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I hope this is a pretty easy question for you guys.

I know for frequency modulation (FM) for fundamental frequency you have the expansion

$$output = J_0(\beta)V\cos(\omega t)\pm J_1(\beta)\cos((\omega \pm \omega_m)t)+... $$

I am wondering what is the equivalent for the third Harmonic? is it

$$output = J_0(\beta)V\cos(3\omega t)\pm J_1(\beta)(3(\omega \pm \omega_m)t)+... $$

or

$$output = J_0(\beta)V\cos(3\omega t)\pm J_1(\beta)((3\omega \pm \omega_m)t)+... $$

thanks for your input.

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That would really depend on where the harmonic distortion is occurring.

If it is downstream of the modulator, then it is the first - ie, the modulation spread is multiplied. In the easy-to-image case where the harmonic distortion were coming from an RF power amplifier this is what you would get.

If it is upstream such that the harmonics are present going into the modulator, then the second, only the center frequency is multiplied.

In simple cases, the FM modulation is applied to the source oscillator, but it is also possible to apply it afterwards, for example by mixing with a modulated local oscillator signal. However, there are likely to be filters following the mixer which may substantially suppress the harmonic content.

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