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8.2 Common fourier transforms

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

Common ctft properties

Time Domain Signal Frequency Domain Signal Condition
a t u t 1 a ω a 0
a t u t 1 a ω a 0
a t 2 a a 2 ω 2 a 0
t a t u t 1 a ω 2 a 0
t n a t u t n a ω n 1 a 0
δ t 1
1 2 δ ω
ω 0 t 2 δ ω ω 0
ω 0 t δ ω ω 0 δ ω ω 0
ω 0 t δ ω ω 0 δ ω ω 0
u t δ ω 1 ω
sgn t 2 ω
ω 0 t u t 2 δ ω ω 0 δ ω ω 0 ω ω 0 2 ω 2
ω 0 t u t 2 δ ω ω 0 δ ω ω 0 ω 0 ω 0 2 ω 2
a t ω 0 t u t ω 0 a ω 2 ω 0 2 a 0
a t ω 0 t u t a ω a ω 2 ω 0 2 a 0
u t τ u t τ 2 τ ω τ ω τ 2 τ sinc ω t
ω 0 ω 0 t ω 0 t ω 0 sinc ω 0 u ω ω 0 u ω ω 0
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 sinc ω τ 2 2
ω 0 2 sinc ω 0 t 2 2 ω ω 0 1 u ω ω 0 1 u ω ω 0 ω ω 0 1 u ω ω 0 u ω ω 0 1 triag ω 2 ω 0
n ∞ ∞ δ t n T ω 0 n ∞ ∞ δ ω n ω 0 ω 0 2 T
t 2 2 σ 2 σ 2 σ 2 ω 2 2

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

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

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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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