Common Fourier Transform Pairs f (t), a> 0,τ> 0 F (ω) e -at u(t) 1 a + jω te -at u(t) 1 a + jω 2 p τ (t)= ( 1, - τ 2 ≤ t< τ 2 , 0, else τ sinc τω 2π Δ τ (t)= ( 1 - 2|t| τ , - τ 2 ≤ t< τ 2 0, else τ 2 sinc 2 τω 4π e -a|t| 2a a 2 + ω 2 e -at sin (ω 0 t) u(t) ω 0 (a + jω) 2 + ω 2 0 e -at cos (ω 0 t) u(t) a + jω (a + jω) 2 + ω 2 0 e -at 2 r π a e -ω 2 /4a sinc τt 2π 2π τ p τ (ω) sinc 2 τt 4π 4π τ Δ τ (ω) 1 a 2 + t 2 π a e -a|ω| δ(t) 1 1 2πδ(ω) u(t) πδ(ω)+ 1 jω e jω 0 t 2πδ(ω - ω 0 ) cos (ω o t) π [δ(ω + ω 0 )+ δ(ω - ω 0 )] sin (ω o t) jπ [δ(ω + ω 0 ) - δ(ω - ω 0 )] sgn(t)= ( t |t| , t 6=0, 0, t =0 2 jω cos(ω 0 t)u(t) π 2 [δ(ω - ω 0 )+ δ(ω + ω 0 )] + jω ω 2 0 - ω 2 sin(ω 0 t)u(t) π j 2 [δ(ω - ω 0 ) - δ(ω + ω 0 )] + ω 0 ω 2 0 - ω 2
