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I was wondering what is the difference between demodulating a signal that is carried with a frequency $F_{\rm carrier}$ and applying a band-stop on a band around $F_{\rm carrier}$ ?

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Filtering is a linear operation on the signal, and as such can alter amplitude an phase of signal components, but not their frequencies.

Demodulation is quite a general concept, but normally includes non-linear operations such as shifting the signal in the frequency domain from the carrier to baseband (or an intermediate frequency) in order to extract the unmodulated signal.

So you are trying to compare very different operations.

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    $\begingroup$ Frequency-shifting is a linear operation, it's just not time-invariant $\endgroup$ – Jazzmaniac Jun 2 '16 at 12:27

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