Engineering Tables/DFT Transform Table

Time-Domain x[n]	Frequency Domain X[k]	Notes
$x_n\equiv \frac{1}{N}\sum _{k=0}^{N-1}{X}_k\cdot e^{i2\pi kn/N}$	$X_k\equiv \sum _{n=0}^{N-1}{x}_n\cdot e^{-i2\pi kn/N}$	DFT Definition
$x_n\cdot e^{i2\pi kn/N}$	$X_{n-k}$	Shift theorem
$x_{n-k}$	$X_k\cdot e^{-i2\pi kn/N}$
$x_n\in 𝐑$	$X_k=X_{N-k}^{*}$	Real DFT
$a^n$	$\frac{1-a^N}{1-a\cdot e^{-i2\pi k/N}}$
$(\genfrac{}{}{0ex}{}{N-1}{n})$	${(1+e^{-i2\pi k/N})}^{N-1}$