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Example AM Spectra

Equation (4.4) can be used to write down the spectral representation of $ x_m(t)$ by inspection, as shown in Fig.4.12. In the example of Fig.4.12, 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.4.6. Note in Fig.4.12 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.$ \,$ (4.3). Phase is not shown.

Recall that $ x_m(t)$ was defined as the second term of Eq.$ \,$ (4.1). 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.$ \,$ (4.1) with $ a_m(t)$ given by Eq.$ \,$ (4.2), the magnitude of the spectral representation appears as shown in Fig.4.13.

Figure: Spectral representation of the sinusoidally modulated sinusoid $ [1+ \sin(40\pi t)]\sin(200\pi t)$ from Eq.$ \,$ (4.1), with $ \alpha =1$ , and $ a_m(t)$ given by Eq.$ \,$ (4.2).

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2016-05-31 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University