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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.


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``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 © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA