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2

You are essentially seeing a Cascade-Integrator-Comb (CIC) response which is identical to a moving average filter (Aliased Sinc function magnitude response) as seen with CIC filter structures. Consider what is happening in units of phase: You start with a white noise signal which is translated from magnitude directly to units of frequency in the FM ...

3

Frequency by definition is the derivative of phase with respect to time (a change in phase divided by the change in time is frequency). You see this with the radian expression for frequency given by $2\pi f$: A frequency of 1 Hz is 1 cycle per second which is $2\pi$ radians per second. So similarly phase versus time is the integral of frequency versus time. ...

10

You need to integrate the modulating signal because frequency is the time derivative of phase. Therefore, the typical relationship from introductory calculus holds: $$\phi_i(t) = \int_{-\infty}^{t} \frac{d\phi_i(\tau)}{d\tau} d\tau = \int_{-\infty}^{t} 2\pi f_i(\tau) d\tau$$ For causal signals, the lower limit on the integral changes to zero:  \phi_i(...

3

I simply use unwrapped atan2(IQ(i)) - atan2(IQ(i-1)) to estimate a discrete derivative, then low pass filter to below 15 kHz. Although with a shallow slope, the 1st order approximation to atan() given by Boschen will work just as well. Your noise might be due to not unwrapping the phase delta, or to not low pass filtering after doing the phase ...

1

At 240 KHz the phase increment per unit sample is very small so your derivative is a high pass filter with a cut-off well above 15KHz. (Review freqz([1 -1]) to see what I mean.) You have a frequency discriminator but its phase slope is very small in converting frequency variation to amplitude variation for your signals of interest. You can try increasing ...

1

This should work if you further filter to 15KHz after your FM discriminator. FM stereo is actually sub-modulated as follows: L+R: 0-15KHz Pilot Tone: 19KHz L-R: 23KHz-53KHz as DSB-SC on a 38 KHz sub-carrier Where L is the left channel and R is the right channel. Update: I found a nice graphic showing this from Keysight with further details on FM ...

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