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Modulation by a Complex Sinusoid
Figure:
System diagram for complex demodulation (frequency-shifting) by
.
|
Figure 9.12 shows the system diagram for complex
demodulation.10.3The input signal
is multiplied by a
complex sinusoid to produce the frequency-shifted result
![$\displaystyle x_c(n) = e^{-j\omega_c n} x(n).$](img1573.png) |
(10.8) |
Given a signal expressed as a sum of sinusoids,
![$\displaystyle x(n) = \sum_{k=1}^{N_x} a_k e^{j\omega_k n}, \quad a_k\in\mathbb{C},$](img1574.png) |
(10.9) |
then the demodulation produces
![$\displaystyle x_c(n) \isdef x(n) e^{-j\omega_c n} = \sum_{k=1}^{N_x} a_k e^{j(\omega_k -\omega_c) n}.$](img1575.png) |
(10.10) |
We see that frequency
is down-shifted to
. In particular, frequency
(the
``center frequency'') is down-shifted to dc.
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