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# Common CTFT Properties

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

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

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