Radar filter process using antenna patterns

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.

Claims

1. A computer-implemented filtering method for attenuating clutter in a radar signal from a target to a radar system, said signal being received by an antenna coupled to said radar system, said method comprising: determining properties of a clutter correlation function R.sub.c of the antenna for continuous clutter of an unknown azimuth angle at a time interval by a continuous relation:
R.sub.c()=g(.sub.p)g*(.sub.p{dot over ()})exp[j{dot over ()}(.sub.B{dot over ()})]d, where .sub.p is a pointing angle of a beam from the antenna, .sub.B is a boresight angle of the antenna, {dot over ()} is an antenna rotation rate, g is an angularly varying 4-way voltage antenna pattern gain, and {dot over ()} is a phase change rate and accounts for a linear motion of the antenna in off-boresight directions, j{square root over (1)} is the imaginary number, and g* is the complex conjugate of said pattern gain g; estimating said clutter-to-noise ratio for a signal correlation matrix: $M_s\left(i,k\right)=\frac{\exp \left[-j\left(i-k\right)T_s\right]}{2{}_{\mathrm{SLB}}}{R}_c\left[\left(i-k\right)T_s\right],$ where i is the row index, k is the column number, azimuth angle .sub.SLB is a side-lobe blanker azimuth angle limit, and T.sub.s is a pulse repetition interval; optimizing said clutter-to-noise ratio by a combination matrix:
(R.sub.c+R.sub.n).sup.1M.sub.s=0, where $R_s=\frac{I}{}I$ is a noise correlation matrix with I as an identity matrix and is a clutter-to-noise ratio; and obtaining a filter signal for the radar system from said combination matrix with which to detect and measure the target.

2. A computer-implemented filtering method for attenuating clutter in a radar signal from a target to a radar system, said signal being received by an antenna coupled to said radar system, said method comprising: determining properties of a clutter correlation function R.sub.c of the antenna for discrete clutter of a known azimuth angle .sub.c by a relation:
R.sub.c()=g(.sub.c.sub.p)g*(.sub.c.sub.p{dot over ()})expj(.sub.c.sub.B{dot over ()}), where .sub.p is a pointing angle of a beam from the antenna, .sub.B is a boresight angle of the antenna, {dot over ()} is an antenna rotation rate, g is an angularly varying two-way voltage antenna pattern gain, is a time offset, and {dot over ()} is a phase change rate and accounts for a linear motion of the antenna in off-boresight directions, j{square root over (1)} is the imaginary number, and g* is the complex conjugate of said pattern gain g; estimating said clutter-to-noise ratio for a signal correlation matrix: $M_s\left(i,k\right)=\frac{\exp \left[-j\left(i-k\right)T_s\right]}{2{}_{\mathrm{SLB}}}{R}_c\left[\left(i-k\right)T_s\right],$ where i is the row index, k is the column number, and azimuth angle .sub.SLB is a side-lobe blanker azimuth angle limit, and T.sub.s is a pulse repetition interval; optimizing said clutter-to-noise ratio by a combination matrix:
(R.sub.c+R.sub.n).sup.1M.sub.s=0, where $R_s=\frac{I}{}I$ is a noise correlation matrix with I as an identity matrix and is a clutter-to-noise ratio; and obtaining a filter signal for the radar system from said combination matrix with which to detect and measure the target.

3. A computing device that executes a filtering process to attenuate clutter in a radar signal from a target to an antenna coupled to a radar system, said device comprising: a clutter correlation processor for determining properties of a clutter correlation function R.sub.c of the antenna for continuous clutter of an unknown azimuth angle at a time interval by a continuous relation:
R.sub.c()=g(.sub.p)g*(.sub.p{dot over ()})exp[j{dot over ()}(.sub.B{dot over ()})]d, where .sub.p is a pointing angle of a beam from the antenna, .sub.B is a boresight angle of the antenna, {dot over ()} is an antenna rotation rate, g is an angularly varying two-way voltage antenna pattern gain, and {dot over ()} is a phase change rate and accounts for a linear motion of the antenna in off-boresight directions, j{square root over (1)} is the imaginary number, and g* is the complex conjugate of said pattern gain g; a ratio estimation processor for estimating said clutter-to-noise ratio for a signal correlation matrix: $M_s\left(i,k\right)=\frac{\exp \left[-j\left(i-k\right)T_s\right]}{2{}_{\mathrm{SLB}}}{R}_c\left[\left(i-k\right)T_s\right],$ where i is the row index, k is the column number, azimuth angle .sub.SLB is a side-lobe blanker azimuth angle limit, and T.sub.s is a pulse repetition interval; a combinatorial processor for said clutter-to-noise ratio by a combination matrix:
(R.sub.c+R.sub.n).sup.1M.sub.s=0, where $R_s=\frac{I}{}I$ is a noise correlation matrix with I as an identity matrix and is a clutter-to-noise ratio; and a signal processor for providing a filter signal for the radar system from said combination matrix with which to detect and measure the target.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) 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:

(2) FIG. 1 is a plan view of an exemplary phased radar antenna array;

(3) FIG. 2 is a graphical view of an antenna pattern gain plot;

(4) FIG. 3 is a graphical view of a clutter time correlation plot;

(5) FIG. 4 is a graphical view of a Doppler frequency clutter spectrum plot;

(6) FIG. 5 is a graphical view of a time correlation target response plot;

(7) FIG. 6 is a graphical view of a Doppler frequency target spectrum plot;

(8) FIG. 7 is a graphical view of an optimum filter frequency response plot;

(9) FIG. 8 is a graphical view of an optimum MTI frequency plot;

(10) FIG. 9 is a graphical view of a frequency weather spectrum plot with clutter;

(11) FIG. 10 is a graphical view of a frequency weather spectrum plot sans clutter; and

(12) FIG. 11 is a tabular view of parameters relating to the antenna correlation.

DETAILED DESCRIPTION

(13) 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.

(14) 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.

(15) 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.

(16) FIG. 1 shows a plan 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 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 .sub.B and azimuth angle . 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.

(17) FIG. 2 shows a graphical view of an antenna patter gain plot 200. 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.

(18) FIG. 3 shows a graphical view of a time correlation plot 300 for clutter response at 10 milliseconds (ms) coherent processing interval (CPI), with the 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 310 in seconds (s), and the ordinate denotes clutter correlation function R.sub.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.

(19) 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.

(20) 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 in seconds, and the ordinate denotes correlation function R.sub.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.

(21) 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 Hz320 Hz.

(22) 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.

(23) 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.

(24) 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.

(25) 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.

(26) 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'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.

(27) 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's two way antenna pattern and CNR.

(28) 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:

(29) (a) Determine the clutter correlation matrix from the two antenna voltage pattern;

(30) (b) Determine the target correlation matrix from the two-way antenna voltage pattern;

(31) (c) Determine the clutter-to-noise ratio (CNR) by direct measurement from the radar or from a suitable clutter model if required;

(32) (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

(33) (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.

(34) Optimum Filters:

(35) 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 priori (via the correlation matrix) subject to reasonable assumptions, including that all the Doppler spread is induced by the antenna motion alone.

(36) 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.

(37) 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.

(38) Clutter Correlation Derivation:

(39) 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 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.

(40) Continuous Clutter:

(41) 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 (i.e., for Doppler processing instead of pulse compression) at the output of the antenna is computed as:
C(t)=c.sub.t()g(.sub.p)d,(1)
where c.sub.t() is the clutter voltage value at azimuth at slow time t, g is an angularly varying two-way voltage antenna pattern and .sub.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 boresight. Using the zero elevation cut is important as the beam rises because the ground clutter enters the radar as side-lobe effects.

(42) 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+ is given as:
c(t+)=c.sub.t+()g(.sub.p{dot over ()})d,(2)
where {dot over ()} is the antenna rotation rate, that also corresponds to the angular speed 160. This angular motion of the antenna is introduced through the antenna rotation rate correspondingly changes the antenna gain as a function of time.

(43) The antenna linear motion enters with clutter voltage value as follows:
c.sub.t+()=c.sub.t()exp[j{dot over ()}(.sub.B{dot over ()})],(3)
where {dot over ()}() 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)}, is the time offset and .sub.B is the boresight angle of the antenna. Note that time-phase ramp due to induced Doppler from linear angular motion is computed as:

(44) $\begin{array}{cc}\left(-{}_B\right)=\frac{2v_{}\left(-{}_B\right)2}{}=\frac{4r\left(-{}_B\right)}{},& \left(4\right)\end{array}$
and azimuth velocity for radial distance r can be expressed as:
.sub.=r sin(.sub.B).(5)

(45) The clutter correlation function 330 of output clutter voltage C(t) can be written as:
R.sub.c()=E{C(t)C*(t+)},(6)
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:
R.sub.c()=E{c.sub.t()c.sub.t()g(.sub.p)exp[j{dot over ()}(.sub.B)]g*(.sub.B{dot over ()})dd},(7)
where and are variables of integration, c.sub.t* is the complex conjugate of clutter voltage value c.sub.t and g* is the complex conjugate of pattern g.

(46) Next, two assumptions about clutter value c.sub.t() are invoked. First, is that the expectation of clutter voltage is zero mean:
E{c.sub.t()}=0,(8)
and secondly, that the clutter voltage value c.sub.t is independent or uncorrelated over angle has unity power to produce:

(47) $\begin{array}{cc}E\left\{c_t\left(\right)c_t^{*}\left(\right)\right\}=\{\begin{array}{c}1,=\\ 0,\end{array},& \left(9\right)\end{array}$
such that the cross correlation in angle of the clutter voltage is zero.

(48) Applying eqns. (8) and (9) to eqn. (7) yields:
R.sub.c()=g(.sub.p)g*(.sub.p{dot over ()})exp[j{dot over ()}(.sub.B{dot over ()})]d.(10)
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'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.

(49) 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 MM correlation matrix R.sub.c of the clutter, where M is the number of pulses in the CPI. This correlation matrix includes elements calculated from eqn. (10) by:
R.sub.c(i,k)=R.sub.c[(ik)T.sub.s],(11)
where i is the row index, k is the column number and T.sub.s is the time between pulses known as the pulse repetition interval (PRI).

(50) Discrete Clutter:

(51) 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:
C(t)=c.sub.tg(.sub.p),(12)
where azimuth angle is now a random variable. The clutter at later time offset is calculated as:
C(t+)=c.sub.tg(.sub.p{dot over ()})exp[j{dot over ()}(.sub.B{dot over ()})].(13)
The correlation function of the discrete clutter from eqn. (6) can be expanded to:
R.sub.c()=E{|c.sub.t|.sup.2g(.sub.p)g*(.sub.p{dot over ()})exp[j(.sub.B{dot over ()})]}.(14)

(52) One can assume that the clutter voltage value c.sub.t is zero mean having variance as unity. The random azimuth angle is assumed to be uniformly distributed. Further, one can assume that the clutter voltage value c.sub.t and azimuth angle are statistically independent. These assumptions obtain:

(53) $\begin{array}{cc}{R}_c\left(\right)=\frac{1}{2}{}_{-}^{}g\left(-{}_p\right)g^{*}\left(-{}_p-\stackrel{.}{}\right)\exp \left[-j\left(-{}_B-\stackrel{.}{}\right)\right]d,& \left(15\right)\end{array}$
where are the limits of the antenna pattern angle. Because the scaling factor of 1/(2) (or written ().sup.1) 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).

(54) Discrete Clutter at a Known Angle:

(55) 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 .sub.c. This improves the filter because the uncertainty of the discrete clutter location has been removed. For this case, eqn. (12) becomes:
C(t)=c.sub.tg(.sub.c.sub.p),(16)
and similarly the clutter at the output of the antenna at clutter at a later time t+ is calculated as:
C(t+)=c.sub.tg(.sub.c.sub.p{dot over ()})exp[j{dot over ()}(.sub.c.sub.B{dot over ()})].(17)
Based on this, the correlation function of the discrete clutter can be written as:
R.sub.c()=E{C(t)C*(t+)}=g(.sub.c.sub.p)g*(.sub.c.sub.p{dot over ()})exp[j(.sub.c.sub.B{dot over ()})],(18)
where C*(t+) 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.

(56) Target Correlation:

(57) Next a similar correlation matrix M.sub.s of the target must be determined. If the beam motion is ignored the signal correlation matrix is determined as:
M.sub.s=ss.sup.H,(19)
where signal vector s=[1 exp(j.sub.d T.sub.s) . . . exp(j.sub.d(M1))].sup.T (transpose of a column matrix). H is the Hermitian conjugate transpose and .sub.d is the target'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 as the azimuth of a radial inbound target.

(58) 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:
s(t)=exp[j(.sub.dt+)]g(.sub.p)exp[j{dot over ()}(.sub.B)],(20)
where .sub.d is the target Doppler angular frequency, is the random phase of the target, and 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 later is:
s(t+)=exp[j(.sub.d(t+)+)]g(.sub.p{dot over ()})exp[j{dot over ()}(.sub.B{dot over ()})].(21)
The correlation function of the signal is determined as:
R.sub.s()=E{s(t)s*(t+)},(22)
where s* is the complex conjugate of the target signal s.

(59) The targets are assumed to be uniformly distributed in the beam. Therefore, angle 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:

(60) $\begin{array}{cc}{R}_s\left(\right)=\frac{\exp \left[-j\left({}_d\right)\right]}{2{}_{\mathrm{SLB}}}{}_{{}_p-{}_{\mathrm{SLB}}}^{{}_p+{}_{\mathrm{SLB}}}\exp \left[-j\stackrel{.}{}\left(-{}_B-\stackrel{.}{}\right)\right]g\left(-{}_p\right)g^{*}\left(-{}_p-\stackrel{.}{}\right)d,& \left(23\right)\end{array}$
where .sub.SLB 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(.sub.d)]. 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:

(61) $\begin{array}{cc}{M}_s\left(i,k\right)=\frac{\exp \left[-j\left(i-k\right)T_s\right]}{2{}_{\mathrm{SLB}}}{R}_c\left[\left(i-k\right)T_s\right].& \left(24\right)\end{array}$

(62) 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.

(63) Complete Interference Correlation:

(64) 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 . Alternatively, one may use clutter models such as LeFurjah's Littoral Clutter Model. Doing this enables the noise correlation matrix to be determined as:

(65) $\begin{array}{cc}{R}_n=\frac{1}{}I,& \left(25\right)\end{array}$
where I is the MM identity matrix and is the clutter-to-noise ratio (CNR). Using the clutter-to-noise ratio then enables the complete interference correlation matrix to be calculated as:

(66) $\begin{array}{cc}{R}_I=R_c+\frac{1}{}I,& \left(26\right)\end{array}$
where R.sub.c is determined from eqn. (11).

(67) Optimum Filter for Known Target:

(68) Schleher provides coefficients for the filter (D. Curtis Schleher, MTI and Pulsed Doppler Radar, 2010, 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:
(R.sub.c+R.sub.n).sup.1M.sub.s=0.(27)
Note that Schleher's interference correlation matrix R.sub.n in his eq. (5.12) is denoted as R.sub.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.

(69) 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.sub.c, the target correlation matrix R.sub.s and the clutter-to-noise ratio as described above. The exemplary filter significantly improves the detection of targets that have Doppler frequencies close to the clutter Doppler spectrum.

(70) 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.sub.c in eqn. (11) is the same as R.sub.c in Schleher's eq. (5.68). Note also that noise correlation matrix R.sub.n in eqn. (25) is the same as R.sub.n in Schleher'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.

(71) 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.

(72) 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.