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1

As per the question, the modulated signal is $x(t + \frac{\phi_{\mathrm{PM}}(t)}{2\pi f_c})$. In the limit of a weak phase deviation ($\Delta\phi\ll1$), we can expand to 1st order: $$x(t + \frac{\Delta\phi \sin(2\pi f_{\mathrm{PM}}t)}{2\pi f_c}) \approx x(t) + \dot{x}(t) \frac{\Delta\phi}{2\pi f_c}\sin(2\pi f_{\mathrm{PM}}t)$$ The first term is simply the ...

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So Tim Wescott's answer was most likely a part of the issue, but during my attempts to correct my IIR filters I realized something even simpler was going on. My LP_RC filter class was calculating its output entirely incorrectly, and in a spectacularly bad way. For some reason, my sleep-deprived mind tried to implement an RC filter by simply multiplying the ...

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I'm about 99.44% sure that you're filtering wrong, in that you're not carrying over any previous filter state. You've kind of fixed this in your differentiation step by saving the previous sample, because the filter that np.diff (effectively) implements uses the previous sample as it's current 'state'. In your filter-and-decimate stage (I assume that's what ...

2

Since the result is small angle FM modulation, for a single tone there will only be two significant sidebands each with magnitude close to $\beta/2$ (for the single sinusoidal modulation case with small $\beta$), where $\beta$ is the modulation index (peak angle). The two sidebands would be centered about the carrier spaced by the modulation rate - ...

2

Note that ideally you would approximate the derivative at time instance $t=nT_s$ by the following central difference quotient: $$\frac{dx(nT_s)}{dt}\approx\frac{x[(n+1)T_s]-x[(n-1)T_s]}{2T_s}\tag{1}$$ Since such a system is non-causal - because you would need to know the signal one time step ahead in order to compute the output - you add a delay of one ...

2

AM Broadcast is large carrier AM, in that the carrier is always present and the amplitude is modulated consistent with the waveform of interest. You can simply hard limit the signal to remove all amplitude modulation, leaving an unmodulated carrier from which it would be easy to derive the precise carrier frequency from any known reference of time. This is ...

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Your bandwidth is between 550 kHz - 1720 kHz, but that is centered around a particular frequency. Now it depends if you want to move the signal to baseband (centered at zero frequency) or some other intermediate frequency. You get to decide this as the system designer. Given a signal centered around the frequency $f_c$, that you wish to modulate to a new ...

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