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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

$\displaystyle x(t) = \cos[\omega_c t + \beta\sin(\omega_m t)], \protect$ (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),4.11i.e., expressing a real-valued FM signal as the real part of a more analytically tractable complex-valued FM signal, we obtain
$\displaystyle x(t) \isdef \cos[\omega_c t + \beta\sin(\omega_m t)]$ $\displaystyle =$ re$\displaystyle \left\{e^{j[\omega_c t + \beta\sin(\omega_m t)]}\right\}$  
  $\displaystyle =$ re$\displaystyle \left\{e^{j\omega_c t} e^{j\beta\sin(\omega_m t)}\right\}$  
  $\displaystyle =$ re$\displaystyle \left\{e^{j\omega_c t} \sum_{k=-\infty}^\infty J_k(\beta) e^{jk\omega_m t}\right\}$  
  $\displaystyle =$ re$\displaystyle \left\{\sum_{k=-\infty}^\infty J_k(\beta) e^{j(\omega_c+k\omega_m) t}\right\}$  
  $\displaystyle =$ $\displaystyle \sum_{k=-\infty}^\infty J_k(\beta) \cos[(\omega_c+k\omega_m) t]$ (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$.

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