Plotting Complex Sinusoids versus Frequency

As discussed in the previous section, we regard the signal

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

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 [42] 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):

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

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 $\theta_x=\pm \pi/2$.