**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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

Choose any two complex numbers and , and form the sequence

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

we see that the signal 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.

Note that the left projection (onto the 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.