Generalized Complex Sinusoids

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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

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

, 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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Mathematics of the DFT
    Sinusoids and Exponentials
       Complex Sinusoids
          Generalized Complex Sinusoids