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== DFT ( Discrete Fourier Transform ) == | == DFT ( Discrete Fourier Transform ) == | ||
| − | The DFT is a finite sum, so it can be computed using a computer. Used for discrete, time-limited signals, or discrete periodic signals. | + | The DFT is a finite sum, so it can be computed using a computer. Used for discrete, time-limited signals, or discrete periodic signals. The DFT of a signal will be discrete and have a finite duration. |
| + | ---- | ||
== Definition == | == Definition == | ||
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'''Inverse DFT (IDFT)''' | '''Inverse DFT (IDFT)''' | ||
*<math>x[n] = \frac{1}{N}\sum_{k=0}^{N-1}{X(k)e^{j \frac{2{\pi}}{N}kn}}, n = 0, 1, 2, ..., N-1</math> | *<math>x[n] = \frac{1}{N}\sum_{k=0}^{N-1}{X(k)e^{j \frac{2{\pi}}{N}kn}}, n = 0, 1, 2, ..., N-1</math> | ||
| + | |||
| + | ---- | ||
| + | |||
| + | |||
| + | == Properties == | ||
| + | |||
| + | '''Linearity''' | ||
| + | For all <math>a,b</math> in the complex plane, and all <math>x_1[n],x_2[n]</math> with the same period N | ||
| + | |||
| + | <math>ax_1[n] + bx_2[n] \longleftarrow zX_1[k] + bX_2[k]</math> | ||
[[ECE438_(BoutinFall2009)|Back to ECE438 course page]] | [[ECE438_(BoutinFall2009)|Back to ECE438 course page]] | ||
Revision as of 10:08, 25 September 2009
DFT ( Discrete Fourier Transform )
The DFT is a finite sum, so it can be computed using a computer. Used for discrete, time-limited signals, or discrete periodic signals. The DFT of a signal will be discrete and have a finite duration.
Definition
DFT
- $ X(k) = \sum_{n=0}^{N-1}{x[n]e^{-j \frac{2{\pi}}{N}kn}}, k = 0, 1, 2, ..., N-1 $
Inverse DFT (IDFT)
- $ x[n] = \frac{1}{N}\sum_{k=0}^{N-1}{X(k)e^{j \frac{2{\pi}}{N}kn}}, n = 0, 1, 2, ..., N-1 $
Properties
Linearity For all $ a,b $ in the complex plane, and all $ x_1[n],x_2[n] $ with the same period N
$ ax_1[n] + bx_2[n] \longleftarrow zX_1[k] + bX_2[k] $ Back to ECE438 course page
