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Table of Common Fourier Transforms

Module by: Melissa Selik, Richard Baraniuk

Summary: Lists time domain signal, frequency domain signal, and condition for twentytwo Fourier transforms.

Time Domain Signal Frequency Domain Signal Condition
ⅇ-atut a t u t 1a+ⅈω 1 a ω a>0 a 0
ⅇatu-t a t u t 1a-ⅈω 1 a ω a>0 a 0
ⅇ-a|t| a t 2aa2+ω2 2 a a 2 ω 2 a>0 a 0
tⅇ-atut t a t u t 1a+ⅈω2 1 a ω 2 a>0 a 0
tnⅇ-atut t n a t u t n!a+ⅈ ω n+1 n a ω n 1 a>0 a 0
δt δ t 1 1
1 1 2πδω 2 δ ω
ⅇⅈω0t ω0 t 2πδω-ω0 2 δ ω ω0
cosω0t ω0 t πδω-ω0+δω+ω0 δ ω ω0 δ ω ω0
sinω0t ω0 t ⅈπδω+ω0-δω-ω0 δ ω ω0 δ ω ω0
ut u t πδω+1ⅈω δ ω 1 ω
sgnt sgn t 2ⅈω 2 ω
cosω0tut ω0 t u t π2δω-ω0+δω+ω0+ⅈωω02-ω2 2 δ ω ω0 δ ω ω0 ω ω0 2 ω 2
sinω0tut ω0 t u t π2ⅈδω-ω0-δω+ω0+ω0ω02-ω2 2 δ ω ω0 δ ω ω0 ω0 ω0 2 ω 2
ⅇ-atsinω0tut a t ω0 t u t ω0a+ⅈω2+ω02 ω0 a ω 2 ω0 2 a>0 a 0
ⅇ-atcosω0tut a t ω0 t u t a+ⅈωa+ⅈω2+ω02 a ω a ω 2 ω0 2 a>0 a 0
ut+τ-ut-τ u t τ u t τ 2τsinωτωτ=2τsincωt 2 τ ω τ ω τ 2 τ sinc ω t
ω0πsinω0tω0t=ω0πsincω0 ω0 ω0 t ω0 t ω0 sinc ω0 uω+ω0-uω-ω0 u ω ω0 u ω ω0
tτ+1utτ+1-utτ+-tτ+1utτ-utτ-1=triagt2τ t τ 1 u t τ 1 u t τ t τ 1 u t τ u t τ 1 triag t 2 τ τsinc2ωτ2 τ sinc ω τ 2 2
ω02πsinc2ω0t2 ω0 2 sinc ω0 t 2 2 ωω0+1uω ω0 +1-uω ω0 +-ω ω0 +1uω ω0 -uω ω0 -1=triagω2ω0 ω ω0 1 u ω ω0 1 u ω ω0 ω ω0 1 u ω ω0 u ω ω0 1 triag ω 2 ω0
∑n=-∞∞δt-nT n δ t n T ω0∑n=-∞∞δω-nω0 ω0 n δ ω n ω0 ω0=2πT ω0 2 T
ⅇ-t22σ2 t 2 2 σ 2 σ2πⅇ-σ2ω2 2 σ 2 σ 2 ω 2 2

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