Powers of

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

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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Powers of $z$

Choose any two complex numbers $z_0$ and $z_1$, and form the sequence


What are the properties of this signal? Expressing the two complex numbers as

we see that the signal $x(n)$ is always a discrete-time generalized complexsinusoid, i.e., an exponentially enveloped complex sinusoid.

Figure 5.9 shows a plot of a generalized (exponentially decaying) complex sinusoid versus time.

Figure 5.9:Exponentially decaying complex sinusoid and its projections.

Note that the left projection (onto the $z$ plane) is a decaying spiral, the lower projection (real-part vs. time) is an exponentially decaying cosine, and the upper projection (imaginary-part vs. time) is an exponentially enveloped sine wave.

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