Discrete Fourier transforms (DFT)

Discrete Fourier Transform

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Discrete Fourier Transform Pairs and Properties (info)
Definition Discrete Fourier Transform and its Inverse
Let x[n] be a periodic DT signal, with period N.
N-point Discrete Fourier Transform	$X [k] = \sum_{n=0}^{N-1} x[n]e^{-j 2\pi \frac{k n}{N}} \,$
Inverse Discrete Fourier Transform	$\,x [n] = (1/N) \sum_{k=0}^{N-1} X[k] e^{j 2\pi\frac{kn}{N}} \,$


$x[n] \ \text{ (period } N)$	$\longrightarrow$	$X_N[k] \ \ (N \text{ point DFT)}$
$\ \sum_{k=-\infty}^\infty \delta[n+Nk] = \left\{ \begin{array}{ll} 1, & \text{ if } n=0, \pm N, \pm 2N , \ldots\\ 0, & \text{ else.} \end{array}\right.$		$\ 1 \text{ (period } N)$
$\ 1 \text{ (period } N)$		$\ N\sum_{m=-\infty}^\infty \delta[k+Nm] = \left\{ \begin{array}{ll} N, & \text{ if } n=0, \pm N, \pm 2N , \ldots\\ 0, & \text{ else.} \end{array}\right.$
$\ e^{j2\pi k_0 n}$		$\ N\delta[((k - k_0))_N]$
$\ \cos(\frac{2\pi}{N}k_0n)$		$\ \frac{N}{2}(\delta[((k - k_0))_N] + \delta[((k+k_0))_N])$

Discrete Fourier Transform Properties
	$x[n] \$	$\longrightarrow$	$X[k] \$
Linearity	$ax[n]+by[n] \$		$aX[k]+bY[k] \$
Circular Shift	$x[((n-m))_N] \$		$X[k]e^{(-j\frac{2 \pi}{N}km)} \$
Duality	$X[n] \$		$NX[((-k))_N] \$
Multiplication	$x[n]y[n] \$		$\frac{1}{N} X[k]\circledast Y[k], \ \circledast \text{ denotes the circular convolution}$
Convolution	$x(t) \circledast y(t) \$		$X[k]Y[k] \$
	$\ x^*[n]$		$\ X^*[((-k))_N]$
	$\ x^*[((-n))_N]$		$\ X^*[k]$
	$\ \Re\{x[n]\}$		$\ X_{ep}[k] = \frac{1}{2}\{X[((k))_N] + X^*[((-k))_N]\}$
	$\ j\Im\{x[n]\}$		$\ X_{op}[k] = \frac{1}{2}\{X[((k))_N] - X^*[((-k))_N]\}$
	$\ x_{ep}[n] = \frac{1}{2}\{x[n] + x^*[((-n))_N]\}$		$\ \Re\{X[k]\}$
	$\ x_{op}[n] = \frac{1}{2}\{x[n] - x^*[((-n))_N]\}$		$\ j\Im\{X[k]\}$

Other Discrete Fourier Transform Properties
Parseval's Theorem	$\sum_{n=0}^{N-1}\|x[n]\|^2 = \frac{1}{N} \sum_{k=0}^{N-1}\|X[k]\|^2$

Discrete Fourier Transform table - Rhea