(New page: == Discrete Fourier Transform (DFT) == ---- == Definition of DFT == '''DFT''' <math>X[k] = \sum_{n=0}^{N-1}{x[n]e^{-j \frac{2{\pi}}{N}kn}}, for \mbox{ }k = 0, 1, 2, 3, ..., N-1</math>...) |
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| − | + | [[Category:discrete Fourier transform]] | |
| + | [[Category:ECE438Fall2010Boutin]] | ||
| + | [[Category:bonus point project]] | ||
| + | [[Category:ECE]] | ||
| + | [[Category:ECE438]] | ||
| + | [[Category:Fourier transform]] | ||
| + | = Discrete Fourier Transform (DFT) = | ||
| + | ---- | ||
| + | A student project for the course [[ECE438]] | ||
---- | ---- | ||
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'''IDFT''' | '''IDFT''' | ||
| − | <math>x[n] = \frac{1}{N}\sum_{k=0}^{N-1}{X | + | <math>x[n] = \frac{1}{N}\sum_{k=0}^{N-1}{X[k]e^{j \frac{2{\pi}}{N}kn}}, for \mbox{ }n = 0, 1, 2, 3, ..., N-1</math> |
| + | |||
| + | X[k] is defined for <math>0 <= k <= N - 1</math> and periodic with period N | ||
| + | |||
| + | X[n] is defined for <math>0 <= n <= N - 1</math> and also periodic with period N | ||
---- | ---- | ||
| + | |||
| + | == Properties of DFT == | ||
| + | |||
| + | '''Linearity''' | ||
| + | |||
| + | <math>ax_1[n] + bx_2[n] \longleftrightarrow aX_1[k] + bX_2[k] </math> | ||
| + | |||
| + | for any a, b complex constant and all <math>x_1[n]</math> and <math>x_2[n]</math> with the same length | ||
| + | ---- | ||
| + | ==Comments/questions== | ||
| + | *Write a comment here | ||
| + | **answer here | ||
| + | ---- | ||
| + | [[ECE438|Back to ECE438]] | ||
| + | |||
| + | [[2010_Fall_ECE_438_Boutin|Back to ECE438 Fall 2010]] | ||
Latest revision as of 07:50, 11 November 2013
Contents
Discrete Fourier Transform (DFT)
A student project for the course ECE438
Definition of DFT
DFT
$ X[k] = \sum_{n=0}^{N-1}{x[n]e^{-j \frac{2{\pi}}{N}kn}}, for \mbox{ }k = 0, 1, 2, 3, ..., N-1 $
IDFT
$ x[n] = \frac{1}{N}\sum_{k=0}^{N-1}{X[k]e^{j \frac{2{\pi}}{N}kn}}, for \mbox{ }n = 0, 1, 2, 3, ..., N-1 $
X[k] is defined for $ 0 <= k <= N - 1 $ and periodic with period N
X[n] is defined for $ 0 <= n <= N - 1 $ and also periodic with period N
Properties of DFT
Linearity
$ ax_1[n] + bx_2[n] \longleftrightarrow aX_1[k] + bX_2[k] $
for any a, b complex constant and all $ x_1[n] $ and $ x_2[n] $ with the same length
Comments/questions
- Write a comment here
- answer here
