Powers of

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). 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.

<< 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 >>


© 1998-2017 – Nicola Asuni - Tecnick.com - All rights reserved.
about - disclaimer - privacy