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}
$](http://www.dsprelated.com/josimages_new/mdft/img599.png)
![$ {\cal A}$](http://www.dsprelated.com/josimages_new/mdft/img600.png)
![$ s$](http://www.dsprelated.com/josimages_new/mdft/img29.png)
![\begin{eqnarray*}
{\cal A}&=& Ae^{j\phi} \\
s &=& \sigma + j\omega.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img601.png)
When , we obtain
![$\displaystyle y(t) \isdef {\cal A}e^{j\omega t} = A e^{j\phi} e^{j\omega t}
= A e^{j(\omega t + \phi)}
$](http://www.dsprelated.com/josimages_new/mdft/img603.png)
![$ A$](http://www.dsprelated.com/josimages_new/mdft/img367.png)
![$ \omega$](http://www.dsprelated.com/josimages_new/mdft/img368.png)
![$ \phi$](http://www.dsprelated.com/josimages_new/mdft/img369.png)
![\begin{eqnarray*}
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].
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img604.png)
Defining
, 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 ...
...{im}\left\{y(t)\right\} &=& A e^{- t/\tau} \sin(\omega t + \phi)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img606.png)
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Sampled Sinusoids
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Analytic Signals and Hilbert Transform Filters