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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

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

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

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

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

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

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``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
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