Plotting Complex Sinusoids versus Frequency

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Plotting Complex Sinusoids versus Frequency

As discussed in the previous section, we regard the signal

$\displaystyle x(t) = A_x e^{j\omega_x t}$

as a positive-frequency sinusoid when $\omega_x>0$ . In a manner analogous to spectral magnitude plots (discussed in §4.1.6), we can plot this complex sinusoid over a frequency axis as a vertical line of length

at the point $\omega=\omega_x$ , as shown in Fig.4.10. Such a plot of amplitude versus frequency may be called a spectral plot, or spectral representation [42] of the (zero-phase) complex sinusoid.

**Figure 4.10:** Spectral plot of a complex sinusoid $A_x e^{j\omega _x t}$ .
$\begin{figure}\input fig/csplot.pstex_t \end{figure}$

More generally, however, a complex sinusoid has both an amplitude and a phase (or, equivalently, a complex amplitude):

$\displaystyle x(t) = \left(A_x e^{j\theta_x}\right)e^{j\omega_x t}$

To accommodate the phase angle $\theta_x$ in spectral plots, the plotted vector may be rotated by the angle $\theta_x$ in the plane orthogonal to the frequency axis passing through $\omega_x$ , as done in Fig.4.16b below (p.

) for phase angles $\theta_x=\pm \pi/2$ .

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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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Mathematics of the DFT
    Sinusoids and Exponentials
       Complex Sinusoids
          Plotting Complex Sinusoids versus Frequency