# Spectrum of FM signal?

If a carrier ($$\f_c\$$) is FM modulated by a single sine wave ($$\f_m\$$), its spectrum is composed of frequencies at $$\f_c + kf_m\$$ ($$\k \in \mathbb{Z}\$$) whose amplitudes are weighted by $$\J_n(\beta)\$$, the $$\k\$$-order Bessel function of modulation index $$\\beta\$$. We could get such spectra :

Now, what would be the spectrum of the carrier if it was modulated by a more complex signal like speech or music ? What would happen in the simple example of a 2 tones modulating signal ?

Short term Fourier transform of the modulating signals would be :

• Harmonic (e.g. : vowel in speech, or a violin note)
• Non harmonic (e.g. : some consonants in speech, some percussive instrument)

In both case, modulating signal will have a more richer frequency content than a single sinusoid.

Will these rich spectra just repeat regularly around $$\f_c\$$ as in the case of a single tone modulating signal ? Why ?

If a carrier is AM modulated by complex signal like speech, we find the spectrum of modulating signal shifted around the carrier (as seen in the spectrogram below) : what would happen with FM modulation ?

• To begin with a simple example, let's say modulating signal has only 2 frequencies ($f_{m1}$<$f_{m2}$) : how would this spectrum repeat around the carrier ? Every $f_{m1}$ or every $f_{m2}$ ? Or a combination of both ? Jan 11 '21 at 14:41
• So let's say modulating signal is a piano note : $f_m$, $2f_m$, $3f_m$, etc... (a fundamental and its harmonics). The spectrum will have frequencies at $f_c$ + k$f_m$, $f_c$ + k$2f_m$, $f_c$ + k$3f_m$, etc...? The spectrum is completely mixed up, how do you retrieve information from this ? Jan 11 '21 at 16:54