Sa (x) = sin(x) / x
sinc function
Table of Fourier Transform Pairs
Function, f(t)
Definition of Inverse Fourier Transform
Fourier Transform, F(w)
Definition of Fourier Transform
1 ¥
f (t ) =
F (w )e jwt dw
ò
2p - ¥
¥
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
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
t
-¥
Signals & Systems - Reference Tables
1
j
sgn(w )
1
pt
u (t )
pd (w ) +
¥
1
jw
¥
å Fn e jnw 0t
2p å Fn d (w - nw 0 )
t
rect ( )
t
tSa(
B
Bt Sa (x) = sin(x) / x
Sa( ) sinc function
2p
2
w
rect ( )
B
tri (t )
w
Sa 2 ( )
2
n = -¥
A cos(
tri(t) = (1-|t|)rect(t/2)
triangle function =
rect(t)*rect(t)
pt
t
)rect ( )
2t
2t
n = -¥
wt
)
2
Sa (x) = sin(x) / x
sinc function
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
a2 +w2
2
2
2
2
e -t /( 2s )
s 2p e -s w / 2
u (t )e -at
1
a + jw
u (t )te -at
1
(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 T
2T
a 0 = ò f (t )dt , a n = ò f (t ) cos(w 0 nt )dt , and
T 0
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)
ò x cos( x)dx
2 x cos( x) + ( x 2 - 2) sin( x)
ò x sin( x)dx
2
2 x sin( x) - ( x 2 - 2) cos( x)
ax
e ax
a
2
ò e dx
ax
éx 1 ù
e ax ê - 2 ú
ëa a û
2 ax
é x 2 2x 2 ù
e ax ê - 2 - 3 ú
a û
ëa a
dx
1
ln a + bx
b
ò xe dx
ò x e dx
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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