### Introduction

LeCroy’s SDA II Serial Data Analysis Package includes revised jitter calculation methods and enhancements to the SDA user interface and feature set.

In SDA II, the jitter calculation methods now include the Spectral Method, which is part of MJSQ, a de facto industry standard for jitter measurements. Users can now choose to use the spectral method or to use LeCroy’s NQ-Scale algorithm, which often yields superior results.

This document gives a qualitative description of the two choices: Dual Dirac Spectral and NQ-Scale. These choices can be found on the Jitter

These algorithms are quite different than the choices offered on SDA series scopes not equipped with the SDA II package. A separate application brief, LAB WM452, describes the methods used in the SDA package.

### Navigating the New SDA Screens

To measure jitter using the new SDA II software package, go to the Analysis menu and select Serial Data. This opens the main configuration screen, which operates as a flow chart. Selecting the button Setup Jitter Measurements opens the Jitter Measure configuration flowchart. (See Figure 1). A complete description of each step can be found in the Operator’s Manual.

### Jitter Measurements and the Dual Dirac Model

It is important to understand that the jitter measurements taken with the SDA II package (as well as the SDA package) are estimates based on the dual-Dirac jitter model and are not a simple calculation such as and RMS or standard deviation measurement. The importance of this fact cannot be understated!

The dual-Dirac jitter model describes jitter as having a dual-Dirac shape, with two Gaussian functions that grow with the bit error ratio. (See Figure 2 and the suggested readings for additional details). Since it is virtually impossible to acquire enough events to directly measure total jitter at extremely small bit error ratios such as the commonly used value of 10^{-12}, a model must be employed to extrapolate beyond the measurements actually taken.

In the dual-Dirac model, total jitter (Tj) is the sum of deterministic jitter (Dj) and random jitter (Rj), with Rj weighted by a multiplier corresponding to the bit error ratio. Note that the model doesn’t stipulate specific algorithms for determining Tj, Rj and Dj. A variety of methods for estimating jitter have been developed and refined over the years, and the methods employed by the SDA II package are the sum of nearly a decade of effort to devise the best algorithms possible.

### SDA II Methodology

The SDA II package uses the Time Interval Error measurement (TIE) as the source of the jitter measurements for the analysis. The measurements are aligned in time to create a “jitter track” and histogrammed as well. The first step in the analysis flow is to calculate data-dependent jitter (DDj). The result is removed from the jitter track to form the RjBUjTrack trace, and from the TIE histogram to form the RjBUjHist Histogram. With DDj removed, these traces are much more amenable to jitter analysis. Figure 3 shows the effect of removing DDj from the jitter Spectrum. In the “after” plot, the only remaining distinct peak is from a 2.6MHz modulation added to the signal. The peak can be seen easily in the first division. What’s left are the RjBUjTrack and RjBUjHist traces that are used by each jitter calculation method as described in the sections below.

### Dual-Dirac Spectral Method

The dual-Dirac Spectral method aims to quickly converge on jitter results that are accurate and that correlate well to other instruments. It stands apart from the Dual-Dirac Fit and NQ Fit methods in that it uses the RjBUjSpect jitter spectrum to determine the sigma value for the Gaussian distributions of the dual-Dirac model. The RjBUjSpect trace, which is the FFT of the RjBUjTrack, is first analyzed for peaks that can be associated with periodic jitter, like the 2.6MHz peak in the lower plot of Figure 3. These peaks are removed, and the remaining spectrum is associated with random jitter. This corresponds to the sigma value for the dual-Dirac Gaussian.

The value of sigma measured from the RjBUjSpect trace is used to fit the tails of the Rj+BUj Histogram, and the optimal mean positions of the Gaussians are determined. The next step is to add back the DDj by convolving it with the Rj+BUj Histogram to form the probability density function (PDF). The PDF is integrated from both sides toward the center to form the cumulative density function (CDF). The space between the sides of the CDF is the total jitter value Tj for that BER. To determine Rj and Dj, 4 points are taken from the CDF and fitted to the Dual Dirac model Tj = Dj + α(BER), where α(BER) is the confidence interval at a confidence level of 1-BER for a single “normal” Gaussian. (e.g. α = 14 for BER = 10^{-12}). The points on the CDF that are used in the fit are in the neighborhood of the user’s selection for BER (often 10^{-12}), and include the selected BER, 1 point above, and two below. For example, for a selected BER of 10^{-12}, the CDF points used for the fit are BER=10^{-11}, 10^{-12}, 10^{-13} and 10^{-14}. (Note: this algorithm is used beginning with firmware version 6.8.0.x. Previously, the fit used 7 points spanning from BER = 10^{-6} to 10^{-15}. Rj and Dj values are determined using the fitting method previously described in the "Dual Dirac Spectral Method" section. The results are posted to the Jitter Measurements table using the notation Tj(1e^{-12}), Dj(sp) and Rj(sp). (The "1e^{-12}" indicates that the user selected the BER level of -12 for the analysis)".

### NQ-Scale Method

The Dual-Dirac Fit method aims to determine values for Tj and Rj that are more accurate that the Dual-Dirac Spectral method by allowing more degrees of freedom when fitting the tails of the Rj+BUj histogram. Instead of using a spectral method to determine the sigma value of the dual-Dirac Gaussian distributions, the tails of the Rj+BUj histogram are fitted to Gaussian distributions that can have both different sigmas and different populations. The mean positions of the Gaussians are also determined as in the Dual-Dirac Spectral method. The remaining steps are the same as in Dual-Dirac Spectral: add back the DDj by convolving it with the Rj+BUj Histogram to form the probability density function (PDF). The PDF is integrated from both sides toward the center to form the cumulative density function (CDF). The space between the sides of the CDF is the total jitter value for that BER. As described previously, 4 points are taken from the CDF and a least squared fit is performed using the dual Gaussian formula: Tj = Rj * (Q@BER) + Dj. The results are posted to the Jitter Measurements table using the notation Tj(1e^{-12}), Dj(nq) and Rj(nq).

### Choosing the Best Method

The two methods offered by the SDA II package make different assumptions about the Gaussian distributions in the dual-Dirac model, and thus calculate different values for Tj, Rj and Dj. This often leads to the question “Which is the correct value?” This is a very good question without an exact answer. The NQ-Scale will typically converge on a more accurate result given its greater flexibility. More statistics are also required with NQ-Scale in order to return a stable result.

The Dual-Dirac Spectral method is known to return incorrect results when there is wide-band bounded uncorrelated jitter (BUj). In other words, deterministic jitter with energy that is spread across a wide frequency band such that it cannot be detected as a peak in the RjBUjSpect trace will be incorrectly interpreted as Rj. Examples include crosstalk from neighboring data channels and non-stationary periodic jitter sources like spreadspectrum clocks. The Dual-Dirac Spectral method also is hindered by its assumption that both Gaussians have the same underlying sigma and population. Nonetheless, the Dual-Dirac Spectral method is fast and returns correct results in the majority of cases. In general when the results for the models are markedly different, using NQ-Scale is highly recommended.

It is important to keep in mind that both methods return estimates of the total, random and deterministic jitter since the algorithms are model-based and are not a simple calculation based on the measured data.

### Additional Reading

- Fibre Channel MJSQ document:

ftp://t11member@ftp.t11.org/t11/pro/fc/mjsq/04-101v5.pdf (t11.org membership required) - A Comparison of Methods for Estimating Total Jitter Concerning Precision, Accuracy and Robustness:

http://www.lecroy.com/tm/Library/WhitePapers/PDF/LeCroy_Jitter_Methods_DesignCon2007.pdf - 6 Tales of Rj and Dj:

http://www.lecroy.com/tm/Library/WhitePapers/PDF/WP_TechBrief_Rj_and_Dj.pdf - LAB WM452: Understanding the Choices for Jitter Calculation Method

http://www.lecroy.com/tm/Library/LABs/PDF/LAB_WM452.pdf