• If I have the spectrum of a FM modulation, does the frequency of the carrier disappear when the modulation index is too big?
  • Does the amplitude of any side band depend from the modulation index?
  • And does the total number of side bands depend on the modulation index?
  • $\begingroup$ This sounds like homework. Did you try to answer these questions? What have you thought so far? $\endgroup$ – Tendero Apr 5 '17 at 1:46
  • $\begingroup$ These are three questions in one. Reduce to one question, show what you've tried so far, and discuss where you encountered problems. If you don't do that: smells like homework you're trying to offload unto us, like Tendero said. $\endgroup$ – Marcus Müller Apr 5 '17 at 6:41

Assuming an FM modulation of a sine wave of frequency $f_m$. Theoretically, the modulated signal is an infinite sum of cosines at the frequencies $f_c + nf_m$ ($n\in \mathbb{Z}$) which the amplitudes depend on the coefficients $J_n(\beta)$ (of course in addition to the amplitude of the carrier wave), where $\beta$ is the modulation index and $f_c$ is the carrier frequency. At this stage, the answer to the question Q.2. is yes. For the other items:

  • Q.1. The PSD of the modulated signal in $f_c$ is determined for $n=0$, then the PSD at that frequency depends on the coefficient $J_0(\beta)$. Knowing that $J_n(\beta)$ trends to zero as $\beta$ trends to infinity, so the PSD of the modulated signal in $f_c$ trends to zero as the modulation index trends to infinity.

  • Q. 3. Theoretically the bandwidth of an FM modulated signal is infinite. But when considering Carson bandwidth rule ($B_C$), the answer to your question is yes since this rule considers the modulation index. $B_C=2(1+\beta)W$ where $W$ is the baseband bandwidth of the modulating signal (assuming a sine wave, $W=f_m$).

  • $\begingroup$ you might want to add the quantitative expression of the Carson rule. $\endgroup$ – robert bristow-johnson Apr 6 '17 at 2:41
  • $\begingroup$ @robertbristow-johnson Updated. Thanks. $\endgroup$ – Sofiane Apr 6 '17 at 5:09

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