Discrete-time Fourier Transform Pairs and Properties
DT Fourier transform and its Inverse
DT Fourier Transform $\,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\,$
Inverse DT Fourier Transform $\,x[n]=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{0}^{2\pi}\mathcal{X}(\omega)e^{j\omega n} d \omega\,$
DT Fourier Transform Pairs
$x[n] \$ $\longrightarrow$ $\mathcal{X}(\omega) \$
DTFT of a complex exponential $e^{jw_0n} \$ $\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \$
(info) DTFT of a rectangular window $w[n]= \$ add formula here
$a^{n} u[n], |a|<1 \$ $\frac{1}{1-ae^{-j\omega}} \$
$\sin\left(\omega _0 n\right) u[n] \$ $\frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right)$
DT Fourier Transform Properties
$x[n] \$ $\longrightarrow$ $\mathcal{X}(\omega) \$
multiplication property $x[n]y[n] \$ $\frac{1}{2\pi} \int_{2\pi} X(\theta)Y(\omega-\theta)d\theta$
convolution property $x[n]*y[n] \!$ $X(\omega)Y(\omega) \!$
time reversal $\ x[-n]$ $\ X(-\omega)$
Other DT Fourier Transform Properties
Parseval's relation $\frac {1}{N} \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 =$

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