# How to Verify S-parameters For Crosstalk Waveforms When you run crosstalk simulations in a PCB layout, you generally only have access to a time-domain crosstalk waveform or a peak amplitude value for the crosstalk waveform. More advanced simulations, typically involving a 2D or 3D field solver, can provide crosstalk data in terms of S-parameters. When you have these two different sets of data, what’s the tool to convert between them and how can you verify the waveform data is correct?

This goes back to some basic definitions, as well as some simple math you can use as a sanity check to verify correspondence between different simulation results. Simulation results taken from the frequency domain then converted to the time domain can appear different from results found directly from a time-domain simulation. Typically, both approaches are needed when evaluating high-speed digital channel capabilities and assessing compliance.

There are two approaches for properly comparing time-domain and frequency-domain results for a high-speed digital system. There is a simple method that involves comparing order of magnitude, and another method that involves direct conversion between time and frequency domains, followed by comparison of the results. We’ll show you how it works in this article.

## Time Domain and Frequency Domain Sanity Checks

Crosstalk can be simulated in the time domain and frequency domain. These methods involve different computational efforts, and the accuracy of the results can be different. While it is known that you can simulate crosstalk in one domain and convert it to the other domain, the converted result might not be accurate as a direct simulation. For example, suppose you calculate a time-domain crosstalk pulse; will this be more accurate or less accurate than converting the frequency domain crosstalk into the time-domain crosstalk?

Determining this correspondence is quite important for simulation-driven workflows. The factors affecting accuracy in each domain are outlined below.

 Time domain Frequency domain Solver method 2D cross-section method Boundary element method Method of moments 3D method FDTD Quasi-TEM Method of lines 2D cross-section methods 3D method FDFD Time/frequency discretization Evenly spaced samples in time without interpolation Adaptive discretization Interpolation may be used Input Voltage waveform Harmonic signal Output Voltage waveform S-parameters

In this table, the main factors involved in determining accuracy are discretization applied in the generated solution. Time-domain solutions can require much more time to complete than frequency-domain simulations, even if the same 2D cross-sectioning or 3D method are used. This is because frequency domain simulations can use adaptive discretization and interpolation to reduce the number of frequency points at which the simulation is being run.

Because of these differences in computational effort, and the potential sacrifice in accuracy in favor of fast simulation, it’s worth comparing crosstalk and S-parameter data to ensure they are consistent. The problem is that crosstalk is often simulated in the time domain, so the result is a voltage waveform, as indicated in the above table. Meanwhile, S-parameters are plotted as a spectrum for a multiport network:

• As a 4-port network (2 single-ended traces) with single-ended excitation
• As a 4-port network (2 differential pairs) with differential-mode and common-mode excitation

For the frequency domain data, the S-parameter data for a 4-port network describing return loss, insertion loss, and crosstalk is given as a matrix as described below: Using the voltage waveform results at each port, these can be compared as a dB value to estimate the corresponding S-parameter results in the above matrix.

### Peak Amplitude vs. S-parameter Magnitude

The fastest way to verify simulation settings is to compare S-parameter values with crosstalk waveforms in the time domain is to compare the input vs. output voltage waveform with the average magnitude of the S-parameters. If the values are similar on an order of magnitude basis, then it means either approach is appropriate for quantifying crosstalk.

To determine a crosstalk metric in terms of peak amplitudes, use the following formula. Simple equation for estimating S-parameter magnitude for crosstalk. V(o,peak) is the crosstalk pulse peak voltage, and V(i,peak) is the input pulse peak voltage. This formula is applicable for both NEXT and FEXT.

Because S-parameters will quantify crosstalk in terms of input and output power, we have to use the factor 20 in the dB calculation instead of a factor 10. This does not exactly describe the conversion between a voltage waveform and S-parameters because it does not include the current or the trace impedance. However, it does give a reasonable order of magnitude result for crosstalk.

Example: Suppose we inject a pulse with 1 V peak; the resulting NEXT waveform amplitude as a -80 mV peak and FEXT has -35 mV peak. The magnitude of the crosstalk in terms of S-parameters would be approximately: If the trace layout is reciprocal (this is often the case), then the results above would hold if you swapped the input and output ports. This is just an order of magnitude results, but a more accurate method requires treating each S-parameter spectrum as a transfer function.

### Pulse Response With Convolution

The other method that can be used to verify the correspondence between time-domain and frequency-domain crosstalk results is to simulate a pulse response using a convolution integral. This would be done with two transfer functions (NEXT and FEXT transfer functions), one for the near-end port and the other for the far-end port.

Normally, convolutions are used with an input pulse spectrum to calculate an output pulse that would be measured at a receiver. Here, we would input a voltage spectrum for an input pulse, and we would use the transfer function for our coupled interconnects to calculate the output pulse response at each port. S-parameters are essentially a transfer function, so they can be used with an input pulse waveform to determine the response at any other port in the system, including crosstalk.

1. Convert S-parameter data for crosstalk (NEXT or FEXT) to time-domain data with an inverse Fourier transform
2. Multiply the output from Step 1 by the time-domain input voltage waveform (converted to an input power)
3. Calculate the convolution between this product of time-domain functions
4. The output from Step 3 is a power waveform; convert this to a voltage using the victim interconnect’s impedance
5. The result from Step 4 is the crosstalk voltage waveform

This type of calculation is typically not performed in EDA tools because they may not include a built-in function for calculating a convolution or inverse Fourier transform. An external scripting program like MATLAB or Mathematica will include built-in functions for this calculation, and this can be performed in tabulated S-parameter data. Simply export the S-parameter data from your EDA program as a CSV file, and import it into your math scripting application to perform this analysis.

### What If the Converted Time-Domain Data is Different From the Simulated Data?

If the data calculated with a convolution or inverse Fourier transform does not exactly match the data from a time-domain crosstalk waveform simulation, this is to be expected to some extent. Ideally, these results should match, but in reality this is not always the case. The adaptive method and any interpolation used in the S-parameter simulation, as well as the windowing function, could distort the calculated time-domain results.

Because windowing has an outsized effect at higher frequencies (effectively truncation), the time domain waveform should have the greatest differences along fastest edge rates. Typically you can calculate a difference error between these two time-domain functions; if the results are small enough then the frequency-domain simulation should accurately reproduce time-domain data and can be considered accurate.

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