Impact of modulation index in FM 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?
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, $W=f_m$).