Patent Publication Number: US-2011071791-A1

Title: Signal-level determining device and method

Description:
FIELD OF THE INVENTION 
     The present invention relates to a method and apparatus for detecting a signal within background noise having impulsive interference, and is particularly, but not exclusively, well suited to detecting a signal reflected from a small object in the presence of interfering signals backscattered by the dynamically disturbed sea surface. 
     BACKGROUND OF THE INVENTION 
     In many practical applications, a signal of interest is corrupted by a mixture of noise with essentially Gaussian characteristics (e.g., thermal noise) and interference of impulsive nature. The probability distribution of such combination will often exhibit so-called ‘heavy’ tails, and various statistical models have been developed to characterize non-Gaussian phenomena. For example, the magnitude of a microwave signal reflected from the sea surface is often characterized in terms of Weibull, log-normal or K distribution. 
     Microwave sensors operating in a maritime environment are expected to reliably detect various small objects of potential interest in the presence of unwanted signals reflected from the sea surface, often referred to as sea clutter. Small objects to be detected include boats and rafts, buoys, various debris and small fragments of icebergs. Some of those objects may pose a significant threat to safe ship navigation, whereas other objects are of interest in search-and-rescue missions, coastal surveillance, homeland security etc. 
     Non-Gaussian sea clutter can negatively affect the detection performance of many sensors, often designed for optimum operation in Gaussian noise. Accordingly, over the many years, different solutions have been offered to the problem of detecting small signals in sea clutter. 
     A representative example of a practical non-coherent system capable of detecting objects in sea clutter is presented in U.S. Pat. No. 7,286,079, 23 Oct. 2007: Method and Apparatus for Detecting Slow-Moving Targets in High-Resolution Sea Clutter. 
     in accordance with the above disclosure, a comparator is used to convert radar returns from each range cell into a stream of binary data. If a reflected signal exceeds a predetermined threshold, the signal is represented by a binary ‘one’; otherwise, the signal is represented by a binary ‘zero’. This operation is depicted schematically in  FIG. 1 . One drawback of such a ‘hard’ decision is evident: signals slightly below the threshold are discarded although they could affect a global detection decision, and therefore useful information is being ignored. 
     Next, a range-extent filter processes the binary data to indicate the presence of clusters comprising ‘ones’ that appear in adjacent range cells. These clusters are regarded as being indicative of a presence of an object extended in range.  FIG. 2  illustrates the operation of a range-extent filtering. However, if such filter is to be efficient, the extent of a hypothetical object must be known a priori. Furthermore, no range-extent filtering is applicable to objects so small as to occupy a single range cell. 
     Finally, persistence in time of a hypothetical object is examined over a predetermined time interval. Object detection is declared if the output of a suitable persistence integrator exceeds a preselected threshold. This operation is depicted schematically in  FIG. 3 . Unfortunately, integration in time cannot recreate information lost in the process of thresholding or hard-limiting. 
     The detection performance of the above system is analyzed in more detail in: S. D. Blunt, K. Gerlach and J. Heyer, “HRR Detector for Slow-Moving Targets in Sea Clutter”, IEEE Transactions on Aerospace and Electronic Systems, vol. 43, July 2007, pp. 965-975. This publication discusses the relevant theoretical background, and also various assumptions and simplifications leading to a practical implementation of the detector. 
     For example, it is shown that the step-like characteristic of a comparator employed by the detector is only a convenient and practical approximation of an optimal non-linear characteristic obtained from a theoretical analysis. Furthermore, it appears that combining the observations in  FIG. 1   a  in a linear manner (as in arithmetic mean) and comparing the result with a threshold does not result in a robust detection procedure required for backgrounds that exhibit noise of impulsive nature. 
     It would therefore be desirable to provide a method and an apparatus for detecting a signal in a noisy background in a robust manner. 
     In particular, it would be desirable, but not essential, for this method and apparatus to be able to detect small objects in spiky sea clutter in a more efficient way than that provided by the prior art techniques. 
     SUMMARY OF THE INVENTION 
     According to the present invention, there is provided a method of processing signal values to detect a signal in a noisy background, the method comprising a signal processor performing processes of: 
     mapping each of a plurality of signal values to a respective point defined by first and second coordinates on an arc of at least part of a circle;
 
calculating a measure of the concentration of the points on the arc; and
 
comparing a value representative of the level of the signal values against a threshold value, wherein at least one of the values compared is generated in dependence upon the calculated measure of the concentration of the points on the arc.
 
     By setting the threshold value in this way, the present inventor has found that an improved detection performance can be achieved even when an arithmetic mean is used as the value representative of the level of the signal values. 
     The present invention further provides apparatus for performing the method above. 
    
    
     
       DESCRIPTION OF THE DRAWINGS 
         FIG. 1  depicts a prior art use of a comparator to convert radar returns from each range cell into a stream of binary data. 
         FIG. 2  illustrates a prior art operation of a range-extent filter performed on the binary data to indicate the presence of clusters. 
         FIG. 3  illustrates a prior art operation of a persistent integrator. 
         FIG. 4  shows the steps of determining the circular mean in accordance with a first example. 
         FIG. 5  is a functional block diagram of a circular-mean calculator CMC constructed in accordance with the first example. 
         FIG. 6  is a functional block diagram of a circular-mean calculator CMC constructed in accordance with a second example. 
         FIG. 7   a  depicts two mapping functions constructed in accordance with a third example. 
         FIG. 7   b  shows the transformation of signal values into angular positions. 
         FIG. 7   c  depicts the mapping of signal values onto a unit semicircle 
         FIG. 8  is a functional block diagram of a modified circular-mean calculator MCMC constructed in accordance with the third example. 
         FIG. 9  depicts decision regions when both the circular mean CM and the circular concentration DC are utilized for signal detection in a first embodiment of the present invention. 
         FIG. 10  is a functional block diagram of a signal detecting system SDT incorporating a circular-mean calculator CMC constructed in accordance with the first embodiment of the invention. 
     
    
    
     EMBODIMENTS 
     An analysis of signal detection in impulsive noise has shown that utilizing a conventional arithmetic mean as a measure of signal values such as those shown in  FIG. 1   a  to compare against a detection threshold cannot provide an optimum detection performance. Therefore, an improved and robust method is needed to detect signals. 
     In accordance with embodiments of the invention, each value p of a set of observed values of a signal (e.g., envelope, magnitude or power) is transformed to a point in two-dimensional space which has two coordinates on orthogonal axes (x,y) defining a position on an arc of a circle. The arc subtends an angle which is equal to, or less than, 180° (or the equivalent in other units). Preferably, the arc is that of a semicircle. A mean value X K  is calculated for the x coordinates and a mean value Y K  is calculated for the y coordinates. The two resulting means are then used to calculate a measure D CC  of the concentration of the points on the semicircle. This measure is then used to set a detection threshold value against which a mean of the signal level values p is compared. The present invention has found that, by setting the detection threshold value in this way, the comparison of the mean with the detection threshold value produces more robust results compared to previous techniques. In particular, the number of erroneous detections caused by impulsive noise is reduced. 
     The mean which is compared against the detection threshold value may be an arithmetic mean or other form of mean. By way of non-limiting example of a further type of mean, the mean values X K  and Y K  of the x and y coordinates of the transformed signal points on the circle may be used to calculate a mean direction θ MD  of the transformed points in the two-dimensional space. This mean direction θ MD  is then used to compare the observed signal values against the detection threshold value. In a first technique, this comparison is performed by applying a reverse transform M −1  to map the mean direction θ MD  back into the signal domain of the observed values, to give what is termed herein a circular mean p CM , and then comparing the circular mean with the detection threshold value, set as described above, in the signal domain. In a second, alternative but equivalent, technique, the comparison is performed by comparing the mean direction θ MD  with the detection threshold value after it has been transformed into the two-dimensional space of the semicircle. 
     Before describing a full embodiment of the present invention, some example methods will be described for calculating the mean values X K  and Y K  that is, the mean values of the x and y coordinates of the signal values mapped onto the semicircle) used in the embodiment to calculate the circular concentration D CC  and, if required, the mean direction θ MD  and/or the circular mean p CM . 
     First Example 
     In accordance with a first example, the mean values X K  and Y K  and the circular mean p CM  is calculated by applying a procedure comprising the following three steps (which are also schematically illustrated in  FIG. 4 ): 
     Step  1 —Observed values p of a signal are mapped onto a unit semicircle with the use of a mapping function M(p). As a result, each value p k  from a predetermined range (PL, PH) of interest will be represented by a corresponding point placed on a unit semicircle at angular position θ k . 
     Accordingly, K observed values of a signal 
       { p   k   }={p   1   ,p   2   , . . . ,p   K-1   ,p   K   };p   k ε( PL,PH )
 
     will be represented by a corresponding set of K angles 
       {θ k }={θ 1 ,θ 2 , . . . ,θ K-1 ,θ K };θ k ε(α,α+π)
 
     where α is an arbitrary initial angle. 
     Step  2 —To determine the x and y coordinates of each mapped point p k  in the two-dimensional space of the semicircle, the sines and cosines of the angles {θ k } are calculated. The calculated values are then averaged separately to obtain two respective means: 
     
       
         
           
             
               
                 Y 
                 K 
               
               = 
               
                 
                   1 
                   K 
                 
                  
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       1 
                     
                     K 
                   
                    
                   
                     sin 
                      
                     
                         
                     
                      
                     
                       θ 
                       k 
                     
                   
                 
               
             
             ; 
             
                 
             
              
             
               
                 X 
                 K 
               
               = 
               
                 
                   1 
                   K 
                 
                  
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       1 
                     
                     K 
                   
                    
                   
                     cos 
                      
                     
                         
                     
                      
                     
                       θ 
                       k 
                     
                   
                 
               
             
           
         
       
     
     If the mean direction or the circular mean is to be compared with the detection threshold value, then the mean direction θ MD  of the set of angles is obtained from 
       θ MD =tan −1 ( Y   K   /X   K )
 
     and the circular mean is calculated in accordance with step  3  below. Obviously, these processes are unnecessary if a mean such as an arithmetic mean of the signal level values p is to be compared against the detection threshold value. 
     Step  3 —The circular mean p CM  of the underlying set of K signal values 
       { p   k   }={p   1   ,p   2   , . . . ,p   K-1   ,p   K   };p   k ε( PL,PH )
 
     is obtained by applying an inverse mapping M −1 (θ) to the mean direction θ MD  to transform it back into the signal domain of the observed signal values p. 
     Subsequently, the circular mean p cm  is compared to the threshold value in the signal domain (or, alternatively, the mean direction θ MD  is compared with the threshold value after it has been mapped into the two-dimensional space of the semicircle). 
       FIG. 4  illustrates schematically the procedure of determining a circular mean p CM  of a given set of signal values. Although in this case, the semicircle shown appears in the first two quadrants (hence, α=0), this particular feature is not essential. 
     In order to facilitate the understanding of the first example and the advantages of using the circular mean p CM , a simple illustrative example is given below. 
     The first step is to transform the measured values of p to values p* that fall into a (0, 180) degree interval. Suppose that the measured values of p are: 
       p 1 =p 2 =p 3 =p 4 =p 5 =260;p 6 =500 
     A linear shifting operation then subtracts 200 from each value to give: 
       p 1s =p 2s =p 3s =p 4s =p 5s =60;p 6s =300 
     A linear scaling operation then divides each value by 2 to give: 
         p*   1   =p*   2   = . . . =p*   5 =30 ;p*   6 =150 
     These transformed values p* of the signal values p now fall into a (0,180)-interval. The transformed signal values p* and their corresponding angular position θ, expressed in degrees, have equal numerical values. 
     It should be noted that this method for shifting and scaling the data is just one example of many methods that can be used for this operation. 
     The arithmetic mean p AM  of the above set of transformed values, p*, is 
         p   AM =(5×30+150)/6=50
 
     As seen, the single relatively large observation p* 6 =150 (e.g., an interfering noise spike) shifts the mean (30) of the remaining five observations, p* 1 -p* 5 , in a significant manner. 
     In the second step of processing in this embodiment, the mean direction θ MD  of the transformed values, p* 1 -p* 6 , is determined from 
     
       
         
           
             
               Y 
               6 
             
             = 
             
               
                 
                   ( 
                   
                     
                       5 
                       × 
                       sin 
                        
                       
                           
                       
                        
                       30 
                        
                       ° 
                     
                     + 
                     
                       sin 
                        
                       
                           
                       
                        
                       150 
                        
                       ° 
                     
                   
                   ) 
                 
                 / 
                 6 
               
               = 
               
                 1 
                 / 
                 2 
               
             
           
         
       
       
         
           
             
               X 
               6 
             
             = 
             
               
                 
                   ( 
                   
                     
                       5 
                       × 
                       cos 
                        
                       
                           
                       
                        
                       30 
                        
                       ° 
                     
                     + 
                     
                       cos 
                        
                       
                           
                       
                        
                       150 
                        
                       ° 
                     
                   
                   ) 
                 
                 / 
                 6 
               
               = 
               
                 
                   3 
                 
                 / 
                 3 
               
             
           
         
       
       
         
           
             
               θ 
               MD 
             
             = 
             
               
                 
                   tan 
                   
                     - 
                     1 
                   
                 
                 ( 
                 
                   
                     3 
                   
                   2 
                 
                 ) 
               
               ≈ 
               
                 41 
                  
                 ° 
               
             
           
         
       
     
     Hence, the mean direction θ MD  of the observed values is equal to 41, whereas the arithmetic mean p AM  in the circular domain was equal to 50. As will be understood, the mean direction θ MD  is a more robust estimator of the mean level of the values p, because it is much less affected by a single larger observation (namely p* 6 ). 
     In the third step of the processing, the present example transforms the mean direction θ MD =41° back to the same domain as the measured signals p using an inverse mapping M −1 . 
     Therefore, the transformed value p CM =41×2+200=282. 
     On the other hand, the arithmetic mean of the signal values p in the signal domain is [(5×260)+500]/6=300. 
     Accordingly, the circular mean p CM  produced by the present example is a more robust indication of the mean level of the values p in the signal domain than the conventional arithmetic mean. This is because the value of the circular mean p CM  is less effected by the spurious value p 6  than the arithmetic mean. 
     An apparatus for performing the above calculation of X K , Y K  and p CM  is shown in  FIG. 5 . 
     The calculator shown in  FIG. 5  comprises the following functional units:
         a shift and scale unit  502 ;   two nonlinear converters, SNL  503  and CNL  504 ;   two tapped delay lines, DLS  505  and DLC  506 ;   two averaging circuits, AVS  507  and AVC  508 ;   an arithmetic unit ATU  509 .       

     A positive input signal PP, such as envelope, magnitude or power, is passed through the sift and scale unit  502  which maps each value of the input signal to a value within a range of values with a span of 180, thereby generating a normalized signal NP. 
     The normalized signal level NP is applied in a parallel fashion to the two nonlinear converters, SNL  503  and CNL  504 . The outputs, SS and CC, of the converters are obtained from the common input NP by utilizing two suitably selected mapping functions; in the considered case 
       SS=sin NP,CC=cos NP 
     As described previously, the values SS and CC define the x and y coordinates of the value NP in two-dimensional space. 
     The signals SS and CC propagate along their respective tapped delay lines, DLS  505  and DLC  506 : K samples of the signal SS are available at the outputs of delay cells, S 1 , S 2 , . . . , SK, whereas K samples of the signal CC are available at the outputs of delay cells, C 1 , C 2 , . . . , CK. 
     The averaging circuit AVS  507  produces at its output a value AS proportional to the sum of its inputs obtained from the cells S 1 , S 2 , . . . , SK. Similarly, the averaging circuit AVC  508  produces at its output a value AC proportional to the sum of its inputs obtained from the cells C 1 , C 2 , . . . , CK. These two values, AS and AC, are utilized by the arithmetic unit ATU  509  to determine a circular mean CM. 
     The calculated circular mean CM is then compared against the detection threshold value in a comparison unit (not shown in  FIG. 5 ). 
     Second Example 
     In the first example described above, signal values p were transformed into angle values θ by employing a linear operation of the ‘shift-and-scale’ type. However, in practical applications, it may be advantageous to apply first a nonlinear (e.g., logarithmic) transformation to the observed signal values in order to adjust their dynamic range non-linearly, and then map such transformed data onto a unit semicircle. An example which performs such processing is described below. 
     For example, a useful nonlinear mapping is of the form 
       θ k   =H (γ log 10    p )
 
     where the clipper function H(·) limits the minimum and maximum values of its argument to −π/2 and π/2, respectively; γ is a scaling factor used to further adjust the dynamic range of the signal being processed. 
     For example, if the range of observed values p extends from 0.01 to 100, then a γ=π/4 would place all values of p within (−π/2, π/2), with values lying on both of the extremities. If a new value of p was subsequently detected that fell outside of the 0.01 to 100 range, then, in order for the same value of γ to be used for the new p value, the clipper function H would be required to limit the new mapped value of p to the (−π/2, π/2) range. In practice, if the p values mostly fell within the 0.01 to 100 range, and rarely were at or beyond these limits, then the clipper function H would not usually be utilised. 
     Applying a logarithmic transformation prior to mapping onto a semicircle may be preferred to using other nonlinear transformations for the following reasons:
         1. A logarithmic transformation, being compressive, suppresses larger observations.   2. Scale parameter of underlying data is converted into a shift parameter thereby simplifying operation of signal normalization because division will be conveniently reduced to simple subtraction.       

       FIG. 6  is a functional block diagram of a circular-mean calculator CMC constructed in accordance with the second example. 
     The calculator comprises the following functional units:
         a logarithmic converter LGI  601 ;   a subtractor SST  602 ;   two nonlinear converters, SNL  503  and CNL  504 ;   two tapped delay lines, DLS  505  and DLC  505 ;   two averaging circuits, AVS  507  and AVC  508 ;   an arithmetic unit ATU  609 .       

     Accordingly, compared with the calculator shown in  FIG. 5  of the first example, the logarithmic converter LG 1   501  and the subtractor SET  602  replace the shift and scale unit  502 . All of the other components remain the same, with the exception of ATU  609 , which performs a different reverse mapping M −1  compared to the first example to take account of the log operation performed on p. 
     A positive input signal PP, such as envelope, magnitude or power, is passed through the logarithmic converter LGI  601  to produce a signal LP being a logarithmic measure of the level of the input signal PP, hence 
       LP=log 10  PP 
     The signal LP is then normalized in the subtractor SET  602  by subtracting from the signal LP a logarithm BL of some reference level BG of interest 
     
       
         
           
             NP 
             = 
             
               
                 LP 
                 - 
                 BL 
               
               = 
               
                 
                   
                     
                       log 
                       10 
                     
                      
                     PP 
                   
                   - 
                   
                     
                       log 
                       10 
                     
                      
                     BG 
                   
                 
                 = 
                 
                   
                     log 
                     10 
                   
                    
                   
                     ( 
                     
                       PP 
                       BG 
                     
                     ) 
                   
                 
               
             
           
         
       
     
     For example, in signal detection problems, the reference level may be the average level of background noise, obtained from long-term observations. The action of subtracting the value BL, from LP maps the value of LP to a point on a unit semicircle having an angular range (−π/2, π/2). It is equivalent to the scaling provided by the factor γ. 
     The normalized signal level NP is applied in a parallel fashion to the two nonlinear converters, SNL  503  and CNL  504 . The outputs, SS and CC, of the converters are obtained from the common input NP by utilizing two suitably selected mapping functions; in the present example 
       SS=sin NP,CC=cos NP 
     The signals SS and CC propagate along their respective tapped delay lines, DLS  505  and DLC  506 : K samples of the signal SS are available at the outputs of delay cells, S 1 , S 2 , . . . , SK, whereas K samples of the signal CC are available at the outputs of delay cells, C 1 , C 2 , . . . , CK. 
     The averaging circuit AVS  507  produces at its output a value AS proportional to the sum of its inputs obtained from the cells S 1 , S 2 , . . . , SK. Similarly, the averaging circuit AVC  508  produces at its output a value AC proportional to the sum of its inputs obtained from the cells C 1 , C 2 , . . . , CK. These two values, AS and AC, are utilized by the arithmetic unit ATU  609  to determine a circular mean CM. 
     The calculated circular mean is then compared against the detection threshold value in a comparison unit (not shown in  FIG. 6 ). 
     Third Example 
     In the above examples, a signal value p is mapped to a point on unit semicircle and then trigonometric operators are applied to determine the two coordinates in the two-dimensional space of the semicircle which define the position of the point. These coordinates are then used to calculate the circular concentration and, if required, the mean direction θ MD  and the circular mean p CM . However, the initial mapping of the signal value p onto the semicircle may be performed in such a way that the mapping directly gives the two coordinates of the resulting point on the semicircle. Accordingly, it is then not necessary to calculate the coordinates by performing the trigonometric operations of the first and second examples. 
     An example of such processing is described below. 
     In general, the mapping of a signal value p to a semicircle can be performed with the use of two mapping functions, S(p) and C(p), constructed in a suitable manner. Because the mapping is required to produce a semicircle, the mapping functions must satisfy the condition 
         S   2 ( p )+ C   2 ( p )= b    
     where b is a constant. In the case of a unit semicircle b=1. 
     In accordance with the third example, a first mapping function S(p) is to be a monotonically increasing continuous function: it assumes its minimum value, −1, for the smallest signal level PL, and reaches its maximum equal to +1 at the largest signal level PH. There are infinitely many such functions, and a suitably selected segment of a sinewave is one of the many choices; another useful function will be discussed in the following. 
     Next, a second mapping function C(p) is obtained from the mapping function S(p) as follows 
         C ( p )=√{square root over (1 −S   2 ( p ))}
 
     Hence, the mapping operation M[S(p), C(p)], utilizing two mapping functions, S(p) and C(p), may be viewed as a non-linear transformation of a one-dimensional p-space into a two-dimensional (S, C)-space. Each signal value p is consequently mapped directly to a point on a unit semicircle, such that one coordinate of the point is defined by S(p) and the other coordinate of the point is defined by C(p). An arithmetic average can then be applied directly to S(p) and C(p) to obtain the parameters for the mean direction. 
     More particularly, the mean direction θ MD  is calculated from 
     
       
         
           
             
               
                 Y 
                 K 
               
               = 
               
                 
                   1 
                   K 
                 
                  
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       1 
                     
                     K 
                   
                    
                   
                     S 
                      
                     
                       ( 
                       
                         p 
                         k 
                       
                       ) 
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               X 
               K 
             
             = 
             
               
                 1 
                 K 
               
                
               
                 
                   ∑ 
                   
                     k 
                     = 
                     1 
                   
                   K 
                 
                  
                 
                   C 
                    
                   
                     ( 
                     
                       p 
                       k 
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               θ 
               MD 
             
             = 
             
               
                 tan 
                 
                   - 
                   1 
                 
               
                
               
                 ( 
                 
                   
                     Y 
                     K 
                   
                   / 
                   
                     X 
                     K 
                   
                 
                 ) 
               
             
           
         
       
     
     Then, the mean direction θ MD  can be compared to the detection threshold value mapped into the space of the semicircle. Alternatively, if required, the circular mean P CM  of the signal values is found by applying an inverse mapping M −1 (θ) to the mean direction θ MD , and the circular mean p CM  is compared against the detection threshold value. The inverse mapping M −1  can be determined either by calculating and evaluating the mathematical function defining M −1  or by using a numerical technique (such as iterative processing) to obtain the required value. 
     The hyperbolic functions, tan h w and 1/(cos h w) can be advantageously exploited by the mapping functions S(p) and C(p). 
     More particularly, since 
     
       
         
           
             
               
                 
                   ( 
                   
                     tanh 
                      
                     
                         
                     
                      
                     w 
                   
                   ) 
                 
                 2 
               
               + 
               
                 
                   ( 
                   
                     1 
                     
                       cosh 
                        
                       
                           
                       
                        
                       w 
                     
                   
                   ) 
                 
                 2 
               
             
             = 
             
               
                 
                   1 
                   + 
                   
                     
                       ( 
                       
                         sinh 
                          
                         
                             
                         
                          
                         w 
                       
                       ) 
                     
                     2 
                   
                 
                 
                   
                     ( 
                     
                       cosh 
                        
                       
                           
                       
                        
                       w 
                     
                     ) 
                   
                   2 
                 
               
               = 
               1 
             
           
         
       
     
     the mapping [tan h w, 1/(cos h w)] will put a point representing a value w on a unit semicircle. 
     For normalization purposes, it will be convenient to use a logarithmic transformation before performing the two hyperbolic transformations, tan h w and 1/(cos h w), so that w=ln(p). In this case, a functional block diagram of a circular-mean calculator has a similar structure as that shown in  FIG. 6 . However, in this case, the nonlinear converters, SNL  503  and CNL  504 , will perform the following mapping operations 
         SS =tan  hNP,CC= 1/(cos  hNP ) 
     It is also possible to combine the logarithmic and hyperbolic transformations thereby simplifying the structure of the circular-mean calculator. Such a simplification can be achieved as follows. 
     If a hyperbolic transformation tan h(w) is preceded by logarithmic transformation, w=ln(p), then 
     
       
         
           
             
               S 
                
               
                 ( 
                 p 
                 ) 
               
             
             = 
             
               
                 tanh 
                  
                 
                     
                 
                  
                 w 
               
               = 
               
                 
                   
                     
                       exp 
                        
                       
                         ( 
                         
                           2 
                            
                           w 
                         
                         ) 
                       
                     
                     - 
                     1 
                   
                   
                     
                       exp 
                        
                       
                         ( 
                         
                           2 
                            
                           w 
                         
                         ) 
                       
                     
                     + 
                     1 
                   
                 
                 = 
                 
                   
                     
                       p 
                       2 
                     
                     - 
                     1 
                   
                   
                     
                       p 
                       2 
                     
                     + 
                     1 
                   
                 
               
             
           
         
       
       
         
           Since 
         
       
       
         
           
             
               C 
                
               
                 ( 
                 p 
                 ) 
               
             
             = 
             
               
                 1 
                 - 
                 
                   
                     S 
                     2 
                   
                    
                   
                     ( 
                     p 
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               C 
                
               
                 ( 
                 p 
                 ) 
               
             
             = 
             
               
                 2 
                  
                 p 
               
               
                 
                   p 
                   2 
                 
                 + 
                 1 
               
             
           
         
       
     
     As shown in  FIG. 7   a  and  FIG. 7   b , the shapes of the two mapping functions depict the relationship between signal values p and the corresponding angular positions θ of points representing the values on the unit semicircle. The relationship is non-linear, and its form of a ‘soft’ limiter results from its mathematical representation 
       θ=tan −1    [S ( p )/ C ( p )]=tan −1  [( p   2 −1)/(2 p )]
 
       FIG. 7   c  shows the unit semicircle with angular positions  9  corresponding to some selected underlying values p of the signal. As seen, for larger values (p&gt;10) of the signal, the corresponding angular positions θ form a cluster close to the limiting value π/2; on the other hand, values of p less than unity generate angular positions θ occupying the whole quadrant (−π/2,0). 
     When the parameters Y K  and X K  have been determined from the observed values {p 1 , p 2 , . . . , p K-1 , p K } according to 
     
       
         
           
             
               
                 Y 
                 K 
               
               = 
               
                 
                   1 
                   K 
                 
                  
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       1 
                     
                     K 
                   
                    
                   
                     
                       
                         p 
                         k 
                         2 
                       
                       - 
                       1 
                     
                     
                       
                         p 
                         k 
                         2 
                       
                       + 
                       1 
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               X 
               K 
             
             = 
             
               
                 1 
                 K 
               
                
               
                 
                   ∑ 
                   
                     k 
                     = 
                     1 
                   
                   K 
                 
                  
                 
                   
                     2 
                      
                     
                       p 
                       k 
                     
                   
                   
                     
                       p 
                       k 
                       2 
                     
                     + 
                     1 
                   
                 
               
             
           
         
       
     
     the circular mean p CM  of the signal values p is obtained from 
     
       
         
           
             
               p 
               CM 
             
             = 
             
               
                 
                   Y 
                   K 
                 
                 
                   X 
                   K 
                 
               
               + 
               
                 
                   
                     
                       ( 
                       
                         
                           Y 
                           K 
                         
                         
                           X 
                           K 
                         
                       
                       ) 
                     
                     2 
                   
                   + 
                   1 
                 
               
             
           
         
       
     
       FIG. 8  is a functional block diagram of a modified circular-mean calculator MCMC  800  constructed in accordance with the third example. In this configuration, the logarithmic converter LGI  601  and the subtractor SBT  602  used in the circular-mean calculator  600  of  FIG. 6  have been replaced by a variable-gain amplifier VGI  801 . The amplifier adjusts its gain in response to a signal BG indicative of the reference level of interest. 
     In this arrangement, the outputs, SS and CC, of the converters, SNL  802  and CNL  803 , are obtained from the common input CP as follows 
     
       
         
           
             
               SS 
               = 
               
                 
                   
                     CP 
                     2 
                   
                   - 
                   1 
                 
                 
                   
                     CP 
                     2 
                   
                   + 
                   1 
                 
               
             
             , 
             
               CC 
               = 
               
                 
                   2 
                    
                   CP 
                 
                 
                   
                     CP 
                     2 
                   
                   + 
                   1 
                 
               
             
           
         
       
     
     Other functions and operations performed by the modified system are similar to those performed by the system of  FIG. 6 , and accordingly will not be described again here. 
     The following example illustrates an advantage of using a circular mean in the detection of signals in background noise. In the example, logarithmic and hyperbolic functions are exploited for mapping. 
     Assume that a random signal of unit power is to be detected in a noisy background comprising a unit-power thermal noise and impulsive interference with occasional spikes exceeding ten times the noise level. Assume also, for illustration purposes, that a detection threshold has been set to 1.9. 
     Suppose that in the no-signal case, six observed values of background noise are 
       p 1 =p 2 = . . . =p 5 =1,p 6 =10 
     The five equal samples may represent thermal noise level, and sample number six may be generated by impulsive interference. In this case, the arithmetic mean p AM  is equal to 2.5. Therefore, if the arithmetic mean p AM  were used as a detection statistic, a false alarm would be declared because the value 2.5 is greater than the detection threshold of 1.9. 
     Suppose now that the circular mean p CM , rather than the arithmetic mean, is used to determine the level of background noise. The parameters Y 6  and X 6  are 
     
       
         
           
             
               
                 6 
                 × 
                 
                   Y 
                   6 
                 
               
               = 
               
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       1 
                     
                     6 
                   
                    
                   
                     
                       
                         P 
                         k 
                         2 
                       
                       - 
                       1 
                     
                     
                       
                         p 
                         k 
                         2 
                       
                       + 
                       1 
                     
                   
                 
                 ≈ 
                 1 
               
             
             , 
             
               
                 6 
                 × 
                 
                   X 
                   6 
                 
               
               = 
               
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       1 
                     
                     6 
                   
                    
                   
                     
                       2 
                        
                       
                         p 
                         k 
                       
                     
                     
                       
                         p 
                         k 
                         2 
                       
                       + 
                       1 
                     
                   
                 
                 ≈ 
                 5.2 
               
             
           
         
       
     
     Hence, the circular mean p CM  is 
     
       
         
           
             
               p 
               CM 
             
             = 
             
               
                 
                   1 
                   5.2 
                 
                 + 
                 
                   
                     
                       
                         ( 
                         
                           1 
                           5.2 
                         
                         ) 
                       
                       2 
                     
                     + 
                     1 
                   
                 
               
               ≈ 
               1.2 
             
           
         
       
     
     As seen, in this case, the circular mean p CM  will not exceed the detection threshold, and hence, no false alarm will occur. 
     Suppose now that a random signal to be detected is present, and let six observed values, in this signal-plus-noise case, be all equal, for example, p 1 =p 2 = . . . =p 6 =2. Obviously, in this case, both the arithmetic mean p AM  and the circular mean p CM  will be equal, p AM =p CM =2, and both will lead to a correct detection decision. 
     Therefore, the main advantage of utilizing the circular mean as a detection statistic follows from its ability to attenuate occasional larger observations. In general, such a property is very useful when processing signals corrupted by noise of impulsive nature. 
     First Embodiment 
     In accordance with a first embodiment of the invention, the circular concentration D CC  of points on the unit semicircle is calculated and used to set a detection threshold value against which the circular mean p CM  is compared and/or the circular concentration is used to adjust the value of the circular mean before it is compared with the detection threshold value. 
     The circular concentration D CC  of points {θ k } that represent values {p k } of a signal level p is determined from 
     
       
         
           
             
               D 
               CC 
             
             = 
             
               
                 
                   X 
                   K 
                   2 
                 
                 + 
                 
                   Y 
                   K 
                   2 
                 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             
               
                 Y 
                 K 
               
               = 
               
                 
                   1 
                   K 
                 
                  
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       1 
                     
                     K 
                   
                    
                   
                     S 
                      
                     
                       ( 
                       
                         p 
                         k 
                       
                       ) 
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               X 
               K 
             
             = 
             
               
                 1 
                 K 
               
                
               
                 
                   ∑ 
                   
                     k 
                     = 
                     1 
                   
                   K 
                 
                  
                 
                   C 
                    
                   
                     ( 
                     
                       p 
                       k 
                     
                     ) 
                   
                 
               
             
           
         
       
     
     and S(p k ) and C(p k ) are the two functions used for mapping. 
     The third example above is the most appropriate for calculating the values X K  and Y K  as these values are output by the initial mapping of the signal values p onto the semicircle in the third example. However, it will be readily apparent to the skilled reader that X K  and Y K  may be calculated after mapping p onto the semicircle as in the first and second examples. 
     When the points {θ k } form a tight cluster on the semicircle, the value of the circular concentration D CC  will be close to one. However, when the points {θ k } are widely dispersed, e.g., due to dominant noise, the value of the circular concentration D CC  will be significantly smaller. Accordingly, the circular concentration D CC  may be used in conjunction with the circular mean to further improve signal detection. 
       FIG. 9  illustrates a potential improvement in detection performance when the circular mean CM is suitably combined with the circular concentration DC to construct a detection-decision region. 
     When the circular mean CM alone is utilized for detection purposes, signal presence will be declared if an observed value of circular mean exceeds the decision threshold TH, irrespective of the circular concentration. In this case, the decision region D 1  has a rectangular shape extending from the line TH. 
     However, when the circular mean CM is used in conjunction with the circular concentration DC, the resulting decision region will be augmented by the region D 2 . When circular concentration values exceed some predetermined threshold C 1 , the decision threshold TH for circular mean may be gradually reduced to a new lower value T 1 . Obviously, the area (D 1 +D 2 ) is greater than the area D 1  alone, and an improved detection performance will be achieved, when the threshold TH is replaced by a decision boundary DB. 
       FIG. 10  is a functional block diagram of a signal detector SDT  1000  constructed in accordance with the first embodiment of the invention which utilizes circular concentration to set the value of the detection threshold. The intended use of the signal detector SDT  1000  is the detection of small objects in sea clutter. Observed samples PP of background noise, or those of signal-plus-noise, are processed in the circular-mean calculator CMC  1001 . The sample level is normalized by utilizing an auxiliary signal EL indicative of the average level of background noise. Such a signal may be obtained, for example, by averaging observations taken over a longer time interval and generated by a plurality of range cells adjacent to a cell under test. 
     The circular-mean calculator CMC  1001  provides two output signals, CM and DC, indicative respectively of the circular mean and circular concentration. Those signals are employed by a decision block DET  1011  to decide whether the observed samples PP have been generated by sea clutter alone or by an object buried in clutter. For this purpose a signal DE that defines the decision boundary is applied to input DB of the decision block DET  1011 . The output signal SD of the block is the global decision regarding the presence or absence of a signal in clutter. 
     It should be noted that, in the embodiment described above, the circular concentration D CC  is used to adjust the threshold value against which the circular mean p CM  is compared (as shown in  FIG. 9 ). However, this is equivalent to adjusting the value of the circular mean p CM  in dependence upon the circular concentration DCC and comparing the adjusted mean against an unchanged threshold value. Accordingly, the embodiment may adjust p CM  instead of the threshold value. Similarly, both the threshold value and the circular mean p CM  may be adjusted in dependence upon the circular concentration D CC . 
     Second Embodiment 
     In the first embodiment described above, the circular mean p CM  is compared against the threshold set in dependence upon the circular concentration D CC . However, instead, a different mean of the observed signal values may be compared with the threshold, such as the arithmetic mean or the mean direction θ MD . 
     Similarly, the mean (of whatever type it is) may be adjusted in dependence upon the calculated value of the circular concentration D CC  and the adjust mean compared against a fixed threshold. 
     Furthermore, both the mean and the threshold could be adjusted in dependence upon the calculated value of the circular concentration D CC . 
     MODIFICATIONS AND VARIATIONS 
     The foregoing description of preferred embodiments of the invention has been presented for the purpose of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. In light of the foregoing description, it is evident that many alterations, modifications, and variations will enable those skilled in the art to utilize the invention in various embodiments suited to the particular use contemplated. 
     For example, in the examples and embodiments described above, the signal values p are mapped to points on a semicircle. However, the use of a full semicircle is not essential and instead a circular arc smaller than a semicircle could be used. More particularly, the signal values p could be mapped to any circular arc subtending an angle of less than or equal to, that subtended by a semicircular arc, namely π radians (180°). 
     In all of the examples and embodiments described above, the mean direction θ MD  is calculated by determining both X K  (that is, the mean of the x coordinates of the points on the semicircle) and Y K  (that is, the mean of the y coordinates of the points on the semicircle) and then using both values to calculate θ MD . However, although not as robust, it is not necessary to determine both X K  and Y K , and instead of calculating the mean direction θ MD  an alternative mean M ALT  can be calculated as: 
       M ALT =X K  or M ALT =Y K    
     More particularly, when the points p are mapped to a unit semicircle covering an angular range 0° to 180° or 180° to 360°, then M ALT  X K  is used because the cosine values employed to calculate X K  have a unique value in this range. That is, there is a 1:1 mapping between the value of the point p on the semicircle with the x-axis. On the other hand, when the points p are mapped to a unit semicircle covering an angular range 90° to 270° or 270° to 90°, then M ALT =Y K  is used because the sine value employed to calculate Y K  have a unique value in this range. 
     The alternative mean M ALT  can then be compared against the detection threshold value after the detection threshold has been mapped into the two-dimensional space of the semicircle. Alternatively, the alternative mean M ALT  can be mapped back into the signal domain and compared against the detection threshold value. For example, in the first and second examples, this mapping back may be performed using an inverse cosine mapping (in the case of M ALT =X K ) or inverse sine mapping (in the case of M ALT =Y K ) to map back to a point on the semicircle, and then applying an inverse mapping M −1 . 
     Other modifications are, of course, possible.