) 2α/(α^2 + k^2) Exponential in k 2πδ(2Ï€k) Gaussian (1/√2Ï€)e^(-k^2/2) Derivative in X f'(x) ikF(k) Derivative in k xF(x) iF'(k) Integral in x ∫f(x’)dx’ F(k)/(ik) Translation in x f(x – a) e^

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Text: Brief Table of Fourier Transforms
Description Function Transform
Delta function in X δ(x)
Delta function in k 2πδ(k)
Exponential in x e^(-α|x|) 2α/(α^2 + k^2)
Exponential in k 2πδ(2πk)
Gaussian (1/√2π)e^(-k^2/2)
Derivative in X f'(x) ikF(k)
Derivative in k xF(x) iF'(k)
Integral in x ∫f(x’)dx’ F(k)/(ik)
Translation in x f(x – a) e^(-ika)F(k)
Translation in k e^(iax)f(x) F(k – a)
Dilation in X f(ax) F(k/a)/a
Convolution f(x)*g(x) F(k)G(k)

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04:52

dip

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Fourier Transform Pairs Table a > 0 jωe^(-at) δ(-t)δ(t)5j6e^(-aω)a/(a^2 + ω^2)e^(-at) δ(t)δ(t)e^(-aω)(δ(ω) + jω/ω^2)2πδ(ω) + u(ω + ω0)cos(ω0t)sin(ω0t)u(t)sgn(t)cos(ω0t) u(t)(2πδ(ω0) + 2πδ(-ω0)) – j(ω + ω0)u(ω + ω0)sin(ω0t)u(t)e^(-at) sin(ω0t) u(t)a/(a^2 + ω^2)(a + jω) + u(ω)4Ï€^2 sinc^2(ω) rect(ω)at cos(ω0t) u(t)rect(t) sinc(ωt)4(F) sinc^2(ωt)6-Ï€nT) n=0{ωnω0) n=0Lδ = 41/(2Ï€)jω/(jω)^2e^jωt

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Complete Table of Fourier Transform Pairs

Function (t) Fourier Transform (F(ω)) Definition of Fourier Transform Definition of Inverse Fourier Transform f(t) F(ω) = ∫f(t)e^(-jωt)dt F(ω) = ∫F(ω)e^(jωt)dω f(t – To) F(ω)e^(-jωTo) f(at) (1/|a|)F(ω/a) f'(t) jωF(ω) F(t) 2πδ(ω) sgn(t) (2/jÏ€)ω cos(ωt) Ï€[δ(ω – ω0) + δ(ω + ω0)] sin(ωt) -jÏ€[δ(ω – ω0) – δ(ω + ω0)] e^(jω0t) 2πδ(ω – ω0)

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