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Time Domain Signal  Frequency Domain Signal  Condition 

$e^{(at)}u(t)$  $\frac{1}{a+i\omega}$  $a> 0$ 
$e^{at}u(t)$  $\frac{1}{ai\omega}$  $a> 0$ 
$e^{(a\leftt\right)}$  $\frac{2a}{a^{2}+\omega ^{2}}$  $a> 0$ 
$te^{(at)}u(t)$  $\frac{1}{(a+i\omega )^{2}}$  $a> 0$ 
$t^{n}e^{(at)}u(t)$  $\frac{n!}{(a+i\omega )^{(n+1)}}$  $a> 0$ 
$\delta (t)$  $1$  
$1$  $2\pi \delta (\omega )$  
$e^{i{\omega}_{0}t}$  $2\pi \delta (\omega {\omega}_{0})$  
$\cos ({\omega}_{0}t)$  $\pi (\delta (\omega {\omega}_{0})+\delta (\omega +{\omega}_{0}))$  
$\sin ({\omega}_{0}t)$  $i\pi (\delta (\omega +{\omega}_{0})\delta (\omega {\omega}_{0}))$  
$u(t)$  $\pi \delta (\omega )+\frac{1}{i\omega}$  
$\mathrm{sgn}(t)$  $\frac{2}{i\omega}$  
$\cos ({\omega}_{0}t())u(t)$  $\frac{\pi}{2}(\delta (\omega {\omega}_{0})+\delta (\omega +{\omega}_{0}))+\frac{i\omega}{{\omega}_{0}^{2}\omega ^{2}}$  
$\sin ({\omega}_{0}t)u(t)$  $\frac{\pi}{2i}(\delta (\omega {\omega}_{0})\delta (\omega +{\omega}_{0}))+\frac{{\omega}_{0}}{{\omega}_{0}^{2}\omega ^{2}}$  
$e^{(at)}\sin ({\omega}_{0}t)u(t)$  $\frac{{\omega}_{0}}{(a+i\omega )^{2}+{\omega}_{0}^{2}}$  $a> 0$ 
$e^{(at)}\cos ({\omega}_{0}t)u(t)$  $\frac{a+i\omega}{(a+i\omega )^{2}+{\omega}_{0}^{2}}$  $a> 0$ 
$u(t+\tau )u(t\tau )$  $2\tau \frac{\sin (\omega \tau )}{\omega \tau}=2\tau \mathrm{sinc}(\omega t)$  
$\frac{{\omega}_{0}}{\pi}\frac{\sin ({\omega}_{0}t)}{{\omega}_{0}t}=\frac{{\omega}_{0}}{\pi}\mathrm{sinc}({\omega}_{0})$  $u(\omega +{\omega}_{0})u(\omega {\omega}_{0})$  
$(\frac{t}{\tau}+1)(u(\frac{t}{\tau}+1)u(\frac{t}{\tau}))+(\left(\frac{t}{\tau}\right)+1)(u(\frac{t}{\tau})u(\frac{t}{\tau}1))=\mathrm{triag}(\frac{t}{2\tau})$  $\tau \mathrm{sinc}(\frac{\omega \tau}{2})^{2}$  
$\frac{{\omega}_{0}}{2\pi}\mathrm{sinc}(\frac{{\omega}_{0}t}{2})^{2}$  $(\frac{\omega}{{\omega}_{0}}+1)(u(\frac{\omega}{{\omega}_{0}}+1)u(\frac{\omega}{{\omega}_{0}}))+(\left(\frac{\omega}{{\omega}_{0}}\right)+1)(u(\frac{\omega}{{\omega}_{0}})u(\frac{\omega}{{\omega}_{0}}1))=\mathrm{triag}(\frac{\omega}{2{\omega}_{0}})$  
$\sum_{n=()} $∞

${\omega}_{0}\sum_{n=()} $∞

${\omega}_{0}=\frac{2\pi}{T}$ 
$e^{\left(\frac{t^{2}}{2\sigma ^{2}}\right)}$  $\sigma \sqrt{2\pi}e^{\left(\frac{\sigma ^{2}\omega ^{2}}{2()}\right)}$ 
triag[n] is the triangle function for arbitrary realvalued $n$ .
$$\text{triag[n]}=\begin{array}{c}\left\{\begin{array}{cc}1+n\hfill & \text{if}1\le n\le 0\hfill \\ 1n\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}0<n\le 1\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}\right)\hfill \end{array}$$
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