RadarLab

User Tools

Site Tools

Trace: • radardetection • radar • thanks • exercises • rcs • introduction
radar:rcs

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revisionPrevious revision
Next revision		Previous revision
radar:rcs [2018/06/08 23:17] – diankaradar: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 ====== ====== 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 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. 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 7: Line 8:
  
 The target radar cross sectional area depends of: The target radar cross sectional area depends of:
 +
 • the airplane’s physical geometry and exterior features, • the airplane’s physical geometry and exterior features,
 +
 • the direction of the illuminating radar, • the direction of the illuminating radar,
 +
 • the radar transmitters frequency, • the radar transmitters frequency,
 +
 • the used material types. • the used material types.
  
Line 213: Line 218:
 <figure label> <figure label>
 {{media:figure16.jpg?450 |}}  {{media:figure16.jpg?450 |}} 
-<caption>Radar cross section for non-spherical object [(cite: )] </caption>+<caption>Radar cross section for non-spherical object [(cite: Fundamentals of Radiolocation)] </caption>
 </figure> </figure>
  
Line 219: Line 224:
 <figure label> <figure label>
 {{ :media:figure23.jpg?450 |}} {{ :media:figure23.jpg?450 |}}
-<caption> RCS of a 5 scatterers system: Polar diagram   </caption>+<caption> RCS of a 5 scatterers system: Polar diagram [(cite: Fundamentals of Radiolocation)]  </caption>
 </figure> </figure>
  
Line 225: Line 230:
 <figure label> <figure label>
 {{ :media:figure24.jpg?450 |}} {{ :media:figure24.jpg?450 |}}
-<caption> RCS of a 5 scatterers system: Histogram   </caption>+<caption> RCS of a 5 scatterers system: Histogram [(cite: Fundamentals of Radiolocation)]  </caption>
 </figure> </figure>
  
Line 266: Line 271:
 {{ :media:figure28.jpg?300 |}} {{ :media:figure28.jpg?300 |}}
 </figure> </figure>
 +This example has been taken from the lecture note.
  
 ===== RCS of Corner Reflector ===== ===== RCS of Corner Reflector =====
Line 277: Line 283:
 <figure label> <figure label>
 {{ :media:figure29.jpg?450 |}} {{ :media:figure29.jpg?450 |}}
-<caption> Corner-reflector   </caption>+<caption> Corner-reflector [(cite: Fundamentals of Radiolocation)]  </caption>
 </figure> </figure>
  
 <figure label> <figure label>
 {{ :media:figure30.jpg?450 |}} {{ :media:figure30.jpg?450 |}}
-<caption> Image of Corner-reflector   </caption>+<caption> Image of Corner-reflector [(cite: Fundamentals of Radiolocation)]   </caption>
 </figure> </figure>
  
-===== Calculation of corner reflectors ===== 
-The specification of an area based on a comparison with an isotropic radiator. Here an ideal spherical conductor is assumed as the reference reflector whose graphical projection onto a plane perpendicular to the direction of the incoming rays (read: its shadow on this plane) has an area of one square meter. These circular area has a diameter of about 1.33 m. From this reference reflector, only a very small area is effective. As a reflector acts a small area (few centimeters only) in the middle of the sphere, which reflect the incoming energy in the direction of the radar exactly. All other subareas distribute the incoming energy evenly throughout the room. It follows that a small handy corner reflector with only a few centimeters of geometric spread can have an effective radar cross section of several square meters. Here one must know that this number of example given 12 m2 actually only says that this little corner reflector reflects the same energy, as 12 spheres per one square meter! 
  
-Such a sphere will, as it is independent of the direction, generating a return signal even at a bistatic radar. A corner reflector can not do that. The corner reflector concentrates the energy that is scattered at a sphere in almost all directions, only in exactly one direction: back to the radar. Therefore, a corner reflector has got an RCS, which is very much larger than its geometrical dimensions. 
- 
-In the calculation of corner reflectors three aspects must be considered. The radar cross section is dependent on: 
- 
- ✔ the frequency-independent calculated viewable area, 
- ✔ the angle of incidence of the radar, and 
- ✔ possible resonances at certain frequencies. 
  
  
Line 451: Line 448:
 </figure> </figure>
  
 +Fig. 4 describes the difference in the scattering of the radar signal in some types of shapes.
 +
 +<figure label>
 +{{ :media:figure22.jpg?450 |}}
 +<caption> Signal dispersion regarding target shape.</caption>
 +</figure>
        
 ==== RCS of complex objects ==== ==== RCS of complex objects ====
Line 470: Line 473:
  
 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. 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 === === Chi-Square of Degree 2m ===
Line 485: Line 493:
 === Swerling I and II (Chi-Square of Degree 2) === === 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} \begin{equation}
Line 501: Line 509:
 \end{equation} \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> <figure label>
Line 507: 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> <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>
 +
 +<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 === === Stealth technology ===
Line 516: Line 678:
 <figure label> <figure label>
 {{ :media:figure17.jpg?450 |}} {{ :media:figure17.jpg?450 |}}
-<caption>Stealth of a aircraft [(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>+<caption>Stealth of a aircraft [(cite: Fundamentals of Radiolocation)] </caption>
 </figure> </figure>
  
Line 530: Line 692:
 <figure label> <figure label>
 {{ :media:figure18.jpg?450 |}} {{ :media:figure18.jpg?450 |}}
-<caption>Northrop B-2 Spirit [(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>+<caption>Northrop B-2 Spirit [(cite: Fundamentals of Radiolocation)] </caption>
 </figure> </figure>
  
Line 537: Line 699:
 <figure label> <figure label>
 {{ :media:figure19.jpg?450 |}} {{ :media:figure19.jpg?450 |}}
-<caption>Joint strike figther[(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>+<caption>Joint strike figther [(cite: Fundamentals of Radiolocation)] </caption>
 </figure> </figure>
  
Line 543: Line 705:
 <figure label> <figure label>
 {{ :media:figure20.jpg?450 |}} {{ :media:figure20.jpg?450 |}}
-<caption>Shaping to reduce the radar cross section </caption>+<caption>Shaping to reduce the radar cross section [(cite: Fundamentals of Radiolocation)]</caption> 
 +</figure> 
 + 
 +<figure label> 
 +{{ :media:figure20.jpg?450 |}} 
 +<caption>Shaping to reduce the radar cross section [(cite: Fundamentals of Radiolocation)]</caption> 
 +</figure> 
 + 
 +<figure label> 
 +{{ :media:figure31.jpg?450 |}} 
 +<caption> Typical value of RCS [(cite: Fundamentals of Radiolocation)]</caption>
 </figure> </figure>
  
Line 556: Line 728:
  
  
 +[(cite: Fundamentals of Radiolocation)]
 +title : Fundamentals of Radiolocation
 +author :  Mauro Leonardi
 +publisher : Lecture note
 +published : 2018
  
- <div class="bar" id="bar_top"> + 
-      <div class="bar-left" id="bar_topleft"> + 
-        <?php tpl_button('edit')?> + 
-        <?php tpl_button('history')?> +
-        <?php print html_btn('exportpdf',$ID,'',array('do' => 'export_pdf')) ?>  <!-- inserted line --> +
-      </div>+
  
  
radar/rcs.1528499852.txt.gz · Last modified: (external edit)