Digital Signal Processing Reference
In-Depth Information
Table 5.3. (
cont.
)
Function
Time-domain waveform
Magnitude and phase spectra
x
(
t
) =
t
e
−
at
u
(
t
)
1
(6) First-order time-rising decaying exponential
function
x
(
t
)
=
t
e
−
at
u
(
t
)
1
X
(
w
) =
<X
(
w
)
(
a
+ j
w
)
2
a
2
p
|
X
(
w
)|
w
t
0
0
−
p
n
!
(7)
N
th-order time-rising decaying exponential
function
x
(
t
)
=
t
n
e
−
at
u
(
t
)
x
(
t
) =
t
n
e
−
at
u
(
t
)
<X
(
w
)
n
!
(
n
+1)
p
X
(
w
) =
a
+ j
w
)
n
+1
(
a
n
+1
2
|
X
(
w
)|
w
t
0
(
n
+1)
p
0
−
2
1
t
>
0
2
(8) Sign function sgn(
t
)
=
x
(
t
) = sgn(
t
)
X
(
w
) =
1
−
1
t
<
0
j
w
|
X
(
w
)|
p
t
2
w
0
p
−
<
X
(
w
)
2
−1
X
(
w
) = 2
pd
(
w
−
w
0
)
(9) Complex exponential function
x
(
t
)
=
e
j
ω
0
t
x
(
t
) = e
j
w
0
t
1
|
x
(
t
)| = 1
<
x
(
t
) =
w
0
2
p
|
X
(
w
)|
w
0
< X
(
w
) = 0
t
w
0
w
0
0
x
(
t
) = cos (
w
0
t
)
X
(
w
) =
p
[
d
(
w
−
w
0
)+
d
(
w
+
w
0
)]
(10) Cosine function
x
(
t
)
=
cos(
ω
0
t
)
p
p
1
<X
(
w
) = 0
t
w
−
w
0
w
0
0
0
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