Digital Signal Processing Reference
In-Depth Information
Table 5.3. Magnitude and phase spectra for selected elementary CT functions
Magnitude spectra are shown as lines and phase spectra are shown as dashed lines
Function
Time-domain waveform
Magnitude and phase spectra
X
(
w
) = 2
pd
(
w
)
(1) Constant
x
(
t
)
=
1
x
(
t
) = 1
2
p
1
|
X
(
w
)|
<X
(
w
) = 0
t
w
0
0
1
(2) Unit impulse function
x
(
t
)
= δ
(
t
)
x
(
t
) =
d
(
t
)
X
(
w
) = 1
1
|
X
(
w
)|
<X
(
w
) = 0
w
t
0
0
X
(
w
) =
pd
(
w
)
+
1
|
X
(
w
)|
(3) Unit step function
x
(
t
)
=
u
(
t
)
x
(
t
) =
u
(
t
)
<X
(
w
)
1
j
w
p
/2
t
ω
0
0
−
p
/2
1
(4) Decaying exponential
x
(
t
)
=
e
−
at
u
(
t
)
x
(
t
) = e
−
at
u
(
t
)
X
(
w
) =
<X
(
w
)
a
+ j
w
p
/2
|
X
(
w
)|
t
w
0
0
−
p
/2
2
a
(5) Two-sided decaying exponential
x
(
t
)
=
e
−
a
t
x
(
t
) = e
−
a
|
t
|
X
(
w
) =
2/
a
a
2
+
w
2
|
X
(
w
)|
<X
(
w
) = 0
t
w
0
0
(
cont.
)
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