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The '''K-distribution''' is a [[probability distribution]] that arises as the consequence of a statistical or probabilistic model used in [[Synthetic aperture radar|Synthetic Aperture Radar]] (SAR) imagery.  The K distribution is formed by [[Compound probability distribution|compounding]] two separate [[probability distribution]]s, one representing the [[radar cross-section]], and the other representing speckle that is a characteristic of coherent imaging.


The model used to represent the observed intensity ''X'', involves compounding two [[gamma distribution]]s. In each case a reparameterisation of the usual form of the family of gamma distributions is used, such that the parameters are:
:*the mean of the distribution, and
:*the usual shape parameter.


==Density==
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The model is that ''X'' has a gamma distribution with mean ''σ'' and shape parameter ''L'', with ''σ'' being treated as a [[random variable]] having another gamma distribution, this time with mean ''μ'' and shape parameter ''ν''. The result is that ''X'' has the following density function (''x''&nbsp;>&nbsp;0):<ref name=Redding>Redding (1999)</ref>
 
:<math>f_X(x;\nu,L)= \frac{2}{x} \left( \frac{L \nu x}{\mu} \right)^\frac{L+\nu}{2}
      \frac{1}{\Gamma(L)\Gamma(\nu)}
      K_{\nu-L} \left( 2 \sqrt{\frac{L \nu x}{\mu} } \right), </math>
 
where ''K'' is a [[modified Bessel function]] of the second kind. In this derivation, the K-distribution is a [[compound probability distribution]]. It is also a [[product distribution]]:<ref name=Redding/> it is the distribution of the product of two independent random variables, one having a gamma distribution with mean 1 and shape parameter ''L'', the second having a gamma distribution with mean ''μ'' and shape parameter ''ν''.
 
This distribution derives from a paper by Jakeman and Pusey (1978).
 
==Moments==
The mean and variance are given<ref name=Redding>Redding (1999)</ref> by
:<math> \operatorname{E}(X)= \mu  </math>
:<math> \operatorname{var}(X)= \mu^2 \frac{ \nu+L+1}{L \nu}  .</math>
 
==Other properties==
All the properties of the distribution are symmetric in  ''L'' and ''ν''.<ref name=Redding/>
 
==Notes==
{{Reflist}}
 
==Sources==
*Redding, Nicholas J. (1999) ''Estimating the Parameters of the K Distribution in the Intensity Domain''  [http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA368069]. Report DSTO-TR-0839, DSTO Electronics and Surveillance Laboratory, South Australia. p.&nbsp;60
*Jakeman, Eand Pusey, P. N. (1978) "Significance of K-Distributions in Scattering Experiments", ''Physical Review Letters'', 40, 546&ndash;550  {{doi|10.1103/PhysRevLett.40.546}}
 
==Further reading==
*Jakeman, E. (1980) "On the statistics of K-distributed noise", Journal of Physics A: Mathematics and General, 13, 31&ndash;48
*Ward, K.D.; Tough, Robert J.A; Watts, Simon (2006) '' Sea Clutter: Scattering, the K Distribution and Radar Performance'', Institution of Engineering and Technology. ISBN 0-86341-503-2
{{ProbDistributions|continuous-semi-infinite}}
{{DEFAULTSORT:K-Distribution}}
[[Category:Radar signal processing]]
[[Category:Continuous distributions]]
[[Category:Compound distributions]]
[[Category:Probability distributions]]

Latest revision as of 17:43, 23 August 2014


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