Projection of Circular Motion

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Projection of Circular Motion

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Projection of Circular Motion

We have

\


Interpreting this in the complex plane tells us that sinusoidal motion is the projection of circular motion onto any straight line. Thus, the sinusoidal motion $\ is the projection of the circular motion$e^{j\ onto the $x$ (real-part) axis, while $\ is the projection of $e^{j\ onto the $y$ (imaginary-part) axis.

Figure 5.7 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 5.7:A complex sinusoid and its projections.
\

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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