Abstract:
A complex signal correlator such as can be implemented in a correlation detector system affords a unique algorithm which estimates the codifference correlation between two complex signals based on the sum and difference of codifference estimates, each codifference estimate having equivalently associated therewith a dispersion estimate. Typical embodiments provide a receiving antenna and a receiver inclusive of the codifference correlator, wherein radio frequency waves are down converted and sampled, the sampled signals are correlated with a reference signal contained in a memory, and the resultant correlation signal is detected and transduced. The inventive correlator is based on an alpha-stable distribution and, in comparison with conventional alpha-stable distribution-based correlators, can more effectively operate in a realm wherein alpha is less than one.

Description:
STATEMENT OF GOVERNMENT INTEREST 
     The invention described herein may be manufactured and used by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor. 
     BACKGROUND OF THE INVENTION 
     The present invention relates to methods and apparatuses for correlating or establishing a statistical correlation of plural random variables, more particularly to methods and apparatuses for doing same in relation to physical phenomena such as impulsive signals or noise. 
     The alpha-stable distribution is an important area for investigation. One reason for the importance of the alpha-stable distribution is that it models impulsive noise. Another reason is that this statistical distribution is expected from superposition in natural processes. The parameter alpha, α, is the characteristic exponent that varies over 0&lt;α≦2. The alpha-stable distribution includes the Gaussian when α=2. For α less than two, the distribution becomes more impulsive, more non-Gaussian in nature, and the tails of the distribution become thicker. This makes the alpha-stable distribution an attractive choice for modeling signals and noise having an impulsive nature. Also, from the generalized central limit theorem, the stable distribution is the only limiting distribution for sums of independent and identically distributed (IID) random variables (stability propert). If the individual distributions have finite variance, then the limiting distribution is Gaussian. For a less than two, the individual distributions have infinite variance. For detailed information, see the following two books, each of which is hereby incorporated herein by reference: C. L. Nikias and M. Shao, Signal Processing with Alpha-Stable Distributions and Applications, John Wiley and Sons, New York, N.Y., 1995; G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman and Hall, New York, N.Y., 1994. 
     Sources that could follow or be modeled by the alpha-stable distribution are abundant and include lightning in the atmosphere, switching transients in power lines, static in telephone lines, seismic activity, climatology and weather, ocean wave variability, surface texture, the slamming of a ship hull in a seaway, acoustic emissions from cracks growing in engineering materials under stress, magnetic avalanche or Barkhausen noise, transition boundary layer flow, etc. Many sources can exist in the area of target and background signatures that affect detection and classification. In underwater acoustics, examples of these sources could include interference to target detection such as ice cracking, biologics, bottom and sea clutter in active sonar, ocean waves near the surface and in the surf zone. They could also include target characteristics such as target strength in active sonar and cavitation. Similar sources in radar and infrared can include: ocean waves in the form of sea clutter and radar cross section (RCS); see R. D. Pierce, “RCS Characterization using the alpha-stable distribution,” Proc. 1996 IEEE National Radar Conference, Ann Arbor, Mich., May, 13-16, 1996, pp 154-159, hereby incorporated herein by reference. A 10 second long example of spiky, horizontally polarized (H-pol), radar sea clutter is shown herein in FIG.  1 . 
     Two known and limitedly successful methodologies of establishing a statistical correlation, based on the alpha-stable distribution, are classical second-order correlation and covariation correlation. Neither second-order correlators nor covariation correlators have proven capable of obtaining consistent estimates of the relationship between two channels of noise characterized by impulsiveness. Furthermore, second-order correlators operate in a realm wherein alpha is equal to two. Covariation correlators operate in a realm wherein alpha is greater than or equal to one and less than or equal to two. Hence, when alpha is less than one (as would generally be the case, for instance, when the noise is extremely spikey, even more than the noise illustrated in FIG.  1 ), neither second-order correlators nor covariation correlators work to provide consistent results. 
     Of interest and incorporated herein by reference are the following United States patents: Nishimori U.S. Pat. No. 5,982,810 issued Nov. 9, 1999; Honkisz U.S. Pat. No. 5,787,128 issued Jul. 28, 1998; Kazecki U.S. Pat. No. 5,365,549 issued Nov. 15, 1994; Baron U.S. Pat. No. 4,860,239 issued Aug. 22, 1989; Horner U.S. Pat. No. 4,826,285 issued May 2, 1989; Zeidler et al. U.S. Pat. No. 4,355,368 issued Oct. 19, 1982Kaelin al. U.S. Pat. No. 4,234,883 issued Nov. 18, 1980; Baario U.S. Pat. No. 4,117,480 issued Sep. 26, 1978; Fletcher et al. U.S. Pat. No. 4,112,497 issued Sep. 05, 1978; Heng et al. U.S. Pat. No. 4,070,652 issued Jan. 24, 1978. 
     SUMMARY OF THE INVENTION 
     In view of the foregoing, it is an object of the present invention to provide an alpha-stable distribution based correlator which can reliably estimate the relationship between two channels of impulsive noise. 
     It is a further object of the present invention to provide an alpha-stable distribution-based correlator which can operate in a realm wherein alpha is less than one. 
     The present invention provides a methodology for correlating at least two remotely and/or locally generated signals, e.g., quantifying a relationship therebetween. An inventive method is provided for producing an output signal which is indicative of a correlation of at least two input signals. The inventive method comprises calculating (i.e., estimating) the codifference correlation with respect to at least two input signals, and producing an output signal commensurate with the codifference correlation. The estimating includes considering the sum and difference of codifference estimates wherein each codifference estimate is equated with a corresponding dispersion estimate. According to typical inventive practice, the calculating includes treating each signal as a complex signal representative of real and imaginary terms characterized by values and defined by real and imaginary axes. 
     Further provided according to this invention is a correlator for correlating a reference signal and a sampler signal. The inventive correlator admits of implementation in a communication system of the type including antenna means for receiving electromagnetic waves, down converter means for down converting a modulated signal received from the antenna, means, sampler means for sampling a down converted. signal received from the down converted means, and memory means for storing a reference signal. The inventive correlator correlates the reference signal (received from the memory means) and the sampler signal (received from the sampling means). The inventive correlator comprises algorithmic means for calculating (i.e., estimating) the codifference correlation based on the sum and difference of codifference estimates, wherein each codifference estimate is equated with a corresponding dispersion estimate. 
     This invention further provides a correlation detection system for use in association with a modulated signal such as produced by an antenna receiving electromagnetic waves such as radio frequency waves. The inventive correlation detection system comprises a down converter, a sampler, a memory and a correlator. The down converter is adaptable to down converting the modulated signal and producing a down converted signal. The sampler is adaptable to sampling the down converted signal and producing a sampled signal. The memory is adaptable to storing a reference signal. The correlator is adaptable to correlating the reference signal and the sampled signal. The correlator includes processor means and is capable of calculating (i.e., estimating) the, codifference correlation between the reference signal and the sampled signal, based on the sum and difference of codifference estimates. Each codifference estimate is taken from equation with a corresponding dispersion estimate. 
     In accordance with the present invention, the measure of a normalized codifference correlation can be indicated as h. An overall estimate h of the normalized codifference correlation between two signals x′ and y′ is found by the following equation, wherein h is equated to a quotient expression which takes the sum and difference of various codifference estimates: 
     
       
           ĥ =[({circumflex over (τ)} x′     R,     −y′     R   −{circumflex over (τ)} x′     R,     y′     R   +{circumflex over (τ)} x′     I,     −y′     I   −{circumflex over (τ)} x′     I,     y′     I   )+ 
       
     
     
       
           i ({circumflex over (τ)} x′     R,     −y′     I   −{circumflex over (τ)} x′     R,     y′     l   −({circumflex over (τ)} x′     I,     −y′     R   −{circumflex over (τ)} x′     I,     y′     R   ))]/(2 α {circumflex over (γ)} x′ ) 
       
     
     The codifference estimates are each formed from its corresponding dispersion estimate, as follows: 
     
       
         {circumflex over (τ)} x′     R,     −y′     R   ={circumflex over (γ)} x′     R   +{circumflex over (γ)} y′     R   −{circumflex over (γ)} x′     R     +y′     R     
       
     
     
       
         {circumflex over (τ)} x′     R,     y′     R   ={circumflex over (γ)} x′     R   +{circumflex over (γ)} y′     R   −{circumflex over (γ)} x′     R     −y′     R     
       
     
     
       
         {circumflex over (τ)} x′     I,     −y′     I   ={circumflex over (γ)} x′     I   +{circumflex over (γ)} x′     I     +y′     I     
       
     
     
       
         {circumflex over (τ)} x′     I,     y′     I   ={circumflex over (γ)} x′     I   +{circumflex over (γ)} y′     I   −{circumflex over (γ)} x′     I     −y′     I     
       
     
     
       
         {circumflex over (τ)} x′     R,     −y′     I   ={circumflex over (γ)} x′     R   +{circumflex over (γ)} y′     I   −{circumflex over (γ)} x′     R     +y′     I     
       
     
     
       
         {circumflex over (τ)} x′     R,     y′     I   ={circumflex over (γ)} x′     R   +{circumflex over (γ)} y′     I   −{circumflex over (γ)} x′     R     −y′     I     
       
     
     
       
         {circumflex over (τ)} x′     I     ,−y′     R   ={circumflex over (γ)} x′     I   +{circumflex over (γ)} y′     R   −{circumflex over (γ)} x′     I     +y′     R     
       
     
     
       
         {circumflex over (τ)} x′     I     ,y′     R   ={circumflex over (γ)} x′     I   +{circumflex over (γ)} y′     R   −{circumflex over (γ)} x′     I     −y′     R     
       
     
     using the real and imaginary terms from the input (or first signal) 
     
       
           x′=x′   R   +ix′   I =( x   R   +n   R )+ i ( x   I   +n   I ) 
       
     
     and the real and imaginary terms from the output (or second signal) 
     
       
           y′=y′   R   +iy′   I =( h   R   x   R   −h   I   x   I   +m   R )+ i ( h   I   x   R   +h   R   x   I   +m   I ) 
       
     
     Using x′ R  as an example, each of the dispersions is estimated from the N data samples            γ   ^       x   R   ′       =         (     C        (     p   ,   α     )       )         -   α     /   p              (       1   N            ∑     k   =   1     N                              x   R   ′          (   k   )            p         )       α   /   p                                
     where          C        (     p   ,   α     )       =       2   p              Γ        (       p   +   1     2     )            Γ        (     1   -     p   α       )             Γ        (     1   2     )            Γ        (     1   -     p   2       )                                    
     and 
     
       
         −1 &lt;p&lt;α   
       
     
     where p can be selected from this interval using a minimum error procedure for a given alpha, and where the normalizing term in the codifference correlator is given by 
     
       
         {circumflex over (γ)} x′ ={circumflex over (γ)} x′     R   +{circumflex over (γ)} x′     I   . 
       
     
     Incorporated herein by reference is McLaughlin,. D. J., et al, “High Resolution Polarimetric Radar Scattering Measurements of Low Grazing Angle Sea Clutter,” IEEE Journal of Oceanic Engineering, Vol. 20, No. 3, July 1995, pp 166-178. An illustrative application of the inventive unnormalized codifference correlator is represented by the calculation of the terms for the averaged Mueller matrix as shown in McLaughlin equation (8). The Mueller matrix is used to describe the polarimetric scattering behavior of a radar target. Here the four complex-valued terms from the polarization scattering matrix (McLaughlin equation (1b)) are taken a pair at a time, and temporally (time) or spatially averaged to cover all sixteen combinations. This operation utilizes the classical second-order correlator (unnormalized). When applied to radar, scattering from spiky sea clutter, the unnormalized codifference correlator in accordance with the present invention would be expected to give a more accurate and consistent measure of the terms for the Mueller matrix. 
     The unnormalized codifference correlator is given by 
     
       
           {circumflex over (q)}   x′y′ =({circumflex over (τ)} x′     R,     −y′     R   −{circumflex over (τ)} x′     R,     y′     R   +{circumflex over (τ)} x′     I,     −y′     I   −{circumflex over (τ)} x′     I,     y′     I   )+ 
       
     
     
       
           i ({circumflex over (τ)} x′     R,     −y′     I   −{circumflex over (τ)} x′     R,     y′     I   −({circumflex over (τ)} x′     I,     −y′     R   −{circumflex over (τ)} x′     I,     y′     R   )) 
       
     
     where 
     
       
         
           x′=x′ 
           R 
           +ix′ 
           I 
         
       
     
     
       
         
           y′=y′ 
           R 
           +iy′ 
           I 
         
       
     
     And, when related to McLaughlin equation (8) by using, for example, the average &lt;S* vh S vv &gt;, the averaged product is replaced by the unnormalized codifference correlator using the polarization scattering elements where 
     
       
         
           x′=S 
           vv 
         
       
     
     
       
         
           y′=S 
           vh 
         
       
     
     For symmetric alpha-stable (SαS) random variables with any characteristic exponent, alpha, from zero to two, the codifference is a measure of bivariate dependence. Signals and noise with this statistical distribution are impulsive in nature and classical correlation methods can give inconsistent answers. In accordance with the present invention, the properties of the codifference are exploited to construct a codifference correlator that can be used for estimating or quantifying the relationship between two complex, isotropic SαS random variables. 
     The traditional covariation correlator only works for alpha from one to two. The traditional classical correlator uses covariance (correlation coefficients and transfer functions) which, for SαS signal and noise, is only defined for alpha of two. The present invention&#39;s codifference correlator functions in a manner somewhat comparable to the covariation correlator and the classical correlator. 
     Besides including alpha from zero to two, a major advantage of the inventive codifference correlator is robustness to uncorrelated SαS noise added to both random variables. A disadvantage of the inventive codifference correlator is a nonlinear relationship or bias with respect to the “amount of dependence,” especially for alpha less than one. Potential applications of the present invention can range from characterizing small radar targets in spiky sea, clutter (e.g., as illustrated in FIG. 1) to the possible estimation of boundary layer velocity and pressure relationships in transitional flow. 
     Under real world circumstances in which a common signal along with uncorrelated, additive noise having an impulsive character is present in each channel prior to measurement, the inventive codifference correlator is capable of obtaining a consistent estimate of the relationship or common signal between the two channels. Known methodologies (e.g., second-order correlators and covariation correlators) cannot provide consistent estimates of this relationship. The present invention&#39;s codifference correlator will also work when the noise is extremely spiky (alpha less than one), and the known methodologies are not defined. 
    
    
     Other objects, advantages and features of this invention will become apparent from the following detailed description of the invention when considered in conjunction with the accompanying, drawings. 
     BRIEF DESCRIPTION OF THE DRAWINGS 
     In order that the present invention may be clearly understood, it will now be described, by way of example, with reference to the accompanying drawings, wherein like numbers indicate the same or similar components, and wherein: 
     FIG. 1 is a graphical representation of spiky, H-pol sea clutter (alpha, α=1.3). 
     FIG. 2 is a diagrammatic representation of an additive measurement noise model. 
     FIG. 3 is a graphical representation of parameter, p, at minimum error. 
     FIG. 4 is a graphical representation of minimum normalized correlator error. 
     FIG. 5 is a graphical representation of estimation bias of h for α=1.25. 
     FIG. 6 is a graphical representation of estimation bias of h for α=1.25. 
     FIG. 6 is a graphical representation of normalized random error for each correlator for a range of alpha. 
     FIG. 7 is a graphical representation of normalized random error for each correlator for a range of alpha. 
     FIG.  8 . is a graphical representation of normalized random error for each correlator as the number of samples is varied. 
     FIG. 9 is a block diagram illustrative of some embodiments in accordance with the present invention, wherein communications apparatus includes an inventive codifference correlator, and wherein a received signal and a reference signal are inventively correlated. 
     FIG. 10 is a block diagram illustrative of some embodiments in accordance with the present invention, wherein communications apparatus includes an inventive codifference correlator, wherein the communications apparatus may (depending on the embodiment) be characterized by any number of electronic channels, and wherein pairs of signals from one or more channels are invertively correlated. 
     FIG.  11  and FIG. 12 are block diagrams illustrating an inventive embodiment involving effectuation of coherent radar. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     CONVENTIONAL CORRELATORS 
     Description of the two aforenoted conventional correlators, viz., the second-order correlator and the covariation correlator, is an appropriate point of departure for description of the codifference correlator in accordance with the present invention. 
     Classical Second-order Correlator (Covariance) 
     The theory of second-order moments is the basis for the classical correlator that involves variance, covariance and the power spectrum. Over the past 50 years, this theory has been the foundation of statistical signal modeling and processing. These methods require signals and noise with a finite variance. The Gaussian assumption is generally made which usually leads to analytically tractable results. For SαS signals and noise, the second moment or variance is infinite and applying second-order moments can result in inconsistent, nonrobust results. The least-squares criterion minimizes the second-order moment of estimation errors and even a small proportion of extreme observations in the data can result in large variations in the results. 
     Using second-order moments, the fundamental measure of bivariate dependence is correlation. With reference to FIG. 2, consider the additive measurement noise model in FIG. 2 where the relationship between x and y is the complex multiplication, y=hx, where the additive noises, n and m, are statistically independent of each other and the input, x, where i={square root over (−1)} and where N samples of x and y are available. For this basic building block, the objective is to quantify the degree of dependence by estimating h. Using cross correlation, h is estimated by the classical second-order correlator.                h   ^     =         1   /   N            ∑     k   =   1     N                       y   k   ′          x   k     ′   *                 1   /   N            ∑     k   =   1     N                       x   k   ′          x   k     ′   *                       (   2.1   )                                
     where * indicates the complex conjugate, and                  E        [       y   ′          x     ′   *         ]         E        [       x   ′          x     ′   *         ]         =     h          σ   x   2         σ   x   2     +     σ   n   2                   (   2.2   )                                
     where σ x   2  is the variance of x, and σ n   2  is the variance of the additive noise on the input. The disadvantage of this method is that it is undefined for SαS signals and noise with α&lt;2. From sampling studies, presented hereinbelow, the classical correlator is seen to give inconsistent results that become more severe as alpha becomes smaller and as measurement noise, n, is added to the input. 
     Covariation Correlator 
     For SαS signals and noise and for 1≦α≦2, the covariation correlator can perform a similar function to the classical correlator in second-order random variables. Using covariation, h is estimated by the covariation correlator                h   ^     =         1   /   N            ∑     k   =   1     N                       y   k   ′          x   k     ′   *                   x   k   ′            p   -   2                 1   /   N            ∑     k   =   1     N                            x   k   ′                    p                     (   2.3   )                                
     where                    E   [       y   ′          x     ′   *                   x   ′                 p   -   2       ]         E        [            x   ′          p     ]         =     h          γ   x         γ   x     +     γ   n                   (   2.4   )                                
     where p&lt;α, and γ x  and γ n  are the dispersion of x and the additive noise on the input. See aforementioned C. L. Nikias and M. Shao, Signal Processing with Alpha-Stable Distributions and Applications, John Wiley and Sons, New York, N.Y., 1995. The dispersion of SαS random variables is defined hereinbelow. For α=p=2, this method reduces to the covariance. This method works well, but it is undefined for α&lt;1. From sampling studies, this correlator gives inconsistent results for α&lt;1, as well as for the case of measurement noise on the input, n, for signal and noise with alpha in the range 1≦α≦2. 
     THE INVENTIVE CODIFFERENCE CORRELATOR 
     In now discussing the codifference correlator according to the present invention, first the codifference will be defined; properties of the codifference will be enumerated through summarization of textbook information. Next, the codifference of the real and imaginary parts of a complex, isotropic SαS random variable will be derived. Then, these results and properties will be combined in a derivation of the inventive codifference correlator as an estimator of bivariate dependence. 
     Codifference 
     The codifference is a measure of bivariate dependence that is defined for SαS random variables with any characteristic exponent, 0&lt;α≦2. In contrast, the covariation is only defined for 1&lt;α≦2, and the covariance is only defined for α=2. The characteristic function for a SαS random variable is 
     
       
         φ( t )=exp( iat−γ|t|   α )  (3.1) 
       
     
     where 0&lt;α≦2 is the characteristic exponent, γ&gt;0 the dispersion, and −∞≦α≦∞ the location parameter (α=0 will be assumed.). See aforementioned C. L. Nikias and M. Shao, Signal Processing with Alpha-Stable Distributions and Applications, John Wiley and Sons, New York, N.Y., 1995. The codifference of two real-valued, jointly SαS random variables x and y is 
     
       
         τ x,y =γ x +γ y −γ x−y   (3.2) 
       
     
     where γ x , γ y  and γ x−y  are the dispersion parameters for x, y and x−y, respectively. See aforementioned G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random_Processes: Stochastic Models with Infinite Variance, Chapman and Hall, New York, N.Y., 1994. The codifference estimate is 
     
       
         {circumflex over (τ)} x,y ={circumflex over (γ)} x +{circumflex over (γ)} y −{circumflex over (γ)} x−y   (3.3) 
       
     
     The codifference has various properties; see aforementioned G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman and Hall, New York, N.Y., 1994. The codifference is symmetric, τ x,y =τ y,x . For α=2, the codifference reduces to the covariance. If x and y are independent, then τ x,y =0 for 0&lt;α≦2. An important relation when x and y are independent is 
     
       
         γ x−y =γ x +γ y   (3.4) 
       
     
     If τ x,y =0, then x and y are independent for 0&lt;α&lt;1. The upper and lower limits on τ x,y  are 
     
       
         τ x,y ≦γ x +γ y   
       
     
     
       
         
           
             
               
                 
                   
                     τ 
                     
                       x 
                       , 
                       y 
                     
                   
                   ≥ 
                   
                     { 
                     
                       
                         
                           0 
                         
                         
                           
                             0 
                             &lt; 
                             α 
                             &lt; 
                             1 
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   2 
                                   
                                     α 
                                     - 
                                     1 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 
                                   γ 
                                   x 
                                 
                                 + 
                                 
                                   γ 
                                   y 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             1 
                             ≤ 
                             α 
                             ≤ 
                             22 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3.5 
                   ) 
                 
               
             
           
         
                 
         
             
         
      
     
     Also, as τ x,y  becomes larger x and y become “more dependent.” 
     The dispersion is related to the fractional lower-order moments 
     
       
         γ x   =C ( p ,α) −α/p   E[|x|   p ] α/p   (3.6) 
       
     
     where −1&lt;p&lt;α, and                C        (     p   ,   α     )       =       2   p          Γ        (       p   +   1     2     )              Γ        (     1   -     p   α       )           Γ        (     1   2     )            Γ        (     1   -     p   2       )                     (   3.7   )                                
     See aforementioned C. L. Nikias and M. Shao, Signal Processing with Alpha-Stable Distributions and Applications, John Wiley and Sons, New York, N.Y., 1995. 
     Using the fractional lower-order moment, the dispersion is estimated by                  γ   ^     x     =         C        (     p   ,   α     )           -   α     /   p              (       1   N            ∑     k   =   1     N                            x   k          p         )       α   /   p                 (   3.8   )                                
     Dispersion estimates are made using equations (3.3) and (3.8). For alpha known, the normalized sampling error for the dispersion is approximately                    N                   Var        [     γ   ^     ]           γ     =       α        p                     C        (       2      p     ,   α     )           C        (     p   ,   α     )       2       -   1                 (   3.9   )                                
     See Robert D. Pierce, “Inconsistencies in parameters estimated from impulsive noise”, Current Topics in Nonstationary Analysis, Proc. Second Workshop on Nonstationary Random Processes and their Applications, San Diego, Calif., Jun. 11-12, 1995, edited by G. Trevino, et al, World Scientific, River Edge, N.J., pp 15-33, (1996), incorporated herein by reference. 
     Referring to FIG. 3, numerical evaluation of the error equation allows selection of the parameter, p, that gives minimum error for a given alpha, p opt . A plot of p opt  is represented in FIG.  3 . To prevent possible computation problems, a limit was placed on p opt  such that |p opt |≧0.06. The error versus alpha curves are shallow, so this restriction has little practical significance. As seen later, the codifference correlator uses a difference normalization such that the normalized codifference correlator error is approximately                    N                   Var        [     γ   ^     ]               2   α        γ       =       α       2   α             p                       C        (       2      p     ,   α     )           C        (     p   ,   α     )       2       -   1                 (   3.10   )                                
     The normalized codifference correlator error relationship is illustrated in FIG.  4 . 
     Codifference of Real and Imaginary Terms 
     The development of the present invention&#39;s codifference correlator is based on the codifference of the real and imaginary terms. Complex, isotropic SαS random variables have the form 
     
       
           z=z   R   +iz   I   =A   1/2 ( g   R   +ig   I )  (3.11) 
       
     
     where g R  and g I  are identically distributed, independent Gaussian random variables, and A is a fully skewed alpha-stable random variable (positive alpha-stable, PaS) with α PαS =α SαS /2; see aforementioned G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman and Hall, New York, N.Y., 1994. Except for α=2, the real and imaginary terms are not independent. 
     Consider the codifference between the real and imaginary terms were μ and υ are real-valued constants 
     
       
         τ μz     R,     υz     I   =γ μz     R   +γ υz     I   −γ z     R−υz1     (3.12) 
       
     
     From equation (3.6), the first two dispersion terms are 
     
       
         γ μz     R   =|μ| α γ z     R    and γ υz     I   =|υ| α γ z     I     (3.13) 
       
     
     and γ z     R   =γ z     I   . Next, 
     
       
         γ μz     R     −υz     I     =C ( p ,α) −α/p   E[|μz   R −υz I | p ] α/p   ,    
       
     
     
       
         E[|μz R −υz I | p   ]=E[A   p/2   ]E[|μg   R −υg I|   p ], and  
       
     
     
       
           E[|μg   R −υg I | p ] 2/p =μ 2   E[|g   R | P ] 2/p +υ 2   E[|g   I | p ] 2/p   
       
     
     so 
     
       
         γ μz     R     −υz     I   =(μ 2 +υ 2 ) α/2 γ z     R     (3.14) 
       
     
     So, 
     
       
         τ μz     R,     υz     I   =(|μ| α +|υ| α −(μ 2 +υ 2 ) α/2 )γ z     R     (3.15) 
       
     
     Codifference Correlator 
     Consider two complex, isotropic SαS random variables, x′ and y′ that have a linear relationship and additive noise in both variables. Again, this basic model is presented in FIG.  2 . The objective is to quantify the degree of dependence represented by h. 
     From the model in FIG.  2 , 
     
       
           x′=x′   R   +ix′   I =( x   R   +n   R )+ i ( x   I   +n   I )  (3.16) 
       
     
     and 
     
       
           y′=y′   R   +iy′   I =( h   R   x   R   −h   I   I   +m   R )+ i ( h   I   x   R   +h   R   x   R   +m   I )  (3.17) 
       
     
     h is estimated by taking the sum and difference of various codifference estimates between the real and imaginary parts. The codifference correlator is 
     
       
           ĥ =[({circumflex over (τ)} x′     R,     −y′     R   −{circumflex over (τ)} x′     R,     y′     R   + 
       
     
     
       
         {circumflex over (τ)} x′     I,     y′     I   −{circumflex over (τ)} x′     I,     y′   I )+ i ({circumflex over (τ)} x′     R,     −y′     I   −{circumflex over (τ)} x′     R,     y′     I   − 
       
     
     
       
         ({circumflex over (τ)} x′     I,     y′     R   −{circumflex over (τ)} x′     I,     y′     R   ))]/(2 α {circumflex over (γ)} x′ )  (3.18) 
       
     
     By treating the numerator and denominator separately, the expected value of h is approximately 
     
       
           E[ĥ]=E [({circumflex over (τ)} x′     R,     −y′     R   −{circumflex over (τ)} x′     R,     y′     R   +{circumflex over (τ)} x′     I,     −y′     I   −{circumflex over (τ)} x′     I,     y′     I   )+ 
       
     
     
       
           i ({circumflex over (τ)} x′     ,     −y′     I   −{circumflex over (τ)} x′     R,     y′     I,   −({circumflex over (τ)} x′     I,     −y′   R −{circumflex over (τ)} x′     I,     y′     R   ))]/(2 α   E [{circumflex over (γ)} x′ ])  (3.19) 
       
     
     where E[γ x′]=γ   x′  is defined as 
     
       
         γ x′ =γ x′     R   +γ x′     I     (3.20)  
       
     
     Note that from equation (3.11), γ x′     R   =γ x′     I   . When estimating γ x′ , the real and imaginary terms would be estimated separately and added together. Next, E[{circumflex over (τ)} s,t ]=τ s,t , so expanding the codifference 
     
       
         τ x′     R,     −y′     R   −τ x′     R,     y′     R   +τ x′     I,     −y′     I   −τx′   I,     y′     I   =γx′   R     y′     R   −γ x′     R     −y′     R   + 
       
     
     
       
         γ x′     I     +y′     I   −γ x′   I   −y′     I   =γ x     R     +n     R     +h     R     x     R     −h     I     x     I     +m     R   −γ x     R     +n     R     −h     R     +h     I     x     I     −m     R   + 
       
     
     
       
         γ x     I     +n     I     +h     I     x     R     +h     R     x     I     +m   I −γ x     I     +n     I     −h     I     x     R     −h     R     x     I     −m     I     (3.21) 
       
     
     and 
     
       
         τ x′     R,     −y′     I   −τ x′     R,     y′     I   −(τ x′     I,     y′     R   )= 
       
     
     
       
         γ x′     R     +y′     I   −γ x′     R     +   y′     I   −γ x′     I     +y′     R   +γ x′     I     −y′     R   = 
       
     
     
       
         γ x     R     +n     R     +h     I     x     R     +h     R     x     I     +m     I   −γ x     R     +n     R     −h     I     x     R     −h     Rx       R     −h     R     x     I     −m     I   − 
       
     
     
       
         γ x     I+n       I     +h     R     x     R     −h     I     x     I     +m     R   +γ x     I     +n     I     −h     R     x     R     +h     I     x     I     −m     R     (3.22) 
       
     
     As an example, examine the difference, τ x′     R,     −y′     R   −τ x′     R,     y′     R   , and apply the properties from equations (3.2) and (3.4) 
     
       
         γ x     R     +n     R     +h     R     x     R     −h     I     x     I     +m     R   −γ x     R     +n     R     −h     R     x     r     +h     I     x     I     −m     R   =γ (1+h     R     )x     R     −h     I     x     I   + 
       
     
     
       
         γ n     R   +γ m     R   −(γ (1−h     R     )x     R     +h     I     x     I   +γ n     R   +γ m     R   )  (3.23) 
       
     
     The purpose of the difference is to cancel the additive noise terms. 
     
       
         γ x     R     +n     R     +h     R     x     R     −h     I     x     I     +m     R   −γ x     R     +n     R     −h     R     x     R     +h     I     x     I     −m     R   = 
       
     
     
       
         γ (1+h     R     )x     R     −h     I     x     I   −γ (1−h     R     )x     R     +h     I     x     I     (3.24) 
       
     
     Next, using the result from equation (3.14) 
     
       
         γ x     R     +n     R     +h     R     x     R     −h     I     x     I     +m     R   −γ x     R     +n     R     −h     R     x     R     +h     I     x     I     −m     R   = 
       
     
     
       
         [((1 +h   R ) 2   +h   I   2 ) α/2 −((1 −h   R ) 2   +h   I   2 ) α/2 ]γ x     R     (3.25) 
       
     
     Note that γ x     R   =γ x     I    and define 
     
       
         γ x =γ x     R   +γx   I     (3.26) 
       
     
     Applying these methods to the remaining terms in equations (3.21) and (3.22), 
     gives the expected value of the codifference correlator. 
     
       
           E[ĥ ]={[((1 +h   R ) 2   +h   I   2 ) α/2 −((1 −h   R ) 2   +h   I   2 ) α/2   ]+i [((1 +h   I ) 2   +h   R   2 ) α/2 −((1 −h   I ) 2   +h   R   2 ) α/2 ]}γ x /(2 α γ x′)   (3.27) 
       
     
     Referring to FIG.  5  and FIG. 6, the nonlinear estimation bias and interaction between the real and imaginary terms can be seen for E[ĥ R ] in FIG.  5  and FIG. 6 for alpha of 1.25 and 0.75, respectively, where h I  is varied from 0.0 to 1.0 in steps of 0.1 with no noise. Due to symmetry, the E[ĥ I ] term will behave similarly. This bias is undesirable and can restrict the usefulness of the present invention&#39;s codifference approach to the region where |h I |&lt;1. This restriction is especially true for alpha less than 1. The present invention&#39;s codifference approach, however, still provides the only consistent method for estimating h for α&lt;1 and for 1≦α&lt;2 when measurement noise is on the input. 
     Additional results can be obtained. A noncoherent estimate of the magnitude of h is                       h   ^          α     =             γ   ^       y   ′     ′         γ   ^       x   ′                       or                          h   ^          noncoh       =       (         γ   ^       y   ′     ′         γ   ^       x   ′         )       1   /   α                 (   3.28   )                                
     By treating the numerator and denominator separately, the expected value of each term is obtained where 
     
       
         γ x′     R   =γ x     R   +γ n     R    and γ x′     I   =γ x     I   +γ n     I     (3.29) 
       
     
     so 
     
       
         γ x′ =γ x +γ n   (3.30) 
       
     
     and 
     
       
         
           
             
               γ 
               
                 y 
                 R 
                 ′ 
               
             
             = 
             
               
                 
                   
                     ( 
                     
                       
                         h 
                         R 
                         2 
                       
                       + 
                       
                         h 
                         I 
                         2 
                       
                     
                     ) 
                   
                   
                     α 
                     / 
                     2 
                   
                 
                  
                 
                   
                     γ 
                     x 
                   
                   2 
                 
               
               + 
               
                 γ 
                 
                   m 
                   R 
                 
               
             
           
         
                 
         
             
         
      
     
     and 
     
       
         
           
             
               
                 
                   
                     γ 
                     
                       y 
                       I 
                       ′ 
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               h 
                               R 
                               2 
                             
                             + 
                             
                               h 
                               I 
                               2 
                             
                           
                           ) 
                         
                         
                           α 
                           / 
                           2 
                         
                       
                        
                       
                         
                           γ 
                           x 
                         
                         2 
                       
                     
                     + 
                     
                       
                         γ 
                         
                           m 
                           I 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   3.31 
                   ) 
                 
               
             
           
         
                 
         
             
         
      
     
     so 
     
       
         γ y′ =( h   R   2   +h   I   2 ) α/2 γ x +γ m   (3.32) 
       
     
     The expected value of the magnitude of h estimator is approximately 
       E[|ĥ|   α ]=[( h   R   2   +h   I   2 ) α/2 γ x +γ m ]/(γ x +γ n )  (3.33) 
     Sampling Studies 
     Sampling studies demonstrate the statistical efficiency of the inventive codifference correlator in the presence of additive noise on the input and output. For this study, the signal, and noise dispersions were: γ x =γ n =γ m =1, and h=(0.45,−0.45). N independent computer generated SαS complex random variables were drawn and the results calculated for each correlator (viz., the second-order correlator, the covariation correlator and the inventive codifference correlator). This process was repeated 50 times and the mean and standard deviation (random error estimate) were calculated for each set of results. The standard deviation was normalized by multiplying by {square root over (N)}. N was varied from 100, 200, 400, 800 and 1600. The normalized error was calculated according to equation (3.10). 
     Reference is now made to FIG.  7 . As alpha was varied from 0.5, 0.75, 1.0, 1.25, 1.5, 1.75 and 2.0, FIG. 7 shows the random error estimates for the output from each correlator. For alpha less than one, the second-order correlator and the covariation correlator are not expected to work, so their results are not shown. The inventive codifference correlator shows stability as N and alpha are varied. The calculated error tends to underestimate the inventive codifference correlator error. Note, however, that the calculated error was derived for dispersion estimates and is intended to give an approximate idea of the inventive codifference correlator error; for this purpose, it works well. The second-order and covariation correlators show error variations over one order of magnitude. 
     With reference to FIG. 8, for alpha of 1.25 and 1.5, the random error estimates are shown in FIG. 8 as a function of N. This demonstrates that the second-order correlator and the covariation correlator show random error increasing with increased sample size. The conclusion is that the second-order and covariation correlators are not statistically efficient, a fundamental requirement of any estimator. For additive noise on the input and output, the second-order and covariation correlators give inconsistent results. By contrast, the inventive codifference correlator gives consistent results. 
     Source Code 
     The following source code is illustrative of the second-order correlator, the covariation correlator and especially the inventive codifference correlator. This source code is written in the programming language called Interactive Data Language (IDL), a product of Research Systems, Inc., 4990 Pearl East Circle, Boulder, Colo. 80301. This source code is presented to demonstrate the implementation of the various correlators, and in particular the inventive codifference correlator. 
     xx is an array of N samples that represents x′ and yy is an array of N samples that represents y′. The second-order correlator from equation (2.1) is written as 
     
       
         sec_order_correl=mean( conj ( xx )* yy )/mean(abs( xx ){circumflex over ( )}2) 
       
     
     where sec_order_correl is the result from the second-order correlator, mean(array_name) is an IDL library function that calculates the sample mean of the named array, and abs(array_name) is an IDL librar function that returns the absolute value of the named array. The covariation correlator from equation (2.3) with p=1 is written as 
     
       
         covar_correl=mean(conj( xx )* yy /abs( xx ))/mean(abs( xx )) 
       
     
     where covar_correl is the result from the covariation correlator, and conj(array_name) is an IDL library function that returns the complex conjugate of the named array. 
     The codifference correlator is presented as an IDL subroutine, codif_cmplx, that takes the arrays xx and yy, and the constant alpha, and returns gam_xx, γ x′ , gam_y; {circumflex over (γ)}y ′ , and cod_xy, the result from the codifference correlator. Comment lines in IDL begin with a semicolon. 
     
       
         
               
               
             
               
             
               
               
             
               
             
               
               
             
               
             
               
               
             
               
               
             
               
             
               
               
             
               
               
             
               
             
               
               
             
               
               
             
               
             
               
               
             
               
             
               
               
             
               
             
               
               
             
               
             
               
               
             
           
               
                   
                   
               
             
             
               
                   
                 pro codif_cmplx, xx, yy, alpha, gam_xx, gam_yy, cod_xy 
               
             
          
           
               
                 ; 
               
             
          
           
               
                   
                 if alpha gt 2.0 then alpha = 2.0 
               
             
          
           
               
                 ; 
               
             
          
           
               
                   
                 find_pp, alpha, pp, norm_gam_err 
               
             
          
           
               
                 ; 
               
               
                 ; find_pp is an IDL subroutine that takes the constant alpha and returns the optimum value 
               
               
                 for p 
               
               
                 ; the norm_gam_err value is calculated according to equation (3.10), this value is returned 
               
               
                 but not used 
               
               
                 ; 
               
             
          
           
               
                   
                 if alpha eq 2.0 then c_fact = 4.*gamma(1.5)/gamma(0.5) 
               
               
                   
                 if alpha lt 2.0 then c_fact = (2.{circumflex over ( )}pp)*gamma((pp+1.)/2.)*gamma(1.−pp/alpha)/$ 
               
             
          
           
               
                   
                 (gamma(1./2.)*gamma(1.−pp/2.)) 
               
             
          
           
               
                 ; 
               
               
                 ; c_fact is the parameter  C(p,α)  from equation (3.7) 
               
               
                 ; gamma() is the IDL library Gamma function 
               
               
                 ; 
               
             
          
           
               
                   
                 gam_xx = ( mean( abs(float(xx)){circumflex over ( )}pp )/c_fact ){circumflex over ( )}(alpha/pp) + $ 
               
             
          
           
               
                   
                 ( mean( abs(imaginary(xx)){circumflex over ( )}pp )/c_fact ){circumflex over ( )}(alpha/pp) 
               
             
          
           
               
                 ; 
               
               
                 ; float() and imaginary() are IDL Library functions that return, respectively, the real and 
               
               
                 imaginary parts or an array 
               
               
                 ; 
               
             
          
           
               
                   
                 gam_yy = ( mean( abs(float(yy)){circumflex over ( )}pp )/c_fact ){circumflex over ( )}(alpha/pp) + $ 
               
             
          
           
               
                   
                 ( mean( abs(imaginary(yy)){circumflex over ( )}pp )/c_fact ){circumflex over ( )}(alpha/pp) 
               
             
          
           
               
                 ; 
               
             
          
           
               
                   
                 gam_xpy_r = ( mean( abs(float(xx)+float(yy)){circumflex over ( )}pp )/c_fact ){circumflex over ( )}(alpha/pp) + $ 
               
               
                   
                   ( mean( abs(imaginary(xx)+imaginary(yy)){circumflex over ( )}pp )/c_fact ){circumflex over ( )}(alpha/pp) 
               
               
                   
                 gam_xpy_i = ( mean( abs(float(xx)+imaginary(yy)){circumflex over ( )}pp )/c_fact ){circumflex over ( )}(alpha/pp) − $ 
               
               
                   
                   ( mean( abs(imaginary(xx)+float(yy)){circumflex over ( )}pp )/c_fact ){circumflex over ( )}(alpha/pp) 
               
               
                   
                 gam_xpy = complex(gam_xpy_r,gam_xpy_i) 
               
             
          
           
               
                 ; 
               
               
                 ; complex(a,b) is an IDL Library function that takes two real variables and returns the 
               
               
                 complex number, a + ib 
               
               
                 ; 
               
             
          
           
               
                   
                 gam_xny_r = ( mean( abs(float(xx)−float(yy)){circumflex over ( )}pp )/c_fact ){circumflex over ( )}(alpha/pp) + $ 
               
               
                   
                   ( mean( abs(imaginary(xx)−imaginary(yy)){circumflex over ( )}pp )/c_fact ){circumflex over ( )}(alpha/pp) 
               
               
                   
                 gam_xny_i = ( mean( abs(float(xx)−imaginary(yy)){circumflex over ( )}pp )/c_fact ){circumflex over ( )}(alpha/pp) − $ 
               
               
                   
                   ( mean( abs(imaginary(xx)−float(yy)){circumflex over ( )}pp )/c_fact ){circumflex over ( )}(alpha/pp) 
               
               
                   
                 gam_xny = complex(gam_xny_r,gam_xny_i) 
               
             
          
           
               
                 ; 
               
             
          
           
               
                   
                 cod_xy = (gam_xpy−gam_xny)/(gam_xx*2.{circumflex over ( )}alpha) 
               
             
          
           
               
                 ; 
               
             
          
           
               
                   
                 return 
               
               
                   
                 end 
               
               
                   
                   
               
             
          
         
       
     
     The codifference correlator subroutine calls the subroutine find_pp that takes the constant alpha and returns the optimum value for p, and the normalized error value, norm_gam_err, as calculated according to equation (3.10). This routine uses a lookup table of alpha and optimum p to interpolate using the alpha parameter passed to the subroutine. 
     
       
         
               
             
               
               
             
               
             
               
               
             
               
             
               
               
             
               
             
               
               
             
               
             
               
               
             
               
             
               
               
             
               
             
               
               
             
               
               
             
               
               
             
               
             
               
               
             
               
               
             
               
             
           
               
                   
               
             
             
               
                 pro find_pp, alpha, pp, gam_err 
               
               
                 ; 
               
               
                 ; given alpha, return optimum p and normalized estimation error for gamma 
               
               
                 ; 
               
             
          
           
               
                   
                 alp = [2.00,1.95,1.90,1.85,1.80,1.75,1.70,1.65,1.60,1.55,1.50,$ 
               
               
                   
                 1.45,1.40,1.35,1.30,1.25,1.20,1.15,1.10,1.05,1.00,$ 
               
               
                   
                 0.95,0.90,0.85,0.80,0.75,0.70,0.65,0.60,0.55,0.50,$ 
               
               
                   
                 0.45,0.40,0.35,0.30,0.25,0.20,0.15,0.10] 
               
             
          
           
               
                 ; 
               
             
          
           
               
                   
                 ppopt = [ 0.992, 0.651, 0.551, 0.483, 0.424, 0.380, 0.337, 0.302, 0.273, $ 
               
               
                   
                 0.239, 0.213, 0.186, 0.166, 0.140, 0.115, 0.096, 0.077, 0.059, $ 
               
               
                   
                 0.041, 0.023,−0.015,−0.012,−0.035,−0.051,−0.068,−0.084,−0.096, $ 
               
               
                   
                 −0.112,−0.124,−0.140,−0.151,−0.159,−0.168,−0.173,−0.175,−0.172, $ 
               
               
                   
                 −0.158,−0.132,−0.096] 
               
             
          
           
               
                 ; 
               
             
          
           
               
                   
                 num_alp = 39 
               
             
          
           
               
                 ; 
               
             
          
           
               
                   
                 pp = 2.0 
               
               
                   
                 if alpha ge 2.0 then goto, L10 
               
               
                   
                 pp = −0.096 
               
               
                   
                 if alpha lt 0.1 then goto, L10 
               
             
          
           
               
                 ; 
               
             
          
           
               
                   
                 val = where(abs(alpha-alp) lt 0.076,count) 
               
               
                   
                 coef = poly_fit(alp(val),ppopt(val),1) 
               
               
                   
                 pp = coef(0) + coef(1)*alpha 
               
               
                   
                 if pp ge 0.0 and pp lt 0.06 then pp = 0.06 
               
               
                   
                 if pp lt 0.0 and pp gt −0.06 then pp = −0.06 
               
             
          
           
               
                 ; 
               
               
                 L10: 
               
               
                 ; 
               
             
          
           
               
                   
                 alph = alpha 
               
               
                   
                 if alpha ge 2.0 then alph = 2.0 
               
             
          
           
               
                 ; 
               
             
          
           
               
                   
                 if alph lt 2.0 then mom_p1 = (2.{circumflex over ( )}pp)*gamma((pp+1.)/2.)*gamma(1.−pp/alph)/$ 
               
             
          
           
               
                   
                 (gamma(1./2.)*gamma(1.−pp/2.)) 
               
             
          
           
               
                   
                 if alph lt 2.0 then mom_p2 = (2.{circumflex over ( )}(2.*pp))*gamma((2.*pp+1.)/2.)*gamma(1.− 
               
             
          
           
               
                 2.*pp/alph)/$ 
               
             
          
           
               
                   
                 (gamma(1./2.)*gamma(1.−pp)) 
               
             
          
           
               
                   
                 if alph eq 2.0 then mom_p1 = (2.{circumflex over ( )}pp)*gamma(1.5)/gamma(0.5) 
               
               
                   
                 if alph eq 2.0 then mom_p2 = (2.{circumflex over ( )}(2.*pp))*gamma(2.5)/gamma(0.5) 
               
               
                   
                 gam_err = alph*sqrt( mom_p2/(mom_p1{circumflex over ( )}2) − 1.)/(abs(pp)*(2.{circumflex over ( )}alph)) 
               
             
          
           
               
                 return 
               
               
                 end 
               
               
                   
               
             
          
         
       
     
     Zero Valuation of Imaginary Values 
     The use of complex valued random variables is a generalization. The inventive codifference correlator can also be expressed with the imaginary values set to zero and all the equations, now simplified, will hold. 
     Estimation of Transfer Functions 
     A property of SαS random variables is that a linear operation will still result in a SαS random variable with the same alpha. Such an operation is the fast Fourier transform, FFT. The codifference correlator can be applied to spectrum analysis or radar Doppler processing where it&#39;s possible to estimate transfer functions by combining it with the Welch method; see P. D. Welch, “The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust., AU-15, June 1967, pp 70-73, incorporated herein by reference. 
     The classical approach uses the second-order correlator for transfer function estimates                    H   ^     so          (     ω   j     )       =         1   N            ∑     n   =   1     N                         Y   n          (     ω   j     )              X   n   *          (     ω   j     )                 1   N            ∑     n   =   1     N                         X   n          (     ω   j     )              X   n   *          (     ω   j     )                       (   5.1   )                                
     and the FFT of the n th  windowed, overlapping data segment with N total segments is                  X   n          (     ω   j     )       =       1   K            ∑     k   =   1     K                       x        (       (     k   -   1   +     ξ   n       )        Δ                 t     )            κ        (     k                 Δ                 t     )                   -                        ω   j        Δ                   t        (     k   -   1     )                         (   5.2   )                                
     at frequency ω j =2π(j−1)/KΔt, where ξ n =(n−1)S and each segment is shifted S samples relative to the last segment. Also, K is the FFT size, κ(kΔt) is the window function and Δt is the time between samples. The Y n (ω j ) is obtained in the same manner. Since the X n (ω j ) and Y n (ω j ) are SαS distributed, the codifference correlator can be applied to estimate the transfer function 
     
       
           Ĥ   cd (ω j )=[(Δ A )+ i (Δ B )]/(2 α {circumflex over (γ)} X(ω   j ))  (5.3) 
       
     
     
       
         Δ A={circumflex over (τ)}   X     R     (ω     j     ),−Y     R     (ω     j     ) −{circumflex over (τ)} X     R     (ω     j     ),Y     R     (ω     j     ) +{circumflex over (τ)} X     I     (ω     j     ),−Y     I     (ω     j     ) −{circumflex over (τ)} X     I     (ω     j     ),Y     I     (ω     j     )   
       
     
     
       
         Δ B={circumflex over (τ)}   X     R     (ω     j     ),−Y     I     (ω     j     ) −{circumflex over (τ)} X     R     (ω     j     ),Y     I     (ω     j     ) −({circumflex over (τ)} X     I     (ω     j     ),Y     R     (ω     j     ) −{circumflex over (τ)} X     I     (ω     j     ),Y     R     (ω     j     ) ) 
       
     
     and 
     
       
         {circumflex over (γ)} X(ω     j     ) ={circumflex over (γ)} X     R     (ω     j     ) +{circumflex over (γ)} X     I     (ω     j     )   (5.4) 
       
     
     where the averages are taken over the N segments. Also, 
     
       
         {circumflex over (γ)} Y(ω     j     ) ={circumflex over (γ)} Y     R     (ω     j     ) +{circumflex over (γ)} I(ω     j     )   (5.5) 
       
     
     so from equation (3.28), a noncoherent estimate of the transfer function is                         H   ^     noncoh          (     ω   j     )            =       (         γ   ^       Y        (     ω   j     )             γ   ^       x        (     ω   j     )           )       1   /   α               (   5.6   )                                
     A form of the coherence function follows                    ξ   ^     XY          (     ω   j     )       =                H   ^     cd          (     ω   j     )                       H   ^     noncoh          (     ω   j     )                      (   5.7   )                                
     Correlation Detection 
     Referring to FIG. 9, diagrammatically illustrated is a correlation detection system (based on modulation system principles) in accordance with the present invention. Basically, the inventive codifference correlator, along with its associated components, effectuates correlation detection based on an inventive cross-correlative mathematical algorithm applied to the combination of a received signal and a locally stored or generated function, typically having some known characteristic of the transmitted wave. In other words, the inventive codifference correlator compares, in point-to-point correspondence, an observed signal with an internally generated reference signal. 
     Electromagnetic waves (e.g., radio waves)  16  are received by antenna  18 , which sends a modulated signal  20  to down converter  22 , which sends a complex down converted signal  24  to sampler  26  (e.g., including or included by a type of analog-to-digital converter). The inventive codifference correlator  30  comprises a memory  32  unit and a processor  34 . Sampler  26  obtains a sequence of instantaneous values of down converted signal  24  (e.g., at regular or intermittent temporal or spatial intervals, wherein the sampling rate is at least twice the highest frequency component of down converted signal  24 ), and accordingly sends a complex sampled signal  28  to processor  34  (of codifference correlator  30 ). 
     Memory  32 , having a complex reference signal  36  stored therein, sends complex reference signal  36  to processor  34  (of codifference correlator  30 ). In inventive practice, memory  32  can include a memory  32  unit made a part of codifference correlator  30  and/or a memory  32  unit which is separate from codifference correlator  30 . That is, the memory  32  unit in which complex reference signal  36  is stored can be either internal to or external to codifference correlator  30 . 
     Processor  34  applies the inventive codifference correlation algorithm to the combination of sampled signal  28  and reference signal  36 . Reference signal  36  is a complex signal represented by real and imaginary components having values and defined by real and imaginary axes. Similarly, modulated signal  20  (and hence, down converted signal  24  as well as sampled signal  28 ) is a complex signal represented by real and imaginary components having values and defined by real and imaginary axes. Codifference correlator  30  thus cross-correlates sampled signal  28  and reference signal  36 , and sends a resultant complex correlation signal  32  to detector (e.g., demodulator or mixer, or threshold or peak detector)  38 . Detector  38  sends a detected signal  40  to sound transducer (e.g., headphone or loudspeaker)  42  or other information sink. 
     A receiver  44  (e.g., of a heterodyne or superheterodyne type) can be considered to include certain of these components, e.g., down converter  22 , sampler  26 , codifference correlator  30 , detector  38  and sound transducer  42 . The ordinarily skilled artisan is well acquainted with implementation of down converter means, sampler means, detector means and sound transducer means in the context of various communications systems. 
     Although the example described with reference to FIG. 9 involves inventive correlation of a received signal with a locally generated signal, it is understood by the ordinarily skilled artisan in the light of this disclosure that the present invention admits of practice involving correlation of two or more electromagnetic signals of diverse kinds and received or originating from diverse sources, such as electromagnetic waves, acoustic waves, pressure waves, etc., as well as from other sensors that sense physical quantities and convert them into electrical signals. 
     Reference now being made to FIG. 10, codifference correlator  30  is connected to any number of a plurality of channels  50 , viz., channel  50   a , channel  50   b , channel  50   c , . . . channel  50   n . Each channel has an antenna  18  means and a filter  54  means such as including a down converter  22  and a sampler  26 . Each antenna  18  means receives waves  16  from a different wave source  52 ; thus, antenna  18   a  receives waves  16   a  from wave source  52   a ; antenna  18   b  receives waves  16   b  from wave source  52   b ; antenna  18   c  receives waves  16   c  from wave source  52   c ; antenna  18   n  receives waves  16   d  from wave source  52   n . Although the term “antenna” commonly denotes a device or group of devices used for receiving electromagnetic waves such as radio waves, it is to be understood with reference to FIG. 10 that “antenna  18  means” can refer to any apparatus or apparatuses which serve to sense physical quantities or receive wave phenomena (e.g., electromagnetic waves, acoustic waves, pressure waves, etc.) and render a signal corresponding thereto. 
     Codifference correlator  30  receives a filtered signal  56  via each channel, viz., filtered signal  56   a , filtered signal  56   b , filtered signal  56   c , . . . filtered Signal  56   n . Codifference correlator  30  can select any pair of filtered signals  56  (e.g., filtered signal  56   a  plus filtered signal  56   b ; or, filtered signal  56   a  plus filtered signal  56   c ; or, filtered signal  56   b  plus filtered signal  56   c ; etc.), and can inventively correlate such pair of filtered signals  54  (e.g., correlate filtered signal  56   a  with filtered signal  56   b ; or, correlate filtered signal  56   a  plus filtered signal  56   c ; or, correlate filtered signal  56   b  with filtered signal  56   c ; etc.). Codifference correlator  30  can perform its algorithmic operation with respect to one pair or any plural number of pairs of filtered signals  56 , and can do so once or any number of plural times. 
     Now referring to FIG.  11  and FIG. 12, the present invention can be propitiously practiced in the context of a coherent radar system. Depicted in FIG.  11  and FIG. 12 is a coherent radar system in accordance with the present invention. The inventive coherent radar system bears some similarities to a typical coherent radar system such as practiced by the U.S. Navy. As shown in FIG. 11, the reference signal to be transmitted is derived from a local oscillator  60 , which is then amplified by power amplifier  62 , which then feeds transmitting antenna  64 , which is initially set for horizontal polarization by the switch  82  set in position 1. When switch  82  is set in position 2, the power amplifier  62  feeds transmitting antenna  94  which is set for vertical polarization. The transmitted signal is in the form of a short burst. 
     The transmitted signal radiates outward, strikes a distant object, and is scattered back to antennas  66  and  68 , which are collocated with transmitting antennas  64  and  94 . Antenna  66  is set to receive vertically polarized waves and antenna  68  is set to receive horizontally polarized waves. The signal received by antenna  66  is amplified by low-noise amplifier  70 . The signal received by antenna  68  is amplified by low-noise amplifier  72 . Then the signals from low-noise amplifiers  70  and  72 , along with the reference signal from local oscillator  60 , go into synchronous detectors  74  and  76  (whereby the corresponding signals from low-noise amplifiers  70  and  72  go into synchronous detectors  74  and  76 , respectively). 
     Each of synchronous detectors  74  and  76  output two signals: an in-phase signal I and a quadrature signal Q. Each I and Q pair can be interpreted as a complex signal, where I is real and Q is imaginary. The corresponding pairs of signals I and Q from synchronous detectors  74  and  76  go into analog-to-digital converters (samplers)  78  and  80 , respectively. 
     As shown in FIG. 12, analog-to-digital converters  78  and  80  are each connected to switch  82  which also switches between the transmitting antennas  64  and  94  in FIG.  11 . According to a first state (switch  82  position “one” or “1”), antenna  64  transmits a short burst signal characterized by horizontal polarization. After that pulse has returned to receiving antennas  66  and  68  and has proceeded through the aforedescribed coherent radar system so as to arrive at analog-to-digital converters  78  and  80 , switch  82  having been set to position “one,” the numerical values for analog-to-digital converters  78  and  80  are stored in memories  84  and  86 . According to a second state (switch  82  position. “two” or “2”), antenna  94  transmits a short burst signal characterized by vertical polarization. After that pulse has returned to receiving antennas  66  and  68  and has proceeded through the aforedescribed coherent radar system so as to arrive at analog-to-digital converters  78  and  80 , switch  82  having been set to position “two,” the numerical values for analog-to-digital converters  78  and  80  are stored in memories  88  and  90 . This procedure occurs N number of times, wherein N is any integer greater than or equal to one. 
     The samples stored in memories  84 ,  86 ,  88  and  90  are taken in pairs by inventive codifference correlator  30  for inventive correlation. Subsequently, correlation signal  37  can be detected by a detector  38  such as shown in FIG.  9 , which can send a detected signal  40  to a sound transducer  42  (e.g., an alarm) such as shown in FIG.  9 . Alternatively, as shown in FIG. 12, correlation signal  37  can be transmitted to and thereby populate averaged Mueller matrix  92 , whereby the positions in averaged Mueller matrix  92  are determined by the pairs of memory signals taken from memories  84 ,  86 ,  88  and/or  90 . See aforementioned article by McLaughlin, D. J., et al entitled “High Resolution Polarimetric Radar Scattering Measurements of Low Grazing Angle Sea Clutter,” IEEE Journal of Oceanic Engineering, Vol. 20, No. 3, July 1995, pp 166-178, incorporated herein by reference. 
     The pairs of memory signals (i.e., signals taken from memory) can be taken from one or two memories among memories  84 ,  86 ,  88  and  90 . That is, the pairs of memory signals can be received from one memory, i.e., auto-correlative (for example, received from memory  84 , only); or, the pairs of memory signals can be received from two memories, i.e., cross-correlative (for the example shown in FIG. 12, received from memories  84  and  86 ). Averaged Mueller matrix  92  ranges over all possible combinations of pairs of memory signals. 
     Other embodiments of this invention will be apparent to those skilled in the art from a consideration of this specification or practice of the invention disclosed herein. Various omissions, modifications and changes to the principles described may be made by one skilled in the art without departing from the true scope and spirit of the invention which is indicated by the following claims.