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FM Spectra

Using the expansion in Eq. (7), it is now easy to determine the spectrum of sinusoidal FM. Eliminating scaling and phase offsets for simplicity in Eq. (5) yields

$\displaystyle x(t) = \cos[\omega_c t + \beta\sin(\omega_m t)]. \protect$ (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,
$\displaystyle x(t)$ $\displaystyle =$ 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$\displaystyle \left\{\sum_{k=-\infty}^\infty J_k(\beta)
e^{j(\omega_c+k\omega_m) t}\right\}$  
  $\displaystyle =$ $\displaystyle \sum_{k=-\infty}^\infty J_k(\beta) \cos[(\omega_c+k\omega_m) t]$ (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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``Sinusoidal Modulation of Sinusoids'', by Julius O. Smith III, (Excerpt from ... ).
Copyright © 2005-12-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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