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Modulation by a Complex Sinusoid

figure[htbp] \includegraphics{eps/modulation}

Figure 8.12 shows the system diagram for complex demodulation.9.2The input signal $ x(n)$ is multiplied by a complex sinusoid to produce the frequency-shifted result

$\displaystyle x_c(n) = e^{-j\omega_c n} x(n).
$

Given a signal expressed as a sum of sinusoids,

$\displaystyle x(n) = \sum_{k=1}^{N_x} a_k e^{j\omega_k n}, \quad a_k\in{\bf C},
$

then the demodulation produces

$\displaystyle x_c(n) \isdef x(n) e^{-j\omega_c n} =
\sum_{k=1}^{N_x} a_k e^{j(\omega_k -\omega_c) n}.
$

We see that frequency $ \omega_k$ is down-shifted to $ \omega_k-\omega_c$. In particular, frequency $ \omega_c$ (the ``center frequency'') is down-shifted to dc.


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About the Author: 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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