SYSTEM ANALYSIS USING RESPONSE SURFACE MODELING - Advanced Signal Integrity for High-Speed Digital Design - page 601

we introduce an example signaling link that we use throughout the chapter to
demonstrate the application and benefit of the RSM approach.

14.2 CASE STUDY: 10-GB/S DIFFERENTIAL PCB INTERFACE

We want to develop a pair of response surface models that estimate the eye
height and eye width for a 10-Gb/s differential link. Our ultimate goal is to
use the models to identify a working solution that accommodates the desired
range of trace lengths, comprehends the expected manufacturing variation, and
meets a defect rate of less than 1000 parts per million (ppm). We have complete
system budgets for timing and noise that require our interconnect channel to
meet minimum specs of 60-mV eye height and 70-ps eye width as determined
from peak distortion analysis in order to achieve a 10 − 12 bit error rate. We use
this example system throughout the remainder of the chapter to illustrate the
model-fitting process, analysis of the model fit, and application to prediction of
design limits and defect rates.
The system characteristics are summarized in Figure 14-1 and Table 14-1. We
will fit our model to the five input variables listed in Table 14-1. The variability
in the models for termination and characteristic impedance represent the expected
variation due to the manufacturing process. The range of the transmission-line
lengths is based on experience with prior systems, which indicates the need to
accommodate a range of trace lengths from 10 to 20 in. Finally, the link uses
a two-tap equalizer with a coefficient range from

− −

0 . 10, which was
determined from results of initial simulations. The equalizer is not adaptive, so the
tap weights must be set prior to operation. The weights are controlled by a 4-bit
digital-to-analog converter, so that the granularity between adjacent taps settings
is equal to 0.0133. The valid equalizer settings are summarized in Table 14-2.
The differential transmission-line characteristics are calculated from the phys-
ical cross-sectional dimensions using a 2D field solver. To generate the eye data,
we first simulate the pulse response in the frequency domain using the causal
transmission-line modeling method (Chapter 8) and frequency-domain equalizer
(Chapter 12) model. After transformation to the time domain, we determine the
worst-case eye width and height for each observation using peak distortion anal-
ysis (Chapter 13), which we show in columns 7 and 8 in Table 14-3.

0 . 30 to

14.3 RSM CONSTRUCTION BY LEAST SQUARES FITTING

The general form of the response surface model is

y = β 0 + β 1 x 1 + β 2 x 2 +···+ β k x k + ε (14-1)

where y is the system response (output), β i the model fit coefficients, x i the
system inputs, k the number of terms in the model, and ε the error in the predic-
tion from the model. The response surface model is a linear function of the fit