Spectrum of FM signal?

Question

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 ?

Answer 1

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).