Abstract:
A search radar including the improvement of apparatus for enhancing the estimation of the target angle within a search scan of the radar is disclosed. More specifically, the improvement apparatus utilizes the generated scan angles and target amplitude measurements correspondingly associated therewith to generate intermediate signals which are representative of the natural logarithm of the target amplitude measurements plus the square of the corresponding scan angle multiplied by a predetermined constant. For each search scan, the apparatus computes separately signals representative of the moments of: the scan angles, the squares of the scan angles, the products of the scan angles and corresponding intermediate signals, and the intermediate signals. In turn, the apparatus operates on the computed moment representative signals in some prespecified mathematical relationship to effect an optimum estimation for the target angle in each search scan of the radar. Simulated test results of the operation of the disclosed apparatus indicate an improvement over the simple centroid method of target angle estimation especially under the adverse conditions of a scintillating target with Rayleigh amplitude distribution.

Description:
BACKGROUND OF THE INVENTION 
     The present invention relates to single-channel search radars, in general, and more particularly, to apparatus for use therein which improves the estimation of the angle of a detected target within the search scan of the radar. 
     Most modern aircraft having weapon delivery systems generally employ a search radar for detecting targets of interest. In searching for a target, these radars usually scan through a spatial area with a plurality of looks or beam search samples. At each look, the radar may derive the amplitude of signals reflected from the target within the spatial area. Thereafter, the search radar may compute an estimated target scan angle from the amplitudes derived through the search scan. In turn, this estimated angle may be used to provide direction to the weapon delivery system for deployment of projectiles toward the target, for example. 
     An illustration of a typical scenario with regard to detecting a target is depicted in FIG.  1 . Suppose that the aircraft, denoted at  10 , is flying along a flight path  12  in the direction indicated by the solid arrow  14 . At a flight position P 1 , the search radar on-board the aircraft  10  may scan a spatial area  16  with its radar beam  18  in search of a target depicted in the figure as the dot  20 . In its search, the beam  18  of the radar may be scanned through a plurality of looks L 1 , L 2 , . . . , L 9  corresponding to a plurality of scan angles θ 1 , θ 2 , . . . , θ 9 . At each look L 1 , the search radar may correspondingly derive an amplitude ai of the radar signal reflected from the target  20 . An idealistic example of a plot of amplitudes a i  for the present example may appear as that shown by the x&#39;s on the dashed line in the graph of FIG.  2 . 
     Referring to FIG. 2, in some search radars, a simple centroiding procedure having the formula Σa i a i /Σa i , for example, has been used to compute the estimated target angle θ t  which, of course, falls between the scan angles θ 5  and θ 6  for the aircraft position P 1  in the present example. Accordingly, as the aircraft  10  moves to another position P 2 , another scan of looks may be performed and corresponding amplitudes computed by the search radar. Similarly, a curve of amplitudes for the search scan at position P 2  may be compiled as that shown by the second dashed line (P 2 ) curve in FIG.  2 . It follows that the computed centroid of this second curve (P 2 ) will be the estimated target angle with respect to the new aircraft position P 2 . 
     While for an ideal case, this simple centroiding procedure appears adequate for accurately estimating the true target angle for weapon delivery, it is evident that in more practical cases, the accuracy of the target angle estimation with this method may be somewhat degraded. For example, under most conditions, the aircraft search radar incurs undesirable noise at the input stages of the search radar itself. It happens that this instrumentation noise is inseparable from the echo signals returned from the target and thus tends to effect relatively large errors in the computation of the amplitude measurements of the target reflections through the various search beam directions. To further complicate matters, there is no guarantee that the beam scanning samples or looks will be scanned symmetrically about the true target angle. Moreover, even greater inaccuracies with the centroidal method can be expected when target scintillation provides further adverse noise sources. 
     Apparently, in view of the practical problems of noise as discussed above, the actual amplitude measurements derived by the search radar are not expected to follow any ideal curve fitting pattern for most practical sets of conditions. For example, the graph of FIG. 3 illustrates a case in which actual amplitude measurements r(θ i ), denoted by X&#39;s, do not coincide with the ideal 1 amplitude measurements s(θ i ) denoted by the dots lying substantially on the dashed line curve. In this case, it is quite apparent that the simple centroid of the actual amplitude measurements will not result in an accurate estimation of the true target scan angle. Consequently, if the calculated simple centroid was used as the true target angle, it would cause an erroneous deployment angle for the weapon delivery system of the aircraft, for example. 
     From the above, it is evident that to be a viable piece of equipment for enhancing the effectiveness of weapon deployment, as one example, the search radar of the aircraft must accurately estimate the true scan angle of the target under even the most adverse conditions of noise with regard to both the aircraft and target flights and the internal operations of the radar itself. To accomplish this, it is felt that more sophisticated apparatus beyond that of a simple centroiding method is needed to process the actual amplitude measurements as derived by the search radar. 
     SUMMARY OF THE INVENTION 
     A search radar, which includes means operative to transmit and receive radar signals for a plurality of predetermined scan angles within a search scan; means operative to generate a plurality of signals representative of the predetermined scan angles; and means operative to generate a plurality of target amplitude measurement signals derived from the received radar signals respectively corresponding to the plurality of predetermined scan angles, is improved by the addition of apparatus for estimating the target angle within a search scan. 
     More specifically, the apparatus comprises a first means operative to compute an intermediate signal for each prespecified angle of the plurality in accordance with a first function based on the generated angle signal and the generated target amplitude measurement signal correspondingly associated therewith; second means operative to compute signals representative of moment relationships of the corresponding plurality of prespecified scan angle signals and intermediate signals for a search scan of the radar; and third means operative to compute a signal representative of the estimated target angle for a search scan in accordance with a second function based on the correspondingly associated moment-related signals of the search scan computed by the second means. In the search scan of the present embodiment, means are provided for detecting the presence of a target within the search scan of the radar from the drive amplitude measurement signals thereof and for generating a target detect signal as a result of the detected condition. In this embodiment, the improvement apparatus includes means operative in response to the generated target detect signal to compute the estimated target angles corresponding to the search scans. 
     In accordance with one aspect of the invention, the first means includes means for computing first signals representative of the logarithm of the derived target amplitude measurement signals; means for computing second signals proportionately representative of the square of the generated scan angle signals; and means for adding corresponding first and second signals to compute the intermediate signals associated therewith. Further, the second means includes means for accumulating separately the first signals, the second signals, the signals representative of the scan angles, and the intermediate signals over the period of a search scan. Still further, the third means includes means for generating first, second, third and fourth product signals representative of the products of: the third signal and moment signal of the second signals, the moment signal of the scan angle signals and the moment signal of the intermediate signals, the moment signal of the first signals and the third signal, and the moment signal of the scan angle signals with itself, respectively; means for generating fourth and fifth signals by subtracting the second product signal from the first product signal and by subtracting the fourth product signal from the third product signal, respectively; and means for generating the signal representative of the estimated target angle for a search scan by dividing the fourth signal by the fifth signal. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is an illustration of a typical scenario of the operation of a search radar on board an aircraft for detecting a target within a search scan; 
     FIG. 2 is a graph idealistically exemplifying a simple centroid approach for estimating the scan angle of the detected target; 
     FIG. 3 is a graph which illustrates a case in which actual amplitude measurements from a target do not coincide with any one ideal curve fitting pattern as utilized by the simple centroid approach; 
     FIG. 4 is a functional block diagram schematic of a search radar suitable for embodying the principles of the present invention; 
     FIG. 5 is a block diagram schematic of a target angle estimator suitable for use in the embodiment depicted in FIG. 4; 
     FIG. 6 is a schematic block diagram depictig the target angle estimator embodiment of FIG. 5 in greater detail; and 
     FIGS. 7,  8  and  9  are graphs exemplifying the improvements attainable with the use of the target angle estimator of FIGS. 5 and 6 in comparison with a simple centroid method. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     Referring to the block diagram schematic of FIG. 4, a conventional antenna shown at  30  may be operative either mechanically, electronically, or a combination of both, to conduct a radar beam through a search scan as exemplified by the description of the embodiment in connection with FIG.  1 . For the present embodiment, the scan angle may be thought of as planar in the azimuth direction with respect to the aircraft. However, it is understood that an elevation direction or any planar combination therebetween may additionally be considered without deviating from the principles of the present invention. 
     In the embodiment of FIG. 4, the antenna  30  may be mechanically positioned through an azimuth scan search, for example, by a conventional gimbal mechanism  32  which in turn, may be governed by a conventional antenna controller  34 . In addition, the search radar embodiment of FIG. 4 may also include the conventional radar elements of a transmitter  36 , a duplexer  38 , a receiver  40 , a stable local oscillator  42 , a range/doppler processor  44 , a linear detector  46 , and some type of a constant false alarm rate (CFAR) detector  48 . In operation, the stable local oscillator  42  provides synchronizing clock signals to both the transmitter  36  and receiver  40  such that range and target echo doppler information can be associated correspondingly with the radar transmissions. 
     Generally, the transmitter  36  provides bursts or pulses of radar energy through the duplexer  38  and antenna  30  into a spatial area in accordance with the angle direction set by the antenna  30 . With respect to the beam pattern of the antenna system  30 , received signals reflected from a target are conducted through the antenna  30 , duplexer  38  to the receiver unit  40  wherein they are appropriately processed and passed along to the range/doppler processor  44 . After the range and doppler information is extracted from the target reflected signals, the signals are conducted to the linear detector  46 . Since these target reflected signals are complex in nature, the linear detector  46  computes the amplitudes thereof using conventional complex arithmetic techniques. The constant false alarm rate (CFAR) unit  48  generally functions cooperatively with the linear detector  46  to determine in a relative fashion if the target amplitude measurements derived by the detector  46  are greater than a relative threshold value such to be considered as being associated with a real target as opposed to merely instrumentation noise, for example. In turn, the CFAR  48 , may provide a target detect signal over signal line  50  in accordance with its relative threshold comparison operations. In addition, the linear detector  46  may provide the actual target amplitude measurements r(θ i ) over signal line  52 . 
     In accordance with the present invention, a target angle estimator unit  60  may be further disposed in the search radar  25  for deriving a true target estimation angle θ t  in accordance with the signals provided over signal lines  50  and  52  which are coupled thereto. In addition, the antenna controller  34  may be linked to the estimator unit  60  utilizing signal lines  61 ,  62  and  63 , for example, in order to provide the information of: the present antenna scan angle; when the scan angle is being moved to a new scan angle; and, when the antenna has completed a search scan, respectively. The estimator  60  employs the information conducted over the signal lines  50 ,  52 ,  61 ,  62  and  63  to derive the estimated true target angle θ t  which it in turn may deliver to an on-board weapon delivery system  65  of the aircraft  10  for use thereby. 
     Theoretically, the target angle estimator  60  performs its computations under the assumption that the target amplitude measurements in the presence of noise may be approximated by the following Gaussian function: 
     
       
           S (θ)= S   O exp(−α(θ−θ t ) 2 )  (1)  
       
     
     where α may be equal, in the present embodiment, to 1.388 divided by the square of the antenna beam width B taken at approximately −3 dB. The principles of operation of the estimator  60  may be based on determining the best fit between the deterministic function, s(θ), and the actual target amplitude measurement, r(θ), derived at the antenna sampling scan angles or looks. In accordance with the optimization operations of the target angle estimator  60 , the peak location of the best-fitted s(θ) is considered the optimal estimate of the true target angle in the search scan. 
     In developing the principles of the present invention mathematically, it was identified that the deterministic function, s(θ) is non-linear and therefore difficult to work with in regard to identifying the best-fitted curve for the actual target amplitude measurements r(θ) of a search scan. It was discovered that the fitting function may be reduced to a polynomial of θ in one case, for example, by performing a natural logarithmic transformation thereon. This logarithmic transformation produces the following equation: 
     
       
           lns (θ)= lnS   O −α(θ−θ t ) 2   (2)  
       
     
     As is well known, there is always some concern in performing a transformation on a non-linear function especially with regard to the errors associated with large amplitudes. However, it is noted that the derivative of ln s(θ) with respect to s(θ) is the inverse of s(θ) as shown by the following equation:                       ln                     s        (   θ   )                s        (   θ   )           =     1     s        (   θ   )                 (   3   )                                
     Thus, the logarithmic transformation of the deterministic function provides the desired effect of suppressing the significance of errors at large amplitudes. 
     It is the purpose of this exemplary mathematical exercise to transform the deterministic function into a linear set of equations or curves to better identify the best-fitted curve and thus the target angle estimation. For this purpose, equation (2) may be expanded as shown below in equation (4) and thereafter reorganized as shown below in equation (5). 
     
       
           lns (θ)= lns   O −αθ 2 +2αθθ t −αθ t   2   (4)  
       
     
     
       
           lns (θ)+αθ 2   =lns   O −αθ t   2 +2αθθ t   (5)  
       
     
     With the reorganization as shown in equation (5), it is readily apparent that those terms to the left of the equation sign are solely a function of θ and may be denoted mathematically as: 
     
       
           y (θ)= lns (θ)+αθ 2   (6)  
       
     
     Moreover, if it is assumed that s O , α, and θ t  are all fixed for each deterministic function or curve, then the first two terms to the right of the equation sign of the reorganization equation (5) may be represented as a constant, say k O , for example, as shown by the following equation: 
     
       
           k   O   =lns   O −αθ t   2   (7)  
       
     
     Similarly, the portion 2αθ t  of the remaining term in the reorganization equation (5) may also be represented as another constant, say k 1 , for example, (i.e. k 1 =2αθ t ). With these mathematical notations in mind then, the reorganization equation 4 may be rewritten in a linear form as shown below: 
     
       
           y (θ)= k   O   +k   1 θ  (8)  
       
     
     Now if it is assumed that there are N+1 discrete scan angles in the search scan of the radar, then each scan angle may be denoted as θ i  with i=0, 1, . . . N. Using the same notation then, the target amplitude measurements with each discrete scan angle θ i  may be denoted as r(θ i ). Furthermore, if the corresponding values of scan angles and target amplitude measurements are substituted into the intermediate equation for y(θ), that is, 
     
       
           y   i   =lnr (θ i )+αθ i   2 ,  (9)  
       
     
     then a family of N+1 linear curves may be obtained for each i having only the constants k 0  and k 1  as shown below:                      y   o         =         k   o         +           k   1          θ   o                 y   1         =         k   o         +           k   1          θ   1               ⋮       ⋮       ⋮                   ⋮             y   N         =         k   o         +           k   1            θ   N     .             }           (   10   )                                
     It was further identified that this set (10) of N+1 linear curves or equations may be placed in matrix form with the following notations:                Y   _     =     [           y   o               y   1             ⋮             y   N           ]             (   11   )                 θ   _     =     [         1         θ   o             1         θ   1             ⋮       ⋮           1         θ   N           ]             (   12   )                 k   _     =     [           k   o               k   1           ]             (   13   )                                
     Using this matrix notation then, the set of linear equations may be rewritten as follows: 
     
       
           {overscore (y)}={overscore (θ)}{overscore (k)}   (14)  
       
     
     It was observed that the optimal value of the matrix {overscore (k)} which yields the best fitted curve for the above matrix equation using the least squares method, for example, may be provided by the following equation: 
     
       
           {overscore (k)}= ({overscore (θ)} τ {overscore (θ)}) −1 {overscore (θ)} τ   {overscore (y)}   (15)  
       
     
     For a better understanding of this mathematical optimization technique reference is made to the text “Applied Optimal Estimation” edited by Arthur Gelb and published by MIT Press (1974), and more particularly the section on Optimal Linear Filtering found therein. 
     Using well-known matrix mathematical techniques, the optimization equation (15) for the matrix {overscore (k)} may be further reduced as follows:                k   _     =       [           k   o               k   1           ]     =         [           ∑     θ   i   2             -     ∑     θ   i                   -     ∑     θ   i               N   +   1           ]           (     N   +   1     )          ∑     θ   i   2         -       (     ∑     θ   i       )     2              [           ∑     y   i                 ∑       θ   i          y   i               ]                 (   16   )                                
     In solving for k 1 , it is found that:                k   1     =           (     N   +   1     )          ∑       θ   i          y   i           -     ∑       θ   i          ∑     y   i                   (     N   +   1     )          ∑     θ   i   2         -       (     ∑     θ   i       )     2                 (   17   )                                
     Since it is known that k 1  is proportional to the estimated target scan angle θ t  by the equation k 1 =2αθ t , then the optimal target scan angle θ t  may be expressed as:                  θ   ^        t     =       k   1       2   ∝               (   18   )                                
     The target angle estimator embodiment will now be described in greater detail using the schematic block diagram embodiments of FIGS. 5 and 6 while keeping the mathematical principles described hereabove in mind. Referring to FIG. 5, signals representative of the scan angle θ i  and target amplitude measurements r(θ i ) may be provided to a logarithmic computer  70  using signal lines  61  and  52 , respectively. A conventional signal generator  72  may also be utilized, in the present embodiment, for providing a signal representative of the constant α to the logarithmic computer  70  over signal line  74 . The primary function of the logarithmic computer  70  may be to compute the intermediate signal y i  for each scan angle θ i  and associated target amplitude measurement r(θ i ) for a search scan of the radar. The computations may be performed in the computer  70  in accordance with a first function based on the generated angle signal θ i  and the generated target angle measurement signal r(θ i ) correspondingly associated therewith. As a by-product of the intermediate signal computation of  70 , a signal θ i   2  may also be produced representative of the square of the scan angle signal θ 1 . 
     The representative signals y i ,θ i   2 , and θ i  may be provided to a moment computer  76  which is operative to compute signals representative of moment relationships of the scan angle signals θ i  and intermediate signals y i  for a search scan of the radar. To assist in the timing relationships for the moment computations of the computer  76  the timing signals denoted as LOOK and SCAN COMPLETE additionally be provided to the moment computer over signal lines  62  and  63 , respectively. The resulting moment signals computed by the moment computer  76 , denoted by the symbols Σθ i , Σθ i   2 , Σθ i y i , and Σy i , for example, may be provided to an optimum angle processor  80  over their appropriately designated signal lines as shown in the schematic embodiment of FIG.  5 . The moment computer  76  may be additionally operative to identify the number of discrete scan angles in the search scan of the radar, the number being denoted as N+1. A signal representative of this scan angle number is also provided to the processor  80  over an appropriately designated signal line. Accordingly, the optimum angle processor  80  is operative to compute a signal representative of the estimated target angle θ t  for a search scan in accordance with a second function based on the correspondingly associated moment related signals of the search scan computed by the moment computer  76  and provided thereto over the designated signal lines. 
     In the preferred embodiment, in order to improve the efficiency of operation of the target angle estimator  60 , the various computation blocks  70 ,  76  and  80  of the target angle estimator  60  may be made operative only in response to the generated target detect signal over line  50 . In this manner, the estimator  60  may not perform unnecessary computations when a target is not present in the search scan of the radar. 
     The target angle estimator  60  is depicted in greater detail in the schematic block diagram embodiment of FIG.  6 . Referring to FIG. 6, the target amplitude measure signal line  52  may be coupled to a conventional logarithmic functional block  81  disposed in the computer  70 , the output of which being coupled to a conventional adder  82 . The signal line  74  carrying the constant a representative signal may be coupled to one input of a conventional multiplier  84  and the output of the multiplier  84  may be coupled to the other input of the adder unit  82 . Moreover, the signal line  61  carrying the scan angle signals θ i  may be coupled to both inputs of another multiplier unit  86 , the output of which may be coupled to the other input of the first multiplier  84 . The output of the multiplier  86  which carries the signal representative of θ i   2  and the output of the adder  82  which carries the signals representative of the intermediate signal y i  may both be coupled to the moment computer  76  over signal lines  88  and  90 , respectively. In addition, the signal line  61  may also be coupled to the moment computer  76 . 
     Disposed within the moment computer  76  may be five conventional summer accumulator combinations denoted in FIG. 6 as S 1 -A 1 , S 2 -A 2 , S 3 -A 3 , S 4 -A 4 , and S 5 -A 5 . In each case, the output of the summer unit is coupled to the input of the accumulator and the output of the accumulator may be fed back to one input of the summer unit. In addition, the outputs of the accumulators A 1 -A 5  may be coupled respectively to the inputs of a set of switches SW 1 -SW 5 . In the present embodiment, the scan complete signal provided over signal line  63  may be coupled to each accumulator and each switch for governing the timely operation thereof in connection with the search scan of the radar. Moreover, the signal lines  61  and  90  may be coupled to he inputs of the summer units S 2  and S 5  and in addition, ay be coupled to the inputs of another conventional multiplier unit  92 . The signal line  88  may be coupled to the input of the summer unit S 3  and the output of the multiplier unit  92  may be coupled to the input of the summer unit S 4 . Another switching device SW 6  may be disposed in the moment computer  76  and may have a signal representative of a unit  1  coupled to its input and may be operated in the open and closed positions by the signal LOOK coupled thereto over signal line  62 . The output of switch SW 6  may be coupled to the input of the summer unit S 1 . Finally, the outputs of the switching units SW 1 -SW 5  may be provided to the optimum angle processor  80  over the signal lines  94  through  98 , respectively. The signal lines  95  through  98  carry the signals representative of the moment relationships of the scan angle signals θ i  and the intermediate signals y i , and the signal line  94  carries the signal representative of the number of scan angles in the search scan (i.e. N+1). 
     Disposed in the processor  80  may be four additional conventional multiplier units  100 ,  102 ,  104  and  106 . The signal lines  94  and  97  may be coupled to the inputs of unit  100 , the signal lines  95  and  98  may be coupled to the inputs of unit  102 , the signal lines  94  and  96  may be coupled to the inputs of unit  104  and the signal line  95  may be coupled to both inputs of the unit  106 . The outputs of the multiplier units  100  and  102  may be coupled to the + and − inputs of another summer unit S 6  disposed in the processor  80 . Similarly, the outputs of the multiplier units  104  and  106  may be coupled to the + and − inputs of another summer unit S 7  of the processor  80 . The output signal lines of the summers S 6  and S 7  may be coupled to a conventional divider unit  108 , the output line  110  of which providing the target angle estimation signal θ t . 
     In a typical operation, then, as the CFAR unit  48  identifies the presence of a target, the target detect signal may be provided to the target angle estimator block  60  over signal line  50  and activate the operation thereof. At the start of a search scan, the signal conducted over line  63  enables the accumulators A 1  through A 5  to begin their accumulation process. At each scan angle sample or look in the search scan of the radar, signals representative of the scan angle θ i  and the target amplitude measurement r(θ i ) are provided over signal lines  61  and  52 , respectively, to the computer  70 . From each of the corresponding sets of signals the logarithmic computer  70  computes the correspondingly associated signals representative of θ i   2  and y i . 
     During a search scan, as each new representative signal of the group θ i , θ i   2 , θ i y i , and y i  are generated, they are, in turn, accumulated in the corresponding accumulators A 2  through AS utilizing the summer units S 2 -S 5 , respectively associated therewith. In addition, with each new scan angle sample or look, switch SW 6  is governed closed by the signal over signal line  62 , for example, and the total number of looks in a search scan is accumulated in A 1  utilizing the summer S 1 . At the end of a radar search scan, the contents of the accumulators A 1  through A 5  contain the moment and other representative signals desirable for computing the optimum target angle estimation θ t . At search scan completion, the switching units SW 1  through SWS may be governed to their closed position by the signal over signal line  63 , for example, to provide the respective signals from the accumulators A 1  through A 5  to the processor  80 . 
     In processor  80  the multipliers  100 ,  102 ,  104  and  106  multiply the signal pairs coupled to the inputs thereof and provide the resulting signal outputs to the appropriate summer units S 6 , or S 7  wherein the predesignated subtraction operation are performed in each case. The signal representative of the difference operation of summer S 6  is divided by the signal representative of the difference operation of summer S 7  in the divider unit  108  to produce a signal representative of the target angle estimation over signal line  110 . Accordingly, a similar set of operations may be performed in connection with each search scan of the radar and likewise an optimal target angle estimation signal {overscore (θ)} t  may be effected for each search scan. 
     A simulation of the principles of the present invention was performed to determine the improvements attainable with the use of the optimum target angle estimator in comparison with the simple centroiding method as discussed in the background section hereinabove. The simulation included a scintillating target with Rayleigh amplitude distribution which may be decorrelated between scan looks. In the examples below, 13 look measurements were used in the target angle calculations and the results were obtained with 1,000 trials having random alignment between the look scan angles and the true target angle. Exemplary results of the simulations are depicted in the graphs of FIGS. 7,  8  and  9 . 
     In FIG. 7, which was conducted under the conditons of three looks per dwell, the RMS target angle error of the optimal estimator (solid line curve)is approximately half of that of the centroidal method (dashed line curve) at 0 dB signal-to-noise ratio (S/N), the improvement being gradually lower at higher S/N ratios. For example, for this example under 20 to 40 dB signal-to-noise ratio conditions, the optimal estimator, as simulated, suffered a slight degradation of 0 to 12%. 
     The simulation results of four and five looks per dwell cases as depicted in the graphs of FIGS. 8 and 9, respectively, show improvement over the entire 0 to 40 dB signal-to-noise ratio region. Under the four looks per dwell simulated conditons, the improvement decreases from 50% at 0 dB to 5% at 25 dB and then back up to 30% at 40 dB S/N ratio. Similar results are observed in the five looks per dwell example wherein the optimal estimator under simulated conditions provides improvement over the centroid method from 40% at 0 dB, down to 15% at 20 dB, and then, back up to 30% at 40 dB target signal-to-noise ratio. 
     With regard to the embodiments described in connection with FIGS. 4,  5  and  6 , it is preferred that the target angle estimator be operative to perform calculations only with the information in those range-doppler cells which satisfy the detection criteria of the CFAR  48 . Therefore, the number of operational calculations may be thus limited. Other advantages include: (1) the calculations involved in the embodied angle estimator  60  are mostly summations and multiplications involving only real numbers; and (2) the natural logarithmic function performed by block  80  in the logarithmic computer  70  may be implemented by a simple look-up table, for example. Moreover, since the present-day digital processing hardware capability is generally well-known by all those skilled in the pertinent art, the implementation of the various other elements as depicted in the block diagram schematic of FIG. 6, for example, may be implemented using straight forward engineering techniques. 
     It should also be pointed out that while the embodiment described in connection with FIG. 6 was used in the present application to describe the principles of the invention, it is understood that modifications may be made to this embodiment or other embodiments may be used to carry out the principles of Applicant&#39;s invention without deviating from those principles. Accordingly, Applicant&#39;s invention should not be limited to any one embodiment, but rather construed in connection with the broad scope and breadth of the claims heretofollow.