I've a fundamental problem with understanding terminology used in DSP. What does it mean to tune a radio (for example, a software defined radio) to a certain carrier frequency $f$? The received signal is an arbitrary waveform and not a sine wave of frequency $f$. I understand AM modulation: you take a sine and manipulate its amplitude. The FM modulation I don't get: the modulated signal is not a sine, so why do we say that we tune a radio to a certain frequency in that case?
there are a lot of misconceptions in the question.
"The received signal is an arbitrary waveform and not [only] a sine wave of frequency $f$."
it's a collection of sine waves at different frequencies. and "tuning" a radio or a filter or whatever is selecting frequency components in the neighborhood of $f$ while rejecting the others.
A mathematical theorem (named after Fourier) says that (non-pathological) signals that are "not a sine" are, for all practical purposes, identical to (or can be accurately described as) the sum of a bunch of pure sinusoids (of various phases and magnitudes).
As practical radios don't have infinitely narrow bandwidths, when tuning to a single pure frequency, you actually get a suitable range of frequencies corresponding to a bunch of pure sinusoids, enough to reasonably describe and thus demodulate your "not a sine" original signal.