102  shows the frequency sensitive mainbeam gain function versus azimuth angle θ shown in  FIG. 1 , section  104  shows the sidelobe gain function azimuth angle θ shown in  FIG. 1 , and section  106  shows the scaling function used here with lighter region representing lower gain levels in dB (decibels). 
     FIG. 2B  shows a diagram  200  of a side view illustrating the non-uniform main beam width for the gain function of  FIG. 2A . The diagram  200  includes sections  202 ,  204  and  206 . Section  202  shows the frequency sensitive main-beam gain function versus azimuth angle θ shown in  FIG. 1 , section  204  shows the side-lobe gain function azimuth angle θ in  FIG. 1 , and section  206  shows the scaling function used here with the lighter region representing lower gain levels in dB (decibels). In the example of  FIG. 2B , the elevation angle is fixed at ninety degrees and the frequency range is between 235 MHz (Megahertz) and 635 MHz (Megahertz). 
   Notice that the mainbeam width is frequency dependent since it is narrow at high frequencies and wider at lower frequencies, and this phenomenon is undesirable since the array focused along a specific direction can project different gain functions depending on the frequency. 
   To address this frequency dependent issue, it is necessary to do prior processing so as to re-focus various frequency components into a reference frequency component. Towards this, using the identity 
                   sin   ⁢           ⁢   ϕ   ⁢           ⁢     cos   ⁡     (     θ   -     θ   n       )         =         sin   ⁡     (     ϕ   +   θ   -     θ   n       )       +     sin   ⁡     (     ϕ   -   θ   +     θ   n       )         2             (   17   )               
in equations (13)-(14) we get
 
                       X   n     ⁡     (   ω   )       =       S   ⁡     (   ω   )       ⁢     ⅇ       -   j     ⁢       ω   ⁢           ⁢   r       2   ⁢   c       ⁢     sin   ⁡     (     ϕ   +   θ   -     θ   n       )           ⁢     ⅇ       -   j     ⁢       ω   ⁢           ⁢   r       2   ⁢   c       ⁢     sin   ⁡     (     ϕ   -   θ   +     θ   n       )               ,           ⁢     n   -   1     ,   2   ,     …   ⁢           ⁢     N   .               (   18   )               
Towards simplifying equation (18) further, consider the periodic term
 
                     ⅇ       -   j     ⁢       ω   ⁢           ⁢   r       2   ⁢   c       ⁢     sin   ⁡     (     ϕ   +   θ   -     θ   n       )           ⁢     =   Δ     ⁢     ⅇ       -   j     ⁢           ⁢     β   ⁡     (   ω   )       ⁢   sin   ⁢           ⁢   ψ         ⁢     
     ⁢   where           (   19   )                   β   ⁡     (   ω   )       =       ω   ⁢           ⁢   r       2   ⁢   c         ,           ⁢     ψ   =     ϕ   +   θ   -       θ   n     .                 (   20   )               
Fourier series expansion of equation (19) gives the identity
 
                     ⅇ       -   j     ⁢           ⁢     β   ⁡     (   ω   )       ⁢   sin   ⁢           ⁢   ψ       =       ∑     k   =     -   ∞         +   ∞       ⁢         J   k     ⁡     (   β   )       ⁢     ⅇ       -   j     ⁢           ⁢   k   ⁢           ⁢   ψ             ⁢     
     ⁢   where           (   21   )                         J   k     ⁡     (   β   )       =       1     2   ⁢           ⁢   π       ⁢       ∫     -   π     π     ⁢       ⅇ     -     j   ⁡     (       β   ⁢           ⁢   sin   ⁢           ⁢   ψ     -     k   ⁢           ⁢   ψ       )           ⁢           ⁢     ⅆ   ψ                       =       1   π     ⁢       ∫   0   π     ⁢       cos   ⁡     (       β   ⁢           ⁢   sin   ⁢           ⁢   ψ     -     k   ⁢           ⁢   ψ       )       ⁢           ⁢     ⅆ   ψ                         (   22   )               
represents the Bessel functions of the first kind and k th  order [G. N. Watson, “A Treatise on the Theory of Bessel Functions”, Second Edition, Cambridge University Press, 1952, Pages 19-22]. Substituting (21)-(22) into (19) we get
 
                         ⅇ       -   j     ⁢       ω   ⁢           ⁢   r       2   ⁢   c       ⁢     sin   ⁡     (     ϕ   +   θ   -     θ   n       )           =       ∑     k   =     -   ∞         +   ∞       ⁢         J   k     ⁡     (     β   ⁡     (   ω   )       )       ⁢     ⅇ       -   j     ⁢           ⁢     k   ⁡     (     ϕ   +   θ   -     θ   n       )                           =       ∑     k   =     -   ∞         +   ∞       ⁢         J   k     ⁡     (     β   ⁡     (   ω   )       )       ⁢     ⅇ     j   ⁢           ⁢   k   ⁢           ⁢     θ   n         ⁢     ⅇ       -   j     ⁢           ⁢     k   ⁡     (     ϕ   +   θ     )                           =       ∑     k   =     -   ∞         +   ∞       ⁢         A     n   ,   k       ⁡     (   ω   )       ⁢     ⅇ       -   j     ⁢           ⁢     k   ⁡     (     ϕ   +   θ     )                             (   23   )               
where we define
   A   n,k (ω)= J   k (β(ω)) e   −jkθ     n     , k=−L,− ( L− 1), . . . 0, 1, . . . , L.   (24) 
Similarly
 
                   ⅇ       -   j     ⁢       ω   ⁢           ⁢   r       2   ⁢           ⁢   c       ⁢     sin   ⁡     (     ϕ   -   θ   +     θ   n       )           =       ∑     k   =     -   ∞         +   ∞       ⁢           ⁢         B     n   ,   k       ⁡     (   ω   )       ⁢     ⅇ       -   j     ⁢           ⁢     k   ⁡     (     ϕ   -   θ     )                       (   25   )               
where we define
   B   n,k (ω)= J   k (β(ω)) e   −jkθ     n     , k=−L,− ( L− 1), . . . 0, 1, . . . ,  L.   (26) 
   The Bessel function coefficients in equations (24)-(26) decay down rapidly and the summations in equations (23) and (25) can be replaced with a finite number of terms (such as ten to fifteen terms). 
     FIG. 3  shows a diagram  300  of a Bessel function as a function of β (along the x axis) for three different order index k. The diagram  300  includes sections  302 ,  304  and  306 . Section  302  shows the solid line for order index value of zero, section  304  shows the dotted line for order index value of 5, and section  306  shows the dashed line for order index value of 10. Bessel functions are well documented in the literature, and can be readily precomputed and stored in a computer processor. 
   Substituting equations (23)-(26) into equations (18)-(19) we get (with S(ω)≡1 in (18)) 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           X 
                           n 
                         
                         ⁡ 
                         
                           ( 
                           ω 
                           ) 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             - 
                             j 
                           
                           ⁢ 
                           
                             
                               ω 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               r 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               c 
                             
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 ϕ 
                                 - 
                                 θ 
                                 + 
                                 
                                   θ 
                                   n 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             
                               - 
                               L 
                             
                           
                           
                             + 
                             L 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               A 
                               
                                 n 
                                 , 
                                 k 
                               
                             
                             ⁡ 
                             
                               ( 
                               ω 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               
                                 - 
                                 j 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 k 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ϕ 
                                     + 
                                     θ 
                                   
                                   ) 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 
                                   - 
                                   L 
                                 
                               
                               
                                 + 
                                 L 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 
                                   B 
                                   
                                     n 
                                     , 
                                     m 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   ω 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       m 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           ϕ 
                                           - 
                                           θ 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 27 
                 ) 
               
             
           
         
       
     
   
   From equations (24) and (26), the coefficients {A n,k (ω)} and {B n,k (ω)} are frequency dependent. Using equation (27) in equation (14), we obtain the circular array output vector to be: 
                         X   ⁡     (   ω   )       =       ⁢     a   ⁡     (     ω   ,   ϕ   ,   θ     )                   =       ⁢       [         ⋮               ∑     k   =     -   L         +   L       ⁢           ⁢         A     n   ,   k       ⁡     (   ω   )       ⁢     ⅇ       -   j     ⁢           ⁢     k   ⁡     (     ϕ   +   θ     )                       ⋮         ]     ·     [         ⋮               ∑     k   =     -   L         +   L       ⁢           ⁢         B     n   ,   k       ⁡     (   ω   )       ⁢     ⅇ       -   j     ⁢           ⁢     k   ⁡     (     ϕ   -   θ     )                       ⋮         ]                   =       ⁢       A   ⁡     (   ω   )       ⁢       s   1     ·     B   ⁡     (   ω   )         ⁢     s   2                     (   28   )               
where the symbol ∘ represents the element wise Schur-Hadamard product that is well known in the literature, and
 
                     A   ⁡     (   ω   )       =     (             A     1   ,     -   L         ⁡     (   ω   )               A     1   ,     -     (     L   -   1     )           ⁡     (   ω   )           …           A     1   ,   L       ⁡     (   ω   )                   A     2   ,     -   L         ⁡     (   ω   )               A     2   ,     -     (     L   -   1     )           ⁡     (   ω   )           …           A     2   ,   L       ⁡     (   ω   )               ⋮       ⋮       ⋰       ⋮               A     N   ,     -   L         ⁡     (   ω   )               A     N   ,     -     (     L   -   1     )           ⁡     (   ω   )           …           A     N   ,   L       ⁡     (   ω   )             )       ,           (   29   )                 B   ⁡     (   ω   )       =     (             B     1   ,     -   L         ⁡     (   ω   )               B     1   ,     -     (     L   -   1     )           ⁡     (   ω   )           …           B     1   ,   L       ⁡     (   ω   )                   B     2   ,     -   L         ⁡     (   ω   )               B     2   ,     -     (     L   -   1     )           ⁡     (   ω   )           …           B     2   ,   L       ⁡     (   ω   )               ⋮       ⋮       ⋰       ⋮               B     N   ,     -   L         ⁡     (   ω   )               B     N   ,     -     (     L   -   1     )           ⁡     (   ω   )           …           B     N   ,   L       ⁡     (   ω   )             )             (   30   )               
where A n k (ω) and B n k (ω) are as defined in equations (24) and (26) respectively. Further let
     s   (θ)=[ e   jLθ   , . . . e   jθ , 1 , e   −jθ   , . . . e   −jLθ ] T .  (31) and     s     1   = s   (φ+θ),    s     2   = s   (φ−θ).  (32) 
   Notice that in equation (28) the frequency dependent steering vector has been broken up into two frequency dependent matrix components A(ω) and B(ω) and two frequency independent steering vector components  s (φ+θ) and  s (φ−θ). 
   To separate out frequency components exclusively, a new strategy in accordance with one or more embodiments of the present invention is developed here as follows: Towards this, one or more embodiments of the present invention need to make use of a matrix result involving the Khatri-Rao product [C. G. Khatri, and C. R. Rao, “Solutions to Some Functional Equations and Their Applications to Characterization of Probability Distributions,” Sankhya: The Indian J. Stat., Series A, 30, pp. 167-180, 1968]: If A=( a   1    a   2  . . .  a   n ), B=( b   1    b   2  . . .  b   n ) represent two matrices then their Khatri-Rao product A⊙B is given by
 
 A⊙B= (   a     1       b     1      a     2       b     2    . . .  a     n       b     n )  (33)
 
where  a   1     b   1  represents the well known Kronecker product of vectors  a   1  and  b   1  [C. R. Rao and M. B. Rao, Matrix Algebra and its Applications to Statistics and Econometrics, World Scientific, Singapore, 1998].
 
   Let A and B represent two m×n matrices and let  a  and  b  represent two n×1 vectors. Then the important identity [S. U. Pillai, K. Y. Li, and B. Himed, “Space Based Radar—Theory and Applications”, Chapter 1, Pages 18-25, McGraw Hill, New York, To be published in December 2007] below is obtained
 
A a ∘B b =(A T ⊙B T ) T ( a     b ).  (34)
 
In equation (34), ∘ represents the element wise Schur-Hadamard product, and ⊙ represents the Khatri-rao product cited in equation (33). Using equation (34) in equation (28) the following is determined
 
                         X   ⁡     (   ω   )       =       ⁢       a   ⁡     (     ω   ,   ϕ   ,   θ     )       ≃       A   ⁡     (   ω   )       ⁢       s   1     ·     B   ⁡     (   ω   )         ⁢     s   2                     =       ⁢           (         A   T     ⁡     (   ω   )       ⊙       B   T     ⁡     (   ω   )         )     T     ⁢     (       s   1     ⊗     s   2       )       =       P   ⁡     (   ω   )       ⁢     (       s   1     ⊗     s   2       )                       (   35   )             where                           P   ⁡     (   ω   )       =       (         A   T     ⁡     (   ω   )       ⊙       B   T     ⁡     (   ω   )         )     T             (   36   )               
is N×(2L+1) 2 . Notice that in equation (35), the frequency dependent steering vector  a (ω, φ, θ) has been separated into a frequency dependent matrix P(ω), and a vector  s   1     s   2  that does not depend upon the frequency. Equations (35)-(36) can be used to refocus the steering vectors  a (ω k , φ, θ), k=1, 2, . . . K to any reference frequency ω o  as follows: From equations (35)-(36), the following is determined
     a   (ω, φ, θ)= P (ω)(   s     1       s     2 )  (37) 
so that
 
                         A   ⁡     (     ω   ,   ϕ   ,   θ     )       =       ⁢     [           a   ⁡     (       ω   1     ,   ϕ   ,   θ     )                 a   ⁡     (       ω   2     ,   ϕ   ,   θ     )               ⋮             a   ⁡     (       ω   K     ,   ϕ   ,   θ     )             ]                 =       ⁢         [           P   ⁡     (     ω   1     )                 P   ⁡     (     ω   2     )               ⋮             P   ⁡     (     ω   K     )             ]     ⁢     (       s   1     ⊗     s   2       )       ⁢     =   Δ     ⁢       ⁢       F   ⁡     (   ω   )       ⁢     (       s   1     ⊗     s   2       )                       (   38   )             where                           F   ⁡     (   ω   )       =     [           P   ⁡     (     ω   1     )                 P   ⁡     (     ω   2     )               ⋮             P   ⁡     (     ω   K     )             ]             (   39   )               
is of size NK×(2L+1) 2 . This gives
   P (ω o )( F *( ω ) F ( ω )) −1   F *( ω )   A   ( ω , φ, θ)≃ P (ω o )(   s     1       s     2 )=   a   (ω o , φ, θ).  (40) 
In rank deficient situations, the inversion in equation (40) is to be interpreted as the pseudo inverse. In such cases, to facilitate the inversion of F*( ω )F( ω ) in equation (40), a small diagonal loading additive term εI, ε&gt;0 may be added to it to generate (F*( ω )F( ω )+εI) prior to inversion. Here I represents the identity matrix of appropriate size. Observe that different frequency components ω o  in equation (38) have been focused into one signal component in equation (40). From equation (40), define the frequency compensating or focusing operator:
   T (ω o ,  ω )= P (ω o )( F *( ω ) F ( ω )) −1   F *( ω )  (41) 
so that equation (40) reads
   T (ω o ,  ω )   A   ( ω , φ, θ)≃   a   (ω 0 , φ, θ).  (42) 
In other words, the matrix T(ω o ,  ω ) refocuses the various frequency terms in  A ( ω , φ, θ) to a single reference frequency ω o . The new gain pattern corresponding to (42) is given by
   G   1 (φ,θ)=| w   ω     o     *T (ω o , ω )   A   ( ω ,φ,θ)| 2 .  (43) 
where  w   ω     o    is as defined in (16) with ω=ω o .
 
     FIG. 4  shows a diagram  400  of a focused wideband gain pattern for a circular array along with the gain pattern at a reference frequency of 635 MHz, such as circular array  2 —in the azimuth direction with elevation angle fixed at ninety degrees. In the example of  FIG. 4 , a wideband signal is split into one hundred frequency bands, each with a bandwidth of 4 MHz, and focused to the reference frequency of 635 MHz using fifteen Bessel function terms. The diagram  400  includes sections  402  and  404 . Section  402  shows the solid line corresponding to the focused array gain pattern and section  404  shows the dotted line corresponding to the reference frequency array gain pattern. Observe that both these diagrams coincide, indicating the effectiveness of the focusing method. 
     FIG. 4  shows the refocused gain pattern in the azimuth direction corresponding to the twenty-five element circular array referred in  FIG. 2  that has one hundred distinct frequency bands all refocused to the frequency (635 MHz) along with the gain pattern at a reference frequency of 635 MHz. Here the elevation angle is fixed at π/2. 
   Notice that the focused wideband gain pattern in  FIG. 4  has the same array gain across frequency and can be used simultaneously to process all frequency components. The treatment above considers the general case for an arbitrary azimuth angle θ and elevation angle φ, and hence a platform, such as an airborne radar platform holding a circular array, such as  2  in  FIG. 1 , can be held steady while the array  2  electronically sweeps an entire joint azimuth-elevation domain. 
     FIG. 5A  shows a diagram  500  of frequency in MHz versus elevation angle, of a wideband gain unfocused beam pattern for a twenty-five element circular array referred in  FIG. 2 , such as circular array  2  with azimuth angle fixed at zero degrees.  FIG. 5A  shows an unfocused beam pattern with frequency dependent gain patterns. The diagram  500  includes sections  502 ,  504  and  506 . Section  502  shows the frequency dependent main lobe region, section  504  shows the side lobe region and section  506  shows the scaling function used here with the lighter region representing lower gain levels in dB (decibels). 
     FIG. 5B  shows a diagram  600  of array gain for a frequency focused array shown in  FIG. 5A  versus elevation angle.  FIG. 5B  shows a diagram of a wideband gain focused beam pattern for a circular array in the elevation direction with azimuth angle fixed at zero degrees.  FIG. 5B  shows a focused uniform gain pattern across all frequencies. All frequencies are focused to 635 MHz using fifteen Bessel function terms. The diagram  600  includes sections  602  and  604 . Section  602  shows the solid line representing the focused gain function in the elevation domain and section  604  shows the dotted line representing the reference frequency gain function at a frequency of 635 MHz. 
   The important breakdown in equation (35) when applied to the clutter data x n (t) in equation (12) is effective in making the clutter data from all azimuth-elevation locations refocus at a common frequency point using the same frequency focusing operator for all locations. 
   To see this, from one or more embodiments of the present invention, the Fourier transform of the total received data at the n th  sensor, such as  4   n  in  FIG. 1 , in equation (12) gives: 
                     X   n     ⁡     (   ω   )       =         S   ⁡     (   ω   )       ⁢     ⅇ       -   j     ⁢           ⁢   ω   ⁢     r   c     ⁢   sin   ⁢           ⁢     ϕ     k   o       ⁢     cos   ⁡     (       θ     i   o       -     θ   n       )             +       S   ⁡     (   ω   )       ⁢       ∑   i     ⁢           ⁢       ∑   k     ⁢           ⁢       α   ik     ⁢       ⅇ       -   j     ⁢           ⁢   ω   ⁢     r   c     ⁢   sin   ⁢           ⁢     ϕ   k     ⁢     cos   ⁡     (       θ   i     -     θ   n       )           .                       (   44   )               
Following equation (14), the transform of the total array output vector  X (ω) takes the form
 
                         X   ⁡     (   ω   )       =       ⁢     [             X   1     ⁡     (   ω   )                   X   2     ⁡     (   ω   )               ⋮               X   N     ⁡     (   ω   )             ]                 =       ⁢         S   ⁡     (   ω   )       ⁢     a   ⁡     (     ω   ,     ϕ     k   o       ,     θ     i   o         )         +       S   ⁡     (   ω   )       ⁢       ∑   i     ⁢           ⁢       ∑   k     ⁢           ⁢       α   ik     ⁢     a   ⁡     (     ω   ,     ϕ   k     ,     θ   i       )                               (   45   )               
where we have used equation (44) for X n (ω), n=1, 2, . . . N. Following equation (37), each frequency dependent steering vector  a (ω, φ k , θ i ) in equation (45) can be written as:
     a   (ω, φ k , θ i )= P (ω)(   s     1 ( i,k )     s     2 ( i,k )= P (ω)   q   ( i,k ),  (46) where     s     1 ( i,k )=   s   (φ k +θ i ),  (47)     s     2 ( i,k )=   s   (φ k −θ i )  (48) and     q   ( i,k )=   s     1 ( i,k )     s     2 ( i,k )  (49) 
where  s (i,k) is defined in equation (31). From equations (45)-(49), we get:
 
   
     
       
         
           
             
               
                 
                   
                     X 
                     ⁡ 
                     
                       ( 
                       
                         ω 
                         m 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         P 
                         ⁡ 
                         
                           ( 
                           
                             ω 
                             m 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         S 
                         ⁡ 
                         
                           ( 
                           
                             ω 
                             m 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         q 
                         ⁡ 
                         
                           ( 
                           
                             
                               i 
                               o 
                             
                             , 
                             
                               k 
                               o 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       P 
                       ⁢ 
                       
                         ( 
                         
                           ω 
                           m 
                         
                         ) 
                       
                       ⁢ 
                       
                         S 
                         ⁡ 
                         
                           ( 
                           
                             ω 
                             m 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           i 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∑ 
                             k 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               α 
                               ik 
                             
                             ⁢ 
                             q 
                             ⁢ 
                             
                               ( 
                               
                                 i 
                                 , 
                                 k 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
                 , 
                 
                   
 
                 
                 ⁢ 
                 
                   m 
                   = 
                   1 
                 
                 , 
                 2 
                 , 
                 
                   … 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     K 
                     . 
                   
                 
               
             
             
               
                 ( 
                 50 
                 ) 
               
             
           
         
       
     
   
   Note that the frequency dependent part P(ω m ) in equation (50) is the same for each term in the summation and hence it can be pulled outside the summation. 
   At this stage the unfocused received data vector in equation (50) at various frequencies can be stacked together as in equation (38) to generate the NK×1 vector 
                           Y   _     ⁡     (     ω   _     )       =       ⁢       (             X   _     ⁡     (     ω   1     )                   X   _     ⁡     (     ω   2     )               ⋮               X   _     ⁡     (     ω   K     )             )     =           (           P   ⁡     (     ω   1     )                 P   ⁡     (     ω   2     )               ⋮             P   ⁡     (     ω   K     )             )       ︸     F   ⁡     (     ω   _     )           ⁢       q   _     ⁡     (       i   o     ,     k   o       )         +                       ⁢         (           P   ⁡     (     ω   1     )                 P   ⁡     (     ω   2     )               ⋮             P   ⁡     (     ω   K     )             )       ︸     F   ⁡     (     ω   _     )           ⁢       ∑   i     ⁢           ⁢       ∑   k     ⁢           ⁢       α   ik     ⁢       q   _     ⁡     (     i   ,   k     )                             =       ⁢         F   ⁡     (     ω   _     )       ⁢       q   _     ⁡     (       i   o     ,     k   o       )         +       F   ⁡     (     ω   _     )       ⁢       ∑   i     ⁢           ⁢       ∑   k     ⁢           ⁢       α   ik     ⁢       q   _     ⁡     (     i   ,   k     )                   ,                 (   51   )               
where F( ω ) is as defined in equation (39) and we have assumed the transmit signal s(t) S(ω) to be flat in the frequency region of interest. Processing as in equations (38)-(41), we can focus all these data vectors to a single frequency ω o  using the focusing matrix T(ω o ,  ω ) in equation (41). This gives
 
                         Z   ⁡     (     ω   o     )       =       ⁢         T   ⁡     (       ω   o     ,   ω     )       ⁢     Y   ⁡     (   ω   )         ≃         P   ⁡     (     ω   o     )       ⁢     q   ⁡     (       i   o     ,     k   o       )         +                       ⁢       ∑   i     ⁢           ⁢       ∑   k     ⁢           ⁢       α   ik     ⁢     P   ⁡     (     ω   o     )       ⁢     q   ⁡     (     i   ,   k     )                         ≃       ⁢       a   ⁡     (       ω   o     ,     ϕ     k   o       ,     θ     i   o         )       +       ∑   i     ⁢           ⁢       ∑   k     ⁢           ⁢       α   ik     ⁢       a   ⁡     (       ω   o     ,     ϕ   k     ,     θ   i       )       .                           (   52   )             Let                           c   ⁡     (       ω   o     ,   n     )       ↔       ∑   i     ⁢           ⁢       ∑   k     ⁢           ⁢       α   ik     ⁢     a   ⁡     (       ω   o     ,     ϕ   k     ,     θ   i       )                     (   53   )               
represent the focused clutter and interference data vector in equation (52).
 
     FIG. 6  shows a block diagram  700  of a frequency focusing method on circular array wideband data x(nT) output from sensors of  FIG. 1  and sent to a computer processor, such as the computer processor  1701  shown in  FIG. 14 . The computer processor subjects the data to a discrete Fourier transform (DFT) at step  702 . The transformed data is then subjected to frequency focusing at step  704  by the computer processor. The data is then subjected to narrowband processing by a narrowband processor  706 , which may be implemented by the computer processor also. At output  706   a , the target detection test statistic is supplied and can be displayed on a display screen. 
   In equation (52), both the target data as well as the clutter data has been focused to a single frequency ω o  using the same frequency focusing matrix T(ω o ,  ω ), and hence narrowband receiver processing schemes can be applied to equation (52) as shown in  FIG. 6 . 
   The optimum narrowband processor for (52) is given by a whitening filter following by the matched filter [J. R. Guerci, Space-Time Adaptive Processing for Radar, Artech House, Boston, 2003]. The optimum filter is given by
 
   w     z   =R   z   −1     a   (ω o , φ k     o   , θ i     o   )  (54)
 
where
 
 R   z   =E{ c   (ω o   , n )   c   *(ω o   , n )}  (55)
 
represents the focused clutter covariance matrix that can be estimated using the neighboring range bins that are adjacent to the target range bin of interest.
 
   The adaptive weight vector in equation (54) is narrowband in nature and it suppresses the undesired clutter and interferences at while detecting the target present at (θ i     o   , φ k     o   ). Notice that unlike subband based schemes, it is not necessary to perform the adaptive processor in (54) to each subband. Instead, in the present invention, data is refocused to one single frequency band as in (52) using a focusing matrix, and then processed in one step at the final stage as shown in  FIG. 6 . 
     FIG. 7  shows a diagram  800  of an unfocused output signal to noise ratio as a function of frequency and azimuth angle θ for a wideband circular array with twelve sensors.).  FIG. 7  shows frequency in MHz on the y axis verus azimuth angle in degrees on the x axis. A wideband signal (235 MHz-635 MHz) is split into 200 frequency bands in the example of  FIG. 7 . The diagram  800  includes sections  802 ,  804  and  806 . The section  802  shows the frequency dependent mainbeam gain function versus the azimuth angle (in degrees) on the x axis. The section  804  shows the frequency dependent sidelobe gain function versus the azimuth angle (in degrees), and section  806  shows the scaling function used here with lighter region representing lower gain levels in dB (decibels). 
     FIG. 7  shows the result of traditional processing when SINR output for each sub band data is plotted as a function of the azimuth angle θ. Observe that the mainbeam width is different at different frequencies indicating the frequency sensitivity of the traditional approaches. 
     FIG. 8  shows a diagram  900  of focused output signal to interference plus noise ratio as a function (SINR) versus azimuth angle θ for a wideband circular array similar to the circular array or  2  of  FIG. 1  using twelve sensors. In the example of  FIG. 8 , a wideband signal as in  FIG. 7  is split into twenty frequency bands. All frequency bands are focused to the frequency of 435 MHz in  FIG. 8 . 
     FIG. 8  shows the output signal to interference plus noise (SINR) given by
 SINR=|   w     z   * a   (ω o , φ, θ)| 2   (56) 
obtained using the focused weight vector in equation (54). In both  FIG. 7  and  FIG. 8 , a twelve sensor circular array is used. Observe that all frequencies have been aligned and only a single mainbeam of constant width is generated at all frequencies. This is unlike the subband processor  802  in  FIG. 7 , where the mainbeam width is frequency dependent generating a wider null at lower frequency. For comparison purposes, the unfocused output as function of the azimuth angle and frequency is shown in  FIG. 7 . From there the output  802  in the mainbeam region is sensitive to the frequency band.
 
   Finally, from  FIG. 6 , as shown in  706 , the optimum weight vector  w   z  in (54) acts of the focused data  z (ω o ) from the range of interest to generate the threshold detector 
                          w   z     *     z   ⁡     (     ω   o     )              ⁢                         H   1     (     Target   ⁢           ⁢   Present     )             &gt;                 &lt;                     H   o     (     Target   ⁢           ⁢   Absent     )           ⁢   η           (   57   )               
where η represents a specific threshold satisfying a certain false alarm. The test in equation (57) maybe repeated over all range bin of interest to detect the target.
 
   To detect moving targets, the radar array in  FIG. 1  transmits a sequence of M pulses and records their returns, thus generating a space-time data vector. A moving target generates a Doppler component. In order to estimate that component, a treatment similar to the above one can be carried out in the Doppler domain. Detailed analysis of clutter data focusing both in space and time, to a single reference frequency and developing efficient algorithms to process the data to detect targets by suppressing clutter are carried out in the next section. 
   The present invention in one or more embodiments also provides a new method and/or apparatus of wideband space-time adaptive processing (STAP) for a circular array. 
   To detect moving targets, the above analysis can be extended to the Doppler domain by analyzing the space-time adaptive processing in the wideband case using circular arrays. When multiple pulses—say M of them—are transmitted at a pulse repetition interval T r , depending on the relative velocity of the target with respect to the sensor platform reference direction, different pulse returns are delayed differently at the receiver reference sensor. In the frequency domain, these delays appear as frequency dependent phase delays, and hence at frequency ω, a temporal steering vector  b (ω, ω d ) can be generated. 
   Assuming V is the platform velocity along the reference direction θ 1  (see  FIG. 1 ) relative with respect to the point of interest at (θ, φ), we obtain the temporal steering vector to be [J. R. Guerci, Space-Time Adaptive Processing for Radar, Artech House, Boston, 2003] 
                   b   ⁡     (     ω   ,     ω   d       )       =     [         1             ⅇ       -   j     ⁢           ⁢   π   ⁢           ⁢     ω   d                 ⋮             ⅇ       -   j     ⁢           ⁢     π   ⁡     (     M   -   1     )       ⁢     ω   d               ]             (   58   )               
where the Doppler frequency ω d  can be shown to be
 
   
     
       
         
           
             
               
                 
                   ω 
                   d 
                 
                 = 
                 
                   
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         VT 
                         r 
                       
                     
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       c 
                     
                   
                   ⁢ 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
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                   ⁢ 
                   
                       
                   
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                       ⁡ 
                       
                         ( 
                         
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                           - 
                           
                             θ 
                             1 
                           
                         
                         ) 
                       
                     
                     . 
                   
                 
               
             
             
               
                 ( 
                 59 
                 ) 
               
             
           
         
       
     
   
   If the platform itself is in motion, then every scattering point in the field of view generates a Doppler component according to equation (59) whose value depends on its angular location. 
   The vector  X (ω)= X   1 (ω) in equation (14) corresponds to the transform of the spatial array sensor outputs due to the first pulse, and by stacking up the returns due to M consecutive pulses  X   k (ω), k=1, 2, . . . M, the MN×1 spatio-temporal data vector 
                   X   ⁡     (   ω   )       =     [             X   1     ⁡     (   ω   )                   X   2     ⁡     (   ω   )               ⋮               X   M     ⁡     (   ω   )             ]             (   60   )               
at frequency ω can be represented as
   X (ω)= S (ω)   b   (ω, ω d )   a (ω, φ, θ)  (61) 
where S(ω) and  a (ω, φ, θ) are as defined in equation (14) and  b (ω, ω d ) represents the temporal steering vector in (58). Here   represents the Kronecker product defined in equation (33).
 
   Proceeding as in equations (14)-(39), the temporal steering vector also can be synthesized as (details omitted)
 
   b   (ω, ω d )= C (ω)   s     1   ∘D (ω)   s     2 =( C   T (ω)⊙ D   T (ω)) T (   s     1       s     2 )= Q (ω)(   s     1       s     2 )  (62)
 
where C(ω) and D(ω) are two M×(2L+1) matrices whose (i, k) th  elements are given by
 
                         C   ik     ⁡     (   ω   )       =         J   k     ⁡     (       (     i   -   1     )     ⁢     γ   ⁡     (   ω   )         )       ⁢     ⅇ     j   ⁢           ⁢   k   ⁢           ⁢     θ   1             ,           ⁢     i   =     1   →   M       ,           ⁢     k   =       -   L     →   L         ⁢     
     ⁢   and           (   63   )                       D   ik     ⁡     (   ω   )       =         J   k     ⁡     (       (     i   -   1     )     ⁢     γ   ⁡     (   ω   )         )       ⁢     ⅇ       -   j     ⁢           ⁢   k   ⁢           ⁢     θ   1             ,           ⁢     i   =     1   →   M       ,           ⁢     k   =       -   L     →   L         ⁢     
     ⁢   where           (   64   )                 γ   ⁡     (   ω   )       =         ω   ⁢           ⁢     VT   r       c     .             (   65   )               
In equation (12),
   Q (ω)=( C   T (ω)⊙ D   T (ω)) T   (66) 
where ⊙ represents the Khatri-Rao product as in (33), and  s   1  and  s   2  are as defined in equation (32). Q(ω) is of size M×(2L+1) 2 . In equation (62), the temporal steering vector has been synthesized as a product of a frequency dependent part Q(ω) and a frequency insensitive part  s   1     s   2 .
 
   Substituting equation (37) and equation (62) into (61) we get the frequency-dependent space-time steering vector 
                           b   ⁡     (     ω   ,     ω   d       )       ⊗     a   ⁡     (     ω   ,     ω   d       )         =       ⁢       Q   ⁡     (   ω   )       ⁢       (       s   1     ⊗     s   2       )     ⊗     P   ⁡     (   ω   )         ⁢     (       s   1     ⊗     s   2       )                   =       ⁢       (       Q   ⁡     (   ω   )       ⊗     P   ⁡     (   ω   )         )     ⁢       (       s   1     ⊗     s   2       )     ⊗     (       s   1     ⊗     s   2       )                     =       ⁢       K   ⁡     (   ω   )       ⁢   u                   (   67   )             where                           K   ⁡     (   ω   )       =       Q   ⁡     (   ω   )       ⊗     P   ⁡     (   ω   )                 (   68   )             and                         u   =       (       s   1     ⊗     s   2       )     ⊗       (       s   1     ⊗     s   2       )     .               (   69   )               
Hence K(ω) is of size MN×(2L+1) 4  and  u  is of size (2L+1) 4 ×1. In (67), we have used another well known Kronecker product identify given by
 AB CD=(A C)(B D) with A=Q(ω), B=D= s   1     s   2 , and C=P(ω).  (70) 
   Once again, the frequency dependent space-time steering vector  b (ω, ω d )   a (ω, ω d ) in equation (67) has been synthesized as a product of a frequency dependent part K(ω) in (68) and a frequency-independent part  u  in equation (69). 
   In the final step, the above synthesis procedure allows focusing all frequency components into a single frequency slot using the procedure described in equations (44)-(52). Observe that as before returns from all scatter point can be refocused simultaneously in the space-time scene as well. 
   Following equations (46) and (61), the transform of the general space-time data vector takes the form: 
                         X   ⁡     (   ω   )       =       ⁢         b   ⁡     (     ω   ,     ω     d   o         )       ⊗     a   ⁡     (     ω   ,     ϕ     k   o       ,     θ     i   o         )         +                     ⁢       ∑   i     ⁢           ⁢       ∑   k     ⁢           ⁢       α   ik     ⁢       b   ⁡     (     ω   ,     ω     d   ik         )       ⊗     a   ⁡     (     ω   ,     ϕ   k     ,     θ   i       )                           =       ⁢         K   ⁡     (   ω   )       ⁢     u   ⁡     (       i   o     ,     k   o       )         +       ∑   i     ⁢           ⁢       ∑   k     ⁢           ⁢       α   ik     ⁢     K   ⁡     (   ω   )       ⁢     u   ⁡     (     i   ,   k     )                             (   71   )               
and stacking up various frequency components as in equation (51) the following is determined:
 
                   Y   ⁡     (   ω   )       =       (           X   ⁡     (     ω   1     )                 X   ⁡     (     ω   2     )               ⋮             X   ⁡     (     ω   K     )             )     =         (           K   ⁡     (     ω   1     )                 K   ⁡     (     ω   2     )               ⋮             K   ⁡     (     ω   k     )             )       ︸     G   ⁡     (   ω   )           ⁢     (       u   (       i   o     ,     k   o       )     +       ∑   i     ⁢           ⁢       ∑   k     ⁢           ⁢       α   ik     ⁢     u   ⁡     (     i   ,   k     )               )                 (   72   )             where                           G   ⁡     (   ω   )       =     (           K   ⁡     (     ω   1     )                 K   ⁡     (     ω   2     )               ⋮             K   ⁡     (     ω   K     )             )             (   73   )               
is of size MNK×(2L+1) 4 . Following equation (40), define
   T (ω o ,  ω )= K (ω o )( G *(ω) G ( ω )) −1   G* ( ω )  (74) 
and apply that to (72) to obtain the frequency focused data
 
   
     
       
         
           
             
               
                 
                   
                     
                       
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                 ( 
                 75 
                 ) 
               
             
           
         
       
     
   
   In rank deficient situations, the matrix inversion in equation (74) is to be interpreted as a pseudo-matrix inversion. Alternatively, in (74), the inversion of G*(ω)G( ω ) can be accomplished by adding a diagonal term εI with ε&gt;0 to it so that G*(ω)G( ω )+εI is full rank and invertible. Observe that the focusing matrix in (74) is valid for returns from all locations. 
   Since G*(ω)G( ω ) is of size (2L+1) 4 ×(2L+1) 4 , its size can be prohibitive in carrying out the above inversion. In that case, the useful inversion identity 
                     (           G   *     ⁡     (   ω   )       ⁢     G   ⁡     (   ω   )         +     ɛ   ⁢           ⁢   I       )       -   1       =       1   ɛ     ⁡     [     I   -         G   *     ⁡     (   ω   )       ⁢       (         G   ⁡     (   ω   )       ⁢       G   *     ⁡     (   ω   )         +     ɛ   ⁢           ⁢   I       )       -   1       ⁢     G   ⁡     (   ω   )           ]               (   76   )               
can be employed. Observe that the matrix inversion in (76) only involves a smaller matrix of dimension MNK×MNK matrix. This gives the frequency focusing operator T(ω o ,  ω ) in (74) to be
   T (ω o ,  ω )= K (ω o ) G *( ω )− K (ω o ) G *( ω )( G (ω) G *( ω )+ε I ) −1   G ( ω ) G *(ω).  (77) Define   A   1   =K (ω o ) G *( ω ),  (78)   A   2   =G ( ω ) G *(ω),  (79) and   A   3 =( A   2   =εI ) −1 .  (80) 
Here A 1  is of size MN×MNK, A 2  and A 3  are of size MNK×MNK, and hence T(ω o ,  ω ) in (74) can be efficiently implemented as
   T (ω o ,  ω )= A   1 ( I−A   3   A   2 ).  (81) 
Observe that equation (79) involves only smaller size matrix multiplications compared to direct implementation of (72), and moreover their entries are data independent. As a result, it can be implemented efficiently prior to actual data collection.
 
   Finally, narrowband STAP processing methods similar to equations (54)-(55) can be applied to equation (75) for final processing. In this manner, the frequency focusing method can be extended into the space-time adaptive processing as well. 
   To be specific, with
 
 R=E{c (ω o ) c *(ω o )}  (82)
 
where c(ω o ) represents the clutter data is as in (75), we have
 
 w=R   −1   s (ω o , ω d     o   , φ k     o   , θ i     o   )  (83)
 
with
 
 s (ω o , ω d     o   , φ k     o   , θ i     o   )=   b   (ω o , ω d     o   )     a   (ω o , φ k     o   , θ i     o   )  (84)
 
represents the optimum narrowband processor. The focused data from each range bin may be processed as in equation (57) to detect targets.
 
     FIG. 9A  shows a diagram  1000  of an unfocused azimuth Doppler output signal to noise ratio (SINR) at a first frequency subband for a wideband circular array such as array or element  2  in  FIG. 1  with twelve sensors and fourteen pulses. A wideband data set (100 MHz-200 MHz) is split into multiple subbands of bandwidth 4 MHz here. A first subband output is shown in  FIG. 9A  and a second subband output is shown in  FIG. 9B . Injected target located at zero azimuth angle and ninety degree elevation angle is moving with velocity of twenty-five meters/second and it appears at different Dopplers at different frequencies (sections  1002  and  1102 ). Clutter to noise ratio is 40 dB. The diagram  1000  includes section  1002  which shows the peak due to the injected target, section  1004  that shows the nulled out sidelobe characteristics, and section  1006  that shows the scaling function used here with lighter region representing lower gain levels in dB (decibels). 
     FIG. 9B  shows a diagram  1100  of an unfocused azimuth Doppler output signal to noise ratio (SINR) at a second frequency subband for a wideband circular array, such as array or element  2  in  FIG. 1  with twelve sensors and fourteen pulses. A second subband output is shown in  FIG. 9B . The diagram  1100  includes section  1102  which shows the peak due to the injected target, section  1104  that shows the nulled out sidelobe characteristics and section  1106  that shows the scaling function used here with lighter region representing lower gain levels in dB (decibels). 
     FIGS. 9A and 9B  show the unfocused azimuth-Doppler output SINR i  pattern for the i th  subband for two different frequency bands (first and last) where the wideband data corresponds to a 100 MHz bandwidth (100 MHz to 200 MHz). Here
 SINR i   =|w*   i   *s (ω i , ω d , φ k     o   , θ)| 2   (85) 
where w i  represents the optimum weight vector for the i th  subband. As mentioned above, the injected target located at zero azimuth angle and elevation angle equal to 90° is moving with a velocity of 25 m/sec, and it appears at different Doppler frequencies in different subbands. From there, the detected target parameters are frequency sensitive. The circular array has twelve sensors and uses fourteen pulses to generate the data over a wide bandwidth of 100 MHz that span from 100 MHz to 200 MHz. The SINR pattern of the focused beam given by
 SINR=| w*s (ω o , ω d     o   , φ k     o   , θ i     o   )| 2   (86) 
using equations (82)-(83) is shown in  FIG. 10 . In this case, for illustration purposes the two extreme frequency bands located at frequencies of 100 MHz and 200 MHz are refocused to their center frequency of the data set (150 MHz) using only seven Bessel function components (L=7 in (63)-(64)). Observe that the target is visible at the correct azimuth and Doppler location.
 
     FIG. 10  shows a diagram  1200  of a focused azimuth-Doppler output SINR for a wideband circular array using twelve sensors and fourteen pulses. The diagram  1200  includes section  1202 , section  1204  and section  1206 . Section  1202  shows the SINR peak due to the injected target, section  1204  shows the nulled out sidelobe characteristics, and section  1206  that shows the scaling function used here with the lighter region representing lower gain levels in dB (decibels). In the example of  FIG. 10 , a space-time data vector such as in equation (72) corresponding to frequencies 100 MHz and 200 MHz from a wideband signal is focused to the center frequency of 150 MHz using fifteen Bessel coefficient terms (L=7) such as in equations (29)-(30). Injected target located at zero azimuth angle and ninety degrees elevation angle is moving with velocity twenty-five meters/second that corresponds to a normalized Doppler frequency of 0.2 at 150 MHz. Clutter to noise ratio is forty decibels. 
   Wideband Linear Array: 
   The above analysis can be easily extended to the linear array case as shown in  FIG. 11  with a an apparatus  1300 . In this case the linear array apparatus  1300  and geometry can be substituted to compute the corresponding delays in equation (9) and the rest of the procedure for frequency focusing is the same as outlined above. For example, for an N element uniformly placed linear array with normalized inter-element spacing equal to d and a first sensor  1302  located as shown, a second sensor  1304  located as shown, a third sensor  1306 , located as shown, and any number of further sensors, up to an N-th sensor  1308 . With the last sensor  1308  as shown in the apparatus  1300 , the time delay for the n th  sensor is computed to be 
   
     
       
         
           
             
               
                 
                   
                     τ 
                     n 
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             n 
                             - 
                             1 
                           
                           ) 
                         
                         ⁢ 
                         d 
                       
                       c 
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ϕ 
                   
                 
                 , 
                 
                   n 
                   = 
                   1 
                 
                 , 
                 2 
                 , 
                 … 
                 ⁢ 
                 
                     
                 
                 , 
                 
                   N 
                   . 
                 
               
             
             
               
                 ( 
                 87 
                 ) 
               
             
           
         
       
     
   
   where θ and φ represent the azimuth and elevation angles respectively, and n can be any integer such as n=1, 2, . . . , N. Equation (87) can be substituted into equations (12)-(86) and simplified accordingly to obtain the corresponding linear array results. In this case proceeding as in (17)-(32) the quantities A n,k (ω), B n,k (ω) defined in (24), (26) takes the form 
                       A     n   ,   k       ⁡     (   ω   )       =       J   k     ⁢     (       (     n   -   1     )     ⁢     β   ⁡     (   ω   )         )         ,     
     ⁢     n   =   1     ,   2   ,     …   ⁢           ⁢   N     ,     k   =     -   L       ,     -     (     L   -   1     )       ,     …   ⁢           ⁢   0     ,   1   ,   …   ⁢           ,   L   ,           (   89   )                     B     n   ,   k       ⁡     (   ω   )       =         (     -   i     )     k     ⁢     J   k     ⁢     (       (     n   -   1     )     ⁢     β   ⁡     (   ω   )         )         ,     
     ⁢     n   =   1     ,   2   ,   …   ⁢           ,   N   ,     k   =     -   L       ,     -     (     L   -   1     )       ,     …   ⁢           ⁢   0     ,   1   ,   …   ⁢           ,   L   ,           (   90   )             with                           β   ⁡     (   ω   )       =       ω   ⁢           ⁢   d       2   ⁢           ⁢   c               (   91   )               
and proceeding as in (58)-(65) the quantities C n,k (ω), and D n,k (ω) defined in (63) and (64) takes the form
 
                       C   ik     ⁡     (   ω   )       =       J   k     ⁢     (       (     i   -   1     )     ⁢     γ   ⁡     (   ω   )         )         ,     
     ⁢     i   =   1     ,   2   ,   …   ⁢           ,   M   ,     k   =     -   L       ,     -     (     L   -   1     )       ,     …   ⁢           ⁢   0     ,   1   ,   …   ⁢           ,   L   ,           (   92   )             and                               D   ik     ⁡     (   ω   )       =         (     -   i     )     k     ⁢     J   k     ⁢     (       (     i   -   1     )     ⁢     γ   ⁡     (   ω   )         )         ,           ⁢     i   =   1     ,   2   ,   …   ⁢           ,   M   ,     k   =     -   L       ,     -     (     L   -   1     )       ,     …   ⁢           ⁢   0     ,   1   ,   …   ⁢           ,   L   ,           (   93   )             with                           γ   ⁡     (   ω   )       =       ω   ⁢           ⁢     VT   r       c             (   94   )               
where J k (β) represents the Bessel function of the k th  order elevated at β as defined in (22). The rest of the procedure remains the same as in the circular array case.
 
     FIGS. 12A  and  FIG. 12B  show diagrams  1400  and  1500  that are focused azimuth-Doppler output SINR for a linear circular array using fourteen sensors and sixteen pulses. Diagram  1400  refers to the top view and diagram  1500  refers to the side view. The diagram  1400  includes a section  1402 , a section  1404 , and a second  1406 . The section  1402  shows the SINR peak due to the injected target, section  1404  shows the nulled out sidelobe characteristics and section  1406  shows the scaling function used here with the lighter region representing lower gain levels in dB (decibels). Similarly, the diagram  1500  includes a section  1502 , a section  1504 , and a section  1506 . The section  1502  shows the SINR peak due to an injected target, section  1504  shows the nulled out sidelobe characteristics and section  1506  shows the scaling function used here with lighter region representing lower gain levels in dB (decibels). In the examples of  FIGS. 12A  and  FIG. 12B , a space-time data vector such as in equation (72) corresponding to frequencies 335 MHz to 535 MHz is focused to the center frequency of 435 MHz using twenty frequency bands. Fifteen Bessel coefficient terms (L=15) such as in equations (29)-(30) are used for frequency focusing here. Injected target located at zero azimuth angle and ninety degrees elevation angle is moving with velocity 40 meters/second. Clutter to noise ratio is 40 decibels. 
     FIG. 13  shows a flowchart  1600  of a method in accordance with an embodiment of the present invention.  FIG. 14  shows an apparatus  1700  which can be used in accordance with the method of the flow chart of  FIG. 13 . The apparatus  1700  includes a computer processor  1701 , shown in dashed lines. The computer processor  1700  may include or may be thought of as including modules  1702 ,  1704 ,  1706 , and  1708 , each of which may have computer memory. 
   The method of  FIG. 13  includes step  1602  in which wideband data, such as x 1 (t), x 2 (t), . . . shown in  FIG. 14 , is collected by a plurality of sensors, such as sensors  4   a - 4 N in  FIG. 1 , using a circular array or element such as circular array or element  2 , with N sensors and M pulse returns, or using a linear array as shown in  FIG. 11  with a plurality of sensors such as  1302 ,  1304 , . . . The wideband data is supplied from the sensors to a computer processor, such as computer processor  1701 , such as to module  1702  of computer processor  1701 , shown in  FIG. 14 . The wideband data is then stored by the computer processor  1701  in a plurality of stacked raw data vectors in computer memory of computer processor  1701 , in module  1702  shown in  FIG. 14 . Each of the plurality of raw data vectors corresponds to one of the M pulse returns. The raw data vectors are generally stacked with the last raw data vector corresponding to the last pulse in time being the first to be output via output  1702   a  of the module  1702 . 
   The method show in  FIG. 13 , further includes step  1604  in which a Fourier transform is performed on each of the plurality of raw data vectors to form a plurality of Fourier transform data vectors. The method may be implemented by module  1704  shown in  FIG. 14  in computer processor  1701 . At step  1606  of  FIG. 13 , the computer processor  1701  in module  1706  may transform the plurality of Fourier transform data vectors into a narrowband data vector by premultiplying the Fourier transform data vectors with a predetermined frequency compensating or focusing matrix. 
   At step  1608 , module  1708  of the a computer processor  1701  may perform traditional space time adaptive methods for target detection on the narrowband data vector to form an output data vector supplied at output  1710 . 
   Although the invention has been described by reference to particular illustrative embodiments thereof, many changes and modifications of the invention may become apparent to those skilled in the art without departing from the spirit and scope of the invention. It is therefore intended to include within this patent all such changes and modifications as may reasonably and properly be included within the scope of the present invention&#39;s contribution to the art.