Powers of

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

It appears that you are using AdBlocking software. The cost of running this website is covered by advertisements. If you like it please feel free to a small amount of money to secure the future of this website.

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

<< Previous page  TOC  INDEX  Next page >>

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.

<< Previous page  TOC  INDEX  Next page >>

 

© 1998-2018 – Nicola Asuni - Tecnick.com - All rights reserved.
about - disclaimer - privacy