Modulation by a Complex Sinusoid

Figure: System diagram for complex demodulation (frequency-shifting) by $-\omega _c$.

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

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

Given a signal expressed as a sum of sinusoids,

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

then the demodulation produces

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

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.