# Chi-square target models

**Swerling models**were introduced byPeter Swerling and are used to describe the statistical properties of theradar cross-section of complex objects.**General Target Model**Swerling target models give the RCS of a given object based on the chi-square probability density function, which has the following form:

:$p(sigma)\; =\; frac\{m\}\{Gamma(m)\; sigma\_\{av\; left\; (\; frac\{msigma\}\{sigma\_\{av\; ight\; )^\{m\; -\; 1\}\; e^\{-frac\{msigma\}\{sigma\_\{av\}$

$sigma\_\{av\}$ refers to the

mean value of sigma. This is not always easy to determine, as certain objects may be viewed the most frequently from a limited range of angles. For instance, a sea-based radar system is most likely to view a ship from the side, the front, and the back, but never the top or the bottom. $m$ is the number of degrees of freedom divided by 2. While the number of degrees of freedom used in the chi-square probability density function is an integer value in statistics, it can assume any positive real number in the target model. Values of m between .3 and 2 have been found to closely approximate certain simple shapes, such as cylinders or cylinders with fins.Since the ratio of the standard deviation to the mean value of the chi-square pdf is equal to m

^{-1/2}, larger values of m will result in less fluctuations. If m equals infinity, the target's RCS is non-fluctuating.**werling Target Models****Swerling target models**are special cases of the Chi-Square target models with specific degrees of freedom. There are five different Swerling models, numbered I through V:**werling I**A model where the RCS varies according to a Chi-square probability density function with two degrees of freedom ($m\; =\; 1$). This applies to a target that is made up of many independent scatterers of roughly equal areas. As little as half a dozen scattering surfaces can produce this distribution. Swerling I describes a target whose radar cross-section is constant throughout a single scan, but varies independently from scan to scan. In this case, the pdf reduces to

:$p(sigma)\; =\; frac\{1\}\{sigma\_\{av\; e^\{-frac\{sigma\}\{sigma\_\{av\}$

Swerling I has been shown to be a good approximation when determining the RCS of objects in aviation.

**werling II**Similar to Swerling I, except the RCS values returned are independent from pulse to pulse, instead of scan to scan.

**werling III**A model where the RCS varies according to a Chi-square probability density function with four degrees of freedom ($m\; =\; 2$). This PDF approximates an object with one large scattering surface with several other small scattering surfaces. The RCS is constant through a single scan just as in Swerling I. The pdf becomes

:$p(sigma)\; =\; frac\{4sigma\}\{sigma\_\{av\}^2\}\; e^\{-frac\{2sigma\}\{sigma\_\{av\}$

**werling IV**Similar to Swerling III, but the RCS varies from pulse to pulse rather than from scan to scan.

**werling V (Also known as Swerling 0)**Constant RCS (m $o$ infinity).

**References*** Skolnik, M. Introduction to Radar Systems: Third Edition. McGraw-Hill, New York, 2001.

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