I am struggling to find or derive the relationship between Doppler frequency and the associated phase shift. Let's say I have a signal transmitted at a given frequency $f_c$. I then down convert from $f_c$ to baseband at the receiver, but due to speed between the transmitter and receiver, I receive a frequency shifted by a Doppler frequency $\Delta f$.
If I wish to generate a matching signal to that being received, I can use
$$ \cos(2 \pi \Delta f t + \phi) + i \sin(2 \pi \Delta f t + \phi) $$
The part I'm struggling to get to grips with is how does $\phi$ relate to $\Delta f$? Is it simply the integral of $\Delta f$ with respect to time?
$$ \phi = \int_0^t \Delta f dt $$