Using model-based design in signal integrity engineering

Colin Warwick, The MathWorks

August 1, 2007

Colin Warwick, The MathWorks

Applying the modified rational function method
We can simplify the design process for the data transmission application using Model-Based Design. The transmission line will be modeled as a modified rational function using MATLAB and RF Toolbox . A modified rational function is a transfer function in the form of a particular type of Laplace transform. The general Laplace transform is the integral over time of a function of time f(t) with a complex sine wave e^-st

In the modified rational function, f(s) consists of residues c_jand poles a_jwhich are complex conjugate pairs. Correspondingly in the time domain, f(t) consists of a direct feed term dδ(t-t_d) plus a set of exponentially decaying sine waves, which begin after the principle delay t_d

The modified rational function has the following advantages:

• We can achieve the same level of accuracy as the IFFT method, with a model that is one or two orders of magnitude simpler.
• Model order reduction can be used to trade off complexity and accuracy through the use of fitting parameters.
• Typical VNA data has a low frequency cutoff at around 20 to 50 MHz, so an extrapolation to DC is needed. With IFFT models, there is nothing to prevent the extrapolated phase from being nonzero, which corresponds to a nonphysical delay.
• This can be avoided by writing a constraint algorithm, but this takes time. In contrast, rational models represent a physical transmission line. So the phase on extrapolation to DC is inherently zero, avoiding the need for constraint algorithms.
• The physical correspondence between the model and transmission line also provides greater insight when building mitigating algorithms. For example, seeing what poles exist on the transmission line is very helpful when building the DFE.

Fitting a rational function to VNA data
The first step is creating a model of the impairment using S4P data from the VNA. In this example, the S4P data includes about 1,500 frequencies from 50 MHz to 50 GHz. After reading the file into MATLAB via the RF Toolbox read function, we use the s2sdd function to extract the equivalent differential two-port behavior of this four-port network. The next step is to compute the transfer function and rational model. The frequency domain transfer function of the two-port parameters is calculated with the "s2tf" function. Then the "rationalfit" function is used to compute the time-domain modified rational function. The end result is that about 24,000 data points are condensed into a simple 48-pole rational function fit.