Digital Signal Processing Reference
In-Depth Information
f
(
t
)
1.0
0.8
0.6
0.4
0.2
time
−
1
1
2
3
(a)
F
(
ω
)
2.0
Re[
F
(
ω
)]
1.5
1.0
Im[
F
(
ω
)]
0.5
Frequency,
ω
−
20
−
10
10
20
−
0.5
−
1.0
1.5
−
(b)
Figure 8-3
(a) Square wave
f
(
t
) with only positive time values; (b) frequency-domain
equivalent spectrum of the square wave. Note that the spectrum has both real and imagi-
nary values.
2. The spectrum of a time-domain waveform with only positive time values
is complex.
3. Since time-domain signals that are physically observable have no imaginary
component, they must be real. Reality in the time domain is guaranteed
when the positive frequencies of the Fourier transform are the complex
conjugate of the negative frequencies [LePage, 1980]:
F(
−
ω)
=
F(ω)
∗
(8-4)
The behavior of equation (8-4) can be observed in Figure 8-3b. For example,
at the frequency
ω
=−
2 rad/s, the spectrum has a value of
F(
−
2
)
=−
0
.
4
+
j
0
.
74
and at
ω
=
2 rad/s, the spectrum has a value of
F(
2
)
=−
0
.
4
−
j
0
.
74
2
)
=
F(
2
)
∗
, the spectrum is for a real time-domain waveform.
Since
F(
−
Search WWH ::
Custom Search