Discrete Fourier Transform (DFT)
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
