SYSTEM ANALYSIS USING RESPONSE SURFACE MODELING - Advanced Signal Integrity for High-Speed Digital Design - page 600

14.1 MODEL DESIGN CONSIDERATIONS

Our understanding of channel noise and jitter modeling, combined with the peak
distortion analysis technique, gives us nearly all the tools we need in order
to design a multi-Gb/s signaling interface. However, we still need a method
that allows us to understand how our design behaves as the physical and elec-
trical characteristics of the system components change. Without the means to
evaluate the signal behavior in a systematic fashion as the system characteris-
tics vary, finding a combination of transmitter, package, PCB, termination, and
receiver characteristics that result in a functional and reliable system becomes an
intractable problem at multi-Gb/s rates.
In addition, we must also account for the impacts of manufacturing large
volumes of high-speed systems. The variation that is an inherent part of all man-
ufacturing processes means that different systems will behave differently, even if
they were built from the same design. For example, the differential impedance of
signal traces on our printed circuit boards may show as much as 20% variation

around the nominal value. This manufacturing variability may manifest itself as a
variation in electrical performance. In particular, we are interested in the impact
of manufacturing variation on eye height and width. If we do not comprehend
these variations in our design, a significant percentage of our systems may fail
to operate properly. The impact of such a poor design is typically increased cost,
due either to returns from customers or to yield loss during manufacture.
Fortunately, the response surface modeling (RSM) technique provides a tool
that gives us the ability to model the behavior of our signaling system as the
circuit and interconnect characteristics vary. RSM works by fitting a statistical
model of the output response as a function of changes in the input variables. For
example, a useful RSM would provide predictions of the eye height and width as a
function of board impedance, termination resistance, line length, and equalization.
Response surface modeling is a broad topic for which entire textbooks exist
[Myers and Montgomery, 1995]. As such, we cannot provide a comprehensive
treatment here. Instead, we endeavor to provide a sufficient overview to allow
the reader to begin applying RSM techniques in the design of signaling links.
The basic idea behind RSM is to apply linear regression techniques to create
a statistical model that predicts the system response (output) as the inputs vary.
We do this by first creating a set of observed outputs in response to specific sets
of input conditions that we determine via a designed experiment. The system
model is a linear equation that is constructed by fitting the observed responses
and inputs via the least squares fitting technique. Once the model is created, we
can use it to predict the output of the system in response to arbitrary combinations
of inputs.
Since the model represents a statistical fit, we expect that there may be some
error in the values predicted. As a result, understanding how well the model
fits the observed data and determining how much uncertainty exists in the pre-
dicted responses become important topics that we must also comprehend in our
analysis. Before describing the creation and use of the response surface model,

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