# Doppler radar electric field expressions

Consider the right handed coordinate system $(x,y,z)$ in which a stationnary radar is positioned at the origin and a metallic plate (initial position $z_0$) that is moving along the $z$ axis with speed $v$ (the plate is perpendicular to the $z$ axis). Consider also the coordinate system $(x,y,z')$ with the $z'$ axis attached to the plate. Let $\vec{E_i}$ and $\vec{E_r}$ be the electric field from the monochromatic EM wave sent and received by the radar. We can write in $(x,y,z')$ :

$\vec{E_i}(z',t) = E_i \cos(\omega_i t - k_i z' + \phi_i) \hat{x} \\\vec{E_r}(z',t) = E_r \cos(\omega_r t + k_r z' + \phi_r) \hat{x}$

My questions :

• In $(x,y,z')$, are $\omega_i$ and $\omega_r$ the same? If so, how can it be proved?
• Same question for $k_i$ and $k_r$? Is the answer the same in $(x,y,z)$?
• In practice, how does the radar decode the reflected wave if there are interferences with the incident one?