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


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Analytic Signals and Hilbert Transform Filters