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Projection of Circular Motion

Interpreting the real and imaginary parts of the complex sinusoid,

\begin{eqnarray*}
\mbox{re}\left\{e^{j\omega t}\right\} &=& \cos(\omega t) \\
\mbox{im}\left\{e^{j\omega t}\right\} &=& \sin(\omega t),
\end{eqnarray*}

in the complex plane, we see that sinusoidal motion is the projection of circular motion onto any straight line. Thus, the sinusoidal motion $ \cos(\omega t)$ is the projection of the circular motion $ e^{j\omega t}$ onto the $ x$ (real-part) axis, while $ \sin(\omega t)$ is the projection of $ e^{j\omega t}$ onto the $ y$ (imaginary-part) axis.

Figure 4.9 shows a plot of a complex sinusoid versus time, along with its projections onto coordinate planes. This is a 3D plot showing the $ z$ -plane versus time. The axes are the real part, imaginary part, and time. (Or we could have used magnitude and phase versus time.)

Figure 4.9: A complex sinusoid and its projections.
\includegraphics[scale=0.8]{eps/circle}

Note that the left projection (onto the $ z$ plane) is a circle, the lower projection (real-part vs. time) is a cosine, and the upper projection (imaginary-part vs. time) is a sine. A point traversing the plot projects to uniform circular motion in the $ z$ plane, and sinusoidal motion on the two other planes.


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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 © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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