Engineering Tables/DTFT Transform Table

Time domain $x[n]$	Frequency domain $X(\omega )$	Remarks
$\delta [n]$	$1$
$\delta [n-k]$	$e^{-ik\omega }$	integer k
$u[n]$	$\frac{1}{1-e^{-i\omega }}$
$e^{-an}$	$\delta (\omega +a)$	real number a
$\cos (an)$	$\frac{1}{2}[\delta (\omega -a)+\delta (\omega +a)]$	real number a
$\sin (an)$	$\frac{1}{2i}[\delta (\omega -a)-\delta (\omega +a)]$	real number a
$\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\left[\frac{(n-M/2)}{M}\right]$	$\frac{\sin [\omega (M+1)/2]}{\sin (\omega /2)}{e}^{-i\omega M/2}$	integer M
$\mathrm{sinc}[(a+n)]$	$e^{ia\omega }$	real number a
$W\cdot {\mathrm{sinc}}^2(Wn)$	$\frac{1}{2\pi W}\cdot \mathrm{tri}\left(\frac{\omega }{2\pi W}\right)$	real number W
$W\cdot \mathrm{sinc}[W(n+a)]$	$\mathrm{rect}\left(\frac{\omega }{2\pi W}\right)\cdot e^{ja\omega }$	real numbers W, a $0<W\le 1$
$\frac{W}{(n+a)}\{\cos [\pi W(n+a)]-\mathrm{sinc}[W(n+a)]\}$	$j\omega \cdot \mathrm{rect}\left(\frac{\omega }{\pi W}\right)e^{ja\omega }$	real numbers W, a
$\frac{1}{\pi n^2}[(-1{)}^n-1]$	$|\omega |$
$\frac{C(A+B)}{2\pi }\cdot \mathrm{sinc}\left[\frac{A-B}{2\pi }n\right]\cdot \mathrm{sinc}\left[\frac{A+B}{2\pi }n\right]$		real numbers A, B complex C