Fourier Transform Properties
2.7
2000/07/25
2002/08/07
Don
Johnson
dhj@rice.edu
Don
Johnson
dhj@rice.edu
Shawn
Stewart
mrshawn@alumni.rice.edu
fourier transform pairs
A table of commonly seen transforms, for reference.
Short Table of Fourier Transform Pairs
s(t)
S(f)
a
t
u
t
1
2
f
a
a
t
2
a
4
2
f
2
a
2
p
t
1
t
Δ
2
0
t
Δ
2
f
Δ
f
2
W
t
t
S
f
1
f
W
0
f
W
Fourier Transform Properties
Time-Domain
Frequency Domain
Linearity
a
1
s
1
t
a
2
s
2
t
a
1
S
1
f
a
2
S
2
f
Conjugate Symmetry
s
t
S
f
S
f
Even Symmetry
s
t
s
t
S
f
S
f
Odd Symmetry
s
t
s
t
S
f
S
f
Scale Change
s
a
t
1
a
S
f
a
Time Delay
s
t
τ
2
f
τ
S
f
Complex Modulation
2
f
0
t
s
t
S
f
f
0
Amplitude Modulation by Cosine
s
t
2
f
0
t
S
f
f
0
S
f
f
0
2
Amplitude Modulation by Sine
s
t
2
f
0
t
S
f
f
0
S
f
f
0
2
Differentiation
t
s
t
2
f
S
f
Integration
α
t
s
α
1
2
f
S
f
if
S
0
0
Multiplication by t
t
s
t
1
2
f
S
f
Area
t
s
t
S
0
Value at Origin
s
0
f
S
f
Parseval's Theorem
t
s
t
2
f
S
f
2