Table of Fourier Transform Pairs - Fermilab
Table of Fourier Transform Pairs
Function, f(t)
Fourier Transform, F(w)
Definition of Inverse Fourier Transform Definition of Fourier Transform
? f
(t)
=
1 2p
?
F (w )e jwt dw
-?
?
? F (w) = f (t)e - jwt dt -?
f (t - t0 )
F (w )e - jwt0
f (t)e jw0t
F (w - w 0 )
f (at)
1 F(w ) aa
F (t)
2pf (-w)
d n f (t) dt n
(- jt)n f (t)
t
? f (t )dt
-?
d (t)
e jw0t sgn (t)
( jw)n F (w)
d n F (w) dw n
F (w ) + pF (0)d (w) jw
1 2pd (w - w 0 ) 2 jw
Signals & Systems - Reference Tables
1
1 j pt u(t)
?
? Fn e jnw0t
n=-?
rect( t ) t
B 2p
Sa(
Bt 2
)
tri(t)
A
cos(
pt 2t
)rect(
t 2t
)
cos(w 0t) sin(w 0t)
u(t) cos(w 0t)
u(t) sin(w 0t)
u(t)e -at cos(w 0t)
sgn(w )
pd (w) + 1 jw
?
2p ? Fnd (w - nw 0 ) n = -?
tSa(w2t )
rect(w ) B
Sa 2 (w2 )
Ap cos(wt ) t (p 2t ) 2 - w 2
p [d (w - w 0 ) + d (w + w 0 )]
p j
[d
(w
-
w0
)
-
d
(w
+
w0
)]
p 2
[d
(w
-
w0
)
+
d
(w
+
w0
)]
+
w
2 0
jw -w
2
p 2j
[d
(w
-w0)
-d
(w
+ w0 )] +
w2
w
2 0
-w2
(a + jw)
w
2 0
+
(a
+
jw ) 2
Signals & Systems - Reference Tables
2
u(t)e -at sin(w 0t) e -a t e -t 2 /(2s 2 ) u(t)e -at u(t)te -at
w0
w
2 0
+ (a
+
jw ) 2
2a a2 +w2
s 2p e -s 2w 2 / 2
1 a + jw
1 (a + jw) 2
? Trigonometric Fourier Series
?
f (t) = a0 + ? (an cos(w 0 nt) + bn sin(w 0 nt)) n =1
where
a0
=1 T
?T
0
f (t)dt
,
an
=
?2 T
T0
f
(t) cos(w 0nt)dt
, and
bn
=
2 T
T
?
0
f
(t) sin(w0 nt)dt
? Complex Exponential Fourier Series
?
? f (t) = Fne jwnt , where n=-?
? Fn
=
1 T
T 0
f
(t)e - jw0nt dt
Signals & Systems - Reference Tables
3
Some Useful Mathematical Relationships
cos(x)
=
e
jx
+ e - jx 2
sin( x)
=
e
jx
- e - jx 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(2x) = cos 2 (x) - sin 2 (x) sin(2x) = 2 sin(x) cos(x)
2cos2 (x) = 1 + cos(2x)
2 sin 2 (x) = 1 - cos(2x)
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
? cos(x)dx ? sin(x)dx ? x cos(x)dx ? x sin(x)dx ? x 2 cos(x)dx ? x 2 sin(x)dx ? eax dx
? xeax dx
? x 2eax dx
?
a
dx + bx
? dx
a2 + b 2x2
sin( x)
- cos(x)
cos(x) + x sin(x)
sin(x) - x cos(x)
2x cos(x) + (x 2 - 2) sin(x) 2x sin(x) - (x 2 - 2) cos(x)
eax a
eax
?x ?? a
-
1 a2
? ??
eax
? ? ?
x2 a
-
2x a2
-
2?
a
3
? ?
1 ln a + bx b
1 tan -1 ( bx )
ab
a
Signals & Systems - Reference Tables
5
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