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Math Table

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−(a⁢t)⁢u⁢t a t u t 1a+i⁢ω 1 a ω a>0 a 0
ea⁢t⁢u⁢−t a t u t 1a−i⁢ω 1 a ω a>0 a 0
e−(a⁢|t|) a t 2⁢aa2+ω2 2 a a 2 ω 2 a>0 a 0
t⁢e−(a⁢t)⁢u⁢t t a t u t 1a+i⁢ω2 1 a ω 2 a>0 a 0
tn⁢e−(a⁢t)⁢u⁢t t n a t u t n!a+i⁢ ω n+1 n a ω n 1 a>0 a 0
δ⁢t δ t 1 1
1 1 2⁢π⁢δ⁢ω 2 δ ω
ei⁢ ω0 ⁢t ω0 t 2⁢π⁢δ⁢ω− ω0 2 δ ω ω0
cos ω0 ⁢t ω0 t π⁢(δ⁢ω− ω0 +δ⁢ω+ ω0 ) δ ω ω0 δ ω ω0
sin ω0 ⁢t ω0 t i⁢π⁢(δ⁢ω+ ω0 −δ⁢ω− ω0 ) δ ω ω0 δ ω ω0
u⁢t u t π⁢δ⁢ω+1i⁢ω δ ω 1 ω
sgn⁢t sgn t 2i⁢ω 2 ω
cos ω0 ⁢t⁢u⁢t ω0 t u t π2⁢(δ⁢ω− ω0 +δ⁢ω+ ω0 )+i⁢ω ω0 2−ω2 2 δ ω ω0 δ ω ω0 ω ω0 2 ω 2
sin ω0 ⁢t⁢u⁢t ω0 t u t π2⁢i⁢(δ⁢ω− ω0 −δ⁢ω+ ω0 )+ ω0 ω0 2−ω2 2 δ ω ω0 δ ω ω0 ω0 ω0 2 ω 2
e−(a⁢t)⁢sin ω0 ⁢t⁢u⁢t a t ω0 t u t ω0 a+i⁢ω2+ ω0 2 ω0 a ω 2 ω0 2 a>0 a 0
e−(a⁢t)⁢cos ω0 ⁢t⁢u⁢t a t ω0 t u t a+i⁢ωa+i⁢ω2+ ω0 2 a ω a ω 2 ω0 2 a>0 a 0
u⁢t+τ−u⁢t−τ u t τ u t τ 2⁢τ⁢sinω⁢τω⁢τ=2⁢τ⁢sinc⁢ω⁢t 2 τ ω τ ω τ 2 τ sinc ω t
ω0 π⁢sin ω0 ⁢t ω0 ⁢t= ω0 π⁢sinc⁢ ω0 ω0 ω0 t ω0 t ω0 sinc ω0 u⁢ω+ ω0 −u⁢ω− ω0 u ω ω0 u ω ω0
(tτ+1)⁢(u⁢tτ+1−u⁢tτ)+(−tτ+1)⁢(u⁢tτ−u⁢tτ−1)=triag⁢t2⁢τ t τ 1 u t τ 1 u t τ t τ 1 u t τ u t τ 1 triag t 2 τ τ⁢sinc2ω⁢τ2 τ sinc ω τ 2 2
ω0 2⁢π⁢sinc2 ω0 ⁢t2 ω0 2 sinc ω0 t 2 2 (ω ω0 +1)⁢(u⁢ω ω0 +1−u⁢ω ω0 )+(−ω ω0 +1)⁢(u⁢ω ω0 −u⁢ω ω0 −1)=triag⁢ω2⁢ 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 =−∞∞δ⁢t−n⁢T n δ t n T ω0 ⁢∑ n =−∞∞δ⁢ω−n⁢ ω0 ω0 n δ ω n ω0 ω0 =2⁢πT ω0 2 T
e−t22⁢σ2 t 2 2 σ 2 σ⁢2⁢π⁢e−σ2⁢ω22 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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