Example AM Spectra

Equation (4) can be used to write down the spectral representation of $x_m(t)$ by inspection, as shown in Fig. 3. In the example of Fig. 3, we have $f_c=100$ Hz and $f_m=20$ Hz, where, as always, $\omega=2\pi f$. For comparison, the spectral magnitude of an unmodulated $100$ Hz sinusoid is shown in Fig. 1. Note in Fig. 3 how each of the two sinusoidal components at $\pm100$ Hz have been ``split'' into two ``side bands'', one $20$ Hz higher and the other $20$ Hz lower, that is, $\pm100\pm20=\{-120,-80,80,120\}$. Note also how the amplitude of the split component is divided equally among its two side bands.

Figure: Spectral magnitude representation of the sinusoidally modulated sinusoid $\sin(40\pi t)\sin(200\pi t)$ defined in Eq. (3). Phase is not shown.

Recall that $x_m(t)$ was defined as the second term of Eq. (1). The first term is simply the original unmodulated signal. Therefore, we have effectively been considering AM with a ``very large'' modulation index. In the more general case of Eq. (1) with $a_m(t)$ given by Eq. (2), the magnitude of the spectral representation appears as shown in Fig. 4.

Figure: Spectral representation of the sinusoidally modulated sinusoid $[1+ \sin(40\pi t)]\sin(200\pi t)$ from Eq. (1), with $\alpha =1$, and $a_m(t)$ given by Eq. (2).