View source for ECE 301 Fall 2007 mboutin Fourier Transform Table - Rhea


{|
! Time Domain !! Fourier Domain
|-
| <math> x(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(j \omega)e^{j \omega t}d \omega </math>
| <math> X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t </math>
|-
| <math> 1\ </math> || <math> 2 \pi \delta (\omega) </math>
|-
| <math> − 0.5 + u(t)\ </math> || <math> \frac{1}{j \omega}\ </math>
|-
| <math> \delta (t) \ </math> || <math> 1\ </math>
|-
| <math> \delta (t-c)\ </math> || <math> e − j \omega c </math>
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| <math> u(t) </math> || <math> \pi \delta(\omega)+\frac{1}{j \omega} </math>
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| <math> e ^{− bt}u(t) </math> || <math> \frac{1}{j \omega + b} </math>
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| <math> cos \omega_0 t </math> || <math> \pi [\delta ( \omega + \omega_0 ) + \delta ( \omega - \omega_0 )] </math>
|-
|cos(ω0t + θ) 	π[e − jθδ(ω + ω0) + ejθδ(ω − ω0)]?
|-
|sinω0t 	jπ[δ(ω + ω0) − δ(ω − ω0)]?
|-
|sin(ω0t + θ) 	jπ[e − jθδ(ω + ω0) − ejθδ(ω − ω0)]?
|-
|rect(\frac{t}{\tau}) 	\tau sinc \frac{\tau \omega}{2 \pi}
|-
|\tau sinc \frac{\tau t}{2 \pi} 	2πpτ(ω)
|-
(1-\frac{2 |t|}{\tau})p_\tau (t) 	\frac{\tau}{2} sinc^2 \frac{\tau \omega}{4 \pi}
|-
\frac{\tau}{2} sinc^2 ( \frac{\tau t}{4 \pi} ) 	2 \pi (1-\frac{2|\omega|}{\tau})p_\tau (\omega)
|}
    Note: sinc(x) = sin(x) / x ; pτ(t) is the rectangular pulse function of width τ

Note: Source courtesy Wikibooks.org