FM Spectra

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

$$x(t) = \cos[\omega_c t + \beta\sin(\omega_m t)], \protect$$ (4.8)

where we have changed the modulator amplitude $A_m$ to the more traditional symbol $\beta$ , called the FM index in FM sound synthesis contexts. Using phasor analysis (where phasors are defined below in §4.3.11),^4.12i.e., expressing a real-valued FM signal as the real part of a more analytically tractable complex-valued FM signal, we obtain

$$x(t) \isdef \cos[\omega_c t + \beta\sin(\omega_m t)] = \left\{e^{j[\omega_c t + \beta\sin(\omega_m t)]}\right\}$$

$$= \left\{e^{j\omega_c t} e^{j\beta\sin(\omega_m t)}\right\}$$

$$= \left\{e^{j\omega_c t} \sum_{k=-\infty}^\infty J_k(\beta) e^{jk\omega_m t}\right\}$$

$$= \left\{\sum_{k=-\infty}^\infty J_k(\beta) e^{j(\omega_c+k\omega_m) t}\right\}$$

$$= \sum_{k=-\infty}^\infty J_k(\beta) \cos[(\omega_c+k\omega_m) t]$$ (4.9)

where we used the fact that $J_k(\beta)$ is real when $\beta$ is real. We can now see clearly that the sinusoidal FM spectrum consists of an infinite number of side-bands about the carrier frequency $\omega_c$ (when $\beta\neq 0$ ). The side bands occur at multiples of the modulating frequency $\omega_m$ away from the carrier frequency $\omega_c$ .