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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 ? 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 ?

Spectrogram of AM modulated signal

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With a more complex modulation signal, the forest of sidebands gets denser, but the overall envelope stays much the same as you have illustrated. For low modulation index, the envelope looks like a broadened line with a peak at the centre frequency. For high modulation index, there are a low and high frequency lowish peak (representing the time the carrier spends dwelling at the extreme of either frequency shift), with a broad lower plateau between (as it zips quickly between the two extremes).

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  • \$\begingroup\$ Thanks : are there some ressources on the subject ? I can only find explanation of single tone FM spectrum, not of richer signals. Will the spectrum of modulating signal repeat around the carrier as with single tone ? In what fashion and why ? \$\endgroup\$
    – Elaws
    Jan 11, 2021 at 14:36
  • \$\begingroup\$ 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 ? \$\endgroup\$
    – Elaws
    Jan 11, 2021 at 14:41
  • \$\begingroup\$ @Elaws It would have components at frequencies of k.fm1 and n.fm2 where k and n are integers. I suspect it would also have components at p.fm1+q.fm2 where p and q are integers, that's my intuition due to the non-linearity of frequency modulation, but I'm not certain. Why not do a simulation and find out? It will repeat at the lowest common denominator of fm1 and fm2. \$\endgroup\$
    – Neil_UK
    Jan 11, 2021 at 16:43
  • \$\begingroup\$ 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 ? \$\endgroup\$
    – Elaws
    Jan 11, 2021 at 16:54
  • \$\begingroup\$ @Elaws It depends what you mean by 'retrieve' and 'information'. If your 'retrieve' method is looking at the spectrogram, then you'll only be able to get the information of occupied bandwidth and a rough estimate of the modulation index. If your 'retrieve' method is demodulating the FM, then you'll have a very good estimate of the information in the original modulation. That's why FM is used, to convey the original modulation to a receiver. The spectrum tells you how wide your IFs need to be to pass all the signal when you design a receiver and demodulator. \$\endgroup\$
    – Neil_UK
    Jan 11, 2021 at 17:19

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