Digital Signal Processing Reference
In-Depth Information
1.0
x(
t
)
y(
t
)
0.8
0.6
0.4
0.2
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Time, ns
Figure 8-13
Pulse response of the transmission line for Example 8-2.
in Example 8-2 (Figure 8-13) is degraded significantly due to the conductor and
dielectric losses. Furthermore, the output pulse is wider than the input pulse
because the frequency components of the digital waveform (as calculated with a
Fourier transform) will travel at different speeds due to the frequency-dependent
dielectric permittivity used to calculate the capacitance. The velocity differences
between each harmonic will distort the waveform by spreading it out in time,
which is known as
dispersion
.
8.2 REQUIREMENTS FOR A PHYSICAL CHANNEL
In this section we introduce specific limitations that channel models of a LTI
system must obey to remain physically consistent with nature. Specifically, the
conditions of
causality, passivity
, and
stability
are described and defined by the
appropriate mathematical conditions that must be met to ensure physical behavior.
The analysis is restricted to linear and time-invariant electrical networks, which
is appropriate for all passive components used in modern bus design, such as
transmission lines, vias, packages, connectors, and so on.
8.2.1 Causality
One seemingly obvious requirement of a model that obeys the laws of nature
is that an output cannot precede its input. In other words, in the real world we
live in, an effect cannot precede its cause. This fundamental principle is called
causality
. Mathematically, a linear time-invariant system is causal only if for
every input all the elements of its impulse response
h
ij
vanish for
t<
0:
h
(t)
=
0
when
t<
0
(8-14a)
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