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

Generalized Complex Sinusoids

We have defined sinusoids and extended the definition to include complex sinusoids. We now extend one more step by allowing for exponential amplitude envelopes:

$\displaystyle y(t) \isdef {\cal A}e^{st}
$

where $ {\cal A}$ and $ s$ are complex, and further defined as

\begin{eqnarray*}
{\cal A}&=& Ae^{j\phi} \\
s &=& \sigma + j\omega.
\end{eqnarray*}

When $ \sigma=0$ , we obtain

$\displaystyle y(t) \isdef {\cal A}e^{j\omega t} = A e^{j\phi} e^{j\omega t}
= A e^{j(\omega t + \phi)}
$

which is the complex sinusoid at amplitude $ A$ , frequency $ \omega$ , and phase $ \phi$ . More generally, we have

\begin{eqnarray*}
y(t) &\isdef & {\cal A}e^{st} \\
&\isdef & A e^{j\phi} e^{(\sigma+j\omega) t} \\
&=& A e^{(\sigma+j\omega) t + j\phi} \\
&=& A e^{\sigma t} e^{j(\omega t + \phi)} \\
&=& A e^{\sigma t} \left[\cos(\omega t + \phi) + j\sin(\omega t + \phi)\right].
\end{eqnarray*}

Defining $ \tau = -1/\sigma$ , we see that the generalized complex sinusoid is just the complex sinusoid we had before with an exponential envelope:

\begin{eqnarray*}
\mbox{re}\left\{y(t)\right\} &=& A e^{- t/\tau} \cos(\omega t + \phi) \\
\mbox{im}\left\{y(t)\right\} &=& A e^{- t/\tau} \sin(\omega t + \phi)
\end{eqnarray*}


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

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``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