TOTAL JITTER MEASUREMENT THROUGH THE EXTRAPOLATION OF JITTER HISTOGRAMS

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1 T E C H N I C A L B R I E F TOTAL JITTER MEASUREMENT THROUGH THE EXTRAPOLATION OF JITTER HISTOGRAMS Dr. Martin Miller, Author Chief Scientist, LeCroy Corporation January 27, 2005 The determination of total jitter is critical to the evaluation of current serial data standards whose high symbol rates leave small jitter margins. The accuracy of jitter measurements is challenged by these smaller margins. Most current jitter measurement techniques estimate the total jitter by breaking the measurement down into random and deterministic components, and then add them together using a multiplier in front of the random component. These methods assume a normally distributed random component and can overestimate the total jitter. The method presented in this paper determines the total jitter by extrapolating the histogram of time interval error measurements without assuming a Gaussian distribution. This method has the unique ability of being insensitive to the distribution of random and deterministic components of the jitter, since the extrapolation is not dependent on separate measurements of these components. Two parts of the distribution are considered separately, specifically the first and last N percentiles of the distribution. These are determined by locating the bins of the distribution that contain N percent of the total population, summing first from the left, and then from the right. In statistics, these horizontal coordinates are called the Nth and (100-N)th percentiles. The choice of N is arbitrary 1 ; however, it does reflect the concern that only the data not central to the distribution can be used to extrapolate. The reason for this is that bounded noise will have the strongest effect in the central part of the distribution. The next step in the process is to represent the non-zero bins in the histogram by the log of their populations. Assuming that the extremes of the distribution behave like a Gaussian distribution s extremes (see box), the functional dependence should be a pure quadratic. Normal (Gaussian) Statistics: The probability distribution function is expressed as EXTRAPOLATING THE DISTRIBUTION The basic requirement is to analyze the histogram of jitter measurements in such a way as to extrapolate the extremes of the distribution, with the ultimate purpose of estimating confidence limits. The confidence limits are normally expressed in terms of an equivalent bit error rate. where µ is the mean of the distribution, and sigma is the Gaussian standard-deviation, So the relationship translated to a logarithmic scale would be: This form is quadratic but with the linear (first order) term exactly zero. Observing this expected behavior for the extremes (or tails) of the distribution, we can obtain an estimate for the functional form, by performing an analytic 2 weighted regression (or Figure 1. A distribution and its extremes 1 N is not necessarily an integer value. It is quite common to use only a small fraction of a percentile (e.g., 0.03% of the population), and the choice of N depends on how populous the recorded distribution is. 2 Analytic as contrasted to numerical, meaning that the solution to the regression is given directly by forming certain sums of the derivatives of the functional form w.r.t. the free parameters. See Appendix for the derivation of the 3- and 2-parameter forms of this fit. TECHNICAL BRIEF THE EXTRAPOLATION OF JITTER HISTOGRAMS 1

population (accordingly adjusted for the logarithmic scale).

2 weighted fit ). The functional form is chosen to be the pure quadratic form with a linear term equal to zero. The weights assigned to each point contributing to the fit are determined according to the strength of the statistical certainty of the bin population, which is proportional to the square root of the population (accordingly adjusted for the logarithmic scale). The logarithm of 1 (as for a population of 1) is zero, and to retain some weight for singleton populations, the weights used are actually: It is not as simple as fitting the left-hand and right-hand sides to the functional form (see box). First, the assumption is made that the effective mean (in the analogy of the Gaussian behavior) is at the median of the original distribution. This is the case only if there is no bounded contribution to the distribution. The fact is, if there is any bounded noise contribution to the distribution, the effective mean for each of the extreme behaviors will be better estimated as displaced toward the extreme in question. For this reason, several fits are made, adapting each time the effective mean (the mu or µ in the equations above) until the best chi squared is obtained. The objective is not to estimate in this fashion the bounded noise, but to obtain the best quality fit to the behavior of the extremes. Using the analytic fits for the right-hand and left-hand extremes, a new composite distribution is formed. The middle 100 2N percent of the population (nominally the bins between the two sets used above) are left unchanged, and the population yielded by fit results is used to replace all of the distribution below the Nth and above the 100-Nth percentiles. This resulting distribution is then normalized to have a total area of one. This final distribution can then be further analyzed. OBTAINING CONFIDENCE LIMITS FROM THE EXTRAPOLATED DISTRIBUTION The confidence interval (with some implied confidence level) for a measurement is the interval within which you can expect the measurement to fall for a fraction of the observations defined by the confidence level. For example, for a theoretically perfect Gaussian distribution, and a confidence level of 99% (or 0.99), the confidence interval for the peak-peak variation of values is ±2.73 sigma. The confidence interval is obtained by (numerically) integrating the Gaussian distribution from the left and determining the displacement from the mean (zero mean for this example) at which the integral attains 0.5% from the left; and, doing the same integration from the right, the displacement from the mean, which attains 0.5% for a total of 1% of the population exceeding those limits. These two limits are called the confidence limits for peak-to-peak variation, and their difference, is called the confidence interval. The same method is used to determine the confidence interval (and limits) for other confidence levels. The confidence interval is related to the total jitter in that this interval is the range of edge positions that a data transition can have for a given confidence level. A bit error will occur for any edge that is outside the expected range defined by the confidence interval so if we know, for example, that 99% of the edges fall within a certain range, then there is a 1% chance of an edge falling outside this range and causing a bit error. In this case the total jitter will be determined at a bit error rate of 1% or SIMULATION RESULTS In order to test the algorithm for total jitter measurement, it is helpful to perform measurements on simulated data signals with controlled amounts of jitter. While simulation is not sufficient for proving the effectiveness of this, or any, measurement, they can show the effectiveness of the measurement exclusive of any system-related effects. Below are two graphs (superimposed) relating measured confidence interval for a theoretical jitter (pure 1 picosecond rms Gaussian) distribution, and a Monte-Carlo jitter distribution (acquired, accumulated, and extrapolated by the methods described above). Figure 2. Analysis of Total Jitter (numerical vs. Monte-Carlo) TECHNICAL BRIEF THE EXTRAPOLATION OF JITTER HISTOGRAMS 2

3 Real systems, of course, do not exhibit purely Gaussian jitter. In general the jitter distributions encountered contain both bounded and unbounded random components as well as deterministic components. Below are several simulation runs using combinations of bounded and Gaussian distributions. The bounded distributions consist of triangular, rectangular (uniform), and dual Dirac. The dual Dirac distribution is significant because it is the model used to represent deterministic jitter in many jitter analysis methods such as the Fibrechannel MJSQ (methods for jitter and signal quality). The uniform and triangular distributions were chosen because they represent examples of the types of random jitter encountered in real systems. The plots show the accumulated histogram of jitter measurements on the simulated data superimposed with the confidence limits plot on a logarithmic scale extending from 1 at the top with each major vertical division at -2 decades. The width is shown at a probability of corresponding to a bit error rate of Figure 3 shows a 1 ps rms pure Gaussian distribution. With a confidence interval at (1-confidence) the total jitter is ~14 ps. The subsequent distributions show this Gaussian combined with the three bounded distributions with a range of 10 ps. The contribution to the total jitter, or confidence 10-12, increases each time by 10 ps. These examples are shown to illustrate that bounded contributors are accounted for correctly in this analysis. A common procedure for measuring the total jitter in a data signal is to first estimate the standard deviation of the random part and, assuming this to be Gaussian, multiply this by the number of standard deviations corresponding to the desired bit error rate. The danger of this approach is that it assumes that the random jitter is purely Gaussian. This assumption can lead to over-estimation of the total jitter in cases where the random jitter is not Gaussian, which is often the case. Figure 4. 1 ps rms + 10 ps bounded triangular, = ps Figure 3. 1 ps rms + 0 ps bounded, = ps To further demonstrate the robust behavior, examples of purely bounded distributions are shown 3 in Figures 7, 8, and 9. Note that the overestimation of the is minimal for these extreme cases. Even for the sum of three 5 ps triangular, the overestimate is only 30% (19.19 ps compared to 15 ps range). See Figure 10. Figure 5. 1 ps rms + 10 ps bounded uniform, = ps 3 The dual Dirac is excluded from this test; however, in the presence of the smallest amount of variation or jitter, it is also well behaved. TECHNICAL BRIEF THE EXTRAPOLATION OF JITTER HISTOGRAMS 3

Figure 6. 1 ps rms + 10 ps bounded dual Dirac, Width @ 10-12 = 23.92 ps Figure 7.

4 Figure 6. 1 ps rms + 10 ps bounded dual Dirac, = ps Figure 7. 0 ps rms + 10 ps bounded triangular, = 5.79 ps Figure 8. 0 ps rms + 10 ps bounded uniform, = 5.72 ps TECHNICAL BRIEF THE EXTRAPOLATION OF JITTER HISTOGRAMS 4

5 APPENDIX A: PURE QUADRATIC FIT We look for an analytic solution to the best fit (minimal nonnormalized chi squared) for fitting data values to the functional form: So this chi squared is given by: And the first partial derivative with respect to α (alpha) is: Figure fs rms + 30 ps bounded dual Dirac, = 31.7 ps and with respect to β (beta) (note the sums are over i = 1 to i = N): Setting these two partial derivatives to zero (in order to find their extreme limits (hopefully the minimum): and And so, by inspection (a solution for beta is found): and And again solving by substitution and inspection, we find the solution for alpha: Figure ps rms + 3x summed 5 ps bounded triangular, = ps TECHNICAL BRIEF THE EXTRAPOLATION OF JITTER HISTOGRAMS 5

6 And further, by inspection, the weighted form of this analytic solutions for alpha and beta are: Setting these two partial derivatives to zero (in order to find their extreme limits, hopefully the minimum): and and Reorganizing these APPENDIX B: SECOND ORDER POLYNOMIAL CURVE FIT We look for an analytic solution to the best fit (minimal nonnormalized chi squared) for fitting data values to the functional form (with quadratic, linear, and constant terms): and so, we express these three conditions as a matrix equation: So this chi squared is given by: The first partial derivative with respect to α (alpha) is: Thus determining the inverse 4 of the left-hand matrix, we can obtain the best fit values for alpha, beta, and gamma, by applying that inverse to the right-hand side vector: and with respect to β (beta; note the sums are over i = 1 to i = N): and with respect to γ (gamma; note the sums are over i = 1 to i = N): Again, it should be clear that for the weighted case, where the function to be minimized is: 4 The details of how to obtain the inverse of a small matrix are not provided here. TECHNICAL BRIEF THE EXTRAPOLATION OF JITTER HISTOGRAMS 6

7 the solutions for the fit parameters are: These choices, while they are somewhat arbitrary, are (in my opinion) reasonable and give some weight to the fact of the unpopulated bins. Another important matter is also to prevent truly bounded distributions from being extrapolated over-pessimistically. That is, what to do about extremal bins that have populations of more than 2, and absolutely nothing observed beyond. Such a distribution would look like this: APPENDIX C: PRAGMATISM, REALITY, AND MAKING THE EXTRAPOLATION ROBUST When the extreme population is translated to the logdomain, you might ask what shall I do with the zero populations? In particular, this is an interesting question when there are holes in the populated bins (i.e., empty bins between populated bins). The procedure that has been adopted is to treat the bins by giving them a probability of 1/2 the reciprocal of the entire population (as an estimate of their probability, since these events have not been observed) and a weight equal to the weight for the nearest (more distant) populated bin. So for the diagram below, the log-domain values for the nonpopulated but interior bins would look something like the orange points. Their weight would be the same as the two rightmost blue points. For this case (whenever the extreme bin has population > 2) the next more extreme bin is associated a population of 1/2 (i.e., a probability of 1/2 the reciprocal of the total histogram population) and a weight associated identical to the extreme bin. In this way, as the weight grows (more hits in the extreme bin) the weight of the next more-extreme bin grows, and the fit is forced to not simply extrapolate the trend of the last few populated bins. Please note: when any Gaussian components are present in the measurement, this case does not occur. It is always the case that the extremes are barely populated, with single hit outliers, and usually spaces between populated bins. Please note: the objective of these details is to prohibit the extrapolation from being overly pessimistic, and in particular, to limit the predictions for truly bounded distributions. TECHNICAL BRIEF THE EXTRAPOLATION OF JITTER HISTOGRAMS 7

8 STRATEGY FOR DETERMINING N The strategy for determining N, or the percentile used at each extreme of the distribution is simple and straightforward. Start with 7% as a default percentile. If possible, reduce the percentile until you have only 1000 events in each extreme, or if you find you only have a range of 3 bins (don t reduce the percentile past 3 bins of range). That s about it for choosing N. I would offer one more comment: It is important to use a sufficient number of histogram bins so you do not find yourself with a number of bins less than 2 or 3 for a 0.1% percentile range. That means in general you should use 200, 500 or even 1000 bins in the histogram. Using too few bins (like 20 or 50) can lead to gross errors in extrapolation results. The diagrams above are simplifications and supplied as a visualization aid only. It should not lead you to believe a 50-bin distribution is a good idea by LeCroy Corporation. All rights reserved. Specifications subject to change without notice Lecroy TECHNICAL BRIEF THE EXTRAPOLATION OF JITTER HISTOGRAMS 8

Index I-1. in one variable, solution set of, 474 solving by factoring, 473 cubic function definition, 394 graphs of, 394 x-intercepts on, 474

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