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