# 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?
• This sounds like homework. Did you try to answer these questions? What have you thought so far? – Tendero Apr 5 '17 at 1:46
• 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. – 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, $$W=f_m$$).