Sign in
Search Online Books
Free Online Books
Chapters
Search Mathematics of the DFT
Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?
Sampled Sinusoids
In discrete-time audio processing, such as we normally do on a computer,
we work with samples of continuous-time signals. Let 
denote the sampling rate in Hz. For audio, we typically have 
kHz, since the audio band nominally extends to 
kHz. For compact
discs (CDs), 
kHz,
while for digital audio tape (DAT), 
kHz.
Let
denote the sampling interval in seconds. Then to
convert from continuous to discrete time, we replace 
by 
, where 
is an integer interpreted as the sample number.
The sampled generalized complex sinusoid is then
![\begin{eqnarray*}
y(nT) &\isdef & \left.{\cal A}\,e^{st}\right\vert _{t=nT}\\
...
...
\left[\cos(\omega nT + \phi) + j\sin(\omega nT + \phi)\right].
\end{eqnarray*}](http://www.dsprelated.com/josimages/mdft/img611.png)
Thus, the sampled case consists of a sampled complex sinusoid
multiplied by a sampled exponential envelope
![$ \left[e^{\sigma
T}\right]^n = e^{-nT/\tau}$](http://www.dsprelated.com/josimages/mdft/img612.png){kind=small}
.
Order a Hardcopy of Mathematics of the DFT
Previous: Generalized Complex SinusoidsNext: Powers of z
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