Engineering Tables/DFT Transform Table

Time-Domain x[n]	Frequency Domain X[k]	Notes
$x_n\equiv\frac{1}{N}\sum_{k=0}^{N-1}X_k \cdot e^{i 2 \pi kn/N}$	$X_k\equiv\sum_{n=0}^{N-1}x_n \cdot e^{-i 2 \pi kn/N}$	DFT Definition
$x_n \cdot e^{i 2 \pi kn/N} \,$	$X_{n-k}\,$	Shift theorem
$x_{n-k}\,$	$X_k \cdot e^{-i 2 \pi kn/N}$
$x_n \in \mathbf{R}$	$X_k=X_{N-k}^*\,$	Real DFT
$a^n\,$	$\frac{1-a^N}{1-a \cdot e^{-i 2 \pi k/N} }$
${N-1 \choose n}\,$	$\left(1+e^{-i 2 \pi k/N} \right)^{N-1}\,$