Radar Detection of Fluctuating and Nonfluctuating Target Cross Sections

A radar system detects targets by transmitting electromagnetic energy into space. Some of this energy is reflected off of targets (scatterers) within the search volume and is referred to as a radar return. An antenna receives the radar return plus noise. This signal is then processed to determine target characteristics such as range and velocity relative to the radar antenna.

Probability of Detection, Pd is the probability that a sample amplitude of the radar return exceeds the threshold voltage. A scatterer may have variations within the radar cross section (RCS) which will lower the probability of detection. This fluctuation of a scatterer is categorized according to five specific cases known as the Swerling Fluctuating Target Models

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Swerling 0 (or Swerling V)
The Swerling 0 type target has a constant RCS with no fluctuation. The probability of detection is found from the Gram-Charlier series. When &lt;math&gt;n_p&gt; \1:&lt;/math&gt; The Gram-Charleir coefficients for the case of a nonfluctuating target are &lt;math&gt;C_3=\frac{SNR+\frac{1}{3}}{\sqrt{n_p}(2SNR+1)^1.5}&lt;/math&gt; &lt;math&gt;C_4=\frac{SNR+\frac{1}{4}}{n_p(2SNR+1)^2}&lt;/math&gt; &lt;math&gt;C_6=\frac{C_3^2}{2}&lt;/math&gt; &lt;math&gt;\omega=sqrt{n_p(2SNR+1)}&lt;/math&gt;

Swerling I
A Swerling I type target has an RCS with constant amplitude over one antenna scan.

The fluctuation occurs from one group of pulses to the next group. The probability of detection is based upon the Chi-Square model with 2 degrees of freedom.

&lt;math&gt;P_d=e^{\frac{-V_t}{1+SNR}}&lt;/math&gt;

Swerling II
A Swerling II type target has a fluctuation which occurs from pulse to pulse. The probability of detection is based upon the Chi-Square model with 2 degrees of freedom.

&lt;math&gt;P_d=1-\Gamma_1(\frac{V_t}{(1+SNR)},n_p)&lt;/math&gt;

This holds for &lt;math&gt;n_p\le \50&lt;/math&gt;

Swerling III
A Swerling III type target has a constant RCS over a single scan but the radar return is created by one large scatterer with a collection of smaller scatterers. The probability of detection is based on the Chi-Square model with 4 degrees of freedom.

When &lt;math&gt;n_p=1,2&lt;/math&gt;

&lt;math&gt;P_d=K_0e^{(\frac{-V_t}{(1+\frac{n_pSNR}{2}})}(1+\frac{2}{n_pSNR})^{n_p-2}&lt;/math&gt;

Swerling IV
Swerling IV type targets are the same as Swerling III but the fluctuation is from pulse to pulse.

When &lt;math&gt;n_p\le \50&lt;/math&gt; The probability of detection is

&lt;math&gt;P_d=1-[\gamma_0+(\frac{SNR}{2})n_p\gamma_1+(\frac{SNR}{2})^2\frac{n_p(n_p-1)}{2!}\gamma_2+...+(\frac{SNR}{2})^n_p\gamma_n_p](1+(\frac{SNR}{2})^-n_p&lt;/math&gt;

Where

&lt;math&gt;\gamma_i = \Gamma_1(\frac{V_t}{1+\frac{SNR}{2}},n_p+i)&lt;/math&gt;