| Time Domain Signal |
Frequency Domain Signal |
Condition |
|
ⅇ-atut
a
t
u
t
|
1a+ⅈω
1
a
ω
|
a>0
a
0
|
|
ⅇatu-t
a
t
u
t
|
1a-ⅈω
1
a
ω
|
a>0
a
0
|
|
ⅇ-a|t|
a
t
|
2aa2+ω2
2
a
a
2
ω
2
|
a>0
a
0
|
|
tⅇ-atut
t
a
t
u
t
|
1a+ⅈω2
1
a
ω
2
|
a>0
a
0
|
|
tnⅇ-atut
t
n
a
t
u
t
|
n!a+ⅈ ω n+1
n
a
ω
n
1
|
a>0
a
0
|
|
δt
δ
t
|
1
1
|
|
|
1
1
|
2πδω
2
δ
ω
|
|
|
ⅇⅈω0t
ω0
t
|
2πδω-ω0
2
δ
ω
ω0
|
|
|
cosω0t
ω0
t
|
πδω-ω0+δω+ω0
δ
ω
ω0
δ
ω
ω0
|
|
|
sinω0t
ω0
t
|
ⅈπδω+ω0-δω-ω0
δ
ω
ω0
δ
ω
ω0
|
|
|
ut
u
t
|
πδω+1ⅈω
δ
ω
1
ω
|
|
|
sgnt
sgn
t
|
2ⅈω
2
ω
|
|
|
cosω0tut
ω0
t
u
t
|
π2δω-ω0+δω+ω0+ⅈωω02-ω2
2
δ
ω
ω0
δ
ω
ω0
ω
ω0
2
ω
2
|
|
|
sinω0tut
ω0
t
u
t
|
π2ⅈδω-ω0-δω+ω0+ω0ω02-ω2
2
δ
ω
ω0
δ
ω
ω0
ω0
ω0
2
ω
2
|
|
|
ⅇ-atsinω0tut
a
t
ω0
t
u
t
|
ω0a+ⅈω2+ω02
ω0
a
ω
2
ω0
2
|
a>0
a
0
|
|
ⅇ-atcosω0tut
a
t
ω0
t
u
t
|
a+ⅈωa+ⅈω2+ω02
a
ω
a
ω
2
ω0
2
|
a>0
a
0
|
|
ut+τ-ut-τ
u
t
τ
u
t
τ
|
2τsinωτωτ=2τsincωt
2
τ
ω
τ
ω
τ
2
τ
sinc
ω
t
|
|
|
ω0πsinω0tω0t=ω0πsincω0
ω0
ω0
t
ω0
t
ω0
sinc
ω0
|
uω+ω0-uω-ω0
u
ω
ω0
u
ω
ω0
|
|
|
tτ+1utτ+1-utτ+-tτ+1utτ-utτ-1=triagt2τ
t
τ
1
u
t
τ
1
u
t
τ
t
τ
1
u
t
τ
u
t
τ
1
triag
t
2
τ
|
τsinc2ωτ2
τ
sinc
ω
τ
2
2
|
|
|
ω02πsinc2ω0t2
ω0
2
sinc
ω0
t
2
2
|
ωω0+1uω
ω0
+1-uω
ω0
+-ω
ω0
+1uω
ω0
-uω
ω0
-1=triagω2ω0
ω
ω0
1
u
ω
ω0
1
u
ω
ω0
ω
ω0
1
u
ω
ω0
u
ω
ω0
1
triag
ω
2
ω0
|
|
|
∑n=-∞∞δt-nT
n
δ
t
n
T
|
ω0∑n=-∞∞δω-nω0
ω0
n
δ
ω
n
ω0
|
ω0=2πT
ω0
2
T
|
|
ⅇ-t22σ2
t
2
2
σ
2
|
σ2πⅇ-σ2ω2
2
σ
2
σ
2
ω
2
2
|
|
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"My introduction to signal processing course at Rice University."