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 :

Spectra of carrier FM modulated by a single sine wave

Now, what would be the spectrum of the carrier if it was modulated by a more complex signal like speech or music ?

Short term Fourier transform of the modulating signal 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 ? What would happen in the simple example of a 2 tones modulating signal ?

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 for such rich signals ?

Spectrogram of AM modulated signal

  • $\begingroup$ STFT behavior would depend on signal component separation and window used, and how quickly the modulators change relative to carrier; relevant, also see section 4.4. $\endgroup$ – OverLordGoldDragon Jan 11 at 14:59
  • $\begingroup$ Thanks @OverLordGoldDragon, my question is much more basic I think : Let's say modulating signal has only 2 frequencies (say $f_{m1}$ < $f_{m2}$). How would the spectrum of this modulating signal repeat around the carier (if it does!) : every $f_{m2}$ ? Every $f_{m1}$ ? Why ? I can't figure it out based on the spectrum of a single tone modulating frequency. It's probably very simple but it would help me a lot understanding richer signals. $\endgroup$ – Elaws Jan 11 at 15:19

Because frequency modulation is a nonlinear phenomenon, there's no good general-purpose answer to your question. You can take any specific baseband signal and figure out its modulated spectrum, but that's it.

For lack of a better technical term, FM modulation takes the frequencies in the parent signal and munges them together, spreading them out by more than the deviation frequency. For a sine wave this is that Bessel function. For a pair of sine waves, it'd be all the harmonics of each sine wave plus all possible combinations of intermodulation ($n f_1 + m f_2\, \forall\, n, m \in \mathbb{Z}$) -- so for the two-sine case there may be some discernible, identifiable lines in the output. For sound with a continuous spectrum, you'll get a continuous spectrum out, but nothing you could just recognize on a spectrum analyzer.

For practical purposes, you're left with rules of thumb, inspired by the above results but limited by our inability to solve the nonlinear differential equations in any general way. If you have a signal with bandwidth $B$, and a maximum frequency deviation of $f_D$, then the bandwidth of the transmitted signal can be limited to $2(B + f_D)$. This isn't satisfying from a theoretical standpoint, but it's backed by over 70 years of practical experience with broadcast and voice communications systems.

  • $\begingroup$ Thanks a lot @TimWescott : In AM, I can clearly find the whole spectrum of the modulating signal on the right and left side of the carrier. In FM, I can see it (with other frequencies added) for a simple one tone modulating signal (so of no practical use, except FM synthesis perheaps?), but not for more complex signals (that is, of practical use !), is this correct ? (to be followed in next comment) $\endgroup$ – Elaws Jan 11 at 17:10
  • $\begingroup$ Let's say I turn on my FM radio and listen to a simple piano note. My modulating signal is the piano note : $f_m$, $2f_m$, $3f_m$, etc... (a fundamental and its harmonics). As opposed with AM modulation where I would find the intact spectrum of the piano note on the right and left side of carrier, it would not at all be the case with FM modulation ? If so, how is it that I can clearly hear the original signal in spite of its modulated spectrum completely messed up (due to nonlinearity as you explained) ? $\endgroup$ – Elaws Jan 11 at 17:11
  • $\begingroup$ Yes, that's correct. I've expanded my answer -- it's not in a mathematically satisfying way, but even in really serious academic works descriptions of FM radio tends to lead to a lot of hand-waving. Major Armstrong managed to invent a solid, high-quality, analog forward error correction system that is quite academic-proof. $\endgroup$ – TimWescott Jan 11 at 17:17
  • $\begingroup$ Thanks @TimWescott : Just to make sure you saw the added comment above with the piano note example. So the spectrum of FM signal is of no practical use when trying to reconstruct the original modulating signal ? The original modulating signal can be fully reconstructed in spite of spectrum of modulated signal completely messed up (= it does not somehow contain the intact spectrum of modulating signal, as with AM). $\endgroup$ – Elaws Jan 11 at 17:27
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    $\begingroup$ Yes. Basically, FM modulation does not work by just moving spectra around like AM or SSB does. Hence my highly technical term "munge". $\endgroup$ – TimWescott Jan 11 at 17:32

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