MATHEMATICAL REQUIREMENTS FOR PHYSICAL CHANNELS - Advanced Signal Integrity for High-Speed Digital Design - page 313

realistic response is expected. Such assumptions, although valid at low frequen-
cies or for very short electrical structures, induce amplitude and phase errors for
digital data rates faster than 1 to 2 Gb/s. In fact, the computer currently being
used to write this chapter was designed using traditional modeling techniques,
which assume frequency-invariant electrical properties of dielectrics and conduc-
tors. Such assumptions, however, cease to produce valid results for higher data
rates. As data rates increase, bandwidth demands skyrocket, form factors shrink,
and new phenomenan that were insignificant in past designs become significant.
When the correct model assumptions are not used, incorrect solution spaces are
determined, lab correlation becomes difficult or impossible, and the design time
is increased significantly. In this chapter we outline some of the most important
techniques used to determine if a physical channel model is adequate for the
design at hand. First, the fundamentals of calculating the channel response and
methods of implementing frequency-domain phenomena in time-domain simu-
lations are explored. Next, the mathematical requirements for a channel that is
consistent with nature are explained, and methodologies for testing these require-
ments are defined. It is not always necessary for a model to obey these physical
rules if the error is small enough; however, it is an important concept for the
modern-day digital designer to understand to ensure full comprehension of the
modeling assumptions.
8.1 FREQUENCY-DOMAIN EFFECTS IN TIME-DOMAIN
SIMULATIONS
Although high-speed digital design is focused largely on the signal integrity of
time-domain digital waveforms, many of the phenomena that heavily influence
the propagation of signals on interconnects are best described in the frequency
domain. Examples covered in previous chapters include skin effect resistance,
surface roughness, internal inductance, and frequency-dependent dielectric prop-
erties. Consequently, it is important for the digital engineer to understand the rela-
tionship between a time-domain waveform and its equivalent frequency-domain
representation. In fact, many modern buses have component specifications in
terms of frequency-domain parameters because it is the most convenient way to
describe the wideband behavior. In this section we outline some of the funda-
mental principles for linear time-invariant systems that will allow the engineer
to translate between the frequency and time domains.
8.1.1 Linear and Time Invariance
A system is linear if the relationship between the input and output of the system
satisfies the superposition property. For example, if the input to the system is the