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Common Fourier Transforms

Module by: Melissa Selik, Richard Baraniuk. E-mail the authors

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

Common CTFT Properties

Table 1
Time Domain Signal Frequency Domain Signal Condition
e(at)ut a t u t 1a+iω 1 a ω a>0 a 0
eatut a t u t 1aiω 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(at)ut t a t u t 1a+iω2 1 a ω 2 a>0 a 0
tne(at)ut 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
ei ω0 t ω0 t 2πδω ω0 2 δ ω ω0
cos ω0 t ω0 t π(δω ω0 +δω+ ω0 ) δ ω ω0 δ ω ω0
sin ω0 t ω0 t iπ(δω+ ω0 δω ω0 ) δ ω ω0 δ ω ω0
ut u t πδω+1iω δ ω 1 ω
sgnt sgn t 2iω 2 ω
cos ω0 tut ω0 t u t π2(δω ω0 +δω+ ω0 )+iω ω0 2ω2 2 δ ω ω0 δ ω ω0 ω ω0 2 ω 2
sin ω0 tut ω0 t u t π2i(δω ω0 δω+ ω0 )+ ω0 ω0 2ω2 2 δ ω ω0 δ ω ω0 ω0 ω0 2 ω 2
e(at)sin ω0 tut a t ω0 t u t ω0 a+iω2+ ω0 2 ω0 a ω 2 ω0 2 a>0 a 0
e(at)cos ω0 tut a t ω0 t u t a+iωa+iω2+ ω0 2 a ω a ω 2 ω0 2 a>0 a 0
ut+τutτ u t τ u t τ 2τsinωτωτ=2τsincωt 2 τ ω τ ω τ 2 τ sinc ω t
ω0 πsin ω0 t ω0 t= ω0 πsinc ω0 ω0 ω0 t ω0 t ω0 sinc ω0 uω+ ω0 uω ω0 u ω ω0 u ω ω0
(tτ+1)(utτ+1utτ)+(tτ+1)(utτutτ1)=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
ω0 2πsinc2 ω0 t2 ω0 2 sinc ω0 t 2 2 (ω ω0 +1)(uω ω0 +1uω ω0 )+(ω ω0 +1)(uω ω0 uω ω0 1)=triagω2 ω0 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 =δtnT n δ t n T ω0 n =δωn ω0 ω0 n δ ω n ω0 ω0 =2πT ω0 2 T
et22σ2 t 2 2 σ 2 σ2πeσ2ω22 σ 2 σ 2 ω 2 2

triag[n] is the triangle function for arbitrary real-valued nn.

triag[n] = 1 + n if - 1 n 0 1 - n if 0 < n 1 0 otherwise triag[n] = 1 + n if - 1 n 0 1 - n if 0 < n 1 0 otherwise

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