User`s guide

APPENDIX B - Tailfit™ Theory
The tail-part of an histogram distribution reflects the random jitter (RJ) process. Physically, random jitter is due to the
random motion of particles within a device or transmission medium. The random velocity of these particles in an
equilibrium state is best described as a Gaussian distribution. Therefore, RJ is naturally modeled by a Gaussian function.
Since multi-temperature particle distribution is possible, a multi-Gaussian distribution function may be needed to model
certain RJ processes.
Based on their definitions, deterministic jitter (DJ) is bounded and random jitter (RJ) is un-bounded. The measured total
jitter histogram represents the scaled-up, total jitter probability distribution function (PDF). On the other hand, the
convolution of RJ PDF with DJ PDF gives the total PDF, if DJ and RJ processes are independent. In most cases, such an
assumption is valid. Therefore the tail part of the distribution is mostly determined by the J, which, in general, has a
Gaussian-type distribution. The random noise can be quantified by the standard deviation (or 1 s rms value) of the
Gaussian distribution. Depending on the error coverage range, the total RJ can be a multiple of the s, determined from
the Gaussian distribution.
In the absence of DJ, a histogram of the jitter should roughly be a Gaussian distribution. Under this condition, there is
only one peak in the distribution that corresponds to zero DJ. The rms RJ is the s value. When both DJ and RJ are
present, the measured jitter distribution will be broadened and non-Gaussian as a whole. On the other hand, both ends of
the distribution should retain Gaussian-type components. These tail component distributions can be used to determine
the RJ number. Because of the DJ, the mean of each tail is no longer the same and multi-peaks can be present in the
histogram.
If there is no bias or statistical sampling noise in the measurement, the two tails, which represent the random process,
should be symmetrical. Since it is not possible to completely randomize measurements and reduce the sampling noise to
zero, the s values for the far left and right Gaussian tails may not be the same. The total RJ value should be the average
of these two.
A fitting algorithm that weights the data record based on the quality of each datum should be used. The bigger the error,
the smaller role it should play in minimizing the difference between the model's expected and measured values. Thus,
goodness-of-fit is used 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 non-linear fitting functions.
The modified least-square fitting is an iterating 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.
When a tail-fit is successfully completed, the calculated tail-fits are plotted on top of the raw histogram and values for
the Deterministic Jitter, Random Jitter, Chi-square goodness of fit and Total Jitter are displayed. You can also view the
resulting Bathtub Plot that is based on the PDF of the raw histogram with extrapolated tails calculated from the tail-fit.
The Total Jitter is extracted directly from the Bathtub Plot. See the following Tailfit Enabled section.
Note: For Histogram Tool only - The Total Jitter Specification (in Time) that is used for this calculation is User Defined,
make sure that reasonable values are assigned for this as well as the Bit Error Probability. For dataCOM Tools, the Total
Jitter Specification is Fixed at 1 UI. The user may still choose the Bit Error Probability at which the TJ is read. Refer to
Technical Bulletin TB 9 "A New Method for RJ/DJ Separation"
© WAVECREST Corporation 2005 Appendix B
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