Why can Quadrature Demodulation demod a Frequency Modulated Signal?

I use a Quadrature Demodulator in my SDR application, which is defined as:

$\angle (S_n, S_{n+1})=arctan(S_{n+1}*\overline{S_{n}})$

So practically its amplitude is the angle between two Samples $S_n$ and $S_{n+1}$ where $n$ refers to a position in a sequence of complex values of I and Q. $\overline{S}$ be the complex conjugate.

I understand that for Quadrature based modulation the phase-changes contain the information. But why can I demodulate frequency-modulated Signals with a Quadrature Demodulator. This modulator is often referred to as FM Demod. Can somebody explain me why that is?

Best, Marius

The two complex samples are the locations (at two successive sampling instants) of the tip of the rotating phasor that represents the analog signal. The information that you need is the angle between phasor positions at these two successive time instants. If $S_{n+1} = r_{n+1}e^{j\theta_{n+1}}$ and $S_n = r_ne^{j\theta_n}$, then you want the value of $\theta_{n+1}-\theta_n$. Now, $S_{n+1}S_n^* = r_{n+1}r_ne^{j(\theta_{n+1}-\theta_n)}$ and if you express these complex numbers in rectangular coordinates, then you can get the desired angle as $$\theta_{n+1}-\theta_n = \arctan\left(\frac{\text{Im}(S_{n+1}S_n^*)}{\text{Re}(S_{n+1}S_n^*)}\right) = \arctan\left(\frac{\text{Re}(S_{n})\text{Im}(S_{n+1}) - \text{Im}(S_n)\text{Re}(S_{n+1})}{\text{Re}(S_{n})\text{Re}(S_{n+1}) + \text{Im}(S_{n})\text{Im}(S_{n+1})}\right)$$ though usually the four-quadrant arctangent function atan2(y,x) is used instead of atan which returns an answer between $-\pi/2$ and $\pi/2$.
In digital communications applications such as DQPSK demodulation, the real interest is not in the actual value of the angle, but in which of the four intervals $(-\pi/4,\pi/4)$, $(\pi/4, 3\pi/4)$, $(3\pi/4, 5\pi/4)$, and $(5\pi/4, 7\pi/4)$ the angle belongs -- in other words, did the phase not change at all, or change by $\pi/2$, or by $\pi$, or by $3\pi/2$ -- and this can be readily determined by comparing the values of the numerator and denominator in the argument of the arctangent above, thus saving computational effort and speeding up the demodulation process. But the more general formula gives the actual angle, and thus can be used to demodulate a general FM signal as well. The output will be a sequence of samples of the original continuous-time signal that modulated the carrier and thus formed the signal that was transmitted. D/A conversion will give the reconstructed signal.