SAR(Synthetic Aperture Radar) 공식들

1. 해상도

1.1. 거리 해상도

\[{\delta}R_s=\dfrac{c}{2B}\text{,}\quad\quad\quad{\delta}R_g=\dfrac{c}{2B\sin\left({\eta}\right)}\]


1.2. 방위 해상도

\[\delta R_a = \dfrac{2\pi}{|\Omega_n|} = \dfrac{2\pi}{2k\sin\theta_n\left(L/2\right)-2k\sin\theta_n\left(-L/2\right)} \approx \dfrac{x_n \lambda}{2L \cos^2\theta_n\left(0\right)} \approx \dfrac{K_a \lambda_c}{2 \Delta \theta}\] \[\delta R_a = \frac{\text{vel}}{\text{PRF}}\]


2. 이득

2.1. 거리 이득

\[G_r = 10 \text{log}_{10}\left(B \times T_p\right) \text{ [dB]}\]


2.2. 방위 이득

\[G_{az} = \dfrac{\text{prf} \times Rs \times \lambda}{2 \left(\delta {R_a}^2\right) v \cos({\theta}_n)}\]


3. Doppler Equations

  • 방위 해상도 Limitation $\dfrac{2\pi}{4k}=\dfrac{\lambda}{4}$

  • Instantaneous Frequency( $K_{un}\left(u\right)$ )

\[K_{un}\left(u\right)=2k\sin\theta_n\left(u\right)\]
  • Doppler Range
\[\left[2k\sin\theta_n\left(-L/2\right), 2k\sin\theta_n\left(L/2\right)\right]\]
  • Doppler Bandwidth Approximation
\[B_{a} = \dfrac{vel \times \cos\left(\theta _ n\right)}{\delta R_a}\]
  • Doppler Center( ${\Omega}_{nc}$ )
\[{\Omega}_{nc}\approx 2k\sin\theta_n\left(0\right)\]
  • Doppler Bandwidth( $ {\Omega}_{n} $ )
\[|{\Omega}_n|=2k\sin\theta_n\left(L/2\right)-2k\sin\theta_n\left(-L/2\right)\approx \dfrac{2kL}{x_n}\cos^2\theta_n\left(0\right)\approx \dfrac{2{V}_{a}^{2}\cos^2(\theta_n(0))}{R_a \lambda_c}T_a\]
Symbol 단위 설명 Symbol 단위 설명
B hz Bandwidth c m/s Speed of light
$\eta$ rad Incidence Angle u m Platform Azimuth Position
$\theta_n$ rad n번째 Pulse의 Aspect Angle, Squint Angle L m Synthetic Aperture Length
$Y_0$ m Azimuth Swath $T_a$ sec Aperture Time
$T_p$ sec Pulsewidth Rs m Slant Range
$\lambda$ m wavelength $\delta R_a$ m 방위 해상도