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Text: Brief Table of Fourier Transforms

Description Function Transform

Delta function in X Î´(x)

Delta function in k 2Ï€Î´(k)

Exponential in x e^(-Î±|x|) 2Î±/(Î±^2 + k^2)

Exponential in k 2Ï€Î´(2Ï€k)

Gaussian (1/âˆš2Ï€)e^(-k^2/2)

Derivative in X f'(x) ikF(k)

Derivative in k xF(x) iF'(k)

Integral in x âˆ«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

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Fourier Transform Pairs Table a > 0 jÏ‰e^(-at) Î´(-t)Î´(t)5j6e^(-aÏ‰)a/(a^2 + Ï‰^2)e^(-at) Î´(t)Î´(t)e^(-aÏ‰)(Î´(Ï‰) + jÏ‰/Ï‰^2)2Ï€Î´(Ï‰) + u(Ï‰ + Ï‰0)cos(Ï‰0t)sin(Ï‰0t)u(t)sgn(t)cos(Ï‰0t) u(t)(2Ï€Î´(Ï‰0) + 2Ï€Î´(-Ï‰0)) – j(Ï‰ + Ï‰0)u(Ï‰ + Ï‰0)sin(Ï‰0t)u(t)e^(-at) sin(Ï‰0t) u(t)a/(a^2 + Ï‰^2)(a + jÏ‰) + u(Ï‰)4Ï€^2 sinc^2(Ï‰) rect(Ï‰)at cos(Ï‰0t) u(t)rect(t) sinc(Ï‰t)4(F) sinc^2(Ï‰t)6-Ï€nT) n=0{Ï‰nÏ‰0) n=0LÎ´ = 41/(2Ï€)jÏ‰/(jÏ‰)^2e^jÏ‰t

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Complete Table of Fourier Transform Pairs

Function (t) Fourier Transform (F(Ï‰)) Definition of Fourier Transform Definition of Inverse Fourier Transform f(t) F(Ï‰) = âˆ«f(t)e^(-jÏ‰t)dt F(Ï‰) = âˆ«F(Ï‰)e^(jÏ‰t)dÏ‰ f(t – To) F(Ï‰)e^(-jÏ‰To) f(at) (1/|a|)F(Ï‰/a) f'(t) jÏ‰F(Ï‰) F(t) 2Ï€Î´(Ï‰) sgn(t) (2/jÏ€)Ï‰ cos(Ï‰t) Ï€[Î´(Ï‰ – Ï‰0) + Î´(Ï‰ + Ï‰0)] sin(Ï‰t) -jÏ€[Î´(Ï‰ – Ï‰0) – Î´(Ï‰ + Ï‰0)] e^(jÏ‰0t) 2Ï€Î´(Ï‰ – Ï‰0)

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