Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Sponsor

DM6467T DaVinci video processor enables H.264 1080p decoding up to 60 fps/eight D1 channel encoding.
Details Here!

Chapters

See Also

Embedded SystemsFPGAElectronics

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

Chapter Contents:

Search Mathematics of the DFT

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

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*}

Order a Hardcopy of Mathematics of the DFT

Previous: Analytic Signals and Hilbert Transform Filters
Next: Sampled Sinusoids
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.


Comments

No comments yet for this page


Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )