Digital Signal Processing Reference
In-Depth Information
Step 2:
Calculate the Hilbert transform of the real part of
H
(
f
) compare it to
the imaginary part of
H
(
f
).
πf
= F
−
1
F
e
−
0
.
00000001
|
f
|
z
cos
10
8
F
1
1
−
4
πf
H
Re
(f )
=
∗
Re[
H(f)
]
3
.
0
×
πω
−
H
Re
The quantity
(
f
) and the imaginary part of
H
(
f
) are plotted in
=−
H
Re
(ω)
, the transmission-line model is
noncausal. The noncausal nature of this model can also be observed by looking
at the impulse response, which is calculated by taking the inverse Fourier
transform of
H
(
f
):
Figure 8-17b. Since Im[
H(f)
]
F
−
1
{
H(f)
}=
h(t)
and is plotted in Figure 8-18. Note how the impulse response rises prematurely.
The theoretical delay of this transmission line can be calculated from the length,
the speed of light, and the relative dielectric permittivity.
√
ε
r
c
10
−
9
s
/
m
=
6
.
66
×
10
−
12
s
/
in.
τ
d
=
169
.
333
×
Since the transmission line is 2 in.
long,
the pulse should arrive at
10
−
12
)(
2
.
0
)
=
τ
d
=
(
169
.
333
339 ps. It is obvious that the waveform has
components arriving much earlier than 339 ps, indicating that the model is
noncausal and is not obeying the limits placed on the speed of light.
×
The causality problems associated with the transmission line in Example 8-4
are caused by the assumption of frequency-independent dielectric properties. In
Chapter 6 we described numerous dielectric models that show how the dielectric
h
(
t
)
×
10
9
3.5
Theoretical arrival time for a
causal model as dictated by the
speed of light in the dielectric
media
3.0
2.5
2.0
1.5
1.0
0.5
t
d
−
500
0
500
1000
Time, ps
Figure 8-18
Noncausal impulse response of the transmission line of Example 8-4.
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