Using the expansion in Eq.(4.7), it is now easy to determine the spectrum of sinusoidal FM. Eliminating scaling and phase offsets for simplicity in Eq.(4.5) yields
where we have changed the modulator amplitude to the more traditional symbol , called the FM index in FM sound synthesis contexts. Using phasor analysis (where phasors are defined below in §4.3.11),4.11i.e., expressing a real-valued FM signal as the real part of a more analytically tractable complex-valued FM signal, we obtain
where we used the fact that is real when is real. We can now see clearly that the sinusoidal FM spectrum consists of an infinite number of side-bands about the carrier frequency (when ). The side bands occur at multiples of the modulating frequency away from the carrier frequency .