Assume 
The E-field transmitted can be written as below
The returned E-field at some time later is
Time it took to travel to and from the object(target) is
Substituting
The received frequency can be calculated by taking time derivative of the quantity in red above and dividing by 2pi
Here,
The blue coloured quantity above is nothing but he Doppler shift frequency.
Here the sign essentially indicates the target moving direction
If the doppler shift is negative then
the target is moving away from the radar and vice
versa
And thus the velocity can be written as
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