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Phasor and Carrier Components of Sinusoids

If we restrict $ z_1$ in Eq.$ \,$ (4.10) to have unit modulus, then $ \sigma=0$ and we obtain a discrete-time complex sinusoid.

$\displaystyle x(n) \isdef z_0 z_1^n = \left(Ae^{j\phi}\right) e^{j\omega n T} = A e^{j(\omega n T+\phi)}, \quad n=0,1,2,3,\ldots \protect$ (4.11)

where we have defined

\begin{eqnarray*}
z_0 &\isdef & Ae^{j\phi}, \quad \hbox{and}\\
z_1 &\isdef & e^{j\omega T}.
\end{eqnarray*}



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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
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