Digital Signal Processing Reference
In-Depth Information
5.5.5 Time differentiation
The time-differentiation property expresses the CTFT of a time-differentiated
signal d
x
/d
t
in terms of the CTFT of the original signal
x
(
t
). We state the
time-differentiation property next.
CTFT
←−−→
If x
(
t
)
X
(
ω
)
then
d
x
d
t
CTFT
←−−→
j
ω
X
(
ω
)
(5.52)
provided the derivative
d
x/
d
t exists at all time t .
Proof
From the CTFT synthesis equation, Eq. (5.9), we have
∞
1
2
π
X
(
ω
)e
j
ω
t
d
ω.
x
(
t
)
=
−∞
Taking the derivative with respect to
t
on both sides of the equation yields
%
&
∞
d
x
d
t
=
d
d
t
1
2
π
X
(
ω
)e
j
ω
t
d
ω
.
'
−∞
Interchanging the order of differentiation and integration, we obtain
∞
∞
d
x
d
t
1
2
π
d
d
t
e
j
ω
t
d
ω =
1
2
π
[j
ω
X
(
ω
)]e
j
ω
t
d
ω.
=
X
(
ω
)
−∞
−∞
Comparing this with Eq. (5.9), we obtain
d
x
d
t
CTFT
←−−→
j
ω
X
(
ω
)
.
Corollary
By repeatedly applying the time differentiation property, it is
straightforward to verify that
d
n
x
d
t
n
CTFT
←−−→
(j
ω
)
n
X
(
ω
)
.
Example 5.18
In Example 5.11, we showed that the CTFT for the periodic cosine function is
given by
CTFT
←−−→
cos(
ω
0
t
)
π
[
δ
(
ω − ω
0
)
+ δ
(
ω + ω
0
)]
.
Using the above CTFT pair, derive the CTFT for the periodic sine function
sin(
ω
0
t
).
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