# Engineering Tables/DTFT Transform Table

Time domain
${\displaystyle x[n]\,}$ where ${\displaystyle n\in \mathbb {Z} }$
Frequency domain
${\displaystyle X(e^{j\omega })}$ where ${\displaystyle \omega \in \mathbb {R} }$
Remarks
${\displaystyle {\frac {1}{2\pi }}\int _{-\pi }^{\pi }{X\left(e^{j\omega }\right)}e^{j\omega n}d\omega }$ ${\displaystyle \sum _{n=-\infty }^{\infty }{x[n]e^{-j\omega n}}}$ Definition
${\displaystyle x[n]={\begin{cases}1,&|n|\leq M\\0,&{\text{otherwise}}\end{cases}}}$ ${\displaystyle {\frac {\sin \left(\omega \left({\frac {2M+1}{2}}\right)\right)}{\sin \left({\frac {\omega }{2}}\right)}}}$
${\displaystyle \alpha ^{n}u\left[n\right]}$ ${\displaystyle {\frac {1}{1-\alpha e^{-j\omega }}}}$
${\displaystyle \delta [n]}$ ${\displaystyle 1\!}$ Here ${\displaystyle \delta [n]}$ represents the delta function
which is 1 if ${\displaystyle n=0}$ and zero otherwise.
${\displaystyle u[n]={\begin{cases}0&{\text{for }}n<0\\1&{\text{for }}n\geq 0\end{cases}}}$ ${\displaystyle {\frac {1}{1-e^{-j\omega }}}+\pi \sum _{p=-\infty }^{\infty }{\delta \left(\omega -2\pi p\right)}}$
${\displaystyle {\frac {1}{\pi n}}\sin \left(Wn\right),\;\;\;\;0 ${\displaystyle X(e^{j\omega })={\begin{cases}1,&|\omega |\leq W\\0,&W<|\omega |\leq \pi \end{cases}}}$ ${\displaystyle X(e^{j\omega })}$ is 2π periodic
${\displaystyle (n+1)\alpha ^{n}u\left[n\right]}$ ${\displaystyle {\frac {1}{(1-\alpha e^{-j\omega })^{2}}}}$