fourier

Table of Fourier Transform Pairs
Function, f(t)
Definition of Inverse Fourier Transform
1
f (t ) =
2p
¥
ò F (w )e
jwt
dw
Fourier Transform, F(w)
Definition of Fourier Transform
¥
F (w ) =
-¥
ò f (t )e
- jwt
dt
-¥
f (t - t 0 )
F (w )e - jwt0
f (t )e jw 0t
F (w - w 0 )
f (at )
1
w
F( )
a
a
F (t )
2pf (-w )
d n f (t )
( jw ) n F (w )
dt n
(- jt ) n f (t )
d n F (w)
dw n
t
ò
f (t )dt
-¥
F (w )
+ pF (0)d (w )
jw
d (t )
1
e jw 0 t
2pd (w - w 0 )
sgn (t)
2
jw
Signals & Systems - Reference Tables
1
Fourier Transform Table
UBC M267 Resources for 2005
Fb(ω)
F (t)
Notes
(0)
Definition.
(1)
fb(ω)
Inversion formula.
(2)
fb(−t)
2πf (ω)
Duality property.
(3)
e−at u(t)
1
a + iω
a constant, <e(a) > 0
(4)
2a
+ ω2
a constant, <e(a) > 0
(5)
Boxcar in time.
(6)
Boxcar in frequency.
(7)
Derivative in time.
(8)
Higher derivatives similar.
(9)
Z
∞
f (t)e−iωt dt
f (t)
1
2π
Z
−∞
∞
fb(ω)eiωt dω
−∞
e−a|t|
β(t) =
1,
0,
a2
if |t| < 1,
if |t| > 1
2 sinc(ω) = 2
sin(ω)
ω
1
sinc(t)
π
β(ω)
f 0 (t)
iω fb(ω)
f 00 (t)
(iω)2 fb(ω)
d
i fb(ω)
dω
d2
i2 2 fb(ω)
dω
b
f (ω − ω0 )
tf (t)
t2 f (t)
eiω0 t f (t)
t − t0
f
k
ke−iωt0 fb(kω)
fb(ω)b
g (ω)
Derivative in frequency.
(10)
Higher derivatives similar.
(11)
Modulation property.
(12)
Time shift and squeeze.
(13)
Convolution in time.
(14)
(f ∗ g)(t)
0, if t < 0
u(t) =
1, if t > 0
1
+ πδ(ω)
iω
Heaviside step function.
(15)
δ(t − t0 )f (t)
e−iωt0 f (t0 )
Assumes f continuous at t0 .
(16)
eiω0 t
2πδ(ω − ω0 )
Useful for sin(ω0 t), cos(ω0 t).
(17)
Z
Convolution:
(f ∗ g)(t) =
Z
Parseval:
∞
∞
−∞
Z
f (t − u)g(u) du =
1
|f (t)| dt =
2π
−∞
2
Z
∞
−∞
∞
−∞
2
f (u)g(t − u) du.
fb(ω) dω.
j
sgn(w )
1
pt
u (t )
pd (w ) +
¥
¥
å Fn e jnw 0t
2p
t
rect ( )
t
tSa(
B
Bt
Sa( )
2p
2
w
rect ( )
B
tri (t )
w
Sa 2 ( )
2
n = -¥
A cos(
pt
t
)rect ( )
2t
2t
1
jw
å Fnd (w - nw 0 )
n = -¥
wt
)
2
Ap cos(wt )
t (p ) 2 - w 2
2t
cos(w 0 t )
p [d (w - w 0 ) + d (w + w 0 )]
sin(w 0 t )
p
[d (w - w 0 ) - d (w + w 0 )]
j
u (t ) cos(w 0 t )
p
[d (w - w 0 ) + d (w + w 0 )] + 2 jw 2
2
w0 - w
u (t ) sin(w 0 t )
2
p
[d (w - w 0 ) - d (w + w 0 )] + 2w 2
2j
w0 - w
u (t )e -at cos(w 0 t )
Signals & Systems - Reference Tables
(a + jw )
w 02 + (a + jw ) 2
2
w0
u (t )e -at sin(w 0 t )
e
w 02 + (a + jw ) 2
2a
-a t
e -t
a2 +w2
2
/( 2s 2 )
s 2p e -s
2
w2 / 2
1
a + jw
u (t )e -at
1
u (t )te -at
(a + jw ) 2
Ø Trigonometric Fourier Series
¥
f (t ) = a 0 + å (a n cos(w 0 nt ) + bn sin(w 0 nt ) )
n =1
where
1
a0 =
T
T
ò0
2T
f (t )dt , a n = ò f (t ) cos(w 0 nt )dt , and
T0
2T
bn = ò f (t ) sin(w 0 nt )dt
T 0
Ø Complex Exponential Fourier Series
f (t ) =
¥
å Fn e
jwnt
, where
n = -¥
Signals & Systems - Reference Tables
1T
Fn = ò f (t )e - jw 0 nt dt
T 0
3
Some Useful Mathematical Relationships
e jx + e - jx
cos( x) =
2
e jx - e - jx
sin( x) =
2j
cos( x ± y ) = cos( x) cos( y ) m sin( x) sin( y )
sin( x ± y ) = sin( x) cos( y ) ± cos( x) sin( y )
cos(2 x) = cos 2 ( x) - sin 2 ( x)
sin( 2 x) = 2 sin( x) cos( x)
2 cos2 ( x) = 1 + cos(2 x)
2 sin 2 ( x) = 1 - cos(2 x)
cos 2 ( x) + sin 2 ( x) = 1
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 )
Signals & Systems - Reference Tables
4
Useful Integrals
ò cos( x)dx
sin(x)
ò sin( x)dx
- cos(x)
ò x cos( x)dx
cos( x) + x sin( x)
ò x sin( x)dx
sin( x) - x cos( x)
òx
2
cos( x)dx
2 x cos( x) + ( x 2 - 2) sin( x)
òx
2
sin( x)dx
2 x sin( x) - ( x 2 - 2) cos( x)
ax
dx
e ax
a
òe
ò xe
òx
ax
dx
2 ax
éx 1 ù
e ax ê - 2 ú
ëa a û
e dx
é x 2 2x 2 ù
e ax ê - 2 - 3 ú
a û
ëa a
dx
1
ln a + bx
b
ò a + bx
dx
ò a 2 + b 2x2
Signals & Systems - Reference Tables
bx
1
tan -1 ( )
ab
a
5
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Engineering Tables/Fourier Transform Table 2
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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( - at2) is its own
Fourier transform. For this to be integrable we must have
Re(a) > 0.
common in optics
a>0
the transform is the function itself
J0 (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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Category: Engineering Tables
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