Signal Integrity Analysis Fundamentals 10 - Overall Jitter Algorithm for Serial Data Testing

In the test of high-speed serial data, jitter test is very important. In the jitter test of serial data, jitter is defined as the deviation between the edge of the signal and its reference clock. For the quantification of jitter measurement value, there are usually two parameters: peak-to-peak value and effective value of jitter. However, as the measurement time increases, the peak-to-peak value of jitter continues to increase, and the jitter value cannot be directly linked to the bit error rate. Therefore, for jitter testing, the peak-to-peak value of jitter is not an ideal indicator to measure the performance of devices and systems.
Total jitter (Tj) is the peak-to-peak value of jitter under a certain bit error rate (BER). In many serial data specifications, it is usually necessary to measure Tj with a bit error rate of 10e-12, abbreviated as Tj@BER=10e-12 . For the measurement of Tj with a BER less than 10e-8, usually only the bit error rate tester BERT can directly measure it. For an oscilloscope, assuming that the high-speed signal is 2.5Gbps PCIe, the duration of a single bit is Unit interval = 400ps, and assuming that the oscilloscope sampling rate is 20G sampling rate, then 1 bit includes 400ps/50ps = 8 sampling points. Analyzing 1M bits at a time requires 8M storage depth. If you want to measure the jitter of 10 bits, you need to let the oscilloscope scan 100 times at 8M storage depth. Since the oscilloscope calculates jitter at 8Mpts, it takes a long time to repeat the test 100 times. Therefore, the overall jitter when the oscilloscope measures a bit error rate less than 10e-12 must be estimated by some algorithm Tj.

Figure 1: TIE jitter diagram and jitter probability density function (PDF)

The algorithm for solving jitter based on oscilloscope is usually observed and analyzed in three fields, namely time domain, frequency domain, and statistical domain. For example, TIE track is the function of TIE jitter in the time domain; by analyzing the jitter spectrum in the frequency domain, periodic jitter Pj and random jitter Rj can be calculated; TIE histogram and Tj's probability density function (PDF) are used to analyze jitter in the statistical domain.
For the calculation of overall jitter, it is usually analyzed from the statistical domain, that is, the jitter histogram, probability density function PDF and cumulative distribution function (CDF) are analyzed. The
probability density function PDF is defined as: For a real random variable X, any function that meets the following conditions 
can be defined as its probability density function:
The process of generating PDF, CDF and bathtub curve from TIE histogram is briefly described in Figure 2 below.
In the first step, the X-axis is the jitter value, and the Y-axis is the number of samples at a certain jitter value. The oscilloscope measures the deviation of each edge of each signal from the reference clock (ie, TIE), and counts the number of edges at a certain jitter value to obtain a histogram of TIE.
In the second step, the histogram is normalized, that is, the number of each square column in the histogram is divided by the total number of samples to obtain the probability of occurrence of each jitter value. In this step, the probability density function PDF of TIE can be obtained.

Figure 2: Jitter Histogram and PDF
In the third step, the PDF histogram is integrated from the left and right sides to the center. Assuming that the signal edge exceeds the distance x relative to the ideal position, it may cause bit errors. The bit error rate is the integral of the PDF from x to ∞ or -∞ (∞ when x is greater than 0 and -∞ when x is less than 0): BER(x) = 
=1-CDF(x). Then, the dark blue histogram in Figure 3 is obtained after taking the logarithm of the Y axis. As shown, due to the small number of test samples, the probability of the shortest histogram (i.e., bit error rate) is only 1%=10e-2. To calculate the BER of 10e-12, it is necessary to interpolate the existing BER histogram;
in the fourth step (as shown in the picture of Step 4 in Figure 3 below), the BER graph after extrapolation is displayed, and the green column is obtained by extrapolation. The width of the inner side of the parabolic BER curve at 10e-12 is measured on the graph to obtain Tj;
in the fifth step, the BER graph after extrapolation (similar to the parabola curve) is divided into two curves at x=0, and the maximum value of the horizontal axis is set to 0.5UI (Unit interval, i.e., the width of one bit), and the minimum value is -0.5UI, i.e., the width of exactly one UI in the horizontal direction. The curve of the left half of the BER graph generated in Step 4 is moved to the right to the far right, and the curve of the right half is moved to the left to the far left, and the bathtub curve Bathtub curve is obtained.

Figure 3: PDF/BER/CDF and bathtub curve of jitter

Of course, in the above Tj solution process, except for the extrapolation part in the BER diagram, the others are calculated based on the measured results, so the extrapolation of the BER diagram is the most critical part of the jitter analysis algorithm of the oscilloscope. The model accuracy of the extrapolation algorithm determines the accuracy of Tj calculation. Since the extrapolation algorithm fits and extrapolates the tail of the measured BER/CDF diagram, it is called the tail-fit algorithm in relevant foreign literature.
In addition to the method of extrapolating the BER diagram to obtain Tj with a small bit error rate, another method in the industry is to extrapolate the PDF to obtain a PDF with BER<10e-12, and then integrate to obtain the BER/CDF and the bathtub curve, so as to calculate Tj. The two algorithms are called the tail-fit method for PDF and the tail-fit method for the BER/CDF.
The following briefly introduces a tail-fit method for PDF. As shown in Figure 4 below:
The first step is to count the histogram of TIE distribution. The more samples are measured, the more accurate the estimated Tj is. The TIE histogram below includes 102.6k samples;
the second step is to take the logarithm of the Y axis (i.e. the number of samples) of the TIE histogram, and convert the Y coordinate into a logarithmic coordinate. After the logarithmic operation, the contour of the histogram is approximately a quadratic equation curve; the
third step is to fit the two tails using the least squares method;
the fourth step is to interpolate the tail of the histogram, and after normalization, a probability density function of BER=10e-16 can be obtained;
the fifth step is to integrate each offset value x: BER(x) = 
=1-CDF, and obtain the BER/CDF curve;
the sixth step is to measure the width of the CDF curve under a certain bit error rate, which is the overall jitter Tj.

 

Figure 4: PDF tail fitting algorithm calculates total jitter

In the tail fitting algorithm, the premise is that the measured jitter samples are sufficient and the jitter histogram includes many low-probability jitter events. Usually, these low-probability jitter samples are distributed in the tail of the histogram. Only when there are enough samples in the tail can the tail fitting and extrapolation be performed accurately.
Summary:
This article briefly introduces the histogram, probability density function, bit error rate BER and cumulative distribution function CDF, bathtub curve of total jitter, and the processing steps of two tail-fit algorithms. Subsequent articles will introduce LeCroy's unique jitter solution algorithm - NQ-Scale algorithm and several jitter decomposition methods in serial data analyzer SDA.
References:
1, Jitter, Noise, and Signal Integrity at High-Speed, Mike Peng Li
2, Fibre Channel – Method Jitter and Signal Quality Specification – MJSQ, T11.2/Project 1315-DT/Rev 14.1, June 5, 2005.