Powers of z Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Powers of z

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

$\displaystyle x(n) \isdef z_0 z_1^n, \quad n=0,1,2,3,\ldots\,. \protect$ (4.10)

What are the properties of this signal? Writing the complex numbers as

\begin{eqnarray*}
z_0 &=& A e^{j\phi} \\
z_1 &=& e^{sT} = e^{(\sigma + j\omega)T},
\end{eqnarray*}

we see that the signal $ x(n)$ is always a discrete-time generalized (exponentially enveloped) complex sinusoid:

$\displaystyle x(n) = A e^{\sigma n T} e^{j(\omega n T + \phi)}
$

Figure 4.17 shows a plot of a generalized (exponentially decaying, $ \sigma<0$) complex sinusoid versus time.

Figure 4.17: Exponentially decaying complex sinusoid and projections.
\includegraphics[scale=0.8]{eps/circledecaying}

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.


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]