PATENT DOCUMENT

Abstract:
The present inventive concept for bistatic or monostatic radar utilization discloses a method and a system for unambiguous angle resolution of a sparse wide-band array antenna by utilizing an ultra wide-band signal, generally a wide-band noise signal which may be continuous bandwidth limited white or colored noise. The noise signal is typically generated and radiated by a transmitting antenna covering the entire reception range of a receiving sparse antenna array. An echo signal is received by the sparse receiving array containing n antenna elements, which are generally positioned with separation distances of several wavelengths without creating strong grating lobes due to the type of generated transmit signal. By means of a selected auto-correlation function defining the wide-band noise signal power spectrum, the convolution of the radiated output signal and the received echo input signal will give the target range information, the individual time-delay setting for each antenna element will give an unambiguous angular resolution.

Full Description:
This application is the U.S. national phase of international application PCT/SE2003/000833 filed 21 May 2003 which designated the U.S., the entire contents of which is hereby incorporated by reference. 
   TECHNICAL FIELD 
   The present invention relates to wide-band antenna arrays, and more particularly to a method and system for unambiguous angle resolution of a sparse wide-band antenna array. 
   BACKGROUND 
   It is complicated or even almost impossible to build classic wide-band array antennas providing good angle resolution. The reason for this is that the separation of the elements is set by the highest frequency, typically half a wavelength at f max . The physical size of the elements is set by the lowest frequency, typically half a wavelength at f min . Thus, there is apparently a contradiction between obtaining a wide bandwidth of the antenna and the array density needed to avoid grating lobes at large scan angles. 
   Another problem is that array antennas having a large number of elements will be undesired complex and expensive. Furthermore rotating antennas are generally undesired as that implies further mechanical complexity. From a strategic point of view there is also a desire to be able to keep peak power as low as possible to minimize the risk of being detected. 
   Typical solutions of today utilize fully filled arrays and rotating antennas combined with range coded long pulses, for instance using a binary phase code. Signal classification is difficult with present solutions and different hardware solutions have to be used for a respective function category, for instance, a radar function or a jamming function. 
   Therefore there is a demand for further improvements of wide-band arrays for obtaining also a better angular resolution produced by a less complex solution compared to the state of the art. 
   SUMMARY OF THE INVENTION 
   A method and a system for unambiguous angle resolution of a sparse wide-band array antenna in a bistatic or monostatic radar configuration utilize an ultra-wide-band signal, which in a typical embodiment may be continuous bandwidth limited white noise or colored noise. The noise signal is generated and radiated for instance by an omnidirectional transmitting antenna covering the entire reception range of a receiving array. An echo signal is received by the receiving array containing n antenna elements. The n antenna elements are generally positioned with separation distances of several wavelengths without creating strong grating lobes. By means of a selected auto-correlation function defining the wide-band noise signal power spectrum, the convolution of the radiated output signal and the received echo input signal will give the target range information, the individual time-delay setting for each antenna element will give an unambiguous angular resolution. 
   A system according to the present invention is set forth by the independent claim  1 , and further embodiments of the invention are set forth by the dependent claims  2  to  9 . 
   Further a method according to the present invention is set forth by the independent claim  10 , and further embodiments are defined by the dependent claims  11  to  17 . 

   
     SHORT DESCRIPTION OF THE DRAWINGS 
     The invention, together with further objects and advantages thereof, may best be understood by referring to the following detailed description taken together with the accompanying drawings, in which: 
       FIG. 1  illustrates an illustrative array antenna according to the present inventive concept; 
       FIG. 2  illustrates a principal embodiment of a wide-band array according to the present invention having the number of receiving antenna elements n=32; 
       FIG. 3  illustrates in a number of graphs a) to f) the auto-correlation function for white bandwidth limited noise using a bandwidth from 200% and down to 6.25%; 
       FIG. 4  illustrates in a graph the convolution of radiated and received signal as a function of signal-to-noise ratio; 
       FIG. 5  illustrates in a block diagram three bearing channels; 
       FIG. 6  illustrates a wide-band pattern for a linear receive array containing a binary number of 1 to 512 elements for a central noise transmitter; and 
       FIG. 7  illustrates a wide-band pattern for a circular receiver array containing a binary number of 4 to 2048 elements for a central omnidirectional noise transmitter. 
   

   DETAILED DESCRIPTION 
   In a typical illustrative embodiment a very broad-banded signal, which may be continuous band-limited white or colored noise is generated and transmitted by an antenna, which for instance is isotropic in its horizontal plane. The echo signal in a bistatic or monostatic configuration is received by n receiver antennas. The n antennas are spaced at distances of several wavelengths, e.g. 10 wavelengths at f min . These antennas being placed at separation distances of several wavelengths do not create dominating grating lobes due to the frequency spread achieved by the large bandwidth. By grating lobes we refer to periodic repetitions of a main lobe. 
   Each antenna may be a vertical array to increase the gain in the horizontal plane. To minimize leakage between the transmitting antenna and the receiver antennas the transmitting antenna preferably is positioned as far as possible from the receiving antennas and for instance at a different height than the receiving antennas.  FIG. 1  illustrates a principal shape with n=3, the transmitting antenna is denoted as  1  and the receiver antennas are denoted  3 . A structure carrying the transmitter antenna is denoted as reference number  5 . A transmitter and n receivers with accompanying signal processing is housed in the electronic unit  7 .  FIG. 2  illustrates a practical embodiment of a circular array with n=32, the transmitting antenna sitting elevated in the middle and the receiving antenna in a ring below. One transmitter, n receivers with accompanying signal processing being housed in the base of the array. 
   The idea of using a continuous ultra broad-band signal like band-limited white noise is, that only the central lobe can be considered additive for all frequencies within the wide band used. Grating lobes resulting from the different frequencies within the broadband signal will not be added in the same way as the grating lobe pattern will for each separate frequency component being a function of the spacing of antenna elements used. This will result in an angle resolution, which may be used, for instance, for radar purposes. This was simulated in a frequency range of 6-18 GHz, which can simply be obtained. The distance to the target is determined by finding the peaks in the convolution between the input and output signal. 
   Principle for Range Resolution 
   Assume an array wherein the main lobe of each antenna element points in a radial direction in the circular case and in a direction θ=0 in the linear case. Further assume that the diagrams of the elements have the simple frequency independent form according to Equation (1): 
   
     
       
         
           
             
               
                 
                   EL 
                   ⁡ 
                   
                     ( 
                     θ 
                     ) 
                   
                 
                 = 
                 
                   | 
                   
                     
                       
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             θ 
                             ) 
                           
                         
                       
                       
                         if 
                       
                       
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
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                               ) 
                             
                           
                           &gt; 
                           0 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         if 
                       
                       
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
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                               ) 
                             
                           
                           ≤ 
                           0 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 1 
                 ) 
               
             
           
         
       
     
   
   The radiated noise signal is assumed to be e t (t). The sum from all n receiver antennas may then for a spot target be written according to Equation (2): 
   
     
       
         
           
             
               
                 
                   
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                     ( 
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                           ⁢ 
                           
                               
                           
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                           ⁢ 
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                 ( 
                 2 
                 ) 
               
             
           
         
       
     
   
   Where R, φ are distance and bearing, respectively, to the target. θ means current antenna search angle. Time corrections τ n  are calculated according to Equation (3): 
   
     
       
         
           
             
               
                 
                   
                     τ 
                     n 
                   
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                     ( 
                     ϕ 
                     ) 
                   
                 
                 = 
                 
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   Convolution between the radiated and the received signal can, according to Equation (4), be written: 
   
     
       
         
           
             
               
                 
                   
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                                             array 
                                           
                                         
                                       
                                       
                                         
                                           
                                             EL 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               
                                                 ( 
                                                 
                                                   θ 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   - 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   
                                                     n 
                                                     · 
                                                     
                                                         
                                                     
                                                     ⁢ 
                                                     
                                                       
                                                         2 
                                                         ⁢ 
                                                         
                                                             
                                                         
                                                         ⁢ 
                                                         π 
                                                       
                                                       
                                                         
                                                             
                                                         
                                                         ⁢ 
                                                         N 
                                                       
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                         
                                         
                                           
                                             Circular 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             array 
                                           
                                         
                                       
                                     
                                     } 
                                   
                                   · 
                                   
                                     
                                       ∫ 
                                       
                                         ω 
                                         1 
                                       
                                       
                                         ω 
                                         2 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         { 
                                         
                                           
                                             ⅇ 
                                             
                                               jω 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   t 
                                                   - 
                                                   
                                                     
                                                       2 
                                                       · 
                                                       R 
                                                     
                                                     
                                                       c 
                                                       0 
                                                     
                                                   
                                                   + 
                                                   
                                                     
                                                       τ 
                                                       n 
                                                     
                                                     ⁡ 
                                                     
                                                       ( 
                                                       ϕ 
                                                       ) 
                                                     
                                                   
                                                   - 
                                                   
                                                     
                                                       τ 
                                                       n 
                                                     
                                                     ⁡ 
                                                     
                                                       ( 
                                                       0 
                                                       ) 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           + 
                                           
                                             ⅇ 
                                             
                                               - 
                                               
                                                 jω 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     t 
                                                     - 
                                                     
                                                       
                                                         2 
                                                         · 
                                                         R 
                                                       
                                                       
                                                         c 
                                                         0 
                                                       
                                                     
                                                     + 
                                                     
                                                       
                                                         τ 
                                                         n 
                                                       
                                                       ⁡ 
                                                       
                                                         ( 
                                                         ϕ 
                                                         ) 
                                                       
                                                     
                                                     - 
                                                     
                                                       
                                                         τ 
                                                         n 
                                                       
                                                       ⁡ 
                                                       
                                                         ( 
                                                         θ 
                                                         ) 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                         } 
                                       
                                       · 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         ⅆ 
                                         ω 
                                       
                                     
                                   
                                 
                               
                             
                             = 
                             
                               
                                 
                                   
                                     σ 
                                     2 
                                   
                                   
                                     
                                       ω 
                                       2 
                                     
                                     - 
                                     
                                       ω 
                                       1 
                                     
                                   
                                 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       { 
                                       
                                         
                                           
                                             
                                               EL 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 ( 
                                                 θ 
                                                 ) 
                                               
                                             
                                           
                                           
                                             
                                               Linear 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               array 
                                             
                                           
                                         
                                         
                                           
                                             
                                               EL 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 
                                                   ( 
                                                   
                                                     θ 
                                                     ⁢ 
                                                     
                                                         
                                                     
                                                     - 
                                                     
                                                         
                                                     
                                                     ⁢ 
                                                     
                                                       n 
                                                       · 
                                                       
                                                           
                                                       
                                                       ⁢ 
                                                       
                                                         
                                                           2 
                                                           ⁢ 
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           π 
                                                         
                                                         
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           N 
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                               
                                             
                                           
                                           
                                             
                                               Circular 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               array 
                                             
                                           
                                         
                                       
                                       } 
                                     
                                     · 
                                     
                                       
                                         ∫ 
                                         
                                           ω 
                                           1 
                                         
                                         
                                           ω 
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           cos 
                                           ⁡ 
                                           
                                             [ 
                                             
                                               ω 
                                               · 
                                               
                                                 ( 
                                                 
                                                   t 
                                                   - 
                                                   
                                                     
                                                       2 
                                                       · 
                                                       R 
                                                     
                                                     
                                                       c 
                                                       0 
                                                     
                                                   
                                                   + 
                                                   
                                                     
                                                       τ 
                                                       n 
                                                     
                                                     ⁡ 
                                                     
                                                       ( 
                                                       ϕ 
                                                       ) 
                                                     
                                                   
                                                   - 
                                                   
                                                     
                                                       τ 
                                                       n 
                                                     
                                                     ⁡ 
                                                     
                                                       ( 
                                                       θ 
                                                       ) 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             ] 
                                           
                                         
                                         · 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           ⅆ 
                                           ω 
                                         
                                       
                                     
                                   
                                 
                               
                               = 
                               
                                 
                                   
                                     σ 
                                     2 
                                   
                                   
                                     
                                       ω 
                                       2 
                                     
                                     - 
                                     
                                       ω 
                                       1 
                                     
                                   
                                 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       { 
                                       
                                         
                                           
                                             
                                               EL 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 ( 
                                                 θ 
                                                 ) 
                                               
                                             
                                           
                                           
                                             
                                               Lin 
                                               . 
                                             
                                           
                                         
                                         
                                           
                                             
                                               EL 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 
                                                   ( 
                                                   
                                                     θ 
                                                     ⁢ 
                                                     
                                                         
                                                     
                                                     - 
                                                     
                                                         
                                                     
                                                     ⁢ 
                                                     
                                                       n 
                                                       · 
                                                       
                                                           
                                                       
                                                       ⁢ 
                                                       
                                                         
                                                           2 
                                                           ⁢ 
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           π 
                                                         
                                                         
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           N 
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                               
                                             
                                           
                                           
                                             
                                               Cir 
                                               . 
                                             
                                           
                                         
                                       
                                       } 
                                     
                                     · 
                                     
                                       
                                         
                                           
                                             
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   [ 
                                                   
                                                     
                                                       ω 
                                                       2 
                                                     
                                                     · 
                                                     
                                                       ( 
                                                       
                                                         t 
                                                         - 
                                                         
                                                           
                                                             2 
                                                             · 
                                                             R 
                                                           
                                                           
                                                             c 
                                                             0 
                                                           
                                                         
                                                         + 
                                                         
                                                           
                                                             τ 
                                                             n 
                                                           
                                                           ⁡ 
                                                           
                                                             ( 
                                                             ϕ 
                                                             ) 
                                                           
                                                         
                                                         - 
                                                         
                                                           
                                                             τ 
                                                             n 
                                                           
                                                           ⁡ 
                                                           
                                                             ( 
                                                             θ 
                                                             ) 
                                                           
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   ] 
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                               sin 
                                               ⁡ 
                                               
                                                 [ 
                                                 
                                                   
                                                     ω 
                                                     1 
                                                   
                                                   · 
                                                   
                                                     ( 
                                                     
                                                       t 
                                                       - 
                                                       
                                                         
                                                           2 
                                                           · 
                                                           R 
                                                         
                                                         
                                                           c 
                                                           0 
                                                         
                                                       
                                                       + 
                                                       
                                                         
                                                           τ 
                                                           n 
                                                         
                                                         ⁡ 
                                                         
                                                           ( 
                                                           ϕ 
                                                           ) 
                                                         
                                                       
                                                       - 
                                                       
                                                         
                                                           τ 
                                                           n 
                                                         
                                                         ⁡ 
                                                         
                                                           ( 
                                                           θ 
                                                           ) 
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                                 ] 
                                               
                                             
                                           
                                         
                                       
                                       
                                         t 
                                         - 
                                         
                                           
                                             2 
                                             · 
                                             R 
                                           
                                           
                                             c 
                                             0 
                                           
                                         
                                         + 
                                         
                                           
                                             τ 
                                             n 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             ϕ 
                                             ) 
                                           
                                         
                                         - 
                                         
                                           
                                             τ 
                                             n 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             θ 
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 4 
                 ) 
               
             
           
         
       
     
   
   Where             denotes the Fourier transform. We have assumed that e t (t) is white bandwidth limited noise from ω 1  to ω 2 .
   Principle for Angle Resolution 
   The principle for resolution in angular direction is that grating lobes of respective frequency will land in different directions. The level of other lobes than the main lobe seen over the entire bandwidth will be expected to be low as grating lobes over the wide bandwidth generally will counteract each other. The radiated noise signal is assumed to be e(t). R ee (τ) is the auto-correlation function of e(t). If we presume that e(t) is white bandwidth limited noise from ω 1  to ω 2  it is easy to prove that the auto-correlation function can be written according to Equation (5) below: 
   
     
       
         
           
             
               
                 
                   
                     R 
                     ee 
                   
                   ⁡ 
                   
                     ( 
                     τ 
                     ) 
                   
                 
                 = 
                 
                   
                     σ 
                     2 
                   
                   · 
                   
                     cos 
                     ⁡ 
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               ω 
                               2 
                             
                             + 
                             
                               ω 
                               1 
                             
                           
                           ) 
                         
                         · 
                         
                           τ 
                           2 
                         
                       
                       ] 
                     
                   
                   · 
                   
                     
                       sin 
                       ⁡ 
                       
                         [ 
                         
                           
                             ( 
                             
                               
                                 ω 
                                 2 
                               
                               - 
                               
                                 ω 
                                 1 
                               
                             
                             ) 
                           
                           · 
                           
                             τ 
                             2 
                           
                         
                         ] 
                       
                     
                     
                       
                         ( 
                         
                           
                             ω 
                             2 
                           
                           - 
                           
                             ω 
                             1 
                           
                         
                         ) 
                       
                       · 
                       
                         τ 
                         2 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 5 
                 ) 
               
             
           
         
       
     
   
   Here σ denotes the noise standard deviation and the mean power of the noise signal is σ 2 . 
   Linear Array 
   For a basic first emrbodiment of the present invention assume a linear array with a length L where the main lobe of each antenna element points in the direction θ=0 and has the simple frequency independent shape according to Equation (6): 
   
     
       
         
           
             
               
                 
                   EL 
                   ⁡ 
                   
                     ( 
                     θ 
                     ) 
                   
                 
                 = 
                 
                   | 
                   
                     
                       
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             θ 
                             ) 
                           
                         
                       
                       
                         if 
                       
                       
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                           &gt; 
                           0 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         if 
                       
                       
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                           ≤ 
                           0 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 6 
                 ) 
               
             
           
         
       
     
   
   The sum of all n receiver antennas then can be written according to Equation (7): 
   
     
       
         
           
             
               
                 
                   
                     E 
                     Σ 
                   
                   ⁡ 
                   
                     ( 
                     θ 
                     ) 
                   
                 
                 = 
                 
                   
                     EL 
                     ⁡ 
                     
                       ( 
                       θ 
                       ) 
                     
                   
                   · 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       e 
                       ⁡ 
                       
                         ( 
                         
                           t 
                           - 
                           
                             
                               2 
                               · 
                               R 
                             
                             
                               c 
                               0 
                             
                           
                           + 
                           
                             
                               τ 
                               n 
                             
                             ⁡ 
                             
                               ( 
                               ϕ 
                               ) 
                             
                           
                           - 
                           
                             
                               τ 
                               n 
                             
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 7 
                 ) 
               
             
           
         
       
     
   
   Where R, φ are distance and bearing to the target, respectively. θ is the current antenna search angle. Time corrections τ n  are calculated according to Equation (8): 
   
     
       
         
           
             
               
                 
                   
                     τ 
                     n 
                   
                   ⁡ 
                   
                     ( 
                     ϕ 
                     ) 
                   
                 
                 = 
                 
                   
                     n 
                     
                       ( 
                       
                         N 
                         - 
                         1 
                       
                       ) 
                     
                   
                   · 
                   
                     L 
                     
                       c 
                       0 
                     
                   
                   · 
                   
                     sin 
                     ⁡ 
                     
                       ( 
                       ϕ 
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 8 
                 ) 
               
             
           
         
       
     
   
   Expected value of the antenna diagram AP(θ, φ) is calculated according to Equation 9:
 
 AP (θ,φ)=10·log { E[E   Σ   2 (θ,φ)]}  (9)
 
   We will now be able to perform the following calculation: 
   
     
       
         
           
             E 
             ⁡ 
             
               [ 
               
                 
                   E 
                   Σ 
                   2 
                 
                 ⁡ 
                 
                   ( 
                   
                     θ 
                     , 
                     ϕ 
                   
                   ) 
                 
               
               ] 
             
           
           = 
           
             
               E 
               ⁡ 
               
                 [ 
                 
                   
                     ( 
                     
                       
                         EL 
                         ⁡ 
                         
                           ( 
                           
                             θ 
                             - 
                             
                               n 
                               · 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 N 
                               
                             
                           
                           ) 
                         
                       
                       · 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           e 
                           ⁡ 
                           
                             ( 
                             
                               t 
                               - 
                               
                                 
                                   2 
                                   · 
                                   R 
                                 
                                 
                                   c 
                                   0 
                                 
                               
                               + 
                               
                                 
                                   τ 
                                   n 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   ϕ 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   τ 
                                   n 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ) 
                   
                   2 
                 
                 ] 
               
             
             = 
             
               
                 E 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         ( 
                         
                           EL 
                           ⁡ 
                           
                             ( 
                             θ 
                             ) 
                           
                         
                         ) 
                       
                       2 
                     
                     · 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           ⁡ 
                           
                             ( 
                             
                               t 
                               - 
                               
                                 
                                   2 
                                   · 
                                   R 
                                 
                                 
                                   c 
                                   0 
                                 
                               
                               + 
                               
                                 
                                   τ 
                                   n 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   ϕ 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   τ 
                                   n 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           
                             ∑ 
                             
                               m 
                               ⁢ 
                               
                                   
                               
                               = 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                             
                               
                                   
                               
                               ⁢ 
                               
                                 N 
                                 ⁢ 
                                 
                                     
                                 
                                 - 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             e 
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 - 
                                 
                                   
                                     2 
                                     · 
                                     R 
                                   
                                   
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       c 
                                       
                                         
                                             
                                         
                                         ⁢ 
                                         0 
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     τ 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       m 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     ϕ 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     τ 
                                     m 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ] 
                 
               
               = 
               
                 
                   E 
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           ( 
                           
                             EL 
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                           ) 
                         
                         2 
                       
                       · 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               e 
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   - 
                                   
                                     
                                       2 
                                       · 
                                       R 
                                     
                                     
                                       c 
                                       0 
                                     
                                   
                                   + 
                                   
                                     
                                       τ 
                                       n 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       ϕ 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       τ 
                                       n 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                             · 
                             
                               e 
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   - 
                                   
                                     
                                       2 
                                       · 
                                       R 
                                     
                                     
                                       c 
                                       0 
                                     
                                   
                                   + 
                                   
                                     
                                       τ 
                                       m 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       ϕ 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       τ 
                                       m 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
                 = 
                 
                   
                     
                       
                         ( 
                         
                           EL 
                           ⁡ 
                           
                             ( 
                             θ 
                             ) 
                           
                         
                         ) 
                       
                       2 
                     
                     · 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           E 
                           ⁡ 
                           
                             [ 
                             
                               
                                 e 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     
                                       
                                         2 
                                         · 
                                         R 
                                       
                                       
                                         c 
                                         0 
                                       
                                     
                                     + 
                                     
                                       
                                         τ 
                                         n 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         ϕ 
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         τ 
                                         n 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         θ 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 e 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     
                                       
                                         2 
                                         · 
                                         R 
                                       
                                       
                                         c 
                                         0 
                                       
                                     
                                     + 
                                     
                                       
                                         τ 
                                         m 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         ϕ 
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         τ 
                                         m 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         θ 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           EL 
                           ⁡ 
                           
                             ( 
                             θ 
                             ) 
                           
                         
                         ) 
                       
                       2 
                     
                     · 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             R 
                             ee 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   τ 
                                   n 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   ϕ 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   τ 
                                   n 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   τ 
                                   m 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   ϕ 
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   τ 
                                   m 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
     FIG. 6  illustrates power normalized to the main beam as function of angle −90 to +90 degrees for variation of number of elements in binary steps from 1 to 512. As can be seen already with 8 elements a good angular resolution is obtained with a surrounding signal level at −10 dB. 
   Circular Array 
   For a second embodiment of the present invention assume a circular array having the main lobe of each antenna element pointing in a radial direction and according to Equation (10) has the simple frequency independent shape: 
   
     
       
         
           
             
               
                 
                   EL 
                   ⁡ 
                   
                     ( 
                     θ 
                     ) 
                   
                 
                 = 
                 
                   | 
                   
                     
                       
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             θ 
                             ) 
                           
                         
                       
                       
                         if 
                       
                       
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                           &gt; 
                           0 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         if 
                       
                       
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                           ≤ 
                           0 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 10 
                 ) 
               
             
           
         
       
     
   
   The sum, from all n receiving antennas then can be written according to Equation (11): 
   
     
       
         
           
             
               
                 
                   
                     E 
                     Σ 
                   
                   ⁡ 
                   
                     ( 
                     θ 
                     ) 
                   
                 
                 = 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       e 
                       ⁡ 
                       
                         ( 
                         
                           t 
                           - 
                           
                             
                               2 
                               · 
                               R 
                             
                             
                               c 
                               0 
                             
                           
                           + 
                           
                             
                               τ 
                               n 
                             
                             ⁡ 
                             
                               ( 
                               ϕ 
                               ) 
                             
                           
                           - 
                           
                             
                               τ 
                               n 
                             
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     · 
                     
                       EL 
                       ⁡ 
                       
                         ( 
                         
                           θ 
                           - 
                           
                             n 
                             · 
                             
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                               N 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 11 
                 ) 
               
             
           
         
       
     
   
   R, φ are distance and bearing to the target, respectively, and θ is the current antenna search angle. Time corrections τ n  are calculated according to Equation (12): 
   
     
       
         
           
             
               
                 
                   
                     τ 
                     n 
                   
                   ⁡ 
                   
                     ( 
                     ϕ 
                     ) 
                   
                 
                 = 
                 
                   
                     D 
                     
                       2 
                       · 
                       
                         c 
                         0 
                       
                     
                   
                   · 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         ϕ 
                         - 
                         
                           n 
                           · 
                           
                             
                               2 
                               ⁢ 
                               π 
                             
                             N 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 12 
                 ) 
               
             
           
         
       
     
   
   Expected value of the antenna diagram AP(θ, φ) is calculated according to Equation (13):
 
 AP (θ,φ)=10·log { E[E   Σ   2 (θ,φ)]}  (13)
 
   Then, we can perform the following calculation: 
             E   ⁡     [       E   Σ   2     ⁡     (     θ   ,   ϕ     )       ]       =       E   ⁡     [       (       ∑     n   =   0       N   -   1       ⁢       e   ⁡     (     t   -       2   ·   R       c   0       +       τ   n     ⁡     (   ϕ   )       -       τ   n     ⁡     (   θ   )         )       ·     EL   ⁡     (     θ   -     n   ·       2   ⁢   π     N         )           )     2     ]       =       E   ⁡     [       ∑     n   =   0       N   -   1       ⁢       e   ⁡     (     t   -       2   ·   R       c   0       +       τ   n     ⁡     (   ϕ   )       -       τ   n     ⁡     (   θ   )         )       ·     EL   ⁡     (     θ   -     n   ·       2   ⁢   π     N         )       ·       ∑     m   =   0       N   -   1       ⁢       e   ⁡     (     t   -       2   ·   R       c   0       +       τ   m     ⁡     (   ϕ   )       -       τ   m     ⁡     (   θ   )         )       ·     EL   ⁡     (     θ   -     m   ·       2   ⁢   π     N         )               ]       =       E   ⁡     [       ∑     n   =   0       N   -   1       ⁢       ∑     m   =   0       N   -   1       ⁢       e   ⁡     (     t   -       2   ·   R       c   0       +       τ   n     ⁡     (   ϕ   )       -       τ   n     ⁡     (   θ   )         )       ·     EL   ⁡     (     θ   -     n   ·       2   ⁢   π     N         )       ·     e   ⁡     (     t   -       2   ·   R       c   0       +       τ   m     ⁡     (   ϕ   )       -       τ   m     ⁡     (   θ   )         )       ·     EL   ⁡     (     θ   -     m   ·       2   ⁢   π     N         )             ]       =         ∑     n   =   0       N   -   1       ⁢       ∑     m   =   0       N   -   1       ⁢       E   ⁡     [       e   ⁡     (     t   -       2   ·   R       c   0       +       τ   n     ⁡     (   ϕ   )       -       τ   n     ⁡     (   θ   )         )       ·     e   ⁡     (     t   -       2   ·   R       c   0       +       τ   m     ⁡     (   ϕ   )       -       τ   m     ⁡     (   θ   )         )         ]       ·     EL   ⁡     (     θ   -     n   ·       2   ⁢   π     N         )       ·     EL   ⁡     (     θ   -     m   ·       2   ⁢   π     N         )             =       ∑     n   =   0       N   -   1       ⁢       ∑     m   =   0       N   -   1       ⁢         R   ee     ⁡     (         τ   n     ⁡     (   ϕ   )       -       τ   n     ⁡     (   θ   )       -       τ   m     ⁡     (   ϕ   )       +       τ   m     ⁡     (   θ   )         )       ·     EL   ⁡     (     θ   -     n   ·       2   ⁢   π     N         )       ·     EL   ⁡     (     θ   -     m   ·       2   ⁢   π     N         )                           
Conformal Array
 
   In yet a third embodiment the antenna array may well be formed as a conformal array and calculated in a similar way corresponding to the above examples of the linear and circular array. 
   Simulations 
     FIG. 3  demonstrates the auto-correlation function for white bandwidth limited noise according to equation (5). As can be seen a bandwidth of 100% is needed to get the sidelobes down to a reasonable level. 
   For the simulations 100% bandwidth and a center frequency of 12 GHz was chosen. In other words the frequency range of the radar was 6-18 GHz. The distance to the target is determined by finding the peaks of the convolution between input and output signal. 
     FIG. 4  illustrates a simulation result when signal-to-noise was varied. In this case the target is an object, 1 meter long, standing still at a distance of 750 meters. The target area as function of distance is an equally distributed random number per distance sample. The sampling rate in this case was 50 GHz. From the response of the convolution the signature of the target can be interpreted in the form of the derivative of target area as function of distance. This may then be used to classify and identify the target by comparison with suitable library data. 
     FIG. 5  illustrates a principal block diagram of three bearing channels. Noise source N, antenna elements, power amplifier PA, filters and A/D converters are common for all bearing channels. By varying the time delays T the direction of the main lobe is selected. In a fully supported system many parallel bearing and distance channels are realized, such that the full solid angular sector continuously can be watched, for example almost ±90° for a linear receiving antenna array and ±180° for a circular receiver antenna array. 
   The principle for the angular resolution utilized is that grating lobes integrated over the large frequency range become rather low, while signals of a main lobe generally will be summed, which of course clearly comes out from the auto-correlation. 
     FIG. 7  illustrates power normalized to the main beam as function of angle −180 to +180 degrees for variation of number of elements in binary steps from 4 to 2048. As can be seen already with 32 elements a good angular resolution is obtained at −12 dB. In this simulation for instance 8 elements correspond to an inter-element distance corresponding to around 8 wavelengths along the perimeter of the array.  FIG. 7  further illustrates that a further increase of elements will not mean any significant further improvement from what is already achieved at element distances well above half a wavelength, i.e. 64 elements corresponding to two wavelengths at the lowest frequency. In standard thinking half a wavelength is generally the limit for avoiding grating lobes periodically repeating themselves in an array according to the state of the art. 
   As is also seen in  FIGS. 6 and 7  the auto-correlation function for the radiated signal appears in the angular direction. Interesting is that the auto-correlation function as a matter of fact can be chosen such that desired characteristics are obtained. The Fourier transform of the selected auto-correlation function gives necessary power spectrum of the radiated signal. 
   It will be understood by those skilled in the art that various modifications and changes could be made to the present invention without departure from the spirit and scope thereof, which is defined by the appended claims.