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--- radar:rcs [2018/06/09 15:09] – dianka
+++ radar:rcs [2026/04/28 18:23] (current) – mauro
@@ Line 1: / Line 1: @@
-TOC is ok for now!   --- //[[webmaster@localhost|DokuWiki Administrator]] 2018/04/24 16:17//
 ====== General definition ======
-Radar cross section is the measure of a target's ability to reflect radar signals in the direction of the radar receiver, i.e. it is a measure of the ratio of backscatter power per steradian (unit solid angle) in the direction of the radar (from the target) to the power density that is intercepted by the target
+Radar cross section is the measure of a target's ability to reflect radar signals in the direction of the radar receiver, i.e. it is a measure of the ratio of backscatter power per steradian (unit solid angle) in the direction of the radar (from the target) to the power density that is intercepted by the target.
 Radar cross section is a measure of power scattered in a given direction when a target is illuminated by an incident wave. RCS is normalized to the power density of the incident wave at the target so that it does not depend on the distance of the target from the illumination source. This removes the effect of the transmitter power level and distance to target when the illuminating wave decreases in intensity due to the inverse square spherical spreading. RCS is also normalized so that inverse square fall-off of scattered intensity due to the spherical spreading is not a factor so that we do not need to know the position of the receiver. RCS has been defined to characterize the target characteristics and not the effects of transmitter power, receiver sensivity, and the position of the transmitter and receiver distance. An other term for RCS is an echo area.
@@ Line 8: / Line 8: @@
 The target radar cross sectional area depends of:
 • the airplane’s physical geometry and exterior features,
 • the direction of the illuminating radar,
 • the radar transmitters frequency,
 • the used material types.
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 This section presents the most commonly used RCS statistical models. Statistical models that apply to sea, land, and volume clutter, such as the Weibull and Log-normal distributions, will be discussed in a later chapter. The choice of a particular model depends heavily on the nature of the target under examination.
+Thus a convenient way to characterize the RCS of a generic target, is to consider it as a stochastic process, sometimes this can lead to even rough approximations. The most commonly used stochastic models are the
+four Swerling models.
+They qualify the variation over time of the RCS assigning to this a function of probability to the firstorder
+density and a trend of the correlation function which decreases rapidly (or slowly) with respect to the time constants (such as the dwell time td and the time Ts) scan.
 === Chi-Square of Degree 2m ===
@@ Line 484: / Line 493: @@
 === Swerling I and II (Chi-Square of Degree 2) ===
-In Swerling I, the RCS samples measured by the radar are correlated throughout an entire scan, but are uncorrelated from scan to scan (slow fluctuation). In this case, the pdf is
+In Swerling I, the RCS samples measured by the radar are correlated throughout an entire scan, but are uncorrelated from scan to scan (slow fluctuation), in other word SwerlingI model assumes that the behavior of the RCS within the dwell time is strongly correlated In this case, the pdf is
 \begin{equation}
@@ Line 500: / Line 509: @@
 \end{equation}
-The fluctuations in Swerling III are similar to Swerling I; while in Swerling IV they are similar to Swerling II fluctuations (see fig.15). Swerlings III and IV are more applicable to targets that can be represented by one dominant scatterer and many other small reflectors.
+The fluctuations in Swerling III are similar to Swerling I; while in Swerling IV they are similar to Swerling II fluctuations (see fig.26). Swerlings III and IV are more applicable to targets that can be represented by one dominant scatterer and many other small reflectors.
 <figure label>
@@ Line 506: / Line 515: @@
 <caption>Radar returns from targets with different Swerling fluctuations. Swerling V corresponds to a steady RCS target case. [(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>
 </figure>
+<figure label>
+{{ :media:figure32.jpg?450 |}}
+<caption>Radar returns from targets with different Swerling fluctuations. Swerling V corresponds to a steady RCS target case. [(cite: Fundamentals of Radiolocation)] </caption>
+</figure>
+=== Application of Swerling models ===
+Considering that the target consists of several independent scatterers, using the central limit theorem, its complex echo has in phase and quadrature (I and Q) components Gaussian distributed with zero mean and variance equal and independent. Therefore, the amplitude
+\begin{equation}
+A = \sqrt{I^2-Q^2}
+\end{equation}
+It is distributed according to Rayleigh and its square that is proportional to the RCS is distributed exponentially. considering :
+\begin{equation}
+S = SNR =\frac{A^2}{2σ_{n}^2}
+\end{equation}
+In the fixed target hypothesis the detection probability was:
+\begin{equation}
+P_{D} =P_{D}(s) = 2e^{-s} \textstyle \int\limits_{\sqrt{-ln(P_{fa})}}^{∞} x.e^{-x^2}.I_{0}(2\sqrt{s}x) dx
+\end{equation}
+\begin{equation}
+P_{D} = \textstyle \int\limits_{0}^{∞} P_D(s)P(s) dx
+\end{equation}
+If we consider a SW1 or SW2 model:
+\begin{equation}
+P(s) = \frac{1}{S_{0}}e^{\frac{-s}{s_{0}}}U(s)
+\end{equation}
+ In the case of targets SW1 and SW2 there is an easier procedure the signals present on the components in phase and quadrature are given by the sum of the $v_{I}$ voltages (t) and $v_{Q} (t)$ associated with the echo produced by the target signal (assumed eg. SW2 type) with two $n_{I}$ signals (t) and $n_{Q}$ (t), representing an additive noise process assumed Gaussian
+\begin{equation}
+I(t) = v_{I}(t) + n_{I}(t)
+\end{equation}
+\begin{equation}
+Q(t) = v_{Q}(t) + n_{Q}(t)
+\end{equation}
+If the target follows a SW2 type model the statistics ofsignals $v_{I}$ and $V_{Q}$ is  also Gaussian and the envelope of the signals is Rayleigh
+\begin{equation}
+σ_{I}^2 = σ_{Q}^2 =σ_{s}^2 + σ_{n}^2 =σ_{n}^2(1+SNR)
+\end{equation}
+\begin{equation}
+SNR =\frac{σ_{s}^2}{ σ_{n}^2}
+\end{equation}
+\begin{equation}
+v(t) = \sqrt{I^2(t) + Q^2(t)}
+\end{equation}
+\begin{equation}
+P(v) = \frac{v}{σ^2}\exp(\frac{-v^2}{2σ^2})U(v)
+\end{equation}
+\begin{equation}
+σ^2 = σ_{s}^2 + σ_{n}^2
+\end{equation}
+\begin{equation}
+P_D =\exp(\frac{-V_{T}^2}{2σ_{n}^2(1+SNR)})
+\end{equation}
+\begin{equation}
+P_D =\exp[\frac{ln(P_fa)}{1+SNR}]
+\end{equation}
+\begin{equation}
+\frac{ln(P_{fa})}{ln(P_{D})} = 1+SNR
+\end{equation}
+with
+\begin{equation}
+P_fa=exp(\frac{-V_{T}^2}{2σ_{n}^2})
+\end{equation}
+<figure label>
+{{ :media:figure33.jpg?450 |}}
+<caption>variation of the probability of detection versus the signal to noise ratio [(cite: Fundamentals of Radiolocation)]</caption>
+</figure>
+<figure label>
+{{ :media:figure36.png?450 |}}
+<caption>Improvement of the SNR versus probabilty of detection [(cite: Fundamentals of Radiolocation)]</caption>
+</figure>
+The received signal, less than a constant factor coming from the radar equation, is equal to
+\begin{equation}
+P(t) = I^2(t) + Q^2(t)
+\end{equation}
+By definition, the voltage of the envelope signal v (t) relative to the echo received is equal to
+\begin{equation}
+v(t) = \sqrt{P(t)} = sqrt{σ}
+\end{equation}
+unless of a multiplicative constant.
+\begin{equation}
+P_{v}(v) = P_{∑}(σ=v^2)\frac{1}{|dv/dσ} = P_{∑}(σ=v^2).2.v
+\end{equation}
+\begin{equation}
+ P_{∑}(σ) = P_{v}(v=\sqrt{σ})\frac{dv(σ)}{σ} = P_{v}\sqrt{σ}\frac{1}{2sqrt{σ}}
+\end{equation}
+For example the power density function of RCS and the one of voltage are given by the following equations :
+\begin{equation}
+P_{∑}(σ) = \frac{1}{σ_{0}}\exp(\frac{-σ}{σ_{0}})U(σ)
+\end{equation}
+\begin{equation}
+P_{v}(v) = \frac{2v}{σ_{0}}\exp(\frac{-v^2}{σ_{0}})U(v)
+\end{equation}
+Some RCS models are derived by approximating a generic target with a set of N scattering elements.The power of the echo signal, and thus the RCS is equal to:
+\begin{equation}
+y = x_{1}^2 + x_{1}^2 + ... + x_{n}^2
+\end{equation}
+where $x_{i}$ are Gaussian variables. If they have zero mean and the same variance, the probability density function of the RCS y has the expression
+\begin{equation}
+f_{y}(y) = \frac{y^{N/2-1}}{(σ_{n}\sqrt{2})^N Γ(N/2)}\exp(\frac{-y}{2σ_{n}^2})U(y)
+\end{equation}
+It is noted that if N is an even, N = 2m number:
+- you get an exponential density function if m = 1. (SW1 AND SW2)
+– If m = 2 the density function of SW3 and SW4 models is obtained
+The previous function can be extended to a parameter m any (non-integer)
+\begin{equation}
+f_{y}(y) = \frac{y^{m-1}}{(σ_{n}\sqrt{2})^2m Γ(m)}\exp{(\frac{-y}{2σ_{}^2})}U(y)
+\end{equation}
+For aircraft targets the typical values are 0.9 m<m<2, while for satellites or cylinders are of the order of 0.3 <m <2.
+There are also other models such as:
+– the one proposed by RICE where it is assumed to have an object in which is identifiable a main scatter surrounded by many small random scatterers.
+– the Log-Normal model, obtained by calculating the exponential of a Gaussian variable having a positive average value.
 === Stealth technology ===