# FM: Do rational ratios of carrier to modulator signals really produce harmonic spectra?

I've read that in FM, all rational ratios of the carrier frequency to the modulator frequency produce harmonic spectrum in the FM signal. Link to what I read.

I don't understand why this is. If for example carrier frequency is 10Hz and modulator frequency is 5Hz, then the spectrum will include the frequencies 15Hz, 25Hz..., which are not integer multiples of 10Hz.

And for some ratios, the frequencies in the sequences come in "pairs", the ratio 10:4 gives the sequence 2,4,8,10,14,16..., which doesn't look like a harmonic series at all.

What is the proof for this?

• FM is non linear, and it does weird things to the spectrum... Carson's own paper is a good start: J.R. Carson, "Notes on the theory of modulation", Proc. IRE, vol. 10, no. 1 (Feb. 1922), pp. 57-64. – MBaz Feb 25 '20 at 20:04

That paper (1973) is about more natural, simple and rich audio synthesis using FM (which later became Yamaha's very populer FM synthesis audio chips).

In that respect, the carrier frequency is taken very low compared to the modulating tone frequency, unlike in communication applications. Hence the the side lobes cross the zero frequency boundary and are reflected back into the positive band using simple trigonometry.

Another alternative to see this is to use positive and negative frequency version (exponential) of Fourier series analysis, to see that negative frequency side lobes penetrate into the positive (vice versa) band when the carrier frequency is insufficiently low.

Note that the resulting Bessel spectra of the single tone modulated FM signal predicts that the side lobes about the carrier will be placed at the frequencies of:

$$f_n = f_c \pm n f_m$$ where $$f_c$$ is th carrier frqeuency, and $$f_m$$ is the single tone modulator. As you can see, if $$f_c$$ is sufficiently low, then for some $$n$$, the component at $$f_n = f_c - n f_m$$ will be less than $$0$$ Hertz and will be reflected into the positive frequency $$f_r = n f_m - f_c$$.

So the FM side lobes that reflect from negative frequencies and the already existing ones do mix up to create the final audio spectrum which can be harmonic or inharmonic depending on the carrier frequency and modulating tone frequency ratios.

This was a genuine way of producing harmonically rich audio synthesis in legacy analog synthesizers.

• Ah nice @Fat32 I didn’t see how the carrier came into play but do now! – Dan Boschen Feb 26 '20 at 1:00