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<?xml version="1.0"?>
<!--
###################################################
##Quadrature Demodulator
###################################################
-->
<block>
<name>Quadrature Demod</name>
<key>analog_quadrature_demod_cf</key>
<import>from gnuradio import analog</import>
<import>import math</import>
<make>analog.quadrature_demod_cf($gain)</make>
<callback>set_gain($gain)</callback>
<param>
<name>Gain</name>
<key>gain</key>
<value>samp_rate/(2*math.pi*fsk_deviation_hz/8.0)</value>
<type>real</type>
</param>
<sink>
<name>in</name>
<type>complex</type>
</sink>
<source>
<name>out</name>
<type>float</type>
</source>
<doc>
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:
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]= A e^{j2\pi( \frac f{f_s} n + \phi_0)}\f
then
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}\\
</doc>
</block>
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