Fourier Transform Table - astro.umd.edulgm/ASTR410/ft_ref3.pdf · Fourier Transform Table x()t X(f)...
1
Fourier Transform Table () x t () X f ( ) X ω () t δ 1 1 1 () f δ 2 ( ) πδω 0 ( ) t t δ − 0 2 j ft e π − 0 j t e ω − 0 2 j ft e π 0 ( ) f f δ − 0 2 ( ) πδω ω − 0 cos(2 ) ft π 0 0 1 ( ) ( 2 f f f f δ δ − + + ) 0 0 ( ) ( π δω ω δω ω − + + ) 0 sin(2 ) ft π 0 0 1 ( ) ( 2 f f f f j δ δ − − + ) 0 0 ( ) ( jπ δω ω δω ω − − − + ) () rect t sin ( ) cf sin 2 c ω π sin ( ) ct () rect f 2 rect ω π () t Λ 2 sin () c f 2 sin 2 c ω π 2 sin () c t () f Λ 2 ω π Λ ( ), 0 t e ut α α − > 1 2 j f α π + 1 j α ω + ( ), 0 t te ut α α − > ( ) 2 1 2 j f α π + ( ) 2 1 j α ω + || , 0 t e α α − > 2 2 2 ( (2 ) f α α π + 2 2 2 ( ( ) α α ω + 2 t e π − 2 f e α − 2 f e α − sgn( ) t 1 j f π 2 jω () ut 1 1 () 2 2 f j f δ π + 1 ( ) j πδ ω ω + () d t dt δ 2 j f π jω 0 ( ) n t nT δ ∞ =−∞ − ∑ 0 0 1 n n f T T δ ∞ =−∞ − ∑ 0 0 1 2 n n T T π δ ω ∞ =−∞ − ∑
Transcript of Fourier Transform Table - astro.umd.edulgm/ASTR410/ft_ref3.pdf · Fourier Transform Table x()t X(f)...
Fourier Transform Table ( )x t ( )X f ( )X ω ( )tδ 1 1 1 ( )fδ 2 ( )πδ ω
0( )t tδ − 02j fte π− 0j te ω− 02j f te π 0( )f fδ − 02 ( )πδ ω ω−
0cos(2 )f tπ 0 0
1 ( ) (2
f f f fδ δ− + + ) 0 0( ) (π δ ω ω δ ω ω− + + )
0sin(2 )f tπ 0 0
1 ( ) (2
f f f fjδ δ− − + ) 0 0( ) (jπ δ ω ω δ ω ω− − − + )
( )rect t sin ( )c f sin
2c ω
π
sin ( )c t ( )rect f 2
rect ωπ
( )tΛ 2sin ( )c f 2sin2
c ωπ
2sin ( )c t ( )fΛ 2ωπ
Λ
( ), 0te u tα α− > 12j fα π+
1jα ω+
( ), 0tte u tα α− > ( )2
1
2j fα π+
( )21
jα ω+
| |, 0te α α− > 2 22
( (2 )fα
α π+ 2 2
2( ( )
αα ω+
2te π− 2fe α−
2fe α− sgn( )t 1
j fπ 2
jω
( )u t 1 1( )2 2
fj f
δπ
+ 1( )j
πδ ωω
+
( )d tdtδ 2j fπ jω
0( )n
t nTδ∞
=−∞
−∑ 0 0
1
n
nfT T
δ∞
=−∞
−
∑
0 0
1 2
n
nT T
πδ ω∞
=−∞
−
∑
![ENSC380 Lecture 28 Objectives: z-TransformUnilateral z-Transform • Analogous to unilateral Laplace transform, the unilateral z-transform is defined as: X(z) = X∞ n=0 x[n]z−n](https://static.fdocument.org/doc/165x107/61274ac3cd707f40c43ddb9a/ensc380-lecture-28-objectives-z-unilateral-z-transform-a-analogous-to-unilateral.jpg)
![Sparse Fourier Transform (lecture 3)people.csail.mit.edu/kapralov/madalgo15/lec3.pdf · Given x 2Cn, compute the Discrete Fourier Transform of x: bxi ˘ X j2[n] xj! ij, where!˘e2…i/n](https://static.fdocument.org/doc/165x107/5fd24444a61a7b54eb23d197/sparse-fourier-transform-lecture-3-given-x-2cn-compute-the-discrete-fourier-transform.jpg)
