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Mathematics of the DFT
    Sinusoids and Exponentials
       Complex Sinusoids
          Sinusoidal Frequency Modulation (FM)
             FM Spectra

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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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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