Fourier Transform Table Time Signal Fourier Transform

∞<<∞− t,1 )(2 ωπδ )(5.0 tu+− ωj/1

)(tu ωωπδ j/1)( + )(tδ ∞<<∞− ω,1

real ),( cct −δ real , ce cjω− 0),( >− btue bt 0,1

>+

bbjω

, realoj toe ω ω real ),(2 oo ωωωπδ −

)(tpτ [ ]πτωτ 2/sinc [ ]πττ 2/sinc t )(2 ωπ τp

[ ] )(1 2 tptττ− [ ]πωττ 4/sinc2

2

[ ]πττ 4/sinc22 t [ ] )(12 2 ωπ ττ

ω p− )cos( toω [ ])()( oo ωωδωωδπ −++

)cos( θω +to [ ])()( oj

oj ee ωωδωωδπ θθ −++−

)sin( toω [ ])()( ooj ωωδωωδπ −−+ )sin( θω +to ( ) ( )j j

o oj e eθ θπ δ ω ω δ ω ω−⎡ ⎤+ − −⎣ ⎦

Fourier Transform Properties Property Name Property Linearity )()( tbvtax + )()( ωω bVaX + Time Shift )( ctx − )(ωω Xe cj− Time Scaling 0),( ≠aatx 0),/(1 ≠aaXa ω Time Reversal )( tx −

real is )( if )(

)(

txX

X

ω

ω−

Multiply by tn …,3,2,1),( =ntxtn …,3,2,1),( =nXddj n

nn ω

ω

Multiply by Complex Exponential real ),( otj txe o ωω real ),( ooX ωωω −

Multiply by Sine )()sin( txtoω [ ])()(2 oo XXj ωωωω −−+

Multiply by Cosine )()cos( txtoω [ ])()(21

oo XX ωωωω −++

Time Differentiation …,3,2,1),( =ntx

dtd

n

n

…,3,2,1),()( =nXj n ωω

Time Integration ∫∞−

t

dx λλ )( )()0()(1 ωδπωω

XXj

+

Convolution in Time )(*)( thtx )()( ωω HX Multiplication in Time )()( twtx )(*)(

21 ωωπ

WX

Parseval’s Theorem (General) ∫∫∞

∞−

∞

∞−

= ωωωπ

dVXdttvtx )()(21)()(

Parseval’s Theorem (Energy)

∫∫

∫∫∞

∞−

∞

∞−

∞

∞−

∞

∞−

=

=

ωωπ

ωωπ

dXdttx

txdXdttx

22

22

)(21)(

real is )( if)(21)(

Duality: If x(t) ↔ X(ω) )(tX )(2 ωπ −x