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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 $ A_x$ 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 [46] of the (zero-phase) complex sinusoid.

Figure 4.10: Spectral plot of a complex sinusoid $ A_x e^{j\omega _x t}$ .
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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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2018-02-13 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University