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Dan Boschen
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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 modulation process.

To demodulate you convert to phase (which is an integration since phase is the integral of frequency) so if you evaluated the frequency response of the resulting phase versus time signal you would expect to see the low-pass frequency response of an integrator, rolling off at 1/f. (The phase for white-FM rolls off at 1/f).

You then compute the differences in phase between samples over N samples to recover the modulating signal-- this is a digital integrator and is the classic "Comb Filter", as with N large the magnitude response represents tooth of a comb.

$$1-z^{-N}$$

So essentially the process matches the cascade of an accumulator (integrator) with a comb, which is identical to a moving average filter over N samples.

To see how these are equivalent:

Moving average as a geometric series: $$ \begin{align} & 1+z^{-1}+z^{-2}+z^{-3} + ... + z^{-N-1} \\ & = \sum_{n=0}^{N-1}z^n \\ & = \frac{1-z^{-N}}{1-z^{-1}} \end{align}$$

accumulator: $\frac{1}{1-z^{-1}}$

comb: $1-z^{-N}$

Also interesting is the poles and zeros of each which also explain how the moving average is identical to a CIC: The comb filter has N total zeros starting at z = 1 and equally spaced on the unit circle (the roots of $1-z^{-N}$). The integrator has a pole at z=1. A moving average filter has N-1 zeros, all at the same locations as the comb filter except for z=1.

From your spectrum it looks like you are using N=1. As you increase N you would expect the envelope of the result to approach a Sinc function consistent with a moving average filter.

Dan Boschen
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