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Actual functions and Fourier transforms in a table
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Signals & Systems - Reference Tables (^) 1
Function, f(t) (^) Fourier Transform, F( )
Definition of Inverse Fourier Transform
f t F e d
j t ( ) 2
Definition of Fourier Transform
F f te dt
j t ( ) ()
f ( t t 0 ) F ( ) e j t^0
j t f ( t ) e^0
f ( t ) ( )
F ( t ) 2 f ()
n
n
dt
d f ( t ) ( j ) F ()
n
( jt ) f ( t )
n n
n
d
d F
t f ( ) d (^0 ) ( )
j
( t ) 1
j t e^0
sgn (t)
j
Fourier Transform Table UBC M267 Resources for 2005
F (t) F̂ (ω) Notes (0)
f (t)
∫ (^) ∞
−∞
f (t)e−iωt^ dt Definition. (1)
1 2 π
∫ (^) ∞
−∞
̂ f (ω)eiωt^ dω f̂ (ω) Inversion formula. (2)
f̂ (−t) 2 πf (ω) Duality property. (3)
e−atu(t)
1 a + iω
a constant, <e(a) > 0 (4)
e−a|t|^
2 a a^2 + ω^2
a constant, <e(a) > 0 (5)
β(t) =
{ 1 , if |t| < 1, 0 , if |t| > 1
2 sinc(ω) = 2
sin(ω) ω
Boxcar in time. (6)
1 π
sinc(t) β(ω) Boxcar in frequency. (7)
f ′(t) iω f̂ (ω) Derivative in time. (8)
f ′′(t) (iω)^2 f̂ (ω) Higher derivatives similar. (9)
tf (t) i
d dω
̂ f (ω) Derivative in frequency. (10)
t^2 f (t) i^2
d^2 dω^2
̂ f^ (ω)^ Higher derivatives similar.^ (11)
eiω^0 tf (t) f̂ (ω − ω 0 ) Modulation property. (12)
f
( t − t 0 k
) ke−iωt^0 f̂ (kω) Time shift and squeeze. (13)
(f ∗ g)(t) f̂ (ω)̂g(ω) Convolution in time. (14)
u(t) =
{ 0 , if t < 0 1 , if t > 0
1 iω
δ(t − t 0 )f (t) e−iωt^0 f (t 0 ) Assumes f continuous at t 0. (16)
eiω^0 t^2 πδ(ω − ω 0 ) Useful for sin(ω 0 t), cos(ω 0 t). (17)
Convolution: (f ∗ g)(t) =
∫ (^) ∞
−∞
f (t − u)g(u) du =
∫ (^) ∞
−∞
f (u)g(t − u) du.
Parseval:
∫ (^) ∞
−∞
|f (t)|^2 dt =
1 2 π
∫ (^) ∞
−∞
∣ ∣ ∣̂f^ (ω)
∣ ∣ ∣
2 dω.
Signals & Systems - Reference Tables (^) 3
u ( t ) e sin( 0 t )
t
2 2 0
0
( )
j
t e
2 2
t^2 /( 2 ^2 ) e
2 2 / 2 2
e
t u te
( )
j
t u tte
( ) 2 ( )
j
Trigonometric Fourier Series
^ ^
1
( ) 0 cos( 0 ) sin( 0 ) n
f t a an nt bn nt
where
T
n
T T n
f t nt dt T
b
f t nt dt T
f tdt a T
a
0
0
0
0 0 0
()sin( )
()cos( ) ,and
Complex Exponential Fourier Series
(^)
T j nt n n
j nt n f te dt T
f t Fe F 0
( ) ,where^1 () ^0
Signals & Systems - Reference Tables (^) 4
cos( )
jx jx e e x
j
e e x
jx jx
sin( )
cos( x y )cos( x )cos( y )sin( x )sin( y )
sin( x y )sin( x )cos( y )cos( x )sin( y )
cos( 2 ) cos ( ) sin ( )
2 2 x x x
sin( 2 x ) 2 sin( x )cos( x )
2 cos ( ) 1 cos( 2 )
2 x x
2 sin ( ) 1 cos( 2 )
2 x x
cos ( ) sin ( ) 1
2 2 x x
2 cos( x ) cos( y )cos( x y )cos( x y )
2 sin( x ) sin( y )cos( x y )cos( x y )
2 sin( x ) cos( y )sin( x y )sin( x y )
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< Engineering Tables Jump to: navigation, search
Signal
Fourier transform unitary, angular frequency
Fourier transform unitary, ordinary frequency
Remarks
10 The rectangular pulse and the normalized sinc function
11
Dual of rule 10. The rectangular function is an idealized low-pass filter, and the sinc function is the non-causal impulse response of such a filter.
12 tri is the triangular function
13 Dual of rule 12.
14
Shows that the Gaussian function exp( - a t
2 ) is its own Fourier transform. For this to be integrable we must have Re(a) > 0.
common in optics
a>
the transform is the function itself
J 0 (t) is the Bessel function of first kind of order 0, rect is the rectangular function
it's the generalization of the previous transform; Tn (t) is the Chebyshev polynomial of the first kind.
Un (t) is the Chebyshev polynomial of the second kind
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