MATHEMATICAL REQUIREMENTS FOR PHYSICAL CHANNELS - Advanced Signal Integrity for High-Speed Digital Design - page 329

0.5

0.4

Noncausal waveform
Causal waveform

0.3

0.2

0.1

0.0

2.0 2.5 3.0 3.5 4.0 4.5 5.0

Time, ns

Figure 8-14 Simulated pulse response for a 20-in. transmission line, showing the out-
puts of a causal and a noncausal model. Note how the noncausal waveform arrives
early.

But more generally, if a system has known delay [ τ ], the system is causal if

h (t) = 0 when t< [ τ ] (8-14b)

An example of a causal and a noncausal pulse response is shown in Figure 8-14,
where a pulse response was simulated by driving a 100-ps-wide digital pulse into
a causal and a noncausal 20-in. transmission-line model. † Note that the noncausal
pulse response has a component that arrives early, which is a telltale sign of a
nonphysical model. In this example, the noncausal transmission-line model was
created using frequency-invariant values of the loss tangent and the dielectric
permittivity. The causal model was created using the infinite-pole dielectric model
presented in Chapter 6.
Unfortunately, it is not always easy to ascertain whether a model is causal
by observing the pulse response. Furthermore, it is not always obvious if it
matters. For example, a very short transmission line may be noncausal, but the
error could be so small that it would not affect the final result significantly.
However, if numerous noncausal models are cascaded together to simulate a bus,
the causality errors could accumulate and significant waveform miscalculations
could be realized. Fortunately, it is not very difficult to create a causal model by
ensuring that the dielectric and conductor properties outlined in Chapters 5 and
6 are followed.
The discussion above prompts an inevitable question: What are the mathe-
matical conditions that specify whether or not a system is causal? To begin, the
frequency response of the model is observed. If the Fourier transform of the

† Don't get the pulse response confused with the impulse response [ h ( t )]. Since pulse responses are
more representative of a realistic digital driver, they are often used to analyze interconnects.