Short Table of Fourier Transform Pairs
|
s(t)
|
S(f)
|
|
ⅇ-atut
a
t
u
t
|
1ⅈ2πf+a
1
2
f
a
|
|
ⅇ-a|t|
a
t
|
2a4π2f2+a2
2
a
4
2
f
2
a
2
|
|
pt=1if|t|<Δ20if|t|>Δ2
p
t
1
t
Δ
2
0
t
Δ
2
|
sinπfΔπf
f
Δ
f
|
|
sin2πWtπt
2
W
t
t
|
Sf=1if|f|<W0if|f|>W
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
t
a
2
s
2
t
|
a
1
S
1
f+
a
2
S
2
f
a
1
S
1
f
a
2
S
2
f
|
|
Conjugate Symmetry
|
st∈ℝ
s
t
|
Sf=S-f¯
S
f
S
f
|
|
Even Symmetry
|
st=s-t
s
t
s
t
|
Sf=S-f
S
f
S
f
|
|
Odd Symmetry
|
st=-s-t
s
t
s
t
|
Sf=-S-f
S
f
S
f
|
|
Scale Change
|
sat
s
a
t
|
1|a|Sfa
1
a
S
f
a
|
|
Time Delay
|
st-τ
s
t
τ
|
ⅇ-ⅈ2πfτSf
2
f
τ
S
f
|
|
Complex Modulation
|
ⅇⅈ2π
f
0
tst
2
f
0
t
s
t
|
Sf-
f
0
S
f
f
0
|
|
Amplitude Modulation by Cosine
|
stcos2π
f
0
t
s
t
2
f
0
t
|
Sf-
f
0
+Sf+
f
0
2
S
f
f
0
S
f
f
0
2
|
|
Amplitude Modulation by Sine
|
stsin2π
f
0
t
s
t
2
f
0
t
|
Sf-
f
0
-Sf+
f
0
2ⅈ
S
f
f
0
S
f
f
0
2
|
|
Differentiation
|
ddtst
t
s
t
|
ⅈ2πfSf
2
f
S
f
|
|
Integration
|
∫-∞tsαdα
α
t
s
α
|
1ⅈ2πfSf
1
2
f
S
f
if
S0=0
S
0
0
|
|
Multiplication by tt
|
tst
t
s
t
|
1-ⅈ2πddfSf
1
2
f
S
f
|
|
Area
|
∫-∞∞stdt
t
s
t
|
S0
S
0
|
|
Value at Origin
|
s0
s
0
|
∫-∞∞Sfdf
f
S
f
|
|
Parseval's Theorem
|
∫-∞∞|st|2dt
t
s
t
2
|
∫-∞∞|Sf|2df
f
S
f
2
|
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"My introduction to signal processing course at Rice University."