AM demodulation documentation

Author: martinber

docs/how-it-works.md

@@ -154,7 +154,7 @@ samples and filter coefficients.
 
 ### AM demodulation
 
-Previously I used a Hilbert filter to get the [analytic signal], then the
+Previously I used a Hilbert filter to get the [analytic signal], because the
 absolute value of the [analytic signal] is the modulated signal.
 
 Then I found a very fast demodulator implemented on [pietern/apt137]. For each
@@ -163,8 +163,6 @@ carrier frequency:
 
 ![AM demodulation formula]({{ site.baseurl }}/images/demodulation.png)
 
-Where theta is the AM carrier frequency divided by the sample rate.
-
 I couldn't find the theory behind that method, looks similar to I/Q
 demodulation. I was able to reach that final expression (which is used by
 [pietern/apt137]) by hand and I wrote the steps on ``extra/demodulation.pdf``. I

extra/demodulation.pdf

@@ -7,14 +7,44 @@
 \begin{document}
 \pagestyle{empty}
 
+Two samples of an AM signal with carrier frequency $\omega_c$.
+
 \begin{align*}
 \begin{cases}
-x[0]=A\sin(\omega t_{0}+\alpha)\\
-x[1]=A\sin(\omega t_{1}+\alpha)
+x[0]=f(t_{0})\sin(\omega_c t_{0}+\alpha)\\
+x[1]=f(t_{1})\sin(\omega_c t_{1}+\alpha)
 \end{cases}
 \end{align*}
 
-If $t_{0}=0$, then $\omega t_{0}=0$ and $\omega t_{1}=\alpha$
+If the samples were taken $\Delta t$ seconds apart, then the sampling frequency
+$f_s$ is $1/\Delta t$.
+
+\begin{align*}
+\begin{cases}
+x[0]=f(t_{0})\sin(\omega_c t_{0}+\alpha)\\
+x[1]=f(t_{0}+\Delta t)\sin(\omega_c (t_{0}+\Delta t)+\alpha)
+\end{cases}
+\end{align*}
+
+If $\Delta t$ is quite small, $f(t_0) = f(t_0 + \Delta t) = A$: 
+
+\begin{align*}
+\begin{cases}
+x[0]=A\sin(\omega_c t_{0}+\alpha)\\
+x[1]=A\sin(\omega_c (t_{0}+\Delta t)+\alpha)
+\end{cases}
+\end{align*}
+
+If $t_0 = 0$:
+
+\begin{align*}
+\begin{cases}
+x[0]=A\sin(\alpha)\\
+x[1]=A\sin(\omega_c \Delta t+\alpha)
+\end{cases}
+\end{align*}
+
+We define $\phi$ as $\omega_c \Delta t = \omega_c / f_s = 2\pi f_c / f_s$.
 
 \begin{align*}
 \begin{cases}
@@ -69,4 +99,6 @@
 y[i]=\frac{\sqrt{x[i]^{2}+x[i-1]^{2}-2x[i]x[i-1]\cos\phi}}{\sin\phi}
 \end{align*}
 
+Where $\phi = 2\pi f_c / f_s$ 
+
 \end{document}