The concept of jitter and how to measure it

As the clock rates in communication systems enter the GHz level, jitter, a critical factor in analog design, has also begun to receive increasing attention in the field of digital design. In high-speed systems, the timing error of the clock or oscillator waveform will limit the maximum rate of a digital I/O interface. Not only that, it will also increase the bit error rate of the communication link and even limit the dynamic range of the A/D converter. Some data show that in systems above 3GHz, time jitter will cause inter-symbol interference (ISI), resulting in an increase in the transmission bit error rate.
Under this trend, designers of high-speed digital equipment have also begun to pay more attention to timing factors. This article introduces the basic concept of jitter to digital designers, analyzes its impact on system performance, and provides common circuit techniques that can minimize phase jitter.
2. The concept of time jitter

Ideally, a perfect pulse signal with a fixed frequency (taking 1MHz as an example) should last exactly 1us, with a transition edge every 500ns. Unfortunately, such a signal does not exist. As shown in Figure 1, the length of the signal cycle will always vary, resulting in uncertainty in the arrival time of the next edge. This uncertainty is jitter.
Jitter is a measurement of the time domain variation of a signal, which essentially describes how much the signal cycle deviates from its ideal value. In most literature and specifications, time jitter is defined as the deviation between the arrival time of a high-speed serial signal edge and the ideal time. The difference is that in some specifications, the slowly changing component of this deviation is called time wander, while the faster changing component is defined as time jitter.

Figure 1 Schematic diagram of time jitter

1. Classification of time jitter 
There are two main types of jitter: deterministic jitter and random jitter. 
Deterministic jitter is caused by identifiable interference signals. This jitter is usually limited in amplitude, has a specific (rather than random) cause, and cannot be statistically analyzed. 
Random jitter refers to timing changes caused by factors that are difficult to predict. For example, temperature factors that can affect the mobility of semiconductor crystal materials may cause random changes in carrier flow. In addition, changes in semiconductor processing technology, such as uneven doping density, may also cause jitter. 
2. Description of time jitter The 
characteristics of jitter can be determined through many basic measurement indicators. Basic jitter parameters include: 
1) Period jitter
measures the width of each clock and data cycle in the real-time waveform. This is the earliest and most direct way to measure jitter. This indicator describes the variation of each cycle of the clock signal.
2) Cycle-cycle jitter 
measures how much the width of the cycle of any two adjacent clocks or data varies. Cycle-cycle jitter can be obtained by applying a first-order difference operation to period jitter. This indicator has obvious significance when analyzing the properties of phase-locked loops.
3) Time interval error (TIE)
measures how much each active edge of the clock or data deviates from its ideal position. It uses a reference clock or clock recovery to provide the ideal edge. TIE is particularly important in communication systems because it illustrates the cumulative effect of periodic jitter in various periods. 
3. Frequency domain representation of time jitter - phase noise
Phase noise is another measure of the timing variation of a signal, and its time jitter is displayed in the frequency domain. Figure 2 uses an oscillator signal to explain phase noise.

If there is no phase noise, the entire power of the oscillator should be concentrated at the frequency f=fo. However, the appearance of phase noise spreads part of the oscillator's power to adjacent frequencies, generating sidebands. As can be seen from Figure 2, at an offset frequency a reasonable distance from the center frequency, the sideband power rolls off to 1/fm, where fm is the difference from the center frequency.
Phase noise is usually defined as a dBc/Hz value at a given offset frequency, where dBc is the ratio of the power at that frequency to the total power in dB. The phase noise of an oscillator at a certain offset frequency is defined as the ratio of the signal power in a 1Hz bandwidth at that frequency to the total power of the signal.

Figure 2 Phase noise diagram

3. Time jitter analysis methods 
1. Statistical characteristics and statistical histograms
Since all signals containing jitter have random components, statistical calculations are widely used in the evaluation of jitter performance. Commonly used statistical parameters include mean, standard deviation, maximum, minimum, peak-to-peak value, etc. Histograms are usually used to vividly describe these statistical characteristics of jitter. The
horizontal axis of the statistical histogram is the size of jitter, and the vertical axis is the frequency of jitter in a certain interval. When the number of measurements is large enough, the histogram is a good estimate of the probability density function of the jitter size. Therefore, when estimating the system bit error rate through jitter, the statistical histogram plays an extremely important role.

Figure 3 Random jitter histogram Figure 4 Periodic jitter histogram

It should be noted that the histogram does not contain the order in which each jitter point occurs, so it cannot be used to display the periodic information in jitter.
2. Jitter-time curve and frequency spectrum of jitter 
Since the statistical histogram cannot display the modulation or periodic component information in jitter, the jitter-time curve can be used to describe the trend of jitter changing over time. The horizontal axis of the curve is the time when the jitter is measured, and the vertical axis is the size of the jitter. In this way, the pattern of jitter changing over time can be clearly seen from the figure.
Since there are components in jitter that change periodically over time, an obvious analytical method is to perform Fourier transform on the jitter-time curve to obtain its frequency domain characteristics.

Figure 5 Jitter-time curve Figure 6 Jitter spectrum

3. Eye diagram 
So far, the eye diagram is still a qualitative and convenient method for analyzing the digital communication process. It can simultaneously provide the amplitude information and time information of the transmission. A series of short segments of the waveform will be superimposed together and aligned with the rated edge position and voltage level. Once the jitter reaches +-0.5UI, the eye will close and the receiver circuit will have bit errors.
It should be noted that the trigger source used when measuring the eye diagram should be a standard clock source with high frequency stability and low jitter, and its indicators directly affect the accuracy of the measurement. If the edge of the test signal is used directly as a trigger, the oscilloscope needs to have a clock recovery function.

Figure 7 Eye diagram of digital signal

4. Time jitter model
In order to better describe jitter, it is necessary to establish a model to analyze the impact of jitter in different situations. Depending on the cause of jitter, the jitter model is generally as follows:

Figure 8 Jitter model

1. Random Jitter (RJ) 
Random jitter is temporal noise without any known pattern. Although random jitter may have various probability distributions in the theory of random processes, it is usually assumed to be a Gaussian normal distribution in the jitter model. There are two reasons for this: first, in many circuits, the main source of random noise is thermal noise, which has a Gaussian distribution; second, according to the central limit theorem, many independent and unrelated noise sources are superimposed to approach a Gaussian distribution. Since random jitter satisfies the Gaussian distribution, its peak value is unbounded. This is an important feature that distinguishes random jitter from deterministic jitter.
2. Deterministic Jitter (DJ) 
Compared with random jitter, deterministic jitter (DJ) is repeatable and predictable time jitter. Therefore, the peak-to-peak value of DJ is bounded, and the position of this boundary can approach the true value as the number of measurements increases. DJ can be divided into several types, each with its own characteristics and corresponding physical mechanisms behind it. 
1) Data Dependent Jitter (DDJ)
Data Dependent Jitter is jitter related to the content of each bit of data. DDJ is usually caused by inter-symbol interference (ISI) when the data stream passes through a channel with significantly limited bandwidth. DDJ usually has two discrete pulse-shaped histograms, and the height of the two peaks is the same (according to the location of the peaks, it can be divided into high-probability DDJ and low-probability DDJ). 
2) Duty Cycle Distortion (DCD) Duty Cycle Distortion 
is the measurement jitter caused by the different positions of the zero crossing points when the duty cycle of the clock signal is not 50%. There are two reasons for its occurrence. First, the slew rate of the rising edge and the slew rate of the falling edge of the signal are different. Second, the decision threshold is too high or too low. DCD usually has two discrete pulse-shaped histograms similar to DDJ, and the height of the two peaks is the same.
3) Bounded Uncorrelated Jitter (BUJ) Bounded uncorrelated
jitter is a general term for time jitter that is not correlated with the jitter measurement moment in time and has a bounded peak-to-peak value in distribution. There are usually three sources: power supply noise. The noise caused by the power supply may affect the bit error rate; crosstalk and external noise. The noise caused by adjacent transmission lines or external electromagnetic interference during the transmission process; periodic noise. The signal periodic jitter (PJ) caused by various periodic noises. For example: switching power supply noise or periodic signals used in testing. Periodic jitter (PJ) with only a single frequency component has a histogram with peaks at both ends and a depression in the middle. 
3. Jitter separation 
In actual testing, the composite time jitter often obtained is a combination of the above two or several jitter models. Using the knowledge of probability theory, we can know that the probability density function of the composite jitter is the convolution of the probability density functions of the random variables that make up the jitter. For example, the probability density function of a DCD jitter and a random jitter is to modulate the random Gaussian distribution onto the two peaks of the DCD. In addition, for periodic jitter (PJ), there is not only the fundamental component, but also often accompanied by higher harmonics.
5. Measurement of time jitter
Below we briefly introduce the existing Jitter measurement technology. According to the different test instruments and test purposes, direct measurement technology can be divided into two categories: 1. Measurement methods for the purpose of obtaining the time domain or frequency domain characteristics of Jitter, such as real-time sampling oscilloscopes, equivalent sampling oscilloscopes, time interval meters, etc.; 2. Measurement methods for the purpose of obtaining the statistical characteristics of Jitter, such as bit error rate meters, time interval analyzers without trigger or external clock mode, and oscilloscopes with statistical analysis functions. Now some instruments have both time-frequency measurement and statistical analysis functions, so they are widely used in Jitter measurement. In addition, time jitter can also be indirectly measured by measuring phase noise. Below we introduce several commonly used test methods.
1. Measuring Jitter with an Oscilloscope 
Using an oscilloscope to measure the jitter of a signal requires that the oscilloscope has sufficient bandwidth, signal-to-noise ratio, resolution, time accuracy, and signal fidelity to reduce the impact of measurement errors. The oscilloscope often uses software clock recovery to recover the ideal edge timing (of course, an external high-quality clock source can also be used as the ideal edge timing). At this time, the oscilloscope can generate an eye diagram by superposition. By analyzing the eye diagram, various parameters of jitter can be obtained. 
When using an oscilloscope for analysis, further jitter analysis is often required to obtain the nature of the bit error. At this time, the input data stream needs to be sent repeatedly according to a certain pattern (usually using a pseudo-random sequence generator) to concentrate the energy of the DDJ component as much as possible. After acquiring such a code stream waveform through an oscilloscope, the following analysis can be performed. 
1) Interpolate the sampled data to restore the sampled waveform, and calculate the decision time of each edge for a certain decision level; 
2) Restore the clock of the input signal through the software lock phase loop method, and calculate the jitter size of each edge respectively; 
3) For places where there is no edge such as continuous 1 or continuous 0, the corresponding jitter is obtained by linear interpolation;
4) Perform FFT on the obtained Jitter-time function to obtain the jitter spectrum. 
Next, you can analyze the jitter spectrum to find the corresponding peak values ​​of DCD, DDJ, PJ, and the noise floor size of RJ. Then separate the components and perform IFFT to obtain the jitter-time function of each component. The specific results here are closely related to the resolution of FFT and the choice of window function. 
At present, many oscilloscope manufacturers provide analysis software that matches the oscilloscope, which can effectively decompose and analyze the jitter according to a certain model. For example: TDS JIT3 provided by Tektronix is ​​a jitter analysis kit for oscilloscopes above TDS5000.
2. Measuring Jitter with a Bit Error Rate Tester 
As mentioned earlier, jitter can cause receiving bit errors. Conversely, if the bit error rate can be measured, the characteristics of jitter should be inferred. The method of measuring jitter with a bit error rate analyzer is based on this idea. The bit error rate analyzer
usually uses two channels, one of which is kept at the center of the eye diagram, and the other channel is used to complete the bit error rate test. In this way, there is no need to know the code stream at the transmitter, so there is no need to repeat the transmission of a certain pattern of encoding. At the same time, it can also solve the synchronization problem well.
The bit error rate analyzer can scan the eye diagram in all directions to obtain a clear outline of the eye diagram, which can provide a lot of valuable data for analyzing jitter. 
3. Indirect measurement 
of jitter through phase noise As mentioned earlier, jitter and phase noise describe the characteristics of the same phenomenon. Therefore, it would be meaningful if the jitter value could be derived from the phase noise measurement result. When measuring a crystal oscillator, the phase noise indicator is often given, which can be used to infer the jitter that the crystal oscillator may bring.

Figure 9 Phase noise diagram

Every oscillator has a phase noise plot, an example of which is shown in Figure 9. This plot shows the phase noise of an oscillator over a frequency band from 12kHz to 10MHz. In the figure, L(f) gives the distribution of the sideband noise as a power spectral density function in dBc. The power at the center frequency is not important, as the jitter only reflects the relative power of the phase noise (i.e. modulation) to the "pure" center frequency. The total noise power N in the sideband can be obtained by integrating the L(f) function over the frequency band of interest (in this case, 12kHz to 10MHz).

The power of the phase modulated noise in this frequency band is calculated, and phase modulation is the cause of jitter. From this, we can also use the following definite integral to deduce the value of RMS jitter.
The following formula can be used to calculate the RMS jitter caused by this noise power:

VI. Summary
This article introduces the definition of time jitter in detail, analyzes the causes of its occurrence, and provides analysis methods and measurement methods. I believe that through this document, users can have a deeper understanding of jitter. I hope this article can be helpful to your actual work. Due to limited knowledge, there are inevitably some omissions in the article. Readers are welcome to contact the author to point them out.

References 
[1] Yang Junfeng, Research on timing jitter in high-speed digital serial communication, PhD dissertation of University of Science and Technology of China, 2005.05.01.
[2] Tektronic, Introduction to TDS JIT3 jitter analysis and timing software, 2004.04.09.
[3] Electronics Enthusiasts, Concept of jitter and jitter measurement method, 2008.11.27.
[4] Electronic Engineering Journal, Concept of phase noise and jitter and its estimation method, 2004.06.30.