Doppler Effect:: When the target is moving relative to radar it will result in, an apparent shift in the carrier frequency of the received signal. This effect is called the doppler effect and it is the basic of Continuous Wave (CW) radar.

If R is the distance of the target from radar station, the total number of wavelengths contained in the two-way path $ = \frac{2R}{\lambda} where \ \lambda = $ wavelength of the transmitted wave. Since one wavelength corresponds to a phase shift of $ 2\pi$ rads, the total phase shift

$ \phi = 2 \pi \times 2 \frac{R}{\lambda} \ \ \ \ i.e. \ \ \ \phi = \frac{4 \pi R}{\lambda}rad$

If the target is in motion, R (the range) and $\phi$ (phase) are continuously changing. a change in $\phi$ with respect to time is equal to frequency. The Doppler angular frequency $(\omega _d)$ is given by.

$(\omega _d) = 2\pi f_d = \frac{d\phi}{dt} = \frac{d}{dt}(\frac{4 \pi R}{\lambda}) = \frac{4 \pi}{\lambda} . \frac{dR}{dt} = \frac {4 \pi}{\lambda}.v_r$

Whre, $f_d = $ doppler frequency shift

$v_r$ = relative velocity of target with respect to radar

$ \therefore \ \ \ \ \ \ \ \ \ \ \ f_d = \frac{2v_r}{\lambda}Hz$

The relative velocity $(v_r) = v \ cos \theta $