View source for CT Fourier Transform (frequency in hertz) - Rhea

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! style="background: none repeat scroll 0% 0% rgb(228, 188, 126); font-size: 110%;" colspan="2" | CT Fourier Transform Pairs and Properties (frequency <span class="texhtml">f</span> in hertz) [[More on CT Fourier transform|(info)]]
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Definition CT Fourier Transform and its Inverse
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| align="right" style="padding-right: 1em;" |  CT Fourier Transform 
| <math>X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt</math>
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| align="right" style="padding-right: 1em;" | Inverse DT Fourier Transform 
| <math>\, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \,</math>
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Pairs [[More_on_CTFTs| (info)]]
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| align="right" style="padding-right: 1em;" | 
| <math> x(t) \  </math> 
| <math>\longrightarrow</math>
| <math> X(f) \  </math>
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| align="right" style="padding-right: 1em;" | CTFT of a unit impulse 
| <math>\delta (t)\ </math> 
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| <math> 1 \ </math>
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| align="right" style="padding-right: 1em;" | CTFT of a shifted unit impulse 
| <math>\delta (t-t_0)\ </math> 
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| <math>e^{-i2\pi ft_0}</math>
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| align="right" style="padding-right: 1em;" | CTFT of a complex exponential 
| <math>e^{iw_0t}</math> 
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| <math> \delta (f - \frac{\omega_0}{2\pi}) \ </math> 
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| align="right" style="padding-right: 1em;" | 
| <math>e^{-at}u(t), \  \text{ where } a\in {\mathbb R}, a>0 </math> 
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| <math>\frac{1}{a+i2\pi f}</math> 
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| align="right" style="padding-right: 1em;" | 
| <math>te^{-at}u(t), \  \text{ where } a\in {\mathbb R}, a>0 </math> 
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| <math>\left( \frac{1}{a+i2\pi f}\right)^2</math> 
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| align="right" style="padding-right: 1em;" | CTFT of a cosine 
| <math>\cos(\omega_0 t) \ </math> 
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| <math> \frac{1}{2} \left[\delta (f - \frac{\omega_0}{2\pi}) + \delta (f + \frac{\omega_0}{2\pi})\right] \ </math> 
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| align="right" style="padding-right: 1em;" | CTFT of a sine 
| <math>sin(\omega_0 t)  \ </math> 
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| <math>\frac{1}{2i} \left[\delta (f - \frac{\omega_0}{2\pi}) - \delta (f + \frac{\omega_0}{2\pi})\right]</math> 
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| align="right" style="padding-right: 1em;" | CTFT of a rect 
| <math>\left\{\begin{array}{ll}1, &  \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \ </math> 
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| <math> \frac{\sin \left(2\pi Tf \right)}{\pi f}  \ </math> 
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| align="right" style="padding-right: 1em;" | CTFT of a sinc 
| <math>\frac{ \sin \left( W t  \right)}{\pi t }  \ </math> 
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| <math>\left\{\begin{array}{ll}1, &  \text{ if }|f| <\frac{W}{2\pi},\\ 0, & \text{else.}\end{array} \right.  \ </math> 
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| align="right" style="padding-right: 1em;" | CTFT of a periodic function 
| <math>\sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t}</math> 
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| <math>\sum^{\infty}_{k=-\infty}a_{k}\delta(f-\frac{kw_{0}}{2\pi}) \ </math> 
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| align="right" style="padding-right: 1em;" | CTFT of an impulse train 
| <math>\sum^{\infty}_{n=-\infty} \delta(t-nT)  \ </math> 
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| <math>\frac{1}{T}\sum^{\infty}_{k=-\infty}\delta(f-\frac{k}{T}) \ </math> 
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Properties
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| align="right" style="padding-right: 1em;" | 
| <math> x(t) \  </math> 
| <math>\longrightarrow</math>
| <math> X(f) \  </math>
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| align="right" style="padding-right: 1em;" | multiplication property
| <math>x(t)y(t) \ </math> 
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| <math> X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta</math>
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| align="right" style="padding-right: 1em;" | convolution property 
| <math>x(t)*y(t) \  </math> 
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| <math> X(f)Y(f) \  </math>
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| align="right" style="padding-right: 1em;" | time reversal 
| <math>\ x(-t) </math> 
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| <math>\ X(-f)</math>
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Other CT Fourier Transform Properties
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| align="right" style="padding-right: 1em;" | Parseval's relation 
| <math>\int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df</math>
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