Modulation by a Complex Sinusoid

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Modulation by a Complex Sinusoid

**Figure:** System diagram for complex demodulation (frequency-shifting) by **$-\omega _c$** .
$\begin{figure}\input fig/modulation.pstex_t \end{figure}$

Figure 9.12 shows the system diagram for complex demodulation.^10.2The input signal is multiplied by a complex sinusoid to produce the frequency-shifted result

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

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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written by 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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Chapters

Spectral Audio Signal Processing
    The Filter Bank Summation (FBS) Interpretation of the Short Time Fourier Transform (STFT)
       The DFT Filter Bank
          Modulation by a Complex Sinusoid