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