Table of Fourier Transform Pairs
Sa (x) = sin(x) / x sinc function
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
sgn(w )
j
pt
u(t)
pd (w) + 1
jw
?
? Fn e jnw0t
n=-?
?
2p ? Fnd (w - nw 0 ) n = -?
rect( t ) t
tSa(w2t )
B 2p
Sa(
Bt 2
)
Sa (x) = sin(x) / x sinc function
rect(w ) B
tri(t)
tri(t) = (1-|t|)rect(t/2) triangle function = rect(t)*rect(t)
A
cos(
pt 2t
)rect(
t 2t
)
Sa 2 (w2 )
Ap cos(wt ) t (p 2t ) 2 - w 2
Sa (x) = sin(x) / x sinc function
cos(w 0t)
p [d (w - w 0 ) + d (w + w 0 )]
sin(w 0t)
p j
[d
(w
-
w0
)
-
d
(w
+
w0
)]
u(t) cos(w 0t) u(t) sin(w 0t) u(t)e -at cos(w 0t)
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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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
Dual of rule 10. The rectangular function is an idealized
11
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.
Shows that the Gaussian function exp( - at2) is its own
14
Fourier transform. For this to be integrable we must have
Re(a) > 0.
common in optics
Retrieved from "" Category: Engineering Tables
Views
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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