+ * This can be used to demod FM, FSK, GMSK, etc. The input is complex

+ * baseband, output is the signal frequency in relation to the sample

+ * rated, multiplied with the gain.

+ *

+ * Mathematically, this block calculates the product of the one-sample

+ * delayed input and the conjugate undelayed signal, and then calculates

+ * the argument of the resulting complex number:

+ *

+ * \f$y[n] = \mathrm{arg}\left(x[n] \, \bar x [n-1]\right)\f$.

+ *

+ * Let \f$x\f$ be a complex sinusoid with amplitude \f$A>0\f$, (absolute)

+ * frequency \f$f\in\mathbb R\f$ and phase \f$\phi_0\in[0;2\pi]\f$ sampled at

+ * \f$f_s>0\f$ so, without loss of generality,

+ *

+ * \f$x[n]= A e^{j2\pi( \frac f{f_s} n + \phi_0)}\f$

+ *

+ * then

+ *

+ * \f{align*}{ y[n] &= \mathrm{arg}\left(A e^{j2\pi\left( \frac f{f_s} n + \phi_0\right)} \overline{A e^{j2\pi( \frac f{f_s} (n-1) + \phi_0)}}\right)\\

+ * & = \mathrm{arg}\left(A^2 e^{j2\pi\left( \frac f{f_s} n + \phi_0\right)} e^{-j2\pi( \frac f{f_s} (n-1) + \phi_0)}\right)\\

+ * & = \mathrm{arg}\left( A^2 e^{j2\pi\left( \frac f{f_s} n + \phi_0 - \frac f{f_s} (n-1) - \phi_0\right)}\right)\\

+ * & = \mathrm{arg}\left( A^2 e^{j2\pi\left( \frac f{f_s} n - \frac f{f_s} (n-1)\right)}\right)\\

+ * & = \mathrm{arg}\left( A^2 e^{j2\pi\left( \frac f{f_s} \left(n-(n-1)\right)\right)}\right)\\

+ * & = \mathrm{arg}\left( A^2 e^{j2\pi \frac f{f_s}}\right) \intertext{$A$ is real, so is $A^2$ and hence only \textit{scales}, therefore $\mathrm{arg}(\cdot)$ is invariant:} &= \mathrm{arg}\left(e^{j2\pi \frac f{f_s}}\right)\\

+ * &= \frac f{f_s}\\

+ * &&\blacksquare

+ * \f}