View source for Student summary Discrete Fourier transform ECE438F09 - Rhea

[[Category:ECE438Fall2009mboutin]]

== 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.

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== Definition ==


'''DFT''' 
*<math>X(k) = \sum_{n=0}^{N-1}{x[n]e^{-j \frac{2{\pi}}{N}kn}},  k = 0, 1, 2, ..., N-1</math>

'''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>

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== 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] \longrightarrow aX_1[k] + bX_2[k]</math>

'''Time-Shifting'''
For all <math>n_0</math> included in Z, and all x[n] with period N

<math>x[n - n_0] \longrightarrow X[k]e^{-j \frac{2{\pi}}{N} n_0 k}</math>

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