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
$x_{n-k}\,$	$X_{k}\cdot e^{-i2\pi kn/N}$	Shift theorem
$x_{n}\in \mathbf {R}$	$X_{k}=X_{N-k}^{*}\,$	Real DFT
$a^{n}\,$	${\frac {1-a^{N}}{1-a\cdot e^{-i2\pi k/N}}}$
${N-1 \choose n}\,$	$\left(1+e^{-i2\pi k/N}\right)^{N-1}\,$

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