DFT Definition
The Discrete Fourier Transform (DFT) of a signal
may be defined by
![$\displaystyle X(\omega_k ) \isdef \sum_{n=0}^{N-1}x(t_n)e^{-j\omega_k t_n}, \qquad k=0,1,2,\ldots,N-1,
$](http://www.dsprelated.com/josimages_new/mdft/img69.png)
![$ \isdeftext $](http://www.dsprelated.com/josimages_new/mdft/img71.png)
![\begin{eqnarray*}
\sum_{n=0}^{N-1} f(n) &\isdef & f(0) + f(1) + \dots + f(N-1)\\...
...mbox{number of time samples = no.\ frequency samples (integer).}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img72.png)
The sampling interval is also called the sampling period.
For a tutorial on sampling continuous-time signals to obtain
non-aliased discrete-time signals, see Appendix D.
When all signal samples
are real, we say
.
If they may be complex, we write
. Finally,
means
is any integer.
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