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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 | 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] \ | + | <math>ax_1[n] + bx_2[n] \longrightarrow aX_1[k] + bX_2[k]</math> |
'''Time-Shifting''' | '''Time-Shifting''' | ||
For all <math>n_0</math> included in Z, and all x[n] with period N | For all <math>n_0</math> included in Z, and all x[n] with period N | ||
| − | <math>x[n - n_0] \ | + | <math>x[n - n_0] \longrightarrow X[k]e^{-j \frac{2{\pi}}{N} n_0 k}</math> |
| + | |||
[[ECE438_(BoutinFall2009)|Back to ECE438 course page]] | [[ECE438_(BoutinFall2009)|Back to ECE438 course page]] | ||
Revision as of 10:12, 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] \longrightarrow aX_1[k] + bX_2[k] $
Time-Shifting For all $ n_0 $ included in Z, and all x[n] with period N
$ x[n - n_0] \longrightarrow X[k]e^{-j \frac{2{\pi}}{N} n_0 k} $
