DJ Variation Error for a Given BER CDF Value

The purpose of the study in this section is to investigate how much variation there is in DJ pk-pk estimation when TJ at BER = 10^–12 is given for these three different DJ PDFs. This is to emulate the case when DJ is estimated from the measured BER CDF function where the exact form of DJ PDF is unknown.

We have found that different BER CDF functions from different DJ PDFs can give rise to the same TJ value at a certain BER level.

When the same TJ value is achieved, the DJ pk-pk values are significantly different for different forms of DJ PDF. The results are plotted in Figure 5.13, and different DJ values are shown in Table 5.3.

Figure 5.13. BER CDFs from different DJ PDFs give rise to the same TJ value at BER = 10^–12. A fixed RJ σ = 0.05 UI is used.

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)

Table 5.2. Comparison of TJ at BER = 10^–12; Jitter Is in UI

DJ PDFs DJ Pk-to-Pk RJ σ TJ at 10^–12 Diff Diff%

Triangular 0.2 0.05 0.844 0 0

Rectangular 0.2 0.05 0.866 +0.022 2.6%

dual-Dirac 0.2 0.05 0.926 +0.082 9.7%

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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)

Table 5.3 indicates that the differences in DJ pk-pk can be as high as 35%, or 0.11 UI, and they all give rise to the same (approximately) TJ at BER = 10^–12. This implies that DJ will be not well constrained if they were estimated from measured or simulated BER CDF with a straightforward dual-Dirac model (that is, without fitting). Moreover, the results in Table 5.3 clearly show that a dual-Dirac PDF has the lowest DJ value, while the triangular DJ PDF has the highest DJ value when the same TJ at BER = 10^–12 is achieved. This tells us that a dual-Dirac PDF can potentially overestimate the actual DJ pk-pk value if it is used to estimate the DJ based on a measured BER CDF. Although trying to cover all the DJ PDF scenarios can be time-consuming, you can intuitively see that this statement generally holds, especially when actual DJ PDF is not a dual-Dirac.

You have learned several very important facts and consequences when the dual-Dirac model is used for applications where DJ PDF is not a dual-Dirac without Tailfit:

z For the same DJ pk-pk value and the same RJ σ, the dual-Dirac method gives the highest TJ estimation among other common DJ PDFs.

z For the same RJ σ, to achieve the same TJ value at a certain BER level, DJ pk-pk values estimated from the dual-Dirac method will be the smallest among other common DJ PDFs.

z For a given BER CDF, the dual-Dirac method can overestimate actual RJ σ and underestimate DJ pk-pk value.

z The error in estimating DJ, RJ, and TJ varies, depending on the exact application. An error up to ~35% in DJ and RJ estimation is clear from a limited simulation, and up to 50% was observed from experiments.^[6]

These results strongly suggest that you must be careful when using the dual-Dirac DJ PDF model for practical jitter PDF and BER CDF analysis unless the DJ PDF is known to be a dual-Dirac.

Table 5.3. The Same TJ Value from Different DJ PDFs

DJ PDFs TJ at 10^–12 RJ σ DJ Pk-Pk DJ Diff DJ Diff%

Triangular 0.926 0.05 0.31 0 0

Rectangular 0.929 0.05 0.27 –0.04 –12.9%

dual-Dirac 0.926 0.05 0.20 –0.11 –35.5%

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Book: Jitter, Noise, and Signal Integrity at High-Speed

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Information Theory Computer Science Mike Peng Li Prentice Hall Jitter, Noise, and Signal Integrity at High-Speed

5.5. Summary

This chapter started by describing why jitter separation is needed, from the viewpoints of understanding the jitter process, as well as the practical values of having jitter components such as DJ and RJ. We then moved to the details of separating jitter into its components based on the jitter PDF in section 5.2. The Tailfit method was introduced, covering theory, simulation, and experimental results. The key is that the tail parts of the PDF are dominated by the random jitter process, and it can best be modeled by a Gaussian distribution.

The Tailfit method gives rise to the parameters defining the tail Gaussian distribution, including means and sigmas, and that in turn yield DJ pk-pk estimation and RJ sigma estimations. DJ PDF determination via deconvolution also was discussed in this section.

Section 5.3 introduced the Tailfit algorithm application and implementation BER CDF as the base function. Two scenarios were considered: the raw BER CDF-based, and the Q-space-based involving the transformation for both base data function and base model from BER CDF space to Q-space. In BER CDF space, Gaussian function becomes an integrated Gaussian, which is essentially a complementary error function, and in the Q-space, the Gaussian function becomes a linear function. The advantages and disadvantages of Tailfit in each space were also given. A quick and conditional accurate linear equation method for estimating TJ was derived under interesting DJ, RJ, and BER conditions. Section 5.4 discussed the accuracy of using the dual-Dirac as the model for the DJ PDF as a straightforward application without involving the tailfit. The results suggest that the error can be significant and caution must be exercised when using dual-Dirac DJ PDF for an arbitrary DJ PDF.

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Book: Jitter, Noise, and Signal Integrity at High-Speed

Section: Chapter 5. Jitter and Noise Separation and Analysis in the Statistical Domain

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Information Theory Computer Science Mike Peng Li Prentice Hall Jitter, Noise, and Signal Integrity at High-Speed

Endnotes

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

2. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, Inc., 1992.

3. Knuth, D. E., Seminumerical Algorithm, Second Edition, Addison-Wesley, 1981.

4. J. Sun, M. Li, and J. Wilstrup, "A Demonstration of Deterministic Jitter (DJ)," IEEE Instrumentation and Measurement Technology Conference (IMTC), 2002.

5. G. Arfken, Mathematical Methods for Physicists, Third Edition, Academic Press, Inc., 1985.

6. M. P. Li and J. Wilstrup, "On the Accuracy of Jitter Separation from Bit Error Rate Function," IEEE International Test Conference (ITC), 2002.

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No part of any chapter or book may be reproduced or transmitted in any form by any means without the prior written permission for reprints and excerpts from the publisher of the book or chapter. Redistribution or other use that violates the fair use privilege under U.S. copyright laws (see 17 USC107) or that otherwise

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6. Jitter and Noise Separation and Analysis in the Time and Frequency