RadarLab

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====== 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.

The size and ability of a target to reflect radar energy can be summarized into a single term, σ, known as the radar cross-section, which has units of m². This unit shows, that the radar cross section is an area. If absolutely all of the incident radar energy on the target were reflected equally in all directions, then the radar cross section would be equal to the target's cross-sectional area as seen by the transmitter. In practice, some energy is absorbed and the reflected energy is not distributed equally in all directions. Therefore, the radar cross-section is quite difficult to estimate and is normally determined by measurement.

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.


======Mathematical definition of radar cross section ======

Electromagnetic waves, with any specified polarization, are normally diffracted or scattered in all  directions when incident on a target. These scattered waves are broken down into two parts. The first part is made of waves that have the same polarization as the receiving antenna. The other portion of the scattered waves will have a different polarization to which the receiving antenna does not respond.
The intensity of the backscattered energy that has the same polarization as the radar's receiving antenna is used to define the target RCS. When a target is illuminated by radio frequency energy, in the far field wavefronts are decomposed into a linear combination of plane waves. 
Assume the power density of the wave incident on a target located at range R away from the radar is $P_{Di}$ The amount of reflected power from the target is

\begin{equation}
P_{r} = σP_{Di}
\end{equation}

σ denote the target cross section. Define $P_{Dr}$ as the power density of the scattered waves at the receiving antenna. It follows that 

\begin{equation}
P_{Dr} = P_{r}/(4πR^2)
\end{equation}

Equating Eqs.(1) and (2) yields
\begin{equation}
σ =4πR^2 P_{Dr}/P_{Di}
\end{equation}

and in order to ensure that the radar receiving antenna is in the far field (i.e., scattered waves received by the antenna are planar), Eq.(3) is modified

\begin{equation}
σ =4πR^2 \lim_{R \to \infty} P_{Dr}/P_{Di}
            
\end{equation}

The RCS defined by Eq.(4) is often referred to as either the monostatic RCS,the backscattered RCS, or simply target RCS.



====== RCS Prediction model =======

Before presenting the different RCS calculation methods, it is important to understand the significance of RCS prediction.Most Radar system use RCS as a means of discrimination. Therefore, accurate prediction of target RCS is critical in order to design and develop robust discrimination algorithms. Additionally, measuring and identifying the scattering centers(sources) for a given target aid in developing RCS reduction techniques. Another reason of lesser importance is that RCS calculation require broad and extensive technical knowledge,thus many scientists and scholars find the subject challenging and intellectually motivating. Two categories of RCS prediction methods are available:
exact and approximate.

Exact methods of RCS prediction are very complex even for simple shape objects. This is because they require solving either differential or integral equations that describe the scattered waves from an object under the proper set of boundary conditions. Such boundary conditions are governed by Maxwell’s equations. Even when exact solutions are achievable, they are often difficult to interpret and to program using digital computers.
Due to the difficulties associated with the exact RCS prediction, approximate methods become the viable alternative. The majority of the approximate methods are valid in the optical region, and each has its own strengths and limitations. Most approximate methods can predict RCS within few dBs of the truth. In general, such a variation is quite acceptable by radar engineers and designers. Approximate methods are usually the main source for predicting RCS of complex and extended targets such as aircrafts, ships, and missiles. When experimental results are available, they can be used to validate and verify the approximations.
Some of the most commonly used approximate methods are Geometrical Optics (GO), Physical Optics (PO), Geometrical Theory of Diffraction (GTD), Physical Theory of Diffraction (PTD), and Method of Equivalent Currents
(MEC). Interested readers may consult Knott or Ruck (see bibliography) for more details on these and other approximate methods


====== RCS Dependency on Aspect Angle and Frequency ======

Radar cross section fluctuates as a function of radar aspect angle and frequency.For the purpose of illustration, isotropic point scatterers are considered.
An isotropic scatterer is one that scatters incident waves equally in all directions.Consider the geometry shown in Fig.1. In this case, two unit isotropic scatterers are aligned and placed along the radar line of sight (zero aspect angle) at a far field range . The spacing between the two scatterers is 1 meter. The radar aspect angle is then changed from zero to 180 degrees, and the composite RCS of the two scatterers measured by the radar is computed.
This composite RCS consists of the superposition of the two individual radar cross sections. At zero aspect angle, the composite RCS is $2m^{2}$ . Taking scatterer 1 as a phase reference, when the aspect angle is varied, the composite RCS is modified by the phase that corresponds to the electrical spacing
between the two scatterers. For example, at aspect angle $10^{o}$ , the electrical spacing between the two scatterers is 

<figure >
{{ :media:figure1.jpg?450 |}}
<caption>RCS dependency on aspect angle. (a) Zero aspect angle, zero electrical spacing. (b) aspect angle, electrical spacing. 45° 1.414λ .>[(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>
</figure>


\begin{equation}
elec_{spacing} = 2*(1.0*cos(10))/λ
\end{equation}

λ is the radar operating wavelength
Fig.2 shows the composite RCS corresponding to this experiment.As indicated by Fig.1 RCS is dependent on the radar
aspect angle.Knowledge of this constructive and destructive interference between the individual scatterers can be very critical when a radar tries to extract RCS of complex or maneuvering targets.
This is true because of two reasons. First, the aspect angle may be continuously changing. Second, complex target RCS can be viewed to be made up from contributions of many individual scattering points distributed on the target surface. These scattering points are often called scattering centers. Many approximate RCS prediction methods generate a set of scattering centers that define the back-scattering characteristics of such complex targets.

<figure >
{{:media:figure2.jpg?450 |}}
<caption> llustration of RCS dependency on aspect angle [(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>
</figure>

Next, to demonstrate RCS dependency on frequency, consider the experiment shown in figure.3 In this case, two far field unity isotropic scatterers are aligned with radar line of sight, and the composite RCS is measured by the
radar as the frequency is varied from 8 GHz to 12.5 GHz (X-band). Figure.4 and Figure.5 show the composite RCS versus frequency for scatterer spacing of 0.1 and 0.7 meters

<figure label>
{{:media:figure3.jpg?450 |}}
<caption>Experiment setup which demonstrates RCS dependency on frequency; dist = 0.1, or 0.7 m. [(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>
</figure>

<figure label>
{{ :media:figure4.jpg?450 |}}
<caption>Illustration of RCS dependency on frequency. [(cite: Radar Systems Analysis and Design Using MATLAB)]
 </caption>
</figure>

<figure label>
{{ :media:figure5.jpg?450 |}}
<caption>Illustration of RCS dependency on frequency. [(cite: Radar Systems Analysis and Design Using MATLAB)]
 </caption>
</figure>

====== RCS Dependency on Polarization ======

The material in this section covers two topics. First, a review of polarization
fundamentals is presented. Second, the concept of target scattering matrix is
introduced.
===== Polarization =====
   
The x and y electric field components for a wave traveling along the positive z direction are given by
    
\begin{equation}
  E_{x} = E_{1}sin(ωt-kz) 
\end{equation}

\begin{equation}
E_{y} = E_{2}sin(ωt-kz+δ)
\end{equation}
where k=2π/λ, ω is the wave frequency, the angle δ is the time phase angle which $E_{x}$ and $E_{y}$, and finally, $E_{1}$ and $E_{2}$ are, respectively, the wave amplitudes along the x and y directions. When two or more electromagnetic waves combine, their electric fields are integrated vectorially at each point in space for any specified time. In general, the combined vector traces an ellipse when observed in the x-y plane.This is illustrated in figure.6
<figure label>
{{ :media:figure6.jpg?400 |}}
<caption>Electric field components along the x and y directions.[(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>
</figure>
The ratio of the major to the minor axes of the polarization ellipse is called the Axial Ratio (AR). When AR is unity, the polarization ellipse becomes a circle, and the resultant wave is then called circularly polarized. When $E_1=0$ and AR = ∞ the wave becomes linearly polarized. The equations 6 and 7 can combined to give the instantaneous total electric field,

\begin{equation}
\vec{E}= \overset{∧}{a}_x  E_1 sin(ωt-kz) + \overset{∧}{a}_y E_2 sin(ωt-kz+δ)
\end{equation}

where $\overset{∧}{a}_x$ and  $\overset{∧}{a}_y$ are unit vector along x and y directions, respectively. At z=0, $E_{x} = E_{1} sin(ωt)$ and $E_{y} = E_{1} sin(ωt+δ)$, then by replacing sin(ωt) by the ratio $E_{x}/E_{1}$ and by using trigonometry properties the equations can be rewritten as

\begin{equation}
(E_x^2/E_1^2) - (2E_x E_y cos(δ)/E_1 E_2) + E_y^2/E_2^2 = sin(δ)
\end{equation}

note that equation 9  has no dependency on ωt.
In the most general case, the polarization ellipse may have any orientation, as illustrated in figure.7. The angle ξ is called the tilt angle of the ellipse. In this case, AR is given by

\begin{equation}
\begin{matrix}
AR= OA/OB   &&  1<AR<∞ 
\end{matrix}        
\end{equation}


   

====== RCS of Simple Objects ======
This section presents examples of backscattered radar cross section for a number of simple shape objects. In all cases, except for the perfectly conducting sphere, only optical region approximations are presented. Radar designers and RCS engineers consider the perfectly conducting sphere to be the simplest target to examine. Even in this case, the complexity of the exact solution, when compared to the optical region approximation, is overwhelming. Most formulas presented are Physical Optics (PO) approximation for the backscattered RCS measured by a far field radar in the direction

<figure label>
{{ :media:figure7.jpg?400 |}}
<caption>Direction of antenna receiving backscattered waves. [(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>
</figure>

=====  Sphere =====
Due to symmetry, waves scattered from a perfectly conducting sphere are co-polarized (have the same polarization) with the incident waves. This means that the cross-polarized backscattered waves are practically zero. For example,
if the incident waves were Left Circularly Polarized (LCP), then the backscattered waves will also be LCP. However, because of the opposite direction of propagation of the backscattered waves, they are considered to be Right Circularly Polarized (RCP) by the receiving antenna. Therefore, the Perpenducular Polarization backscattered waves from a sphere are LCP, while the horizontal Polarization backscattered waves are negligible.
The normalized exact backscattered RCS for a perfectly conducting sphere is a Mie series given by

\begin{equation}
\tfrac{σ}{πr^2}=(\tfrac{j}{kr})\sum_{n=1}^∞ (-1)^n (2n+1)[(\qquad \frac{krJ_{n-1}(kr)-nJ_n(kr)}{krH_{n-1}^1 -nH_n^1(kr) }) - (\dfrac{J_n(kr)}{H_n^1 (kr)})]
\end{equation}

where is the radius of the sphere,k=2π/λ, λ is wavelength $J_n$ is the spherical Bessel of the first kind of order n, and $H_n^1$ is the Hankel function of order n, and given by 
\begin{equation}
H_n^1(kr) = J_n(kr)+jY_n(kr)
\end{equation}

$Y_n$ is the spherical Bessel function of the second kind of order n. Plots of the normalized perfectly conducting sphere RCS as a function of its circumference in wavelength units are shown in figures 8.a and 8.b.
In the figures, three regions are identified. First is the optical region(corresponding to the large sphere), in this case ,

\begin{equation}
\begin{array}{lcr}
σ = πr^2 & r»λ
\end{array}
\end{equation}

Second is the Rayleigh region (small sphere). In this case,

\begin{equation}
\begin{array}{lcr}
σ≈9πr^2(kr)^2 && r«λ
\end{array}
\end{equation}
The region between the optical and Rayleigh regions is oscillatory in nature and is called the Mie or resonance region.

<figure label>
{{ :media:figure8.a.jpg?450 |}}
<caption>Normalized backscattered RCS for a perfectly conducting sphere. [(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>
</figure>

<figure label>
{{ :media:figure8.b.jpg?450 |}}
<caption>Normalized backscattered RCS for a perfectly conducting sphere using semi-log scale. [(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>
</figure>

The backscattered RCS for a perfectly conducting sphere is constant in the optical region. For this reason, radar designers typically use spheres of known cross sections to experimentally calibrate radar systems. For this purpose, spheres are flown attached to balloons. In order to obtain Doppler shift, spheres of known RCS are dropped out of an airplane and towed behind the airplane whose velocity is known to the radar

===== Radar Cross section for non-spherical object =====

The back-scattering radar cross-section area (RCS) of a non-spherical object, moving with respect to the radar
should be considered variable over time due to the continuous variations of the target attitude.
The RCS variations can be taken into account by treating the RCS as a stochastic process.
The complete characterization of a stochastic process requires knowledge of the density of joint probabilities of each order.Since we do not have such data, we limit ourselves to consider a stochastic description of the phenomenon based on the moments of the first and second order (averages and correlation functions).

\begin{equation}
σ =\left\vert V^2(2) \right\vert k = \left\vert \textstyle \sum_{k=1}^N α_{i}\exp[j\frac{4π}{λ}δ_{i}] \right\vert^2
\end{equation}

<figure label>
{{media:figure16.jpg?450 |}} 
<caption>Radar cross section for non-spherical object [(cite: Fundamentals of Radiolocation)] </caption>
</figure>


<figure label>
{{ :media:figure23.jpg?450 |}}
<caption> RCS of a 5 scatterers system: Polar diagram [(cite: Fundamentals of Radiolocation)]  </caption>
</figure>


<figure label>
{{ :media:figure24.jpg?450 |}}
<caption> RCS of a 5 scatterers system: Histogram [(cite: Fundamentals of Radiolocation)]  </caption>
</figure>

===== Example =====
Let’s considering a simple case in which there are only two scatterers (N = 2). Let’s assume that the two scattering elements are identical and non-interacting, in the far field

<figure label>
{{ :media:figure25.jpg?200 |}}
</figure>

\begin{equation}
δ = l*sin(θ)
\end{equation}

\begin{equation}
∇φ =\frac{4π}{λ}lsin(θ)
\end{equation}

\begin{equation}
∇φ =φ_{2}-φ_{1} =\frac{4πδ}{λ}
\end{equation}

\begin{equation}
\left ( \frac{A}{2} \right )^2 = a^2cos^2(\frac{∇φ}{2}) =\frac{a^2}{2}(1+cos(∇φ)) =\frac{a^2}{2}[1+cos(4π \frac{l}{λ}sin(θ))]
\end{equation}

\begin{equation}
σ_{tot} = 2σ_{1}[1+ cos(\frac{4πl}{λ}sin(θ))]
\end{equation}

<figure label>
{{ :media:figure26.jpg?200 |}}
</figure>

<figure label>
{{ :media:figure27.jpg?300 |}}
</figure>

<figure label>
{{ :media:figure28.jpg?300 |}}
</figure>
This example has been taken from the lecture note.

===== RCS of Corner Reflector =====

Corner reflectors are used to generate a particularly strong radar echo from objects that would otherwise have only very low effective Radar cross section (RCS). A corner reflector consisting of two or three electrically conductive surfaces which are mounted crosswise (at an angle of exactly 90 degrees). Incoming electromagnetic waves are backscattered by multiple reflection accurately in that direction from which they come. Thus, even small objects with small RCS yield a sufficiently strong echo.

The single areas of the corner reflector should be large compared to the radar wavelength. The larger a corner reflector is, the more energy is reflected. It can also be constructed in resonance with the radar wavelength. This will increase the echo signals again. To reduce losses caused by the earth's curvature the corner-reflectors are mounted as high as possible at the top of a mast at small boats. If too much wind load would disturb, then spherical or cylindrical disguised versions are used.

If should be backscattered in a three-dimensional directions, the corner reflector must be constructed of three reflecting surfaces. If should be backscattered in three-dimensionally distributed directions, then the corner reflector must be made of three reflecting areas. It is then called trihedral corner reflector. It will take place three reflections then before the beam is reflected back in the original direction

<figure label>
{{ :media:figure29.jpg?450 |}}
<caption> Corner-reflector [(cite: Fundamentals of Radiolocation)]  </caption>
</figure>

<figure label>
{{ :media:figure30.jpg?450 |}}
<caption> Image of Corner-reflector [(cite: Fundamentals of Radiolocation)]   </caption>
</figure>





=====  Ellipsoid =====

 An ellipsoid centered at (0,0,0) is shown in fig.10. It is defined by the following equation:

\begin{equation}
\left ( \frac{x}{a} \right )^2 + \left ( \frac{y}{b} \right )^2 + \left ( \frac{z}{c} \right )^2 =1
\end{equation}

One widely accepted approximation for the ellipsoid backscattered RCS is given by

\begin{equation}
σ = \qquad \dfrac{π a^2 b^2 c^2}{(a^2(sinθ)^2 (cosφ)^2 + b^2(sinθ)^2 (sinφ)^2 + c^2(cosθ)^2)^2}
\end{equation}

<figure label>
{{ :media:figure9.jpg?450 |}}
<caption>Ellipsoide [(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>
</figure>

When a=b the ellipsoid becomes roll symmetric. Thus, the RCS is independent of φ, and eq (16) is reduce to 

\begin{equation}
σ =  \qquad \dfrac{π b^4 c^2}{(a^2(sinθ)^2 +c^2(cosθ)^2)}
\end{equation}

and for the case when $a=b=c$,

\begin{equation}
σ=πc^2
\end{equation}

Note that equation 18 defines the backscattered  RCS of the a sphere. This should be expected, since under the condition $a=b=c$ the ellipsoid becomes a sphere.  

===== Circular flat plate =====

Figure.11 shows a circular flat plate of radius , centered at the origin. Due to the circular symmetry, the backscattered RCS of a circular flat plate has no dependency on φ. The RCS is only aspect angle dependent. For normal incidence (i.e., zero aspect angle) the backscattered RCS for a circular flat plate is

\begin{equation}
\begin{array}{lcl}
σ=\frac{4π^3r^4}{λ^2}  &&  θ=0
\end{array}
\end{equation}

<figure label>
{{ :media:figure10.jpg?450 |}}
<caption> Circular flat plate. [(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>
</figure>
For non-normal incidence, two approximations for the circular flat plate backscattered RCS for any linearly polarized incident wave are

\begin{equation}
σ=\frac{λr}{8πsin(θ)(tan(θ))^2}
\end{equation}

\begin{equation}
σ=πk^2r^4[\frac{2J_i(2krsin(θ))}{2krsin(θ)}]^2 (cos(θ))^2
\end{equation}

where $k=2π/λ$, $J_1(β)$ is the first order spherical Bessel function evaluated at . The RCS corresponding to equation 19 through equation 21.it is shown in figure12

<figure label>
{{ :media:figure11.jpg?450 |}}
<caption>Backscattered RCS for a circular flat plate. Solid line
corresponds to Eq. (21). Dashed line corresponds to Eq. (20). [(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>
</figure>

==== Truncated cone ====

Figure 13 and figure 14 show the geometry associated with a frustum. The half cone angle α is given by 

\begin{equation}
tanα=\frac{r_2-r_1}{H}=\frac{r_2}{L}
\end{equation}

Define the aspect angle at normal incidence (broadside) as $θ_n$. Thus, when a frustum is illuminated by a radar located at the same side as the cone’s small end, the angle $θ_n$ is 

\begin{equation}
θ_n =90^o-α
\end{equation}

Alternatively, normal incidence occurs at

\begin{equation}
θ_n =90^o+α

\end{equation}

At normal incidence, one approximation for the backscattered RCS of a truncated cone due to a linearly polarized incident wave is

\begin{equation}
 σ_{θ_n} = \frac{8π {z_2^{3/2}-z_1^{3/2}}^2}{9λsin{θ_n}}{tanα(sinθ_n-cosθ_n tanα)}
\end{equation}

<figure label>
{{ :media:figure12.jpg?450 |}}
<caption>Truncated cone (frustum). [(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>
</figure>

<figure label>
{{ :media:figure13.jpg?450 |}}
<caption>Definition of half cone angle. [(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>
</figure>

Where λ is wavelength, and $Z_1$ $Z_2$ are defined in figure12. Using trigonometric identities, Eq.(25) can be reduced to

\begin{equation}
 σ_{θ_n} = \frac{8π(Z_2^{3/2} -Z_1^{3/2})}{9λ}{\frac{sinα}{cosα^4}}
\end{equation}

For non-normal incidence, the backscattered RCS due to a linearly polarized incident wave is

\begin{equation}
σ = \frac{λZ tanα}{8πsinθ}[\frac{sinθ-cosθ tanα}{sinθ tanα + cosθ}]^2
\end{equation}

where Z can be either $Z_1$ or $Z_2$ depending on whether the RCS contributionis from the small or the large end of the cone. Again, using trigonometric identities Eq.(26) (assuming the radar illuminates the frustum starting from the large end) is reduced to

\begin{equation}
σ =\frac{λZ tanα}{8πsinθ}(tan(θ-α))^2
\end{equation}

When the radar illuminates the frustum starting from the small end (i.e., the radar is in the negative z direction in figure.(13), Eq.(27) should be modified to 

\begin{equation}
σ =\frac{λZ tanα}{8πsinθ}(tan(θ-α))^2
\end{equation}

==== Cylinder ====
   
Figure14 shows the geometry associated with a cylinder. Two cases are presented: first, the general case of an elliptical cylinder; second, the case of a circular cylinder. The normal and non-normal incidence backscattered RCS for an elliptical cylinder due a linearly polarized incident wave are, respectively, given by

\begin{equation}
σ_{θ_n}= \frac{2πH^2 r_2^2 r_1^2}{λ(r_1^2(cosφ)^2 +r_2^2(sinφ))^1.5}
\end{equation}

\begin{equation}
σ= \frac{λ r_2^2 r_1^2 sinθ}{8π(cosθ)^2(r_1^2(cosφ)^2 +r_2^2(sinφ))^1.5}
\end{equation}

For a circular cylinder of radius , then due to roll symmetry, Eqs.() and (), respectively, reduce to 

\begin{equation}
σ_{θ_n}=\frac{2πH^2r}{λ}
\end{equation}

\begin{equation}
σ=\frac{λrsinθ}{8π(cosθ)^2}
\end{equation}

<figure label>
{{ :media:fugure14.jpg?450 |}}
<caption>(a) Elliptical cylinder; (b) circular cylinder. [(cite: Radar Systems Analysis and Design Using MATLAB)]</caption>
</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 ====
 A complex target RCS is normally computed by coherently combining the cross sections of the simple shapes that make that target.In general, a complex target RCS can be modeled as a group of individual scattering centers distributed over the target. The scattering centers can be modeled as isotropic point scatterers (N-point model) or as simple shape scatterers (N-shape model). In any case, knowledge of the scattering centers’ locations and strengths is critical in determining complex target RCS. This is true, because as seen in the previous section, relative spacing and aspect angles of the individual scattering centers drastically influence the overall target RCS. 
Complex targets that can be modeled by many equal scattering centers are often called Swerling 1 or 2 targets. Alternatively, targets that have one dominant scattering center and many other smaller scattering centers are known as Swerling 3 or 4 targets.

==== RCS Fluctuations and Statistical Models ====

In most practical radar systems there is relative motion between the radar and an observed target. Therefore, the RCS measured by the radar fluctuates over a period of time as a function of frequency and the target aspect angle.
This observed RCS is referred to as the radar dynamic cross section. Up to this point, all RCS formulas discussed in this chapter assumed stationary target, where in this case, the backscattered RCS is often called static RCS.
Dynamic RCS may fluctuate in amplitude and/or in phase. Phase fluctuation is called glint, while amplitude fluctuation is called scintillation. Glint causes the far field backscattered wavefronts from a target to be non-planar. For most radar applications, glint introduces linear errors in the radar measurements, and thus it is not of a major concern. However, cases where high precision and accuracy are required, glint can be detrimental. Examples include precision instrumentation tracking radar systems, missile seekers, and automated aircraft
landing systems.
Radar cross-section scintillation can vary slowly or rapidly depending on the target size, shape, dynamics, and its relative motion with respect to the radar. Thus, due to the wide variety of RCS scintillation sources changes in the radar cross section are modeled statistically as random processes. The value of an RCS random process at any given time defines a random variable at that time. Many of the RCS scintillation models were developed and verified by experimental measurements.


 

=== RCS Statistical Models - Scintillation Models ===

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 ===

The Chi-square distribution applies to a wide range of targets; its pdf is given by

\begin{equation}

f(σ)= \frac{m}{Γ(mσ_av)}(\frac{mσ}{σ_av})^{m-1}  \exp ^{-mσ/σ_a}  \qquad σ≥0  

\end{equation}

where Γ(m) is the gamma function with argument m, and $σ_{av}$ is the average value. As the degree gets larger the distribution corresponds to constrained RCS values (narrow range of values). The limit $m \rightarrow ∞$ corresponds to a constant RCS target (steady-target case).

=== 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 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}
f(σ) = \frac{1}{σ_{av}} \exp(-\frac{σ}{σ_{av}})   \qquad σ≥0  
\end{equation}

where σ_{av} denotes the average RCS overall target fluctuation. Swerling II target fluctuation is more rapid than Swerling I, but the measurements are pulse to pulse uncorrelated. This illustrated on fig.15. Swerling II RCS distribution is also defined by equation 34. Swerlings I and II apply to targets consisting of many independent fluctuating point scatterers of approximately equal physical dimensions.

=== Swerling III and IV (Chi-Square of Degree 4) ===

Swerlings III and IV have the same pdf, and it is given by

\begin{equation}
f(σ)= \frac{4σ}{σ_{av^2}} \exp(-\frac{2σ}{σ_av}) \qquad σ≥0
\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.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>
{{ :media:figure15.jpg?450 |}}
<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 ===
With the term “stealth” (being stealthy, clandestine) it is referred the military technology that aims to make an airplane or missile (or nay other object) a "nearly invisible" to enemy radar or any other form of revelation (eg. thermal, ect.).
Stealth technology (or LO for "low observability") is not a single technology. It is a combination of technologies that attempt to greatly reduce the distances at which a person or vehicle can be detected; in particular radar cross section reductions, but also acoustic, thermal, and other aspects.
At the basis of the stealth ability of an aircraft there is the combination of the effects due to
particular materials and an appropriate shape of the object.

<figure label>
{{ :media:figure17.jpg?450 |}}
<caption>Stealth of a aircraft [(cite: Fundamentals of Radiolocation)] </caption>
</figure>

=== Radar cross-section (RCS) reductions ===

Almost since the invention of radar, various methods have been tried to minimize detection. Rapid development of radar during World War II led to equally rapid development of numerous counter radar measures during the period; a notable example of this was the use of chaff. Modern methods include Radar jamming and deception.

The term "stealth" in reference to reduced radar signature aircraft became popular during the late eighties when the Lockheed Martin F-117 stealth fighter became widely known. The first large scale (and public) use of the F-117 was during the Gulf War in 1991. However, F-117A stealth fighters were used for the first time in combat during Operation Just Cause, the United States invasion of Panama in 1989.[22] Increased awareness of stealth vehicles and the technologies behind them is prompting the development of means to detect stealth vehicles, such as passive radar arrays and low-frequency radars. Many countries nevertheless continue to develop low-RCS vehicles because they offer advantages in detection range reduction and amplify the effectiveness of on-board systems against active radar homing threats.
Example of radar cross section reduction:

 - The reflection echo radar by a B-2 bomber Stealth frontally disposed is equal to -40 dBm2 ie 10-4 m2.

<figure label>
{{ :media:figure18.jpg?450 |}}
<caption>Northrop B-2 Spirit [(cite: Fundamentals of Radiolocation)] </caption>
</figure>

- The JSF (Joint Strike Fighter) is seen with a RCS (Radar Cross Section) slightly higher, about -30 dBm2.

<figure label>
{{ :media:figure19.jpg?450 |}}
<caption>Joint strike figther [(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: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>




[(cite: Radar Systems Analysis and Design Using MATLAB)]
title : Radar Systems Analysis and Design Using MATLAB
author :  Bassem R. Mahafza
publisher : by Chapman & Hall/CRC
published : 2000


[(cite: Fundamentals of Radiolocation)]
title : Fundamentals of Radiolocation
author :  Mauro Leonardi
publisher : Lecture note
published : 2018