Digital Signal Processing Reference
In-Depth Information
The time-integration property of the Fourier transform will now be proved.
Consider the convolution of a generic waveform
f(t)
with a unit step function:
q
f(t)
*
u(t) =
L
f(t)u(t - t)dt.
-
q
The unit step function
u(t - t)
has a value of zero for
t 6 t
and a value of 1 for
t 7 t.
This can be restated as
1,
t 6 t
b
u(t - t) =
t 7 t
,
0,
and, therefore,
t
f(t)
*
u(t) =
L
f(t)dt.
(5.25)
-
q
The convolution property yields
1
jv
f(t)
*
u(t)
Î
f
"
F(v)
B
R
pd(v) +
,
(5.26)
5
6
where
f
u(t)
= pd(v) + 1/jv
from Table 5.2. (The derivation of this transform
pair is provided in Section 5.3.)
By combining Equations (5.25) and (5.26), we write
t
F(v)
jv
f(t)dt
Î
f
"
+ pF(0)d(v).
L
-
q
The factor
F
(0) in the second term on the right follows from the sifting prop-
erty (2.42) of the impulse function.
The time-integration property of the Fourier transform
EXAMPLE 5.12
Figure 5.13(a) shows a linear system that consists of an integrator. As discussed in Section
1.2, this can be physically realized electronically by a combination of an operational ampli-
fier, resistors, and capacitors. The input signal is a pair of rectangular pulses, as shown in
Figure 5.13(b). Using time-domain integration, we can see that the output signal would be
a triangular waveform, as shown in Figure 5.13(c). We wish to know the frequency spec-
trum of the output signal. We have not derived the Fourier transform of a triangular wave;
however, we do know the Fourier transform of a rectangular pulse such as is present at the
input of the system. Using the properties of linearity and time shifting, we can write the
input signal as
t
+
t
1
/2
t
1
t
-
t
1
/2
t
1
B
R
B
R
x(t) = A rect
- A rect
