# 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

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.

• 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) – Elaws Jan 11 at 17:10
• 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
• 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. – TimWescott Jan 11 at 17:17
• 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). – Elaws Jan 11 at 17:27
• Yes. Basically, FM modulation does not work by just moving spectra around like AM or SSB does. Hence my highly technical term "munge". – TimWescott Jan 11 at 17:32

Yes the signal would be grouped around the carrier just as in the simpler AM case: With AM we get two sidebands for every frequency tone in the modulated signal. For AM of a more complicated waveform, we can use the Fourier Series decomposition of that waveform to predict the average frequency spectrum as the superposition of each of the sideband pairs given by each Fourier single tone component.

FM is no different in this regard, there are just multiple sidebands (as $$\beta$$, or modulation index, is increased) for each tone. Each tone in the Fourier Series decomposition will produce a certain number of significant sidebands (as determined using Bessel functions based on the respective modulation index for each tone), and the resulting more complicated waveform will have an average frequency spectrum as the superposition of each of these tones.

Note with FM that the number of significant sidebands for a single tone is given by $$\beta$$ as $$f_{dev}/f_{mod}$$, where $$f_{dev}$$ is the frequency deviation and $$f_{mod}$$ is the modulation rate. So for a more complicated waveform with several tonal components, the lower frequency components will have a higher modulation index and therefore many more significant sidebands than the highest frequency components.

The modulation index $$\beta$$ also happens to be the peak angle of the equivalent phase modulation that occurs and I find it very insightful to use complex phasor diagrams to show the equivalence of AM and FM for the case of a single pair of sidebands (small angle or $$\beta$$ referred to as "narrowband FM"). This also directly shows why the additional sidebands are needed, since with only two sidebands, as the level of the sidebands increase we would induce both AM and FM if we were restricted to not have additional sidebands.

This detail is demonstrated below with the AM case as a large carrier and one small pair of sidebands, with the rotating phasors in time representing the higher frequency side and (phasor rotating counter-clockwise relative to the carrier) and lower frequency sidebands (phasor rotating clockwise), which when all summed together will stay on the real axis with only the amplitude of the result going up and down (amplitude modulation!). But if the same phasors are changed to sum in quadrature with the carrier as in the FM case for very small angles we can similarly predict the sideband level and frequency using the same vector diagram, where the two vectors would add to a peak angle $$\beta$$ while minimally changing the amplitude, and each sideband level is $$\beta/2$$ relative to the carrier given $$\sin(\beta) \approx \beta$$ for small angles. (Also for clarity, in the FM case the sidebands shown are directly proportional to the peak phase of the underlying phase modulation (PM), so our observation of that diagram with time is showing the resulting PM and if we took the derivative of that phase versus time we would directly see the underlying FM of that same waveform.)

The spectrums shown above are the magnitude spectrums which would be identical for the two cases (the phase for each, not shown, is what would be different). But here is the observation with this which removes all mystery of the requirement for multiple sidebands for FM in general as predicted by Bessel functions. Note that if we increased the sideband level (by increasing $$\beta$$) in the above diagrams, if we restricted the spectrum to only have one single pair of sidebands, the resulting signal MUST have both AM and FM components in the FM implementation: the single sideband case would sum to be in quadrature with the carrier, which would deviate for large angles from the circular path that would maintain a constant amplitude (no AM components). The additional significant sidebands as predicted by use of Bessel functions for large angles, are exactly what is needed to provide a summation that stays on the constant amplitude trajectory and results in an FM waveform with no incidental AM components!

Wideband FM is a form of "spread spectrum" (with associated processing gains), in that the original spectrum is simply spread out by an increased factor related to $$\beta$$, but will grouped around the carrier frequency with symmetric sideband components similar to AM (just with more bandwidth than the original waveform for the frequency components where the modulation index is large). Given the larger "spreading" of lower frequency components and associated processing gain using the spread-spectrum analogy, less SNR is needed for those components as compared to the higher frequency components of the signal. For this reason pre-emphasis is often used in FM transmission where the higher frequency components are amplified in compensation. (This is typically and equally explained as an enhancement of high frequency noise in FM demodulation; in the end it is the signal to noise ratio that we care about).