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Radar Equation

The power P_r returning to the receiving antenna is given by the radar equation:

$P_r = {{P_t G_t A_r \sigma F^4}\over{{(4\pi)}^2 R_t^2R_r^2}}$

where

P_t = transmitter power
G_t = gain of the transmitting antenna
A_r = effective aperture (area) of the receiving antenna
σ = radar cross section, or scattering coefficient, of the target
F = pattern propagation factor
R_t = distance from the transmitter to the target
R_r = distance from the target to the receiver.

In the common case where the transmitter and the receiver are at the same location, R_t = R_r and the term R_t² R_r² can be replaced by R⁴, where R is the range. This yields:

$P_r = {{P_t G_t A_r \sigma F^4}\over{{(4\pi)}^2 R^4}}.$

This shows that the received power declines as the fourth power of the range, which means that the reflected power from distant targets is very, very small.

The equation above with F = 1 is a simplification for vacuum without interference. The propagation factor accounts for the effects of multipath and shadowing and depends on the details of the environment. In a real-world situation, pathloss effects should also be considered.