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FT Table

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Fourier Transform Table
x(t )
X (ω )
X (f )
δ (t )
1
δ (t − t0 )
e j 2π f0t
cos(2π f0t )
sin(2π f0t )
1
1
δ (f )
2πδ (ω )
e − j 2π ft0
δ (f − f0 )
1
δ (f − f0 ) + δ (f + f0 )
2
1
δ (f − f0 ) − δ (f + f0 )
2j 
e − jω t0
2πδ (ω − ω0 )
π δ (ω − ω0 ) + δ (ω + ω0 )
− jπ δ (ω − ω0 ) − δ (ω + ω0 )
 ω 
sin c 

 2π 
rect (t )
sin c(f )
sin c(t )
rect (f )
 ω 
rect 

 2π 
sin c 2 (f )
 ω 
sin c 2 

 2π 
Λ (f )
 ω 
Λ

 2π 
1
α + jω
Λ(t )
sin c 2 (t )
e −α t u(t ), α > 0
1
α + j 2π f
te −α t u(t ), α > 0
1
1
(α + j 2π f )
e −α |t |, α > 0
2
2α
2α
2
(α + (2π f )
e −π t
sgn(t )
e −α f
1
jπ f
2
u(t )
∑
n =−∞
2
2
(α + (ω )2
e −α f
2
jω
2
1
1
δ (f ) +
j 2π f
2
δ (t − nT0 )
1
T0
∞
∑
n =−∞

δ f −

2
πδ (ω ) +
j 2π f
d
δ (t )
dt
∞
(α + jω )2
1
jω
jω
n 

T0 
1
T0
∞

∑ δ  ω −
n =−∞
2π n 

T0 
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