Quadrature Demod

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 rate, multiplied with the gain.

Mathematically, this block calculates the product of the one-sample delayed-&-conjugated input and the undelayed signal, and then calculates the argument (a.k.a. angle, in radians) of the resulting complex number:

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

Let x be a complex sinusoid with amplitude A>0, (absolute) frequency $f\in \mathbb {R}$ and phase $\phi _{0}\in [0;2\pi ]$ sampled at f_s>0 so, without loss of generality,

$x[n]=Ae^{j2\pi ({\frac {f}{f_{s}}}n+\phi _{0})}$

then

$y[n]=\mathrm {arg} \left(Ae^{j2\pi \left({\frac {f}{f_{s}}}n+\phi _{0}\right)}{\overline {Ae^{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)\$

$=\mathrm {arg} \left(A^{2}e^{j2\pi {\frac {f}{f_{s}}}}\right)$

$A$ is real, and so is $A^{2}$ , and hence only scales, therefore $\mathrm {arg} (\cdot )$ is invariant: = arg $\left(e^{j2\pi {\frac {f}{f_{s}}}}\right)={\frac {f}{f_{s}}}$

This block does not support C++ output, so it cannot be used when the output language of a flowgraph in GRC is C++.

Parameters

Gain: Gain setting to adjust the output amplitude. Set based on converting the phase difference between samples to a nominal output value. Default: "samp_rate/(2*math.pi*fsk_deviation_hz)".