# Introduction to Radar Scattering Application in Remote Sensing and Diagnostics: Review
## 3 Radar Fundamental Equations and Parameters  
Radar range equation is helpful to investigate the radar characteristics and to determine the range resolution. It is also useful for designing and manufacturing of radar systems; the radar-range equation may be expressed as signal-to-noise ratio (SNR), and defined as follows [28]:  
\[SNR=\frac{P_{t}G^{2}\lambda^{2}\sigma}{(4\pi)^{3}kT_{0}BFLR^{4}} \tag{1}\]  
The parameters introduced in the equation above are described in Table 3.  
In Figure 5, some Matlab simulations are shown where symbols in Table 3 (e.g., frequency, peak power, bandwidth, and target cross section) are modified to provide the reader with an intuition on the effects of the different parameters on the radar range equation, here expressed in terms of SNR. The range resolution, mentioned above in this paragraph, is the radar system ability to differentiate between two or more targets on the same bearer but at different ranges. The range resolution depends on the width of the transmitted signal, the target types and sizes, and the efficiency of the receiver and indicator elements. Pulse width is the primary resolution factor of a pulse radar. A well-designed radar  
Figure 4: Radiation pattern for the linear array in Table 2.  
system should be able to distinguish targets separated by half the pulse width-time \(\tau\), when all other factors are at their maximum efficiency. The range (R) equation for a pulse radar may be written as:  
\[R=\frac{c_{0}\tau}{2},\]  
where \(c_{0}\) is the speed of light in a vacuum [29; 30; 31]. Another fundamental parameter is the radar operating frequency: as seen, radars radiate electromagnetic energy which, when reflected, allows us to detect and locate targets. Radars operating at different wavelengths (or energy bands, see Table 4, where the association between frequency ranges and energy bands is reported [2]) have different purposes: for instance, a C-band radar can penetrate through clouds and dust particles on the surface of the Earth, while an L-band radar can measure the GPS and soil moisture. X and Ku bands are commonly used for satellite communications. For certain applications, multiple energy bands have been explored: [PERSON] et al. analyzed radar backscatter intensity from forest and they found the greatest intensity at P-band; the intensity would then decrease with the increase of frequency [32]. Different tree elements (leaves, branches, stems, etc.) using different frequency bands were studied by [PERSON] [33] (see Table 5). A plant crowded with a large number of leaves and twigs can be investigated with higher frequency than what can happen for wilted plants. Figure 5: Matlab numerical simulation of the radar equation for different parametric values in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{**Parametric Values of the Radar Equation**} \\ \hline \# & **Symbol** & **Description** & **Numerical Values for Input** & **Unit** \\ \hline
1 & \(P_{l}\) & Peak power & \(2.16\times 10^{6}\) & W \\
2 & \(f\) & radar center frequency & \(5.6\times 10^{9}\) & Hz \\
3 & \(c\) & Speed of light & \(3.0\times 10^{8}\) & ms\({}^{-1}\) \\
4 & \(\lambda\) & Wavelength = \(\frac{c}{f}\) & 0.0600 & m \\
5 & \(k\) & Boltzmann’s constant & \(1.38\times 10^{-23}\) & JK\({}^{-1}\) \\
6 & \(G\) & Antenna gain & 90.0 & dB \\
7 & \(\sigma\) & Target cross section & 0.1 & m\({}^{2}\) \\
5 & \(B\) & Bandwidth & \(8.0\times 10^{6}\) & Hz \\
6 & \(F\) & noise figure & 3.0 & dB \\
9 & \(L\) & radar losses & 6.0 & dB \\
10 & \(T_{0}\) & Antenna temperature & 290.0 & K \\
11 & \(R\) & Target range & \(R_{min}=25.0\times 10^{3}\), \(R_{max}=165.0\times 10^{3}\) & km \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the radar equation. Radar incident angle is also a parameter which affects remote-sensing applications: it is defined as the angle between the direction of the incident radiation and the perpendicular to the target surface. [PERSON] et al.