Fourier Transform Table - ws.binghamton.edu personal page/EE301_files/DTF… · DTFT Properties...

2
DTFT Table Time Signal DTFT < < n , 1 −∞ = Ω k k ) 2 ( 2 π δ π = " , 2 , 1 , 0 , 1 1 , 2 , 3 ..., , 1 ] sgn[n Ω j e 1 2 ] [n u −∞ = Ω Ω + k j k e ) 2 ( 1 1 π δ π ] [n δ < Ω < , 1 , 3 , 2 , 1 ], [ ± ± ± = q q n δ , 3 , 2 , 1 , ± ± ± = Ω q e jq 1 | | ], [ < a n u a n 1 | | , 1 1 < Ω a ae j , re o j n o e Ω Ω al real , ) 2 ( 2 o k o k Ω Ω Ω −∞ = π δ π + = = otherwise q q q n n p q , 0 , 1 , 0 , 1 , , 1 , , 1 ] [ ( ) [ ] ( ) 2 / sin sin 2 1 Ω Ω + q [ ] n B B π π sinc −∞ = + Ω k B k p ) 2 ( 2 π cos( ) o n Ω [ ] −∞ = Ω Ω + Ω + Ω k o o k k ) 2 ( ) 2 ( π δ π δ π cos( ) o n θ Ω + [ ] −∞ = Ω Ω + Ω + Ω k o j o j k e k e ) 2 ( ) 2 ( π δ π δ π θ θ sin( ) o n Ω [ ] −∞ = Ω Ω Ω + Ω k o o k k j ) 2 ( ) 2 ( π δ π δ π sin( ) o n θ Ω + [ ] −∞ = Ω Ω Ω + Ω k o j o j k e k e j ) 2 ( ) 2 ( π δ π δ π θ θ

Transcript of Fourier Transform Table - ws.binghamton.edu personal page/EE301_files/DTF… · DTFT Properties...

Page 1: Fourier Transform Table - ws.binghamton.edu personal page/EE301_files/DTF… · DTFT Properties Property Name Property Linearity + ax n bv n [ ] [ ] Ω +aX bV Ω( ) ( ) Time Shift

DTFT Table Time Signal DTFT

∞<<∞− n,1 ∑∞

−∞=

−Ωk

k)2(2 πδπ

⎪⎩

⎪⎨⎧ −−−−

=,2,1,0,1

1,2,3...,,1]sgn[n Ω−− je1

2

][nu ∑∞

−∞=Ω− −Ω+

− kj k

e)2(

11 πδπ

][nδ ∞<Ω<∞−,1 …,3,2,1],[ ±±±=− qqnδ …,3,2,1, ±±±=Ω− qe jq

1||],[ <anuan 1||,1

1<

− Ω− aae j

, reoj noe Ω Ω al

real ,)2(2 ok

o k Ω−Ω−Ω∑∞

−∞=

πδπ

⎪⎪⎩

⎪⎪⎨

+−−=

=

otherwise

q

qqn

npq

,0

,1,0,1,

,1,,1

][ …

… ( )[ ]( )2/sin

sin 21

ΩΩ+q

[ ]nBBππ sinc ∑

−∞=

+Ωk

B kp )2(2 π

cos( )onΩ [ ]∑∞

−∞=

−Ω−Ω+−Ω+Ωk

oo kk )2()2( πδπδπ

cos( )on θΩ + [ ]∑∞

−∞=

− −Ω−Ω+−Ω+Ωk

oj

oj keke )2()2( πδπδπ θθ

sin( )onΩ [ ]∑∞

−∞=

−Ω−Ω−−Ω+Ωk

oo kkj )2()2( πδπδπ

sin( )on θΩ + [ ]∑∞

−∞=

− −Ω−Ω−−Ω+Ωk

oj

oj kekej )2()2( πδπδπ θθ

Page 2: Fourier Transform Table - ws.binghamton.edu personal page/EE301_files/DTF… · DTFT Properties Property Name Property Linearity + ax n bv n [ ] [ ] Ω +aX bV Ω( ) ( ) Time Shift

DTFT Properties Property Name Property Linearity ][][ nbvnax + )()( Ω+Ω bVaX Time Shift

integerany

],[

q

qnx −

integerany ),( qXe jq ΩΩ−

Time Scaling 0),( ≠aatx 0),/(1 ≠Ω aaXa Time Reversal ][ nx −

real is ][ if )(

)(

nxX

X

Ω

Ω−

Multiply by n ][nnx )(Ω

ΩX

ddj

Multiply by Complex Exponential real ],[ onj nxe o ΩΩ

real ),( ooX ΩΩ−Ω

Multiply by Sine ][)sin( nxnoΩ [ ])()(2 oo XXj

Ω−Ω−Ω+Ω

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

oo XX Ω−Ω+Ω+Ω

Summation ∑−∞=

n

i

ix ][ ∑∞

−∞=Ω− −Ω+Ω

− kj kXX

e)2()0()(

11 πδπ

Convolution in Time ][*][ nhnx )()( ΩΩ HX Multiplication in Time ][][ nwnx

∫−

−Ωπ

π

λλλπ

dWX )()(21 (conv.)

Parseval’s Theorem (General) ∫∑−

−∞=

ΩΩΩ=π

ππdVXnvnx

n

)()(21][][

Parseval’s Theorem (Energy)

∫∑

∫∑

−∞=

−∞=

ΩΩ=

ΩΩ=

π

π

π

π

π

π

dXnx

txdXnx

n

n

22

22

)(21|][|

real is )( if)(21][

Using CTFT Table to find Inverse of a DTFT X(Ω): x[n] = ??

Form and look up )()()( 2 ωωω πpX=Γ )()( ωγ Γ↔tThen get

nttnx

== )(][ γ