Abstract:
A computer-implemented method is provided for filtering clutter from a radar signal received by an antenna. The method includes determining a transient clutter voltage at first and second times separated by a time interval, determining a clutter correlation for the time interval, and dividing a received signal correlation by the clutter correlation. In alternate embodiments, the clutter correlation can be combined with a noise correlation and the sum divided by the signal correlation.

Description:
STATEMENT OF GOVERNMENT INTEREST 
       [0001]    The invention described was made in the performance of official duties by one or more employees of the Department of the Navy, and thus, the invention herein may be manufactured, used or licensed by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor. 
     
    
     BACKGROUND 
       [0002]    The invention relates generally to radar filtering. In particular, the invention relates to discrimination from clutter of received radar signals by distinguishing antenna patterns. Such clutter from low-velocity sources can obscure the target, compensation for this effect being the inventive focus. 
         [0003]    Radar systems employ Doppler processing to discriminate targets from clutter. This process operates satisfactorily for targets having high Doppler frequency that contrast with clutter typically having zero or low Doppler frequency. However, this does not hold for Doppler frequencies close to the clutter. Detection of slow moving targets necessitate having an accurate estimate of the clutter&#39;s Doppler spectrum because the detection process endeavors to filter out the clutter power based on background Doppler spectrum. Clutter presents undesirable radar return signals and thereby constitutes noise. 
         [0004]    Any error in the knowledge of the clutter Doppler spectrum degrades the detector&#39;s ability to distinguish targets. This degradation is negligible for fast moving targets but can be quite significant for targets whose Doppler frequency approaches the clutter Doppler spectrum. The classic approach to the problem of determining the clutter spectrum for the benefit of improved target detection is some sort of on-line clutter estimation scheme coupled with a detector. 
         [0005]    There have been many technical papers that incorporate this approach by estimating the clutter spectrum and including this information in their detector structure, such as R. S. Raghavan, “Statistical Interpretation of a Data Adaptive Clutter Subspace Estimation Algorithm”,  IEEE Transactions on Aerospace and Electronic Systems,  48 (2), 1370-1384 (April 2012); Peng-Lang Shui, Yan-Ling Shi, “Subband ANMF Detection of Moving Targets in Sea Clutter”,  IEEE Transactions on Aerospace and Electronic Systems,  48 (4), 3578-3593 (October 2012). However, these approaches are complicated and must compensate for the non-stationarity of clutter. 
         [0006]    This leads to the problem of obtaining sufficient training data, while taking into account real world issues of said training data being corrupted due to radio frequency interference (RFI). The exemplary approach described in the disclosure enables the radar designer to estimate the clutter spectrum accurately using knowledge of the antenna pattern alone. Using the accurate estimate of the clutter spectrum enables providing an optimum filter with the addition of an estimate of clutter-to-noise ratio (CNR), which can be accurately measured on-line or estimated with a clutter model. 
         [0007]    Exemplary embodiments improve weather prediction using radar. Weather radars produce the three weather determinations based on analysis of clutter, as described by D. J. Doviak, et al.,  Doppler Radar and Weather Observations  2 nd  edition, Academic Press (1993). These are: (1) Weather signal power of the zeroth moment of the Doppler spectrum. (2) Mean Doppler velocity of the first moment of the power-normalized spectra. (3) Spectrum width, the square root of the second moment about the first of the normalized spectrum. This is a measure of the velocity dispersion within the resolution volume. 
         [0008]    Clutter can seriously degrade the accuracy of the weather moments produced by weather radars. The largest amplitude clutter that weather radars must contend with is ground clutter. The classic approach to the problem of clutter in weather radar involves filtering out the clutter using Doppler processing. These schemes rely on assumptions of the clutter correlation matrix or Doppler spectrum. (Correlation and spectrum are both related by the Fourier transform. Thus, knowing one enables computing the other.) 
       SUMMARY 
       [0009]    Conventional Doppler radar filtering techniques yield disadvantages addressed by various exemplary embodiments of the present invention. In particular, a computer-implemented method is provided for filtering clutter from a radar signal received by an antenna by providing clutter temporal correlation properties solely from the antenna combined with clutter modeling to determine clutter-to-noise ratio, and forming a clutter filter to optimally remove clutter while preserving the desired signal. 
         [0010]    The method includes determining a transient clutter voltage at first and second times separated by a time interval, determining a clutter correlation for the time interval, and dividing a received signal correlation by the clutter correlation. In alternate embodiments, the clutter correlation can be combined with a noise correlation and the sum divided by the signal correlation. In other embodiments, a computer-implemented device is provided to execute the exemplary operations. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0011]    These and various other features and aspects of various exemplary embodiments will be readily understood with reference to the following detailed description taken in conjunction with the accompanying drawings, in which like or similar numbers are used throughout, and in which: 
           [0012]      FIG. 1  is an elevation view of an exemplary phased radar antenna array; 
           [0013]      FIG. 2  is a graphical view of an antenna pattern gain plot; 
           [0014]      FIG. 3  is a graphical view of a clutter time correlation plot; 
           [0015]      FIG. 4  is a graphical view of a Doppler frequency clutter spectrum plot; 
           [0016]      FIG. 5  is a graphical view of a time correlation target response plot; 
           [0017]      FIG. 6  is a graphical view of a Doppler frequency target spectrum plot; 
           [0018]      FIG. 7  is a graphical view of an optimum filter frequency response plot; 
           [0019]      FIG. 8  is a graphical view of an optimum MTI frequency plot; 
           [0020]      FIG. 9  is a graphical view of a frequency weather spectrum plot with clutter; 
           [0021]      FIG. 10  is a graphical view of a frequency weather spectrum plot sans clutter; and 
           [0022]      FIG. 11  is a tabular view of parameters relating to the antenna correlation. 
       
    
    
     DETAILED DESCRIPTION 
       [0023]    In the following detailed description of exemplary embodiments of the invention, reference is made to the accompanying drawings that form a part hereof, and in which is shown by way of illustration specific exemplary embodiments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention. Other embodiments may be utilized, and logical, mechanical, and other changes may be made without departing from the spirit or scope of the present invention. The following detailed description is, therefore, not to be taken in a limiting sense, and the scope of the present invention is defined only by the appended claims. 
         [0024]    In accordance with a presently preferred embodiment of the present invention, the components, process steps, and/or data structures may be implemented using various types of operating systems, computing platforms, computer programs, and/or general purpose machines. In addition, those of ordinary skill in the art will readily recognize that devices of a less general purpose nature, such as hardwired devices, or the like, may also be used without departing from the scope and spirit of the inventive concepts disclosed herewith. General purpose machines include devices that execute instruction code. A hardwired device may constitute an application specific integrated circuit (ASIC), digital signal processor (DSP), a field programmable gate array (FPGA) or other related component. 
         [0025]    Because exact knowledge of the clutter spectrum is not known, simple assumptions are made in designing the filters to remove the clutter that reduces performance. The exemplary approach described herein enables a radar designer to accurately estimate the clutter spectrum using characteristics of the antenna pattern alone. Based on this information, an optimum clutter filter can be designed. In addition other more sophisticated techniques of clutter elimination can further improve performance by using an exemplary clutter correlation spectrum determined herein. The direct advantages of exemplary embodiments are enabling the weather radar designer to maximize clutter reduction minimize distortion of a weather signal. 
         [0026]      FIG. 1  shows an elevation view  100  of an exemplary phased radar array  110 . An antenna panel  120  connects to a rotation axis  130 . The panel  120  has a phase center  140  and the rotation axis  130  is separated from that phase center  140  by radial distance r  150 . The axis  130  turns at angular speed w  160 . From the phase center  140 , the panel  120  projects an antenna array normal vector  170  and beam pointing vector  180  for radiation direction, respectively from normal angle θ B  and azimuth angle θ. Because the phase center  140  and the rotation axis  130  do not coincide, and the direction of radiation and the antenna normal angle  170  also do not coincide, there will be an instantaneous linear velocity of the antenna in the direction of the beam pointing angle  180 . 
         [0027]      FIG. 2  shows a graphical view of an antenna pattern gain plot  800 . The abscissa denotes azimuth angle  210  in degrees, and the ordinate denotes gain  220  in decibels. A legend  230  identifies two-way gain  240 , receive gain  250  and transmit gain  260  in increasing order, and peaks at normal (i.e., zero azimuth). The two-way antenna gain  240  modulates the clutter and target amplitudes as the antenna panel  120  rotates. 
         [0028]      FIG. 3  shows a graphical view of a time correlation plot  300  for clutter response at 10 milliseconds (ms) coherent processing interval (CPI), number of pulses at one-hundred. Conditions for the plot  300  include antenna rotation rate of 30 revolutions-per-minute (rpm) about the axis  130 , and offset distance  150  of 1 meter (m) for the phase center  140 . The abscissa denotes time offset τ  310  in seconds (s), and the ordinate denotes clutter correlation function R c    320  in decibels (dB). The inverse parabolic response curve  330  extends in time domain from +0.01 s to +0.01 s and in power range from −2 dB to 0 dB. 
         [0029]      FIG. 4  shows graphical view of a Doppler frequency clutter spectrum plot  400  for CPI with antenna rotation. The Doppler spectrum is obtained by performing a Fourier transform of the clutter response time correlation function in graph  300 . The abscissa and ordinate respectively denote frequency  410  in Hertz and power  420  in decibels. The response curve  430  shows minimum side-lobe powers of about −44 dB beyond ±400 Hz, which is determined by the window function. The peak response of the clutter spectrum  440  is offset from zero due to the rotation of the antenna panel  120  as illustrated in view  100 . The spectral spread of the clutter spectrum  450  is due to the modulation of the antenna two-way gain  240  and the finite length of the CPI being 10 ms. 
         [0030]      FIG. 5  shows graphical view of a time correlation target response plot  500  for target spectrum response at 10 ms CPI, number of pulses at one-hundred, and with antenna rotation previously described. The abscissa denotes time offset  510  τ in seconds, and the ordinate denotes correlation function R s    520  of the target correlation in decibels. The inverse parabolic response curve  530  extends in time domain from −0.01 s to +0.01 s and in response range from −2 dB to 0 dB and is a function of the antenna two-way gain  240 . 
         [0031]      FIG. 6  shows graphical view of a Doppler frequency target spectrum plot  600  for CPI with antenna rotation. The Doppler spectrum is obtained by a Fourier transform of the target response time correlation function in plot  500 . The abscissa denotes frequency  610  in Hertz, and the ordinate denotes power  620  in decibels. The response curve  630  shows maximum response  640  determined by the target range rate and the antenna motion. Spectral width  650  is determined by the antenna pattern of the two-way gain  240  and the finite CPI length of 10 ms. The main lobe for the spectral width  650  extends symmetrically from +460 Hz±320 Hz. 
         [0032]      FIG. 7  shows graphical view of an optimum filter frequency response plot  700 . The optimum filter maximizes the signal-to-interference ratio (SIR) for a target of known Doppler response where the interference is the sum of the receiver noise and the clutter signal. The abscissa denotes frequency  710  in Hertz, and the ordinate denotes power  720  in decibels. A response curve  730  shows a minimum peak  740  of about −85 dB at a frequency near 0 Hz corresponding to the maximum clutter Doppler spectrum  440  in view  400 . The maximum response  760  corresponds to the target maximum Doppler response  640  in view  600 . 
         [0033]      FIG. 8  shows graphical view of an optimum moving target indicator (MTI) filter frequency response plot  800 . The optimum MTI filter seeks to maximize the SIR for targets of unknown Doppler. The abscissa denotes frequency  810  in Hertz, and the ordinate denotes power  820  in decibels. The response curve  830  shows Doppler rejection region  840  with amplitude reduction over maximum filter response of at least 40 dB, and maximum rejection  850  (corresponding to maximum clutter response  440  of over 120 dB. 
         [0034]      FIG. 9  shows a graphical view of a frequency weather spectrum plot  900  with clutter. The abscissa denotes frequency  910  in Hertz, and the ordinate denotes power  920  in decibels. The response represents a desired weather target signal  930 . Spikes include a confounding clutter signal  940  and an average noise response  950  of 21 dB less than the desired signal  930 . Note separate peaks for clutter  940  at 0 Hz and weather target  930  at 60 Hz, and that clutter  940  and target  930  have similar values at 0 dB. 
         [0035]      FIG. 10  shows a graphical view of a frequency weather spectrum plot  1000  using exemplary embodiments to mitigate the clutter. The abscissa denotes frequency  1010  in Hertz, and the ordinate denotes power  1020  in decibels. The first response  1030  has an unaltered weather signal while the second response  1040  shows the clutter signal significantly diminished at about −12 dB, with average noise power unchanged; the noise threshold  1050  being at −21 dB. One can observe a reduction of about 10 dB of the clutter  1040  from the unfiltered clutter  940  that clearly distinguishes over peaks for the weather targets  930  and  1030 . 
         [0036]    Various exemplary embodiments provide improvements in the ability of radars to detect slow moving targets in the presence of clutter. An additional objective of the exemplary embodiments to improve the ability of weather radars to detect and measure weather phenomena by mitigating the negative effects of ground clutter. This disclosure describes a process to maximize the signal to interference ratio for slow-moving targets by applying the characteristics of the radar&#39;s two-way antenna pattern gain  240  and the clutter-to-noise ratio (CNR). The antenna pattern can be measured during manufacture of the antenna. 
         [0037]    The CNR can be measured from the radar or predicted using a clutter model such as the Littoral Clutter Model, as provided by George LeFurjah et al., “A Robust Integrated Propagation and Site Specific Land Clutter Model”,  IEEE Radar Conference,  (2007) 1-4244-0283-2. In addition, exemplary embodiments present a process to estimate with high accuracy the ground clutter correlation matrix/spectrum applying the knowledge of the radar&#39;s two way antenna pattern and CNR. 
         [0038]    Exemplary embodiments reveal the radar antenna as having a two-way voltage pattern sufficient to design an optimum filter, thereby maximizing the probability of detection of targets in the presence of clutter, and additionally for maximizing the weather signal and improving the estimates of weather moments. Hence, features of this exemplary technique include: 
         [0000]    (a) Determine the clutter correlation matrix from the two antenna voltage pattern;
 
(b) Determine the target correlation matrix from the two-way antenna voltage pattern;
 
(c) Determine the clutter-to-noise ratio (CNR) γ by direct measurement from the radar or from a suitable clutter model if required;
 
(d) Using information from (a) and (b) to develop an eigen-filter for application to the radar signal processor (ahead of the detector) for improved target detection; and
 
(e) Using information from (a) and (c) to apply clutter correlation properties to the MTI filter or other more sophisticated clutter mitigation schemes for improved weather moment estimation.
 
         [0039]    Optimum Filters: 
         [0040]    Artisans of ordinary skill recognize that antenna motion induces a Doppler spread on the clutter spectrum. By determining the clutter spectrum the optimum filter, i.e., the filter that maximizes signal-to-interference ratio (SIR) can be determined. This description shows that the clutter spectrum can be calculated à priori (via the correlation matrix) subject to reasonable assumptions, including that all the Doppler spread is induced by the antenna motion alone. 
         [0041]    While clutter sources such as trees and other objects at fixed locations can have internal motion, the largest scatterers tend to be buildings, towers and mountains. These scatters primarily produce very large direct current (DC) clutter with negligible nonzero Doppler relative to background. Because the largest clutter amplitudes most adversely affect target detection, one may assume that the nonzero Doppler spectrum of scatterers can be ignored. 
         [0042]    Conventionally, clutter is generally modeled as either distributed or discrete background noise. Distributed clutter is continuous over range and angle, and can have random values while appearing continuously. By contrast, discrete clutter only occurs at specific ranges and angles. Discrete clutter can occur at any range or angle, and can develop at any range or angle with random amplitude. In this derivation, the disclosure explains that the optimum filter is the same for distributed and discrete clutter. 
         [0043]    Clutter Correlation Derivation: 
         [0044]    Artisans of ordinary skill will recognize that antenna motion induces a Doppler spread on the clutter spectrum. By determining a more accurate clutter spectrum existing and future clutter mitigation schemes can significantly improve their performance. Thus, the clutter spectrum can be calculated a priori (via a correlation matrix) subject to some reasonable assumptions. The first assumption imposes all the Doppler spread being induced by the antenna motion alone, thereby ignoring smaller clutter sources, such as from trees in favor of larger clutter sources, such as buildings, bridges, towers and mountains. These scatters primarily produce very large DC clutter with negligible nonzero Doppler. Because the largest clutter amplitudes affect target detection the most, the nonzero Doppler spectrum of scatterers can be ignored. This disclosure demonstrates the important result that the spectrum correlation is identical for distributed and discrete clutter. 
         [0045]    Continuous Clutter: 
         [0046]    Continuous or distributed clutter appears at all angles with random amplitudes and is not resolvable in angle. Thus, transient output clutter voltage C(t) observed at slow time t at the output of the antenna is computed as: 
         [0000]        C ( t )=∫ c   t (θ) g (θ−θ p ) dθ,   (1)
 
         [0000]    where c t (θ) is the clutter voltage value at azimuth θ at slow time t, g is an angularly varying two-way voltage antenna pattern and θ p  is the pointing angle  180  of the beam at the start of the CPI.  FIG. 11  shows a tabular listing  1100  as a Table for the definitions of variables and symbols. The integration is conducted over the entire antenna pattern, which represents the zero elevation cut and may or may not be aligned with antenna bore-sight. Using the zero elevation cut is important as the beam rises because the ground clutter enters the radar as side-lobe effects. 
         [0047]    The integral in eqn. (1) sums up the back-scatter from all the scatters modulated by the antenna gain (described as pattern g) as a function of angle. The output clutter voltage at some later time t+τ is given as: 
         [0000]        C ( t +τ)=∫ c   t+τ (θ) g (θ−θ p −τ{dot over (θ)}) dθ,   (2)
 
         [0000]    where {dot over (θ)} is the antenna rotation rate, that also corresponds to the angular speed w  160 . This angular motion of the antenna is introduced through the antenna rotation rate correspondingly changes the antenna gain as a function of time. 
         [0048]    The antenna linear motion enters with clutter voltage value as follows: 
         [0000]        c   t+τ (θ)= c   t (θ)exp [ j τ{dot over (φ)}(θ−θ B −τ{dot over (θ)}],  (3)
 
         [0000]    where {dot over (φ)}(θ) is the phase change rate and accounts for the linear motion of the antenna in directions that are off-boresight in view  100 , j≡√{square root over (−1)}, τ is the time offset and θ B  is the boresight angle of the antenna. Note that time-phase ramp due to induced Doppler from linear angular motion is computed as: 
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         [0000]    and azimuth velocity for radial distance r can be expressed as: 
         [0000]        v   θ   =wr  sin(θ−θ B ).  (5)
 
         [0049]    The clutter correlation function  330  of output clutter voltage C(t) can be written as: 
         [0000]        R   c (τ)= E{C ( t ) C *( t +τ)},  (6)
 
         [0000]    where E represents expectation operator and C* denotes the complex conjugate of clutter voltage C. Next, using eqns. (1), (2) and (3), eqn. (6) can be rewritten as 
         [0000]        R   c ( T )= E{∫∫c   t (α) c   t *(β) g (α−θ p )exp [ j τ{dot over (φ)}(β−θ B −τθ)] g *(β B −θ B τ{dot over (θ)}) dαdβ},   (7)
 
         [0000]    where α and β are variables of integration, c* t  is the complex conjugate of clutter voltage value c t  and g* is the complex conjugate of pattern g. 
         [0050]    Next, two assumptions about clutter value c t  (θ) are invoked. First, is that the expectation of clutter voltage is zero mean: 
         [0000]        E{c   t (θ)}=0,  (8)
 
         [0000]    and secondly, that the clutter voltage value c t  is independent or uncorrelated over angle has unity power to produce: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    such that the cross correlation in angle of the clutter voltage is zero. 
         [0051]    Applying eqns. (8) and (9) to eqn. (7) yields: 
         [0000]        R   c ( t )=∫ g (θ−θ p ) g *(θ−θ p −τ{dot over (θ)})exp [− j τ{dot over (φ)}(θ−θ B −τ{dot over (θ)})] dθ.   (10)
 
         [0000]    This represents an important result showing ability to compute an accurate estimate of the clutter correlation function  330  based solely on knowledge of the antenna characteristics using eqn. (10). Moreover, eqn. (10) is independent of the clutter&#39;s distribution and only requires that the clutter has a zero mean and is independent for different azimuths. The distributions need not be the same for different azimuths. 
         [0052]    Because the clutter Doppler spectrum can be obtained by the Fourier transform of the time correlation function, eqn. (10) can be used to determine the Doppler spectrum of the clutter (as shown in graph  400 ). To determine optimum filter, one should establish an M×M correlation matrix R c  of the clutter, where M is the number of pulses in the CPI. This correlation matrix includes elements calculated from eqn. (10) by: 
         [0000]        R   c ( i,k )= R   c [( i−k ) T   s ],  (11)
 
         [0000]    where i is the row index, k is the column number and T s  is the time between pulses known as the pulse repetition interval (PRI). 
         [0053]    Discrete Clutter: 
         [0054]    Discrete clutter is produced by a single scatter whose azimuth and amplitude are random. Under this clutter model, the clutter voltage observed at slow time t at the output voltage of the antenna is computed as: 
         [0000]        C ( t )= c   t   g (θ−θ p ),  (12)
 
         [0000]    where azimuth angle θ is now a random variable. The clutter at later time offset τ is calculated as: 
         [0000]        C ( t +τ)= c   t   g (θ−θ p −τ{dot over (θ)})exp [ j τ{dot over (φ)}(θ−θ B −τ{dot over (θ)})].   (13)
 
         [0000]    The correlation function of the discrete clutter from eqn. (6) can be expanded to: 
         [0000]        R   c (τ)= E{|c   t | 2   g (θ−θ p ) g *(θ−θ p −−τ{dot over (θ)})exp [− j τ{dot over (φ)}(θ−θ B −τ{dot over (φ)})]}.  (14)
 
         [0055]    One can assume that the clutter voltage value c t  is zero mean having variance as unity. The random azimuth angle θ is assumed to be uniformly distributed. Further, one can assume that the clutter voltage value c t  and azimuth angle θ are statistically independent. These assumptions obtain: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0000]    where ±Δθ are the limits of the antenna pattern angle. Because the scaling factor of 1/(2Δφ) (or written ½(Δθ)−) can be ignored, eqn. (15) reduces as equivalent to eqn. (10). Thus, the clutter correlation function and the optimum filter are identical for distributed and point clutter. Because real world clutter is neither purely distributed or point clutter the filter derived herein remains the optimum filter (subject to the previous assumptions). 
         [0056]    Discrete Clutter at a Known Angle: 
         [0057]    For the case that the location of the interfering point clutter is known, a better filter can be developed. In this case the clutter is at angle θ c . This improves the filter because the uncertainty of the discrete clutter location has been removed. For this case, eqn. (12) becomes: 
         [0000]        C ( t )= c   t   g (θ c −θ p ),  (16)
 
         [0000]    and similarly the clutter at the output of the antenna at clutter at a later time t+z is calculated as: 
         [0000]        C ( t +τ)= c   t   g (θ c −θ 9 −τ{dot over (θ)})exp [− j τφ(θ c −θ B −τ{dot over (θ)})].  (17)
 
         [0000]    Based on this, the correlation function of the discrete clutter can be written as: 
         [0000]        R   c (τ)= E{C ( t ) C *( t +τ)}= g (θ c −θ p ) g *(θ c −θ p −τ{dot over (θ)})exp [− j τφ(θ c −θ B −τ{dot over (θ)})],  (18)
 
         [0000]    where C*(t+τ) is the complex conjugate of the antenna clutter at the later time. Note that the correlation function differs from eqn. (9), although this is a function of the antenna pattern. 
         [0058]    Target Correlation: 
         [0059]    Next a similar correlation matrix M s  of the target must be determined. If the beam motion is ignored the signal correlation matrix is determined as: 
         [0000]        M   s   =ss   H ,  (19)
 
         [0000]    where signal vector s=[1 exp(jw d T s ) . . . exp(jw d (M−1))] T  (transpose of a column matrix), H is the Hermitian conjugate transpose and w d  is the target&#39;s angular Doppler frequency. A more accurate manner to calculate signal correlation matrix takes into account the motion of the antenna. To accomplish this, one can define angle θ as the azimuth of a radial inbound target. 
         [0060]    Because the position of the target within the beam is unknown, this can be treated as a random variable and used to calculate the signal correlation matrix. To begin with, the signal s is modeled in continuous time as: 
         [0000]        s ( t )=exp [ j ( w   d   t +φ)] g (θ−θ p )exp [ j τ{dot over (φ)}(θ−θ B )],  (20)
 
         [0000]    where w d  is the target Doppler angular frequency, co is the random phase of the target, and θ is the azimuth angle of the target denoting a random variable. The amplitude of the target is a scaling factor that can be ignored. The target signal s at some time offset r later is: 
         [0000]        s ( t +τ)=exp [− j ( w   d ( t +τ)+φ] g (θ−θ p −τ{dot over (θ)})exp [ j τ{dot over (φ)}(θ−θ B −τθ].   (21)
 
         [0000]    The correlation function of the signal is determined as: 
         [0000]        R   s (τ)= E{s ( t ) s *( t +τ)},  (22)
 
         [0000]    where s* is the complex conjugate of the target signal s. 
         [0061]    The targets are assumed to be uniformly distributed in the beam. Therefore, angle θ is a uniformly distributed random variable. One can also note that targets outside the beam are blanked by the side-lobe blanker (SLB), the signal correlation 520 can be determined as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         R 
                         s 
                       
                        
                       
                         ( 
                         τ 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           exp 
                            
                           
                             [ 
                             
                               - 
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     
                                       ω 
                                       d 
                                     
                                      
                                     τ 
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                         
                           2 
                            
                           
                             θ 
                             SLB 
                           
                         
                       
                        
                       
                         
                           ∫ 
                           
                             
                               θ 
                               p 
                             
                             - 
                             
                               θ 
                               SLB 
                             
                           
                           
                             
                               θ 
                               p 
                             
                             + 
                             
                               θ 
                               SLB 
                             
                           
                         
                          
                         
                           
                             exp 
                              
                             
                               [ 
                               
                                 
                                   - 
                                   jτ 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     ϕ 
                                     . 
                                   
                                    
                                   
                                     ( 
                                     
                                       θ 
                                       - 
                                       
                                         θ 
                                         B 
                                       
                                       - 
                                       
                                         τ 
                                          
                                         
                                           θ 
                                           . 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                             g 
                              
                             
                               ( 
                               
                                 θ 
                                 - 
                                 
                                   θ 
                                   p 
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                               g 
                               * 
                             
                              
                             
                               ( 
                               
                                 θ 
                                 - 
                                 
                                   θ 
                                   p 
                                 
                                 - 
                                 
                                   τ 
                                    
                                   
                                     
                                       
                                           
                                       
                                        
                                       θ 
                                     
                                     . 
                                   
                                 
                               
                               ) 
                             
                           
                            
                           
                              
                             θ 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where θ RB  is the angular limit of the SLB function. Note that eqn. (23) is similar to eqn. (10), and absent any side-lobe blanker limit on target detection, this becomes exactly eqn. (10) modified by the target Doppler frequency factor exp [−j(w d τ)]. U.S. Pat. No. 4,959,653 provides an adaptive side-lobe blanker for an antenna. The signal correlation matrix is determined from the correlation function similar to eqn. (20) as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       M 
                       s 
                     
                      
                     
                       ( 
                       
                         i 
                         , 
                         k 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         exp 
                          
                         
                           [ 
                           
                             
                               - 
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     i 
                                     - 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                              
                             
                               T 
                               s 
                             
                           
                           ] 
                         
                       
                       
                         2 
                          
                         
                           θ 
                           SLB 
                         
                       
                     
                      
                     
                       
                         
                           R 
                           c 
                         
                          
                         
                           [ 
                           
                             
                               ( 
                               
                                 i 
                                 - 
                                 k 
                               
                               ) 
                             
                              
                             
                               T 
                               s 
                             
                           
                           ] 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0062]    The target time correlation determined by eqn. (23) is shown in graph  500 . Correspondingly, the target spectrum computed by eqn. (23) by the Fourier transform is shown in graph  600 . Without using the antenna patterns, a completely accurate target correlation function would not be possible to determine. 
         [0063]    Complete Interference Correlation: 
         [0064]    In order to form a filter or otherwise mitigate the effect of clutter, a measure of the clutter amplitude as compared to the receiver noise is needed. In order to accomplish that objective, one may perform a direct measurement from the radar to determine the clutter-to-noise ratio γ. Alternatively, one may use clutter models such as the Littoral Clutter Model by George LeFurjah et al., “A Robust Integrated Propagation and Site Specific Land Clutter Model”, IEEE Radar Conference, 2007. Doing this enables the noise correlation matrix to be determined as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       R 
                       n 
                     
                     = 
                     
                       
                         1 
                         γ 
                       
                        
                       I 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where I is the M×M identity matrix and γ is the clutter-to-noise ratio (CNR). Using the clutter-to-noise ratio γ then enables the complete interference correlation matrix to be calculated as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       R 
                       I 
                     
                     = 
                     
                       
                         R 
                         c 
                       
                       + 
                       
                         
                           1 
                           γ 
                         
                          
                         I 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where R c  is determined from eqn. (11). 
         [0065]    Optimum Filter for Known Target: 
         [0066]    Schleher provides coefficients for the filter (D. Curtis Schleher,  MTI and Pulsed Doppler Radar , pp. 283-284, Boston, Mass., Artech House) whose coefficients are equal to the eigenvector element values for the eigenvector associated with the largest eigenvalue for the combined matrix: 
         [0000]      ( R   c   +R   n ) −1   M   s .  (27)
 
         [0000]    Note that Schleher&#39;s interference correlation matrix R n  in his eq. (5.12) is denoted as R c  in eqn. (26). Optimum in this context means that the eigen-filter produces the highest signal-to-interference ratio output of all possible filters. 
         [0067]    The graph  700  shows the frequency response of the optimum filter for the clutter spectrum in graph  400  and the target spectrum in graph  600 . The exemplary filter is possible due to the ability to determine the clutter correlation matrix R c , the target correlation matrix R s  and the clutter-to-noise ratio I as described above. The exemplary filter significantly improves the detection of targets that have Doppler frequencies close to the clutter Doppler spectrum. 
         [0068]    Schleher (pp. 295-302) derives the optimum filter that maximizes the signal-to-interference ratio for the condition that the target Doppler speed is unknown. To apply this discovery to the exemplary filter one can observe that clutter correlation matrix R, in eqn. (11) is the same as R, in Schleher&#39;s eq. (5.68). Note also that noise correlation matrix R, in eqn. (25) is the same as R, in Schleher&#39;s eq. (5.68). Applying, eqn. (11) to the Optimized MTI Processor enables one to design a more accurate and better performing filter than possible with previous approach. Note that eqn. (11) for the clutter correlation matrix employs either eqn. (10) or eqn. (18) as applicable. 
         [0069]    Weather radar processing can be improved in other manners using exemplary embodiments. The plot  900  shows the Doppler spectrum of weather radar signals including clutter  940 , weather  930  and noise  950 . Using eqn. (10) or eqn. (18), one can calculate the clutter spectrum using the Fourier transform, as shown in plot  400 . Combining this with a direct measurement of the clutter amplitude or estimating from a clutter model enables one to calculate the clutter spectrum in reference to the noise level, and thereby subtract this from the Doppler spectrum, as illustrated in plot  1000 . Here, the clutter spectrum  1040  illustrates the clutter residue that is significantly attenuated by the exemplary technique while the weather spectrum  1030  and noise floor  1050  are not affected. Therefore, the exemplary techniques improve the ability of the radar to estimate weather phenomena. 
         [0070]    While certain features of the embodiments of the invention have been illustrated as described herein, many modifications, substitutions, changes and equivalents will now occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the embodiments.