Schaefer's dichotomy theorem: Difference between revisions

Revision as of 18:36, 2 October 2013

Swerling models were introduced by Peter Swerling and are used to describe the statistical properties of the radar cross-section of complex objects.

General Target Model

Swerling target models give the radar cross-section (RCS) of a given object using a distribution in the location-scale family of the chi-squared distribution.

p(\sigma )={\frac {m}{\Gamma (m)\sigma _{av}}}\left({\frac {m\sigma }{\sigma _{av}}}\right)^{m-1}e^{-{\frac {m\sigma }{\sigma _{av}}}}I_{[0,\infty )}(\sigma )

where $\sigma _{av}$ 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 degree of freedom divided by 2. The degree of freedom used in the chi-squared probability density function is a positive number related to the target model. Values of $m$ between 0.3 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-squared distribution is equal to $m$ ^-1/2, larger values of $m$ will result in smaller fluctuations. If $m$ equals infinity, the target's RCS is non-fluctuating.

Swerling Target Models

Swerling target models are special cases of the Chi-Squared target models with specific degrees of freedom. There are five different Swerling models, numbered I through V:

Swerling I

A model where the RCS varies according to a Chi-squared 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.

Swerling II

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

Swerling III

A model where the RCS varies according to a Chi-squared 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 {4\sigma }{\sigma _{av}^{2}}}e^{-{\frac {2\sigma }{\sigma _{av}}}}

Swerling IV

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

Swerling V (Also known as Swerling 0)

Constant RCS ( $m\to \infty$ ). also known as infinite degree of freedom

References

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

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+{{one source|date=January 2014}}
+'''Swerling models''' were introduced by [[Peter Swerling]] and are used to describe the statistical properties of the [[radar cross-section]] of complex objects.
+==General Target Model==
+Swerling target models give the radar cross-section (RCS) of a given object using a distribution in the location-scale family of the [[chi-squared distribution]].
+:<math>p(\sigma) = \frac{m}{\Gamma(m) \sigma_{av}} \left ( \frac{m\sigma}{\sigma_{av}} \right )^{m - 1} e^{-\frac{m\sigma}{\sigma_{av}}}
+I_{[0,\infty)}(\sigma)</math>
+where <math>\sigma_{av}</math> refers to the [[mean]] value of <math>\sigma</math>. 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. <math>m</math> is the [[Degrees of freedom (statistics)|degree of freedom]] divided by 2. The degree of freedom used in the chi-squared probability density function is a positive number related to the target model. Values of <math>m</math> between 0.3 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-squared distribution is equal to <math>m</math><sup>-1/2</sup>, larger values of <math>m</math> will result in smaller fluctuations. If <math>m</math> equals infinity, the target's RCS is non-fluctuating.
+==Swerling Target Models==
+'''Swerling target models''' are special cases of the Chi-Squared target models with specific degrees of freedom. There are five different Swerling models, numbered I through V:
+===Swerling I===
+A model where the RCS varies according to a Chi-squared probability density function with two degrees of freedom (<math>m = 1</math>). 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
+:<math>p(\sigma) = \frac{1}{\sigma_{av}} e^{-\frac{\sigma}{\sigma_{av}}}</math>
+Swerling I has been shown to be a good approximation when determining the RCS of objects in aviation.
+===Swerling II===
+Similar to Swerling I, except the RCS values returned are independent from pulse to pulse, instead of scan to scan.
+===Swerling III===
+A model where the RCS varies according to a Chi-squared probability density function with four degrees of freedom (<math>m = 2</math>). 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
+:<math>p(\sigma) = \frac{4\sigma}{\sigma_{av}^2} e^{-\frac{2\sigma}{\sigma_{av}}}</math>
+===Swerling IV===
+Similar to Swerling III, but the RCS varies from pulse to pulse rather than from scan to scan.
+===Swerling V (Also known as Swerling 0)===
+Constant RCS (<math>m\to\infty</math>).
+also known as infinite degree of freedom
+==References==
+* Skolnik, M. Introduction to Radar Systems: Third Edition. McGraw-Hill, New York, 2001.
+[[Category:Signal processing]]
+[[Category:Radar]]

Schaefer's dichotomy theorem: Difference between revisions

Revision as of 18:36, 2 October 2013

Contents

General Target Model