# 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 ?

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 ?

• 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. – OverLordGoldDragon Jan 11 at 14:59
• 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. – Elaws Jan 11 at 15:19

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.
• 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) ? – Elaws Jan 11 at 17:11