Tailfit Method for PDF

Let us look at a jitter PDF (or histogram) for edge transitions. Such a jitter PDF reflects the mixture of DJ and RJ processes associated with the edge transitions. Here we assume that the timing reference used to obtain the jitter PDF is the ideal clock. This information has been available for many years, but, to our knowledge, no theory or method had been established to decompose the total jitter (TJ) PDF into DJ and RJ components until ^[1]. The metrics that had been used to quantify jitter are simple and straightforward statistical pk-pk value and 1 σ standard deviation, based on the entire PDF that has both DJ and RJ components. With the knowledge of jitter model introduced in Chapter 4, "Jitter, Noise, BER (JNB), and Interrelationships," it is clear that the correct way to quantify jitter is to use the right metric for each jitter component. For example, use pk-pk value for DJ because it is bounded, and use 1 σ standard deviation for RJ because it is unbounded and random. We assume that RJ is white and a Gaussian is a good model for it.

The following sections present a method/algorithm to decompose a total jitter PDF into DJ and RJ PDFs through the Tailfit method.

5.2.1.1. Theory for Total Jitter PDF and Its Relationship to DJ and RJ PDFs

An obvious consequence of DJ and RJ interaction through the convolution process is that the tail part of the PDF reflects the random jitter process. If the random jitter is due to the random motion of electrons or holes in a semiconductor, the random velocity of these particles in an equilibrium state is best described by a Gaussian distribution. This gives another justification for using the Gaussian model to describe the random jitter. Because multitemperature particle distribution is possible, a multiple-Gaussian distribution function may be needed to model certain random jitter processes. A single Gaussian jitter PDF is defined as follows:

Equation 5.1

where Δt is the jitter and µ and σ are the Gaussian mean and standard deviation, respectively.

From an observational point of view, the measured or simulated total jitter histogram represents the scaled-up total jitter PDF. The tail part of the distribution should largely be determined by the random jitter process, which, in general, has a Gaussian type of distribution.

The random jitter can be quantified by the 1 σ (or rms) standard deviation of Gaussian distribution, and DJ can be quantified by the

pk-pk value.

In the absence of DJ, the total jitter PDF should be a Gaussian. Under this condition, only one maximum in the distribution corresponds to zero DJ. The rms of the PDF equals σ in this case. When both DJ and RJ processes play together, the resulting jitter PDF is

broadened and is no longer a Gaussian as a whole. On the other hand, both ends of the distribution should still keep the Gaussian type of tails, because DJ PDF is bounded. These tail part distributions can be used to deduce the RJ distribution parameters. Because of the DJ, the mean of each tail is no longer at the same location, and multi-peaks can be present in the PDF. The time distance between far-left peak position and far-right peak position gives rise to the DJ pk-pk value. Figure 5.1 shows such a broadened total jitter PDF in the presence of both DJ and RJ.

Figure 5.1. Total jitter PDF and Tailfits.

If there is no bias and statistical sampling noise or the asymmetry in rising and falling edges in the measurement, the two tails, which represent the random process, should be symmetrical. Because it is not possible to completely randomize measurements and reduce the sampling noise to zero, and because the rising and falling edges are different in a practical system, the σ values for the left and far-right Gaussian tails may not be the same. The total RJ σ value should be the average of these two, and DJ pk-pk is the distance between two peaks of far-left and far-right Gaussian tails:

Equation 5.2

and

Equation 5.3

The average gives a typical value or mostly likely value for RJ. If a conservative RJ estimation is desired, the overall RJ should be the larger one among the left and right RJ rms—namely, σ_t = max(σ_r, σ_l).

5.2.1.2. Implementation Algorithm

Identifying the tail parts of the PDF, and then fitting them with the Gaussian function, are the key to DJ and RJ separation with a given measured/simulated jitter PDF or histogram. It is impossible to tell where the tail part of the PDF is without studying each individual data and its relationship with the neighboring data. The easiest way to identify a tail part is through the graphical display of the PDF and picking up the tail parts via visual inspection. The disadvantage of such an approach is that it lacks repeatability, and it cannot be adopted for automated testing. Therefore, the requirements for a search algorithm should be as follows:

z It can find the true tail part quickly, accurately, and repeatedly.

z It has to be automatic (no user intervention or visual inspection are required).

The fitting procedure should be able to deal with the statistical fluctuation and factor this into the fitting routines. The tail part has the lowest event counts, and statistical uncertainty can be high. A simple, straightforward, least-square fit algorithm would not work well, because the statistical error would propagate into the fitting parameters. This in turn gives rise to large errors in DJ and RJ estimation. A more advanced nonlinear fitting algorithm is needed to meet these requirements.

5.2.1.2.1. Tail PDF Identification

One of the key characteristics of a Gaussian tail is its monotonicity. That means that the left side of the tail monotonically increases, and the right side of the tail monotonically decreases. Due to the presence of DJ, monotonicity will break and this in turn will create local maximums near the left and right parts of the tail. Without DJ, only one maximum corresponds to the mean of the distribution.

A difficult issue that a tail-search algorithm faces is statistical fluctuation. In the presence of statistical fluctuations, the monotonicity of a real Gaussian distribution is no longer true, and using the raw fluctuated data to find the local maximum points for both left and right tails will be extremely difficult, if not impossible. The solution should be to first filter out the fluctuation noise and then use the smoothed PDF to locate the maximum points. There are two general ways to achieve this. One is through direct time domain averaging.

Another is to use Fourier Transformation (FT) to get the spectrum, apply a low-pass filter, and do inverse FT (IFT). For time-domain averaging, you need to determine how many data points to use, because this determines the smooth level or, equivalently, the cutoff frequency of the low-pass filter. In the FT/IFT approach, you must determine the filter's bandwidth. The number of averaging points and filter bandwidth may need to be adjusted, depending on the fluctuation in noise frequency and amplitude. In other words, a rule-based artificial-intelligence algorithm must be used to enable the smoothing algorithm to deal with a wide range of fluctuation amplitudes and frequencies. This is an important requirement to guarantee that smoothing washes away only the unwanted fluctuation noise, not the true feature of the jitter histogram.

As soon as the smoothed measured PDF is obtained either through time domain averaging or time-frequency domain FT-low-pass filtering-IFT, the maximum locations can be found by calculating the first- and second-order derivatives of the jitter PDF. The only maximum points of interest are the first maximum from the far left and the first maximum from the far right.

5.2.1.2.2. Tailfitting

You should use a fitting algorithm that weights the data record based on the quality of the data. The bigger the error, the smaller the role it should play in minimizing the difference between the model expected value and the measured value. Thus, we need to use χ² as a gauge to determine how good the fit is. The fitting function is Gaussian, and the fitting algorithm is nonlinear, so it can handle both linear and nonlinear fitting functions. For details on χ² theory, refer to ^[2].

χ² fitting is an iterative process, in contrast to linear equation solving in the case of linear least-squared fitting. The final answer is obtained when the iteration converges. For this reason, initial values of the fitting parameters are needed. A primitive way to do this is to try different initial values and see whether they converge to the same final values. If the initial guessed values are far from the final actual values, they may either take longer to converge or get stuck at a local χ² minimum and never converge to the final global χ² minimum point. Calculations should be carried out to estimate the initial fitting parameters by using the tail parts of PDF so that the initial fitting parameters are close to the final converging values. This also makes the iteration converge rapidly and avoids stuck-in-local minimum (pivot). We would like to emphasize that the χ² method has proven to be very robust.

5.2.1.3. Monte Carlo Simulations

Another complication for a measured/simulated PDF is the sampling noise associated with it in practical application. The goal is to best determine the DJ and RJ PDF and associated parameters in the presence of the sampling noise. A good PDF-based jitter separation method should be immune to sampling noise. This section evaluates the χ² Tailfit method via Monte Carlo simulation to determine its accuracy in dealing with a "noisy" measured PDF.

5.2.1.3.1. PDF with Statistical Noise

We started with a known bimodal PDF, which is represented by two added Gaussian distributions superimposed with random noise.

This makes the overall PDF close to that of a practical one. For example, in the presence of a PJ (approximated by a dual-Dirac), RJ (Gaussian), and sampling noise, the overall PDF is two separated Gaussians superimposed with sampling noise. Mathematically, such a PDF can be represented by the following equation:

Equation 5.4

where N₁ and N_r are the peak values, µ_l and µ_r are means, and σ_l and σ_r are standard deviations for two Gaussian distributions. ran(t) is a random-number-generating function based on the Monte Carlo method. It has a mean of 0 and a standard deviation of unity. N_n is the amplitude for the random number envelopes. For Monte Carlo-based random-number generation, refer to ^[3].

A good search and fitting algorithm should return the fitted parameters that are consistent with those predefined in the simulation. A critical test is as follows: Can an accurate fitting parameter be obtained in the presence of significant statistical fluctuations? In other words, N_n is a significant portion of N₁ or N_r. Otherwise, no accurate parameters can be obtained, because all the real-world measurements are subject to statistical fluctuation.

5.2.1.3.2. Fitting Results

There are two scenarios that we need to treat differently. The first is when two Gaussian distributions are well separated—that is, when µ_r – µ_l > >σ_l + σ_r. Under such conditions, the two distributions are not mixed, and the tail parts up to the point of the first maximum are essentially uncontaminated. Therefore, we could use both left and right tail data from the lower value to the first maximums. This enhances the tail data usage, and the Gaussian model is better constrained. This can correspond to the case when DJ >> 2 σ in the jitter analysis, as shown in Figure 5.2.

Figure 5.2. Two well-separated Gaussians: N_n = 0, with no statistical fluctuation (a), and N_n = 30, with significant statistical fluctuation (b). The lower panels show the overlay of the original PDF with the tailfit Gaussian.

Source: M. P. Li, J. Wilstrup, R. Jesson, and D. Petrich, "A New Method for Jitter Decomposition Through Its Distribution Tail Fitting," International Test Conference (ITC), 1999. (© 1999, IEEE)

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In both cases, the simulation results suggest that the fitted parameters are consistent with the predetermined parameters to within 2.8 %, even when the statistical fluctuation reaches 15% of the total PDF peak.

The second case is when two Gaussian distributions are not well separated—namely, µ_r – µ_l < σ_l + σ_r. Under such condition, the contamination of two distributions could extend to the tail parts. As a result, you should use only the lower parts of the tails for the fitting to minimize contamination. A conservative way is to use the tail part from the lowest event count to half of the N_l or N_r. This can correspond to the case when DJ < 2 σ in the jitter analysis applications.

Figure 5.3 shows the results corresponding to two mixed Gaussian PDFs (µ_r – µ_l < σ_l + σ_r ). In each case, nonfluctuated (N_n = 0) and fluctuated (N_n 0) scenarios are considered. In both cases, the fitted parameters are consistent with the predetermined parameters to within 4%, even when the statistical fluctuation reaches 15% of the total PDF peak.

Figure 5.3. The same as Figure 5.2, but the two Gaussians are very close to each other. N_n = 0, with no statistical fluctuation (a), and N_n = 30, with significant statistical fluctuation (b). The lower panels show the overlay of the original PDF with the tailfit

Gaussian.

Source: M. P. Li, J. Wilstrup, R. Jesson, and D. Petrich, "A New Method for Jitter Decomposition Through Its Distribution Tail Fitting," International Test Conference (ITC), 1999. (© 1999, IEEE)

[View full size image]